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IN MEMORIAM
FLORIAN CAJORI
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/ A
NEW TREATISE
ON THE
USE OF THE GLOBES,
AND
Practical Astronomy ;
OR
A COMPREHENSIVE VIEW
OF
THE SYSTEM OF THE WORLD.
IN FOUR PARTS.
I. An extensive collection of Astro- X or a star, at any given time, with
noraical and other Definitions. the method of representing the
II. Problems performed by the TER- spherical triangles on the globe, &c,
RESTRIAL GLOBE, including 1 those >:IV. A comprehensive account of the
relative to Geography, Navigation, A SOLAR SYSTEM, with the elemen-
Dialling, See. with many new and {> tary principles, and most valuable
important problems and investiga- > modern discoveries in Astronomy
tions, particularly useful to the Na- X to the present time. The nature
vigator and Practical Astronomer. and motion of COMETS, OF THE
III. Problems performed by the CE- y FIXED STARS, ECLIPSES, THE
LESTIAL GLOBE, including those >: THEORY OF THE TIDES, LAWS
of finding the longitude at sea, new A OF MOTION, GRAVITY, &c. with
methods of finding the latitude, i"> DIAGRAMS elucidating the de.-
with only one altitude of the sun, V mnstrations.
The whole serving as an introduction to the higher Astronomy and Natural
Philosophy, is illustrated with a variety of important notes, useful remarks,
Sec. and each problem with several examples. The necessary astronomical
instruments are poiin-cQ uuv, iuul the muat usciui taDles are inserted in
the work.
DESIGNED FOR THE INSTRUCTION OF YOUTH,
AND PARTICULARLY ADAPTED TO THE UNITED STATES.
BY J. WALLACE,
Jllcmber of the JVeiv^York Literary Institution, &c.
Quid munus Reipublicce majus aut melitis afftrre posximus, quam si Jnventutem
bene erudiamus ? CJCERO,
NEW-YORK :
Printed and published by SMITH 8c FORMAN,
AT THE FRANKLIN JUVENILE BOOKSTORES
195 and 213 Greenwich-Stree f .
1812.
M308234
DISTRICT OF NEW-YORK, ss.
Be it remembered, That on the sixth day of January, in the thirty -sixth
year ofthe Independence of the United Slates of America, JJMES W*IL-
LJ1CE> of the said district, hath deposited in this office the title of a book,
the right whereof he claims as Author, in the words and figures fol-
lowing', to wit : c A new Treatise on the Use of the Globes, and Practical
Astronomy ; or a comprehensive view of the System of the World. In four
parts. I. An extensive collection of astronomical and other definitions. II.
Problems performed by the Terrestrial Globe, including those relative to
geography, navigation, dialling, &c. with many new and important problems
and investigations, particularly useful to the navigator and practical astrono-
mer. III. Problems performed by the Celestial Globe, including those of
finding the longitude at sea, new methods of finding the latitude, with only
one altitude of the sun, or a star, at any given time, with the method of re-
presenting the spherical triangles on the globe, &c. IV. A comprehensive
account of the Solar System, with the elementary principles, and most va-
luable modern discoveries in Astronomy to the present time. The nature
and motion of Comets, of the Fixed Stars, Eclipses, the theory of the Tides,
Laws of Motion, Gravity, &c. with Diagrams elucidating the demonstrations.
The whole serving as an introduction 'to the higher Astronomy and Natural
Philosophy, is illustrated with a variety of important notes, useful remarks,
&c. and each problem with several examples. The necessary astronomical
instruments are pointed out, and the most useful tables are inserted in the
work. Designed for the instruction of youth, and particularly adapted to
the United States.' By J. Wallace, Member of the New-York Literary Insti-
tution, &c. Qifid munus lleipublicx mqjus aut melius afferre possimus, quam
si Juventutem bene erudiamus, &c. Cicero.
In conformity to the act of the Congress of the United States, entitled
* An act for the encouragement of learning, by securing the copies of Maps,
Charts, and Books to the authors and proprietors of such copies, during the
times therein mentioned.' And also, to an act, entitled * An act, supple-
mentary to an act, entitled ' An act for the encouragement of learning, by
securing the copies of Maps, Charts, and Books, to the authors and propri-
etors of such copies, during the times therein mentioned, and extending the
benefits thereof to the ails of designing, engraving, and etching historical
and other prints.'
CHARLES CLINTON,
Clerk of the District of New-York.
PREFACE.
. MAN cannot but behold with gratitude and delight, the multi-
plied benefits and amazing; objects which surround him on all
sides, contributing equally to his wants and pleasure. This plea-
sure, however, is greatly increased in proportion as the nature,
utility, and number of these objects .ire known and understood ;
and this knowledge is only attained from a cultivation of those no-
ble powers with which the mind of man is gifted, and which so
eminently distinguish him from the brute creation.
The savage that ranges our forests in common with the brute ;
that at the same fountain satisfies his thirst, and eats of nature's
fare, whatever his tuste or appetite craves ; that seems no way dis-
tinguished from the animals with which he associates, than by the
figure of his species ; has still within him the seeds of those noble
acquirements which exalt and dignify human nature. Yes, this
same savage enjoying similar advantages with a Cicero, a Demost-
henes, or a Newton, might become their rival ; but those seeds,
from a \vantofr cultivation, must remain for ever buried in oblivion.
Such is the picture of uncultivated man, whom, in his wild and
savage state, the mines of Peru cannot enrich, or whose wants
the most fertile regions of the earth cannot lessen In the
midst of profusion he is indigent, and in the unequal conflict with
those animals, whose master he was destined to be, must often be-
come a prey to their superior strength and ferocity.
It is evident, then, that an acquaintance with the elements of
science is intimately connected with our necessities, no less than
with our future progress, advancement, and eminence ; and that in
proportion as we neglect the acquirement of this knowledge, we
approximate to the state of th# rude, uncultivated savage It is
well known, that Great-Britain and France respectively owe more
to the successful cultivation and application of the sciences, than
they do to the valour of their armies, or to the strength of their
marine.
Among all the branches of science within the compass of hu-
man acquirements, there are few that unite greater importance
and utility, than that which exhibits and explains the phenom-
ena of the earth, our destined habitation, and more pleasure,
than that which traces the evolutions of those immense orbs that
decorate the heavens, and investigates the unerring laws by which
they are regulated and governed : for there is nothing which so
much excites our attention and curiosity, which unites in itself so
much grandeur and magnificence, and which produces in the soul
so much sublimity and admiration, as the contemplation of those
prodigies which that immense vault surrounding the habitation of
man exhibits to our view. And if there be some in whom this grand
spectacle excites no emotion, it is because they are too much ab-
sorbed in those artificial wants or necessities which they create to
themselves ; -veluti fiecora, as Sallust suys, qua natura firona^ af
quc vcnfri obedcntia finxlt \
iv PREFACE.
It is in the heavens that the Creator has chiefly manifested his
greatness and majesty It is here that the Sovereign Wisdom
shines with the greatest lustre, and that the sublime ideas of order
and harmony reign. In this immense host of celestial bodies all is
prodigy and magnificence : all is regularity and proportion : all
announce a power infinitely ferule in the production of beings, in-
finitely wise in their arrangement and destination.
But this magnificent spectacle is not thus exposed to our con-
stant view, to be the object of an idle admiration, or a fruitless con-
templation ; it is much more connected with the wants and ad-
vantages ot the inhabitant of the earth. It is in the heavens that
we have found the means of arresting time in the rapidity of its
course : of regulating our seasons, and fixing those interesting
epochs, from which the Historian and Chronologer date the most
important events The form, the extent, the exact position of the
different parts of the earth we inhabit, and its situation in the im-
mense expanse, is attained only by the assistance of Astronomy.
If we now traverse the ocean with so much security and skill, it is
principally owing to this science which has furnished the means of
ascertaining our place, at any time, in this trackless element. Thus
by the interposition of the heavens the most distant nations hold
their correspondence : extensive deserts, immense oceans, seas, and
unknown countries are explored, and their riches transported to
other countries destitute of these resources. In a word, it is to
this science that Columbus owed the greatest discovery that human
ingenuity has ever made, and that he has been able to add a new
world to the old.
It is not only in enlarging the sphere of human knowledge, and
contributing to the wants and convenii ncies of man that Astronomy
is useful ; it has also dissipated fflfe alarms occasioned by extraor-
dinary celestial phenomena, and destroyed many of the errors aris-
ing from our true relation with nature. Such are the obligations
we have to this science ; such the benefits which it has conferred
on society ; such the services it has rendered the human mind.
This sublime science then, claims a right to our esteem and res-
pect, and without doubt, there is not among human sciences another,
more worthy to engage our attention, and better calculated to oc-
cupy and amuse our leisure moments.
It is no objection to it that it has often been made the unwilling 1
instrument of impiety in the hands of the impious, or of an absurd
science in the hands of the Astrologer ; for the greatest benefits
conferred on man are susceptible of abuse. To put a stop to these
growing evils, Emperors have passed their edicts and enforceti
their decrees, to expel those impious pretenders from cities that
became the scenes of their folly and impiety, and some who de-
served a better fate were unhappily involved in their number. The
irreligious Philosopher and the impious of the day, will ascribe
many of these unhappy occurrences to the religious prejudices and
ignorance of those times; but with no more reason than those
PREFACE v
have, who charge this science with supporting impiety, though of
all others the least calculated to afford it any support If history
has any truth in it, history affirms that it was in houses dedicated
in those days to piety and religion, that the most precious remains
of science were preserved, and that it is from them they have been
principally handed ddwn to the present time.
To trace this science to its origin, and point out the various al-
terations and improvements it has received, the long series of dis-
coveries which it presents, and the illustrious authors who have
contributed to them, would far exceed the limits of a preface. It
will be sufficient to observe, that the origin of Astronomy coiri-
nunces its date with that of Agriculture and of Society itself.
There is still an immense difference between the first view of the
heavens, and the view by which, at present, we comprehend the past
and future state of the system of the world. It is, however, to the
improvements in the past and present age, that we are principally
indebted for this developement of the most important and curious
discoveries in this system ; and such of those authors as have been
most successful, and have particularly excelled in this respect, have
been consulted in the following compendium. Their works have
been also pointed out to direct the choice of the student, and exhi-
bit their superior advantages and excellence.
Among the inconveniencies attending our public places of edu-
cation, it is no small one, that many of those works which are the
standard of elegance and perfection, are inaccessible both to the
Student and Master, in consequence of the difficulty of procuring
them from Europe, and their too great expense to be introduced
into Schools. To remedy, in some measure, this inconvenience,
the author of the present work has^mdertaken to draw up an entire
course of Mathematics and NaUmB Philosophy (if his avocations
will suffer him to continue) principally for the use of the Students
belonging to the New-York Literary Institution. And conceiving
that this course, undertaken more from necessity than choice,
would asssist him no less than others occupied in the education
of youth, he has been induced, principally from this motive, to
make this introduction public.
The present treatise on the Use of the Globes and Practical As-
tronomy is complete in itself, and detached from the contemplated
course, the author having immediate and urgent necessity for its
use ; and being a subject uniting extensive utility with pleasure and
ornament, no pains have been spared, in calculating it for these im-
portant objects, as far as his hurry in drawing it up would allow.
Each problem is illustrated with several examples, and their de-
monstration or calculation, cc. given in notes at the bottom, in or-
der to make it more fully answer the end of an elementary trea-
tise on practical astronomy, and to adapt it to academies and places
of public education in general, where this branch of science is now
considered as one of the most entertaining and necessary.
vi PREFACE.
Many new and important problems will be found in this, inaddi-
~tion to :hose f'ounn in oiher treatises; and which likewise are per-
formed on the globes, by methods generally entirely new, and found
.in no other treatise ; which cannot but render this work extremely
tinte resting to those who are capable of relishing the beauties of
science, and of appreciating its value. Many important Tables
arr: inserted in the course of the work, as well as figures to illus-
trate the demonstrations, &c. and it is no small recommendation
to it that these figures were cut by the celebrated Dr. Anderson.
There arc also given, besides a complete account of the Solar Sys-
tem, the elements and laws of the planet's motions, their phe-
nomena, their principles, Sec a full investigation of the nature and
motion of Cornets, the doctrine of Eclipses, the Tides, the Gene-
ral laws of Motion, Gravity, Sec enriched with many discoveries
and late improvements from H^rschel, Vince, Maskelyne, La
Lade, Laplace, Delambre, &c.
The work bein>; printed close, and the notes (which are of con-
siderable length) being in small type, this treatise must contain
-more matter than any other of the size and nature in print; so
that in one volume f moderate size, besides the Treatise on the
Globes, an entire course of Astronomy is given, including both the
calculations, and the geometrical and physical part ; and the au-
thor does not believe that he has omitted any thing of importance,
that has any particular relation to these subjects.
The teacher will immediately perreive that the work is calcu-
lated for three distinct classes of students. The first is, of those
who are supposed to be unacquainted with the principles of Mathe-
matics, and who may read the definitions and all the problems on
the globes, contained in the ^ and 3d parts. The second class,
who are supposed to have sorffi knowledge of Geometry and Trig-
onometry, may read the notes on the definitions and problems on
the g;obes, and perform the problems by calculation ; they may
also read some select parts of the 4th part, particularly those rela-
tive to the order and motion of the planets* in the solar system.
The third class, supposed to be somewhat acquainted with the ele-
ments of the Conic Sections, Algebra, and the first principles of
Fluxions, may continue the 4th part This last class, by finishing
the elements of Fluxions, will obtain any further knowledge in
Physical Astronomy that may be necessary, being the most proper
place for fully investigating this abstruse subject.
The author in presenting this work to the public, is equally re-
gardless of its censure or praise, as his object is neither emolu-
ment nor celebrity. His whole aim in the undertaking 'vas to
Jighten the burden of the Teacher and to improve the student If
by comprising in a comparatively small compass ail that is useful
and necessary either on the Globes or in Astronomy, he succeed
in this, his object will be fully attained.
Distance from the press and hurry in the execution^ have pro-
duced some few errors, most of which are found in the errata at
the end.
CONTENTS.
PART I.
Containing the description of the Globes, Astronomical and other Definitions,
t/ie method of calculating and adjusting the Calendar, table of Climates, a
full description of all the Constellations, number of Stars, Clusters, Nebula,
&c. as inserted on t/te neiv British Globes, ivith observations on their origin t
the origin and nature of the Heathen worship, with some useful refections,
&c. page 1 to 50.
FART II.
Or problems performed by the Terrestrial Globe, contains different methods of
finding the latitudes and longitudes of places, of finding the hour of the day
or night at any time, difference of times, hours, seasons, &?c. according to.
the change of latitude or longitude, Antxci, Periceci, Antipodes, Sun's Ion"
gitude, declination, rt. ascension, rising, setting, length of days and nights,
&c. in every latitude, the manner of exhibiting these phenomena ivith the
different seasons on the globe, their calculation, &c the equation of time
with its investigation and tables, the method of finding morning and evening
twilight in the different latitudes, -with observations, &c. method of finding
the different climates, their breadth, 6?c. methods of finding the distance of
places, with extensive tables of the lengths of a degree, &c. and the measures
used in different countries, together with the late French measures, &c.
Problems in Navigation, Dialling, &c. and many other particulars sv/wc/i
itnll be seen by consulting this part, 50 to 192.
PART III.
Contains the methods of finding the right ascensions, declinations, latitudes and
longitudes of the sun, stars, planets, &c. and the methods of making the ob-
servations, application of pendulum clocks to this purpose, the manner of ad-
justing them, &c. of finding- the rising and setting of the stars, or planets^
time of their passing the meridian, method of finding their distance, and from
thence the longitude at sea, various methods of finding the latitude, of finding
a meridian line, variation of the compass, moon's southing, &c. time of high
water, with the necessary tables, Ac/ironical, Cosmical and HcUucal rising, set"
ting, &c. of the stars, of describing the apparent paths of the planets or comets
among the fixed stars, of the precession of the equinoxes, &c. 192 to 247.
PART IV. CONTAINS
Ch. I. Of the sun, his phenomena, &c. diameter, spots, rotary motion, distance,
Kepler's laws, nature of the Centre of Gravity, atmosphere, &c. 250 to 257-
Ch. II. Of JMercury, his phenomena, motions, diamet r, distance, magnitude^
methods of finding these, c#c. of describing a planet's orbit, of finding its
nodes and inclination, of finding a planet's heliocentric and geocentric places,
conjunctions of the inferior planets, phases of the planet s, &?c. 257 to 270.
viii CONTENTS.
Ch. HI. Of Venus, her motions, phenomena, &c. her spots, phases, her par ai-
lax, &c. 270 to 280.
Ch. IV. Of the earth and its satellite the moon, the earth's globular figure ,
measure of a degree on its surface, its diurnal motion, general refections on
its cause, an idea of the universe, the earth's annual revolution, aberration
f light, length of tJie sidereal year, of the solar or tropical year, the
substance of Newton's discoveries relative to the planets' motions, theory
of the earth or planets-' motions, &c. 280 to 320. Of the moon, her
mean motions, plienomena, method of finding her apparent diameter, 6fc.
Parallax of the moon and planets, moon's phases, atmosphere, mountains,
fcfc. 320 to 346.
Ch. V. Of Mars, his motions and pJienomena, &c. Theory of the planets' mo.
tions, &c. 346 to 357.
Ch. VI. Of the new planets Ceres, Pallas, Juno and Vesta, 357 to 359.
Ch. VII. Of Jupiter, his motions and phenomena, &?c. 359 to 363. Of his Sat-
ellites, their periodic times, distances, &c. ho-w found, their immersions and
emersions, their rotary motion, their edipses, laws of their inequalities, their
configurations, &c. 363 to 377.
Ch. VIII Of Saturn, his motions, phenomena, &c. method of finding the oppo-
sition oftJie superior planets, of Ids ring and its phenomena, his satellites and
their periods, &c. 377 to 388.
Ch. IX. Of Uranus or HerscJiel and his satellites, 388 to 392.
Ch. X. Of the nature and motion of Comets, of their tails, table of their ele-
ments, -with examples of the method of calculating them, &c. the names of the
authors who have calcidated them, &c. 392 to 436.
Ch. XI. Of the fixed stars, double stars, their phenomena, a full investigation
of their aberration, the nature of the nebulous appearances in the heavens,
their number, distance of the stars, &c. 436 to 448.
Ch. XII. Of Solar and Lunar Eclipses, method of calculating them, &c. 448
to 458.
Ch. XIII. Of the Tides, their phenomena, laws, method of calculating them, &fc.
458 to 463.
Ch. XIV. Of the General Lavas of Motion, Forces, Gravity, &c. motion of
bodies on inclined planes, in curves, &c. properties of the pendulum, masses
and densities of the planets determined, forces of gravity at their surface,
their perturbating forces, &c. 463 to 486,
PRACTICAL ASTRONOMY, &c.
IN THE
DEFINITION AND USE OF THE GLOBES.
'
PART I.
DEFINITIONS, UV.
1. A GLOBE or SPHERE, is a round solid body, having every
part of its surface equally distant from a point within it, called the
centre. It is formed by the revolution of a semicircle round its
diameter, which remains fixed.
2. The terrestrial globe* is an artificial representation of the
earth, having the different countries, empires, kingdoms, chief
towns, seas, rivers, Sec. truly represented on it, according to their
relative situations on the real globe of the earth.
* If a map of the world be accurately delineated on a spherical ball, its
surface will represent the surface of the earth. For the highest hills are so
inconsiderable with respect to the bulk of the earth, that they take off' no
more from its spherical figure, than grains of sand do from the spherical
figure of an artificial globe. The diameter of the earth is about 7964 miles.
Chimborazo, one of the Andes, considered the highest mountain in the
world, is about 20,282 feet or nearly 4 miles high. The radius or semi-
diameter of the earth is about 3982 miles, and that of an 18 inch globe
9 inches : hence we have this proportion 3982m : 3986m :: 9 in. : 9.009 in.
Now by taking the radius of the artificial gl<3be from this, the remainder
.009 =s=Y^y= ==T *-j of an inch, nearly, which is the elevation of the highest
peak of the Ancles on an 18 inch globe. That the globe of the earth is
spherical, or nearly so, appears 1. From its casting- a spherical shadow on
the modn, whatever be its position, when it is eclipsed. 2. From our seeing
the further, the higher we are elevated on its surface. 3. From our first
seeing the tops of mountains, the masts of vessels, Sec. when we advance
towards them in any direction. 4. From its having been sailed round from
east to west byseveraj. persons ; and that in whatever direction a ship sails,
the stars are elevated above the horizon as many degrees as the vessel sails
towards them, and those behind depressed in like manner. Thus in sailing
from the equator towards the north pole one degree, the pole star is elevat-
ed 1; in sailing 2, the pole is elevated 2, &c. so that if thsi-e were a star
exactly in the pole, its height would always indicate the number of degrees
a place is from the equator or its latitude. This phenomenon could not
possibly take place unless the globe was round. 5. From the length of
pendulums vibrating in the same time in different parts of the world, being 1
always as the force of gravity (Emerson's Tracts, part 1. prop. 27.) that is,
as the distance from the earth's centre (Newton's Principia. b. 3, prop. 6.)
But the increase of gravity or weight in passing from the equator to the
poles is as the square of the sine of the lat. (Newton, b. 3, prop. 20.) so
that the equator is something higher than the poles, the diameters being s*
2 DEFINITIONS, We.
3. A great circle of a sphere is any circle on its surface, whos?
centre is the same as the centre of the sphere. Its plane divides
the sphere into two equal segments called hemispheres.
Note. The plane of a circle is the surface included -within its circumference.
4>. A lesser circle is that whose centre is different from the cen-
tre of the sphere. I plane Avid* th globe into two unequal
segments.
5. The axis of a sphere is the fixed straight line about whicfe
the generating semicircle revolves. The axis of the earthy is an
imaginary straight line passing through its centre, and upon which
it is supposed to turn. The axis of the artificial globe is a line
which passes through its centre from north to south, and is repre-
sented by the wire on which it turns.*
6. The poles of a great circle of the sphere, are the two points
equally distant from any part of the circumference of that circle.
The poles of the earth are the extremities of its axis, at the earth's
surface ; one of which is called the north or arctic pole : the other
the south or antartic pole The celestial poles are the imaginary
points in the heavens corresponding to the terrestrial poles, or the
extremities of the earth's axis produced to the heavens.f
7. The diameter of a sphere is any straight line which passes
through the centre, and is terminated both ways by the surface of
the sphere.
8. The circumference \ of a sphere is any great circle described
on its surface.
230 to 229 And the pendulum indicates not only this small difference, but
even the difference made in the height of mountains ; for a pendulum that
vibrates seconds in a valley, will not vibrate seconds exactly when carried
to the top of a mountain. Now if the semi-diameter of the equator be 3982,
the polar semi-diameter will be 3964.6. For 230 : 229 :: 3982 : 3964.6
nearly. Hence the radius or semidiameter of the earth at the pole, is
shorter than the semidiameter at the equator by 17 miles nearly. But
this difference is so imperceptible on the largest globes, that it is not
thicker than the paper and paste on the surface. For suppose the diam-
eter of a globe at the equator be 18 inches, then 230 : 229 :: 9 :
|| .the polar semidiameter ; therefore the difference is -y^ of an inch,
the flatness of an 18 inch globe at each pole ; a difference less than the 23d
part of an inch. Hence though the earth be not strictly speaking a globe,
yet no other figure can give so exact an idea of its shape. And a lecturer
who informs his hearers that it is in the form of a turnip or orange, gives a
very false idea of its true figure. Though 7964 be generally assumed for
1.he earth's diameter, it is however probably something less.
* The diurnal motion of the earth on its imaginary axis is from west to
east, and is the cause of the apparent motion of the heavens from east to
west. This phenomenon of the earth is not unlike that of a large vessel
carried along the current of a river, in which the passengers imagine them-
selves at rest, and that the banks and objects on shore, which are at resl^
are actually in motion.
f The poles of the earth nre the same as those of the equator. The poles
are 90 distant from the great circle to which they belong.
t The circumference of every circle is divided into 360 equal parts called
degrees, each degree into 60 equal parts called minutes, each minute into
DEFINITIONS, er c . 3
9. The equator* is a great circle of the earth equidistant from
the poles, which divides the globe into two equal hemispheres,
northern and southern.
10. Latitude of a place f on the terrestrial globe, is its distance
from the equator north or south.
60 equal parts called seconds, &c. The length of a degree is therefore differ-
ent in different circles, and on the equator is 60 geographical or 69 English
miles nearly. It varies in the respective parallels of latitude towards each
pole, in the direct proportion of the cosine of the latitude, or which is the
eame as the semidiameter of the respective circles. The utility of finding-
Ike length of a degree, in order to determine tjie magnitude and figure of
the earth is apparent, and may be rendered familiar to a learner thus ; sup-
pose the latitude of New- York be 40 43', and that a person travels due north
until the latitude be found 41 43', then he will have travelled a degree, and
the distance between the two places will be its length. Mr. Richard Nor-
wood in 1635 measured the distance between London and York, and found
it equal 905751 feet London measure, and observing the difference of lati-
tude to be 2 28' found that 1 degree was equal 367196 feet. M. Picard
found by a trigonometrical survey, that the distance of the " Pavilion de
Malvoisine" south of Paris, to the steeple of the cathedral of Amiens, re-
duced to the meridian, was 78907 toises. He found also by astronomical
observation, that the distance of these places was 1 22' 58 tf ; hence
1 22 r 58" : 78907 :: 1 : 57064 toises the length of a degree. The as-
sumed distance (in the late French measures) from the equator to the north
pole, established on the measure ofia degree of the meridian equally distant
from both, is 30794580 feet, which divided by 90 gives 342162 feet or 57027
toises. Now as 5280 feet make a mile, therefore 367196-4-5280=69.54
(or 69) miles nearly, which multiplied by 360 produces 25034 the circum-
ference of the earth ; but the circumference of a circle is to its diameter as
355 to 113 ; hence 355 : 113 :: 25034 : 7965 miles the earth's diameter
according to Norwood's measure. Again ; as 811 French feet are equal to
864 English feet, or 107 to 114 nearly, hence 107 F.f. : 114 E.f. :: 342162
F.f. : 364546 English feet, which divided by 5280 gives 6.9.04 English
miles, the length of a degree, according to the late French measure. Now
342162X360=123178320 French feet the circumference of the earth, and
811 : 864 :: 123178320 : 131228188 English feet =24853.82 miles the cir-
cumference, and 355 : 113 :: 24853.82 : 7911.2, the diameter in English
miles. According to Picard the circumference is 24871.5 miles, and diam-
eter 7916.8 miles. It was Picard's measure that Sir Isaac Newton has fol-
lowed in his principia, making the number of toises in a degree =57060
by taking the distance between Malvoisine and Amiens 1 22' 25". See his
principia book, 3 prop. 19.
* The equator, so called from its dividing the earth into two equal parts,
is, when referred to the heavens, fermed the equinoctial, because when the
sun appears in it, the days and nights are equal all over the world, viz. 12
hours each. This circle is also by mariners called the line. On this line is
found the rt. ascension, oblique ascension, oblique descension, ascensional
difference, longitude of places, semidiurnal and nocturnal arches, planetary
hour, distinction between north and south latitude of places, difference o'f
longitude, most exact and equal measure of time, &c.
f Difference of latitude is the nearest distance between any two parallels
of latitude shewing how far the one is to the north or south of the other, and
difference of longitude is the nearest distance between any two meridians
either east or west If the latitude be in the northern hemisphere, it is call-
ed north latitude, if in the southern, south latitude. The greatest latitude
that a place can have N, or S. is 90, and the greatest longitude E. or W,
4 DEFINITIONS, fcV.
11. Longitude of a place, is its distance from the first meridian,
reckoned on the equator towards the east or west.
12. Parallels of latitude, are small circles drawn on the terres-
trial globe, through every ten degrees of latitude parallel to the
equator
'i3 The tropics* are two lesser circles parallel to the equator,
at the distance of 23 28' from it ; the northern is called the
tropic of Cancer, the southern the tropic of Capricorn.
14. The /War circles are two lesser circles, parallel to the equa-
tor, at the distance of 66 32' from it, or 23 28' from each pole.
15. A zonef is a portion of the surface of the earth contained
between two lesser circles, parallel to the equator ; they are^f e in
number, one torrid, two temperate, and two frigid.
16. The torrid zone \ is the space contained between the two
tropics, and is 46 56 ; broad.
17. ihe temperatt zones are the spaces between the tropics
and polar circles, in both hemispheres. They are each 43 4' broad.
18. The frigid zones are the spaces included within the polar
circles.
19. Amphiscii \\ are the inhabitants of the torrid zone, so called
because they cast their shadows both north and south at different
times of the year.
20. Heteroscii is a name given to the inhabitants of the tempe-
rate zones, because they cast their shadows at noon only one way.1[
2 1 . Periscii are those people who inhabit the frigid zones, be-
cause their shadows, during a revolution of the earth on its axis,
are directed towards every point of the compass.
* So called from the Greek word trepo, to turn, because when the sun
comes to either tropic s it begins to return again towards the other.
f So called from zone or zona, a girdle, being extended round the globe
in that form. It is similar to the term climate, for pointing out the situation
of places on the earth, but less exact, as there are only five zones, whereas
there are 60 climates, as will be seen in its proper place.
$ This zone was called by the ancients Torrid, because they conceived
that being exposed to the perpendicular or direct rays of the sun, the heat
must be so great, and the country so barren and parched, as to render it
entirely uninhabitable. But this idea has long since been refuted, The sun
is perpendicular twice in the year to every part of this zone.
These zones were called temperate by the ancients, because, meeting
the sun's ra}^s obliquely, they enjoy a moderate degree of heat, the sun be-
ing never perpendicular to my part of them. The breadth of the temperate
zones increases a little every year, whilst that of the torrid and frigid zones
decrease in the same proportion, owing to the animal decrease of the ob-
liquity of the ecliptic.
|| When the sun is vertical or in the zenith, which happens twice a year,
they are then called ascii, or shadowless, because at that time they have no
shadow.
5f Thus the shadow of an inhabitant of the north temperate zone always
falls to the north at noon, because the sun is then directly south ; and an in-
habitant of the south temperate zone casts his shadow towards the south at
noon, because the sun is due north at that time. These distinctions are
however rather trifling.
DEFINITIONS, &c. 5
22. The anted* are those who live under the same meridian,
and in the same latitude, but on different sides of the equator.
23. Periceci f are those who live in the same latitude, but in op-
posite longitudes.
24. Antipodes \ are those inhabitants of the earth, who live dia-
metrically opposite to each other.
25. Meridians are great circles passing through the poles, and
cutting the equator at right angles.
* The antccci have the same hours, but contrary seasons of the year; thus
when it is noon with one, it is noon with the other, Sec. But when it is sum-
mer with one, it is winter with the other, &c. consequently the length of the
days with one, is equal to the length of the nights with the other ; the sun when
in the equinoctial rises and sets to the one at the same time that it rises ancl
sets to the other, &c. Those who live at the equator have no antoeci.
f The perioeci have their seasons of the year at the same time, and also
their days and nights of the same length with each other ; but when it is
noon with the one it is midnight with the other, and when the sun is in the
equinoctial, he rises with one when he sets with the other. Those who live
under the poles have no perioeci. Their difference of longitude is 180.
t The antipodes have both their latitude and longitude different, and con-
sequently both their seasons and hours ; so that when it is summer with one
it is winter with the other ; when it is twelve o'clock in the day with one it
is twelve at night with the other. They have like seasons, and the same
length of days and nights, but at different times. When they stand, their
feet are towards one another, and their heads opposite. Hence that part of
the heavens which appears over the head of one, seems to be beneath or un-
der the feet of the other ; and therefore, when we speak of up or do-tvn, we
speak relatively and only with regard to ourselves ; for no point, either in the
heavens, or on the surface of the earth is above or below, but only with res-
pect to ourselves. Upon whatsoever part of the earth we stand, our feet is
always nearly directed towards the centre, and our head towards the sky ;
in the latter case we say up t in the former do-urn.
$ These are so called from the Latin word meridies, midday, because
when the sun is on any of these meridians, it is then noon or 12 o'clock, in
all places under that meridian. Every place on the globe is supposed to
have a meridian passing through it, though on most globes there are but 24,
the deficiency being supplied by the brass meridian, which is therefore called
the universal meridian. They are drawn through every 15 of the equinoc-
tial, and are therefore sometimes called hour circles, the reason of which is
evident; for if 360, the number of degrees in a circle, be divided by 24, the
hours in one day, the quotient 15 will give the number of degrees corres
ponding to each hour. Geographers assume one of these meridians as the
first, commonly that which passes through the metropolis of their own coun-
try, but the general practice is, to reckon longitude from the meridian oi'
Greenwich observatory in England. The brazen meridian is divided into
360 equal parts, called degrees, these are again supposed to be divided into
60 equal parts, called minutes, and these into 60 equal parts, called seconds.
&.C. to thirds, fourths, fifths, &c. On the globes, however, the degrees aiv
seldom subdivided into fewer parts than quarters. In the upper semicircle
of the brass meridian, the degrees are numbered from to 90 from the equa
tor towards the poles, and are generally used in finding the latitude of places
On the lower semicircle they are numbered from to 90, reckoning from t !
poles towards the equator, und are principally used in elevating either of the
poles to the latitude, .c ^-
6 DEFINITIONS, Vc.
26. The brazen meridian (or universal meridian) is the brass
circle in which the artificial globe turns.
27. Thejirst meridian is that from which geographers begin to
count the longitude of places.
28. Hour circles,* or horary circles, are the same as the meri-
dians ; they are supplied by the brass meridian, the hour circle and
its index.
29. The hour circle or index, is a small circle of brass fixed to
the north pole, and on which the hours of the day are marked.
30. The ecliptic t is that great circle in which the earth per-
forms its annual motion round the sun, or in which the sun seems
to move round the earth once bi a year.
31. Signs of the ecliptic are the 12 equal parts into which it is
divided. The signs and the days on which the sun enters them are
These circles are drawn through every 15 of longitude reckoning fr
meridian, for the reason given above, but on Gary's globes they
from
any meridian, tor tne reason given above, but on Vary's globes tney are
drawn through every 10, as on a map, though without answering any useful
purpose. As 15 correspond to an hour, 4 minutes of time must correspond
to each degree, 2 minutes to half a degree, 1 minute to one quarter of a de-
gree, &c. (see Keil's astr. lect. 18.) On some globes the index, which points
out the hours, has two rows of figures on it, others but one. On Bardin's new
British globes, there is an hour circle at each pole numbered with two rows
of figures. On Gary's there is but one hour circle placed under the brass
meridian at the north pole, marked with only one row of figures, and is
therefore more convenient, as it answers every purpose to which a circle of
this kind can be applied, without that confusion generally arising from two
rows of figures. On Adams' common globes there is but one index ; but on
his improved globes the hours are counted by a brass wire with two indexes
placed over the equator. On many of the globes fitted up by Jones, the hour
circle is calculated to slide on the brass meridian, for the conveniency of
pointing- out the bearings of places, &c. These circles are however of little
consequence, as the equator and quadrant of altitude will answer every pur-
pose to Xvhich they can be applied.
f The ecliptic (so called, because the eclipses of the sun and moon can
happen only in the plane of this circle) makes an angle of 23 28' with the
equinoctional, one half being 1 in the northern hemisphere, and the other in
the southern. The spring and autumn signs being- in the northern hemis-
phere, are therefore called northern signs ; the other six, or the summer and
winter signs, being in the southern, are for the same reason called southern
sig?is. The spring and autumnal signs are likewise called ascending' signs,
because when the sun is in any of these signs,his declination is increasing-;
the summer and winter signs are called descending signs, because when the
sun is in any of them, his declination is decreasing. Each of these signs is
divided into 30, &c. and in whatever sign and degree the sun is, that point
is called the sun's place. The day of the month corresponding to the sun's
place is likewise commonly marked on this circle. The equinoctial point
aries is that point from which the sun's place or longitude is reckoned, with-
out any regard to the constellations themselves, which, on account of the
precession of the equinoctial points, are now a whole sign advanced from
west to east, or according to the order in which the signs are reckoned. Be-
sides the sun's place or longitude, his apparent and annual motion, stars
longitude, poetical rising and setting-, increase and decrease of days, culmi-
nating- degree, eclipses of the sun and moon, distinction of north and south
latitude of the stars, &c. arc dso found on this circle.
DEFINITIONS, fcV. 7
as follows, according as they are represented on Gary's globes.
The beginning of each day is to be taken.
Sfiring signs.
9jP Aries, the ram, 21st of March.
8 Taunts, the bull, 20th of April.
n Gemini, the twins, 21st of May.
Autumnal signs.
aQt libra, the balance, 23d of Sept.
}\ Scorpio, the scorpion, 23d of Oct.
$ Sagittarius,tbc archer, 22d of Nov.
Summer signs.
25 Cancer, the crab, 21st of June.
SI Leo, the lion, 23d of July,
njj Virgo, the virgin, 23d of August.
Winter signs.
V? Capricornus, the goat, 22d of Dec.
Zg Aquarius, the water-bearer, 20th
of January.
X Pisces,, the fishes, 18th of Feb.
32. The equinoctial points* are Aries and Libra, -where the eclip-
tic cuts the equinoctial.
33. The solstitial points\ are Cancer and Capricorn.
34. The colures\ are the two meridians passing through the
equinoctial and solstitial points. The one called the equinoctial,
and the other the solstitial colure.
35. The horizon is a great circle,* which separates the visible
half of the heavens from the invisible. It is distinguished into
two kinds, the sensible and the rational.
* The point aries is called the vernal equinox, and the point libra the
autumnal equinox. When the sun is in either of these points, the days and
nights are equal on every part of the globe.
f When the sun is in or near these points, the variation in his meridian,
or greatest altitude, is scarcely perceptible for several days, because the
ecliptic, near these points, may be considered nearly parallel to the equinoe-
tial, and hence in these points, the sun does not perceptibly vary his decli-
nation for some days. When the sun enters the beginning of cancer, all the
Inhabitants on the north side of the equator have their longest day % and those
in the southern hemisphere their shortest. When he enters Capricorn, the
inhabitants of the northern hemisphere have their shortest day, and those iit
the southern their longest. The learner must notice, that when the sun en-
ters cancer, all the inhabitants within the north polar circle have constant day,
and those within the south polar circle constant night, but wh$n the sun
enters Capricorn, the reverse happens. They are called solstice* from the
circumstance of the sun's standing' still, or having no motion when he is in
either of these points, hence said to be stationary (solis static.)
$ These colures divide the ecliptic into four equal parts, and mark the
four seasons of the year. In the time of Ifipparckus the equinoctial colure
ies supposed to have passed through the middle of the constellation aries.
Hipparchus was born at Nicsea, a town of Bythinia in Asia minoir, about 75
miles S. E. of Constantinople, now called Isnic ; he flourished between the
154th and 163d olympiads, or between 160 and 135 years before Christ. He
foretold eclipses, and as Pliny remarks, was the first who dared to number
the stars for posterity, and reduce them to a standard. He gave a catalogue
of 1022 stars, and rendered many other important serviccs\o astronomy.
Horizon takes its name from the Greek word orizon CfniensJ because
it defines or bounds our view. The sensible horizon extends only a few
miles ; thus at the height of 6 feet, the utmost extent of our view on the
earth, or sea, would be 2.42 miles ; at 20 feet 4.4 geographical! miles, &c.
In general, if h be the height of the eye above the surface of the sea, and d
the diameter of the earth in feet, then ^d^Ti^h will nearly' shew the
greatest extent to which a person can see, or the diameter of t he sensible
horizon, the centre being- supposed at the eye. (Euclid, 36 prop , 3b.) This
DEFINITIONS, &e.
36. The sensible or apparent horizon is that circle that termi-
nates our view, where the sky, and the land or water, appear to
meet.
37. The rational or real horizon, is an imaginary circle, whose
plane passes through the centre of the earth, parallel to the plane
of the sensible horizon.
38. The wooden horizon is that circular plane circumscribing
the artificial globe, which represents the rational horizon on the
real globe.
39. The cardinal points of the horizon, are the east, west, north
and south points.*
40. The cardinal points in the heavens, are the zenith the nadir,
and the points where the sun rises and sets
4 1 The cardinal points of the ecliptic are the equinoctial and
solstitial points, which mark out the four seasons of the year ; and
the cardinal signs are ^ Aries, 05 Cancer, =^ Libra, and itf
Capricorn.
42. The Zenith is a point in the heavens exactly over our heads,
and is the elevated pole of our horizon.
determines the apparent rising, setting-, &c. of the sun, stars, planets, &c.
The rational horizon determining their real rising, setting 1 , &c. The wooden
horizon respecting the rational horizon on the real globe of the earth, is di-
vided into several concentric circles. On Bardin's new British globes they
are arranged in the following order : the 1st circle marked amplitude, is
numbered from the east towards the north and south, from to 90, and
from the west towards the north and south in the same manner. The 2d
circle marked azimuth, is numbered from the north and south points of
the horizon towards the east and west from to 90. The 3d circle repre-
sents the 32 points of the compass. The degrees belonging to these may be
found iai the circle of amplitude. The 4th circle contains the twelve stgnx
of the Zodiac, The 5th, the degrees corresponding to each sign, each com-
prehending 30. The 6th contains the day of the month corresponding- to each
degree, &c. of the sun's place in the ecliptic. The 7th contains the equation
of time, the sign -f- shews that the clock is faster than the dial by so many
minutes, the sign that it is slower, and the number of minutes in the dif-
ference is expressed opposite the corresponding- days of the month. The
8th circle contains the twelve calendar months of the year, &c. These cir-
cles are in the same order on Gary's globes, except that of the equation of
time, which is represented on a vacant part of the globe between the tro-
pic's, nearly in the shape of the figure g. The days of the month being
marked in the curve of the figure, and the time or equation on a small scale
drawn through that point where the curve of the figure intersects, in a di-
rection pa rallel to the equator.
Though the rising and setting of the stars respect the rational horizon, and
the place of observation reduced to the earth's centre, yet it holds true of
the sensible horizon, the spectator being placed on the earth's surface, on
account of the great distance of the fixt stars, the semidiameter of the earth
being no i nore than a point at that immense distance.
* The east is that point of the horizon where the sun rises when in the
equinoctia I, and the -west is the point directly opposite on the plane of the
horizon, < >r where the sun sets when the clays and nights are equal : the
south is 9C ) distant from the east or west, and is that point towards which
the sun appears at noon, to those situated in north latitude, and the north
is that poi nt of the horizon directly opposite to the south.
v DEFINITIONS, tff. 9
43. The nadir is a point in the heavens opposite to the zenith,
or directly under our teet, and is the depressed pole of our horizon.
44. The mariners compass is a representation of the horizon,
which is divided into 32 equal parts, and is so called from its being
used to ascertain the course of a ship at sea.
45. The -variation of the comfiass* is the deviation of its point?
from the corresponding points in the heavens, or the angle formec|
between the true and magnetic meridian, and is reckoned towards
the east or west.
46. Azimuth or -vertical circles are imaginary circles passing
through the zenith and nadir, cutting the horizon at right angles.f
47. The azimuth of any object in the heavens is an arch of the
horizon, contained between a vertical circle passing through the
object, and the north or south points of the horizon.
48. The firime -vertical is that azimuth circle, which passes
through the east and west points of the horizon. :$
49. The altitude of any object in the heavens is an arch of a
vertical circle, contained between the centre of the object and the
horizon.
50. The zenith distance of any celestial object is an arch of a
vertical circle, intercepted between the centre of the object and
the zenith.
5 1 . The meridian altitude, or meridian zenith distance, is the
altitude or zenith distance, when the object is on the meridian.
52. The/zo/ar distance of any celestial object, is an arch of the
meridian, contained between the centre of that object and the pole
of the equinoctial.
53. The quadrant cf altitude is a thin slip of brass, one edge of
which is divided into degrees, &c. equal to those of the equator,
arid is used to find the distances of places, &c. on the earth, and the
distances, altitude, Sec. of bodies in the heavens,.
54. The amplitude of any object in the heavens, is an arch of
the horizon contained between the centre of the object, when ris-
ing or setting, and the east or west points of the horizon. Or it
is the number of degrees which the sun or a star rises from the
east and sets from the west.
*. See the note to definition 54 and problems 49 and 50, part 2d.
f- The altitudes of the heavenly bodies are measured on these circles t
they may be represented by the quadrant of altitude screwed in the zenith
of any place and moving- the other end along- the wooden horizon of the
lobe. These circles are always at rig-lit angles to the horizon.
\ This is always at right angles both with the brazen meridian and ho*
rizon.
In our summer the sun rises to the north of the east and sets to tha
north of the west ; and in the winter it rises to the south of the east and
sets to the south of the west. The amplitude and azimuth are in point of
utility, much the same ; the amplitude shewing- the bearing- of any object
when it rises or sets, from the east or west points of the horizon, and the-
azimuth the bearing- of any object when it is above the horizon, cither from
the north or south points thereof. They are generally useful in determining
B
10 DEFINITIONS, &c.
55. Time* is that succession in the existence of beings, which'
have a beginning and will have an end, and is measured by the mo-
tion of some moving body. It is distinguished into years, months,
weeks, days, hours, minutes, Sec.
56. Time is either absolute and relative, true and apparent, or
mathematical and common. Absolute, true, and mathematical time,
6f itself and from its own nature flows equably, without regard to
any thing external, and by another name is called duration ; rela-
tive, apparent, and common time, is some sensible and external
measure of duration, by means of motion, whether accurate or un-
equal, and is commonly used instead of true time.
57. The equation of time\ is the difference between the abso-
lute and relative time, or it is the difference of time shewn by a
well regulated clock and a correct sun dial.
58. Apparent noon is the time when the sun comes to the meri-
dian, viz. 12 o'clock, as shewn by a correct sun diai.
the variation of the magnetic needle. For if th6 observed and true ampli-
tudes be both north or both south, their difference will be the variation ; but
if one be north and the other south, their sum will be the variation. In like
manner if the true and observed azimuth, be both east or both west, their
difference will be the variation ; if otherwise, their sum will be the variation.
The variation is easterly, when the true bearing is to the right hand of the
magnetic bearing, but westerly when to the left hand ; the observer being 1
supposed to look directly towards the point representing the magnetic
bearing*.
* What time is in itself, or what its physical essence is, no philosopher
can fathom or define, but this we know, and it is the most important know-
ledge for us, if reflected on, that it hurries us to that eternity in which time
has no existence, and that every moment may be the last" momentum a
quo tota peiidit ceternitas" If then it be necessary to consider time, as it re-
gulates our seasons, is it not more necessary to consider it, as it relates to
an immortal existence towards which it imperceptibly hurries us. Truths
of this nature are better calculated to expand our ideas, and point out to us
that state which has no termination or limits, and in which we are destined
to enjoy a dignified existence, than those whose objects are as fleeting us
time itself; for as soon as futurity begins to expand its extensive prospects,
then we see the vanity of what the world sets such a value on, and learn
to value those things alone which are immortal.
f The equation of time arises from t\\r> principal causes, the sun's unequal
motion in the ecliptic, describing the southern signs in less time than the
northern, the difference amounting to about eight days ; and from the obliqui-
ty of the plane of the ecliptic to that of the equator. For the space between
two meridians, or hour lines on the ecliptic will not, in consequence, be al-
ways the same as the space between the same meridians on the equator,
the difference being sometimes greater, sometimes equal, and sometimes
less ; but as the sun in consequence of this difference, takes sometimes less,
sometimes more than 24 hours, in revolving from any meridian, until his
return to the same again, it thence follows that the hours shewn by a well
regulated clock, must be different from those shewn by a true sun dial, and
hence the equation of time. If the sun performed its annual revolution in
the plane of the equator, there would be no equation except what arises
irom the difference in his annual motion, (see prob. 22, part 2d, Keil lect
25, Ferguson, chap. 13, or Mayor's tables, published by Nevil Maskehm-.
and note to prob. 8.)
DEFINITIONS, csV. 11
59. True or mean noon is the middle of the day, or 12 o'clock,
^ shewn by a well regulated clock, adjusted to go 24 hours in a
mean solar day.
60. An hour is a certain determined part of the day, and is
either equal or unequal. An equal hour is the 24th part of a mean
natural day, as shewn by well regulated clocks, &c. unequal hours
are those measured by the returns of the sun to the meridian, or
those shewn by a correct dial. Hours are divided into 60 equal
parts called minutes, a minute into 60 equal parts called seconds, a
second into 60 equal parts called thirds, &c.
6 1 . A true solar day>* is the time from the sun's leaving the
meridian of any place on any day, till it returns to the same me-
ridian on the next day. Or it is the time elapsed from 1 2 o'clock
at noon, on any day, to 12 o'clock at noon on the next day, as shewn
by a correct sun dial.
62. A mean solar dayrf is the space of time consisting of 24
hours, as measured by a clock or time-piece.
63. The astronomical or natural day,\ is the time from noon to
noon, as shewn by a correct dial, and also consists of 24 hours.
* A true solar day is subject to a continual variation, arising from the ob-
liquity of the ecliptic and the unequal motion of the earth in its orbit ; the
duration thereof sometimes exceeds and sometimes falls short of 24 hours,
as taken notice of in the note on the equation of time. The variation is the
greatest about the 1st of November, when the solar day is 16' 15" less than
24 hours, as shewn by a well regulated clock.
| There are in the course of a year as many mean solar days as there are
true solar days, the clock being as much faster than the sun dial on some
days of the year, as the sun dial is faster than the clock on others, as may be'
seen by consulting the analemmaor the circle on which the equation of time
is marked on the globes. Thus the clock is faster than the sun dial from the
24th of December to the 15th of April, and from the 16th of June to the 31st
of August ; but from the 15th of April to the 16th of June, and from the 31st
of August to the 24th of December, the sun dial is faster than the clock.
When the clock is faster than the sun dial, the true solar day exceeds 24
hours ; and when the sun dial is faster than the clock, the true solar day is
less than 24 hours ; but when the clock and sun dial agree, viz. about the
15th of April, 16th of June, 31st of August, and 24th of December the true
solar day is exactly 24 hours. (See the table annexed to problem 21.)
t This is called a natural day, being of the same length in all latitudes,
It begins at noon, because the increase and decrease of days, terminated by
the horizon are very unequal among themselves ; which inequality is like-
wise augmented by the inconstancy of the horizontal refractions (see 183
Perguson's Astronomy) and therefore the astronomer takes noon, or the mo-
ment when the sun's centre is on the meridian, for the beginning of the day.
The hours are reckoned in numerical succession from 1 to 24. Navigators
begin their computation at noon 24 hours before the commencement of the
astronomical day, reckoning their hours froml to 12; the first 12 hours arc
marked A. M. (ante meridiem} or forenoon, the second P. M. (post meri-
diem } or afternoon. All the calculations in the nautical almanac are adapt-
ed to astronomical time. The declination, 8cc. there calculated, is adapted to
the beginning of the astronomical day, or to the end of the sea day; it being
at the end of the sea day, that mariners want the declination, to determine:
their latitudes.
12 DEFINITIONS, &v.
64. The artificial day, is the time elapsed between the sun's ris-
ing and setting, and is variable according to the different latitudes
of places. Night, is the time from sun setting to sun rising, and
varies in like manner.
65. The civil day,* like the astronomical or natural day, consists
6f 24 hours, but begins differently, according to the customs of dif-
ferent nations
66. A siderceal day, is the interval of time from the passage of
any fixed star over the meridian, till it returns to it again ; or it is
the time which the earth takes to revolve once round its axis, and
consists of 23 hours, 56 minutes, 4> seconds.
Note. Though -we suppose the earth to turn on its axis once in 24 fiours from
H>est to east, yet its eocact revolution is as above, making about 366 revolutions
in 365 days. But as the sun advances about 1 in its orbit daily,* which cor-
responds to about 4? of time, the day is properly taken 24 hours, because the
earth has to advance 1 more on its axis to have tJie sun over the same meridian
as on the preceding day.
* The ancient Babylonians, Persians, Syrians, and most of the eastern na-
tions, began their day at sunrising, which custom is followed by the modern.
Greeks. The ancient Greeks, Jews, &c. began their day at sunsetting, and
this custom is observed by the modern Austrians, Bohemians, Silesians, Ital-
ians, Chinese, &c. The Arabians beg-in their day at noon like the astrono-
mers. The more ancient Jews, tog-ether with the ancient Egyptians, Ro-
mans, &c. began their day at midnight, and this custom is followed by the
9nglish, French, Germans, Dutch, Spanish, Portuguese and Americans.
The famous astronomers Hipparchus, Copernicus, and some others, began
their day in like manner from midnight. Those wko begin their day at
aim rising, have this advantage, that their hours tell them how much time is
Already past since sun rising; and they who reckon their hours from sun
setting, know how long it is to sun setting ; and hence they may proportion
their journies and labours for that time. But both have this inconvenience, that
their midday and midnight happen on different hours, according to the seasons
<Jf the year. The Babylonians, &c. reckoned 24 hours in order, from sun ris-
ing to its rising again, hence called Babylonish hours. In Italy, See. where they
reckon their day from sun setting 1 , they likewise reckon 24 hours in order :
these hours are hence called Italian hours. The Jews and Romans formerly
divided +he artificial days and nights each into 12 equal parts ; these are
termed Jeivish hours, and are of different lengths, according to the seasons of
the year. This method of computation is now in use among the Turks, and
the hours are styled the first hour, 2d hour, &c. of the day or night, so that
midday always falls upon the 6th hour of the day. These hours were also
called planetary hours, because in each of these hours one of the seven planets
was supposed to preside over the world. The first hour after sun rising- on
Sunday was allotted to the Sun, the next to Venus, the 3d to Mercury, and the
rest in order to the Moon, Saturn, Jupiter and Mars. By this means, on the
first hour of the next day, the moon presided, and so gave the name to that
day ; and thus the seven days had names given to them respectively, from
the planets that were supposed to govern on the first hour.
a The earth revolves round the sun in 365 days nearly, and the ecliptic
consists of 360, hence 365^ D : 360 :: ID : 59' 8" 2, the daily mean mo-
tion of the earth in its orbit, or the apparent mean motion of Uie sun, in a
day.
DEFINITIONS, We. 13
67. A week,* is a system of seven days, each of which is dis-
tinguished by a different name.
68. A month, is properly the space of time the moon takes to
perform one revolution round the earth, and is either astronomical
or civil.
69 The astronomical month, is the time in which the moon
passes through the zodiac (or that zone in which are the 12 con-
stellations or signs, through which the sun passes.) This month is
either periodical or synodical.
70. The periodical month, is the time intervening, in a revolu-
tion of the moon, from her leaving any point until her return to
the same, and is equal to 27 days, 7h. 43' 5".
7 1 . The synodic month, or lunation, is the time between the
moon's parting with the sun at a conjunction, until her return
again, or the time between two new moons, and is equal to 29d.
12h. 44' 3".
72. The chdl months, are those which are framed for the uses
of civil life, and are different, as to their names, number of days,
and times of beginning, in different countries.!
* A week is the most ancient collection of days that ever was, as is evi-
dent from the sacred writings. The Jews always made use of this collection,
and every other nation since the establishment of Christianity, wherever it has
been received. All nations that have any notion of religion, set apart one
day in seven for public -worship. The Jews observe Saturday, or the seventh
day of the week, for their Sabbath or day of rest, being that appointed in the
3d commandment under the law. But the day solemnized by Christians, is
Sunday or the first day of the week, being- that in which our Saviour rose from
the grave, and on which the apostles afterwards used more particularly to as-
semble together, to perform divine worship. The French had adopted a cal-
endar entirely new, soon after the abolition of royalty in 1792 ; " But from the
rapidity with which every thing is there returning to the ancient customs (says
a late writer) it is probable that in a short time it will be discontinued." This
has accordingly since taken place, which shews no less the truth of the asser-
tion, than the futility of the project.
f Thus the first month of the Jewish year fell, according- to the moon, in
our August and September, old style ; the 2d in September and October, &c.
The first month of the Egyptian year began on the 29th of our August: the
first month of the Arabic and Turkish year began the 16th of July : the first
month of the Grecian year fell according to the moon in June and July ; the
2d in July and August, &c. For a further account of these, see Ferguson's
Astronomy, chapter 21.
The names, both of the weeks and months, which are no\v adopted in civil
use, originated among our heathen ancestors, and, as well as the names of
the constellations in the heavens, alluded to some part of their idolatrous wor-
ship, or to their gods. The months, as will be shewn in the, following note,
for the most part, took their names from some of these gods. The Jews,
before the Babylonish captivity, reckoned their twelve months in numerical
order, 1st. 2d. 3d. &c. Sometimes, however, besides these ordinal names,
they distinguished some of their months by particular appellations, alluding
1o the season, &c. Thus the 1st month was called Mib (Exod. 13, v. 4.)
tiie 2d. Zins (3 Kings 6, v. 1 & 37.) the 7th. Ethanim (3 Kings 8, v. 2.) the
8th, liul, &c. Where the first Mib signifies the month ofnw corn, See. This
month ans'-vers to the Niacm of the ancient Syrians (whose months, with
little variation, the Hebrews adopted after their captivity.) The Phanemoth
14 DEFINITIONS, &c.
73. That space of time in which the sun describes one sign, or
30 of the ecliptic, is also called a solar month) and is about 30^
days.
74. A year* is properly the space of time measured by a revo-
lution of the sun in the ecliptic, and is of several kinds.
of the Chaldeans and Egyptians, the Elaphebolion of the Athenians, the Xan-
thikos of the Macedonians and other Grecian states, with the Maccabees and
till Syria, the Mnharram of the Arabs and Turks, the Jttartius of the Pagan
Romans and our J\tarch. (See Censorinus, &c.) July and August, among
the old Romans, were called Quintilis and Sextilis, and hence, except a few
of their first months, they agreed with the ancient Jewish reckoning.
The days of the week were also called, by all the idolatrous nations, after
the names of the planets, and these after the names of their pretended gods.
Thus the first day was called Dies Solis (the sun being the principal lumina-
ry) the 2d. Dies Lunae ; the 3d. Dies Martis ; the 4th. Dies Mercurii ; the
5th: Dies Jovis ; the 6th. Dies Veneris, and the 7th. Dies Saturni. The Saxons
called the days of the week by the name of the idol which on that day they
particularly worshipped. Thus the first day was called Sunday, from their
worshipping the sun on that day ; the 2d. JMonday, from their worshipping
Diana, or the moon ; the 3d. Tuesday, from their idol Tuisco or Tew, the
Saxon name of Mars ; the 4th. Wednesday, from Woden or Odin, another of
their idols ,- the 5th. Thursday, from tteeir idol Thor, the Saxon name of Ju-
piter ov Jove ; the 6th. Friday, from Friga or Frigidag, supposed to be the
Venus of the Saxons, ond the 7th. Saturday, Saeter or Saetor, an idol by them
then worshipped. For this reason some reject their names, but in general
all the modern Europeans adopt them, to avoid that confusion in their calen-
dar resulting from the introduction of new names. The Jews, however, called
the days of the week Sabbaths. In St. Mark, c. 16, v. 2, the first day of the
week is called Una Sabbatorum, and in v. 9, Prima Sabbati. In St. John, c.
20, v. 1, the first day of the week is called Una Sabbati. In St. Luke, c. 18,
v. 12, twice in the week is called Bis in Sabbato, &.c. The Latin church
called the days of the week Ferix (holidays or days of rest.) Thus Feria
prima (Sunday) Feria secunda (Monday) &,c. Feria being the same as Sab-
bath, which signifies rest or cessation. These latter denominations of the
days of the week, evidently allude to the six days, in which God created the
world, ordering the 7th to be a day of rest, from which the others, therefore,
took their name. And hence this division of the week is anterior to all others.
* Or a year is that period of time in which all the variety of seasons
return and afterwards begin anew. The ancient Haitians divided the year
into twelve calendar months, to which they gave particular names, as
follows : January from Janus, the most ancient king of Italy, to whom the
people dedicated this month. February from the latin word Febnto, to
purify. In this month the ancient Romans, particularly the priests of Pan,
made use of purifications and sacrifices for the ghosts of the dead.
March from their god Mars, to whom this month was kept sacred. April
from Aperio, to open or unfold, because in this month the spring begins
t disclose all the beauties of the vegetable creation ; or as some suppose
from the Greek appellation of Venus. May from Maia, or Jllaius, a hea-
then goddess, to whom this month was kept sacred. June from the hea-
then goddess Juno, or as some say from Juvenis a youth, as nature, in
this month, appears in the vigour and bloom of youth. July from Julius
Ctfsar, the Roman general. August from Augustus Cesar, the first Roman
emperor. September, October, November, and December, from Septem,
Octo, Novem, and Decent; these months in the Roman calendar, being
the 7th, 8th, 9th, and 10th months, their year beginning on March an
ending- on February. (See Adams Roman Antiquities, page 327.)
DEFINITIONS, bV. 15
75. A solar or tropical year, is the time measured by the sun in
the ecliptic, in passing from one equinox or solstice, until it re-
turns to the same again, and is equal to 365 days, .5 hours, 48 mi-
nutes, 48 seconds.
76. A sydereal year is the space of time which the sun takes, in
passing from any fixed star, till it returns to the same again, and
consists of 365 days, 6 hours, 9 minutes, 8 seconds.
77. The civil year is the common or political year, established
by the laws of a country, and is either lunar or solar.
June, September, and November, has each 30 days, February in common
;ars has 28 days, but in leap year 29. Each of the rest has 31 days.
he year is also divided into four quarters, viz. Spring, Summer, Jiutumu
and Winter. These quarters are properly made when die sun enters into
the equinoctial and solstitial points of the ecliptic ; but in civil uses they
are differently reckoned, according 1 to the customs of several countries.
The year used by the ancient Grecians and Romans did not exactly
agree with the motion of the sun, and hence their Winters, Summers,
and in general their seasons, were found every year to differ considera-
bly, so that the same seasons did not uniformly happen in the same
months, as in a well regulated calendar should be the case. They there-
fore proposed the rising- and setting of the stars instead of this erroneous
calendar, not doubting that the sun returning to the same place in respect
of the same fixed star, did not come to the same place again in respect to
the equinoxes or solstices ; that is when the same star should rise or set
cosmically, achronically, or heliacally, the same seasons should again
return. And hence we find such use made by the antients, of what is
called the poetical rising and setting of the stars. In time, however, this
method was found erroneous, and is now almost entirely out of use, owing
to the precession, or rather recession of the equinoctial points, not then taken
notice of. The difference between a solar and sidereal year being 20 24',.
and as the sun returns to the equinox every year before it returns to the
same point in the heavens, the equinoctial points have therefore a slow
motion backwards. The sun apparently describing the whole ecliptic or
360 in a tropical year ; hence we have this proportion, 365^7 : 360 :: Id
: 5S' 8" 2. the daily mean motion of the earth, or the apparent mean mo-
tion of the sun in a day, and therefore the sun in 20' 24" of time, describes
5(H". For Id : 5S' 8" :: 20' 24" : 50i" the precession of the equinoxes,,
nearly corresponding to what JVevvton makes it, as derived from physical
causes, (prop. 39, b. 3, of his principia.) This slow motion of the sun.
in receding from the equinoctial points, every year, is called the preces-
sion of the equinoxes, and is performed from east to west, contrary to the
order of the signs, which is from west to east. Now 50$" : lyr. :; 360 :
25791 years, the time in which the equinoctial points would perform one
revolution. Hence in 2149 years the stars would appear to recede 30
or 1 sign backwards. In the time of Hipparchus, the equinoctial points,
were fixed in Aries and Libra ; but now these signs are 30 to the east-
ward, &c. so that Aries is now in taurus, taurus in gemini, &c. hence the
rising and setting of the stars, at particular seasons of the year, as describ-
'ed by later writers, such as Hcdod, Eudoxus, Pliny, &c. do not answer
their description.
The anticipations of the seasons is not however owing- to the precession
of the equinoctial and solstitial points in the heavens, which can only effect
the apparent motions, places and declinations of the fixt stars ; but to tb:
difference between the civil and solar year, which is 11' li" (commonly
reckoned IV S'.) From this difference it would happen that in 129 years.
16 DEFINITIONS, err.
78. The lunar year is the time measured by twelve synodic re-
volutions of the moon, and consists of S54 days, 8 hours, 48 mi-
nutes, and 36 seconds.
there would be a difference of one day in the time of the equinoxes, and
therefore a difference of 10 days in 1260 years, the time between the
JVicene Council, A- D. 325 and 1585, when. Pope Gregory, 15th reform-
ed the Julian calendar. Hence the vernal equinox happening the 21st of
March in the time of the council of Nice, in the year 1582 it was found
to happen on the llth. That the equinox might therefore be reduced to
its former place, 10 days were suppressed in the month of October in the
year 1582, and the 5th day was called the 15th, and thus the llth day of
the March following, being the time of the equinox, became the 21st as
in the time of the council of Nice, which fixed the time of keeping Easter.
At the time that Caesar with the assistance of Sosigines reformed the cal-
endar, the vernal equinox happened on the 25th of March. Besides the
above, another correction was found necessary, that the same seasons
might be kept to the same times of the year, as one day every fourth
year, according to the Julian intercalation, was found too much. For this
purpose the bissextile day in February, at the end of every century of
years not divisible by 4, was to be omitted, and these years reckoned as
common years (every 4th year according to the Julian or civil account
being bissextile or leap year.) Thus the 17th, 18th, 19th centuries, or the
years 1700, 1800, 1900/&C. which according to the Julian account would
be leap years, were to be reckoned as common years, (17, 18, 19, &c.
not being divisible by 4) and to retain the bissextile day at the end of
those centuries divisible by 4, as the 16th, 20th, 24th, &c. or the years
1600, 2000, 2400, &c. By this correction the difference between the
civil and solar accounts will differ no more than 2 hours in 400 years, and
in less than 5082 years will not amount to a whole day, at the end of
which time a new correction for this day will become necessary. With-
out these changes, the seasons in time would be entirely reversed with re-
gard to the months of the year. It was Julias Cjesar who first ordained
that one day should be added to February, every 4th year, by causing the
24th to be reckoned twice ; and because this 24th day was the 6th
CsextilisJ before the Kalends of March, there were in this year two of
these scxtiles, and hence this year was called bissextile. This being
corrected, \vas thence called the Julian year, afterwards the Gregorian,
from the farther corrections of Pope Gregory. This Gregorian year is
now received in almost every country where truth or exactness is regard-
ed, and from hence is called the Civil year. The Civil year thus correct-
ed took place in different countries of Europe at different times, and was
not adopted in England until A. D. 1752, at which time a correction of
11 days became necessary, the 3d of September being called the 14th.
This is now called the New Stile, as the Julian is called Old Stile.
The year 1700, happening between the time of the correction by Pope
Gregory, and that made by the British, this year in the Gregorian ac-
count was considered as a common year, and thus a day was omitted,
which in the Julian was not ; and as the Gregorian account omitted 10
days in the beginning, the English omitted 11, to make theirs agree with
the former. And moreover, as the year 1800 was a common year, there
is now 12 days difference between the old and the new style- It is almost
needless to mention, that in 1752, the United States were British colonies,
and hence the corrected account or new style was here adopted at the
same time, and is the account now in use.
The beginning of the year in different countries is no less various than
its form : but a further detail would be inconsistent with the plan of thi?
DEFINITIONS, fcV. 17
79. The civil solar year, or Julian year, is a period of 365 clays,
6 hours, but the common years contain only 365 days, and every
4th year or bissextile 366 days.
80. A cycle,* is a period of time, after which the same pheno-
mena of the celestial bodies begin to occur again, in the same order.
abridged introduction. Those who wish to see more on this subject, may
consult Gregory, Keil, Ferguson, Ewing, or Vince's astronomy, or vol. 3
of Ozinam's Mathematical Recreations.
* In the cycle of the sun the return of the days, &c. does not differ 1
in 100 years, and the leap years begin their course again with respect to
the days of the week on which the days of the month fall. In the cycle of
the mo'on the new and full moons return, &c. within If hours of the time
in which they happened on the same days of the month, 19 years before ;
hence in 312 years this difference increases to a whole day, so that this
cycle can only hold for that time, and hence for the next 312 years the
golden number ought to be placed one day earlier in the calendar. This
correction is however made at the end of whole centuries, and hence at
the end of 300 years the new moon is advanced 1 day for 7 times succes-
sively, that is, during 2100 years. To account for the odd 12^ years, they
deferred putting the moon forward to the end of 400 years, making a pe-
riod of 8X312^ = 2500 years. The indiction was established by Constan-
tine in the year 312. The year of our Saviour's birth according to the
vulgar era, was the 9th year of the solar cycle, or the 1st year of the lu-
nar cycle ; and the 312th year after his birth was the 1st year of the
Roman indiction. Therefore to find the year of the solar cycle add 9 to
any given year of Christ, an.d divide the sum by 28, the quotient will
give the number of cycles elapsed since his birth, and the remainder the
cycle for the given year ; if nothing remains, the cycle is 28. To find the
lunar cycle, add 1 to the given year of Christ and divide the sum by 19,
the quotient is the number of cycles in the interval, and the remainder
the cycle for the given year : if nothing remains, the cycle is 19. For the
indiction, subtract 312 from the given year of Christ, divide the remain-
der by 15, the remainder after this division is the indiction for the pre-
sent year.
The ancients formed the cycle of the moon, by taking any year for the
cycle and observing all the days in which the new moon happened through
the year, and placing the number 1 against each day ; in the 2d year of
the cycle they placed the number 2 against each day in which the new
moon happened as before ; the 3d year the number 3, &c. through the
whole 19 years. These numbers corresponding to one cycle, were fitted
to the calendar to point out the new moons in every future cycle, and
from their great use were written in gold, and thence called gulden mini'
bers. The whole day gained in 312 years, which since the" council of
Nice in 325 has since been neglected, causes the golden numbers to be 5
days higher in the old style, or 7 days lower in the new, than they were
at the abovementioned council, and ought to be so placed in the calendar.
Since 1800 there are 12 days difference between the old and new style.
The golden number is not, however, so well adapted to the Gregorian as
the Julian calendar. The golden number is the same as the lunar cycle,
and found in the same manner. Thus to find the golden number for 1812,
1812-}- l~i- 19=95 and 8 over; hence 8 will be the golden number. Any
other year will answer as well as the current year, by adding its own
golden number to it, and proceeding as above with the difference between
both years.
In the calendar it is usual to mark the seven days of every week with
the first 7 letters of the alphabet, calling the first of January A, the 2<i
C
18 DEFINITIONS, &c.
81. The cycle of the sun, is a period of 28 years, which being
completed, the days of the month return in the same order to the
same days of the week ; the sun's place to the same signs and de-
grees in the ecliptic, &c.
82. The cycle of the moon, or metonick cycle, called also the
golden number, is a revolution of 19 years, which being completed,
the new and full moons return to the same days of the month, 8cc.
B, the 3d C, the 4th D, the 5th E, the 6th F, the 7th G, the 8th A again,
and so on through the year ; and whatever letter corresponds to the first
Sunday of January, will answer to every Sunday in a common year, and
is therefore called the dominical letter.
A common year contains 52 weeks and one day, therefore the first and
last days of a common year fall on the same day of the week ; hence if
any year begins on Sunday, the next will begin on Monday, &c. but this
order is interrupted by the leap years ; February having one day more
than in common years : so that the dominical letter for March and the
rest of the year, will be the letter preceding that which swerved for Janu-
ary to the 24th of February ; leap years having therefore two dominical
letters. The dominical letter is thus found : to the given year add |th
of it, for the leap years contained in it (neglecting the fractions if any)
and from the sum subtract 7 for the 18th century (or from 1800 inclusive
to 1900) 8 for the 19th and 20th centuries, 9 for the 2lst century, 10 for
the 22d century, 11 for the 23d and 24th centuries ; because the three
years 2100, 2200, and 2300 will not be leap years, &c. divide the remain-
der by 7, and the remainder after this division will give the dominical let-
ter, reckoning from the last G towards the first A. If remains, the
dominical letter will be A ; if 1 remains, the dominical letter will be G ;
if 2 remains, the dominical letter will be F, 8cc. thus ^ ?' C J *?' ?' ' ^'
V, O, O, t, 3, 2, 1,
where the figures or remainders, correspond to the dominical letters
above them. Hence to find the dominical letter for 1807, it will be 1807
-} . 7=2251 (rejecting the remainder) which being divided by 7,
will leave a remainder of 4 corresponding to D the dominical letter re-
quired, and counting back from D to A, thus (I)) Sunday, Saturday, Fri-
day, Thursday, which corresponds to A, the day on which the year began.
To find the dominical letter for 1812, proceeding as above, we shall find
a remainder of 4, which corresponds to D, but as 1812 is a leap year, it
has two dominical letters, that is E, the letter preceding D, counting from
G, which answers for January and February to the 24th, and D the rest
of the year. To find the dominical letter for 1910, we have 1910-f-^l?
8=2379, which divided by 7 leaves a remainder of 6 corresponding to B
the dominical letter for that year, \yhich therefore will begin on Satur-
day. The dominical letters for 1996, a leap year, are GF ; this year will
therefore begin on Monday, &c. (see a table of the dominical letter to
the year 4600 in Ferguson's Astronomy, pa. 398, 8th ed.)
The difference between a solar and lunar year, which is 10 days 21h. 0'
12" (defs. 75 and 78) or nearly 11 days, constitutes the epact When the
solar and lunar years begin together, the epact for that year is or 29 ;
the 2d year the epact is 11 ; the 4th, 33 : but when the epact exceeds 30,
an incercalary month is added, making the lunar year consist of 13 months,
and hence at the beginning of the 4th year the epact is 3, the 5th, 14, &c.
all the varieties happening in 19 years, or one lunar cycle, except the cor-
rections made at the end of centuries, &c. to allow for which the follow-
ing rule must be observed in calculating the epact for any year from 1800
DEFINITIONS, &c. 1<J
83. The cycle ofindiction^ is a period of 15 years, but has no
reference to the celestial motions.
84. The Dionysian period, is the number of years that arises
by multiplying the cycles of the sun and moon together, and con-
sists of 532 years.
to 1900 Multiply the golden number for the given year by 11, and the
product divide by 30, then subtract 11 from the remainda^ the last re-
mainder will be the epact If 11 cannot be subtracted, SOjEbst be added
to the remainder, and then 11 subtracted as before. Tbfll to find the
epact for 1810 ; the golden number for 1810 is 6, this multiplied by 11
fives 66, which divided by 30, leaves a remainder of 6; hence 6-J-30-
1=25 the epact required. To find the epact for 1812, the golden num-
ber is 8 ; hence 8xU-v-30 leaves a remainder of 28, and 2811 = 17 the
epact required. The epact may be found thus, without the golden num-
bers. Divide the given year by 19, multiply the remainder by 11, the
product will be the epact if it does not exceed 29, but if it exceeds 29,
divide 30 into it, and the remainder will be the epact. Ozanam in his
math, recreations, gives the following rule : multiply the golden number
by 11, and take the number of days retrenched by the reformation of the
calendar, from the product ; that is, 11 days if the year be between 1700
and 1800, 12 if betwjen 1800 and 1900, 13 if between 1900 and 2100, &c,
divide the remainder by 30, the remainder after this division will be the
epact. Between 1800 and 1900 this gives the epact 1 day less than the
former methods, between 1900 and 2100 two days less, &c. The former is
however used in the present calendar. But this method will sometimes
differ from the true epact (which is the age of the moon for any year, on
the 1st of January exclusively, or at the end of the preceding year ; or the
number of days since the last mean new moon) the annual epact being too
great.
From the dominical letter, the day of the week on which any day of a
given month falls, may also be found. When the days of the week are
marked by the seven first letters of the alphabet, the letter A is always at
the first day of January and October ; B at the first of May ; C at the first
of August ; D at the first of February, March and November ; E at the
first of June ; F at the first of September and December ; and G at the
first of April and July. Hence each letter in the following order, A, D,
D, G, B, E, G, C, F, A, D, F, marks the first day of each month in the
year : and the same letters mark the 8th, 15th, 22d, and 29th days of the
month. If the dominical letter be A, the first of January and October
will be Sundays ; the first of May marked B, will be Monday ; the first
of August marked C, will be Tuesday ; the first of February, March, and
November, being marked with D, will be Wednesdays ; the first of June
marked E, will be Thursday; the first of September and December mark-
ed F, will be Friday ; and the 8th, 15th, 22d, and 29th of the month, will
be on the same days of the week. In the same manner may these days
be found when the dominical letter is any other besides A ; and hence
any day of the year- Tables for this purpose are given in most books of
practical astronomy.
Besides the annual epacts, there are monthly epacts commonly called
the numbers of the months, which are the moon's age on the first day of
every month when the solar and lunar years begin together, and are thus
found : divide the number of days between the first of January and the
first day of any month by 29$, the remainder will be the number for that
month. Thus the epact for January is 0, for February nearly 2, for March
in common years 0, but in leap years 1, &c,
20 DEFINITIONS, &c.
85. The Julian period, is the number of years that arises from
the product of the cycles of the sun, moon and indiction, viz.
28X19x15=7980 years-
86. Positions of the sphere, are its situations witL respect to the
horizon, and are principally three, right, parallel and oblique.
The number for the month being given, the moon's age on any day is
thus found : to the epact for the year add the days of the month, and the
number for the month, the sum, if it does not exceed thirty, is her age ;
but if it exceed thirty, take 30 from it, and the remainder is the moon's
age, if the month has 31 days ; but in months of 30, subtract only 29, ex-
cept when it is leap year. Thus to find the moon's age on the 28th of Jan-
uary, 1811 ; here the epact 6+28=54 and 3430=4 days the moon's
age, one day less than in the nat. almanac. For the 28th of April, 1811,
we have 6-f-2+28=36 and 36 29=7 agreeing with the nat. aim. For
1812, to find the moon's age on the 20th of April, we have epact 17-J-3+
20=40 and 40 30 (the year being bissextile) =10 days, agreeing also
with nat. aim. Whenever accuracy is required, recourse must however
be had to Astronomical calculation.
If the moon's age be multiplied by 5 and divided by 6, the quotient is
the hours, and the remainder multiplied by 12 the minutes, nearly, -when
the moon comes to the meridian, reckoning from noon'*; if to this be added
the time of tide on the days of new and full moon at that place, the sum
will give the time of high water there. The tides at any place happen
always when the moon is in the same position with respect to the meri-
dian of the place. Thus at London it is always high water when the
moon is S. W- or 3 hours past the meridian ; at New-York when she is
S. E. or 3 hours before noon ; at Sandy-Hook when she is E S. E. or
4 hours before noon, &c. (See the tab^ pa. 142 of Hamilton Moore's
Navigation, 10th ed.)
This rule is sufficiently exact for common use, and no rule of calcula-
tion can be given that will always produce an exact answer, as the time
of high water depends so much on the winds, swell of the sea, &c. at the
time. The mean motion of the moon from the sun in a day is 12 11'
26" 7. For according to Mayer, the sun's daily mean motion is 59' 8" 3,
and that of the moon 13 10' 35" the difference of which is the above.
Now 15 : lh. or 60' :: 12 11' 26" 7 : 49' 8. Hence if the moon's age
be multiplied by 49.8 and divided by 60, or multiplied by 5 and divided
by 6 as above f t_L_ being nearly equal ) the quotient will give the
moon's southing nearly. For more accurate methods see the note to ex.
8, prob 18, part 3d, or prob. 39, part 3.
From the above rules the method of finding on what day Easter Sunday
falls in any year, is very simple. At the Council of Nice, Easter Sunday
was fixed on the first Sunday after the full moon, which happens on or
next after the 21st of March, and therefore it must always fall between
the 21st of March and 25th of April. The method is this ; find the day
of full moon on or next after the 21st of March, and then find what day
of the week the full moon is on, and the next Sunday will be Easter Sun-
flat/. The moon's age being given, subtract it from the day of the month ;
or the day of the month increased by 30, the remainder will give the day
on which new moon falls. If to this 7^ days (or rather 29^-r-4) be added,
the mean time of, first quarter is given ; add 15 days (or 29$-j-2) for
mean full moon and 22 % days nearly, for the third quarter. Thus to find
Easter Sunday in the year 1811, the moon's age on the 21st of March was
27 days, hence 21+30 27=24 the day of the month on which new moou
takes place, and 24-f 15=39 and 3931=8 ; hence full moon fell on the
DEFINITIONS, Vc. 21
87. A right sphere* is that position of the earth, where the equa-
tor passes through the zenith and nadir, the poles being in the ra-
tional horizon.
88. A parallel sphere} is that position of the earth, where one
pole is in the zenith and the other in the nadir ; in which case
the equator coincides with, and all its parallels are parallel to
the horizon.
89. An oblique sphere^ is when the rational horizon cuts the equa-
tor obliquely, which is the case with ali parts of the earth, except
those under the poles and the equinoctial.
8th of April, which being- Monday, the Sunday following, or the 14th of
April, was Easter Sunday. To find Easter in the year 1812. Here the
moon's age on the 21st of March will be 9 days (this being leap year)
and 21- 9=12 the day on which new moon will fall ; hence 12-j~15=27
the day on which full moon will fall (the Nautical Aim. gives it 16 f after
12 on the night of the 27th) which being Friday, the 29th of March will
therefore be Easter Sunday, &c.
The Nautical Aim. gives the moon's age one day later than here, pro-
bably making the epact the age of the moon on the 1st of January inclu-
sively, or astronomical time, the above calculation being adapted to civil
time'
The feast of Easter regulates the moveable feasts of tlue whole year.
Thus the 1st Sunday after Easter is Lota Sunday ; Rogation days com-
mence 35 days after Easter ; Ascension Thursday is the Thursday follow-
ing, or the '40th day after Easter ; the feast of Pentecost, commonly call-
ed Whitsuntide, is 10 days after, or the 50th day after Easter ; on the Sun.
day after, or 56 days after Easter, the feast of the Holy Trinity is cele-
brated ; and the Thursday following, or 11 days after Pentecost, or 60 days
after Easter, is the feast of Corpus Christi. The 9th Sunday before Easter,
or 63 days before it, is called Septuagssima, the 8th or following Sunday,
which is 56 days before Easter, Sexagesima, the 7th or 49 days before
Easter is called Quinquagesima, and the Wednesday following, dsh Wed-
nesday, the 1st Sunday of Lent is called Quadragesima, the 5th Sunday of
Lent is called Passion Sunday, the 6th or the Sunday before Easter, Palm
Sunday. The other Sundays in Lent and those after Easter are called by
other names, as Reminescere, Laetare, Judica, Misericordia, Jubilate, &c,
Jldvent Sunday does not depend on Easter, but on the feast of St. Andrew,
which is on the 30th of November, being the nearest Sunday to this feast.
Christinas day is always on the 25th of December. The first of January
or new year's day, is the feast of the Circumcision of our Lord, the 6th of
January is the feast of the Epiphany, or manifestation of Christ to the
Gentiles, &c.
* The inhabitants who have this position of the sphere, live at the equa-
tor. It is called a right sphere, because all the parallels of latitude cut
the equator at right angles, and the horizon divides them into two equal
parts, making equal day and night.
f The inhabitants who have this position of the sphere (if there be any)
live at the poles. It is called a parallel sphere, because all the parallels
of latitude are parallel to the horizon. In this position of the sphere the
sun appears constantly above the horizon for six months.
t So called from the parallels of latitude cutting the horizon obliquely.
In this position of the sphere, the days and nights are of unequal lengths,
the parallels of latitude being divided unequally by the rational horizon.
22 DEFINITIONS, We.
90. A Climate* in a geographical sense, is a part of the surface
of the earth, contained between two lesser circles, parallel to the
equator ; and of such a breadth, that the longest day in the parallel,
nearest the pole, exceeds the longest in that nearest the equator, by
half an hour in the torrid and temperate zones j or by one month
in the frigid zones.
* There are therefore 24 climates between the equator and each polar
circle, and 6 climates between each polar circle and its pole. The climates
between the polar circles and the poles were, in a great measure, unknown
to the ancient geographers ; for Ptolomy does not give an exact computa-
tion of the parallels as far as the polar circles itself. They reckoned only
seven climates north of the equator. The middle of the first northern cli-
mate they made to pass through Meroe, the metropolis of the Ethiopians,
built by Cambyses, on an island in the Nile, of the same name, nearly under
the tropic of cancer ; the second through Syene, a city of Thebais, in Upper
Egypt, near the cataracts of the Nile; the third through Alexandria; the
fourth through Rhodes ; the fifth through Rome, or the Hellespont : the sixth
through the mouth of the Borysthenes, or Dnieper ; and the seventh through
the Riphxan mountains, supposed to be situated near the Tanais or Don
river. The southern parts of the earth being in a great measure unknown,
the climates received their names from the northern, and not from any par-
ticular places. Thus the climate which was supposed to be at the same
distance southward, as Meroe was northward, was called Jlntidiatneroes,
or the opposite climate to Meroe ; Jlntidiasyenes, was the opposite climate to
Syenes, &c. The following table exhibits the climates from the equator to
the poles, with their latitudes, breadth, &c. The twenty-four first are the
climates between the equator and polar circles ; the six last those between
the polar circles and poles.
For the method of constructing this table, see the note to problem 29, part
In tables of this kind, it is usual to give the names of the principal places
situated in these respective climates, but these the learner may easily find
on the globes (by prob. 3.) or without the globes on a map. Although it
appears that all places situated in the same parallel of latitude are in the
same climate, yet we must not infer from thence that they have the same
atmospherical temperature. Large tracts of uncultivated lands, sandy de-
serts, elevated situations, woods, morasses, lakes, winds, &c. have a con-
DEFINITIONS, b^c. 23
91. The right ascension of the sun or a star, is an arch of the
-equinoctial between the first point of aries and the meridian, or cir-
cle of declination, which passes through the centre of the star,
and is reckoned from west to east, round the globe. Declination is
their distance from the equinoctial north, or south.
92. Oblique ascension is an arch of the equator between the be-
ginning of aries and that point of the equinoctial which rises with
the sun, or a star, in an oblique sphere, and is reckoned as the
right ascension.
93. Oblique descension^ is that degree of the equinoctial, which
sets with the sun or a star.
siderable effect on the atmosphere. In New-Britain the climate, even about
the mouth of Haye's river, between Lake Winipeg and Hudson's Bay, and
in only lat. 56 or 57 N. is, during winter, so excessively cold, that the ice
on the river is seven or eight feet thick. Port wine freezes into a solid
mass, and even brandy coagulates, which only happens with a cold of 7 of
Farenheit ; and what is contrary to the ordinary course of nature, the cold
seems to increase every year, in these northern regions. (See Martin's essay
towards a natural and experimental history of the various degrees of heat in
bodies.) This shews that the seasons owe much of their mildness to culti-
vation. The climate between Edinburgh and Aberdeen, in Scotland, is the
same as the above, but no such extremes of heat and cold are perceived there.
Fn Canada, in about the latitude of Paris, and the south of England, the
winter is so severe from the latter end of November to April, that the St.
Lawrence and other rivers are frozen over, and the snow all this time lies
generally about 5 feet deep. During a great part of the summer, on the
western coast of America, it is extremely hot, and what is more astonishing,
and in which we cannot sufficiently admire the wise dispensations of Provi-
dence, is, that in the higher latitudes, such as 59 and 60 degrees, the heat
of July is frequently greater than in lat. 51, which heat seems necessary for
the growth and maturity of corn, &c. during their short summer, See. (See
Winterbotham's America or Kerwin's ingenious work, entitled an estimate
of the temperature of different latitudes.) The heat on the western coast of
Africa, after the wind has passed over the sandy desert, is almost suffocat-
ing; but after the same current of air has passed over the Atlantic ocean, it
is cool and refreshing to the inhabitants of the Caribbean or West-India
islands. On the eastern coasts of America, and even beyond the Allegany
mountains, the seasons are not so variable or subject to so great extremes
of heat and cold as on the western. It cannot be doubted but mountains have
a great effect on the temperature of the countries to which they belong, by
stopping the course of certain winds (as the Allegany stops part of the trade
winds, and probably increases the force of the N. W. and other winds re-
flected from their sides) by forming barriers to the clouds, by cooling the
atmosphere from the snow on their summits, or by reflecting the sun's rays
from their sides, and likewise by serving as elevated conductors to the elec-
tricity of the atmosphere. Hence on the Alps, the Andes, &c. the travel-
ler experiences, even in summer, all the four seasons of the year. The cli-
mates in the United States are by late geographers divided into four princi-
pal regions, &c. (See Spafford's geog. ch. "6.) The winds having not only
a considerable influence on the seasons, but also produce many other phe-
nomena, as currents in the ocean, &c. A general theory of them deduced
from facts would therefore be a desideratum. Our limits are too contracted
to specify any in our present undertaking : in the philosophical part of our
course, we shall consider this subject more fully. See a piece written by
he author of this introduction,, signed J. W. and published in the Mercantile
24 DEFINITIONS, &c.
94?. The ascensional difference is the difference between ' ihe
right and oblique ascension or descension, and shews how long the
sun rises or sets, before or after the hour of six.
95. The dx o'clock hour line is that great circle passing through
the poles, which is 90 distant, on the equator, from the meridian
or 12 o'clock hour circle *
96. Culminating point of the sun, or star, is that point of its
orbit, which, on any given day, is the most elevated ; or that point
in which it is at 12 o'clock, or when on the meridian.
97. Angle of position, between two places, on the terrestrial
globe, is an angle at the zenith of one of the places, formed by the
brass meridian and the quadrant of altitude, passing through the
other place, and is measured on the horizon.
98. Rhumbs are the divisions of the horizon into 32 points, call-
ed the points of the compass. A rhumb line is the way a ship de-
scribes, while she sails on any point of the compass, and cuts all
the meridians in the same angle. t
99. Course is the angle which the rhumb, or ship's way, makes
with the meridian.
100. CrefluNculum, or twilight, is that faint light which we per-
ceive before the sun rises, and after it sets.J:
Advertiser of Nov. 1st, 1809, in New-York, where the theory of some im-
portant phenomena of this nature is experimentally illustrated. Observa-
tions, similar to that made by Dr. Franklin, on the course of a N.E. storm,
(Philosophical letters, pa. 38.) would throw much light on this subject. Its
velocity was 100 miles an hour.
* The sun and stars are on the eastern half of this circle 6 hours before^
they come to the meridian, and on the western half 6 hours after they have
passed the meridian.
f A rhumb line, properly speaking, is a spiral curve drawn on the earth's
surface, as above described, and which, if continued, will never return on it-
self so as to form a circle, except it happens to be due east or west, or due
north and south; these can never be right lines on any map, except the
meridians be parallel to each other, as in Mercstor's and the plane chart,
unless the parallels and meridians : hence it is only on these charts that the
bearing can be easily found. By the compass if a place A bear due east
from a place B, the place B will bear due west from A : but if the bearing 1
on the globe should be measured by the quadrant of altitude, as some di-
rect, this would not be the case, for the angle thus measured on the globe
by the quadrant, is the angle of position between the places.
* The twilight is supposed to end in the evening or begin in the morn-
ing, when the sun is 18 below the horizon. The twilight in the morning-
and evening, arises both from the refraction and reflection of the pun's
rays by the atmosphere. Some suppose that the reflection principally
arises from the exhalations of various kinds, with which the lower parts
of the atmosphere are charged, for the twilight lasts until the sun is
further below the horizon in the evening than it is in the morning when
it begins ; and in summer it is longer than in winter, which phenomena
seem to confirm the above supposition. The greater heat also from the earth
may have its share in producing this effect, by resisting the rays of light
and changing their direction ; many phenomena and experiments might
be adduced to prove this latter supposition : and it is to this medium of
heat or light t that the particles of the atmosphere, very probably, owe
DEFINITIONS, Vc. 25
101. Refraction is that change in the elevation of any celestial
object, caused by the earth's atmosphere.*
102. Parallax, is the difference between the altitude of any ce-
lestial object, seen from the earth's surface, and the altitude of the
same object, seen at the same time from the earth's centre ; or the
angle under which the semidiameter of the earth would appear as
seen from the object.
103. Eclipse of tlie sun, is an occultation of the whole or a part
of the face of the sun, occasioned by the moon's interposition
between it and the earth.
104i. Eclipse of the moon is a privation of the light of the moon,
caused by the earth's interposition between the sun and moon.
Ip5. Diurnal arch is the arch described by the sun, moon or
stars, from their rising to their setting. The sun's semidiurnal
arch, is the arch described in half the length of the day.
106. JVocturnal arc//, is the arch described by the sun, moon,
or stars, from their setting to their rising.
107. Circles of perpetual apparition, are those in an oblique
sphere as much distant from the elevated pole, as the place itself
is from the equator. They are the greatest of all those that con-
stantly appear, and are such that all the stars inclosed within them
never set.
108 Circles of perpetual occupation, are those opposite the for-
mer, and within which all the stars that are contained never rise.
tlieii* elasticity. Its remaining in gross bodies in a latent state, may
cause them to produce the same effects in proportion to its density, See.
These properties, if fully investigated, might lead to important discove-
ries.
* When a ray of light passes out of a vacuum into any medium (or
fluid substance) it is found to deviate from its right line course towards a
perpendicular to the surface of the medium, through which it enters ;
and if the medium be of different densities, the ray will continually de-
flect from its former direction, and describe a curve. Froi its being 1 thus
continually broken, it is said to be refracted. Now as the earth is sur-
rounded by a body of air called the atmosphere, into which the rays of light
enter from a vacuum, or at least a very rare medium, and as in approach-
ing- the earth's surface the density of the atmosphere continually increa-
ses, the rays entering it obliquely will therefore be refracted, and des-
cribe a curve ; and hence the apparent place of the body from which the
light proceeds, must differ from its true place. But where the ray enters
perpendicularly, there can be no refraction, and the less oblique the ray
is, the less will be the refraction ; (from a principle in optics, that the
angle of incidence is equal to that of reflection;) hence it happens, that
at noon the refraction of the sun is the least, because it has then its great-
est altitude, and the nearer the horizon it is, the greater will be its re-
fraction. At the horizon the refraction is the greatest, and this is called
the horizontal refraction : and hence also the refraction is least in the
torrid zone and greatest at the poles. The property of refraction being-
to elevate the body from which the light proceeds, it must therefore be
subtracted from the observed altitude. From this property it follows,
that the sun and moon will sometimes appear of an oval figure near the
horizon ; for the lower limbs being more refracted than the upper, the
D
26 DEFINITIONS, bV.
109. The Jixed stars* are so called from their being observed
to keep, nearly, the same apparent distance, with respect to each
other.
perpendicular diameters will be less than the horizontal, which is rot af*
i'ected by retraction: for the diameter of the sun being supposed 32', then
the mean refraction of the lower limb when it just touches the horizon,
will be 33', but the altitude of the upper limb being- then 32', its refrac-
tion is only 28' 6", the difference of which is 4' 54", the excess of the di-
ameter parallel to the horizon above the vertical diameter. The refraction
is also" variable according to the different densities of the air, and hence
we can sometimes see the tops of mountains, steeples, &c. which at other
times are invisible, though we stand in the same place.
The ancients were not unacquainted with these effects of refraction.
Ptolemy mentions a difference in the rising and setting of the stars in
different states of the atmosphere, but makes no allowance for it in his
computations. Jlrchimedes observed, that in water the refraction was in
proportion to the angle of incidence. AUtasen y an Arabian, in the Hth
century, found the distance of a circumpolar star, from the pole, to be
different when observed above and below the pole, and such as ought to
arise from refraction. For suppose a circumpolar star passes through
the zenith, and its distance from the pole be then observed, this will be its
true distance, if its distance be again observed when on the meridian be-
low the pole, this latter distance will be its distance affected by refraction,
the difference between which and the former will be the refraction at the
lower altitude. Snettius first observed the relation between the angles of
incidence and refraction ; but Tyco Brake was the first who constructed a ta-
ble, though incorrect, for that purpose. Cassini in 1660 published another
more correct, and Mayer in his tables has given another much more ac-
curate. Modern astronomers have bestowed much attention on this sub-
ject, the niceties in the present improved state of astronomy requiring the
greatest accuracy.
* By star, in astronomy, is understood any body which shines in the hea-
vens whether it emits or reflects light ; the latter are called planets or wan-
derers, because they do not observe the same position among themselves :
the former are called fixed, for the reason above given ; though strictly-
speaking, they have several motions among themselves, which only a lapse
of many ages can render preceptible (see part 4th) the principal is, that
motion caused by the precession of the equinoctial points, their longitudes
from thence increasing yearly 5Q\''. This likewise causes a variation in their
rt. ascentions and declinations: their latitudes are also subject to a small
variation. The nutation or change of the earth's axis, the aberration of
light, &c. have some effect in changing the places of the stars.
The fixed stars are divided into six classes, from their apparent various
magnitudes. Those that appear largest (occasioned probably from their
being nearer to us than the first) are called stars of the 1st magnitude; the
next to them in lustre, stars of the 3d magnitude; and so on to the 6th,
which are the smallest that are visible to the naked eye. Besides these,
there are an inconceivable number which are not visible without the
help of a telescope, and these are called telescopic stars. The distinction
of stars into six classes or degrees of magnitude, is commonly received by-
astronomers : there is however no rule for classing the stars but by the esti-
mat on of the observer; for in reality there are as many orders of stars as
there are stars, few of them being exactly of the same bigness and lustre ;
and hence some astronomers reckon those stars of the first mngiiitude 3 which
others reckon to be of the second.
DEFINITIONS, fcfr. 27
1 10. The fioetical rising and setting of the stars, is that particular
rising and setting of the stars referred to the sun by the ancient
poets ; whence called poetical. Thus, when a star rose at sun
setting, or set with the sun, it was said to rise and set archroni-
catty : when a star rose with the sun, or set when the sun rose, it
was said to rise and set cosmically : when a star first became visible
in the morning, after having been so near the sun as to be hidden
by the splendour of his rays, it was said to rise heliacally : and
when a star first became invisible in the evening on account of its
nearness to the sun, it was said to set heliacally.
1 1 1. A Constellation* is a collection of stars in the heavens, re-
presented on the surface of the celestial globe, "and contained
within the out lines of some assumed figure, as a ram, a lion, a
be&r, a dragon, &c.
1 i2. The Zodiac is a space which extends about 8 on each side
of the ecliptic, within which, the motion of all the planets (ex-
cept Ceres and Pallas lately discovered) are performed. It is so
called from the figure of the animals described in it, to represent
the twelve signs, commonly called the 12 signs of the zodiac,
(zodion in Greek signifying an animal.)
* The division of stars into constellations is of great antiquity, as Job
makes mention of Orion, Arcturus and the Pleiades- In the writings of Ho-
mer, Hesiod, Sec. many of the constellations are mentioned. The ancients
took the figures which represent them from the fables of their religion, and
the moderns still retain them to avoid the confusion resulting- from intro-
ducing new ones ; as this, or some similar division of the stars is necessary,
in order to direct a person to any part of the heavens which he wants to
point out, or in which any particular str is situated. The whole heavens is
almost thus divided into constellations. Those stars which could not be
brought into any particular constellation, were called unformed stars. These
constellations are ranged in order on the surface of a celestial globe, and
their names are to be learnt by inspection, as also their forms and disposi-
tion. On Bardin's globes all the fig-tires, See. are painted, but on Cary's there
are only the boundaries or limits of the constellations given. The Ilevd. Mr.
Wbllaston has published in 1789 a general catalogue of the stars, arranged
in zones of north polar distance, and adapted to January 1, 1790.
The following tables contain all the constellations on the New
British Globes, with the number of stars, double stars, clusters,
clusters and nebulae, and nebulae, according to the latest observa-
tions. The number of the stars in each constellation in the first
column is taken from Flamstead, except those marked thus*.
Those in the othefr columns are taken from Cary's celestial globe,
according to their respective degrees of magnitude, See The con-
stellations in the zodiac are 12. The northern constellations on
Bardin's globe are 34, on Cary's 30. The southern constellations
on Bardin's are 47, on Cary's 45 : amounting in all, on Bardin's
globe to 93 ; on Cary's to 87. The respective magnitudes are de-
noted by the numbers 1, 2, 3, &c. double stars by two dots, thus .. ;
clusters, thus *%. ; clusters and nebulae, thus & ; nebulas, thus |f .
All those less than the 6th mag. are only visible with the telescope >
28
DEFINITIONS, We.
J\"o. of j
dumber of stars, double stars, clusters,
CONSTELLATIONS IN THE
starsfr.
Flam-
&c. from Cary's globes, with their res-
pective magnitudes,
stead, "
&c. j
234
5
6
7
8
.
tlu
clu
--
c.n
neb
1. Aries. The ram.
66
1 1
o
6
22
15
4
6
11
2. Taurus. The bull.
141 I
1 4
8
23
60
51
21
43
17
3
1
3. Gemini. The t-wins.
85 1
2 4
7
13
27
29
6
9
16
5
4. Cancer. The crab.
83
8
U
48
30
5
11
5
2
5. Leo. The lion.
95 2
2 6
15
12
47
17
14
7
11
1
15
6. Virgo. The -virgin.
110 1
6
10
10.
71
16
12
5
12
2
50
7. Libra. The balance.
51
1 3
12
4
27
6
12
4
2
8. Scorpio. The scorpion.
44
211 10
4
29
3
4
3
4
9. Sagittarius. The archer.
69
5
10
12
59
23j 16
9
1
5
10
10. Capricornus. The goat.
51
3
3
7
44
8
15
8
5
1
11. Aquarius. The -water-bearer.
108
4
7
28
59
9
8
i
6
1
1
12. Pisces. Tlie fishes.
113 |
1
5
28
63
19
15
10
14
I
NORTHERN CONSTELLATIONS.
1. Ursa minor. The little bear.
24
1 2
4
6 4
5(
c
2. Ursa major. Tlie great bear.
87 1
3 7
13
31
37
13
1
7
8
3. Draco. The dragon.
80
4 7
12
25
32
^
4. Cepheus.
35
r
6
13
00
3
1C
5. Cassiopeia,
55
Aj
6
8
38
5
g
2
6. Camelopardalus. The camelopard
58
6
25
42
q
1C
7. Auriga, The charioteer.
66 1
1
9
20
26
5
1
14
4
3
8. Lynx. The lynx.
44
3
15
25
12
9. Leo minor. The little lion.
53
1
5
10
39
i
3
10. Canes venatici. The greyhounds
25
1
1
7
15
c
J
1
11. Coma berenices. Beiierice's hair
43
13
13
17
6
;
7
1
1
21
12. Bootes.
54 1
7
10
18
30
'
13
14
2
13. Corona borealis. Tlie northen
croin.
21
1 1
5
9
i
7
1
14. Hercules.
113
9
1936
46
12
li
2
15. Lyra. The harp.
21 1
9
2J 6
12
1C
1
16. Cygnus. Tlie sivan.
81
1 6
1116
49
1
25
3
1
17. Vulpecula et anser. The fox and
the goose.
35
5
13
21
2
4
1
18. Sagitta. The arrow.
18
4
15
12
2
1
19. Delph'mus. The dolphin.
18
5
1
2
11
2
1
2
20. Equuleus. The little horse.
10
4 1
t
4
21. Pegasus. Thefying horsff:
89
3 3
914
51
11
1
8
2
22. Lacerta. The lizard.
16
3
^_
7
3
1
23. Andromeda.
66
3 2
12
15
34
cy
11
8
24. Triangulum. The triangle.
16
3
1
7
4
4
1
25.* Musca borealis. The northern Jly.
4
1
2
1
2
26. Perseus et Caput Medusae. Head
of Medusa.
59
2 4
10
14
31
c
1
1
16
1
27. Serpens. The serpent.
64
1 9
5
3
to
8
4
1
28. Ophiucus vel scrpentarius.
74
1 5
10
9
42
25
10
1
3 8
3
5
29. * Taurus Poniatouski. Ponia-
touski's bidl.
16
3
1 12
}
5
2
30. Aquila. The eagle, and Antinous.
71 jl
9
7
1438
3
2
1
5
5,
3
DEFINITIONS,
29
SOUTHERN CONSTELLA-
TIONS.
JVo. of
tarsfr.
Flam-
Vc.
Vumber of stars, double stars, clusters,
&?c. from Cary's Globes, with their
respective magnitudes.
2f
3
4
5
6 7
tu
9
hi
*
'4
C.W
&
neb
S
1. Cetus. The whale.
97
2
7
L3
11
669
3 7
1
3
2. Eridanus. The river Po.
84
(
11
27
JO
57 2
8
1
3
3. Orion.
78
4
3
15
j8
36 2
> 1 28
6
o
1
4. Monoceros. 7%e rmicorn.
31
7
7
12 1
5
14
2
5. Canis minor. TAe little dog.
14
1
3
9 1
3 3
6. Hydra.
60
1
13
16
45 2
1 9
3
7. Sextans. The sextant.
41
1
6
36 1
LI
1
2
8. Crater et hydra. The cup, &c. ,
31
10
9
14 1
4
1
3
9. Corvus. The crow.
9
2
2
2
1
.0. Centaurus. The centaur.
35
1
6
10
14
100
1
1
1
5
1. Lupus. The wolf.
24
3
3
18
29
2. * Norma. The ride or square.
12
3
26
3. * Circinus. The compasses.
4
1
1
8 1
14. * Triangulum australe. The
southern triangle.
5
1
2
1
16
1
.5. Ara. The altar.
9
3
3
1
30
1
1
.6 * Telescopium. The telescope.
9
3
C
30
2
.7. Corona australis. The south, cro-wn.
12
5
10
.8. Indus. The Indian.
12
1
1
2
54
.9. * Microscopium. The microscope.
10
1
12
20. Piscis australis. The southernjish.
24
I
2
5
9
19
21. Grus. The crane.
13
1
2
o
6
41
22. Toucana. The American goose.
9
1
o
xi
5
58 1
23. Phoenix.
13
1
1
3
7
63
24. * Apparatus sculptor is. The
sculptor's apparatus.
12
5
29 1
25. * Fornax chemica. The furnace.
26. * Horologium. The clock.
14
12
2
2
43
39
1
27. * Cela sculptoria. The engraver's
tools.
16
4
18
28. Lepus. The hare.
19
o
r
n
13
2
2
29. Canis major. The great dog-.
31
1 4
o
to
11 ]
1 C
4
30. Columbu. The dove.
10
1
1
2
4
53 1
* j
L3
31. * Equuleus pictorius. The paint-
er's horse or easel.
8
1
39
32. Argo navis. The ship Argo.
64
"\ i
t *i
9
12
37
289 6
2 1
11
2
2
33- * Pixis nautica. The mariner's
1
compass.
4
9
4
13
34.* Antliapneumatica. The air pump
3
o
18
35. * Crux. The cross.
5
1 2
1
1
1
12
36.* Musca australis. The southern fly
5
4
17
1
37. * Apus vel avis Indica. The bird
of Paradise.
11
9
16
38. * Pavo. The peacock.
14
1
C)
c
/J
80
39. * Octans. The octant.
43
1
6
64
40. Hydra, or the water snake.
10
o
gl
3
38
1
41. * Reticulus. The net.
10
1
C)
;
17
42. * Dorado. The sword fish.
6
1
1
i
24
43. "* Piscis volans. 1 'he flying fah
8 i
6
8 3
J44. * Chameleon. 1 10 1
6
35
.]45. * Mons mensje formis. The ta-\
1
\ bit; mountain. $ 30
31
1
30 DEFINITIONS, &c.
Some of the stars in the above tables inserted in the column of double
stars, are marked triple, quadruple, &c. on the globe, but this distinction
was thought unnecessary in the tables, as the globes may be consulted.
These double stars, &c. are useful in trying the goodness and magnifying
power of telescopes. The constellations on Bardin's globes, and not marked
on Gary's, will be found among the following observations. Changeable
stars not taken notice of in the following remarks, may be seen on the globe,
and likewise those which have disappeared, as the letters indicating them are
not inserted on the globe. See part 4th.
The following observations on the different constellations collected from
various authors, may not be uninteresting.
The constellations in the zodiac appear to relate to the motion of the
sun, or to refer to the climate and agriculture of those nations to whom the
zodiac owes its origin, and are therefore Chaldean or Egyptian Hieroglyph-
ics, intended to represent some remarkable occurrence in each month.
Thus the spring signs were distinguished for the production of those ani-
mals which were held in the greatest esteem, viz. the sheep, the black cat-
tle, and the goats ; the latter being the most prolific, were represented by
the figure of Gemini, afterwards so called from the two brothers Castor and
Pollux, placed in this constellation by the Greek philosophers. The retro-
grade motion of the sun in the tropic of Cancer, was represented by a Crab,
which is said to go backwards. The heat that usually follows in the next
month, is represented by the Lion, an animal remarkable for its fierceness,
and which at this season was frequently impelled, through thirst, to leave
the sandy deserts, and make its appearance on the bank of the Nile. The
sun entered the 6th sign Virgo, about harvest time, which season was there-
fore represented by a virgin, or female reaper, with an ear of corn in her
hand. When the sun enters Libra, the days and nights are equal all over
the world, and seem to be an equilibrio like the arms of a Balance. Au-
turnn, which produces fruits in great abundance, brings with it its variety
of diseases. This season is represented by that venomous animal the Scor-
pion, which wounds with its sting in its tail as it recedes. The fall of the
leaf was the season for hunting, and the stars which marked the sun's path
at this time, were represented by a huntsman or Jlrcher, with his arrows
and we&pons of destruction.
The Goat, which delights in ascending some mountain or precipice, is
the emblem of the winter solstice, when the sun begins to ascend from the
southern tropic, and gradually to increase in height for the ensuing half
year.
Jtffiiarius, or the water-bearer, is represented by the figure of a man.
pour ing out water from an urn, an emblem of the dreary and uncomfortable
season of winter.
The last of the zodiacal constellations was Pisces, or a couple of fishes
tied back to back, representing the fishing season. The severity of the
winte"- is over, the flocks do not afford sustenance, but the seas and rivers
are open and abound with fisb.
Jlries is thought by some to be the ram, whose fleece was of gold, that
carried Phryxus and his sister Helle through the air on his back, when
they fled to Colchis from the persecution of their step-mother Ino. The
fable of the flight of Phryxus to Colchis on a ram, is explained by some, who
observe, that the ship on which he embarked was called by that name, or
carried on her prow the figure of that animal. The fleece of gold is ex-
plained by the immense treasures which he carried from Thebes. He was
afterwards murdered by his father-in-law JEtes, who envied him this trea-
sure, which gave rise to the famous Argonautic expedition under the com-
mand of Jason. (See Lempriere.) Others imagine that this constellation
was f rst fo.-med from Jupiter appearing to Hercules, or as some say to
Bacchus, in the deserts of Lybia in Africa, in the form of a ram ; and shew-
DEFINITIONS, fct. 31
ed him a fountain, when, with his army, he suffered extremely for want of
water. Whence the temple of Jupiter Ammon was erected in this place.
& Arietis, 2d mag. is the principal star in this constellation.
Taurus. Some say that this was the' animal, under the figure of which
Jupiter carried away Europa, daughter of Agenor, king 1 of Phaenicia, to the
Island of Crete ; from, whom Europe, according to some, has derived its
name. (See Lempriere or Chompre.) She is supposed to have lived 1552
years before the Christian ^ra. See Ovid's Met. lib. 8. The meaning of
this allegory according to some, is, that the ship in which Europa was car-
ried, was in the shape of this animal, or according to others, that the mas-
ter was called Taurus, &c. &c. Aldebaran, 1 Mag. the Pleiades and the
Hyades are in this constellation.
Gemini. In this constellation are two remarkable stars called Castor (1)
or Apollo, and Pollux or Hercules (2) They were sons of Jupiter by Leda,
the wife of Tyndarus, king of Lacedaemon. They accompanied Jason in his
expedition to Colchis. See their history in Lempriere ; also Ovid, lib. 6.
Cancer. Some say this was made a constellation by Juno, as he went by
her order, and bit the foot of Hercules when he attacked the Lerneari
Hydra, and was killed by him.
Leo, is supposed to be the famous lion killed by Hercules on mount
Cithxron. This huge monster preyed on the flocks of Amphitryon, his
supposed father, and laid waste the adjacent country, until at length killed
by this hero. Others suppose it to be the Nemean line killed by Hercules
(which was his first labour) and which Juno placed among the stars.
llegulus (1) and ft or Deneb, 2 mag. are in this constellation.
Virgo. This constellation, according to some, took its rise from *4strea
n, daughter of Astreus, king of Arcadia ; or according to others, of Titan,
Saturn's brother by Aurora. Some make her daughter of Jupiter and
Themis, and others consider her the same as Rhea, wife of Saturn. She
was called Justice, of which virtue, according to some, she was the goddess.
She lived upon the earth as the poets mention, during the golden age, which
they often call the age of Astrea, but the wickedness and impiety of man-
kind droye her to heaven in the iron ages, and she was placed among the
constellations of the zodiac under the name of Virgo. She is represented
as a virgin, with a stern but magestic countenance, holding a pair of scales
in one hand, and a sword in the other. The scales is the Libra in the next
sign : others give the office to Themis. See Libra. Others, in fine, con-
sider Erigone as the Virgo we here speak of. She was daughter of Ica-
rius, an Athenian, who hung herself when she heard that her father was
killed by some shepherds whom he had intoxicated, and was changed into
the constellation Virgo. Icarius, as some say, was changed into Bootes, and
the Dog Maera, by which Erigone was led to discover where her father was
buried, was changed into the star Canis or Sirius. Spica virginis and vin-
demiatrix 1 and 3 mag. are the principal stars in this constellation.
Libra, from Themis (filia Caeli et terrre) the goddess of Justice ; she is
also represented with a balance in one hand, and a bandage on her eyes,
and sometimes with a sword in the other hand. Jupiter made her the god-
dess of law and peace, and placed her balance among the constellations. She
had an oracle in Baeotia, near the river Cephisus. Others consider Astrea as
the goddess of Justice, &c. See Virgo. Zuben-el-Chamali of the 2 mag. is
the principal star.
Scorpio. According to Ovid, Orion died of the bite of a Scorpion, which
the earth produced to punish his vanity in boasting that there was not on
earth any animal which he could not conquer : on account of which, Jupiter
.placed the Scorpion in the heavens. See Ovid's Delp. pa. 42, notes, &c.
Liompriere, Orion, When Orion sets, Scorpio rises. Antares, 1 m. is in
this const.
32 DEFINITIONS, &c.
Sagittdrius, took its name from Chiron, the famous Centaiu* (half man
and half horse) so called from his skill in chirurgery. He was the son of
Philyra and Saturn, who changed himself into a horse to escape the inqui-
ries of his wife Rhea. He was famous for his knowledge of music, medi-
cine, and shooting-. He was the master of Achilles, JEsculapius, Hercules,
Jason, Peleus, Eneas, &c. Hercules, when in pursuit of the Centaurs,
wounded him in the knee with a poisoned arrow. Chiron, on account of
the excruciating pain, 'begged of Jupiter to deprive him of immortality : he
was therefore placed by the god among the constellations under the name
of Sagittarius. Hesiod in Scuto. Ovid, lib. 2, 8cc. See Newton's Chro-
nology. Some take this to be Crotus a son of Pan and Eupheme, the nurse
of the muses.
Capricornus. This is supposed to be the goat Amalthea which fed Jupi-
ter with her milk, and with whose skin he afterwards covered his shield.
Jupiter placed this goat among the constellations, and gave his shield to
Pallas, who placed upon it Medusa's head, which turned all those who fixed
their eyes upon it into stones. Some maintain that it represented Pan, who
changed himself into a goat, at the approach of Typhon. Pan was the god
of shepherds, of huntsmen, and of all the inhabitants of the country. Homer
makes him the son of Mercury by Dryope ; some give him Jupiter and
Calisto for parents. Lucian, Hyginus, &c. maintain that he was the son of
Mercury and Penelope, the daughter of Icarius and wife of Ulysses, 8cc.
Though extremely deformed, the multiplicity of his amours was 'little infe-
rior to those of Jupiter, and tvas therefore thought worthy to be ranked among
the gods. The worship, and the different functions of Pan, are derived from
the mythology of the ancient Egyptians. This god was one of their eight
great gods, who ranked before the other twelve whom the Romans called
consentes. His statues represented him as a goat, which is the emblem of
fecundity, and they looked upon him as the principle of all things ; his horns
represented the rays of the sun ; his ruddy complexion and vivacity express-
ed the brightness of the heavens ; the star which he wore on his breast was
the symbol of the firmament, and his legs, feet, and tail, being those of a
goat, denoted the inferior parts of the earth as woods, giants, &c. When
the gods fled into Egypt, in their war with the giants, Pan changed him-
self into a goat, an example that was followed by all the rest. As Pan usual-
ly terrified the inhabitants of the neighbouring country, it is from him that
kind of fear which is only ideal, has received the name of panic fear.
Jlquariiis. This is the famous Ganymede, a beautiful youth of Phrygia,
son of Tros, King of Troy, or according to Lucian, son of Dardanus. He
was taken up to heaven by Jupiter as he was tending his father's flocks on
mount Ida, and he became the cup bearer of the gods in place of Hebe ;
hence made the constellation Jlquarius or the water-bearer. The principal
star in it is Scheat, 3 mag.
Pisces. These are thought to be the fishes which carried Venus and Cu-
pid over the Euphrates, when they fled the pursuit of the giant Typhon ;
others think that they were the dolphins who carried Amphitrite to Neptune,
and others that they were the dolphins that carried the famous musician Arion
to Tsenarus, having leaped into the sea when the sailors attempted to mur-
der him for his riches. These two last opinions, however, rather relate to
the constellation Delphinus.
THE NORTHERN CONSTELLATIONS.
Ursa Minor and Ursa Major, are said to be Calisto and her son Jlrcas.
Calisto or Helice, was daughter of Lycaon, king of Arcadia, and one of
Diana's attendants. She was seduced by Jupiter ; and Juno in revenge
changed her into a she bear, and her son Areas into a little bear : but Ju-
piter, fearful of their being hurt by 'he huntsmen, made them constellations.
Some considep Areas the same as Bootes- (See Bootes.) The ancients also
DEFINITIONS, &c. 33
Represented each of these constellations under the form of a wagon drawn
by a team of horses ; and the country people, at the present, sometimes call
Ursa Major by the name of Charles's Wain . in some places it is called the
plough, which it resembles. There are two remarkable stars in Ursa Major
called the pointers, because an imaginary line drawn through them, will
pass over the pole star in the tail of the little bear. These stars are the
hindmost in the square of the wain, or dub he and $. The great bear is also
sometimes called Maenalis Ursa, from a mountain in Arcadia. In this con-
stellation is the pole star, 2d mag.
Draco. There are various accounts given of this constellation; some re-
present it as the watchful dragon which guarded the golden apples in the
garden of the Hesperides, near mount Atlas in Africa, and was slain by
Hercules, being his eleventh labour. Juno, who presented these apples to
Jupiter on the day of their nuptials, took Draco up to heaven, and made a
constellation of it as a reward for its fidelity. These Hesperides who were
three sisters, are considered by Varro as having immense flocks of sheep,
and that the ambiguous Greek word melon, which signifies an apple and a
sheep, gave rise to the fable, and that Draco was their shepherd. Others
that this Draco, in the famous war with the giants, was broug'ht into combat
and opposed to Minerva, who seized it in her hands, and threw it, twisted as
it was, into the heavens round the axis of the earth, before it had time to
Unwind its foldings. Others imagine that it was the dragon killed by Cad-
mus when in search of his sister Europa, and which he had slain by the as-
sistance of Minerva. This was the dragon from whose teeth, sowed in a
plain, armed men suddenly sprung from the ground, &c. Some suppose the
dragon to be a king whom Cadmus conquered, and the teeth his soldiers.
See Lempriere. The principal star is Rastaben, 2 mag.
Cepheus was a king of Ethiopia, father of Andromeda, by Cassiopeia. He
was one of the Argonauts who accompanied Jason to Colchis in quest of the
golden fleece. In this const, the principal star is Alderamin, 3 mag.
Cassiopeia was the wife of Cepheus and mother of Andromeda. She
boasted herself to be fairer than Juno and the Nereides. Neptune at the
request of these, punished the insolence and^vanity of Cassiope by sending a
huge sea monster to ravage Ethiopia. The wrath of Neptune could only be
appeased by exposing Andromeda, tied to a rock, to be devoured by this
monster ; but Perseus mounted on Pegasus, with the head of Medusa,
changed this monster into a rock, delivered Andromeda, and obtained of
Jupiter that Cassiope might have a place among the stars. Some suppose
Cetus (see Cetus) to be the monster sentto devour Andromeda. Shedir or
Scheclar, 3 mag. is the principal star.
Camelopardalus is one of the new constellations formed by Hevelius : this
jrii'ivil is described in various natural histories. (See Monoceros.)
vluriga is represented on the celestial globe by the figure of a man in a
kneeling or sitting posture, with a gcat and her kids in his left hand, and a
bridle in his right. There are various accounts given of this constellation.
Some suppose it to be Erichthonivts, the fourth king of Athens, and son of
Vulcan and Minerva, who is said to have invented chariots, and the manner
f harnessing horses to draw them ; some, however, take this Erichthonius to
be Bootes. Others think that Auriga was Mirtilus, a son of Mercury and
Phcetusa. He was charioteer to (Enomaus, king of Pisa, in Elis, and so ex-
perienced in riding and in the managing of horses, that he rendered those of
CEnomanus the swiftest in all Greece; his infidelity to his master proved
fatal to him at last, but being a son of Mercury, he was made a constella-
tion after his death. It has been supposed that the goat and her kids refer
to Amalthsea, daughter of Melissus, king of Crete, who, in conjunction with
her sister Melissa, fed Jupiter with goat's milk. It is moreover said, that
Amalthaea was a goat called Olenia, from its residence at Olenus, a town of
Peloponesus. Capella 1 mag. is the principal s^r.
E
34 DEFINITIONS, We.
Lynx, orie of Hevelius' constellations, composed of the unformed star?
of the ancients, between Auriga and Ursa Major.
Leo Minor, was formed by Hevelius out of the unformed stars of the
ancients, and placed above Leo, the zodiacal constellation. Mythologists
are not agreed whether the latter be the Nemaean lion slain by Hercules,
as this constellation was among the Egyptian hieroglyphics long before
this exploit of Hercules. Nemea was a town of Argolis, in Peloponnesus.
Canes Venatici, these are Jlsterion et chara, the two greyhounds held in.
a string by Bootes: they were formed by Hevelius out of the unformed
stars. Cor Caroli a double star of the 3d magnitude is in this constellation,
Coma Berenices, is composed of the unformed stars between the lion's
tail and Bootes. Berenice was the wife of Ptolemy Evergetes. When
Ptolemy went on a dangerous expedition, she vowed to dedicate her hair
to the goddess Venus if he returned in safety. Some time after Conon,
an astronomer of Samos, to make his court to Ptolemy, publicly report-
ed that the queen's locks were carried away by Jupiter, and were
made a constellation. Conon, according to Lempriere, flourished 247
years before Christ. He was intimate with the celebrated Archimedes.
E-uergetes was a surname signifying benefactor, which Ptolemy re-
ceived from the Egyptians, on account of carrying back 2500 statues of
their gods, which Cambyses had carried away into Persia when he con-
quered Egypt. This title was also given to Philip of Macedonia, to anti-
gonus Doson, to the kings of Syria and Pontus, and to some of the Ro-
man emperors.
Bootes, also called Bubulcus, is supposed to be Icarus the father of Eri-
gone, who was killed by shepherds for inebriating them. Others maintain
that it is Areas or Arctophylers, son of Jupiter and Calisto. (See ursa
minor, &c.) Bootes is represented as a man in a walking posture, grasp-
ing in his left hand a club, and having his right hand extended upwards,
holding the cord of the two dogs Asterion and Chara, which seem to be
barking at the great bear ; hence Bootes is sometimes called the bear
driver, and the office assigned him is to drive the two bears round the
pole. Arcturus 1 and Mirach 2 mag. are the principal stars.
Cororia borealis is a beautiful crown of seven stars, given by Bacchus,
the son of Jupiter, to Ariadne, the daughter of Minos, second king of
Crete and Pasiphae. Bacchus is said to have married Ariadne after she
was basely deserted on the Island ofNaxos, by Theseus, king of Athens,,
whom she had delivered from the labyrinth of Crete, after having con -
quered the Minotaur. After her death the crown which Bacchus had
given her, was made a constellation. This crown is called by Virgil
Gnossia. Stella, because Ariadne was born at Gnossus. The principal star
in it is Alphecca 2 magnitude-
Hercules is repi-esented on the celestial globe with a club in his right
hand, the three headed dog Cerberus in his left, and the skin of the Ne-
mxan lion, thrown over his shoulders. This Hercules was the son of
Jupiter and Alcmena, and generally called the Theban. He was a scholar
of Chiron the centaur, and accompanied Jason in the Argonautic expedi-
tion. Ninus, son of Belus, was however the Hercules of the Chaldeans,
He founded the Assyrian monarchy, and was therefore the Jupiter of the
Assyrians- He caused the same honours to be paid his father Belus in
Babylon, as to the creator of the universe, and established his worship
wherever he extended his conquests. The Grecians afterwards followed
the example, and worshipped him under the title of Jupiter ; and the hea-
then world formed to themselves no less than 300 gods of this name.
The worship of Jupiter became universal. He was the Ammon of the
Africans, the Belus of Babylon (for Ninus after his death was called Her-
cules, the son of Jupiter) the Osiris of Egypt, &c. An account of his wor-
DEFINITIONS, &c. 35
ship is given in the sacred writings. See the remarks at the end of the
constellations. Ras. Algethi 3 mag. is situated in the head of Hercules.
Lyra was at first a tortoise, afterwards a lyre, because the strings of
the lyre were originally fixed to the shell of a tortoise. It is asserted
that this is the lyre which Apollo or Mercury gave to Orpheus, jnd with
which he descended the infernal regions in search of his wife Euridice,
Orpheus after death received divine honours ; the muses gave an honour-
able burial to his remains, and his lyre became one of the constella-
tions. Pythagoras and his followers represent Apollo playing upon a
harp of seven string's, by which is meant (as appears from Pliny, lib 2. c.
22, Macrobius 1. c- 19. and Censorinus c. 11.) the sun in conjunction with
the seven planets, for they made him the leader of that septenary chorus
and moderator of nature, and thought that by his attractive force he act-
ed upon the planets in the harmonica! ratio of their distances ; and
hence they called the sun Jupiter's prison, alluding to the force with
which he retains the planets in their orbits. Pythagoras made this
discovery from observing, as he passed by a smith's shop, that the sounds
of the hammers were more acute or grave, in proportion to their weight,,
and from thence found that the sound of strings were as the weights sus-
pended, &c. see Macrobius lib. 2. insomn. scip. c. 1. or Doctor Gregory's
preface to his astronomy. Pythagoras was born at Samos ; he travelled in-
to Egypt and Chaldea, where he gained the confidence of the priests, and
was initiated into their mysteries, learned their symbolic writing, the na-
ture of their gods, &c. and acquired a knowledge of the true system of
the world. Pythagoras distinguished himself by his discoveries in geo-
metry, astronomy, and mathematics in general, and invented the demon-
stration of the 47 prop, of the 1st book of Euclid. He was the first
who supported the doctrine of Metempsychosis, or the transmigration of
souls into different bodies. He founded a sect in Italy called the Italian.
The most learned and eloquent men of the age, the rulers and the legis-
lators of all the principal towns of Greece, Sicily, and Italy, boasted in
being the disciples of Pythagoras. See Lempriere. Lyra 1. and @ a
quadruple star of the 3 mag, are the most remarkable stars in this con-
stellation.
Cygnus is fabled by the Greeks to be the swan, under the form of which
Jupiter deceived Leda, or Nemesis, the wife of Tyndaris, king of Sparta.
Leda was the mother of Pollux and Helena, who was the cause of
the Trojan war ; and also of Castor and Clytemnestra. The two former
were deemed the offspring of Jupiter, and the others claimed Tyndarus
as their father. Some, however, supposed that this constellation derived
its name from Cycnus, a son of Mars by Pelopea, who was killed by
Hercules. Others that it was Cycnus, whom Achilles smothered, his
darts having no effect on him ; but was immediately changed into a swan,
&c. Arided 2. Albireo 3, and two stars that sometimes are invisible, at
other times of the 3d mag. are the most remarkable stars.
rulpecula and Anser was made by Hevelius out of the unformed stars.
Sagitta the arrow, is supposed to be one of the arrows of Hercules,
with which he killed the eagle or vulture that perpetually knawed the
liver of Prometheus, who was tied to a rock on Mount Caucusus by order
of Jupiter.
J)elpfumu 9 the dolphin, was placed among the constellations by Neptune,
because by means of a dolphin he obtained Amphitrite, his wife.
Equulus, the little horse, sometimes called eqnisectio, the horse's head,
is supposed to be the brother of Pegasus ; some take him to be the horse
which Neptune struck out of the earth with his trident, when he disputed
with Minerva for superiority, and which some confound with Pegasus.
Peffasiis, a winged horse, sprung from the blood of Medusa after Per-
seus had cut off his head. He received his name from his being 1 borr,
36 DEFINITIONS, & e .
according to Hesiod, near the sources (Pege) of the ocean. According 1
to Ovid, he fixed his residence on mount Helicon, where, by striking- the
earth with his foot, he produced a fountain called Hippocrane. He be-
came the favourite of the muses ; and being- tamed by Neptune or Mi-
nerva, he was given to Bellerophon, son of Glaucus, king- of Ephyre, to
conquer the Ghimxra, a hideous monster that continually vomited flames ;
it had three heads, that of a lion, a goat, and a dragon. The fore parts
of its body were those of a lion, the middle that of a goat, and the hinder
parts those of a dragon. It lived in L} cia in the reign of Jobates, by
whose orders Bellerophon was sent to destroy it The Chimsera is sup-
posed to be a burning mountain in Lycia, whose top was the resort
of lions, on account of its desolate wilderness ; the middle, which was
fruitful, was covered with goats, and at the bottom, the marshy ground
abounded with serpents ; and that Bellerophon was the first who made
his habitation on it. Plutarch says that it was the captain of some pirates
who adorned their ship with the images of a lion, a goat, and a dragon.
Afcer the destruction of this monster, Bellerophon attempted to fly to
heaven. This presumption was punished by Jupiter, who sent an insect
to torment Pegasus, which occasioned the melancholy fall of his rider. *
Pegasus continued his flight up to heaven, and was placed by Jupiter
ainoi'p- the constellations. From the Chimsea and Orthos, a dog with
two heads which belonged to Geryon, and which Hercules killed, sprung-
the sphinx and lion of Nemea. In this const, are the stars Markab 2,
Sheat Alperas 2, and Algenib, 3d mag.
Lacerta, the lizard, was added by Hevelius to the old constellations.
Andromeda is represented on the celestial globe by the figure of a wo-
man almost naked, having her arms extended, and chained by the wrist
of her right arm to a rock. She was the daughter of Cepheus, king of
Ethiopia, by Cassiope. Cepheus, by the advice of the oracle of Jupiter
Hammon, exposed Andromeda tied to a rock, near Joppa, now Jaffa in
Judea (according to Pliny) to be devoured by a sea monster, to preserve
his kingdom, but she was rescued by Perseus. (See Cassiopeia.) Pliny
says that the skeleton of the huge sea monster, to which she had been
exposed, was brought to Rome by Scaurus, and carefully preserved. The
fable of Andromeda and the sea monster, is explained," by supposing that
she was courted by the captain of a ship who attempted to carry her
away, but was prevented by another more successful rival. The star mark-
ed a 2, Mirach 2, and Almaach 2, are the principal in this constellation.
Triangulum. A triangle is a well known figure in geometry ; it WAS
placed in the heavens in honour of the most fertile part of Egypt, being
called the delta of the Nile, from its resemblance to the Greek letter of
that name A- The Nile, anciently called Jigyptus, flows through the
middle of Egypt, in a northerly direction, and when it comes to the town
of Cercassorum, it divides itself into several streams, and fails into the
Mediterranean by seven channels or mouths ; the island which these
several streams form is called Delta. The invention of geometry is usual-
ly ascribed to the Egyptians, so called, as some think, from JEgyptus son
of Belus and brother of Danaus, and it is asserted that the animal inun-
dations of the Nile which swept away the bounds and landmarks of es-
tates, gave rise to it, by obliging the Egyptians to consider the figure and
quantity belonging to the several proprietors. The Nile yearly overflows
the country, and it is to those regular inundations that the Egyptians are
indebted for the fertile produce of their lands. It begins to rise in the
month of May for 100 successive days, and then decreases gradually for
the same number of days- If it does not rise as high as 16 cubits, a fa-
mine is generally expected ; but if it exceeds this "by many cubits, it is
q>f the most dangerous consequences ; houses are overturned, the cattle
are drowned, and a great number of insects are produced from the mud 3
DEFINITIONS, We. 37
which destroy the fruits of the earth. As it very seldom rains in Egypt,
the cause of the Niles overflowing", is the heavy rains which regularly
fall in Ethiopia in the months ot 'April and May, and which rush down
in torrents upon the country, and lay it all under water.
Mwca Borealis, This is a new constellation supposed to be formed in
opposition to Musca australis, which see.
Perseus et Caput Mudusce. Perseus is represented on the celestial globe
with a sword in his right hand, the head of Medusa in his left, and wings at
his ancles. Perseus was the son of Acrisius and Danac. He was no sooner
born, than he was thrown into the sea, with his mother Danae, but being
driven on the coast of the island of Seriphos, one of the Cyclades, they
were found by one Dictys, a fisherman, and carried to Polydectes, the king
of the place, who intrusted them to the care of the priests of Minerva's tem-
ple. Here he promised the king to bring him the head of Medusa. To
equip him for this arduous task, Pluto, the god of the infernal regions, lent
him his helmet, which had the power of rendering its bearer invisible. Mi-
nerva, the goddess of wisdom, furnished him with her buckler, which was
as resplendent as glass ; and he received from Mercury wings, and the te -
laria, with a short dagger made of diamonds ; or, as some say, he received
the telaria or herpe from Vulcan, which was in form like a scythe. Thus
equipped he began his expedition, and traversed the air, conducted by the
goddess Minerva. Having discovered Medusa, he cirv. off her head, and
from the blood which dropped from it in his passage through the air, sprung-
innumerable serpents, which is said to have ever since infested the sandy
deserts of Lybia : from the same blood sprung- Chrysaor with his golden
sword, and the horse Pegasus, which immediately flew to Mount Helicon,
though Ovid says Perseus was mounted on him when he freed Andromeda.
Medusa was one of the three Gorgons, who were daughters of Phorcys and
Ceto; their names were Stheno, Euriale, and Medusa ; all immortal except
the latter. They were represented with snakes on their heads, instead of
hair, with yellow wings and brazen hands ; their bodies were also covered
with impenetrable scales, and their very looks turned all those who beheld
them into stones. Medusa, according to some, was celebrated for the beauty
of her locks, but having, with Neptune, violated the temple of Minerva, that
goddess changed her locks into serpents. Perseus in gratitude to Minerva,
placed her head on her segis or shield, where it retained the same petrifying
power as before. It was afterwards placed among the constellations. (See
Lempriere.) Diodorus and others suppose that the Gorgons were a warlike
race of women near the Amazon, whom Perseus, with the help of a large
army, totally destroyed. The principal stars in it are a 2, and Algol 2.
Serpens is also called Serpens Ophiuci, being grasped by the hands of
Ophiucus. (See Serpentarius.)
Serpentarius, also called Ophiucus or Opheus, is supposed by some to be
Hercules, who before he had completed his eighth month, squeezed two
serpents to death, which Juno sent to devour him. Some, however, take
him to be JEsculapius, son of Apollo by Coronis. He w r as taught the art of
medicine by Chiron, and \vas physician to the Argonauts. He was consider-
ed C M .skilful in the medical power of plants, that he was called the inventor
as well as the god of medicine. jEsculapius was represented with a large
beard, holding in his hand a staff, round which was wreathed a serpent ; his
hand being supported on the head of a serpent. Serpents were more
particularly sacred to him, not only as the ancient physicians used them in
iheir prescriptions, but because they were the symbols of prudence and
i'x-esight, so necessary in the medical profession. Cicero reckons three of
this name, (de nat. Deor. 3., c. 22.) The most remarkable star in this
constellation is Ras Alhagus.
-Jlquiia et -Jlnthwus. Aquila is supposed to be Merops, a king of the
island of Cos., one of vas changed into an eagie (Ovid,
38 DEFINITIONS, &c.
met. 1.) and placed among the constellations. Antinous was a youth of
Bithynia, in Asia Minor, a great favourite of the emperor Adrian, who
erected a temple to his memory, and placed him among- the constellations.
Antinous was armed by Hevelius with a botv and arrow. Jlltair or Jltair 1,
is situated in this constellation.
Taurus Poniatotuski was so called in honour of Count Poniatowski, a
polish officer of great merit, who saved the life of Charles XII. of Sweden,
at the battle of Pultowa, a town near the Dnieper, about 150 miles south
east of Kiow ; and a second time at the island of Rugen, near the mouth of
the river Oder.
Scutum Sobieski was so named by Hevelius, in honour of John Sobieski,
king- of Poland. Hevelius was a celebrated astronomer, born at Dantzick ;
his catalogue of fixed stars was entitled Firmamentum Sabieskianum, and
dedicated to Sobieski.
J\fons JVTaenalus, the mountain Mzenalus in Arcadia, was sacred to the god
Pan, and much frequented by shepherds : it was covered with pine trees,
whose echo and shade has been much celebrated by all the ancient poets :
it received its name from Maenalus a son of Lycaon. It was made a con-
stellation, and placed by Hevelius under the feet of Bootes.
Cor Caroli is a star in the neck of Chara, and was so denominated by
Sii Charles Scarborough, physician to king Charles II. in honour of king-
Charles I.
Triangulum Minus. This constellation was made by Hevelius, of the
unformed stars between Triangulum Boreale and the head of Aries.
THE SOUTHERN CONSTELLATIONS.
Cetus is pretended by the Greeks to be the sea monster, which Neptune,
brother to Juno, sent to devour Andromeda. In this are Menkar 2. Baten
Kailos 3, and Mira, which is sometimes of the 2d. mag. and sometimes in-
visible. The period of its variations is 334 days.
Eridanus, the river Po, called by Virgil the king of rivers, was placed in
the heavens for receiving Phaeton, whom Jupiter struck with thunderbolts,
when the earth was threatened with a general conflagration, through the
ignorance of Phaeton, who had presumed to be able to guide the chariot of
the sun. According to the poets, while Phseton was unskilfully driving the
chariot of his father, the blood of the Ethiopians was dried up, and their
skins became black, a colour which is still preserved by the greater part of
the inhabitants of the torrid zone. The territories of Libya w r ere also
parched up, according to the same tradition, on account of their too great
vicinity to the sun, and ever since Africa, unable to recover her original
verdure and fruitfulness, has exhibited a sandy country and uncultivated
waste. Phaeton, according to the Mythologists, was a Ligurian prince, who
studied astronomy, and in w T hose age the neighbourhood of the Po was
visited with uncommon heats. From his love of astronomy, he was called
a son of the sun, or Phoebus and Clymene ; or as others say, of Aurora and
Tithonus or Pausanias. The river Po is sometimes called Orion's river.
The most remarkable star in it is Achernar 1.
Orion is represented on the globe by the figure of a man with a sword in
his belt, a club in his right hand, and the skin of a lion in his left ; he is
said by some authors to be the son of Neptune and Euriale, and that he had
received from his father the privilege and power of walking over the sea
without wetting his feet. Others make him son of Terra, like the rest of
the giants, according to Diodorus. Orion was a celebrated hunter, supe-
rior to the rest of mankind, by his strength and uncommon stature. He
built the port of Zancle, and fortified the coast of Sicily against the fre-
quent inundations of the sea, with a mound of earth called Pelorum, on
which he built a temple to the gods of the sea. Others say that Jupiter,
Neptune, and Mercury, as they travelled over Bxotia, met with great hospi-
DEFINITIONS, csV. 39
tality from Hyrieus, a peasant of the country, who was ignorant of their
dignity and character. When Hyrieus had discovered that they were
gods, he welcomed them by the voluntary sacrifice of an ox. Pleased
vrith his piety, the gods promised to grant him whatever he required,
and the old man who had lately lost his wife, and to whom he had made
a promise never to marry again, desired them, that, as he was childless,
they would give him a son without obliging him to break his promise.
The gods consented, and ordered him to bury in the ground the skin of
the victim ; nine months after he dug the skin, and found a beautiful
child, which he called Orion, ab urina, quia Dii urinam in pellem redderanf,
ex qua procreatus. Ovid' says that the name was changed from Urion to
Orion. Orion was buried in the island of Delos,and after his death was
made a constellation. According to the ancient poets, this constellation
never rises or sets without great storms, and hence he is called nimbosita
Orion by Virgil, and tristis Orion by Horace. Authors who explain this
fable say, that Orion was a great astronomer and disciple of Atlas. The
stars Betelgeux 1, and Rigel 1, are in this constellation.
JWonoceros, the unicorn, was composed by Hevelius, according to most
authors, of those stars which the ancients had not comprised within the
outlines of the other constellations. According to Doctor Gregory (astr.
b. 2, pr. 22 ) this constellation, together with the Camelopard, was first
described by Bartschius on his globe of four feet diameter, and afterwards
retained by Hevelius.
CanisMmor, according to the Greek fables, was one of Orion's hounds....
Some suppose it to refer to Anubis, an Egyptian god, with the head of a
dog ; others to Diana, the goddess of hunting ; others to Acteon, who was
changed by Diana into a stag and devoured by his own dogs, &c. Others
are of opinion, that the Egyptians were the inventors of this constellation,
and as it rises before the dog star Sirius, which in the dog-days was so
much dreaded, it is properly represented as a little watchful creature,
giving notice of the others approach ; hence the Latins have called it
Antecanis, the star before the dog. The most remarkable star in it is
Procyon or Algomeiza 1.
Hydra is the water serpent which, according to the fable of the poets,
infested the neighbourhood of the laka Lerna in Peleponesus. It had
several heads, and as soon as one was cut off, two immediately grew in
its place, if not prevented by fire. It was one of the labours of Hercules
to kill this monster, which he effected with the assistance of lolaus, king
of Thessaly. It was in the gall of this Hydra that Hercules dipt his ar-
rows, the wounds inflicted by which were incurable and mortal. The
general opinion is, that the Hydra was a multitude of serpents which in-
fested the marshes of Lerna. Cor Hydrae is in this constel. a triple star
2 mag. There is another constellation of the same name near the south
pole.
Sextans, called also Sextans Urani<e, is a mathematical instrument well
known to mariners. This constellation was formed by Hevelius, of the
unformed stars between Leo and Hydra. Urania was one of the muses,
daughter of Jupiter and Mnemosyne, who presided over astronomy. She
was represented as a young virgin, dressed in an azure coloured robe,
crowned with stars, and holding a globe in her hands, and having many
mathematical instruments placed around.
Crater, according to the mythologists, is the cup or pitcher of Aqua-
rius. Alkes, 4 mag. is its principal star.
Corvus, according to the Greek fables, was made a constellation by
Apollo. This god being jealous of Coronis, the daughter of Phlegyas and
mother of JEsculapius, sent a crow to watch her behaviour ; the bird per-
ceived her criminal partiality to Ischys, the Thessalian, and acquainted
Apollo with her conduct, Some think that this Corvus was the daughter
40 DEFINITIONS, Vc.
of Coronseus, king of Phocis, changed into a crow by Minerva, when flying 1
before Neptune.
Centaums. The Centauri were a people of Thessaly, half men and half
horses. The ancient people of Thessaly were famous for their skill in
taming horses, and their appearance on horseback was so uncommon a
sight to the neighbouring states, that at a distance they imagined the man.
and horse to be one animal. When the Spaniards landed in America,
and appeared on horseback, the Mexicans had the same ideas ; and sim-
ilar ideas were excited when they saw their vessels expand their wings
ami fly along the surface of the ocean. Plutarch and Pliny are however
of opinion, ihut such monsters have really existed. The battle of the
Centaurs with the Lapithse is famous in- history. This constellation is by
some supposed to represent Chiron, the Centaur ; but as Sagittarius is
likewise a Centaur which some contend to be Chiron, it is probable that
Theseus is represented by this constellation. Among the principal stars
in this constellation, the most remarkable is a double star marked a 1 of
the 1st, and a 2 of the 4th mag.
Lupus is supposed to be Lycaon, king of Arcadia, celebrated for his
cruelties. He was changed into a wolf* by Jupiter, because he offered
human victims on the altars of the god Pan. (See Ovid, met. 1.) Some
suppose that this king, to try the divinity of Jupiter, who once visited
Arcadia, served up human flesh on his table, and that it was Areas, sou
of Calisto, who thus became the victim of his impiety, and was served up
for Jupiter ; for which horrid crime Jupiter punished him by metamor-
phosing him into a wolf.
Norma, the square, a well known instrument, is a new constellation
made of the unformed stars between Lupus and Ara.
Circiims, the compasses, an instrument known for its extensive utility,
is a new constellation formed near the Centaur, in allusion to the neigh-
bouring constellations.
Triangulum Jlustrale is a new constellation formed near the constella-
tions Circinus and Norma. The three foregoing constellations are placed
near Ara, the altar, the two former being useful to the practical artist,
and the latter being the foundation of many important sciences ; Euclid,
as is well known, commencing his elements with this figure. The oracle
of Apollo, at Delphos, being* consulted about a raging pestilence which
desolated Athens, answered, that the pestilence would cease, if his altar,
which was of a cubical form, were doubled. To execute the orders of
the oracle, a knowledge of the properties of such solid bodies became
necessary, and this gave rise to a great part of the geometry of solids.
Jlra, the altar, is supposed by some to be the altar on which the gods
swore before their combat with the giants ; but from the observation on
the last constellation, it is more probable that it was Apollo's altar at
Delphos.
Tekscopium, a well known optical instrument, is a new constellation
formed near Ara.
Corona Jluztralis, anew constellation formed near Sagittarius.
Indus, the indian, is a new constellation formed to commemorate the
original inhabitants of the new world.
JWicroscopium, an optical instrument for distinctly viewing minute ob-
jects, is a new constellation formed between Sagittarius and Piscis Aus-
tralis.
Piscis Australis, is supposed by the mythologists to be Venus, who
transformed herself into a fish, to escape from the giant Typhon. (See
Pisces.) Fomalhaut 1, is in this constel.
G.-MS, the Crane, a new constellation formed hear Piscis Australis, in
allusion, perhaps to the cranes, against which the Pygmies, a race of
dwarfs said to be no more than one foot high, was accustomed yearly to
DEFINITIONS, tfc, 41
make war; or perhaps in allusion to their princess Gerana; who was
changed into a crane, for boasting- herself fairer than Juno.
Toucana or Touchan, the American goose ; a new constellation near
Indus.
Phccnix, the Phenix, a new constellation near Eridanus. This was the
name of a fabulous bird worshipped among the Egyptians.
Apparatus sculptoris, is a new constellation between Cetus and Phoenix.
Fomax chemica, a new constellation formed out of Eridanus.
Horologium, a recent constellation formed near Eridanus.
Cela sculptoria, Columba Noachi, or Noah's dove, Equuleus pictorhis,
Pixis nautica, and Jlntlia pneumatica, are all new constellations.
Lepus, the hare. This constellation is formed near the great dog, which,
from the motion of the earth, seems continually to pursue it.
Canis major, according- to the mythologists, is one of Orion's hounds,
The Egyptians, who carefully watched the rising of this constellation,
and by it formed their judgment of the swelling of the Nile, called the
bright star Sirius, the centinel or watch of the year ; and represented it
according to their hieroglyphic manner of writing, under the figure of a
dog. The Egyptians called the Nile Siris, and hence, some are of opinion,
they derived the name of their deity Osiris. This Osiris was son of Jupiter
and" Niobe, and king of Egypt, who is said not only to have civilized his
own subjects, but also to have civilized and polished many other nations.
To perform this task, he left his own kingdom, accompanied by his bro-
ther Apollo, and by Anubis, Macedo and Pan ; and left the management
of affairs to Isis or lo, his wife, and his faithful minister Hermes or Mer-
cury ; and the command of his troops at home to Hercules. At his re-
turn, he was murdered by his brother Typhon, and his body was thrown
into the Nile. The Egyptians worship him under the titles of Apis; Sera-
pis, &c. Historians take him to be JWizraim, eldest son of Cham, the third
son of Noah, who, in the division of the world, received Africa for his
lot. He was worshipped by the Egyptians under the title of Hammon or
Jupiter Hammon. The dog star Sirius, is of the 1st mag. and the most remark-
able, not only in this constellation, but in the heavens, being the largest
and brightest, and therefore considered the nearest to us of all the fixed
stags.
Jlrgo Navis, the ship Argo, which carried Jason and his Argonauts to
Colchis, in quest of the golden fleece. Some say that this vessel was built
at Argos, from whence it derived its name ; others derive it from one Ar-
gos, who first proposed the expedition ; others because it carried Gre-
cians, commonly called Argives ; and others, from the Greek word argos 9
swift. She had 50 oars, and, according to some, abeam in her prow, cut
in the forest of Dodona, which gave oracles to the Argonauts. After the
expedition, she was drawn ashore at the isthmus of Corintb, and conse-
crated to the god of the sea ; and afterwards made a constellation by the
poets. Canopus 1 mag. is the principal star. There is another marked
D remarkable only with the telescope, from its containing- 9 other stars
in the neighbourhood of a nebeculous cluster, 8cc.
Crux, the cross. There are four stars in this constellation forming a
cross, by which mariners, sailing in the southern hemisphere, readily find
the situation of the antartic pole, by means of the stars and y which,
nearly point in this direction. This is a new constellation, and formed, no
doubt, in honour of that instrument on which the Son of God redeemed
mankind. Venerable Bede gave Christian names to the signs of the zo-
diac, and Julius Schillerus has followed the example in his Gcttym Stella-
Piim, or Starry Heavens, published in 1627. But later astronomers think
that the ancient names ought to be retained, to avoid confusion, and to
preserve the ancient astronomy. They will also afford so many monu-
ments of the folly and stupidity of the mqst enlightened nations, in the *r-
F
42 DEFINITIONS, tfc.
tide of religion, when carried away by the current of their passions, and
the violence of human depravity. In these constellations we find some of
the gods of the heathens, it is true, placed in the heavens, to whose mon-
strous vices so much incense has been offered, under the veil of an ab-
surd religion, but a veil, however, too transparent to hide their guilt,
The most abandoned men became the most powerful divinities ; worship-
ped with sacrifices, addressed with prayers and supplications, and their
vices xt length became the objects of adoration. The light of the gospel
dispelled these dark shades of infidelity ; the wisdom of the Son of God
raised man from this state of abasement and folly; the arms which he
made use of, was the cross on which he suffered ; and wherever this is
planted, these idols must fall. The humility of the cross is an antidote
to the pride of man, the sufferings of the cross an antidote to his pas
sions.
Musca australis, a new constellation, formed between Crux and Apus.
Jlpus, vel Jlvis Indica, a new constellation, near Pavo. This bird is a
native of the Molucca Islands.
Pavo, the peacock, one of the new constellations near the south pole.
This is often called Junonia avis, Juno's bird.; this goddess being repre-
sented as drawn through the air in her chariot by these birds. Juno, be-
ing the wife of Jupiter, became queen of the gods, of the heavens, &c. and
is famous for her severity to the mistresses and illegitimate children of
her husband Jupiter, whom she at length abandoned on account of his
debaucheries. This goddess employed Argus, who had an hundred eyes,
two of which only slept at a time, to watch lo, one of Jupiter's mistresses,
She afterwards put the eyes of Argus on the tail of the peacock.
Octans, the octant, a well known marine instrument, one of the new con-
stellations surrounding the antartic pole.
Hydra, a new constellation near the south pole, in which the remarka
ble nebecula minor is situated.
jReticulus, the net, is a new constellation, formed between HorologiuiTi
and Dorado. It is sometimes called Reticulus rhomboidalis.
Dorado, or Xiphias, the sword fish, a new constellation, formed round
the pole of the ecliptic.
Piscis volan8 t the flying fish, a new constellation formed out of the ship
Argo.
Chameleon, a new constellation near the south pole.
Jlfons mensi formis, or table mountain, a mountain at the Cape of Good
Hope, well known to mariners. In this constellation the greatest of the
nebula, called by sailors the Magellanic clouds, is situated.
Ilobur Caroli, or Charles' oak, was so called by Dr. Halley in honour of
the tree in which Charles II. saved himself from his pursuers, after the
battle of Worcester. It was made out of the unformed stars between the
ship Argo and Centaur. This constellation is not on Gary's globe. Dr.
Halley went to St. Hellena in the year 1676, to make a catalogue of such
stars as do not rise above the horizon of London.
The reader may not be a little surprised to find us dwell so long upon
these fables of the ancients,this masquerade of folly, which exhibits nothing 1
but a chaos of fiction, without order or connection ; where we find the same
heroes presented under different names, the same actions at different
times, and though sometimes true, at other times destitute of foundation,
and worthy only of contempt. A slight acquaintance with these chimeras
of the poets, and the mythology of the Pagans, was however deemed, not
only necessary in understanding these authors, but also, being the founda-
tion of much of the ancient astronomy, considered as not foreign to a
work of this nature, and as useful in conveying an idea of the pretended
wisdom of those ancient nations, who were the inventors of them, or who
misapplied their original use and meaning.
DEFINITIONS, fcV. 43
The mystical OP allegorical sense of these fables in a philosophical or
historical view, conveyed an obscure explanation of some of the ordinary
operations of nature, or the inventions or exploits of some of these pre-
tended gods. In a religious sense, they served as a cloak for vice, and in
a political sense, they served to keep a superstitious people in subjection,
to those whose interest it was to conceal their mysteries.
The different parts of nature were portioned out to those whose know-
ledge was the greatest, or who were most successful in investigating the
properties of these parts, and applying that knowledg-e to the advantage
of mankind : and lest these persons, who were afterwards converted into
deities, should be thought mortal, their names were changed, and others
were given to t;hem expressive of their rank among the gods. Uranus,
Auranos, or the Heavens, was considered by them as the oldest of the
gods ; and Tithea, Tellus, Terra, or the Earth, his wife, by whom he had
the Titans. The chief of these was Saturn or Time, who is said to have
disputed superiority with his father ; as these heathens probably thought
nothing anterior to time. He married Ops or Terra t also called Rhea or
Cybele. She was therefore called the mother of the gods, who were
nothing else in reality but sons of the earth, or mortals. Saturn, consi-
dered then the most remote of the planets, was called after the name of
this god ; and hence the planet Herschel has, for the same reason, ob-
tained the name of Uranus, from modern astronomers, being more remote
than Saturn. Saturn is represented as a cruel god, who devoured his
own children, in allusion to time, which at length destroys every thing ;
and hence human sacrifices were offered to him, by the ignorant and su-
perstitious Pagans. Jupiter, however, the most illustrious of his off-
spring, escaped his fury, and afterwards dethroned him for attempting to
take away his life, and thus became sole master of the empire of the
world, which he divided with his brothers- He reserved the heavens and
the earth for himself, which, according to the poets, he filled with his
natural children, as he became a Proteus to gratify his passions. The
empire of the air he gave to Juno, his wife, that of the sea to Neptune,
and constituted Pluto king of the infernal regions. He was called Jupi-
ter, or Jove, in allusion to the Jehovah of the Jews ; as the Chaldeans,
who were so recently descended from Noah's son, could not be entirely
ignorant of the supreme being. The planet Jupiter is so called from this
god, being the largest of the planets, and the next in order after Saturn.
The three next planets, Mars, Venus and Mercury, the offspring of Jupi-
ter, were emblems of war, pleasure and science, which this god was so
famous for. The sun was called the prison of the gods, which shews thai
they had some idea of the force of gravity which retains the planets in
their orbits ; it was therefore an emblem of Jupiter, who held all the other
deities in subjection. Science, which conveyed the mysteries of these
gods and their pretended knowledge, was indicated by Mercury, who,
from his rapid velocity in its orbit, represented their messenger, and
hence he was painted with wings, 8cc. The goddess of love, or rather
lust, was represented by Venus, from its beautiful appearance, and its
remaining alone with the sun, the emblem of Jupiter, when all the other
luminaries disappeared. War was represented by Mars, from his fiery
or bloodlike appearance, &c.
That the Chaldeans were the first who under these fictitious titles deified
their kings, their warriors, their philosophers, &,c. is now universally allow-
ed. It is well known that Cham, one of Noah's sons, received Africa for
his portion, and made Egypt the chief seat of his residence. His name
signifies Cater or Niger, and Chamo, signifies Terra Cham or Egyptus, so
called therefore from Cham. His offspring Chanaan was cursed on account
of liis immodest behaviour to his father. Nemrod, his grandson, founded
the Babylonish empire, and is supposed to be the Saturn described above,,
44 DEFINITIONS, bv.
his cruel nature, agreeing with that of Saturn, who devoured his own chil?
dren ; as Nemrod hunted his subjects, as others hunt wild beasts. Uranus,
or Coelum, is supposed to allude to Cham, as these proud nations wished to
derive their origin from heaven, and not to acknowledge their gods as the
children of men. Belus, according to most authors, was the son of Nem-
rod, and second king of Babylon. Belus signifies Dominans, from Bel,
which in the Syriae language signifies the sun, the ruler of the solar sys-
tem. He was the first among the Chaldeans that cultivated astronomy, and
hence was honoured with the title of Jupiter or Jove, from his superiority
in the knowledge of this science, which obtained him so eminent a station
in the heavens. His son Ninus set up his father's image, and caused his
people to worship it. Ninus was therefore probably the first that attempt-
ed to pay those honours to a man (and an impious man too) that was only
due to God himself. Other nations followed the example, each bestowing
the same honours and marks of distinction on their founders, and ranking
them in the number of the gods. Hence Ninus, the Hercules of the Chal-
deans, being a great warrior, becomes the Jupiter of the Assyrians, whose
empire he founded. The Greeks and Romans had likewise their Jupiters,
their Hercules's, their Junos, their Venuses, their Mercuries or Minervas,
&c. which the mythologists so often confound with each other, and hence
the confusion in the accounts that we have of them. The tower of Babel,
begun by Nemrod, was converted into a temple, in which Belus was wor-
shipped. The priests of Belus applied themselves to the study of astrono-
my, and placed in the heavens, among the number of their deities, all those
that distinguished themselves either by their valour, their knowledge, or
their vices, or that supported them in their superstition. Hence arose the
numerous gods of the Chaldeans, the Egyptians, the Grecians, &c. and
hence the rapid increase of idolatry almost all over the world ; every na-
tion being desirous of claiming an alliance with, and of boasting their de-
scent from the gods. I shall mention here one circumstance that will throw
some light on the nature of these pretended deities, and account for their
vast number and increase. The famous tower of Babel was composed of
eight pyramidal towers raised one above another, in the highest of which
was a magnificent bed, where the priests daily conducted a woman, who,
as they said, was honoured with the company of the god. (For more parti-
culars see Joseph, ant. Jud. 10. Herodot. 1. c. 181, &c. Strabo. 16. Ar-
rian. 7. Diodorus 1, See.) Hence so many sons of Jupiter, so many he-
roes, so many Gods, Sec. The pretended worship of these priests, their re-
ligious ceremonies, &c. were all calculated to support and gratify their in-
famous passions ; and there was no place, from which modesty was more
industriously banished than from these ceremonies. 'I'hey even gave every
vice its own god, to support the worship of it. We need not, therefore, be
surprised at the rapid increase of idolatry, or at the description given of
this impious Babylon by different authors. (See Curtius. lib. 5. c. 5.) The
sacred writings also point out its abominations, and exhibit it, as an exam-
ple for posterity, of the folly of those who abandon their maker, and even
their reason, to gratify their passions ; and of the ridiculous pride of man
in desiring to be honoured as God.
The history of these gods became at length so obscure, and the human
mind so blind and corrupted, that the sun, moon, stars, &c. and at length
serpents, crocodiles, onions, &c. became objects of veneration and worship.
And this worship, extravagant as it may seem, became the worship of the
learned as well as the ignorant, except among the few whom God selected
from among these idolaters, who retained a knowledge of him, an esteem
for the dignity of human nature, a recollection of the glorious end to which
the true religion points the hope of man, and a reverence for that being
alone, who called all. other beings out of nothing. This chosen people was
Abraham and his posterity.
DEFINITIONS, e#. 45
113. The Galaxy, -via Lactca, or Milky way, is a whitish lu-
minous tract, which seems to encompass the heavens, sometimes
in a double, but generally in a single path, varying in breadth from
about 4 to 25 degrees.*
1 14. Bayer's Characters. This is a useful invention of denot-
ing the stars in every constellation, by the letters of the Greek and
Roman alphabets ; setting the first letter in the Greek alphabet ,
What a contrast would the true knowledge of God, handed down by these
venerable patriarchs and prophets, and displayed in its full splendour in the
gospel, afford, compared with the abominable mysteries of the Pagans (and
the impious absurdities of those who in modern times have copied their ex-
ample) traced to their origin, if modesty could permit the reading of them,
or our contracted limits a more ample detail.
Who could believe that the most enlightened part of modern Europe,
could afford anew, these scenes, so degradingto civilized man, and to human
nature ; that amid the acclamations of the polite, the accomplished, the en-
lightened citizens of Paris, the prostitutes of this city should be carried
fuidis corpdribus on triumphal cars, together with several youths in the same
savage degradation ! That dressed and decorated as became the solemnity of
the occasion, the citizens should inarch in solemn procession, accompanied
with the greater part of the youth of the city, crowned with chaplets of
flowers, emblematic of their being disciples of the goddess of reason, whom
they conveyed in such pomp to be worshipped in her tempts. (Sometimes
the Cathedral church cte notre dame !) When this age of science and of
Christianity, affords so humiliating a picture of the abuse of reason, and of
man, under the dominion of false philosophy, have we not reason to exclaim
with the philosophic Cicero, O tempora ! O mores ' ! !
Every age then affords monuments as testimonies to the value of that re-
ligion, in which nothing is left to the vanity of human speculation^but by
its own divine constitution conducts man to a greatness above his nature,
to that dignified and immortal existence, alone worthy the nobility of a ra-
tional being, the greatness and hopes of an immortal soul monuments that
will for ever decide the question in favour of those amiable virtues emana-
ting from the practice of the Christian religion, and the spirit which it
breathes ; when contrasted with the folly of impious or philosophic man,
adoring those idols which are the objects of his brutal passions, or overturn-
ing those laws which forbid the commission of his crimes.
* The miiky way comes properly under the head of constellations, being
composed of an infinite number of small stars, which causes that whiteness
from which it derives its name. It passes through Cassiopeia, where it is
nearest to the north pole, then through Perseus, Auriga, Taurus, the feet
of Gemini, Orion's Club, Monoceros, part of Canis Major, the ship Argo,
Robur Caroli, Crux, the feet of the Centaur, Musca Australis (where it ap-
proaches nearest to the south pole) Circinus, Norma, Ara, and Scorpio, where
jt divides into two parts. The eastern branch passes through the tail of
Scorpio, the bow of Sagittarius, Scutum Sobieski, the feet of Antinous,
Aquita, Sagitta, Vulpecula, and Cygnus. The western branch passes through
the tail of Scorpio, the right side of Serpentarius., Taurus Poniatowski,
Sagitta, Anser and Cygnus, where it meets the foregoing branch, and ends
in Cassiopeia, where J\Ianilius begins the description of it.
Manilius Caius was a celebrated mathematician and poet of Antioch, who
wrote a poetical treatise on Astronomy, of which five books are extant,
treating of the fixed stars. The age in which he lived is not known, though
some suppose that he flourished in the augustan age.
There are other lesser divisions of the galaxy, which may be seen in
IJevelius's firmament,
46
DEFINITIONS,
to the principal star in each constellation, to the second in mag-
nitude, and so on in order j and when the Greek alphabet is finish-
ed, the first letters a, b, c, &c. of the Roman alphabet is used.*
1 1 5 Nebulous or cloudy, is a term applied to certain fixed stars,
smaller than those of the 6th magnitude, which only shew a dim,
hazy light, like little specks or clouds. Nebula is when several
of these form a Cluster, ,f
116. The Solar Systcm\ is that part of the universe which
consists of the sun, filanetS) and comets.
* John Bayer, of Augsburg-, in Swabia, published in 1603, an excellent
work entitled Uranometria, being 1 a complete celestial atlas of all the con-
stellations, in which the stars are denoted as above. Succeeding astrono-
mers have adopted this useful method of describing the stars, and enlarged
it by adding the numbers 1, 2, 3, &c. in order, when any constellation
contains more stars than can be marked by both alphabets. These figures
are also sometimes placed above the Greek letter, especially where double
stars occur ; for though many stars may appear single to the naked eye, yet,
v/hen viewed through a telescope of considerable magnifying power, they
appear double, triple, &c. Thus in Dr. Zach's Tabulae Motum Solis, we
find / Tauri, Tauri, y Tauri, ^ Tauri, $* Tauri, &c.
The following Greek alphabet is inserted for the use of those who are
unacquainted with the letters ; the capitals are however seldom used.
Name.
Sound.
Name.
Sound.
A a,
Alpha
a
N v
Nu
n
B0C
Beta
b
S |
X
x
Fyf
Gamma
g
o
Qmicron
o short.
A 3
Delta
d
n
P
P
E E
ILfisilon
e short.
p P
Rho
r
z
Zeta
z
So-r
Sigma
s
H 1
Eta
e long.
Trl
Tau
t
s 6
Thcta
th
Y t
Ujfisilon
u
I
Iota
i
o p
Phi
ph
K *
Kaftfia
k
X %
Chi
ch
A A
Lambda
1
%p4/
Psi
ps
M p
Mu
m
n<
Omega
o long.
| Dr. Herschel has discovered no less than 1250 of these nebulae : there
were only 103 known to former astronomers. He has shewn that the milky
way is a continued nebulae. There are two remarkable nebulas near the
south pole, called by sailors the magellanic clouds, which resemble in
brightness the milky way. The number of stars in these nebulae exceed
conception. 70 stars have been reckoned in the Pleiades, no less than 2500
in the constellation Orion ; and Herschel, in some of his observations on the
milky way, found that by allowing 15' for the diameter of his field of view,
a belt of 15 long, and 2 broad, which he had often seen pass before his
telescope in an hour's time, could not contain less than 50,000 stars, large
enough to be distinctly numbered.
$ By system is meant a lucid body, with some number of opake bodies
situated within the sphere of its influence, and round which the others re-
volve.
By the universe we understand the whole material creation. The Greeks
called it Topan, signifying every, thing, and the Latins Inane, the void. See
the fourth part of this work.
DEFINITIONS, Vc. 47
117. The sun is that lucid body, situated nearly in the centre
of the solar system.
118. Planets are opake* bodies similar to our earth, which
perform their motions round the sun, in certain periods of time.
They are divided into primary and secondary.
119. The primary planets f are those which regard the sun as
their centre of motion. There are 9 primary planets, distinguish-
ed by the following characters, and names, according to their
proximity to the sun, viz. $ Mercury, ? Venus, (B Eaith,
5 Mars, J-J Juno, -f- Pallas, o Ceres, ^ Vesta, 11 Jupiter,
*2 Saturn, $ Herschel or Uranus.
120. The secondary planets, called Satellites or moons, are
those bodies which are attendants on the primary planets, and re-
gard them as the centres of their motion ; as the moon which re-
volves round the earth, the Satellites of Jupiter which revolve
round Jupiter, &c. There are 18 secondary planets, of which the
Earth has one, Jupiter four, Saturn seven, and Uranus six.
121. The orbit of a planet is the imaginary path which it de-
scribes round the sun. The earth's orbit is the ecliptic. The
real motion of all the planets in their orbits round the sun is from
west to east, or according to the order of the signs on the ecliptic.
122. The nodes are the two opposite points, where the orbits
of the primary planets cut the ecliptic, and where the orbits of
the secondaries cut the orbits of their primaries. That node is
called ascending, where the planet passes from the south to the
* Opake bodies are such bodies as do not shine by their own light, or
which only reflect the light received from another body, as the planets
which reflect the light received from the sun.
t The planets are so called from Planeta, a wanderer, because they
change their positions in the heavens, with regard to the other bodies,
which are called fixed, for a contrary reason. Uranus, Juno, Pallas, and
Ceres, were recently discovered, and obtained their names in conformity
to the names given to the other planets by the ancients.
Uranus was discovered by Dr. Herschel, in 1781. La Place, in B.I.
c. 9, vol. 1, of his Astronomy, observes, that Flamstead, at the end of the
last century, and Mayer and Le Monnier, in this, had observed it as a
small star.
On the 1st of January, 1801, M. Piazzi, astronomer royal at Palermo,
in Sicily, discovered Ceres, generally called Ceres Ferdinanda (or rather
Fernanded) the latter name being added in honour of Ferdinand IV. king
of the Two Sicilies. It is of the 8th mag. and consequently invisible to
the naked eye, nor is it confined within the ancient limits of the zodiac.
It is called by some astronomers an asteriod.
On the 28th of March, 1802, Dr. Olbers, of Bremen, discovered Pallas,
and on the 29th of March, 1807, at 21 min. after 8, mean time, he dis-
covered another, which he called Vesta. This last, in size, appears like
a star of the 5th mag.
On the first of September, 1804, Mr. Harding, of Lilienthal, in the
duchy of Bremen, discovered the planet Juno. It appears like a star of
the 8th mag. These four last satellites are all so nearly at equal dis-
tances from the sun, that it is not as yet ascertained, with certainty, which
of them is nearest to or most remote from it.
48 DEFINITIONS, &c.
north side of the ecliptic ; and the opposite point, where the planet
appears to descend from the north to the south, is called the de-
scending or south node. The ascending node is marked thus 2,
and the descending node thus S The straight line which joins
the nodes ib called the line of the nodes.
123. Asfiect of the stars or planets, is their situation with respect
to the sun or each oth^r. There are five aspects, viz. cS Conjunc-
tion^ when they have the same longitude, or are in the same sign
and degree with the sun ; sfc Sextite, when they are two signs or
a sixth part of a circle distant ; D Quartile, when they are distant
three signs, or a fouth part of a circle ; A Trine, when they are
four signs, or a third part of a circle from each other ; and 8 Ofi-
position, when they are six signs, or half a circle from each other.
The Conjunction and Opposition, particularly of the moon, are
called the Syzygies,* and the quartile aspect the Quadrature.
124. The apparent motion of the planets is either Direct, Sta-
tionary, or Retrograde. Direct is when a planet appears to a
spectator on the earth to perform its motion from west to east, or
according to the order of the signs. A planet is Stationary when,
to an observer on the earth, it appears, some time, in the same
point of the heavens ; and Retrograde when it apparently goes
backward or contrary to the order of the signs.
125. Aphelion, or Aphelium, is that point in the orbit of a planet
which is furthest from the sun. This point is also called the higher
Apsis.
126. Perihelion, or Perihelium, is that point in the orbit of a
planet, which is nearest to the sun. This point is called the lower
Apsis.
127. Apogee, or Apogseum, is that point in the orbit of a planet,
the moon, Sec. which is furthest from the earth.
128. Perigee, or Perigaeum, is that point in the orbit of a plan-
et, the moon, Sec. which is nearest to the earth.
1 29. Apsis of an orbit, is either its aphelion or perihelion, apo-
gee or perigee, and the straight line which joins the higher and
lower apsis is called the line of the Apsides.
130. Eccentricity of the orbit of any planet, is the distance be-
tween the sun and the centre of the planet's orbit.
131. Geocentric latitudes and longitudes of the planets, are their
latitudes and longitudes as seen from the earth.
132. Heliocentric latitudes and longitudes of the planets, are the
latitudes and longitudes as they would appear to a spectator, situa-
ted in the sun.
133. True Anomaly of a planet is its angular distance at any
time, from its aphelion or apogee. Mean Anomaly is the angular
distance at the same time, and from the same point, if it had mov-
ed uniformly with its mean angular velocity.
* So called from the Greek word Suzugia, Conjunctio, Zugos signify"
ing 1 jugum, a yoke, or pair.
DEFINITIONS, &c. 49
134. Equation of the centre is the difference between the true
and mean anomaly ; this is sometimes called the firosthapheresis*
135. The mean place of a body is the place where it would
have been if it had moved with its mean angular velocity (on sup-
position that the body in motion does not move with an uniform,
angular velocity about the central body.) The true place of a
body is the place where the body actually is at any time.
136. Equations, are corrections which are applied to the mean
place of a body to get its true place.
137. Argument, is a term used to denote any quantity by which
another required quantity may be found. Thus the argument of a
planet's latitude is its distance from the node, because it is upon
that the latitude depends.
138. The elongation of a planet from the sun, is its angular dis-
tance from the sun when seen from the earth j or the angle form-
ed by two straight lines drawn from the earth, the one to the sun,
and the other to the planet.
139. The curtate distance of a planet from the sun or earth, is
the distance of the sun or earth from that point of the ecliptic
where a perpendicular to it passes through the planet.
140. A Digit, is the twelfth part of the apparent diameter of the
sun or moon.
141. Disc, is the face of the sun or moon, such as they appear to
a spectator on the earth ; the sun and moon appearing as circular
planes, though they are in reality spherical bodies.
142. Occupation of a star or planet, is when they are hidden
from the sight by the interposition of the moon or some other
planet.
143. Merration, is an apparent motion of the celestial bodies,
occasioned by the earth's annual motion in its orbit, combined with
the progressive motion of light.
PROBLEMS
PERFORMED BY THE
TERRESTRIAL GLOBE,
i
PART II.
PROB. I.
To Jind the latitude and longitude of any given jilace.
JRule. BRING the given place to the graduated side of the
brazen meridian, which is numbered from the equator ;* the de-
gree over the place is the latitude (definition 10.) and the degree
on the equator, cut by the brass meridian, is the longitude (def. 1 1.)
Example 1 . What is the latitude and longitude of Washington
city ? t
Answer. Lat. 38 53' north. Longitude 77 14' west.
* Whenever a place is brought to the brazen meridian, the graduated
edge which is numbered from the equator towards the north or south pole
is always understood, unless the contrary be mentioned.
j- The latitude and longitude of this city is not, as yet, so correctly as-
certained as might be expected, and therefore it cannot be made the basis
of any accurate or important calculations or tables. This being therefore a
point of such public utility, its importance must appear evident to every Ame-
rican citizen, who has the most superficial knowledge of these matters, and
ieels an interest, not merely for science, but for the reputation and growing
importance of his country.
Various methods are given by different authors, for finding the latitudes
and longitudes of places on the earth, the substance of which is given in the
course of this work, with some new and important methods not published
in any other treatise, together with the principles on which they are found-
ed. See problems 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, Sec.
part 2d. and problems 19, 20, 21, 22, 23, 24, 25, 26, 27, &c. part 3d. with
the notes, &c. to these respective problems. For finding a star or planet's
transit over the meridian, see probs. 8 and 39, part 3d.
The principal difficulty in any of the methods for finding the latitude, is
to find the correct altitude, and when necessary, the time of the body's tran-
sit over the meridian. To obtain these requisites, various instruments have
been contrived ; the most useful of which on land, for want of a good hori-
zon, are the astronomical and mural quadrants, and a good transit instru-
ment. These are fully described in Vince's treatise on Practical Astrono-
my. But at sea, the best instruments are Godfrey's quadrant, commonly
called Hadley's quadrant, and a good sextant, which is preferable to the
foregoing instrument. These are sufficiently described in the various books
on Navigation, particularly in M'Kay, Norie, Hamilton Moore, Bowditch's
American Practical Navigator, &c.
There is also a circle of reflection or repeating circle (now sometimes
called the astronomical circle) described in the latin part of Mayer's tables,
published by Nevil Maskelyne, and after him by different authors, which
PROBLEMS, We.
51
2. What is the latitude and longitude of New-York ?
Ans. Lat. 40 42' 40" N. and long. 74 I 7 W.
Note. The long, of New-York, from a solar eclipse observed in June,
1806, is found to be 74 1' west of Greenwich ob. See pa. 60 of the Nauti-
cal Almanac, 1811 or 1812, published in New-Brunswick, New-Jersey, un-
der the direction of Mr. John Garnett.
3. Required the latitudes and longitudes of the following places :
Amsterdam,
Aleppo,
Algiers,
Baltimore,
Barcelona,
Batavia,
Bencoolen,
Berlin,
Boston,
Breslaw,
Buenos Ayres,
Cadiz,
Cairo, (Grand)
Calcutta,
Canton,
Cape of Good Hope,
Charlestown,
Constantinople,
Copenhagen,
Dantzic,
Delhi,
Dresden,
Dublin,
Edinburg,
Fez,
Funchal,
Greenwich, (obs.)
Halifax,
Hamburg,
Ispahan,
Lexington,
Lima,
Lisbon,
London,
Madrid,
Paris,
Pekin,
Petersburg,
Philadelphia,
Quito,
Rome,
Stockholm,
Tripoli,
Vienna.
at sea is a very useful and accurate instrument, and by a method easily
practised, may be used with equal advantage on land, being lately much
improved and adapted for astronomical observations in general. The me-
thod consists in bringing the image of the sun, &c. (in the same manner as
it is brought to the edge of the horizon at sea) to coincide with its image
reflected from a basin of water, quicksilver, molasses, or any reflecting
surface parallel to the horizon ; half the sum of the angle thus found, will
be the altitude required.
Thus let S represent the sun's place, A the
reflecting surface, placed parallel to the hori-
zon BH, C the place of the spectator, and Cd
the height of his eye. Now it is evident that
the arch SB, being described from the centre
A, will be the altitude of S above the horizon.
If the image of the sun be then brought by the
instrument to the point B, the altitude will be
pointed out by the index ; but to have the image
of the sun (now at B) to coincide with the reflected image at A, it is evi-
dent that it must be brought to the point t, where it meets with the reflect-
ed ray As ; and an eye placed in any part of the line As, where the ray ivS
reflected, will observe the image of the sun, brought by the instrument to t,
in this direction. But as the angle of incidence SAB is equal to the angle
of reflection sAH (prop. 9. cor. 2. Emerson's tracts, or his optics B. 1.
prop. 10.) that is to BAt (15 Eucl. 1. B.) the arch St, which is the mea-
sure of the angle observed by the instrument, is double of SB ; therefore
half of St, without any allowance for the height of the eye, above the re-
flecting surface, will be the altitude required.
Though it be evident that no allowance is to be made for the height of
the eye above the reflecting surface, as an eye placed in any part of the line
As will observe both images coincide in the direction At, there is, however,
rigorously speaking, an allowance to be made for the height of the sp^-t.
52 PROBLEMS PERFORMED BY
2. Find all the places on the globe which have no latitude.
3. What is the greatest latitude a place can have ?
tor or bason at A above the level of the sea, &c. or the true horizontal lev-
el ; it being evident that the more elevated the point A is above this level,
the less will the altitude be. For if a be taken as the reflecting surface*
the arch Su, which is double of Sr, will be less than St.
But this will make so small a difference even on the tops of the highest
mountains, that it maybe always neglected except when very great accura-
cy is required. For in the suii's altitude the whole semidiameter of the
earth will only make a difference of 8" 83 even at the horizon ; hence the
semd. of the earth : the height of the mountains, &c. :: 8" 83 : the correc-
tion of the horizontal alt. which will also diminish in proportion to the height
of the object. And even for the altitude of the moon thus found, scarce
any allowance must be made for the height of the mountain, &c. though
its mean horizontal parallax amounts to 57* 39''. For 3956 miles the earth's
$em. diam. : 1 mile, the supposed height of a mountain :: 57' 39" : 0" 82,
not amounting to V even at the horizon, when the height of the mountain
is 1 mile ; and this will diminish in the proportion of Had : cosine apparent
altitude. Hence this method may be successfully practised in any situation
on land, care being taken that the reflecting surface be not agitated by the
wind, &c. to prevent which several contrivances may be made use of, which
the skill and ingenuity of the observer will suggest.
A good observer ought to be well acquainted with the elementary princi-
ples of Geometry, Astronomy, Mechanics, and Optics, to be able to adjust
his instruments with skill, and in every circumstance to apply them to the
test advantage,- and allow for every defect or error, &.c. that may take
place. Hadley's quadrant or the sextant, will answer equally well on land,
as the above instrument, when the altitude of the object does not exceed
half the number of degrees marked on them. The learner must also take
notice that the glass through which the sun's reflected image, in the water
or on the reflecting surface, is observed, must be coloured to preserve the
eye from injury, and to render the sun's image more distinct by destroying
the effects of irradiation.
There are likewise various methods given for finding the longitude, the
most useful of which are the following : 1st. Having the time of the moon's
southing at Greenwich, or at any other place whose longitude is known, ob-
serve the time of the moon's southing at the place of observation, by help
of a correct meridian line, and a good clock or time piece, exactly adjusted
to mean time, or 24 hours in a day ; (see notes to prob. 29 and 39, part 3d.)
find the difference of these times, then say as the difference of the times of
the southing of the sun and moon in 24 hours : this difference :: 360 : diff.
longitude ; this difference added to the longitude of the known place, if
the time of southing there be later, or subtracted if sooner, gives the longi-
tude of the place of observation. Where exactness is required, the motion
of the moon from the sun in 24 hours, must be taken from the Nautical Al-
manac, or from correct astronomical tables, the latest and best of which
are Mr. de Lambre's new tables of the sun, &c. and Mr. Burg's new tables
of the moon, published in 1806 by the French board of longitude, and since
translated and published by Vince. The difference of the times of the sun
and moon's southing, at a medium, is about 48 minutes daily. The south-
ing of any celestial body is found by a transit instrument, or by suspending
two plummets in the meridian line, the lower end of each being immersed
in a basin of water to prevent their swinging 1 . When the object comes in ;i
straight line with the threads of these plummets, viewed with the naked
eye, 3 small hole made with a pin through a sheet of paper, or with a tele-
scope, it is then on the meridian.
THE TERRESTRIAL GLOBE. 53
4. What is the greatest longitude a place can have ?
5. Find all those places which have no longitude.
The longitude maybe also found by the meridian transit of a fixed star,
allowance being- made for the sun's right ascension, &c.
Solar and lunar eclipses afford another method, by converting 1 the differ-
ence of times between their beginning-, middle, or ending-, as observed in
the place whose longitude is required, and the same times calculated or
rather observed in any other, whose longitude is correctly ascertained ; the
difference of times converted into degrees, &c. of longitude, will give their
difference of longitude, from which the required longitude is found.
Eclipses of Jupiter's satellites are however much preferable on land, be-
cause they happen almost every day, and the times of their happening- are
more correctly and easily found. This time should be found very accurate-
ly, as an error of one second in time, will produce an error of fifteen seconds
iii the longitude. (For 4 min. : 1 :: 1" : 15") The first satellite is the
most proper for determining the longitude. Its emersions are not however
visible from the time of Jupiter's conjunction with the sun to the time of
his opposition, and its immersions are not visible from Jupiter's opposition
to his next conjunction. The positions or configurations of Jupiter's sa1,el-
lites as they appear at Greenwich, are laid down in page 12th of the month
in the Nautical Almanac, for every night when visible. The times of their
eclipses happening at the meridian of Greenwich, are found in page 3d of
the Nautical Almanac for every month. These eclipses must be observed
with a good telescope, and a well adjusted pendulum clock, that beats se-
conds or half seconds. The telescopes proper for observing the eclipses of
Jupiter's satellites, as Nevil Maskelyne remarks, are common refracting-
telescopes from 15 to 20 feet, reflecting telescopes of 18 inches or 2 feet,
local length, and telescopes of Mr. Dolland's construction, with two object
glasses from 5 to 10 feet, or which are still more convenient, those of 46
inches focal length, and 3f inches aperture, constructed with three object
glasses, which are as manageable as reflecting telescopes, and perform as
much as those which he makes of 10 feet with two object glasses. The
manner of adjusting the clock, and an explanation of the configurations of
the satellites, will be given in part 4th.
At sea the principal method of finding the longitude, is by computing the
distance between the moon and the sun, or some principal fixed star ; from
which their altitudes being given, the required longitude is found. The
principles of this method are demonstrated in the notes to prob. 28 and
29, part 3d. The reader is also referred to M'Kay's treatise on the longi-
tude, his treatise on Navigation, to Norie or Bowditch's improvements on
Hamilton Moore, or to the Nautical Almanac for 1812, revised by John Gar-
net, New-Jersey. Mayer jn the beginning of his tables gives also a method
of finding the same under the title of JWethodus Lpngitudinum promota et
Jl.dditame.ntum. Mr. Delamar in Philadelphia, has likewise lately favoured
the public with some very useful methods in the practice of this important
problem.
Another method of finding the longitude is with a timepiece. If the
time piece could lie depended on, this method would be by far the most
easy and expeditious. For if set to mean time in any place whose longitude
is known, it would point out the difference of times between this place and
any other place to which a person arrives, which converted into de-
grees, &.c. would be the difference of longitude required. The watch
should be kept going, and not changed the whole time during the journey
or voyage ; or if changed, allowance should be made for it.
The last method I shall take notice of here is that of the variation chart.
introduced by the celebrated Uoctor Haley. On this chart are drawn curve
lir;'-s which represent the variation in almost every degree of longitude.
54 PROBLEMS PERFORMED BY
6. Find all places those which have neither latitude nor longi-
tude.*
7. Find all those places that have the greatest longitude.
8. Find those places that have the greatest latitude and longi-
tude.
9. Find those places that have all possible degrees of longitude,
reckoned from the same meridian.
PROB. 2.
To find any place on the globe, the latitude and longitude of which
are giv en.
Rule. FIND the longitude of the given place on the equator, f
(problem 1 .) and bring it to the brazen meridian ; then under the
given latitude found on the brass meridian is the place required.
(Def. 10.)
As this chart is so familiar to mariners, it needs no further description here.
This would also be an easy and expeditious method of finding the longi-
tude, could the variation and its yearly change be once exactly ascertained
in the different parts of the world.
An attempt has likewise been made of rendering the magnetic needle
useful in finding- the latitude. In the year 1580 it was discovered by one
Robert Norman, a compass maker in England, that the needle had a cer-
tain inclination in a contrary direction to the inclination of the earth's axis ;
this discovery being communicated to others, it was found that at the equa-
tor it has no inclination, being there parallel to the plane passing through
the earth's axis or to the horizon ; but that it depresses one end if we re-
cede from the equator towards either pole ; the north end, if we advance
towards the north, and the south end if w r e go towards the south. This in-
clination was found to vary in proportion to the distance from the equator,
and was therefore thought to correspond to the latitude : but the poles of
the earth varying from those of the needle, and continually changing, be-
sides many other latent causes, prevent its theory from being sufficiently
known, and render the task of making the necessary experiments to ascer-
tain the phenomena, rather discouraging in the present state of science, so
that nothing conclusive can as yet be deduced from it. The reader will
find more particulars on this subject in Cavallo's treatise on Magnetism.
* As all places lying under the equinoctial or on the equator have no la-
titude, and all places situated on the first meridian have no longitude,
therefore that point on the globe where the first meridian intersects the
equator, has neither latitude or longitude. Again, as the latitudes of pla-
ces increase as their distance from the equator increases, and their longi-
tudes increase as their distance from the first meridian increases, it follows
that the greatest latitude a place can have is 90, and the greatest longi-
tude 180, which being half the circumference of the globe, no two places
can be at a greater distance from each other than 180.
f On Adam's globes there are two rows of figures above the equator.
When the place lies to the right hand of the meridian of London, the longi-
tude must be reckoned on the upper line ; when it lies to the left hand, it
must be reckoned on the lower. On Gary's globes, on which are also two
lines, the longitude being reckoned from the meridian of Greenwich,
\vhen the longitude is east, the upper line is used, but when west, the low-
er. The figures under the equator in this globe indicate the half hours and
quarter^ and the dots the minutes. There are two rows of hours on each
THE TERRESTRIAL GLOBE. 55
Example 1. Find that place whose latitude is 32 25 ; 40" N.
Longitude 63 35' 40" west from Greenwich.
Ans. The most northerly part of the Island of Bermudas.
3. Find those places whose latitudes and longitudes are as fol-
lows:
Latitudes. Longitudes. Latitudes. Longitudes.
52 22' 45" N. 4 45' 30" E. II 42<> 23' 15" N. 70 58' 0" W.
31 11 20 N. 30 15 30 E. I 4 56 10 N. 52 16 W.
52 S2 30 N. 13 26 E. 41 00 N. 28 22 SO E.
30 2 30 N. 31 15 15 E. 51 28 39 N.
33 55 15 S. 18 29 E. 12 1 15 S. 76 55 30 W-
48 50 14 N. 2 19 E. II 39 55 16 N. 116 21 30
PROB. 3.
To find all these places that are in the same latitude or longitude
ivith any given place.
Rule. BRING the given place to the brazen meridian,* and mark
the point over it ; then all those places under the same edge of the
meridian, between both poles, are in the same longitude ^def. 11.)
and all those places passing under the mark are in the same lati-
tude, (def. 10.)
Example 1 . Find all those places that have the same, or nearly
the same latitude or longitude as New-York.
Ans. Montreal, Heneaga island, the western part of St. Domin-
go, St. Martha, Guamanga, La Conception, 8cc. in S. America, are
nearly in the same longitude ; and Madrid, Naples, Constantinople,
Pekin, Sec. nearly in the same latitude.
2. What places have the same or nearly the same latitude as the
following places : London, Petersburg, Rome, Philadelphia, and
Lima?
3. Find all those places that have nearly the same longitude
as the following places : Paris, Archangel, Naples, Boston, and
Mexico, in North America.
4. What inhabitants of the earth have their days of the same
length as those of Philadelphia ?
5. What inhabitants of the earth have the same seasons of the
year as those of London ?
side of the equator, reckoned all round from aries both ways ; the one to-
wards the right hand or east, is carried to XXIV, to this the minutes, &c. are
adapted. The other, which is reckoned towards the left hand or west, is
counted only to XII, and then begins I, II, III, &c. again. These lower
lines are useful in finding the difference of times between any two meridi-
ans, or for shewing how much sooner or later the time is in one place than
in another. One is fitted for astronomical, the other for civil time. Bardin's
new British globes have also two rows of figures above the equator, but the
lower line is always used in reckoning the longitude.
* It will answer equally as well to bring the given place to the horizon,
wd count the degrees from the east or west points, &c.
56 PROBLEMS PERFORMED BY
6. Find all those places that have their longest day the same a:
at Petersburg.
7. When it is noon at Baltimore, what inhabitants of the earth
have the same hour ?
8. When it is noon, midnight, or any other hour in Boston, find
all those places that have the same hours respectively.
PROB. 4.
To find the difference of latitude and difference of longitude between
any two filaccs.
FOR THE DIFFERENCE OF LATITUDE.
Rule. FIND the latitude of both places (prob. I.) and the num-
ber of degrees between them, reckoned on the brazen meridian,
will be the difference of latitude. (Note to def. 10.)
Or, find both latitudes ; (prob. 1.) then if they be of the same
name, that is both north or both south, their difference is the dif-
ference of latitude ; but if they be of different names, that is one
north and the other south, their sum will be the difference of lati-
tude.
FOR THE DIFFERENCE OF LONGITUDE.
Rule. Find the longitude of both places (prob. 1.) and the num-
ber of degrees between them, reckoned on the equator, will be
their difference of longitude. (Note to def. 10.)
Or, find both longitudes as before, then if they be of the same
name, that is both east or both west, their difference is the differ-
ence of longitude ; but if they be of different names, their sum
will give the difference of longitude.
Note. If this last sum should exceed 180, take it from 360, and the re-
mainder will be the difference of longitude.
For the difference of longitude in time. Bring one of the places
to the brazen meridian, and set the hour index to 1 2 ; then bring
the other place to the meridian, and the hours, Sec. passed over by
the index, will be the difference of longitude in time, as required.
The same may be found more correctly on the equator, by taking
the sum or difference of the times corresponding to the longitudes
on the equator, instead of the longitudes themselves.
Example \ . What is the difference of latitude and difference of
longitude between Philadelphia and Greenwich observatory ?
Ans. Diff. lat. lloSl' 45". Diff. long. 75 8' 45".
Note 2. If one of the places have no latitude or no longitude, the lati-
tude or longitude of the other will be the difference.
2. What is the difference of latitude and difference of longitude
between Paris and Greenwich observatories ?
Ans. Diff. lat. 2 38' 25". Diff. long. 2 19'.
3. What is the difference of longitude in time and degrees be-
tween Paris and Gottingen observatories i
Ans. Diff. of longitude in degrees 7 32' 45". In time 30 11'.
Note 3. For exactness in these problems, consult the table of latitudes
and longitudes at the end of the work.
THE TERRESTRIAL GLOBE. 57
4?. Find the greatest difference of latitude and difference of lon-
gitude between any two places
5. Required the difference of latitude and longitude between the
following places ?
London and New-York, Edinburgh and Baltimore,
Ooa and Rome, Washington City and Jerusalem,
Constantinople and Quito, Hamburg and New-Orleans,
Peiersburg and Vienna, Cape of Good Hope and Canton,
Dublin and Boston, Calcutta and Philadelphia,
Charlestown and Pekin, Havanna and Gibraltar.
Pekin and Lima,
Note 4. The difference of lat. or difference of long, between two places
being 1 given, and if one of the places be also given, the other is given by ad-
ding or subtracting- this difference, according- as the place is north or south,
east or west of the given place.
Thus, for the latitude. 1. If the latitude of Washington city be 38 53'
N. and the difference of lat. between it and New-York be 1 4V, then 38
53'+ 1 4<y=40 42', the lat. of New-York, being northward of the given
place.
2. The lat. of Washington being 38 53' N. and the diff. between it and
that of St. Domingo 20 33', then 38 53' 20 33'= 18 20', the lat. of St.
Domingo, being southward of Washington.
3. The lat of Washington being as above, and the diff. of lat. between it
and Lima, in south lat. being 50 54', then 50 54' 38 53'= 12 1', the lat.
of Lima. The difference of lat. in this case being the sum of both latitudes,
the lat. of either is evidently found by taking the lat of the other from the
cliff, of lat.
Again, for the longitude. 1. If the longitude of Washington city be 75
14' 22" W. of Greenwich observatory, and the diff. of long, between it and
New- York be 1 13' 22", required the long, of New-York. Here 75 .14' 22"
1 13 r 22"==: 74 1', New-York being eastward of Washington.
2. The longitude of Washington city being given as above, and the diff. of
long, between it and New-Orleans being 14 52' 8", then 75 14' 22"+ 14
52' 8"= 90 6' 30", the long, of New-Orleans, being west of Washington.
3. If the long, of Washington be as above, and the difference of longitude
between it and Paris observatory be 79 33' 22" ; then 79 33' 22" 77 H/
22"=2 19 f . Paris being situated in east long, and the sum of their longi-
tudes from Greenwich being the difference of longitude.
PROB. 5.
To find the antoeci, periced^ and antipodes ofanyfilace.
Rule. BRING the given place to the brass meridian, and find
its latitude (prob. I .) then under the same meridian, in the same
degree of latitude, in the opposite hemisphere, you will find the
antoeci. (Def. 22.) The globe remaining in the same position, set
the index to 12, and turn the globe on its axis until the other 12
comes to the meridian (or until the index points to it) then under
the latitude of the given place you will find the perioeci (def. 23.)
and under the same meridian, in the same degree of latitude, but
in the opposite hemisphere, you will find the antipodes, (def, 24.)
.H
8 PROBLEMS PERFORMED BY
Or thus,
Place both poles in the horizon, and bring the given place to
the eastern part of the horizon ; then, if the place be in north lati-
tude, observe how many degrees it is to the northward of the east
point of the horizon ; the same number of degrees reckoned to the
southward of the same point will give the antoeci ; an equal num-
ber of degrees counted from the west point of the horizon towards
the north will shew the perioeci ; and the same number of degrees
counted towards the south from the west, will point out the anti-
podes.
If the place be in south latitude, the same rule will serve, by
feading south for north, and the contrary. This method is the
same in effect as the above.
Example 1. Required the antoeci, perioeci, and antipodes of Ber-
mudas ?
Ans. The antoeci is in Paraguay, a little S. E. of Cordova, or N.
W. of Buenos Ayres ; the perioeci is near Yongyong, N. W. of
Nankin, in China ; and the antipodes is near Binning's land, in
the S. W. pan of New Holland.
2. Required the antoeci, perioeci, and antipodes of the following
places : Constantinople, Rome, London, Cape of Good Hope,
Quito, Buenos Ayres, Kingston, and Skalholt.
3. A person sailing in lat. 5l south, and long. 180. Where
was his antipodes ?
Note. Those places situated on the equator have no antoeci, and their pe-
I'iceci are their antipodes ; and those places at the poles have no perioeci, and
their antoeci are their antipodes.
4. Required those places whose seasons are directly contrary to
those of New-York (that is summer with one being winter with
the other, &c ) but whose hours are the same (that is mid -day with
one being midnight with the other, Sec.)
5. Required those places whose seasons are the same as those of
Philadelphia, but hours contrary ?
6. Required those places whose seasons and hours are contrary
to those of Washington city ?
For the three last problems see notes to definitions 22, 23, 24.
PROB, 6.
The hour of the day at any particular place being given, to find the
corresponding hour (or what o'clock it is at that time) in any
other place.
Rule. BRING the place where the time is given to the brass
meridian, set the index to the given hour ; then turn the globe
till the other place comes to the meridian, and the index will point
out the time required.
Or, Having brought the given place to the meridian, as before,
set the index to 1 2 ; then bring the other place to the meridian,
and the hours passed over by the index will be the difference of
THE TERRESTRIAL GLOBE. 59
time between both places. If the place where the hour is sought
be to the east of the other, the time there is so much later, if to
the west, the time is so much earlier. Hence, in the former case,
you add the diff to the given time, in the latter you subtract Thus
a place 1 5 to the eastward of another, has the sun on its meridian
an hour earlier than the latter place ; therefore 12 o'clock in the
former place is but 1 1 o'clock in the latter ; and 12 o'clock in the
latter place is 1 o'clock in the former, Sec.
WITHOUT THE HOUR CIRCLE.
Find the difference of longitude between the two places (prob.
4.) and turn it into time by allowing 15 for every hour, and 4
minutes of time to every degree, Sec. The difference of longi-
tude in time will be the difference of time between the two places,
with which proceed as above.
Examfilc 1 . When it is 7 o'clock in the morning at Philadel-
phia, what hour is it at London ?
Ans. Twelve o'clock at noon ; the difference of time being five
hours nearly, and London to the east of Philadelphia.
Or, The difference of longitude between both places is 75 1 3'.
Now 75-r-l5 = 5 hours and 13'x4=52, hence 5h. 0' 52"+7h.=r:
12h. 0' 52", or 52" after 12 o'clock.
Note. Degrees of longitude multiplied by 4 produce minutes of time,
and minutes multiplied by 4 produce seconds of time, &c. and minutes and
seconds of time divided by 4 give degrees and minutes of longitude, &c.
2. When it is 7 o'clock in the morning in London, what o'clock
is it at Philadelphia ?
Ans. 2 o'clock in the morning, or 7h. 5h. 0' 5 2"= In. 59' 8".
3. When it is 2 o'clock in the afternoon at Greenwich observa-
tory, what o'clock is it at Baltimore ?
Ans. 8h. 52' 40" in the morning. The difference of longitude
is 76 50', which multiplied by 4=307' 20"=5h. 7' 20". 12-f 2
= 14; hence 14h. 5h. 7' 20"=8h. 52' 40", or 52' 40" after 8
in the morning at Baltimore. The same answer will be found, if
5h. 7' 20", the difference of longitude in time, be counted back-
wards from 2 o'clock in the afternoon, as Baltimore is to the west
of Greenwich observatory.
4. When it is noon at Paris, what hour is it at Quito ?
5. When it is 10 o'clock in the morning at Kingston in Jamai-
ca, what hour is it at Petersburg ?
6. When it is 1 o'clock in the afternoon in Washington city,
what o'clock is it in Canton ?
7. When it is midnight in New-York, what o'clock is it in Lon-
don, in Madrid, in Rome, in Vienna, in Calcutta, and in Botany
Bay?
8. My Watch being well regulated at Dublin, and when I arrived
at Philadelphia it was 5 hours faster than the clocks there. I want
to know whether it gained or lost during the voyage, and how
much ?
60 PROBLEMS PERFORMED BY
9. Are the clocks in Philadelphia faster or slower than those a?
Calcutta, and how much ?
10. Being at sea in lat 10 45' N, my watch, which was adjust-
ed for the meridian of Greenwich, was by observation found to be
4h. 4 minutes too slow. Required the place of observation and
its longitude ?
Ans. Trinidad in the West-Indies.
1 1 . Being at sea in the year 1 806, on the 1 6th of June, I observ-
ed the beginning of an eclipse of the sun at lOh. 16' 12" in the
forenoon, apparent time, arid found that by an Almanac calculated
for New-York, in longitude 74 I' W. from Greenwich, the be-
ginning of the eclipse there, happened at 9h. 39' A. M. app
time. The latitude of the place of observation was 32 15' N.
required the place and its longitude ?
Ans. The place is near the western part of the Island of Ber-
mudas, and its longitude 64 43' W. from Greenwich. See the
following problem.
PROB. r.
The hour of the day being given in any filace, tojftnd all the filacea
on the globe where it is then noon, or any other given hour.
Rule. BRING the place to the meridian, and set the index to the
given hour in that place ; turn the globe until the index points
out any other given hour ; all the places that are then under the
brazen meridian, are those places required.*
Note. This method is attended with some confusion, if there be more
rows of figures than one on the hour circle ; to remedy which, the following*
methods are given. The same must be observed with respect to the pre-
ceding 1 and some of the following problems.
* Tliis rule is manifest from what is said in the preceding 1 problem, from
which, or from this prob. the following 1 observations are evident.
1. If a ship set out from any port and sail round the earth eastward untii
she arrives at the same port again, the people in that ship will gain one en-
tire day, in their reckoning, at their return. If they sail westward, they
will lose one day or reckon one day less.
2. Hence if two ships sail from the same port, the one eastward, and the
other westward, until they arrive again to the place from which they de-
parted, they will differ two days in their reckoning ; the one reckoning one
day less, the other one day more than those who remained in the port. If
they sail twice, they will differ four days from each other, and two from
those who remained in the port. If three times, six days, &c.
3. If the vessels sail the one northward and the other southward, no dif-
ference will appear in their reckoning, nor will they differ from those who
reside at the port ; the difference in time being Jin proportion to the change
made in their longitude east or west.
4. As the distance of meridians near the poles is very small, an inhabitant
situated within 5 or 6 miles, &c. of either, may make the same changes in
his actual account of time, as the ship mentioned in the first remark, or may
keep pace with the sun during his apparent diurnal revolution round the
earth. Whoever is curious to see more of these remarks, may consult s.
small mathematical miscellany, published by Samuel Fuller.
THE TERRESTRIAL GLOBE. 61
Rule 2. Bring the given place to the meridian, and set the in-
dex to 12 ; then, if the hour at the required places be earlier than
the hour at the given place, turn the globe eastward until the in-
dex has passed over as many hours as are equal to the given differ-
ence of time ; but if the hour at the required places be later than
the hour at the given place, turn the globe westward until the in-
dex has passed over the given difference of time ; and in each
case all the places required will be found under the brass meri-
dian.
WITHOUT THE HOUR CIRCLE.
Rule 3. Find the difference of longitude in time (prob. 4.)
reduce it to minutes, Sec. these minutes divided by 4 will give de-
grees of longitude ; if there be a remainder after dividing by 4,
reduce it to seconds, and add the seconds in the difference of lon-
gitude, if any : this sum again divided by 4, will give minutes or
miles of longitude. Now, if the hour at the required places be
earlier than the hour at the given place, the required places lie as
many degrees to the westward as are equal to the difference of
longitude ; but if the hour at the required places be later than the
hour at the given place, the required places lie as many degrees
to the eastward of the given place, as are equal to the difference of
longitude.
Note. Whenever we direct the globe to be turned, we mean on its axis,
either east or west.
Examples. 1. When it is 12 o'clock in the day at London,
\vhere is it 8 o'clock in the morning, at that time ?
Answer. If London be brought to the meridian, and the index
set to 12 o'clock (or 12 o'clock brought under the meridian) the
globe being then turned until 8 o'clock comes under the meridian,
or until the index points to 8, all the required places will then be
under the meridian ; as, Cape Canso, Martinico, St. Lucia, Trini-
dad, &c. the mouth of the river Oronoko, a part of Amazonia,
Paragay, &c. the Falkland Islands, Sec.
Or, bring London to the brazen meridian, and set the index to
12 as before ; turn the globe eastward until the 8 o'clock hour
line comes under the meridian, or until the index has passed over
4 hours. Then under the brass meridian all the places required
will be found as above. Or, (without the hour circle.) The dif-
ference of longitude between London and the required places, is
4 hours or 240 minutes, which divided by 4 gives 60<> the differ-
ence o f longitude. (=4X15.) Now as the hour at the required,
places is earlier than at London, they lie 60 westward of it.
Hence all the places situated in 60 west longitude from London,
are the places required and will be found by prob. 3, as above.
2. When it is 2 o'clock in the afternoon at London, where is it
.half past 5 in the afternoon ?
Ann, The places sought will be found as above (in method 1 st)
to be the Caspian sea, western part of Novazembl*^ the island of
eastern part of Madagascar, &c
62 PROBLEMS PERFORMED BY
Turning the globe westward (the time being later than at "the
given place) until the index has passed over 3 hours, London
being brought to the brazen meridian, the places will be found
as above by 2d method j or, by the 3d method, the difference of
longitude in time being 3J hours or 52 30'. The required pla-
ces, therefore, lie so many degrees to the east of London.
3 When it is 5 o'clock in the afternoon at Madrid, where is it
noon ?
4. When it is half past 5 in the morning at Pekin, where is it
noon f
5. When it is noon at Delhi, where is it 6 o'clock in the morn-
ing ?
6. When it is 5 o'clock in the morning at Philadelphia, where
is it 5 o'clock in the evening ?
7. When it is noon at New- York, where is it midnight ?
8 Being at sea in lat. 42 north, when it was 9 o'clock in the
morning by the time piece, which shews the hour at Washington
city (77 43' W. long.) and finding by a correct celestial obser-
vation, that it was 1 1 o'clock in the morning at the ship, in what
longitude was the vessel ?
9. When it is 10 o'clock in the morning in New-York, find all
those places that have the same hour. (Prob. 3.)
Note. On Gary's globes the hours are marked on the equator to every
minute, particularly on his large globes, and adapted to the meridian of
Greenwich observatory. Hence any place being 1 brought to the meridian,
the hours, Sec. on the equator will point out the time to the minute that the
'sun will come to the meridian of the place sooner or later than to the meri-
dian of Greenwich. Any other place may be taken instead of Greenwich,
and the hours will answer equally as well, by taking the difference of times
in each place.
PROB. 8.
The day of the month being given to find the surfs place, or his lo?i-
gitude in the eelifitic y and his declination.
Rule. Look for the given day in the circle of months on the
horizon, and opposite to it in the circle of signs, are the sign and
degree which the sun is in that day. (Def. 31.) Find the same
sign and degree in the ecliptic, on the surface of the globe, and
bring the degree of the ecliptic thus found to that part of the bra-
zen meridian which is numbered from the equator towards the
poles, then that degree of the meridian which is over the sun's
place, is the declination required. (Def. 91.)
OR BY THE ANALEMMA.*
Bring the analemma to the brass meridian, and the degree, cut
on it, exactly above the day of the month, is the sun's declination ;
* The Analemma is properly an orthographic projection of the sphere on
the plane of the meridian, and is a useful invention for shewing by in-
spection the time of the sun's rising and setting, the lengths of days and
nights, the points of the compass on which the sun rises and sets, the be-
THE TERRESTRIAL GLOBE. .63
turn the globe until the point of the ecliptic corresponding to the
day passes under this degree of the sun's declination, that point
will be the sun's place or longitude for the given day.
Xote 1. If the sup's declination be north, and increasing 1 , the sun's place
will be between Aries and Cancer. If the declination be decreasing-, his
place will be between Cancer and Libra. It' the declination be south, and
increasing 1 , it will be between Libra and Capricorn. If decreasing, between
Capricorn and Aries.
The sun's longitude and declination are given in the 2d page of every
month in the Nautical Almanac for every day in the month. The method of
accurately calculating them for any time, is given in prob.,3d and 9th of
Mayer's tables, published by Nevil Maskelyne.
The sun's place, 8cc. on the latest globes, viz. Bardin's and Gary's, is
adapted to the year 1800.
Example 1 . What is the sun's longitude and declination on the
22d of February ?
Ans. The sun's place is 41 in Pisces, declination 10 S.
2. What is the sun's place and declination on the 1 5th of April ?
Ans. 25|- e in V, declination 10 N.
3. Required the sun's place and declination for the first day of
each month.
Required the sun's place on the following days :
January 10,
February 1 3,
March 15,
April 2,
May 10,
June 15,
July 12,
August 5,
September 30,
October 20,
November 22,
December 31.
ginning and end of twilight, &c. but the Analemma on the globe is a nar-
row slip of paper, the length of which is equal to the breadth of the torrid
zone. It is pasted on some vacant place on the globe, between the two tro-
pics, and is divided into months and days of the month, corresponding to
the sun's declination for every day in the year. It is divided into two parts ;
the right hand part begins at the winter solstice or December 22d. and is
reckoned upwards towards the summer solstice or June 21st. where the
left hand part begins, which is reckoned downwards in a similar man-
ner, or towards the winter solstice. On Cary's globes the Analemma
somewhat resembles the figure g, being drawn in this shape for the
convenience of shewing the equation of time by means of a straight
line which passes through the middle of it. It begins at the tropic of
Cancer with the 24th of December, at which time there is no equation of
time, thence towards the opposite tropic January, February, &c. during
which months the clock is faster than the sun, to the 15th of April, at which
time the clock and sun are equal, and therefore no equation of time ; from
thence it continues April, May, &c. during which the clock is slower than
the sun to the 16th of June, nearly, at the tropic, at which time the clock
and dial are again equal, thence returning in the order of the months from
July to the 31st of August, the clock is too fast ; at the 31st of August the
equation is again nothing from thence to the 24th of December, reckoning
towards the southern tropic, the clock is too slow, &c. And if any day on
the Analemma be brought to the brass meridian, the degree cut on the' line
which crosses the middle of the Analemma, will show how much the clock
is fast or slow. The equation of time is placed on the horizon of Bardin's
globes corresponding to the respective days of the month. (See note to def
f>7> 61, and 62, and also, prob. 22.)
64 PROBLEMS PERFORMED BY
Note 2. The sun's place being 1 given, the day of the month corresponding
is found in the outer circle or calendar of months, &c. on the horizon. On
Gary's globes the days are likewise marked on the ecliptic.
Note 3. The declination being given, the corresponding months and days
are found by observing the two points of the ecliptic that come under the
declination, which will be the sun's place corresponding.
5. On what day of the month docs the sun enter each of the
signs ? (See clef. 3 i .)
Note 4. The earth's place, as seen from the sun, among the fixed stars,
is always in the sign and degree opposite the sun's place. Thus when the
sun is 10 in aries, the earth is 10 in libra, and so of any other. The sun
in reality having no motion (at least to produce this phenomenon) but the
earth by revolving on its axis every 24 hours from west to east, causes an
apparent diurnal motion of all the heavenly bodies from east to west : (Dr.
Kiel, lect. 26.) In like manner by revolving round the sun in a year, the
sun seems to pass over the same signs in the heavens which the earth has
passed, -ffnd in the same direction. But the sun being in the centre, it is
plain that in whatever sign the earth is, as seen from the sun, the sun must
be in the sign diametrically opposite as observed from the earth. The phy-
sical causes, &c. of these phenomena, is given by Newton in his principia,
and after him by the writers on physical astronomy and the laws of centre-
petal forces, as Dr. Gregory in his astronomy, McLaurin in his fluxions and
view of Newton's philosophy, Emerson in his fluxions, tracts, and in his as-
tronomy, Simpson, and others ; and lately in France, the celebrated La
Grange, De la Place, La Lande, De Lambre, &c. In Germany, Mayer in
his theory of the moon, &c. Mr. Burg, of Vienna, has lately in his tables
constructed principally on the observations of Nevil Maskelyne, much im-
proved this subject.
Note 5. The declination for every day is given in page 2 of the month in
the Nautical Almanac, or it may be found by having the sun's meridian alt.
given, see note to prob. 42, from which and the obliquity of the ecliptic, the
sun's place or longitude is found by Napier's rule, thus : As sine greatest
decl. : sine present decl. :: Rad. .- sine longitude from aries. This longitude
is likewise given in page 2 of the N. A. as also the sun's rt. ascension, the
equation of time, &c. (For the sun's greatest declination or obliquity of the
ecliptic, see note to prob. 49.) To find the sun's longitude at any time dif-
ferent from noon, say as 24h. is to the hour from noon reckoned by the me-
ridian of Greenwich, so is the daily variation of the sun's longitude to a
fourth number, which added to the longitude at noon, gives the longitude
for the given time ; if the time be that of a place differing in longitude from
Greenwich, it must be reduced to it. In like manner proportion may be
Hjade for any of the other articles in the Nautical Almanac,
PROB. 9.
To rectify the globe for the latitude, zenith 9 and sun's place ; and
to filace it agreeably to the corresponding situation of the earth) or
the four quarters of the 'world.
1. FOR the latitude. If the latitude be north, elevate the north
pole as many degrees above the horizon as are equal to the latitude*
* The reason of this method is evident from the latitude of the place being
always equal to the height of the elevated pole above the horizon. (D.
Gregory's astr. b. 2, prob. 7.) On any part of the earth we shall always- see
one half of the heavens, or 90 from the vertex to the horizon in every di-
THE TERRESTRIAL GLOBE. 65
but if the lat, be south, elevate the south pole, until the degrees
upon the meridian below the pole cuts the horizon, and then the
globe is rectified for the latitude.
2. For the zenith. Having elevated the pole to the latitude,
the same number of degrees reckoned from the equator towards
the elevated pole will give the zenith or vertex of the place. (To
this point the graduated edge of the quadrant of altitude is fixed.)
3. Bring the sun's place in the ecliptic (prob. 8.) to the meri-
dian, and set the hour index to 12 at noon, or bring the upper 12
to the graduated edge of the brazen meridian, and the globe is then
rectified for the sun's place.
4 Lastly, by means of the mariners* compass attached to the
globe, let the intersection of the planes of the meridian and hori-
zon be placed in the meridian line by the compass (allowing for
variation, if necessary) so that the elevated pole of the globe may
point towards the elevated pole of the world. Then the different
points of the compass on the globe will point to the corresponding
bearings on the earth, &c.
Note. The same method will answer for the Celestial globe.
Examfile 1 . On the 1 Oth of May it is required to rectify the
globe for the lat. 40, the sun's place, zenith, &c. to fix the quad-
rant of alt. and to place the meridian north and south, as on the
globe of the earth.
2. Rectify the globe for the lat. of Washington city, the zenith,
and sun's place, on the 1st of June, &c.
PROB. 10.
The month and day of the month being given, to find those places on
the globe to which the sun will be vertical^ or in the zenith^ on that
day.
Rule. FIND the sun's declination for the given day (prob. 8.)
and mark it on the brass meridian ; then the globe being turned
on its axis, all those places which pass under this mark will have
the sun vertical on tb*t day.*
reckon, if our view be not intercepted by hills, &c. Therefore to an ob-
server on the equator, the poles of the heavens would appear in his hori-
zon ; and if he advance from the equator towards either of the poles, he
-will see that pole towards which he advances rise as many degrees above
the horizon as he advances towards it from the equator : so that to an in-
habitant at the poles, the corresponding pole would appear in his vertex.
The 2d rule is evident ; for the height of the pole is equal to the dis-
tance of the equator from the vertex, both being equal to the complement
of the latitude, or what the latitude wants of 90. In applying the 4th
rule, the globe must be placed on a plane parallel to the horizon.
* The reason of this rule is evident, for the distance of the sun from
the equinoctial or his declination, is equal to the distance of those places,
from the same or their latitude, and therefore the sun, on that day, must
pass over the parallel of latitude passing through those places.
I
66 PROBLEMS PERFORMED BY
OR BY THE AtfALEMMA.
Bring the analemma to the brass meridian, then the degree
over the given day is the sun's declination, with which proceed as
above.
Example 1 . Find all the places on the earth to which the sun
will be vertical on the 1 5th of April.
Ans. It will be nearly vertical to Carthage, Porto Bello, Carora,
Barcelona, &c. in South-America ; to the island of Trinidad ; to
all that part of Africa under the parallel of 1 ; the northern ex-
tremity of the island of Ceylon ; the mouths of the Cambodia or
Japanese river (nearly) Parago, Negros, Sec. in the Philippine
islands, &c.
2 Find all those places where the sun is vertical on the 9th of
May.
Ans. St. Anthony, one of the Cape Verd islands, Antigua, St.
K-itts in the West-Indies, Acapulco, Anatajan in the Ladrone
islands, Manilla in the Philippines, the southern parts of Pegu,
Golconda, the southern parts of the great desert in Africa, &c.
Note. In solving this problem, it is more natural to turn the globe from
west to east, as in the last example, because those places to the eastward
have the sun first on their meridian, and thence in order towards the west-
3. Find all the places to which the sun will be vertical on the
following days, viz. 21st of March, 21st of June, 23d of Septem-
ber, and 22d of December.
4. Find all the places of the e*rth where the inhabitants have no
shadow when the sun is on their nveiidian on the 1st of May. (See
Upte todef. 19.)
PROB. 11.
A place being given in the torrid zone, tojind those two days of the
year on which the sun will be vertical there.
Rule. BRING the given place to i\\e brass meridian, and mark
the degree of latitude that is exactly ovtr it ; turn the globe on its
axis, and observe what two points of the ecliptic pass under that
latitude : These points will be the sun's place corresponding to
By finding where the sun was vertical on any day, the limits of ihe tor-
rid zone were discovered by the ancient geographers. For, knowing that
an object will project no shadow where the sun is vertical, they observed
the most northerly place where objects cast no shadow when the sun's de-
clination north was greatest ; the distance of which place from the equa-
tor, gave them the limits of the northern tropic, and consequently half the
breadth of the torrid zone. But in accurately determining the aforesaid
place, though their method was correct, they found themselves, notwith-
standing, considerably embarrassed, as on the same day no shadow was
cast for a space of no less than 300 stadia; the reason of which is, that
the apparent diameter of the sun being about 31^'of a degree at this time,
seemed to extend itself over as much of the surface of the earth, and to
be vertical to every place within that space. But this difficulty might be
easily overcome by taking the middle of the space in which objects were
found to project no shadow, as this would give the place where the sun's
centre was then vertical, and consequently the tropic required.
THE TERRESTRIAL GLOBE. 67
the two days required, which days are found on the horizon exactly
opposite to the sun's place. (See notes 2d and 3d. prob. 8.)
OR BY THE ANALEMMA.
Bring the analemma to the brass meridian, upon which, exactly
under the latitude of the given place (found by prob. 1.) will be
the two days required.
Example 1 On what two days of the year will the sun be verti-
cal at Batavia, in the island of Java ?
Arts. On the 4th of March, and on the Sth of October.
2. On what two days of the year will the sun be vertical at the
following places :
Kingston in Jamaica, St. Helena, Borneo,
St. Domingo, Gondar in Abyssinia, Manilla*
Fort Royal in Martinico, Goa, Otaheite,
Barbadoes, Columbo in Ceylon,, Owhyhee,
St. Antonio, Cape verd islands, Achen, Lima.
3. If the sun be vertical at a certain place on the 15th of April,
how many days will elapse before he is vertical a second time at
that place ?
frote. The sun's declination on those days is equal the latitude of the
respective places. (See note to prob. i.)
PROB. 12.
The day of the month and the hour at any place being given, to find
inhere the sun is "vertical at that hour.
Rule. FIND the sun's declination (prob. 8.) bring the given
place to the brass meridian, and set the index to the given hour 5
then turn the globe westward if the hour be given in the forenoon,
or eastward if the hour be given in the afternoon, until the index
points to 1 2 ; the place then exactly under the sun's declination is
that required.
Or : Having found the sun's decl. as before, bring the given
place to the brass meridian, and set the index of the hour circle to
12 (or bring 12 to the meridian.) Then turn the globe as above
directed, as many hours as the given time is from noon, and the
place under the sun's declination will have the sun that moment
in the zenith.
Or : Find the longitude of the given place (by prob. 1 .) and rec-
kon, on the equator (eastward if the time be given in the fore-
noon, or westward if the time be given in the afternoon) as many
degrees as are equal to the given time from noon, converted into
degrees (see the note to prob. 6.) this will give the longitude of
the place required. Then having the latitude of the place (being
equal to the sun's declination) and the longitude, the place is given
by prob. 2.
Example l. When it is 15 minutes after 8 in the morning at
New-York, on the 30th of April, where is the sun, at that time,
vertical ?
68 PROBLEMS PERFORMED BY
Ans. Cape Verd. Here the globe must be turned towards the
west, the time being given in the forenoon.
Or by the last method The given time before 12 o'clock is
S hours 45 minutes, which converted into degrees (by note to
prop. 6.) gives 56 15', the difference of longitude, or the number
of degrees the place is eastward from New-York, the hour being
given in the forenoon. This difference therefore, subtracted from
the longitude of New-York, which is 74 I' W. (see note 4,
prob 4.) gives 17 46', the longitude of the place required; and
under the sun's decimation for the present day is the latitude,
(Note to prob. 10.) Hence the place is given, by prob. 2, and
corresponds to Cape Verd nearly.*
2. When it is 4> o'clock in the afternoon at London, on the 18th
of August, where is the sun vertical ?
Ans. Here the given time is 4 hours past noon ; hence the globe
must be turned eastward until the index has passed over 4 hours,
then under the sun's declination you will find Barbadoes, the place
required.
3. When it is half past one o'clock at the Cape of Good Hope,
on the 5th of February, where is the sun vertical ?
Ans. At St Helena.
4. When it is 20 minutes past 5 o'clock in the afternoon at Phi-
ladelphia, on the 18th of May, where is the sun vertical ?
5. When it is 8 minutes past 8 in the morning at Petersburg?
on the 6th of June, where is the sun vertical ?
PROB. 13.
To find the time of the sun's rising" and setting, and the length of the
day and trig/it at any place.
Rule.\ ELEVATE the north or south pole to the latitude of the
place (according as it is north or south) by prob 9. Bring the
sun's place for the given day (found by prob. 10.) to the brass me-
* If the time in Example 1st. was given, 8 hours, 13 minutes, 32 sec-
onds, the longitude of Cape Verd would come out 17 33', as it ought.
But the hour circles, in general, are not divided into parts less than a
quarter of an hour, and therefore such exactness was unnecessary. On
Cary's large glebes, however, the index is divided into parts, each cor-
responding to 5 minutes. The index or hour circle on these globes hav-
ing but one row of figures, and being placed under the brass meridian,
renders them much more convenient and less liable to perplex beginners.
When more exactness is required, the hours, and the degrees of longi-
tude corresponding, &c. should be found on the equator, on which every
quarter of a degree corresponds to one minute of time. It is upon a 21
inch globe of Cary's, that most of these problems have been tried.
f The reason of elevating the pole thus, may be seen in the note to
prob. 9. The reason of bringing the sun's place to the meridian, and of
setting the index to twelve, is because the sun is always on the meridian,
or north and south (see note to def. 25.) at 12 o'clock. The reason of
bringing tlie sun's place to the eastern part of the horizon, to find his
rising, is because the eun rises towards the east, and the contrary reason
THE TERRESTRIAL GLOBE. 69
ridian, and set the hour circle (or the index) to twelve ; then turn
the globe eastward until the sun's place comes to the eastern part
of the horizon, and the index will shew the time of sun rising. In
like manner bring the sun's place to the western part of the hori-
zon, and the index will shew the time of sun setting ; then double
the time of sun rising, it will give the length of the night ; and
double the time of sun setting, it will give the length of the day ;
or, the time of sun rising taken from twelve, will give the time of
sun setting, and vice -versa. And the length of the night taken
from twenty-four, will give the length of the day, and the contra-
ry, 8cc. Also, half the length of the day, gives the time of sun
setting, and half the length of the night, the time of sun rising.
(See the note at the bottom.)
By the same rule the length of the longest day in all places not
in the frigid zones may be found. For in north latitudes, the
longest day is when the sun is in the beginning of cancer, that is
on the 2 1 st of June ; and in south latitudes, the longest day is when
the sun enters Capricorn, which is on the 2,2d December.* There-
fore to find the longest day, in the northern hemisphere, not ex-
ceeding 24 hours, bring cancer to the meridian, and proceed as in
the rule.
OR,
Rule 2. Find the sun's declination (prob. 8.) and elevate the
north or south pole, according as the latitude is north or south, as
many degrees above the horizon as are equal to this declination ;
bring the given place to the brass meridian, and set the index to
twelve ; turn the globe eastward until the given place comes to
the eastern part of the horizon, and the number of hours passed
over by the index will be the time of sun setting ; whence the
time of sun rising, and the length of the day and night is found as
above.
Note. The time of stm rising 1 and setting- may be found independent
of the globe by the following- rule : To the tangent of the latitude add the
tangent of the declination, the sum rejecting- radius will be the log. cd.
sine of an arch, which reduced to time, will be the time of sun rising, the
lat and decl. being of the same name, or the time of its setting if of dif-
ferent names. (See note to prob. 8, part 3d. where the demonstration
of tliis rule is given.)
holds for bringing it to the western part, because the sun sets there. It
is likewise evident that the index passes over as many hours as the time
from 12 o'clock to the sun's rising or setting. Now as the sun rises as
many hours before twelve as it sets after twelve, it is evident that the
time of sun rising subtracted from 12, must give the time of sun setting,
and vice versa. Again, as the hours are reckoned from 12 o'clock in the
night or midnight, it is plain that the hour indicating sun rising must also
indicate half the length of the night, and that therefore its double must be
the length of the night ; for the same reason the hour indicating sun set-
ting must also indicate half the day, as the hours are reckoned from 12
o'clock or midday, and therefore its double must be the length of the day.
* The longest day in both hemispheres, is placed on the 21st of June
and 22d of December. But as the sun varies from the equinoxes about
70 PROBLEMS PERFORMED BY
OR SY THE ANALEMMA.
Elevate the pole to the latitude of the given place, as above ;
bring the middle of the analemma* or the 1 6th of June, or 25th
of December, corresponding to it, to the brass meridian, and set
the index of the hour circle to twelve ; turn the globe westward
until the day of the month on the analemma comes to the western
part of the horizon, and the number of hours passed over by the
index will be the time of the sun's setting, Sec. ; which being giv-
en, the rest is easily found as above. On Gary's globes the given
day must be brought to the brass meridian, and not the middle of
the analemma except on the above days.
Note. If the day of the month on the analemma be brought to the
eastern or western part of the horizon respectively ; those hour circles
placed under the brass meridian, and with only one row of figures, will
always point out the hour of sun rising- or setting, the prob. being per-
formed as in the last rule.
Examples. 1 . What time does the sun rise and set at New-
York on the 10th of May, and what is the length of the day and
night ?
Ans. The sun rises at 4h. 56m. or 56 minutes after 4 o'clock,
and sets (12h. 4h. 56m. =) 7h. 4m. or 4 minutes after seven,
and therefore the length of the day (7h. 4m. X2) is 14 hours, 8
minutes, and hence 24h. 14h. 8m. (=4h. 56m. X2) =9h. 52m.
2. What time does the sun rise and set at Dublin on the llth
of March, and what is the length of the day and night ?
Jlns, Sun rises 6h. 20min. sets 5h. 40min. length of the day
is 1 Ih. 20min. and length of the night 12h. 40min.
3. What time does the sun rise and set at New- York on the
2 1st of June, and what is the length of the day and night ? (On
this day the sun enters Cancer, and makes the longest day in the
northern hemisphere, 8cc.)
Am. Sun rises 4h. 32m. sets 7h. 28m. longest day 14h. 56m.
shortest night 9h. 4m.
4. At what time does the sun rise and set at the following pla-
ces, and what is the length of the day and night, on the respective
days mentioned ?
Washington City, 1 st of May, Cape Horn, 1 st of June,
London, 1 7th of July, Petersburg, 20th of Oct.
Pekin, 10th of May, Constantinople, 1st of Jan.
50i" yearly, or 1 in 71.6 years (see note to def. 74, &c.) in the course of
some time the longest day will not happen on these days, the sun reced-
ing- backwards from cancer and Capricorn 50^", as above, every year Now
as the sun's mean motion in tiie ecliptic is 59' &".2, we have this propor-
tion, 1 : 71.6 years :: 5S' S ff .2 : 70.56 years, or 70 years 6 months 22 days,
the time in which the equinoctional points will recede one day from their
present place in the ecliptic.
* One of the meridians passes through the middle of the analemma on
Gary's globes, and this meridian passes through the 16th of June and
24th of December, at which times the clock and sun are equal. (Sec
the remark on the analemma prob. 8, part 2d.)
THE TERRESTRIAL GLOBE. 71
5. At what time does the sun rise and set at every place on the
surface of the globe on the 21st of March, and likewise on the
23d of September ?
6. Required the length of the longest day and shortest night at
the following places :
Washington, Paris, Botany Bay,
London, Vienna, Boston,
Dublin, Madrid, Charleston,
Edinburgh, Prague, Buenos Ayres,
Petersburg, Copenhagen, Cape of Good Hope.
7. At what hour does the sun rise and set, at any time of the
year to all the inhabitants of the equator, and what is the length
of the day and night ?
8. Required the length of the shortest day and longest night, at
the following places :
Philadelphia, Quito, Rome,
London, Mexico, Baltimore,
Quebec, St. Helena, Georgetown, near?
Lima, Lisbon, Washington City. )
9. How much longer is the 21st of June at Halifax than at
Mexico ?
10. What is the difference between the 22d of December at
Boston, and Cape Horn ?
1 1 . At what time does the sun rise and set at the South Cape,
in Spitzbergen, on the 31st of March," and 30th of April I
Note. On the 30th of April in the last example, the learner will easily
perceive that the sun does not set at all on that day, as his place, during
an entire revolution of the globe on its axis, remains the whole time
above the horizon.
PROB. 14.
The month and day of the month being given^ to Jind those places
where the sun does not set^ and likewise where he does not rise on
the given day ; or to Jind where the sun begins to shine constantly
without setting, and also where he begins to be totally absent.
Rule* FIND the sun's declination for the given day. (prob. 8.)
Count the same number of degrees towards the equator from the
* The reason of these rules is very clear. For on the 21st of March and
23d of September the sun is on the equinoctial, and therefore enlightens the
globe exactly from pole to pole : hence as the earth turns round its axis,
which terminates in the poles, every place on the surface of the globe will
equally go through the light and the dark, and thus make equal day and
night in every part of the earth. But as the sun declines from the equator
towards either pole, he will enlighten as many degrees round that pole as
are equal to his declination from the equator, so that no place within that
distance of the pole will then go through any part of the dark, and conse-
quently the sun will not set to any part of this space. Now as the sun's de-
clination is northward from the 20th of March to the 23d of September, he
nxust constantly shifle round the north pole during that time, and from thence
72 PROBLEMS PERFORMED BY
north and south poles, then all those places that pass under the
degree where the reckoning ends, are the places required. If the
declination be north, then to those places near the north pole and
under the declination, the sun will not set, and to those places at
the same distance from the south pole, the sun will not rise, and
the contrary if the declination be south.
Or : The globe may be elevated according to the sun's declina-
tion ; then, when turned on its axis, to those places which do not
descend below the horizon, in that frigid zone near the elevated
pole, the sun does not set on the given day, and to those which do
not ascend above the horizon in that frigid zone adjoining to the
depressed pole, the sun does not rise on the given day.
Note 1. Both these methods are the same in effect ; the latter, however,
seems to be more natural, the former more convenient. The learner will
also observe, that when the decl. of the sun becomes equal to the comple-
ment of the lat. (or what it wants of 90) and they are both of the same
name, the sun does not descend below the horizon, but at midnight passes
the meridian again, so as to touch the horizon exactly at the north or south
(according- to the lat.) and thus continues to circulate, gradually rising higher
above the horizon (in proportion as his decl. exceeds the comp. of the lat.)
until he arrives to his greatest declination, from which he continues to de-
scend in like manner, until he again reaches the horizon ; that is, when his
decl. becomes equal to the comp. of lat. as before. But when the decl. be-
comes less than this, he will descend below the horizon in proportion, &c.
This problem is performed by the analemma in the same man-
ner, the only difference being to find the sun's declination by it,
and then proceed as above.
Examples. 1 . Find all those places in the north frigid zone,
where the sun does not set on the ^Oth of May, and those places
in the south frigid zone, where he does not rise on the same day.
gradually to that space included between the pole and the arctic circle I'M
proportion as his declination increases, so that on the day in which he is in
Cancer, his declination being then 23 28', he will shine so many degrees
beyond the pole, or to the polar circle, and therefore to all that space Avithin.
the north polar circle the sun will not set during that day (21st June.) In
like manner, from the 23d of September to the 21st of March inclusively, the
sun constantly shines rovnd the south pole, &c. And hence the same phe-
nomenon will take place at this pole as at the north. So that when the sun
is on the equator, it evidently shining as far as both poles, will here make
the days equal to those in any other part of the hemisphere then enlighten-
ed, or equal to 12 hours, for the reason given above (or because the sun be-
ing over the equator, all the parallels from the equator to the poles are di-
vided equally, and must therefore be 12 hours enlightened, and 12 hours in
the dark, as the earth performs her revolution in 24 hours.) Now the sun
-advancing towards the south until his declination is supposed 10, it is evi-
dent that the space within 10 of the south polar circle must be enlightened
the whole time of the sun's revolution ; and equally as evident that the space
within 10 of the north pole must be in the dark : so that when it is day in
the one, it is night in the other, and the contrary. But as the sun takes half
a year from its crossing the equator until its return again, therefore at each
pole the day and night must each be half a year long. (We do not here
speak of natural days of 24 hours, but of artificial days, or the time which
the sun remains above the horizon ;) and from the poles to the polar circles.
the length of their day is proportional to the sun's declination.
THE TERRESTRIAL GLOBE. 73
Ans. All places within 20 of the north pole (or within the lat.
70) will have constant day ; and those (if any) within 20 of
the south pole, will have constant night.
2. Whether does the sun shine over the north or south pole on
the 30th of September, and where is there constant day and con-
stant night on that day ?
3. What inhabitants of the earth have their shadows directed to
every point of the compass during a revolution of the earth on its
axis, on the 10th of June. (See def. 21.)
4. How far does the sun shine over the south pole on the 2d of
February, and what places have perpetual darkness on that day ?
Note 2. By perpetual darkness is only meant the absence of the sun, and
not of that faint light called twilight, Aurora Borealis, 8cc. With regard to
perpetual darkness, &c. the learner will easily observe, that every part of
the world partakes of an equal share, and consequently of an equal share
of day-light. The wisdom of the Creator is here displayed in a wonderful
manner, by causing the twilight, Aurora Borealis, &c. to supply the ab-
sence of the sun during the long winter nights near the poles (as will
be seen hereafter) and thus enabling the inhabitants to carry on their
work, which they would otherwise be unable to perform. during this gloomy
season.
PROB. 15.
The month and day of the month being given at any filace (not in
the frigid zones) to find what other day of the year is of the same
length
Rule. FIND the sun's place in the ecliptic for the given day,
(by prob. 8.) bring it to the brass meridian, and observe the degree
over it ; turn the globe on its axis, until some other point of the
ecliptic comes under the same degree of declination, and the day
of the month corresponding, found on the horizon, will be the day
required.
OR BY THE ANALEMMA,
Look for the given day of the month, and opposite to it will be
the day required.
OR WITHOUT A GLOBE,
Find how many days the given day is before the longest day, the
same number of days will the required day be after it, and the con-
trary. The same may be found by counting the number of days
between the 2 1 st of March and the given day, and reckoning the
same number from the 22d of September backwards, and on the
same side of the equator ; the day on which the reckoning ends,
is that required, and the contrary if the given day be after the
longest.*
* The reason of this rule is evident, as any two days of the year which
are of the same length, will be equally distant from the longest or shortest
day, or from the days corresponding to the sun's entry into Aries and Li-
bra ; the sun's declination, to which the length of the day ia proportional^
being equal on both these days.
K
tf4 PROBLEMS PERFORMED BY
Examfile 1 . What day of the year is of the same length as
the 15th of April?
Ans. The 27th of August.
2. What day of the year is of the same length as the 20th of
August ?
3. If the sun rises at 4h. 20m. in the morning at Dublin, on
the 9th of May, on what other day of the year will it rise at the
same hour ?
4. If the sun set at seven o'clock in the evening at London, on
the 24th of August, on what other day of the year will the sun
set at the same hour ?
5. If the sun's meridian altitude be 90 at Barbadoes, on the
24th of April, on what other day of the year will the meridian al-
titude be the same ?
6. If the sun's meridian altitude be 51 35' at London, on the
25th of April, on what other day of the year will the meridian
altitude be the same ?
PROB. 16.
The length of the day at any filace being given to find the sun's de-
clination and day of the month,
Rule 1 . BRING the given place to the brass meridian, and set
the index to twelve, turn the globe on its axis until the index has
passed over as many hours as are equal to half the length of the
day, keep the globe from revolving on its axis, and elevate or de-
press one of the poles until the given place exactly coincides with
the horizon ; then the distance of the elevated pole from the hori-
son will be the sun's declination ; this declination being marked on
the brass meridian, the two points of the ecliptic, which pass un-
der it, correspond to the days required, and may be found on the
circle of months on the horizon.
Note. It is more convenient to turn the globe eastward, as the brazen
meridian is graduated on that side, and as the learner should generally
stand at that side in performing his problems.
OR,
Rule 2 Bring the meridian passing through Aries* to the
brass meridian, elevate the pole to the latitude of the given place,
and set the index to twelve ; turn the globe eastward until the in-
dex has passed over as many hours as are equal to half the length
of the day, and mark where the. meridian passing through Aries
is cut by the eastern part of the horizon ; bring this mark to the
brass meridian, and the degree over it is the sun's declination,
with which proceed as above.
* Any meridian will answer the purpose as well as this ; but as this on
Cary's and most globes is graduated like the brass meridian, the point cut.
by the horizon will be the sun's declination, and therefore there is no neces-
sity of bringing it to the brass meridian. The meridians passing through
libra, Cancer, and Capricorn, are also marked on Cary's globes, and may
therefore be used in the same manner;
THE TERRESTRIAL GLOBE, 75
THE SAME BY THE ANALEMMA.
Bring the middle, or the meridian* passing through the middle,
of the analemma to the brass meridian, elevate the pole to the lati-
tude, and set the index to twelve ; turn the globe eastward until
the index has passed over as many hours as are equal to half the
length of the day, then observe the point in the middle (or in the
brass meridian, passing through the middle) of the analemma, that
is cut by the horizon, the days opposite to it are those required,
and if the point be brought to the brass meridian, the degree over
it will be the sun's declination.
Example 1 . What two days of the year are each fourteen hours
long at New-York ?
Ana. The 6th of May and the 6th of August.
2. What two days of the year are each 1 6 hours long at London ?
3. What two days of the year are each 9 hours long at Boston?
4. On what two days of the year does the sun set at 7 o'clock at
Copenhagen ?
Note. Having the sun's rising or setting, the length of the day, &c. is
given by prob. 13.
5. On what two days of the year does the sun rise at 4 o'clock
at Petersburg ?
6. What is the sun's declination when the sun rises at 5 o'clock
in Washington city ?
7. What two nights of the year are each 10 hours long at Am-
sterdam ?
is. Required the sun's declination and day of the month, when
the length of the day is 14 hours, 44 min. at Georgetown, District
f Columbia, latitude 38 55' north ?
PROB. 17.
The length of the longest day at any place, not within the polar cir-
cles^ being given, to find the latitude of that place ; or which is the
same, to Jind in what latitude the longest day is of any given
length less than 24 hours.
Eule. BRING the beginning of cancer or Capricorn to the me-
ridian (according as the latitude is north or south) and set the in-
dex to twelve ; turn the globe westward on its axis, until the index
has passed over half the number of hours given ; then elevate or
depress the pole until the sun's place (viz. cancer or Capricorn)
* On Cary's globes, the analemma resembling the figure 3, tlie meridian,
passes through the point of intersection, as also through those days at the
lop and bottom on which the equation of time is nothing. But on other
globes, the analemma being a narrow slip of paper, is drawn parallel to the
meridian, and therefore either of its sides, or rather the line passing through
the middle, will answer. The days cut by the horizon (as Keith Bays in
his treatise on the globes) are not the cays required, but those days cor-
responding to the point cut on this line, and opposite to each other, as is
evident, unless the middle of the analemma on Bardin's globes be made
use of.
76
PROBLEMS PERFORMED BY
comes to the horizon, and that elevation of the pole will shew the
latitude. This method will answer for any other day, the sun's
place being used instead of cancer or Capricorn.
OR BY THE ANALEMMA.
Bring the analemma to the brass meridian, as before directed,
and set the index to twelve ; turn the globe westward until the in-
dex has passed over half the number of hours, the day of the month
being made to coincide with the horizon, by elevating or depressing
the pole, this elevation will then shew the latitude.
Example I . In what degree of north latitude, and at what places
is the length of the longest day 1 6 hours ?
dm. In latitude 49, and in all other places that have this lat.
the day will be of the same length.
2. In what degree of south lat- is the longest day 17 hours ?
3. In what lat. north does the sun set at 5 o'clock on the 10th of
April ?
4. There is a town in Norway, where the longest day is five
times the length of the shortest night, what is its name ?
5. In what latitude north is the 20th of May 1 6 hours long ?
6. In what lat. north is the night of the isthof August 10 hours
long?
A TABLE, shewing the length of the longest day in almost every
degree of latitude from the equator to the pole.
-#
N
1
1
'.
Q
}
Latitude.
I
3
Latitude.
*-
1 $
V
H. M.
H. M.
H. M.
D. H. M. ?
c
12
42
15 4
59
18 10
74
96 17 C
S 6
12 20
43
15 12
60
18 30
75
104 1 4 S
$ 12
12 42
44
15 18
61
18 54
76
110 7 27 S
12 58
45
15 26
62
19 20
77
116 14 22 >
S 20
13 12
46
15 34
63
19 50
78
122 17 6 $
S 24
13 26
'47
15 42 :
64
20 24
79
127 9 55 S
? 27
13 42
48
15 52
65
21 10
80
134 4 58 J
S 30
13 56
49
16
66
22 18
81
139 1 36 J
S 32
14 6
50
16 10
66%
24
82
145 6 43 S
J 34
14 16
51
16 20
D. H. M
83
152 2 6 S
J 35
14 22
52
16 30
67
24
84
156 3 3 J
\ 36
14 28
53
16 42
68
42 1 16
85
161 5 23 ?
S 37
14 34
54
16 54
69
54 16 25
86
166 11 23 S
J 38
14 38
55
17 8
70
64 13 46
87
171 21 47 J
S 39
14 44
56
17 22
71
74
88
176 5 29 J
S 40
14 52
57
17 36
72
82 6 36
89
181 21 58 S
S 41
14 58
58
17 52
73
89 4 58
90
187 6 39 S
THE TERRESTRIAL GLOBE. 77
The tength of the longest day from the equator to the polar circles is
found by the following 1 proportion :
As radium : to tangent 23 28' :: so is tangent latitude : to sine of the
ascensional difference, which converted into time, will give the time the
sun rises or sets before or after six o'clock, from which the length of the
longest day is given by prob. 13. For the reason of this rule see note to
prob. 49.
The length of the longest day from the polar circles to the poles, may be
found thus :
Find the complement of the latitude, which consider as the sun's decima-
tion, find the sun's places corresponding to this declination (north and south,
as directed, prob. 19.) from a Nautical Almanac, with which proceed as
directed in that prob. or without the Almanac the sun's longitude may be
thus found :
As sine of the greatest declination 23 28',
To sine of the present declination,
So is Radius,
To sine of the sun's longitude.
The day corresponding to this longitude may be found in the Nautical
Almanac, and the hours, minutes, &c. by allowing 59' 8" 3 for the sun's
daily motion, 2' 27" 8 for every hour, 2" 5 for a minute, &c. or rather by
taking the difference between the corresponding and preceding day, from
the Nautical Almanac for the sun's daily motion, and then allowing pro-
portional parts.
The longitude given by the preceding rule may be reckoned from Aries
the contrary way, and also from Libra both ways, to find the four places of
the sun required in prob. 19. By applying these principles, the above ta-
ble taken from Fuller's treatise on the globes, may be rendered more cor-
rect, it being rather old. Tliis author and some others have given this ta-
ble, without any principles of calculation.
PROB. 18.
The latitude and day of the month being gi-uen^ to find how much
the sun's declination must increase or decrease^ to make the day
an hour longer or shorter than the given day.*
Pule. ELEVATE the pole to the given latitude ; bring the sun's
place, for the given day, to the brass meridian, and set the hour
index to twelve ; turn the globe westward until the sun's place
conies to the horizon, and observe the hours passed over by the
index ; then if the days be increasing, turn the globe westward un-
til the index has passed over half an hour more ; the point of the
ecliptic then cut by the horizon, will correspond to the sun's place,
where the day is an hour longer, &c. and hence the decl. is found,
(prob. 8.) But if the days be decreasing, turn the globe eastward
until the index has passed over half an hour, the point of the eclip-
* Note. The prob. may become general for any time corresponding to
the length of the longest day in any place, if instead of half an hour, the
g-lobe be turned until the index passes over half the time that the required
day is to be longer or shorter than the given day, and then proceeding as
before. The latitude of the place, on the longest or shortest day, must ad-
mit of the given increase or decrease in the day required, otherwise the
rules will not hold. Those places within the polar circles arc not here con-
sidered.
78 PROBLEMS PERFORMED BY
tic then cut by the horizon, will shew the sun's place when the
day is an hour shorter than the given day.
OR,
Find the sun's decimation for the given day, and elevate the pole
to that declination ; bring the given place to the brass meridian,
and set the index to twelve, turn the globe eastward until the given
place comes to the horizon ; then if the days be increasing, con-
tinue the motion of the globe eastward, but if decreasing, west-
ward, until the place comes a second time to the horizon, the last
elevation of the pole will shew the sun's declination required.
OR BY THE ANALEMMA,
Proceed as above, only instead of the sun's place, bring the ana-
lemma to the brass meridian, and use the day of the month on the
analemma instead of the sun's place.
Example 1 . How much must the sun's declination vary, that the
day at New- York may be increased one hour from the 13th of
March, 1810.
Ans. On the 1 3th of March the sun's decl. is 3 south, and the
sun sets at 50 minutes past 5. Now when the sun sets at 20 min.
after 6, his declination will be found to be about 5 4-0' north, near-
ly, answering to the 4th of April. Hence the sun must cross the
equator, and make his declination 5 40' N. and in 22 days the
day has increased one hour.
2. How much must the sun's decl. vary that the day at London
may decrease one hour, in length, from the 26th of July \
Ans. The sun's decl. on the 26th of July (1807 or 1811, &c.) is
19 38' north, and the sun sets at 49' past seven (see note 1 to
prob. 14-.) When the sun sets at 19' after 7, his declination will
be found to be 14 43' north, answering to the 13th of Aug. Hence
the declination has decreased 5 55', and the days have decreased
1 hour in 1 8 days.
3. How much must the sun's declination vary from the first of
Oct. that the day at Petersburg may decrease one hour ?
4. How much must the sun's decl. vary from the 10th of April,
that the day at Skalholt may increase two hours. (See the note at
the bottom of page 77.)
PROB. 19.
A place being given in the north frigid zone, to find the length of the
longest day and longest night there, or (which is the same) to Jind
what number of days of 24 hours each the sun constantly shines
upon it, how long he z'o- absent ; likewise the Jirst and last day of his
appearance, and the number of days of 24 hours each that he will
there rise and set.
Rule. FIND the complement of the latitude, or what it wants of
90, and reckon an equal number of degrees from the equator ojt
tjie brass meridian north and south, and mark the points where
the reckoning- ends : then bring the first quadrant of the ecliptic,
THE TERRESTRIAL GLOBE. 79
or that from aries to cancer, to the brass meridian, and observe
what point of it passes under the above mark ; this point will give
the sun's place when the longest day commences, or the first day
on which the sun will constantly shine without setting. The globe
being then turned westward until some point in the second quarter
of the ecliptic coincides with or falls under the sam'S mark, this
point will give the sun's place when the longest day ends, and the
day corresponding to it will be the last day on which the sun will
constantly shine without setting ; the number of natural days be-
tween these two, will be the length of the longest day in the given
place. The motion of the globe being continued westward, mark
the next point of the ecliptic, in the 3d quadrant that comes under
the mark on the brazen meridian, south of the equator ;. this point
will give the sun's place corresponding to the last day of his ap-
pearance above the horizon or the beginning of the longest night ;
next find that point in the 4th quadrant of the ecliptic that cornea
under the mark south of the equator, and it will be the sun's place
when the longest night ends ; lastly, the number of days between
the end of the longest day and the beginning of the longest night*
together with the number of days from the end of the longest
night to the beginning of the longest day, will be those days on
which the sun will rise and set alternately every 24 hours.
Note. Though it be more natural to have the globes rectified for the
latitude, and that the points of the ecliptic cut the meridian at the horizon
in the north and south points, in the same order as above, yet the above
method is more convenient i# practice. The learner will easily perceive
that both methods are the same in effect, but the reason of the rule will
appear more evident from the latter method, as the rising 1 , setting-, &c. of
the sun, will be seen on the horizon of the globe in the same manner as in
the horizon on the earth, corresponding 1 to the place whose latitude is giv-
en. The application of this method is left to the learner. The problem is
not applied ^to the south frig-id zone, this zone being- uninhabited (at least
we know of none) however the rule is g-eneral, reading- south for north, &c.
and proceeding 1 as above.
The time when the long-est day or nig-ht begins being 1 known, their end
maybe found, as the beginning- and end of cither are equally distant from
the solstice that intervenes, that is, the beginning- of the longest day is the
same number of natural days from the succeeding solstice that the end of the
longest day is after it, &c. The number of days which the sun alternately
rises and sets, is also found by adding the length of the longest day and
longest night together, and taking their sum from 365 days.
The reason of reckoning the complement of the latitude from the equator
is evident, as it must always be equal to the sun's declination, when the
longest day commences and ends there. For when there is no declination,
then the longest day commences or ends at the pole. When there is 10
north declination, then the longest day commences or ends in the parallel
of 80 distant 10 froin the pole, because the whole of that parallel will then
be in the illnminated hemisphere, &c.
OR BY THE ANALEMMA.
If the place be in the north frigid zone, the two days on the
analemma, that pass under the complement of the latitude north
of the equator on the brass meridian, will be the beginning and
80 PROBLEMS PERFORMED BY
end of the longest day, and those two days that pass under the
complement of the latitude south of the equator, will be the be-
ginning and end of the longest night ; from which the rest is given
as above. The contrary will answer for the south pole
Example \ . What is the length of the longest day and longest
night at the North-cape, in the island of Maggeroe, latitude 71
10' north ; the first and last day of his appearance, and the num-
ber of days that he rises and sets there ?
Ans. The complement of the lat or what it wants of 90, is 18
50', this being marked on both sides of the equator on the brass
meridian, the four points of the ecliptic that pass under it will cor-
respond to the 15th of May, 28th of July, 14th of November, and
26th of January. Consequently the longest day begins on the 1 5th
of May, and ends on the 28th of July, and is therefore 74 natural
days long (that is the sun does not set during 74 revolutions of
the earth on its axis.) The longest night commences on the 14th
of November, and ends on the 26th of January, and is therefore 73
days long. The sun will rise and set alternately from the 26th of
January to the 1 5th of May, which is 109 days from the end of the
longest night to the beginning of the longest day ; and also from
28th of July to the 14th of November, which is also 109 days
from the end of the longest day to the beginning of the longest
night. The learner will observe, that on the 26th of January the
upper edge of the sun will just touch the horizon, and again de-
scend below it ; the next day it will advance a little higher, &c.
increasing the day by little and little, until the sun crosses the
equator, when the day and night will be exactly equal ; then after
crossing the equator, the day will become longer than the night,
and will continue increasing, in proportion to the sun's declination
until the 14th of May, at which time the day will be exactly 24
hours. The same observation will hold, -vice versa^ with regard to
those days on which the sun rises and sets from the 28th of July
to the 14th of November. The length, therefore, of the longest
day is 74 days, of the longest night 73*, and the number of days
that the sun rises and sets is 2 ' 8, making in all 365 days.
2. What is the length of the longest day and longest night in
the northeast land in Spitzbergen, under the parallel of 80 ; when
* Here there is a difference of one day between the longest day and long-
est night, owing- to the obliquity of the ecliptic and the eccentricity of the
earth's orbit. (See notes to def. 57 and 61.) Fuller and Keith in their re-
spective treatises on the globes, make this difference amount to 4 days, and
each reckons the lat. of the north cape 71 30 But accuracy in these mat-
ters cannot be obtained on globes ; the learner, however, should be cau-
tioned against errors, and not to trust too much to instruments where cal-
culation, See. is not applied. The above prob. has been tried on one of
Gary's globes of nearly two feet diameter. Some authors make the lat. of
this cape 71 38' ( Vyst's Geogr. pa. 47 of the introduction, where he makes
a difference of 7 days between the longest day and longest night.) If more
accuracy be required in this problem, tables' of the sun's motion, &c. may
be consulted. See Voice's Tables lately published.
THE TERRESTRIAL GLOBE. 81
o they begin and end, and how many days does the sun rise and
set there I
Ana. The longest day begins on the 15th of April, and ends on
the 27th of August. The longest night begins on the 1 8th of Oc-
tober, and ends on the 22d of February. Hence the rest is easily
found as above.
3. What is the length of the longest day and longest night at
Ice-cape in Novazembla, lat 761 N. and how many days does
the sun rise and set there ?
4. What is the length of the longest day and longest night at
the north and south poles ; when they do commence, and what is
the difference between the length 'of the summer and winter half
year at both poles ?
PROB. 20
Any number of day snot exceeding 186|- in north) or 17 8^ in south
latitude, being given, to Jind in what latitude the sun does not set
during that time.
Rule. COUNT half the number of days from the 21st of June,
or the 22d of December (according as the place is in north or
south lat.) eastward or westward on the horizon, and find the
sun's declination corresponding to the days where the reckoning
ends (by prob 8.) the same number of degrees reckoned from
either pole, on the brass meridian, will give the latitude required.
OR BY THE ANALEMMA.
Count half the number of days from the 21st of June or 22d of
December, &c. towards the equator, the sun's declination corres-
ponding to the day on which the reckoning ends, will be the com-
plement of the latitude.
Note. In the same manner can be found the latitude, in which the sun
does not rise for any time less than 178 natural days, in north, or 187 in
south latitude (the longest absence of the sun at the poles consisting of so
many days) by reckoning- half the number of natural days from the 22d of
December, in north, or 21st of June, in south latitude, and proceeding as
above. If the end of the longest night and beginning of the longest day be
given, and the number of days between them be found, reckon half that
number from aries on either side, and. the sun's declination, corresponding
to the place where the reckoning ends, will be the complement of the lati-
tude where the sun will rise and set before and after the vernal equinox only
so many days. In the same manner we may proceed reckoning from libra
for the autumnal equinox, the end of the longest day, and the beginning of
the longest night being given. The longest day at the north pole reckoning
from the 21st of March inclusive to the 23d of September, is 187 days ; but
on account of refraction the sun will remain longer visible above the horizon.
Example 1. IN what degree of north latitude, and in what
places, does the sun continue above the horizon during 134 natural
days?
Ans. Half the number of days being 67, which reckoned to-
wards the east from the 21st of June, will answer to the 15th of
April, or reckoned towards the west, to the 27th of August ; on
L
82 PROBLEMS PERFORMED BY
either of which days the sun's declination is 10 north; conse?
quently the latitude is 80<> north, and the places all those passing
under this parallel of latitude.
2 Where is the longest day 74 days or 1776 hours ?
3 In what degree of north lat. does the sun continue above the
horizon 90 days or 2 1 60 hours ?
4. In what degree of north lat is the longest night 1752 hours
long ? See the last note for this and the following prob.
5. In what degree of north lat. does the sun alternately rise and
set no more than 54 days, both before and after the vernal
equinox ?
PROB. 21.
To find hoto much any number of days in one month) is longer or
shorter than the same number in another month.
Rule. FIND the sun's place for the beginning and ending of the
given days in one month, bring these places to the brazen meridian,
and mark the corresponding degrees on the equator cut by the brass
meridian ; the two points, on the equator, will be the sun's right as-
cension, and the number of degrees between these points converted
into time (by the note to prob. 6.) will give the length of the days
in that month. In the same manner find the right ascension of the
sun for the beginning and ending of the given days in the other ,
inonth, and convert it into time as before ; if this time agrees
with the former, the given days in one month are equal to the
same number in the other ; but if not, their difference in time
(or in degrees converted into time) will show the difference.
Example 1 How much longer is the number of days from the
1st to the 20th of August inclusively, than the same number from
the 1st to the 20th of September?
Ans, The right ascensions corresponding to the beginning and
20th of August, are 1- 0| and 149| the difference of which is
19, and the right ascensions corresponding to the beginning and
20th of September, are 1 59| and 1 77| respectively, the difference
of which is 1 8. Now the difference between this and the former
is 1, which in time is 4 minutes, the excess of the given num-
ber of da> s in August above those in September.
Note. The right ascension is here reckoned from the beginning- of the 1st
day in each month, or from-the lust day in the preceding month.
2. Find how much longer or shorter the month of January is
than that of May ?
3. How much longer or shorter is the month of September than
that of November ?
THE TERRESTRIAL GLOBE. 83
PROB. 22.
To find the part of the equation, of time which de fiends on the ob-
liquity of the ecltfitic, and also the true equation*
Rule. FIND the sun's place in the ecliptic, and bring it to the
brass meridian ; count the number of degrees from aries both on
the equator, and the ecliptic to the brass meridian, the difference
converted into time is the equation depending on the obliquity of
the ecliptic. If the number of degrees on the ecliptic exceed those
* The difference between a well regulated clock and a good sun dial, will
always give the equation of time ; it is therefore necessary we should jwint
out the manner of regulating these instruments and the principles on wliich
this difference depends, the equation of time being necessary, not only in
civil affairs, but also in almost every part of practical astronomy, and abso-
lutely necessary in the important problem of determining the longitude, &c.
A pendulum clock is the best measure of time as yet discovered ; but front
the expansion or contraction of the materials by heat or cold, which varies
the length of the pendulum, from which, as well as from the imperfection of
the workmanship and other accidental causes, the time indicated by the best
clocks must be subject to irregularity. Hence it becomes necessary that WQ
should be able, at any time, to ascertain how much it is too fsw* or too slow,
and at what rate it gains or loses ; and for this purpose we must compare
it with some motion which is uniform, or whose variation, if it be not uni-
form, can be ascertained. Now as the earth revolves uniformly on its axis,
the apparent diurnal motion of the fixed stars must be uniform, and is there-
fore considered as most proper to ascertain the variation above mentioned,
If a clock be therefore adjusted to go 24 hours from the passage of any iixed
star over the meridian, until it returns to the same meridian again, it is saicl
to be adjusted to sidereal time, and its rate of going may, at any time, be de-
termined, by comparing it with the transit of any fixed star, and observing
whether the interval be exactly 24 hours ; if not, the difference will be what
it has gained or lost during that time ; or if the apparent right ascension of
a known star when it passes the meridian, be observed, and this right ascen-
sion be compared with die right ascension shewn by the clock, the differenca
will be the error of the clock. In this latter case the clock must begin its
motion from Oh. 0' 0" at the moment that the first point of aries is on the
meridian ; then, when any star comes to the meridian, the clock will shew
the apparent rig-lit ascension of the star, allowing 15 for Ih. because, when
subject to no error, it will then shew how far the point aries is from the
meridian. The error, if any, is found and allowed for as follows: let the
apparent right ascension of aldebarcw, for example, be 4h. 23' 50" at thq
same time that its transit over the meridian is observed by the clock to be
4h. 23' 52", then the error of the clock is 2 7 more than it ought to be. If
similar observations be made with other stars, and the mean error taken,
the error, at the mean time of all the observations, will be more accurately
found. These observations being repeated every day,' we shall get the rate
of the clocks going, or how fast it gains or loses. It may not be unworthy
of .cmark here, though not its proper place, that the error of the clock,
an . 'ie rate of its going being thus ascertained, if the time of the true tran-
sit of any star or planet be observed, and the error of the clock, for the time,
corrected, the right ascension of the star or planet will be given ; this being
the method by which tKc right ascension of the sun, moon and planets are
regularly found in observatories.
i.!' v, e adjust a clock with the sun, or to go 24hoxirs from the time the sun
leaves the meridian on any day until he returns to the same meridian ag'ain,
which interval is a true solar day, the clock will soon vary from tb* sim.
$4 PROBLEMS PERFORMED BY
on the equator, the sun is faster than the clock ; if equal, the clock
and sun agree ; if less, the clock is faster than the sun.
TO FIND THE TRUE EQUATION.
Look on the horizon, in Barden's globes, and corresponding to
the day of the month, you will find the equation required. On Ca-
even in the supposition that it is good and goes uniformly, and will not indi-
cate 12 when the sun comes on the meridian. This inequality depends on
the two following causes. The 1st arises from the obliquity of the ecliptic,
or its inclination to the equator. Let the sun and an imaginary star com-
mence their motion at dries, and move through equal spaces in equal times,,
the sun in the ecliptic, and the imaginary star in the equinoctial ; at the
end of every degree let both bodies be brought under the brass meridian on.
the globe, and it will be found that they will never come to the meridian
together, except at the time of the equinoxes and on the longest and
shortest days, or the solstices ; and that from the equinox to the next tropic
or solstice, the apparent time, or the time shewn by the sun, precedes the
true, or the time shewn by the imaginary star, or a clock regulated to mean
solar time, because then the degrees in the ecliptic exceed those on the
equator (at the equinox, the ratio is that of radius to cos. of the obliquity of
the ecliptic) and that from the solstices to the equinoxes, the true time
precedes the apparent (for a contrary reason) the proportion of the degrees
on the equate? to those on the ecliptic being as R. to cos. obi. at the sol*
stices. The 2d cause on which this inequality depends, is the unequal mo-
tion of the earth in its orbit, being slower in its aphelion or greatest dis-
tance from the sun, and quicker in its perihelion or least distance. This
part of the equation of time is found from the distance of the sun at any-
time from the apogee, reckoning round with the sun (or from the point
where it commences its motion, and not to this point) being its mean ano-
maly. For as the earth describes equal areas in equal times by lines drawn
from the sun to its orbit (see prob. 1. sec. 2. Newton's prin. B. I. Emers,
centerp. forces prob. 11. Greg. astr. prob. 11 or part 4th of this treatise)
the sun's motion is therefore slower in its apogee, and increases in velocity
to its perigree where it is swiftest, and from thence decreases until it comes
to its apogee again. If we now suppose the sun to revolve round the earth,
instead of the earth round the sun, the effect being here the same, and the
explanation easier on this supposition ; and that the sun departs from its
apogee or aphelion, whilst an imaginary star departs from thence at the
same time with the mean angular velocity of the sun (or to perform its mo-
tion in the equinoctional, supposed here to coincide with the ecliptic, as its
obliquity is not considered) so as to describe an equal arch every 24 hours,
it will then be evident that the imaginary star will gain on the real sun,
and every day advance more to the east, and therefore that the sun will
come to the meridian first ; and hence that the apparent time will precede
the true. But before the sun comes to its perihelion, or perigee, it is plain,
from the above general law, that it will move quicker than the imaginary
star, but will not be able to overtake it until they are both in conjunction.^
which will take place in the perigee after the sun has performed half its re-
volution, and hence from the apogee to the perigee, the apparent time ivill
precede the true. Now the sun and star departing tog'ether from the peri-
gee, the sun's velocity will be greater than the star's, so as to advance more
to the eastward, and therefore to come later to the meridian, until they arc;
both in conjunction again, or in the apogee, the point from whence their
motion commenced, and hence from perigee to apogee the true time -will
precede the apparent. There arc then but two points in which their mo-
tions will be equal between the apogee and perigee (see Emerson's centr
forces cor. 9 prop. 16 sect. 2d.) and but two points in which they will
THE TERRESTRIAL GLOBE. 85'
ry's globes, bring the day of the month on the analemma to the
brass meridian, under which on the scale drawn through the ana-
lemma parallel to the equator, you will find the number of minutes,
&c. required. The scale indicates whether the equation be fast or
slow.
come to the meridian together, viz. at the perigee and apogee ; and hence
while the sun is describing- the first 6 signs of the anomaly, the imaginary
star, which shews the mean time or the time by the clock, being more to
the east than the sun (which shews the apparent time, or time shewn by a
sun dial) comes to the meridian later, and shews the apparent time greater
than the mean. Hence in these 6 signs, to find the equation depending on
this cause, the difference between the mean and apparent time must be ta-
ken, which subtracted from the apparent time, will give the mean time, or
added to the mean time, will give the apparent. While the sun is in the
last 6 signs of its anamoly, the mean noon precedes the apparent, for the
reason given above, and hence the difference between the motion of the
imaginary star and the sun, or the difference between the sun's mean and!
true motion, converted into time, must be added to the apparent time tb
give the mean, and vice versa.
Now as both the above causes conspire to make the inequality before no-
ticed, or the equation of time, it is evident that when both are faster or slow-
er, their sum is the true equation of time ; but when one is faster and the
other slower, their difference is the true equation.
To compute this equation, let
APLS be the ecliptic, ALm the
equator, A the beginning of aries, A
P the sun's apogee, S any given **
place of the sun ; draw Sv perpen- s
dicular to the equator, and take Att=AP. Now when the sun departs from
P, let the imaginary star depart from n with the sun's mean motion in longi-
tude or in right ascension, or at the rate of 59* b" 2 in a day (365^d. : Id.
:: 360 : 5$ 8" 2) and when n passes the meridian, let the clock be adjusted
to 12 as described above ; take 7wnn=Ps, and when the star comes to tn,
the sun if it moved uniformly with its mean motion, would be at s, but let
S be the sun's place at that time, and let S, and consequently r, be on the
meridian, then as the imaginary star at that instant is at m, mv is the equa-
tion of time. Let a be the mean equinox, or the point where the equinox
would have been if it moved uniformly backwards with its mean velocity,
and draw az perpendicular to AL ; then z on the equator would have coin-
cided with a if the equinox had moved uniformly ; therefore the mean right,
ascension from z must be reckoned. Now mv^Av Am ; but Att?=Az-f-
zm and Az=AaXcos. aAz far the small triangle aAz being considered as
right lined, it will be rad. : cos. aAz (23 28') :: Aa : Az or using the nat.
sines, &c. 1 : .9172919 :: Aa : Az, hence Az=,A X .9172919 or cos. aAz,
but .9172, &c.rt=^.i nearly, whence Am=s*i aAz-J-m; and therefore 7nT'=
Av zm ^L Aa. Now Av is the sun's true right ascension, zm the mean,
right ascension, or me an longitude, and \^ Aa (Az) is the equation of the,
equinoxes in right ascension, hence the equation of time is equal to the differ-
ence of the sun's true right ascension and its mean longitude connected by the
equation of the equinoxes in right ascension. When Am is less than AT/-, meau,^
time precedes the apparent, but the apparent precedes the mean when Am
is greater. For as the earth revolves on its axis in the direction Av, or in the
order of right ascension, that body whose right ascension is least, comes to
the meridian first ; that is, when the sun's true right ascension is greater
than its mean longitude corrected as above, the equation of time must be
tfdded to the apparent, to get the mean time, and when it is less, it must be
86 PROBLEMS PERFORMED BY
Examjilc 1. What is that part of the equation of time which de-
pends on the obliquity of the ecliptic on the 1 7th of July, and what
is the true equation ?
Ana. The degrees on the ecliptic is less than those on the equa-
tor by two nearly, which in time is 8 minutes, and hence the son,
subtracted. But to convert mean time into apparent, we must subtract in '
the former case, and add in the latter.
The following tables were constructed on the above principles. The first
gives the equation resulting from the obliquity of the ecliptic alone, the 2d
the equation depending on the eccentricity of the earth's orbit or the sun's
mean anomaly, and the 3d the true equation, or the equation resulting front
both these causes.
TABLE I.
S Sun faster than the clock in S Correction ofTa.l. ^
$ s?
* Q
<Y>. 0.
=2=. 6.
v j -i
O *
HI- 7.
n. 2.
t 8
12 J; Sun's
32 s place : Cor.
Sun's >
place C
S o
/ 0"0
&2-J'6
3' 44" 7
30 S ^^
&*r %
T 1
19 8
8 33
8 34 5
29 c
C"000
J
S 2
39 7
8 42 *
8 23 6
28 S 3
001
27 I
S 3
59 5
8 52 2
8 12 1
27 S 6
003
24 S
S 4
1 19 2
909
800
26 > 9
004
21 S
S 5
I 38 8
98^
7 47 1
25 s 12
005
18 >
t Q
1 58 3
9 16 3
7 33 7
24 \ 15
007
15 J;
S 7
2 17 7
9 23 1
7 19 6
23 S 18
008
12 S
S 8
2 37
9 29 2
750
22 J 21
009
9 S
5 9
2 56 1
9 34 6
6 49 8
21 s 24
010
6 J
s 10
3 15
9 39 4
6 34 1
20 S 27
Oil
3 t
S 11
S12
> 13
3 33 7
3 52 1
4 10 3
9 43 5
9 46 8
9 49 5
6 18
612
5 43 9
19 S tf^
18 J
17S 3
(T012
013
TTJ^ "y^ tj
o/ r*
S 14
4 28 2
9 51 5
5 26 2
16 S 6
013
24 ^
$15
4 45 9
9 52 7
5 8 1
15 J 9
014
21 S
S16
5 3 "2
9 53 3
4 49 5
14 ^ 12
014
18 S
$17
5 20 2
9 53 1
4 30 5
13 S 15
014
15 >
V 18
5 36 8
9 52 1
4 11 1
12 S 18
014
12 J
S 19
5 53 1
9 50 5
3 51 4
11 ? 21
014
9 S
S 20
690
9 48 I
3 31 4
10 J 24
014
6 S
S 21
6 24 5
9 45
3 11 1
9S 27
014
3 S
5 22
6 39 6
9 41 2
2 50 6
8 S /
SLw c
S23
6 54 2
9 36 6
2 29 8
7
013
;j ?
S24
783
9 31 3
288
6} 3
012
27 V
> 25
7 22
9 25 3
I 47 6
5 S 6
o on
24 S
? 26
7 35 2
9 18 6
1 26 2
4 S 9
010
2 ;
<27
7 47 9
9 11 2
148
3 J 12
009
18 C
S 28
8 1
930
43 2
2 < 15
008
15 S
S 29
8 11 8
8 54 2
21 6
1 S 18
006
12 S
<30
8 22 6
8 44 7
000
J 21
005
9 ?
*
J OA
n nm
6?
S22
1TJ7. 5.
SI 4.
r& 3.
&b S 27 10 002
V
3 S
\*
3CH.
/VVV 10
"5 9.
^ S 000
S
_D /y ^
THE TERRESTRIAL GLOBE.
as depending on the obliquity of the ecliptic is 8', or rather 7', 49",
slower than the clock The true equation is 5 min. 41 seconds
slower : hence the equation depending on the sun's mean anomaly
or the sun's distance from the apogee is 7% 49", 5', 4l",=3',
8", sun slower, &c
2. On what days of the year is the true equation of time noth-
ing, and also the equation depending en the obliquity of the eclip-
tc
TABLE II.
S Sun faster than the clock if his anomaly be
X
L &C
0,.~
i..- :2 S .-
3s.
4s.
5s.
s
r
c
S
iy o f/ o
3' 46" 8\ 6' 35"7
7' 41" 7: 6' 44" 1
3' 55"2
30 S
> 1
079
i 53 66 39 7
7 41 8:
6 40 1
3 48 1
29 >
S 2
15 8
404
6 43 7
7 41 7
6 36
3 40 9
28 ^
S 3
23 7
471
6 47 5
7 41 5
6 31 7
3 33 6
27 S
S 4
31 6
1 13 8
6 51 2
7 41 2
6 27 3
3 26 3
26 S
5
39 4
4 20 4
6 54 8
7 40 8
6 22 8
3 18 9
25 >
6
47 3
4 26 9
6 58 2
7 40 2
6 18 2
3 11 5
24 5
7
55 1
4 33 3
715
7 39 4
6 13 5
340
23 S
8
130
4 39 6
7 4 7
7 38 5
686
2 56 4
22 S
9
1 10 8
4 45 9
778
7 37 5
636
2 48 8
21
10
1 18 6
4 52 1
7 10 8
7 36 S
5 58 5
2 41 1
20 s
11
1 26 A
4 58 2
7 13 6
7 35
5 53 3
2 33 4
19 S
S 12
1 34 1
,S 4 2
7 16 3
7 33 6
5 48
3 25 6
18 S
k ^
1 41 8
5 10 1
7 18 8
7 32
5 42 6
2 17 8
17 >
S ^
1 49 4
5 16
7 21 3
7 30 3
5 37 1
299
16 ^
S 15
1 57 1
5 21 7
7 23 5
7 28 4
5 31 4
220
15 S
S 16
247
5 27 4
7 25 7
7 26 4
5 25 7
1 54
14 S
? 17
2 12 4
5 32 9
7 27 7
7 24 2
5 19 8
1 46
13
S 1{ *
2 19 9
5 38 4
7 29 6
7 21 9
5 13 9
1 38
12
^ >9
2 27 4
5 43 7
7 31 4
7 19 5
578
1 30
11
20
2 34 9
5 49
7 33
7 17
517
1 21 9
10
21
2 42 3
5 54 1
7 34 5
7 14 3
4 55 4
I 13 8
9
23
2 49 7
5 59
7 35 9
7 11 5
4 49 1
1 5 6
8
S.3
2 57 1
6 4
7 37 1
785
4 42 6
57
7
> 24
3 4 3
6 8
7 38 1
754
4 36 1
49
6
2J
3 11 5
6 13
7 39 I
722
4 29 5
41
5
26
3 18 6
6 18
7 39 9
6 58 8
4 22 8
32
4
27
3 25 8
6 22
7 40 5
6 55 3
4 16
24
3
28
3 32 8
6 27
7 41
6 51 7
4 9 1
16
2
? 29
3 39 8
6 31
7 41 4
6 48
422
8
1
C 30
3 46 8
6 35
7 41 7
6 44 1
3 55 2
000
S
11* -h
10s +
W +
a.*
7s +
6 S +
Ue^.\
iS'zw slower them the clock if his anomaly be
4
88
PROBLEMS PERFORMED BY
3, What is the equation of time depending on the obliquity of
the ecliptic and the eccentricity of the earth's orbit, respectively,
on the 27th of October ?
Note Here as before, the equation depending on the ecliptic, and like-
Wise the true equation being- found, their difference will give the equation
depending- on the earth's eccentricity, &c.
4 What is the equation of time when the sun is in the begin-
ning of taurus ?
TABLE in.
*
s
Jan.
Feb.
March.
April.
May
June.
July.
Aug.
Add
Jidd
Add
Add
Sub.
Sub.
Add
Add
$ 1
J 32" 9
16' 5] "3
i>'o6"5
3' 55",
3' b3
2' 36"5
3' 21"!
df 56" 1
S 2
4 i 3
13 59 3
12 24 2
3 37 4
3 13 8
2 27 4
3 32 6
5 52 4
S 3
4 39 4
14 6 fc
12 11 3
3 19 3
3 20 6
2 18
3 43 8
5 48 1
s 1
4 57 1
14 13 u
ll 58
313
3 26 9
2 8 1
3 54 8
5 43 2
5 24 5
14 18 6
11 44 3
v> 43 5
3 32 6
t 57 8
455
5 37 7
$ 6
5 51 5
14 23 5
11 30 1
2 25 9
3 37 7
I 47 2
4 15 8
5 SI 6
$ ?
6 18
14 27 5
11 15 6
286
3 42 3
1 36 3
4 25 8
5 25
S 8
6 44 1
14 30 8
11 6
1 51 5
3 46 2
1 25
4 35 4
5 17 &
S o
797
14 33 3
10 45 3
1 34 6
3 49 6
1 13 5
4 44 7
5 10
510
7 34 8
14 34 9
10 29 7
1 18
3 52 4
I 1 7
4 53 6
516
s 11
7 59 3
14 35 8
10 13 7
117
3 54 6
49 7
520
4 52 6
S12
8 23 2
14 36
9 57 5
45 7
3 56 2
37 5
599
4 43
S13
8 46 6
14 35 3
9 40 9
29 9
3 57 2
25 2
5 17 4
4 32 9
514
994
14 33 9
9 24 1
14 5
3 57 7
12 7
5 24 4
4 22 2
$15
9 31 5
14 31 8
970
subO 6
3 57 7
addO(J
5 30 9
4 10 9
$H
9 52 9
14 28 9
8 49 7
15 3
3 57
12 8
5 36 9
3 59 1
J17
10 13 6
14 25 2
8 32 1
29 7
3 55 8
25 6
5 42 4
3 46 8
$18
10 33 6
14 20 8
8 14 4
43 7
3 54 1
38 5
5 47 3
3 33 9
$19
10 52 9
14 15 7
7 56 4
57 S
3 51 90 51 4
5 51 6
3 20 5
520
11 11 4
14 9 9
7 38 3
1 10 6
3 49 1 1 43
5 55 4
367
S2;
11 29 2
14 3 4
7 20
1 23 5
3 45 8 1 17 2
5 58 6
2 52 3
? 22
U 46 1
13 56 I
716
1 35 9
3 42 Oil 30
&13
2 37 5
$23
12 2 3
13 48 3
6 43 1
1 47 9
3 37 7 1 42 8
634
2 22 2
S24
12 17 7
13 39 9
6 24 6
1 59 4
3 32 811 55 6
649
265
25
12 32 2
13 30 8
659
2 10 5
3 27 4 2 82
658
1 50 4
J 26
12 45 9
13 21 1
5 47 2
2 21 1
3 21 6 ; 2 20 8
6 6 1
1 34
3 ~7
12 58 8
13 10 8
5 28 5
2 31 2
3 15 3|2 33 2
6 5 1>
1 17 1
S 28
13 11
12 59 9
598
2 40 8
3 8 5(2 45 4
651
59 9
J 29
13 22 3
12 48 5
4 51 2
2 49 9
3 1 2 2 57 5
637
42 3
2 30
13 32 8
4 32 6
2 58 4
2 53 4 3 94
617
24 4
$.31
13 42 4
4 14 1
2 45 21
5 59 2
062
In the 1st table the signs of the 1st and 3d quarters of the ecliptic are at
the top, and the degrees at the left hand. The signs of the 2d and 4th
quarters are at the bottom, and the degrees at the right hand. When the
sun is in the former signs, it is faster than the clock, but when in the latter,
.slower. Thus when the sun is in 15 of & or KI it is 9' 52" 7 faster than
the clock, or the apparent time is faster than the mean ; but when the sun
3s in 20 of 05 or V? it is 6' 34" 1 slower than the clock. The 2d table is
applied in like manner, according to the respective sign and degree of the
sun's anomaly. Thus when his anomaly is 2 signs 13, or when the sun is
THE TERRESTRIAL GLOBE.
89
PROB. 23.
To skew at one -view the length of day and night, in all places
ufion the earth, at any given time ; and to explain how the
-vicissitudes of day and night are really made, by the. motion of
the earth on its axis, in 24 hours, the nun standing' still.
THE sun being at an immense distance from our globe, the rays
of light emitted from it may therefore be considered as parallel,
and hence it will always illuminate one half of the globe, or that
TABLE HI. Continued.
^r^^^r^^^^r^r^^^r^^^^^ 2g 13 o distant from his apojree.
g
Q
Sept.
Oct.
J\"ov.
Dec S the equation of time depending
- <| on this cause is 7m. 18" 8, when
Sub ^ in 9s. 29 the equation is 6m. 39'<
Sub.
Sub.
Sub
1
G' 12" 3
1C/21"4
16' .4" 8
1(/3S"2S The 3d table is constructed
2
31 1
10 40 2
16 15 4
iO 15 ^ from the two first, and contains
3
4
50 1
1 9 3
10 58 6
11 16 7
16 15 3
16 14 3
512^ the equation of time for leap years,
9 26 7 Vj but particularly calculated for the
5
1 28 8
11 34 4
16 12 5
9 1 7 S year 1812. From this however
6
1 48 6
11 51 8
16 9 9
2 ^ the equation of time may be neurly
7
285
12 8 7
16 6 4
8 10 1 ^ found for common years as fol-
8
2 28 7
12 25 3
16 2 1
7 43 6 s, i ovvs : i s t. When the equation m-
9
2 49
12 41 4
15 57
16 7 S creases and the clock is faster than
10
395
12 57 1
15 51 1
6 49 3 Jj the sun, or the sign is add. taka
31
3 30 1
13 12 4
15 44 3
6 21 6 c the difference between the equa-
12
3 50 9
13 27 2
15 36 7
5 53 5 tion of the given and preceding
13
4 11 7
13 41 5
15 28 3
5 25 1 S days : then add of this differ-
14
4 32 7
13 55 3
15 19
4 56 3 ? ence for the 1st year after leap
15
4 53 7
14 8 5
15 9
4 27 3 <> year, for the 2d. and for the
16
5 14 8
14 21 2
14 58 1
3 58 1 S 3d. from the 1st of January until
17
18
5 35 9
5 57
14 33 4
14 45
14 46 3
14 33 8
& the equation begins to decrease,
2 59 2 ^ but at other times of the year, in,
19
6 18 1
14 55 9
14 20 4
3 29 5 ^ this case subtract } of the differ-
20
6 39 2
15 6 2
14 6 2
1 59 6 S ence for the 1st. ^ for the 2d. and
21
702
15 15 9
13 51 2
1 29 7 S | f or the 3d year after leap year.
22
7 21 1
15 24 9
13 35 4
59 8 <J 2d When the clock is faster or
23
7 41 9
15 33 3
13 18 7
29 8 c the equat. to be added, and the
24
25
826
8 23 1
15 40 9
15 47 8
13 1 3
12 43 1
add 2 S equation decreases ; take the differ-
30 2 S ence of the equations for the giv-
26
8 43 4
15 54
12 24 1
1 1 ^ en and preceding" days as before,
27
935
15 59 4
12 4 4
1 29 8 <, then add this difference for the
28
9 23 4
16 4
11 43 9
1 59 4 S 1st. $ for the 2d. and for the
29
9 43 &
;6 7 9
11 22 7
2 28 9 S 3d. year after leap year.
30
10 2 3
16 11
11 0'8
2 58 i ^ 3d. When the equation increas-
31
16 13 3
3 27 1 ^ es and the clock is slower, or the
equat. to be subtracted ; take $ of
the difference, See. found as above, and subtract it for the 1st year, ^ for
the 2ct. and f for the 3d. after leap year. 4th. and last case, when the
equation decreases, and the clock is slow (or the equat. to be subtracted)
add i- of the difference, 8cc. for the 1st year, for the 2cl. and | for the
3d. after leap year. Tiius to find the equat. of time on the 14th of January,
1811, being the 3d. after leap year : from table 3d. above the diff, bei. \veen
the equat. on the 13th and 14th days, is 22" 8, i of which is 5" 7, which
in this case is to be added to 9' 9" 4, the sum is </ \-J' 1 the cquui-ort
for the 14th of Jan. 1811, agreeing to the decimal with that given in the
M
90 PROBLEMS PERFORMED BY
hemisphere turned towards it, while the other will remain in dark-
ness. If the gol>e be therefore elevated according to the sun's de-
clination, it is evident that the sun will illuminate all that hemis-
phere which is above the horizon, that the wooden horizon itself
will be the circle terminating light and darkness : and that all
those places below it are wholly deprived of the solar light. The
globe being fixed in this position, those arches of the parallels of
latitude which are above the horizon, are the diurnal arches (def.
105) and shew the length of the day in all those latitudes, at that
time of the year, corresponding to the declination for which the
globe was rectified ; the remaining parts of those parallels, which
are below the horizon, are the nocturnal arches (def 106) which
shew the length of the night in those places at the same time (The
length of the diurnal arches may be found by reckoning how ma-
ny hours are contained between any two meridians or any two pla-
ces cutting the same parallel of latitude, in the eastern and western
parts of the horizon. Or if these two places be brought to- the
brazen meridian respectively, and marking the two points then cut
on the equator by the meridian, the number of degrees between
nautical almanac for 1811. Again to find the equat. for the 23d of June,
1811. Here the differ, is 12* 8, of which is 9" 6, hence V 42" 8 9"
6=1' 33" 2 the eq. required ; that given in the naut. aim. is 1' 33" 8.
These methods are therefore sufficiently exact for almost any purpose. The
other rules are applied in the same manner.
The rule given by Me. Kay in his complete Navigator in the explanation
of table 29th. holds only in a few particular cases, thougii given as general.
It is therefore only calculated to lead into innumerable errors.
The equation of time being applied to the apparent time (or time shewn
by a dial, &c.) according to its title in the table, will give mean time, or
the time shewn by a watch or clock ; but the contrary method is to be
used to turn mean time into apparent. Thus, on the 20th of March the
equation is 7' 38" 3 additive- to the apparent time, which shews that the
sun or dial is slower than the clock or watch, and therefore 12 o'clock by
the dral is I2h. 1' 38" 3, by the watch ; on the contrary, 12 o'clock by the
watch is llh. 52' 21" 7, by the sur* or dial ; and it is worthy of remark,
that this is the instant when the sun's meridian altitude ought to be ob-
served, in order to find the latitude.
The best tables of the equation of time will not hold for rnaay years.
For the sun's apogee has a progressive motion, the equinoctial points a
regressive motion, the obliquity of the ecliptic continually varies, and
even the sun*s longitude at noon, at the same place, is different for the
same days on different years, and as it is for apparent noon the equation
is computed, it must therefore be competed anew every year, when great
exactness is required.
The sun's apogee, according to Mayer, in the year 1716, was 8 e of can-
cer, and in 1771, 9 of cancer, which gives its motion equal to 1 in 55
years, or nearly 1' 5" 45, every year. Delambre (tab 1.) makes the sun's
apogee, in 1800, 3s. 9 29' 3'', and in 1820, 3s, 9 49' 4tf ', the difference
between which is 20' 43'', the mot. of the apogee in 20 years, or 1^2" 15,.
yearly. For more information on the equation of time, consult Delambre's
tables, pa. 14, 15, 16, 17, and pa. 30, 31, 32, &c. notes ; and the articles
there referred to in La Lande's astronomy. It may be necessary to add,
that the 1st table is calculated for the obliquity 23 27' 54", and that the
THE TERRESTRIAL GLOBE. 91
these two points, on the equator, converted into time (note to
prob. 6 ) will give the length of the day, and the number of de-
grees between the same points reckoned on the other part of the
equator, converted into time in like manner, will give the length
of the night. The globe being again fixed in the same position as
before, all those places that are in the western semicircle of the
horizon will have the sun rising (For the sun standing still in
the zenith or vextex, or over that degree of the brazen meridian
corresponding to its declination, appears easterly, and 90 distant
from the zenith of all those places that are in the western semicir-
cle of the horizon (definitions 42 and 50) and consequently is then
rising in those places.) If we now take any particular place on
the globe and bring it to the meridian, and then bring 1 2 marked
on the hour circle to the meridian (the index must be set to the
lower 12, if it be fixed on the outside of the brass meridian) the
globe being then turned on its axis, until the aforesaid place comes
to the western side of the horizon, the index will then shew the
time of sun rising in that place. Turn the globe again from,
west to east, and the index will shew the progress made in the day,
every hour, and in every place on the globe, by the real motion of
the earth on its axis. When they come under the brass meridian
they have their noon, and the sun has then its greatest altitude be-
correction annexed to this table is calculated for a second in the variation
of this obliquity ; and that the 2<1 table is calculated for the year 1800,
and may be adapted to any other year, by diminishing 1 1" 2 for every 100
years, according to Delambre Nevil Maskelyne, in the nautical aim. for
1813, assumes the mean obliquity of the ecliptic for the beginning of that
yew 23 27' 51" 3, and its mean secular diminution 42" 6.
All the elements, 8cc. from which these tables are calculated, such as
the sun's mean longitude, mean anomaly, obliquity of the ecliptic, true
right ascension, &c. are easily found from Mayers tables, or rather from
Delambre's, translated by Vince. These tables, rendering 1 these calcula-
tions, and that of the nautical almanac itself, a mere arithmetical or me-
chanical operation, which any schoolboy can become acquainted with in a
few weeks The learner must however notice, that, as the equation of
time is computed for apparent noon, or when the sun is on the meridian,
and as the time of apparent noon in mean solar time, for which we com-
pute, can only be known by knowing the equation of time, it follows, that
to compute The equation on any day, the equation must be assumed the
same as it \VSLS four years before on that day, from which it will differ but
very little. This will give the apparent time sufficiently exact for the
purpose of computing the equation. Where great exactness is required,
the operation may be repeated. Thus, if it be required to find the equation
for the 10th of March, 1816 ; the equation for the 10th of March, 1812,
being 10' 2" 7, to be added to apparent noon, to give the corresponding
mean time ; hence the computation must be made from the tables for
March 10th, at Oh. !(/ 29" 7. In the month of January and February, in
leap year, one day must be taken from the given time before the compu-
tation is made. When the equation 4 years before is not given, the equa-
tion may be computed accurately enough for noon mean time, particular-
ly if the operation be repeated.
That the learner may understand the reason of the above, he must ob-
serve, that as a meridian of the earth, when it leaves m.(in the foregoing
92 PROBLEMS PERFORMED BY
ing equal to the number of degrees between the place and the hori-
zon, reckoned the nearest way. The motion of the globe being
still continued easterly, the sun will seem to decline westward,
until, as the places successively come to the eastern part of the
horizon, the sun appears to set in the western part.*
Example \ To find the length of the day and night in all places
on the earth on the 16th of April, and to shew how they are caus-
ed, &c.
fig-.) returns to it in 24 hours, it may be considered, when it leaves that
point as approaching a point 360 distant from it, at which it arrives in
24 hours The relative velocity, therefore, with which a meridian accedes
to or recedes from m, is at the rate of 15 an hour, and consequently
when the meridian passes through T>, the arc vm reduced into time at the
rate of 15 to an hour, will give the equation at that instant. The equa-
tion* of time is therefore computed for the instant of apparent noon, or
when the sun is ;n the meridian.
If the place be situated east or -west from the meridian of Greenwich,
allowance must be made for the difference of longitude in time.
The above note, &c. though rather long, must not, however, be unwel-
come to students who desire accuracy on that subject, and wish to pene-
trate deeper than the surface of those branches of science which they
make their study, in order to become useful to the community and to
themselves.
* From the foregoing solution it will appear, that all those places upon
the earth which differ in latitude, have their days of different lengths, ex-
cept when the sun is in the equinoctial, being longer or shorter, in propor-
tion to that part of the parallel that is above the horizon; if the entire of
the parallel be above the horizon, it is evident that there is constant day,
but if no part appear above the horizon, that there is constant night. Those
places that are in the same latitude have their days of the same length, but
commencing sooner or later, according as the places differ in longitude. It
will likewise appear, that the arches of those parallels which are above the
horizon in north latitude, are equal to those below the horizon in south lati-
tude, and therefore when the inhabitants of north latitude have the longest
day, those in south latitude have the longest night, and vice versa, the arches
of the parallels in south latitude which are above the horizon being equal to
those in north latitude which are below. In this problem, as in all others
where the pole is elevated to the sun's declination, the sun is supposed to
be fixed, and the earth to revolve on its axis from west to east. If a small
brass ball fixed upon a strong wire, be contrived to screw on the brazen me-
ridian like the quadrant of altitude, this ball placed over the sun's declina-
tion at a considerable distance from the globe, will represent the real sun,
and assist the young student in more easily comprehending the problem.
Such contrivances, however, to boys of genius, who at one glance can com.
prehend such problems, will appear childish and unnecessary, but the teach-
er, from experience, will find that there are few who does not stand in need
of such. The learner will perceive that the 2d problem above is calculated
to shew the positions of the earth with regard to the sun, that are most re-
markable, such as the eqidnoxes and solstices, from which will be seen, at
one view, when the days and nights are equal all over the world, and the
comparative lengths of the longest and shortest. When the sun is in the
equinoxes, the poles are placed in the horizon, the sun having then no de-
clination, but when the sun is in either solstice, the pole is elevated 23 2h',
the north pole if it be the summer solstice, and the south pole if the ,wmtr
solstice.
THE TERRESTRIAL GLOBE. 93
Ans. On the ICthot April the sun's declination is 10 north,
the north pole bring thereiore elevated 10, the sun will then illu-
minate all those places above the horizon, Sec. being fixed over
the meridian at 10 of declination. The globe being then fixed
in this position, the arches of the respective parallels of" latitude or
the diurnal arches will be as follow : In the parallel of 20 N. there
are 12 J meridians on Bardin's, or 18-f on Gary's globes, which
answer to 188 on the equator, or to 12 hours 32 minutes. In the
parallel of 40 N. there are 197|, which are equal to 19| meri-
dians of Cary's, or 13 of Bardin's globes, or 13 hours and 10 mi-
nutes. In the parallel of 60 N. there are 211, which corres-
pond to 21 T ^ on Cary's, or 14^ on Bardin's globes, or 14 hours
4 minutes ; and so on. for any other latitude. It is plain, also, that
if the above degrees be taken from 360, the meridians on Cary's
globes taken from 36, or those on Bardin's, or the hours, from
24, the remainders will give the degrees, meridians, or hours re-
spectively, corresponding to the length of the night in each of the
above respective latitudes. In the same manner the length of the
day is found in any parallel of south latitude ; thus in the parallel
of 60 south, there are 145= 14 iner. on Cary's, or 9| on Bar-
din*s=9 h. 40 min. &c. If we now bring any particular place, as
Washington, to the brazen meridian, then all those places in the
eastern horizon will have the sun setting when it is nocn at Wash-
ington, all those in the western will have the sun then rising, and
all those places under the brass meridian will have noon ; and the
height of the sun in these respective places under the meridian,
will be equal to their distance from the nearest horizon ; thus, in
latitude 10 N. the sun is vertical ; in any latitude less than 10 N.
,the sun appears north ; in 20, 40, and 60 north, the meridian
altitude is 80, 60, and 40 respectively, the sun appearing due
south ; and in 60 south latitude, the meridian altitude is 20, Sec.
The index being now set to 12, and Washington brought to the
western part of the horizon, the hours past over by the index, or
the hours pointed out by it taken from 12, will shew the time of
sun rising there, namely, half past five nearly ; the globe being
then turned eastward, the index will shew the progress of day and
night in each place. Thus, when the index has passed over two
hours, to all those places that were at the western horizon with
Washington, it will be then two hours after sun rise, and to those
places that were at the eastern horizon, it will be two hours after
sun set, Sec. ; when Washington comes to the eastern horizon, the
sun will then set at half past six, nearly. The length of the night
will be pointed out by the index, if the motion of the earth be con-
tinued eastward, until Washington again appears in the western
part of the horizon, Sec.
Note. If the meridian be drawn through every 15 of the equator, twice
the number reckoned from the horizon to the brass meridian will give tbo
of the day, &c.
94 PROBLEMS PERFORMED BY
2. Required the length of the day and night, See. as above, in
all places on the globe, on the following days ; 10th of March,
21st do. 19th of May, 21st of June, 23d of July, 23d of Septem-
ber, and 2 1st of December ? (See the next prob )
PROB. 24.
To explain in general the alteration of seasons^ or length of the days
and nights^ in every part, of the earthy caused by the earth's an-
nual motion in the eclifitic.
Rule. RECTIFY the globe for every degree of the sun's decli-
nation from the equinoxes (or any other point of the ecliptic) un-
til the sun returns to the same point again, the different portions
of the parallels of latitude, which are above the horizon, corres-
ponding to each degree of elevation, will give the length of the
day in each respective latitude, as in the last problem.
Note. The last problem is only a particular case of this ; what is required
here in general being there required only for one day, and the method of
performing this, therefore, differs in nothing from that given in the last
problem but in the different elevations of the pole. We shall here there-
fore more particularly solve the problem for the most remarkable positions,
the equinoxes and solstices, as the method of performing the problem for any
other point in the ecliptic, corresponding to any day in the year, or to any
degree of the sun's declination, is the same as in the last problem, only ob-
serving, that when the declination is south, the south pole must be elevated.
For the Equinoxes. At this time the sun having no declination,
the two poles of the globe must be placed in the horizon ; then
the point aries on the equator being brought to the eastern, part of
Ihe horizon, the point libra will be in the western point, and the
sun will appear setting to the inhabitants of Greenwich, and to all
the places under the same meridian (if the first meridian pass
through Greenwich) from this position let the globe be gradually
turned on its axis towards the east, the sun will then appear to
move towards the west, and to be setting, as the different places
successively enter the dark hemisphere ; the motion of the globe
being continued until Greenwich comes to the western edge of the
horizon, the moment it emerges above the horizon, the sun will
then appear to be rising in the east. If the motion of the globe be
-continued eastward, the sun will appear to rise higher and higher,
and to move towards the west ; when Greenwich comes to the
brass meridian, the sun will appear at its greatest height, and af-
ter Greenwich has passed the meridian, the sun will continue its
apparent motion westward, and gradually diminish in altitude, un-
til Greenwich comes to the eastern part of the horizon, when the
sun will again be setting. During the revolution of the earth on
its axis, every place on its surface has been twelve hours in the
dark, and twelve hours in the enlightened hemisphere, and there-
fore the days and nights are equal all over the world. For all the
parallels ot latitude are divided into two equal parts by the horizon^
THE TERRESTRIAL GLOBE. 95
and in every degree of latitude there are 90 between the eastern
part of the horizon and the brass meridian, or nine meridians on
Gary's, which are equal to six on Bardin's globes, and correspond
to six hours, which is half the length of the diurnal arch ; and
hence the length of the day in every latitude, when the sun is in
the equinoxes, is twelve hours. The meridian altitude of each
place, found as in the foregoing problem, will be exactly equal to
the complement of their latitudes. Thus the meridian altitude oi:
the sun at Greenwich will be 38 31' 2 1 7 ', at Philadelphia 50 3' 6",
at Boston 47 36' 45", at Washington city 51 7', (in each of
the above places the sun will appear south, when on the meridian)
at Quito 89 46' 43", at the Cape of Good Hope (the town)
56 4' 45". In both these places the sun will appear north when
on the meridian. At the equator the sun's altitude is 90, and is
there consequently vertical. But at the poles the sun having no
altitude, will therefore appear in the horizon, and as its altitude
varies very little during the space of 24 hours, it will appear to
glide the whole day along the edge of the horizon, until it comes
to the same point again ; but as its declination increases, it will
rise gradually above the horizon, describing a kind of spiral in the
heavens, until it reaches its greatest altitude 23 28', from which
it returns in the same gradation until it appears again in the hori-
zon at the next equinox. At the contrary pole it will descend be-
low the horizon in the same manner, and at the equinoxes the sun
will rise and set at six o'clock to all the inhabitants of the earth
except at the poles,
As the sun now advances from aries towards the next tropic, of
summer solstice, if we gradually elevate the north pole, according
to the progressive alterations made in the sun's declination, by his
motion in the ecliptic, we shall find the diurnal arches of all those
parallels that are in the northern hemisphere, continually increase ;
and those in the southern hemisphere continually decrease, in the
same proportion as the days increase and decrease in those respec-
tive places. We shall likewise see the entire of those parallels
where constant clay begins round the north pole, gradually elevat-
ing above the horizon, whilst those round the south pole, where
constant night commences, are depressed in the same manner.
Let us for example observe the sun when his declination is 10
north, the same phenomena will take place as described in the
solution of example the 1 st. in the foregoing problem. Moreover,,
the globe remaining in this position, the meridian altitude of the
sun in all those places whose latitude is north, will be equal to the
complement of the latitude, or what it wants of 90, added to the
sun's declination, this being their distance from the nearest hori-
zon ; and the meridian altitude of the sun in all those places hav-
ing south latitude, will be the complement of the latitude made
less by the sun's declination. To those in 10 north latitude, the
sun will appear vertical, and to the southward of those whose lati-
tude is more than 10 north* But to those, in south latitude, the
96 PROBLEMS PERFORMED BY
sun will appear to the northward, and likewise to those within the
parallel of 10 north. Thus the meridian altitude of the sun at
Greenwich will now be 48 31' 21", at Philadelphia 60 3' 6",
at Washington 61 7', at Quito 79 46' 43", at the Cape Town
46 4' 4j". Hence it appears that as the sun's declination in-
creases northward, the meridian altitude ot the sun, to those in
north latitude, increases, and to those in south latitude lessens
in the same proportion. Thus when his decimation is 20 north,
the meridian altitude at Greenwich is 58 31' 21", but at Quito
is 69 46' 43". Now as the sun's greatest declination cannot ex-
ceed 23 28', his greatest altitude at Greenwich cannot exceed
61 59' 21", at Philadelphia 70 3i> 6", &c. On the contrary,
the least meridian altitude at Quito cannot be less than 60 1 8' - 3",
nor at the town of the Cape of Good Hope less than 32 36' 45",
Sec. The contrary rule must be observed when the declination is
south. From what has been here said, it appears how the sun's
meridian altitude may be found at any place, on any given day, by
having his altitude on some preceding day, and a correct table of
the sun's declination. But to proceed : The globe remaining still
in the same position, we shall find that the lower part of the 80th
parallel of longitude just touches the horizon, and that therefore
all the space between this and the pole is in the illuminated hem-
isphere, or has constant day, the beginning of constant day light
being then at this parallel. From this parallel to the equator, and
from thence to the 80th parallel of south latitude, the days grad-
ually shorten ; the upper part of this parallel just touching the
horizon, therefore total darkness commences there, and all the
places between this and the south pole have constant night. It
holds likewise universally, that whatever be the length of the day
in north latitude, the night will be equally long in the same lati-
tude south, vice -versa, and that at the equator the days and nights
are always equal. In the same manner we may reason with regard
to any other degree of the sun's decimation until he is advanced to
the tropic of cancer.
For the summer solstice. The summer solstice, in north lati-
tude, happens on the 2 1st of June. On this day the sun enters
cancer, at which time his declination is greatest, being 23 a 8'.
The globe being elevated to this declination, bring cancer to tne
brass meridian, over which, in the point where cancer intersects
it-, let the sun be supposed to be fixed at a considerable distance
from the globe, whilst the globe remains in this position, the r qui-
noctial point aries will appear in the western part of the horizon,
and the opposite point libra in the eastern . hence, the equator
being divided into two equal parts, the one half in the illuminated
and the other half in the darkened hemisphere, it will therefore
appear that the clay and night at the equator is of the same length,
that is, 12 hours long each. From the equator to the arctic cir-
cle, the diurnal arches will exceed the nocturnal, or the days will
be longer than the nights. All the parallels of latitude within this
TTHE TERRESTRIAL GLOBE. 97
north polar circle, will be above the horizon, and therefore all the
inhabitants within it will have no night* From the equator to the
antarctic or south polar circle, the nocturnal arches will exceed the
diurnal, or the night will be longer than the day. All the paral*
lels of lat within the south polar circle will now be below the ho-
rizon, and the inhabitants, if any, will have twilight or dark night.
If instead of cancer, aries or the first meridian be brought to the
brass meridian, the meridians passing through the respective pla-
ces at the horizon, will (on the equator on which the hours are
marked) point out half the length of the day If no meridian on
the globe passes through any given place, its meridian may be
found by bringing the place to the brass meridian. The globe re-
maining in this position, the sun will have its greatest or least me-
ridian altitudes according as the places are situated N or S of the
tropic of cancer in the enlightened hemisphere ; this altitude may
be found as before. From this position, if the north pole be
now continually depressed as the sun's declination lessens, until
both poles are again in the horizon, the days and nights decrease or
lengthen until the sun arrives at the equinoctial again, in the same
gradations as before, from aries to Capricorn, Sec.
From the autumnal equinox to the winter solstice (which to the
inhabitants of north latitude happens on the 22d of December, at;
which time the sun enters Capricorn) the same alteration of seasons,
of day and night, Sec. will take place, and in the same gradation, to
all the inhabitants of the southern hemisphere, as was observed be-
fore to have taken place, while the sun performed his apparent mo-
tion in the ecliptic, from aries to cancer. By now elevating the south
pole in the same manner for the sun's declination as before the-
north pole was elevated, the same phenomena will appear in suc-
cession, until the sun advances to Capricorn. Here as at the sum-
mer solstice, the days at the equator will be twelve hours long, but
the equinoctial point aries will now be in the eastern part of the
horizon, and libra in the western. From the equator to the south,
pole, the seasons, Sec. will be as before in the summer solstice, in
the northern hemisphere (exclusive of the variation made by the
earth's distance from the sun being now at her nearest) the greatest
mefidian altitudes of the sun will now be in south latitude, and the,
least in north ; the reverse of what they were when the sun was in
cancer. Thus at Greenwich the sun's greatest altitude will be 15
2' 21", instead of 61 59' 2l", Sec. Hence it appears, that the
difference between the sun's greatest and least meridian altitudes
at any place in the temperate zone, is equal to the breadth of the
torrid zone, viz. 46 56'. The difference between the sun's great-
est and least altitude at the poles, is equal to the sun's greatest de-
clination ; his altitude or depression at either pole being always
equal to the declination, Moreover the sun's altitude at the pole
never varies, as in other latitudes, for at the poles the sun while
visible is always on the meridian, Sec. &c.
N
98 PROBLEMS PERFORMED BY
PROB. 25.
To place the globe in the same situation with respect t& the poles of
the equinoctial^ as our earth is to any of its inhabitants, so as to
shew at one view the length of the days and nights in any particu-
lar place, at all times of the year*
Rule. RECTIFY the globe according to the latitude of the place ;
then those parts of the parallels of declination which are above the
horizon, are the diurnal arches, and those parts which are below
the horizon are the nocturnal arches. Hence the length of the
days and nights at any time of the year may be determined, as in
the preceding problems, by finding the number of hours contained
between the two extreme meridians, which cut any parallel of de-
clination, in the eastern and western parts of the horizon.
We shall exemplify this rule, by placing the globe in its more
remarkable positions, such as the right, oblique and parallel. (See
definitions 87, 88 and 89.)
For the right sphere. Here the given place must be on the
equator, and the globe rectified for of latitude, or both poles of
the globe must be placed in the horizon, then the north pole on
the globe will correspond to the north pole of the heavens, and all
the heavenly bodies will appear to revolve round the earth from
east to west, in circles parallel to the equinoctial, according to their
different declinations, t When the sun is in the equinoctial, he will
* In this problem and in all others, where the pole is elevated to the lati-
tude of a given place, the earth is supposed to be fixed, and the sun to move
round it from east to west. When the given place is brought to the brass
meridian, the wooden horizon is the true rational horizon of that place, but it
does not separate the enlightened part from the dark, as in the two preced-
ing problems ; however, there is nothing unnatural in elevating the pole to
the latitude of the place on the earth. For, as Keith remarks, in the note to
prob. 22d of his treatise on the globes, this is placing the globe in its true
situation respecting the heavens and the fixed stars. The pupil who wishes
to make himself master of the globes, must endeavour to comprehend why
he sometimes elevates the pole to the latitude of the place, and at other
tames to the sun's declination. A little perseverance and diligence will soon
remove every difficulty, and he should be well convinced, that, notwith-
standing the exertions of the most eminent masters, without close applica-
tion and attention on liis side, nothing but a superficial knowledge of any
subject can be obtained.
j- The ecliptic being drawn on die terrestrial globe, young students are
often led to imagine that the daily apparent motion of the sun round the
earth is performed in the same oblique manner. To correct this false prin-
ciple, we must suppose the ecliptic to be transferred to the heavens, where,
it properly points out the sun's apparent annual path among the fixed stars,
As the sun, in receding from or advancing towards the equinoctial every day,
alters, a little, his declination ; if we therefore suppose all the torrid zone to
be filled up with a spiral line or thread, having as many turns, or a screw
having as many threads as the sun is days ingoing from one tropic to another,
and these threads at the same distance from one another on the globe as the
sun alters his declination, in one day, in all those places over which it passes ;
this spiral line or screw will represent the apparent paths described by the
.sun round the earth everyday, in passing from one tropic to another. Thus,
THE TERRESTRIAL GLOBE. 99
be vertical to all the inhabitants situated upon the equator, and his
apparent diurnal path will be over that line ; when the sun has any
declination, as 10 N. for example, his apparent diurnal path will
be from east to west, nearly along that parallel. When he comes
to the tropic of cancer, his diurnal path in the heavens will be along
that line, and he will be vertical to all the inhabitants on the earth
in latitude 23 28' north. The inhabitants on the equator will al-
ways have 12 hours day and i2 hours night, notwithstanding the
variation of the sun's declination, from north to south, or from
south to north, because the parallel of latitude, which the sun ap-
parently describes for any day, -will always be cut into two equal
parts by the horizon ; all the stars will here be 12 hours above the
jiorizon from rising, and 1 2 hours below it from their setting ; and
in the course of a year an inhabitant on the equator may see all the
stars in the heavens. Those which are at or very near the pole,
will always nearly remain in the same point of the heavens, and
the circles which the stars describe in their apparent diurnal revo-
lutions, will be greater in proportion as they are distant from the
poles, or approach nearer the equator. The sun's greatest meri-
dian altitude at the equator will be 90, and the least 66 32', the
altitude of any celestial body being here always equal to the com-
plement of its declination, or its distance from the equinoctial.
Hence the stars that are situated in the equinoctial, will be always
vertical to the inhabitants of the equator, and the meridian altitude
of any others will be always equal to the complement of the paral-
lel of latitude where they are vertical, or to the complement of their
declination. Those at the pole, in a right sphere, will have no al-
titude, for the same reason, and will therefore appear in the horir
zon. (Here we do not consider the effects of refraction, parallax,
Sec.) During one half of the year an inhabitant of the equator will
see the sun due north at noon, and during the other half it will be
due south when on the meridian.
For the oblique sfihere.* If from the right position we gradual-
ly elevate the pole (the north for example) according to the dif-
if the thread be fastened at the point Capricorn, and wound round the globe
towards the right hand or the equator, by turning 1 the globe from east to
west, until we arrive at cancer, it will point out the paths described by the
sun daily, from the winter to the summer solstice. But if the thread be
wound towards the left hand from cancer, until we come to Capricorn ag-ain,
it will describe the sun's path from the summer to the winter solstice, or the
remaining 1 half year. But, as the inclinations of those threads to one another
are but small, especially near the tropics, we may suppose each diurnal
path to be one of the parallels of latitude drawn, or supposed to be drawn
upon the globe, as above.
* Every inhabitant of the earth, except those who live upon the equator
and at the poles, has an oblique sphere, and hence the globe must be recti-
fied for every latitude accordingly. But by elevating and depressing 1 the
poles, for every latitude according to the situation of the places which are
given, the student may imagine that the earth's axis moves northward or
southward just as the pole is elevated or depressed. This is however amis-
take, as the earth's axis has no motion, except a kind of librrUory motion.,
100 PROBLEMS PERFORMED BY
ferent latitudes from the equator towards the elevated poles, the
lengths of the diurnal arches will continually increase, until we
come to a parallel of latitude as far distant from the equator as the
place itself is from the pole. This parallel will just touch the ho-
rizon, and all the celestial bodies that are between the parallel of
declination cbrresponding to this in the heavens and the pole of
the equinoctial, never descend below the horizon. While we thus
gradually elevate the globe to the different latitudes, the diurnal
arches in the southern hemisphere continually diminish in the
same proportion, that those in the northern hemisphere increased,
until we corne to that parallel which is so far distant from the equi-
noctial southerly, as the place itself is from the north pole. 1 he
upper part of this parallel in the heavens, just touches the horizon,
and ail the stars that are between it and the south pole, never ap-
pear above the horizon. Here all the nocturnal arches of the
Southern parallels are exactly of the same length as the diurnal
arches of the corresponding parallels north ; and hence every
place on the surface of the earth equally enjoys the benefit oi the
sun, in respect of time, the days at one time of the year being ex-
actly equal to the nights at the opposite season .
Thus, the latitude of Washington city being 38 53' N. if
Washington be brought to the meridian, and the north pole ele-
vated 38 53' above the horizon, then the wooden horizon will be
which is called its nutation, and. which has no relation to that conceived above,
nor can it be represented by elevating 1 or depressing the pole. During the
earth's annual revolution round the sun, the axis always remains parallel to
itself, and the poles always point to the same star or point in the heavens, the
whole semidiameter of -he earth's orbit causing 1 no sensible change or devia-
tion in the earth's axis. Dr. Bradley having- found from a series of accurate
observations on the stary dvaconis, that its annual parallax or the angle under
which the semidiameter of the earth's orbit would appear as seen from y
draconis, did not amount to a single second. The precession of the equi-
noxes, the aberration of light, &c. causes also some small change in the
earth's axis, &c. ; but nothing compared to that motion conceived in elevating-
the poles, &c. In going* from the equator towards either pole, our honzoi*
varies .; thus when we are on the equator, both poles are in the horizon, the
northern point of the horizon representing the north pole, being opposite the
north pole in the heavens, &c. If we advance 10, for example, northward,
the north point of our horizon is 10 below the pole, &c. Now the wooden
horizon on the terrestrial globe is immoveable, otherwise it ought to be ele-
vated or depressed, and not the poles ; but whether the pole be elevated or
the horizon depressed, the appearance will be exactly the same. Though
the wooden horizon be the true horizon of the place for which the pole is
elevated, it does not however separate the enlightened hemisphere from the
dark. For instance, when the sun is in aries, and Washington at the meri-
dian, all the places on the globe above the horizon beyond those meridians
which pass through the east and west points thereof j reckoning towards the
north, are in darkness, although they are above the horizon, and all places
below the horizon between the same meridians and the southern point of
the horizon, have day light, notwithstanding they are below the horizon of
Washington. Thus the meridians passing through 14 E. and 166 W. lo>.
gltude from Greenwich, will be the boundaries of lig-ht and darkness.
HE TERRESTRIAL GLOBE. 101
the true horizon of Washington : and if the artificial globe be pla-
ced north and south, by a mariner's compass or a meridian line, it
\vili have exactly the same position, with respect to its axis, as the
real globe has in the heavens. Now if we imagine lines to be
drawn through every degree of the sun's place parallel to the equa-
tor, or rather through every 59' 8" 3, the sun's apparent daily
mean motion, these lines will give the sun's diurnal path on any
given day.* By comparing these diurnal paths with each other,
they will be found to increase in length from the equator north-
ward, and to decrease from the equator southward, therefore when
the sun is north of the equator, the days are increasing in length,
and when south of the equator, decreasing. When the sun is in
the tropic of cancer, the day is nearly 14 h. 40 min at the equa-
tor 12, and when the sun is at the tropic of Capricorn, the day has
decreased to 9 hours 20 min. nearly. The meridian altitude of the
sun, fpr any day, may be found by reckoning the number of degrees
from the parallel in which the sun is on that day, towards the hori-
zon, on the brass meridian. Thus when the sun is in that parallel
which is 1 north of the equator, his meridian altitude at Wash-
ington will be 61 7', and is equal to the complement of the lati-
tude added to the sun's declination. If the declination be south,
it must be subtracted from the complement of the latitude to find
the sun's meridian altitude. The lower part of that parallel of de-
clination which is 5 1 7 7 from the equinoctial northerly, just touches
the horizon ; and all the stars between this parallel and the north
pole, never set at Washington. In like manner the upper part of
the southern parallel of 5 i e 7', just touches the horizon, and all
the stars that lie between this parallel and the south pole are never
visible in this latitude. If we now rectify the globe for the lati-
tude 66 32 7 north, we shall find, that when the sun is in cancer,
he just touches the horizon on that day, without setting to the in-
habitants of the arctic circle, remaining 24 hours complete above
the horizon ; and when he is in Capricorn, his centre would just
appear in the horizon were it not elevated on account of refraction,
about an entire diameter above the horizon, and would not rise
again for the space of 24 hours. When the sun is in any other
point of the ecliptic, the days are longer or shorter according to
his distance from the tropics. All the stars that lie between the
tropic of cancer and the north pole, never set in this latitude ; and
those between the tropic of cancer and the south pole never rise.
If we elevate the globe still higher, the circle of perpetual appari-
tlon (see def. 107 will be nearer the equator on one side, as will
that of perpetual oculation (see def. 108) on the other. If for
example the globe be rectified for the lat. 80 N. the sun's decli-
nation being 10 N. he will then begin to revolve above the hori-
* On Adams' gU-bes such lines are drawn through every degree of the
meridian v kuin the torrid zone, parallel to the equator ; these will nearly
Represent the sun's diurnal path on any given day.
102 PROBLEMS PERFORMED BY
zon without setting, in an oblique direction, just touching it in the
north point, and at the south being elevated 20 above it ; during
his progress from this point in the ecliptic to the tropic of <ZD, and
his return again to the same degree of declination, he never sets.
In like manner when his declination is 10 S. he is just seen at
noon in the south point of the horizon, and during his progress
from this point to the tropic of Capricorn, and his return again to
10 S. decl- he remains below the horizon ; the time of his being
invisible or below the horizon, being as long as the time he appear-
ed visible at the opposite season of the year.
For the parallel sphere. The north pole being now elevated 90
above the horizon or to the zenith, then the equator or equinoctial
will coincide with the horizon, and the parallels of latitude will
consequently be parallel to it ; those in the northern hemisphere
being above, and those in the southern hemisphere below it. When
the sun enters aries. that is on the 2 1 st of March, he will be seen
by the inhabitants of the north pole (if there be any) to glide along
the edge of the horizon, and as his declination increases, he will
increase in altitude until he comes to the tropic of 55, forming a
kind of spiral as before described, from this tropic until his return
to the autumnal equinox on the 23d of September, his altitude will
again also gradually decrease in proportion as his declination de-
creases. From the vernal to the autumnal equinox, or during the
summer half year, the sun will therefore appear above the hori-
zon, and make constant day ; and consequently the stars and plan-
ets will be invisible during that time. The sun's altitude at any
time, or at any hour of the day, will be always equal to his decli-
nation, and his greatest altitude cannot exceed 23 28', at which
time he will have arrived to the tropic of cancer. When the sun
just enters the sign libra, he will again appear to glide along the
edge of the horizon, after which he will entirely disappear until
his arrival again at aries or the vernal equinox ; hence during six
months, from the autumnal to the vernal equinox, there will be
constant night at the north pole. But though the inhabitants at
this pole will lose sight of the sun at the autumnal equinox (or a
short time after, on account of refraction, &c. which is very great
near the poles) yet the twilight will continue for nearly two months,
for the sun will not be ) 8 below the horizon until he enters the
20th of scorpio (as may be seen by observing on the globe the sun's
place corresponding to 18 on the brass meridian below the hori-
zon, as the quadrant of altitude cannot be conveniently screwed
over the pole) so that dark night will only continue from the 12th
of November to the 29th of January, during which time the sun
will be more than 18 below die horizon ; and even then the light
of the moon and of the aurora borealis, increased by the reflection
from the snow, supplies, in a great measure, the absence of the
sun in this inclement region. The inhabitants at the north pole
can, at any time of the year, only see those stars that are situated in
the northern hemisphere, and the greater part of these will be ifc-
THE TERRESTRIAL GLOBE. 103
visible except from about the 12th of November until the 29th of
January, during which time the sun will be 13 or more below the
horizon. The planets when they are in any of the northern signs,
will also be visible, and together with the stars will appear to have
a diurnal revolution round the earth from east to west, as the sun
appeared to have when above the horizon.
The moon is likewise above the horizon during fourteen revolu-
tions of the earth on its axis or half a lunation, and at every full
moon which happens from the autumnal to the vernal equinox, the
moon is in some of the northern signs, and therefore visible at the
north pole ; for the moon being in that sign which is diametrically
opposite to the sun, at the time of full moon, and the sun while in
the southern signs being below the horizon, the moon must there-
fore be above the horizon at this time, while in any of the northern
signs. When the sun is at his greatest depression below the ho-
rizon, which happens when he is in Capricorn, the moon is then
full at cancer. The new moon being in Capricorn, herons? quarter
will be in aries t and the third in libra. Now the beginning of aries
being the rising point, cancer the highest, and libra the setting
point, the moon rises at her first quarter in aries, has her greatest
height when full in cancer, and sets in her last quarter in libra,
being visible for fourteen days, or during her passage from aries to
libra. Thus the north pole is supplied one half of the winter time
with constant moon light in the sun's absence, and the inhabitants
there are only deprived of her light from her 3d to her 1st quarter,
while she gives but little light, and can therefore be but of little or
no service to them.
Many other useful observations might be added here, were not
these three last problems already rather long, and therefore dis-
couraging to beginners. But their utility in giving a general idea
of the seasons, 8cc. in every part of the world, deserve their parti-
cular attention, as they will here learn, in the most easy and enter-
taining manner, how these happen from the regular motion of the
earth and its various positions. The following observations, de-
duced as collaries from the preceding problems, may be no less
worthy the readers perusal.
1. When the north pole was in the zenith, the equator just
touched the horizon, and as the pole was depressed, the equator
was raised the same number of degrees above the horizon, whence
it follows that the elevation of the equator above the horizon is al-
ways equal to the complement of the latitude or what it wants of
90o.
2. The sensible horizon of a place changes as often as we change
the place itself.
3. Every place on the earth, in respect to time, equally enjoys
the benefit of the sun's light, and is equally deprived of it : that
is, the whole time that the sun is above the horizon of any place, is
equal to all the time taken together, that he is above the horizon
of any otherj being about 6 months annually, ^he same may be
104 PROBLEMS PERFORMED BY
said of the lime that he is below the horizon, and the days at one
time is equal to the nights at the opposite season in any place. Here
the effect of refraction, twilight, aurora borealis, See. is not consider-
ed, nor the difference of time in which the sun is passing through
the northern and southern signs, being seven days longer in the
former than in the latter These latter causes being considered,
the inhabitants at or near the north pole, have in consequence more
light in the course of a year than any other inhabitants on the earth.
But what it gains in duration, it loses in the intensity of the sun's
light or heat, from the same causes and the obliquity of the sun's
rays, and the quantity and the density of the atmosphere through
which they have to pass. (See Simpson's Fluxions, vol. "2. prob.
32 and 33.) For the article aurora borealis, Sec. see Ree's Cyclo-
pedia, a new edition of which is now printed in New-York, or the
Encyclopedia. Simpson makes the proportion of the heat receiv-
ed at the equator, to that received at the pole during one year, as
17 to 7, nearly.
4. In all places of the earth, except under the poles, the days
and nights arc each 1 2 hours long at the equinoxes, that is on the
2 1st of March and 23d of September, at which time the sun has
no declination.
5. In all places situated on the equator, the days and nights are
always equal, viz. '2 hours each.
6. In all places between the equator and the poles, the days and
nights are never equal, but when the sun enters the equinoctial
points <Y> and =2=.
7. In all places lying under the same parallel of latitude, the days
and nights at any particular time of the year are always equal ;
that is, the days in one place are equal to the days in the other, at
the same time, &c.
8. The nearer any place is to the equator, the less is the differ-
ence between the days and nights, and the more remote the greater.
The increase of the longest days does not however bear any re<-
gular proportion to the increase ol the latitude. For it the longest
days increase equally, that is half an hour, an hour, Sec. the lati-
tudes increase unequally, as is evident from consulting a table of
climates. (See the table in the note to def. 90 )
9. The twilight is shortest at the equator, and increases from
there to the poies, where it continues the longest.
10. To all places situated within the torrid zone, the sun is ver-
tical twice a year, to those under each tropic once, but to those in
the temperate and frigid zones it is never vertical.
1 1. In all places between the equator and polar circles, the sun
rises and sets alterna ely every twenty-four hours
12. At all places between the polar circles and the poles, the
sun appears a certain number of natural days without setting, and
at the opposite season of the year disappears for nearly the same,
length of time ; and the nearer the place is to the pole, the longer
the sun continues without setting, and the contrary.
THE TERRESTRIAL GLOBE. 105
13. Between the end of the longest day and beginning of the
longest night, in the frigid zones, and between the end of the lon-
gest night and beginning of the longest day, the sun rises and set^
alternately every 24 hours, as at other places on the earth.
14 At all places situated exactly at the polar circles, the sun
when he is in the nearest tropic, appears 24 hours without setting,
but when in the opposite tropic, he does not rise for the same
length of time ; but at all other times of the year rises and sets as
in other places.
15. In all places situated in the northern hemisphere, the lon-
gest day and shortest night take place when the sun is in the nor-
thern tropic ; and the shortest day and longest night when the sun
is in the southern tropic. The contrary must be observed with
respect to those situated in the southern hemisphere.
1 6. All places situated under the same meridian as far as the
globe is enlightened, have noon or any other hour at the same time,
and those situated on the same parallel of latitude have the same
seasons and are in the same climate.
1 7. At the north pole none of the stars ever rise or set, but
move round it in circles parallel to the horizon, and have therefore
always the same altitude. (The small yearly variation of about
50", owing to the procession of the equinoxes, is not here taken
notice of.)
18. At all places on the earth, except the poles, all the points
of the compass may be distinguished in their horizon : but from,
the north pole every place is south, and from the south pole every
place north. Consequently there is no distinction of noon or meri-
dian at the poles, or rather the sun is constantly on the meridian
during six months in each. And although the winds in any other
place may blow from any point of the compass, at the poles they
can only blow from one ; that is, at the north pole from the south,
and at the south pole from the north.
19. When the sun's declination is greater than the latitude of
any place, the sun will come twice to the same azimuth or point
of the compass in the forenoon, and twice to a like azimuth in the
afternoon, at that place : that is, the sun will go back twice every
day while his declination continues to be greater than the latitude,
which can only happen between the tropics, or in the torrid zone.
Thus suppose the globe rectified for the lat. of Port Royal, in Ja-
maica, which is 1 8 north, and the sun in any point of the ecliptic
between 21 of taurus and 9 of leo, suppose the beginning of
gemini, and the quadrant be set to any degree between 12 and 2 1 9
from the east northward on the horizon, at 1 8 for example, the
globe being then turned westward on its axis, the sun will rise in
the horizon about 3| north of the quadrant, and thence ascending
will cross it towards the south at an elevation of about 1 1 , and
thence advancing until his azimuth be about 80 nearly, from the
north, from which azimuth circle it will return again towards the
north, until, at an elevation of about 82, and consequently before
106 PROBLEMS PERFORMED BY
it comes to the meridian, it will again cross the quadrant, and pass
over the meridian about 2 north of Port Royal. In like manner
if the quadrant be set about 18 north of the west, the sun will
pass over the edge of it twice as it descends from the meridian to-
wards the horizon, in the afternoon.
20. At all places situated on the equator, the shadow at noon, of
any object placed perpendicular to the horizon, falls towards the
north for one half of the year, and towards the south the other half.
The nearer any place is to the torrid zone, the shorter the meri-
dian shadows of objects will be. When the sun is perpendicular,
there is no shadow, and when his attitude is 45, the shadow of any
perpendicular object is equal to its height (Euclid, 6 prob. I B.)
as the sun inclines towards the horizon, the shadows lengthen, &c.
21. At the equator the sun always rises in the east, and sets in
the west points of the horizon ; but the more distant any place
(situated in the temperate or torrid zones) is from the equator, the
greater will be the rising and setting amplitude of the sun, or his
distance from the east or west points of the compass. At the
poles the sun always performs his revolutions round the horizon as
before remarked.*
* The utility of such general observations as the foregoing 1 , will be readily
perceived by those who retain a relish for the study of history, geography,
&c. and take a pleasure in contemplating- the wisdom of the Creator in all
the phenomena which nature exhibits. They might be rendered much
more extensive and interesting, but these few remarks will enable the in-
genious student to pursue them at his leisure. I shall however make here
one more remark, which will open an extensive field for speculations of this
nature, to those who have a taste or inclination for them. That in all the
primary planets similar phenomena take place, as they have almost all been
found, to revolve on their axis, and to have an atmosphere the same as the
earth ; but the axis of Mars and Jupiter are not inclined to the planes of
their orbits, and hence in these the seasons will be always the same, that is,
a constant spring. In all the others the seasons, &c. will vary as on the
earth. The terrestrial globe with a few additional circles, or even a move-
able ecliptic, might be contrived so as to exhibit the greater part of their
phenomena; but the instrument best calculated for this purpose is an orrery t
the use of which is so generally known, that it is unnecessary for me, in a
treatise calculated for the globes alone, to enter into any description of it.
(See its description in the Philadelphia edition of the Encyclopedia, article
Orrery or Astronomy, in Low's Encyclopedia, printed in New-York, or Ilee's
Cyclopedia ; also in Ferguson's Astronomy or Fuller's Treatise on the Globes.)
The famous Ilittenhouse, of Philadelphia, has considerably improved this
useful instrument. Those made by Messrs. Wm. & S. Jones', London, being
on a small scale, are recommended for cheapness and utility. Besides the
appearances of the superior planets, the stationary and retrograde appear-
ances of the inferior planets are neatly illustrated by them. A learner who
is but slightly acquainted with the elementary principles of mathematics,
will, however, have little or no use for suh instruments, calculated only to
help the conceptions of beginners, as at one glance he can conceive infinitely
more than such machines can represent, and calculate the phenomena which
they exhibit to a degree of exactness at which he can with no instrument
ever arrive. Such readers are referred to the 2d vol. of Dr. Gregory's as-
tronomy, where the elements of comparative astronomy are given by this
THE TERRESTRIAL GLOBE. 107
PROB. 26.
The day of the month being given, to find when the morning and
evening twilight* begins, its duration and end, at any place on
the globe.
Rule. RECTIFY the globe for the latitude, zenith, and sun's
place (prob 9.) and screw the quadrant of altitude upon the brass
meridian over the given degree of latitude, and set the hour index
able master. It is from principles alone, and not from any machinery, that
a learner can obtain a complete or general knowledge of any branch of
science.
* This phenomenon is caused by the reflection of the sun's rays which
fall on the higher parts of the atmosphere after sun setting- or before he
rises. If there were no atmosphere, the sun would shine immediately be-
fore his setting as bright as at noon, but the moment after his setting, we
should have as great darkness as at midnight. This is one of the innu-
merable instances in which the wisdom of the Creator appears in provid-
ing for the conveniences of man, in this element alone. The height at
which the atmosphere is supposrd capable of reflecting the sun's light, so
as to render it visible to us, is, u\ a medium, about 49 or 50 miles. Now,
if a straight line drawn from an object, situated at this height, to the sun,
just touches the surface of the earth, the sun at that instant will be 18*
below the horizon (see the demonstration in Keil's astronomy, lect 20, or
in Keith's treatise on the globes, note, pa. 107) which is the limit of the
sun's depression below the horizon to have any of its light reflected to
us. This particle of the sun's light in passing from that part of the at-
mosphere where it is first reflected, will be continually bent from the right
line in which it would otherwise proceed, were the atmosphere equally
dense, as is fully demonstrated by the writers on optics, and this proper-
ty, called the refraction of light, increases the twilight, as the property of
refraction is always to elevate the object from which the light is reflect-
ed ; and whether it be the light of the sun, moon, or stars, whether it be.
native or reflected, intense or weak, &c. the refraction is the same, pro-
vided the medium through which it passes remain the same. But as the
atmosphere continually varies, particularly towards the poles, the limits
given above will also vary in the same proportion. And the variation,
even during one day, and in the same place, is so sensible, that the even-
ing twilight is found to continue longer than the morning twilight, owing
to the expansion of the atmosphere during the day, and consequently to
its greater height. The more oblique the sun's rays, the greater the re-
fraction or reflection. When the rays fall perpendicular, then there is no
refraction, because the rays, if reflected at all, are reflected back in the
same direction. All these properties ave accounted for from mechanical
principles, and may be easily applied to the motion ef light in the atmos-
phere with the assistance of a thermometer and barometer, the law of its
expansion being given. Dr. Coles has demonstrated that if the altitudes
of the air be taken in arithmetical proportion, its rarity will be in geome-
trical proportion. But to enter into any investigation of these principles,
would far exceed our prescribed limits 'in this introduction (see Mayer's
tables, prob. 13, and Scholia.) we shall however give an example which
shews how much the refraction is affected by the density of the atmos-
phere. In the year 1682, the Dutch navigators who wintered in Nova-
xembla, in lat. about 75 north, saw the sun 17 days before he could have
been seen were there no atmosphere, or were it not endowed with this
refractive power. It is owing to this property in the atmosphere, of re-
flecting the sun's light, that the sky is always illuminated while the svu
108 PROBLEMS PERFORMED BY
to twelve, then turn the globe westward until the sun's place comes
to the western edge of the horizon, the hours passed over by the
index will give the time of sun setting or the beginning of the
evening twilight ; continue the motion of the globe westward un-
til the sun's place coincides with 1 8 on the quadrant of altitude
below the horizon, or the opposite point be 18 above the horizon
in the eastern part of it, the time passed over by the hour circle
after sun setting, will be the duration of evening twilight, and the
index will point out the time of its ending. In like manner, if the
sun's place be brought to the eastern horizon, the beginning, du-
ration and ending of morning twilight may be found, its beginning*
being when the sun is 1 8 below the horizon, and ending the same
as sun rising.
Note. When the sun's place does not extend 18 below the horizon, oir
the opposite point in the ecliptic, 18 above it, the twilight will continue
the whole night.
OR THUS,
Find the sun's decimation for the given day (prob. 8) and ele-
vate the north or south pole, according as the decimation is north
or south, to this declination ; screw the quadrant of altitude in the
shines, for without this property, or were there no atmosphere, the whole
lieavens, except that part in which the sun appeared, would be as dark as
at midnight, and the smallest stars, as in a clear night, would be visible;
rtor would our artificial lights be of any service to us during the absence
of the sun. M. De Saussure when on the top of Mount Blanc, which is
elevated 5101 yards above ihe level of the sea, and where the atmosphere
must therefore be more rare than on the surface of the earth, says, that
the moon shone with the brightest splendour in the midst of a sky as black
as ebony. (Append, vol. 74, Monthly Review.) The sun's atmosphere
likewise shines after the sun is set, and increases the light reflected by
our atmosphere. In the northern regions, the sun, when visible, rises and
sets with a large cone of yellowish light, the stars appear of a fiery red-
ness, owing to the density of the atmosphere, and the aurora borealis
spreads a thousand different lights and colours over the whole firmament.
Taking 18 at a medium for the limits, beyond which the sun being de-
pressed below the horizon, there is no twilight, the prob. may be solved
by spherical trigonometry, thus : The comp. of the lat the compl. of the
sun's declination, and the arch formed by the quadrant of alt. between the
sun's place, and the zenith (being always equal 90-f-l8 =a 108) form a
triangle, the three sides of which are given to find the angle included by
Ihe meridian passing through the zenith, and the meridian passing through
the sun's place depressed below the horizon as above, which converted in-
to time, will give the end of evening twilight, reckoning from 12 o'clock.
The time of sun setting (found by prob. 13 or the annexed note) taken
from the above, will give the duration of evening twilight, the rest is
found in the same manner. See the proportions for calculating this and
similar probs. investigated in lect. 20th of Keil's astronomy, or more cor-
rectly in article 2206 and 2241 of La Land's astronomy, 3d edit. See also
Gregory's astronomy, b. 2. probs. 39 and 41 and Scholium. The reader is
also referred to P. S. Laplace's astronomy, b. 1, ch. 14, vol. 1, where he
\vill find many important and interesting observations on this subject This
useful work, together with the same author's Celestial Mechanics^ are trans-
lated into English, by J. Pond, F. R. S. now Astronomer Royal, an<l pub
lisltfd in Loiujon, in 1809.
THE TERRESTRIAL GLOBE. 109
zenith, bring the given place to the brass meridian, and set the in-
'dexto 12 ; turn the globe eastward until the given place comes to
the horizon, and the hours passed over by the index will shew the
time of sun setting, or the beginning of evening twilight ; continue
the motion of the globe eastward until the given place coincides
with 1 8 on the quadrant of altitude below the horizon, or until
the opposite point of the ecliptic be 1 8 above the western part of
the horizon, the time passed over by the index, from sun setting,
will be the duration of evening twilight, &c. The morning twi-
light is nearly of the same length, and found in the same manner.
OR BY THE ANALEMMA.
Elevate the pole to the latitude as before, and screw the quad-
rant of altitude in the zenith ; bring the middle of the analemma
(corresponding to the 16th of June, &c. on Gary's globes) to the
brass meridian, and set the index to 12 ; turn the globe westward
until the given day of the month on the analemma comes to the
western part of the horizon, and the index will shew the beginning
of the evening twilight ; continue the motion of the globe west-
ward until the day of the month coincides with 1 8 on the quadrant,
as before, and the index will point out when twilight ends, the tim$
between the beginning and ending of which is the duration. To
find the morning twilight, bring the day of the month to the east-
ern part of the horizon, and proceed as before.
Example \. Required the beginning, end, and duration of
morning and evening twilight at Washington city, on the 21st of
March ?
Ans. The sun sets at 6 o'clock, and rises at 6. The evening
twilight ends at half past seven, and the morning at half past four,
'its duration being therefore one hour and a half.
2. What is the duration of twilight at London on the 19th of
April ; what time does dark night begin, and what time does day
break in the morning ?
Ans. The sun sets at 2 minutes past 7, and rises 58 min. aftev
four ; the duration of twilight is 2 hours 17 min. and hence even-
ing twilight ends at 19 min. past nine, and morning twilight be*
gins, or day breaks, at 41 min. past two.
3. Required the beginning, end, and duration of morning and
eyening twilight at Philadelphia, on the 1st of August ?
4. Required the beginning, end, and duration of morning and
evening twilight at Buenos Ayres on the 10th of March ?
5. Required the same as above, in the following places, on the 1st
of January, New-York, Lima, Cape of Good Hope, and Canton ?
6. Required the beginning and end of morning and evening twi-
light at the north pole ofl the. 1st of March, and likewise on the 2<3
of February ?
HO PROBLEMS PERFORMED BY
PROB. 27.
To find the beginning, end, and duration of constant day or twilight
at any place*
Rule. IF the complement of the latitude be greater than 1 8,
subtract 1 8 from it, and the remainder will be the sun's declina-
tion (north if the place be in the northern hemisphere, &c.) when
total darkness ceases. But if the complement of the latitude be
less than j 8, their difference will be the sun's declination, of a
contrary name with the latitude, when the twilight begins to con-
tinue ail night. Observe what two points on the ecliptic corres-
pond to this declination, the day of the month corresponding to
that point in which the sun's declination is increasing, will be that
on which constant twilight commences, and the day corresponding
to that point in which the sun's declination is decreasing, will be
the last day or end of constant twilight.
Note 1. When the sun has 18 south declination, constant twilight com-
mences, &c. at the north pole, as is plain from the above, the days corres-
ponding to which arc found as in the rule.
Note 2. If after subtracting- 18, the remainder be greater than 23 28',
the sun's greatest declination, there can be no constant twilight at that place,
as is evident. Hence between the latitude 48 32' and the equator, there
can be no constant twilight.
Examples. \ . When do the inhabitants of London begin to have
constant day or twilight, and how long does it continue ?
An*. The latitude of London being 51 31' N. hence 90
5lo 31' 18==20 29' the sun's declination, the two days corres-
ponding to which (found by note 3, prob. 8.) are the 23d of May
arid 20th of July So that on the 23d of May constant twilight be-
gins, and on the 20th of July it ends ; hence its duration is nearly
two months.
2. What is the duration of twilight at the north pole, and also
the duration of dark night there ?
Ans. The two points corresponding to the sun's declination 1 8
south (see note 1.) are the 12th of November and 29th of Janua-
ry, between which days the sun is \ 8 below the horizon ; hence
the duration of total darkness is 78 days ; the twilight continues
from the 23d of September (the time of the autumnal equinox
when the sun first disappears) to the 12th of November, the be-
ginning of total darkness, being 50 days ; and from the 29th of
January (the last day on which total darkness ceases) to the 2 1st
of March (the vernal equinox, when the sun again begins to ap-
pear and the longest day commences) being 5 1 days. Hence
there are 186 days constant day, 10! days twilight, and only 78
clays dark night at the north pole, and even during this short period,
the moon and aurora borealis shine with uncommon splendour, as
oefore remarked.
* See the Trigonometrical Solutions of this prob. in lect. 20th, KeiFs As-
tronomy. See also Vince's Astronomy, 8vo. articles 94, 96, 97 and 98.
THE TERRESTRIAL GLOBE. HI
3. Can there be constant day or twilight at Washington city at
any time of the year ?
Ans. The lat of Washington city being S8 53' N. hence 90
38 53'=51 7' the complement, and 51 7' 18=33 7', which
being greater than 23 28', there can never be constant clay or
twilight in this latitude.
4. "When do the inhabitants of Petersburg cease to have constant
day or twilight, and how long does their dark night continue ?
5 How long do the inhabitants of the north cape in Lapland,
enjoy the benefit of constant twilight, and how long does their dark
night continue ?
6. When does constant twilight begin and end, and what is its
duration in Ice Cape, the most northern part of Nova Zembla ?
PROB. 28.
The month, day, and hour of the day at any filace being given, to find
all those places on the earth, where the sun is then rising, setting,
where it is noon, that particular place where the sun is -vertical,
where it is daylight, twilight, darknight, midnight, where the twi-
light then begins and where it ends, the height of the sun in any
part of the illuminated hemisphere, also his depression in the oo~
scure hemisphere.
Rule, ELEVATE the north or south pole to the sun's declina-
tion for the given time, according as it is N. or S. (probs. 1 and 8)
bring the given place to the brass meridian, and set the index to
12 ; then if the given time be before noon, turn the globe west-
ward, if in the afternoon, eastward as many hours as the given
time precedes or is after noon : the globe being kept in this posi-
tion ; then all those places along the eastern edge have the sun
setting, those under the brass meridian above the horizon, have
noon, that particular place under the sun's declination on the brass
meridian, has the sun vertical, all those places within 1 8 below the
western edge of the horizon, have morning twilight, those within.
1 8 below the eastern edge of the horizon, have evening twilight.
In the former, twilight begins 18 below the horizon, and ends at
the horizon in clear day : in the latter twilight begins at the hori-
zon, and ends 1 8 below it in dark night. The sun's altitude in
any place in the enlightened hemisphere, is equal to the height of
that place above the horizon, reckoned on the brass meridian, on
which its meridian altitude is found, or on the quadrant of altitude
screwed in the zenith. Its depression below the horizon, is equal
to the depression of the place, or the altitude of its antipodes ; to
those between the eastern part of the horizon and the meridian,
the sun will appear westward, having crossed their meridian ; to
those between the meridian and the western part, the sun will ap-
pear towards the eastward, not having as yet passed their meridian.
This problem may be solved by taking the globe out of the
frame, and fastening a strong thread to the latitude of the place on
112 PROBLEMS PERFORMED BX
the brass meridian j then suspending it in the sun shine, an'd'
bringing the place to the brass meridian or to the zenith, and fixing
the meridian north and south, by a meridian line or compass, the
elevated pole of the globe will then point to the elevated pole in
the heavens, and the whole globe will correspond, in every respect?
to the position of the earth itself, in respect of the sun.
If then a pin, or bit of wire be erected perpendicularly (on a hol-
low basis of wood, cork, wax, &c.) in the middle of the enlight-
ened hemisphere, it will project no shadow, which shews that the
*un is vertical to that place. If this place be brought under the
brass meridian, the degree over it will be its latitude, and the sun's
declination at the given hour. Then all those places under the
brass meridian have noon or midnight, according as they are in the
illuminated or dark hemisphere.* Those on the westward (or
right hand when our face is turned towards the sun if situated N. of
the place where it is vertical) have their morning, for with them
the sun is ascending from the east, and those in the semicircle
bounding light and darkness westward, will have the sun rising,
those towards the eastward have evening, for with them the sun is
descending towards the west, and those situated between the en-
lightened and dark half of the globe eastward, have the sun setting.
All those countries within the sun shine have day, all those in the
shade have night or twilight. On the east side of the globe is
seen those places where night comes on, and on the west where
the darkness is dispelled by the approaching day. In those places
round the elevated pole, where it is sun shine while the globe is
turned round, there is constant day until the sun decreases in de-
clination. At the opposite pole within the polar circles, where it
is dark while the globe is revolved on its axis, there will be con-
stant night until the sun decreases in declination. The number of
degrees that the sun shine reaches beyond either pole, will be his
ileclination N. or S. The difference of longitude between any
place situated in the semicircle, separating the enlightened from
the darkened hemisphere, and any other place on the globe, re-
duced to time, will give the time before or after sun rising or sun
-setting, according to the situation of the place. If any place be
* When we here speak of dark hemisphere, we mean that in the shade OP
on which none of the sun's rays fall but by reflection ; it being- evident that
no part of the globe, thus suspended in the sun shine, can be in the dark,
which evidently shews the great utility of the reflecting 1 principle in the at-
mosphere : for otherwise that part of the artificial globe on which the direct
rays of the sun do not fall, would be as dark as if placed in a dungeon
where no ray of light could have access : but the twilight on a small globe,
for this reason, cairnot be ascertained by this method, and the limits of light
jind shade is very doubtful. The experiment succeeds best in a darkroom,
Avhere the sun's rays are admitted through a hole in the window shutters.
The best time for performing the prob. is when the sun is rising or setting,
or on the iix-iidian, as the shadow of the brazen meridian will not then pre-
vent the light of the sun from illuminating the hemisphere over which it is
perpendicular. Noon is however preferable.
THE TERRESTRIAL GLOBE. 113
brought to the brass meridian, the number of degrees between it
and the circle bounding; light and darkness, will be the sun's meri-
dian altitude that day ; if the index be set to 12, and the globe
then turned on its axis until any given hour comes under the meri-
dian, the nearest distance in degrees between the given place and
the shaded hemisphere, will give the sun's altitude for that hour.
If pins be erected perpendicularly on different parts of the globe,
their shadows will be projected the same way as the shadows of
the inhabitants of those respective places ; some pointing to the
north, some to the south, some to the west, others to the east, &CT
and some projecting no shadow at all.
If a narrow slip of paper be placed round the equator, and di-
vided into twice 12 hours, beginning at the meridian of your place,
and counting westward : the 6 o'clock mark being brought under
the brass meridian, the sun at noon will then shine on this meridi-
an, and the hours marked on the paper at the east or west part of
the equator, or at the circle bounding light and shade, will indicate
the time of the day in that place : thus at 12 o'clock the two 12's
will be in the circles bounding light and shade ; at one, the two
one's, &c. In the evening, or at night, it may be seen in the same
manner, if the moon shines, what nations are itluminated by her
light, where she is rising and setting, the various projections of
her shadow, where she is vertical, 8cc. and to which of the poles
she does not set that night. Her meridian alt. or alt. for any giv-
en hour, may be found in the same manner as the sun's found
above, with this difference, that when the given place is brought
to the brass meridian, the index must be set to the time of her
passing the meridian that night.
JExamfile 1. When it was 15 minutes after 5 o'clock in the
morning, at Washington city on the 16th of April, 1811, where
was the sun then rising, setting, Sec. &c.
Ans. On the 16th of April the sun's declination was 9 58' 4-2"
N. or 10 nearly, therefore elevate the north pole 10 above the
horizon* and as the given time is 5 hours 15 minutes, in the morn-
ing, 12 h. 5 h. I5m.=6h. 45m. what it wants of noon ; hence
the globe must be turned westward until the index has passed over
6 h. 45 m. The globe being fixed in this position, then,
The sun is rising from the north west to the south east parts of
Hudson's Bay, at Burlington, N. Jersey, the eastern part of the
Island of St. Domingo, near Porto Cabello in S. America, St.
Christopher on the Amazon R. near Sta. Cruz. Buenos Ayres,
east of the Falkland Islands, &c.
Setting, between the Lena and Indighirka rivers in Siberia, near
Chynian in Chinese Tartary, the mouth of the Blue River east
of Nanking, the eastern part of the Islands of Borneo and Java,
the western extremity of N. Holland, &c.
Abo/z, at the eastern part of Spitzbergen, North Cape, the north-
ern extremity of the Gulf of Bothnia, Revel on the Gulf of Fin-
hind, the eastern part of Prussia, Gailicia, Hungary, 8cc. the wes-
P
H4 PROBLEMS PERFORMED BY
tern part of the Archipelago, the middle of Negropont, east of
Athens, the middle of the desert of Lybia, Bornou, Mossel Bay
in Cuffraria, east of the Cape of Good Hope, Sec.
Vertical, in lat. 10 N. long. 24 E. about the middle of the
Ethiopic mountains.
Morning twilight, at Prince William's sound, near Beering's
bay, part of the Stony Mountains, Gulf of Mexico, Merida in New-
Spain, Gulf of Papagaya, along the Pacific ocean, Sec.
Evening twilight, at Gore's island, south of Beering's strait,
Beering's island, east of the Japonese islands, the western part of
New -Guinea, Van Diemen's and Nuyt's land in New Holland, the
Southern ocean, Sec.
Midnight, at the western extremity of N. America, Owhyhee
Island, Pacific ocean, west of the Society islands, Sec.
Day, in all Europe, Africa, and Asia, except a small portion of
the eastern part : in Labrador, Newfoundland, Nova-Scotia, New-
Brunswick, part of Canada, and the New-England states in N.
America, all that part of the West-Indies and South America,
comprehended between the eastern part of St. Domingo and
Buenos Ayres, Sec. towards the east, Sandwich land, Sec.
Night. The remaining part of North and South America be-
low the circle of twilight, Kamtschatka, the Carolinas, New-Gui-
nea, New-Britain, Sec. the eastern part of New-Holland, New-
Hebrides, New-Zealand, Sandwich islands, the greater part of the
Pacific ocean, Sec.
The sun's alt. at Petersburg is nearly 40, at Cairo 69, at Cal-
cutta 281, at London 441, Sec.
Those inhabitants situated at the northern extremity of the Island
of Ceylon, at Cochin, Sec. will see the sun due west : those in the
same parallel of lat. west of the brass meridian to the horizon, will
see it due east ; from Tartary, Persia, Sec. it will appear towards
the S. W. from Madagascar towards the north west ; from the
western part of Candia island clue south ; from Mossel Bay in
Caffraria due north ; from Lisbon, Fez, Sec. towards the S. E.
from St. Helena, Sec. towards the N. E. Sec.
2. When it is four o'clock in the afternoon at Washington on
the 21st of January, where is the sun rising, setting, &c. Sec.
Ans. The sun's declination being nearly 20 south, the south
pole must be therefore ele/ated 20 above the horizon: and as
the given time is 4 o'clock in the afternoon, the given place being
brought to the brass meridian, and the index set to 1 2, the globe
must be turned eastward 4 hours. Then the sun will be rising at
Berring's straits, Berring's island, Van Diemen's land, &c.
setting in Hudson's Bay, James' island, the western extremity of
Nova-Scotia, the eastern part of South America, Sec. Noon at the
eastern extremity of King G. 3d's Archipellago, east of Marquesas*
island, Sec. vertical lat. 20 south, long 136 W. near Whitsun-
day, island of Wallis, Sec. The other places are easily found by
following the directions in the rule.
THE TERRESTRIAL GLOBE.
3. When it is 6 o'clock in the morning at London, on the long-
est day, where is the sun then rising, setting, &c.
4. When it is 12 o'clock at Philadelphia, on the 10th of Decem-
"her, where is the sun then rising, setting, &c.
5. When it is 10 o'clock in the afternoon at Cape Horn, on the
21st of March, where is the sun then rising, setting, 8cc
6. When it is midnight at Washington on the 4th of July, where
i the sun rising, setting, &c.
PROB. 29.
To Jind in what climate any given place on the globe is situated.*
Rule. IF the place be not in the frigid zone, find the length of
the longest day in that place (prob. 1 3, or the rule annexed) from'
which subtract twelve hours ; the number of half hours in the re.-
mainder will shew the climate. (Def. 90.)
* This problem may be calculated by the following- proportion ;
As tangent of the sun's greatest declination
To radius or sine of 90,
So is sine of the sun's ascensional difference
To tangent of the latitude not within the polar circle^,
Thus suppose the ascensional diff.=2> 45?
As tangent of 23 28' - 9 63761
To radius 10
So is sine of 3 45' - - 8 81560
To tangent lat. 8 45' - 9 17793
As the ascensional difference converted into time always shows how much
before or after six the sun rises or sets, and as at the end of the 1st climate
the sun rises a quarter of an hour before 6, or sets after 6 ; and in every cli-
mate forward to the polar circles the sun rises % of an hour earlier and sets
^ later than in the preceding : and moreover, as the longest day is found by
doubling the time of sun setting, it will therefore follow, that if 6 hours be
taken from half the length of the longest day, the remainder converted into
degrees will give the ascensional difference. Hence the ascensional differ-
ence for the first climate, or where the day is 12^ hours long, is 15 minutes
of time (before and after 6, which makes the half hour) equal to 3 45' ; for
the 2d. climate 30 minutes =7 30'; for the 3d. 45 min.=ll 15'; for the
4th. 1 hour =15, &c. Hence the reason of taking 3 45' above. From
these principles the climates from the equator to the polar circles in the ta-
ble annexed to def. 90 were calculated, the remaining part of which cor-
responding to rule 2, above, may be constructed as follows :
The beginning and end of the longest day being equally distant from the
solstice intervening (see note prob. 19.) reckoning half the number of days
which the sun shines constantly without setting, from the 21st of June, both
before and after it ; find the sun's declination corresponding to those two
days in the Nautical Almanac, or in a table of the sun's declination, half the
sum of which taken from 90, will give the latitude. The reason of which
is plain, as the complement of the latitude is always equal to the sun's de-
clination when the longest day begins or ends within the polar circles. (See
note to prob. 19.) And as this declination is equally distant from the point can-
cer, in which the sun is on the 21st of June, the method is evident. From the
116 PROBLEMS PERFORMED BY
2. If the place be within the polar circles, find the length of the
longest day at the given place (by prob. i 9) and if that be less
than 1 month or 30 days, the place is in the twenty-fifth climate
or the first within the polar circle ; if more than 1 month and less
than 2 months or 60 days, the place is in the 26th climate or 2d
within the polar circles, &c.
Examfile 1. In what climate is Washington city, and what
other remarkable places are situated in the same climate ?
7 Ans. The longest day at Washington city is 14 hours 44 min.
hence 14 h. 44m. I2h.=2h. 44m. which multiplied by 2=
5 h. 28m, or 6 half hours nearly: hence Washington is in the
6th climate north of the equator. And as the breadth of this cli-
mate extends from latitude 36 3i' to 41 24' N. all those plaV
ces within these two parallels are in the same climate, viz in the
United States, Richmond, Baltimore, Philadelphia, Lexington,
Frankfort, Trenton, New-York, New-Haven, Sec. In Europe,
Lisbon, Madrid, most of the islands in the Mediterranean, Naples,
Ancient Greece, the islands in the Archipellago, Constantinople,
Sec. In Asia, Bursa, Smyrna, the southern parts of the Caspian
Sea, Samarcand, Pekin in China, the southern parts of the Japan
Isles, &c.
2. In what climate is the North Cape in the island of Maggerqe,
latitude 71 iO' north ?
Ans. The length of the longest day is 74 natural days, which dir
vided by 30, gives 2 months 1 4 days for the quotient, and hence
the place is in the 3d climate within the polar circle, or the 27th
climate reckoning from the equator. As the breadth of this cli-
mate extends from 69 33' to 73 5' (see the note and table an-
nexed to definition 90) the space contained within these paral-
lels in Greenland, Baffin's Bay, the northern part of Siberia, the
southern part of Novazembla, &c. is in the same climate.
3. In what climate is Dublin, and what other places are situated
an the same climate ?
variation of the sun's declination, it is plain that no table can answer exactly
for every year, as the declination for that year ought to be taken from the
Nautical Almanac, or from tables constructed for leap years and the three
following 1 years. A mean of these decimations has been taken in construct-
ing 1 the 2d part of the table alluded to, in order to have it correspond to every
year as near as the nature of the problem can admit. Ricciolus (an Italian
astronomer and mathematician, born at Ferrara, in 159JB) in his Jlstronomia
Jteformata, published in 1665, makes an allowance for the refraction of the
atmosphere, in his tables of climates. He reckons the increase of days by
half hours from 12 to 16 ; by hours from 16 to 20 ; by 2 hours from 20 to
24 ; and by months in the frigid zones ; making the number of da) r s in each
month in the north frigid zone something- more than those in the south. But
the refraction of the atmosphere is so variable, as to render such a table of
no material advantage. In fact the division of places by their parallels of
latitude, and length of their longest days, &c. being the most accurate and
useful, renders all others, such as zones, climates, &c. of little comparative
advantage.
THE TERRESTRIAL GLOBE. 117
4. In what climate is that part of N. E. land in Spitzbergen>
situated in 80 lat. N. ?
5 In what climate is Cape South, in New Zealand ?
6. In what climate is Cape Horn situated ?
PROB. 30.
To find the breadths of the several climates from the equator to
the fioles.
Rule. 1. FOR the northern hemisphere, elevate the north pole
23 28' above the horizon, bring cancer to the meridian, and set
the index to 1 2 ; turn the globe eastward on its axis until the index
has passed over a quarter of an hour ; observe that particular point
of the meridian passing through libra which is cut by the horizon,
and at the point of intersection make a mark with a pencil ; con-
tinue the motion of the globe eastward until the index has passed
over another quarter of an hour, and make a second mark as be-
fore j proceed in this manner until the meridian passing through
libra will no longer cut the horizon, the several marks * brought
to the brass meridian will point out the latitudes where each cli-
mate ends, from the equator to the polar circles ; the difference of
which will give the breadth of each climate.
2 For the climates from the polar circles to the poles. Find
the latitude corresponding to the length of the longest day in each
climate, namely, one month, two months, &c. (by prob. 20.)
These will be the latitudes where each climate ends, and hence
their difference will be the breadth of each climate. (See note to
def. 90.)
Examfile \ . What is the breadth of the 6th climate, and what
remarkable places are situated within it ?
Ans. The 6th climate extends from 36 31' to 41 24' N. the
difference of which is 4 53', which is the breadth required, and
all places situated within this space, are in the same climate. See
Example I of the preceding prob.
2. What is the breadth of the 27th climate ?
Ans. The 27th climate is situated between 69 33' and 73* 5',
hence its breadth is 3 32'.
3. What is the breadth of the 2d, 5th, 9th, 23d, 25th, and 30th
climates respectively, and what remarkable places are situated with-
in each of them ?
* On Gary's and Adam's globes the meridian passing through libra being-
divided into degrees, &c. in the same manner as the brass meridian ; the
Itorizon will therefore cut this meridian in the several degrees answering- to
the end of each climate, and hence on these globes the above marks become
Unnecessary.
118
PROBLEMS PERFORMED BY
PROB. 31.
To find the distance between any two places on the globe.
Rule. As the shortest distance between any two places on the
earth is an arch of a great circle contained between the two places*
(the earth being considered a sphere) therefore, lay the graduated
edge of the quadrant of altitude over the two places so that the di-
vision marked may be on one of them, the degrees on the quad-
rant between the two places will give their distance. If these de-
grees be multiplied by 60, the product will give the distance in
geographical miles : or by 691 f the product will give the distance
in English or American miles.
Or, Extend a pair of compasses between any two places ; this
extent applied to the equator will give the number of degrees be-
tween them.
If the distance between the places should exceed the number of
degrees on the quadrant, stretch a piece of thread or narrow rib-
band over them ; this extent applied to the equator, from the first
meridian, will shew the number of degrees between them, or both
places may be brought to the horizon, and the degrees on the hori-
zon will give their distance. This method will answer when their
distance is greater or less than 90.
* See Emerson's Trigonometry, cor. 2. prop. 13. b. 3.
| According to the late French adopted measures, the length of a degree
in English miles is 69.04 or 69^ j miles. See note to definition 8. To give
the learner some idea of the variation in the length of a degree on the
earth's surface, the following tables, &c. are inserted.
P Mean la-
c titudes.
Toises
Observer's names and year.
Places. j>
S 0' S.
56750
Condamine, Bouguer and Go- S
s
din, in 1736 and 1743.
Near Quito. S
c 33 18 S.
57037
De la Caille, 1752.
Near the Cape of Good >
c
Hope. ^
S 39 12 N.
56888
Mason & Dixon, 1764 & 1768.
In North America. S
? 43 N.
56979
Boscowich & Le Maire, 1755.
In Italy. S
? 44 44 N.
57069
Beccarius, 1768.
Lombardy. ^
S 45 N.
57028
Cassini, the Father, 1739 and
From Collioure to Pa- <J
1740.
ris observatory, thence S
S
to Dunkirk, distance S
?
in all 8 31 ; ll"f
S 47 40 N.
57091
Liesganig, 1768.
Germany, near Vienna. S
49 23 N.
57074
Maupertuis and Cassini, 1739
S
5
and 1740.
t
3 52 44
57300
Norwood, 1635.
Between London and t
S
York.diff.lat.obs^^S 7 . S
S 66 20
S.
57422
Maupertuis, Camus, Clairaut,
1736 and 1737.
At the extremity of the S
Gulf of Bothnia. Jj
THE TERRESTRIAL GLOBE.
119
Example 1 . What is the nearest distance between Bermudas
and Ferro Island in the Canaries ?
39| or, 39J
Distance in degrees
Geographical miles 2385
39|X69=2742|
English miles.
69
351
234
2762Eng. miles.
27821-
2762|
Length of a degree from the
theory of gravity. Ntnvton's
principia, prop. 20, b. 3.
Newton takes it foi-
M. Boue-uer, S granted in forming his
'
Lat.
Length of
a deg. on
the merid.
toises.
56637
56642
56659
56687
56724
56769
56823
56882
56945
56958
56971
56984
56997
57010
Lat.
deg.
47
46
48
49
50
55
60
65
70
75
Length o/j
a deg on {
the merid. <
toises.
57022
57035
57048
57061
57074
57137
57196
57250
57295
57332
57360
57377
57382
an ingenious
mathematici-
an of France,
thus corrects
theforegoing
table.
table, that from lat.
^ to 49, is 57060 toises,
V) according to Picard ;
S he found that in Paris
S the length of a pendu-
c lum to vibrate seconds
Toises,
56753
56776
56843
56946
57072
57139
57206
57332
57435
57530
57525
is 3 feet 8^ lines, or ra-
ther 8-f lines, and that
under the equator a pen-
dulum vibrating in the
S same time will be 1,087
S
length of pendulums be-
S ing as the force of gravi-
S ty^ that is as the versed
sine of double the lat. or
as the square of the right
sine of the same, the
construction of the table
85 57377 ]> > is manifest. This is on
57382 <? ^ supposition that the
S earth is homogeneous,
that it is not is well
known. (See Clairault's treatise on the figure of the earth.) Other observ-
ations were made by the following persons : By Eratosthenes, of Cyrene, in
Egypt, 270 years before Christ, who makes a deg. =250900 stadia, "or 66493
toises ; by Hipparchus, of Rhodes, 140 years A. C. who makes it 275000
stadia, or 73142 to;ses ; Possidonius,of Alexandria, in Egypt, 60 years A. C.
240000 stadia, or 63833 ; Strabo and Ptolemy, in Egypt, in the year 135,
makes it 180000 stadia=47875 toises ; Almaimon, an Arabian king, with
his mathematicians, in the plains of Mesopotamia, in 800, makes it 43279
toises ; Fernell, from Paris to Amiens, in 1550, makes it 56746 ; Snell, in
Holland, makes it 55021 toises ; Ricciolus and Grimaldus, in Italy, in 1661.
make it 629000 toises ; Picard, in France, in 1670, makes it 57060 toises ;
Cassini the younger, in 1700, makes it 57292 toises ; La Place, from an arch
measured at the equator, and another between Dunkirk and Mountjoy, de-
termines that the polar diameters of the earth is less than the equatorial, by
#$ part of the fatter, and that a fourth part of the ecliptic meridiau=5130740
PROBLEMS PERFORMED BY
'2. What is the nearest distance between the island of Barbadoes
and St. Helena ?
Ans. The distance in degrees is 60|- : hence 60x60 "630
geographical miles, or 60^x70 ^5l=4204| English miles The
reason of this last method is evident, as 70 ^s=69|. For al-
though 69 be not correct, this number is however generally used
for the length of a degree. 69 should however be always used in
preference. See the note to def. 8, and prob. 35.
toises. The toise being- used for the measure in Peru, and reduced to a
temperature of 16| of a mercurial thermometer, divided into 100 from the
freezing point to that of water boiling 1 , under a pressure equivalent to a co-
lumn of mercury, 76 centimetres or 30 inches English measure in height.
Within these few years an arch has been measured extending from Dunkirk
to Barcelona, and the degree whose middle is lat. 45 has by this means
been found =57029 toises. At the equator likewise some of the members
of the academy of sciences has found a deg. of the meridian 5=56753 toises,
and in Lapland, about the lat. 66 20', they found it to be 57458. From all
which it is evident, that the degrees of the meridian gradually increase
from the equator to the poles.
From these latter data and the rules of mensuration, the equatorial di-
ameter is found equal 3271267, and the polar 3261461 toises, the differ-
ence being 9769 toises =58614 French feet, equal 56647-j^ English feet, or
something less than 10| English miles. (3 feet English being equal to 3
feet 1 inches French, and 6 feet equal one toise, or 12 inches English or
American equal 12 inches 9 lines 3 points French measure, 12 points being
equal to a line, 12 lines an inch, &c.)
From the difference observed in the length of a degree of the meridian in
the above tables, it is evident that the surface of the earth is of no regu-
lar form. It is known that there are rivers on its surface, which run from
their sources from 2000 to 3000 miles and upwards, before they discharge
themselves into the sea, and that frequently these rivers have cataracts or
water falls of considerable height ; (the falls of Niagara, for example, is 273
feet perpendicular height, including 65 feet fall in the chasm, and 58 for the
half mile above the cataract.) Therefore it must be admitted that the part
of the earth at the fountain head of such rivers is considerably higher than
where they discharge themselves into the ocean ; because if the earth were a
perfect plane, or a perfect sphere, the water at the fountain head of rivers
could not possibly flow to any other place, or in any direction in preference
to another, except westward from the motion of the earth on its axis. There-
fore to ascertain its figure to as much exactness as the nature of the tiling
can admit, the exact length of a degree in the various parallels of latitude,
as well as a degree on the meridian, the elevation of countries above the
level of the sea, the motion and direction of rivers, of currents in the ocean,
&c. should likewise be ascertained. And yet if it be admitted that at the
equator the earth is higher than at the poles, we may ask how it happens
that the greatest rivers in the world flow towards or parallel to the equator,
and generally in an easterly direction, while many others have their direc-
tion towards the poles. Thus the Amazon flows towards or parallel to the
equator, while the La Plata flows towards the south pole. The Oronoke
in some places directs its course westward, then northward, and afterwards
eastward, declining in each winding from the equator. The Mississippi
and all the rivers in the United States, and within that latitude as far as the
southern ocer.n, flow almost universally towards the equator, while the St.
Lawrenre and others direct their course towards the north. In the eastern
THE TERRESTRIAL GLOBE.
3. What is the nearest distance between the island of Barbadoes
and Bermudas, in Geographical and English miles ?
4. What is the shortest distance between Washington city and
London ?
hemisphere, the Senegal and Gambia flow westward, the Niger takes its
course in an easterly direction parallel to the equator, while the Nile, the
sjjurce of which is nearer the equator, flows towards the north ; on the con-
trary, the Euphrates, the Indus, the Ganges, &c. and all the rivers in India
beyond the Ganges, direct their course towards the equator. Those in China,
&c. eastward. The Danube, Dniper, Don, Volgo, &c. likewise flow towards
the equator, whilst the Oby, Enissey, Lena, &.c. which have their sources
further south, direct their courses towards the north. Hence the great ir-
regularity in the surface of the land indicated by these rivers. Nor is the
sea without a similar or perhaps greater, as any one that considers the phe-
nomenon of the gulf stream alone, may clearly perceive. First, it directs its
course towards the north as far as the banks of Newfoundland, thence east-
ward towards the Azores, again it directs its course towards the equator,
changes its direction about the Cape Verd islands, where, impelled by the
trade winds, in a direction parallel to the equator west, it is forced again
into the Gulf of Mexico, its mean velocity being about 3 miles an hour.
From these phenomena it is evident, that the earth can have no regular-
figure, and as it deviates but little from a circular figure (see note to def. 1)
our calculations are not the less certain when we consider it as such. St.
Pierre in his Studies of Nature shews that the above measures of a degree,
&c. tend as much to prove the excess of the polar diameter above the equa-
torial as the contrary. Hence, simple as this problem may appear in theory,
on a superficial view, yet when applied to practice, the difficulties which oc-
cur are almost insurmountable. And granting even that the earth is a per-
fect sphere, much of the difficulty would still remain. For in sailing across
the ocean, or in travelling through extensive and unknown countries,"without
any other guide than the compass, with such a guide it is plain that we can-
not take the shortest rout, as measured by the quadrant of altitude (an arch
of a great circle being the shortest distance between two places on the
globe, as before observed) because the rhumb lines must always cut the
meridians in the same angles, and this cannot happen in sailing or travelling-
by the compass, unless the places be situated directly north and south of
each other, or upon the equator.
To render these observations more intelligible to the young student, 1st.
Let two places be situated in lat. 50 N. and differing in longitude 48 50',
which will nearly correspond with the lands end and the eastern coast of
Newfoundland. Now there are given the complement of the lat. =40, and
the angle formed by the two meridians passing- through both places equal
48 50', to find the distance between both places in an arch of a great cir-
cle ; but as the triangle formed by both complements of lat. which are
equal, and the required arch, is therefore Isosceles, a perpendicular let fall
from the given angle bisects the base, and also the vertical angle (Emer-
son's Trigonometry, prob. 14. cor. 2. 13. 2.) Hence there is given the hy-
potenuse ^=40 and angle at the vertex =24 25' to find the base or half the
distance ; and therefore by Baron Napier's rule (see Simson's Trig, at the
end of his Euclid. Emerson's, prob. 28. B. 3. or Keith's.) Racl. x Sine
of the distance = Sine 40* X Sine 24 25' (being the cosines of the
opposite extremes.) And therefore rad. : sine 40 :: sine 24 25' : sine
base or half the distance =15 24' 33" 3, hence the whole distance =30
49' 6" 6=1849.11 geographical miles, or 2127.7 English (allowing 69.04
miles to a deg. See note to de.f. 8) But if a ship steer from the lands end
directly westward in lat. 50 N. until her difference of longitude be 48 50',
Q
122 PROBLEMS PERFORMED BY
5. What is the shortest distance between Washington city anui
the junction between the Mississippi and Missouri ?
6. What is the extent of Europe in English miles from Cape
Matapan in the Morea, lat. 36 35' N. to the north cape in Lap-
then by parallel sailing, rad. : co. sine 50 :: diff. longitude 2930 miles :
the distance 1883.4 geographical, or 2167.16 English miles, which last
distance is greater than the former, on the arch of a great circle by 34.29
geographical, or 39.46 English miles. Those who are acquainted with
Spherical Trigonometry and the principles of Navigation, particularly great
circle sailing, know that it is impossible to conduct a ship exactly on the
arch of a great circle, except, as before observed, on the equator or meri-
dian ; for in this example, she must be steered through all the different
angles from N. 70 49' 30" W. to 90 ; and continue sailing from thence
through all the same variety of angles until she arrives at the intended place,
where the angle will become 70 49' 30", as at first. For as the comple-
ments of 50, together with the distance, form an Isosceles triangle, as before
observed, the angles at the base being equal, is found by Napier's rule thus :
rad. X sine co. 40= tangt. co. 24 25' X tangt. co. course. Hence co,
tangt. 24 25' : co. sine 40 :: rad. co. tangt. of the angle at the base =
70 49^ 30" as above. But as this is the angle which the ship's way makes
with the meridian, it is equal to the course required. In the same manner
may the course be found for any other point in the parallel between the
land's end and the meridian which bisects the distance. Thus, if instead
of 24 25',- the distance from the vessel to this perpendicular be 18 (for ex.)
the course will then be 76 I 7 22", &c.
2. Suppose it were required to find the shortest distance between the Li-
zard, lat. 49 57' N. Ion. 5 21' W. and the Island of Bermudas, lat. 32 25'
N. and Ion. 63 35' W. Here there are given the complements of both lati-
tudes, and the difference of longitude 58 14' (which is equal to the angle
formed by the two meridians passing through the
two given places) to find the distance in an arch
of a great circle. Let PL=co. lat. Lizard =40 3',
PB=co lat. Bermudas=57 3^ the angle LPB =
58 14/ (P being the pole) to find the distance LB.
Draw LI perp. to PB, then by Napier's rule rad.
X co. sine LPI=co. tang. PL X tang. PL Hence
co. tang. 40 3' : rad. :: co. sine 58 14' : tang. PI
=23 52^ 15", therefore IB =PB PI=33 42' 45". P
The sides LI, LB may be found in like manner or shorter, thus : Co. s,
PI 23 52' 15" : co. sine BI 33 42' 45" :: co. sine PL 40 3' : co. sine LB
(Emerson's Trig. cor. 2. prop. 28. b. 3) =45 52' 5" the shortest distance be-
tween the two places, equal 2752-fV geographical or 3166.73 English miles
(allowing 69.04 miles to a deg.) Now for a ship to sail on this arch, be-
tween the Lizard and Bermudas, she must sail from the Lizard S. 89 10 ;
46" W. (being the angle which the ship's way LB makes with the meridian
PL, or the angle PLB) and gradually lessen this course so as to arrive at
Bermudas on the rhumb bearing S. 50 46' 2" W. which is the angle that
the ship's way makes with the meridian PB passing through Bermudas, or the
angle PBL. But this, though true in theory, is impracticable, and there-
fore the course and distance must be calculated by Mercator's sailing. The
direct course by the compass being S. 67 59' 43" W. and the distance upon
that course 2807.68 geographical or 3230.7 English miles, which is greater
than the former by 55.6 geographical or 63.97 English miles?.
The shortest distance between any two places, the lat. and long, of which
are given, may be found in the same manner as LB has been calculated
above, and that whether the perpendicular falls within or without LB,IA,
THE TERRESTRIAL GLOBE. 123
land, lat. 71 10' N. the places being situated nearly due north
and south ?
Note. 1. Here the difference of lat. is nearly equal the distance. See
the notes at the bottom.
7. What is the shortest distance between Quito, in longitude
77 55' W. and Macapa, in longitude 51 20' W. both situated
nearly under the equator ?
Note. 2. Here the difference of longitude is nearly the distance.
8. What is the shortest distance between the town of St. Do-
mingo and Cape Horn ?
9. W^hat is the breadth of North-America from Sandy-Hook, in
lat. 40 N. and that part of the coast of New-Albion in the Pacific
ocean in the same parallel ? (See the notes and table to prob. 35.)
10. Suppose the track of a ship to Batavia be from New-York
to Bermudas, thence to St Anthony, one of the Cape Verd islands,
thence to St. Helena, thence to the Cape of Good Hope, thence
to the Isle of France or Mauritius, thence to the headland west-
ward of Bantam, thence to Batavia ; how many English miles from
New- York to Batavia, on these different courses ?
PROB. 32.
A place being given on the globe tojind all those places that are at
the same distance from it as any other given place.*
Rule. EXTEND the quadrant of altitude between both places,
so that the division marked may be on the given place from
although it be more convenient to have the perp. always fall on the longest
side as in the above calculation. But when we want to find the distance be-
tween any two places whose lat. and long, are known, in order to travel or
sail from one place to the other, on a direct course by the compass, the fol-
lowing methods must be used :
1. If the places be situated on the same meridian, or have the same lon-
gitude, their difference of latitude (found by prob. 4) will be the nearest
distance between them in degrees, and the places will be exactly north and
south of each other.
2. If the places be situated on the equator, their difference of longitude
will be the nearest distance in degrees, and the places will be exactly east
and west of each other.
3. If the places be situated in the same parallel of lat. they will be direct-
ly east and west of each other, and their difference of longitude (found by
prob. 4) multiplied by the number of miles which make a degree in the
given lat. (see the table annexed to prob. 35) will give the distance.
4. If the places differ both in their latitudes and longitudes, the distance
between them, and the point of the compass on which a person must travel,
or a vessel sail from the one to the ther, must be found by Mercator's sail'
ing, as in navigation.
* A general solution to this prob. may be obtained from prob. 2d. sect. 1.
vol.2, of Simpson's Fluxions. Thus let A = fluxion of the angle of position
between the given place and that required. B = flux, of the diff. long,
and F = flux, of the complement of the lat. of the place required. D =
comp. lat. of the given place, C = the angle of position between the re-
124 PROBLEMS PERFORMED BY
which the distance is reckoned, move the quadrant round, keep-
ing on the quadrant in its first position, all those places that pass
under the degree of distance, observed to stand over the other
place, will be the places required. The globe may be also recti-
fied for the given lat. and the quadrant screwed in the zenith, &c.
Or, Wilh the extent between both places, describe, with a pair
of compasses, a circle, the centre of which is the first given place,
all those places in the circumference of this circle, will be those
required.
If the extent between both places should exceed the length of
the quadrant, or the extent of a pair of compasses, stretch a piece
of thread over both places, with which describe a circle as before.
Or both places being brought to the horizon, turn the globe on the
notch in the direction of the meridian, and the horizon will point
out the places required.
Examfile 1 . Find all those places that are at the same (or near-
ly the same) distance from Washington city, as Port au Prince in
St. Domingo.
Ans. Kingston in Jamaica, Cape Catouche in New-Spain, An-
tonio in Mexico, the mouth of Haye's river, the island and strait
of Bell Isle, at the mouth of the river St. Lawrence, St. John's
in Newfoundland, &c.
2. Required all those places that are at the same distance from
Paris as London ?
3. Find all those places that are at the same distance from Lon-
don as Paris ?
4. Find all those places that are at the same distance from Con-
stantinople as Naples ?
quired place and the given place ; then, co. sec. D : sin. B :: A : F and
sin. F : co. tang. C :: F : B ; that is 1st. as the secant of the given latitude,
to sine of the difference of longitude (assumed at pleasure) so is the altera-
tion in the angle of position between the given place, and that required, to
the alteration in the complement of lat. of the place required. 2cl. As the
co. sine of the lat. required, to co. tangent of its angle of position with the
given place, so is the alteration in the lat. of the place required, to the al-
teration in the difference of longitude. From either of which theorems the
required places may be found. If the difference of longitude be assumed,
and any degree of position, the difference of lat. is given, and therefore the
place itself. (Theo. 1.) If the diff. of lat. be assumed and any degree of
position, the diff. of long, is found. (Theo. 2.) The difference of latitude
N. or S. or the diff. of long. E. or W. can never exceed the distance be -
tween the two given places. The section alluded to in Simpson's Fluxions,
which treats of the fluxions of spherical triangles, is extremely useful in a
great variety of cases in Practical Astronomy, Geography, and Navigation,
which are calculated with much more labour bv other methods.
THE TERRESTRIAL GLOBE. 125
PROB. 33.
Given the latitude of a place and its distance from a given place > to
find that place the latitude of which is given*
Rule. IF the distance be given in English or geographical
miles, reduce them to degrees (by allowing 60 geographical, or
69 English milesf to a degree) then place on the quadrant of
alt. upon the given place, and move the other end eastward or
westward, according to the position of the place east or west, until
the degrees of distance cut the given parallel of lat. under the
point of intersection is the place required.
Or, Having taken the degrees of distance from the equator with
a pair of compasses, with this extent, and one foot of the compas-
ses on the given place, with the other intersect the given parallels,
the point of intersection will be the place required, and will be
east or west as before. If none of the parallels on the globe pass
through the given place, you may describe one with a fine pencil,
by holding it over the lat. and turning the globe on its axis. This
prob. may be also performed by means of the horizon.
Example 1. A place in lat. 32 25' N is 2752^ geographical
miles westward from the Lizard in England, required the place ?
Ans. 2752| -7- 60 == 45 521' ; this on the quadrant will extend
from the Lizard to Bermudas in the given parallel west ; hence
Bermudas is the place required.
2. A place in W. long, and 13 N. lat. is 3660 geographical
miles from London, required the place ?
3. A place in lat. 60 N. is 1320i English miles from London,
and is situated in east longitude ; required the place ?
4. A place in lat. 53 34?' N. is distant from Washington city
4002 Eng. miles ; required the place ?
PROB. 34.
Given the longitude of a place and its distance from a given place to
Jind that place , the longitude of which is given.
Rule. CONVERT the degrees into minutes as before, and apply
on the quadrant of alt. to the given place, as in the foregoing
prob. move the other end northward or southward (according as
the required place lies north or south of the given place) until the
degrees of distance cut the given longitude, and under the point of
intersection you will find the place required.
Or, Bring the longitude of the place required to the brass meri-
dian, then take the degrees of distance from the equator with a pair
* The reason of this and the following 1 problem is too evident to need any
explanation.
f Though we sometimes make use of 69| Eng. miles to a degree, yet the
learner is advised to make use of 69, or where greater exactness is requir-
ed, of 69.04, in preference. See note to def. 8, or notes to prob. 35 and 06
following 1 .
126 PROBLEMS PERFORMED BY
of compasses, and with one foot in the given place, under the
point where the other cuts the brass meridian, you will find the
place required.
If the given place be west of the place required, the meridian
passing through the place required, instead of the brass meridian,
must be used; but if no meridian pass through it, one maybe
described with a fine pencil by bringing its longitude under the
brass meridian.
Example 1. A place in north lat. and long. 70 58' W. is 817
English miles north eastward of Charlestown in South Carolina ;
required the place ?
Ans, 817-7-69! or 1634-i- 139=1 1| nearly ; hence the place is
Boston.
2. A place in north lat. and 77 1 ; west long, is 3675 Eng. miles
towards the south west from Greenwich observatory ; required
the place ?
3. A place in longitude 5 49' W. is distant from Barbadoes
3630 geographical iiiiles south eastward ; required the place ?
4. A place in longitude 63 36' W. is distant from Ferro island
276v| English miles, and lies in a north westerly direction from
it ; required the place ?
PROB. 35.
Tojind how many miles make a degree of longitude in any given
parallel of latitude.
Pule. LAY the quadrant of alt. parallel to the equator between
any two meridians in the given lat. which differ in longitude 15
for Bardin's, or 20 for Gary's globe, the number of degrees in-
tercepted between them multiplied by 4 for Bardin's, or by 3 for
Gary's, will give the length of a degree in geographical miles.
Or, Take the distance between two meridians which differ in
long. 1 5 for Bardin's, or 20 for Gary's globes, in the same par-
allel of lat with a pair of compasses ; apply this distance to the
equator, and observe how many degrees it makes, with which pro-
ceed as before. The distance between 10 on Gary's globe mult,
by 6, will likewise give the miles required.
Examjile \ . How many geographical and American miles make
a degree in the latitude of Philadelphia ?
Ans. The lat. of Philadelphia is nearly 40<> N. The distance be-
tween two meridians, in that lat. which differ 15 in long, is 11|,
or 15-1 if the meridians differ 20. Now 11^x4- or l5-Jx3=46
geographical miles for the length of a degree of longitude in the
latitude of Philadelphia, which multiplied by 1.16 (because 60 :
69| :: 1 : 1.1 6 nearly) gives 53.36 English miles, the length of
a degree; or 15 : 11J :: 69 : 53,36.*
* The reason of this is evident from this principle, that the distance or
num. of degrees contained between any two meridians on the equator, is to a
similar arch or distance between the same meridians in any parallel of lat.
THE TERRESTRIAL GLOBE.
127
2. How many miles make a degree in the parallels where the
following places are situated :
Boston, Washington, Savannah, Kingston in Jamaica, London,
Paris, Petersburgh, Skalholt, North Cape, and the most northern
part of Spitzbergen ?
as the length of one degree on the equator, to the length of a degree in this
parallel. For similar arches have the same proportion to the whole circum-
ferences (Lemma 2. Simson's trig, or Emerson's geom. cor. to prop. 19. b. 4.)^
and therefore to one another. (Eucl. b. 5. prop. 15.) Thus in the above lat. of
Phil, two places which differ 15 on the equator, differ only 11 such de-
grees in this parallel ; hence 15 : 1H :: 60^: 46', or 15 : 60m. :: 11^ : 46,
but 15 : 60 :: 1 : 4, hence the reason of multiplying 11^ by 4, &c. If in-
stead of 15 we take 20, then 20 : 60 :: 1 : 3, or 10 : 60':: 1 : 6, &c. But
since the quadrant of alt. will measure no arch truly but that of a great cir-
cle, and that a pair of compasses will only measure the chord of the arch,
and not the arch itself, it follows that the preceding rule is not mathemat-
ically true, though sufficiently correct for practical purposes. When greater
exactness is required, recourse must be had to calculation or the following
table.
S Deg.
Geoff.
Eng.
Deg.
Geoff.
Eng. }
Deg.
Geoff.
Eng.
iDeff.
Geoff.
Eng.' ^
\Lat.
miles.
niles.
Lat.
miles.
miles.
Lat.
miles.
miles.
Lat.
miles.
miles. S
60 00
69 07
23
55 23
63 51
46
41 68
47 93
69
21 50
24 73 S*
\ 1
59 99
69 06
24
54 81
63 03
47
40 92
47 06
' 70
20 52
23 60 ?
S 2
59 96
69 03
25
54 38
62 53
48
40 15
46 16
71
19 53
22 47 S
V 3
59 92
68 97
26
53 93
62 02
49
39 36
45 26
i 72
18 54
21 32 >
S 4
59 85
68 90
27
53 46
61 48
50
38 57
44 35
73
17 54
20 17 J
S 5
59 77
68 81
28
52 97
60 93
51
37 76
43 42
74
16 54
19 02?
S 6
59 67
68 62
29
52 48
60 35
52
36 94
42 48
75
15 53
17 86 S
S 7
59 55
68 48
30
51 96
59 75
53
36 11
41 53
76
14 52
16 70 S
S 8
59 42
68 31
31
51 43
59 13
54
35 27
40 56
77
13 50
15 52 <
S 9
59 26
68 15
32
50 88
58 5l
55
34 41
39 58
78
12 48
14 35?
S 10
59 09
67 95
33
50 32
57 87
56
33 55
38 58
79
11 45
13 17 S
> 11
59 89
67 73
34
49 74
57 20
57
32 68
37 58
80
10 42
11 98 S
? 12
58 69
67 48
35
49 15
56 51
58
31 79
36 57
i 81
9 38
10 79 ?
S 13
58 46
67 21
36
48 54
55 81
59
30 90
35 54
' 82
8 35
9 59?
S 14
58 22
66 95
37
47 92
55 10
60
30 00
34 50
83
7 31
8 41 S
S 15
57 95
66 65
38
47 28
54 37
61
29 09
33 45
84
6 27
72lS
S 16
57 67
66 31
39
46 63
53 62
62
28 17
32 40
85
5 22
6 00?
S 17
57 38
65 98
40
45 96
52 85
63
27 24
31 33
86
4 18
4 81 ?
S 18
57 06
65 62
41
45 28
52 07
64
26 30
30 24
87
3 14
3 61?
S 19
56 73
65 24
42
44 59
51 27
65
25 36
29 15
88
2 09
2 41 >
? 20
56 38
64 84
43
43 88
50 46
66
24 40
28 06
89
1 05
1 21 ?
S 21
56 01
64 42
44
43 16
49 63
67
23 45
-26 96
90
00
00 ?
S. 22
55 63
63 97
45
42 43
48 78
68
22 48
25 85
S
The length of a degree is here given= 69.07 English miles. But as has
been shewn in the note to def. 8, 69.04 is more correct ; moreover an arch
about the lat. 45, which is a mean of that which has been lately measured
from Dunkirk to Barcelona, gives the length of a degree equal 57029 toises
=342174 French feet, or 364559.2 English feet (see note def. 8) 69.04
English miles as above ; and as 45 is a mean between the lat. at the equa-
tor and at the poles, 69.04 E. miles is properly taken as the mean length of
,1 degree. This table having- been found ready calculated in Keittt*s Trea-
128 PROBLEMS PERFORMED BY
PROB. 36.
Tojind at what rate fier hour the inhabitants of any given place arc
carried from west to east) by the re-volution of the earth on its axis.
Rule. FIND how many miles make a degree of longitude in
the given latitude (by the preceding prob. or the table annexed)
which multiplied by 1 5 for the answer.*
tise on the globes, pa. 173, it was thought unnecessary, for so trifling a dif-
ference, to repeat the calculation, as any one may perform it at pleasure.
Thus in the lat. 60 a degree =30 geog. miles, or 30X69.04_ gg-Qj^o^ 5 o
English miles, differing only yjj^ of a mile from that given in the table, for
there it ought to be 34.53. Hence also appears, that in practice we may consider
69 English miles in a degree, which will make the calculations much easier ;
for X =r34.5, differing only -j^ miles from the truth. If we make use
of 69i 6 nules 30 geog. miles will equal ^^2^ ~ = 34.75 E. miles, &c.
The above table is thus calculated, radius : co. sine lat. of any parallel ::
any given portion of the equator, as 1 : to a similar portion of the given pa-
rallel. For let EQ represent the equator, P the
pole, B any given place on the meridian QP ;
then the arch BQ is the lat. of B, and BP its com-
plement, and BC drawn perpendicular to the semi-
diameter PA, will be the co. sine of the lat. of
B,to the radius AQ. Now as similar arches are
as the radii of the circles of which they are arches
(Emerson's geom. b. 4. prop. 8.) Therefore AQ : T*
CB :: as any part of the circumference EQ : to a
similar part of the parallel DB.
All the properties of parallel sailing will follow from the same principle
(the earth being considered as a perfect sphere.) Thus the difference of
longitude between any two places, in an arch of the equator, and the dis-
tance between these places, if under the same parallel, is a similar portion of
that parallel ; hence as rad. : co. s. lat. :: diff. long. : distance; and by inver-
sion co. s. lat. : R. :: dist. : diff. long. Also as diff. long. : dist. :: R. : co. s.
lat. whence it likewise follows, that co. s. of any given lat. : co. s. of any
other lat. :: as any given position of the first parallel : any given portion of
the second ; and lastly, any portion of a given parallel : a similar portion of
any other : co. s. lat. of the 1st : co. s. of the 2d.
By considering the three first terms of any of the above proportions as
qiven, the 4th is given, and can be performed on the globes in the same man-
ner as the above. The learner can therefore here find an agreeable exercise
in performing all the cases in parallel sailing by help of the globe alone. To
give the problems here in detail, would be contrary to our intended brevity
in this introduction.
* The reason of this rule is evident ; for if n = the number of miles in a
de. then 24 hours : 360 X n :: 1 hour : ??2 *L! 15 n, the rule as
above. This rule is on a supposition that the earth turns on its axis from
west to east in 24 hours ; but it has been observed in clef. 66, that it re-
volves on its axis in 23 h. 56m. 4s. but this trifling diff. is scarce worth
observing. The learner must not however forget, that it takes exactly 24
hours, for any place on the earth's surface, to perform its diurnal revolution
from under any meridian, or any point in the heavens, until it comes exact-
ly under that mer. or point again. (Note to def. 66.)
From the following tables the length of a degree given in the preceding
table, may be reduced to the measures of other countries, which will often.
THE TERRESTRIAL GLOBE.
129
JExamfile 1. At what rate per hour are the inhabitants of Phila-
delphia carried from west to east by the revolution of the earth on.
its axis f
be of use to the learner, particularly in Geography, History, constructing 1
Maps, &c. The first of these tables is collected from Dunn's Atlas ; the
second from Danville as given by Ozanam, vol. 3. of his Mathematical rec-
reations ; the third from vol. 1st. of Ozanam. (Montucla's Edition,)
TABLE I.
60
69
20
50
15
15
26
18
25
12
"60
174
66
50
65
Geographical miles.
Or rather 69 5 V Eng. miles.
Marine leagues.
Scotch miles.
Miles 14 poles, Irish.
Dutch miles or 20 Marine.
Common leagues of Germany
Common miles of Lithuania.
Statute miles of Prussia.
Hungarian miles.
Common French leagues.
Leagues of Switzerland.
Italian miles.
Common Spanish leagues.
Turkish miles.
do. Berri.
Common miles of Piedmont,
Venetian miles.
21$ Turkish agash.
18 Parasan s of Persia.
56* Arabian miles.
-50 do. in some places.
1<> 5 Wersts of Russia.
37 Indian coss ' s -
250 Common lis of China.
33 Jeribi. great measured coss
of India.
34 Japanese leagues.
75 Ancient Roman miles.
80 Grecian miles.
671 Stadia of Herodotus.
533 Egyptian stadia.
22$ Italian or 30 Arabian travel
hng leagues.
60 > 000 Geometrical paces.
S
- >
Jlncient and modern measures.
Jlncient and modern measures.
Olympic stadia
Lesser stadia
Least do.
Egyptian schene^
Parasang of Persia
Roman mile (milliare)
Stadia of Judea or Rez.
Mile do. (or Berath)
League of ancient Gaul
German league (rast.)
Arabian mile
French mile
Fr. small league 30 to a deg.
Do. mean 25 to a degree
Do. great 20 to a degree
German miles 12- to a deg.
Do. 15 to a degree
Swedish mile
Danish mile
Toises.
94
75^
50*
3024
2268
756
76
569^
1134
2268
1084
1000
1902
2283
2853
4536
3800
5483
3930
Toises.
English mile or 1760 yards 826
Scotch mile 1147
Irish mile 1052
Spanish league of 5000 vars 2147
Do. common 17? to a deg. 3261
Italian or Roman mile 768
The mile of Lombardy 848|
Venetian mile 992
The league of Poland 2850
The ancient werst of Russia 656
Modern do.
Turkish agash
The little coss of India
The great coss
The gau of Malabar
The Nari or Nali do.
The lis of China
The pu equal 10 lis
130 PROBLEMS PERFORMED BY
Ans. The lat. of Philadelphia is nearly 40, in which parallel a
degree of long = 46 geogr. or 53.36 English miles (prob. 35)
hence 46x 15 = 690 and 53.36X 15 = 800.4 therefore the inhabi-
tants of Philadelphia are carried 690 geographical or 800 English
miles per hour.
Note. The geometrical pace is 5 Roman feet, 1000 of which = 1 mile
= 8 stadia = 400 cubits of 1 foot 9^ inches English. ; 30 stadia = a para-
sang = 2188| English feet. The toise is 6 French feet. But as the foot
differs in various countriesj the following table will give the learner an idea
of this variation. As the length of the toise is given in Paris feet, we shall
compare all the other feet to it. It is divided into 12 inches, each inch into
12 lines, and each line into 10 parts ; hence the foot will be 1440 of these
parts. We shall therefore consider the foot of other countries under both
these denominations, that is, in parts, and in inches, lines, &c.
ANCIENT FEET.
% Parts. Ft. in. li. pts.
The ancient Roman foot = 1306 = 10 10 6
Grecian and Ptolemaic 1364 11 4 4
Grecian Phyleterien 1577 1117
Archimedes, or probably that of Syracuse 8c Sicily 986 826
The Drusian 1473 1033
Macedonian 1567 1107
Egyptian 1920 1400
Hebrew 1637 1177
Natural 1100 920
Arabic 1480 1040
Babylonia 1546 1 10 6
or 1534 1094
MODERN FEET-
The foot of Paris 1440 1000
Amsterdam 1253 10 5 3
Antona and Eccles. states 1732
Altorf(Underwald) 1047 887
Anvers or Antwerp 1270 10 7
Augsburg 1313 10 11 3
Avignon and Aries 1200 10
Aquileia (Venice) 1524 1084
Basle 1276 10 7 6
Barcelona 1340 11 2
Bologne 1682 1202
Bourg. (Bress and Bugey, Switz'd.) 1392 11 7 2
Berlin 1340 11 2
Bremen 1290 10 9
Bergame 1933 1413
Besaneon 1372 11 5 2
Brescia 2108 1568
Brue-es 1013 853
Brussels 1219 10 1 9
Breslaw 1520 1 8
China, tribunal of mathematics, 1523 1083
The imperial foot 1420 11 10
Cologne 1220 10 2
Chambery (and Savoy) 1496
Copenhagen 1448 11 9 8
C2966 2086
Constantinople | 1575 ills
THE TERRESTRIAL GLOBE. 131
J3y the table. In lat 40 a degree of long. = 45.96 geogr. or
52.85 English miles. Hence 45.96x15=689.4 and 52.85X15 =
792.75, consequently the inhabitants in this parallel are carried
6891 geogr. or 793 English miles per hour, by the earth's revolu-
tion on its axis. The latter result is the most correct.
MODERN FEET.
Parts. Ft. in. U. pts.
The foot of Cracow =
1580
= 1 1
2
Dantzic
1247
10
4
7
Dijon
1392
11
7
2
Delft (Holland)
739
6
1
9
Denmark
1415
11
9
5
Dordrecht
1042
8
8
2
- Edinburgh
1485
1
4
5
Ferrara (Italy)
1779
1 2
9
9
Florence
1345
11
2
5
Francfort on the Maine
1260
10
6
Franche Comte
1483
1 1
2
3
Genoa (Le Palme)
1098
9
1
8
Geneva
2592
1 9
7
2
Grenoble and Dauphin
1512
1
7
2
Halle (on the Elbe, Up. Sax.)
1320
11
Harlem
1267
10
6
7
Hambourg
1260
10
6
Heidelberg (Palat.)
1220
10
2
Inspruck (Cap. of Tyrol)
Ley den
1488
1382
1
11
4
6
8
2
Leipsic
1397
11
7
7
Liege
1276
10
7
6
Lisbon
1287
10
8
7
Leghorn
1340
11
2
Lombardy, &c.
1926
1 4
6
London
135l|
11
3
1
Lubeck (Holstein)
1260
10
6
Lucca (Italy)
2615
1 9
9
5
Lyons and Lyonnois, Ferez, &c.
1512
1
7
2
Lorraine
1292
10
9
2
Madrid
1237
10
3
7
Malta (Le Palme)
1207
10
7
Marseilles and Provence
1100
9
2
Malines
1017
8
5
7
Mentz
1335
11
1
5
Mastricht (on the Meuse) and the low
countries
1238
10
3
8
*M Cpied decimal
Man Ipiedaliprand
1155
1926
9
1 4
7
5
6
Modena
2812
1 11
5
2
Monaco
1042
8
8
2
Montpellier (Le Pan)
1050
8
9
Moscow
1255
10
5
5
Mantua (La Brasse)
2055
1 5
1
5
Munich
1280
10
8
Naples (Le Palme)
1164
9
8
4
"Nuremberg j J^^.,,
1346
1226
11
10
2
2
6
6
Padua
1899
1 3
9
9
132 PROBLEMS PERFORMED BY
2. At what rate per hour are the inhabitants of the following;
places carried from west to east, by the earth's revolution on its
axis ?
Boston, New-York, Washington city, Quito, Cape Horn, Mad-
rid, London, Petersburg, Skalholt, Spitzuergen.
Note. If the velocity per hour be multiplied by any given number of
hours, the velocity for that time is given.
MODERN FEET.
Parts. Ft. in. It. pts.
The foot of Parma = 2526 =1906
Pavia 2080 1540
Prague 1336 11 1 6
Palermo 1010 850
The Riiine or Bhinland 1382 11 6 2
Riga 1260 10 6
Borne (Le Palme) 990 830
Rouen (as Paris) 1440 1 0,
Seville (Andalusia) 1340 11 2
Stetin (in Pomerania) 1654 1194
Stockholm 1450 1010
Sien (common foot) ( 1674 1 1 11 4
Toledo 1237 10 3 7
Turin (Piedmont) 2265 1 6 10 5
Trent 1622 1162
Valladolid 1227 10 2 7
Warsaw 1580 1120
. Venice 1537 1097
Verona 1510 1070
Vienna, in Austria, 1400 11 8
Vienne (Dauphine) 1430 11 11
Vicenz a (Estates of Venice) 1535 1095
Wessel 1042 882
TJlm (Swabia) 1117 937
TJrbino (Italy) 1570 1110
Utrecht 1001 841
Zurich 1323 11 3
Here the learner sees that there are few countries but differ in the
length of their foot; and as almost all of them reckon 12 inches (digits)
to their foot, the inch must be no less variable. Thus 1440 .- 135 if (or
864 : 811) :: 12 inches : 11 in. 3 li. if pts. Paris measure, the length of an
English or American foot. On the contrary 1351-| : 1440 :: 12 in. : 12 in. 9
li. 4 pts. the length of a Paris foot in English measure, and so on for any
other country. Hence the English inch, foot, &c. is -f| of the Paris, and
the Paris inch, foot, &c. is |TT of the English, &c. According to the latcs
French measures, the old French foot =12.78933 English inches, and
their metre equal 0,513074 toises =39.371 English inches, or 3 281 feet.
5130740 toises being the length of 90 or a quarter of the meridian. The
metre is also equal to 443.296 lines, or 3 feet 11.296 lines, or 0.841 aunes
of Paris. See tables of the comparison between the ancient and modern
measures in France, published in Paris, by order of the Minister of the
Interior, &c. &c. or La Place's System of the World, vol. 1. b. 1 ch. 12.
where this subject is handled with that accuracy for which this author is
THE TERRESTRIAL GLOBE. 153
3. How many geographical miles are the inhabitants of Madrid
carried, in 24 hours, more than those of Petersburg, by the earth's
revolution on its axis ?
4. At what rate per hour are the inhabitants of the north pole
(if any) carried, by the earth's diurnal motion on its axis ?
PROB. 37.
To find the bearing of one place from another.
Rule. IF the places be situated on the same rhumb line, that
rhumb line is their bearing ; but if not, lay the quadrant of alti-
tude over both places, and the rhumb line, that is the nearest of
being parallel to the quadrant, is their bearing.
Note 1. As the parallels of lat. are real east or west, and the meridians
north and south rhumb lines, hence those places situated in the same parallel
are east or west, and those on the same meridian north and south, from
each other.
Or, If the globe have no rhumb lines* drawn on it, make a
small mariner's compass, and apply the centre of it to any given
Eirticularly remarkable. The treatise on arithmetic by Theveneau, or
'arithmetique par le Cen. Prevost-Saint-Lucian may be consulted. It re-
mains only to add, that the new French linear measures are the millime-
tre, centimetre, decimetre, metre, decametre, hecatometre, chilometre,
myrcometre, each of which is 10 times the value of the foregoing ; the
millimetre being- 0.039371 English miles. The centimetre is printed wrong
in the table in the beginning of the English edition of La Place's Astrono-
my, 9.39371 being given for 0.39371. The decimetre =0.30784, the cen-
timetre 0.36941 inches, and the millimetre 0.44330 lines, all the other
measures being in proportion. The length of a degree in the above tables
should be corrected by these measures, in the same manner as 69^ Eng-
lish miles was changed to 69.04 &c. Thus 69 : 69.04 :: 50 Scotch m. :
49.669 m. The length of the metre above given was found from the arch
of the meridian contained between Dunkirk and Barcelona, the length of
which is 9 40' 25".6, equal to 551584.70 of the iron toise used at the equa-
tor at the temperature 16, the quadrant being divided into 100, com-
pared with the arch measured in Peru. The above arch from Dunkirk
to Barcelona in Spain, is the result of Delambre and Mechain's measures ;
the astronomical and trigonometrical observations being made with a re-
peating circle, which gives great precision in the measure of angles. The
above 9 4(7 25 '6, is given 6 40' 26*2, in the English translation of La
Place. Hence, whoever uses this work should take the precaution of
making the calculations over again, as there are many errors in the Eng-
lish measures. The other different measures, &c. of the respective coun-
tries will be given in the Treatise of Arithmetic, at present nearly ready
for the press. The utility of the above table in a mere an tile view, no less
than in a geographical, will atone for its uncommon length, considering
our contracted limits. The length of the note to prob. 31st precluded the
tables from being inserted there as their proper place.
* Neither Gary's or Bardin's globes have any rhumb lines on them. On
Adams' globes there are two compasses drawn on the equator, one on a
vacant place in the Pacific ocean, between America and New Holland ;
the other in the Atlantic, between Africa and South America. Each point
of either of these compasses will serve as rhumb lines. The compass
may be easily made by describing- a circle on a sheet of paper, or on a
134 PROBLEMS PERFORMED BY
place, so that the north and south points may coincide with some
meridian, the other points will shew the bearings of all the cir-
cumjacent places, to the distance of more than 1000 miles, if the
central place be not far distant from the equator.
Note 2. The latitudes and longitudes of two places being given, the
course and distance are given, by this and prob. 31.
Examfile \ . What is the bearing between the Lizard and the
island of Madeira ?
Ans. S. S. VV.
card, &c. with a radius of any convenient lerfgth, and then dividing its
circumference into 32, or each quadrant or 4th part into 8 equal parts,
and annexing to each part its appropriate name found on the horizon of
the globes. Any two lines drawn through the centre, at right angles to
each other, may be first considered the E. W. N. & S. lines. These points
may be again divided into halves, quarters, &c. or each quadrant into 90,
Sec. The bearing is however found much more correct from J\fercator > s
sailing, by the following proportion ; Meridional difference of latitude :
radius :: difference of longitude : tangent course. Here the diff. lat. and
difT long, are both considered as given. The meridional parts are ready
cu aated in books on navigation, in tables constructed for that purpose.
See McKay's Treatise on Navigation, Hamilton Moore's, improved by Bow-
ditch, Norey, requisite tables, or Robertson's Navigation. When no ta-
ble of meridional parts is at hand, the defect may be supplied by the fol-
lowing rules : The length, of the meridian line for the given lat. accord-
radius
ing to Wright's projection is equal to 7915705 X log. tang . % com , i at .
(Emerson's Math. prin. of Geography, Art. Navigation, cor. 3. prop. 4.)
Or thus, To half the given lat. add 45, and find the logarithmic tangent
of the sum, and divide it by 7915.7, the quotient will be the meridional
parts required, for the sphere. If the meridional parts for the spheriod be
required, they may be found thus ; As rad : sine lat. :: 30 : x ; then this
number subtracted from the meridional parts for the sphere, will give the
meridional parts answering to the spheriod. (Simpson's Fluxions, vol. 2.
sect. 11. prob. 29, and cor.) But the course may be found thus independ-
ent of the meridional parts. In the Philosophical Transactions, No 219,
it is demonstrated, that the meridional line on Mercator's chart, is a scale
of the logarithmic tangents of the half complements of the latitudes ; that
these logarithmic tangents, of Mr. Briggs^ form, are a scale of the dif-
ferences of longitude, upon the rhumb which makes an angle of 51 38' S"
with the meridian ; and that the differences of longitude on different
rhumbs, are to one another, as the tangents of the angles which these
rhumbs make with the meridian. Hence,
As the difference of the log. tangts. of the complements of the lati-
tudes of any two places,
To the difference of longitude between these places,
So is the tangt. of 51 38' S" (log. 10.1015104)
To the tangent of the course.
All the cases in JMercator's sailing may be solved by this rule. See Ew-
Ing's Synopsis, 4th ed. pa. 243. In his Mercator's sailing there are given
other methods for finding the meridional parts, taken from Robertson's
Navigation, C 2d ed. pa. 532. In Emerson's treatise above quoted, there
are given a variety of examples in sailing on the principles both of the
sphere and spheriod, with the investigation of all the rules, &c.
The learner will remark, that Bowditch's Treatise on Navigation has
been lately improved and published in England.
THE TERRESTRIAL GLOBE. 135
2. What is the bearing between the Lizard and St. Mary's, one
of the western islands ?
' Ana. S. 47 54/ W.
3. Required the bearing* between Washington city and any of
the following places :
Amsterdam, Annapolis, Berlin, Boston, Brussels, Charlestown,
Copenhagen, Dublin, Edinburgh, Lisbon, Madrid, Paris, Phila-
delphia, Rome, Stockholm, Vienna, and New- York ?
4. Required the bearing between Philadelphia and Madrid ?
5. Required the bearing between Lima and Washington city ?
PROB. 38.
'To find the angle of position between any two places.
Rule, RECTIFY the globe for the lat. of one of the given places
(prob. 9) bring that place to the brass meridian, and screw the
quadrant of alt. in the zenith, or degree over the given place ; then
extend the graduated edge of the quadrant over the other given
place, and the degrees on the horizon, between the graduated edge
of the quadrant and the brass meridian, reckoning towards the ele-
vated pole,* will be the angle of position between that place, for
which the globe was rectified, and the other given place.
Note. All those places that are under the edge of the quadrant, have the
same angle of position.
* Some choose to reckon the angle of position towards the nearest part
of the brass meridian, as in the 2d example above, where 89 is given in
place of 91. It is of little consequence which method is used.
The angle of position between two places is a different thing from their
bearing, the latter being determined by a spiral Line or loxodromic (from
loxos, oblique, and dromos, a course) called a rhumb line, which makes
equal angles with all the meridians through which it passes ; but the
former by a great circle passing through the zenith of a given place, and
another whose position from the former is required, in the same manner
as the azimuth is found in astronomy. That the angle of position is there-
fore essentially different from the bearing, except when the places are on
the equator, or upon the same meridian, is sufficiently manifest. , But sim-
ple as it may appear, it has been the cause of various disputes among
writers on the globes; some contending that the angle of position between
two places is very different from their bearings, while others suppose that
they are the same. Hence a further illustration of this matter may be
deemed necessary for young students. By attending to the method given
in the above rule of finding the angle of position, we shall find that the part
of the quadrant of alt. included between both places, forms the base of a,
spherical triangle, the two sides of which are the distance of both places
from the elevated pole, or the complements of the latitudes of the two
places, when they are on the same side of the equator, or the complement
of one of the places and the latitude of the other added to 90 (or its
complement taken from 180) when the places are on different sides of
the equator ; and that the vertical angle, included by both sides, and
formed at the elevated poles, is their difference of longitude, the angles
at the base of the triangle, being the angles of position between the two
places respectively, which are therefore easily calculated by the method
given in the note to prob. 31. Thus,
136 PROBLEMS PERFORMED BY
Examfile 1 . What is the angle of position between Washington
city and Dublin ?
An*. Nearly 48 from the north towards the east. The follow-
ing places have nearly the same angle of position from Washing-
ton, viz. Harwick in England, Antwerp, Cologne, Temeswar,
Aleppo, &c.
2. What is the angle of position between London and Prague ?
Ans. Nearly 91 reckoning from the north towards the east, or
89, reckoning from the south towards the east. The southern
1. When the two places are situated on the same parallel of latitude.
Let the two places be the Land's end, and the eastern coast of New-
foundlasd (as given in the note referred to above) the lat. of which is
nearly 50 N. and diff. of long. 48 50' . the complement of both latitudes
is therefore 40, and hence the triangle is Isosceles, the sides being- each
40, and the vertical angle 48 5C/, from which the base or the arch of
nearest distance measured by the quadrant of alt. will be =30 49' 6"6,
and the angles at the base, or the angles of position each e=70 49' 30"
(the triangle being Isosceles.) If we now take the middle point in the
arch of distance, on the quadrant of alt. its distance from the elevated
pole will be 37 23', and hence its lat =52 37' N. and the meridian pas-
sing through this point will be at right angles to the arch of distance be-
tween the two places. An indefinite number of points being now taken
along the edge of the quadrant of alt. between the two places, the angle
of position between the Land's end and each of these points, will be 70
4S f 30" from the north westward But if it were possible for a ship to
sail by the compass on the arch of a great circle passing through these
places, indicated by the edge of the quadrant of alt. her latitude would
continually increase from the Land's end until she had sailed half her dis-
tance to the other place, or from 50 to 52 37' N. and her course would
vary from 70 49' 30" to 90. But in sailing the other half of the distance
to the eastern coast of Newfoundland, her lat. would continually decrease
from 52 37' N. .to 50 N. and her course must vary from 90 to 70 4S r
30" westward. But were the ship to sail along- the parallel of lat. passing
through both places, her course would then be invariably due west.
Hence it follows, that when the places are situated in the same parallel of
lat. their angles of position can never represent their true bearing by the
compass, unless when they are on the equator, where their angle of po-
sition would be invariably 90 degrees.
2. If the two places differ both in latitudes and longitudes.
Let L represent a place in lat 50 N. B a place in lat. 13 30' N. (see
the fig. in note to prob. 31) and let their difference of longitude BPLss
52 58', the angle of position between L and B will be found by Spher.
Trigonometry =S,68 57' W. and the angle of position between B and L
will be N 38 5' E. ; whereas the direct course by the compass from L to
B, by Mercator's sailing, is S. 50 6' W. and from B to L it is N. 50 6'B.
If any number of points be taken in the arch LB, the angle of position
between L and each of these points will be invariable, being each 68
57', while the angles of position between each of these places and L are
continually diminishing. If a ship were, therefore, to sail from L in a
S. 68 57' W. course by the compass, she would never arrive at B ; and
were she to sail from B on a N. 38 6' E. course by the compass, she
would never arrive at L
Hence the angle of position, in any case, can never represent the bear-
ing, except, as before remarked, the places be on the equator, or on the
meridian.
THE TERRESTRIAL GLOBE. 137
extremity of the Caspian sea, the mouths of the Indus, Bombay,
the southern extremity of the island of Ceylon, 8cc. have nearly
the same angle of position.
3. What is the angle of position between Dublin and Washing-
ton city ?
Ans. 78 nearly, reckoning from the north westward.
4. Required the angles of position between New-.York and the
following places :
Petersbu^gh, Copenhagen, London, Paris, Constantinople, Cairo,
Cape Verd, Bermudas, St. Domingo, Cape Nicholas do. New
Orleans, Mexico, and the East Cape in Bhering's strait.
PROB. 39.
The distance of two places situated on the same meridian, and their
angles of position with a third place, being given, to find that
place, with its nearest distance from each of the other two.
Rule. RECTIFY the globe for the lat of the first place, and
screw the quadrant of alt. in the zenith, bring the given place to
the meridian, extend the quadrant to the degree on the horizon
which is equal to its position from the third, draw a line along the
graduated edge of the quadrant ; then elevate the pole to the lat.
of the second place, bring it to the meridian, and screw the quad-
rant over it, which extend, as before, to the degree on the horizon
which is equal to its position from the third ; the intersection of the
quadrant with the line drawn before, will give the third place re-
quired ; the distance of which from the former two is found as ia
prob. 31st.
Example \ . The distance between Madrid and Edinburgh, situ-
ated on the same meridian, is 15* 33'=933 geographical miles;
and the angle of position with a third is 5 3|- nearly, and of the
latter 66 from the south nearly; required the place, with its
nearest distance from each of the former ?
Ans. The required place is Vienna, its distance from Madrid
=922, and from Edinburgh 847 geographical miles, nearly.*
* This prob. may be calculated as follows : Let M re-
present Madrid, E Edinburgh, and V the required
place ; then there are given the distance ME between
the given places == 933 miles, the angle MEV = the
angle of position between Edinburgh and the place V<
required = 66, and EMV the angle of position be-
tween Madrid and the place V = 53, to find the
sides VE,VM, the distance of the required place from
each of the former. Hence by spherical trigonometry,
we have as cosine of half the sum of the two angles of position : cosine of
half their difference :: tangt. of half the distance of the given places E and
M : tangt. of half the sum of the sides VE,VM (the required distances.)
And sine of half the sum of the angles of position : sine of half their differ-
ence :: tangt. of half the given distance (EM) : tangt. of half the difference
of the required distances ; then to half the sum add half the difference and,
S
138 PROBLEMS PERFORMED BY
2. The distance between Halifax, in lat. 44 40' N. and the
north-east part of Margarita island, in the West-Indies, lat. 11 10'
N both situated nearly under the same meridian, being equal to
33 ->6' or 2016 miles, and the angle of position of the former
with a third being equal 62 from the south, and of the latter with
the same place 75 ; required the place, with its nearest distance
from each of the foregoing places ?
PROB. 40.
Given the course and distance, to find the latitude and longitude
come to, the place left being known.
Rule. MARK the given rhumb in the lat. of the place left,
bring that mark to the meridian which passes through the long,
left, convert the distance sailed into degrees, take one degree
from the equator, in a pair of compasses, and turn it over on the
rhumb as often as there are degrees in the given distance, and
where the reckoning ends will be the place required, whose lat.
and long, is found as in prob. 1.
Note. If the rhumb does not pass through the given place, find the lon-
gitude of the place where the reckoning- ends, and the number of degrees
between this and the longitude of the first mark on the rhumb line, will be
the difference of longitude, whence the long, come to is found by note 4.
prob. 4. Where no rhumb lines are given, a small mariner's compass made
on paper, will answer.*
Examjile 1. A ship from Cape Clear, in lat. 51 18' N. and
long. 11 15' West, sails S. E. J S. 480 miles ; required the lat.
and long, come to ?
Ans. The place required is in lat. 45 22' N. and long. 3 9' W.
2. A ship from New- York or Sandy-Hook light-house, in lat.
40 28' N. long. 74 7' W. sails E. N. E. 1200 geographical miles ;
required the lat. and longitude the ship is in ?
you have the side opposite the greater angle of position given or the side
VM, and from half the sum, take half the difference, and you have the side
opposite the lesser angle given, or the side VE.
Note. That half the sum of the required sides will be of the same affec-
tion as half the sum of the given angles, and the contrary.
In the above prob. the bearing may be made use of instead of the angle
of position, when the given distance is small.
* If a small compass, made of paper, be used, it may be always easily
placed N. and S. by the meridians on the globe, or rather the brass meri-
dian ; but as it may be difficult to place the centre on the given place exact-
ly, a quarter or half of the compass will answer better.
The solution of the prob. on the principles of Mercator's sailing, is as
follows :
Had. : cos. course :: distance : difference of latitude, and
Had. : tangt. course :: meridional diff. of lat. : difference of longitude.
The difference of latitude and difference of longitude being thus given,
the latitude and longitude arrived at may be found by the method given in
note 4. prob. 4th. part 2.
To enter into the investigation of the principles on which the above pro-
portions are founded, would be foreign to our intended plan,
THE TERRESTRIAL GLOBE. 139
3. A ship from the Lizard, in lat. 49 57' N. long. 5 21' W.
sails S. 47 51' W. 1162 miles ; required the lat. and long, of the
place the ship is in ?
PROB. 41.
Soth latitudes and course given, to find their distance and difference
of longitude.*
Rule. TURN the globe on its axis until the given rhumb cuts
the brazen meridian in the lat. left, there mark the rhumb under
the given degree of lat. and observe the degree of the equator cut
by the brass meridian ; then turn the globe until the same rhumb
cuts the meridian in the lat. come to, under which on the rhumb
make a mark as before ; the number of degrees between these
two marks, reckoned on the equator, will give their difference of
longitude ; and the distance is found by taking a degree of the
equator in a pair of compasses, and extending it on the rhumb, be-
tween the two marks, as often as possible, the number of degrees,
thus measured, being converted into miles, will give the distance
required.
Note. If the globe has no rhumb lines described on it, a compass made
of paper may be used as in the foregoing- problems ; in which case the rhumb
will cut the meridian passing through the given place, or the lat. left, with-
out first turning the globe. The shorter the radius of such a compass is,
the more correct will the distance be ; in which case it will be often neces-
sary to find different centres in the same rhumb, &c.
Example 1. A ship from the Lizard, in lat. 49 57' N. makes
her course S. 39 W. and then by observation is in lat 45 31' N.
required her distance run and longitude in ?
Ans. The difference of longitude being 5 21' W. and the long,
of the Lizard equal 5 15' W. hence the long, is 10 36' W. and
the distance is 342 miles.
2. A ship from Bayonne, in lat. 43 29' N. and long. 1 30' W.
sails N. W- \ N. until by observation she is in lat. 51 31' N. re.-
quired the distance run and longitude come to.
* The proportions for calculating this prob. are as follow :
Rad. : secant course :: difF. lat. : distance,
Rad. : tang, course :: meridional difF. lat. : difF. long.
In the same manner may other problems in navigation be performed on
the globes, the above being well understood.
For the sake of readers not in the habit of using the Nautical Almanac, it
may not be improper to remark, that in pa. 96 of the Naut. Aim. for 1813,
revised by John Garnett, there is a table for correcting the middle latitude,
which renders the calculation by this method more expeditious, and as ac-
curate as in Mercator's sailing. There is also in pa. 168 of the same alma-
nacs, for 1812 and 1813, a table, shewing, very nearly, the difference be-
tween a ship's direct course in a great circle, and that found by Mercator's,
or mid. lat. sailing. (See the note to prob. 38, part 2.)
140 PROBLEMS PERFORMED BY
PROB. 42.
To find the meridian altitude of the sun, on any day, at any giv-
en place.
Rule. ELEVATE the pole to the latitude of the given place,
bring the sun's place to the meridian, and the degree over it will
be the declination, the number of degrees reckoned from which
to the horizon, will give the meridian altitude required.*
Or, Elevate the pole to the sun's declination, bring the given
place to the brass meridian, and the number of degrees between
it and the horizon, will be the meridian altitude required.!
OR BY THE ANALEMMA.
The globe being rectified to the latitude, bring the given day
found on the analemma to the brass meridian, the number of de-
grees between which and theJiorizon will be the alt. required.
Example 1. What is the sun's meridian altitude at New- York,
on the i Oth of May ?
66 6'.
* See note to prob. 1. and prob. 25. f See prob. 24.
If the learner be not accustomed to use the quadrant or sextant of reflec-
tion, and yet wish to perform this prob by observation, he may use a com-
mon quadrant with a plummet. These are to be had at the instrument
makers, with lines, sometimes drawn on them, for finding the hour of the day,
the sun's azimuth, &c. or they may be easily made of wood or slate suffi-
ciently correct, where exactness is not required. It would however be bet-
ter to have a quadrant, or rather semicircle, immoveable in the place of the
meridian, and divided into degrees and their lesser parts, according- to art
(using either the nonius, as in Hadley's quadrant, or a scale divided diagon-
ally) and having an index moveable on its centre, furnished with telescopic
sights. But whoever wishes to use the improved astronomical circle, will
have, with a good telescope and watch, all the astronomical apparatus ne-
cessary.
Note. The complement of the lat. added to the sun's declination, where
they are of the same name (that is both north or both south) or subtracted
when they are of different names (that i? one north and the other south)
will give the sun's meridian altitude.
The declination being a necessary requisite for solving this prob. is found
in the 2d page of every month in the Nautical Almanac. (See the Nautical
Almanac, published with important additions, under the direction of John
Garnett, New-Brunswick, New-Jersey, where the daily difference of declina-
tion is given to reduce it to the meridian of Greenwich, &c.) The declina-
tion may be obtained by knowing the meridian altitude and latitude of the
Elace, for as the co. lat. -{- sun's decl. = sun's mer. alt. (when the decl. and
it. are of the same name) hence in this case, sun's decl. =r mer. alt. co.
lat. Again, when the decl. and lat. are of different names, co. lat. decl.
= mer. alt. hence declination = co. lat. mer. alt. This latter only
takes place when the complement of the lat. is greater than the declination ;
when less, the contrary sign must be used. The decl. and mer alt. being
given, the lat. may be found from the same equations ; thus, in the 1st equa-
tion, co. lat. = mer. alt. sun's decl. &c. In the same manner from va-
rious other problems, a variety of conclusions may be drawn, with only a
slight knowledge of the nature of equations in Algebra.
The Nautical Almanac is also latelv published by E. M. Blunt, in New-
York.
THE TERRESTRIAL GLOBE. 141
2. What is the sun's meridian altitude at Washington city, on
the 21st of June ?
Am. 740 35'.
3. What is the sun's meridian altitude at Philadelphia, when
the days and nights are equal ?
4. What is the sun's greatest altitude at New-York ?
5. What is the sun's meridian alt. at Quito, on the 22d of De-
cember ?
6. What is the sun's greatest meridian alt. at Cape Horn ?
PROB. 43.
Tojind the sun* 8 altitude by placing the globe in the sunshine.
Rule. MAKE the plane of the horizon on the globe truly level
or horizontal, then erect a needle perpendicularly over the north
pole, or in the direction of the axis of the globe, and having turned
the pole towards the sun, move the brass meridian until the needle
casts no shadow ; then the arch of the meridian between the pole
and the horizon, will give the sun's altitude. (See prob. 28.)
Or in general. Turn the north or south pole towards the sun,
erect a needle, as before directed, towards the earth's centre on that
part of the brass meridian where it will cast no shadow, and the
degrees between it and the horizon will be the altitude required.
PROB. 44.
TV find the sun's altitude for any time at any given filace, inde-
pendent of the foregoing method.
Rule. RECTIFY the globe for the latitude, screw the quadrant
of altitude in the zenith, bring the sun's plabe for the given time
to the brazen meridian, and set the index to 12 ; turn the globe
on its axis until the index points out the given hour, extend the
graduated edge of the quadrant of altitude over the sun's place,
and the degree cut on it will be the sun's altitude.*
Or, Elevate the pole to the sun's declination, screw the quad-
rant of alt. in the zenith, bring the given place to the brass meri-
dian, and set the index to twelve ; then if the given hour be in the
forenoon, turn the globe westward, but if in the afternoon, east-
ward, as many hours as the time is before or after twelve ; extend
the quadrant of alt. over the given place, and the degree cut on it
will be the sun's altitude.!
* The reason of this method is evident from what is said in prob. 25.
j The reason of tkis rule is clear from what is delivered in prob. 24.
The prob. may be solved in numbers thus ; the lat. day, and hour being-
given.
1 Rule. Here, to find the altitude, the re are given the complement of the
lat. the hour angle (or the angle formed between the brass meridian and
the meridian passing through the sun's place) and the complement of the
sun's declination. The learner will perceive that with these the comple-
142 PROBLEMS PERFORMED BY
Example 1 . What is the sun's alt. at New-York, on the 10th of
May, at 6 o'clock in the morning ?
Ans. \ l^ nearly.
2. What is the sun's alt. at Washington city, on the. 2 1st of
June, at 3 o'clock in the afternoon ?
Ans. 54.
3. What is the sun's altitude at Philadelphia, on the 21st of
March, at 10 o'clock in the morning ?
4. What is the sun's altitude at Quito, on the 1st of January, at
1 o'clock in the afternoon ?
PROB. 45.
FO find all those filaces where the sun has the same altitude as
any given place, at any given time.
Rule. FIND where the sun is vertical at the given time (by
prob. 12) mark this place, and find its distance from the given
place (by prob. 31) find all those that are at the same distance
from it as the given place (by prob. 32) these will be the places
required.
Example 1. When it is 15 minutes after 8 in the morning at
New-York, on the 30th of April, required all those places where
the sun, at that moment, will have the same altitude as in New-
York ?
Ans. The place where the sun is then vertical being Cape Verd,
those places that are at the same distance from it as New- York are
Quebec, Moskitto Cove in west Greenland, the middle of the gulf
of Bothnia and Finland, near Mecca, the middle of Abyssinia, Cape
Volta in Caffraria, the western part of St Domingo, Cumberland
Harbour in Cuba, St. Salvador in the West Indies, &c.
ment of the sun's altitude will form a triangle, whose two sides and the
included angle are given, to find the base, or the complement of the sun's
altitude. Hence rad : cos. hour angle :: cot. latitude : tang, or, the seg-
ment between the pole, and a perpendicular from the zenith on the meri-
dian passing through the sun's place ; then cos. x : s. lat. :: cos. remain-
ing segment (comp. decl. less a:) to sine altitude required. (See Emer-
son's Trig. b. 2. sec. 4. case 8. or Simson's Euclid general, prop, case 4,
Sec. Sp. Trig.)
2 Rule. Here the brass mer. the meridian passing through the given
place, and the quadrant of alt. form a spherical triangle, the two equal
sides of which, or the complement of the decl. and co. lat. and the in-
cluded angle (or hour angle) are given, to find the third side, or comple-
ment of the sun's altitude, which is found exactly as above. The hour
angle is converted into degrees by allowing 15 for every hour.
* As a ray of light from the sun, conceived at an infinite distance from
the earth, will make equal angles with the tangent touching the globe at
each of the above places, which represents their horizon, and that the
altitude of the sun is its height above the horizon, hence the reason of
the rule is evident. The same reasoning is applicable to several other
problems where the sun's alt. is required.
THE TERRESTRIAL GLOBE. 143
2. When it is 4 o'clock in the afternoon at London on the 18th
of August, find all those places where the sun will then have the
same altitude as in London ?
3. Find all those places where the sun will have the same alti-
tude as at Philadelphia, at 12 o'clock the 2 1st of March ?
PROS. 46.
To find the sun's altitude at any place in the north frigid zone,
where the sun does not descend below the horizon, when it is mid-
night at any place in the temperate or torrid zones, on the same
meridian.
JRule. ELEVATE the pole to the lat. of the place in the frigid
zone, bring the sun's place to the brass meridian, and set the in-
dex to twelve ; turn the globe on its axis until the other twelve
comes to the meridian, and the number of degrees between the
sun's place and the horizon, counted on the brass meridian towards
the elevated pole, will be the altitude required.
Or, Elevate the pole to the sun's declination for the given day ;
bring the place in the frigid zone to that part of the brass meridian
which is numbered from the pole towards the equator, and the num-
ber of degrees between it and the horizon will be the sun's altitude.
Example \ . What is the sun's alt. at the South Cape in Spitz-
bergen, in lat. 76 N. when it is midnight at Naples, on the 10th
of May ?
Ans. 4 degrees.
2. What is the sun's altitude on the 21st of June at the North
Cape in Lapland, when it is midnight at Adrianople in Turkey in
Europe ?
3. What is the sun's altitude at the northwest part of Spitzber-
gen, latitude nearly 80, when it is midnight at Cagliari in Sar-
dinia ?
PROB. 47.
To place (he terrestrial globe in the sunshine, so as to represent the
natural position of the earth.
Rule. PLACE the globe north and south by the mariner's com-
pass (allowing for variation, if any, see note to prob. 49) or by a
meridian line,* bring the place where you are situated to the me-
ridian, and elevate the pole to its latitude ; then the globe will
correspond in every respect to the situation of the earth itself.
All the circles, &c. on the globe will correspond to the same
imaginary circles, Sec. in the heavens ; and each town, kingdom,
state, &c. will point out the position of the real one which it re-
presents, Sec. See probs. 24 and 25.
* The method of drawing a meridian line is shewn in several of the fol-
lowing problems, but more particularly in prob. 75, part 2-
144 PROBLEMS PERFORMED BY
PROB. 48.
The latitude and day of the month being gi-ven^ to find the hour
of the day when the sun shines.
Rule. 1. PLACE the wooden horizon of the globe truly level or
parallel to the horizon of the place, and the brazen meridian due
north and south ; elevate the pole to the lat. bring the sun's place
to the brass meridian, and set the index to 12 ; fix a needle perpen-
dicularly over the sun's place in the ecliptic, turn the globe on its
axis until the needle casts no shadow, and the index will point out
the hour
Or, The globe being placed horizontally, due north and south,
and rectified for the lat. as before ; then if a long pin be fixed per-
pendicularly on the brass meridian, in the direction of the axis, and
in the centre of the hour circle, and 12 on the hour circle be
brought to the meridian, the shadow of this pin will point out the
hour of the day.
Note. If the place be in north lat. and the decl. be N. the sun will shine
over the north pole ; but if the declination be more than 10 south (nearly
the radius of the hour circle) the sun will not shine upon the hour circle at
the north pole.
Or, The equator being divided into 24 equal parts from the
point aries, on which place the number 6, and then westward on
the other points 7, 8, 9, >0, 11, 12, 1,2, &c. to 6, which will fall
on the point libra, 7, 8, &c. to 12, then again 1, 2, &c. to 6 ;* then
place the globe horizontal, north and south, and rectify as before ;
bring aries to the meridian ; observe the circle which is the bound-
ary between light and darkness, if westward of the brass meridian,
and it will intersect the equator in the given hour in the morning ;
but if eastward, it will intersect the equator in the given hour in
the afternoon.
Or, Having placed the globe as before, and the point aries being
brought to the meridian ; tie a small string round the elevated
pole, stretch its other end beyond the globes, and move it so that
the shadow of the string may fall upon the depressed pole ; its
shadow on the equator will then give the hour.
* The antartic circle on Adams' globes is thus divided, by which the
problem may therefore be solved.
The altitude of the sun (which is equal to the number of degrees between
the needle placed as above when it casts no shadow and the horizon, reckon-
ing on a verticle circle) and the lat. and day of the month being 1 given, the
solution by spherics may be as follows : _^______ _ ____
Cos. decl. X cos, lat. : R2 :: sine j ^X co. deci. + co. lav. + to. <dt. X
sine ^ X co. decl. -f- co. lat. co. aii. : cos. h 2 (h being" the hour angle)
which converted into time, will give tho time from apparent noon, (See the
note to prob. 11. part 3.)
THE TERRESTRIAL GLOBE, 145
PROB. 49.
The latitude of the place and day of the month being given, to find
the sun's amplitude, right ascension^ oblique ascension^ oblique
(tesccnsion^ ascensional difference, and time of rising and set ting*
Rule. ELEVATE the pole to the given latitude ; bring the sun's
place to the brass meridian, and the degree cut on the equator, rec-
koned from aries eastward, will be the sun's right ascension. The
globe being then turned on its axis, until the sun's place eomes to
the eastern part of the horizon, the degree of the equinoctial
cut by the horizon, reckoning from aries as before, will be the
sun's oblique ascension, and the degree cut on the horizon, rec-
koning from the east, will be the sun's amplitude at rising. The
globe "being now turned again on its axis, until the sun's place
comes to the western part of the horizon, the degree on the equi-
noctional cut by the horizon, reckoning from aries eastward, as
before, will be the sun's oblique descension, the degree cut on
* To perform this prob. by calculation, the learner will first perceive that
the sign and degree of the sun's place reckoned from aries, or the sun's lon-
gitude, the obliquity of the ecliptic (or its inclination with the equator) or
the sun's declination, are requisite to find the sun's rt. ascension. The sun's
rt. ascension, longitude, and declination, forming a right angled spherical
triangle. Now the obliquity of the ecliptic may be found thus : let the sun's
least distance from the vertex about the summer (or winter) solstice be ob-
served ; this distance subtracted from the lat. of the place, when the place
is nearer to the pole than the sun is, or added when the sun is nearer, will
give the greatest declination of the sun, or the obliquity required, allowance
being made for refraction, &c. particularly if the observation be made at the
winter solstice. If the solstice should not take place when the sun is on the
meridian (as it generally happens) allowance must be made. The error,
however, is not wortli observing here, as it never arises to more than 4"
when greatest, that is, when the solstice happens at midnight, being equal
to what the sun's declination, 12 hours before or after the solstice, wants of
its greatest declination. Professor Mayer in his Solar and Lunar Tables,
gives a method of calculating- this obliquity, having found from observations
made with an excellent mural quadrant, at both solstices, in 1756, 57, and 58,
that the mean obliquity of the ecliptic in the beginning of 1756, was 23 28'
16", and the decrease in 100 years is about 46" ; whence the mean obliquity
for any other year, month, or day, may be easily found bv proportion. Thus
the mean obliquity for the beginning of 1811 is 23 27' 50" 7 ; now to find
the true obliquity, the nutation, &c. must be found as directed in prob. 4 of
Mayer's, which is here = 9" 6 ; so that the true or apparent obliquity for
the beginning of 1811 was 23 27' 41" l,agreeingnearly with the Nautical Al-
manac for 1811. The greatest nutation according to Mayer is 9"6. From a
like calculation it will be found, that the obliquity varies considerably in the
space of one year. For on the first of January, 1811, according to the Nau-
tical Almanac, the obliquity was 23 27' 41"8 ; on the 1st of April 23 27'
42"7 ; on the 1st of July 23 27' 41"8 ; on the 1st of October 23 27' 42'V ;
and on the 31st of December 23 27' 41*9. (See prop. 34 of Emerson's Cen-
tripetal forces. The reader is also referred to La Grange or De La Place's
Physical Theories, or to Mayer's, printed in London in 1770, under the di-
rection of Nevil Maskelyne, A. R.) For the beginning of 1811, N. Maske-
lyne makes the mean obliquity 23 27' 5G"9, and corrects it by his folio ta-
bles 31 and 32. For the beginning of 1813, he makes it 23 27' 51"3, and
makes the secular variation 42"6. For more information on this subject^ &c-
T
146 PROBLEMS PERFORMED BY
the horizon, reckoning from the west point of it, will be the sun's
amplitude at setting, and the difference hetween the sun's right
ascension and oblique ascension, or clescension, or which is the
same, the time between the index at either of these positions and
the hour of six, is the ascensional difference, which in the former
case must be converted into time (by prob. 6) then if the sun's
declination and the lat. of the place be both of the same name, that
is botn north or both south, the sun rises before, or sets after six,
by a space of time equal to the ascensional difference ; but if the
latitude and sun's declination be of contrary names, that is one
north and the other south, the ascensional difference will shew how
long the sm rises after six, or sets before six.
Note 1. The ascensional difference reduced into time, and added to or
subtracted from 6 o'clock, gives the length of half the day or semi-diurnal
arch, the complement of which to a semicircle, or to 12 hours, will give the
length of half the night or semi-nocturnal arch : or the time of the sun's con-
tinuance above the horizon, may be found by reckoning the number of hours
on the upper p&rt of the hour circle between the places where the index
pointed when the sun's place was at the eastern and western parts of the
horizon, or by prob. 13- See also probs. 23, 24, 25.
Note 2. From this prob. and prob. 8, the learner will observe that the me-
thod of finding the sun's right ascension and declination in the heavens, is
the same as finding the latitude and longitude of a place on the earth, with
this difference, that the rt. as. is reckoned quite round the globe.
Consult Mason's Tables of 1780, Wargentine's Tables published at the end
of the Nautical Almanac for 1779, La Land's Astronomy, 3d edition, for
1792, where accurate tables of the sun, moon, and planets, and of the eclip-
ses of Jupiter's satellites are given ; these being constructed principally by
Delambre on the best observations, and on the Physical Theories of M. La
Grange and M. De La Place, founded on Newton's Theory of Gravity. But
the. late lunar tables of Mr. Burg of Vienna, constructed principally on the
observations of Maskelyne, is looked on by this astronomer as the most cor-
rect. Mayer's tables and precepts of calculation are given in the Philadel-
phia edition of the Encyclopedia. These observations being useful to di-
rect the study and choice of the young astronomer, we think it necessary to
caution him, at the same time, against several remarks found in some of
these works, tending to favour impiety, and impose on superficial minds,.
We shall make it a particular study in our intended course, to point out the
dangerous tendency and falsehood of such principles, assumed, for the most
part, without a shadow of proof. And thus we hope to be able to present
our young students with the most valuable observations and improvements of
past ages, without any danger to the more valuable deposit which, as Chris-
tian s, enlightened by truths far more important, more consoling and sublime,
they are in possession of. For truth is always consistent with itself. But
to proceed.
Having now obtained the obliquity of the ecliptic, or the sun's greatest
declination, and the present declination being obtained by note to prob. 42
(see prop. 8.) the rt. ascension is found by this proportion, Rad. ; co. t. sun's
greatest declination :: tangt. present decl. : sine rt. ascension. (Napier's
rules.) Now to find the oblique ascension, amplitude, Sec. the learner will
observe, that the globe, being placed as above directed for finding the am-
plitude, &.c. the amplitude reckoned on the horizon, the sun's declination,
and the ascensional difference, form a right angled spherical triangle, and
the inclination of the plane of the equator \vith the horizon being equal to
the complement of the lat. is also equal to the opposite vertical angle (Kmev-
THE TERRESTRIAL GLOBE. 147
Examfile \. Required the sun's amplitude, right ascension, ob-
lique ascension, oblique descension, ascensional difference, and
time of rising and setting at New-York, in lat. 40 42' 40", on the
2 1st of June?
Ans. The sun's amplitude at rising and setting is 30l, the right
ascension is 90, oblique ascension 68, ascensional difference 22 ?
or Ih. 28'. Hence 6 In. 28' = 4h. 32' = time of sun rising,
and 6 -f- Ih. 28' = 7h. 28', time of sun setting.
son's Trig-, cor. 2. prop. 3. b. 3) or the angle formed by the arches express-
ing the amplitude and the ascensional difference ; hence we have these pro-
portions ; Rad. : tangt. lat. :: tangt. decl. : sine of the ascensional difference;
which subtracted from the right ascension, when the declination is north,
or added when the decl. is seuth, will give the oblique ascension when the
place is in north lat. but when the declination is north, the as. diff. must be
added, &.c. when the lat. is south. The oblique descension, 8cc. is found in
like manner. Again, to find the amplitude, it will be cos. lat. : sine decl. ::
rad. : sine ampl. (Napier's rule, as above.)
It may not be improper to remark here, that the point of the compass on
which the sun rises and sets being known, the magnetic amplitude is given,
being equal to the distance from this point to the east or west points of the
horizon respectively ; and that the difference between this magnetic ampli-
tude and the true amplitude found above, is the variation of the compass, if
both be of the same name, that is both north or both south ; but if they be
of different names, that is one north and the other south, their sun is the
variation. To know whether the variation be east or west, this rule must be.
observed ; the observer's face being turned towards the sun ; then if the
true amplitude be to the right hand of the magnetic, the variation is easter-
ly, but if to the left hand, westerly. The valuation may be also found by
taking the sun's alt. in the morning, and at the same time its bearing, and
likewise in the afternoon when its alt. is the same ; the middle point will be
the meridian, the difference between which and the N. and S. points of the
compass, will be the variation. If in place of taking equal altitudes of the
sun, the points of the compass on which it rises and sets be observed, then
half the difference will be the variation as before. The instrument calcu-
lated to make this observation with, though not generally very exact, is an
azimuth compass, for the description and use of which the reader is referred
to McKay's Complete Navigator, or the Encyclopedia. The astronomical
circle answers the purpose of an azimuth compass, transit instrument, theo-
dolite in surveying, &c. and is extremely exact. In New-York the variation
for 1810 was about 3 west. (See definition 45. ^ In Washington city the va-
riation for 1811 is nearly ; along the coast of the United States the varia-
tion is decreasing. As the declination of the sun at rising or setting differs
from his declination at noon, found in the Nautical Almanac, and in the
former is used in finding the amplitude by calculation, the following propor-
tion is necessary ; as 24 hours is to the hours from sun rising, so is the daily
variation of declination to a fourth number, which must be added or sub-
tracted according as the declination is increasing or decreasing, &c. In the
same manner proportion may be made for the right ascension, &c. Allow-
ance must also be made if the meridian differ from that of Greenwich.
The longitude of the sun is easily found by prob. 3, 8 or 10 of Mayer's
tables, &.c. his hourly motion by prob. 6, rt. ascension by prob. 7, declina-
tion by prob. 9, sun's parallax by prob. 12, and ref' action by prob. 13 ; th
t.wo last articles being necessary in finding the correct alt. of the sun. How-
ever the learner is desired to make use of Burg- ami Delumbre's tables trans-
lated and corrected by Vince, and lately published in England, being the
most valuable now extant. These are the. tables, a,t present, principally used
la calculating the Nautical Almanac.
148 PROBLEMS PERFORMED BY
2. What is the sun's amplitude, right ascension, oblique as-
cension and descension, ascensional difference, and time of rising
and setting at Washington city, on the 10th of May i
3. On the 21st of December, what is the sun's amplitude, right
ascension and declination, oblique ascension and descension, sun's
rising and setting, and length of the day and night at London ?
Note. At the vernal equinox the sun has no amplitude, rt. ascension or
declination, no oblique ascension or descension, and therefore no ascensional
difference ; it rises and sets at six, making the days and nights each equal
12 hours all over the world.
PROB. 50.
The latitude, day, and hour being given, to find the sun's azimuth
and his altitude.
Rule. RECTIFY the globe for the lat. zenith, and sun's place
(prob. 9) then the number of degrees between the sun's place and
the vertex is the sun's meridional altitude. The index being then
set to 12, turn the globe eastward* if the time be in the forenoon,
or westward if the time be in the afternoon, as many hours as the
time is before or after 1 2 o'clock ; the quadrant of altitude being
then extended over the sun's place, the degrees cut by it on the
horizon, reckoning from north to south, will give the azimuth, and
the degrees from the horizon to the sun's place, reckoned on the
quadrant of alt. will give the sun's altitude.
OR BY THE ANALEMMA.
Rectify the globe as before ; bring the middle of the analemma
to the brass meridian, and set the hour circle to 12 ; then the
globe being turned as before, bring the graduated edge of the
quadrant of alt. to coincide with the day of the month on the ana-
lemma, and the number of degrees on the horizon, cut by the
quadrant, as before, will be the azimuth, and the number of degrees
from the horizon, where the day of the month cuts the quadrant,
will be the altitude.
Examfile \ . What is the sun's altitude and his azimuth at New-
York, on the I Oth of May, at 9 o'clock in the morning ?
Ans. The alt. is 45, and the azimuth 107-^ from the north,
or 72^ from the south.
2. What is the sun's altitude and azimuth at Boston, on the 10th
of June, at 6 o'clock in the morning, and also his meridian alti-
tude ?
* Whenever the pole is rectified for the lat. the proper motion of the
globe is from east to west, and the sun is on the east side of the brass me-
ridian in the morning 1 , and on the west in the afternoon ; but when the pole
is elevated for the sun's decimation, the motion is from west to east, the
place being- on the west side of the meridian in the morning 1 , and on the.
r-QSt side in the afternoon.
THE TERRESTRIAL GLOBE. 149
3. What is the sun's azimuth and altitude at St. Domingo, at 7
o'clock in the morning, and also at a quarter past 10, on the 10th
of June?*
4. Required the time of the sun's appearing twice on the same
azimuth, both in the forenoon and in the afternoon, at Barbacloes,
on the 20th of May ?
5. Being at sea, in lat. 57 N. on the 13th of August, I observ-
ed that the azimuth of the sun was 40 14' from the south, at half
past 8 o'clock in the morning, what was the sun's alt. his true
azimuth, and the variation of the compass? (See the notes.)
6. On the 14th of January, in lat. 33 52' S. at half past three
o'clock in the afternoon, the sun's magnetic azimuth was observed
to be 63 51' from the north ; required the true azimuth, variation
of the compass, and the sun's altitude I
PROB. 51.
fliven the latitude, day and hour^ as in the last firod. to find the de-
pression of the sun below the horizon-) and his azimuth at any hour
of the night.
Rule. RECTIFY tbe globe for the latitude, zenith, and sun's
place, as before j take that point in the ecliptic exactly opposite to
* Whenever the declination of the sun exceeds the lat. and both are of
the same name, the sun will appear twice in the forenoon, and twice in the
afternoon, on the same point of the compass, at all places in the torrid zone ;
and will cause the shadow of an azimuth dial, to go back several degrees.
In this example the sun's azimuth at 6 is N. 68 E.; at 7, N. 71 E.; at
past 7, 72 from the north ; at 8, 73 ; at 8, 73; at 9, 73^; at 9, 72 ;
at 10, 71* ; at 10$, 71 ; at 11, 61 ; and at 11, 39 from the north.
To perform the prob. by calculation, the learner will observe on the globe,
that the complement of the latitude reckoned on the brazen meridian, the
complement of the altitude reckoned on the quadrant of alt. and the com-
plement of the sun's declination reckoned from where the quadrant cuts the
ecliptic in the sun's place, to the pole (the globe being rectified, &c. as above,
and turned until the index points at the given hour) form a spherical trian-
gle ; that the angle formed by two of these sides, i. e. the comp. of the lat.
and the comp. of the decl. is the hour from noon converted into degrees, &c.
and that the azimuth is the angle formed by the comp. of the lat. and com-
plement of the altitude. This being premised, the azimuth is found as fol-
lows : conceive a perpendicular arch to be drawn from the sun's place on
the brazen meridian (the globe being rectified, Sec. as above) then will rad. :
cos. hour angle :: co. tangt. decl. ::"tangt. x, a fourth arch or segment of
the base (or base produced) between the pole and perpendicular on the mer.
which being therefore given, the remaining segment, between the zenith and
perpendicular, is given, which call y, then sine x : sine y :: co. tangt. of the
hour angle : co. tangt. of the azimuth south ; which if reckoned from the
north, is greater or less than a quadrant or 90, according as the perpendi-
cular falls north or south of the zenith. (See Emerson's Trig. b. 3. part 4,
case 7.) To find the altitude. Sine azim. : s. 90 : decl. :: s. hour angle :
cos. altitude. (See prob. 9. part 2.) From this prob. the variation of the
compass may be obtained, being the difference between the true* azim. and
tfie magnetic, or azim. observed by a compass.
150 PROBLEMS
the sun's place, and find its altitude and azimuth as in the preced-
ing prob. and these will be the depression and azimuth reqniied.
Example. What is the sun's depression and azimuth at New-
York, on the 12th of November, at 9 o'clock at night ?
Ans. The alt. will be the same as in ex. 1, of ihx preceding,
and the azimuth likewise the same ; but reckoned contrary., that
is, 724 from the north, or 107-| from the south. In the same
manner may any of the examples in the foregoing problem be
changed and performed.
PROB. 52.
Given the latitude, the sun's filace and altitude, to find the sun's
azimuth and the hour of the day.
Rule. RECTIFY the globe for the lat. zenith, and sun's place,
and set the index to twelve ; turn the globe eastward or westward
(according as the altitude is given in the forenoon or afternoon)
until the sun's place coincides with the given degree of altitude on
the quadrant ; then the hours passed over by the index, will shew
the time from noon, and the quadrant will point out the azimuth
on the horizon, as before.
OR BY THE ANALEMMA.
The pole being elevated for the lat. and the quadrant screwed in
the zenith as before ; bring the middle of the analemma to the
brass meridian, and set the index to 12 ; turn the globe as before,
moving the quadrant, at the same time, until the day of the month
coincides with the given altitude ; the hours passed over by the in-
dex will give the time, and the azimuth will be found on the hori-
zon as before.
Rxamjile 1. At what hour of the day in the forenoon of the 21st
of June, is the sun's altitude 30 at New-York, and what is his azi-
muth ?
Ans. The time from noon is 7 hours 20 minutes, and the azi-
muth 83-^ from the north towards the east.
Note 1. This prob. is performed more accurately with the hours on the
equator than with the hour circle. On Cary's twenty-one inch globes the
hours, quarters and single minutes are marked on the equator, and the halt'
minutes may he also distinctly pointed out. In performing- the problem, the
learner should make the on the quadrant coincide with the horizon, by
drawing the end of the quadrant tight with one hand, adjusting it at the
same time to the lat. and turning the globe with the other.
2. At what hour on the 21st of March, in the afternoon, is the
sun's altitude 221, and what is his azimuth ?
3. On the 10th of May the sun's altitude at Washington city
was observed 40 25' ; required the hour of the day and sun's azi-
muth, the observation being made in the forenoon ?
4. In New-York, on the l Oth of March, having observed thai
the shadow of a perpendicular object was exactly equal to its
height, it is required from hence, to find the hour of the day when
the observation was made, supposing it to have been made m the
THE TERRESTRIAL GLOBE. 151
morning, the point of the compass on which the shadow was pro-
jected, and the sun's azimuth ?
Note 2. The length of the shadow of perpendicular objects is equal to
their heights when the sun's alt. is 45, as appears from the 6th and 13th
problems of the 1st book of Euclid ; and the point of the compass is shewn
by the quadrant of alt. or azimuth circle, for though in reality this be the
sun's position, yet in the small compass of our horizon, it agrees accurately
enough with the bearing-.
PROB. 53.
Given the latitude^ the sun's place and his azimuth, to Jind his alti-
tude and the hour of the day.
Rule. RECTIFY the globe for the lat. screw the quadrant of alt.
in the zenith, bring the sun's place to the brass meridian, and set
the index to twelve ; then the quadrant being set to the azimuth
on the horizon, turn the globe until its graduated edge meets the
sun's place, the degree cut on the quadrant will be the altitude,
and the index will point oat the hour.*
* Here the complement of the altitude, the complement of the latitude,
and the sun's declination subtracted from or added to 90, according as it
is of the same or of a different name from the latitude, will form a triangle,
and the acute angle included between the brass meridian and the quadrant
will be the azimuth ; if this fall within the triangle, it will be the angle in-
cluded between the comp. of the alt. and comp. of the lat. but if it be with*-
out the triangle, its supplement, or what it wants of 180, will be the angle
included by the above sides. There are therefore given two sides, and the
angle opposite one of them, to find the third side, which is the comp. of the
alt. and the angle opposite to it, or the hour angle included between the
brazen meridian, and the meridian passing through the sun's place, which
may be thus found, first letting fall a perpendicular from the pole on the
quadrant of altitude produced, if necessary ; Rad. : cos. azimuth :: tangt. co.
lat. : tangt. of a 4th arch, which call x ; then sine lat. : cos. x .-. 90 -^
decl. : cosine of another arch which call y, then the difference between x
and y will be equal to the complement of the alt. when the perpendicular
falls without the triangle, and their sum .when the perpendicular falls
within. Moreover, if 90 1 decl. and the angle formed by the arch a:
and co. latitude be of the same affection (that is each less or each greater
than 90) y will be less than a quadrant or 90; but if these angles be of
different affections, that is one less and the other greater than a quadrant,
v will be greater than 90 ; all which the learner will easily understand on
the globe. Now to find the hour angle it will be sine 90 t decl. : s. azi-
muth :: sine co. altitude : sine hour angle from 12.
Thus, in the 1st example above, Rad. : cos. azim. 65 :: co. tan. lat. 40
45' : tang, x 26 S' ; and s. lat. 40 43' : cos. 90 23 2&' = 66 32' : cos.
y 56 43', hence y x = 56 43' 26 9> = 30 34', the complement of
the alt. and therefore the alt. is 59 26', as above. Now to find the hour
angle we have sine 66 32' : sine az. 65 :: cos. alt. 59 26' : sine hour angle
from 12 = 30 &' or 2h. Om. 36 seconds ; hence 12h. 2h. (/ 36" = 9h. 5 7
24", or 59 min. 24 seconds after 9 in the morning.
The learner must take notice that the decl. is added to 90 when in the
triangle, the opposite angle is greater than 90 or the supplement of the azi-
muth, but subtracted if the op. angle be less than 90, or equal to the azi-
muth, or according as the arch of the meridian between the pole and the
^un's place, is greater or less than 90.
152 PROBLEMS PERFORMED BY
This prob. and the following may be performed by the analem-
ina, nearly in the same manner as the foregoing.
Example \ . On the 2 i st of June, in lat 40 43' N. the sun's
azimuth in the morning was 65 from the south ; required his alt.
and the hour of the day, when the observation was made ?
Ans. Alt. 59 26', and the time 9h 59' 24".
2. On the 4th of July, in lat. 38 53' N. the sun's azimuth in the
morning was 70 from the south ; required the alt. and hour ?
3. In lat. 5 1 4: N. the sun's azimuth from the south, in the even-
ing, was 40, on the 22d of Dec required the alt. and hour ?
PROB. 54.
Given the sun's altitude and azimuth^ to find the sun's place and
the hour of the day, the latitude being known.
Rule. RECTIFY the globe for the latitude, screw the quadrant
of altitude in the zenith, and set the graduated edge of the quad-
rant to the given azimuth on the horizon ; then turning the globe
on its axis, that point of the ecliptic which cuts the altitude will be
the sun's place, the quadrant being kept in the same position ;
bring the sun's place to the brazen meridian, and set the index to
twelve, then turn the globe again until the sun's place cuts the
quadrant of alt. and the index will point out the given hour *
Example 1. In lat. 40 43' N. the sun's altitude in the forenoon,
being 59 26', and his azimuth from the south 65 ; required the
sun's place, and the hour of the day ?
Ana. The sun's place is the beginning of cancer, and the hour
nearly 10 o'clock.
* To perform this prob. by calculation, there are given the complement
of the latitude on the brass meridian, the complement of the altitude OH
the quadrant of alt. and the angle included by these sides, which is equal
to the sun's azimuth if acute or less than 90, or its supplement if obtuse
or greater than 90, to find the opposite or third side which is always =
903t the declination (from which the declination will be given) and the
hour angle or the angle included at the pole, between the brass meridian
and the meridian passing through the sun's place ; the declination being
therefore given, and the obliquity of the ecliptic, or sun's greatest de-
clination, the sun's place is given by note 5, prob. 8.
Now to find the declination we have these proportions, having let fall a
perpendicular as in the preceding prob. then, rad. : cos. azim. :: tang, co.
lat. : tang, .r, and the difference between the complement of the alt and
.r = y. Whence cos. x : s. lat. : cos. ?/. : cos. 90 ~ decl. Then as
the compl. alt. -f- *> and the angle included by the given sides are ol
the same or different affection, 90r decl. is greater or less than a quad-
rant. The declination being from thence given, we have sine obliq.of the
ecliptic or greatest decl. : s. present decl. :: rad. : s. longitude from
aries ; if the sun's place be nearer libra, the result will be the same, reck-
oning the degrees from libra, or taking the supplement of what the above
proportion gives.
The hour angle is found as in the preceding problem.
THE TERRESTRIAL GLOBE. 153
2. In lat. 51 1 N. the sun's alt. in the forenoon was 40, and his
azimuth 60 from the south ; required the sun's place, and the
hour of the day ?
3. In latitude 60 N. the sun's alt. in the morning was 9, and
his azimuth 70 from the north ; required the sun's place, and
hour of the day when the observation was made ?
Note. In these and similar problems, there are two days of the year
which will answer these conditions, both equally distant from the longest or
shortest day.
4. In lat. 30 S. the sun's alt. in the morning was 28, his azi-
muth being 80 from the south ; required the sun's place, and the
hour of the day ?
5. In lat. 30 N. required the two days of the year in which the
sun's altitude in the afternoon will be 30, and his azimuth 79
from the north, and the hour when the observation is to be made ?
PROB. 55.
The day of the month being given, to find the surfs altitude^ azimuth-)
the latitude of the place, and hour of the day, by placing the globe
in the sunshine.
Rule. PLACE the globe upon a truly horizontal plane, in a north
and south direction, by the compass (or a good meridian line) fix
a needle perpendicularly over the sun's place in the ecliptic for the
given day (found by prob. 8.) bring it to the brass meridian, and
set the index to twelve, move the globe until the index casts no
shadow, in any direction ; then the degree of the brass meridian
cut by the horizon is the latitude, the index will point at the hour,
and the quadrant of alt. being applied to the zenith and extended
over the sun's place, the degree then cut by the sun's place will
be the altitude, and the azimuth will be found on the horizon as
before.
PROB. 56.
The latitude of the place being given, to find the sun's declination,
his place in the ecliptic, his altitude, azimuth, and hour of the day,
by placing the globe in the sunshine, as above.
Rule. PLACE the globe horizontally, and also north and south,
as above, and elevate the pole to the given latitude ; then the num-
ber of degrees which the sun shines beyond the north pole, is his
decimation north. If the sun do not shine beyond the north pole,
his decimation is as many degrees south as the enlightened part is
distant from the pole ; if the sun shine exactly as far as the pole,
the sun is then on the equinoctial line, and consequently has no
declination. The sun's declination being thus found, his longitude
is given, and the day of the month corresponding (by prob. 8. note
3.) next fix a needle perpendicularly in the parallel of the sun's de-
clination for the given day, and turn the globe on its axis until the
needle casts no shadow ; the globe being then fixed in this position.
U
154 PROBLEMS PERFORMED BY
screw the quadrant of alt in the zenith, bring the graduated edge
to coincide with the sun's place, or the point where the needle is
fixed, the degree cut hy the needle will be the sun's altitude, and
the degree on the horizon will give the azimuth. The hour may
be found as in the preceding prob.
PROB. 57.
The latitude ofthefilace and the day of the month being given, to find
when the sun is due east or west.
Pule. ELEVATE the pole to the given lat. screw the quadrant
of alt. in the zenith, bring the sun's place, for the given day, to
the brass meridian, and set the index to twelve, move the quadrant
of tilt until on it coincides with the east point of the horizon ?
the quadrant being held in this position, turn the globe on its axis
until the sun's place comes to the graduated edge of the quadrant ;
the hours passed over by the index will be the time from noon
when the sun is due east, and at the same time from noon he will
be due west.*
Or, This may be performed by the anaJemma in the same man-
ner, only instead of bringing the sun's place to the meridian, you
bring the analemma there, and then the day of the month on the
aualemma to the graduated edge of the quadrant.
Example 1. On the 2ist of June, in latitude 40 43', required
when the sun is due east or west ?
An* The sun is due east at 41 min 8 seconds after 8 in the
morning, and due west at 18 min. 52 st-c. after 3 in the afternoon.
2. In latitude 5-| on the 19th of May, at what hour will the
sun be due east and also due west ?
Ans. The hour angle from 12 is 4h. 54m. the time that the sun
is west ; hence ^2h. 4h. 54m = 7h. 6m. the time that the sun
is due east. The alt. may be found at the same time as in prob.
5,9. Here it is 25 26'.
3. At what hours will the sun be due east and west at Washing-
ton city, on the 21st of June and 22d of December, and what will
his alt. be at the same time, on the 2 1st of June ?
* Here the brass meridian, quadrant of alt. and the meridian passing-
through tlie sun's place, form a right angled triangle, two sides of which
are given, viz. the complement of the latitude, and the distance from ihc
elevated pole to the sun's place, or 90 1, decl. (For the day of the month
being given, the .declination is given probi'S.) to find the included angle or
hour angle, which converted into time, will give the hour from noon, at
which the sun is due east or west. Hence from Napier's rule, we have this
proportion ; Rad. : co. tan. lat. :: co. tangt. 90 ~t. decl. : cosine hour angle
from noon. Thus, in ex. 1. Rad. : co. tun. 40 45' :: co. tang. 90 23 28'
= 66 32' : cosine hour angle = 59 43' 3h. 18m. 52 seconds, the time
when the sun is west ; and therefore 12h. 3h. 18m. 52s. = 8h. 41m. 8s.
when the sun is due east.
The alt. may be found by this proportion ; Rad. : sine 90 zt decl. :: sine
hour from noon : cosine alt. Thus in ox. 1. Rad. : s. 66 32' :: s. 59 45' :
cos. 37 37', the alt. required.
THE TERRESTRIAL GLOBE. *
155
4. At what hours will the sun be due east and west, at every
place on the surface of the globe, on the 21st of March and 23d of
September ?
5. At what hours is the sun due east and west at Lima, on the
22d of December ?
PROB. 58.
The declination and meridian altitude of the sun being given, to find
the latitude ofthefilace*
Rule. MARK the declination on the brazen meridian ; then
count as many .degrees from this mark on the brass meridian, as is
equal to the given latitude, reckoning towards the south, it the sun
was south of the observer, or towards the north, if the sun was
towards the north ; bring the degree where the reckoning ends,
to coincide with the horizon, and the number of degrees the ele-
vated pole is from the horizon, will be the latitude required.
Or, The latitude may be thus found without a globe : subtract the
altitude of the sun's centre (corrected for dip, or height of the eye,
and refraction, if necessaryt) from 90, the remainder is the zenith
* The reason of this prob. is evident from the operation.
f The height of the eye above the level of the horizon, or the sea, and re-
fraction, both tend to elevate the sun above its true height, and therefore
the sum of both must be subtracted from the observed altitude. The fol-
lowing table will answer the learner's purpose sufficiently. If more exact-
ness be required, McKay's Treatise on Navigation, Mayer, or other authors
may be consulted.
ll
dip.
Alt.
refr.
alt.
refr.
alt.
refr.
alt.
refr.
Paral.
in alt.
Sun's semid. S
<s 1
I'O
C'|33' 0"
ji
13' 6"
17
5' 4"
37
1'16"
"
Jan. 1 16'18" t
S 2
1 4
5 32 10
31
12 27
18
2 54
38
1 13
4
Q
25
16 16 S
c ^
1 7
10 31 22
4
11 51
19
2 44
39
1 10
8
9
Feb. 1
16 15 S
S 4
1 9
15 30 35
4
11 18
20
2 35
40
1 8
12
9
25
16 11 >
S 5
2 1
2029 50
4*
10 48
21
2 27
41
1 5
16
8
Mar. 13
16 7^
S 7
2 5
25:29 6
4|
10 20
22
2 20
42
1 3
20
8
Apr. 1
16 2 S
9
2 9
30' 28 23
5
9 54
23
2 14
43
1 1
24
3
25
16 5 5 S
S 12
3 3
35127 41
5
9 8
24
2 8
44
59
28
8
May 1
15 54
S I 5
3 7
40'27
6
8 28
25
2 ' 2
45
57
32
7
25
15 49 <
S 18
4 1
50|25 42
6*
7 51
26
1 56
46
55
36
7
Junel3
15 46 S
Jj 21
4 4
1 24 29
7
7 20
27
1 51
48
51
40
6
July 25
lo 48
c 25
4 8
1 15 22 47
8
6 29
28
1 47
50
48
; 44
6
Augl3
15 50?
S30
5 2
1 30
21 15
9
5 48
29
1 42
55
40
48
6
Sept. 1
15 54 Z
S 35
5 6
1 46
19 51
10
5 15
30
1 38
60
33
;52
5
25
16 .OS
> 40
6
2
18 35
11
4 47
31
1 35
65
26
; 56
5
Oct. 1
16 oS
k 50
6 7
2 15
17 26
12
4 25
32
1 31
70
21
60
4
25
16 8
?60
7 4
2 30
16 24
13
4 3
33
1 28
75
15
64
4
Nov. 1
16 10 v
S70
8
2 45
15 27
14
3 45
34
1 24
80
10
68
3
25
16 15S
I; 80
8 5
3
14 36
15 3 30
35
1 21
85
U 5
80
2
Dec. 1
16 16 S
<: 9-J
9
3 15
13 49
16 ! 3 17
36
I 18
90
J
90
25
16 18 !>
The learner will observe that there are here four tables, separated from
"ach other by the double lines. The 1st contains the dip of the horizon in
156 * PROBLEMS PERFORMED BY
distance, which is north, if the zenith be north of the SUH, or south?
if the zenith be south ; take the sun's declination out of the Nauti-
cal Almanac, or any good table, for the time and place, and ob-
serve whether it be north or south ; then if the zenith distance and
declination be both north or both south, add them together ; but if
the one be north and the other south, subtract the less from the
greater, and the sum or difference will be the latitude, of the same
name with the greater.
Note 1. If the alt. be taken by reflection from a basin of water, &c. al-
lowance must be made for refraction. (See note to prob. 1.)
Example \. On the 17th of October, 1805, the meridian alt. of
the sun's centre was 28 51', the observer being north of the sun ;
required the lat of the place of observation ?
Ans. Here the declination is 9 15' south, which being marked
on the meridian, and 28 5 1' reckoned from this mark towards the
south, the reckoning will end at 38 6', which being brought to
the horizon, the north pole will then be elevated 5i 54', which
shews the lat. to be so many degrees north.
BY CALCULATION.
90 28 51' S. (the sun's alt. at noon) = 61 9' N. the zenith
distance, from which the sun's declination 9 15' S. being subtract-
ed, leaves 51 54' N. the lat. required.
minutes and decimal parts, for the feet in the 1st column corresponding to
the height of the eye. The dip is a vertical angle contained between a hori-
zontal plane passing through the eye of an observer, and a line from his eye
to the visible unobstructed horizon. As this increases the alt. it must be
subtracted ; but added, if a back observation with a Hadley's quadrant or
sextant be used.
The 2cl table contains the refraction in alt. of any celestial body corres-
ponding to the degrees and min. of altitude given in the table. It is adapt-
ed to 29.6 inches of the barometer, and 50 of Fahrenheit's thermometer ;
as tliis increases the alt. of objects, it must likewise be subtracted. It also
affects the distances of the sun and moon, or stars, and must therefore be
allowed for. If the atmosphere, &c. should vary, allowance is to be made
when great precision is necessary. (See tab. 32 of Mayer, or Delambre in
his tables annexed to La Land's Astronomy, where the hor. refr. is 6"2 less
than in Mayer, Delambre making it 32'5S"8.
The 3d table contains the sun's parallax in alt. that is, the difference be-
tween the sun's places as seen from the surface, and the centre of the earth
at the same time. This table, except the two last numbers, is calculated to
every 4th degree. As the parallax always diminishes the apparent altitude,
it must be added to the observed alt. to "find the true, or the alt. observed
from the earth's centre.
The 4th table contains the sun's semidiameter in minutes and seconds,
corresponding to the days of the month opposite, the semidiameter being
the angle under which it appears, as seen from the earth, is necessary to re-
duce the observed alt. of the sun's upper or lower limb to that of its centre.
It is also useful to astronomers to ascertain the exactness of the scale of their
micrometers, by comparison with the measure of the sun's horizontal diam-
eter. This is practised principally in solar eclipses, when the distance of
the cusps, or the versed sine of the uneclipsed part, lias been measured with
the micrometer. It is likewise used in finding the distance of the sun and
moon's centres, when their nearer limbs are brought in contact, &.c. When-
great accuracy is required, proportional parts for the dip, refraction^ parai ,
lax, and semidiameter, may be taken.
THE TERRESTRIAL GLOBE. 157
2. On the 30th of May, 1808, the meridian alt. of the sun's cen-
tre was observed to be 49 25', the observer being south of the
stin ; required the latitude ? Ans. 18 45' S.
BY CALCULATION.
90 49 25' S. = 40 35 ; S. the zenith distance, the difference
between which and the sun's declination 21 50' N. is 18 45' S.
the lat. sought.
Note 2. The table of the sun's declination, and its change for periods of
four years, is given before the table of the lat. of places at the end of the
book.
3. On the 10th of May, 1808, the sun's meridian alt. was obr
served to be 40 south of the observer ; required the latitude ?
4. On the 1 2th of July, 1810, the sun's meridian alt. was ob-
served to be 50 30' north of the observer ; what was his latitude ?
5. On the 24th of February, 1809, the meridian alt. of the sun's
lower limb was 38 40', the observer being north of the sun, and
height of his eye equal 18 feet; required the latitude of the place
of observation ?
By help of the foregoing table, this may be performed thus :
Obs. alt. sun's lower limb 38 40' S. 90 38 51'= 51 V N. zenith dist.
Sun's semidiameter -f- 16 Sun's declin. 24th Feb. 9 3G' S.
Dip for 18 feet 4 Zenith distance 51 9 N,
Refraction 1
Latitude 41 39- N.
True alt. sun's centre 38 5i'
6. On the iOth of December, 1810, the upper limb of the sun
was observed appearing in the south part of the horizon, height of
the eye 1 6 feet ; required the latitude ?
BY CALCULATION.
Obs alt. sun's upper limb 0' S. 90 -|- 53' = 90 53' N. zenith distance
Semidiameter 16 22 54 S. declinatio.u
Dip for 16 feet 4
Refraction 33 67 59' N f latitude.
Depression of the sun's cen. 53' S.
7. May 10th, 1808, in longitude 60 W. the meridian alt. of the
sun's lower limb, by a back observation, was 40 10', the observer
being north of the sun, and height of the eye 27 feet ; required
the latitude ?
BY CALCULATION.
Obs. alt. sun's up. limb 40 10' Sun's decl. 10th May
Semidiameter 16 Variation of decl/
Dip -f 5
Refraction 1 Reduced declin.
Zenith dist. 90 39 58'
True alt. sun's centre 39 5fc/
Latitude 6744'N.
* When the longitude is different from that of Greenwich observatory,
the difference of longitude must be converted into time, and reduced to
that of Greenwich (by prob. 6 ) the variation of declination, during ihis
time, may then be found by this rule ; as 24 hours : hour frojrri noon
158
PROBLEMS PERFORMED Bl
8. At a certain place where the clocks are three hours slower
than at Greenwich^ the meridian alt. of the sun's lower limb on the
21st of March, was observed to be 32 15', the observer being
north of the sun, and the height of his eye 1 7 feet ; required the
place ?
9 Suppose that on the 4th of June, 1812, a ship in longitude
53 E. was distant about three quarters of a mile from land, at noon,
the lower limb of the sun being brought down to the line of sepa-
ration between the sea and land, the alt. was i6 a 19% the observer
being south of the sun, and height of his eye 20 feet ; required the
latitude ?
BY CALCULATION.
Obs. alt sun's lower limb 46
Semidiameter
Dip*
llefraction
True alt. sun's centre 46 16'
Sun's declination 22 27' N.
Variation of decl. 1
Reduced decl.
Zenith dist.
Latitude
22 26 N.
43 41 S.
21 15' S.
reckoned by the meridian of Greenwich :: the daily variation of the sun's
declination : a fourth number, which must be added to,^ or subtracted
from the decl. for the given day at GreenVich, according- as the reduced
time is before or after twelve, and the declination increasing- or decreas-
ing-. If the time be in the forenoon, and the deck increasing, the variation
must be subtracted; but if the time be in the afternoon, the variation must
be added; again, if the reduced time be in the forenoon, and the declina-
tion decreasing 1 , it must be added, but the contrary, if the reduced time
be in the afternoon. Thus in ex. 7. the difference of long = 60 = 4
hours, and as the place is W. of Greenwich, the time reduced to the me-
ridian of Greenwich is 4 o'clock in the afternoon. Now the decl. for the
10th of May is 17 39', and for the llth 17 54', their diff. is 15' increas-
ing; hence 24 h. : 4h. .: 15' : 2' 3o" which must be added, because the
time is in the afternoon and the decl. increasing-, (see table 26 in M'Kay's
Navigator.) When great exactness is required, the decl. must be taken
from the Nautical Almanac, where its daily variation is also given.
If the land intervenes, and the ^
S
5
sun's limb be brought in contact
with the line of separation of the sea
and land, the dip will be considera-
bly increased, and will become great-
er in proportion as the land is ap-
proached. In this case the distance
to the water's edge is to be found ;
with this distance and the height of
the eye above the level of the water,
the dip is found from the annexed
table, or it may be calculated as fol-
lows :
In the annexed figure, let A re-
present the place of the observer,
AB the height of his eye above the
level of the horizon, BI) the diame-
ter of the earth = 7911.2 Eng. miles
or 4171136 feet, (note to def. 8) E
ist. of
nd iii
a mil
Height above the sea in feet. S
5
10 15
20
25 30
35|40 S
J
11'
22' 34'
45'
56' 68'
79>,9 :.- %
*
6
11 17
22
28 34
39 4.) S
-2
4
8
12
15
19 23
27 So
1
4
6
9
12
15 r,
20 (23 ?
1 i
3
5
7
9
12 14
1619 s
1
3
4
6
8
10 11
14 15 S
2
2
3
5
6
8 10
11J12 S
2 f
2
3
5
6
7 8
9 10 J
3
2
3
4
5
6 7
88w
3 it
2
3
4
5
6! 6
7 7S
4
2
3
4
4
5i 6
7 7S
5
2
3
4
4
5J i
66?
6
2 3
4
4
5 5
66?
THE TERRESTRIAL GLOBE. 159
PROB. 59.
The length of the longest day at any place , not within the polar circles,
being given, to find the latitude of the place.
Rule. BRING the beginning of cancer or Capricorn to the brass
meridian (according as the place is in N or S. latitude) and set
the index to twelve ; turn the globe westward until the index has
the place where the sea and land are separated, AE (or
BE, which is nearly equal to it) the distance from the
observer to E ; draw BH from B perpendicular to BD,
to represent the horizon, and AK parallel to it. Then
the angle HLF = KAE (29. E. 1.) is the dip or depres-
sion of the object below the horizon, and is equal ACG,
each of the lust being- complements of CAG to 90 ; AE
being- produced to F, and from C the centre CG being-
drawn perpendicular to EF. Again AB X AD=AE X AF
(Eucl.cor. prop. 36. b. 3) and therefore AE : AB :: AD :
AF; (16 E. 6.) hence EF = AF AE is given, and as
CG bisects EF in G (3 E. 3) AG = AE-f-EG is given.
Now in the right angled triangle ACG, the two sides AC, AG are given, to
find the angle ACG, the dip required. Thus AC : AG :: Had. : sine ACG.
The following method is no less accurate, and in practice extremely sim-
ple ; AL differing very little from BE, the distance of land in miles, and the
angle ALB = KAE or HLF, that is equal the dip required, we have this pro-
portion, AL : AB :: Rad. : sine of the dip. Thus let the observer's distance
BE or AL = 2 sea or geographical miles = 12151 feet nearly (a sea or geo-
graphical mile being = ^80X69.04 __ 88X69 04 __ 6 Q75.52 feet, allow-
ing 69.04 Eng. miles to a deg. See notes to def. 8 and prob. 35) and
height of his eve AB = 20 feet. Then,
As AL 12151 , - 4.0846120 This rule in words is, as the distance
To AB 20 - - 1.3010300 from the observer to -where the sun
So is radius, 10.0000000 touches the horizon, is to the height of
his eye, so is radius, to the sine of the
To sine dip. 5' 4(/' 7-2164180 angle of depression or dip.
The dip may be also found thus : join CE, then in the triangle ACE
there are given the three sides AC, AE ( = BE nearly) and EC to find
the angle BAE, whose complement is the depression required.
But the most correct manner of calculating the dip is as follows : Let.
EM be drawn perpendicular to AC. Now as the length of the arch BE,
which is the measure of the angle ACE is given, in geographical or sea
miles, each of which corresponds to a minute of a degree, the angle ACE
is therefore given, arid hence CM, ME are each given, and as CA is
given, AM is therefore given ; hence in the triangle AME the angle
AEM, which is tlie true dip, is likewise given.
The difference between AM and AB, or B\I, is what the surface of the
earth at E falls below the horizontal or true level at B ; and hence this
latter solution may be useful to the practical surveyor, engineer, &c. This
level varies as the square of the distance nearly. (See quest. 4 in the
Diary for 1795 published by Hutton.)
If from the centre B, with the distance BI, a circle SHI?j be describ-
ed, and S be the sun's place, then SH will represent his true altitude, HJ.
the dip, when the visible horizon is unobstructed, as in the 1st table, H
the dip when the sun's limb is brought in contact with any other part of
the visible horizon, in a vertical circle, as at E : when tin = US the dip
is equal to the true altitude, in which case if the sun's image be made to
coincide with its reflected image from the water or any horizontal reflect-
160 PROBLEMS PERFORMED BY
passed over as many hours as are equal to half the length of the
day ; elevate or depress that pole until the sun's place (cancer or
Capricorn) comes to the horizon ; the elevation of the pole will
then shew the latitude.*
Note 1. The prob. may be performed in the same manner for any other
day, by bringing 1 the sun's place to the meridian, and proceeding as above.
Or, Bring the middle of the analemma to the brass meridian,
and set the index to 1 2 ; turn the globe westward until the index
points out the hours, &c. as before ; elevate or depress the pole
until the day of the month coincides with the horizon ; this eleva-
tion will give the lat. required.
ing- surface, the angle measured on the quadrant or sextant will be double
the true alt. of the sun ; the angle of incidence being equal to the angle
of reflection, or the angle formed by a line from S. to E. and ME pro-
duced, will be equal the angle AEM, &c. See note to prob. 1. Hence
we have here several methods of finding the lat. on land near the sea
shore, a river, lake, &c. or from any reflecting surface.
If no land intervenes, then the angle ACI is the dip, as given in the 1st
table, AI being drawn to the extremity of the visible unobstructed horizon,
in which case EF will vanish, E will coincide with I, CG will become = CJ
= the semid. of the earth at right angles to the tangent AI at the point I.
(cor. 16. Eucl. 3.) whence we have this proportion ; /YC : CI :: Rad. : sine
ACI the dip required ; or, as AC : CI :: Rad. : sine CAI the complement of
which is the required depression.
More useful observations might be made here, but our contracted lim-
its would not permit : what we have said is however sufficient to give the
learner an idea how useful geometrical principles are in inquiries of this
nature, and how necessary their study is for those who wish to be more
than superficially acquainted with the nature and foundation of the most
useful arts and inventions in general.
* This prob. is calculated thus : the complement of the sun's declina-
tion, the lat. reckoned from the elevated pole to the horizon, and the
included angle, or the supplement of the hour angle or half the length
of the day, form a right angled spherical triangle, the circular parts be-
ing the sun's declination, the comp. of the included angle between the
brass meridian and the meridian passing through the sun's place, and
latitude, of which this angle is the middle part (see Simson's Trig, at the
end of his Euclid, pa. 26) then by Napier's first rule, rad. X co. sine of
the included angle at the pole tangt. decl. X tang, latitude, whence
16 E. 6.) tangt. sun's decl : rad :: co. sine of the angle included be-
tween the meridian passing through the sun's place and brass meridian :
tangt. latitude. Thus in ex. 1. the hour angle = 7 h. 30 min. ;= 112 30' ;
the supplement of this = 180. 112 30' = 67 3(/ = the angle includ-
ed between ths meridian passing through cancer and the brass meridian,
vhc sun's declination being here greatest = 23 28' ; hence tangt. 23 28' :
rad. :: co. sine 67 30' : tang, latitude 41 24' required.
in the same manner may the lat. be calculated for any other time, tak-
ing the sun's declination for the given day, instead of his greatest decli-
nation 23 28'.
The reason of the rule is evident ; for when the globe is rectified to the
lat. and the sun's place brought to the meridian, if then the globe be
turned on its axis until the sun's place coincide with the horizon, the in-
dex in this revolution will pass over half the length of the day ; hence,
-sice versa, if the distance between the brass mer. and where the sun cuts
the horizon be made equal to half the length of the diurnal arch, by ele-
vating or depressing the pole, the elevation thus found must be the lati-
tude. The s-ime reasoning will answer for any other prob. performed in
a similar manner.
THE TERRESTRIAL GLOBE. 161
tixamfile \ . In what degree of north lat. and at what places is
the length of the longest day 1 5 hours ?
Ans. In lat. 41 42', and at all those places situated on or near
that parallel.
Note 2. The prob. may be performed much more correctly by help of the
equator than of the hour circle.
2. In what degree of south lat. and at what places is the long-
est day 131 nours ?
3 In what degree of north lat, is the length of the longest day
twice the length of the shortest night ?
4. In what degree of south lat. is the longest day three times the
length of the shortest night ?
5. In what lat. is the 10th of May 14 hours 15 min. long ?
Note 3. The minutes are marked on the equator of Gary's large globes.
The lat. may be here N. or S.
6. In what lat. north does the sun set at 7 o'clock, on the 10th
of August ?
7. In what lat. south does the sun rise at 5 o'clock, on the 31st
of January ?
8. In what lat. N. is the night of the 4th of July 9 hours long ?
PROB. 60.
Given the sun's declination and amplitude^ to Jind the latitude.
Rule. ELEVATE the north pole to the complement of the am-
plitude, screw the quadrant of alt. in the zenith, and bring the be-
ginning of aries to the brass meridian ; then bring the degree on
the quadrant of alt. which is equal to the declination, to coincide
with the equator, and the degree cut on the equator, reckoned,
from aries, will be the latitude required.*
* Here the complement of the decl. on the quadrant, the complement of
the altitude reckoned on the brass meridian, and the lat. reckoned on the
equator, form a right angled triangle, and hence the lat. is thus found ;
S. ampl. : rad. :: s. decl. : cos. latitude ; thus, in the 1st ex. S. ampl. 32 :
rad. :: s. decl. 23* 28' : cos. lat. 41 1'.
tercepted between the horizon and the meridian passing through the sun*
place, form a right angled sp. triangle, and the angle included between the
equator and amplitude, in this triangle, is equal to the complement of the
latitude, whence in this triangle it will be S. ampl. : R. :: s. decl. : cos. lat.
the same as above. If we now conceive the sides of the triangle which re-
present the decl. and ampl. to be produced to the brass meridian, another
triangle formed from the compl. of the decl. the compl. of the amplitude,
and the latitude, reckoned from the pole to the horizon, will be delineated
on the globe, from which the first rule above is formed. The second rule
is manifest from the globe being rectified to the lat. &c.
From the triangie formed at the horizon by rectifying the globe, &c. other
methods may be deduced of solving the prob. Thus, bring the beginning-
of aries to the brass meridian ; from aries on the equator reckon as many
degrees as are equal to the decl. then with the amplitude in the compasses,
and one foot in this point, cross the equinoclional sohye, und mark the de-
w
162 PROBLEMS PERFORMED BY
OR THUS,
Find the sun's place corresponding to the given declination, anQ
bring it to the eastern or western part of the horizon (according as
the eastern or western amplitude is given) elevate or depress the
pole until the sun's place coincides with the given amplitude on the
horizon ; the elevation of the pole will be the lat. sought.
Example 1. The sun's amplitude was observed to be 32 from
the east towards the north on the 2 1st of June ; required the lat. ?
Am. 41 N nearly.
2. In May when the sun's declination was 18 north, the rising
amplitude was observed to be 30^ from the east towards the
north ; required the lat. ?
Ans. 540 N.
3. When the sun's declination was 10 south, his setting ampli-
tude was 1 3 from the west southward j required the latitude ?
PROB. 61.
Given two observed altitudes of the sun, the time between them, and
the sun's declination, to find the latitude.
Rule. TAKE the complement of the first alt. from the equator
(or any great circle on the globe which is divided into degrees, &c.)
in your compasses, and with one foot in the sun's place, and a fine
pencil, or a pen with ink, in the other, describe an arc on the sur-
face of the globe ; then bring the sun's place to the meridian, and
set the index to the hour at which the first altitude was taken, or
mark the degree of the equator under the brass meridian ; turn
the globe eastward until the index has passed over as many hours
as are equal to the time between the two observations, or until the
equator has passed over as many degrees, &c. as the elapsed time
converted into degrees ;* (by prob. 6) then under the sun's de-
gree cut on it ; elevate this pole to the degree, and screw the quadrant of
alt. in the zenith ; extend the quadrant over the degree of decl. marked on
the equator, and the degree then cut on the horizon, reckoned from the
nearest pole, will be the complement of the latitude, as is evident. Or if the
decl. be marked on the colure, from aries, and with the ampl. in the com-
passes as before, and with one foot on the colure where the reckoning ends,
cross the equator with the other, and mark the degree thus cut ; bring this
degree to the brass meridian; screw the quadrant over it, and bring both
poles to the horizon ; extend the quadrant over the degree of decl. on the
colure ; the degree then cut on the horizon will he the complement of the
lat. as before. This method follows from the same principle.
* The elapsed time is found to a minute of time on Cary's large globes,
on the equator. Any two points in the parallel of the sun's declination for
the time, distant from each other by the interval of the elapsed time, may be
taken. If the declination varies much during the elapsed time, the comple-
ment of the second alt. must be set off from the parallel in which the sun is
nt that time, the variation of the declination being found by the proportion in
the latter part of the note to prob. 49.
To perform this prob. by calculation, join the zenith, or point where the
two circles representing the complements of the nit. intersect each other,
und the apparent places cf the sun at each alt. represented on the paia'.iel of
THE TERRESTRIAL GLOBE.
163
clination on the brass meridian mark the place on the globe ; take
the complement of the second altitude in your compasses, and
with one foot in this mark, describe another circle intersecting the
former ; the point of intersection will be the zenith of the place,
which being brought to that part of the brazen meridian which is
numbered from the equator towards the poles, will give the lati-
tude required.
Note. The respective altitudes may be corrected in this prob. as in the
foregoing problems, when necessary.
Example 1 . June 4, 18:0, in north latitude, the corrected alti-
tude of the sun at 29 minutes past iO in the forenoon, was 65 24',
and at 3 1 niin. past 1 2, the correct alt. was 74 8 r ; required the
latitude ?
Ans. The sun's declination was 22 27' N. the elapsed time was
2h. 2m. = 3Qo so' ; the complement of the first alt. was 24? 36',
of the 2d, 15 52', and the lat. sought 36 57' N.
declination, through each of the sun's places, and the zenith, let meridians be
diMwn with a line pencil, by means of the brass meridian, then will the figure
be projected on the surface of the globe. In this figure it will be seen that
the complement of the latitude and the sun's declination at each alt. added
to or subtracted from 90, form with the complements of the altitudes, two
triangles respectively, and that the angles at the pole, included by the com. of
the sun's decl. and com. of the lat. reckoned on the meridian passing- through
the zenith, are the hour angles counted from 12. From either of which tri-
angles, two sides and an angle being given, the third side, which is the
complement of the latitude, may be found by the following proportions, first
letting fall a perpendicular on the meridian passing through the zenith or
given lat. from the sun's place at either of the altitudes. Rad. : cos. hour
angle before or after 12 :: tangent 90 ^t the sun's decl. : tangt. x or distance
between the pole and the perpendicular; and sine 90 it. the decl. : cos x. -.-.
sine alt. : y the distance between the zenith and perpendicular ; then xy
= complement of the lat. required. (See Emerson's Trig. b. 3. sect. 4. case
3, oblique spher. tri.) I'o know whether the declination is to be added or
subtracted ; when the lat. and decl. are of the same name, add ; when they
are of different names, subtract. In the above solution either of the lati-
tudes above will answer. (See prob. 61.) If the watch be not adjusted so
as to give the time at which each of the observations was made, though suf-
ficiently correct to measure the elapsed time, then the following method will
answer. In the annexed figure, let P represent the pole, Z the zenith, EQ
the equator, HO the horizon, A and B the two
places of the sun when the altitudes were taken ;
then in the triangle BPA there are given AP,
BP the complement of the sun's decl. and the
angle BPA the elapsed time, or the time be-
tween the two observations converted into de-,-,-.
grees to find the side AB, and the angles ABP -"-I
or BAP. In the triangle AZB there are given
AZ the complement of the first alt. BZ the
com pi. of the second alt. and the side AB, to
find the angles ABZ or BAZ, and from thence
the angles ZAP or ZBP ; then in the triangle
E
more simple. If the altitudes be equal, or AC = BD, and the sun's decline-
164 PROBLEMS PERFORMED BY
2. Given the sun's declinatiort 12 16' N. his alt. in the fore-
noon c<t lOh 24m. was 49 9', and at Ih. 14ra. in the afternoon his
alt. was 51 59'; required the latitude ?
Ans. 47 20' north.
3. The sun's declination being given 1 1 17' N the alt. at 10h.
2m. in the forenoon was 46 5.?', and at 1 Ih. 27m. in the forenoon
the secona alt was 54 9' ; required the latitude ?
Ans. 46<> 27 ; north
4. Being at sea when the sun was on the equator, I observed
that at 1 o'clock, P. M the correct alt. of the sun's centre was 28
53', and at 3 o'clock, P. M. the alt. was 20 42' ; required the
latitude ?
Ans. The lat. was 60 north.
5. If on February 24, 181 1, at 38m. past 12 in the afternoon,
the correct alt. of the sun's centre be observed 36 5', and 46m.
past 2, the alt. be 24 49', the lat. is required ?
6 Oct. 17, 1820, at 32m past 12, the alt. of the sun's lower
limb being supposed equal to 28 32', and at 41m. after 2, the 2d
alt. equal 9 25', height of the eye 12 feet; required the lati-
tude? An*. 51 31'N.
PROB. 62.
Given the sun's declination^ his altitude, and the hour of the day> to
Jind the latitude.
Rule. FIND the sun's place in the ecliptic ; from this place
with the complement of the alt. describe an arc (as in the last
prob.) bring the sun's place to the brass meridian, and set the in-
dex to 12 ; then if the time be in the forenoon, turn the globe
eastward, but if in the afternoon, westward, as many hours as the
given hour is before or after twelve ; the degree on the brazen
meridian, cut by the arc before described, will be the latitude re-
quired.
The examples given in the foregoing prob. will answer this,
taking one of the altitudes and time corresponding, instead of both
altitudes. If the lat. found by making use of both altitudes sepa-
tion'remain nearly equal (aA = 6B) then the middle time between the ob-
servations is the time of his being on the meridian. If this prob. be per-
formed on shipboard, and that the ship is under sail, an allowance must be
made for the alteration in lat. (See McKay, Blunt, Norey, Moore, or Robin-
son's Navigation, or the principles of navigation in Emerson's Math. Princ.
of Geog. prop. 17, or Citizen Dulague's Lessons on Navigation, revised by
Cit. Prudhomme of Rouen.) McKay remarks, that Hues in his Treatise on
the Globes, published in 1594, solved this prob. on the globes ; the substance
of Keith's solution in prob. 52 of his Treatise on the Globes is the same as
the above, and differs little from Fuller's solution of the same prob. in his
Treatise on the Globes, published in Dublin, in 1732, prob. 35 astron. See
other authors mentioned by McKay in pa. 158 of his Navigator, Amer. edit.
The methods given in most of the books on Navigation, are but approxima-
tions, and consequently the answers obtained by such methods generally dif-
fer something from those found by the above, and it is very seldom that the
altitudes and times are given correct, the examples mostly given at random.
Iqeing seldom truly limited.
THE TERRESTRIAL GLOBE.
165
rately in the same example, agree with each other, the question is
truly stated, otherwise not.
In ex. 2, prob. 61, the lat. is found from both
altitudes separately, thus : let P, in the annexed
figure represent the pole, A and S, the sun's ap.
places at each of the altitudes, PZB the meridian
passing through the zenith Z or the given place, 1?
AZ the complement of the first altitude = 40 51', |
APZ the hour from noon= 12h. lOh. 24m. = \
Ih. 36m. = 24, and AP the complement of the
'sun's declination = 77 44'. For the second alt.
SZ = its comp. 38 1', SP comp. decl. = 77 44', and SPZ hour angle
-after noon = Ih. 14m. = 18 30'.
Rad. 10.0000000 Cos. PA 77 44' 9.3272811
Cos. APZ 24 9.9.607302 Cos. PB 76 37' 9.3644852
Tangt. PA 77 44' 10.6626887 Cos. AZ 40 51' 9.8787656
Tang. PB 76 37'
10.6234189
Cos. BZ 34 3G 1
PB BZ = ZP = 42 7' . . 90 42 7' = 47 53',
the lat. for 2d observation.
Rad. 10.0000000
Cos. SPZ 18 30' 9.9769566
Tang. PS 77 44' 10.6626887
Cos. PS 77 44'
Cos. P6 77 i>'
Cos. SZ 38 1'
Tan. Pd 77 5'
9.6396473
Cos. bZ 34 C
5' and 90 43 5' = 46 55'.
19.2432508
9.9159797
9.3272811
9.3493429
9.8964334
19.2458763
9.9185952
The examples given to illustrate the last problem, were selected from
McKay and Hamilton Moore's Treatises on Navigation. To point out the
errors unnoticed in most of these, calculated as directed in those authors,
the preceding calculation was given. Mariners and others will therefore
judge for themselves, whether the common method of double altitudes, at-
tended with so much uncertainty, so much labour and tedious calculations,
be of any real advantage. The method given in this prob. for finding the
lat. will answer every purpose of the double altitudes, with only a single al-
titude and the time being given.
For a further illustration of this valuable problem, the following examples
are given.
In north latitude, when the sun's declination was 13 45' X. his alt. at 81i.
39' 33" in the forenoon was 36 53', and at 9h. 39' 33", or.one hour after, the
alt. was 45 53' ; the lat. by either of the Observations independent of the
hther, is required ?
Latitude from the 1st altitude.
Here AP = 76 15', AZ = 53 7', the hour g
angle, APE = 12h. 8h. 39' 33" 3h. 20m.
27s. = 50 9 7' nearly ; hence,
Rad. 10.0000000
Cos. 50 7' 9.8070114
Tang. 76 15' 10.6113688
Tang:. PB 69 7<
10.4183802
& :R
Cos. 76 15'
Cos. PB 69 7'
Cos. 53 1'
Cos. BZ 25 49'
Z I
9.3760034
9.5520184
9.7782870
19.3303054
9.954302ft
166 PROBLEMS PERFORMED BY
Hence 69 7' 25 49' = 43 18' and 90 43 18' = 46 42', the lati
tude required.
Latitude from the 2c? altitude.
Here SP = 76 15', SZ = 44 7', the hour angle SJtf= I2h. 9h. 39m
33s. = 2h. 20m. 27s. == 35 7' nearly.
Rad. 10.0000000 Cos. 76 15' 9.S760034
Cos. 35 7' 9.9127440 Cos. Pb 73 2V 9.4571618
Tang. 76 15' 10.6113688 Cos. SZ 44 7' 9.8560784
Tang. P5 73 21' 10.5241128 19.3132402
Cos. Zb 30 3' 9.937^368
Hence 73 21' 30 3' = 43 18' and 90 ZP 43 18'= 46 42', the
latitude required the same as above.
This last example is taken from Dulague's Lessons on Navigation, revised
by Prudhomme (pa. 196) the answer by the double alt. being there given
as above. The learner will there find more examples where the time is
given to seconds, without which this prob. by any method, will seldom suc-
ceed, as 30" of time, if rejected, correspond to 7' of a degree, and this is
the principal reason why the examples in McKay, Moore and others, will
not agree with accurate calculation. The learner will perceive that the lat.
is here found by either of the altitudes, whereas the common method re-
quires both. Whence the principal advantages of this method are, 1st.
That there is nothing to do with elapsed time or variation of declination, ex-
cept in reducing the declination to the meridian of the place. 2d. That
there is no alteration of latitude from the vessel's sailing during the time
between the observations. 3d. That the operation consists in the two sim-
ple proportions given above (which become more simple when the sun is
due east or west, when on the equator, when the observation is made at 6
o'clock, &c. or by tables which may be easily adapted to it.) 4th. That the
observation may be made at any time of the day, &c. The same method
will answer by having the time and the altitude of any star when the de-
clination is known, the time of its passage over the meridian being also giv-
en, which is found by prob. 8, part 3d. for the difference of time will be
the hour angle APB or SP6, in the above fig. A or S being the star's place.
The principal difficulty in the practice of this prob. consists in determin-
ing the time exactly, and hence the observer ought to be provided with a
good time piece well regulated ; when this is not the case, the method by
double altitudes with the intermediate time, will then become useful, as the
watch requires no regulation to measure the elapsed time. See the fore-
going problem.
PROB. 63.
Given the sun's amplitude and ascensional difference^ tojind the lat-
itude and sun's declination*
Rule. ELEVATE the pole as many degrees above the horizon
as are equal to the ascensional difference ; screw the quadrant oi'
* The learner will observe, that when the prob. is performed by the first
rule, the ascensional difference reckoned on the brass meridian, the ampli-
tude pn the quadrant, and the declination on the equator, form a right an-
gled triangle ; and that the angle included between the quadrant of alt. and
brass meridian, or between the amplitude and as. diff. is equal to the com-
plement of the latitude : hence, by Napier's rules, we have these propor-
tions ; Rad. : co. tangt amplitude :: tang, ascen. diff. : sine latitude. And
for the decl. Cosine as. diff. : cosine amplitude :: rad. : cosine declination.
THE TERRESTRIAL GLOBE. 167
alt. in the zenith, and bring the beginning of aries to the brass me-
ridian ; then number on the quadrant of altitude from the com-
plement of the sun's amplitude, and move the quadrant until that
number cuts the equator ; the degree then cut on the horizon,
reckoning from the east or west points, will be the latitude, and
the degree cut on the equator will be the sun's declination.
Or, If the day of the month be given, find the sun's place in the
ecliptic (by prob. 8) bring this place to the brass meridian, and
then mark the sun's right ascension on the equator ; reckon as
many degrees from this on the equator as are equal to the ascen-
sional difference, and elevate or depress the pole until this point
comes to the horizon ; the elevation of the pole will then be the
latitude required. The decl. will be found as in prob. 8.
Note. In north lat. when the sun is in the first and second quarter of the
<4cl5ptic, the oblique is less than the right ascension, in which case the as,
diflr. is to be subtracted from the right ; but when the sun is in the 3d and
4th quarters, the oblique as. is greater, and hence the as. diff. is to be added
to the right as. to find the oblique. When the sun has north decl. he is in
the 1st or 2d quarters of the ecliptic, but in the 3d or 4th when he has south.
The ascensional difference is always equal to the time the sun rises or
sets before or after 6 o'clock, converted into degrees.
Example 1. On the 10th of May, the sun's amplitude at rising
was 23 45', and the ascensional difference 16; required the lati-
tude and declination i
Ans. Lat. 40 40' N. decl. 17 47' nearly.
2. On the 4th of April, the sun's amplitude was 8, and the as-
censional difference 4| ; required the latiu.de and declination ?
3. On the 21st of June, the sun's amplitude was 39, and the as-
censional difference 31^; required the latitude ?
4. On the I Oth of August, the sun's amplitude was 22, and
he rose 3 minutes before 5 ; required the latitude ?
5. On the 20th of October, the sun sets 17 minutes after 5, his
setting amplitude being 1 6 ; required the latitude ?
PROB. 64.
Given the sun's declination and hour at east) tojind the latitude.
Rule. ELEVATE the pole to the sun's declination, and screw
the quadrant of alt. in the zenith ; then reduce the time after 6, of
the sun's being due east, into degrees and minutes, and reckon the
same number on the horizon from the east towards the south ;
bring the quadrant of altitude to that degree on the horizon, and
If we now perform the prob. by the second rule, it will be seen that when
the sun's place is made to coincide with the horizon, and likewise the ob-
lique ascension, by elevating or depressing the pole, &c. that the amplitude,
ascensional difference, and sun's declination, form a right angled triangle, si-
milar to that before described, and that the inclination of the equator with
the horizon, or the angle formed by the amplitude and asccn. diff. is equal
to the complement of the lat. hence the calculation will be exactly the same
as above. From this lust method the reason of the foregoing method i<*
therefore manifest.
168 PROBLEMS PERFORMED B\
the degree then cut an the quadrant by the equator, will be the
complement of the latitude required.*
Note. Unless the sun's decl. be of the same name with the latitude,
he cannot be seen in the east.
Example 1. On the 10th of May, the sun was observed in the
cast at 7 o'clock in the morning ; required the latitude ?
Ans. 51 N. nearly
2. On the 4th of July, I observed the sun due east at 8 o'clock
in the morning ; required the latitude ?
3. On the 21st of June, the sun was observed due east at 7
hours 30 minutes in the morning ; required the latitude ?
PROB. 65.
Given the surfs declination and azimuth at six^ to Jind the latitude
and altitude.
Rule. ELEVATE the pole above the horizon as many degrees as
are equal to the complement of the given azimuth, screw the
quadrant of altitude in the zenith, bring the beginning of aries to
the brass meridian ; then from on the quadrant reckon the com-
plement of the sun's declination, and bring that degree to the equa-
tor ; the degree then cut on the horizon by the quadrant, reckoned
from the N. or S. will be the complement of the latitude, and the
degree of the equator cut by the quadrant will be the sun's altitude
at six.f
* Here the learner will observe, that the complement of the alt. on the
quadrant, the degrees corresponding- to the time after the sun was due
east, and that part of the equator between the edge of the quadrant
and the horizon, form a right angled spherical triangle, and that the an-
gle formed by the equator and horizon, is the complement of the declina-
tion : hence to find the lat we have, from Napier's rule, this proportion ;
Tang, sun's decl. : Had. :: sine hour after 6 : Co. Tan. lat. Thus in ex. 1.
Tan. decl. 17 3i/ : R. :: sine 15 : Co. T. lat =50 52'.
Or thus, If we suppose the globe elevated to the lat. the quadrant of alt.
screwed in the zenith, and extended over the sun's place so as to cut the
east point of the horizon ; then the angle formed by the brazen mer and
the merid. passing through the sun's place, is equal to the time from the
sun's being due east to 12, converted into degrees, the complement of
which to 90 is the distance from the meridian passing through the sun's
place, to the east point of the horizon, reckoned on the equator, or the
time from 6 until the sun is due east . this time being therefore given,
and likewise the sun's decl. the angle formed by the equator and the
quadrant of alt. at the east point, being equal to the latitude, is likewise
given. The right angled triangle thus formed is solved by the above pro-
portion, which shews the truth of the method.
f The .globe, quadrant, &c. being placed as directed in the rule, it will
then be seen, that the complement of the azimuth, reckoned on the brass,
meridian from the equator, the sun's decl reckoned from the zenith on
the quadrant, and the sun's alt. reckoned from aries on the equator, to
the point where it is intersected by the quadrant, form a right angled
triangle ; and that the angle formed by the complement of the azimuth
reckoned on the brass meridian, and the decl. reckoned on ihe quadrant,
is equal to the latitude : Whence by Napier's theorems ve have these
proportions ; Had. : co. tang. decl. :: co. tang. azim. : cos. latitude. And
THE TERRESTRIAL GLOBE. 169
' Mxamjilc \. The sun's azimuth at 6 o'clock, on the 2 1st of June,
was 71 30' from the north ; required the latitude and sun's alti-
tude ?
Ans. Lat. 40 N. and the sun's altitude is 1 5 nearly.
2. The sun's azimuth on the 10th of May, at 6 o'clock in the
morning, was 78-1 frona the north ; required the latitude and al-
titude ?
3. The sun's azimuth at 6 o'clock in the evening, on the 20th of
April, was 82i from the north ; required the latitude and altitude ?
PROB. 66.
Given the sun's declination and altitude at east, to find the latitude.
Rule. ELEVATE the pole to the complement of the sun's given
altitude, screw the quadrant in the zenith ; bring the beginning of
aries to the brass meridian ; reckon on the quadrant of altitude
from the horizon upwards, or from 0, as many degrees as are
equal to the sun's declination ; bring the point where the reckon-
ing ends to the equator, and the degree cut on the equator will be
the complement of the latitude sought.*
Sine azim. : rad. :: cos. decl. : cosine altitude. Thus in the first example
above, Rad. : co. t. 23 28' :: co. t. az. 71 30' : cos. lat. = 39 35'. Again
S. az. 71 30' : rad. :: co. s. decl. 23 28' .- cos. alt. 14 42'.
If we suppose, as in the foregoing problem, the globe to be elevated to
the latitude, the sun's place being then brought to the meridian, the index
set to 12, and the quadrant of altitude screwed in the zenith, and that
the globe is turned on its axis eastward, until the index points to six ;
and that the quadrant is extended over the sun's place, we shall then have
the azimuth on the horizon, the complement of which with the sun's de-
clination, and altitude marked on the quadrant, will form a right angled
spherical triangle as before. And as the meridian passing through the
sun's place always coincides with the east point of the horizon at 6, the
angle formed by the sun's declination and complement of the azimuth, on
the horizon, will be always equal to the latitude. The triangle thus
formed being in every respect equal to that described above, is therefore,
calculated in the same manner, and shews no less the truth of the rule
than the method of forming it.
The learner may also observe, that in the triangle formed according to
the above rule, of the sun's decl. azimuth at 6, altitude at 6, or the lati-
tude of the place, any two being given, the rest are given, and may bft
found by the globe in the same manner as the above. Thus the sun's decl.
and altitude at six being given to find the latitude, we have this propor-
tion ; Sine decl. : rad. :: sine all. : sine latitude. As these right angled
spherical triangles are solved by Napier's Theorems, being the most gen-
eral and the simplest that has ever been discovered, the learner should be
well acquainted with their use, and also with the practice gf the 16th pro-
position of the 6th book of Euclid, in alternating, inversing, &c. the differ-
ent proportions, as with very little labour man\ new and useful conclusions
will thus arise, and on which the learner may try his own invention, in ex-
ercising them on the globes.
* Here the quadrant of alt. the brass meridian, and the equator, form
a right angled spherical triangle, the three sides of which are the com-
plement of the sun's alt. on the brass meridian, the complement of the de-
clination on the quadrant, and the complement of the latitude en the equa-
X
170 PROBLEMS PERFORMED BY
Or, With a pair of compasses take as many degrees as are equal
to the given alt. from the equator ; bring the beginning of aries to
the horizon ; with one foot of the compasses in the point aries ex-
tend the other towards the zenith ; then elevate or depress the
pole until the other point of the compass extends to the parallel of
the sun's declination ; the elevation of the pole will then be the
latitude required ; which will be of the same name with the sun's
cleclinatian. Or the quadrant of alt. being extended from aries (or
the east point of the horizon) through the point where the compass
cuts the parallel of decl. will point out the lat. on the equator.
tor; to find the latter of which, we have tins proportion ; Sine altitude :
Had :: sine -decl. : sine lat. Thus in ex. 1, sine 29 : Rad :: s. 17 3!>' : s.
lat. =38 45'.
To understand how the rule lias been formed, the pole must be elevated
to the supposed latitude, the quadrant of alt. screwed in the zenith, and
then brought lo coincide with the east point of the horizon ; the sun's place
corresponding to his declination being then brought to coincide with the
graduated edge of the quadrant, it will be seen that the meridian passing
through the sun's place or the sun's declination, the number of degrees
from where this meridian cuts the equator to the east point of the hori-
zon, and the sun's alt. form a right angled triangle, and that the angle
contained by the equator and quadrant is equal to the lat. Hence we have
this proportion; s. alt. : Rad. :: s. decl. : s. lat. being exactly the same as
the above. Moreover the sides of this latter triangle, representing the de-
clination and altitude, being produced to the brass meridian, will form
another right angled triangle, whose sides will be the complement of the
altitude, the compl. of the decl. and the compl. of the latitude ; this latter
triangle being that formed by the above rule, shews how the rule itself
was formed. What is here said will be useful in assisting the learner's
invention in investigating new rules or problems, and the teacher will find
the advantage of thus teaching Practical Astronomy on the globes.
The second rule is but representing the triangle, found by the latter
method on the globe. It may likewise be represented on the brass meri-
dian, the equator, and quadrant of alt. thus ; from the point aries count
on the meridian passing through it, as many degrees as are equal to the
decl. from the point where the reckoning ends, with the number of de-
grees equal to the alt. in the compasses, extend the other leg to the equa-
tor, and tlTe same triangle as above will be formed; and the angle formed
by the equator and decl. will be the lat. as before. Hence we have ano-
ther method of solving the prob. by the globe as follows : bring both poles
to the horizon, screw the quadrant of alt. over the point aries ; turn the
globe westward until the point cut on the equator by the compasses (with
the extent of the alt.) comes to the brass meridian ; extend the quadrant
over the sun's decl. marked on the equinoctial colure, and the degree cut
by the quadrant on the horizon, reckoning from the nearest pole, will be
the lat. required. The decl. might also be set off from aries on the equa-
tor, and with one foot of the compasses in this point, intersect the equi-
noctial colure with the other (the compasses being extended to the lat.)
the triangle thus formed will be the same as the foregoing, and the angle
formed by the colure and sun's alt. will be equal to the lat. If then the
pole be elevated to the degree of the colure cut by the compasses, the
quadrant screwed in the zenith and extended over the decl. marked on
the equator (aries being brought to the brass mer.) the degree then cut
on the horizon, reckoned from the nearest pole, will be the lat. requined,
which will alwavs be of the same name with the decl.
THE TERRESTRIAL GLOBE. 171
Example 1. The sun's altitude when east was observed equal to
^9, and his declination 17 39' N. required the latitude I
Ans. Lat. 38 43' north.
2. On the 21st of June, the sun's alt. when east was observed
equal to the sun's declination ; required the latitude ?
3. On the 20th of April, the sun's alt. when west was 10 ; re-
quired the latitude ?
4. The sun's altitude when east, on the 20th of October, was
observed equal to 15; required the latitude ?
PROS. 67.
Given the sun's azimuth and altitude at six, to find the latitude and
sun's declination.
Rule. RECKON on the equinoctial colure, from the equator, as
many degrees as are equal to the altitude, and on the equator from
aries as many degrees as are equal to the complement of the azi-
muth ; bring the point on the equator where the reckoning ends to
the brass meridian ; bring both poles to the horizon ; screw the
quadrant of alt. in the zenith, and extend it over the degree marked
on the colure ; then the degree cut on the horizon reckoned from
the west, will be the lat. and the number of degrees on the quad-
rant, between the colure and the equator, will give the declination.
Note. If the degrees be reckoned on the equator from aries westward,
then the degree cut on the horizon, reckoned from the east, will be the la-
titude. This is the most convenient manner of performing the prob.
Or, Reckon the degrees equal to the compl. of the azimuth on
the colure as before, and the degrees equal to the alt. on the equa-
tor from aries eastward ; elevate the pole to the given amplitude ;
screw the quadrant in the zenith, and extend it over the degree
marked on the equator ; the degree then cut on the horizon, reckon-
ing from the brass meridian, will give the latitude ; and the num-
ber of degrees on the quadrant between the colure and equator,
will be the declination as above.*
Example 1. The sun's altitude at 6, being equal 12, and his
azimuth from the south 761 ; required the latitude and sun's de-
clination ? Ans. 42 19' N.
Note. While the sun is in the northern signs, it always rises before 6 in
northern latitudes, and does not rise until after 6 in southern latitudes ;
the contrary is to be observed when he is in the southern signs ; hence it
is always known whether the latitude be north or south.
* To understand the above rules, let the globe be elevated to the sup-
posed latitude ; screw the quadrant in the zenith, and bring its graduated
edge to coincide with the azimuth on the horizon ; then the alt. on the
quadrant will cut the parallel of the sun's decl. for the given time ; and as
the mer. passing through this point coincides with the east or west point
of the horizon, its inclination with the horizon will be the lat. required.
Hence the alt. com. of the azimuth, and declination, form a right angled
triangle, and the angle formed by the decl. and co. azimuth, is the" alt.
required. This triangle transferred to the equator and brazen meridian,
ivives the above rules. The proportion for calculating- the lat- is the fol-
lowing; Tang. alt. : Had. :: Co Sine azimuth : Co. Tangt. latitude. Thus
172 PROBLEMS PERFORMED BY
2. The sun's altitude at 6, was observed equal 17, and his
azimuth 74 from the north ; required the latitude and sun's de-
clination ?
3. Required the latitude, when the sun's observed alt. at six was
IS 1 *, and his azimuth from the north 70?
PROB. 68.
Given the sun's declination^ altitude^ and azimuth) to Jind the lati-
tude and the hour of the day.
Rule. ELEVATE the pole to the given altitude, screw the quad-
rant of altitude in the zenith, and bring it to coincide with the given
azimuth on the horizon ; then turn the globe on its axis until the
sun's place comes to the graduated edge of the quadrant, and the
degree cut on it will be the latitude required.*
Example 1. On the 10th of May, the sun's alt. being 44, when
his azimuth from the south was 75 ; required the latitude ?
Ans. The sun's declination for the given day being 17 39 ; , cor-
responding to 20o of taurus, the latitude is therefore 40 27' N.
2. On the 2 1 st of June, the sun's alt. was 50, when his azimuth
from the south was 63 ; required the latitude ?
3. The sun's alt. on the 20th of October, was 36, and his azi-
muth from the south 26 ; required the latitude ?
PROB. 69.
Given the sun's declination and amplitude^ to Jind the latitude.
Rule. COUNT from the beginning of aries on the equator as
many degrees as are equal to the sun's declination, and mark the
point where the reckoning ends ; then with the sun's amplitude in
in ex. 1, Tang. alt. 12 : Rad. :: Co. S. azimuth 76 30' : Co.Tangt. lat. 42
IS'. The declination may also be found thus ; Rad. : cos. alt. :: s. azimuth :
cos. decl. In ex. 1, R. : cos. 12 :: s.76^ : cos. dec). 17 5S'.
* The reason of this rule is thus shewn ; having- found the latitude as
above, elevate the pole to this latitude; screw the quadrant of altitude in the
zenith, and set it to the given azimuth on the horizon ; then the compl. of
the latitude on the equator, the complement of the altitude on the quadrant,
and the distance between the sun's place and the elevated pole, form an ob-
lique spherical triangle, and the angle included by the compl. of the lat. and
compl. alt. is the given azimuth, or its supplement. Now the triangle formed
by the above rule being similar and equal to this in every respect, with this
difference, that the complement of the lat. is reckoned on the quadrant, and
the complement of the alt. on the brass meridian, shews the truth of the
rule, and whence it is derived. The method of calculating the prob. is as
follows : let fall a perpendicular from the pole on the co. lat. or quadrant of
alt. produced, &c. then R. : cos. azimuth :: co. tang. alt. : tang, x, and sine
alt. : cos. x :: cos. 90 zt decl. : cos. y. Then if the perpendicular falls with-
in, the difference between x+y will be the compl. lat. But if x + y be
greater than 180, x -f- y = compl. lat. See note to prob. 53.
Thus in ex. 1. Rad. : cos. 75 :: co. tang. 44 : tang. x = 15 35'; and sine
alt. 44 : cos. x 15 35' :: cos. 90 17 35'= cos. 72 21' : cos. y 65 8'.
Hence y x=- 49 35' sps compl. lat. and therefore the latitude is 40 27', as
required,
THE TERRESTRIAL GLOBE. 173
the compasses and one foot on this mark, intersect the equinoctial
colure with the other ; elevate the pole to the degree cut on the
colure ; screw the quadrant of alt. in the zenith, and extend it over
the degree marked on the equator ; the degree then cut on the
equator, reckoning from the nearest pole, will be the complement
of the latitude required.*
Or, Reckon on the equinoctial colure from aries, as many de-
grees as are equal to the declination, with the sun's amplitude in
the compasses, and one foot on the colure where the reckoning
ends, intersect the equator with the other ; then bring both poles
to the horizon ; screw the quadrant in the zenith, and extend it
over the degree cut on the colure ; the degree then cut on the ho-
rizon by the quadrant, will be the latitude required.
Example 1. On the 31st of May, the sun's rising amplitude
from the east towards the north, was 30 ; required the latitude ?
Ans. The sun's declination being 21 56', the latitude is found
equal 41 40' N.
2. When the sun's declination was 23 28' N. his rising amph
was 40 from the east towards the north ; required the latitude ?
3. When the sun's declination was 20 South, his rising ampli-
tude was 23 30' from the east towards the south ; required the
latitude ?
PROB. 70.
The day and hour being givcn^ tvhen a solar eclifoe will happen^ to
find where it will be -visible.
Rule. FIND the place where the sun is vertical at the given
hour, by prob. 12; then at all places within about 35f of this
place, the eclipse may be visible, especially if it be a total eclipse.
Note. Where exactnes is required, the centre of the penumbra or shade
should be found ; then if from this centre with the distance or number of
degrees corresponding to the semidiameter of the penumbra, a circle be de-
scribed on the globe, all those places within this circle will have the eclipse
visible at that time ; the nearer they are to the centre of the penumbra, the
greater will the eclipse be,
* This and the following rule is found in the same manner as the rules
given in the foregoing problems, thus ; elevate the pole to the supposed
latitude, or the lat. found as above, and bring- the sun's place to the hori-
zon ; then the sun's declination, amplitude, and that part of the equator
included between the horizon and the meridian passing through the sun's
declination, will form a right angled triangle, similar and equal in every
respect to that found by the above rules, &c. ; and the angle formed by
the equator at the horizon, and the sun's amplitude, will be the comple-
ment of the latitude. Hence this proportion, Sine ampl. : Rad :: sine
decl. : cosine latitude. Thus in ex. 1, Sine 30 : Rad. :: sine decl. 21
5& : Co. s. lat. 41 40.
To know whether the lat. be N. or S. if the sun's amplitude increase
svhen in the northern signs by elevating the north pole, or decrease when
in the southern signs, the lat. is N. If the contrary take place, the lati-
tude is south.
f Keith in his Treatise on the Globes, in solving this problem, says, thai
,a)l places within 70 of the place where the sun is vertical, may have thr-
174 PROBLEMS PERFORMED BY
Example 1. On the 3d of April, 1791, there was an eclipse of
the sun ; its beginning was at Oh. 17m. middle at Ih. 46m. and end
at 3h. 9m. as observed in Greenwich. Where was this eclipse
visible ?
Ans. It was visible in every part of Europe, a great part of Asia,
Africa and America. It was annular along the central track of the
penumbra, as in Iceland, for example, at their 12 o'clock. It was
no where total, because the sun's apparent diameter exceeded the
moon's at that time.
If the central track of the penumbra be represented on the globe,
and lines be drawn at each side of it at the distance of half the di-
ameter of the penumbra, the portion of the earth involved in the
shadow, during the eclipse, will be represented on the globe.
2. On the 17th of September, ISli, there will be an eclipse of
the sun, its beginning will be at 12h. 35m. greatest obscuration at
2h. 17m. and end at 3h. 51m. apparent time at New-York ; where
will this eclipse be visible ?
For more examples, consult the Nautical Almanacs or Ephemerides. See
also part 4th. where the subject is more fully treated.
PROB. 71.
The day and hour being given when a lunar eclipse will happen, to
find all those places on the globe to which the same will be -visible.
Rule. FIND the sun's place for the given time, elevate the pole
which is most remote from the sun to this declination, bring the
place where the hour is given to the brass meridian, and set the
eclipse visible if it be total ; but this is evidently false, as the semidiameter
of the penumbra or distance from the centre of the shade, is only about 35.
This may be shewn as follows :
Let the mean apparent diameter of the sun be taken equal 32' 3'*, the
mean apparent diameter of the moon equal 31' 7". Let BAG be the
earth, L the moon,
AIB half the angle
of the cone (sup-
posing- the moon to
be in its node, and
the centres of the
sun,moon and earth
to be in a straight
line) this will be
equal to the semidiameter of the sun = 16' 1" nearly. Xow the semidiameter of
the moon being- about .2692 semidiameters of the* earth, we have Sine 16' 1" :
rad. :: .2692 : 57.781 semidiameter of the earth = LI. The mean distance
of the moon from the earth is also given (see part 4) equal 60.3 of the
earth's semidiameters ; hence TT = 57.781 -j- 60.3 = 118.081. But TB =
1 : TI, 118.08 :: sine TIB, 16' 1" (7.6682967) : TBI or IBN = 33 17' 23".
But as IBN = ITB-J-TIB; hence 33 17' 23"=16 f l"=33 1' 22". the dou-
ble of which is the arch CB=66 2' 44", the portion of the earth's surface
covered by the penumbra. When the sun is nearest the earth, and the moon
in her apog-eon or greatest distance from the earth, this arch is then about
70 50', therefore the truth of the above remark is evident. If the centre
of the sun, moon, and earth be not in a right line, or if the moon be not. in
its node, then the shadow will fall obliquely on the earth, as represented by
THE "TERRESTRIAL GLOBE. 175
index to 12 ; then if the given time be in the forenoon, turn the
globe westward, but if in the afternoon, eastward, as many hours as
the time is before or after noon ; then the place exactly under the
sun's declination will be the antipodes of that place where the moon
is vertically eclipsed. The globe being kept in this position, set the
index to 42, and turn the globe until the index has passed over 12
hours ; then all those places above the horizon will have the eclipse
visible, to those places along the western edge of the horizon the
moon will rise eclipsed, to those along the eastern edge, she will
set eclipsed,* and to that place directly under the zenith or that deg.
on the brass mer. 90<> from the horizon, the moon will be vertical-
ly eclipsed.f
Note. As lunar eclipses continue for a considerable time, they may be
visible in more places than one hemisphere of the earth ; for, owing- to the
earth's motion on its axis during the time of the eclipse, the moon will rise
in several places after the eclipse began ; -hence if the prob. be performed
for the beginning and ending of the eclipse, the limits of those places where
the eclipse will be visible, will be determined.
Example \. On the 10th of March, 1811, the beginning of an
eclipse of the moon, at New-York, was at 1 3m. after 1 2 at night, and
the end 48 min. after 2 in the morning, apparent time ; where was
it visible ?
*dns. The sun's decl. being 4 26" S. to which the north pole
being elevated, and New-York brought to the meridian ; then the
time being taken 1 In. 47m. before noon, and the prob. performed
as directed in the rule, it will be found that the eclipse will then be
visible in all America, almost the whole of Europe, and about one
half of Africa. The moon will rise eclipsed in the beginning,
near Bhering's strait, the Fox islands, and west of Sandwich islands.
She will set eclipsed between Wardhus and the North Cape, in
Revel, Riga, the middle of Prussia, and Hungary, in Cephalonia,
the middle of Congo in Africa, See. and she will be vertical near
Santa Fe in New Granada, 4 26' north from the equator. If the
prob. be performed for 2h, 48' in the morning, or 9h. 12' before
noon, we shall find that the moon will rise eclipsed in the eastern
part of Siberia, the sea of Jesso, east of the Ladrone islands, near
the middle of the New Carolinas, between Solomon's isles and
New Zealand, 8cc. That she will set eclipsed at the eastern part
of Iceland, eastward of the Azores, at St. Antonio, in the Capo
the plane CB, in which case the distance of the sun from the moon being 1
found, the rest can be easily calculated. When the moon is not in her node,
then SLo passing through the moon's centre will differ from Sa-T passing-
through the earth's, as in the figure.
The reader is referred to part 4th for a fuller elucidation of these proper-
ties in the doctrine of eclipses.
* The rising and setting of the moon eclipsed is for the given particular
hour. See the answer to ex. 1st. ^
f Keith in his solution to this prob. says, that the moon will be vertically
eclipsed under the sun's declination, which is evidently an error, as it ought
to ( be under the degree of declination which is equal to that of the sun, rec-
koned on the brass meridian on the contrary side of the equator. This mjs-
t.ake would produce an error equal double the sun's declination.
176 PROBLEMS PERFORMED BY
Verd islands, between Sandwich land and the island discovered by
La Roche. The eclipse will be vertical in about 117 W. long,
and in lat, 4 26' N. And hence the eclipse will be visible in all
that space between where the sun rose eclipsed at the end of the
eclipse, and where she appeared setting at the beginning of the
eclipse, which space is considerably more than a hemisphere. In
all those places between where she rose eclipsed at the beginning
and ending of the eclipse, she will successively rise eclipsed ; in
those places between where she appeared eclipsed at setting in the
beginning and end of the eclipse, she will set successively eclipsed,
and consequently in almost all Europe, 8cc. In the parallel 4 26 f
N. of the equator, and from long. 74 W. to long. 117 W. she
will be successively vertically eclipsed.
2. On the 22d of August, i812, the beginning of an eclipse of
the moon will be at ih. 10', the middle at 2h. si^', and the end at
4h, 47', Greenwich apparent astronomical time ; where will the
eclipse be visible, &c. at each of these times ?
More examples may be found in the Naut. Aim. or in common almanacs.
PROS. 72.
To find when an eclifise of the sun or moon is likely to hajifien in
any year.
Rule 1. FIND the place of the moon's nodes, the time of new
moon, and the sun's place at that time by the Naut. Aim. or an
ephemeris ;* then if the sun be within 17 31' 27", or nearly 17
of the moon's node, there will be an eclifise of the sun.
2. Find the place of the moon's nodes, the time of full moon, anil
the sun's place or longitude at that time, by the Naut. Aim. or
any good ephemeris ; then if the sun's longitude be within 1 1 34"
of the moon's node, there will be an eclifise of the moon.
Example 1. In 1812, on the 26th of February, at 17 hours 51
minutes astronomical time, or at 5 1m. after 5 in the morning of
the 27th civil apparent time at Greenwich, there will be full moon,
at which time the place of the moon's node will be 5s. 8 9', and
the sun's longitude 11s. 7 33', or X 7 33'; will there be an
eclipse of the moon at that time ?
* This prob. may be solved, though not very accurately without an
ephemeris, thus : If the place of the moon's nodes be given for any parti-
cular year, its place for any other year may be easily calculated ; the mean
annual variation according to Mayer, being- 19 IS' 45"1, or according to
La Land, its diurnal motion being 3' 10" 638603696. The time of new and
full moon may be found by the note to definition 80, and the sun's place
for the given time may be found on the globe.
Dr. Halley remarks, that in the period of 223 lunations, there are 18
years 10 or 11 days (according as there are 5 or 4 leap years) 7h. 43-f;
that if we add this time to the middle of any eclipse observed, we shall
have the return of a corresponding one, certainly, within Ih. 3C'; and that,
by the help of a few equations, the like seizes may be found for several
periods : hence the time when an eclipse may be expected, can be easily
found by this rule. For other methods see part 4th.
THE TERRESTRIAL GLOBE. 177
Ans. The moon's node being 5s. 8 9', the opposite node will be
1 Is. 8 9' ; hence Us. 8 9' 1 Is. 7 33' = 36', the distance of
the sun from this node ; hence there will be a total eclipse ; for
when the sun is in one of the moon's nodes at the time of full
moon, the moon is in the opposite node, and the earth is directly
between them, in which position the shadow of the earth will fall
directly on the moon, and produce a total and central eclipse.
2. On the 3d of November, 1808, there was a full moon, at
which time the place of the moon's nodes was 7s. 12 18', and the
sun's longitude 7s. 10 55 ; ; was there an eclipse of the moon at
that time ?
3. On the 17th of September, 1811, there will be a, new moon
at 57m after 6 in the afternoon, Greenwich time, at which time
the place of the moon's node will be 5s. 16 27', and the sun's lon-
gitude 5s. 23 56' 39" ; will there be an eclipse of the sun at that
time ?
dns. The distance of the sun from the node being 7 29' 39",
there will therefore be an eclipse.
4. On the 5th of September, 1812, at 22m. after 7 in the after-
noon at Greenwich, there will be a new moon, at which time the
place of the moon's node will be 4s 28 i', and the sun's longitude
6s. 13 0' 35" ; will there be an eclipse of the sun at that time ?
5. On the 27th of March, 1812, at 12h. 16m. astronomical time
at Greenwich, there will be a full moon, at which time the place
of the moon's node will be 5s 6 35', and the sun's longitude 7
11' 14" ; will there be an eclipse of the moon at that time ?
PROS. 73.
The time of an eclifise of any of the satellites ofJu/iiter being gi-ven^
tojind those filaces on the earth where it ivill be "visible.
Rule. FIND the place where the sun is vertical at the time of
the eclipse; (found by prob. 12, part 2.) bring this place to the
brass meridian, elevate the pole to the place where the sun is ver-
tical, and set the hour circle to 1 2 ; then,
1. If Ju fitter be in consequentia or rise after the sun,* draw a
line with a black lead pencil, or with ink, along the eastern edge of
the horizon ; this line will pass over all those places where the sun
is setting at the given time ; take the difference between the right
ascension of the sun and Jupiter, and turn the globe westward on
its axis, until as many degrees of the equator pass under the brass
meridian as are equal to the difference ; keep the globe from turn-
ing on its axis, and raise or depress the pole until its elevation be
equal to Jupiter's declination ; the globe being again fixed in this
position, draw a line with a pencil along the eastern edge of the
horizon ; the eclipse will then be vi&ible to every place between
these lines, that is from sun setting until the time of Jupiter's set-
ting-
* Jupiter rises after the sun, or is an evening 1 star when his longitude
is greater than the sun's longitude.
Y
178 PROBLEMS PERFORMED BY
2. If Jufiiter be in antecedentia or rise before the sun* Having
rectified the globe as before, 8cc. draw a line along the 'western
edge of the horizon ; this line will pass over all those places where
the sun is rising at the given hour ; then elevate the pole accord-
ing to Jupiter's declination, and turn the globe eastward on its axis,
until as many degrees of the equator have passed under the brass
meridian, as are equal to the difference between the sun's and Ju-
piter's right ascension ; the globe being fixed in this position, draw
a line along the western edge of the horizon ; then the space con-
tained between this and the former line, will comprehend all those
places upon the earth where the eclipse will be visible, between the
time of Jupiter's rising and the rising of the sun.
Example . On the 1st of May, 1812, there will be an emersionf
of the first satellite of Jupiter at 14?' 33" past 6 in the afternoon, at
Greenwich ; where will the eclipse be visible ?
Ans. In this ex. the longitude of Jupiter will exceed that of the
sun, and therefore Jupiter will rise after the sun, or be an evening
star. His declination will be 23 30' north, and his longitude 3
signs So i.8' by the Nautical Almanac ; the sun's longitude will be
Is. I ! 9' 43", and his rt. as 38 43' 48", and decl. 1 5 1 1' 24" N.
Jupiter's rt. as- may be found by the note to prob. 2, part 3, or
more easily, but not so correctly, by the Globe. For if his longi-
tude in the ecliptic be brought to the brass meridian, his place will
be under the degree of his declination nearly,! and his right ascen-
sion will be found on the equator. In this ex. Jupiter's rt. as. will
be found by the globes nearly 93|, his lat. being 4' north. The
eclipse will be visible in all those places in Europe, Asia and Afri-
ca, eastward of a great circle passing through those places where
the sun sets at the given time, at Cagliari> Florence, Prague, Dant-
* Jupiter rises before the sun, or is a morning star, when his longitude
is less than the sun's longitude.
f The emersion and immersion of a satellite, are terms used principally
in the Nautical Almanac, to signify the appearance or disappearance of the
satellites of Jupiter, &c. The immersion is the instant of the disappear-
ance of a satellite by entering into the shadow of its primary planet, and
the emersion is the instant of its appearance or emerging from the shadow.
They generally happen when the satellite is at some distance from the
body of Jupiter, except near the opposition of Jupiter to the sun, when the
satellite approaches nearer to his body. The immersions and emersions
take place on the west side of Jupiter, before his opposition to the sun,
but on the east side after the opposition. If a telescope be used which re-
verses objects, this appearance will be directly contrary. Before the op-
position the immersions only of the first satellite are visible, and after the
opposition the emersions only. The same is generally the case with re-
gard to the 2d satellite ; both the phenomena of the same eclipse are fre-
quently observable in the two outer satellites. The longitude from Green-
wich is found, by taking the difference between the observed time and
that found in the ephemeris, and converting it into degrees, &c. In part
4th this subject will be more fully entered into. The eclipses of Jupiter's
satellites are set down in the lower part of pa. 3, in the Naut. Almanac.
This is on supposition that Jupiter performs his motion in the ecliptic*
and as he deviates but little from it, the solution by this method, on th&
globe* *vill be sufficiently accurate.
THE TERRESTRIAL GLOBE. 179
2ac, Revel, &c. to another great circle passing through the western
part of Madagascar, the eastern part of Arabia, Little Thibet, the
middle of Siberia, &c. nearly, and therefore not visible in Green-
wich, &c.*
2. On the 1 8th of January, 1811, there was an emersion of the
1st satellite of Jupiter at 32' 10" after 5 o'clock in the evening ;
where will it be visible? Jupiter's longitude being is. 2i 24', lat,
52' S. and decl 17 17' N. and the sun's right ascen. 299 54' 33",
and decl 20 37' 15" 5 south.
3. On the 19th of August, 1812, there will be an immersion of
Jupiter's 1st satellite at 40' 26" after nine o'clock in the morning,
Jupiter's longitude being then 3s. 26 40', lat. 15' N. and decl. 21
5', and the sun's right ascension 148 22' 45", and decl. 12 47' 32"
north ; where will it be visible ?
4. On the 21st of December, 1812, at 31' 34" after one in the
morning, there will be an immersion of Jupiter's 2d satellite, his
longitude being then 4s. 8, lat. 36' N. and decl 18 5 i' N. and
the sun's right ascension 269 2' 24", and decl. 23 27' 21"; where
will this immersion be visible ?
PROB 74.
To explain the phenomenon of the Harvest Moon.
Definition 1 . IN north latitude, the full moon which happens at
or is nearest to, the autumnal equinox, is called the harvest moon.f
* To know if an eclipse of any of the satellites of Jupiter will be visible
at any place, the Naut. Aim. directs to find whether Jupiter be 8 above
the horizon of the place, and the sun as much below it. This observation
may be easily applied to the above examples.
f As the sun is in virgo, and libra in our autumnal months, and as the
moon can never be full but when she is opposite to the sun, therefore the
moon is never full in the opposite signs pisces and aries, but in these two
months. This remarkable rising- of the moon is not observed but in har-
vest, or in September and October, when the moon is in pisces and aries.
For although the moon is in these signs twelve times in the year, it is only
about the autumnal equinox that her orbit is nearly parallel to the horizon,
so that there is very little difference in her rising- for several nights. In win-
ter these signs rise at noon, at which time the moon is in her first quarter,
being only a quarter of a circle distant from the sun, so that when the sun
is above the horizon, 'the moon's rising is neither perceived or regarded.
In spring these signs rise with the sun, because the sun's place is then in
them; but as the moon changes, or is new moon in them at that time, she
is therefore invisible. In summer these signs rise about midnight, and the
sun being then three signs or a quarter of a circle before them, the moon
is in them about her third quarter, at which time she rises so late, and
gives so little light, that her rising passes unobserved. In autumn these
signs being opposite the sun's place, rise when the sun sets ; and as the
moon is then in opposition or at the full, her rising becomes very remark-
able.
This phenomenon becomes more remarkable the farther the place is from
the equator, if not beyond the polar circles ; for in this case the angle
which the ecliptic makes with the horizon, gradually diminishes when
pisces and aries ris^.
180 PROBLEMS PERFORMED BY
D(f. 2. The harvest moon in south latitude, is the full moon
which happens at or near the vernal equinox.*
Rule 1 . For north latitude. Elevate the north pole to the lati-
tude of the place ; mark the point aries in the ecliptic, and also
every 1 2f on each side of that point, until there be 1 2 or 1 3 marks ;
bring that mark nearest to pisc.es to the eastern part of the horizon,
and set the index to 12 ; the globe being then turned westward on
its axis, until the other marks come to the horizon successively ;
then the intervals of time between the marks coming to the hori-
zon, will shew the difference of time between the moon's rising
every day. These marks being brought to the western edge of the
horizon in the same manner, the diurnal difference of time between
the moon's setting may be found.
Thus in New-York, the diurnal difference of time in the moon's
rising, in pisces, aries, SCG is 26', 27', 28', 29', &c. and the diurnal
difference in her setting is 60', 61', 62', 63', &c. The cause of
this difference arises from the different angles which the ecliptic
makes with the horizon ; for those parts or signs which rise with
the smallest angles, set with the greatest, and the contrary. Thus
in the above case, the point aries makes only an angle of 25| with
the horizon, when it rises, but when it sets, it makes an angle of
724. In equal times whenever this angle is least, a greater por-
tion of the ecliptic rises than when the angle is larger ; therefore
when the moon is in those signs which rise or set with the smallest
angles, she rises or sets with the least difference of time, and with
the greatest difference in those signs which rise or set with the
greatest angles, from which the whole is evident-!
'2. For south latitude. Elevate the south pole to the latitude of
the given place ; mark the point libra and every 12 degrees of the
* In northern latitudes the autumnal full moons are in pisces and aries,
and the vernal full moons in virgo and libra ; but in southern latitudes the
reverse takes place, the seasons being- contrary. Now as virgo and libra
rise at as small angles with the horizon, in southern latitudes, as pisces and
aries in northern latitudes, the harvest moons are therefore as regular on
one side of the equator as on the other, only that they happen at contrary
seasons of the year.
f The reason that 12 is marked, is because the moon gains nearly 12 G
on the sun every day ; for the moon's daily mean motion is 13 1C' 35 , and
the sun's 5S' 8" 3, the difference of which is 12 11' 26" 7. The solution
is on supposition that the moon remains constantly in the ecliptic, which is
accurate enough for illustrating- the prob. otherwise the moon's place may
be marked on the globe at the time of full moon, and a few days before
and after it, by having- her lat. and long-, or rt. ascen. and decl. given ;
which may be found from the Nautical Aim. or any good ephemeris. See
the investigation of this prob. given in page 128 of Vince's Astronomy, 8vo.
or in ch. 16, pa. 203 of Ferguson's Astronomy, 8th ed.
$ As there is a complete revolution in the moon's nodes in about 18 years
S months, all the varieties of the intervals of the rising- and setting- of the
moon will happen within that time. The following table extracted from
pa. 216 of Ferguson's Astron. will shew in what years the harvest moon's
are the least, and the most beneficial with regard to the times of their
rising, from 1811 to 1861. The columns of years under L, are those in
which the harvest moon are the least beneficial, because they fall about the
THE TERRESTRIAL GLOBE. 181
ecliptic, preceding and following that point, as before ; bring that
mark which is nearest to virgo to the eastern edge of the horizon,
and set the index to 1 2 ; then turn the globe westward until the
other marks come to the horizon successively, and observe the
hours passed over by the index (or rather on the equator) the in-
tervals of time between the marks coming to the horizon, will be
the diurnal difference of time between the moon's rising, &c. If
these marks be brought to the western edge of the horizon in like
manner, the diurnal difference of the moon's setting may be found,
&c. as in the preceding part of the problem.
PROB. 75.
To draw a meridian line ujion a horizontal plane*
Rule 1. FROM a correct altitude of the sun find the time of the
day (by the note to prob. 48, or 52, part 2d ) and set a well regulat-
ed watch to that time ; then suspend a plumb line so that the sha-
descending- node ; and those under M are the most beneficial, because
they fail about the ascending- node.
L
L
L
L 1VI
M
M
M
M
1811
1827
1833
1848
1816
1822
1837
1843
1858
1812
1828
1834
1849
1817
1823
1838
1853
1859
1813
1829
1844
1850
1818
1824
1839
1854
1860
1814
1830
1845
1851
1819
1825
1840
1855
1861
1815
1831
1846
1852
1820
1835
1841
1856
1826
1832
1847
/ , i
1821
1836
1842
1
1857
i . 1
In this instance of the harvest moon, as in innumerable others, which
astronomy points out, the beneficence and wisdom of that intelligent be-
ing- who presides over the universe, discover themselves. Here we see
the wandering- course of the moon so ordered, as to bestow more or less
light on those parts of the earth, where their circumstances and seasons
render it more or less necessary and serviceable. Wherever we cast our
eyes, we see that all is formed and regulated with design, that every
thing- has its particular use, that every thing proclaims the boundless wis-
dom, and witnesses the most attentive kindness of the maker. If at times
(says De Feller, in his Philosophical Catechism) those visible beings oc-
casion some physical evil, the reason and understanding- given to man,
supply him with means to escape the evil or to remedy it ; and besides,
what are those physical evils compared with the benefits attending- them,
the services they render, and the virtues they occasion? Even Voltaire
himself, the great champion of every error, acknowledges, that " In
the system that admits of a God, there are only difficulties to get over ; but
in all other systems there are absurdities to swallow." ' What idea, says
De Feller, could make amends for that of God ; an idea so vast in itself, and
so rich, that begets and fosters so many others, that of duty, that of jus-
tice, that of charity ! And what shall we say of the great, the exalted
sentiments that flow from those ideas; the voice of conscience, the study
of the law of God, the knowledge in detail, and upon principle of his com-
mandments, &c. of the many obligations of a good Christian, of the pious
practices that occupy the soul, and with unction ineffable, render it happy
in every situation of life. Heavens ! what a void must not the loss of all
.his leave behind it, in the soul and the life of man ! and is it not quite
natural, we should become triflers and fools, in the same proportion as we
become irreligious?"
182 PROBLEMS PERFORMED BY
dow of it may fall on the plane, and when the hour hand of the
watch is at 1 2, the shadow of the plumb line will be the true me-
ridian.*
Rule 2. Describe several concentric circles on a horizontal plane
(as a board, Sec.) In the centre of these circles fix a pin or
straight wire perpendicular to the plane ; observe, in the forenoon,
when the extremity of the shadow exactly touches the respective
circles, beginning with the outermost, and mark these respective
points on the circles. In the afternoon mark where the extremity
of the shadows cuts the same circles as before, beginning with the
innermost ; then with a pair of compasses bisect the arch between
the two marks on any of the circles ; a line drawn from the centre
to that point, will be a true meridian line.f If the pin be not per-
pendicular, let the circles be described from the top of it, and pro-
ceed as before4
Rule 3. Find when the pole star and the star alioth in the great
bear are in the same plumb line, or have the same azimuth, by
means of two plummets suspended at a considerable distance, with
their ends in vessels of water, to keep them steady ; the line be-
tween the two plummets or this line produced, will be the meri-
dian, sufficiently exact. Any two stars that have the same right
ascension, will answer the same purpose.
Rule 4. Having the time of the northing of the star alioth, or
the northing or southing of any other star or planet from astrono*
* The most proper time of the year to perform both this and the follow-
ing rule, is about the solstices, because then the sun's declination does not
sensibly vary for several days.
As there are various methods of finding 1 the 'hour of the day (most of which
are given in parts 2d and 3d of this work, particularly in the notes) and con-
sequently of regulating 1 the watch ; hence there are as many different me-
thods of performing this prob. Moreover from the true and magnetic azi-
muth or amplitude being given, the variation of the compass may be found,
and hence a meridian line may be traced out by the compass. This prob.
being necessary for fixing dials, is therefore given in this place.
f The board may be a foot or more in breadth, and the circles about a
quarter of an inch or $ an inch from each other. The pin or wire ought to
he about -g- of an inch thick, with a round blunt point, or well defined head like
the head of a pin, and of such a length that the shadow may fall within the in-
nermost circle, at least four hours, in the middle of the day. This method is
not, however, capable of very great accuracy, as the shadow is scarcely ever
well defined. If however the mean of the several meridians, so found, be
taken, the meridian thus found will be sufficiently accurate for all common
purposes.
i In describing these circles, one end of a wooden ruler may be placed on
the top of the wire, and with a sharp pointed iron pin, or wire, in the other
end of the ruler, circles may be described. The same prob. may be perform-
ed by means of a small hole in a window-shutter, through which the sun
shines, circles being described on the floor, with the hole as their common
centre. Or if the casement of a window on which the sun shines at noon be
perpendicular to the horizon, the shadow cast by it, on the floor, will trace
out the meridian. Various other methods and contrivances will easily pre-
sent themselves.
THE TERRESTRIAL GLOBE. 183
mical tables (see prob. 8. and 39, part 3d ) A line drawn from the
observer towards the respective star or planet, at the time of its
northing or southing, will trace out the meridian required. Or
when the pole star and alioth, or any two stars having the same or
opposite rt. ascensions, be in the same azimuth, set the watch to
the time of their northing or southing, according as they are north
or south of the observer, and next day at 12 o'clock, by the same
watch or clock, draw a meridian line by the shadow of a plumb
line hung in the sun.*
GNOMONICAL OR DIALLING PROBLEMS
SOLVED BY THE GLOBES.
fundamental principles^ observations, ^c.
DIALLING, or the art of making dials, is founded entirely on as-
tronomy. For the lines on a dial which shew the hours, are the
intersections of the respective circles with the plane of the dial ;
and the projection of these hour lines, is the same as the projection
of the sphere upon the plane of the dial. And hence the con*
struction of dials depends on the projection of the sphere, particu-
larly the gnomonic projection, where the circles are projected into
right lines. The principles of this projection Emerson has given
at large, in his treatise on the projection of the sphere (see his
Tracts.) As the art of measuring time is of the greatest impor-
tance, so the art of dialling, until the invention of clocks and
watches, was held in the greatest estimation. And although at
present we are furnished with these machines, yet as the best of
them are often out of order, and that in general, they are liable to
stop and go wrong, that unerring instrument, a true sun dial, will
be always useful to correct and regulate them.
Suppose the globe of the earth as represented in the annexed
figure, to be transparent, with hour circles or meridians, &c. drawn
upon it, and that it revolves round a real axis NS, which is opake,
and casts a shadow ; then it is evident that the shadow of this axis
will fall upon every particular meridian or hour line, when the sun
corner to the plane of the opposite meridian, and will therefore
shew the time in all those places on that meridian. Now if any
* To take away the star's rays, look through a small hole in a thin plate,
or piece of paper. Any line drawn parallel to the meridians found as above,
will also be meridians ; hence when those found above will not answer, others
may be easily drawn. When the meridian line is intended to be the basis
of any nice astronomical calculations or observations, it must be traced out,
very accurately, by the 1st rule, then two poles may be erected at a consi-
derable distance from each other, with marks to render them visible, &o.
For making- these observations, the astronomical circle or circle of reflection.
is the most proper instrument. An artificial horizon may be made with mo-
lasses, quicksilver, &.c. When this circle cannot be had, a Hadley's quad
rant, or rather sextant, will answer,
place must be selected.
184 PROBLEMS PERFORMED BY
opake plane be imagined to pass through the centre of this trans-
parent globe, the shadow of half the axis NC will fall upon either
side of this intersecting plane.
Let HORI represent the plane
of the horizon, RN the elevation
of the pole or lat. of the place ;
then while the sun is above the
horizon, the shadow of the axis
CN will fall upon the upper side
of the plane HOR. When the
edge of the plane of any hour cir-
cle as K, I, &c. points directly to H|
the sun, the shadow of the axis,
being coincident with this plane,
will mark the respective hour
lines on the plane of the horizon
HOR ; and hence the hour line
on the horizontal plane, is a line
drawn from its centre, to the point where this plane intersects the
meridian, opposite to that on which the sun shines. Now as the
sun's apparent motion about the earth's axis is at the rate of 1 5, an
hour (nearljr) let the shadow of the earth's axis be supposed to be
projected into the meridian opposite to that in which the sun is,
and then this meridian will move at the rate of 15 an hour. (See
Emerson's Dialling, sect. 1. prop. 1. and cor. &c.) Let ZNRSH
represent a meridian on the surface of the earth, SCN the earth's
axis, Z the place of the spectator, being also the pole of the hori-
zon HORI ; let the meridians NlS, NiS, &c. be drawn so as to
make angles with the mer. NRS of 1 5, 30, &c. respectively ; then
supposing NR the meridian into which the shadow of CN is pro-
jected at 12 o'clock, Nl, N2, N3, Sec. are the meridians into
which the shadow is projected at 1,2, 3, 8cc. of the clock, and these
shadows will be projected on the plane HORI into the lines CR,
Cl, C2, C3, Sec. and the angles NCl, NC2, &c will be the angles
between the 12 o'clock line CR, and the hour lines of 1,2, 3, &c.
Hence in the right angled triangle NRl, we have NR the lat. of
the place (See the last note to problem 19, part 3d. art. 1.) and
the angle RNl = 15, and therefore, by Nafiier's rule, Co.
tang. 15 (or h^ur angle) : rad. :: s. lat. NR : tang. Rl the hour
arch, which is the measure of the angle RCI ; or Rad : sine NR
( the lat,) :: tang. 15 : tang. Rl ; or tang. 30 : tang R2 or angle
RC2, &c. (Simson's Spher. Trig, annexed to his Euclid, prop. 17.)
in the same manner by either of these proportions, the arch R3,
R4, &c. may be found, and hence a table of the hour angles for
any lat. may be easily calculated. In the foregoing elucidations we
made the earth's axis the gnomon, and considered the shadow as
projected upon the plane HORI. But as it is the same thing
whether a dial be drawn upon any given plane, or upon the plane
of the great circle of the sphere which is parallel to it. (Emerson's
THE TERRESTRIAL GLOBE.
185
Dialling, cor. 3. prop 1. sect. 1.) Let the plane arbh be drawn
parallel to HORI at Z, this plane will represent the sensible hori-
zon, on which let ZP be drawn parallel to CN, and let its shadow
be projected on the plane arbh in the same manner as the shadow
of CN is projected on the rational horizon HORI, and the hour
angles rl, r2, &c. be calculated in the same manner ; this will be
an horizontal dial, ZP will be its gnomon, and the lines Zl, Z2,
&c. the hour lines required. For at the immense distance of the
sun, ZP may be considered as coinciding with CN. If at the other
side of r, arches rll, rlO, r9, &c. be made equal to rl, r2, r3, Sec.
respectively, and lines be drawn from Z to these points ; these
will be the hour lines for the forenoon. Moreover, the hour lines
may be continued from 6 towards H as far as may be necessary, by
laying off from 6 to 7, the same distance as from 6 to 5, &c. and also
from I towards H, by producing 5C, 4C, SC, &c. The reason of
which see in Emerson's Dialling.
Again, Let ABD be an opake
vertical plane, or great circle, per-
pendicular to the plane HBNZ
(and therefore to the horizon atZ)
and passing through the centre of
the globe. Here the globe being
supposed transparent as before,
while it revolves on its axis SN,
it is evident that the shadow of
the part of the south end of the
axis CS, considered as opake,
will always fall on the plane ABD,
and mark out the hour as in the
horizontal dial Then for the
same reason as before, if the an-
gles BS1, BS2, BS3, &c. be 15,
30, 450J & c t he shadow of SC will be projected into the lines CI,
C2, C3, Sec at the hours of 1, 2, 3, Sec. of the day ; and the angles
BCl, BC2, &c. will be measured by the arches Bl, B2, &c. Hence
in the right angled triangle BSl,BS = ZNco. lat. or dist. ofZ
from the pole N, and the angle BSl = 15 ; therefore by Napier's
rule as before, we have Co. tang. 15 (or hour angle) ; rad. :: cos.
lat. :: tang Bl the hour arch. Or by Simson's Spher. prop. 17,
Rad. : sine SB, the co. lat. :: fang. 15 : tang. Bl. In the same
manner B5, B3, &c. may be found, making use of 30, 45, &c.
respectively, in place of 15. Now if Zabc be a place coinciding
with the plane \BD, and at be parallel to CS, at will project its
shadow on the plane Zabc in the same manner as CS on the plane
ABD, for the same reason as for the horizontal dial ; hence the
hour angles from the 12 o'clock line, are computed by the same
proportion. This is a -vertical south dial. Hence also it appears
that the surface of every dial whatever, is parallel to the horizcij of
some place or other upon the earth, in which place it w uld be-
come a horizontal dial ; and that if a dial be taken to any other lati-
Z
186 PROBLEMS PERFORMED BY
tude different from that for which it was made, it will indicate the
apparent time truly, if placed parallel to its former situation.
Moreover as the angle SCB = tsZ (29 Eucl. 1.) = the arch ZN =
co. lat. and that the lat of H or R is = co. lat. of 25, it follows that
the elevation of the stile or gnomon above the dial's surface, when
it faces the south> is always equal to co. lat. of the place, or equal
to the lat. of the place whose horizon is parallel to its surface. In
the same manner the hour lines may be calculated, when the sha-
dow is projected upon a plane, in any other position. See sect. 1
of Emerson's Dialling.
Remark. It appears from the above observations, as the whole
earth is but a point in comparison of the heavens, that if a small
sphere of glass be placed on any part of the earth's surface, having
an opake axis parallel to the axis of the earth, and such lines upon
it, and such a plane within it as above described ; it will shew the
hour of the day as correctly as if it were placed at the earth's cen-
tre, and the whole body of the earth were as transparent as crystal,
PROB. 76.
To make a horizontal dial for any latitude.
Rule. ELEVATE the pole as many degrees above the horizon as
are equal to the given latitude ; bring aries to the brass meridian,
and set the index^to 12 ; then turn the globe westward until the in-
dex has passed over 1, 2, 3, 4, 5 and 6 hours successively, and
mark the degree cut on the horizon by the equinoctial colure, at
each respective hour, reckoning from the north or south points,
these will be the distances of the hour lines from noon until 6
o'clock at night : And as the hour of I and 11,2 and 10, 3 and 9,
Sec. are equally distant from noon, the hour arches for 1, 2, 3, &c.
in the afternoon, will serve for those of 11, 10, 9, Sec. in the fore-
noon Or the globe may be turned eastward until the index has
passed over II, 10, 9, &c. on the index, and the degrees cut by the
colure on the horizon marked as before. The hours on the equator
will answer rather better than those on the hour index. If the
half hours, quarters, &c. be brought to the meridian, in the same
manner, the colure will mark the hour arches, Sec. corresponding
on the horizon.*
* The reason of this rule is evident from what is said in the preceding ob-
servations. For the latitude of the place, the hour arch on the horizon, and
the arc of the colure between the elevated pole and the horizon, form a right,
angled triangle, similar to that described in the observations on the horizon -
tal dial, from which the rule is derived.
From the above rule it appears, that there is no absolute necessity of hav-
ing meridians drawn through every 15 on Cory's globes, as the hour index.,
or the hours marked on the equator, are abundantly sufficient, though some-
late authors imagine the contrary, owing to their manner of solving tho
problem. If however the meridians be drawn through every 15, the whole
of the haur arches maybe seen atone view on the horizon, without any mo
tion of the globe on its axis, which is much more convenient.
THE TERRESTRIAL GLOBE.
187
Examfile \ . To make a horizontal dial for New- York in latitude
40 43' N.
The pole being elevated to the lat. and the point aries brought to
the meridian ; then the hour arches from 12 will be 9 55', 20 38',
33 7', 48 29', 67 40' and 90 for the hours I, II, &c to VI res-
pectively ; or reckoning from the east towards the south, they will
be 22o 20', 41 31', 56 53', 69 22', 80 5' (the compl. of the for-
mer) for the hours VI, V, See. to I, reckoning from VI o'clock
backwards to XII.*
2. To make a horizontal dial for London in lat. 5 l N.
3. To make a horizontal dial for Philadelphia in lat. 39 57' N.
tt
* It is not necessary in the above solution to give the distances further
than VI, for the distances from XII to VI in the forenoon, are the same as
from XII to VI in the afternoon ; and if the hour lines be continued through
the centre of the dial, they will point out the opposite hours.
The following table calculated for the lat. 40 43' by the following- pro-
portion, Jiad. : sine lat. :; tang, hour angle tang, hour arch (see the forego-
ing- principles, &c.) contains the hour arches, halves, quarters, &c. from
XII to VI-
y
it
'All
J
II
s "t*
\l
A
if
J12
!
K li
S **
S II
345 ;
7 30
11 15
15
18 45
22 30
26 15
30
2 27'
4 54
7 24
9 55
12 29
15 7
17 50
20 38
2*
III
3*
IV
33 45'
37 30
41 15
45
48 45
52 30
56 15
60
23 33'
26 35
29 46
33 7
36 38
40 22
44 19
48 29
3
4*
42
V
5*
5*
VI
63 45'
67 30
71 15
75
78 45
82 30
86 15
90
52 54'
57 35
62 SO
67 40
73 2
78 35
84 15
90
The horizontal dial may be constructed geometrically, as follows
Take any point C for the centre, and draw CS or
C12 for the meridian or 12 o'clock hour line, C6 at
right angles to it, for the 6 o'clock hour line ; take
Cc equal the thickness of the stile or gnomon, and
draw cs parallel to CS. Now from the centre C or c
with the radius of some line of chords, describe a cir-
cle ; then on the arch of this circle lay off the hour
arches 9 55', 20 38', &c. on each side of the 12
o'clock hour line, together with the half hours and
quarters, as in the above table. Lastly, make the an-
jle 80* = the lat. =^=40 43' for the stile, which is to, be placed perpendicu-
lar to the plane of the dial, the dial is then finished. To erect it, the line
C12 must be placed in the meridian (found by the last prob.) so that the
12 may point towards the north ; then the gnomon Ct will exactly point to
the north pole. The dial plane must be placed horizontal with a level, and
then the dial is fit for use.
If the edge of the stile has no considerable breadth, and is in the same
place with the substile CS, no allowance is necessary, so that C may be taken
us the centre of both semicircles^ &c. Various hours, ornaments, tables of
;he equation of time, &c. may be inserted on those dials, and different in .
PROBLEMS PERFORMED BY
PROB. 77.
To make an erect south dial for any latitude.
Rule. ELEVATE the pole to the complement of the latitude
(the south pole if the lat. be north, &c.) bring the point aries to
the brass meridian, and set the index to 12 ; then turn the globe
eastward or westward until each hour, &c. on the equator or on
the hour circle, comes successively to the meridian ; then the
colure passing through aries will point out, on the horizon, the dis-
tance of the respective hour arches, or hour lines from the me-
ridian.
Or, If the meridian be drawn through every 1 5, as on Bardin's
globes, aries being brought to the brass mer. the meridians pas-
sing through every 1 5 will point out the hour arches on the ho-
rizon.*
Example 1. To make an erect south or vertical dial for New-
York, lat. 40 43' N.
The south pole being elevated 49 17', and aries brought to the
mer. then the globe being turned on its axis, the colure will inter-
sect the horizon in the following degrees : 1 1<> 29', 23 38', 37 10',
52 42', 70 32', and 90 for the hours I, II, &c. to VI ; or XI, X,
Sec. to VI'; or if you count from the east towards the south, they
will be 0, 19 28', 37 18', 52 50', 66 22', and 78 31', for the
hours VI, V, &c. to I ; or VI, VII, &c. to Xl.f
2. To make an erect south dial for Washington city, lat. 38
53' north.
struments, scales, &c. used in their construction, for which the learner is
referred to Emerson's Dialling, Ferguson's Lectures, Ozanam's Mathemati-
cal Recreations, Jones's Dialling, &c.
Emerson in his Dialling (schol. to prob. 7, sect. 2) remarks, that if a ho-
rizontal dial be made for a place in the torrid zone, to shew the hour by the
top of the perpendicular stile tS, whenever the sun's declination exceeds the
lat. of the place, the shadow of the gnomon will go back twice in the day,
once in the forenoon, and once in the afternoon. (See the note to ex. 3.
prob. 51. part 2.) The greater the difference between the lat. and the sun's
decl. the further will the shadow go back. The same will take place with
respect to any star, &c. the declination of which is greater than our latitude.
In the 38th chap, of Isaias, it is related, that the shadow on the dial of
_2e/mz was brought back ten lines or degrees, in confirmation of a promise
made by Isaias to Ezechias, king of Juda, that his life should be prolonged
15 years. This was truly, as it was then considered, a miracle, being con-
trary to the established laws of nature ; as Jerusalem, where the dial was
erected, was not in the torrid zone, and therefore the shadow could not
possibly go back from any natural cause. \Vhatever incredulity may object
to this, it is certain that that Being who framed the laws of nature, can sus
pend their operations, or change them at his pleasure ; as in the present
case, to reward the piety of a virtuous prince, and exhibit to the world the
efficacy of fervent and humble prayer.
* The reason of both these methods will appear from the observations in
the beginning.
f The arches for the half hours, quarters, &,c. may be found by the set
method above in the same manner as the hour lines. The following table
contains the hour arches, halves and quarters, from XII to VI. It is calcu-
THE TERRESTRIAL GLOBE.
189
PROB 78.
To find the hour lines on the plane of any dial) by one position of the
globe.
Rule. IF the meridians do not pass through every 15, draw
them with a black lead pencil or ink, by bringing every 1 5 on the
equator to the brass meridian, &c. then elevate the pole as many
degrees above the horizon as are eqUal to the latitude ; bring aries
to the brass meridian ; all the other meridians will then cut any
circle representing the plane of a dial, in the number of degrees on
that circle, that each respective hour line is distant from the 12
o'clock hour lines passing through the same circle.
PROB. 79.
To find in ivJiat part of the earth any dial plane, which in not hori-
zontal in a given latitude^ will become horizontal.
As every plane, whatever be its situation, is parallel to the hori*
zon of some place on the earth, hence a dial, though not horizontal
in one place may become so in another, and the horizontal dial
made for the latitude of this place, will be the same as the former ;
thus, For an erect direct solith or north dial* Find the co. lat. of
latecl by the following proportion ; R : Cos. lat. .-.- tang, hour angle : tang, hour
arch The hour angles are omitted, being- the same as those in the table in
the note of the foregoing- prob. The reason of the rule is given in the pre-
ceding observations.
m
12i
S\ I*
2 50'
III
I,
20
23
26
30
33
37
^
IV
4
fe4l
50'
39
36
42
57
20
*i
V
5A
VI
The geometrical construction of this dial being the same as for a horizontal
dial, made for the complement of the lat. or in ex. 1, for lat. 49 17', the me-
thod given for constructing the horizontal, dial (note to prob. 76) will answer
for this. The point C, or the centre, may be taken near the top of the plane
of the dial. The forenoon hours are to be numbered towards the west, and
the afternoon hours towards the east ; and the angle for the stile must br
made ~ co. lat. See the preliminary remarks. The two dials described in
this and the foregoing prob. are the most useful, and therefore the most
common.
To find whether a wall be due south for a vertical south dial, erect a gno-
mon perpendicularly to it, and hang a plumb line from it ; then the watch
being adjusted to apparent time, if when it points out 12 o'clock the shadow
of the gnomon coincides with the plumb line, the wall is due south.
* An erect "vertical dial is that which is drawn on a plane perpendicular to
the horizon ; but a direct diql fa^es the east, west, north, or south points of
i\\r* horizon.
190 PROBLEMS PERFORMED BY
the given place, then the horizontal dial made for any place in this
lat. having its longitude the same as the given place, will be the
erect direct north or south dial required
Thus an erect dial under the pole will be an horizontal dial under
the equinoctial ; an erect dial in lat. 30 will be a horizontal dial in
lat. 60, Sec. and the contrary.
For an erect decliner* The globe being rectified to the given
latitude, bring the given place to the brass meridian ; reckon the
declination on the horizon, frorn the north or south ; that place on
the globe, opposite the point where the reckoning ends, will be the
place required. Thus an erect decliner, which in New-York de-
clines 60 from the south towards the east, will be horizontal in
lat 22 S. long. 3 W. If it declines 90 from the south towards
the east or went, it will become a horizontal dial on the equator in
long, 18 E. or 162 W. &c. If the plane declines eastward, the
sun will come later to the meridian of it, than to the meridian of
the place where it becomes a horizontal dial ; or sooner if the plane
declines westward, by as many degrees or by as many hours, mi-
nutes, Sec. as are equal to the difference of longitude.
For a direct recliner ,\ the complement of the plane's reclination
will be the latitude, where it becomes a horizontal dial in the same
longitude.
For a declining recliner. Rectify the globe for the given lat.
bring the given place to the brass meridian ; screw the quadrant of
alt. in the zenith, bring the quadrant to coincide with the degree
of the plane's declination on the horizon, and count from the hori-
zon on the quadrant the degrees of the plane's reclination, under
which mark the place on the globe, this will be the place required,
where the declining recliner will become a horizontal dial, and its
latitude and longitude may be found by the globe. The difference
of longitude changed into time, will give the difference that the
sun makes between the two meridians.
The learner can have no difficulty in applying the above princi-
ples when understood, to any kind of dial and for any latitude. For
more information, the treatises on dialling before referred to, may
be consulted. The demonstration of the above properties are given
in sect. 1. Emerson's Dialling.
* A declining dial is a dial that faces none of the cardinal points, but de-
clines towards the east or west, and an inclining dial is that whose plane
makes oblique angles with the horizon ; the inclination is the angle which it
makes with the horizon.
| The reclination of a plane is the angle it makes with a vertical plane, or
the number of degrees it leans from you, being the plane's distance from the
zenith ; and the declination of a plane is an arch of the horizon contained be-
tween the plane and the prims vertical, or between the meridian and plane
perpendicular to the dial plane, and is always reckoned from the south oi
north.
THE TERRESTRIAL GLOBE.
191
The inclination, recltnation and declination of a plane, may be thus
found.
1 . To find the inclination of a plane. Let
AB be a plane inclined to the horizon HR ; to
this plane let the quadrant CDE be applied so
that the plummet CF may touch its edge, or
the surface of the quadrant ; then the arc DF
will be the measure of the plane's inclination
or the angle BAH. Draw AG perpendicular
to HR, then because CF is parallel to AG,
the angle ACF = CAG (29 E. 1.) but DCE
II Aft
= GAH, both being
right angles, therefore the remainders are equal or DCF= BAH.
Q. E. D.
2. To find the reclination of a plane. Let
AB be the reclining plane, and AG perpen-
dicular to HR, the hor. will represent the
prime vertical ; then the angle BAG will be
the plane's reclination. Draw 1C perp. to
AB, and apply the quadrant CDE to this
perp. then will the arc DF, or the angle
DCF, be the measure of the reclination. For H
in the rt. angled A AIK the angles AKI -f I AK = 90" = DCE ;
but because CF is parallel to GA, the angle ECF = AKI (29 E. 1)
therefore DCF = K AI. Q. . JD.
3. To find the declination of any plane. Let
ABDE be a piece of board, &c. whose surface is
a rt. angled parallellogram, on which let the cir- Gr
cle x-v be described. On the centre C let a per-
pendicular pin or wire be erected, and place the
plane AD in a horizontal position, with the side JJ
AB applied to the dial plane ; observe when the
shadow of the top of the pin is afi> in the fore- ^~ 13
noon, and at x in the afternoon, on the same day, (see prob. 75)
let the diameter FG bisect x-v, and draw FH parallel to AB, it is
evident that the angle GFH will be the angle of declination re-
quired.
Or, If the square box of an azimuth compass be applied to the
dial plane (i. e. the north side of the box to a south plane, and the
south side to a north plane) so that the box be kept horizontal ;
then the needle will point out the plane's declination, regard being
had to the variation. For other methods see Emerson, See. When
great exactness is required, the problems on the azimuth, &c. will
afford the learner various astronomical methods, &c.
PROBLEMS
PERFORMED BY THE
CELESTIAL GLOBE,
PART III.
mmmmmmimmmt^fmm
PROB. 1.
To find the right ascension and declination of the sun or a star.*
Rule. BRING the sun's place in the ecliptic, or the star, to the
brass meridian ; then the degree which is over the sun's place, or
the star, on the meridian, is the declination, and the degree of the
equinoctial, cut by the brass meridian, reckoning from aries east-
ward, is the right ascension.
* The right ascensions and declinations of the moon and the planets, must
be found from astronomical tables or from a good ephemeris, as they cannot
be represented on the globe on account of their continually changing their
places in the heavens. In the 4th page of the month in the Nautical Alma-
nac, their longitude, latitude, declination, and passage over the meridian,
are given ; and hence as their longitude is given, their it. ascension may be
easily found, by calculation. The declination of the sun or a star may be
thus observed; when the sun is nearer to the equator than the place, the dif-
ference between the complement of the altitude (or the zenith distance) of
the sun or star wJl give the declination. (See notes to prob. 8 and 48, part
2.) The decimation being given, the proportion for determining the sun's
rt. ascension is given in the note to prob. 49, part II.
To find the rig-ht ascension of a star by observation. With a good pendu-
lum clock adjusted, that the hand may run through the 24 hours in the time
that a star leaving the meridian will come to the same meridian again
(which time is equal to 23h. 56' 4" 1, taking for unity the mean astronomical
day, being less than the natural day by the space the sun moves through in
the mean time eastward.) The clock being thus adjusted, when the sun is
in the meridian, set the hand or index to 12 ; observe when the star comes
to the meridian, and then observe the time shewn by the clock ; this time,
or the hours, minutes and seconds described by the index, turned into de-
grees and minutes of the equator, will give the difference between the right
ascension of the sun and star ; this difference added to the right ascension
of the sun, will give the right ascension of the star.
If the dial plate on the clock, instead of being divided into 24 hours, be
divided into 360, and their sexagesimal parts, and if at the moment the
sun is on the meridian, the index be placed to the number of degrees and
minutes the sun's rt. ascension then consists of, the index will then point
out the right ascension of the star, when it comes to the meridian. By know-
ing the right ascension of one star, we may from it find the rt. ascensions of
all the others which are visible, by finding the difference of the time of their
coming to the meridian, which converted into degrees and minutes of the
equator, will give the difFerence of their right ascensions. Or the declination
and right ascension of one star being given to find the right ascension of an-
other whose distance from the Termer and its declination are given. As the
PROBLEMS, UV. 193
Or, Place both poles in the horizon, bring the sun's place, OP
star, to the eastern part of the horizon ; then the degree cut on the
horizon, from the east, northward or southward, will be the decli-
nation, north or south, and the degree on the equinoctial, from
aries to the horizon, reckoning as before, will be the right ascension.
Example 1. Required the right ascension and declination of
Aldebaran in Taurus ?
Ans. Right ascension 66 6 ; 45", declination 16 5' 50".
2. Required the right ascension and declination of the following
stars ?
at, Acherner in Eridanus :;;,: a. Belelgeux in Orion
a Alioth in the Great Bear :;;:: y Bdlatrix in Orion
a Arcturus in Bootes :;!:: a Cafiella in Auriga
y Algenib in Pegasus :: a Menkar in Cetus
y Algorab in the Crow :;';:: a Procyon in the Little Dog
a Antares in Scorpio :;;:: a Regulus in Leo
Atair in the Eagle j;!ii a Syrius in Canis Major
PROB. 2.
77*? rz-A ascension and declination of the sun, a star, the moon, a
planet, or a comet being given, tojind its place on the globe.*
Rule. BRING the given right ascension to the brass meridian,
and under the given declination, you will find the place of the
planet or star required.
complements of the declinations and the given distance between the two
stars form a spherical triangle, and that the angle at the pole included by
the circles of declination passing through both stars, is equal to the differ-
ence of their right ascensions ; this angle being therefore grven, the differ-
ence of the star's right ascensions is given, and must be added to, or sub-
tracted from, the right ascension of the given star, according as the circle of
declin. passing through it, is east or west of the circle passing through the
other. If it cannot be subtracted, 360 must be added to the other, and then
the difference of rt. ascensions must be subtracted from the sum.
The rt. ascension and decl. of a star may be also thus found ; having the
latitude of the place, the hour from noon, and the sun's right ascension,
together with the altitude and azimuth of the star given. For there are
given the complement of the alt. the complement of the latitude, and the an-
gle included by these sides ; being the star's azimuth, or what it wants of
180. Hence the star's decl. and the arch of the equator between the brass
meridian and the circle of decl. passing through the star, is given, by the
rule in the note to prob. 54, part 2 ; and as the hour of the day is given,
the arch of the equator intercepted between the meridian and circle of decl.
passing through the sun, is likewise given. The sum of these arches if the
sun and star be on different sides of the meridian, or their difference if on
the same side, will give the difference of the right ascensions of the sun and
star, from which the right ascension of the star is known. The decl. is
found as in the note prob. 54. Many other methods could be given, but
our contracted limits would not permit. (See Gregory's Astronomy, b. 2.
sect. 5. Keil's Astronomy, Lecture 19th. Vince's Astronomy, or notes to
prob. 8th. part 2d.)
* As the latitudes and longitudes of the planets are given in pa. 4 of the
month in the Nautical Almanac, their right ascension* and declinations front
Aa
194 PROBLEMS PERFORMED BY
This may be performed on the horizon as the foregoing.
Note. The star's right ascension may be given in time, or in degrees,
both being marked on the equinoctial.
Example \ . Required the star whose declination is 30 40' 39" S.
and right ascension 22h. 46m. 33s. or 341 38' 15" I
Ans. Fomalhaut in the southern fish.
2. On the 2 1 st of April, 1 8 i 1 , the moon's right ascension was
9 58' 9", and her declination 2 53' north ; required her place
on the globe for that time ?
Ana. In pisces between the star and the equator.
i hence may be thus found ; the complement of the lat. (or the lat. -f- 90) the
compl. of the decl. and the distance between the poles of the equator and
ecliptic form a spherical triangle, two of the sides of which are given, viz.
the distance of both poles, and the compl. of the lat. Sic. and likewise the an-
gle included by them at the pole of the ecliptic, being equal to the distance
of the given planet in longitude, from the colure, reckoning on the ecliptic.
Hence to find the decl. Rad. : cos. angle at the pole of the ecliptic :: tangt.
co. lat. (or 90+ lat.) : tangt. x from the pole of the ecliptic ; then the sum.
or difference of x and 23 28' =y, and cos. x : cos. y :: sine lat. : sine decl.
The declination being then given, the right ascension is thus found ; Co. sine
decl. : sine angle at the pole of the ecliptic, or distance of the star in longi-
tude from the solstitial coiure :: cos. lat. or sine 90 -f- lat. : sine angle at the
pole of the equinoctial, or the angle formed by the colure and compl. of the
decl. This ang'le being equal to the distance, in right ascension, from the
given planet to the colure, whence the right ascension is easily found. (See
the note to the following problem.)
The declinations of the planets are given in pa. 4 of the Nautical Almanac,
but not their right ascensions. In the same manner may the right ascen-
sions and declinations of any of the fixed stars be obtained, from their lati-
tudes and longitudes being given.
On the 1st of May, 1811, the latitude of Jupiter, as seen from the earth,,
was 31' south, and his longitude 2s. 5 49' or 65 49', from which his rt,
ascension and decl. is thus found ; Rad. : cos. angle at the pole of the eclip-
tic, 90 65 49'= 24 11' :: tang. 90 31' or 89 b 29' : tang, a-, 90 34' ; ami
cos. x 90 34' or 89 26' : cos. y y 90 34' 23 28' = 67 6' :: sine lat. 31' :
sine decl. 20 47' N. agreeing with the Nautical Almanac. Again, cos. dec!,
20 47' : sine 90 -65 49' = 24 11' :: sine 90 31' or 89 29' : cos. right as-
cension 64 1'.
The annual variation of the stars in right ascension and decl. owing to th<r
precession of the equinoxes, which, according to La Place, is 50" 1 annually,
and also to the nutation of the earth's axis, is not considered in these prob-
lems. See La Place's Astronomy, vol. 1. b. 1. ch. 11. or his treatise of Ce-
lestial Mechanics, where the laws of these phenomena are investigated, and
agree with observation, as Nevil Maskelyne remarks, to surprising exactness.
The learner will also find all the problems relative to the latitude, longi-
tude, right ascension and declination of the planets or stars, solved from ac-
curate tables in Mayer. In the 3d edition of La Land's Astronomy, accu-
rate tables of the places of the planets, and of Jupiter's satellites, are given.
In the above calculations where greater exactness is required, the seconds,
&c. must be used, for which purpose Taylor's tables of logarithmic sines and
tangents will be useful, as they arc calculated to seconds ; Gardner's or
Hutton's may also answer.
For the annual alteration of declination and right ascension of a fixt star
through the precession of the equinox, or alteration of longitude, see the
theorems in Simpson's Fluxions, vol. 2. sect. 1. prop. 2.
THE CELESTIAL GLOBE. 195
3. Required those stars, whose right ascensions and declinations
are as follow ?
RIGHT ASCENSIONS.
DECLINATIONS.
in time.
in degrees.
12h.
15m.
38s.
=
183
54'
30"
69
59'
26"
S.
13
14
40
198
55
10
6
43
S.
16
17
9
244
17
15
25
58
23
S.
18
30
10
__
277
32
30
38
36
25
N.
20
34
36
308
39
44
34
21
N.
22
46
33
_
141
58
15
30
40
39
S.
22
54
5
=3=
343
<6
15
27
8
N.
22
54
48
343
42
14
8
3
N.
2
57
=
44
15
13
57
40
N.
52
8
-
13
2
89
9
52
N.
4. On the first of December, 1 8 1 0, the moon's right ascenskm
was 320 28', and her declination 11 45' S. ; required her place
on the globe ?
5. On the 1st of May, 1805, the declination of Venus was 11
41' N. and her right ascension 31 30' ; find her place on ihe
globe ?
6. On the 1 9th of January, 1805, the declination of Jupiter was
19 29' S. and his right ascension 238 ; required his place on
the globe I
PROB. 3.
To find the latitude and longitude of a given star.*
Rule. BRING the north or south pole of the ecliptic to the me-
ridian (according as the star is on the north or south side of the
ecliptic) elevate the pole 66i<> above the horizon j screw the
* The latitudes and longitudes of the planets must be found from astrono-
mical tables, from the Nautical Almanac, or from any good ephemeris ; or
their right ascensions and declinations being given (see the table at the end
of this work) their latitudes and longitudes may be found as follows : Bring
the star to the brass meridian, and draw with a pen and ink, or rather with
a fine pencil, a circle of declination from the pole of the equator through it ;
then if no circle of latitude pass through the star, draw by the help of the
quadrant of altitude a circle of latitude from the pole of the ecliptic through
it, as before, and intersecting the former at the given star ; then the comple-
ment of the declination, the compl. of the latitude of the star (or the lat. -f-
90) and the distance between the pole of the equator and ecliptic, which is
equal to the sun's greatest declination, or the obliquity of the ecliptic, will
form a spherical triangle, two sides of which, viz. the comp. of the decl. of
the star, and the distance of both poles are given, together with the angle
formed at the pole of the equinoctial between the arch of the solstitial co-
lure passing through both poles, and the arch representing the compl. of
the decl. being equal to the distance of the star in right ascension from the
colure or its supplement ; from which the complement of the latitude is
found thus : Conceive a perpendicular arch let fall from the given star to
the colure ; then it will be Rad. : cos. angle at the pole of the equinoctial
formed by the colure, and compl. decl. :: co. tang. decl. : tang, x, the dis-
tance between the pole of the equinoctial and the perpendicular, the stim or
196
PROBLEMS PERFORMED BY
quadrant of altitude in the zenith, keep the globe from revolving
oh its axis, and move the quadrant until its graduated edge comes
over the given star ; then the degree on the quadrant, cut by the
star, will be its latitude, and the sign and degree cut by the quad-
rant on the ecliptic, will be its longitude.
Or, Place one end of the quadrant on the given pole of the
ecliptic, and move the other end until the star comes to its gradu-
ated edge ; then the number of degrees reckoned on the quadrant,
between the ecliptic and the star, will be the latitude, and the
number of degrees on the ecliptic, reckoning eastward from the
point aries to the quadrant will be the longitude.
Example 1. Required the latitude and longitude of Aldebaran f
in Taurus ?
An&. Lat. 5 29' S. longitude 2 signs 7 or 7 in Gemini.
2. Required the latitudes and longitudes of the following stars ?
a, Altair in the Eagle j:< y, Rastaben in Draco
, Scheat in Pegasus jj > Arcturus in Bootes
a, Fomalhaut in S. Fish ;; , Rigel in Orion
d table of the longitudes of the nine principal fixed stars ^
made use of in the Nautical Almanac , for determining S
the longitude, calculated for the beginning ef the year ^
1809, with their latitudes for the mid. of the same year. S
Longitude beg.
0/1809.
Annual Latitude middle
increase. 1 of 1809.
Jinnual S
variation. S
9 , Arietis
Aldebaran
Pollux
Regulus
Spica virg.
Antares
^ a, Aquilas
Fomalhaut
\ a, Pegasi
Is. 4 59' 31" u
27 7 10 4
3 20 34 44 5
4 27 10 27 8
6 21 10 31 7
87 5 44 1
9 29 4 55 8
11 1 10 21 6
11 20 49 33 6
ou" 271
50 204
49 470
50 004
.50 059
50 141
50 870
50 717
50 133
9 57' 37" 5 N.
5 28 48 7S.
6 40 15 7N.
27 35 5N.
2 2 13 8S.
4 32 22 3S.
29 18 59 4N.
21 6 26 7S.
19 24 46 9 N.
4-0" 180 ^
317 S
4-0 280?
4-0 200?
4-0 080 S
4-0 167 >
-f-o 372$
4-0 013 S
-4.0 163,;
difference of which, and the distance between both poles, will give y, the
distance between the pole of the ecliptic and the same perpendicular ; then
cos. x : cos. y :: sine decl. : sine latitude.
Now to find the longitude, we have this proportion ; Cos. lat. of the star
or sine 90 -J- lat. : sm e angle at the pole of the equinoctial :: co. sine decl. .1
sine angle at the pole of the ecliptic, formed by the solstitial colure and
compl. lat which will give the distance in longitude reckoned on the eclip-
tic from the circle of lat, passing through the given star to the next solsti-
tial colure, if the angle be greater than 90, but its supplement if less ;
from which the longitude of the star is given. This will appear plainer by
having the globe with the figure delineated on it as directed above ; the fol-
lowing example will render the method more evident. The right ascension
of Aldebaran being given, 66 6' 45" or nearly 66 7', and his declination
16 5' 50" or 16 6' nearly, for the year 1800 ; his latitude and longitude are
required. Here we have Had. : cos. 90 66 7'*= 23 53' :: eo. tang-,
THE CELESTIAL GLOBE. 197
PROB. 4.
The latitude and longitude of the moon, a star, or a planet, being
given, to find its filace on the globe
Rule. BRING the north or south pole of the ecliptic to the brass
meridian, according as the latitude is north or south, elevate the
pole 66 ; screw the quadrant in the zenith, over the elevated
pole, and extend it over the given longitude in the ecliptic ; then
under the given latitude, on the graduated edge of the quadrant,
you will find the star, or the place of the moon or planet.
Examfile 1. Required the star whose longitude is 3s. 22 56",
and latitude 15 58' south ?
Ans. Procyon in the little dog.
2. On the 1st of May, 181 1, at noon, the moon's longitude was
4s. 20 35' 30", and her latitude 2 57' 20" ; required her place
on the globe ?
5. Required the stars which have the following longitudes and
latitudes ?
Longitudes Latitudes. Longitudes. Latitudes.
3s. 11 14' 390 33 ; S. r:;;; 3s 17 22' 10 4' N.
6 20 57 2 3 S. i:;ii n 25 25 41 N.
9 28 51 29 18 N. i;:;! 2 6 53 2 59 S.
4. On the 1st of December, 18 11, the longitudes and latitudes
of the planets will be as follows ; required their places on the
globe I
Longitudes. Latitudes.
% Mercury 8s. 15 23' 1<> ,35' S.
9 Venus 8 21 1 35' S.
% Mars 10 10 31 1 24' S.
11 Jupiter 342 16' S.
h Saturn 8 26 43 O 55' N.
$ Herschel 7 20 21 18' N.
Note f In the above and in most other examples, the geocentric places of
rhe planets are made use of, or their places as seen from the earth's centre,
being more convenient for a spectator on the earth than their heliocentric or
true places, as seen from the centre of the sun. The learner must also take
notice, that the planets' places are given for the meridian of Greenwich.
16 6 f :: tangt. x == 72 29'. In this case, therefore, y => 72 29' + 23 28'
= 95 57'. Whence Cos. x 72 29' : cos. y 95 57' or its suppl. 84 S' ::
sine decl. 16 6' : sine lat, 5 29' as above. Again, for the longitude it will
be, sine 90 -f- 5 29' == 95 29' or sine suppl. 84 31' : sine angle at the pole
of the equinoctial 180 23 53' 156 7' or its sup. 23 53' :: cosine dec).
16 & : sine angle at the pole of the ecliptic = 23. Now as the right as-
cension is in the first quadrant from aries, the longitude is in the same ;
hence 90 23 = 67 = the longitude of Aldebaran agreeing with that
given in the Nautical Almanac, reduced to the year 1800.
In this manner the places of the stars in general are calculated, and a
catalogue of them is made. In like manner the latitude and longitude being
given, the right ascension and declination are found, Sre the note annexed
to the foregoing- problem.
198 PROBLEMS PERFORMED BY
PROB. 5.
The latitude of a place being given, to find the amplitude of any
its oblique ascension, and descension, its ascensional difference, and
time of its continuance above the horizon.*
Rule. ELEVATE the pole to the given latitude, bring the given
star to the eastern part of the horizon ; then the degrees between
the star and the east point of the horizon will be its rising ampli-
tude, and the degree of the equinoctial cut by the horizon will be
the oblique ascension. The globe being kept in this position, set
the hour index to 1 2 ; then turn the globe westward, until the
given star comes to the brass meridian, and the hours passed over
by the index will be the star's semidiurnal arch, or half the time of
its continuance above the horizon ; the degree cut on the equinoc-
tial by the brass meridian, will be the star's right ascension, the
difference between which and the oblique ascension is the ascen-
sio?ial difference. The setting amplitude, and oblique descension,
are found by continuing the motion of the globe, until the star comes
to the western part of the horizon, &c.
Example \ . Required the rising and setting amplitude of Pro-
cyon, its oblique ascension and descension, ascensional difference,
and diurnal arch, at New-York ?
Ans. The rising ampl. is 7 to the north of the east, the setting
ampl. 7 north of the west ; oblique ascension 1071, oblique de-
scension 117; right ascension being 112 12', the ascensional
diff is therefore 5 nearly ; the semidiurnal arch is 6h, 20m. and
hence the time of its continuance above the horizon is 12 hours
40 minutes.
2. Required the rising and setting amplitude of Sirius at Phila-
delphia, also its oblique ascension and descension, ascensional and
descensionai difference, and the time of its continuance above the
horizon ?
3. Required the rising and setting amplitudes of Aldebaran,
Arcturus, Rigel, Regulus and Deneb ; together with their oblique
ascensions and descensions, ascensional differences, and their semi-
diurnal arches at London ?
PROB. 6.
The latitude, day of the month, and hour being given ; to place the
globe in such a position as to represent the heavens, at that time,
as seen from the given place ; in order to Jind out the relative
situations and names of the constellations and visible stars.
Rule. ELEVATE the pole to the given latitude ; place it north
and south (by the compass, allowing for variation, if any, or by a
meridian line) bring the sun's place in the ecliptic to the brass
meridian, and set the index to 12 ; then if the time be in the af-
* For the method of calculating this problem, see the notes to problem
49, part 2.
THE CELESTIAL GLOBE. 1D9
ternoon, turn the globe westward, but if in the morning, eastward,
as many hours as the given time is after or before noon, the globe
being fixed in this position ; then every star on the globe will cor-
respond to the same star in the heavens, and a perpendicular erect-
ed over any of them, will point out the same star in the heavens.
By this means the constellations and remarkable stars may be
easily known. All those stars which are on the eastern side of the
horizon are then rising ; all those on the western side are setting ;
all those under the brazen meridian are on the meridian of the
place at the given hour ; those stars between the south point of the
horizon and the north pole, have their greatest altitude, if the lati-
tude be north, but those stars between the north point of the hori-
zon and the south pole, are at their greatest altitude if the latitude
be south. That star in the zenith, if any, is vertical, and if the
sun's place be brought to the brass meridian, below the horizon,
all those stars above the horizon whose declinations are equal to
the given latitude, will be vertical successively, and visible in the
given place.
Note. The globe should be taken into the open air or near a large win-
dow, on a clear night, where the view on the surrounding- horizon is not
intercepted by different objects ; a small observatory erected on the top of
a house where the roof is flat, or nearly so, would best answer the purpose,
PROB. 7.
The latitude of a filace, day of the month) and hour being given ; to
find what stars are rising, setting-, on the meridian^ c3*c.
Rule. RECTIFY the globe for the given latitude ; bring the
sun's place to the meridian, and set the index to 12 ; then if the
time be in the forenoon, turn the globe eastward, but if in the af-
ternoon, westward, until the index has passed over as many hours
as the time is before or after noon ; then all the stars at the east-
ern semicircle of the horizon will be rising, those at the western
semicircle will be setting, those under the graduated edge of the
i>rass meridian, above the horizon, will be culminating or on the
meridian ; all those that are above the horizon will be visible, and
those below it invisible, at the given time and place ; if the globe be
turned on its axis from east to west, those stars that do not descend
tfelow the horizon, never set at the given place, and those which
do not come above the horizon, never rise. These circles of per-
petual apparition or occultation may be found by describing circles
on the globe, parallel to the equinoctial, at a distance from it equal
to the complement of the latitude.
JExamfile I. At 10 o'clock in the evening in New-York on the
10th of May, required those stars that are rising, setting, on the
meridian, &c. ?
Ans. Altair in the eagle is rising ; Spica in virgo, the two stars
mizar and alcor in the tail of the great bear, and in Cassiopeia, are
"nearly on the meridian ; Procyon is about 6 above the western
200 PROBLEMS PERFORMED BY
point of the horizon, the star marked y in getnini, is nearly set-
ting, &c.
2 On the 16th of November, at 4 o'clock in the morning, at
New-York, what stars are rising, setting, on the meridian, &c.
Ans. Arcturusis after rising about 5 above the K. N E part
of the horizon, Procyon is on the meridian, Pollux is near the me-
ridian, and Castor after passing it ; in Andromeda, and mirac in
Cetus, are near the western part of the horizon, Sec.
3. On the 9th of February, when it is 9 o'clock in the evening
at London, what stars are rising, on the meridian, setting, &c. ?
4. Required those stars that never set in the latitude of New-
York, and at what distance from the equinoctial is the circle of
perpetual apparition ?
5. Required those stars that never rise at Cape Horn, and those
that never set at Copenhagen ?
6. Required those stars that are always above the horizon at the
north pole and also those that cannot be seen there ?
7. How far must an inhabitant of New-York travel southward
to lose sight of Arcturus ?
8. In what parallel of latitude do those reside to whom Sirius h
never visible but when in their horizon ?
9 In what latitude do those reside to whom Aldebaran is always
vertical when on their meridian ?
Note. When the decl. of the star is equal to the lat. of the place, the
star will be always vertical in that lat. when on the meridian.
PROB. 8.
To Jind at what hour any star or planet will rise^ come to the
meridian, and set at any given place*
Rule. ELEVATE the pole to the given latitude, bring the sun's
place in the ecliptic to the brass meridian, and set the index to 12 ;
then bring the star or planet's place to the eastern part of the hori-
* The apparent time of the transit of any star over the meridian is thus
found ; subtract the sun's right ascension in time at noon from the star's
right ascension in time, increased by 24 hours, if necessary ; the remain-
der is the apparent time of the star's passing- the meridian nearly ; from
which the proportional part of the daily increase of the sun's right ascen-
sion, for this apparent time from noon (corrected by the longitude you
are in or difference of longitude from Greenwich) being subtracted, the
remainder will be the correct time of the star's passing the meridian.
The apparent time of a star's rising or setting is found by applying its
semidiurnal arch answering to its declination, and the latitude of the
place, by subtraction or addition, to the time of its transit over the meri-
dian. The semidiurnal arch is thus found ; the complement of the lati-
tude of the place, the complement of the star's declination, and the dis-
tance from the vertex to the point where the star rises or sets (whicli is
always equal 90) forming a quadrantal triangle, are given, to find the an-
gle at the pole of the equinoctial formed by the brass meridian, or meri-
dian of the place, and the circle of declination passing through the star,
which will be the semidiurnal arch required. Or the decl. of the star, its
THE CELESTIAL GLOBE. 201
Son, and the index will point out the time of the star's rising ;
turn the globe westward until the star, or planet's place comes to
the brass meridian, and the index will shew the time of the star's
coming to the meridian of the place ; continue the motion of the
globe westward until the star or planet's place comes to the west-
ern part of the horizon, and the index will shew the time of its
setting.
Note 1. The trme may be more accurately found on the equator, always
reckoning the hours between the meridian passing through the sun's place
and the brass meridian, for the time before or after noon when the star or
planet rises, sets, &c.
If the sun's place be to the east of the brass meridian, the star or planet
will rise before noon, but if to the west, the star or planet will rise in the
afternoon.
Example 1 . At what time will Sirius rise, come to the meri-
dian, and set at New-York, on the 2d of November ?
Ans. It will rise at 9 o'clock in the evening, come to the meri-
dian at 2 in the morning, and set at 7 in the morning.
2 On the 13th of May, 1811, the longitude of Jupiter was 2
signs 8 3 1 ', and his latitude 30' south ; at what time did he rise,
culminate, and set at Greenwich, and whether was he a morning
or an evening star ?
Ans. He rose at 5 o'clock in the morning, came to the meridian
at 5 min. after 1 in the afternoon, and set about 10 minutes after 9
at night. Jupiter was here an evening star, because he set after
the sun.
amplitude, and what the semidiurnal arch reckoned on the equinoctial ex-
ceeds or wants of 90 (according as the decl. is of the same or a different
name with the lat.) and the angle in this triangle included by the equi-
noctional and amplitude, or the inclination of the equinoctial to the hori-
zon, is the comp. of the lat. of the place ; whence by Napier's rule, Rad. :
tatigt. lat. :: tang-, decl. : sine of an angle which added to or subtracted
from 90, according- as the star's decl. is of the same or a different name
from the latitude. (This gives the investigation of the note in prob. 13,
part 2d.)
The apparent time of a planet's passing the meridian may be found
thus ; let the planet's right ascension, the preceding noon or midnight,
be calculated from its longitude and latitude (by note to prob. 2) and
turned into time ; subtract the sun's right ascension in time, the same
noon or midnight from it, the remainder will be the time of the planet's
passing the meridian nearly, which call x ; take the difference of the
sun's daily or half daily variations in right ascension in time, if the planet
be progressive in right ascension, or the sum if it be retrograde, which
call y ; then say as 24h. ^ y or 12h. : y . 24h. or 12h. (according as
the daily or half daily variation is used) :: x : to the time of the planet's
passing the meridian. The sign -f- is to be used if the planet's progres-
sive motion in right ascension be greater than the sun's ; in any other case
the sign is to be made use of. Where accuracy is required, the 2d differ-
ences of the right ascension should be allowed for, and the difference of
longitude, if for any other meridian different from that of Greenwich.
See the method of allowing for these differences at the end of the Nauti-
cal Almanacs for 1811, 1812 or 1813, published by Mr. John Garnett,
See also Emerson's differential method in his Conic Sections,
B b
202 PROBLEMS PERFORMED BY
Note 2. When a planet rises or sets after the sun, it is then an evening
star, but \vhen it rises or sets before the sun, it is a morning star.
3. At what time does Aldebaran rise, set and come to the meri-
dian of Philadelphia, on the 4th of July ?
4. On the 1 st of October, 1811, the longitude of Venus will be
6s. 4 36', and her latitude 1 22' N. at what time will she rise>
set, and come on the meridian of Greenwich, and whether will she
be a morning or an evening star ?
5. On the first of June, 1812> the longitude of Mars will be
2s. 27 37', and his latitude 54.' north ; required the time of his
rising, coming on the meridian, and setting at Greenwich ?
6. The longitude of Saturn on the first of November, 1813, will
be 9s. 14 24', and his latitude 6' N. ; required the time of his
rising, culminating, and setting at Greenwich ?
PROB. 9.
To find on what day of the year a given star will be ufion the
meridian , at any given hour.
Rule. BRING the given star to the meridian, and set the index
to 12 ; then turn the globe westward or eastward, according as the
time is in the forenoon or afternoon, as many hours as the given
time is from noon ; the brass meridian will then cut the ecliptic
in the sun's place corresponding to the time required, which may
be found on the horizon.
Example 1 . On what day of the month does Sirius come to the
meridian of New- York, at 4 o'clock in the morning ?
Ans. The time being 8 hours before noon, the globe must there-
fore be turned 8 hours towards the west, the point of the ecliptic
then intersected by the brass meridian, will be 12 of Scorpio,
answering nearly to the 4th of November.
2. At what time of the year will Regulus in Leo come to the
meridian of Philadelphia, at 9 o'clock at night ?
Here the time being 9 hours after noon, the globe must there-
fore be turned 9 hours towards the east ; then the ecliptic will be
intersected by the brass meridian in 15j of Aries, corresponding
to the 5th of April, nearly.
3. At what time of the year does Procyon come to the meridian
of London, at 4 o'clock in the afternoon ?
4. At what time of the year does Arcturus come to the meri-
dian of Dublin, at 10 o'clock at night ?
5. At what time of the year does Alcyone in the Pleiades come
to the meridian of Washington city at noon,* or when the sun is
on the meridian ?
* If the given star comes to the meridian at noon, the sun's place will
be found under the brass meridian without turning the globe ; if the star
comes to the meridian at midnight, the globe may be turned eastward or
westward until the index has passed over 12 hours. When the time is
given for the meridian of any other place, it must be reduced to that of
the given place by prob. 6, part 2.
THE CELESTIAL GLOBE. 203
6. At what time of the year does Lyra in the harp, come to the
meridian of Boston, at midnight ?
7. On what day of the month, and in what month, does Spica
in Virgo come to the meridian of New-York, when the sun is on
the meridian of Constantinople ?
8. Having observed Aldebaran in Taurus pass the meridian of
George Town College, on the Potomac, when it was 8 o'clock in
the morning, by a time piece set to the meridian of Greenwich ob-
servatory ; required the month and day when the observation was
made ?
PROB. 10.
Given the latitude, day of the month, and hour, to Jind the alti*
tude and azimuth of any given star*
Rule. RECTIFY the globe for the given latitude, screw the
quadrant in the zenith, bring the sun's place for the given day to
the brass meridian, and set the index to 12 ; then if the given
The reason of the rule is evident, as the sun always comes to the meri-
dian at 12, and that it is distant from the meridian, on which the sun is
at the given time, as many hours as are equal to the time that the star
culminates before or after noon. This time is equal to the difference of
the sun and star's right ascensions ; when their right ascensions are equal,
they are on the meridian at the same time, that is at 12 o'clock. Hence
the problem may be easily solved by calculation.
* Here the day of the month being given, the sun and stars right ascen-
sions and declinations are given (by prob. 1. part 2,) Moreover the comple-
ment of the latitude of the place, the compl. of the star's altitude, and the
compl. of his declination, or his distance from the elevated pole of the equi-
noctial, form a spherical triangle. Now the sun and star's right ascensions
being given, their difference is given, or the distance between the sun and
star reckoned on the equator ; and as the distance of the sun from the meri-
dian is given in time, and consequently in degrees, being equal to the hour
from noon, the distance of the star from the same meridian is also given, being 1
equal to the remainder of the difference of their right ascensions, or equal
to the angle formed at the pole by the brass meridian, or meridian of the
place, and the circle of declination passing through the star. Therefore in
the above triangle there are given two sides, viz. the cornp. of the latitude,
and comp. of the declination of the star, and the angle included by these
sides, or the distance of the star in right ascension from the meridian, to find
the third side or compl. of the alt And as the angle included by the compl.
of the lat. of the place and the compl. of the alt. of the star, is the star's azi-
muth or its supplement, according as the north or south point of the horizon,
from which the azimuth is reckoned, is of the same or a different name from
the elevated pole. Hence to find the azimuth we have the following pro-
portions ; Rad. : cos. angle at the pole, or dist. of star in rt. ascension from
the mer. :: tangent dist. of the star from the pole, or 90 ltdecl. : tangt. x t
the distance from the pole of the equinoctional to the perpendicular let fall
from the star to the meridian of the place, the sum or difference between
which and the distance between the zenith and pole (according as the perpen-
dicular falls towards the zenith or in a contrary direction from the pole) cull
2i ; then sine x : sine y :: co. tang, angle at the pole, or distance of the star in
right ascension from the meridian : co. tang, azimuth. Again, sine azim. :
sine distance of the star from the pole, or 90^ decl. ;: sine an^-le nt the
pole : cos. altitude.
204 PROBLEMS PERFORMED BY
time be in the morning, turn the globe eastward, but if in the af-
ternoon, westward* as many hours as the time is before or after
noori ; keep the giobe in this position, and move the quadrant of
altitude until its graduated edge coincides with the centre of the
gi>;en star ; the degree then cut on the quadrant, reckoned from
the :;o:izon, Avill be the altitude, and the degree on the horizon, cut
by the quadrant, reckoning from the north or south, will be the
azimuth required.
Example i Required the altitude and azimuth of Arietis at
Philadelphia, when it is 5 o'clock in the morning of the 23d. of
September ?
Ann. The alt. is 47, and the azimuth nearly 78^ from the
south towards the west.
2. Required the altitude and azimuth of Altair in the Eagle, at
New- York, when it is 9 o'clock in the evening of the 2 1st of June ?
Am, The alt. is 20 22', and azimuth 83 23' from the south
towards the east.
3. Required the altitude and azimuth of Lyra in the harp, at
Washington city, at 3 o'clock in the morning of the 2 1st of March ?
4. Required the altitude and azimuth of Procyon on the 10th of
February, at 9 o'clock in the evening at London ?
5. On what point of the compass does the star Algol in Perseus
bear at New-York, on the 10th of August} at 10 o'clock in the after-
noon, and what is his altitude ?
Noie. The points of the compass are reckoned the same way as the
azimuth, allowing- 11 15' to each.
PROB. 11.
The latitude, day of the month, and the altitude of any known star
being given^ to find the hour of the night ^ and the star's azimuth.
Rule. RECTIFY the globe for the latitude, screw the quadrant
of altitude in the zenith, bring the sun's place for the given day
to the brass meridian, and set the hour index to 12 ; bring the
Thus in ex. 2. the sun's right ascension = 90 his distance from the me-
ridian or noon = 9 hours = 135. Altair's right ascension in time = 19h.
41' 1" = 295 15' nearly. Whence Altair's distance from the meridian in
right ascen. = 295 15' 9G J -f Lo = 70 15', the angle at the pole
formed by the co. lat. and circic of declination passing' through the star, and
the star's declination is 8 21' 8" N. or 8 21' nearly ; hence rad. : cos. 70
15' :: tang. 90 8 21' =81 39' or cot. 8 21' : tang, x 66 31'. Whence
66 31' 49 17' (co. lat.) = 17 14' = y ; then s. x 66 31' .- s. y 17 14'
:: cot. 70 15' : co. tang, azimuth 83 23'. Again, s. azim. 83 23' : s. dist.
of the star from the pole, or cos. decl. 8 21' :: s. 70 15' : cos. alt. 20
22^' nearly.
The learner will observe, that the places of the stars on our newest globes
are calculated for the year 1800, to which we have therefore adapted most
of our calculations, &c.
The calculation of prob. 50, part 2d. is performed in the snme manner as
the above. The variation of the compass may be obtained from this prob, in
the same manner as in prob. 50 above alluded to.
THE CELESTIAL GLOBE. 205
quadrant of altitude to the side of the brass meridian, east or west on
which the star was situated when observed, turn the globe west-
ward until the centre of the star cuts the given altitude on the
quadrant ; then the hours which the index has passed over, will
shew the time from roon when the star has the given altitude, and
the quadrant will intersect the horizon in the required azimuth *
Example 1. The star Altair in the Eagle on the 2 1st of June,
at New- York, was observed to be 20 224' above the horizon, and
east of the meridian ; required the hour of the night and the star's
azimuth ?
jins. The sun's place being brought to the meridian, and the
globe turned westward until the star cuts 20 22' east of the me-
ridian, the index will then have passed over 9 hours, and the star's
azimuth, indicated by the quadrant, on the horizon, will be 83 23' ,
from the south towards the east.
2. The altitude of Arietis was observed 47 at Philadelphia, on
the 23d of September, the star being west of the meridian ; re-
quired the hour and the star's azimuth ?
Ans. Here the globe being turned westward until the star cuts
the given altitude on the quadrant, west of the meridian, the index
will have passed over 17 hours corresponding to 5 o'clock in the
morn, and the azim. from the south towards the west is 79 nearly.
* The prob. being* performed as directed in the rule, the complement
of the latitude of the place, the complement of the star's altitude, and the
complement of its declination, will form a spherical triangle, and as the
three sides are given, the angles are therefore given. Now the angle
formed by the quadrant of alt. and the brass meridian is equal to the
star's azimuth, and the angle formed at the pole, by the circle of declina-
tion passing- through the star and the brass meridian, is the distance of the
star from the same meridian ; and as the sun and star's right ascensions
are given, their difference is therefore given, from which, if the distance
of the star from the meridian be taken, the remainder is the distance of
the sun from the meridian, which converted into time, will give the hour
required. Thus in example 1. The comp. of the lat. = 49 17', the
comp. of the alt. = 69 38', and the comp. of the star's decl. = 81 39'.
Hence by spherical trigonometry we shall have this proportion, tang. ^ co.
lat. 24 3*8' .- tang, of half the sum of the com. of the decl. (81 39') and com.
of the alt. (69 37 tf) = 75 38' :: tang-, of half the difference of the co.
of the decl. and co. of the alt. 6 0|' : tang, x == 41 53'. (See Em-
erson's Trig. b. 3. sect. 4. case 11.) Hence 41 53' -f- 24 38' (half the
co. lat. nearly) = 66 3 IS then by Napier's 1st rule, R : tang. 66 31' ::
tang. decl. of the star 8 21' : cos. 70 15', the angle formed'at the pole
by the brass meridian and circle of declination passing through the star,
or the star's distance from the meridian. Now the sun's right ascen. is
90, and the star's 295 15', the diff'. is therefore 205 15', from which the
distance of the star from the meridian being taken, the remainder 135 ( '
dist. of the sun from the mer. = 9 hours'; hence the time is 9 o'clock
in the evening.
To find the azimuth, the sines of the sides of spherical triangle belug-
as the sines of the angles opposite to them, it will be sine co. alt. 69 3/r,'
: sine co. decl. 81 39' :: sine 70 15' : sine azimuth 83 23^' nearly, as
required. The azimuth or its supplement will be found by this latter
proportion.
206 PROBLEMS PERFORMED BY
3. The altitude of Lyra at Washington city, on the 21st of
March, was observed 59 east of the meridian ; required his azi-
muth and the hour ?
4. The altitude of Deneb in the Lion's Tail, on the 28th of De
cember, at London, was observed 40 when -east of the meridian ;
required its azimuth and the hour ?
PROB. 12.
The latitude, day of the month, and azimuth of a star being given, to
find the hour of the night and the star'* altitude.
Pule. ELEVATE the pole to the given latitude, screw the
quadrant in the zenith, bring the sun's place in the ecliptic for the
given day to the brass meridian, and set the hour index to 12 ;
bring the graduated edge of the quadrant to coincide with the giv-
en azimuth on the horizon, and keep the quadrant in this position ;
turn the globe westward until the given star comes to the gradu-
ated edge of the quadrant, then the hours passed over by the in-
dex will be the time from noon, and the degrees on the quadrant,
reckoning from the horizon to the star, will be the altitude.*
Examfile \. On the 21st of June at New-York, the azimuth of
Atair in the Eagle was observed to be 83 23J', from the south to-
wards the east ; required the hour of the night and the star's alti-
tude ?
jins. The globe being turned on its axis, the index will pass
over 9 hours, corresponding to 9 o'clock in the evening, and the
star's altitude will be 20 22'.
2. On the 23d of September at Philadelphia, the azimuth of a
Arietis was 79 from the south towards the west ; required the
hour of the night and the star's altitude ?
3. On the 8th of October, the azimuth of the star marked & in
the shoulder of Auriga, was 49 from the north towards the east ;
required its altitude at London, and the hour of the night ?
* In this, as in the two preceding problems, the compl. of the lat. the
compl. of the star's alt. and compl. of his decl. form a spherical triangle,
two sides of which, viz. the co. lat. and the co. decl. are given, and the
angle opposite the co. decl. is the azimuth or its supplement ; hence
the other parts of the triangle may be found, by case 2, b. 3, sect. 4,
Emerson's Trig. Thus in ex. 1. s. co. decl. 81 39' : s. co. lat. 49 17' ::
s. azim. 83 23' s. oc 49 33' = the angle opposite the co. lat. ; then s.
810 */-" o , 81 39- +49 17' O ,
96 37' _ 49 33'
- - - = 23 32' : co. tang, half the hour angle, or half the dis-
tance of the star from the meridian 35 & ; hence the whole distance is
70 16', and therefore the hour of the night is found as in the last note.
96 37' is the supplement of the azimuth nearly, or 180 83 23 ; = 96
37' = the angle opposite the co. decl. Now to find the altitude, it will
be s. x. 49 33' : s. 70 16' : s. co. lat. 49 17' : s. co. alt. 69 38'; hence
the alt. is 20 22'. The results in this and the foregoing notes would ex-
actly agree if the seconds were retained, but as the calculations are giver.
only to illustrate the prob. such nicety was considered unnecessary*
THE CELESTIAL GLOBE. 207
PROB. 13.
The latitude^ day of the month, and two star's having the same
azimuth, being given y to find the hour of the night*
Rule. ELEVATE the pole to the given latitude, screw the quad-
rant of altitude in the zenith, bring the sun's place in the ecliptic
to the brass meridian, and set the hour index to 12 ; turn the globe
westward on its axis, until the two given stars coincide with the
graduated edge of the quadrant of altitude ; and the hours passed
over by the index, will be the time from noon. The common azi-
muth will be found on the horizon.
Example 1. At what hour at New-York, on the 22d of Septem-
ber, will Capella in Auriga and Castor in Gemini, have the same
azimuth, and what will that azimuth be ?
Ans. In turning the globe westward, &c. the index will pass
over 1 3 hours before the stars coincide with the quadrant, they
will therefore have the same aximuth at a quarter past one in the
morning, and the azimuth will be 63i from the north towards the
east.
2. At what hour at London on the 1st of May will Altair in the
Eagle, and Vega in the Harp, have the same azimuth, and what
will that azimuth be ?
3. At what hour will Arcturus and Spica Virginis, have the
same azimuth at Paris, on the 20th of April ?
4. At what hour will Arcturus and a Zuben el C. of Libra, have
the same azimuth at Boston, on the 21st of June ?
5. At what hour at Philadelphia will Procyon and Sirius have
the same azimuth, on the 2 1 st of March ?
Note. When the two stars have the same right ascension, they will
have the same azimuth when on the meridian, and as the star's passing-
the meridian is found by prob. 8, the hour is therefore given. If a cor-
rect table of those remarkable stars which have the same right ascension
were given, and the times of their passing the meridian of any remarka-
ble place, as that of Greenwich or Paris observatory, this would afford an
easy method of finding the hour of the night, as every star is on the me-
ridian of any place at the same hour. It would also afford a method of
* This prob. may be thus calculated ; let S, s
be the two stars, P the pole, Z the zenith, EQR
a portion of the equator, and Q the sun's place.
In the triangle SPs there are given SP, sP the
complements of the star's declinations, and the
angle SPs, the difference of the star's right as-
cension; hence the angle at S and s, and the
side Ss, are given, and therefore the angle ZSP,
the supplement of PSs, is given. Now in the ^P
triangle PsZ or PSZ, there are given sP or SP, the angles at s or S, and
ZPthe complement of the latitude; hence the angle PZS, =acthe suppL
of the azimuth, and therefore the angle QZS the azimuth are given, and
likewise the angle sPZ or SPZ, the distance of the stars from the meri-
dian PQ is given ; but as the angle *PR or SPR, the difference between
the sun and the stars right ascensions respectively* are given, therefore
208 PROBLEMS PERFORMED BY
PROS. 14.
The latitude, day of the month, and two stars that have the same,
altitude, being given, to Jind the hour of the night.*
Rule. RECTIFY the globe for the latitude, zenith and spin's
place, (prob 9 part I) turn the globe westward until the two giv-
en stars coincide with the given altitude on the quadrant, or until
the two stars be at the same distance from the horizon, if the alti-
tude be not given ; then the hours passed over by the index will
be the time from noon, when the two stars will have that altitude.
Example \ . At what hour at New- York, on the 20th of July, will
Belelgeux in Orion, and Castor in Gemini, have each 5 of alti-
tude ?
Ans. At 45 min after 3 in the morning.
2. At what hour at London on the 2cl of September, will Markab
in Pegasus, and in the head of Andromeda, have each 30 of alt. ?
3. At what hour at Philadelphia on the 1 8th of January, will Al-
tair in the Eagle, and Fomalhaut in the southern Fish, have each
12 of altitude ?
4 At what hour at Dublin, on the 1 5th of May, will n Benetnach
in the tail of the Great Bear and y, in the shoulder of Bootes, have
each 5 6 of altitude ?
the remaining- angle QPR, the distance of the sun from the meridian is
given, which converted into time, will give the hour required.
In like manner when two stars in one azimuth are given, and the alti-
tude of either being given, the latitude of the place may be easily found
on the globe, or by calculation thus ; if the altitude of S be given, its com-
plement SZ is given, and in the triangle SPs, SP, sP, and the angle SPs are
given, hence the angle PSs and its suppl. PSZ are given. Again, in the
triangle ZSP, SP, SZ, and the angle at S are given, hence ZP, which
is the compl. of the latitude, is given. See Emerson's Algebra, prob. 160.
page 448.
* The prob. may be thus solved by
Trigonometry, the altitude being giv-
en ; let S, s" be the two stars, P "the
pole, and Z the zenith ; then in the tri-
angle SPZ or sPZ, the three sides are
given, and the angle ZPS, or ZPs, the
distance of the stars S, or 5, respective-
ly, from the meridian, from which the
hour may be found as in the notes to
the last problems. From the solution
it is evident, that the alt. of one star alone is sufficient to determine the
hour when the lat. is given. When both altitudes are given, the lat. and
hour may be found thus ; in the triangle SPs, SP, sP, the complements of
the stars decl. and the angle SPs, the difference of their right ascensions
are given, and hence the side Ss, and the angle PS,?, are given. Again, in
the triangle ZSs, the three sides are given, and therefore the angle ZSs
is given, consequently the angle ZSP is given, and therefore ZS *.he co.
alt. and SP being given, ZP the co lat. is also given When neither of
the altitudes are given, the solution becomes rather tedious and trouble-
some.
THE CELESTIAL &LOBE. 209
5. At what hour, at George Town on the Potomac, will Aldeba-
ran, and Algol in Perseus, have each 17^ of altitude, on the 31st
of March ?
PROB. 15.
Given the azimuth of a known star, the latitude of the filace y and the
hour ; to find the starts altitude^ and the day of the month.*
Rule. RECTIFY the globe for the latitude, screw the quadrant
of altitude in the zenith, bring the graduated edge of the quadrant
to the given azimuth on the horizon, turn the globe until the star
coincides with the quadrant, and set the index to 12 ; then if the
time be in the forenoon, turn the globe westward, but if in the af-
ternoon, eastward, until the index has passed over as many hours
as the given time is from noon ; the degree then cut on the eclip-
tic by the brass meridian will correspond, on the horizon, to the day
of the month required. The altitude of the star when brought
to the graduated edge of the quadrant, will be the degree on it, cut
by the centre of the star.
Example 1 , At Washington city at 9 o'clock at night, the azi-
muth of Aldebaran was by observation 89 from the south towards
the west ; required its altitude and day of the month ?
Ans, Its altitude is 26, and the day is the 21st of March ; as the
time is 9 hours past noon, the globe must be turned as many hours
towards the east, &c.
2. At Philadelphia at 5 o'clock in the morning, the azimuth of
Arietis was 79 from the south towards the west ; required its al-
titude and the month and day when the observation was made ?
Ans. As the time wants 7 hours of noon, the globe must be
turned 7 hours westward ; the altitude of the star will be found
47, and the time the 23d of September.
3. At London, at 10 o'clock at night, the azimuth of Spica was
observed 40 from the south towards the west ; required its alti-
tude and the day of the month ?
* Here the compl. of the lat. the co. of the decl. of the star, and the co.
of its altitude, form a spherical triangle, two sides of which, viz. the co,
lat. and co. decl. of the star, and an angle opposite one of them, that is
the angle opposite the co. decl. being the azimuth of the star or its sup-
plement, or what it wants of 180 ; from which the distance of the
star from ihe meridian will be found exactly as in the note to prob. 12 of
the preceding. Ana as the hour is given, the distance of the sun from the
meridian, or the angle formed by the circle of declination passing through
the sun, and the brass meridian, is given, being equal to the time front
noon converted into degrees ; hence the distance of the star from the me-
ridian being added to that of the sun, will give the difference of their rt.
ascensions, and as the rt. as. of the star is given, the right ascension of the
sun will be therefore given. Now as the obliquity of the ecliptic is given,
the sun's longitude may be easily found by Napier's rule, and hence the
corresponding day may be found from an ephemeris or the globe. The
application of these remarks is left as an exercise for the learner, in cal-
culating the above examples.
Cc
PROBLEMS PERFORMED BY
4. At Dublin at 2 o'clock in the morning, the azimuth of ft Pe-
gasus or scheat, was 70 from the north towards the east ; requir-
ed its altitude and day of the month ?
PROB. 16.
The latitude of the filace, day of the month, and hour of the day be-
ing given,) to find the Nonagesimal degree* of the eclifitic, its al-
titude and azimuth, and the Medium Cceli, &c.
Rule. RECTIFY the globe for the latitude, zenith and sun's
place (by prob. 9.) then if the given time be in the forenoon, turn
* The nonagesimal degree of the ecliptic, so called from its being the
90th degree reckoning from the horizon on the ecliptic, is the most ele-
vated point of the ecliptic above the horizon, and is measured by the an-
gle which the ecliptic makes with the horizon at any elevation of the pele,
and is equal to the distance between the zenith of the place and the pole
of the ecliptic. It is frequently made use of in the calculation of eclipses.
The medium caili, or midheaven, is that point of the ecliptic which is on
the meridian.
From the 22d of December to the 21st of June, the nonagesimal degree
of the ecliptic is east of the meridian ; and from the 21st of June to the
22d of December, it is west of the meridian.
The globe being rectified as above, then the day of the month being-
given, the sun's right ascension for the given time, may be found in the
Nautical Almanac (see notes to problems 42 and 49) and therefore its
distance from the equinoctial point, which is above the horizon, is given ;
moreover, as the hour is given, the sun's distance from the meridian of the
place, or the brass meridian, is given, and hence the distance from this
meridian to the next equinoctial point is given. Now as the obliquity of
the ecliptic is given (note to prob. 49) the degree cut on the ecliptic by
the brass meridian,' or the medium cosli or midheaven will be given (by Na-
pier's rule.) Again, as the number of degrees from the elevated equi-
noctial point to the brass meridian is given, its complement, or the dis-
tance from the equinoctional to the horizon, on the equator, is given, and
the inclination of the equator with the horizon is the complement of the
latitude. Hence in the spherical triangle formed by the equinoctial or
equator, the ecliptic, and the horizon ; two angles, viz. ( the obliquity of the
ecliptic, and the co. lat. and one side, that is the dist/on the equinoctial to
the horizon from the elevated equinoctial point, and therefore the angle
opposite the given side, is given (by Napier's rules, or by case 10, s. 4,
b. 3, Emerson's Trig.) the suppl. of which, or the inclination of the eclip-
tic with the horizon is the nonagesimal degree required. Thus in ex. I,
the sun's rt. ascension is 90, and is 90 distant from libra, the elevated
eq. point ; and as the hour from noon is 3h. 45' = 56 15', its compl. 33
45', is the distance of the meridian from the point libra. Hence by Na-
pier's rule, Tang. 33 45' : r. :: cos. obi. eclip. 23 28' : co. tang. 36 4',
the distance from libra to the medium cccli, reckoning backwards, which
therefore corresponds with 23 56' of leo. Again, 90 33 -45' = 56 15'
dist. from libra on the equator to the horizon, and 90 51^ = 38 co.
lat. ; hence, letting fall a perpendicular from the point libra, on the hori-
zon, it will be U. : tang. 3S :: cos. 56 15' : 66 1C', the angle formed at
libra by the equinoctial and perpendicular ; from which the obliquity of
the eclip. being taken, the rem. 42 42' is the angle formed at libra by
the ecliptic and perpendicular ; then S. 66 : s. 42 42' :: cos. 38^ : cos.
54 32', the inclination of the ecliptic to the horizon, or the nonagesima?
degree required.
THE CELESTIAL GLOBE. 211
the globe eastward, but if in the afternoon, westward, until the in-
dex has passed over as many hours as the time is before or after
noon ; reckon 90 on the ecliptic from the horizon, eastward or
westward, the point where the reckoning ends will be the nona-
gesimal degree, and the degree of the ecliptic, cut by the brass
meridian, will be the medium coeli : the graduated edge of tfce
quadrant being brought over the nonagesimal degree, will point
out its altitude, ai.d its azimuth will be then seen on the horizon.
Example 1. On the 2 1st of June, at 45 minutes past 3 o'clock
in the afternoon at London, required the point of the ecliptic which
is the nonagesimal degree, its altitude and azimuth ; the longitude
of the medium cceli, and its altitude, &c.
Ana. The nonagesimal degree is 10 in Leo, its altitude is 54{,
and its azimuth 22 30' from the south towards the west or
S. S. W. The midheaven or point of the ecliptic under the brass
meridian is nearly 24 in leo, and its altitude above the horizon is
52. The right ascension of it is 146. The rising point of the
ecliptic is '.0 in scorpio, and the setting point 10 in taurus. If
the quadrant of alt. be extended over the sun's place, the sun's alt.
will be found equal 38|, and his azimuth 78 J from the south to-
wards the west, or W. by S nearly.
2. At New-York on the 10th of May, at 10 o'clock at night, re-
quired the point of the ecliptic, which is the nonagesimal degree,
its altitude and azimuth ; the point of the ecliptic, which is the mid-
heaven, &c. &c.
.3. At Philadelphia on the 25th of October, at 4 o'clock in the
morning, required as in the last example, &c.
4. At George Town College on the Potomac, in lat. 38 55'
N. required the nonagesimal degree of the ecliptic, the medium
cceli, &c.
PROB. If.
Given the latitude^ day^ and hour, together with the altitude and az
imuth of a star, tojind the star*
Rule. RECTIFY the globe for the latitude, zenith and sun's
place, as before, and turn the globe eastward or westward (accord-
ing as the time is in forenoon or afternoon, as many hours as the
time is from noon ; keep the globe in this position, and bring the
graduated edge of the quadrant to the given azimuth on the hori-
* As the star is given, when its right ascension and declination are given ;
hence the right asc. and decl. may be thus calculated. The prob. being
performed as directed in the rule, it will be found that the comp. of the
lat. the comp, of the star's decl. and the comp. of its alt. form a spherical
triangle, two sides of which, viz. the co, lat and co. alt. are given, and the
included angle being the star's azimuth or its supplement, from whence
the other side, or the co. decl. and the angle included by this and the
brass meridian, or the distance of the star from the meridian passing
through the zenith of the place, will be given ; now as the hour angle is
given, the distance of the sun from the meridian is given, and hence,
tlje distance from the sun to the star, reckoning on the equator, or the
212 PROBLEMS PERFORMED BY
zon ; then under the given altitude on the quadrant, you will find
the star required.
Example 1. At New-York on the 21st of June, at 9 o'clock in
the -afternoon, the altitude of a star was 20 22', and its azimuth
83 23' from the south towards the east ; required the star I
JLns. Altair in the Eagle
2. At Washington city on the 2 1st of March, at 9 o'clock at
night, the altitude of a star was 26, and its azimuth 89 from the
south towards the west ; required the star ?
3. At Philadelphia at 5 o'clock in the morning of the 23d of
September, the altitude of a star was 47, and its azimuth 79 from
the south towards the west ; required the star ?
4. At London, on the 22d of December at 4 o'clock in the
morning, the altitude of a star was 50, and its azimuth 37 from
the south towards the east ; required the star's name ?
PROB. 18.
The latitude and day of the month* being given, tojind the- meridian
altitude of any star or planet.
Rule. RECTIFY the globe for the given latitude ; then,
For a star. Bring the given star to the meridian, and the
degrees between the star and the horizon will be the altitude re-
quired.
For the moon or a planet. Find the latitude and longitude, or
the right ascension and declination of the planet, for the given time,
in the Nautical Almanac, a good ephemeris, or from astronomical
tables, and mark its place on the globe (as in prob. 4th or 2d)
bring this place to the brass meridian, and the number of degrees
between the point on the meridian over it, and the horizon, will be
the altitude required.
OR WITHOUT THE GLOBE.
The declination of the star or planet at the time of its passing
the meridian, added to or subtracted from the complement of the
lat. according as they are of the same or a different name, will give
the meridian altitude required. .
Example 1. What is the meridian altitude of Aldebaran in tau-
rus, at the New-York Literary Institution, York or Manhattan
Island, lat. 40 46* N. ?
difference of their right ascensions is given, and as the sun's right ascen-
sion is given, therefore the star's right ascension is also given. The ap-
plication of these principles to the above examples, must now be familiar
to the learner, and is left for his exercise.
* The day of the month need not be attended to when the meridian alt.
of a star is required, as the meridian altitudes of the stars on the globe,
are invariable in the same latitude. Their places may be taken out of the
ephemeris for noon without any sensible eiTor. Their annual variation
in decl. &c. should, however, be allowed for, where accuracy is required ;
and the right asc. and decl. of the planets, reduced to the given time and
place, in the same manner as the moon's, in the following notes.
THE CELESTIAL GLOBE. 213
Am. 65 20'. Or comp.of the lat. is 49 14', decl. of Aldebaran
for 1800 was 16 5' 43" or 16 6' nearly; hence the comp. of
the lat. + 16 6' = 65 20', as before.
2. What is the meridian altitude of Procyon at London ?
3. What is the meridian altitude of Arcturus in Bootes, at
Washington city ?
4. On the 1st of June, 1812, the longitude of Jupiter will be
3s. 9> 20', and latitude 7' north, or his declination will be 23 15'
north ; what will his meridian altitude be at Philadelphia ?
5. On the 1st of August, 1811, the longitude of Mars was 8
signs 8', and his latitude 3 3' south, or declination 23 10'
south i required the meridian alt. at Greenwich ?
6. On the 1st of April, 1810, the longitude ofsaturn, was 8s.
15 17', and lat. 1 45' north; what was his meridian altitude at
Paris observatory ?
7. On the 2 2d of December, 1812, at the time of the moon's
passing over the meridian of Greenwich, her right ascension will
be 142 53' 13", and declination 14 37' north ; required her
meridian altitude at Greenwich ?*
Ans. Comp. lat. = 38 31' 20" N. -f decl. 14<> 37 7 N. = 53<>
8' 20" mer. alt. required.
8. What will the moon's meridian altitude be at New-York, in
longitude 74 0' 45" W. from Greenwich, lat. 40 42' 40" N. on
the 16th of October, 1812 ; her right ascension at the time of
passing the meridjant being 339 37' 56", and declination 9 33'
south ?
Ans. The com. lat. = 49 17' 20' N. decl. = 9 33' S. =
39 44' 20" mer. alt. required.
* By the Nautical Almanac the moon will pass over the meridian of
Greenwich obs, on the 22d of December, 1812, at 30 min. after 3 in the
morning" (or on the 21st of December, 15h. 30m. astronomical time.)
140 6 59' 1" j)'s rt. asc. at midnight, Dec. 21st. Decl. 15 4' N.
147 30 35 do. at noon, Dec. 32d. Decl. 13 32 N.
6 31 34 increase in 12 hours. Decrease in 12h.
As 12h : 3h. 30m. :: 6 31' 34" : 1 54' 12" ; hence 140 59'
12" = 142 53' 13" moon's rt. asc. at 15h. 30m.
Again, 12h. : 3h. 30m. :: 1 32' : 27' ; hence 15 4' 27' = 14 37', the
moon's decl. at 15h. 30m.
If greater accuracy be required, consult the theorems at the end of the
Nautical Almanac for 1812 or 1813, published by J. Garnett.
j" To find the time of the moon's passing the meridian of a given place, dif-
ferent from that of Greenwich. Take the difference between the time of
the moon's passing- over the meridian of Greenwich on the given day, and
the day preceding- or following-, according- as the place is to the east or
west of Greenwich ; then say, as 24 hours is to this difference, so is the
difference of longitude in time, to a number of minutes and seconds,
which must be added to the time of the moon's passage over the meridian
of Greenwich, if the place be west, or subtracted if east of Greenwich ; as
the moon in the latter case will come to the meridian sooner than in the
former. Thus on the 16th of Oct. 1812 (exam. 8) the moon will come to
$be meridian of Greenwich at 9 hours, and on the 17th at 9h. 55m. the
214 PROBLEMS PERFORMED BY
PROB. 19.
The meridian altitude of a known star or jilanet being, given to
Jind the latitude.
Rule. BRING the given star, or the place of the planet,* to the
brass meridian, count the number of degrees in the given altitude
(corrected t) on the brass meridian, from the star or planet's place,
towards the south part of the horizon if the latitude be north, or
towards the north part of the horizon if the latitude be south, and
mark where the reckoning ends ; elevate or depress the pole un-
til this mark coincides with that part of the horizon towards which
the altitude was reckoned ; then the elevation of the pole above
the horizon will be the latitude required. \
difference is 55m. Hence 24h. : 55m. :: 4h. 56' 3" (the difference of lon-
gitude in time) : 11' 18" 4, which as the given place is west of Green*
wich, must be added to 9 hours ; whence 9h. Hm. 18.4 seconds, is the time
the moon will come to the meridian of New- York on the 16th of Oct. 1812.
Now to find her right ascension and decl. corresponding to this time ;
first reduce it to that of Greenwich, and proceed as above. Thus 9h. 11 '
18" 4 + 4h. 56' 3" = 14h. 7' 21" 4, the time at Greenwich ; then,
338 23' 20" } 's rt. asc. at midn. Oct. 16th. Decl. 9 56' S.
345 25 4 do. at noon, Oct. 17th. Decl. 7 46 S
7 I 44 increase in 12 hours. Decrease in 12h. 2 10
As 12h. : 2h. 7' 21" 4 :: 7 l f 44" : 1 14 r 35" 9 ; hence 338 23' 2G" -f 1
14' 35" 9 = 339 37' 55" 9, moon's rt. as. at 14h. 7' 21" 4 at Greenwich.
Again, 12h. : 2h. 7' 21" 4 :: 2 1(X 22' 59" 6 ; hence 9 50 22' 59" 6=
9 33' 0" 4, the moon's decl. corresponding to 2h. 7' 21" 4 after midnight.
The above method of calculating the time of the moon's coming to the
meridian is a sufficiently near approximation ; more accurate methods are
given in the Nautical Almanacs for 1812 and 1813, revised by John Gar-
nett, New-Brunswick.
Having- the time of the moon's coming to the meridian or southing-, the
hour of the night by the moon shining on a sun dial, may be found thus ;
count how many hours and minutes the shadow on the dial wants of 12
o'clock, subtract them from the time of her southing for the hour of the
night. But if the shadow be after 12, add these hours and minutes on the
dial to the time of her southing, rejecting 12 if it exceed it, and you have
the hour of the night.
Accurate methods are also given in the Nautical Almanacs for 1812 and
1813, for finding the moon's decl. rising and setting. The longitude from
her meridional distance, &c. See the note to prob. 19.
* The places of the planets when on the meridian may be calculated from
the Nautical Almanac, when accuracy is required, in the same manner as
that of the moon in the notes to prob. 18. But as their places varyjess than
that of the moon, they may be taken from the almanac for noon, without any
sensible error, by only taking proportional parts for the daily variation.
j- The observed altitude of a star may be corrected for dip and refraction,
by the tables given in the note to prob. 58, part 2.
} It is evident that when the star is brought to the brass meridian, and
the pole elevated or depressed until the star is at the same distance from the
nearest part of the horizon as its altitude was observed to be, the globe
will then be in the same position with regard to the horizon, as the earth it-
self; the height of the pole above the artificial horizon of the globe, being
equal to the height of the real pole above the horizon of the earth; but the
THE CELESTIAL GLOBE.
Or, The declination of the star being given, or that of the plan-
et (from the Nautical Almanac) reduced to the time and meridian
of the place of observation (by note to prob. 1 8) then the sum or
height of the pole above the horizon, is always equal to the latitude, therefore
the truth of the rule is evident. The demonstration of this property with
the others that follow, may not be unworthy the readers perusal.
1. The height of the pole above the
horizon is equal to the latitude. Let
HR represent the horizon, EQ the
equator, Z the zenith, and P the pole ;
then ZE = the lat. and PE = ZR be-
ing each equal 90; hence ZP, which
is common to both, being taken
away, EZ will remain = PR. Again,
2. Half the sum of the greatest and least's,
alt. of a circumpolar star will give the
latitude. For if vw be the circle de-
scribed by the star, v being the great-
est, and w tire least alt, then as Pv=
Pw, PR = Rv+ Rw X i 3. The lat.
may be also thus found; Let eOt be the
ecliptic ; then when the sun comes to
e, eH will be its greatest meridian alt. its decl. being then greatest ; but
wken the sun comes to t, ts being the parallel described on that day, sH
will then be the least mer. alt. Now Ee being = Es, Lie 4- H* X $ = HE
the co. lat. 4. The inclination of the ecliptic to the equator, is equal to half the
diff. of the sun's greatest and least meridian altitudes. For EOc- the obliq. of
the ecliptic = X He Us, or se = Ee. 5. Tto alt. of that point of the
equator -which is on the meridian, or the angle which the equator makes -with
the horizon, is equal to the compl. of the latitude. For EH is the measure of
the angle EOH, because EO and HO are each = 90, and EH is the compl.
of EZ, which is equal the lat. 6. As the apparent time is generally found
by the alt. of some celestial body, hence if this latitude be wrong, the time
must also be wrong. Now the error in alt. being given, the error in time may
be thus found ; let mn be parallel to the horizon, and nx represent the error
in alt. then the body being supposed at m instead of x, as the time is calcu-
lated on supposition that there is no error in the declination, the angle
mPx, or the arc qr, measures the error in time. Now the triangle nmx being
small, the sides may be considered as right lines ; then by trig, it will be
nx : xm :: sine nmx : R. and xm : qr :: cos. rx : R. (by note to prob. 35,
part 2.) hence multiplying the corresponding terms, and cancelling arm from
the two first, nx : qr :: sine nmx X cos. rx : R 2 (Emerson's Geom. proper.
nx X ^R 2
prop. 18.) Therefore qr = ^ ne nnix x cos . ~ x \ but the angle PxZ = nmx,
nxm being the compl. to both ; also sin. PxZ or nmx : sin. PZ :: sin. xZP -.
sin. Pa: or cos. rx, (Emerson's Trig. b. 3, prop. 29) hence sin. nmx X cos.
rx = sin. PZ X sin. xZP ; and therefore qr =nx X R 2 -i, sine PZ X sin .
vTZP = nx X R 2 -r- cos. lat. X sin. azim.
The error is therefore least at the prime vertical, or the vertical circle
which cuts the meridian at right angles ; and hence all altitudes for the pur-
pose of obtaining the time, ought to be taken as near this circle as possible.
The following ex. will illustrate this latter rule. In lat. 40 43', if the error
in alt. at an azimuth 50 be <2, then qr = 2 X I 2 -r- ,7579 X ,766= 3' 445
of a degree = 13" 78 in time. The perp. ascent of a body is likewise quick-
est when on the prime vertical; for nx varies as the sine of the azim, when
7 r and the lat, are given, and the azimuth is then 90.
216 PROBLEMS PERFORMED BY
difference of the zenith distance and decl. of the star or planet?
according as they are of the same or a contrary denomination, will
be the latitude required.
Note. The true alt. taken from 90 gives the zenith distance, which
is north, if the observer be north of the star or planet, otherwise s^.t th.
If the object be in the opposite meridian, or between the elevated pole
and the horizon, at the time of observation, then the sum of the true alt.
and the compl. of the declination or polar distance, will be the latitude.
If the alt. be negative, or the centre of the object be below the horizon,
it must be subtracted from the polar distance to find the latitude.
Examjile 1. In what degree of north latitude is the meridian al-
titude of Aldebaran 65 20'?
Ann. In lat. 40 46' N.
2. In what degree of north lat. is the meridian altitude of Mirach
in Bootes 70 ?
3. In what degree of north lat. is Procyon 90 above the hori-
zon, or vertical when it culminates ?
4. In what degree of north lat. will the meridian altitude of Ju-
piter be 58 on the first of June, 1813, its longitude being then
4s. 5 19 ; , and lat. 42' north, or declination 19 38' north ?
7. The time -when the appar. diurnal motion is perp. to the horizon is thus
found ; let vw be the parallel described by the star, and let the vertical
circle Z/& touch it at y ; then when the star comes to y, its motion will be per-
pendicular to the horizon. Now PyZ being a rt. angle, we have by Napier's
rule, R. X cos. ZPy = tang. Py X cot. PZ, or R X cos. distance of the star
from the meridian = cot. decl. X tang. lat. and hence R. : cot. decl. :: tang,
lat. : cos. hour angle, or star's distance from the meridian. The time of the
star's coming to the meridian being known, the time required will from,
thence be given. 8. As the time of the sun's semidiameter passing the me-
ridian serves to reduce an observation of a transit of the preceding or sub-
sequent limb over the mer. to that of the centre, when only one limb was
observed, the following method of finding the time in which the sun passes
the meridian or horizontal wire of the telescope, may have its use. Let mx
be the diameter of the sun in seconds = d" estimated on the arch of a great
circle ; then the seconds in mx considered as a lesser circle, must be in-
creased in proportion as the radius is diminished, the angle being inversely
as the radius where the arc is given ; hence Px or cos. decl. rx : R. :: rf"
the seconds in mx of a great circle : to the seconds in mx of a lesser circle
eu = the seconds in or, or in the angle qPr ; therefore qPr = d" ~ cos. decl.
(rad. being 1.) = d" X sec. decl. = the time of the sun's passing over a
space equal to its diameter, or of passing the mer. Hence 15" in space (be-
ing 1" in time) : d" X sec. decl. in space :: 1'^ in time : the seconds of pass-
diameter in rt.
32' or 1920",
2",88. The
same will hold for the moon if d" be its diam. If nx (in the foregoing part of
the note) = d" the sun's diam. qrd" X R 2 ~- cos, lat. X sin. azim. hence
time of describing qr or of the sun's ascending or descending 1 perpendicu-
d" R2
larly a space = its diam. will be ^ X ^s. lat. X sin. azim. If d " = S "
1980' 7 , the horizontal refraction, then 1980" -i- 15" 132"; hence 132'''
X R 2 "*- cos. lat. X sine azim. is the acceleration of the sun by refraction,
at rising, &c. (See Vince's Astron. 8vo.) Other useful principles could be
deduced from the foregoing, but our limits would not permit their insertion.
THE CELESTIAL GLOBE. 217
5. In what degree of latitude will the meridian altitude of the
moon be 53 8' 20" south of the observer, on the 22d of Decem-
ber, 1812, astronomical time ?
PROB. 20.
Given the day of the month and the hour when any known star
rises or sets^ to Jind the latitude of the filace.
Rule. BRING the sun's place in the ecliptic for the given day
to the brass meridian, and set the hour index to 1 2 ; then turn the
globe eastward or westward, according as the time is in the fore-
noon or afternoon, as many hours as the time is from noon ; ele-
vate or depress the pole until the centre of the star coincides with
the horizon ; then the elevation of the pole will be the latitude
required.*
Example 1. In what latitude does Altair in the Eagle rise at 10
o'clock in the evening, on the 10th of May?
Ans. 41 35'.
2. In what latitude does Mirach in Bootes rise at half past 12
'clock at night, on the lOth of December ?
3. In what latitude does Rigel in Orion set at 4 o'clock in the
morning, on the 2 1 st of December ?
4. In what latitude does $ Capricorni set at 1 1 o'clock at night,
on the 10th of October ?
* The prob. being performed as directed in the rule, then the co. lat. co.
decl. of the star, and its distance from the vertex (e=90) form a quadrantal
spherical triangle. Now as the sun and star's right ascensions are given,
their difference is given, and as the hour is given, the sun's distance from
the meridian is given, therefore the star's distance from the meridian or the
angle formed, at the pole, by the co. lat. and the co. decl. of the star, is
likewise given ; and as the star's decl. is given, and the dist. of the star from
the zen. = 90, there are two sides, and the angle opposite one of them giv-
en to find the third, which is the co. lat. required. This may be solved more
easily as follows ; the co decl. of the star (or the decl. -f- 90) the lat. reck-
oning on the brass meridian, from the elevated pole to the horizon, by pro-
ducing one of the sides of the former triangle, and the arc of the horizon,
between the star and the point of the horizon, north or south, corresponding
to the elevated pole, form a right angled sp. triangle, the hyp. of which,
viz. the co. decl. of the star (or the decl. -f- 90) and the angle formed at
the pole between this side and the brass mer. (being the supplement of the
distance of the star from the meridian) are given to find the lat. Thus in
ex. 1. at 10 o'clock in the evening, the sun is 10 hours or 150 distant from
the mer. Now the sun's right ascension on the 10th of May (suppose 1812)
will be 3h. 8' 38" 4, and Altair's rt. as. 19h. 41' 38" 3, the difference of
which is nearly 16h. 33' ; hence 16h. 33' lOh. = 6h. 33' = 97 3(X 45"
Altair's distance from the mer. the supplement of which is 180=97 30' 45"
= 82 29' 15", the angle formed at the pole, and as the star's decl. will
then be 8 23', the com. of which is 81 37', hence the comp. of the angle
at the pole or 7 30' 45" being middle part, and the decl. or 8 23', and the
lat. adjacent extremes, we have by Napier's rule, r. X sine. 7 30' 45" =
tang. decl. 8 23' X tang. lat. therefore tanrr. dec!, 8 23' : R sine, 7
30' "45" : tun. lat. 41 35'.
Dd
218 PROBLEMS, PERFORMED BY
PROB 21.
The altitudes of two known stars being given, to find the latitude
of the place.
Rule. WITH one foot of a pair of compasses extended on the
equinoctial to the complement of each star's altitude successively,
and placed in the centre of each star respectively, describe arches
on the globe with a black lead pencil, fixed in the other leg of the
compasses ; these arches will intersect each other in the zenith ;
the zenith being then brought to the brass meridian, the degree
over it will be the latitude required.*
Example I Being at sea, I observed the altitude of Aldebaran
to be 51 45', and at the same time that of Castor in Gemini equal
76 40' ; required the latitude ?
jins. With an extent of 38^ (=90 5l|) taken from the
equinoctial (or any great circle on the globes which is divided into
degrees) and one foot of the compasses in the centre of Aldeba-
ran, describe an arc towards the north ; then with 13 20' (90-
76 40') in the compasses, and one foot in the centre of Castor,
describe another arc crossing the former ; the point of intersection
will be the zenith of the place, which being brought to the brass
meridian, will give the latitude 42 nearly.
2. The altitude of Capella being observed 30, and at the same
time that of Aldebaran 35, the latitude being north ; required the
latitude ?
3. The altitude of Markab in Pegasus, was 30, and that of AI-
tair in the Eagle at the same time was 65 ; required the latitude
supposing it north ?
4. In north latitude the altitude of Procyon was observed to be
50, and that of Betelgeux in Orion, at the same time was 58 ;
required the latitude ?
5. In south latitude the altitude of Betelgeux was 67^, and
that of Aldebaran 60| ; required the latitude ?
PROB. 22.
Two known stars being observed, the one on the meridian, and the
other on the east or west part of the horizon^ to Jind the latitude
of the place.
Rule. BRING the star which was observed on the meridian of
the place, to the brass meridian ; keep the globe from turning on
its axis, and elevate or depress the pole until the other star comes
* Let Z be the zenith, P the pole, S, s the places of the star (see the fig.
in note to prob. 14) then in the triangle $PS, there are given the sides *P,
SP the co. declinations, and the angle *PS, the diff. of rt. ascensions ; hence
S.s the distance of the stars, and the angle 6-SP are given. And in the triangle
ZS&, all the sides are given to find the angle ZSs ; hence the angle PSZ is
given. Then in the triangle ZSP two sides ZS and SP, and the included
angle are given, and therefore ZP is given, which is the CO. lat, required.
THE CELESTIAL GLOBE. 219
cO ihe eastern or western part of the hoilzon ; the elevation of the
pole will then be the latitude required.*
Exam-file 1 . When Procyon was on the meridian, Arcturus in
Bootes was rising ; required the latitude ?
Ans. 23 north.
2. When the two pointers of the Great Bear marked a and /?,
r Dubhe and & were on the meridian ; Vega in Lyra was rising ;
required the latitude ?
3. When $ Leonis was on the meridian, the Pleiades were set-
ting ; required the latitude ?
4. When the star marked & in Gemini was in the meridian, i in
the shoulder of Andromeda was setting ; required the latitude ?
5. In what latitude is a. or Canopus, in the ship Argo, rising,
when a in Phoenix is on the meridian ?
6. In what latitude is Achernar in Eridanus on the meridian,
when Procyon is rising ?
PROB. 23.
Given two altitudes of a star, and the time between them) to find
the latitude.
Rule. TAKE the complement of the first altitude in a pair of
compasses, from the equinoctial (or any other great circle on the
globes which is divided into degrees, &c.) and with one foot on
the given star, describe an arch with the other (having a pencil
fixed in it) in a contrary direction to that in which the star was ob-
served ; then bring the star to the brass meridian, and set the in-
dex to 12 (or any other hour) or mark the point on the equinoctial
cut by the brass meridian ; turn the globe eastward on its axis
until the index, or the point marked on the equinoctial, has pass-
ed over as many hours as are equal to the time elapsed between
the two observations, allowing 15 2' 28" to every hour, or add-
ing 9" 85 of time to every hour,f and mark the point on the par-
allel of the star's declination then under the brass meridian ; take
the complement of the next altitude in the compasses, and with
one foot in this point, describe with the other an arch intersecting
the former ; the point of intersection will be the zenith of the place,
which being brought under the brass meridian, will give the lati-
tude required. \
* The prob. being performed as directed in the rule, then the distance
between the star which is at the horizon and the mer. towards the elevated
pole, reckoning- on the horizon, the lat. on the mer. or dist. of the pole from
the horizon, and the star's co. decl. form a rig-lit angled sp. A? one side of
which, viz. the co. decl. and the angle at the pole, included between the
brass meridian and circle of decl. passing- through the star, which is at the
hor. being equal to the supplement of the difference of the star's rt. ascen-
sions, are given ; hence the third side is given, and may be found by Na-
pier's rule.
f A sidereal day being 23h. S& 4", hence 23h. 56' 4" : Ih. :: 360 : 15 2'
27" 9, &c. 2' 27" 9 = 9.85 seconds of time.
t This prob. may be performed by trigonometry in the same manner as
prob. 61, part 2, (see the note to this prob ) thus ; A and B being the places
220 PROBLEMS PERFORMED BY
Examfile 1. On the 31st of April in the afternoon, the altitude
of Procyon was observed to be 44 30', and one hour after its alti
tude was 51 31', it being southward of the observer; required
the latitude I
Ann Here the complement of the first altitude is -J-5 30' with
this extent in the compasses, and one foot in the centre of the
given star, describe an arch towards the north ; then the given star
being brought to the meridian, and the index to 12; turn the
gloue eastward 1 hour 9" 83 or 15 2' 28" on the equinoctial,
and mark the point under the declination of Procyon on the brass
meridian ; from this point as a centre, and with the complement of
the 2d altitude = 38 30' in the compasses, describe a second
arch intersecting the former ; the point of intersection brought to
the brass meridian will give the latitude 4 i N. nearly.
2. In north latitude on the 1st of April, in the evening, the alti-
tude of Sirius was observed to be 30, and one hour 15 minutes
after his altitude wa$ 19J ; required the latitude of the place of
observation \
PROB. 24.
Given one altitude of a star, and the time at which the altitude
wan taken,) to jfind the latitude.
Rule. WITH the complement of the given altitude in the com-
passes, taken from the equinoctial (or any other great circle on the
globe divided into degrees, Sec.) and one foot in the centre of the
given star, describe an arch with the other, in a direction contrary
to that in which the star appeared when observed ; bring the sun's
place in the ecliptic to the brass meridian, and set the hour index
to 2 ; then if the time be in the forenoon, turn the globe east-
ward, but if in the afternoon, westward, as many hours as the time
is before or after noon ; the degrees then cut by the arch on the
brass meridian, will be the latitude required.*
Note. The observed altitude must be previously corrected for the height
of the eye and refraction. The same method will also answer for any planet,
its declination being given; the observed altitude being- first corrected for
the height of the eye or dip of the horizon, refraction and parallax. Set
the note to prob. 19.
of the star at the respective altitudes (see the note to prob. 61, above al-
luded to) then in the isosceles sp. A APB, AP or BP the co. decl. of the
star, and the /_ APB, being 1 the elapsed time reduced into degrees (by al-
lowing 15 2' 28''' to every hour) are given, hence the PAB and the side
AB is givepi ; and in the sp. A ABZ, AZ, BZ, the com. of the altitudes and
the side AB are given, hence the /_ BAZ is given, from which the angle
PAB being taken, the angle PAZ will be given ; and in the triangle AZP,
PA, AZ, and the angle at A are given ; hence the side ZP, which is the
complement of the latitude required, is given.
* This prob. may be calculated in the same manner ns prob. 62, part 2.
For the time of the star's passage over the meridian may be found by prob.
8, part 2, the difference between which and the time at which the alt. was
taken, will be the distance of the star from the mcr. or the hour angle, or
THE CELESTIAL GLOBE. 221
Example 1. At 9 o'clock in the evening on the 21st of March,
the altitude of Deneb in Leo, was 47 ; required the latitude of
the place of observation ?
Ans. 4-i 50' N. nearly.
2. At 10 o'clock in the .evening on the 22d of December, the
altitude of Procyon was 27 in north latitude ; required the lat. ?
3. On the 1st of January, 1810, in north latitude, the correct al-
titude of Jupiter at 9 o'clock in the evening was 37 15', its latitude
being 1 18' S. and longitude Os. 15 48'; required the lat.?
4. On the first of November, 1810, the altitude of Venus at 6h.
30' in the evening was 10 35', being towards the S. W. of the
observer, height of the eye being 30 feet above the level of the
horizon ; the longitude of Venus being 8s. 24 30', and latitude
4 7' S. ; required the latitude of the place, allowing also for
parallax and refraction ?
5. In north latitude on the 17th of October, 1810, the altitude
of the lower limb of the moon, taken by a Hadley's quadrant, at 3
o'clock in the morning, was 61. Her right ascension being 87
31', and declination 18 14' N. ; required the latitude, allowing
for semidiameter, refraction, parallax and dip of the horizon ;
height of the eye being 20 feet ?
PROB. 25.
The altitudes of two stars having the same azimuth, and that azi-
muth being given, to find the latitude of the place.*
Rule. PLACE the graduated edge of the quadrant of altitude
over both stars, so that each star may be exactly under its given al-
the angle APZ, &c. See the fig. in the note to prob. 62. But as this time is
ndereal time, it must be converted into degrees, by allowing 1 15 2' 27" 9. to
an hour. (See the note to the foregoing prob.)
Or, The prob. may be calculated as follows ; find the sun's rt. asc. from
the Nautical Almanac or some good table, and likewise the star's rt. ascen ,
(see the table at the end of this work) their sum or difference will be the
distance between the sun and the star reckoning on the equator (see note to
prob. 12, part 3) convert the given time of observation into degrees, allow-
ing 15 to an hour ; this will give the sun's dist. from the mer. in degrees,
the difference between which and the dist. between the sun and star will
give the dist. of the star from the mer. or the angle APZ, &.c. as above.
This angle being given, and the star's decl. and alt. the calculation is the
same as that in the note, prob. 62, as above quoted. Tims in ex. 1, the
sun's rt. ascen. (the year being supposed 1812) will be 28" 4 in time, or /'
6" of a degree, and the star's rt. as. 174 52' ; hence their diff. is nearly
174 45', and the dist. of the sun from the mer. being 9 hours or 135, the
dist. of the star is therefore 39 45'. Now AP the co. decl. = 74 22' 40",
A7 the co, alt. 43, and the angle APZ = 39 45' ; hence R. : cos. 39 C
45' :: tang. 74 22' 40" : tang. PB 70 1', and cos. 74 22' 40" : cos. 70 1' : .
cos. 43 : cos. EZ 21 51', and 70 1' 21 51' = 41 50' the co. lat. there-
fore the lat. is 41 50' as above.
* In calculating this prob. it will be seen that the altitude and azimuth of
either of the stars alone would be data sufficient to solve the prob. For in
the triangle PZS or PZs (see the fig. in the note to prob. 13, part 3) ZS or
222 PROBLEMS PERFORMED BY
titude on the quadrant ; the quadrant being held in this position,
elevate or depress the pole until the division marked on the
quadrant, coincides with the given azimuth on the horizon, the
elevation of the pole will then be the latitude.
Example 1. The altitude of Aldebaran was observed 45 45'
when that of Sirius was 30', their common azimuth, at the same
time, being 67 from the south towards the east or E. S. E.
nearly ; required the latitude ?
Ans 40 46' N. nearly.
2. The altitude of Arcturus was observed to be 40, and that of
Cor Caroli 68 ; their common azimuth at the same time being
71 from the south towards the east ; required the latitude ?
3. The altitude of Dubhe was 40, and that ofyin the back of
the Great Bear 29 ; their common azimuth at the same time
being 30 from the north towards the east ; required the latitude ?
4. The altitude of Vega or a in Lyra, was observed to be 70,
and that of a in the head of Hercules 39^ ; their common azi-
muth at the same time being 60 from the south towards the west ;
required the latitude ?
PROB. 26.
Given two known stars having the same azimuth^ and that azimuth
being given, together with the altitude of one of the stars, to Jind
the latitude of the filac e .*
Rule. PLACE the graduated edge of the quadrant of altitude
over both stars, so that the star whose alt was taken may be un-
der the same altitude on the quadrant ; then proceed as in the
foregoing problem.
Note. This problem being- similar to the foregoing-, the examples there
given will answer this, taking 1 one of the altitudes instead of both.
PROB. 27.
Given the altitudes of two known stars observed at different times,
and the interval of time between the observations, to Jind the la-
titude.
Rule. WITH the complement of the first altitude (taken from
the equinoctial) in the compasses, and one foot in the centre of the
star whose altitude was first taken, describe an arch ; bring the
star whose altitude was next taken to the brass meridian, set the
Zs, the co. alt. SP or sP, the co. decl. of the stars, and the common azim.
SZP or .sZP are given, therefore the side ZP is given, the compl. of which
is the lat. required. Or the prob. may be calculated without the azimuth,
thus ; in the triangle SPs, Ss the difference of the co. altitudes, and SP, P
are all given, hence the angle PSs is given, and therefore its supplement
ZSP is given. Again in the triangle ZSP, ZS and SP, and the included an-
gle are given to find ZP, which is therefore given.
* For the solution of this prob. by trigonometry, see the note to the fore-
going problem.
THE CELESTIAL GLOBE. 223
index to 12 ; mark the star's declination on the brass meridian,
and also the point cut by the meridian on the equinoctial, or the
star's right ascension ; turn the globe eastward on its axis until
the index has passed over as many hours as are equal to the inter-
val of time between the two observations, together with 9" 83,
added for every hour of solar time, or until the point marked on
the equinoctial passes over as many degrees as are equal to the
same interval, allowing 15 2' 28" for every hour ; then mark the
point on the globe under the degree of the star's declination on the
brass meridian, from this point as a centre, with an extent in the
compasses (taken from the equinoctial as before) equal to the
complement of the second altitude, describe another arch intersect-
ing the former ; the point of intersection will give the zenith,
which being brought to the brass meridian, the degree over it will
be the latitude required.*
Example 1 . In north latitude December 20th, 1 806, the true
altitude of Menkar in Cetus was 43 38', and Ih. 18m. after,
the altitude of Rigel was 29 5 1 ; ; required the latitude ?
Ans. With an extent of 46 22' ( 90 43 38') taken as
directed in the rule, and one foot of the compasses in the centre of
Menkar, describe an arch with a fine pencil fixed in the other ;
then Rigel being brought to the meridian, and the index set to 12,
or mark the equinoctial as directed ; turn the globe eastward Hi.
18' -f- 13", the equation = Ih. 18' 13", the interval or sidereal
time, or until the point marked on the equinoctional has passed
over 19 33' 12" (for Ih. : 15 2' 28" :: Ih. 18' : 19 33' 12")
mark the point on the globe under 8 26', the decimation of Rigel,
* Let A, B be the two known stars, Z
the zenith, and P the pole. Now if the
time at which either of the observations
was made be given, the altitude of one of
the stars will be sufficient to determine
the lat. For let the alt. of A and the time
at which it was taken, be given ; then the
distance of the star from the meridian, or A.
the angle APZ will be given (see the note
to prob. 24, part 3) and AZ the co. alt. and
also AP the co. decl. of the star are given ;
therefore ZP the co. latitude is given.
If only the interval of time between the observations be given, the prob.
may be thus calculated ; let A be the place of the star A, when the first
observation or its alt. was taken, and a its place when the second observa-
tion was made, or when the alt. of B was taken ; hence the angle APa will
be the elapsed time, or the interval between the two observations, which is
converted into degrees by allowing 15 2' 27" 9 to every hour. But the an-
gle aPB being equal to the difference of the star's right ascensions, is there-
fore given, and hence the angle APB = aPB PA is given. Now in the
triangle APB, the two sides AB, BP = the co. dccls. and the included angle
are given, therefore the side AB and the angle BAP are given. Again, in
the triangle AZB, there are given AZ, BZ the co. alts, and the side AB,
kence the angle BAZ is given, and therefore PAZ = BAZ BAP is given.
Lastly, in the triangle PAZ, PA, AZ, and the included angle are given, and
herefore PZ is given, the compl of which is the lat. required.
PROBLEMS PERFORMED BY
from this point with an extent of 60 9' (= 90 29 51') de-
scribe another arch as before, inserting the former ; the point of
intersection being brought to the meridian, will give the latitude
49 o21' nearly.
2 In north latitude on the 2 1 st of March, 1 8 1 0, at 9 o'clock at
night, the correct altitude ofCorHydrae, or Alphard, was 4l
15'. its right ascension being 139 33' 42", and declination 7 5(V
20" S and one hour after the altitude of Deneb was 57^, its right
ascension beins 74 50' 22", and declination 15 38' 5" N. ; re-
quired the latitude ?
PROB. 28.
To find the distance* of the stars or planets from each other
in degrees^ &c.
Rule. EXTEND the quadrant of altitude between any two stars
or the given places of the planets, so that the division marked
may be on one of them, the degrees on the quadrant between that
and the other star or planet's place will shew their distance, or the
angle which these stars or planets subtends as seen by a spectator
on the earth, or rather as seen from the earth's centre
* In order to find the correct distances of celestial objects, it is necessary
to determine their altitudes, which may be accurately found by an astrono-
mical quadrant (for the description and use of which, see Vince's Practical
Astronomy, which contains the description of the construction and use of all
astronomical instruments, 1 vol. 4to.) or by a good Hadley's quadrant, a sex-
tant, or by a repeating circle or circle of reflection. The apparent altitudes
being thus found, the true alt. may be found by allowing 1 for the height of
the eye and refraction if a fixed star ; for the height of the eye, refraction,
and parallax, if a planet ; and if the sun or moon for their semidiameter, ac-
cording as the tipper or lower limb was taken. A small allowance is also to
be made for the aberration of light, as will be shewn in part 4.
From the observed and corrected altitudes, and the observed distance be-
tween the two objects, the true distance may be thus computed ;
Let Z be the zenith, S the apparent
place of the sun or a star, s the true place,
M the apparent place of the moon ; m its
true place ; then in the triangle ZSM,
there are given SM, tke apparent distance,
SZ, ZM the complements of the apparent
altitudes to find the angle SZM. Let fall
from M the perpendicular MP or Mp ;
then,
Tang. ZS : tang. + MS :: tang.
ZM ceMS
2 --: tang. x.
If % ZS be greater than x, the perpendicular falls within as MP ; if less,
without as Mp. Then -J ZS -f- x ZP or Zp, and ZS ce.r = SP or Sp.
Moreover if ZS -f- SM be less thun 180, the perpendicular falls nearest to
the lesser side, but if ZS -f- SM be greater than 180, the perpendicular
falls nearest the greater side ; this being premised, then by Napier's rtUe,
Bad. : tang. ZP or Zp :: co. tang. ZM : co s . sine angle at Z
THE CELESTIAL GLOBE. 225
The same may be performed by a pair of compasses, as is man-
ifest.
Exa.mp.le 1. What is the distance between Betelguex in Orion,
and Castor in Gemini ?
Ans. 37|.
2. What is the distance between Procyon and Capella ?
'3. On the 2d of January, 1812, at midnight, the moon's right
ascension will be 151 39' 59 ", and her distance from the north
pole 79 17* ; required her distance from Spica Virginis ?
Ana. 5 1 20'.
4. On the 19th of March, 1812, Jupiter will be exactly in the
ecliptic, and his longitude will be 2s. 27 24' ; required his dis-
tance from the sun, and from each of the following stars, viz. AI-
debaran, Rigel, Betelgeux, Sirius, Procyon, Castor, Pollux, Ca-
pella, the Pleiades ; and also from the moon, whose right ascen-
sion at midnight will be 89 45' 2 1", and declination 18 34' N. ?
Now in the triangle sZm, there are given the angle Z, and sZ, mZ, to find
sm the true distance ; hence it will be,
Rad. : cos. Z. :: tang. Zm. : tang. Zx the distance between Z and a perp.
let fall from m ; then Zs wj Zx = sx. And cos Zx : cos. Zm :: cos sx : cos.
sm, the correct distance required. As sx and the angle at Z are of the
same or different affection, sm is greater or less than a quadrant.
The learner will also observe, that the mark 02 denotes the difference of
those quantities between which it is placed.
Example. On June 29, 1793, the complement of the sun's apparent alt. of
his apparent zenith distance ZS was 70 56' 24", the comp. of the moon's
app. alt. or app. zenith distance ZM was 48 53' 58", their app. distance
SM was 103 29' 27", and the moon's horizontal parallax was 58' 35" ; their
true distance calculated by the above method, will then be 103 3 18", as
required.
The sun's place being given, together with the moon's lat. and long, their
distance may be thus found : The cliff, of long, of the sun and moon, the
moon's lat. and the dist. between the sun and moon (represented on the
globe) will form a right angled spherical triangle, the sides of which are
given, viz. the diff. of longitude and moon's lat. to find the third side, which
is the true distance between the sun and moon. Whence, making use of
Baron Napier's rule, we have this proportion ; Rad. : co. sine diff. long. ::
co. sine moon's lat. : co. sine true distance required. The rule given in page
147 of the Nautical Jllmanac for 1813, is, in substance, the following ; log.
co. sine diff. long, between the sun and moon -f- log. co. sine moon's lat. = log. co.
vine trite distance^ which is evidently an error, as appears from the above
proportion, as it ought to be, log. co. s. diff. long. + log. co. s. moon's lat.
rod. = log. co. s. true distance.
The Nautical Almanac above alluded to, is that revised by Mr. John Gar-
nett, New-Jersey, a work which deserves every encouragement, from its ex-
tensive utility, and the many important additions made by the Editor. (The
error above noticed must evidently have been an oversight, or omission in the
printer.) The Nautical Almanac for 1813 is the first distinguished with that
considerable advantage of having the accurate Lunar tables of Mr. JBurg,
and the late improved Solar tables of M. De Lambre, made use of in its cal-
culation. These tables are corrected and improved by the Rev. Samuel
Vince of Cambridge, and published in English. He has adapted them to
astronomical time, those of Mr. Burg and De Lambre being adapted to civil
time ; so that the year, in the latter, commences at the midnight with which
the last day of the former year ends, and in Vincc's 12 hours later, &c.
Ee
226 PROBLEMS PERFORMED BY
PROB. 29.
Given the true distance of the moon from the sun or a star, and the
time at which the observation was made, tojind the corresponding
time at Greenwich, and the longitude of the place of observation.
Rule. MARK the moon's path on the globe for the noon and
midnight preceding and following the time of observation (by
probs. 2 and 4. part 3d ) and also the moon's places for every three
hours during this interval of time, by taking proportional parts : if
it be the distance of the moon from the sun that is given, mark the
sun's place in the ecliptic, corresponding to the times in which
the moon's respective places were marked in its orr.it ; then find
the true distances of the moon from the sun or star, which are
next greater, and next less than the true distance deduced from
observation (either with a pair of compasses applied to the sun's
places or the centre of the star, and to the corresponding places
of the moon at the same time ; or taken from the Nautical Alma-
nac) and the difference of these distances (which call D) will give
the access of the moon to, or recess from, the sun or star in three
hours ; then take the difference between the moon's distance at
the beginning of that interval, and the distance deduced from ob-
servation (which call d} and say, D : d :: 3h, : to the time the
moon is approaching to, or receding from, the sun or star by the
quantity d ; which added to the time at the beginning of the inter-
val, gives the apparent time at Greenwich, corresponding to the
given correct distance of the moon from the sun or star ; the dif-
ference between which and the apparent time at the place of ob-
servation, will be the difference of longitude in time, which may
be easily reduced into degrees, Sec.
Example I. Suppose that on the 14th of May, 1812, in latitude
40 42' 40" N. at 6h. 3' 57" apparent time in the afternoon, the
correct distance of the sun and moon's centres was 52 30' 40 ;/ ;
For other methods of finding the distances of celestial objects, see Vince's
Complete System of Astronomy, or McKay's Treatise on Navigation, &c.
As the learner may be at a loss to determine the parallax of any of the
celestial bodies, the following* remarks may be necessary.
Observations prove that the diameter of any of the fixed stars is less than
1"6, and therefore that they have no sensible parallax. The parallax of the
sun resulting from the observations of the transit of Venus in 1761 and 1769,
is 8" 8. The horizontal parallax, of the moon is given in pa. 7 of the month
in the Nautical Almanac for every noon and midnight, or it may be calcu-
lated from probs 16 and 17 of Mayer's tables. A table of the reduction of
latitude and moon's horizontal parallax, for the spheroidical figure of the
earth, is also given in page 12 of July in the Nautical Almanac for 1812.
In general the distance of a phenomenon, from the earth : to the semi-
diameter of the earth :: cos. apparent altitude of the body : sine of the pa
rallax. See Gregory's Astronomy, b. 2, sect. 7, or Vince's Astronomy, 8vo t .
ch. 6, where several methods are given. The parallax varies inversely as
the distance. De Lambre, in his calculations, makes use of 8" 6 for the sun's
Jiorizontal parallax. See his tables annexed to the 3d edit, of La Land's
Astronomy. The distances, &c. of the planets will be given in part 4,
THE CELESTIAL GLOBE. 227
required the apparent time at Greenwich, and the longitude of the
place of observation ?
Ans. True distance of the moon from the sun is 52 30' 40"
Do. by Naut. Aim. on May 14th. at 9h. 51 29 25
Do. by do - - at midnight 53 1 17
d 9= 1 115
D = 1 31 52
Hence 1 31" 52" : 1<> 1' 15" :: 3h. : 2 hours, which added
to 9 hours gives 11 hours, therefore 1 Ih 6h. 3' 57" = 4h.
56m. 3s. which in degrees is 74 0' 45", the longitude of the
place of observation, and is west of Greenwich, as the time in
Greenwich is later.
2. Suppose that on the 15th of May, 1812, in latitude 39 57' N.
at 9 o'clock in the afternoon apparent time, the distance of the
moon's centre from Regulus was 26 24/ 27" ; required the appa-
rent time at Greenwich, and the longitude of the place of obser-
vation ?
Ans. True dist. of the moon from Regulus by obs. 26 24' 27"
True dist. by Naut. Aim. on May 1 5 at midnight 2729 I
Do. at 15 hours 25 52 56
d = 1 4 34
D = 1 26 5
Then 1 36' 5" : 1 4' 34" :: 3h. : 2h. ; 57" 5, which
added to 12 hours, gives I4h. 0' 57" 5 ; hence I4h 0' 57" 5
9h. = 5h. 0' 57" 5, the diff. of longitude in time, which in de-
grees is 75 t4' 22", west of Greenwich.
3. On June 29, 1793, in latitude 52 12' 35", the sun's altitude
in the morning was by observation 19 3' 36", the moon's altitude
was observed to be 41 6' 2", the sun's declination at that time
was 23 14' 4", and the moon's horizontal parallax 58' 35" ; to
find the apparent time at Greenwich, and the longitude of the place
of observation ?
Ans. True dist. of the moon from the sun (note to prob. 28) 103 3' 18 V
Do. by Naut. Aim. on June 29, at 3h. 103 4 58
Do. - - - - on June 29, - 6h. 101 26 42
d=0 1 40
D=l 38 16
Now 1 38' 16" : 1' 40" :: 3h. : Oh. 3' 3" which added to 3h.
gives 3h. 3m. 3s. the apparent time at Greenwich.
Now to find the apparent time at the place of observation, we
have the sun's alt. 19 3' 36", its refraction 2' 44", and parallax
8", hence its altitude was 19 1', and therefore its true zenith dist.
was 70 59' ; also the co. decl. was 66 45' 36" ; hence by the
note to prob 48, part 2d, or by the globes, the hour angle is found
equal 88 37' 44" in time equal 5h 54' 30" 9, the time before
apparent noon, or I8h. 5' 29" 1 on June 8th. Hence 29d. 3h.
3m. 3s. the app. time at Greenwich, less 28d, 18h, 5m. 29s, the
228 PROBLEMS PERFORMED BY
app. time at the place of obs. gives 8h. 57' 34" = the diff. of
meridians or diff. long, in time, which in degrees is 134 22' 31",
the long, of the place of obs. west of Greenwich.
PROB. 50.
To find what stars lie in or near the moon's path, or what stars
the moon can eclipse, or make a near approach to.
Rule. FROM the longitude and latitude of the moon, or her
right ascension and declination, taken from the Nautical Almanac,
or any good ephemeris, mark the moon's places on the globe, for
several days (by problems 2d. and 3d. part 3d.) then by extending
the quadrant of altitude or a thread, over these places, you will
nearly find the moon's path, and consequently those stars that lie in
her way, or that she can make a near approach to.*
Example \ . What stars were in or near the moon's path on the
16th, 17th, 18th, and 19th of May, 1810 ?
16th. 3) 's right ascension 206 47' declination 9 42' S.
17th. 220 43 - - J3 14 S.
18th. - 235 22 - - 16 3 S.
19th. - . 250 38 - 17 53 S.
Ans. Zuben el Chamali, Zuben ha Krabi, & in Libra, &c.
2. On the 4th, 5th, 6th, 7th, and 8th of September, 1812, what
stars will be near the moon's way ?
4th. j's longitude 4s. 27 3' 40" latitude 11' 3" S.
5th. - - - 5 9 17 19 56 22 N.
6th. - - 5 21 22 14 2 22 N.
7th. - - - 6 3 20 6 - 2 58 22 N.
8th. - - - 6 15 12 51 3 48 9 N.
PROB. 31.
The latitude of a place being given, to Jind the time of the year at
which any known star rises or sets achronically, that is, when it
rises or sets at sun setting.
Rule. ELEVATE the pole to the latitude of the place, bring the
given star to the eastern part of the horizon, and then mark the
point of the ecliptic at the western edge, or the point of the eclip-
tic that sets when the star rises, the day of the month correspond-
ing to this point will give the time when the star rises at sun set,
or when it begins to be -visible in the evening. The globe being
* The situation of the moon's orbit for any particular day may be found
thus ; find the place of the moon's ascending- node in the Nautical Almanac ;
mark that pluce and its antipodes (being the descending- node) on the globe ;
take the middle between these two points, and make two marks 5 8' 49' ;
( the inclination of the lunar orbit to the plane of the ecliptic), on the north
and south sides of the ecliptic ; so that the northern mark may be between
the ascending and descending node, and the southern between the descend-
ing and ascending node ; a thread extended through these four points,, witt
shew the position of the moon's orbtl.
THE CELESTIAL GLOBE. 229
then turned westward on its axis, until the star comes to the west-
ern edge of the horizon, observe the degree of the ecliptic cut
by the western part of the horizon as before, the day of the month
answering to that degree, will shew the time When the star sets
with the sun, or when it ceases to apfiear in the evening.
Example 1 . At what time does Arcturus rise and set achroni-
cally at Ascra* in Boeotia ; the latitude of Ascra, according to
Ptolemy, being 37 45' N. ?
Ans. When Arcturus is in the eastern part of the horizon, the
twelfth degree of Aries will be at the western, which answers to
the 1st of April,! the time when Arcturus rises achronically ; Arc-
turus will set achronically on the 30th of November.
* Ascra is a small village in Boeotia at the foot of Mount Helicon, where
Hesiod lived, and was probably born, hence called Ascreus. (Virgil Eel. 6,
70, and Georg. 2, 176.) Strabo says that he came with his father Dio from
Cuma, a city of Eolis, opposite to Lesbos, now called Taio Nova. Some are
of opinion that he lived before the time of Homer, as his style is more rude
and simple ; others that he was cotemporary with Homer (this is the com-
mon opinion. Rollin's Anc. Hist. b. 5, art. 9) and others that he lived after
Homer. Velleius Paterculus, who lived in the time of the three first Roman
emperors, says, in his abridgment of the Roman History, that he lived 120
years after the time of Homer. However, as Carolus Rureus Soc. Jesu, the
learned commentator on Virgil, remarks, * Lis adhuc pendet.' (See New-
ton's Chronology.)
f If we allow for the star's refraction, which at the horizon is about 3S f ,
the time will nearly correspond to the 31st of March, and then Arcturus
would rise achronically in lat. 37 4i/ N. about 99 days after the winter sol-
tice. Hesiod, in his Opera et Dies, lib. 2, verse 185, says,
When from the solstice sixty wintry days,
Their turns have finished, mark, with glitt'ring rays,
From ocean's sacred flood, Arcturus rise,
Then first to gild the dusky evening skies.
Hence (supposing Hesiod to be correct) there is, between the time of
Hesiod and the present time, a difference of 39 days in the achronical rising
of this star ; and as a day answers to about 59' 8" of the ecliptic (note to def,
66) 39 days will answer to 38 26' 12", and therefore the winter solstice in
the time of Hesiod was in 8 26' 12" of aquarius. Now the precession of the
equinoxes being about 50^" in a year, we have 50" : 1 year :: 38 26' 12" :
2753 years nearly, since the time of Hesiod ; so that (the places of the stars
on the globe being adapted to the year 1800) he must have lived 953 years
before Christ by this mode of reckoning. Homer, according to most chro-
nologies, lived 907 years before Christ. Lempriere, in his Classical Diction-
ary, says that Hesiod lived at the same time. Herodotus however (lib. 2,
c. 53) says that Homer wrote 400 years before his time, that is 340 years
after the destruction of Troy, which happened 1184 years before Christ, so
that, according to Herodotus, Homer lived 844 years' before Christ.
The above calculation was made without reflecting that the same allov\--
ance should be made for the sun's refraction, which would make the time
nearly correspond to the 30th of March, giving the astronomical rising of
Arcturus about 98 days after the winter solstice ; differing from the same in
Uesiod's time 38 days ; which answers to 37 27' 4", or 7 27' 4* of Aqua-
rius. Hence 50i" : ly. :: 37 27' 4" : 2683 years ; therefore 2683 1300=
883 years. This might be rendered more accurate by strict calculation.
Keith in his treatise on the Globes, makes the time of Hesiod 990 years be
tb're Christ. From the whole we see that there 5s a strong probability" of his be-
230 PROBLEMS PERFORMED BY
2. At what time of the year does Aldebaran rise and set achron-
ically at Athens, in latitude 38 5' N ?
3. When does Sirius rise achronically in New-York, and at what
time of the year does it set achronically ?
4. When does Procyon rise at London when the sun is setting,
and when does he set at sunset ?
PROB. 32.
The latitude of the place being given, to find the time of the year at
which any known star rises or sets cosmically, that is, rise* or
sets at sun rising.
Rule. ELEVATE the pole to the given latitude, bring the given
star to the eastern part of the horizon ; then the day of the month
corresponding to the degree of the ecliptic cut by the eastern part
of the horizon, will give the time when the star rises with the sun ;
bring the star to the western part of the norizon, the sign and de-
gree of the ecliptic, then intersected by the eastern part of the
horizon as before, will point out on the horizon the time when the
star sets cosmically, or at sun rising.
Example 1. At what time of the year do the Pleiades set cos-
mically at Miletus* in Ionia, in lat. 37 N. according to Ptolemy,
and at what time of the year do they rise with the sun there ?
ing cotemporary with Homer. What we have here said may to some seem
rather tedious, but the learner will derive much information from it in simi-
lar calculations, as the ancients made frequent use of the poetical rising and
setting of the stars.
This and the following- problem may be
solved by trigonometry, as follows ; let HO
represent the horizon, HZO the mer. JEQthe
equinoctionul, EC the ecliptic, ^ the point
aries, or the intersection of the equinoctial and
ecliptic, S the point of the ecliptic which rises
with the star, and o the point of the equator ;
then in the triangle f oS, we have TO, the
oblique ascen. of the star, the angle at T, the
obliquity of the ecliptic,and the angle ^oS,
the height of the equator above the horizon
(being equal to the co. lat. Note to prob. 19)
or its supplement; hence ^S is given, and
therefore the point S of the ecliptic, which
rises with the star, or the star's long-, is given, the time corresponding to
which, found by the Xautical Almanac, or the globe, will be the time when
the star rises cosmically. The angle TSo, is the angle which the ecliptic and
horizon make at the rising point. When the sun is in the sign and degree
opposite the point S, the star will then rise achronically. In a similar the
time is found when the star sets cosmically or achronically..
* Miletus, the birth place of Thales, was situated in Asia Minor, on
the coast of the Egean sea, near the borders of Caria. south of Ephesus,
and southeast of the island of Samos, or six miles to the south of the mouth
of the river Mccander. This city, as Pliny remarks, was sometimes railed
Pithyusa, Anactoria and Lelegis; now called Melaxo or Melasso. It was
formerly famous for its wool.
THE CELESTIAL GLOBE. 231
Ans. The Pleiades rise with the sun in lat. 37 N. on the llth
of May, and they set at the time of sun rising on the 2 1st of No-
vember.*
Thales laid the first foundation of philosophy in Greece, and founded
the Ionian school, where he taught the sphericity of the earth, the ob-
liquity of the ecliptic, and the true causes of the eclipses of the sun and
moon. He had exactly foretold the time of the eclipse of the sun, that
happened in the reign of Astyages, king of Media, of which mention is
made in Rollin's Anc. Hist. He also discovered the solstices and equi-
noxes, divided the heavens into five zones, and recommended the division
of the year into 365 days. When he travelled into Egypt, he discovered
an easy and certain method of determining the exact height of the pyra-
mids, by observing the time when the shadow of a perpendicular body
was equal in length to the body itself. His life, as well as that of the
other wise men, is written by Diogenes Laertius.
* Pliny (in his Natural History, b. 18, c. 25) says, that Thales deter-
mined the cosmical setting of the Pleiades to be 25 days after the au-
tumnal equinox. Supposing this observation to be made at Miletus, there
will be a difference of 35 days in the cosmical setting of this star ; hence
Id. : 59' 8" of the ecliptic :: 35d. : 34 29' 40", which in the time of Thales will
make the equinoctial colure pass through 4 2S' 40' of scorpio ; and 50i":
ly. : 34 29' 40" : 2471 years since the time of Thales Hence Thales
lived 2471 1800 =? 671 years before Christ, by this mode of reckoning.
This time will however be lessened, by allowing for refraction, &c. Sir
I. Newton, in his chronology, makes it 596 years before Christ; most
chronologers make it 600 years. According to Lemprierc in his Classical
Dictionary, he died in the 96th year of his age, about 548 years before the
Christian era. The remarkable eclipse predicted by Thales happened in
the 545th year before Christ. See Ferguson's Astronomy, page 25.
Some affirm that Thales taught that one intelligent being presides over
and governs the universe. Many of the heathen philosophers came to the
knowledge of this truth by the light of reason alone ; but among the
whole there was not one that did not worship the ridiculous Gods of his
country with the vulgar, the knowledge of whom in other respects they
so much despised, so that we find Socrates, the wisest among them, at his
dying moments ordering his friend to sacrifice a cock to Escidapi-us.
This is another unanswerable argument in favour of that knowledge
which the Christian possesses of the true God, contrasted with that which
a proud, shallow, presumptuous philosophy affords. To him that is desti-
tute of this knowledge, the glimmering ray of human science will yield
but little assistance, in dissipating the darkness that surrounds him Its
frigid rules is but a feeble support to resist the violence of human de-
pravity. No light, therefore, but that of the gospel, could dispel the
darkness of infidelity in which man was involved ; no power but that of
religion could resist his lawless passions.
The notions which the heathens had of a providence of immortality, and
other truths of a similar nature, were, according to most authors, the ef-
fects of a tradition as old as the world, and derived from revelation ; but
its feeble light was almost extinguished among them, and hence arose
their superstition and folly. It is true, the impiety of philosophers have
too often been attributed to the sciences which they profess, but with as
much reason as the immorality of some Christians is attributed to the re-
ligion which they pretend to practice. Uoth evils originate in the cor-
ruption of the human heart.
The province of philosophy, rightly understood, is extensive and 5m
portant. There is no employment more innocent none more ingenious,
*ncl to those who have a taste for science, more amusing none, in fine A
232 PROBLEMS PERFORMED BY
2. At what time of the year will Procyon rise with the sun, and
also at what time will he rise when the sun rises, at Washington
city ?
3. At what times of the year will Regulus rise with the sun, and
also set when the sun rises at Petersburg in Russia ?
that lead to more important and useful discoveries, than guided by ma-
thematical principles and the result of laborious experiments, patiently
to investigate the phenomena and laws of nature, and apply them to those
useful purposes for which they are adapted. Conducted by this science,
wg contemplate with pleasure and advantage those stupendous bodies that
exist around us, guided by more stupendous but unerring laws our fa-
culties and conceptions are thereby expanded, and our mind has some-
thing to employ it that bears no small proportion to those noble powers
which it possesses. We delight in the beauty and universal harmony
which we perceive in these new scenes of wonder which every moment
open to our view, from the smallest atom from the blade of grass which
we trample under foot, to the remotest world which we contemplate. No-
ble and extensive as the human intellect is, between these two extremes
its shallow line of reason is soon exhausted. Here man, however, traces
the omnipotence, the wisdom and goodness of a being infinitely his own
superior ; from his works he feels a desire of being more acquainted with
him, and this being, having destined man for himself, nourishes the de-
sire. Thus far philosophy may conduct us, thus far its horizon extends,
beyond which there is nothing but shadows and delusion. Religion opens
here a brighter prospect, and like the sun which banishes the light of all
other luminaries, will have no other light to conduct us but its own.
Here futurity developes its extensive prospects, immortality, and not this
short span of existence eternity, not time, arrest our attention. Here
omnipotence itself stoops to our assistance infinite wisdom guides, and
informs us of the true source of our condition and misery, and the good-
ness of this omnipotent being generously affords and points out the reme-
dy. He exhibits the happiness of heaven as a reward for the good, and
the torments of hell as punishments for the wicked He enlightens and
gives us a more intimate knowledge of his nature and of our duty towards
him calls himself by the endearing name of Father, and calls us by the
loving and exalted title of children ; thus indicating our dignity, and the
point in which alone our true nobility consists. These truths are estab-
lished on divine authority, confirmed by miracles by human testimony as
far as human testimony could go, and by their own internal evidence and
necessity. From the nature of truth they are immutable, and not subject
to presumptuous innovation or versatile fancy. They are truths not in the
province of philosophy to teach truths far more important, interesting,
and noble, and a mind possessed of all other wisdom and knowledge un-
der the sun, that views them not in this light, is, after all, neither intelli-
gent or wise.
We have already seen (remark after the constellations) how civilized and
learned men can act when withdrawn from the salutary restraint of religion; or
when guided by a false philosophy, and under the influence of unrestrained,
licentious passions. Their deeds bear awful testimony to the feeble .efforts
of reason in establishing 1 vain systems of fancied superiority, ia opposition to
that established by unerring wisdom, whose benignant influence argues its
origin, and shews that no reformation is wanting to it, that no substitute
ran ever supply its place
CELESTIAL GLOBE. 233
PROB. 33.
To find the time of the year when any given star rises or sets helia-
cally, or when a star becomes first -visible after emerging from the
sun's rays, in the morning before sun rising, or invisible in the
evening, on account of its nearness to the sun.
Rule. ELEVATE the pole to the latitude of the given place ;
screw the quadrant in the zenith ; bring the quadrant to the east-
ern edge of the horizon, and move it until the star intersects it
1 2 below tlie horizon, if the star be of the first magnitude ; 1 3
if the star be of the 2d rnag. ; 14 if the star be of the 3d, &c.*
The point of the ecliptic cut by the quadrant, will shew the day of
the month on the horizon, when the star rises heliacally. The
given star being brought to the western edge of the horizon, and
the quadrant of alt. being moved until intersected by the star as
before ; the point of the ecliptic then cut by the quadrant will
give the day of the month when the star sets heliacally.
Examfile 1 . At what time of the year does sirius, or the dog
star, rise heliacally at Alexandria, in Kgypt, in lat. 31 11^' N. ;
and when does it set heliacally at the same place ?
Ans. On the 4th of August f and 2 3d of May.
* According to Ptolemy, stars of the first mag. are seen rising 1 and setting
when the sun is 12 below the horizon, stars of the 2d mag-, when the sun
is 13 below the horizon, &c. reckoning- 1 for each mag. For the brighter
a star is when above the horizon, the less the sun will be depressed before
it becomes visible.
\ The ancients reckoned the dog days from the heliacal rising 1 of sirius,
and their continuance to be about 40 days. Hesiod remarks, that the hottest
season of the year (or the dog days} ended about 50 days after the summer
solstice. In the note to prob. 31, it is shewn that the winter solstice, in the
time of Hesiod, was in about 8 26' 12" of aquarius, and consequently the
summer solstice was in the same degree of leo. Now from the above it ap-
pears that sirius rises heliacally when in 12 of leo (corresponding- to the
4th of Aug.) and as 59' 8" or 1 nearly corresponds to a day, sirius rose
heliacally about 4 days after the summer solstice ; and if the dog days con-
tinued 40 days, they ended about 44 days after the summer solstice. In our
almanacs the dog 1 days begin on the 3d of July, which is 12 days after the
summer solstice, and end on the llth of August, which is 51 days after the
summer solstice ; their continuance is therefore 39 days.
The dog days of the moderns have therefore no reference to sirius or the
Jog star, for as it varies in its rising and setting 1 according 1 to the latitude of
places, it could therefore have no influence, or indicate no change in the
temperature of the atmosphere. However as this star rose heliacally at the
commencement of the hottest seasons in Egypt, Greece, &c. in the infancy
of astronomy ; and at a time when astrology 7 referred almost every thing 1 to
the influence of the stars, it was natural for those people to imagine that
the heat, See. was the eifect of this star's influence, &c. A few years ago
the dog days were reckoned in our almanacs from the cosmical rising- of pro-
eyon, viz. on the 30th of July, and continued to the 7th of September ; but
are now very properly altered, and made to depend on the summer solstice,
and not on the variable rising 1 of any particular star whatever.
The solution of the prob. by trigonometry, may be as follows ; let S (see
the fig 1 , in the note to prob. 31, part 3) be the point of the ecliptic which
rises with the star, and let be the place of the sun in the ecliptic, so that
Ff
234 PROBLEMS PERFORMED BY
2. At what time does Aldebaran set heliacally at New-York ?
3. At what time of the year does Arcturus rise heliacally at
Washington city ?
4. At what time does Procyon rise and set heliacally at London ?
5. How many years will elapse from IS 10, before Shius will
rise heliacally on Christmas day, at Cairo, in Egypt, allowing the
precession of the equinoxes to be 501. seconds ?
PROB. 34.
T/ie latitude of the place and day of the month being given, tojind
all those stars that rise and set achronically, cosmically, and
heliacally.*
Rule. RECTIFY the globe for the latitude : then,
1. For the achronical rising and setting. Bring the sun's place
in the ecliptic to the 'western edge of the horizon ; then all the
stars along the eastern edge will rise achronically, and those along
the western will set achronically.
2 . For the cosmical rising and setting. Bring the sun's place
to the eastern edge of the horizon ; then all the stars along that
edge of the horizon will rise cosmically, and those along the west-
ern edge will set cosmically.
3. For the heliacal rising and setting. Screw the quadrant of
alt. in the zenith, turn the globe eastward until the sun's place
cuts the quadrant 12 below the horizon ; then all the stars of the
1st magnitude along the eastern edge of the horizon will rise helia-
cally ; and by continuing the motion of the globe eastward until
the sun's place intersects the quadrant in 13, 14, 15, &c. below
the horizon, you will find all the stars of the 2d, 3d, 4th, Sec. mag-
nitudes, which rise heliacally on that day. By turning the globe
the arc QR of the circle of depression maybe 12 or 13, &c. according as
the star is of the 1st, 2d, &c. magnitude ; then in the right angled triangle
vSRQ RS0 the angle formed by the ecliptic and horizon (note to prob.
31) and the side RQ = 12 or 13/&c. are given, and therefore the side SQ
is- given, which added to <Y^S, gives the arc ^0, and the point Q or the
sun's place when the star rises heliacally. The star's heliacal setting may
be found in like manner.
* The principal use of this and the foregoing problems, is to illustrate
several passages in the ancient writers, as Hesiod, Virgil, Columella,
Ovid, Pliny, &c. These different risings and settings of the stars were
called poetical, because principally used in the writings of the poets. The
knowledge of these poetical risings and settings of the stars was much
esteemed by the ancients, as it served to adjust the times set apart for
their civil and religious duties, and to mark the seasons proper for the
several parts of husbandry, the time of the overflowing of the Nile, &c.
their knowledge of astronomy being too limited to adjust the length of
the year, &c. The knowledge which the moderns have acquired of the
motions of the heavenly bodies, renders such observations unnecessary.,
as an almanac answers every purpose of the husbandman.
This problem being the reverse of the three foregoing, the solution by
trigonometry is performed in a similar manner, and is left for the learn-
er's exercise.
THE CELESTIAL GLOBE. 235
westward in a similar manner, and bringing the quadrant to the
western part of the horizon, you will find those stars that set helia*
cally.
Example 1. What stars rise and set achronically, cosmically,
and heliacally at New-York on the 4th of December ?
Ans. For the achronical, &c- Aldebaran, 8cc. will rise achronic-
ally. Arcturus, Sec will set achronically.
For the cosmical, 8cc in Lupus will rise cosmically, Sec. An-
tares will be near the eastern horizon. Algol in Perseus will set
cosmically, &c. Betelgeux will be near the western horizon.
For the heliacal, &c. Arided in Cygnus will rise heliacally,
& in Serpens will set heliacally, &c.
2. What stars rise and set achronically at Petersburg on the 10th
of May ?
3. What stars rise and set cosmically at Washington city on the
5th of April ?
4. What stars rise and set heliacally at Philadelphia on the 4th
of July ?
5. What stars rise and set achronically, cosmically, and heliacal-
ly at London on the 7th of October ?
PROB. 35.
The latitude ofthejilace and the day of the month being given, tojind
what planets will be above the horizon, after sun setting.
Rule. RECTIFY the globe for the given latitudes, bring the
sun's place to the western part of the horizon, or to JO or 12 de-
grees below it ;* then all the planets whose places are in the hem-
isphere above the horizon, will be visible after sun setting, whose
places may be found in an ephemeris for that day and month ; if the
motion of the globe be continued westward until the sun's place
comes within 10 or 12 of the eastern part of the horizon, all the
planets that were above the horizon during this motion, will be
visible, and fit for observation on that night.
Example \ . Were any of the planets visible at New-York when
the sun had descended 10 below the horizon, on the 1st of Janua-
ry, 1811, their latitudes and longitudes being as follow :
TT . . 1 V . -I T . . 1 TT
Latitudes.
% 2 6 7 S.
9 4 12 N.
% 1 29 N.
Longitudes :;';::
9s. 23 20' I::!:
9 4 48 g
6 28 32 ;:;?
Latitudes.
%. GO 57' S
k 1 17 N.
$ 21 N.
Longitudes.
Is. 21 45'
8 20 20
7 17 41
3) 's latitude at midnight 1 0' 58" S. long. Os. 11 33' I
Ans. Mercury was near the horizon, Jupiter and the moon
were visible.
* The planets are not visible until the sun is a certain number of de-
grees below the horizon, and these degrees are variable according to the
apparent magnitudes and brightness of the planets. Mercury becomes
visible when the sun's depression is about 10 ; Venus when the sun is 5
below the horizon ; Mars when the sun is at 11 3CX; Jupiter at 10; Sa>
turn at 11 ; and Herschel at 17^.
236 PROBLEMS PERFORMED BY
2. What planets will be above the horizon at New-York on the
1st of December, 1812, and what planets will be visible during
that night, or before the sun is within 10 of the eastern part of
the horizon ; their latitudes and longitudes being as follow :
Latitudes.
2 22' S.
9 2 9 N.
Z 1 2N.
3)'s lat. at mic
Longitudes. ::|:i Latitudes.
8s 26 59' >;; y QO 32' N.
6 28 21 g \i 31 N.
6 24 45 : : ;:;j $ 14 N.
night 4<> 59' 27" N. long. 7s.
Longitudes.
4s, 9 0'
9 6 52
7 24 40
11 24' 27".
PROB. 36.
The latitude, day, and hour being given, to Jind 'what planets will be
-visible, or above the horizon, at that hour
Rule. RECTIFY the globe for the latitude ; bring the sun's
place to the meridian, and set the index to 2 ; then turn the globe
eastward or westward, according as the time is in the forenoon or
afternoon, as many hours as the time is before or after 12 ; fix
thf: globe in this position, and from the latitudes and longitudes of
the planets found in the Nautical Almanac, see whether any of
them be in the hemisphere which is above the horizon ; such
planets will be visible.
Example 1. Were any of the planets whose places are found
from example 1st of the foregoing prob- visible at 9 o'clock in the
evening, at New-York, on the : st of January, 1811?
Ann Jupiter and the moon were visible.
2. Will any of the planets whose latitudes and longitudes are
given in ex. 2, last prob. be visible at Philadelphia at 4 o'clock in
the morning of the 1st of December, 1812 ?
PROB 37.
Given the latitude of the place and day of the month, to Jind how
long Venus rises before the sun when she is a morning star,* and
how long she sets after the sun when she is an evening star.
Rule. RECTIFY the globe for the given latitude ; find the lati-
tude and longitude of Venus in an ephemeris, and mark her place
on the globe ; bring the sun's place for the given day to the brass
meridian ; then if the place of Venus be to the eastward of the
meridian, she is an evening star, or rises after the sun, but if to
the westward, she is a morning star, or she rises before the sun.
When Venus is an evening star. Turn the globe westward
until the sun's place comes to the western part of the horizon, the
index will then shew the time of sun setting ; the motion of the
globe being continued westward until Venus comes to the edge of
* Venus is a morning- star from inferior to superior conjunction, and ai
evening star from superior to inferior conjunction.
THE CELESTIAL GLOBE. 237
the horizon, the index will shew when Venus sets ; the differ-
ence between which and the time the sun sets, will shew how long
Venus sets after the sun.
When Venus is a morning star. Find the time of sun rising
(prob. IS. p 2d.) and also the time that Venus rises (prob. 8,
part 3d.) the difference between these times will shew how long
Venus will rise before the sun.
Note. The same rule will answer to shew when any of the planets
rises before the sun and sets after him ; and how long.
Example \ On the i st of November, 1810, the longitude of
Venus was 8s. 24 30' lat. 4 7' south ; will she be a morning
or an evening star ? If she be a morning star, how long will she
rise before the sun at New-York ; if an evening star, how long
will she shine after sun set ?
Ans. Venus was an evening star ; the sun set at 5h. 10' or 10
minutes after 5, and Venus 7h. 20 min or 20 min. after 7 ;
hence Venus set 2h. 10' after the sun.
2. On the 19th of November, 18 12, Jupiter's longitude will be
4s. 9, or 9 in Leo, latitude 29' north ; will Jupiter be a morning
or an evening star ? If a morning star, how long will he rise be-
fore the sun ; if an evening star, how long will he shine after
sun set ?
3. On the 13th of October, 1812, the longitude of Venus will
be 5s 3 40', and latitude 36' south ; will Venus be then a morn-
ing or evening star ? If a morning star, how long will she rise
before the sun ; if an evening star, how long will she shine after
sun set?
4. On the 1st of April, 1812, Saturn's longitude will be 9s.
7 50', and latitude 52' N. will he be a morning or an evening
star, &c.
PROB. 38.
Tojind all those places on the earth to 'which the moon will be nearly
vertical on any given day.
Rule. TAKE the moon's latitude and longitude for the given day
from an ephemeris, and mark the place corresponding to them,
on the globe (by prob. 4.) bring the place to the brass meridian,
and observe the degree over it ; then all those places having the
same or nearly the same latitude, will have the moon vertical when
on their respective meridians.
Or, Thone places whose latitudes are equal to, and of the same
name with her declination for the given time (found in the Nauti-
cal Almanac) will have the moon successively vertical on the given
day.
Example 1. On the 20th of December, 1810, the moon's lon-
gitude at midnight was 6s. 20, and her latitude 1 $' N. ; requir-
ed the places to which she will be nearly vertical that day t
238 PROBLEMS PERFORMED BY
Ans. From the moon's latitude and longitude, her declination is
found nearly 7 south ; hence the places are the Sunda Isles, Solo-
mon's Isles, Sana in Peru, Olinda in Brazil, Congo, and Lower
Guinea in Africa, &c.
Note 1. When the declination is given, the prob. may be performed by
the terrestrial g-lobe alone, being the same as prob. 10, part 2.
Note 2. The places where any of the planets will be vertical, are found
in the same manner.
2. On the 1st of November, 1812, the moon's longitude at mid-
night will be 6s. 14 54' 25", and her latitude 3 46' 32" ; re-
quired the places to which she will be nearly vertical when on
their meridian that day ?
3. On the 9th of November, 1811, the moon's declination at
midnight will be 8 N. ; required where she will be vertical that
day ?
4. On the 7th of December, 18 11, Jupiter's longitude will be
3s. 3 20', and latitude 15' south ; to what places will he be ver-
tical that day ?
5. On the 21st of October, 1813, the declination of Herschel
will be 19 11' south ; where will he be vertical on that day ?
6. Required those places where the moon will be vertical when
she has her greatest north declination, and also her greatest decli-
nation south ?
7. Required those places to which Venus will be vertical when
she has her greatest north declination, and required that declination ?
PROB. 39.-
To find the time of the moon's southing, or coming to the meridian
at any given filace, on any given day of the month.
Rule. ELEVATE the pole to the given latitude ; find the moon's
latitude and longitude, or her right ascension and declination from
the Nautical Almanac or a good ephemeris, and mark her place
on the globe (by prob. 2 and 4) bring the sun's place to the brass
meridian, and set the hour index to 12 ; turn the globe westward
until the moon's place comes to the meridian, and the hours pass-
ed over by the index, will shew the time from noon when the moon
will be on the meridian.
OR WITHOUT THE GLOBE.
Find the moon's age by the note to def. 80, which multiply by
.81* ( T VV) ancl tne product will be the hours and decimal parts of
an hour, which multiply by 60 for minutes.
OR CORRECTLY, THUS,
Take the difference between the sun and moon's right ascension
in 24 hours ; (found by the Nautical Almanac) then say as 24 h.
* The synodic revolution of the moon being- according to La Place, 29d.
12h. 44' 2'' 8, or nearly 29f days, we have as 29|d. : 24h. :: Id. : .81h. near-
ly. Hence the reason of the rule is manifest.
THE CELESTIAL GLOBE. 239
less this difference to 24 h. so is the moon's right ascension at
noon diminished by the sun's, to the time of the moon's southing.
Note 1. When the sun's right ascension is greater than the moon's, 24
hours must be added to that of the moon before you subtract.
Example 1. At what time on the 10th of March, 1811, did the
moon pass over the meridian of Greenwich,* the moon's right
ascension being 172 10' 48", and her declination 2 55' N. at
noon ?
Ans. By the globe the moon came to the meridian at half an
hour after 12 at night.
By the note to def. 80. The moon's age is 16 days ; hence 16 X -81 =
12.96 hours = 12h. 57m. by this method.
By using the Nautical Jllmanac
Sun's rt. as. at noon, 10th March, = 23h. 19' 46" 3
Ditto llth do. = 23 23 27
Increase of motion in 24 hours 3 40 7
Moon's rt. asc. at noon, 10th March, = 172 10' 4b"
Ditto llth do. =183 41 27
Increase in 24 hours 11 30 39
and 11 30' 39" = 47' 2" 6 (note to prob. 6, part 2) hence 46' 2" 6 3' 40" 7
= 42* 22" nearly, the moon's motion in 24 hours exceeds the sun's. Moon's
right asc. 172 10' 48" = llh. 28' 43" 2, to which 24h. being- added, we have
35h. 28' 43" 2, from which the sun's rt. asc. 23h. 19 7 46" 3, being taken,
leaves 12h. 8' 57" nearly. Now 24h. 42' 22" = 23h. 17' 38" : 24h. :: 12h.
fc f 57" : 12h, 31m. the true time of the moon's passage over the meridian at
night, agreeing with the Nautical Almanac.
2. At what hour on the 5th of February, 1810, did the moon
pass over the meridian of Greenwich, the moon's right ascension
being 161 9', and declination 4 48' north?
3. At what hour on the 3 1st of December, 1811, will the moon
pass over the meridian of Greenwich, her declination at noon be-
ing 17 7' N and right ascension 120 34' 6"? (Her right as-
cension the following day will be 133 23' 41" ; that of the sun's
in time on the 31st 18h. 39' 7" 5, and on the 1st of January, 1812,
18h. 43' 32" 8.)
4. Onthe22dof August, 1812, at what hour will the moon
pass the meridian of Greenwich ; her right ascension being 329
37' 48", and declination 12 13' south?
Note 2. To find the time of the moon's southing or coming to the meri-
dian of any place, different from that of Greenwich. See the note to ex. 8,
prob. 18th, pail 3d.
* The time of the moon's transit in Greenwich is found calculated in page
7 of the month in the Nautical Almanac.
240 PROBLEMS PERFORMED BY
PROB 40.
The latitude of the jilace, day of the month, and time of high water
at the full and change of the moon being giuen^ to find tht tune of
high water on the given day.
Rule. FIND the moon's southing by the foregoing- prob. (or the
note to prob. 1 8th, part 3d ) to which add the time of high water
at the full- and change of the moon,* and the sum will give the
time ot high water in the afternoon. If the sum exceed 12 hours,
subtract 2 h 24 min. from it, and the remainder will give the
time of high water in the morning ; but if the sum exceed 24 h.
take 24 h. 48 min. from it, and the remainder will give the time
of high water in the afternoon.
Or, Find the moon's age, and opposite to it in the following ta-
ble ( 1st) take out the time in the right hand column correspond-
ing to it, to which add the time of high water at the full and change
of the moon, and the sum will shew the time of high water in the
afternoon. If the sum exceed 12 h. or 24h. proceed as above.
OR THUS,
Find the moon's horizontal parallax,! and the time of her com-
ing to the meridian, for the given time and place ; from the fol-
lowing table (2d) take out the correction corresponding to this
time, and apply it, as the table directs, to the result ; add the time
of high water at the full and change of the moon (found as direct-
ed above) and the sum will be the time of high water in the after-
noon. If the sum exceed 12 or 24 hours, proceed as directed in
the first rule.
Example 1 . Required the time of high water at London bridge
on the Sth of June, 1811. The moon's right ascension at that
time being 277 21' 2 ", and her declination 13 26' south ?
Ans By the globe. The moon came to the meridian at 14
hours, or in the morning at 2h.
Time of high water at the full and change at London 3
Time of high water in the morning 5 hours.
By rule 2d. The moon's age is 18, the time answering to which
in table 1, is 13h. 54'
Time of high water at full and change at London, 3
Sum 16 54
Subtract 12 24
Time of high water in the morning- 4 30
* This is given in the table of the latitude and longitude of places found
at the end of this work.
j- The moon's horizontal parallax for noon and midnight, and the time of
her coining to the meridian of Greenwich, is found in page 7 of the month
in the Nautical Almanac, and the latter time may be i educed to any other
meridian by the note toprob. 18. The moon's parallax maybe reduced to any
other meridian by taking proportional parts, and allowing- 15 for every hour..
THE CELESTIAL GLOBE,
241
Jiy the Nautical Almanac. The moon will come to the meridian
at , 14 hours, her horizontal parallax at midnight will be
59' 35" ; hence for 14 hoars and parallax 60', the corres
ponding- correction from tab. 2 to be subtracted, is
From
Difference
Time of high water at full and change,
Subtract
Time of high water in the morning,
Oh. IS'
14
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PROBLEMS PERFORMED BY
2. Required the time of high water at London on the JOtb of
May, 1812, the moon's right ascension being 41 50' 15", and
declination li 16' north?
3 Required the time of high water at New- York on the 4th of
July, 1811, the right ascension of the moon at Greenwich being
256 39' 6", and her declination 17 49' south. (The mooa
passes the meridian of Greenwich on the 4th of July, at lOh. 41%
and on the 5th at 1 Ih. 39'. Her horizontal parallax being nearly 59'.)
4. Required the time of high water at Boston on the 1st of Oc-
tober, 1813, the right ascension of the moon at Greenwich being
64 26' 59", and declination 19 50' south; also her passage
over the meridian of Greenwich on the 1st of October, being at
5h. 19', and on the 2d. at 6h. 8', and her hor. parallax at noon
on the 1st of October, at Greenwich being 54' 17", and at mid-
night 44' 26" ?
PROB. 41.
To describe the apparent fiath of any given planet or comet,
among the fixed starts*
Rule. FIND the planet's geocentric latitude from an ephemeris
or from page 4th of the month in the Naut. Aim. or if a cornet^
In table 1st if !' be added for each hour to the daily correction, the
sum will be nearly ihe correction for that time.
In table 2d the distance of the sun from the earth might be also allowed
for, but being- so small, it is here neglected. See McKay's Treatise on
Navigation, tables 5th and 6th. The principles on which these tables are
constructed, will be given in part 4th, article tides. See also chap. 5, b.
1, of McKay's Navigation .
* To perform this prob. on a plane or on paper. Draw a straight line to
represent the ecliptic, and divide it into any number of convenient equal
parts. At the ends of this line draw perpendiculars, and on each of them
bet off eight or ten of those equal parts northward and southward of the
ecliptic ; through every one of these parts draw straight lines parallel to the
ecliptic, and others perpendicular to these, through the divisions on the.
ecliptic, these lines will represent the zodiac. Then mark the geocentric
lat. and long, (as above) on this zodiac, beginning at the right hand of the i
ecliptic line, and proceeding towards the left, as the stars appear in a con-
trary order in the heavens, to what they appear on the surface of the globe ;
because in the heavens we see the concave part, and are supposed in the
centre of the sphere, but on the globe we see the convex, and are suppos-
ed to be situated without the sphere of the stars, &c. To describe the
principal fixed stars and the constellations near which the planet or comet,
passes, the sides of the map or the degrees of lat. and also the degrees, &c.
on the ecliptic, must be extended so as to take in their latitudes and longi-
tudes. Their lat. and long, must be set off, in a similar manner, from the
right to the left. In this manner you will have a complete representation
of the heavens with the positions of the several stars, constellations, &c. as
they appear to a spectator on the earth. Hence this manner of delineating
the stars is useful in learning their places, &.c.
The places of the stars may be laid down in like manner, from their right
ascensions and declinations, by drawing a portion of the equinoctial in plac
of the ecliptic, &c.
THE CELESTIAL GLOBE, 243
find its place by observation or from tables constructed for that
purpose ; mark those places on the globe for every month, or for
several days in each month ; these marks connected will be the
path required.
Examfile. Describe the path of the planet Jupiter for the year
1812, the latitudes and longitudes being as follow :
Longitudes. Latitudes.
Jan. 1st. 3s. 1' GO 11' S.
19th. 2 27 53 8 S.
Feb. 1st. 2 26 48 6 S.
March 25th. 2 27 58 1 N.
Apr. 1st. 2 28 44 1 N.
13th. 3 20 O 3 N.
. 25th. 3 2 13 4 N.
May 1st. 3 3 15 4 N.
13th. 3 5 28 O 6 N.
25th. 3 7 52 7 N.
June 1st. 3 9 20 7 N.
i3th. 3 1 1 54 8 N.
25th. 3 14 34 9 N.
July 7th. 3 17 15 <) 10 N.
19th. 3 19 56 O 12 N.
Aug. 1st. 3 22 50 13 N.
13th. 3 25 26 14 N.
25th. 3 27 56 O 16 N.
Sept. 7th. 4 28 17 N.
25th. 4 3 33 O 20 N.
Oct. 7th. 4 5 26 O 22 N.
25th. 4 7 32 Q 25 N.
Nov. 7th. 4 8 32 27 N.
25th. 4 9 4 31 N.
Dec. 7th. 4 8 50 33 N.
25th. 4 7 37 37 N.
Ans. January 1st. Jupiter will be near D in Gemini. On the 1 9th
near a small star marked h. On March 25th. its motion will be re-
trograde. On the 13th of April its motion will be again forward,
and so continue to the 25th of November, when its motion will
again become retrograde. On July 7th. it will be near ^in Gemini.
On Oct. 7th. it will be near or Asellus Australis in Cancer, &c,
PROB. 42.
To illustrate the firecession of the equinoxes*
Rule. ELEVATE the north pole 90 above the horizon, the
equinoctial will then coincide with the horizon ; bring the pole of
* The sun returning to the equinox every year, before it returns to the
SMne point in the heavens, shews that the equinoctial points have a retro-
grade motion from east to west. The cause of this motion was unknown
until Newton (prob. 39, b. 3. of his principia) had proved, that it is produced
244 PROBLEMS PERFORMED BY
the ecliptic (or that point on the globe where the circular lines
meet) to coincide with that part of the brass meridian which is
numbered from the pole towards the equinoctial, and mark the
point ovrr it ; consider this mark as the pole of the world, and let
the equinoctial be considered as the ecliptic, and the ecliptic as the
equinoctial ; then turn the globe gradually on its axis from east
to west, and the equinoctial points will move the same way, and
ivill describe one revolution round the globe, in the same time
that the pole of the world (here represented by the pole of the
ecliptic) will describe a circle round the pole of the ecliptic (here
represented by the extremity of the earth's axis.) Now as the
equinoctial points move backwards^ or from east to west, at the
rate of 50^" in a year, or 1 in 71 64 years, this circle will be de-
scribed in 2579 ! years. (See the note to def. 74.) In this time the
pole of the heavens will also describe a circle, the semidiameter
by the combined actions of the sun and moon on the protuberant matter
about the earth's equator ; and that this protuberant matter was caused by-
the revolution of the earth on its axis, which gives the earth the figure of
an oblate spheriod, flat towards the poles, and elevated towards the equa-
tor. (Prob. 19. b. 3. prin. see Perm's System of the Physical World, pa. 61.
see also the demonstration of this curious phenomenon in prob. 26. sect. 3.
Emerson's Fluxions.) Thus (as La Place remarks) every part of nature is
linked tog-ether, and its general laws connect phenomena with each other,
which, in appearance, have not the most remote analogy.
Hipparchus was the first (as Ptolemy informs us ch 1 b. 7. of his Alma-
gest) who observed this motion of the stars of the zodiac, by comparing his
own observations with those which JLristillus and Timocharis had made in
Alexandria about 165 or 160 years before ; and Ptolemy, by comparing his
observations with those of Hipparchus, found that all the stars had a simi-
lar motion, and as well as Hipparchus estimates it at 1 in 100 years. (See
his Almagest, ch. 2d and 3d.j In the year 128, before J. C. Hipparchus
found the longitude of Spica Virginis to be 5s. 24, and in 1750 its long-,
was found 6s. 20 21', the diff. of which is 26 21'. In the same year he
found the longitude of Co?' Leonis to be 3s. 29 50', and in 1750 it was
4s. 26 21', the difference of which is 26 31'. The me?n of these two
gives 26 26' for the increase of long, in 1878 years, or 50" 4</* ye a'y for
the precession. Jllbategnius, from the places of regulus, observed bv J\$ene.-
laus and himself at the distance of 785 years, makes the precession 1 in
66 years. (Chap 52 of his book of the knowledge of the stars.) By compar-
ing the observations of Jllbategnius in the year 878 with those made in 1738,
the precession is found to be 5i" 'J". Uhigh Heigh (in the preface to his
tables) makes the precession 1 in 70 solar years. Tycho Brahe (in his
Progymn, b. 1.) makes it 1 25' in 100 years, or 51" yearly. (Tycho's tables,
as also the Rudolphine, are given at the end of Nicholas Mercator's Astron-
omy, latin ed. 1676.) From a comparison of 15 observations of Tycho with
as many made by De la Caille, the precession is found to tie 50" 0'^
Copernicus, who considered this motion unequal, makes the mean equal
1 23' 40" 12"' in 100 years, and Ricciolus 1 23' 20" in the same time.
Street in his Jlstromrtia Carolina makes it 1 20'. Bulialdus in his Jistronomia,
Philolaica 1 24' 54", and HeveKm 1 24' 46" 50"', in 100 years. The first
star of Aries, marked /? was at the beginning of the year 1701, according
to Hevelius, in 29 0' 58^ of Aries. La Land, from the observations </
De La Caille, compared with those in Flamsteadt s catalogue, mokes tA^
secular precession 1 23' 4V, or 50" 2-5 in a year.
THE CELESTIAL GLOBE. 245
of which is equal to the obliquity of the ecliptic or 2 3 28', and
hence it varies its position a little every year. It from the pole of
the world or the above mark on the brass meridian, the comple-
ment of the lat. be reckoned upwards (which for New-York, for
example, is 49 17') and the point where the reckoning ends be
marked, this mark will be exactly over the lat. New-York will
therefore be under 64 11' on the brass meridian, reckoning- from
the southern point of the horizon or from the equator And when
1 2895-J- years, being half one entire revolution of the equinoxes, are
completed (which may be known by turning the globe half round,
or until the pole comes under the opposite degree on the meri-
dian or Aries from the western to the eastern point of the horizon)
that point of the heavens which is now 2 21' north of the zei.ith
of New- York, will be the north fiole, as may be seen by the pole's
distances from the mark over 64 1 1' on the meridian This is
on supposition that the obliquity of the ecliptic, &c. will not alter
this period, &c.
In the same manner it will be found, that in all those places 2
21' north of New-York, or in lat. 43 4', the north pole will, in
the same time be in their zenith, and all those places still farther
north, will have the pole in their zenith before thb period or
before 12895i years would be expired.
Hence likewise we see that the pole is advancing towards the
present equinoctial, &c.
For a further illustration of this prob. see Keil's Astronomy,
lect. 8, or La Place's Astronomy, b. 4, ch. 1 3.
Besides this motion of the equinoclional points, or the pole, there is
another called the nutation, which depends on similar principles. If the
pole of the equator be supposed to move upon the circumference of a
small ellipsis, tangent to the celestial sphere, whose centre, which may
be regarded as the mean pole of the equator, describes uniformly every
year (154" 63) 50" 1 of the parallel of the ecliptic on which it is situated;
the greater axis of this ellipses, always tangent to the circle of latitude,
and in the plane of this great circle, will, according to La Place, subtend
an angle of about (62" 2) 20" 15, and the lesser axis an angle of (46" 3)
15". La Place determines the situation of the real pole of the equator up-
on this ellipsis thus ; let a small circle on the plane of this ellipsis be sup-
posed concentric with it, and its diameter equal to the greater axis of
the ellipsis ; let a radius of this circle move uniformly with a retrograde
motion, so as to coincide with that half of the greater axis nearest to the
ecliptic, every time that the mean ascending node of the lunar orbit coin-
cides with the vernal equinox. From the extremity of this moveabie ra-
dius, let a perpendicular fall upon the greater axis of the ellipsis ; the
point where this perp. cuts the circumference of the ellipsis, will be tin-
place of the real pole of the equator ; this motion of the pole is called its
nutation. Dr Maskelyne, by examining the observations of Bradley, the
first discoverer of this nutation, makes the quantity (62" 2) 20? 15 ( == 58"6)
= 18" 98, which differs but (3" 6) 1" 16 from the result found by the tides'
This phenomenon being better determined by the tides, induced La Place-
to take (5b" 2) IS" 85 as more correct. See the laws of these motions in
ch. 13, b. 4, of his Astronomy.
The phenomena of the precession and nutation throws great light on the
%ure of the earth, supposed elliptic, as its ellipticity or compression
246 PROBLEMS,
Remark. The period of the revolution of the equinoctial
points, or 25791 years, was called by the ancients a filatonic year,
and they imagined that when this period would be completed, the
world was to begin anew, and the same series of things return
over again. This idle notion, however, had no other foundation
than in their imagination. Whenever we find this sportive faculty
permitted to wander unrestrained, its excursions are, in general,
more characteristic of extravagant fictions, than the imaginary ex-
ploits of a Don Quixote and his squire Sancho ; though many of
our prime philosophers have always shewed an unaccountable dis-
position to give into extravagancies not warranted by reason or com-
mon sense, if they only served to support some favourite opinion
or fancied system. In this case truth must always suffer, and be-
come the victim of folly and prejudice. Hence as McLaurin re-
marks, " False schemes of natural philosophy may lead to atheism,
or suggest opinions concerning the deity and the universe, of most
dangerous consequence to mankind ; and have been frequently em-
ployed to support such opinions." (View of Newton's Philoso-
phy See an account of the Indians or Brahman's division of
time, &c. in Bartolomeo's voyage to the East-Indies, ch. 9.)
appears from them not to exeed -5^5-; Bouguer making- it T TS- Delam-
bre, table 94, makes it -5-^; most astronomers before him making it yjo'
And although the meniscus, or protuberance at the equator, was supposed
solid, 8cc. in these investigations, the fluidity of the ocean will not change
the conclusions, as La Place proves in this remarkable theorem : " What-
ever be the law of the depth of the ocean, and whatever the figure of the sphe-
roid which it covers, the phenomena of the precession and nutation tvill be the
same as if the ocean formed a solid mass -with the spheroid." The mean ob-
liquity of the ecliptic would be constant, if only the sun and moon acted
on the earth, but from the action of the other planets, this is subject to
constant variation, and the same cause produces in the equinoxes a direct
annual motion of (G"5707) 0"1849. From the actions of the sun and moon
alone, the precession, according to La Place, would be (155" 20) 50*2848,
which diminished by the above quantity gives 50" 1 nearly.
The variation in the motion of the equinoxes changes the duration of
the tropical year, the latter diminishing as the former augments; so that
at present the actual length of the year is ( 12") 3" 888 less than in the time
of Hipparchus. Bnt this variation has its limits, and La Place finds that
they would be about (500") 5-' 42" (a) but that the action of the sun and
moon reduces it to (120 ff ) 38"88. (See Emerson's Centripetal Forces, prop.
34, sect. 3.) The length of the tropical year at present is 365d. 5h. 48/
48", from which the length of a sidereal year may be found by this pro-
portion ; (taking the precession of the equinoxes 50" 25) 360 50" 25 :
360 :: 365d. 5h. 48' 48" : 365d. 6h. 9' 1H" the length of a sidereal year.
The mean length of the day, according to this theory, may be supposed
constant, as La Place has shewn in the chapter above quoted, and re-
marks that this is an important result for astronomy, as it is the measure
of time, and of the revolutions of the heavenly bodies.
For more information on this curious subject, consult Simpsons Miscel-
laneous Tracts,- D'Alambert's Recherches sur la Precession des Equinoxes,-
Ruler, Mem. de Berlin., torn. 5, 1749 , La Place's Celestial Mechanics, &c.
("a) The translator of La Place makes the adore 500'' = 27 '.
ELEMENTS
OF
ASTRONOMY.
PART IV.
OF THE SOLAR SYSTEM.
HAVING in the preceding part of this work given the learner a
comprehensive view of the most useful and interesting parts of
practical astronomy, we shall in this fourth part endeavour to give
him a general idea of those bodies that exist around us, and of the
admirable laws by which they are connected and governed.
The view of the heavens the innumerable bodies that, from
our distant habitation, appear on a serene night, like so many lamps
that ornament the firmament the fixed appearance of some, while
others seem destined to no permanent station, must at all times
have attracted the attention of mankind, and engaged the most
learned in the investigation of their nature.
At first sight their confused, and sometimes insulated appear-
ance, exhibit no traces of order or regularity ; but on examination
we are astonished to find, amid such magnificent profusion and
awful grandeur, such harmony, such order and connexion ; and
must conclude that a fabric so immense, and at the same time so
well proportioned in all its parts so complicated, and yet directed
by laws so simple, evidently points out the wisdom of an architect
as far above our limited conceptions, as that power necessary to
eall the universe into existence.
Who has ever considered without emotion those operations and
laws which combine and regulate the distant parts of the world,
and so admirably display the greatness, the watchful providence of
that intelligent Being who presides over the magnificent scene.
What sublime and awful grandeur does this august temple of the
Deity exhibit. Thousands of worlds obey his voice and observe
his laws. Here the mind is struck with man's little schemes of
insignificant and fleeting greatness, when it sees that kingdoms,
nations, and the whole earth itself, dwindles into an atom, when
compared with the majestic greatness of the heavens. While, on
this extensive scale, we contemplate nature in all her perfections,
while we behold every part so exactly corresponding to its end, the
moral disorder which we witness in man, from an abuse of the
freedom and the reason which he possesses, strikes us- more forci-
248 OF THE SOLAR SYSTEM.
bly, points out our weakness, humbles our pride, and force us to
have recourse to and rely on that beneficent Being who sometimes
permits partial evils for greater good. Such extensive views of
the immense grandeur and magnificence of creation call the mind
from all its little cares and vanities to higher destinies and nobler
views, and brings us more and more acquainted with that Being,
whom, from his works alone, we learn to venerate and obey.
If then (as De Feller remarks, Philosophical Catechism, vol. 1,
pa. 138) " If the thought of God's existence and of our own im-
mortality enlivens all nature ; if xvithout that thought all would be
drowned in silence, and in the disconsolate prospect of death and
eternal naught, it is chiefly in the sublime regions of the stars that
it displays that enlivening power. Bright and powerful luminaries,
it is that thought that heightens and dignifies the lustre you shed,
it is by that thought that you dispel the horror of the midnight
hour, that you adorn the heavens and charm the earth. While
you fix my eyes by the radiance and purity of your beams, the
liveliness of my faith, the sweetness of my hope, excite in my soul
the most delightful emotions !. ..Cheerless philosophy, where you
see nothing but sparks, scattered at random over the vast expanse,
I see, I hear, the most eloquent, the most indefatigable panegyrists
of the Deity.'* The Atheists' books, says the same author, with
all that philosophy can do to render them fashionable, are cold,
melancholy performances, they rise only when they borrow the
language that confutes their errors. It has been attempted (says
a modern writer) to represent the Atheist as a sage, with whom
ufion the extinction of faith, reason is become omnipotent (New
Philosophical Thoughts) would it not be better to define him a
man over whom reason and faith have lost all their power ? Will
it not even be too much for him to be allowed a place in the class
of human beings ? Like us, I know, he raises his looks towards
heaven ; but like the brute, whose eyes are fixed on the earth, he
can discover no connection between it and the Supreme Being.
Heaven has given him that sublime countenance that bespeaks in-
tellect, and perhaps he was made, like man, to possess it to a cer-
tain degree, but like the brute, he no where can perceive any
traces of it. With the faculty of thinking, he received at his birth,
privileges far superior to that of instinct : but is it not the animal
senses alone he takes for his guides ? Like man, in short, he en-
joys the gift of speech ; but like the brutes, he either never ex-
amined nature, or nature never answered him. The sun may go
on illuminating the world, and rolling from east to west his reful-
gent orb : to the brightness of day may succeed the sable majesty
of night, ushered in by thousands of radiant stars, hailing with ju-
bilee the greatness of their maker ; the Atheist is deaf to the heav-
enly concert, though sounding from end to end of the glorious
march : millions of living creatures may people our woods and
lawns, may soar into the region of the air, may breathe in the
ocean's deep gulfs, and perpetuate their various species from age
OF THE SOLAR SYSTEM. 249
to age : never will they be able to raise his thoughts to the author
of life. The constant and regular return of the winter frosts and
vernal blooms, of glowing summers and mild maturing autumns,
announce to mankind, the God of wisdom and providence ; but
order and regularity tell him no more than chaos and confusion
Earth may renew her clothing ; she may dress herself out in her
richest attire ; he will gather her fruits, arid ascribe them to
chance. Insensible in the midst of the great theatre of the uni-
verse ; he never will hear that potent voice, that cries out so dis-
tinctly : It is God that made us. (Ip.se fecit nos y et non ipsi nos.)
Is he then that being that was destined to contemplate nature ? Is
he, while his heart is begirt with ice, and his mind palsied with
the apathy of stupidness, is he, I say, qualified to judge of the or-
der, the variety, the riches of all kinds she displays to his view ;
and by the beauty, the magnificence, and the aggregate of the
work, to raise his thoughts to the power and wisdom of its author ?'
The deeper we penetrate into the works of this great Author,
the contemplation of which forms the delight of thinking beings,
the more our thoughts are elevated above this little spot on which
we are at present confined the more we despise the trifling pur-
suits of mortals the more we pride ourselves in our relation with
him, whose goodness has performed so much for us who gave
us such a distinguished place in the scale of beings endowed us
with faculties capable of knowing him, and contemplating such
stupendous prodigies of his immense power. Hence we are led
to form this pleasing deduction, that minds capable of such deep
researches, possessed of qualities so noble and extensive, could not-
be the ephemeral productions or victims of a day, like the vegeta-
ble that perishes, but destined to a nearer approach, to a more ex-
tensive knowledge of the great Author of the universe, when time
throws off those shackles that retain us in our banishment, and our
immortal part is called to a country and state of existence worthy-
its nature, and the wisdom and goodness of its Creator.
Hence the knowledge of this august and amiable Being is the
only science worthy the noble powers and faculties of the soul of
man his service the only service which is truly dignified and
honourable and the enjoyment of him the only possession that
can satisfy the boundless desires and the noble and aspiring pas-
sions implanted in man ; a strong argument that he is destined,
one day, to enjoy that happiness, when this transitoiy and imper-
fect view of nature and every thing that occupies and deceives us
here, passes away, and our immortal country and state of existence,
presents to our view its more sublime and extensive prospects.
These are the pleasing, the important deductions we should draw
from viewing the majesty of the heavens the noble sentiments
that should guide and influence our conduct sentiments which
suit well the character of a Christian, who knows the indispensible
obligations he is under of serving God, and the greatness of the
glory which accrues from his service.
Hh
250 OF THE SOLAR SYSTEM.
Convinced of and impressed with these truths, the contempla-
tion of the works of creation will be no less delightful than useful
in conducting us to a knowledge of that infinite Being whom we
are so apt to forget. A knowledge which draws the line of dis-
tinction between rational beings and the brute creation.
CHAP. I.
OF THE SUN.
WITH a liberty common to all mankind, the astronomer, like a
skilful architect, who, to examine the workmanship of a building,
inspects the different apartments, and takes his point of view from,
different stations ; selects his position in various parts of the uni-
verse, and there considers the phenomena. Among the innume-
rable bodies that appear in the heavens, the sun claims the chief
place in our system and our first consideration.
To give the young student an idea of the magnitude of this im-
mense body, we find by observation, that if its centre coincided
with the centre of the earth, it would fill the whole orbit of the
moon, and its surface extend as far again.* The sun is situated
near the centre of the orbits of all the planets, and revolves on its
axis in 25 days, 14 hours, 8 minutes. This revolution is deter-
* The diameter of the sun, as seen from the earth, appears at a medium
under an angle of 32' or 1920", and the parallax of the sun or the angle un-
der which the earth's semidiameter appears as seen from the sun, is found
from the latest transit of Venus, equal to 8" 8, and hence the whole diame-
ter would appear under an angle of 17" 6. Therefore the proportion be-
tween the sun's diameter and that of the earth, is as 1920" to 17" 6 or
V? V s== 109 nearly ; and as the magnitudes of spherical bodies are as the
cubes of their diameters (Euclid prop. 18. b. 12. or Emerson's Geom. prop.
18. b. 7.) 1093 = 1295029, the number of times the sun is greater than
the earth. Again, 109 X 7911 = 862299, the number of miles in the sun's
diameter, that of the earth being 7911 nearly. (See note to def. 8.)
Now to find the distance of the moon.
Let MC represent its mean distance from
the centre of the earth C, AC, or CB the
earth's semidiameter, and the angle AMC
the moon's horizontal parallax, which at
the mean radius of the earth, according
to La Land, is 57' I". Hence,
As sine 57' 1" - - 8.2197069
To Radius - - 10.
So is AC 3956 - - 3.5972563
To CM 238533 - - 5.3775494
The diameter of the sun being 862299 miles, and the distance of the moon
from the earth only 238533 miles, it follows that the sun's diameter is near-
ly 4 times this distance, and hence the truth of the above assertion-
OF THE SOLAR SYSTEM. 251
mined from the motion of certain spots * on its surface, which first
appear on the eastern extremity, then advance towards the mid-
dle, and at length to the western edge, where they disappear.
When they have been absent nearly as long as they were visible,
they appear again as at first, finishing their entire circuit in 27
days, 12 hours, 20 minutes. Hence to an observer placed inihe
* Black spots of an irregular form are observed at the surface ofthe sun,
whose number, magnitude, and position, are very variable. They are often
very numerous, and of considerable extent ; sometimes, though rarely, the
sun has appeared pure and without spots for several years tog-ether.
It is doubtful by whom these spots were first discovered. Scheiner, a
German Jesuit and professor of Mathematics in logolstadt, Galileo in Rome,
David Fabriciusy and Harriot, each seemed to have discovered them about
the year 1611, and as telescopes were then in use, it is probable that each
might make the discovery independent of the other. These spots are
supposed to adhere to the sun's body, and hence its rotation has been dis-
covered.
M. Cassini determined the time of rotation from observing the time in
which a spot returns to the same situation on the sun's disk, or to the cir-
cle of latitude passing through the earth. This time from a great number
of observations, he determines to be 27d. 12h. 20', and the mean motion of
the earth in that time being 27 7' 8", we have this proportion ; 360 -f 27
7' 8" : 360 :: 27d. 12h. 20' : 25d. 14h. 8' the time of the sun's revolution
on its axis, the motion ofthe spots being supposed uniform. Their motion
is from west to east.
When the earth is in the nodes of the sun's equator, and consequently in
its plane, the spots appear to describe straight lines : this happens about the
beginning of June and December. As the earth recedes from the nodes,
the path of a spot grows more and more ecliptical until the earth is 90
from the nodes, which takes place about the beginning of September and
March, at which time the ellipsis has its lesser axis the greatest, and is then
to the greater axis as the sine of the inclination of the solar equator to ra-
dius. Hence the inclination of the solar axis to the plane of the ecliptic is
found to be 7^, or rather this is the angle which the axis ofthe sun makes
with the axis of the ecliptic, or a perpendicular to its plane which passes
through the sun's centre. Most of the spots appear always within the com-
pass of a zone, whose breadth, measured on the solar meridian, extends
between 29 42' and 30 So 7 ; they have sometimes, however, been seen
29 36', and in July 5th, 1780, M. de la Land observed one 40 distant
from the solar equator.
There have been various opinions respecting the nature of these spots.
Scheiner supposed them to be solid bodies revolving round the sun, near its
surface. But if they were not on the sun's surface, they would be longer
visible than invisible, which is not .the case. Moreover, if they revolved
about the sun like the planets, their motion would necessarily be in a plane
passing through its centre, which seldom happens. Galileo compared them
to smoak and clouds, as they varied their figures, increased and sometimes
disappeared. Hevelius appears to be of the same opinion in his Comejo
vraphia, pa. 360. The permanency of most of the spots is however an ar-
gument against this hypothesis. J\L de la Hire supposes that they are
solid bodies which swim on the sun's surface, and which are sometimes im-
mersed in the liquid of which he conceives the sun's surface composed.
La Lande supposes that the sun is an opake body covered with a liquid
fire, and that the spots arise from the opake parts like rocks, which are
sometimes raised above the surface by the alternate flux and reflux of the
liquid igneous matter ofthe sun. .0,-. Wilson., professor of Astronomy '.<
52 OF THE SOLAR SYSTEM.
sun, all the planets and fixed stars will appear to revolve from east
to west in the space of 25f of our days nearly, exclusive of the
motion which the planets have in their orbits. The northern pole
of this revolution will be in that place which an observer, situated
on the earth, would refer to the tenth degree of pisces, and in lati-
tude between 83 and 84 north, near the stars or K in Draco.
The south pole will be in the 10th degree of Virgo, in latitude be-
tween 83 and 84> south, near the star a in equuleus pictorius.
Besides these apparent motions arising from the motion of the sun
on its axis, the planets will be observed to have regular and proper
motions in their orbits round the sun, and all from the west to-
wards the east. These motions are performed nearly in the same
plane, at least not differing more than 7 or 8. An observer in
the sun will also find, that if he compares any planet with the fixed
stars, it will sometimes go slower and sometimes swifter, he will
therefore conclude that it is sometimes nearer and sometimes fur-
ther off. This will also appear more evidently from the change of
its apparent diameter. He will likewise observe some perform
their revolution in less time than others, and if he has any idea of
the laws of gravity,* he will refer this difference of velocity to its
proper cause, which is the different distances of these respective
bodies. But these distances he cannot so easily observe as if si-
tuated on the earth, as on the latter, from its motion, hecan select
Several stations at great distances from each other.
We have observed that the motion of the planets round the
sun is not uniform? being subject to very perceptible inequalities,
the laws of which form one of the most important objects of as-
tronomy,
Glasgow, opposes La Land, and is of opinion that the spots are excavations,
or deep caverns in the luminous matter of the sun, the bottom of which
forms the dark spot or umbra formed in the middle. (See the Phil, trans
1774 and 1783.) La Place in his Astr. vol. 1. b. 1. c. 2. remarks that they
are eruptions in the sun's body, of which our volcanoes form but a feeble
representation, as they are almost always surrounded by a penumbra, which
is enclosed in a cloud of light more brilliant than the rest of the sun, and in
the midst of which the spots are seen to form and disappear. Dr. Hal ley
was of opinion that the spots are formed in the atmosphere of the sun. Dr.
jfferschel supposes the sun to be an opake body, surrounded by a very gross
atmosphere. He says that if some of the fluids which enter into its compo-
sition, should be of a shining- brilliancy, while others are merely transparent,
any temporary cause which may remove the lucid fluid, will permit us to
see the body of the sun through the transparent ones. See the PMl. trans-.
for 1795. Some of these spots have been observed whose diameter exceed
6 or 7 times that of the earth. Dr. Herschel on April 19, 1779, saw a spot
which measured 1' 8", 06 in diameter, which in length is near 30600 miles.
(For 8", 8 : 1' 8*, 06 :: 3956 : 30596.) This was visible to the naked eye.
For the phenomena of the spots as described by ScJieiner and Jfevelius, see
"Vince's Astronomy, 8vo. pa. 136. Besides the dark spots upon the sun,
there are also parts of the sun called Facul*, Lvcili y &c. which are brighter
than the general surface ; these abound most in the neighbourhood of spots,,,
or where spots had been recently observed.
* These general laws will be given after the solar system.
OF THE SOLAR SYSTEM. 253
Kepler conceived the ingenious idea of comparing the figure of
the orbit of the planets with that of an ellipsis,* in one of the foci
of which he placed the sun ; and the innumerable observations we
have since had, besides the important and numerous discoveries of
Newton, leave no doubt concerning the truth of the hypothesis.
Besides this useful discovery, Kepler has made two others no
less important, and which all observations and physical reasoning
concur to establish, viz. that
The planets, by radii drawn from the sun to their respective cen-
tres, describe areas round the sun, proportional to the time of de-
scribing them ; and that
The squares of the times in which the planets revolve round the
sun* are to each other as the cubes of the greater axis of their orbits.
Newton, in his discoveries, has extended the same laws to the
secondary planets revolving round their respective primaries.
The sun is not, however, exactly the centre of the planets* mo-
tions, but rather the centre of gravity of the whole system ;f but
* If the two ends of a thread be tied tog-ether, and placed round two pins
fastened in a sheet of paper, on a table ; then the thread being- uniformly
stretched by a black lead pencil, carried round by an even motion of the
hand, will trace out an ellipsis, and the points where the pins were fixed are
called its foci ; the longer diameter is called the transverse, and the shorter
the conjugate. In describing the ellipsis the ends of the string might be
fastened to the pins. The nearer the pins are, the more the ellipsis will ap-
proach a circle, and the further they are apart, the more eccentric will the
ellipsis be. When the ends of the thread are fastened to the pins, the
length of the thread ought to be equal to the length of the transverse di-
ameter.
It is a known property of the ellipsis, that the sum of two lines drawn
from the foci to meet in any point of the cure, is equal to the transverse di-
ameter ; hence other constructions by compasses, &c. will arise. For which
and the properties of this figure, see Simpson, Emerson, Hamilton, Milnes,
Simson, Vince, or other writers on the conic sections.
f The centre of gravity of a system of bodies is that point round which, if
the bodies were suspended, they would remain in equilibrio in any position.
(They may be conceived to be suspended by inflexible levers, or in am'
other manner, from this centre.)
This centre may be thus found ; let the line ACB -u
be supposed an inflexible lever, considered without
weight, and Jet the two bodies A and B be suspend- .
cd on the ends of the lever ; take the point C in AB /j^ C--
~
so that AC : BC :: B : A, then C will be the centre x^
of gravity between A and B. For if the bodies A X^ T>\
and B be made to vibrate about the immovable point \^
C, A and B will describe the arches Aa, Bb, which Eh^
will be as the velocities of the bodies, and also as the
radii AC, CB of the circles ; and hence their velocities are as the radii.-
Therefore vel. A : vel. B :: AC : CB, or as B : A (by supposition) whence
A X vel. A = B X vel. B. But as these products represent the quanti-
ty of motion of these bodies (see the laws of motion, next section) which
being equal, the bodies will therefore remain in equilibrio round the centre
C, which is therefore the centre of gravity required. Again, if A and B be
mow supposed to act in C, and E another body, the centre of gravity of these
bodies will divide CE in D, sp that CD : DE ;: E : A -f B, &c.
254 OF THE SOLAR SYSTEM.
as this centre is generally within the body of the sun, and can never
be at the distance of more than the length of a solar diameter from
its centre, the sun is therefore generally considered by astronomers
as the centre of the solar system. The sun is however agitated by
a small motion round the true centre of gravity, owing to the va-
rious attractions of the surrounding planets.
As the sun revolves on its axis, his figure, like the rest of the
planets, is supposed not to be strictly in the form of a globe, but of
an oblate sjiheroid (a figure formed by the revolution of an ellipsis
round its shortest axis or conjugate diameter) and therefore flat
towards the poles.
An observer placed in the sun can have no vicissitude of day and
night, yet the fixed stars and planets will make unequal arches
above and below the horizon, as they decline towards either pole,
as it happens to the inhabitants of the earth. He will see no sha-
dow or eclipse, but to an eye on the surface of the sun, when the
planet is in the horizon, its satellite will sometimes appear in the
penumbra of it, or the penumbra of the satellite may appear cast
on the disk of the primary planet, which will be known by the co-
lour being something duller. But the whole diameter of a prima-
ry planet will appear so small to an observer in the sun, unless as-
sisted by very powerful telescopes or some other substitute, that
these appearances can scarce be observed.
At the sun the diameter of Saturn (as laid down by Dr. Gregory
in book 6th of his Astronomy) subtends an angle of 1 8", that of
Jupiter of about 40", that of Mars only 8", of Venus an angle of
28", and of Mercury 20". Hugens makes the diameter of Jupiter
near 54", and that of Saturn without his ring 27".
The apparent diameters of bodies diminishing as the distance in-
creases, an observer on the earth will therefore form an estimate
of the relative change of the sun's distance ; in like manner an ob-
server in the sun will form an idea of the planets' variation in their
respective distances ; and as the sun's apparent diameter is greater
in the beginning of January than in the beginning of July, it fol-
lows that the sun is nearer to our earth in the winter than in the
summer. The greatest apparent diameter of the sun, as given in
the Nautical Almanacs for 1811 and 1812, on the 1st of January,
is 32' 35" 6, and the least on the 1st of July is 51' 31", the mean
between which is 32 ; 3" 3. La Place in his Astronomy (b. 1, c. 1)
makes the apparent diameter of the sun, when the velocity of the
earth is greatest, equal 6035" 7,* when the velocity is least, equal
Hence if by this method we find the centre of gravity between the siui
and Mercury, between this centre and Venus, between this centre again
and the earth, and so on to the remotest planet (their quantities of matter
being given) the last centre will be the centre of gravity of the whole sys-
tem, and the focus of all the planetary orbits. But this will seldom differ
much from the sun's centre.
* In La Place's astronomy the quadrant is divided into 100, each te-
gree into IOC', and each minute into 100", &c. Hence a degree in the
sexagesimal arithmetic adopted in this country is equal 1-J of this cen
OF THE SOLAR SYSTEM. 255
5 836" 3, and its mean diameter 5936". He says that these quan-
tities should be diminished a few seconds to allow for the effect of
irradiation^ which dilates a little the apparent diameters of lumi-
nous bodies.
The mean apparent diameter of the sun being taken about 32',
and if we take the sun's mean distance from the earth to be 95
millions of miles,* its real diameter will be 862299 miles, as before
tesimal division of the quadrant, and therefore =' 11111-Jr" ; conse-
3t "> 96 > -
., =32' 3", 26, agreeing with the Nautical Almanac. We shall
L(j(J(J{J(J
in the ensuing- part of this work, when we have a necessity for quoting-
La Place, use his own measures, and give their value in the margin, as
the reduction of them in the English translation is replete with faults.
The above three numbers are thus given in the margin of the English
translation, 32' 36", 6, 2b' 49", and 30.42, 25.
As the apparent diameter of the sun is greatest near the winter solstice,
least near the summer solstice, and nearly at a mean in the equinoxes, its
apparent diameter may be easily found at any other time sufficiently cor-
rect by proportion.
The apparent diameters of the planets are found by a micrometer \>\&ced
in the focus of a telescope ; or the apparent diameter of the sun may be
measured in a dark room, by means of the projection of his image through
a circular aperture. From these apparent diameters, and the respective
distances from the earth, their real diameters may be determined thus :
Let M in the foregoing figure represent the earth, AB the sun's diame-
ter, the angle AMC the apparent semidiameter of the sun = 16', and MC
the distance of the sun from the earth ; to find AB the true diameter, it
will be,
Rad. - - - 10.0000000
To tang. 16' . 7.6678492
So is 23464.5 - 4.3704112
To 109.2095 - 2.0382604
Hence 109.2095x2 == 218.419, which multiplied by 3956, gives 86406
5.564, the diameter of the sun ; the cube of which divided by the cube of
7911, will give the number of times the sun is greater than the earth.
In the above calculation 23464.5 is taken in place of 23405 (see ch. 2)
hence 23464.5x3956 = 92825562 miles.
* If in the figure for determining the distance of the moon, &c. (pa 250)
we suppose M to be the sun, AC the earth's semidiameter, and the angle
AMC the sun's horizontal parallax 8" 65, the distance MC is thus found ;
As tang. AMC 8" 65 5.6219140
To rad. or sine ACM ..... 10.0000000
So is one semid. of the earth AC - - 0.0000000
To 23882.84 semidiameters - - - 4.3780860
We have here taken the sun's parallax at its mean distance 8" 65 ac-
cording to Mr. Short, who has taken incredible pains in calculating it
from the best observations made on the transits of Venus in 1761, an ac-
count of which is given in the philosophical transactions for 1762 and
1763. But from the transit of Venus in 1769, compared with the
^he parallax is found to be 8" 8
256 OF THE SOLAR SYSTEM.
determined, and its magnitude 1295029 times that of the earth i
the diameter of the earth being only 791 1 miles, the sun's diame-
ter will be 109 times as great.
Besides the planets and fixed stars, an observer in the sun will
discover other bodies of a different nature, called comets, on account
of their hairy appearance, as seen from the earth* These are carried
in very eccentric orbits round the sun, sometimes approaching
near his body, and at other times going off to immense distances
from it. These comets will appear to be carried among the fixed
stars, some in one direction and some in a contrary, in orbits
whose planes are very much inclined to that of the ecliptic, al-
though they regard the sun, or rather the centre of gravity of the
solar system, as the focus of their motions.
What we have here said is upon supposition that an observer in
the sun is not prevented by its atmosphere (which from late expe-
riments we have reason to conclude is very gross*) from seeing as
Mr. Short by taking- the semidiameter of the earth 3985 miles, makes
the earth's mean distance 95173127 miles. But if the earth's semidiam-
eter be 3956, the earth's mean distance from the sun will be 23882.84 X
3956 = 94480515 miles. Hence in round numbers it is sometimes taken
about 95000000 miles, although it is probably less. La Place taking the
parallax 8" 8, makes the sun's mean distance 23405 times the radius of
the earth ; hence 23405x3956=92590180 English miles, the sun's mean
distance from the earth. There must, however, be some mistake in La.
Place's calculation. (See ch. 2d.)
* Bouguer, by some curious experiments on the intensity of light on
different parts of the sun's disk, found that this light was more intense at
the centre than near the limb. Two equal and very small portions of the
sun's surface seen from the earth, one at the centre of the disk, and the
other near its edge, appear to occupy different spaces, which are to each
other as radius to the co. sine of the arc of the great circle, which sepa-
rates these two parts on the sun's surface ; this makes the intensity of
light increase in this proportion inversely from the centre to the edge ot"
the sun's disk. Bouguer has however found the reverse. In comparing-
the light of the centre with that of a point distant from the limb by a
quarter of the semidiameter, he found the intensities of these two lig-hts
in the proportion of 48 to 35. This difference indicates a thick atmos-
phere round the sun, which weakens its iig-ht.
It follows from the preceding- results and from the experiments of Bou-
guer, that the intensities of the light of a star seen from the surface of
the sun at the zenith, is reduced 0.24065, and that the sun deprived of
its atmosphere would appear 12J- times more luminous. A horizontal
stratum of air at the temperature of or zero, the thermometer being:
divided into 100 from the freezing- to the boiling point of water, and under
the pressure of a column of Mercury, O me ' 76 ought to have 53548 me "
(metres) of thickness, to weaken light in the same degree as the sun's
atmosphere. This is on supposition that at equal densities the transparen-
cy of the sun's atmosphere is the same as that of the air, but of this we
are ignorant. Uouguei 's experiments deserve also to be repeated in dif-
ferent aspects of the solar disk.
La Place in his astronomy (b. 1, c. 2.) remarks, that the faint light
which is visible particularly about the vernal equinox, a little before the
rising or after the setting- of the sun, asid which is called zodiacal light ',
is supposed to be produced from the reflexion of this atmosphere* Tire
OF THE SOLAR SYSTEM. 257
far, and as freely as an observer on the earth sees on a clear night,
when the moon does not shine. If this be not the case, he may be
unable to trace several of the phenomena which we have mention-
ed ; but if he sees them at all, he will observe them as we ha,ve
described them.
CHAP. II,
OF MERCURY.
WE will now transfer our solar observer from that station where
the motion of the planets are regular, to the earth, his own habita-
tion, where the phenomena will appear something different. From
this point of view the planets will not observe the equal description
of areas in equal times as round the sun, but will sometimes ap-
pear to move towards the east, at other times towards the west,
and sometimes to remain stationaiy or without any motion.
Let a body revolve in the pe-
riphery of the circle A, B, C,
D, E, F, &c. and move through
equal arches AB, BD, DE, EF, O
&c. in equal times ; and let an
eye, in the plane of this circle,
view the motion of the body
from O. When the body moves from A to B,
its apparent motion will be measured by the
arch LM, or the angle LOM ; while it moves
from B to D, its apparent motion is determined
by the arc MN, which is less than the former,
though described in the same time, when it
comes to E, it will still be observed in the same .-
point N ; hence during the time that it describ-
ed the arch DE, it was stationary at N. The motion of the body
being continued in its orbit, when it comes to F, it will appear in,
L, and to have gone backward or retrograde the arc NM, and
when it comes to G, it will appear in L, where it appeared before
when in A. In like manner when it comes to H, it will appear in
P, and at I it will appear at Q, where it will seem stationary^
while it describes IK. At K it will again go forward as before,
and with unequal motions describe the arch QN. This unequal
motion is evidently owing to the eye being placed at O, without
the orbit ABD, Sec. of the body, while the revolving body regards
C as the centre round which it regularly revolves. If the eye at
fluid which transmits it to us is extremely rare, since the stars are visi-
ble through it ; its colour is white, and its' apparent figure that of a cone
whose base is applied to the sun. The length of the zodiacal lightsome-
times subtends an angle of 100, but the atmosphere of the sun does not
extend to so great a distance, and cannot therefore be the cause of this
light. The true cause is still unknown.
n
258 OF THE SOLAR SYSTEM.
O be in motion during the revolution of the body in the orbit ABD>
See it will retard or accelerate its motion according as it moves
in the same or a contrary direction. This explains the phenomena
of Mercury or Venus' motions in their orbits, which are within the
orbit of the earth.
If the eye be now placed within the orbit of the body at O, as in
the 2d fig. but not in the centre ; while the body describes the arch
AB, it will seem to move quicker than while it describes its equal
CD, which is more distant, because the angle AOB, by which it
forms an idea of the motion of the body, is greater than DOC. In
this case, however, the body will never appear stationary or retro-
grade, but will always appear to move forward, though with very
unequal motions. But if the point O be in motion, then the phe-
nomena will be different. If the motion of the eye at O be equal
to the motion of the body and in the same direction, the body will
appear stationary ; if the motion be greater, the body will appear to
go backward, if less^ forward, Sec.
This last case explains the phenomena of the motions of Mars,
Jupiter, Saturn, Herschel, &c. round the earth, in orbits which are
therefore without the orbit of the earth. The two former planets
are therefore called inferior or rather interior planets, and the lat-
ter sufierior or rather exterior planets.
Hence these appearances prove that the planets do not regard the
earth as their centre of revolution, but that they in reality revolve
round the sun, as we have before described. Two of the planets,
Mercury and Venus, never recede from the sun beyond certain
limits, the others are occasionally separated from him by all the
angular distances possible.
Of all the planets Mercury is nearest to the sun, and the least of
those whose magnitudes are accurately known. He performs his
periodical revolution round the sun in 87 days, 23h. 15m. 43"6.*
His greatest elongation is 28 20', and least 17 36', the mean of
which is 22 5S ; , and his distancef from the sun is 35933619,76
miles.
* For the method of finding the planet's periodical revolutions, see the
following- note.
t The distance of Mercury, or any other planet from the sun, may be
found by Kepler's rules given in chap. 1 ; thus, the squares of the periodic
times being always as the cube of their mean distances ; or which is the
same, divide the square of the time in which any planet revolves round
the sun by the square of the time of the earth's revolution, the cube root
of the quotient will give the relative distance of the planet from the sun,
which multiplied by the earth's mean distance from the sun, will give the
planet's mean distance required.
For Mercury. The earth's periodic revolution is 365d. 5h. 48' 48" r=
31556928'', the square of which is 995839704797184 (a constant divisor for
all the planets) and 23464.5 the distance from the earth to the sun in
semidiameters, will be a constant multiplier. (Tang. 8" 8 (log. t =
5.6295869) : rad. (log. 10.0000000) :: 1 setnidiameter (log. = 0.0000000,
: 23464.5 semid. (log. 4.3704131) see notes, ch. 1) 87d. 23h. 15m. 43" 6
= 7600543" 6, the square of which is 57768263015500.96, which divided
OF THE SOLAR SYSTEM. 259
by the former square, gives .058009600076417, the cube root of which is
.38711, nearly, the distance of Mercury from the sun, supposing- the dis-
tance of the earth from the sun to be an unit or 1. Hence .38711 X
23464 5 = 9083.34, which mult, by 3956, the radius of the earth, gives
35933693.04 miles, the mean distance of Mercury from the sun. (See
table 102 of Delambre,, pa. 113, &c Paris edition.)
The distance of Mercury or any inferior planet from the sun, may also
be found by their elongations. If S represent the sun, E
the earth, and M Mercury, and EM a tangent to Mercu-
ry's orbit; then the angle SEM will be the greatest
elongation of the planet from the sun, which angle, if the
orbit were circular and the sun in the centre, would be __,
found by saying ES : SM :: Rad. : sine SEM. But as Jyi
the orbits are elliptic, the angle EMS will not be a right
angle, unless the greatest elongation happen when the plan-
et is in one of its apsides. The angle SEM is also sub-
ject to variation in proportion to the variation of SE and
SM. The greatest angle SEM happens when the planet
is in its aphelion, and the earth in its perigee, and the least, when the
planet is in its perihelion, and the earth in its apogee. M. de la Lande
finds these elongations equal 28 20' and 17 36' respectively, the mean
of which is 22 58'. But Laplace makes the greatest and least elonga-
tions of Mercury = 32 and 18, or in our measures = 28 48' and 16
12' respectively, the mean of which is 22 3(/ Now in the triangle SEM
taking the angle SEM = 22 3(/, the distance of the earth from the sun
SE = 23464.5 of the earth's semidiameters, and SME is a right angle ;
hence rad. : SE 23464.5 log. = 4.3704113 :: sine 22 30' = 9.5828397 :
log. 3.9532510 = 8979.5 nearly, which multiplied by 3956 gives 35522902
miles, the distance of Mercury from the sun by this method ; but an er-
ror of a few minutes in the elongation will make a considerable differ-
ence ; for taking 22 58* instead of 22 3(/, we find 9155.8 semidiametera
nearly, = 36220344.8 miles. (See La Land's Astronomy, 3 ed. 1792. art.
1142.)
The distances SE, SM being given, the angle SEM and MSE are also
given, the former of which is the greatest elongation of the planet, and
the latter the angle of commutation or heliocentric distance of the planets
(or which is the same, the common mutation or angular distance of any
two of the planets among themselves, as seen from the sun.) But this is
on the supposition of circular orbits ; however the prob. may be solved
nearly in the same manner, the elliptic figure being considered. For the
angle SME being given, and the distance SM,
and moreover the angles AS, ASM, MSE
(nearly equal to the same in circular orbits)
the angle ASE will be known, and also SE
will be known in magnitude, from which the
rest will be given as before. Here the great-
est elongation changes according to the dif-
ferent distances of the point M from the aphe-
lion of its orbit ; foritisgreatestin a in its aphe-
lion, less in p its perihelion, and a mean in the
mean longitude. It is also various, the place M
of the inferior planet remaining the same, according as the superior is situa-
ted in E or e, &c. Laplace remarks that the length of an entire oscilla-
tion of Mercury, or return to the same position relatively to the sun, varies
from 106 to 130 days, that the mean arc of its retrogradation is about
15 (= 13 30*) and its mean diameter 23 days ; but that in different
retrogradations there is a great difference in these quantities.
By observing two heliocentric places of an inferior planet, its periodic
time may be nearly found, though it is more accurate to observe the
260 OF THE SOLAR SYSTEM.
The eccentricity of Mercury's orbit* is estimated at one-fifth of
its mean distance from the sun Vince makes the eccentricity of his
orbit 7955,4 parts of the mean distance of the earth from the sun,
planet when twice successively, in the same node, that is when the planet
has no latitude, the time between these observations will be the planets'
periodic time, as the motion of the nodes will vary but little during this
short period. The conjunctions of the inferior planets are also proper for
discovering their periods, for then they appear in the same point of the
heavens to an observer, either in the sun or on the earth ; if the planet
be nearer than the sun, it will appear in opposite points. The superior
planets when in opposition to the sun, also appear in the same point of the
heavens or the ecliptic, when seen from the earth, as if observed from
the sun, and hence their geocentric and heliocentric places agree. If any
of them be then observed, and the time marked, and the same observa-
tion be made when it comes to its next opposition, the arch which the
planet seen from the sun, has in the elapsed time described, will be thus
discovered; then say as that arch : the whole circumference :: the time
between the two oppositions : a fourth which will give very nearly the
periodic time of the planet. The planet can never be observed in the
ecliptic from the earth, which is in the plane of the ecliptic, except when
it is also in the ecliptic, and consequently in its node.
The daily mean motion of Mercury, according to Delambre, is 4 5' 33'-'
qr 4 5' 34", his mean hourly motion 1(X 14", in a minute 10", and in a
second 10"', & c .
The hourly motion of Mercury in miles may be thus found ; taking the
jnean dist. from the sun =35933693.04, this multiplied by 2, gives 71867
386.08 = diameter of Mercury's orbit, which multiplied by 3.1416, gives
225778580.109; hence as 87d. 23h. 15' 43" : 225778580.109 :: lh. :
106940 miles, the hourly motion of Mercury.
* In order to describe a planet*s orbit, or to find its position and ec-
centricity, the planets' heliocentric place or its place as seen from the
sun, and its distance from the sun, must be obtained. Dr. Halley gives
the following ingenious method of finding these requisites, with no other
data Chan the periodic time of the planet. .
Let KLB be the orbit of the earth, S the sun, P
the planet, or rather the point in the plane of the
ecliptic on which a perpendicular let fall from the
planet meets that plane ; when the earth is in K, ob-
serve the planet's geocentric longitude (this is calcu-
lated in pa. 4 of the month in the Nautical Almanac)
and having the theory of the earth, its place in the
heavens, or the apparent longitude of the sun is giv-
en (found also in pa. 2 of the month in the Nautical
Almanac, and its hourly motion, pa. 3.Y and hence
the angle PKS is given. The planet after complet-
ing an entire revolution, returns again to the same
point P, at which time the earth being supposed at L, observe the angle
PLS the planet's elongation from the sun. Now the times of the observa-
tions being given, we have the places K, L of the earth in the ecliptic
given, and consequently the angle LSK and the sides LS and SK ;
wherefore we shall have the angles SKL and SLK, and the side LK.
The angles PKS, PL.S being likewise known, the remaining angles PKL,
!PLK will be known. Hence in the triangle PKL, two angles and the side
LK being given, the side PL is given ; and having the side PL and LS,
and the angle PLS, the angle LSP is given, which determines the heiio->
centric place and its distance from the node according to the ecliptic,
and likewise the side SP is [given,, But as the tangent of the geqcentrjtc
OF THE SOLAR SYSTEM. 261
supposing this distance 1 00000. Laplace taking half the greater
axis of the earth's orbit or its near distance = 1 .000000, makes
half the greater axis of Mercury's orbit or his mean distance =
0.387100, and the proportion of his eccentricity to this mean dis-
tance for the beginning of the year 1750 = 0.2055 13, and the secu-
latitude, is to the tangent of the heliocentric, so is the curtate distance of
the planet from the sun, to its curtate distance from the earth. By ob-
servations the planet's geocentric latitude is found, wherefore its helio-
centric latitude is given. (These are given in the Nautical Almanac.
The method of finding these will be given in chap. 4.) The heliocentric
lat.of the planet being thus found, and also its curtate distance from the
sun, its true distance can be easily found. (See the note, page 264, &c.)
Three heliocentric places of the planet and the corresponding distances
from the sun being thus found, we shall find from thence the form ot the
orbit and the position of the apsides, by describing an ellipsis that will
pass through three given points.
Let the three given places of the
planet be L, M and N, and S the sun's
place or the given focus ; join LN, and
produce it to B, so that SL : SN :: LB
: NB ; also join MN, and produce it to
D, so that SM : SN :: MD : ND ; join
13 D, and from S and L let'-'fall the per-
pendiculars SE, LF, and divide SE in
P, so that LF : LS :: EP : PS, and also
make EA : AS in the same ratio. Then AP will be the axis, and the mid-
dle point C the centre, A and P the vertices, and Cs being taken =>= CS, s
will be the other focus : whence the ellipsis may be easily described.
From M and N let fall the perpendiculars MG, NH on DB ; then by
construction SL : SN :: LB : NB, that is by sim lar triangles, as LF : NH,
and permutatio SL : LF :: SN : NH. Again by construction SM : SN ::
MD : DN, that is by sim. triangles, as MG : NH, hence perm ut. SM : MG
:: SN : NH ; also SP : PE :: SL : LF (by constr.) that is as SN NH, or
as SM : MG, and therefore BD is the directrix of the ellipsis, in which
are the points N, M, L, and whose focus is S, and vertices A and P. (Emer-
son's Conic Sections, b. 1, prob 29, or Mimes, part 4, prop. 9.) This
prop, is demonstrated nearly in the same manner in Emerson, prop. 85,
b. 1, Vince's Ast c. 13, Keif's Ast lect. 26, Gregory's Ast. prop 29, b. 3,
or Newton's Principia, prop. 21, b. 1. Schol. Newton remarks, that when
EP is greater than, equal to, or less than PS, the figure thus described
will be either an ellipsis, a parabola or an hyperbola ; the point A in the
first case falling on the same side of the line BD, as well as the point P ;
Jn the second going off to an infinite distance, and in the third falling on
the other side of the line BD.
As a calculation is preferable to any construction ; it may be drawn from
the foregoing investigation. Thus, to find NB we have, by division, SL
SN : SN :: LB NB or LN : NB, but the three first terms are given, be-
cause the points N, S and L are given in position, or NS, SL and the an-
gle NSL are given, and hence NL is given, NB is therefore given =
N X LN. . M _ SN . SN .. MD ND or
MN : ND = SN X M ^> ( Euc i. j^ 5 ) PS an( j AS are found in the
o ivi - o IN
same manner. Moreover in the two triangles NSL, NSM, two sides and
the included angles are given (being the distances of the planet from the
sun, and the degrees between its observed places in its orbit) hence NL,
NM are given, and the angles LNS, SXM, and therefore the angle LXM
262 OF THE SOLAR SYSTEM.
lar increase of this proportion or its increase for 100 years s=
0.000003369.* The place of Mercury's aphel. for the beginning of
1750, according to Fmcc,was 8s. 13 33' 58", and its motion in lon-
gitude in 100 years 1 33' 45". Mercury's greatest equation, ac-
cording to the same author, is 23 40'. (See def 133, p 2.) Ac-
cording to Laplace, the longitude of the perihelium in 1750 was
(8l.7401) 73 33' 57"9, and its secular direct motion (1735"5) 9'
22" 3. Delambre, tab. 97, makes the place of Mercury in 1810, 9s.
23<> 32', the place of his aphelion 8s 1 4 30' 1 4", the place of its node
Is. 16 4' 1", the motion in longitude of the aphelion in 100 years
1 33' 45", and of the nodes 1 12' 10", both increasing; hence
his place at any other time may be found. (These places are set
down to mean time in Delambre's tables.) The mean longitude
of a planet seen from the sun, is found by adding its mean motions
to the efioch, or its place for any given year. The longitude of
the aphelion taken from the mean longitude of the planet, will give
its mean anomaly, and the contrary, &c. In finding the planet's
place or his longitude reckoned from the apparent equinox, the
nutation (see note to prob. 42, part 3) must be applied as given by
Delambre, table 11, page 29. His apparent diameter is very va-
riable. It is a minimum or the least when the planet in a morning
immerges into the solar rays, or when in the evening it disengages
itself from them ; it is at its maximum, or the greatest, when it
immerges into the sun's light in an evening, or when it again be-
comes visible in the morning. Its mean apparent diameter, ac-
cording to Laplace, is (21" 3) 6" 9 or nearly 7", and his apparent
and its vertical opposite angle BND, hence ND and NB being given, the
angle BDN is given, therefore in the right angled triangle DHN, the hy-
pot. DN, and the angle at D are given, hence NH is given. Join SH then
in the triangle SHN, Nil, NS are given, and also SNH (= suppl LNS
BNH or 180 LNS BNH) hence SH the angle, NHS, NSH and SHE
are given ; therefore in the right angled triangle SHE, SE is given, hence
we know SA and SP ; for HN : NS :: EA : AS and by division UN NS :
NS :: EA AS or ES : AS = ^215?. Again, HN : NS :: EP : PS and
JtziS ".W o
KTO vy pO
(Eucl. 18, 5.) HN -f. NS : NS :: EP -f- PS or ES : PS=- A -y Hence
HN~f~NS
PS and SA being given, their sum or AP is given, and half their differ-
ence is the eccentricity SO- Lastly in the triangle SsL, we have the sides
Ss, SL (for AP SLandsL sL. Emerson's Conies, prop. 1, b. 1) to find the
angle LSA. the distance of the aphelion from the observed place L; in the
same manner the distance of the aphelion may be found from the observed
places M and N.
In the year 1740, on July 17, August 26, and September 6, M- de la
Cuille found three distances of Mercury (the mean distance being 10000)
as follows ; SL 10351,5, SN = 11325,5, SM = 9672,166, the angle LSI*
= 3s. 27 0' 36", NSM = 44 40' 4". From whence its eccentricity (by
calculating as directed above) is found = 2099.75, the place of its aphe-
lion 8s. 13 51' 14", and the greatest equation = 24 3' 5". The above
scheme must be fitted for these distances by the rules given for construct-
ing the ellipsis. (See other methods in Vince's Complete System of As-
tronomy, and also his Elements of Astronomy, 8Vo.)
* See tables 101, 102, &c. of Delambre.
OF THE SOLAR SYSTEM. 263
diameter as seen from the sun is about 1 7 " 8 nearly. His real
diameter is therefore about 3105 miles,* and his magnitude nearly
1 6| times less than that of the earth.
It was no doubt difficult at first to recognize the identity of
the two stars which were alternately seen in the morning and in
the evening, to depart from and return to the sun, but as the
one never shewed itself until the other disappeared, it was found
to be the same planet which thus oscilated on each side of the
sun ; its position, apparent diameter, and retrograde motion,
confirming this conjecture, and agreeing with the laws of its
motion afterwards discovered. In general these laws are very
complicated, they do not take place in the plane of the ecliptic,
sometimes the planet departs from it, 4 30' being its greatest
geocentric latitude, or its greatest distance from the ecliptic, as
seen from the earth ; but this distance as seen from the sun
amounts to 7, being his greatest heliocentric latitude, and equal
to the inclination of the plane of its orbit to that of the ecliptic.
Its secular variation, according to Laplace, is (55" 09) 17" 50.
The place of the nodes being the point of intersection of the
orbit of the planets and the ecliptic ; the place of Mercury's as-
cending node,f or its longitude at the beginning" of 1750, was
Is. 15 20 ; 42" 8, or taurus 15 20' 43" nearly. Hence the de-
* The mean distance of the earth from the sun being 23464.5 semidiame-
ters, and Mercury's mean distance 9083.3214 semidiameters, the difference is
14381.1786, the distance of Mercury from the earth ; and as the magni-
tudes of bodies vary inversely as their distances nearly, we have by inverse
proportion 23464.5 : 6" 9 :: 9083.3214 : 17" 8 nearly, the apparent diameter
of Mercury as seen from the sun. Now taking- the mean apparent diameter
of the sun 32', and its real diameter = 864065.5, we have 3i' : 6" 9 ::
864065.5 : 3105.23 miles the diameter of Mercury.
The diameter of Mercury may be also thus found ; let S in the annexed
figure represent the sun's place, SC Mercury's mean distance =: 9083.32
semidiameters of the earth nearly, and the ang-le GSM, the apparent semi-
diameter of Mercury as seen from the sun = S" 9 nearly ; then
Sine SMC = 89 59' 54" 5 10.0000000
SineCSM = S"9 5.6247021
So is 9083.32 semid. 3.9582446
To .38278 1.5829467
Hence .38278 X 3956 = 3028.5 miles, differing- a little from the above.
The mean diameter of Mercury is measured when he has his greatest
elong-ation from the sun, his greatest diameter may pethus found by the in-
verse rule of proportion; 23464.5 : 6" 9 :: 14381.2 ::'!!" 2 nearly, and his
least thus ; 23464.5 : 6" 9 :: 32547.8 (= 23464.5 -f 9083.3) : 4" 9 nea>ly.
Now if the cube of the diameter of the earth be divided by the cube of
the diameter of Mercury, the quotient will give the number of times the
earth's magnitude exceeds that of Mercury. For the magnitudes of bodies
are as the cubes of their diameters. (Eucl. B. 12 p. 18.) Hence 31053 :
79113 :; 1 : , _ = 16.5 nearly, the number of times the earth is greater
than Mercury.
f The nodes and inclinations of the orbits of the planets, &,c. may be
thus determined. First to find the position of the line of the nodes.
264
OF THE SOLAR SYSTEM.
scending node was in scorpio 1 5 20' 43". The motion of the
nodes is found by comparing their places at two different times,
from whence the motion of Mercury's nodes in 100 years is found
to be 1 12' 10" according to Vince. Laplace makes the side-
real and secular motion of the node upon the true ecliptic diminish
(2332.90 seconds) 12 min. 35" 8.
In the annexed figures (adapted to an inferior and superior planet) let
S represent the sun, T* the t
earth's orbit, Nn the line of
the nodes of the planet. Let
the earth be in T. from which
let the planet P be observed,
when in the ecliptic, and
therefore in its node P ; next V S ii>
after one revolution let the
planet be observed again in the same node, the earth being in t ; draw the
straight lines ST, PT, S*, P*. Then in the A (triangle) ST\r, there are
given the L. (angle) *ST by the theory of the earth (see chap. 4) and time
between the observations (the motion of the earth in this time is also
found in the Nautical Almanac, that of the sun being given) and STP the
observed elongation of the planet from the sun, and ST the distance of the
earth from the sun ; therefore Sa: is found, and also xt ; S* the distance
of the earth from the sun at the second observation being given (this dis-
tance can always be found from page 3 in the Nautical Almanac, or prob.
5 of Mayer's Tables, the log. there given being adapted to the mean dis-
tance 1 ; their index is increased by 10 when it is negative, which must be
allowed for.) Again, in the A txP t the j_ txP = TcrS is given, and the
/^SP the elongation in the second obs. is given, and likewise tx t hence
Vt is given. Lastly, in the A SfP. *S, *P and the AP*S are given, and
therefore SP is given, which is the planet's distance from the sun when in
its node: the 2L*SP is also given, and therefore the position of the point
P is given, the point t being the place of the earth as seen from the sun
at the second obs. and hence the position of n is gives, and therefore the
position of the line of the nodes Nw is given. -.
If the place of the nodes found by the ancients be compared with that
found by the moderns, its motion will be given.
If a planet be observed twice in any point of its orbit, as seen from
the earth, the place of the planet as seen from the sun, and its distance
from the sun, are found in the same manner-
Having the motion of the nodes and the pe-
riodic time of the planet, and moreover its
place being given for any year, its place for any
other year may be easily found. Now the posi-
tion of the line of the nodes being given, the
inclination of the planet's orbit to the plane of the
ecliptic, can be found thus ; let S represent the sun,
NB/ithe ecliptic among the fixed stars, NA/z the
planet's o\*bit, as seen from the sun among them,
and NSn the line of the nodes. The earth comes
twice in the year to this line, and as the mean time of its coming to it is
given, let the geocentric place A of the planet P be observed at this time
(or found in page 4 of the Nautical Almanac, or calculated by Delambre's
Astronomical Tables, translated by Vince. See notes to prob, 1 and 3,
part 3) and let the latitude thus found be AB, an arch perpendicular to
the ecliptic, and longitude 'Y'B. Now as the longitude of the sun ^N is
known, the difference of these longitudes NB is known. '-
Hence in the
OF THE SOLAR SYSTEM.
265
The latitude of Mercury is greater when retrograde and nearest
the earth, and less when direct and remotest from it. Moreover
if it be in its lower conjunction, or most retrograde and nearest
the earth, and at the same time in or near one of its nodes, it will
be directly between the observer and the sun. If it be at a consi-
derable distance from a node, it will pass the sun to the northward
right angled spher. A ANB rt. angled at B, AB, NB are ^iven, and there-
fore the /.ANB the measure of the inclination required will be known.
The inclination of the plane of the planet's orbit to the plane of the eclip-
tic being- thus found by observation, the heliocentric place of the planet and
his distance from the sun may be found, whenever the planet is in oppoti-
iioji to, or conjunction with the sun thus ; let S repre-
sent the sun, T the earth, P the planet in its orbit,
NB the ecliptic among- the fixed stars, NA the inter-
section of the planet's orbit with the sphere of the fixed
stars, N the node ; then SN will be the line of the
nodes, the sun being in the plane of the orbit of each
planet. Let A and B be the planet and earth's places
respectively, as seen from the sun among the fixed
stars ; and as the planet is either in opposition to die
sun, as in fig. 1st. or in conjunction with it, as in fig.
2d. the arch AB will then be the circles of latitude,
and therefore perpendicular to the ecliptic. Hence in
the spher. rt. angled A ABN, the /_ANB, and BN
(found as above) are given, therefore AB and AN are given ; but AB is the
heliocentric lat. and AN is the distance of the planet in its orbit from the
node N, as seen from the sun, and therefore the heliocentric place A of
the planet P is given. Moreover in the A PST there are given ST from
the theory of the earth, and the /. PTS the lat. by observation or geo-
centric lat. or its complement to a semicircle, and also the heliocentric lat.
PST, therefore PS and PT, the distances of the planet from the sun and
earth respectively, are given.
The same may be found for any other aspect of the planet thus ; let P
be the planet, NS the line of its nodes, and
the angle PTB be its apparent or geocentric
latitude as seen from the earth in T ; let the
plane of this latitude be produced until it cuts
the plane of the orbit of the planet in PN, and
the plane of the ecliptic in the right line BTN.
Draw PB perp. to NB, and erect TO prep, to
the same, then (prop. 38, Euclid 11.) TO will
be perpend, to the plane of the ecliptic, be-
cause the plane of the lat. PNB is perp. to the
plane of the ecliptic. Let fall TE perp. to NS
or NS produced ; join OE, which is perpendi-
cular also to NS, and the /_ OET will be equal
to the inclination of the planes of the orbit of
the planet and of the ecliptic. In the A NST
there are given ST, the A TSN by the theory
of the earth, and the given place of the node," and the /_ NTS by observa-
tion, being the elongation of the planet from the sun, computed in the eclip-
tic, or its complement to two rt. angles, therefore TN, NS and the /_^ TNS
will be given. In the A TEN rt. angled at E, NT and the 2_ TNE are
given, and hence TE is given. In the A OTE rt. ang. at T, TE and TEO
the inclination of the planes of the orbit of the planet and of the ecliptic
(found as above) being given, OT is given. In the A OTN rt. ang, at T
OT and TN are given, hence the /_ ONT i3 given. In P2sT are riven*
Kk
266 OF THE SOLAR SYSTEM.
or southward. But if it be most direct, and at its greatest distance
from the earth, and at the same time in or near a node, it will be
covered by the sun ; if otherwise situated than near its node, it
will pass on one side of the sun. When it is nearer the earth and
near its node, it is seen in the interval of its disapparition in the
evening, and reapparition in the morning, projected like a black
spot on the disk of the sun on which it describes a chord.
These motions of the inferior planets may be thus explained ;
let ABC be the orbit of an inferior planet,
and S the sun, the circle LMO the zo-
diac in the heavens ; let the earth be now
supposed in T, and the inferior planet in Qi
A, near its superior conjunction with the
sun i a spectator in T will then evidently
see the planet at A in the point L. If the
earth had no motion, the inferior planet,
while describing the portion AB of its or-
bit, would appear to have described the portion of the zodiac LM.
But in the mean time the earth is in motion, so that when the in-
ferior planet is in B, the earth is in the point of its orbit H, from
which the planet in B appears in N. Hence Venus has appa-
rently moved further eastward from the earth's motion. But when
the planet comes to C, the earth has moved on to G, and then the
planet is seen in the point of the zodiac O, GO touching its orbit
in C, in which position its apparent motion is nearly equal to the
apparent motion of the sun or direct. From this position let the
planet move from C to A, and the earth in the same time from G
to K, from which the planet will be observed in the zodiac in P,
but as it was before observed in O, it will appear to have gone
retrograde or backwards in the zodiac, through the arch OP, or
to have moved westward contrary to the order of the signs ; and
as the planet was direct in C, there must be some point of its or-
bit between C and A where it appeared stationary, or without
any motion. Let the planet now be in E, and the earth at the
NT, the L. TXP and PTX the geocentric lat. of the planet or its comple-
ment to two rig-ht angles, therefore NP is given. In the A XPB rt. ang-,
at B, the side NP and /. PNB being- given, PB and XB will also be given.
In BXS, XB and XS and the BXS are given, hence XSB the heliocen-
tric longitude of the planet computed from its node, and the side SB are
given. Then in the A PUS, rt. ang-. at B, PB, BS heing- known, PS the
distance of the planet from the sun, and the angie PSB, which is its helio-
centric latitude, will be given. Lastly, in the A PXS all the sides being-
given, the angie XSP is known, being- the heliocentric distance of the plan
et in its orbit computed from the line of the nodes XS. The mean distance
of the earth from the sun may be taken as the measure in finding- the plan*
tt's distance.
If by this method we find out two other heliocentric places of the planet
and tlie distances from the sun, having- likewise the focus of its orbit, which
is the sun's centre, an ellipse may be described passing- through the given
points, as before- shewn, which will be the orbit of the planet. The learner
will notice that SB is called the wtatv distance of the planet from the sun .
OF THE SOLAR SYSTEM. 267
same time in F, from this point the planet will be seen in Q, and
will appear to have moved further backward from P towards the
west. But when it is seen again in a line that just touches its or-
bit, its motion will be direct or towards the east, between which
and the former place the planet will be stationary as before. The
earth having now come to D, and the planet to C, it will seem to
have described the arch QR, and its motion to be quicker east-
ward. Hence when the planet is in its superior conjunction with
the sun, its motion is always eastward, or according to the order
of the signs, but when it is in its inferior conjunction, its motion
is 'westward or contrary to the order of the signs. In the former
case it will seem to go forward, in the latter backward or in a con-
trary direction.*
* Here might be shewn how to find the position of a planet when sta-
tionary, the time of the station, &c. but as these are subjects of curiosity
rather than matters of any real practical utility, the learner is referred to
lecture 27 of Keil's Astronomy, or to ch. 15 of Vince's Astronomy, 8vo.
Vince in his Astronomy observes, that the place and time of the oppo-
sition of a superior or the co?ijunction of an inferior planet, are the most
important observations for determining- the elements of their orbits, be-
cause at that time the observed is the same as the true longitude, or that
seen from the sun ; whereas if observations be made at any other time,
the observed must be reduced to the true longitude, which requires the
knowledge of their relative distances, which, at that time, are supposed
not to be known.
The conjunctions of the inferior planets may be thus determined; find
the diurnal motion of the planet from the sun, and also the diurnal angular
motion of the earth ; the difference of these motions is the relative diur-
nal motion, or the quantity by which the planet recedes every day from the
earth, as observed by a spectator in the sun. Thus the mean motion of
the earth in a day is"360 divided by 365d. 5h. 48m. 48s. or 365.242d.c-
59' 8" 3, and that of Mercury 360 -f,87d. 23h. 15m. 43"6 or 87.96925d.=
4 5' 32*4, the difference of which is nearly 3 V 24". Hence 3 6' 24" :
360 :: 1 day : 115.88 days, the time wherein Mercury having left the
earth will return to her again, or the time between two conjunctions of
the same kind. The mean conjunctions of Venus is found in like manner
thus ; her daily mean motion is 360 -f- 22d. 16h. 49' lG"6or 224.7008*14d.
= 1 36' 7"6 and 1 36' 7"6 59' 3"3 = So* 59"3. Hence 36' 5S"3 : 360 :?
1 day : 583.96 days. This will also give the time between any two simi-
lar stations as two mean oppositions, &c.
These mean conjunctions, &c. are computed by ihe planets' mean
motion, or on the supposition that they move equably in circular orbits ;
but as they really move in elliptic orbits, in which their motions are con-
stantly variable, it may happen that the true conjunctions may differ a few
days from the mean ; however by having- the mean conjunction given, the
true conjunction is thus found ; compute by astronomical tables the true
places of the earth and the given planet, at the time of mean conjunction,
found as above, from which their angular distance as seen from the sun
will be given. Now the angular motions of the planets being given for
any time, for example for four hours, the difference of these motions will
give the access of the planet to, or its recess fiom, the earth in four
hours. Then as this difference of motion : angular distance of the earth
and planet :: 4 hours : the time between the mean conjunction and thp
true conjunction required.
268
OF THE SOLAR SYSTEM.
These transits* of Mercury are real annular eclipses of the sun,
from which we discover that the planet borrows its light from it.
When we observe it with a good telescope, it presents phases to
tis similar to those of the moon, directed in the same manner to-
wards the sun, the variations of which, according to its relative
position and the direction of its motion, throw great light on the
nature of its orbit.
In his superior conjunction, or at the opposite side of the sun,
that side of Mercury which is towards it, is likewise towards the
earth, and when visible he appears nearly round. He never ap-
pears quite round, as he is either hidden by the sun's body, or the
splendour of his rays, and therefore to us invisible ; sometimes he
appears in the form of a half moon, and sometimes a little more
or less than half his disk is seen. When he is in his inferior con-
junction, or between the sun and the earth, the whole of his en-
lightened side is turned from the earth towards the sun, and hence
he appears when seen on the sun's surface, like a dark spot, as has
been observed befofe.f
The above proportion for finding the mean conjunction, &.c. may be
thus expressed in general ; let P = the periodic time of a superior pla-
net, p = that of an inferior, t = the time required; then, proceeding- as
above, we have t = ^-. And this will hold general for any two simi-
1 p
lar stations. (Sete the note, page 24.)
* The method of finding these transits and some other things of a simi-
lar nature, will be given in the next chapter.
f To exhibit these phases of the
planets at any time ; let S be the sun,
E the earth, P an inferior planet, aPb
the plane of illumination perpendicu-
lar to SP, cPd the plane or vision per-
pendicular to EP ; draw oe perp. to
cd; then ca is the breadth of the visi-
ble illuminated part, whose breadth
to the eye is ce, the versed sine of cPa
or SPZ, SPc being the complement of
each. Now the circle cad \v\l\ be pro-
jected into the right line cd, as its
plane passes through the eye, but the circle which is the boundary between
the iHuminated and darkened part of the planet, being seen obliquely, will
be an ellipsis (Vince's Con. Sect. p. 36, or Emerson's Project, of the Sphere,
sect 1, prop. 4.) hence if cmdn represent the projected hemisphere of the
planet, Avhich is next to the earth, mn, cd two diameters perpendicular to
each other ; make ce equal the versed sine of SPZ or cPa, and describe the
ellipsis men, then mcnem will represent the visible enlightened part, us it ap-
pears at the earth ; and from the property of the ellipsis (Emerson's Con.
Sect. b. 1, prop. 73 and cor. 2) Pm or PC : Pe :: the semicircle men : the
semi-ellipsis men; and by the nature of proportion Pm : PC Pe or ec :: the
area men : men men or mcne. That is the semidiameter : the versed sine of
cPa :: half the -whole disc : the risible enlightened part.
Hence the planets Mercury and Veniis, will have the same phases from
their inferior to their superior conjunction, as the moon has from the new to
the full ; and the same from the superior to the inferior conjunction as the
moon from the full to the new. M.irs will appear gibbous in quadratures, as
OF THE SOLAR SYSTEM. 269
As Mercury is always in the neighbourhood of the sun, and that
his apparent diameter is so very small, he is seldom seen, and only
appears a little after sun-set and again a little before sun rise. The
light and heat which this planet receives from the sun, is computed
to be about seven times greater than the light and heat which is re-
the angle cPa or its complement will differ considerably from a right angle,
and its versed sine therefore from the diameter. , In Jupiter, Saturn and
Herschel, the angle SPZ never differs so much from 180 as to make these
planets appear gibbous, and hence they always appear full orbed.
If P represent the moon ; then as EP is small compared with SE, SP,
these lines will be veiy nearly parallel, and the ^ SPZ nearly equal SEP ;
hence, the visible enlightened part of the moon varies nearly as the versed sine
t,f her elongation from the sun.
The following prob. of finding the position of Venus when brightest, on
supposition that her orbit, together with that of the earth, were circular,
was proposed by Dr. Halley, and solved by him in the Phil. Trans, numb.
349, and may be equally applied to Mercury.
Draw S/- perp. to EPZ, and take a => SE, b = SP, x = EP, i/=Pr ; thqn
b y = the versed sine of SPr (to the rad. SP, as will be evident by de-
scribing a circle from the centre P with the distance PS, &c.) which, from
the above, varies as the illuminated part ; and as the intensity of light varies
inversely as the square of its distance (see the following note) the quantity
of light received at the earth varies as -^^ == -- ; but (Euclid. 12p.
0:2 #5 a-2
2 b,) a 2 2 4- ;r2 4- 2xy ; hence y = *~~ *?""** s this value of t/ be-
ing substituted in the above expression, we get the quantity of light
b a z #2 x z 2bx 2 4- 2 4-rc2
= - - - = - _JL - a maximum, whose
fl xion 4. '2xx -f 2o-3 . SxZxX^bx a* + b* -f
~~
4. 4?x4x 12 b x^x 4-
a* b* X
--- - - - - - = 0. Hence, clearing it of frac-
4x6
tions, and dividing by 2x%x, we get a 2 b% X 3 4>bx .r 2 = 0, by
transp. a-2 4. 4bx = 32 3b*, which solved gives x = ^/3a^^-b2 2b.
If we apply this equation to Mercury, a = 1, b = .3871 nearly (as cal-
culated pa. 259) and hence x 1.00058 ; then by trigonometry a : x+b :'
x ^ : ^ ~ . .85131 nearly, = the difference of the segments of the
base SE made by a perpendicular from P ; then -- \-'- = the great-
er segment, and -- .42565 = .07435. Hence the following proportions :
As b = .3871 : .07435 :: radius : cosine ESP = 78 55' 35", and x
1.00058 : .925G5 :: rad. : cos. SEP = 22 14' 9". But the angle ESP at the
time of the planet's greatest elongation is about 67, Sec. Hence Mercury
Is brightest between his greatest elongation and superior conjunction, and
a<-. this time his elongation will be 22 14' 9",
270 OF THE SOLAR SYSTEM.
ceived by the earth,* the solar disk as seen from Mercury being se-
ven times greater than it appears to us But the light and heat OB
this planet are more or less intense in proportion to its distance
from the sun. This distance is very variable, the orbit of Mercury
being more eccentric than any other planet.
The accelerating gravity of Mercury towards the sun is also se-
ven times greater than on the earth.
It has not as yet been discovered by observation whether Mercury
revolves upon its axis, and therefore we are ignorant whether it
has the vicissitude of day and night, and still more so of their length.
But as all the other primary planets perform this motion on their
axis, from analogy it is extremely probable that Mercury is subject
to the same law. We are also ignorant whether it has different
seasons, because these depend upon the inclination of the axis of
its rotation to the plane of the orbit, which it describes about the
sun ; but this is also unknown.
To an observer in Mercury, all the primary planets that we know
would be superior, and appear as Mars, Jupiter, &c. do to us ; and
it is unknown to us whether the inhabitants of Mercury (if any) see
any inferior planet ; if not, the argument deduced from the phases
of such planets to establish the true system of the world, will be
wanting to them : for these phenomena clearly prove, that the pla-
nets move round the sun ; but although we can observe no planet
inferior to Mercury, it does not however follow that there are none,
for Mercury itself is seldom seen by us : and a planet that would be
much inferior to it, would never be visible on account of its nearness
to the sun.
CHAP. III.
OF VENUS.
VENUS, the next planet in order, offers the same phenomena as
Mercury, with this difference, that its phases are much more sen-
sible, its oscillations or elongations much more extensive, and their
period more considerable. Her orbit, including that of Mercury,
her periodic time must be greater. According to the latest and
best observations, the sidereal revolution of Venus round the sun
is224d. 16h. 49m. 10.5888 sec.f Her greatest elongation, accord-
ing to La Land, is 47 48', and least 44 57' ; her greatest, accord-
ing to Laplace, is (53) 47 42', and least (50) 45 ; the mean of
* Light or heat, so far as it depends on the sun's rays, decreases in pro-
portion as the square of the distances of the planets from the sun. (Fergu-
son's Astro, art. 169, Smith's Optics, b. 1. art. 57, or Emerson's Optics, b.
1, prob. 6, and corollaries.) The same may be easily proved of any virtue
or fluid substance flowing- from or to a centre. See Gregory's Astronomy,
t>. 1. prop. 48.
f The method of finding the periodic time is given in the notes in chap
'.?. prob. 9. S^e also chap. 7.
OF THE SOLAR SYSTEM. 271
these last is 46 21'. The mean length of its entire oscillation is
584 days.* Here it may be asked why Venus necessarily remains
a longer time to the eastward or westward of the sun than the whole
time of her entire revolution ; but when we consider that the rela-
tive motion of Venus is greater than her absolute motion, because
while Venus is moving round the sun, the earth is performing its
motion round the sun the same way, the question is therefore ea-
sily answered. The retrogradations of Venus commence or end
when the planet, approaching the sun in the evening or receding
from it in the morning, is distant from it according to Laplace 32,
in our measures 28 48'.f The mean arc of its retrogradation Is
about ( 1 8) 1 6 42'. The distance $ of Venus from the sun is found
from its elongation equal 67165759.2 miles, and from its periodic
time 67435662 67 nearly.
The eccentricity of the orbit of Venus, according to Vince or
La Land, is 498, the mean distance of the earth from the sun being
100000 of these parts. Hence her eccentricity in miles is 46227 1 .3
nearly
According to Laplace Venus's mean distance from the sun is
0.723332,1) the earth's being 1 or an unit ; proportion of the eccen-
tricity of the semimajor axis for the beginning of the year 1750
When Venus appears -west of the sun, she rises before him in the morn-
ing, and is called the morning star : when she appears east of the sun, she
shines in the evening after sun set, and is then called the evening star :
being alternately morning and evening star each 292 days.
f Here we must again notice, that these numbers are given wrong in the
English edition of Laplace's Astronomy, 28 48' being given 27 48', and
16 48!, 16 12,'.
* The distance of Venus from the sun may be found in the same manner
as that of Mercury, in chap. 2d. Thus in the triangle SEM (pa. 259) let
M now represent Venus, and the angle SEM be taken equal 46 21'.
Hence rad. : s. 46 21' :: 23464.5 : SM 16978.2 the distance of Venus from
the sun in semidiameters of the earth, which multiplied by 3956 gives
67165759.2 miles.
The same by the periodic times, &c. 224 d. 16 h. 49 m. 10.6 sec. =
19414150" 6 the square of which is 376909243519480.36, which divided by
995839704797184 (see the note pa. 259) gives 378473843879 nearly, the cube
root of which is .7254. Hence .7264. X 23464.5 X 3956 =, 67435662.6748
miles, the mean distance of Venus from the sun by Kepler's rule and ex-
tremely near the above, considering the great difference in the principles
of calculation, and that an error of a few seconds in the elongation will
make a considerable difference. This is a strong proof of the truth of Kep-
ler's laws and of the copernican system. The dist. of Venus from the sun,
and her periodic time being given, her hourly motion may be found as in
the note pa. 9. for Mercury, thus ; 67435662.67 X 2 = 134871325.34 =
the mean diam. of her orbit, which multiplied by 3.1416 ghes 423711755.688
miles its circumference ; then 224 d. 16 h. 49' 10" 6 : 1 h :: 423711755.688
-. 78569 miles the hourly motion of Venus.
For 100000 : 498 :: 92825662 miles the earth's mean distance from
the sun : 462271.29876.
|| If .723332 be multiplied by 92825563 miles, the earth's mean distance
(see note, pa. 255) the result will be 6714,5699,4 nearly, the mean distance,
'>f Venus according to Laplace
272 OF THE SOLAR SYSTEM.
= 0.006885 ; and the secular diminution of this proportion
0.000062905.
The place of Venus's aphelion "for the beginning of 750, ac-
cording to Vince, was 10s. 7 46' 42", and its motion in longi-
tude for 100 years 1 21'. Its greatest equation is 47 ' 2(/'
According to Laplace the longitude of the fierihelion for 1750 was
(141 9759) equal 127 46' 4l" 9 or 7 46' 41" 9 in Leo, the
sidereal retrograde motion in 100 years (699" .07) equal 3' 46 "4.*
Delambre makes the place of Venus in the beginning of t bOO,
4s. 25 9' 1", of her aphelion 10s. 8 36' 12", "and of her node
2s. 14 52' 8", and makes the secular variation of the aphelion
1 21', and of her node 5 I' 40". Her daily mean motion, accord-
ing to the same author, is 1 36' S", her hourly motion is 4', her
motion in one minute 4", and in one second 4"', &c
The apparent diameter of Venus continually varies, which
proves that her distance is no less variable. Her distance from the
earth being the least at the moment of her transit over the sun's
disk, her apparent diameter will then be the greatest, and will de-
crease until she arrives at her superior conjunction, where her di-
ameter will be the least. The position of the earth in its orbit will
also vary it a little. By having the apparent diameter and the planet's
distance from the earth at any time, its apparent diameter cor-
responding to any other distance may be easily found, as it varies
nearly in proportion to the distance. The greatest diameter of Ve-
nus at a medium, is about 58". f Her real diameter is therefore
7301 .7 miles, and her magnitude is in proportion to that of the earth
as 1 : 1.2742.
* Newton in the Scholium to prob. 14, b. 3. of his Prlncipia, says, that by
the theory of gravity the aphelions of the planets near the sun, from the ac-
tion of those more remote, move a little in consequentia in respect of the
fixed stars, and that in the sesquiplicate proportion of their several dis-
tances from the sun ; so that if the aphelion of Mars in the space of 100
years be carried 33' 20", in consequentia in respect of the fixed stars, the
aphelions of the earth, of Venus, and of Mercury, will, in a hundred years,
be carried forwards 17' 40", 10' 53", and 4 r 16" respectively. See this
property demonstrated in Emerson's comment on the Principia, pa. 83.
-j- Mr. Bliss at Greenwich in 1761, June 6th, from three good observa-
tions of Venus on the sun's disk, finds its diameter 58" Mr. Short in
London makes it 58", and the diameter of the sun 31' 35" 24" The
above observations were made nearly at the same time. Laplace makes
the diameter of Venus (177") 57" 3 at the moment of her transit. Hence
at a medium the diameter is taken equal 58''. Now the mean distance of
the earth from the jun being 23464.5 semidiameters, and that of Venus
16978.2 semidiameters ; hence the difference 6486.3 sernidiam. is the dis-
tance of Venus from the earth, and therefore inversely, 169782 : 6486.3
:: 5b" : 2-" 1, the apparent diameter of Venus as see'n from the sun.
(The distance of Venus from the earth being- here taken to correspond
with its greatest diameter.) And again, 234-64.5 6486.3 :: 58" : 1C",
her apparent diameter at the distance of the sun from the earth or her
inean apparent diameter. Laplace makes this (51" 54) 16" 6. Now
31' 33" 4 (the sun's up. diam.) 16" .-: 864065.5 (the sun's real diam. in
miles) : 7301.7 miles the diameter of Venus.
@F THE SOLAR SYSTEM.
Venus does not perform her revolution round the sun exactly in
the plane of the ecliptic, but sometimes deviates from it several de-
grees* At the beginning of 1 750, the inclination of the plane of her
orbit to that of the ecliptic, was, according to Laplace (3<>.7701)
3 23' 35", and the secular variation of this inclination to the true*
ecliptic ( 1 3"80) 4"47 increasing. The distance of this planet from
the ecliptic, as seen from the earth or its geocentric latitude, will
sometimes exceed the inclination of its orbit. In the Nautical Al-
manac for 1812, Aug. 13, it is made 7 41'.
The longitude 9r place f the ascending node of Venus was 74
26' 18", or 14 26' 18" in Gemini, and the descending node was
therefore in the opposite sign and degree. The motion of the nodes
in 100 years, according to Vince, is 51' 40". Laplace makes
the sidereal and secular motion of the node on the true ecliptic
( 5673"60) 30'38"2 decreasing.f
All the primary planets, except Mercury and Uranus or Her-
schel, are found to have a rotary motion on their axis, or like the
earth a diurnal motion ; and from analogy we conclude that these
planets observe the same universal law, though at present not
within the reach of observation ; no telescope possessing sufficient
magnifying power to exhibit this phenomenon in these two planets,
Or the diameter may be found by trigonometry, in the same manner as
that of Mercury has been found pa. 263, using the angle 11" in place of
8" 9, and taking Venus's mean distance 16978,2 semidiameters of the
earth; thus,
As sine (90 II") 89 59' 49" , 10.0000000
To sine 11" - - - * - - 5.7269676
So is 16978.2 semid. - 4.2298916
To .90544 semid. - 1.9568592
Hence .90544 X 2 X 3956 = 7163.84 miles the diameter of Venus,
by this method, which would more nearly agree with the above if the de-
cimal parts of the seconds, &c. were retained.
Now the cube of the earth's diameter divided by the cube of the diam-
eter of Venus, will give the proportion of their magnitude thus 7 o ai ,.-
= 1.2742 or log. 79113 log. 7301.73 = 0.1052575, the number cor-
responding to which is 1.2742.
* The true ecliptic is the ecliptic corrected, or when allowance is mad*
for the secular variation. See the note to prob. 49, part 2d. In the same
manner the obliquity of the planet's orbit to the plane of the ecliptic, at
any time, and its secular variation being given, its obliquity at any othet
time may be found, as is evident. See the method of finding it by obser^
vation, &c. part 4th. ch. 2d. pa. 264, note.
f The secular motion of the nodes being given, their place for any time
may be found ; and the periodic revolution of the planet round the sun
being also given, and its distance from the node, its distance at any other
time may from thence be easily found. But the inclination of its orbit to
that of the ecliptic being given, its place in the true ecliptic or longitude,
and its distance from it or latitude may be easily found, in the same man-
ner as the right ascension and declination of the sun is Calculated ; bv
tlie solution of a right angled spherical triangle.
Jbi
274 OF THE SOLAR SYSTEM.
the one being too near, and the other too far removed from the
sun. The cause of this interesting phenomenon is not yet discov-
ered ; but from the numberless improvements in natural sciences,
it is very probable that in a short time it will develope itself, and
probably from its connection with the law of universal gravity, it
will throw new light on that intricate subject.
Galileo in 1611, was the first that observed this phenomenon in
Venus. In 1666 M. Cassini discovered a bright spot upon her
straight edge when dichotomised, similar to those on the moon's
surface, and found the time of its sidereal motion to be 23h. 16'.
In. 1726 Bianchim, from some observations on Venus, asserted in
his Hesperi et Phosphori nova phenomena^ that the time of her rota-
tion was 24- days, that her northjiole answered to the 20th degree
of Aquarius, and was elevated 1 5 or 20 above its orbit, and that
her axis continued parallel to itself. M. Cassini) the son, makes it
about 23h. 2>.>'. Schroeter,* from several continued observations of
the variation of her horns, and of some luminous points towards the
edges of the dark parts, has confirmed Cassini's result, which had
been disputed before He fixes the duration of her rotary motion
at 23 h. 2 1' 7" 2, and like Cassini has found that the equator of Ve-
nus makes a considerable angle with the ecliptic. He has also con-
cluded the existence of high mountains on her surface, from his ob-
servations, and the law by which her light varies from her enlighten-
ed to her dark side. (Phil. Trans. 1795.) He supposes the planet
surrounded with an extensive atmosphere, the refracting power of
which differs but little from that of the earth. The cusps or horns
appeared sometimes to run 15 19' into the dark hemisphere, and
hence he computes that the height of the atmosphere, to refract such
a quantity of light, must be 15156 Paris feet, or 16146.4 English.
But this must depend on the nature and density of the atmosphere,
of which we are ignorant. (Phil. Trans. 1792.) Dr. Herschel
agrees with M. Schroeter, that Venus has a considerable atmos-
phere : he has published in the Phil. Trans, for 1793, a long series
of observations on this planet, from which he concludes, 1 . That
the planet revolves on her axis, but that the period and the position
of the axis are uncertain ; 2. That the planet has a considerable
atmosphere ; 3. That there are probably hills and inequalities up-
on her surface, although he has not been able to see much of them,
owing, perhaps, to the density of her atmosphere ; and 4. That
this planet is somewhat larger than the earth, instead of being
less, as former astronomers imagined.
* Schroeter, a learned astronomer of Lilienthal, in the Duchy of Bre-
men. Among- others he has published a new work on the height of the
mountains of Venus, some of which he makes upwards of 23000 toises,
which is more than seven times the height of Chimbora9o, in South
America. He says that in the moon there are mountains 1000 toises higher
than ChimboraCo. But Dr. Herschel, considers the height of lunar
mountains in general as greatly overrated, and estimates them at no
more than half a mile perpendicular height.
OF THE SOLAR SYSTEM. 275
M. De la Hire bserved with a telescope 16 feet long:, mountains
in Venus higher than the moon ; but the difficulty of observing
those as well as the spots, particularly in northern latitudes where
the atmosphere is so dense, renders the result very doubtful.
Venus surpasses in brightness all the other planets and stars, and
is sometimes so brilliant as to be seen in the day with the naked
eye.* The light and heat which she receives from the sun, are
about double to what the earth receives f
The inclination of the axis of Venus to that of her orbit, is, ac-
cording to most astronomers, about 75, which is 51 32' greater
than the inclination of the equator and ecliptic. This is a singular
circumstance, and must cause a great variety in the seasons of Ve-
nus. Ferguson remarks that the north pole of her axis inclines
towards the 20th degree of Aquarius, our earth's to the beginning
* The equation investigated pa. 269, being* here applied to Venus, we
have a 1, b .7254 (pa. 271) then x = </3a* + b* __ 26 = 1.8778
1.4508 = .427 ; hence the angle ESP (Venus being- supposed at P, see the
fig. page 268) = 22 8' 26*. For a : b + x :: bx : b * ~ ^ = .343876
nearly, = the diff. of the segments of the base a or SE, made by a perp.
from P ; then " + = .6719 nearly, = the greater segment ; and
" _ L 438 = .3281 nearly, = the lesser segment. Hence the following
proportions ; As b = .7254 : .6719 :: rad. : co. sine ESP = 22 8' 26"; and
x = .427 : .3281 :: rad. : co. sine SEP = 39 47' 27" r= the elongation of
Venus from the sun, when brightest. The angle ESP at the time of the
planet's greatest elongation is 43 4(7, according to Vince ; and therefore
Venus is brightest between her inferior conjunction and greatest elongation.
The angle SPZ is also equal ESP -J- SEP (Eucl. 1, prop. 32) = 61 55' 53"
or 61 56 f nearly, the versed sine of which is 0.53 nearly, radius being uni-
ty ; hence the visible enlightened part : the whole disk :: 0.53 : 2 (note to
pa. 268, or art. 195, Vince's Ast.) Venus therefore appears a little more than
4-th illuminated, and answers to the appearance of the moon when 5 days
old. Pier diameter is here about 39", and therefore the enlightened part is
about 10" 25. At this time Venus is bright enough to cast a shadow at
night. This appearance of Venus takes place about 36 days before and af-
ter her inferior conjunction with the sun. For suppose Venus in conjunction
with the sun, and when seen from the sun to depart from the earth at the
rate of 37' in 1 day (Vince) we have 37' : 22 8' 26" :: Id. : 36 days nearly,
the time from conjunction until Venus is brightest. De Lambre makes the
daily mean motion of Venus 1 36' 8", and that of the earth = 59* 8"3 ;
hence 1 36 f 8" 59' 8f' 3 = 36' 59* 7, nearly equal 37', as above.
Vince remarks, that when Venus is brightest, and at the same time is at
her greatest north latitude, she can then be seen with the naked eye at any
time of the day, when she is above the horizon ; for when her north latitude is
the greatest, she rises highest above the horizon, and therefore is more easily
seen, the rays of light having to pass through a smaller portion of the at-
mosphere, in proportion as Venus is elevated. 'I 'his takes place once in 8
years, Venus and the earth returning to the same parts of their orbits after
that interval of time.
f This is found by dividing the square of the earth's distance from the
snn, by the square of the distance of Venus from the sun. Sqe the note, pa. 20.
276 OF THE SOLAR SYSTEM.
of Cancer. Consequently the northern parts of Venus have sum-
mer in the signs, where those of our earth have winter, and vice
versa. That the artificial day at each pole of Venus is 1 121 of
our natural days.* The sun's greatest declination on each side of
the equator of Venus amounts to 75 jf hence her tropics are only
15 from her poles, and her polar circles 15 from her equator.
The tropics of Venus are therefore between her polar circles and
"her poles, contrary to what those of the earth are.
The day in Venus making so considerable a part of her year, the
fun will therefore change his decl. so much in one day, that if it
be vertical to any place in the tropic, the next day it will be about
26 from it ; and in one day he will remove from the equator about
36^. So that the sun changes his decl. 14 more at a mean rate
in one day on Venus, than in a quarter of a year on the earth. t
If the inhabitants about the north pole of Venus fix their south
or meridian line, through that part of the heavens where the sun
has his greatest altitude, or north declination, and call those the
east and west points on the horizon, which are 90 from that point
where the meridian cuts the horizon ; then the following remark-
able phenomena will take place. The sun will rise 22| north of
the east, and advancing 1)2-1 (90 -{- 22i) as measured on the
liorizon, he will cross the meridian at an altitude of 12i ; then
Jnaking an entire revolution without setting, he will cross it again
at an alt. of about 48| ; at his next revolution he will come to his
greatest alt. and decl. and cross the meridian at an altitude of 75 ;
'and distant from the zenith of the place 15. Again, he will de-
scend in the same spiral manner, first crossing the meridian in an
angle of 48!, next in an angle of 12|, and advancing from thence
1121, he will set 22^ north of the west ; so that after having
made 4| revolutions above the horizon, he will descend below it to
Exhibit similar phenomena at the south pole.
The polar inhabitants of Venus, like those of our earth, have
but one day and one night, each of half a year long, or one half of
Venus's annual revolution. On Venus, however, the difference
* Or rather half of her annual revolution (see page 270) the sun being
Visible at her poles during this time.
f Whatever problems we have performed on the terrestrial globe relative
to the sun's greatest decl. 23 28', &c. may be applied to Venus, on suppo-
sition that the greatest declination is 75.
t If any point be taken on the equator of our common globes, and another
be taken, in a lesser circle drawn 15 from either pole, at the distance oi
90 east or west of the former, and through these points a great circle be
drawn, with the quadrant of alt. by which it may be also divided into de-
grees ; this will represent the ecliptic of Venus, and hence the above phe-
nomena maybe easily pointed out on the globe.
The great variation i\\ the sun's decl. seems to be providentially ordered,
to prevent the great effects of the sun's heat, which on Venus is twice as
great as on the earth (pa. 275) as he can shine perpendicularly on the same
place but a short lime, and pn tlrat rccpurrt the heated places have time tq
cooL
OF THE SOLAR SYSTEM.
Between the heat of summer and the cold of winter, and also be-
tween midday and midnight, is much greater than on the earth, the
sun's daily variation and the change of his declination and altitude
being much greater on Venus than on the earth. When the suu
is in the equinoctial, or over the equator of Venus, one half of his
disk appears above the horizon of the north pole, and one half
above the horizon of the south pole, his centre being in the hori-
zon of both poles ; and when he descends below the horizon of
one, lie ascends above the horizon of the other in the same propor-
tion. Hence, in the course of a year, each pole has one spring,
one autumn, a summer as long as both, and a winter equal in
length to the other three seasons.
At the polar circles of Venus, the seasons are nearly the same
as at the equator, the distance between both being 1 5 ; but the
winters are not so long, nor the summer so short. The same
seasons also happen twice a year.
At Venus's tropics, the sun continues about 1 5 of our weeks
without setting, in summer, and as long without rising in winter.
While his decimation is more than 1 5, he does not set to the in-
habitants of the adjacent tropic, nor rise to the inhabitants of the
other. The seasons are also, at her tropics, nearly the same as at
her poles ; the difference being similar to that at the polar circles..
At her equator the days and nights are equal, each being about*
1 1 J hours long. The diurnal and nocturnal arches are here how-
ever very unequal, particularly when the sun's declination is great-,
est ; for at this time his meridian alt. is sometimes double his
midnight depression, and at other times the reverse. At her
equator there are two winters, two summers, two springs, and two
autumns every year, owing to the obliquity of the sun's rays when
his declination is greatest, being equal to that in the latitude 5 1*
32' (75 23o 28') on the earth at the winter solstice. But
every winter at the equator is double the length of the summer*
the four seasons returning twice in that time, that is in 9i days.
From the quick change in the sun's declination, the sun's ampli-
tude at rising and setting will differ considerably ; and hence no
place has the forenoon and afternoon of the same day equally long,
unless at the equator or at the poles.
Where the sun crosses the equator or equinoctial of Venus in
any year, he will have 9 d.ecl. from that point on the same day
and hour the following year ; and will cross the equator 90 more
to the west. This phenomenon will make the equinoxes of Venus
a quarter of a day, or about 6 of our days later every year, and
hence in four annual revolutions the sun will pass vertically in the
same places,* Sec. &c.
* Many other observations could be made here, but the abore are suf-
iicient to enable the learner to pursue the subject at his leisure, and, in a
similar manner, to examine the phenomena of the other planets, and exhibit
them on the globe, independent of an orrery or any other instrument. Tlir
investigation of -|hese curious phenomena, and their representation on the
278 OF THE SOLAR SYSTEM.
Venus, when viewed through a telescope, exhibits all the pha-
ses of the moon from the crescent to the enlightened hemisphere,
though she is seldom observed perfectly round. Previous to the
rising of the sun in the morning, when she begins to disengage
herself from the sun's rays, she is seen under the form of a cres-
cent, at which time her apparent diameter is at its maximum, be-
ing then nearer to us than the sun, and almost in conjunction with
him. In proportion as she recedes from the sun, her crescent aug-
ments and her apparent diameter diminishes. When she departs
from the sun about 45, she returns towards him again, during which
time her enlightened hemisphere is increasing, and her apparent
diameter diminishing, until she is again immersed, in the morning,
in the solar rays. She is then further from us than the sun ; the
hemisphere which is turned towards the sun is also towards us,
and therefore Venus appears full. Her apparent diameter is then
a mininum or the least possible. Here Venus disappears for some
time, after which she re-appears in the evening, and produces, in.
a contrary order, the same phenomena as before. Her crescent
diminishes, and her apparent diameter increases as she advances
from the sun, and her enlightened hemisphere is turned from the
earth. At about 45 distance from the sun, she returns again to-
wards the sun? her crescent diminishing, and apparent diameter
increasing, until she again plunges into the sun's rays,
These phenomena evidently prove that Venus's orbit is within
that of the earth, and that she revolves round the sun, which is near-
ly in the centre of her orbit. These results obtained from observa-
tions of the phases and apparent diameter of Venus combined with
the earth's annual motion round the sun, explain also, in a natural
manner, the alternate, direct, and retrograde, motion in longitude
of this planet, also her complicated motion in latitude ; and the
same is true of Mercury. (See page 257.)
During the interval between Venus's disappearance in the even-
ing and her re-appearance in the morning, she is sometimes seen
moving on the disk of the sun, in the form of a dark round spot.
Dr. Jfalley remarks, that when at St. Helena, observing the
stars about the south pole, he had an opportunity of observing
Mercury passing over the sun's disk, which he observed with the
greatest degree of accuracy, by means of a telescope 24 feet long,
and found the time of the ingress and egress without being subject
*o an error of 1". The lucid line intercepted between the dark
limb of the planet and the bright limb of the sun being visible to
the naked eye, and the small dent made in the sun's limb by Mer-
cury's entering the disk, appearing to vanish in a moment. From
globes, must afford no small pleasure to those who are well acquainted with
what we have delivered in parts 2d and 3d ; as these phenomena afford
ample scope for inquiry and investigation, and every moment present new
scenes of wonder to the mind. These inquiries likewise bring us more ac-
quainted with the variety of those curious and admirable laws displayed in
t!r mechanism of th,^ uni verse? '
OF THE SOLAR SYSTEM. 279
this he concluded that the sun's parallax might be accurately de-
termined by such observations, from the difference of the times of
the transit over the sun at different places upon the earth's sur-
face,* provided Mercury were but nearer to the earth, and had a
greater parallax from the sun. But the difference of these paral-
laxes, and therefore the difference of times is so small, that the
difference of the parallaxes is always less than the solar parallax
sought ; and hence Mercury was considered unfit for this pur-
pose. Venus was therefore selected, for its parallax being near-
ly 4 times as great as the solar parallax ; and therefore produc-
ing a considerable difference between the times at which Venus
will be seen to pass over the sun at different parts of the earth, so
that the accuracy of the conclusion will be proportionably increas-
* For the method of finding the horizontal parallax of Venus by observa-
tion, and from thence by analogy, the parallax and distance of the sun and
of all the planets from him. See Ferguson's Astronomy, ch. 23, art. 2.
The following will give the learner a sufficient
idea of the nature, cc. of a parallax. Let C be
the earth's centre, A the place of a spectator on
its surface, V any object, ZH the sphere of the
fixed stars, to which the places of all the planets,
&c. are referred, Z the zenith, and H the sensi-
ble horizon; through the object V conceive the
lines AVf, CVw to be drawn, then t is the place
of the object as seen from the surface of the
earth or its apparent place, and t its place as seen
from the earth's centre C, or its true place, and
the arch tu, the distance between the apparent
and true place, is the parallax of the object, and
is measured by the angle tVii. The apparent place of the object observed
at the horizon is H, and its true place is s, the horizontal, parallax is there-
fore Ha measured by the angle Hw, and is greater than any other parallax
tu. At the zenith Z, the parallax is nothing, for here there is no difference
between the true and apparent place. That the angle AVC measures the
parallax is thus shewn ; the angle ZAV = /CV -f AVC (32 E. 1) AVC is
therefore the difference in the zenith distances or places of the body, as
seen from the centre and surface of the earth, or the angle under which the
diameter of the earth appears, as seen from the object or body V. Now
to find this angle we have this proportion, CV : VA :: sine VAC : sine AVC
rtN CA X sine VAC
(Simson's Trig. prop. 2) = . Now as CA is constant, the
O V
earth being supposed a sphere, the sine of the parallax varies as the sine of
the apparent zenith distance directly, and the distance of the body from the cen-
tre of the earth inversely, or as . ..... ^. Hence appears why the parallax
C V
is greatest at the horizon, and nothing at the zenith. If the object be at au
indefinitely great distance, it has no parallax ; the apparent places of the
fixed stars are not therefore altered by it. The parallax depresses an ob-
ject in a vertical circle, t being the apparent and u the true place. The.
parallax varies as the sine of the apparent zenith dist. or 1 : x :: y : xy, so
being the appar. zen. dist. and y the horizontal parallax, radius being 1.
To ascertain the parallax at all altitudes, it must therefore be found for
some given alt. For different methods by which this i performed, see-
Keil's Ast, lect: 31 or Vince's Ast. 8vo. ch. G, pa. 54, tc.
280 OF THE SOLAR SYSTEM-
ed, and not liable to any error greater than a small part of a secoifd.,
(See Motte's abridgment of the Phil. Trans, vol. 1. pa. 243.)
The transits of 1 76 1 and 1769 affording every opportunity of put-
ting these observations into practice, astronomers were therefore
sent from England, France, &c. to the most proper parts of the
earth, to observe both those transits, and the results of their obser-
vations give the parallax to a great degree of accuracy.
If the plane of the orbit of Venus coincided with the plane of
the ecliptic, she would pass directly between the earth and the sun
at each inferior conjunction, and would then appear like a dark
spot on the sun for about 7} hours, but like the moon's, Venus's
orbit only intersects the ecliptic in the nodes, and therefore one
half of it is on the north and the other half on the south side of the
ecliptic. Hence Venus can never be seen on the sun but at those
inferior conjunctions which take place in or near the nodes of her
Orbit. At all other conjunctions she passes either above or below
the sun, and is invisible, her dark side being then turned towards
the earth.
The mean time from conjunction to conjunction of Venus being
known (see pa. 275*) and the time of one mean conjunction, the
time of all the future mean conjunctions will be given. If those
which happen near the node be therefore found,f and the geocen-
tric latitude of the planet be- then computed ; if it be less than the
apparent semidiameter of the sun, there will be a transit of the
planet at that time.
* The conjunctions of any number of planets, in circular orbits, may be
thus calculated ; Let ~L = the time in which two superior planets
would meet from one conjunction to the next (see the note pa. 268.) and let
V equal to the periodic time of an inferior planet, taking- the planets in
order, then " y a like process, as in the note pa. 268. _ X V -r- -p^~~ V
PpV
In ^ ce manner fr w ^ k f 01111 ^ * r a fourth, that-
p'~~~
p/>v
_ __
P-p X Q divided by J>/)V _ Qxp# _- x15 ^ ' = the time
required, Q being- taken in order after V. Whence the general law is manl-
iest. In the same manner oppositions, &c. may be calculated.
This calculation may also be applied to elliptic orbits, provided the for-
mula for the daily revolution be substituted for the daily angular velocity.
f Vince in his Astronomy determines the periods when such conjunctions
happen, in the following- manner ; let P = the periodic time of the earth,
p = that of Venus or Mercury ; now that a transit may happen again at the
same node, the earth must perform a certain number of complete revolutions
in the same time that the planet performs a certain number, for then they
must come into conjunction ag-ain at the same point of the earth's orbit, or
nearly in the same position with respect to the node. Let the earth perform
a- revolutions, while the planet performs y revolutions, then will Vx = pn ;
bcMice y -5- j = / ^ p. Now P =: 365.256, ajid for Jfercum /; = 87.968 ;
OF THE SOLAR SYSTEM. 281
The .following new method of comfiuting the effect qf fiarallajc^ in
accelerating or retarding the time of the beginning and end of a tran-
sit of Venus or Mercury over the sun*s disk, was given ~by the late
Nevil Maskelyne, astronomer royal in Greenwich, and inserted here
for the sake of those who wish to know the printifiles of this interesting
'phenomenon, &c. The Jig. relates fiarticularty to the transit of Venus
which haftfiened in 1769.
I
Let SN repre-
sent the sun, C its
centre, P the north
pole of the equinoc-
tial, PC a meridian
passing through the
sun, Z the zenith of
the place ADB^
the relative path of
Venus, 5 being the
place of the de-
scending node ; A
the geocentric place
of Venus at her
ingress, B at her
egress, and D at her nearest approach to the sun's centre, as seen
from the centre of the earth, and b the apparent place of Venus at
P =s 365 jab =s= ^y resolving it into its continued fractions. (See Trail's
Algeb. Appendix No. 1, &c.) J, ^ 6 T , ^, $, f&> ~% 6 T , &c. that is I, 6, 7,
13, &c. revolutions of the earth are equal to 4, '25, 29, 54, &c. revolutions
of Mercury, approaching- still nearer to equality the further the series is
carried. The first period, or that of t year, is not sufficiently exact ; the
period of 6 years will sometimes bring on a return of the transit at the
same node; that of 7 years will bring on the transit's return more frequent-
ly ; that of 13 still more frequently, &c. There was a transit of Mercury
at his descending node in May, 1786 ; hence all the years at which the
transit may be expected to happen at that node, will be found by con-
tinually adding 6, 7, 13, 33, 46, &c. to it. In November, 1789, there was
a transit at the ascending node ; hence by adding the same numbers to
this year, the years at which it may be expected to happen, at the same
node, will be found. The following years 1799, 1832, 1845, 1878, 1891, &c.
are the years for the transit to happen at the descending node ; and 1802,
1815, 1822, 1835, 1848, 1861, & c . at the ascending 'node. For Venus
00,1 * U P 224 ' 7 8 235 7l3 9
* = 224.7; hence - = 3 ~ = -, , _, & c . Hence the pe -
riods are 8, 235, 713, &c. years. The transits at the same node will
therefore sometimes return in 8 years, but oftener in 235, and still oftener
in 713 years. In 1769 there was a transit of Venus in June, at the de-
scending node ; the next transits at the same node will be in 2004, 2012
2247, 2255, 2490, 2498, 2733, 2741 and 2984. In 1639 a transit hap'
pened at the ascending node in November ; the next transits at the
same node will be in 1874, 1882, 2117, 2125, 2360, 2368, 2603, 2611,
2846 and 2854. These transits are found by continually adding the pe-
riods, so as to find the years when they may be expected, and then com-
M m
282 OF THE SOLAR SYSTEM.
the egress to an observer whose zenith is Z ; draw baZ, then a is-
the true place of Venus, when the ajifiarent place is at d, and bn
is the parallax in altitude of Venus from the sun ; and the time of
contact xvill be diminished by the time which Venus takes to de-
scribe cB. Through b draw xybnJL parallel to AB, meeting ZB
produced in E, and Bn, Ax tangents to the circle, and let CD
be perpendicular to AB. Now in the figure 06EB, the sides may
be considered as rectilinear, on account of their smallness ; and
from the magnitude of ZB compared with Ba, BE may be consider-
ed as parallel to a, and hence c6EB may be considered as a paral-
lelogram ; aB may be therefore taken equal 6E. Now 6E = Ew
-f- bn y according as E falls without or within the sun's disk BNS ;
and En : EB :: sine EBn = cos. CBZ (because CBn is a right an-
gle: 18 Eucl. 3) : sine BnE<= sine nBD or nB?$ (29 Eucl. ))=cos.
CBD ; (Simson's Trig. prop. 2) hence En x cos CBD == EB x
cos. CBZ ( 1 6 Eucl. 6) and therefore En = . Now
as bn X ny Bn 2 (36 Eucl. 3) hence bn - = - - very
ny AB
nearly ; but Bn : BE :: sine BEn = sine ZBD (29 Eucl. 1) : sine
BnE = cos. CBD ; hence Bn = - TTTTR , and squaring both
cos.
., _ BE*x sine ZBD 2 BE 2 x sine ZBD
Sldes B * =B _____ ; therefore , n
ABxcos.CBD*
= the horizontal parallax of Venus from the sun ; then, as
the parallax varies as the sine of the apparent zenith distance, we
have Rad. or 1 : sine Zb :: fi : BE (note pa. 279) = p. x sine Z
=/* x sine ZB ; hence cB'= 6E = En -f bn (and substituting the
fi X sine ZB X cos. CBZ
value of BE) = -f- fi 2 X sine ZB 2 x
cos. CBD
AB S x e cr?BD* = * X Sine ZB X COS - CBZ X sec - CBD **
x S i ne ZB* X ZBD X sec. (For thc ^^ of
AB
arch is as the cos. reciprocally, or sec. = *) Hence the parallax
consists of two parts, one of which varies as //, and the other as
puting- tire shortest geocentric distance of Venus from the sun's centre at
that time, when their conjunction takes place; if this distance be less than
the semidiameter of the sun, there will be a transit.
The periods of the planets made use of above, are as given by Vince. La-
place makes the sidereal revolutions of Mercury, Venus, and the earth.
87.969255, 224/00817, and 365.256384 days respectively.
* The radius being- a mean proportional between the cos. and sec. (Cor.
% def. 9, Simson's Trig.) hence Had.* = cos. X sec. and rad. being 1, we
have sec. = . See b. 1, prop. 1, Emerson's Trigonometry.
OF THE SOLAR SYSTEM. 283
P 2 t the remaining quantities being the same. Let those quah-
iities multiplied by fl = v, and those multiplied by A 2 = w ;
let t" = the time in which Venus, by her geocentric relative
motion, takes to describe the space fa and let m be the relative
horary motion of Venus; then to find this motion we have
/? v ~* f\C\C\^
m : ft :: 1 hour or 3600" : t" = ^ . Hence to find
the time of describing aB, we have ft : fi X v + 1^ X w :: r :
f V -{_ j/j W j where by substituting the values of -v and w, the time of
describing aB, or the effect of parallax in accelerating or retarding .
the time of contact is given ; the upper sign is to be used when
CBZ is acute, and the lower sign when obtuse. If CBZ be a
right angle very nearly, but obtuse, it may happen that nE may be
less than nd, in which case nE is to be taken from nb, according to
the rule. The principal part, nE of the effect of parallax, will
increase or diminish the planet's distance from the sun's centre,
according as the angle ZBC is acute or obtuse ; but the small part
bn of the parallax will always increase the planet's distance from
the centre ; hence the sum or difference of the effects, \vith the
sign of the greater is to be taken, with respect to the increase or
decrease of the planet's distance from the sun. The second part
of the correction in the transits of Venus for 1761 and 1769, did
not exceed 9" or 0" of time, where the nearest approach of Ve-
nus to the sun's centre was about 10'. In the transit of Mercury -,
the first part of the parallax will be sufficient, unless the nearest
distance be much greater.
If the mean horizontal parallax of the sun be taken =r 8"83,
then it appears by calculation from the above expression, that the
total duration at Wardhus was lengthened by parallax 11' 16"88,
and diminished at Otaheite by 12' 10"07 ; the computed difference
of the times is therefore 23' 26"95, but the observed difference
was 23' 10" (see Vince's Complete System of Astronomy, ch. 25.)
The correct parallax may be therefore accurately found as fol-
lows : as the observed difference of the total durations at Ward-
hus and Otaheite is 23' 10", and the computed difference, from
the above given parallax, is 23' 26" 95, the true parallax of the
sun is less than the assumed. Let the true parallax be to the
assumed as 1 r to 1 ; then, from the foregoing expression, the
first part of the computed parallax will be lessened in the ratio of
1 r : 1, and the second part in the ratio of 1 r* : 1, or of
1 2r : 1 nearly. All the first parts in the above expression, viz.
406"05, 287"05,341' / 48, 382"47,inall 1417"05,cornbine the same
way to make the total duration longer at Wardhus than at Otaheite.
With respect to the second parts, the effects at Wardhus were
7"31 and 8"91 ; and at Otaheite 1"63 and 4"49, in all
10"10. Hence 1417"05 X f^r 10"10 X 1 2r = 1390",
the excess of the total duration at Wardhus above that at Otaheite,
or H17"05 10" 10 1390"= 1417"05 ~ 20"20 Xr,andr =
284 OF THE SOLAR SYSTEM.
T ax 0.0121. Therefore the sun's mean horizontal paral-
1396"85
lax = 8"83 X l 9.0 12 1 = 8"7231 6. The mean horizontal pa-
rallax of the sun is therefore assumed = 8J".
Hence the semidiameter of the earth : its distance from the
sun : : sine 8|" : rad : : 1 : 23575. Now the semidiameter of the
earth) according to the late French measures, being 3956 miles ;
j^ence the earth's distance from the sun is 23575 X 3956 =
9 S2 62700 miles. (See note to def. 8 )
The effect of the parallax being determined, the transit affords
an easy method of finding the difference of the longitudes of two
places where the same observations were made ; thus, compute
the effect of parallax in time, and reduce the observations at each
place to the time, if seen from the centre of the earth, and the
difference of time is the difference of longitudes required. For
ex. the times at Wardhus, at which the internal contact would take
place at the earth's centre, are 9h. 40' 44" 6, and 21h. 38' 25" 07,
the difference of which is 12h. 2' 19" 53 = 180 34' 53", the dif-
ference of longitude between Wardhus and Otaheite. From the
mean of 63 results from the transits of Mercury, Mr. Short found
the difference of longitude between Greenwich and Paris = 9' 15",
and from the transit of Venus in 176 1 =s= 9' 10". Mayer makes it
9' 16", Delambre 9' 20", and from trigonometrical calculation 9'
,18" 8, in time.
The transit of Venus also affords an accurate method of finding
the place of the node. For from the observations of Mr. Rittcn-
Iwuse at Norriton or Norristown, within 18 miles N. W. of Phila-
delphia, the least distance CD was observed to be 1 0* 1 0" ; hence
drawing CV perpendicular to C5, cos DCV = 8 '28' 54" : rad. : :
CD iO' 10" : CV 10' 17" the geocentric latitude of Venus at the
time of conjunction ; and 0.72626 (the dist. of Venus from the
sun*) : 0.28895 (her distance from the earth) : : CV 10' 17" : 4' 5"
the heliocentric latitude CV of Venus.f (Now considering C&V
a rt. angled triangle, we have tang. V&C 3 23' 35" (see pa 273)
2 rad. : : CV 4' 5" : C& 1 8' 52", which added to 2s 13 26' 34 /v
the place of the sun, gives 2s. 14 35' 26" for the place of the as-
cending node of the orbit of Venus. For the beginning of 1 800
Delambre makes it 2*. 14 52' 8", and its secular variation 5 1' 40"
(see pa. 272.)
Vince determines the time of the ecliptic conjunction as fol-
lows ; let the difference of longitudes (d) of Venus, and the sun's
centre be found for any time (t) ; and also the apparent geocentric
hourly motion (m) of Venus/rom the sun in longitude ; then say
in : d : : 1 hour : the interval between the time (t) and the con-
junction, which interval is to be added to or subtracted from t,
* This distance, &c. differs a little from that given in the notes pa. 271,
ieh see, the above being taken from Vince.
f The angle Subtended by CV is inversely as the distance from CV.
OF THE SOLAR SYSTEM. 285
according as the observation was made before or after the conjunc-
tion. In the transit for 1761 at 6h. 31' 46", apparent time at Pa-
risj M. de la Land found d = 2' 34' ; 4, and m = 3' 57"4 ; hence
3' 57"4 : 2' 34"4 : : Ih. : 39' l", which subtracted from 6h 31' 46",
because at that time the conjunction was past, gives 5h 52' 45"
for the time of conjunction from this observation. The latitude
at conjunction may be also thus found. The horary motion of
Venus in lat. being 35"4 ; hence 60' : 39' i" : : 35"4 : 23", the
motion in lat. in 39' l", which subtracted from 10' 1"2, the ob-
served lat. at 6h 31' 46", gives 9' 38"2 for the latitude at the
time of conjunction.
CHAP. IV.
OF THE EARTH,
AND ITS SATELLITE, THE MOOK.
WE now come to describe the earth, that part of the System by
far the most interesting for us, as it is that which we are destined
to inhabit, and of the phenomena of which we are therefore more
intimate observers.
Its figure, as composed of land and water, has been already
proved to be spherical or nearly so ;* but here it becomes neces-
sary to enter a little more into the detail of those arguments on
which this important truth is founded.
The first notions that mankind pliably formed of the earth
were, those which arose from the immediate suggestion of the
senses ; but by comparing the phenomena together, and examin-
ing the nature of the senses themselves, correcting and assisting
them ; and by a proper application of geometrical and mechanical
principles, the scheme of nature soon appeared very different from
that which is presented to a vulgar eye. At first sight the surface
of the earth appears of an unbounded extent, the clouds, me-
teors, moon, planets, sun and stars of every degree of magnitude,
appear in one azure surface, concave towards the earth, which
latter was therefore taken as its centre. It is to sight, as M'Laurin
remarks,! that we owe our knowledge of the different parts of the
system, those objects that are near us falling under the other
senses only : but admirable as this sense is, it has its imperfec-
tions. Vision depends upon the picture of external objects form-
ed on the retina^ together with a judgment of the understanding
acquired by habit and experience ; which is so immediately con-
nected with the sense, that it is impossible, by an act of reflection*
to trace it, or when it is erroneous, suddenly to correct it. If vi-
sion depended on the picture only, then equal pictures upon the
* See the note to def. 2, and also to prob. 31, part 2.
| View of Newton's Philosophy, 4to. pa. 223.
* Retina is the dark coat at the bpttom of the eye, on which objects are
painted.
286 OF THE SOLAR SYSTEM.
retina would suggest ideas of equal magnitudes of the objects ;
and if the smallest fly was so near that it could cover a distant
mountain from it, the fly ought to appear as large as the moun-
tain. But, by habit, \ve have acquired a faculty of correcting this
opinion, or idea of apparent magnitudes or distances ; and this,
with a quickness of thought almost inconceivable. Hence we see
how many fallacies may arise in vision : for as often as we are mis-
taken in our notion of distance, so often must a corresponding
error be produced in our idea of the magnitude of the object.
Thus we would imagine that the moon was of no greater mag-
nitude than about two feet in diameter, if we were not certain,
from other sources, that her magnitude is immensely greater.
Those objects that are seen in the same direct line, would, if at a
considerable distance from the observer, appear equally distant,
as both coincide on the retina ; those that are beyond the reach of
distinct vision, would also appear equally distant, as the clouds,
the moon, the sun, &c. did we not perceive that the clouds, in
passing between us and the moon, concealed it from our view ?
and that the moon obscures the sun when in conjunction with him.
Moreover, a distant body that is in motion, but in a direct line from
the eye to the object, will appear at rest, and a body at rest may
appear in motion, from our imperceptibly advancing from it, as is
the case with passengers in a vessel sailing along the shore, or
from land, or of the earth with regard to the sun. Hence as our
knowledge of the system must be founded upon the real figures,
magnitudes and motions of the bodies of which it is composed,
so, rejecting the prejudice produced by our senses, we must, from
the apparent phenomena, produce such an account of the real
act as may be consonant to reason, and the nature of the objects
under consideration.
The simplest proof that we have of the globular figure of the
earth, is from her shadow projected on the moon in a lunar eclipse.
For this shadow being always bounded by an arc of a circle, it
hence follows, that the earth, which projects the shadow, must be
of a spherical figure, since no other figure but that of a sphere,
when turned in every position with respect to a luminous body,
can cast a circular shadow ; and that if there were any considera-
ble irregular protuberance on the surface, or any remarkable an-
gle, this would necessarily, at some time, appear by the shadow.
We have another proof equally evident in our seeing the further
the higher we are elevated. For if the earth was a plane, we
could see as far on its surface when no object intervened, as at
1000 miles above its surface ; but at one or two miles above the
earth's surface we can see much farther, even with the naked eye,
than on its surface, and this is true of any part of the earth ;
moreover the proportion between the distance seen and the height
of the spectator above the surface of the earth, will answer to no
figure but that of a sphere, or an oblate spheroid, elevated a
OF THE SOLAR SYSTEM. 287
little towards the equator.* Likewise the calculation of eclipses,
of the places of the planets, and in general, all astronomical cal-
culations, are made on the supposition of the earth's spherical
figure, and all answer the times, when accurately calculated.
When an eclipse of the moon takes place, it is observed by those
who live eastward sooner than by those who live westward ; and
astronomers have found by frequent experience that, for every 1 5
difference of longitude, an eclipse begins so many hours sooner to
those who live eastward, and later to those who live westward ;
but eclipses would happen at the same time at all places, were the
earth a plane, nor could one part of the world be deprived of the
light of the sun while another enjoyed the benefit of it. The ele-
vation of the stars, and particularly the N. pole star, in travelling
or sailing towards the north, in N. lat. and the depression of
those towards the south, clearly prove that the earth is circular
from N. to S. and the voyages of the circumnavigators sufficiently
prove that the earth is round from east to west. The first who
attempted to circumnavigate the globe was Magellan, a Portu-
guese, who, on the 10th of August, 1519, sailed from Seville, in
Spain ; in 1522 his ship returned again to St. Lucar, near Seville,
on the 7th of September, not having altered its direction, during
this time, towards the north or south, except as compelled by the
winds or intervening land. Sir Francis Drake was the next, who,
in 1577, performed the voyage in 1056 days; afterwards Thomas
Cavendish performed it in 777 days, in the year 1586. Lord An-
son, Captain Cook, La Perouse, &c.
These arguments clearly prove that the earth is round or nearly
so, though common experience shews us that, mathematically or
strictly speaking, it is not a sphere, from the mountains, valleys,
&c. seen on its surface ; but these no more prevent its spherical
figure, than grains of dust on an artificial globe, (note to def. 1.)
Moreover from the properties of the pendulum, this truth has
received further confirmation, (see note to def. I.) though it ap-
pears that the earth is not truly spherical, but rather in the form
of a spheroid, and this is also confirmed by the different measures
made use of to determine this important point. (See the note to
prob. 31, part 2d.)
The earth being thus discovered to be globular, and from the
discoveries of the circumnavigators and others, that it had inhabi-
tants on every side of it, it followed, that some must have their
heads directed towards that part of the heavens where the feet of
the others would point, if the line were continued through the
earth, or that their feet must be directly opposite to each other,
while each one considered himself upright : but this difficulty
* Although St. Pierre endeavoured to prove that the earth is more ele-
vated towards the poles, yet his reasoning 1 is fallacious (as is evidently
proved in the Philadelphia edition of the" translation of his works) ancj.
contrary to the known laws of the general gravitation of matter and the
effects of the rotarv motion of the earth.
288 OP THE SOLAR SYSTEM.
vanished as soon as the general laws of gravity were discovered,
and that it was found that all heavy bodies tended to the centre of
the earth, by a force equal to the quantity of matter in them, and
therefore that on every part of the surface of the earth bodies were
kept in their natural positions, without a possibility of falling off,
by means of this law.
No sooner was this law and the globular figure of the earth dis-
covered by analysis^ than innumerable phenomena and important
discoveries were unfolded by the direct method of synthesis.
Hence followed the whole doctrine of the sphere, the motion of
the planets, comets, &c. and those admirable laws set forth in the
writings of Newton and his commentators.
This globe being circumscribed and limited as it is, it was na-
tural that some should undertake accurately to discover its dimen-
sions. It is probable that the first attempts were made at a period
anterior to those of which history has preserved the record, and
that the results have been lost in the physical and political changes
which the earth has experienced. The labours of the moderns
have, however, been, no doubt, more successful on this head,
from the accuracy of their instruments ; the following, collected
chiefly from Laplace, will exhibit the most accurate result of their
observations.
The elevation or depression of the stars, gives the angles which
verticals, elevated at the extremity of the arc passed over, form
at their point of contact ; for this angle is evidently equal to the
difference of the meridian altitudes of the same star, less the angle
which the arc described would subtend at the centre of the star ;
and we are certain from observation, that this is insensible. It
was then only requisite to measure this arc. It would be a long^
and tedious operation to apply our measures to so great an extent ;
it is a much more simple process to connect its extremities by a
chain of triangles, to those of a base of 12 or 15,000 feet, and
considering the precision with which the angles of these triangles
may be determined, its length can be obtained very exactly. It
is thus that the arc of the terrestrial meridian which crosses
France from Dunkirk to Mountjoy, near Barcelona, has been
measured ; that part of this arc whose amplitude is equal to the
hundredth part of a right angle, and whose central point corres-
ponds to (5l.|) 46^-, is equal to 100179 metres.
Of all the re-entering, or curve lined, &c. figures, the spherical
is the most simple, since it only depends on a single element,
the size of its radius ; and hence for the facility of calculation,
this form was attributed to the earth But the figure of the earth
is the result of those laws, which modified by a thousand circum-
stances, might alter it sensibly from a sphere. Inevitable errors
of observation left doubts on this interesting phenomenon, and the
Academy of Sciences, in which this great question was anxiously
agitated, judged with reason, that the difference of the terrestrial
degree, it it really existed, would be principally manifested in the
OF THE SOLAR SYSTEM. , 289
comparison of the degrees at the equator and towards the poles.
Academicians from France, and others from different parts of
Europe and America, have measured degrees of the meridian in
different parts of the world, and their measures incontestibly prove
that the earth is not perfectly spherical, from the increase of the
degrees from the equator to the poles (see note to prob. 31.) The
ellipse being next to the circle, the most simple of the re-entering
curves, the earth was considered as a solid formed by the revolu-
tion of an ellipsis about its shorter axis ; and its compression in
the direction of the poles, is a necessary inference from the ob-
served increase of the meridional degrees from the equator to the
poles. The radii of these degrees being in the direction of grav-
ity, are, by the law of the equilibrium of fluids, perpendicular to
the surface of the sea, with which the earth is in a great measure
covered. They do not tend as in a sphere, to the centre of the
ellipsoid. They have neither the same direction nor the same
length as radii, drawn from this centre to the surface, and whichj
except at the equator, and at the poles, cut it every where ob-
liquely. The point of contact of two adjoining verticals, is the
centre of the small terrestrial arc which they comprise between,
them ; if this arc were a straight line, these verticals would be
parallel, or would only meet at an infinite distance ; but in pro-
portion as they are curved, they meet at a distance so much the
shorter. Thus the extremity of the shorter axis being the point
where the ellipse approaches more to a straight line, the radius of
a degree at the pole, and consequently the degree itself, is of its
greatest length ; but the contrary takes place at the extremity of
the greater axis of the ellipse. At the equator, where the curv-
ature is the greatest, the degree in the direction of the meridian
is the shortest. Passing from the second to the first of these ex-
tremes, the degrees augment, and if the ellipse is but little flat*
tened, their increase is very nearly proportional to the square of the
sine of the latitude.
The measure of two degrees in the direction of the meridian,
is sufficient to determine the two axis of the generating ellipse,
and consequently the figure of the earth, supposing it elliptic. If
this be the hypothesis of nature, the same proportion should bo
found very nearly between the two axes, the degrees of France,
of the north, and of the equator, being compared two by two ; but
this comparison gives differences which it is difficult to attribute
to errors of observation alone. The excess of the axis of the
equator above that of the pole, taken as unity, is called the com-
pression or ellipticity of the elliptic spheroid ; now the degrees of
the north, and of France give T {^ for the ellipticity of the earth,
but the degrees of France and at the equator, give 7 ^j, and hence
it appears that the earth differs sensibly from an ellipsoid. There
is even reason to believe that it is not a solid of revolution, and
that its two hemispheres are not equal at each side of the equator.
The degree measured by La Caille, at the Cape of Good Hope,
Nn
290 OF THE SOLAR SYSTEM.
in the southern latitude of 33 18' 32"4, has been found to be
greater than the degree in France, Pennsylvania, and Italy ; but it
ought to be smaller than ail these degrees, if the earth were a re-
gular solid of revolution, formed of two similar hemispheres.
Every result leads us to conclude that it ib not the case. (See
Laplace.)
The figure of the earth being extremely complicated, it is im-
portant to multiply the measures of it in every direction, and in as
many places as possible We may always, at every point of its
surface, suppose an osculatory ellipse, which sensibly coincides
with it, to a small extent round the point of contact.
Terrestrial arches measured in the direction of the meridian
and perpendicular to it, compared with observations of latitudes,
and of the angles which the direction of the extremities of these
arches form with their respective meridians, will give us the na-
ture and position of this ellipsoid, which may not be a solid of
revolution, and which varies sensibly at great distances
The operations which DeLambre and Mechain have executed in
France to obtain the length of the metre, determine very nearly
the osculatory ellipse of that part of the earth, the latitude being
observed at three intermediate points. Two bases of more than
12000 metres have been measured, the one near Melun, the other
near Perpignan And the correctness of the observations is con-
firmed from this circumstance, that the base at Perpignan deduced
from that at Melun, by a chain of triangles which unites them,
does not differ ^ of a metre from its measurement, though the
distance which separates the two bases is more than 900,000
metres.
The principal results of this important operation, as given by
Laplace, are as follows :
OBSERVED LATITUDES.
Decimal. Sexagesimal.
Mountjoui 45 958281 41^2i'45' /
Carcassone 48 016790 43 12 54
Evaux 51 309414 46 10 42
Pantheon at Paris 54 274614 48 50 49
Dunkirk 56 706944 51 2 10
Arc of the terrestrial meridian comprised between Mountjoui and
Carcassone 205621.3 metres, Evaux 534714.5 metres, Panthe-
on 831536.4 metres, and Dunkirk 1075058.5 metres.
The comparison of these results evidently indicates a diminu-
tion in the terrestrial degrees fron> the pole to the equator ; but
the law of this diminution seems very irregular If however the
ellipsoid, which satisfies these measures nearer than any other, be
required, it is sufficient only to alter the observed latitudes about
(4|") 0"324.
The compression is then j^ the semiaxis of the pole parallel
to that of the earth, is 63440 11 metres, and the degree corres-
ponding to the mean parallel} is 99983.7 metres, An error qf
OF THE SOLAR SYSTEM. 291
(4i") o"324, though very small, is not admissible, considering
the great precision of the observations ; but this ellipsoid may at
least be considered as osculatory to the surface of the earth in
France at (51) 45 54' of latitude, and suppose that it coincides
with it to an extent of (5 or 6) 4 e 30' or 5 24' round the point
of osculation. It gives !007i6.9 metres, for the degree perpen-
dicular to the meridian, at (56 3 '44) 50 40' 58"6 latitude, and
by a very exact operation, lately performed in England, it has
been found to be 100700 5 metres. This agreement proves that
the action of the Pyrenees and other mountains in the south of
France, has had very little influence on the latitudes observed at
Evaux, Carcassone and Mountjoy, and that the great compression
of the osculatory ellipse depends on attractions much more ex-
tended, the effect of which is felt in the north as well as the south
of France, and even in England. Italy, and Austria : for the de-
grees, which have been carefully measured, are very nearly the
same as on the ellipsoid.
Whatever be the nature of the terrestrial meridians, it is evi-
dent, as their degrees diminish from the poles to the equator,
that the earth is flattened in the direction of its poles, or that the
axis of the poles is less than the diameter of the equator. To
explain this, let the earth be supposed a solid of revolution ; then
k is evident, that the radius of the degree at the north pole, and
the series of all the radii from the pole to the equator, which by
the supposition continually diminish, form, by their consecutive
intersections, a curve which at first touches the axis of the pole,
and afterwards separates from it, its convexity being constantly
turned towards it, and raises itself towards the pole, until the ra-
dius of the meridional degree takes a direction perpendicular to
the first ; it will then be in the plane of the equator If this ra-
dius of the polar degree be supposed flexible, and that it involves
successively the arc of the curve which we have considered, its
extremity will describe the terrestrial meridian, and the part of it
intercepted between the meridian and curve, will be the corres-
ponding radius of the degree of the meridian. This curve is
what geometricians call the evolutc. of the meridian* Let us now
consider the intersection of the diameter of the equator with the
axis of the pole, as the centre of the earth. The sum of the two
tangents to the evolute of the meridian drawn from this centre,
the one following the axis of the pole, and the other the diameter
of the equator, will be greater than the arc of the evolute which
they include between them. Now the radius drawn from the cen-
tre of the earth to the north pole, is equal to the radius of the
polar degree, less the first tangent ; the semidiameter of the
equator is equal to the radius of the degree of the meridian at the
equator, more the second tangent. The excess of the semidiam-
eter above the terrestrial radius of the pole, is then equal to the
sum of these two tangents, less the excess of the radius of the
polar degree above the radius of the degree of the meridian at the
292 OF THE SOLAR SYSTEM.
equator ; this last excess is the arc itself of the evolute, which is
less than the sum of the extreme tangents. The excess then of
the semidiameter of the equator above the radius, drawn from the
centre of the earth to the north pole, is positive. It can be prov-
ed in the same manner that the excess of the semidiameter ot the
equator above the radius, drawn from the centre of the earth to
the south pole, is positive The whole axis of the poles is there-
fore less than the diameter of the equator, or, which comes to the
same thing, the earth is flattened in the direction of the poles.
Considering every portion of the meridian as a small arc of its
osculatory circumference, it is easy to see that the radius drawn
from the centre of the earth to that extremity of the arc which is
nearest to the pole, is less than the radius drawn from the same
centre to the other extremity. From whence it follows, that the
terrestrial radii increase from the poles to the equator, if, as all
observations indicate, the degrees of the meridian augment from
the equator to the poles. The difference of the radii of the de-
grees of the meridian from the pole to the equator, is equal to
the difference of the corresponding terrestrial radii, more the ex-
cess of twice the evolute above the sum of the two extreme tan-
gents, which excess is evidently positive : thus the degrees of
the meridian increase from the equator to the poles in a greater
^proportion than the diminution of the terrestrial radii. These
demonstrations are equally applicable, if the northern and southern
hemispheres were not equal and similar, and it is easy to extend
them to the supposition of the earth not being a solid of revolution.
It is however remarkable, that the observations made in the
northern hemisphere, give the evolute of the meridian from (43
to 73) 38 42' to 65 42', very little different from that of an
ellipsoid of T -^ compression, and of which the mean degree is
99983.7 metres. For this ellipsoid nearly satisfies the measures
lately made in France, the degrees measured in Italy and Lap-
land, and that which has been measured in England perpendicu-
lar to the meridian. It also represents the degree of the meri-
dian measured in Austria at (53) 47 42' of latitude, and which
Liesganig has found to be 100U4/J metres. Finally, it agrees
with the degree of the longitude measured in France at (48 4')
43 33' 36" latitude, and of which Cassini and La Caille have
fixed the length at 72003.5 metres.
Curves have been constructed at the principal places in France,
on the line which has been considered as the meridian of the ob-
servatory of Paris, traced in the same manner as this line, with
this difference only, that the first side, always toigent to the sur-
face of the earth, instead of being parallel to the plane of the
celestial meridian of Paris, is perpendicular to it. It is by the
length of these curves, and by the distance from the observatory
to the points where they meet the meridian, that the position of
these places is determined. This labour, the most useful to Ge~
which has yet been performed, is a model (as Laplace
OF THE SOLAR SYSTEM. 293
remarks) which every enlightened nation will, no doubt, hasten
to imitate.
We shall now proceed to another point no less worthy of at-
tention, that is, the diurnal motion of the earth on its axis^ a phe-
nomenon which has been so clearly elucidated by the astrono-
mers of the last and present age, that, though contrary to the
direct testimony of our senses, the variety of strong and forcible
arguments in confirmation of this motion, must effectually dis-
sipate every doubt, and gain the assent of every impartial in-
quirer.
When we reflect on the diurnal motion to which all the hea-
venly bodies are subject, we cannot but recognize one general
cause which moves and regulates them, or causes them appa-
rently to revolve round the earth. If we consider that these bo-
dies are insulated, with respect to each other, and placed at very
different distances from the earth, that the sun and the stars are
at much greater distances from it than the moon, and that the
variations in the apparent diameters of the planets, indicate great
alterations in their distances ; and that moreover the comets tra-
verse the heavens freely in all directions, it will be difficult to con-
ceive that it is the same cause which impresses on all bodies a
common motion of rotation. But since the heavenly bodies pre-
sent the same appearance to us, whether the firmament carries
them round the earth, considered as immoveable, or whether the
earth itself revolves in a contrary direction ; it seems much more
natural to admit this latter motion, and to regard that of the hea-
vens as only apparent.
The earth is a globe whose diameter is only 7911.2 English
miles, as we have shewn in the note to def. 8, part 1 ; the sun,
as we have seen, is incomparably larger ; the earth then, which
is but a point in comparison of the sun, must turn on its axis in
a certain time, or else the sun, stars, 8cc. revolve round the earth
in nearly the same time. Is it not then infinitely more simple to
attribute to the globe we inhabit, a motion of rotation on its own
axis, than to suppose in masses so immense and so remote as the
sun and stars, such an extremely rapid motion as would cause
them to revolve in one day round the earth ?
But let us suppose that the sun does actually revolve round the
earth. Now it is a known principle in the laws of motion (which
will be shewn afterwards) that if any body revolve round another
as its centre, it is necessary that the central body be always in the
plane in which the revolving body moves, whatever curve it de-
scribes (Emerson's Astr. p. 11.) ; therefore the diurnal path of
the sun, in moving round the earth in a day, must always describe
a circle which will divide the earth into two equal hemispheres.
But this never happens but at the equinoxes, when the sun rises
exactly in the east and sets exactly in the west ; for in our sum-
mer the sun rises to the north of the east, and sets to the north
of the west, and when on the meridian, it is nearer to us than
294 OF THE SOLAR SYSTE1&
the equator, its declination being north ; in the winter it rises to
the south of the east, and sets to the south of the west- and when
on the meridian, is further from us than the equator, and there-
fore in both cases its diurnal path divides the globe into two une-
qual parts ; consequently the sun does not move round the earth.
Moreover we have seen that the pole of the equator seems to
move slowly round that of the ecliptic, from whence results the
precession of the equinoxes. If the earth be immoveable, the
pole of the equator must be likewise immoveable, as it always
corresponds to the same point of the terrestrial surface ; the
ecliptic therefore moves round these poles, and in this motion
carries all the heavenly bodies with it. Thus the whole system,
or rather the whole universe, composed of so many bodies, dif-
fering from each other in their magnitudes, moiions, and distan-
ces, would be again subject to a general motion, which disap-
pears, and is reduced to a simple appearance, if we suppose the
terrestrial axis to move round the poles of the ecliptic.
It is no argument againet the earth's diurnal motion, that we
are not sensible of it ; a person on the earth can no more be sen-
sible of its undisturbed motion on its axis, than a person in the
cabin of a ship, on smooth water, can be sensible of the ship's
motion when it sails along, or turns gently and uniformly round.
Carried on with a velocity which is common to every thing that
surrounds us,* we are in the case of a spectator placed in a ship
that is in motion. He fancies himself at rest, and the shores,
the Mils, and all the objects placed out of the vessel, appear to
him to move. But on comparing the extent of the shore, the
planes, and the height of the mountains, with the simllness of
his vessel, he recognizes that the apparent motion of these ob-
jects, arises from his own real motion. The innumerable stars
which occupy the celestial regions, are, relatively to the earth,
what the shores and the hills are to the vessel ; and the sdme rea-
sons which convince the navigator of the reality of his own mo-
tion, prove to us the motion of the earth.
These arguments are likewise strengthened by analogy. We
find that the sun, and those planets on which there are visible
spots, turn round their axis ; and this motion is always from west
to cast, similar to that which the diurnal motion of the heavens
indicate in the earth.
There is one effect of the motion of bodies on their axis,f
which will enable us to judge with certainty whether this rotation
takes place with regard to the earth. By the laws of the gravi-
tation of matter, we comprehend that the centrifugal force which
tends to remove every particle of a body from its axis of rotation,
should flatten the earth at the poles, and elevate it at the equator ;
for as the equatorial parts move with the greatest velocity, they
* Lnplace's Astr. vol. 1, B. 2, ch. 1.
i Ferguson's Astronomy, Art. 116.
OF THE SOLAR SYSTEM. 295
therefore recede furthest from the axis of motion, and in-
crease the equatorial diameter. That our earth is really of such
a figure, we have sufficiently proved in the foregoing articles.
This proof receives additional strength from the doctrine of pen-
dulums, as is sufficiently proved in Prop. 20. B. 3, Newton's
Principia (see note to def. 2.) And as the earth is therefore high-
er at the equator than at the poles, the fluid parts, or the sea,
which naturally seeks its level, would rush towards the polar re-
gions, and leave the equatorial parts dry, if the centrifugal force
at the equator did not prevent it.
This centrifugal force, or the tendency that bodies receive
from the earth's rotary motion, should likewise diminish the
force of gravity, or the weight of bodies at the equator ; and
hence at the poles the gravity is the greatest, owing to this force
being nothing, and moreover that, from the flatness of the earth
at the poles, bodies are nearer to the earth's centre, where the
force of the earth's attraction is accumulated. We find, front
experience, that a pendulum which vibrates seconds near the
poles, vibrates slower near the equator, which shews that it is
lighter or less attracted there ; for as we have remarked before
(note to def. 2) the length of pendulums vibrating in the same
time, in different parts of the world, are as the force of gravity,
or weight of bodies on the earth's surface. Every thing then
leads us to conclude, as Laplace remarks, that the earth has really
a motion of rotation, and that the diurnal motion of the heavens
is merely an illusion produced by it. An allusion similar to that
which represents the heavens as a blue -vault, to which all the
stars are fixed, and the earth as zfilane on which it rests.
But some are apt to imagine* that if the earth turns eastward
(as it must from the phenomena) a ball thrown perpendicularly
upwards in the air, must fall considerably westward of the place
it was projected from ; but as the gun or whatever it was pro-
jected from, partakes of the earth's motion, it must fall exactly
in the same place. A stone dropped from the top of the main-
mast of a ship, will fall on the deck, if it meet with no obstacle,
as near the foot of the mast when the ship sails as when it has no
motion ; but persons on shore would observe the ball to describe
a curve, if the vessel was sailing, as it partook of two motions,
one in the direction of gravity, and the other in the direction of
the vessel. (See the laws of motion, after this system.) If an inverted
bottle full of liquor be suspended from the ceiling of the cabin,
and a small hole be made in the cork to let the liquor drop
through on the floor, the drops will fall just as far forward on the
floor when the ship sails as when she is at rest.
It is moreover objected from the Scriptures, that at the com-
mand of Joshua the sun stood still, and that therefore it must
have had a previous motion. But those who bring forward this
* Ferguson's Astr, Art. 12-1. See also Hell's Astr. Lect. 2.
296 OF THE SOLAR SYSTEM.
objection, know little of the spirit of the Scriptures, and as little
of the idiom of language. The Scriptures were not given us to
teach us profound lessons of philosophy or astronomy, but to teach
us how to lead a virtuous lite ; and what is more common, even
in the writings of the most accurate philosophers and astronomers,
than these expressions, the nun rises, the sun sets^ though they
know at the same time that the sun has no such motion at all ; and
were they to make use of more correct expressions, they would be
as unintelligible to the generality of men, as Joshua would, in a
similar case, be to the Jewish people*.
* Although the motion of the earth on its axis be established beyond the
possibility of doubt, its cause has, however, never been investigated ; nor
can it be deduced from any law consequent of the gravitation of matter.
We shall, however, offer a few remarks on it, rather as conjectures than as
principles, on which any new theories or systems could be established. The
reasoning from strict analogy is conclusive, and the reasoning' from analogy
in this case, favours a good deal our remarks. It is a general law observed
,by all the planets, that they perform their revolutions round the sun, al-
ways in one and the same direction, that is, in the direction in which it re-
volves on its axis ; that all the primary planets, as far as observations could
be made on them, are found to have gross atmospheres surrounding them,
and are also observed to have a motion on their axis, all in the same direc-
tion as the motion of the earth on its axis ; that the secondary planets are
found to have little or no atmosphere, and also that they do not perform
similar revolutions on their axes, as they are found always to keep the same
side turned towards their primary planets. Does it not then follow, that
there is some regular cause for these phenomena, which are so constant
and so regular ? It seems to have no connection with the laws of gravity,
and the cause of gravity itself being occult, prevents our forming any just
notions of it. The only agent that we can observe is light or heat (for heat,
ealoric, &c. whether latent or not, I look upon to proceed from the same
principle, though differently modified ; this is proved from the writings of
many able chymists.) For from the laws and nature of light, the phenom-
ena of the motion of bodies can be accounted for on mechanical principles.
It is well known, that as a body, it acts on others, and has a momentum
proportional to its velocity and quantity of matter ; but as its velocity is so
very great, and its particles so exceedingly small, this momentum is not
easily appreciated : in consequence of this law, it displaces those particles
of the atmosphere on which its influence is exerted, and causes a rarefac-
tion ; it also repels bodies, but as it is bent into a curve, or refracted in pass-
ing from one medium into another, its force, or momentum, may be thus
exerted on the side of a body opposite to that from which it was emitted,
and thus cause an attraction, or motion of the body towards that from which
the effluvia of light is emitted, similar to those phenomena produced by the
electric fluid, &c. &c. This being premised, let us now consider the phe-
nomenon of the sun's daily apparent motion. We shall find that that part
of the atmosphere, over which the sun is perpendicular, is more rarefied
than any other, and as the different parts of the earth over which the sun
is perpendicular, pass successively under the sun in a direction from ivest
to east, the whole hemisphere on the east side of the sun, will have its at-
mosphere more rarefied than the hemisphere west of the sun ; and hence
on the east side, the rays will act more directly and meet with less resist-
ance, than when acting through the dense atmosphere on the west, and
therefore a motion of the earth on its axis must be the consequence of this
tin .-ence of action This explication is strengthened from analogy, be-
cause all the primary planets have a similar, rotation, and are also found to
OF THE SOLAR SYSTEM.
The diurnal revolution of the earth on its axis being thus es-
tablished, we shall proceed to that of if a annual motion round the
sun. For we must suppose the sun, accompanied with the planets
and satellites, in motion round the earth ; or the earth, with the
other planets, cc. to revolve round the sun. The appearances of
the heavenly bodies, as seen from the earth, are the same in both
hypotheses ; but the latter is to be preferred, for the following
reasons.
have an atmosphere ; and the secondary planets, which have no such mo-
tion, are found to be destitute of an atmosphere, or if they have any, it is
so rare as to be insensible as well as its effects. But the rotation above,
described could not take place, unless the earth had received a primary
impulse in the direction in which it revolves, and as this direction is the
same in all the planets, the impulse could not be fortuitous, but must be
regulated by some constant and regular law. Now as the sun, which emits
the light, is found to have a motion of rotation in this direction, it is in this
that the cause of the direction of the planets' rotation is to be looked for.
Already an extensive field for speculation is open to our view, and innume-
rable questions present themselves. The sun has an atmosphere and a
similar motion : is this motion produced in a similar manner ? What gives
this direction to the sun's rotary motion? Whence the sun's light, or why
not long since exhausted, being emitted into spaces from which it can never
return ? These are questions too arduous to be discussed in the compass of
a note ; but we cannot pass them over in silence. AVith regard to the first,
we must, according to one of the first rules in philosophy, assign the same
causes to the same natural effects, as far as possible. And hence whatever
produces the motion of the earth, a similar cause must produce that of the
sun. But whence this cause ? Here we are embarrassed, and can produce
no satisfactory solution. This we know, that as all bodies gravitate towards
each other, in proportion to the quantity of matter which they severally
contain, all the bodies in the universe would tend to that part where the
attraction was the greatest, or to one general centre, unless counterbalanced.
!y a centrifugal force ; and the stars must have such a centre, or we must
admit of creation in infinitum. That the stars have, in reality, some such
centre, and also a periodic revolution around it, analogy and observation
both concur to prove. Many of the stars are found to have motions, which
cannot be accounted for from any other cause ; and others are at too great
a distance, in the immense expanse, to have their motions sensible. From,
analogy, we see the distant Herschel, from which the sun appears not
much greater than a star, or the still more distant comet, no less than
Mercury, regard the sun as the centre of their motion ; is it not then ra-
tional to conclude, that the sun and stars observe a similar law, and revolve
about some common centre ? The secondary planets revolve round their
primaries as their centres; and both primaries and secondaries regard the
sun again as their center, and hence the sun, together with both primaries,
and secondaries, may regard another body as their centre, &c. It is no
objection to say, that such a centre, or such an immense body, has never
been observed, for the whole solar system, at the distance of some of the stars,
would appear no greater than a point. If then this centre has really an ex-
istence, its magnitude must be immensely great beyond conception, or its na-
ture must be different from all those bodies that we have any knowledge of.
For in the solar system, the economy of this system requires, that the sun,
should be much greater than all the other bodies, that they might regard him
us their centre ; and this we find to be the case. Hence if the stars have a
common centre, the magnitude of the body, placed in this centre, must
exceed that of all the stars put together ; for the aggregated of their attrat**
00
298 OF THE SOLAR SYSTEM.
The masses of the sun, of Jupiter, Saturn, and Herschel, are
considerably greater than that of the earth; hence it is much
more simple to have the latter revolve round the sun, than that
the whole solar system should revolve round the earth. Moreo-
ver, by the laws of centripetal forces (given after this system) if two
bodies revolve round each other, they perform their revolutions
round their common centre of gravity (Newton's Principia, B. 1.
tions on it in one direction, lessened by their aggregate in the opposite,
acts as one body, and a greater body or mass of matter can never revolve
round a lesser, according to the present laws of nature. And as the sun
dispenses light, &,c. to the solar system, and regulates the motions of those
bodies that exist around it, it is probable that this immense body supplies
the whole system of the universe with that light, &c. with which every part
of it is replete, and regulates, as a main spring, the motions of the whole
machine. What an idea does this give of the universe ! But an idea the
most simple and consonant to the present laws of nature, and uniting unity
and simplicity in the design, with magnificence and awful grandeur in the
execution. An idea, which shews the universe to be the work of one intel-
ligent, sublime Being, who formed and presides over the magnificent struc-
ture. And so far is this system from being at variance with the account
which Moses has given us of the works of creation, that it seems to ema-
nate from the pen of the sacred historian. We read in the 1st ch. of Gen.
that " in the beginning God created heaven and earth," where, by the word
earth, is meant, according to most interpreters, all that opake matter which
enters into .the formation of the different bodies of the universe. Next
God created light, (v. 3) afterwards the firmament, (v. 7) or all that space
in which bodies are placed ; then were those lights made in the firmament
of heaven to be for signs, and for seasons, &c. (v. 14) and to give light upon
the earth, (v. 15) or those stars which we call fixed, because their motions
are not sensible, and which, no doubt, are destined to perform the same
functions as our sun. We see, moreover, that the matter of which the sun
is composed, is dense and heavy like that of the earth, that it must receive
a supply of light from some source to preserve the same uniform splendour,
if we except the effects of some spots on its surface (which also shew that
parts of it are opake) and that therefore it is only calculated to reflect or
emit that light, heat, &c. more copiously : an effect which the planets, from
their opacity and contexture, as well as their inferior magnitude, are not
calculated to produce. We see, moreover, that those bodies which are
nearest the sun, receive most of its light and heat, and that those that are
at the greatest distance from him, receive very little more than we do from
some of the lai-gest of the stars. Now, reasoning from analogy, if we
suppose the body above described to occupy the centre of the universe, and
to be the fountain of light and heat to the whole universe, those bodies that
are nearest this, must enjoy more of its light and heat; others that are more
remote, may be at a loss to know, unless by analogy, whether there be such
a body in the universe, as is the case with us ; and others may be so re-
mote, as to receive little of its light or heat, and thus remain almost buried
in a continual night. If we carry our ideas a little further* we shall find
nothing but an immense void, where a ray of light has never penetrated.
This is ultimately the view of nature which the present system of philoso-
phy, or the general gravitation of matter, developes a system, which swal-
lows up, from its immensity, the feeble powers of our reason, and leaves us
nothing to build on but conjecture. This is then the utmost stretch of phi-
Unsoplnj ; it may unfold this system, but it can go no further. Forever
would it leave us ignorant of our destination, and the great end of our be-
ing-, did not the author of nature dissipate our doubts, and point them out
OF THE SOLAR SYSTEM. 299
. 57 ; or prop. 20, sect. 2, Emerson's Centr. Forces) and it is
evident, that if the two bodies be of equal magnitude and density,
the centre of gravity will be equidistant from each body (see the
note p. 253 ;) but if they be of different magnitudes, the centre
of gravity will be proportionally nearer the greater body. If the
earth, therefore, remain at rest while the sun revolves round it,
its magnitude must be vastly greater than that of the sun ; it being
'to us by means worthy his infinite wisdom. The magnificence which we
behold in creation, is worthy the Great Author ; but all this magnificence is
one day to vanish and hence, by his example, he would point out the van-
ity of all created things, and call our attention to higher destinies and re-
gions. It is not then the magnitude or multiplicity of those orbs that should
challenge our estimation ; they may excite our astonishment, and produce
an awful respect for their Creator ; but they are nothing more than inani-
mate heaps of matter, incapable of knowing that being that called them
into existence, and destined one day to perish. Of all the beings then that
the universe exhibits to our view, we find only man possessed of that im-
mortal principle destined to survive the wreck of matter, and capable of
knowing, serving, and enjoying that great Being that called it into exist-
ence. One immortal soul is therefore more precious in the sight of its Cre-
ator, than all those vast orbs that roll their immense masses through the
expanse of heaven ; and hence we need not wonder that he has done so
much for its preservation. To examine the question, whether those bodies
are inhabited as well as our earth, would be a futile as well as useless in-
quiry, as it is evident we can never, in our present state of existence, know
any thing of the matter. The Creator (as De Feller remarks) undoubtedly
could, for his own glory, and to display the treasures of his wisdom and
power, do great and beautiful things without any reference to man, or to
any rational creature. This is the opinion of many learned writers, and
particularly of St. Augustine, St. Thomas, Petavius, Leibnitz, &c. and the
sacred writings declare, that Universe, propter sennet ipsum operatus est J)om-
inus. Prov. 164. "God," says Hugens (Plurality of Worlds, ch. 8) "is
himself the spectator of the works he has created ; and who can doubt, but
that he who made the eye can see very well, and delights in doing so ? In-
quire no further. Is it not for that that lie created man, and all that is
contained in the universe ?" Pythagoras's music of the celestial spheres, is
an allegorical expression of the pleasure which intellectual beings take in
viewing them. Young, in his Night Thoughts (Night 9th) considers the
stars as grand refulgent thrones, on which the ministers of the Eternal sit
in majestic state, executing throughout the universe the decrees of his love
or his vengeance. But to spare the mind of man from amusing itself with
vain systems and philosophic dreams, religion saves for it this useless waste
of time, and calls it to more important studies. It points out all that is ne-
cessary as regards our own destination. To reject this knowledge, because
it does not point out the destination of beings which it would be useless for
us to know, and of whose existence we are perfectly ignorant, would be
folly in the extreme. To say that the Christian religion is unphilosophical,
is equally frivolous. It advances no one principle contrary to the known
established laws of nature ; and while absurd systems of philosophy and
false schemes of nature inundated the world, a few expressions in the sa-
cred writings contained more sound sense, and genuine philosophy, than all
those reveries. Let any of these philosophers solve those important ques-
tions found in the book of Job, or shew that they import any thing contrary
to the true system of nature. " Hast thou considered the breadth of the earth ?
Tell me if thou knoivest all things ? Where is the ~vay where light divelleth, and
-yhere is the place of darkness? (ch. 38, v. 18, 19). It cannot be here the
500 OF THE SOLAR SYSTEM.
contrary to the laws of nature for a heavy body to revolve round a
lighter one, as its centre of motion ; for the lighter one must be
at a greater distance from the common centre of gravity, and must
have a greater velocity to counterbalance the attraction of the
other. Now the sun is found, from observation, not only to ex-
ceed the earth in magnitude, but so far to exceed the magnitudes
df all the planets in the solar system, that the common centre of
gravity of the whole is almost constantly within its body, so that
its motion round the common centre of gravity of the whole sys-
tem, is scarcely perceptible to the nicest observers. The earth,
therefore, and all the planets, must revolve round the sun.
The analogy of the earth with the other planets (as Laplace re-
marks, Astr. B. 2, ch. 3) confirms the hypothesis of its annual
revolution. Like Jupiter it revolves on its axis, and is accompa-
nied by a satellite. An observer on the surface of Jupiter would
conclude that the solar system was in motion round him, and the
magnitude of that planet would render this illusion less improba-
ble than for the earth. Is it not, therefore, reasonable to suppose,
that the revolution of the solar system round us, is likewise only
an illusion ? Let us examine the phenomena of the earth and the
darkness produced by the absence of the sun, or the light caused by his
presence, that is meant, for then the question would be too trifling-, being-
proposed by God himself. It is in this book that we find this remarkable sen-
tence, " He (God) stretched out the north over the empty space, and hang-
eth the earth ripon nothing" (ch. 26, v. 7.) What more philosophical than
the latter part of this sentence ? We also find in it, speaking of the wicked,
this no less remarkable sentence : " He shatt drive him out of light into dark-
ness, and shall remove him out of the -world" (ch. 18, v. 18.) We find the
same idea, expressed on a similar occasion, in St. Matthew. " Jlnd the un-
profitable servant cast ye out into the exterior darkness. There shall be -weeping
and gnashing of teeth" (ch. 25, v. 30. see also ch. 8, v. 12. ch. 22, v. 13.)
Until philosophers point out that this exterior darkness, &c. has got no ex-
istence until they bring 1 us intimately acquainted with the extremes of
nature, their arguments against the sacred writings, drawn from their
knowledge of a little corner or point of the universe, must not only be in-
conclusive, but ridiculous and vain. It is with a view of shewing the folly
of shallow philosophers, who pass the bounds of their knowledge to attack
incontestible truths, that the above remarks have been made to shew
how little we know as yet of the system of the universe, and the design of
the Author of this sublime structure, in its formation and to induce those
who are in possession of the most sublime philosophy that man can learn,
I mean the Christian religion, to appreciate that sacred treasure, and de-
spise those vain systems that have no other support but the imaginations of
their authors.
The remarks that have been made on the system of the universe are not
novel. They are deductions strictly drawn from Newton's Philosophy.
And either this Philosophy, now universally received, must be false, or the
general conclusions cannot be denied. The remarks are therefore princi-
pally calculated for those who are versed in the principles of this Philoso-
phy, as a superficial view of the system of nature, may produce notions
hypoth
or strict calculation.
OF THE SOLAR SYSTEM. 301
planets from the sun's surface. All these bodies will appear to
move from west to east; this identity, therefore, indicates a mo-
tion of the earth ; but that which proves it, evidently is the law
which exists between the times of the revolutions of the planets
and their distances from the sun. They perform their motions
round it slower in proportion as their distances are greater, and in
such a manner, that the squares of the periodic times are firofior-
tional to the cubes of their mean distances (Principia, Phenomenon
5, B 3.) From this remarkable law, the length of a revolution
of the earth, supposing it in motion round the sun, to correspond
with the earth's distance, should be exactly a sidereal year, as is
really the case ; this, therefore, is an incontestible proof that the
earth moves like the other planets, and is subject to the same
laws. Another argument, still more incontestible, is the follow-
ing, that the force of gravity, which balances the centrifugal force
in the other planets, and retains them in their respective orbits,
should likewise act on the earth ; and that the earth must, there-
fore, oppose to this action the same centrifugal force. Hence the
consideration of the celestial motions, as observed from the sun,
leaves no doubt of the real motion of the earth.
An observer on the surface of the earth, has another evident
proof in the phenomenon of the aberration (Laplace, B. 2, ch 3)
which is a necessary consequence of it. Roemer, about the end
of the 1 7th century, observed that the eclipses ef the satellites of
Jupiter happened sooner about the oppositions of this planet, and
later towards the conjunctions ; this led him to conjecture, that
light was not transmitted instantaneously from those bodies to the
earth, but took a perceptible interval of time to traverse the di-
ameter of the sun's orbit. Now, Jupiter being nearer to us in his
oppositions than conjunctions, by a distance equal to the sun's
orbit, the eclipses ought therefore to happen to us sooner in the
first case than in the latter, by the time which the light takes to
traverse the sun's (or rather the earth's) orbit ; and the retardation
of these eclipses so exactly correspond to this law, that it is im-
possible to refuse assent to it. It is therefore found, that light
takes about 8' 7 "5 in passing from the sun to the earth, at its
mean distance.
A star near the constellation Draco, that passed near the zenith,
was observed by Messrs. Molyneux, Bradley, and Graham, with an
instrument contrived by the latter, with a view of discovering its
parallax. They soon discovered that the star did not always ap-
pear in the same place in the instrument, but that its distance from
the zenith varied, and that the difference of its apparent places
amounted to 21" or 22". This star was y draconis, near the pole
of the ecliptic. They made similar observations on oiher stars,
and found a like apparent motion in them, proportional to the la-
titude of the star. This motion was by no means such as could
result as the effect of a parallax ; and it was some time before
they could discover any method of accounting for this new and
302 OF THE SOLAR SYSTEM.
strange phenomenon ; but Dr. Bradley, at length, resolved
variety in a satisfactory manner, by the motion of light and the
annual motion of the earth compounded together. For as the
earth describes 59' 8" of her orbit in a day = 3548", and that
light comes from the sun to us in 8' 7 "5, we have this proportion,
24 hours or 86400" : 8 ; 7"5 or 487"5 : : 3548" : 20" very near,
the aberration of light or the change in the star's place ; and this
is what Dr. Bradley has made it. And hence it affords as sensi-
ble a demonstration of the motion of the earth round the sun, as
the increase of degrees and the force of gravity in passing from
the equator to the poles, afford of the revolution of the earth on
its axis. We shall give the principles of the aberration of light
more at large when we come to treat of the fixed stars.
It is objected against the annual motion of the earth, in its orbit
round the sun, that if it really had such a motion, the annual pa-
rallax of the stars, or the angle under which the diameter of the
earth's orbit would appear, as seen from a fixed star, should make
a considerable difference in the position of the star observed at two
different times ; but it is never found to make the least difference,
though observed with our nicest instruments. To understand this
more clearly, as the axis of the earth keeps always parallel to it-
self, it would follow, that if it pointed to any star at one time of
the year, in six months after it ought to point to another, distant
from the former by the angle under which the whole diameter of
the earth's orbit appears from the star ; but it is found not to de-
viate a single second from its former position. Now this objec-
tion, the most forcible that has been brought against the earth's
motion, vanishes when we come to consider the immense distance
of the fixed stars, of which we may form an idea thus : if we
should suppose the distance between us and a fixed star to be di-
vided into 1000 equal parts, and that a spectator, after having
passed over 999 of those parts, should view it from the last divi-
sion, or at T <yVo P art f tne whole distance, it would not appear
larger than to the naked eye ; because a telescope that magnifies
1000 times, though it will render it brighter, will not sensibly
magnify its diameter. Herschel's 20 feet telescope magnified
460 times, and his 40 feet telescope magnified some thousand
times ; and Herschel confirmed the above assertion. Hence the
immense distance of the fixed stars, indicated by these observa-
tions, so far from being an objection against the earth's annual
motion, rather confirms it.
The earth's annual motion round the sun being thus established,
we shall now shew how the quantity of this motion is estimated.
We know, as the earth regards the sun as the centre of its mo-
tion, that in whatever part of the heavens the earth actually is at
any time, the sun must be directly in the opposite point at the
same lime ; and that therefore, if the sun's place be observed in
the heavens, the opposite point is the place of the earth ; now as
the earth advances round the sun 5 the sun will seem to perform a
OF THE SOLAR SYSTEM. 303
similar motion in the heavens, and hence, if we compute this ap-
parent motion of the sun among the fixed stars, it will give the
earth's real motion round the sun.
From comparing the sun's right ascension every day, with the
right ascensions of the fixed stars lying to the east and west, the
sun is found constantly to recede from those on the west, and ap-
proach those on the east ; its apparent annual motion is therefore
found to be from west to east, and as the earth performs a similar
motion in an opposite part of the heavens, the earth's real motion
must also be from west to east. The interval of time from the
sun's leaving any fixed star, until its return to the same star again,
is called a sidereal year, being the time in which the sun com-
pletes its apparent revolution among the fixed stars, or in the
ecliptic. But the sun, after it leaves either of the equinoctial
points, returns to it again sooner than it returns to the same fixed
star ; and this interval is called a solar or tropical year, because the
time of the sun's leaving one equinox until its return to it again,
is equal to the time from its leaving one tropic until its return
again. This solar or tropical year, is that on which the return of
the seasons depends.
To find the length of a sidereal year. On any clay when the
sun passes the meridian, take the difference between its right as-
cension, when on the meridian, and that of a fixed star ; call this
difference (a) and when the sun returns to the same part of the
heavens the next year, compare its right ascension with that of
the same star, for two days, that is, when their difference ( d) of
right ascension is less, and also when greater (e) than the differ-
ence (a) before observed ; then e d is the increase of the sun's
right ascension in the time t ; and as the increase of right ascen-
sion may be considered uniform for a small time, we have e d
: a d : : t : T the time in which the sun's right ascension is
increased from the sun's place, when the difference d of his right
ascension, and that of the star, was less than a. This time T
being therefore added to the time of the observed right ascension
of the sun, when the quantity d was found, will give the time
when the sun is at the same distance a from the star, as when ob-
served the former year ; and the interval of these times is there-
fore a sidereal year. About March 25, June 20, September 1 7, and
December 20, is the best time for these observations, as the sun's
motion in right ascension is then uniform.
If, instead of repeating the second observations the following
year, there be an interval of several years, and if the observed
interval of time, when the difference of the right ascensions of
the sun and star was found to be equal, be divided by the number
of years, the length of the sidereal year will be given more ex-
actly.
On April 1st, 1669, at Oh. S' 47" mean solar time, M. Picard
observed the difference of longitude between the sun and Procyon
to be 3s, 8 59' 36", which is the most ancient observation of this
304 OF THE SOLAR SYSTEM.
kind, the accuracy of which can be depended on. (See Hist. Ce-
leste, par M. le Monmer, p, 37.) On April 2d, 1745, M de la
Caillc found, by taking their difference of longitude on the 2d and
Sd, that at llh. 10 ; 45", mean solar time, the difference of their
longitudes was the same as at the first observation. Now as the
sun's revolution is nearly 365 days, it is manifest that it made 76
complete revolutions, in respect to the same fixed star, in the
interval between the two observations, or in 76 years, Id. 1 lh.
6 ; 58". In these 76 years, there were 58 of 365 days, and 18
bissextiles of 366 days each ; hence in the interval, there were
27759 d. 1 lh.. 6' 58", which being divided by 76, the quotient is
365 d 6h. 8 ; 47", the length of a sidereal year. According to
Laplace, the length of a sidereal year is 365 25638-i days, or
365 d. 6h. 9 ; ll"5776, which is the most accurate, being com-
puted from the best observations.
To find the length of the Solar or Tropical Year. Take the me-
ridian altitude a of the sun, on any day when it is nearest to the
equinox ; then the following year, let its meridian altitude be taken
on two days, as follow : one when its altitude m is less than a,
and next when its altitude n is greater than a ; then n m is the
increase of the sun's declination in 24 hours. Also when the de-
clination has increased by the quantity a m, from the time when
the meridian alt. m was observed, the declination will then be-
come a ; and as the increase of the declination may be considered
as uniform for one day, we have this proportion, n m : a m
: : 24h. : the interval from the time the sun was on the meridian
on the first of the two days, until the sun has the same declina-
tion a, as at the observation the foregoing year ; hence this time
being added to the time when the sun's altitude m was observed*
will give the time when the sun's place in the ecliptic had the
same situation, in respect to the equinoctial points, which it had
at the time of observation the preceding year ; the interval of
these times is the length of a tropical year.
If, as in the method for determining the sidereal year, there be
an interval of several years between the observations, and that the
interval between the times when the declination was found to be'
the same, be divided by the number of years, the length of the
tropical year will be obtained more exactly.
On the 20th of March, 1672, M. Cassini, the father, observed
the meridian alt. of the sun's upper limb to be 41 43', at the
Royal Observatory at Paris ; and on March 20, 1716, M Castim,
his son, observed the mer. alt. of the upper limb to be 41 27' iO",
and on the 2 1st, to be 41 5 1' ; the difference of these two latter
altitudes was then 23' 50", and of the two former 15' 50" ; hence
23' 50" : 15' 50" : : 24h ; : 15h. 56' 39"; therefore, on March
20, 716, the sun's decimation, at I5h. 56' 39", was the same as
on March 20, 1672. Now the interval between these two obser-
vations was 44- years, of which 36 consisted of 365 days, and IO
of 366 each, that is in all, 16070 days ; hence the whole interval
OF THE SOLAR SYSTEM. 605
between the equal declinations was 16070d. I5h. 56' 39", which
divided by 44, gives 365 d. 5h. 49' 0" 53'", the length of a tro-
pical year, from these observations. The length of a tropical
year, or the return of the sun to the same equinox, from the best
observations, as given by Laplace in his Astronomy, is 365-242222
or 365 d. 5h. 48' 47"98. Hence the sidereal year exceeds the
tropical by 0.014162 of a day, The equinoxes have therefore a
retrograde motion in the ecliptic, or in a direction contrary to that
of the sun, by which they describe every year an arc equal to the
mean motion of the sun in the interval of 0.014162 of a day ;
hence 1 d. : 0.014162 d. : : 59' 8"3 : 50" 151, the precession of
the equinoxes.
The precession of the equinoxes being given, and also the length
of a tropical year, the length of a sidereal may be easily found, as
shewn in the note to prob 42, part 3d.
There is another year, called by astronomers the anomalistic
year, and is the time from the sun's leaving his apogee until his
return to it again. Now the progressive motion of the apogee in
a year, according to Vince, is 1 1"75, and hence the anomalistic
must exceed the sidereal year by the time the sun takes in moving
over 1 1 "75 of longitude at its apogee; but the sun's motion in
longitude, when in its apogee, is 58' 13" in 24 hours; hence
58' 13" : 1 1"75 : : 24h. : 4' 50"63S4, which added to 365 d. 6h.
9' 11"5776, gives 365 d. 6h. 14'2"216, the length of the anom-
alistic year. The motion of the apogee here given, is that deter-
mined by M. de la Lande, from the observations of M. de la Hire^
and those of Dr. Maskelyne, agreeing also with Cassini's determi-
nation. M. Laplace makes the sidereal and secular motion of the
earth's perihelion (367 l"63) 19' 49" 60812, which gives 11"896
yearly Delambre makes the mot. of the apog. in a year 62" which
includes the precession, cc. Mayer makes it 66",
The longitude of the earth's perihelion at the beginning of
1750, according to Laplace, was (309o 579') 278 37' 15"96. The
mean longitude of the earth, reckoning from the mean vernal
equinox at the epoch of the 31st December, 1749, at noon, mean
time at Paris, was (31 IQ 12 18) 280 0' 34/'632.
In accounting for the cause of the planets' revolutions round
the sun, philosophers had recourse to various hypotheses. The
ancients invented their solid orbs, and Descartes vortices ; but
both were imaginary fictions, void of proof. Newton was the first
that built his explanations on actual experiment and observation,
and fully investigated the laws of motion resulting from the grav-
itation of matter. He seemed possessed of all that could qualify
him for this arduous task ; and the innumerable mathematical the^
orems and inventions which he discovered in his inquiries will, pro-
bably for ever, remain the greatest monument of human ingenuity.
The substance of Newton's discoveries, relative to the cause of
the planets' motions, we shall give here, principally collected from
Cote's preface to Motte's translation of the Principia. That we
306 OF THE SOLAR SYSTEM.
may begin our reasoning from what is most simple and nearest to
us, let us first consider what is the nature of gravity with us on
the earth. All agree, that every circumterrestrial body gravitates
towards the earth ; that no bodies really light are to be found, as
experience shews ; that those bodies which are relatively light,
are not really so, but apparent only, and arising from the prepon-
derating gravity of the contiguous bodies, or the fluids in which
they are immersed ; that ail bodies gravitate towards the earth,
and the earth in like manner towards bodies ;* that the action of
gravity is mutual and equal on both sides ; that the weights of bo-
dies, at equal distances from the centre of the earth, are as the
quantities of matter in the bodies ;f this is proved from the equal
acceleration of all bodies that fall from a state of rest by the force
of their weights, the resistance of the air being taken away ; and
this is yet more accurately proved from the doctrine of pendu-
lums ; that the attractive forces of bodies, at equal distances, are
as the quantities of matter in the bodies ;| and that therefore the
attractive force of the entire bodies arises from, and is compound-
ed of, the attractive forces of the^parts, so that terrestrial bodies
must attract each other mutually, with absolute forces, that are as
the matter attracting. This being the nature of gravity on the
earth, the following will shew what its nature is in the heavens.
Every body perseveres in its state, either of rest or of uniform-
ly moving in a right line, unless it is compelled to change that
state by other forces impressed ;l( and hence it follows, that bodies
that move in curve lines, and therefore continually deflect from
the right lines that are tangents to their orbits, are, by some con-
tinued force, retained in those curve lined paths. Now, as the
planets move in curve lined orbits, there must be some force ope-
rating, by whose repeated actions they are perpetually made to de-
flect from the tangents.
It is a mathematical principle, the demonstration of which we
shall give when we come to treat of the doctrine of centripetal
forces, that all bodies that move in a curve line described in a
plane, and which by a radius drawn to any point, whether quies-
cent, or any how moved, describe areas about that point propor-
tional to the times, are urged by forces directed towards that
point.U Since then all astronomers agree, that the primary plan-
ets describe about the sun, and the secondary planets about their
respective primaries, areas proportional to the times, it follows,
that the forces by which they are deflected from the rectilinear
tangents, and made to revolve in curve lined orbits, are directed
towards the bodies that are situated in the centres of the orbits,
* Principa Scholium after the laws.
I Principia, P.. 3, prop. 6.
t Principia, B. 1, prop. 69, cor. 3, and prop. 7,.B. ".
$ Principia, cor. 1, prop. 7j 15. 3.
y Principia, Law 1, B. 1.
*[ Principia, B. 1, prop. 2.
OF THE SOLAR SYSTEM. 307
This force may therefore not improperly be termed centripetal in
respect of the revolving body, and in respect of the central body
attractive , whatever cause it may be imagined to arise from.
The following is also mathematically true ; that is, if several
bodies revolve with an equable motion in concentric circles, and
the squares of the periodic times be as the cubes of the distances
from the common centre ; the centrifugal forces will be recipro-
cally as the squares of the distances.* Or if bodies revolve in or-
bits that are nearly circles, and the apsides of the orbits rest, the
centripetal forces of the revolving bodies will be reciprocally as
the squares of the distances ; and both these cases hold in all the
planets, as observations fully testify.
From what has been hitherto said, it is evident that the planets
are retained in their orbits by some force perpetually acting upon
them ; that that force is always directed towards the centre of
their orbits ; that its efficacy is augmented in proportion as the
centre is approached, and diminished as its distance increases from
the centre ; and that it is augmented in the same proportion as the
square of the distance is diminished, and diminished in the same
proportion as the square of the distance is augmented. Now, by
making a comparison of the centripetal forces of the planets and
the force of gravity, we shall find that they are, in effect, of the
same kind ; for to have them of the same kind, it is only neces-
sary that both observe the same laws, &c
Let us therefore first consider the centripetal force of the moon,
being nearest to us. The rectilinear spaces which bodies, let fall
from rest, describe in a given time at the very beginning of the
motion, when the bodies are urged by any forces whatsoever, are
proportional to the forces (as will be shewn after the solar sys-
tem .f) Hence the centripetal force of the moon, revolving in its
orbit, is to the force of gravity at the earth's surface, as the space
which, in a very small particle of time, the moon, deprived of all
its centrifugal force, and descending by its centripetal force to-
wards the earth, would describe, to the space which a heavy
body would describe, when falling by the force of its gravity near
the earth, in the same given particle of time. The first of these
spaces is equal to the versed sine of the arch described by the
moon in the same time,! because that versed sine measures the
translation of the moon from the tangent, produced by the centri-
petal force, and may therefore be computed, the periodic time of
the moon, and its distance from the centre of the earth being
given. The last space is found by experiments of pendulums,
* Principia, B. 1, prop. 4, cor. 6, and B. 3, prop. 2, also cor. 1, prop.
t--)j 15* !
f Principia, B. 1, Lemma 10, cor. 3.
* Principia, B. 1, sect. 2, prop. 1, cor. 4.
The mean distance of the earth from the moon being taken =
^38533 miles (see the note p- 250 ; ) hence, the moon's orbit being- nearly
circular, its diameter is 477066 miles, and its circumference 477066 x
308 OF THE SOLAR SYSTEM.
as shewn by Mr. Hugens* Therefore by making a calculation,
we shall find that the first space is to the latter, or the centripetal
force of the moon revolving in her orbit, to the force of gravity at
the surface of the earth, as the square of the semidiameter of the
earth, to the square of the semidiameter of the orbit. But by what
has been shewn before, the very same ratio holds between the
centripetal force of the moon, revolving in its orbit, and the cen-
tripetal force of the moon near the earth's surface. Therefore,
3.1416 = 1498750.5456. Now the moon's sidereal revolution is 27 d. 7h.
43' 11" = 236059F; hence this proportion 2360591" : 1' or 60" : : 1498750.
5456 miles : 38.0942 miles the moon will describe in V, which is nearly
equal to the tangent of 1' to her orbit. But as the secant of the same arc
of 1' less the radius, gives her distance fallen towards the earth in her
orbit by the power of gravity in 1', we have rad. 2 or 2385332 X tang.z or
38.0942* = secants = 56837993540.16807364, the square root of which
is 238533.0030418 miles nearly, from which the semidiameter of the orbit
being subtracted, and the remainder .0030418 being multiplied by 5280
(the feet in a mile) gives 16.060704 feet, the moon's descent in 1' in her
orbit towards the earth, by the force of gravity. This may be also cal-
culated from Cor. 9, prop. 4, B. 1, Principia. If we now divide the
moon's distance from the earth 238533 miles, by 3956 miles, the earth's
semidiameter, there will result 60.29 nearly, the moon's distance in semidi-
ameters of the earth. Now as the gravity of the moon increases in propor-
tion as the square of her distance from the centre of the earth decreases
(Emerson's Tracts, prop. 13, p. 23;) hence her gravity at the surface of
the earth would be 60.29 X 60.29 = 3634.8841 times greater, and there-
fore at the earth's surface the moon would fall towards the earth, or de-
flect from its tangent 3634.8841 X 16.06 = 58376.238646 feet in 1'; hence
as the spaces described by falling bodies are as the squares of the times
of falling (Emerson's Tracts, prop. 13, p. 23) the same power would
carry the moon 60 X 60 = 3600 times less space in 1" than in 1', and
therefore 58376.238646 -r- 3600 = 16.215 feet, the space the moon would
fall in 1" at the earth's surface. This might also be computed from the
equal description of areas, whatever be the distance of the body. Now
to compare this with the gravity of terrestrial bodies, found by the pen-
dulum: by a very accurate experiment, Borda has found that the length
of pendulums vibrating- seconds, at the Observatory at Paris, and reduced
to a vacuum, is 0.74 I88f metres, or 29.208833077 English or American
inches. The seconds here used by Borda must be those adopted in the
French measures ; and as they divided the day into 10 hours, the hour
into 100', and the minute into IOC" ; hence the number in the day, French
measure, is 10000C", but in our division of the day it is 8640C", conse-
quently 7 86400 : 100000 or 108 : 125 : : 1* (English or American) : 1"1574
Paris nearly. Now as the lengths of pendulums, describing similar arch-
es, are as the squares of the times of vibration (Emerson's Tracts, prop.
25) we have this proportion; 1": 1"1574* or V 33957476 : : 29.208833077
inches : 39.127315559 inches nearly, the length of a pendulum vibrating
seconds, in our measures, in the latitude of Paris, half of which is
19.5636577795 inches. And as the square of the diameter of a circle : the
square of its circumference : : half the length of a pendulum : the space
described by a falling body in the time of one vibration (Emerson's
Tracts, prop.' 24, cor. 5) hence this proportion ; 12 : 3.14162 or 9.86965056
: : 19.5636, &c. : 193.086465959, &c. inches = 16.09 feet nearly, the space
* Newton's Principia, B. 3, prop. 4
OF THE SOLAR SYSTEM. 309
the centripetal force near the surface of the earth, is equal to the
force of gravity ; and hence these two forces are identically the
same. For if they were different, these forces united would cause
bodies to descend to the earth with twice the velocity produced by
the force of gravity alone. Hence it is evident, that the force
which retains the moon in its orbit, is the force of terrestrial grav-
ity extending to it. And it is reasonable to suppose that this vir-
tue should extend to vast distances, as we find no sensible diminu-
tion of it on the tops of the highest mountains. Now as the moon
gravitates towards the earth, so the earth, on the other hand,
gravitates towards the moon, which is also confirmed from the
phenomena of the tides arid the precession of the equinoxes ;
which arise from the actions of the sun and moon on the earth.
Hence also we discover by what law the force of gravity decreases
at great distances from the earth : for as gravity does not differ
from the moon's centripetal force . and that this is reciprocally
proportional to the squares of the distances ; it follows, that it is in
that same ratio the force of gravity decreases. We do not con-
sider here the small deviations arising from the actions of the sun
and planets.
In like manner the same reasoning may be applied to the pri-
mary planets. The revolutions of the primary planets round the
sun, and of the secondary planets round their respective primaries,
are phenomena of the same kind with the revolution of the moon
round the earth ; and as it has been found that the centripetal
forces of the primary planets are directed towards the centre of
the sun, and those of the secondaries to their respective primaries,
that bodies fall In a second on the earth's surface, differing but little
from that calculated from the moon's motion. Owing to the earth's cen-
trifugal force, the gravity of bodies is diminished at the earth's surface,
in advancing from the poles to the equator, nearly as the versed sine of
double the lat. (Newton's Principia, prop. 20, B. 3) and the moon's grav-
ity is also diminished by the sun's action. Laplace gives this diminution
__!__ part. B. 4, ch. I. Newton makes this something 1 different. Prop. 3,
li. 3, Principia. When these quantities are allowed, the forces will come
out very nearly equal ; and hence the force of gravity on the earth's sur-
face, is the same force which retains the moon in her orbit.
dj^ \Ve must remark here, that the division of the day above men-
tioned, is taken from p. 162, of Laplace's Astronomy, vol. 1, as translated
by J. Pond / and that this must be the division used by Itorda, as no other
would correspond to the length of this pendulum. This circumstance I
have discovered in making the above calculations ; and hence, wherever
Laplace makes use of seconds of time, it must be the above division that
he adopts, though he no where' mentions it. This is very necessary to be
known by those who raake use of Laplace's works, as his translator, J.
Pond, takes no notice of it- He makes the seconds of time the same as
the seconds of a degree, according to the division of the quadrant adopt-
ed in France, which, besides the other errors in reducing- the French
measures, is a source of error, in the translation, through the whole
.vork. See p. 173, vol. 1. of the translation, ?tc.
310 OF THE SOLAR SYSTEM.
in the same manner as that of the moon is directed towards the
earth ; and that moreover all these forces are reciprocally propor-
tional to the squares of the distances from the respective centres,
as we have shewn to be the case with the moon ; we must there-
fore conclude, that the nature of all these forces is the same.
Therefore, as the moon gravitates towards the earth, and the earth
again towards the moon, so also the secondary planets gravi-
tate towards their primaries, and the primary planets again towards
their secondaries ; and in like manner, the primary planets to-
wards the sun, and the sun again towards the primary planets.
Hence the action of gravity is mutual between the sun and all the
planets ; for the secondary planets, while they accompany the pri-
mary, revolve at the same time with the primary round the sun.
And Newton further confirms this general gravitation of matter
from the inequalities of the moon, &c. the theory of which is
clearly explained in the 3d Book of his Principia Hence also we
conclude, from analogy, that the gravitation of matter is universal)
and that therefore the whole solar system gravitates towards the
fixed stars, and the fixed stars towards the solar system. The
motions of the comets evidently shew, that the action of the sun,
or its attractive virtue, is propagated on all sides to prodigious dis-
tances ; for from the discoveries of the penetrating Newton, it is
now evident, that the comets describe conic sections round the
sun, having their foci in the sun's centre, and by radii drawn to
the sun, describe areas proportional to the times ; and also that
the forces by which they are retained in their orbits, respect the
sun, and are reciprocally proportional to the squares of the distan -
ces from his centre.*
The foregoing conclusions are grounded on this axiom or rule,
laid down by Newton in the beginning of the 3d Book of his Prin-
ciples, and now received by all philosophers, viz. that " to the.
same natural effects we must, as far as possible, assign the same
natural causes." For no one can doubt, if gravity be the cause of
the descent of a stone in Europe, that it is also the cause of the
like descent in America. If in Europe the attraction of the earth
be propagated to all kinds of bodies, and to great distances, can
any one doubt that the same happens in America or in China, &c. ?
It this rule were not admitted, then nothing could be affirmed of
the properties of bodies in general. The nature of particular
things being known from observations and experiments, from
these, as from certain data, \ve judge of the nature of such bodies
in general. And hence, as we find that all bodies, whether on the
earth or in the heavens, are heavy, as far as we can make any
experiments or observations on them, we must therefore allow,
that gravity is found in all bodies universally. In like manner all
bodies, that come under our observations, are extended, movea*
* Principia, B.3, prop. 40, and corollaries.
OF THE SOLAR SYSTEM. 311
ble and impenetrable ; and from thence we conclude, that all bo-
dies, even those we have made no observations on, are extended,
moveable, and impenetrable. In like manner all bodies, that we
have made any observations on, are found to be heavy ; hence we
conclude, that weight is a universal property of all bodies in gen-
eral. Hence the gravity or weight of the fixed stars can no more
be denied, though not as yet precisely observed, than their exten-
sion, mobility, or impenetrability, which no one will deny, though
these qualities are no less out of the reach of observation.
It is thus that, from strict analogy, we can apply the knowledge
which we obtain of bodies from experiments and observations on
the earth, to those that we can have no such access to ; and as
Newton remarks (Rule 4th, B. 3, Prin.) " In experimental phi-
losophy, we are to look upon propositions, collected by general
induction from phenomena, as accurately or very nearly true, not-
withstanding any contrary hypotheses that may be imagined, till
such time as other phenomena occur, by which they may be either
made more accurate, or liable to exceptions/*
As the earth performs its motion round the sun in an orbit
which is not circular but elliptical, having the sun in one of the
foci, it follows, that the earth must at some times approach near-
er to the sun than at others, and will therefore take more time in
describing that half of its elliptic orbit, in whose focus the sun is,
than the other, in consequence of that general law, first observed
by Kepler, that is its describing equal areas round the sun in equal
times. It is in our winter that the earth is in that part of its orbit
where its velocity is greatest ; and hence astronomers observe,
that the earth is more rapid in the winter half of its orbit, than in
the summer by about 7 or 8 days. It follows also, that in the win-
ter we are nearer the sun than in summer, although in winter the
season is colder and more inclement ; but this phenomenon is ea-
sily explained from the sun's rays falling more perpendicularly on
us in summer than in winter,* from their acting on the same
place a longer time,f the days being longer in summer than in
winter, and from their passing through a more dense and exten-
sive part of the atmosphere. That the sun is actually nearer to
us in winter than in summer, is also proved from the increase of
his apparent diameter in this season,^: as observed by all astrono-
mers. The unequal motion of the earth in its orbit, will be more
fully explained afterwards.
* Thus the heat in the torrid zone, is not caused from that part of the
earth being nearer the sun, but from the sun's rays being darted perpen-
dicularly on it, and through a comparatively small portion of the atmos-
phere.
f In the northern regions, the accumulation of the sun's heat is so
great during their short summer, that it is sufficient for vegetation, &c.
which takes place in these inclement regions, much quicker than in more
emperate latitudes.
t See the table, p. 155.
312 OF THE SOLAR SYSTEM.
The earth's axis makes an angle of 23 2S ; with a perpendicular
to the plane of the ecliptic, or its orbit, and keeps always<s.he same
oblique direction during its annual course ; the north pole is there-
fore turned towards the sun during one part of the earth's revolu-
tion, and the south pole is turned towards it, in like manner, dur-
ing the other ; and this is the cause of the different seasons, as
sfiring, summer, autumn^ and winter. If a small ball of wood, or
any other substance, be procured, having the ecliptic, the equator,
the tropics, polar circles, and a few meridians, delineated on it,
and also a small wire passing through the poles of the equator ;
if this bail be carried round a lighted candle placed on a table, ei-
ther in a circle, or in the curve of an ellipsis, having the candle
placed in one of the foci, and the axis of the earth, during the
motion, be always kept parallel to itself, the enlightened part of
the earth will exhibit the different seasons in a pleasing and satis-
factory manner. If the 12 signs be delineated on the ellipsis or
circle, a line drawn from the ball through the candle will point
out, on the opposite side of the curve, the sun's place, corres-
ponding to that of the earth, pointed out by the ball.*
We shall now give the theory of the earth's motion^ or rather
the theory of the planets' motions in general, in elliptic orbits,
about their common focus. This has been given by various au-
thors, as Sir Isaac Newton, in his Principia, Dr. D Gregory,
Keil, and others, in their respective treatises on astronomy ; but
the most concise is that given by Vince in his Astr. from which
we shall collect most of what we shall give on this subject.
As the orbits, which are described by the primary planets re-
volving round the sun, are ellipses, having the sun in one of the
foci, and as each planet describes about the sun equal areas in
equal times, it is from these principles that we shall deduce such
consequences, as will be found necessary in our inquiries respect-
ing their motions From the variation in the planets' distances
from the sun, and their describing equal areas in equal times, it
is evident that they must move with unequal angular velocities
round the sun. The principal proposition, therefore, on which
the planets' theory depends, is the following. The fieriodic time
of a jilanct* the time of its motion from its ajihelion, and the eccen-
tricity of its orbit being given ; to Jind its angular distance from
its afihelion, or its true anomaly, and its distance from the sun.
This problem was first proposed by Kepler, and has therefore ob-
tained the name of Kepler's Problem. (See Keil's Astr. Lect. 23
and 24.) Kepler knew no direct method of solving the problem,
and therefore performed it by long and tedious trials.
i
* See part 2d, where this subject is fully elucidated on the globes,
OF THE SOLAR SYSTEM.
313
N
Let AEB be the ellipse described by
the planet round the sun at S in one of
its foci, AB the greater axis, EC half
the lesser axis, A the aphelion, B the
perihelion, P the planets' place, AVB
a circle, C its centre. Let NPI be
drawn perp. to AB, join PS, NS and
NC, on which produced let fall the
perp. ST. Now let a body be supposed
to move uniformly in a circle from A
to Q with the mean angular velocity of
the body in the ellipsis, whilst the bo-
dy moves in the ellipse from A to P ;
then the angle ACQ is the mean, and
the angle ASP the true anomaly ; the
difference of these two angles is called
the equation of the planets' centre, or firostafiheresis. Let fi ==
the periodic time in the ellipse or circle (the periodic time being
equal in both by supposition) and t = the time of describing AP or
AQ ; then, as the bodies in the ellipse and circle describe equal
areas in equal times about S and C respectively, we have
area AQC : area of the circle : : t : /*, and
area of the ellipse : area ASP : : ft : t ; also
area of circ. : area of ellip. : : area ASN : ASP ;*
hence area AQC : area ASP : : area ASN : area ASP ; therefore,
area AQC = area ASN ; take away the area SNC which is com-
mon to both, and the area^ QCN = SNC ; but QCN = 1QN X
CN ; therefore ST = QN. Now if t be given, the arc AQ will
be given ; for as the body in the circle moves uniformly, it will be
ft : t : : 360 : AQ. In this manner, the mean anomaly for any
given time may be found, the time when the planet was in the
aphelion being given ; and therefore, if ST or NQ be found, the
/_ NCA, which is called the eccentric anomaly, will be given, from
whence, by one proportion, as we shall presently shew, the L. ASP
the true anomaly will be given. The prob. is therefore reduced to
this ; to find a triangle CST such, that the L C -f- the degrees of
an arc = ST may be equal to the given L ACD. M. de la Caillc^
in his Astronomy, gives an expeditious method of performing
this by trial, as follows : Find the arc of the circumference of
the circle AQB that is equal to CA, by saying as 3.1416 : 1 ::
180 : 57 17' 44-"8 = the number of degrees in an arc = CA ;
hence CA : CS : : 57 17' 44"8 : the degrees of an arc = CS,
Now assume the - SCT, multiply its sine into the degrees in CS,
and to the product add the JL SCT, and if the sum be equal to the
given angle ACQ, the supposition was right ; if not, add or sub-
tract the difference to or from the first supposition, according a**
Yince's Conic Sect. 2d cd. prop. 7 of the ellip. cor. 3 and 4.
Qq
314 OF THE SOLAR SYSTEM.
the result is less or greater than ACQ ; this operation being re-
peated, a few trials will give the accurate value of SCT. The
degrees in ST may be most readily obtained by adding the loga-
rithm of C{5 to the log. of the sine of the angle SCT, and lessen-
ing the index by 10, the remainder will be the log. of the degrees
in ST. Having thus found the value of the arc AN, or the angle
ACN, we shall now show how to find the angle ASP.
Let v be the other focus, and let AC = 1 ; then SP 2 Pi* =
fcS* -f 2 vS X -vl (Eucl. 1 2 prop 2 B ) = (uS + 2 vl) X i>S =
(2 Cv + 2 vl) X 2 SC a= 2 CI X 2 SC ; hence SP -f Pv : 2 CI : :
2 SC : SP Pv (because the difference of the squares of two
quantities, is equal to the rectangle of their sum and difference)
or 2 : 2 CI : : 2 SC : SP ^- 2 SP,* or dividing by 2, 1 : CI : :
SC : SP 1 ; hence SP == i -f CS x CI = 1 -f CS x cos.
ACN. But ~' ^ tang } ASp2 fsee Vince s Trig>
I *-p cos. ASP
art. 94) also SP, or 1 -f CS x cos. ACN : rad. = 1 : : SI, or CS
-f CI or CS -f cos. ACN : cos. ASP (Vince's Trig. art. 125) ==
CS -{- cos. ACN , 1 cos. ASP ^
1 + CSX cos. ACN' Hence tang, i ASP* (- 1 + cos . AS p^=
1-fCSxcos.ACN CS cos.A^N 1 CS+cos. ACNxCS i
l-fCSxcos.ACN+CS-f-cos.ACN! -f CS+cos. ACN+GS+T
SB cos ACN x SB 1 cos ACN SB
X ^
SA + cos.ACNxSA l + coa.ACN
O T> 1 |
X -- (Vince's Trig, art, 95.) hence SA^ : SB*:: tang. ACN:
oA
tang -^ ASP, therefore we get ASP the true anomaly required.
Exam file. Required the true place of Mercury -, on August 26,
1740, at noon, the equation of the centre, and his distance from
the sun.
According to la Cattle, Mercury was in its aphelion on Aug. 9,
at 6h. 37'. Hence on Aug. 26, it had passed its aphelion 16d.
17h. 23 ; ; therefore 87 d. 23h. 15' 32" (his periodic rev.) : 16 d.
I7h. 23': : 360 : 68 26' 28" the arc AQ or mean anomaly.
Now, according to la Caille, CA : CS :: 101 1276 : 21 1 165 :: 57*
17' 44 ;/ 8 : 1 1 's7' 50" = 43070 seconds, the value of CS reduced
to the arc of a circle, the log. of which is 4. 634 174-9. Also 68
26' 28" == 246388"; and assuming the L SCT = 60 = 2 1 6000",
the operation to find the L, ACN will be as follows :
^= 2 AC ; but us At' in this case = 1, AB, or SP -f Pv =
OF THE SOLAR SYSTEM.
4.6341749
9.9375306 log. of - - 216000 = a
4.5717055 37300
253300
246388
6912 =
i.6341749 r
9.9287987 209088 = a b =* 58* 4' 48^ c
4.5629736 36557
245645
246388
743 = d
4.6341749
9.9297694? 209831 = c -f d = 58* 17' 1 1" ft
4.5639443 ...... 36639
246470
246388
82 ==/
4.6341749 ,
9.9296626 20^749
4.5638375 36630
246379
246388
9 =//
Hence, as the difference between the value deduced from the
assumption and the time value, is now diminished about nine times
every operation, the next difference would be 1"; hence It -f g
1" = 58 15 ; 57" the true value of the angle ACN the eccentric
anomaly. Hence, from the proportion laid down above, the true
anomaly is found by logarithms, thus :
OF THE SOLAR SYSTEM.
Log. tang. 29o 7' 58i" -. - - 9.7461246
ilog. SB = 800111 ----2.9515751
12.6976997
i log. SA = 1222441 - - - 3.0436141
Log. tang. 24 16' 15" - - - 9.654-0856
Hence the true anomaly is 48 32' SO'. Now the aphelion A
was in 8s. 13 54' 30", therefore Mercury's true place was 10s.
2 27* ; hence, from what we have shewn above, 68 26' 28"
48 32' 30" == 19 53' 58", the equation of the centre. Also SP
= 1 + CS x cos. L. ACN = 1.10983, the distance of Mercury
from the sun, the radius of the circle, or the planets' mean dis-
tance being unity. Vince remarks, that the above method of com-
puting the eccentric anomaly, appears to be the most simple and
easy of application of all others, and capable of any degree of
accuracy. From the 'same method, we are able to compute, at
any time, the place of a planet in its orbit, and its distance from
the sun.
As the bodies Q and P were supposed to depart from A at the
same time, and will coincide again at B, AQB, APB being each
described in half the time of a revolution ; and as the planet moves
with its least angular velocity at A ; for as AC is its greatest dist.
from the sun, the arc which it describes must be proportionally
smaller, to have the areas described in the same time equal ;
therefore from A to B, or in the Jirst six signs of anomaly, the
angle ACQ will be greater than ASP, or the mean will be greater
than the true anomaly ; but from B to A, in describing the other
half of its orbit, or in the last six signs, as the planet at B moves
with its greatest angular velocity, being nearest the sun, the true
will be greater than the mean anomaly. When the equation is
greatest in going from A to B, the mean place is before the true
place by the equation, and in the remaining half of the orbit, the true
place is before the mean place by the equation ; hence from the
time the equation is greatest, until it becomes greatest again, the
difference between the true and mean motions is twice the equa-
tion. From apogee to perigee, the true and mean motions are the
same.
There is another method ascribed to Seth Ward^ Professor of
Astronomy, at Oxford, which, though less accurate than the me-
thod given above, yet, as in many cases it serves as a useful ap-
proximation, and renders the calculation more simple and easy,
we shall here briefly explain it.
OF THE SOLAR SYSTEM.
317
Ward assumed the angular velocity about the
focus -u to be uniform (the sun being supposed in
the other focus S) and therefore made it represent
the mean anomaly. Let vP be produced to r, and
make Pr = PS ; then in the triangle Svr, rv -f-
Sv : TV Si; : : tang. \ of the angles i>Sr -f- vrP\
: tang, half their difference i>Sr -yrS (Emer-
son's Trig prop. 6, B. 2, or Simson's prop. 3) but
(rv -f v$) = 1 AB -f -vS = AS ; (for rv =
SP -f P V *= AB) and f (rv vS) = AB t>S
= SB ; likewise tang. the sum of the angles i>Sr
-f T>rS = tang, i A Ai>P (Eucl. p. 32, B 1) and
tang. & diff. of the angles vSr vrS = (Pr being = PS) tang. f
diff. of the angles vSr PSr (Eucl. 5, B. 1) = tang, i Z. ASP ;
hence the afihelion distance : perihelion dist. : : tang. of\ the mean
anomaly : tang. % the true anomaly. This is called the si?nfile cllip.-
tic hypothesis. In the earth's orbit, which is nearly circular, the
error is never greater than 17" ; the error is greater in the orbits
of Mars and Mercury , and hence Bulialdus corrects this theory to
adapt it to these planets (see KeiPs Astr Lect. 24.) In the orbit
of the moon, the error may amount to 1' 35". By Ward* a hypo-
thesis, the computed place is more backward than the true, for
90 from the aphelion and perihelion, and for the other part it is
more forward.
^That Ward's' hypothesis of the uniformity of the angular velo-
city about the focus v is not true, may be shewn as follows :
From the centre S at the distance SV =
AC x CE^ for so that SV may be a mean pro-
portional between the semi-transverse AC, and
semi-conjugate CE of the ellipse AEB) de-
scribe the circle zV ; then the area of this cir-
cle will be equal the area of the ellipse (see B
Vince'e Con. Sect. Ellip. pr. 7, cor. 5) let a bo- y
dy be supposed to move with a uniform motion
through the periphery of the circle, in the same *
time that the planet performs one revolution in
the ellipse ; and let the body and the planet
commence their motion at the same time, the
planet from A and the body from z, so that the planet may de~
scribe AP in the same time that the body describes z-v ; then the
angle zSz is the mean, and ASP the true anomaly. Take fi inde-
finitely near P and join SP, and draw Pr, fio, perpendicular to S/*,
FP respectively ; then Pr = fio ; but the angle PFji varies as
pp = rj~, ; but the area PS/j is given in a given time ; therefore
* The angle PF/> increases in proportion as po increases, all other cir-
cumstances remaining the same ; and it diminishes in proportion as PF
increases, po remaining constant, or is inversely as PF ; hence when both
vary, the angle PF/> varies as pp
318 OF THE SOLAR SYSTEM.
Pr varies as ps ; hence the angle PF/z, described in a given time,
varies as PF -J- PS> which is not a constant quantity. Now the
angle PF# : L. PSfi : : PS : PF : : PF x PS - PS 2 ' and as equal
areas are described in equal times in the circle and ellipse about
/ S, the angular velocity in the circle, about S, is equal s/v 2 .
Therefore the angular velocity about S is greater or less than the
mean angular velocity, according as PF X PS is less or greater
than vSV 2 , or than AC X CE. Also, the angular velocity about S,
is the same in similar points of the ellipse in respect to the centre,
or at equal distances from the centre. From the above investiga-
tion, the greatest equation of the centre may be found, the dimen-
sions of the orbit being given. For while the angular velocity of
the body in the circle, is greater than that of the planet in the el-
lipse, about S, the equation will increase, the planet and body
commencing their motion from A and z together ; when the an-
gular velocities are equal, the equation is then greatest ; and this
i __i __
takes place when b.p 2 = SV 2 AC x Cri> or when AC X CE
= SP 2 ; hence SP is given. Let this value of SPbe represented
by S V, then as SV is known, FV (= 2 AC S V) will be given ;
and as SF is given, we can find the angle FSV, the true anomaly.
Hence (see the fig. p 3 1 3) by what we have shewn in determining
the angle ASP (p. Si 4) SB : SA : : tang, i true anomaly : tang.
$ eccentric anom. ACN, or tang. | SCT ; and as SC is given, ST
or its equal NQ is likewise given ; now to convert this into degrees,
we have this proportion, rad or 1 : NQ : : 57 17' 44"8 : the de-
grees in NQ. which added to or subtracted from the angle ACN,
gives ACQ the mean anomaly ; the difference between which and
the true anomaiy is the greatest equation. The equation at any
other time may in like manner be found, SP being given.
The greatest equation being given, the eccentricity, and there-
fore the dimensions of the orbit may be found. For, as is plain
from the last article, the equation is greatest when the distance is
a mean between the semi-transverse and semi-conjugate of the
elliptic orbit, and therefore in orbits nearly circular, the body must
be nearly at the extremity of the conjugate or minor axis, and hence
the angle NCA or SCT will be nearly a right angle, ST will be
therefore nearly equal to SC ; and also the angle NSA nearly
equal PSA. Now the angle NCA NSA (dr PSA) = SNC,
and QCA NCA = QCN ; these being added, we have QCA
. PSA = QCN -f SNC = 2 QCN nearly (NC being nearly pa-
rallel to QS) that is, the difference between the true and mean
anomaly, or the equation of the centre, is nearly double the arc
QN 7 , or double ST, or very nearly twice SC. Hence 57 17' 48"S
r half the greatest equation : : rad. 1 : SC the eccentricity. If the
* This and similar -properties will be demonstrated in the laws of motion.
ha
|
*?
OF THE SOLAR SYSTEM. 319
orbit be considerably eccentric, compute the greatest equation to
this eccentricity ; then as the equation varies nearly as SC, we
have this proportion ; as the computed equation : the eccentricity
found : : given greatest equation : true eccentricity.
Thus, if with la Caille, we suppose that Mercury's greatest
uation is 24 3' 5" (see p. 262) then 57 17' 44"8 : 12* i ; 32""
1 : .209888 the eccentricity very nearly. Now the greatest
equation, computed from this eccentricity, is 23 54 ; 28"5 ; hence
23 54' 28"5 : 24 3' 5" : : .209688 : ,21 1 165 the true eccentri-
city. Delambre makes the eccentricity of Mercury 79855.4, his
mean dist- being 387 10. By taking the mean distance of the earth
from the sun 100000, Vines makes the eccentricities and greatest
equations of the planets as follow; Mercury eccen. 7955 4, great-
equat. 23 40' ; Venn* eccen. 498, gr. eq. 47 ; 20" ; the Earth
eccen. 1681.395, gr. eq. 1 55' 36"5 ; Mars eccen. 14183.7, gr.
eq. 10 40' 40"; Jupiter eccen. 25013.3, gr. eq 5 30' 38"3 ;
Saturn 53640.42, gr eq. 6 26' 42"; and Herschet eccen. 90804,
gr. eq. 5 27' 6". M. Delambre^ in his tables annexed to LaLande's
Astr. 3d ed. makes the greatest equations of the planets for the
respective years, as follow; Mercury 23 3^' 39" tab. 101 ; Venus
47' 20", tab. 108, year 1780 ; Earth 1 55' 2"4, tab. 5, year 1780 ;
Mars 10 40' 39", tab. 115, year 1770; Jupiter 5 30' 37"7, tab.
124, 1750 ; Saturn.6 26' 4. "7, tab. 147, 1750 ; Herschel 5 2l'
2"7, tab. 165, year 1780. In these tables, Delambre gives the
equation and its secular variation, for every degree of the planets'
mean anomaly. Laplace, taking the mean distance of the earth
from the sun = 1 , makes the proportion of the eccentricities of
the semi-major axes, for the beginning of the year 1750, as fol-
lows ; Mercury 0.205513, Venus 0.006885, the Earth 0.016814,
Mars 0.093808, Jupiter 0.048877, Saturn 0.056223, Uranus or
Herschel 0.046683. He gives the secular variation of this pro-
portion as follows ; the sign indicates a diminution. Mercury
0.000003369, Venus 0.000062905, the Earth 0.000045572,
Mars 0.000090685, Jupiter 0.000134245, Saturn 0.000261553,
Herschel 0.000026228.
The eccentricity and true anomaly being given, the mean ano-
maly may be readily found by a direct solution, as follows ; the
eccentricity being given, the ratio of the transverse and conjugate,
or the major and minor axes, which is the ratio of NI : PI (Em-
erson's Conic Sect. prop. 19, B. 1) is given; for as AC, CS are
given (see the fig, p. 313) we have GC = (SG 2 SC 2 ) * =
(AC -f SC X AC SC) . Hence the angle ASP being given,
we have PI : NI : : tang. ASP : tang. ASN ; therefore in the tri-
angle NCS, NC, CS, and the angle CSN are given, and therefore
the angle SCN is given by Trig, the supplement of which is the
angle ACN or SCT ; hence in the rt. angled triangle STC, SC
and the angle SCT are given, therefore ST, which is equal NQ,
is given ; this arc, being the measure of the equation, may be
320 OF THE SOLAR SYSTEM.
found by this proportion ; rad. : ST : : 57 17' 44"8 : the degrees
in NQ, which added to ACN, gives ACQ, the mean anomaly.
The mean hourly motion of a jilanet being given, the hourly motion
in its orbit may be found in the following manner :
The planete' hourly motion in its orbit, is found immediately
from what we have shewn above in the correction of Ward's The-
ory ; for it appears from thence, that the angles PS^, VS? de-
scribed by the planet at P in the ellipse, and the body V in the
circle in the same time, are as SV 2 : SP 2 , or as AC X CE : SP 2
(see the fig p. 317) hence PS/z = VS* X AC X CE -7- SP 2 , which
is the hourly motion of a planet in its orbit^ the angle VSf being
the mean motion of the planet in an hour. For greater accuracy
SP must be taken at the middle of the hour. Tables of the plan-
ets' hourly motions in their orbits may be thus easily computed.
OF THE MOON.
THE moon being the nearest celestial body to the earth, and,
next to the sun, the most remarkable and interesting in our sys-
tem ; interesting- not only from its resplendent appearance, but
also from its Various phases, which afford us a measure of time
so remarkable, that it has been primitively in use among all peo-
ple. Ancient histoiy testifies, that the Hebrews, the Greeks, the
Romans, and in general all the ancients, used to assemble at the
time of new or full moon, to testify their gratitude for its manifold
uses. It is no wonder, therefore, that the ancient astronomers
were always attentive to discover its motions ; and their observa-
tions, handed down to succeeding astronomers, enable them to
settle her mean motion more accurately than could be done by
modern observations alone. It was from the observations of some
ancient eclipses, that Dr. Halley discovered an acceleration in her
mean motion
The proper motion of the moon in her orbit, is, like the sun or
rather the earth, from west to east ; and her place being compared
with the fixed stars in one revolution, she is found to describe an
orbit inclined to the ecliptic ; her motion also appears not to be
uniform ; and the position of her orbit, and the line of its afisides,
are observed to be subject to a continual change. These and
other phenomena we shall explain in the following remarks.
The mean motion of the moon is found thus : observe her place at
two different times, then the mean motion during this interval is
given . on supposition that the moon had the same situation with
regard to her apsides during each observation ; if not, it will be
sufficiently exact, if the interval of the times be very great.
Hence the moon's places, at a small interval of time from each
other, being compared, we get the mean time of a revolution
nearly ; and then at a greater interval, the mean time of her revo-
OF THE SOLAR SYSTEM. 321
Uition is obtained more correctly. The moon's place may be de*
terrnined directly from observation, or deduced from an eclipse.
M. Cassiniy in his Astronomy, p. 294 (as Vince remarks) ob-
serves that on Sept. 9, 1718, the moon was eclipsed, the middle
of which eclipse happened at 8h. 4', when the sun's true place
was 5s 16 40'. Having compared this with another eclipse, the
middle of which was observed at 8h. 32', on August 29, 1719,
when the sun's place was 5s. 5 47', the interval gives 354 d 28',
in which the moon made 12 revolutions and 349 7' ever ; hence
354 d. 28' being divided by 12, rev. -f- 349 7' part of a revolution,
or 354.0125 days divided by 12.96947685 = 27 d 7h. 6' 7 for the
time of one revolution. From two eclipses, in 1699 and 1717,
the time was found to be 27 d. 7h. 43' 6".
The moon was observed to be eclipsed at Paris, on Sept. 20,
1717, the middle of which eclipse was observed at 6h. 2'. And
Ptolemy remarks* that a total eclipse of the moon was observed
at Babylon, on March 19th, 1720 years before Christ, the middle
of which was at 9h. 30' at that place, or 6h. 48' at Paris The
interval of these times was 2437 years, 147 days less 46', of which
609 were bissextiles; this being divided by 27 d, 7h. 43' 6", gives
a little more than 32585.5 revolutions. Now the difference of the
sun's places, and therefore of the moon's,* as observed at both
observations, was 6s. 6 12' ; therefore the moon had made 32585
revolutions, 6s. 6' 12' in the interval of 2437 y. 174 d. 46',
which gives 27 d. 7h. 43' 5" for the mean time of one revolution.
This determination is veiy exact, as the moon was at each time,
very nearly, at the same distance from her apside. Her mean di-
urnal motion is therefore 13 10' 35 ;/ , and her mean hourly mo-
tion 32' 56" 27"'5. M. de la Land makes her mean diurnal
motion 13 10' 35" 02784394. Ddambre in his tables (tab. 28)
has it 13 1C' 35". This is the mean time of a revolution with
respect to the equinoxes.
The annual precession of the equinoxes being 50|", or nearly
4" in a month ; hence the moon*s mean revolution must be great-
er with respect to the fixed stars, than with respect to the equinox,
by the time in which she describes 4" with her mean motion,
which is about 7". Hence the time of a sidereal revolution of the
moon is 27 d. 7h. 43' 12". Lafilace makes the length of her si-
dereal revolution at the commencement of 1750 = 27 d. 321661-
18036, or 27 d. 7h. 43' 11 "5 nearly.
The acceleration of the moon, before taken notice of, though
but little sensible, since the most ancient recorded eclipse will be
developed in progress of time, as Laplace remarks, though an
immense number of ages would be necessary to determine it by
observations. The discovery of its cause has, however, antici-
* The place of the moon at the eclipse is here taken the same as that
of the sun, which is not accurate unless when the. eclipse is central ; for
-.his long interval it is, however, sufficiently accurate.
Rr
322 OF THE SOLAR SYSTEM.
pated this immense length of time, and shewn that this accelera*
tion is periodical.
Laplace, whose penetration has enabled him to discover -most
of the secuter variations in our system, from his profound investi-
gations, and strict application of the laws of gravity, has elucidated
this as well as many other intricate subjects, in so satisfactory a
manner, that much of our observations on the moon will therefore
be collected from him, as time would not permit, at present, our
entering deeply into their investigation.
The moon moves in an elliptic orbit, in one of whose foci the
tarth is situated. Her radius vector, or a line drawn from her to
the earth, describes about this point equal areas in equal times.
The eccentricity of her orbit is 0.0550368, her mean distance from
the earth being taken as unity ; which gives for the greatest
equation of her centre (7 0099) 6 18' 32"076. The lunar peri-
gee has a direct motion, that is, in the same direction as the mo-
tion of the sun, and the length of its sidereal revolution is 3232.
46643 days or 8 y. 312d. llh. 11' 39"552.
If the place of the moon be observed as often as possible during
a whole revolution, and the true and mean motions be compared,
the difference will be double the equation. If there should hap-
pen to be found two observations, where the difference of the true
and mean motions is nothing, the moon must then have been ia
her apogee in the one, and in her perigee in the other, as is evi-
dent from the theory of the planets* motions, given in chap. 4.
Mayer makes the greatest eccentricity 0.05503568, and thegreai-
cst equation corresponding 6 18' 31 "6. In his last Tables, pub-
lished by Mr- Mason, under the direction of Dr. Maskelyne, he
makes it 6 18 ; 32". Delambre, in his Tables (tab. 50) makes it
60 18' 31 "6.
The place of the apogee may be thus determined, from M. Cas-
sini's observations; the greatest equation = 5 1' 44"5 ; hence
570 17' 48"8 : 2 30' 52"25 : : AC = 100000 : CS == 4388 (see
p. 319) = the moon's eccentricity at that time. This eccentricity
ts, however, subject to a variation, being the greatest when the
apsides are in the syzygies, and least when in the quadratures.
Now let v be the focus in which the earth is situated (see the
small fig. p. 317) then taking BSP for the mean anomaly, Bt)P
being the true anomaly, their difference SPv (Eucl. 32, B. 1) is
the equation of the orbit, which equation is here 37' 50"5 ; and as
PS = Pr, the angle vrS = 18' 55"25 ; hence (Trigonom.) 7>S =
8776 : -vr 200000 : : sine vrS = 18' 55"25 : sine vSr or its
supplement BSr, = 7 12' 20", from which let -urS = 18' 55"25
be taken, and we have Bi>P = 6 53' 25" = the distance of the
moon from its apogee ; to this let the true place of the moon ~
2s. 19 40' be added, the sum gives "2s, 26 33' 25" for the place
of the apogee on December 10, 1685, at lOh. 38' 10" meantime
at Paris. Hence this may be considered as an epoch of the place
of the apogee.
OF THE SOLAR SYSTEM. 323
The mean motion of the apogee may be thus determined ; let its
place be found at different times, and let the difference of these
places be compared with the interval of the time between. In
performing this, the observations must be first taken at a small
distance from each other, as we might be deceived in a whole re-
volution ; then those observations at a greater distance may be
compared. Thus the mean annual motion of the apogee is found,
according to Mayer, = 40" 39' 50" or Is. 10 39' 50 /; , its monthly
motion = 6' 41", its hourly mot. = 17", &c. Delambre makes
its annual motion = Is. 10 39' 50"5,* in a month 6' 41", See-
To determine the place of the moon's nodes. The moon's place
is directly opposite to the sun in a central eclipse of the moon, and
hence the moon must then be in her node ; if the true place of
the sun be then found by calculation, or rather by observation, the
opposite sign and degree, &c. in the ecliptic, will be the true
place of the moon, and consequently the place of her node.
M. Cassini, in his Astr. p. 281, says, that on April 16, 1707,
a central eclipse was observed at Paris, the middle of which took
place at 3h. 48' apparent time. Now the sun's true place calcu-
lated for that time, was Os. 26 19' 17" ; hence the place of the
moon's node was 6s. 26 19' 17". The moon, at that time, passed
from north to south lat. and therefore this was the descending
node. The place of the node is always ready calculated in the
astronomical tables, with its mean motion.
The place of the nodes may be also determined as in p. 264.
The mean motion of the nodes may be determined, by finding the
place of the nodes at different times, from which its motion, in
the interval, will be given ; the greater the interval the more ac-
curate will the motion be discovered. Mayer and Delambre, both
make the mean annual motion of the nodes 19<> 19' 43". The
motion of the nodes is westward, contrary to the order of the
signs. The length of their sidereal revolution, according to La-
place, is 6793.3009 days, or 18 y. 223d. 7h. IS' 17"76. Their
motion is subject to several inequalities, of which the greatest,
according to Laplace, is proportional to the sine of double the
angular distance of the sun from the ascending node of the lunar
orbit, and at its maximum amounts to (1 8105) 1 40' 46"02.
The moon's nodes being those points where the lunar orbit cuts
the orbit of the earth, or the ecliptic ; and the angle formed by
the planes of these orbits, the same as the inclination of the orbit
of the moon to the ecliptic, we shall now shew how to find this
inclination. When the moon is 90 distant from her nodes, it is
evident that she has then her greatest latitude, and that this lati-
tude will measure the inclination of her or'oit, in the same manner
as the sun's greatest declination measures the inclination of the
* As Delambre (tab. 27) only gives the moon's mean motion and mean
anomaly for entire years, it is necessary to remark, for the learner's sake,
that it' the mean anomaly be taken from the mean motion, the remainder
vili give the motion of the apogee.
324 OF THE SOLAR SYSTEM.
equator and ecliptic. Hence if, when the moon is 90 from her
nodes, her right ascension and declination be observed, and from
thence her latitude be computed (ste the note to prob. 3, part 3d)
this will be the inclination of her orbit tor that time. If similar
observations be made for every distance of the sun from the earth,
and for every position of the sun in respect of the moon's nodes,
the inclination at those times will ue observed. It appears, from
these observations, that the inclination of the moon's orbit to the
ecliptic is variable, and that the least inclination is about 5, which
takes place when the nodes are in the quadratures ; and the great-
est about 5 18', which is found to happen when the nodes are in
the syzygieS' The inclination is also found to depend on the sun's
distance from the earth. Lafilace makes the inclination (5. 7 i 88)
5 8' 48"9. He makes the greatest inequality in its variation
(0 1631) 8' 48"444, and remarks, that it is proportional to the
cosine of the same angles on which the inequality of the motion
of the nodes depends.
The moon would always describe the same ellipse, in her revo-
lution round the earth, if this revolution were not disturbed by the
action of the sun ; the principal axis of her orbit would remain
invariable, her periodic times would be the same, and the inclina-
tion of her orbit to the ecliptic, as well as the place of her nodes,
would remain fixed ; but from the sun's action, her motions be-
come subject to so many irregularities, that to establish her theory,
and calculate her place truly, is one of the greatest difficulties in
physical or practical astronomy. These irregularities are, how-
ever, evidently connected with the sun's position
The moon's motion being examined for one month, it will be
found that it is subject to an irregularity which sometimes amounts
to 5 or 6, but that every 14 days this irregularity disappears.*
If these observations be continued for different months, it will
also appear that the points where the inequalities were the great-
est, were not stationary, but advanced forwards about 3 in a month,
so that, in respect to the apogee, the moon's motion was about y-^
less than her absolute motion ; and hence the apogee's progressive
motion has been discovered. This./?r*f inequality, or equation of
the orbit) was determined by Ptolemy^ from th/ee lunar eclipses
observed at Babylon, in the years 719 and 720 before J. C. by the
Chaldeans ; he found it amounted to 5 1 ' when greatest. But
he soon found that this would not account for all the irregularities
of the moon, as her distance from the sun, observed both by Hifi-
parchus and himself, sometimes agreed with this inequality and
sometimes did not. He found that thisjirst inequality would give
the moon's place sufficiently correct, when the apsides of her orbit
were in the quadratures ; but that when the apsides were in the
syzygies, he discovered that there was a further inequality of 2|,
which, in this case, made the whole inequality amount to about
* Yince
OF THE SOLAR SYSTEM, 325
7|. This second inequality is called the evection, and arises from
the variation of the eccentricity of. the moon's orbit Hence Pto-
lemy found that the moon's inequality varied from 5 to 7f, and
at a mean was therefore 6 20'. Mayer makes it 6 1 8' 3 1"6. It
is therefore extraordinary, how Ptolemy, in a point of so delicate
a nature, should have determined this to so great a degree of
accuracy
Lafilace makes the evection (14902) 1 20' 28"248, and re-
marks, that it is proportional to the sine of double the mean an-
gular distance of the moon from the sun, minus the mean angular
distance of the moon from the perigee of its orbit. In the oppo-
sitions and conjunctions of the moon with the sun, it is confounded
with the equation of the centre, which it constantly diminishes,
and hence the ancient astronomers, who only determined the ele-
ments of the lunar theory by means of eclipses, with a view of
predicting these phenomena, always found the equation of the
centre less than the truth, by the whole quantity of the evection.
There is another inequality observed in the moon, which is
called the -variation ; this inequality disappears in eclipses, or in
the conjunctions and oppositions, and could not have been discov-
ered from the observation of those phenomena. It also disappears
in those points where the sun and moon are distant from each
other 90. It is at its maximum, and amounts to 35' 40"992,
when their mutual distance is 45 ; from whence it is inferred,
that it is proportional to the sine of double the mean distance from
the sun.
The last inequality which we have to observe, is that known by
the name of the annual equation, caused by the moon's motion
being accelerated when that of the sun is retarded, and the con-
trary. The law of this inequality is exactly the same as that of
the equation of the centre of the sun,* but with a contrary sign ;
at its maximum it is (0 2064) 1 I' 8"736. This inequality in
eclipses becomes confounded with the equation of the centre of
the sun, and in calculating the instant of these phenomena, it is
indifferent whether these two equations be considered separately,
or tiie annual equation of the lunar theory be suppressed to augment
the equation of the sun's centre. This is one principal reason
why the ancient astronomers gave too great a value to this last
equation, and assigned too small a value to the equation of the
sun's centre affected by the evection.
The following exhibits, at one view, the re-volutions of the moon t
of its apogee and nodes, as determined by M. de la Lande.
d. h. ' '
Tropical revolution 27 7 43 4,6795
Sidereal revolution 27 7 43 11,525?
Synodic revolution 29 12 44 2,8283
* The equation of the sun's centre at its maximum, according- to La-
place, was in 1750, equal (2 1409) 1*55' ntf'516. The method of finding
this is given in p. 313.
326 OF THE SOLAR SYSTEM.
d. h. ' "
Anomalistic revolution 271318 33,949
Revolution in respect to the node - - 27 5 5 35,605
Tropical revolution of the apogee 8y. 311 8 34 57,6177
Sidereal revolution of the apogee 8 312 11 II 39,4089
Tropical revolution of the node 18 228 4 52 52,0296
Sidereal revolution of the node 18 223 7 13 17,744
Diurnal motion of the moon? , n9rs/t Q ,
in respect to the equinox $ ) S5 > 02
Diurnal motion of the apogee - - 641,069815195
Diurnal motion of the node - - 3 10,638603696
Ne-uil Maskelyne finds, from the new Tables of M. Burg and
Delambre, that the mean longitude of the moon, at the middle of
the year 1813, including the secular equation and new equation of
180 years, will be 9s. 25' 48"2 ; mean anomaly with secular
equation 7s. 25 59', and the supplement of the node with the
secular equation 7s. 8 13' 15"6.
The apparent diameter of the moon varies in a manner analogous
to her motions. This diameter may be measured at the time of
full moon, by a micrometer placed in the focus of a telescope, or
it may be measured by the time of its passing over the vertical
wire of a transit telescope ; but this must be within one or two
hours of the time of full moon, before the visible disk is sensibly
changed from a circle. The diameter may be thus found from
the time of its passing over the meridian ; let d" = the moon's
horizontal diameter, c = sec. of her declination, and m = the
length of a lunar day, or the time from the moon's passage over
the meridian on the day of observation, to the time of her passage
over the meridian, on the next day. Then (art. 8 of the last note
lo prob. 19, part 3, p. 216) cd" = the moon's diameter in right
ascension ; therefore 360 : cd" : : m : the time (r) of passing the
meridian; hence d" = 360 x *-. If the time be observed when
cm
the limb of the moon comes to the meridian, the time when the
centre comes to it can be found, by adding to or subtracting from,
the time when the first or second limb comes to the meridian, half
the time of the moon's passage over the meridian.
Albategnius made the diameter of the moon vary from 29' 30"
to 35' 20", and hence found the mean = 32' 25". Copernicus
found the diameter to vary from 27' 34" to 35' 38, and therefore
the mean to be 31' 36". Kepler made the mean diameter 31' 22",
M. de la Hire made it 31' 30". M. Cassini made the diameter
from 29' 30" to 33' 38". La Lande^ from his own observations,
found the mean diameter = 31' 26", and the extremes from 29'
22", when the moon is in apogee and conjunction, to 33' 31"
when in perigee and opposition. The mean diameter here taken
is the arithmetic mean between the greatest and least ; the diam-
eter at the mean distance being 31' 7". Delambre^ in his tables
("table 91) gives the extreme horizontal diameters 29' 30" and 3S'
OF THE SOLAR SYSTEM. 327
30" respectively ; and also its augmentation in every degree of
altitude corresponding to its respective horizontal diameters, and
corrects these altitudes of refraction by tab. 93 (see articles 1510,
2247, and 2248, La Lande's Astr. 3d ed.) Lafilace makes the
apparent diameter at the moon's greatest dist. = (5438") 29' 21",
912, and at her least dist. = (6207") 33' 31"068. Taking the
apparent diameter at her mean distance =31' 7", her real diam-
eter is found to be 2159 07 miles,* and her magnitude is about
^y of the magnitude of the earth.
When the moon appears in the horizon, she is then an entire
semidiameter of the earth more distant from a spectator on the
earth's surface, than when she appears in the zenith ; hence it
follows, that her apparent diameter must augment in proportion
as her altitude increases from the horizon. Let C be the centra
of the earth, A the place of a spectator on its
surface, Z his zenith, M the moon ; then, as
the sides of triangles are as the sines of their
opposite angles, we have sine CAM or its suppl.
ZAM : sine ZCM : : CM : AM = CM X sine
ZCM -*- sine ZAM ; but the apparent diameter
varies inversely as its distance > hence the appa-
rent diameter will vary as sine ZAM divided by
sine ZCM, the moon's distance from the centre
of the earth being supposed constant. Now in
* In the fig. p. 250, let M represent the place of the earth, AB the
31' 7"
moon, the angle AMC half its semidiameter = =*= 15' 35"5 ; hence
the angle MAC = 90 IS! 33*5 = 89 44' 2#'5. MC is the moon's
distance from the earth = 238533 miles (see p. 250.) Now conceive a
straight line to be drawn from M to B, then in the triangle AMB it
will be,
As sine /_ A = 89 44' 2C"5 - - - 9.9999956
To sine AMB 31' 7" 7.9567133
So is the dist. MB =- 238533 - - - 5.3775494
13.3342627
To AB the moon's diam. 2159 07 - - 3.3342671
The diameter might also have been thus calculated ; rad. : sine 15' 33? 5
-. : MA : AC the moon's semidiameter. Or it might be calculated in the
same manner as the diameter of Mercury has been calculated in the first
pan of the note, p. 263. 3
Now the cube of the earth's diameter = 7911 divided by the cube of
the moon's diameter = 2159.07 will give the proportion of their mag-
nitudes ; thus, log. 7911 log. 2159.07 = 11.6946942 10.0028013
= 1.6918929 ; the number corresponding to this log. is 49.19, which
shews that the magnitude of the earth is something more than 49 times
that of the moon.
Keith, in his Treatise on the Globes, has calculated the diameter of
the moon in a similar manner, but makes the angle at A, from data the
same as the above, with respect to the angles, = 89 55' 44''26 w , and
hence all his conclusions, resulting from these erroneous premises, must
he false*
328 OF THE SOLAR SYSTEM.
the horizon, taking sine Z \M -r- sine ZCM as equal to unity or
one, we have this proportion ; I : sine ZAM ~- sine ZCM, or
s. ZCM : s. ZAM,* or cos. true alt MCH (a) : cos. apparent ait.
MAA () : : the horizontal diameter to the diameter at the appa-
rent alt b. Hence the '^oriz diam. : its increase : : cos. a : cos.
b a = 2 sine 3 a + i. 6 X sine ia %& rad. being 1 (Emer-
son's Trig. B 1, prop 3, cor. 4, or Vince's Trig. an. 111.)
Hence (16 Eucl. 6) the increase of the semidiameter = hor.
sne a
semid. X -- v + - 2 .. .... . From this expression, a
cos. a
table of the increase of the semidiameter for any horizontal diam-
eter, may be easily constructed ; and for any other horizontal se-
midiameter, the increase will vary in the same proportion.
The moon's parallax is the next subject that requires our con-
sideration. Various methods have been given by authors, but the
following are the principal.
First method. Let the meridian altitudes of the moon be taken
when she is at the greatest north and south latitudes, and let these
altitudes be corrected for refraction ; then if there were no paral-
lax, the difference of these corrected altitudes would be equal to
the sum of the two latitudes of the moon ; hence the difference
between the sum of the two latitudes, and the difference of the
altitudes, will be the difference between the parallaxes at the two
altitudes. Now from thence to determine the parallax itself, let
S be the sine of the greatest, and * of the least, apparent zenith
distance, and P, /^, the sines of the corresponding parallaxes ;
then, as the parallax varies as the sine of the zenith distance/
when the distance is given (see the note, p. 279) we have S : * : :
P :fi- t henceS s : * :: Vflifi = ^~ ~~ ( l
the parallax at the greatest altitude. As the above calculation is
on supposition that the moon is at the same distance in both ob-
servations, which will generally not be the case ; one of the obser-
vations must be reduced to what it would have been had the distance
been the s- me as the other, the parallax being inversely as the
distance (note, p 279 ) If the moon pass through the zenith of
one of the observers, the difference between the sum of the two
latitudes and the zenith dist at the other observation, will be the
parallax at that altitude.
* For 1 X sine ZAM = s ^-^f x sine ZCM therefore, Sec. (16
s. ZCM
Eucl. 6)
OF THE SOLAR SYSTEM.
329
Second method, for any planet. Let the
planet P be observed from two places A, B,
in the same meridian ; then the angle APB
is the sum of the two parallaxes at both
places. The parallax APC or sine APC =
hor. par. x sine PAL (p. 279) and parallax
BPC or s. BPC = hor. par. X sine PBM ;
hence hor. par. X (s. PAL -f s. PBM) =
APB ; therefore hor. par. = APB divided
by the sum of these two sines. If the me-
ridians of the places differ, the variation of
the planets' declination, in the interval of the passages over the
meridians of the two observations, must be known.*
Third method, answering for any
planet. Let EQ be the equator, P
the pole, Z the zenith, -v the true
place of the planet, and r the appa-
rent place as depressed by the pa-
rallax in the vertical circle ZA ;
let the circles of declination Pi>a,
Pr, be drawn ; then ab is the pa-
rallax in right ascension, and rs in
declination. Now -vr : -vs : : rad.
1 : sine i>rs, or Zx>P (see Vince's
Trig. art. 125) and ~us : ab : : cos.
va : rad. I (see the note to prob. 35, part. 2) hence -vr : ab : : cos.
va : sine ZuP ; therefore ab = vr X sine Zt>P -r- cos. va ; but vr
= hor. par. X s. i^Z (note, p. 279) and the sides of spher trian-
gles being as the sines of the angles opposite to them, s. i>Z : s.
ZP : : s. ZPi> : ZvP = s. ZP X s. ZPz; ~ s. vZ. Hence, by
substitution ab = hor.fiar. X s. ZP X s. ZPu. -f- cos. va. There-
fore, for the same star, the parallax in right ascension varies as
the sine of the hour angle, where the hor. par. is given. The hor.
parallax is also = ad X cos. va -~- s ZP X s. ZPz>.
The apparent place b on the equator, is to be east of c, the true
place, for the eastern hemisphere, or that hemisphere east of the
meridian, and therefore the right ascension is increased by the
parallax ; but in the western hemisphere, b lies to the west of c,
and therefore the right ascension is diminished. Hence, if the
right ascension be taken before and after the meridian, the whole
* Ex. On Oct. 5, 1751, M. de la Cattle observed Mars to be 1' 25"8 be-
low the parallel A in Aquarius, at the Cape of GoodJIope, and to be 25
distant from the zenith. On the same day, at Stockholm, Mars was ob-
served to be 1 57T below the parallel of A and his zenith distance to
be 68 14'. Here then the angle APH = 3i"9, and the sines of the ze-
nith distances being 1 0.4226 and 0.9287, the horizontal parallax was 25"tV
If the ratio of the distance of tke earth from Mars and the sun respec-
tively, be given, the sun's hor. parallax will therefore be given, the pa-
rallaxes of the planets being inversely as their distances. (Xat*, p. '?" ' .
330 OF THE SOLAR SYSTEM.
change of parallax in right ascension, between the two observa-
tions, is the sum * of the two parts before and after the meridian,
and therefore = = x S the sum of the sines of the two hour
cos. -va
angles, and the hor fiar. = s x cos. va -~ sin. ZP X S. There is
no parallax in rt. as. on the mer. for then the value of ab y shewn
above, is nothing, as the angle ZvP vanishes. As the spher. A
ur* is rt. angled at s, rs, the parallax in decl. may be easily found
by JVafiier's rules.
In the application of the above investigations, the rt. as. of the
planet when it passes the mer. compared with that of a fixed star,
must be observed, as then there is no parallax in rt. as. Let the
din*", of their rt. ascensions be again observed, 6 hours after, and
let the change of the diff. d between the apparent rt. ascensions
of the planet and star, during that time, be observed. Again, to
obtain the planets' true motion in rt. as. let its rt. as. be observed,
when it passes the mer. for 3 or 4 days ; then, if in this interval
of time, its motion in rt. as. between taking the rt. ascensions of
the star and planet on and off the mer. be equal to rf, the planet
has no par. in rt. as. but if it be not = </, the diff. is the parallax
in rt. as. ; from which, by what is shewn above, the hor. par. will
be given. If one of the observations be made before the planet
,comes to the meridian, and the other after, a greater diff. will be
obtained.*
As the right ascen. and decl. is thus affected by the parallax, it
is evident that the lat. and long, of the moon and planets must, in
like manner, be affected by it ; and as the determination of this,
in respect to the moon is, in many cases, particularly in solar
eclipses, of great importance, we shall here shew how to com-
pute it, on supposition that the lat. of the place, the time, and
therefore the sun's rt. as. the moon's true lat. and long, with her
hor. parallax, arc givcn.f
* Ex. Mars was very near a star of the 5th mag 1 , in the eastern
shoulder of Aquarius, on Aug. 15, 1719, at 9h. 18 f in the evening, and
in 1C/ 17" lie followed the star ; on the 16th, at 4h. 21' he followed the
star in 10' 1" ; hence in that interval, the appar. rt. as. of Mars had in-
creased 16" in time. But from observations made in the mer. for several
days after, Mars, from its proper motion in that time, approached the
star only 14"; therefore the effect of parallax, in the interval of the ob-
servations, was 2." in time, or 30" in motion. Now the decl. of Mars was
15, the co. lat. 41 10', and the two hour angles 49 15', and 56 39";
hence the hor. par. = 30" X cos. 15 -j. sin. 41 10> X (sin. 49 15' 4.
sin. 56 39') = 27$". But the dist. of the earth from Mars, was to its
dist. from the sun, at that time, as 37 : 100, whence the sun's parallax
comes out = 10"17, but this is too great by nearly 1$".
f The following- solution is principally taken from Vince.
OF THE SOLAR SYSTEM. 3S1
Let HZR be the meri-
dian, T EQ the equator, fi
its pole; T DC the ecliptic,
P its pole, T the beginning
of Aries, HQR the horizon,
Z the zenith, ZL a vertical
circle, or secondary to the
horizon passing through the
true place r, and apparent H B L Q C
place t of the moon ; draw P/, Pr, which produce to s, and draw
the small circle ts^ parallel to o~u ; then rs is the fiarallax in fat.
and ov the parallax in longitude.* Draw the great circles TP,
PZAB, Pfide, and ZW perp to P* ; then as TP = 90, and also
T/i = 90, T is the pole of Pete (see def. 6, or Simson's Spher,
Trig, annexed to his Euclid, cor. to prop. 3) and hence rfT = 90 ;
therefore d is one of the solstitial points Cancer or Capricorn ;
draw Z^perp. to Pr, and join ZT,/*T. Now TE or the
T/;E or Z/iT, is the rt. as. of the midheaven, which is known
(see the note to prob. 1 6, part 3) PZ = AB (being each the
comp. of AZ) the alt. of the highest point A of the ecliptic above
the horizon, or nonagesimal degree, and TA, or the angle TPA
is its longitude ; also Zp, = co. lat. of the place, and the /- ZfiW
is the diff between the rt. as. of midheaven T/*E and Tr. Now
in the rt. angle A Z/zW, rad. X cos -. fi = tang. fiW X cot.
fiZ (Napier's rule) hence (16 Eucl. 6) cot. fiZ : rad. : : cos./i :
tang, fi VV ; or by logarithms,
log. tang. fiW = 10, -f log. cos. fi log. cot. fiZ ;
therefore PW = fiW i/*P, where the ufifier sign takes place
when the sign of the midheaven is less than 180, and ihe lower
sign when greater. Also in the triangles WZ/z, WZP, we hav
kin. Wfi : sin. WP : : tang. WPZ : tang. WflZ (Vince's Trig.
art. 231, or Simson's prop. 26) : : cot. WfiZ : cot WPZ, or tang.
APT (the tangents being reciprocally as the cotangents, Emer-
son's Trig. prop. 1, cor. 4, or Vince, art. 82) therefore,
log. tang. APT == ar. co. log. sin. W/jf -f log. sin. WP -f log. cot.
W/*Z 10
or, log. tang. APT = log. sin. WP -f log. cot. WfiZ log. sin.
and as To, or TPo, the true long, of the moon is given, APo, or
ZP;r is therefore given. Also in the triangle WPZ ; cos. WPZ,
or sin. APT : rad. : : tang. WP : tang. ZP (Simson's Spher.
prop. 20, or Vince's, art. 219) therefore,
* See Keil's Astronomy, Lect. 21, or Gregory's Astr. B. 2, sect. 8,
where this subject is also fully investigated.
f The arithmetical complement of any logarithm is what it wants of 10,
or 20, and is used to avoid subtraction ; thus the ar. com. of 2.6963564
is 7.3036436. Hence in the above proportions the ar. com. log. s. Wp
being added and 10 subtracted, is the same as subtracting log. "sin. \V/>,
as is evident. However, there seems to be more perplexity, particularly
for beginners, in using tfce ar. co. than the simple log.
332 OF THE SOLAR SYSTEM.
log. tang. ZP = 1 0, -f log. tang. WP - log. sin. AP V .
Again, in the triangle ZPr, ZP, Pr, and the angle P are given,
whence the angle ZrP or srt may be thus found ; in the rt. angled
triangle ZP^, ZP and the angle P are given ; hence (by Napier's
rule) rad X cos. ZPx == cot. PZ X tang. Px \ which resolved by
logs, gives
tog. tang Px = 10, -f cos. ZP.r log. cot. PZ ;
hence rx is given ; therefore sin. rx : sin. Px : : tang. ZP:r : tang,
2>rx or trs (Simson's Spher. prop. 26) which in logarithms is,
log. tang. 7*rx sa ar. co. log. sin. rx -f hg* sm. Px -f log. tang.
ZPx 10}
also in the rt angled triangle Zrx, we have (by Napier's rule)
rad. X cos. Zrx = cot Zr x tang rx ; therefore,
log. cot. Zr = 10, -f- log. cos. Zrx log. tang. rx.
with this true zenith dist. Zr, let the parallax be found (note, p.
279) as if it were the apparent zenith distance, and the true pa-
rallax will be given nearly ; let this par. be therefore added to the
true zenith dist. and the apparent zenith dist. will be given nearly,
to which let the parallax be again computed (p. 279) and the true
parallax rt will be obtained extremely near ; then in the rt. angled
triangle rst, which may be considered as plane, we have rad. :
cos. r : : rt : r.s, the parallax in latitude (Simson's Trig. prop. 1)
hence, log rs = log. rt -f log cos. r 10 = log. par. lat. Also
rad : sin. r : : rt : ts ; therefore log. ts =5 tog. rt -f log. sin. r
10 ; hence cos. tv : rad. : : ts : ov, the paral. in longitude (see the
note to prob 35, part 2.)
Ex. On January l, 1771, at 9h. apparent time, in lat. 53 N.
the moon's true longitude was 3s. 18 27' 35" and lat. 4 5 ; 30" S.
and her horizontal parallax 61' 9 ;/ j to find her parallax in lat. and
long.
The sun's rt as. by the Tables, was 282 22' 2", and his dist.
from the mer. = 9h x 15 = 135 ; also the rt. as. of the mid-
heaven was 57 22' 2" ;* hence the whole operation for the solu-
tion of the triangles will be as follows.
In the triangle ZfiW.
Z/iW = 32o 37' 58" . . . 10, -f- cos. 19.9253864
Zfi = 37 cot. 10.1228856
= 32 23 57 cot. 9.8025008
-f //P =*= 23 28' = P W = 55o 51' 57".
| 360 282 22' 2" = 77 37' 58"; hence 185 77 37' 58" = 57 22*
2". ZpW = 90 57 22' 2" = 32 37' 58".
OF THE SOLAR SYSTEM. 333
In the triangles W/zZ, WPZ.
32 23' 57" - - - - ar. co. s. 270.9855
PW = 55 51 57 sin. 9.9178865
= S2 27 58 - .... cot. 10.1935941
APT = 67 29 8 Ian. 10 3824661
oPT D's long. 108 27' 35" APT 40 58' 27".
In the triangle WPZ.
WP =555l / 57" - - - 10, -f tan. 20.1688210
APT = 67 29 8 sin. 9.9655700
ZP = 57 56 36 tan. 10.2032510
In the triangle WPZ.
40 58' 27" . . . 10, + cos. 19.8779503
ZP = 57 56 36 cot. 9.7967445
50 19 33 tan. 10.0812055
Pr = 90 -f 4 5' 30" = 94 5' 30", hence 94 5' 30"
Px = 43 45' 57" = rx.
In the triangles ZP.r, Zr\r.
rx =43 45' 57" - - - ar. cos. s- 0.1600743
Vx = 50 19 33 sin. 9.8863144
ZP.r = 40 58 27 tan. 9.9387676
= 44 1 16 - .... tan. 9.9851563
In the triangle Zrr.
Zrr =44 1' 16" - - - 10, + cos. 19.8567795
rx = 43 45 57 tan. 9.9812846
Zr = 53 6 10 cot. 9.8754949
*Zr = 53 6' 10" ...... sin. 9.9029362
Hor. par. 61' 9" = 3669" - - - log. 3.5645477
rt uncorrected = 2934" =x 48' 34" log. 3.4674839
App. zen. dist. Zff = 53 55 ; 4" nearly sin. 9.9075042
Hor. par. 3669" - - - - - log. 3.5645477
* See the latter part of the note, p. 279.
Zt = Zr 53 & 10" - rt 48' 54" =s 53 5B f 4T.
334 OF THE SOLAR SYSTEM,
"Par. rt corrected = 2965" = 49' 25" log. 3 47205 1 9
& 1 trs or Zrx 44 i' 16" - - - - cos. 9 8567795
-s par. in latitude = 2132" = 35' 32" log. 3 3288314
V* corrected = 2965" - - - - - log. 3.4720519
J trs = 44 1' 16" sin. 9 b4l9369
= 2061" = 34' 21" log. 3.3139888
The true latitude ro being = 4 5' 30" S. hence,
Appar. lat. tv = ro -f rs = 4 41' 2" cos. 9.9985472
?s=*2061" 10, -f- log. 13,3139888
o-v par. in longitude = 2067" = 34' 27" log. 3.3 1 544 1 6
Note 1 . The value of tv is ro + rs, according as the moon has
N. or S. lat.
JVbte 2. The order of the signs being from west to east, from A
towards C is eastward, and from A towards T is westward; now
as the parallax depresses the hedy from r to t, it increases the lon-
gitude from o to v ; but if the point o had been on the other side
of A, ov would be the contrary way ; hence when the body is to
the cast of the nonagesimal degree, the parallax increases the
longitude ; and when to the west) it diminishes the longitude.
Ex. 2, On June 29, 1813, at 7h. 3' 57" apparent time, in the
evening, at New- York, lat. 40 42' 40", the moon's true longitude
will be 4s. 1 11' 36", and latitude 52' 10" S, and her horizontal
parallax 59' 16" ; required her parallax in lat. and longitude ?
The sun's it. as. will be 6h. 33' 40"3 in time = 98 25' 4"5, by
the Naut. Aim and his distance from the mer. will be 7h. 3' 57"
= 105 59' 15"; also the rt. as. TE of the medium cceli will be
204 24' 19"5.*
From Mayer's tables the moon's greatest parallax (or when she
is in her perigee and in opposition) is 61' 32" ; her least parallax
(or when in her apogee and conjunction) is 53' 52" in the lat. of
Paris. The arithmetical mean of these is 57' 42" ; but this is not
the parallax at the mean dist. as the par. varies inversely as the
dist. the par. at the mean. dist. is therefore 57' 24", an harmonic
mean between the two.f Lajilace makes the moon's par. at her
dist. from the earth, which is an arith. mean between the two ex-
tremes = (10676") 57' 39"024, so that at the same dist. at which
the moon appears to us to subtend an angle of (5823") 3 1' 26"652,
* To find the rt. as. of midheaven ; the sun's rt. as. 98 25' 4''5 -f- his
dist. from the mer. 105 55' 15" = 204 24' 19"5.
f Harmonic ratio, is when a quantity is divided into three parts, so that
the whole is to one part, as the second part to the third. When the se-
cond and third are equal, it is called harmonic proportion continued- Em
crson's Doctrine of Prop. sect. 2, def. 14.
OF THE SOLAR SYSTEM. 335
the earth would appear under an angle of (21322") 1 55' 18"048.
M. de Lambre re-calculated the parallax from the same observa-
tions from which Mayer calculated it, and found that it did not
exactly agree with Mayer's* He made the equatorial parallax
57' ll"4. M. de IzLande makes it 57' 5" at the equator, 56'
53"2 at the pole, and 57' 1" for the mean radius of the earth,
supposing the diff. of the equatorial and polar diameters to be 7 J^
of the whole. From the formula of Mayer (at the end of his ta-
bles) the equatorial parallax is 57' 1 1"4.*
During the course of a lunation, or synodic revolution, the moon
constantly exhibits very singular phenomena, which we call her
phases. At the moment that the moon passes between the earth
and the sun, in her revolution round the earth, which she regards
as her centre, the enlightened half of her is then entirely turned
towards the sun, and the other dark half is towards the earth ; in
this case, the moon will therefore be invisible to us, and this po-
sition of the sun and moon is termed the conjunction or new ?noon,
The moon remains invisible during 3 or 4- days ; because for a day
or two both before and after conjunction, her crescent is so small,
and her light so obscured by the sun's rays, that she escapes the
nicest observation. After disengaging herself in the evening from
the rays of the sun, she re-appears towards the east with a slender
crescent, convex towards the sun, which increases with her dis-
tance ; in about 7|- days after the conjunction it becomes a semi-
circle, at which time she will come to the meridian about 6 o'clock
in the evening, when the moon is 90 distant from the sun ; mov-
ing still eastward, she becomes an entire circle of light in about
14^ days, when she is in opposition with the sun, at which time
she will come to the meridian at midnight ; hence in this position
she appears full, and is therefore called/w// moon. When she af-
terwards approaches the sun, this luminous circle is changed into
a crescent, which diminishes according to the same degrees it had
increased before, until, in the morning it becomes immersed in
the solar rays. The lunar crescent being always turned towards
the sun, evidently indicates that it is from the sun the moon re'
ceives her light. The law of the variation of her phases we have
given in p. 269, and the method of delineating the phases may be
easily collected from what is given in p. 268. These phases are
renewed at every conjunction, and their return depends on the
excess of the moon's synodical motion above that of the SUP.,
which excess is called the synodical motion of the moon. The
length of the synodic revolution of the moon, or the period of her
* M. John Machin, Astron. Prof. Gresh. Col. has, at the end of Matte 1 s
translation of the Principia, given the laws of the moon's motions ac-
cording to gravity; that is, her variations, her inequalities during 1 a revolu-
tion, &C. tbe motion of the nodes, the inclination of the plane nf her orb. : i i f >
that of the ecliptic, the variation f the areas described about the mn, the
motion of her apogee, the "variation of the eccentricity of her orbit, the equa-
tion of the apogee, equation of the *crntr?, and oilier thing's of a similar
nature.
336 OF THE SOLAR SYSTEM.
mean conjunction, according to Laplace, is 29.530588 days (see
pages 325, 326) it is to the tropical year nearly as 19 : 235, that
is, 19 solar years for about 235 lunar months.
It is in those points of the moon's orbit called the syzygies, that
she is in conjunction or opposition with the sun ; in the first point
she is new, in the second full. In the quadratures^ when she is
distant from the sun 90 or 270, reckoning in the direction of her
proper motion, or in her first and second quarters, we see half of
her enlightened hemisphere, strictly speaking, we see a little
more, for when the exact half is presented to us, the angular dis-
tance of the moon from the sun is a little less than 90. At this
instant, as Laplace remarks, the enlightened being separated from
the obscure part of the moon by a straight line, the radius drawn
from the observer to the centre of the moon, is perpendicular to
that which joins the centres of the moon and sun : so that in the
triangle formed by the straight lines which joins those centres and
the eye of the observer, the angle at the moon is a right one ;
hence the distance of the earth from the sun, may be determined
in parts of that of the moon from the earth. This method is,
however, very inaccurate, from the difficulty of fixing with preci-
sion the instant when half of the lunar disk is enlightened ; how-
ever, it is to this method we owe the first just notions that were
formed of the immense magnitude of the sun, and of his distance
from the earth. An observer will moreover observe, that from
new to full moon the phases are horned, half moon, and gibbous^
and as the enlightened or convex side of the moon is always turned
to the sun, the crescent, or irregular side will appear towards the
east, or, if the spectator be in north lat. towards the left. From
the full to the change, the phases appear in this order, gibbous^
half ?noo?i, and horned ; in these positions, the convex or enlight-
ened side will appear towards the east, and the horns or crescent
towards the west, or to the right hand. The earth exhibits to the
moon similar phases ; when she is new to us, the earth is full to
her, and when she is in her first quarter to us, the earth is in her
third quarter to her, See. In consequence of this, one half of the
moon will have no darkness at all, the earth affording her a much
greater light in the sun's absence than she does to us ;* while the
other halt' has about 14| days darkness and 14| days light alter-
nately. As the moon's axis is almost perpendicular to the eclip-
tic, she has scarce any difference of seasons.
In north lat. all the full moons, in the winter, happen when the
moon is on the north side of the equinoctial ; as they always hap-
pen when the moon is directly opposite the sun. While the moon
passes from Aries to Libra, she will be visible at the north pole,
and from Libra to Aries she will be invisible there ; hence, at the
north pole there is alternately a fortnight's moonlight and a fort-
" As the surface of the earth is about 13 times greater than that of
the moon, it affords 13 times more light to the moon than the moon does
to the earth.
OF THE SOLAR SYSTEM. 337
sight's darkness The same phenomena will take place at the
south pole, in our summer, during the sun's absence.
The explanation of the moon's phases naturally leads us to that
of eclipses ; but as this subject merits a separate chapter, we shall
give it in the following part of the work. The influence of the
moon on the waters of the ocean shall also be explained, when the
laws of gravity, &c. on which it depends, are first investigated.
Before the first and after the last quarter, but principally about
the time of new moon, we can sometimes distinguish that portion
of the lunar disk which is not enlightened by the sun ; this feeble
light is called lumiere cendree, and is caused by the light reflected
from the illuminated hemisphere of the earth on the moon's disk.
This is evident from its being more perceptible at the new moon,
when the greatest part of the earth's enlightened hemisphere is
turned towards the moon. According to Dr. Smith, the propor-
tion of moonlight to daylight, at the full moon, is 90000 to 1 .
Emerson^ in his Optics (B. 1, prop. 20) makes it as 96000 to I.*
But Bouguer has found, by experiment, that it is as 300000 to l.f
This is the reason why the light of the moon, collected in the fo-
cus of the largest mirrors, produces no sensible effect on the
thermometer
The moon's disk is greatly diversified with spots or inequalities,
which have been accurately described. Through a telescope,
those spots have the appearance of hills, valleys, &c. They, how-
ever, shew us that the moon always presents to us very nearly the
same hemisphere, and that she revolves upon her axis in a period
equal to her revolution round the earth. J From the bestobserva-
* Emerson shews (Optics, B. 1, prop. 20, cor. 1) that moonlight is to
daylight as half the square of the moon's radius, to the square of the
moon's distance, when she is full. And in the quadratures as the
square of the moon's radius to the square of ihe moon's distance; and
shews that from the same principle, the light of any other body, com-
pared with daylight, may be found.
t In the English edition of Laplace, this is given as 300 : 1.
i Each of the moon's spots have been distinguished by a proper name,
principally from the most noted astronomers, philosophers, and mathe-
maticians, or from their respective appearances. Thus Sinus roris, Mare
frigomm, Oceanus procellarum, 7e?ra siccitatis, Pains nimborum, Copernicus,
Keplerius, Gnmaldi, Galileo, HerscheVs volcano, &c. Many astronomers
have given maps of the face of the moon : but the most celebrated are those
of Hevelius in his Selenographia, in which he has represented the different
phases of the moon during an entire revolution. Florentine, Langrenus,
Grimaldus, and Iticciolus, have each distinguished himself in describing-
the lunar spots, &.c. Langrenus and Rictiolus denoted the spots by the
names of the principal philosophers, mathematicians, &o giving- the
names of the most celebrated characters to the largest spots, fferelius
marked them with the geographical names of places on ihe earth. The
former distinction is, however, generally followed, though Mayer prefers
Hevelius's figures: see Keii's Astr. lect. 10. The best arid mos. complete
representation of the moon's disk, is that drawn on Mr. Russefs hmar
globe, published a few years ago. This globe not only shews the libra-
tion of the moon in the most perfect manner, but is also a complete pic-
ture of the mountains, pits, shades, &c. on her
T t
338 OF THE SOLAR SYSTEM.
tions, these spots are found to be produced by the mountains ana-
valleys on the moon's surface. This is evident from the irregu-
larity of that part of her surface which is turned from the sun ;
for if her surface was perfectly level or smooth, the illuminated
part of her disk, at the quadratures, would be separated from the
dark by a straight line ; at all other times, this line would appear
of an elliptic form, convex towards the enlightened part of the
moon, in the 1st and 4th quarters, and concave in the 2d and 3d ;
but these lines, so far from appearing regular and well defined,
particularly when the moon is viewed through a telescope, that
they always appear notched and broken in innumerable places.
In all situations of the moon, the elevated parts always cast a tri-
angular shadow with its vertex turned from the sun ; on the con-
trary, the cavities are always dark on the side next the sun, and
illuminated on the opposite side ; moreover, when the sun be-
comes vertical to some of these parts, there is no shadow percep-
tible ; hence these are mountains, and those, that are dark on the
side next the sun, are cavities ; for these appearances are exactly
conformable to what we observe of hills and valleys on the earth.
It is not, therefore, singular that the edge of the moon, which is
always turned towards the sun, is regular and well defined, and
that no indented parts are seen on her surface at the time of full
moon ; for the shining spots on her surface would not be percep-
tible, did not the shade or dark part separate them from the illu-
minated part of the disk ; but in the above circumstances, there
is more of the dark part turned towards a spectator on the earth,
all being equally and more strongly enlightened. The dark parts
by some have been thought to be seas ; but the irregularity of the
line between the enlightened and dark parts, shews that there can
be no very large tracts of water, as such a regular surface would
necessarily produce the line perfectly free from any irregularity.
On the dark part of the moon's disk, near the confines of the lu-
cid part, some bright spots are perceptible ; these shining spots
are supposed to be the summits of high mountains, which are en-
lightened by the sun's rays, while the adjacent valleys, near the
enlightened part, are entirely dark. On this supposition, astron-
omers have determined the height of some of these mountains ;
the method of performing which we shall presently shew.
Continued observations on the lunar disk, have discovered some
small changes in these appearances, so that the same side of the
moon is not always exactly turned towards the earth, the spots that
lie near the edge or limb, successively appearing and disappearing
by periodical oscillations, which have been distinguished by the
name of the libration of the moon.
This phenomenon arises from four principal causes. 1. From
the observer not being placed at the centre of the earth, but at its
surface. Galileo, who first observed with a telescope the moon's
spots, discovered this circumstance ; he observed a small daily
variation, arising from the motion of the spectator about the earth's
OF THE SOLAR SYSTEM. 339
centre, which caused a little of the moon's western limb to disap-
pear, from her rising to her setting, and brought into view a small
portion of the eastern limb. For the visual ray, drawn from the
eye of an observer to the moon's centre, determines the middle
of the visible hemisphere, and it is evident, that from the effect
of the lunar parallax, this radius cuts the surface of the moon at
different points, according to her alt, above the horizon. 2. Ga-
iileo likewise observed that the north and south poles of the moon,
and the part of the surface that are near them, alternately appeared
and disappeared ; this is called the libration in latitude, and is caused
irom the axis of the moon not being perpendicular to the plane of
her orbit, it making an angle of about 1 43' with a perp. to the
plane of the ecliptic. In supposing this axis to maintain its pa-
rallelism diTing the moon's revolution round the earth, it inclines
more or less to the radius vector of the moon, as observed from
the earth ; and the angle which is formed by these two lines, is
therefore acute during one half of the revolution, and obtuse dur-
ing the other half. 3. The third cause, is the unequal angular
motion of the moon about the earth, and her uniform motion about
her axis, which makes a little of the eastern and western parts
alternately appear and disappear, the period of which is a month ;
this is called the Iteration in longitude* 4. The fourth cause of
the libration arises from the attraction of the earth upon the moon,
in consequence of its spheroidical figure.
* The libration in longitude would not take place, if the moon's angu-
lar motion about the earth were equal to her angular motion about her
axis. For if T be the earth, abed the moon at v and e ; let avc be perp.
to Tt>, then abc is that hemisphere of the moon at v which is next the
earth. Now when the moon comes
to <?, if she had no motion on her axis,
bed would be parallel to bvd, and the
same face would not be turned to-
wards the earth. But if b was brought
to coincide with the line Te, by the
moon's rev. on her axis in the direc-
tion abc, the same face would remain
turned towards the earth ; in this case
the moon would have revolved, on
her axis, the angle beT, which is equal
to the alternate angle eTt>, the angle
which the moon has described about
the earth.
The same face of the moon is al-
ways turned towards the earth, when
in the same point of her orbit, and hence, from what we have now
shewn, the time of her rev. in her orbit, is equal to the time of her rev.
on her axis. But as her angular motion eTv about the earth is unequal,
while that on her axis is equal, these two angles cannot continue equal,
and hence, from the above, the same face cannot continue towards the
earth, but in the intermediate points, must vary sometimes a little more
to the east, and sometimes to the west. The greatest libration in longi-
tude is therefore nearly equal to the equation of the orbit, or at its max-
imum about 7$ ; this would be accurately so, if the mooa's axis were
340 OF THE SOLAR SYSTEM.
It is an extraordinary circumstance, as Vince remarks, that the
time of the moon's revolution on her axis, should be equal to that
in her orbit ; and still more extraordinary, that all the secondary
planets should observe the same law Sir Isaac Newton has com-
puted fPrin. B 3, prop. 37) from the altitude of our tides, that
the alt. of the moon's tides must be 93 feet, and that therefore the
figure of the moon is a spheroid, whose greatest diameter pro-
duced, would pass through the centre of the earth, and exceed
the diameter perp. thereto by 186 feet. Hence it is, says he, that
the same face of the moon always respects the earth - t nor can the
body of the moon possibly rest in any other position, but would
always return by a libntory motion to this situation, from the
earth's attraction. But it has be<-n shewn (p. 338) that there can
be no large tracts of water on the moen's surface, and hence Neiv-
ton's supposition cannot account for this phenomenon. The sup-
position of Dr. Mairan is, that the hemisphere of the moon next
the earth is more dense than the opposite one, in which case the
same face would be kept towards the earth, from the earth's at-
traction. We have pointed out a more probable cause '(note, p.
296, &c ) from the moon's having little or no atmosphere.
Whether the moon has an atmosphere or not, is a question,
however, that has long been agitated by various astronomers.
Schroeter, of Liiienthal, in the duchy of Bremen, endeavours to
establish the existence of an atmosphere from the following con-
siderations. 1. He observed the moon when 2| days old, in the
perp to her orbit ; for the equal, of the orbit, or the diff between the
true and mean motion, is equal to the diif. of her mot. about her axis,
and her true motion, which is the libration. As there is no equation of
the orbit in apogee and peng-ee, the same face will then be turned to-
wards the earth. Let T, in the above small fig represent the earth, M
the moon, P6 its axis, not perp. to the phtne of the orbit ev, then at e the
pole P will be visible to the earth, and at v the pole p will be visible ;
hence as the moon revolves about the earth, the poles will alternately
appear and disappear, which explains the libration in latitude. Our sea-
sons are caused in a similar manner from the obliquity of the ecliptic.
From what we have here shewn, it is evident, that one half of the moon
is never visible at the earth ; and that the time of its rotation about its
axis being- one month, the length of the lunar days and nights will be
each nearly a fortnight, being subject but to a small variation, as the
moon's axis is nearly perp, to the ecliptic.
Jfevelius observes, that the libration in lat. was the greatest when the
moon was at her greatest north lat the spots which are near the northern
limb being- then nearest to it ; and that the spots receded from that
limb, as the moon advanced from thence, until she came to her greatest
lat. S, where the spots near the southern limb were then nearest to it.
He found this variation to be about 1' 45', the moon's diam. being 3(/ .
It therefore follows, that when the moon has her greatest lat. a plane
passing through the earth and moon, perp. to the plane of the moon's
orbit, will pass through the moon's axis ; the moon's equator must
therefore intersect the ecliptic in a line parallel to the line of the nodes
of the moon's orbit , so that in the heavens, the nodes of the moon's or-
bit coincide with those of her equator (see Vince's Astr.)
OF THE SOLAR SYSTEM. 34]
evening soon after sun-set, before the dark part was visible, and
continued to observe her until it became visible. The cusps or
horns, appeared tapering in a very sharp, faint prolongation, each
exhibiting its further extremity faintly illuminated by the sun's
rays, before any part of the dark hemisphere was visible ; after-
wards the whole dark limb appeared illuminated. The prolonga-
tion of the cusps beyond the semicircle, Schroeter thinks, must
arise from the refraction of the sun's rays by the moon's atmos-
phere. He also computes the height of the atmosphere, and finds
it = 1356 Paris feet, when it is capable of refracting light enough
into the dark hemisphere to produce a twilight, more luminous
than the light reflected from the earth, when the moon is about
32 from the new ; and that the greatest height, capable of re-
fracting the solar rays, is 5376 feet 2. At an occultation of Ju-
piter's satellites, the 3d disappeared, after having been about 1" or
2" of time indistinct ; the 4th became indiscernible near the limb ;
this was not observed of the other two. PhiL trans. i792 There
is another argument brought forward to prove the existence of a
lunar atmosphere, taken from the appearance of a luminous ring
round the moon, in the time of solar eclipses ; this has been par-
ticularly observed in the total eclipse of the sun in 1706, and in
another in t717, when, during the time of total darkness, certain
streaks of light were seen to dart from different places of the
moon, during the time of total darkness. These were imagined
to be flashes of lightning, and hence the existence of clouds and
vapours) and an atmosphere, has been inferred. These flashes are
also, by some, supposed to be connected with such appearances,
as Dr. Herschel has concluded,* to be volcanoes, which have also
been considered as a proof of the lunar atmosphere.
On the other side it is urged, that as the moon constantly ap-
* On April 19, 1787, Dr. Herscliel discovered three volcanoes in the
dark part of the moon ; two of which appeared to be almost extinct, but
the third shewed an actual eruption of fire, or luminous matter, resem-
bling' a small piece of burning 1 charcoal, covered by a very thin coat of
white ashes ; it appeared about as bright as such a coal would be seen
to glow in faint daylight. The adjacent parts of the volcanic mountain
seemed faintly illuminated by the eruption. Ulloa, in an eclipse of the
sun, discovered a similar eruption, several years ago ; it appeared like
a star near the moon's edge. Another eruption appeared on May 4, 1783.
Phil trans. 1787. On March 7, 1794, a few minutes before 8 o'clock in
the evening-, a brig-ht spot was observed, with the naked eye, on the
dark side of the moon, by Mr. Wilkins, an eminent architect of Norwich ;
he conjectured thnt he saw it about five minutes. London, Phil, trans.
1794. On April 13, 1793, and on Feb. 5, 1794, the celebrated Mr. Pi-
azzi, of Palermo, observed a bright spot on the dark part of the moon,
near Jiristarclms. Several other astronomers have observed the same
phenomenon. Laplace remarks (Astr. ch. 4, B. 1) that the crown of pale
light which has been perceived round the lunar disk, is probably the
solar atmosphere, for that its extent cannot accord with that of the
moon, as we are assured by eclipsas of the sun and stars, that the lunar
atmosphere is nearly insensible.
342 OF THE SOLAR SYSTEM.
pears with the same brightness, when there are no clouds in our
atmosphere, she cannot be surrounded with an atmosphere, at
least like ours, which is so variable in its density, and so frequent-
ly obscured by clouds and vapours. And Vince remarks, that ii
there were much \vater on her surface, or an at a osphere, as
conjectured by some astronomers, the clouds and vapours might
easily be discovered by the telescopes we have now in use ; but
no such phenomena have ever been observed. Lafilace says, that
the atmosphere which we may suppose to surround the moon, in-
flects the luminous rays towards her centre, and if (as should be
the case) the atmospherical strata are rarer in proportion as they
are removed from the surface, these rays, in penetrating into them,
will be inflected more and more, and will describe a curve con-
cave towards her centre. An observer in the moon will not cease
to see a star until it is depressed below the horizon, an angle call-
ed the horizontal refraction. The rays emanating from this star,
seen at the horizon, after having first touched the moon's surface,
will continue to describe a curve similar to that by which they ar-
rived ; thus an observer placed behind the moon, relatively to the
star, wiil see it in consequence of the inflection of the lunar at-
mosphere. The diam. of the moon is not sensibly augmented by
the retraction of its atmosphere ; and hence a star, eclipsed by the
moon, would appear eclipsed later than if this atmosphere did not
exist ; an ! would, for the same reason, sooner cease to be eclipsed.
Thus the effect of a lunar atmosphere would be principally per-
ceived in the eclipses of the sun and stars by the moon. Very-
exact and numerous observations have scarcely indicated a suspi-
cion of this influence ; and, according to Laplace, the horizontal
refraction, at the surface of the moon, does not exceed (5") "62.
At the surface of the earth, this refraction is at least 1000 times
greater.* The lunar atmosphere, therefore, if any exist, must
be extremely rare, and even superior to that produced in the best
air-pumps.
Some remark, that if we reason from analogy, the advocates for
an atmosphere have the advantage over those who contend that
there is none ; but the reverse is, in reality, the case. For it is
not analogy to compare the phenomena of a secondary planet with
those of a primary : the phenomena of the moon, being compared
with those on our earth. Whereas, reasoning from strict analogy,
we should compare her phenomena with those similar phenomena
* The horizontal refraction on the earth is about 33' = 980 ff (see p.
155.) Newton has shewn (cor. 5, prop. 37, B. 3) that the accelerative
gravity, or weight of bodies, on the surface of the moon, is about three
times less than on the surface of the earth, and as the expansion of the
air is reciprocally as the weight that compresses it ; hence the moon's
supposed atmosphere, being pressed or attracted towards the moon's
centre, by a force only one third of that which attracts our air towards
the earth's centre, it follows, that the lunar atmosphere is only one third
as dense as that of the earth, and is therefore, from the laws of refrac-
tion, too rare to produce any sensible refraction.
OF THE SOLAR SYSTEM.
343
in the secondary planets. Now the secondary planets, from the
observations of the most skilful astronomers, are found, like the
moon, to have little or no atmosphere ; and hence the probability,
from analogy, is in favour of those who contend that there is no
atmosphere *
We have shewn that the bright spots which are sometimes vis-
ible on the moon's disk, are the tops of high mountains, the sha-
dow of these, projected on the planes, varying with the sun's po-
sition ; upon or near the edge of the enlightened part of the disk,
we see these mountains forming an indented border extending
beyond the line, whick separates the illuminated and dark part ;
by a quantity from which, being measured, their altitude may be
determined. The method used by Hrudius, Riccioli^ and others,
to determine this alt. is the following. Let SLIM be a ray of light
from the sun, passing the moon at L,
and touching the top of the mountain S
at M ; then the space between L and
M will appear dark. Now by means of
a micrometer, the ratio of LM to the
moon's diameter, or its half LC, may
be determined ; hence LC being given,
LM is therefore given ; then (47 Eucl.
1) CM = ^CL2 -f LM 2 is given ; but
CP = CL, hence PM = CM - CP =
the height of the mountain, is given.
in his Astr. lect. 1 0, remarks, that Ricdoli observed the il-
luminated part of the mountain St Catharine, on the 4th day after
new moon, to be distant from the confines of the lucid surface
about T ^ part of the moon's diameter, or -i part of her semidiam-
ter LC ; hence LC being = 8, and LM == 1, we have CM =
64 -f 1* =v/ 6 "" 5 8.062, and hence PM = .062 ; therefore PC
or LC : PM : : 8 : 062. Now taking the moon's semidiameter
= 1079 miles (p. 66 or p. 344) the height of this mountain will
be = = 8. 1 miles nearly. Galileo makes LM = T ^
8
of LC, from which the height of the mountain will be 5.07 miles;
and ffevelius makes LM = T ' T of LC, from which the mountain's
height = 3.15 miles.
The foregoing method^ as Dr. Herschel observes (Phil, trans.
178 1) is only applicable when the moon is in her quadratures ; he
has therefore given the following general method. Let E repre-
sent the earth (see the last, fig.) draw Erara and bo perp. to the
moon's rad. RC, and br paral. to era, also me perp. to Sra ; then,
to an observer at E, the line mb will not measure its full length,
* We have been rather diffuse on this article ; but the importance of
the subject deserves to be strictly examined, as it may lead to important
results relative to the planets' motions on their axes, &<-. see the note.
p. 296.
344 OF THE SOLAR SYSTEM.
being projected into the line br or on, which will therefore be the
lines observed by the micrometer ; but when the earth is in quad-
ratures at e, the line bm will measure its full length. From the
observed quantity bm or o?z, when the moon is not in her quadra-
tures, to find bm we have the following proportion : the triangles
brm, bCo, are similar ; hence bo : dC :: br : bm = . ButC
bo
is the radius of the moon, and br or on the observed dist. of the
moon's projection ; also bo is the sine of the angle RC = o6S =
the distance, or elongation of the moon from the sun, very nearly,
which may be easily found by calculation, or from the Nautical
Almanac ; hence bm = br divided by the sine of the elongation,
radius being unity, from which m/z, the height of the mountain, is
found as before.
In June, 1780, at 7 o'clock, Dr Herschel measured bm or br,
and found it = 4O"625, for a mountain in the south-east quadrant ;
the moon's elongation was 125 8', the sine of which = .8104 ;
Jience 40"625 -i- .8 104 = 50" 13, the angle under which dm would
appear, if seen directly. Now C6, the moon's semidiameter, was
16' 2"6, and if, with Fince, we take its length = 1090 miles,*
we have 16' 2"6 : 50" 1 3 : : 1590 : bm 56.78 miles ; hence wjt
= 1 4-7 miles, the height of the mountain.
Dr. Herschel has determined the height of several other moun-
tains, and thinks that the height of the lunar mountains is in gen-
eral greatly overrated, and that, a few excepted, they do not ex-
ceed half a mile in their perpendicular elevation. He observes,
that it should be examined whether the mountain stands upon lev-
el ground, that the measurement may be exact : as a low tract of
ground, between the mountain and the sun, will make its alt.
greater, and elevated places will make it lower than its true height.
* The moon's semidiameter may be thus calculated. In the fig 1 , p. 250,
let M represent the earth, AB the moon, AC its semidiameter ; then
MC her dist. from the earth being 1 "238553 milt-s (o 250) and her appar,
diam. at her mean dist. from the earth = 3i' '(" (p 326) or semidiam.
or the angle AMC = 15' 33"5, we have this proportion :
As Cos. 15' 33"5 - - - 9.9999956
To Sine 15' 33"5 - - - 7.6554652
So is 238533 .... 5.3775494
13.0330146
To AC 1079 miles - - 3.0330190
"Having the diameter of the moon given = 1079 X 2 = 2158 miles, its
magnitude from this = ~i- = log-. 7911 3 log-. 2158 3 = 1.6925400
the number corresponding to which, is 49.26 nearly, the number of times
the earth is greater than the moon See this also calculated p. 327.
f For more information on the moon's phenomena, the reader is refer-
red to J\'ewton's Principia, B. 3, iMplace's Celestial Mechanics, Mayer's
Lunar Theory, or the late Tables of M. liurg as published by Vince.
OF THE SOLAR SYSTEM. 345
Schroeter, on the contrary, asserts, that there are mountains in the
moon much higher than any on the earth, and instances one above
1000 toises higher than Chimbara9o.* But, as we have seen above,
a small error in taking the angle with the micrometer, will produce
a great error in the height of a mountain.
Before we finish this chap, it may not be improper to say a few
words on the comparative astronomy of the moon. In one of her
hemispheres the inhabitants in the moon (if any) constantly see
the earth, but in the other never, except, from her libration, those
who are situated near the limits of her disk. To those the earth
sometimes rises a little above the horizon, and sometimes appa-
rently moving backward, subsides below it. In the hemisphere, from
which the earth is visible, it seems as it were fixed to the same point
of the heavens, except a small oscilatory motion from the moon's
libration, whilst in the space of a natural day, the sun and stars
move towards it from the east, and then advance from it towards
the west, the earth's atmosphere concealing the stars, some time,
from a lunar observer. To such of the lunarians as live near the
middle of the disk, the earth appears continually vertical ; but to
some it appears to decline towards the north, and to others towards
the south ; to some towards the east, and to others towards the
west ; the declination being in proportion to their dist. from the
middle of the disk. To the inhabitants who reside near those
limits which divide the visible from the invisible hemisphere of
the moon, one half of the earth will appear always in theii horizon
like a stupendous flaming mountain, its orb appearing to them
nearly 14 times larger than the moon's disk appears to us ;f but
those that dwell in a circle of the moon, passing from the pole*
through the middle, will have the earth always in their meridian,
Sec. &c | In her compound motion round the earth and sun, her
path is every where concave towards the sun, as McLaurin has
shewn in his account of Newton's discoveries B. 4. ch. 5. The
force, as he remarks, that bends the course of a satellite into a
curve, when the motion is referred to an immoveable plane, is, at
the conjunction, the difference of its gravity towards the sun, and
of its gravity towards its primary ; when the former prevails over
the latter, the force that bends the course of the satellite tends to-
wards the sun ; hence the concavity of the path is towards the sun,
and this is the case of the moon, as he proves, in the ch. above
* This author has also lately published a new work, on the height of
the mountains of Venus, some of which he makes 23000 toises in perp.
height, an alt. seven times more than that of Chimboraco.
f Laplace remarks, that at the same distance at which the moon appears
under an angle of 5823" (in his measures) the earth would subtend an angle
of 21352" ; hence the ratio of their diameters are nearly as 3 to 11 ; and as
the areas of circles are as the squares of their diameters (2 Eucl. 12) WQ
have 11 divided by 3 = 13-f nearly.
t For more information on this curious subject, consult Gregory's Astev
B. 6, prop. 9,
Uu
346 OF THE SOLAR SYSTEM.
.quoted * McLaurin further remarks, that when the gravity towards
the primary exceeds the gravity towards the sun, at the conjunc-
tion, then the force that bends the course of the satellite tends to-
wards the primary, and therefore towards the opposition of the
sun ; therefore the path is there convex towards the sun : and
this is the case with Jupiter 3 s satellites. When these two forces
are equal, the path has, at the conjunction, what is called by math-
ematicians a point of rectitude ;f in which case, however, the path
is concave towards the sun throughout.
CHAP. V.
OF MARS.
. MARS is the next planet in order after the earth ; he always ap-
pears of a dusky red colour, and though sometimes apparently as
large as Venus, yet he never appears so bright. From his red and
dull appearance it is very probable that he is encompassed with a
gross, cloudy atmosphere \
We have shewn that the orbits of Mercury and Venus are with-
in the orbit of the earth ; that these planets seem to accompany
the sun, like satellites, their mean motion round the earth being
the same as that of the sua ; that they never recede from the sun,
so as to be seen in opposition, or even a quadrant or 90 from him,
and are only visible a few hours in the morning before the sun ri-
ses, and a few hours in the evening after he sets ; and that they
are sometimes seen to pass over the sun's disk in the form of a
dark round spot, which phenomena evidently prove that they are
situated within the earth's orbit. But Mars, being the first planet
situated without the orbit of the earth, exhibits to the spectator dif-
ferent appearances. He is sometimes seen in conjunction with
the sun, but he is never seen to transit or pass over his disk. He
recedes from the sun to all possible angular distances, is some-
times in opposition, comes to the meridian at midnight, or rises
when the sun sets, and sets when the sun rises ; at this time he
* See also Rowers Fluxions, 2d ed. pa. 225, A Treatise on Ast. by O. Gre-
gory, art. 458, or Ferguson's Ast. art. 266. See also Dr. D. Gregory's Ast.
b. 4, where the theory of the secondary planets are fully established ; or
B. 3, Newton's Prin.
j- See Simpson's Fluxions, vol. 1. sect. 8.
t Emerson, in his Optics (B. 1. prop. 12. cor. 4) has shewn, that the
red rays are the strongest, are the least refracted, or turned out of their
way, and penetrate furthest into a resisting- medium. And that the rest of
the colours grow weaker in order, the violet being- the weakest. This is
also proved from the observations of those who dive into the sea ; for the
deeper they g-o the redder the objects appear, the other rays being- reflect-
ed back. And we see that the sun and moon always appear ruddy in the
horizon, where their light has to pass tliroug-h a greater portion of the at-
mosphere, than when they are in the zenith.
OF THE SOLAR SYSTEM. 347
appears brightest being nearest the earth. The disk of Mars,
when viewed through a telescope, changes its form and becomes
sensibly oval, according to his relative position from the sun ;
sometimes appearing round, at other times gibbous, but nearer
horned. These phenomena evidently shew, that Mars moves in an
orbit more distant from the sun than that of the earth (see p. 257)
and that it is from the sun he receives his light. When viewed!
from the earth, he appears to move sometimes from west to east)
at other times from east to west, and sometimes he appears sta-
tionary ; 'which shews that he does not regard the earth as his
centre of motion, not observing the equal description of areas j
(see pages 258, 306 and 307) but when viewed from the suii
Mars observes this law, and always performs his motion from west
to east, which proves that he regards the sun as the centre of hi3
motions.
The mean length othe sidereal revolution of Mars is 686,979579
days, or I y. 32 Id. 23 h. 30' 35" 6. His synodic revolution is
1 y. 321 d. 22 h. 17' 56".* His motion is very unequal. In the
morning when he begins to be visible* it is then direct and its ve-
locity the greatest ; it becomes gradually slower until the planet's
elongation from the sun is about 136 48', where he becomes
stationary ; after which his motion becomes retrograde, increas-
ing in velocity until the planet is in opposition with the sun. At
this time his velocity is a maximum, after which it diminishes and
again becomes nothing, when Mars is distant from the sun 136
48' as before. His motion then becomes again direct, after hav-
ing been retrograde during 73 days ; in which interval, the arc of
retrogradation, described by the planet is about 1 6 1 2', immerg-
ing at length, in the evening into the sun's rays. Mars renews
these singular phenomena at every opposition, with considerable
variations, however, in the extent and duration of his retrograda-
tion s.
The rotation of Mars on its axis is found, from some spots on
his surface, to be from west to east. M Causing in 1666, dis-
covered some well defined spots, from which he determined the
time of the rotation to be 24 h. 40'. M. Miraldi determined the
time of rotation to 24 h. 39'. Dr. Herschel makes the time of
rotation = 24 h. 39' 21" 67, without the probability of a greater
error than 2" 34 ; he remarks that the spots are permanent, and
* The mean motions of the planets being 1 given from the ast. tables, the
length of their synodic, or tropical year, can be easily found, by saying-, as
the mean motion for a year to 360, so is 365 days to the synodic revolu-
tion. Thus for Mars, the mean motion in a year, according- to "Ddambre (tab,
112) is 6s. 11 17' 10" ; hence 191 If 10" : 360 :: 365d. : 686d. 22h. 17' 56"
nearly. Independent of the tables the length of a tropical year may be
found, by saying-, as 360 to 360 less the precession of the equinoxes, during-
the time, so is the length of the planet's sidereal revolution, to the length of
its synodic or tropical revolution. And on the contrary the tropical rev. be-
ing- given, the sidereal is found as shewn before, pa. 246. See also pa. 303,
304, &e.
348 OF THE SOLAR SYSTEM.
that this planet has a considerable atmosphere. Lajilace makes
this motion on its axis 1.02723 clays = Id Oh. 39' 12" 672, and
on an axis inclined to the ecliptic in an angle of (66 33') 59 41'
49" 2 Dr Htrschel (fihil. trans. 1784^ makes the axis of Mars
inclined to the ecliptic in an angle of 59 -2', and to his orbit in
an angle of 61 18' , and the north pole to be directed to 7 47'
of Pisces upon the ecliptic, and 19 28' on his orbit. He makes
the ratio of his diameters as 16 : 15 ; but Dr. Ma^kelyne^ who
carefully observed Mars at the time of opposition, could perceive
no difference in his diameters.
According to Vince, the relative mean distance of Mars from
the sun is 15236 K that of the earth being 100000. This distance
may be found, by Refills rule, from the periodic time (see p 253*)
or by the note p 260 Kepler has also given the following me-
thod (in his works, de motihus stellx Marfis.) Let S (fig. p. 260)
represent the sun, P Mars, L, K, two places of the earth, when
Mars is in the point P of his orbit. When the earth was at L,
Kepler observed the difference between the longitude of the sun
and that of Mars, or the angle PLS ; in the same manner he ob-
served the angle PKS Now the places L, K, of the earth in its
orbit being known (which for any given time may be found from
Astr. tables, or the Naut Aim pa. 11, being the point opposite
the sun's place) the distances Ls, KS, and the angle LSK, will
be given ; hence in the triangle LSK, LS, SK, and the angle
L.SK, are given, to find LK and the angles SLK, SKL ; hence
the angles PLK, PKL, and the side LK, are given, to find PL ;
and lastly, in the triangle PLS, PL, LS, and the angle PLS are
given, to find PS the distance of Mars (or of any other planet)
from the sun-t
* The learner will take notice, that the mean dist. of a planet from the
sun, is equal to half the transverse, or greatest diameter of its orbit. For
the mean dist. is expressed by a line drawn from the focus of the orbit,
to the extremity of the conjugate axis, and this line is always equal to half
the transverse. (Emerson's Conic Sect. b. 1, prop. 2.)
j- Kepler also determined the angle PSL, or the diflT. of the heliocentric
long", of Mars and the earth. From the above method, from his observa-
tions on Mars in the aphelion and perihelion (as lie had before determined
the position of the line of the apsides) he determined the dist. of Mars from
the sun in his aphelion to be 166780, and in the perihelion 138500, the
earth's mean dist. from the sun being- 100000 ; the mean dist. of Mars was
therefore 152640, and the eccentricity of his orbit 14140. He also deter-
mined three distances of Mars as follow ; 147750, 163100 and 166255 ; he
calculated these same distances, on supposition that the orbit was circular,
and found-them equal 148539, 163883 and 166605 ; and therefore the errors
of the circular hypothesis were 789, 783 and 350 respectively. From these
observations, Kepler, relying on Tycho Brakes' observations, first concluded
that the orbit of Mars must be oval. After this discovery Kepler discovered
the relation between the mean distances and the periodic times of the plan-
ets, and thus laid the foundation of the Principia of Newton, and conse-
quently the foundation of all Physical Astronomy. The angle under which
the semidiameter of the earth's orbit, or the parallax of its annual orbit, ns
seen from Mars, being eauai to the angle LPS, is, at a medium equal 41 9 ,
tbe greatest being 47 24 .
OF THE SOLAR SYSTEM. 349
Lqfrface makes half the greater axis of the orbit of Mars, or his
mean distance equal 1.5^3693 the earth's being 1. He also makes
the proportion of the eccentricity of the greater axis 0.093808, and
the secular augmentation of this proportion 0.000090685 Vince
makes this eccentricity 14183.7, the earth's mean dist. from the
sun being taken 100000 j and also makes the greatest equation
equal 10 40' 40". Delambre (table 116) has given the log. of
the mean dist. of Mars from the sun, for every degree of his mean
anomaly, allowing the secular corrections, &c. He makes his
greatest equation (tab. 115; 10 40' 39".
Lafitace remarks that the variations in the afifiarent diameter of
Mars are very great ; he makes it in its mean state about (30")
9" 72, and when greatest (90") 29" 16 when the planet ap-
proaches his opposition. M. Mollet in his Etude de del (p. 223)
makes his mean diameter 1 i-i". Some make the greatest appar.
diam. 2s". Lafilace further remarks, that when the app. diam.
is greatest, the parallax of Mars becomes sensible, being then
nearly double that of the sun, or equal to 17" 6. M la CaiUe
makes his horizontal parallax at the time of opposition 23" 6
(see p 329.) From other observations he makes the hor. par. =
27 \" (3d. method of determining the paral. pa. 330.) The same
law vvhich exists between the parallax of the sun and Venus, exists
also between that of the sun and Mars, as we have shewn pa. 279,
and this last parallax had given a near approximation to the sun's
paral. before the transit of Venus had more accurately determin-
ed it (See pa. 330.)
From the periodic time, the mean distance* of Mars is
142088087.7 miles ; his real diameter 4887 miles,f and his mag-
* The planet's synodic rev. eo 686d. 22h. 17' 56" (pa. 34-7) = 59350676",
the square of which is 3522502741656976, which being divided by the sq.
of the earth's per. rev. = 995839704797184 (pa. 258) gives 3-537208573
sube root of which is 1.5307 nearly, the dist. of Mars from the
nearly, the cube
sun, that of the earth being- an unit or 1; hence 1.5307 X 23464.5 (see pa.
258) =35917.11015 dist. of Mars in semidiam. of the earth; therefore
35917.11015 X 3956 == 142088087.7 miles the mean dist. of Mars from the
sun.
f Now taking the hor. par. at the time of opposition 23"6 as given by
La CaiUe. Then in the fig. pa. 250, let M represent Mars, and AB the
earth, and we have,
As sine AMC 23"6 - - - 6.0583927
To radius or sine 90 - - 10.0000000
So is AC = 1 semidiam. - 0.0000000
To MA = 8741.93 semidiam. 3.9416073
Hence Mar's dist. from the earth at his oppos. is 8741.93 semid. of the
earth, and taking- his appar. diam. at oppos. =s 29"16, it will be 23464.5 :
8741.93 :: 29"16 : 10*86, the app. diam. of Mars as seen from the earth at a
dist. equal that of the sun; therefore 32' : 10"86 :: 864065.5 (the sun's diam.
pa. 255} : 4887 miles, tlie diameter of Mars.
350 OF THE SOLAR SYSTEM.
nitude is something more than \ of that of the earth.* His velo*
city in his orbit round the sun is about 54152 miles an hour /f
and his light and heat is in proportion to that on the earth as 0.43
to 1 nearly4 The distances of the exterior planets may also be
found from the parallax of the earth's annual orb, as shewn before.
The mean Longitude of Mars at the commencement of 1750,
reckoning from the mean vernal equinox at the epoch of the 31st
of December, 1749, at noon, mean time at Paris, was, according
to Lafilace (24 4219) 21<> 58" 46" 956. Longitude of the per-
ihelion at the beginning of 1750 (358 3005) 321 28' 13" 62.
The sidereal and secular progressive motion of the perihelion
(4834" 57) 26' 6" 4 The inclination of his orbit to the plane
of the ecliptic (2 0556) 1 51' ; and its secular retrograde varia-
tion (4/' 45) 1" 44. Longitude of the ascending node upon the
ecliptic (52 9377 > 47 38' 38" .'48 ; audits sidereal and ecu-
lar retrograde motion upon the true ecliptic (7027" 41) 37' 56"
88. Vince makes the place of the aphelion for 1750, 5s. 1<> 28'
14", and its secular motion in longitude 1 51' 40". In 1760
(Bissexiile) Delambre makes the mean place of the aphelion 5s.
1 39' 34", and of the node Is. 17 43' >8". In 18 (com.
year) he makes the place of the aphel in 5s. 2 35' 24", and of
the node 'is. 18 6' 38". According to him the annual mean,
motion of Mars is 6s 11 17' 10", of the aphel. 1' 7", and of the
node 28" ; the daily mean motion is 31' 27".
The learner should take notice, that all the epochs in the As-
tronomical tables are reckoned from noon on December 3 1 , for
ommon years, and from January 1st for the bissextiles.
* To determine his magnitude we have 79113 divided by 48873 =5 log 1 .
79113 4887 3 r= .6275673,^ the number corresponding to which is 4.234,
wh ch shews that the earth is more than four times as large as Mars. It is
proHblv greater in proportion, as we have taken Mars' greatest diam. as
given by Laplace ; if 25" were taken the proporti on would be greater.
f The mean dist. 142088087.7 X 2 X 3.1416 = 892767872 miles nearly,
the circumference of the orbit of Mars ; and therefore 59350676" : Ih. or
3600" :: 892767872m. : 54152 miles.
4 The effects of light and heat being reciprocally proportional to the
squares of the distances from the centre from which they are propagated ;
hence the sq. of the earth's distance from the sun divided by the sq. of the
<iist. of Mars, the quot. will be the comparative heat, &.c.
Thus in the fig. pa. 260, in the right angled triangle SLP, taking the
angle LPS at a medium = 41 (see pa. 348) and SL, the earth's mean dist.
from the sun = 23464.3 semidiam. SP, the dist. of Mars from the earth is
found = 35766 semid. of the earth.
In Mercury and Venus as the sidereal rev. had been taken in place of the
tropical, to correspond with the tropical rev. of the earth, in finding their
dist. from the sun (pa. 258 and 271) this will therefore alter a little the dis-
tances thus determined. When the sidereal rev. of the planets is used, the
dd. rev. of the earth should also be used, which is nearly 365d. 6h. 9 ; 11"
=c 31558151", the sq. of which is 995916894538801, a constant divisor, for
the sidereal rev. of the planets. The examples of different authors, who
could scarce be suspected of the above mistake, was the cause of this inad-
vertent error, which the reader, knowing the principle, may correct, at his
leisure.
OF THE SOLAR SYSTEM. 351
Now the places of the aphelion and nodes being given, and the
planet's true anomaly (found from the theory of the filanet's motions,
given in ch. 4. see pa. 312) its distance from the node, which is
called the argument of latitude, is given ; from which we can find
the central distance of the planet, and its curtate distance from
the sun, or the dist from the sun to that point where a perp. let
fall from the planet meets the ecliptic. From these data the geo-
centric place of the planet may be easily
found. Let TO be the earth's orbit,
and T the earth's place ; DNP the orbit
of a planet, and P the planet's place ; S
the sun ; nSN the line of the nodes
From P let fall on the plane of the eclip-
tic the perp. PB, join SB and produce it
until it meets the planet's place reduced
to the ecliptic, found by the arch PN,
(the planet's dist. from the node) and the
inclination of the orbit to the plane of the
ecliptic, which are given (for Mars see
page 350) but the place of the earth
seen from the sun, or the point opposite
the sun's place, is given ; and hence the angle between them, er
the angle TSB, which is called the angle of commutation, is given.
Then in the triangle STB, ST the earth's dist from the sun, and
SB the curtate dist of the planet, are given (see pa. 265) the an-
gle TSB is given, being the elongation of the planet from the sun,
pr the arc of the ecliptic intercepted between the sun's place, 'and
that of the planet reduced to the ecliptic ; as also TB the curtate
dist. from the earth. And as the sun's place is given, the place
ot the planet, as seen from the earth, is likewise given. Again in
the triangles PSB, PTB, right angled at B, tang. PSB : tang.
PTB :: TB : SB :: sine of the commut. TSB : sine of the elong.
STB. Hence as sine of the commutation : sine of the elong :: tang,
heliocentric lat. : tang, geocentric lat. In this manner the geocen-
tric place, or the place seen from the earth, and the lat. of a planet
is found for any time. (See Dr. Gregory's Astr. b. 3. where
several curious and important points relative to the theory of the
primary planets, are fully investigated )
The following observations principally collected from Delambre
and la Lande, will be of use to the learner.
The mean longitude of a planet, seen from the sun, as also that
of the sun and moon, is found by adding the epoch to the mean
motions.*
* Thus if it be required to find the rnoorSs mean tongitude, necn anomaly,
and place of t^ node y on the I9;h of May, 1819, at 5 o'clock. In the
noon, at New-York. If we make use of belambre's tables, th< times i:
reduced to the meridian of Paris, and asthediff. of long-, is 76 2^' 1 /, the
352 OF THE SOLAR SYSTEM.
The longitude of the ajihelion taken from the planet's mean lon-
gitude, will give the mean anomaly. If the sun's longitude, in-
creased by 20", on account of the aberration (pa. 302) be taken
from the reduced heliocentric long, if the remainder exceeds 6
signs, subtract it from 12 signs ; half this commutation, or suppl.
of 12 signs, is called the semi-commutation. Also if the difference
between the log. of the sun's dist. from the earth, and that of a
planet's dist. from the sun being found, and so be added to the
caracteristic of the remainder, let this diff. be found in the log.
tangents, and from the corresponding angle let 45 be subtracted
(la Lande's Astr. Art. 3850) the log. tang, of the remainder add-
ed to the log. tang, of the semi-commutation, will give the log.
tang, of an angle, which added to the semi-commutation for the
superior planets, or subtracted from the semi-commutation for the
inferior, the sum or remainder will give the planet's elongation
respectively (La Lande, Art. 114<2.)
The geocentric longitude is found by adding the elongation to
the sun's long, when the commutation is less than 6 signs, or sub-
tracting it when the commut. is greater.
La Lande (art. 1146) gives the following rule for finding the
planet's dist. from the earth ; as sine elong. : sine commut. ::
planet's dist. from the sun reduced to the ecliptic : the dist. from
the earth on the plane of the ecliptic : this distance divided by the
cosine of the geocentric, lat. gives the direct distance from the
earth to the planet. Also the diameter of the planet for the mean
dist. of the sun, being divided by the dist. of the planet from the
earth, will give its actual and apparent diam. seen from the earth.
(La Lande, art. 1391 and 1384.) And the sun's parallax being
divided by the same dist. of the planet from the earth, gives the
hor. parallax of the planet. (La Lande, Art. 1631.)
Vince shews how to reduce the places of the planet's seen from
the earth to their places seen from the sun, as follows.
Let T be the place of the earth (see the iig pa. 35 i ) P the planet.
S the sun, T the point aries ; let PB be drawn perp. to the ecliptic,
and TS be produced to /. Let the longitude of the sun seen at (, at
the time of observation be computed (note to prob. 1 and 3, part III)
and the opposite point in the ecliptic is the long, of the earth at T,
or the angle TST ; compute also the long, of the planet, or the
angle TSB (pa. 195) and the difference of these two angles is the
angle of commutation TSB. The place of the planet in the eclip-
difference of time is 5h. 5 f 22"8 ; hence the corresponding time in Paris is
2 Oh. 5' 22", which is the time for which we must calculate.
Moon's long.
Epoch for 1819 (tab. 26) 10s. 2639' 31"3
May 19 (tab. 28) 1 1 31 9
Motion for lOh. (t. 29) 5 29 24 6
Motion for 5' 2 44 7
Motion for 2i"8 12 5
Me tin cmom.
7s. 18 8' 37*4
16 2 02
5 26 37 5
2 43 3
124
Slip, ofthenode.
11s. 4o 11' 25"0
7 21 38 8
1 194
07
Mean long. reg. 3 43 2 1
8 9 40 10 8
11 11 34 21 9
OF THE SOLAR SYSTEM. 353
tic being observed, and the sun's place being known, we have the
angle BTS, the elongation in respect of the longitude; hence the
angle SBT, which is the measure of the diff of the planet's pla-
ces, as seen from the earth and the sun, is given ; therefore the
geocentric place of the planet being known, its heliocentric will be
known. Moreover tang. PTB : rad. :: BP : TB (Simson's
Trie:, prob 1.) and rad. : tang. PSB :: JBS : BP ; hence tang.
PTB : tang. PSB :: BS : TB :: sine STB : sine TSB ; that
is, sine of the elong in long. : sine diff. long, of the earth and
planet :: tang, geocentric lat. : tang, heliocentric lat. When the
lat. is small SB : TB :: PS : PT very nearly, which, in oppo-
sition, is very nearly as PS : PS ST. Or the values of PS
and ST can be computed with more accuracy (see the method in
pa. 314) than we can compute the angles STB, TSB. The cur-
tate dist. ST of the planet from the sun, is found by this propor-
tion rad. : cos. PST :: PS : SB.
The place of the planet's node may be thus determined ; find its
heliocentric lat. immediately before and alter it has passed the
node, and let n, m> be the places in the orbit, and , a, the cor-
responding places reduced to the ecliptic ; then the triangles u^N
?aN, which may be considered as rectilinear, being similar, we
have nb : ma :: N : Na ; hence nb -f ma : nb :: Nb -f- Na
(ab) : Nb (5 Eucl. 18) or alternately, nb -{-ma : ab :: nb : N,
that is, the sum of the two latitudes : diff of the longitudes ::
either lat. : dist of the node from the long, corresponding to that
lat. Or it will be very nearly as accurate to take both latitudes
from the earth, when the observations are made in opposition. If
the dist. of the observations exceed 1, this rule will not be suffi-
ciently accurate, in which case we can compute by spherical trig,
thus, in the rt. angled spher. triangle aNw, we have, by Napier's'
rule, rad. X sineaN = tang ma. X cot. L N ; hut rad. being =
1, and sine aN =* sine ab bn ; hence, sine ^ = cot. N ;
tang ma
and (Napier) rad. X sine Nb = tang, nb X cot. N ; hence cot.
XT sine JN * " i _ _ _" * _ T
N - ; but sine ab No = sine ab x cos. No sine
tang, no
N x cos. ab (Vince's Trig. art. 101, or Emerson's b. 1. prob. 6.)
sine ab x cos. Nb - sine No x cos ab sine Nb
tang, ma
f s. ab X cos. Nb s Nd
tung. nb
s. No X cos. ab
>
tan. ma t. nb
tan. ?
. ab x tan 720
s. NoXtan. wa-}-cos. aoxtan. nb s
: hp.nc.e. -
X tan. ma tan. wa-f-cos. a^xian. nb
= rr: = (Emerson's Trig. b. 1. prob. 1. cor. 4 ) taner. N6.*
cos Nb
* Mr. Bngge having- observed the rt. as. and decl. of Saturn, from thence
found the following heliocentric longitudes and latitudes (see prcb. 3, part
3, and the above rule for the heliocentric lat. Ste,)
W w
354 OF THE SOLAR SYSTEM.
The inclination of the orbit may be thus determined ; bn the
lat. of the planet, and 6N its dist. from the node on the ecliptic
being given, we have sine AN : tan. nb :: rad. : tan. L^ N ; or
by Nafiier's theorem tang, bn : rad. :: s AN : cot. N the incL
required. The observations must not however be taken near the
node, in determining the incl. as a very little error in the lat. will
make a considerable error in the inclination.*
Or the incl. may be thus found Find the angle PSB, as shewn
above, then the place of the planet and that of its nodes being giv-
en, BN is given ; hence (Simson's Spher Trig. prob. 17.) sine
BN : tang. PB :: rad, : tang. PNB the incl. of the orbit. Or by
Napier's rule, rad X s BN =* tan. PB x cot. PNB ; hence tan.
PB : rad. :: s BN : cot PNB
On March -7, 1694, at 7h. 4' 40", at Greenwich, Mr. Flam-
stead determined the right ascension of Mars to be 115 48' 55"
and his decl. x!4 10' 50" N. hence the geocentric long found as
directed above, was Cancer 23 26' 12", and lat. 2 46' 38"
Let P represent the place of Mars, B his place reduced to the
ecliptic, and S, T, the places of the sun and earth respectively ;
then the true place of Mars, by calculation, as seen from the sun,
-was Leo 28 44' 14", and the sun's place Aries 7 34' 25" ;
hence the diff. of these places or the angle BTS = 105 51' 47" ;
and the earth's place being Libra 7 34' 25", if the place of Mars
be taken from it, the remainder is the angle TSB = 38 50' 11";
hence from the above, sine 105 5t' 47" : sine 38 50' 1 1" ::
tang. PTB = 2 46' 38" : tang. PSB = 1 48' 36". Now the
place of the node was Taurus 17 15', which subtracted from Leo
1784. Jlppar.
time.
ffelioc. long".
Helioc. lat.
July
12 at
I2h.
3'
1"
J 9s.
20
37'
25"
3 r
13" N.
20
11
29
9
S 9
20
51
53
s
2
41
Aug.
1
10
38
25
21
13
17
so
1
34
8
10
9
9
21
26
2
S
56
21
9
14
59
S 9
21
49
27
s
2
27
8
50
19
S 9
22
12
v
27 S.
31
8
33
47
S9
22
7
32
S
50
Sep.
5
8
13
45
S 9
22
16
28
s o
1
21
15
7
33
45
S 9
22
34
32
1
59
Dec.
8
6
4
23
S 9
23
16
15
so
3
35
From the obs. on August 21 and 27, the triangles being- considered as
plane, N6 = 4*" 5, from the observations on the 21 and 31, N6 =c 42" 5 ;
and from those on Aug. 21, and Sept. 5, N6 = 4G" ; the mean of these
gives N6 = 42". Mr. Biigge makes N6 = 41", either from taking the
mean of more obs. or computing from spher. hence the heliocentric place
of the descending node was 9s. 21 50' b" 5. On Aug. at 9h. 12' 26" true
time, Saturn's hel. long, was 9s. 21 49' 27", and on 27, at 8h. 49 / 23" true"
time,, it was 9s. 22 0' 12"; hence in 5d. 23h. 56' 57" Saturn moved 10* 45 y
in long, therefore 10' 45" : 41" :: 5d. 23h. 36' 57" : 9h. 7' 44" the time of
describing 41" in long, which being added to Aug. 21, 9h. 12' 26", gives
Aug. 21, l8h. 20' 10'', the time when Saturn was in its node.
* The obs. on July 20 (see the last note) gives the angle 2 38' 15"
that on Oct. 8, gives it = 2 22' 15" ; the mean of these is 2 30' 14" the
inclination of Saturn's orbit to the ecliptic from ftiese observations..
OF THE SOLAR SYSTEM. 355
28 44' 14", gives 101 29' 14" = BN the dist. of Mars from
his node ; hence sine BN 101 29' 14" : tang. PB 1 *8' 36" ::
rad. : tan. PNB = 1 50' 50" the inclination of his orbit. Mr,
Bugge makes the incl 1 50' 56" 56 for March, 1788. M. de
la Landc makes it i 51' for 1780.
If we conceive lines to be drawn from t to P and B. it is evident
that the angle PTB will be much greater than PtE ; and that
also the angle PTB is greater than PSB, while T is nearer to P
than S, and less when further from P, as at t ; hence when the
earth is in T, the geocentric lat. of Mars is greateMhan his helio-
centric, but when the earth is in t, the heliocentric is greater than
the geocentric ; hence the visible lat. of Mars vary according to his
various positions ; so that, other circumstances remaining the same,
his latitude is greater the nearer he approaches hi> opposition with
the sun, and becomes less as he approaches his conjunction. The
same reasoning may be applied to the other superior planets Jupi-
ter, Saturn, Herschel, and also the newly discovered planets Ceres,
Pallas, Juno and Vesta.
When the planets are in opposition to the sun, they rise when
the sun sets, and set when he rises ; after they depart from the
opposition, they appear to the eastward of the sun, and after sun
set they are visible in the evening until their conjunction with the
sun, when they rise and set with him. As they recede from the
sun, after their conjunction, they are visible only in the morning'
before sun rise, for they set in the evening before the sun. When
they come to their opposition again, these phenomena will appear
in order as before, &c. In their oppositions their appar. diame-
ters appear much larger than in conjunctions, being nearest the
earth in one position, and furthest in the other ; the diff in their
distances, in these two positions, being equal to the diameter of the
earth's orbit ; and, as this bears a considerable proportion to the
dist. of Mars, being about Jive times nearer the earth in opposition
than in conjunction, his app. diam. will be 5 times greater in the
one than in the other, and hence, as his visible disk and lustre in-
creases as the squares of the app. diam, he will in oppos. be twen-
ty times larger and brighter than in conjunction.
All the superior planets observed from the sun, will appear to
move regularly the same way, though with unequal angular mo-
tions, arising from their different distances, but yet so as to observe
the general law of describing equal areas in equal times, round
the sun. But when observed from the earth, their appearances arc
very different : they sometimes move forward, or direct, that is
from west to east, at other times retrograde, or from east to
west, and at other times they appear immoveabie or stationary.
356
OF THE SOLAR SYSTEM.
(See pa. 258 ) Let TEe be the orbit
ot the earth, A MO the orbit of a supe-
rior planet, as Mara* and S the sun's
place. No'v let the earth be at T when
M irs is at M, then Mars and the sun
will b- in conjunction ; but if the earth
be at t. when Mars is at M, they will be
in opposition. If the earth remained
fixed at F, the motion of Mars would
appeur always direct ; but the ear in
moving in its orbit quicker than Mars
in his. the latter will therefore be over-
taken by the earth, and afterwards the
earth wiii advance before him, so that
he will seem to move backwards. If
we now suppose the earth at E when
Mars is at M, he will be seen in the heavens among the fixed stars
at a, and for some time before and after he has been at E, he will
appear nearly in the same point a, that is, he will be sta wnary.
While the earth moves in its orbit through the space Ete, if
Mars remained without any motion at M, he would appear to move
in a retrograde direction, through the arc aPrd, among the fixed
stars, and would again appear stationary at b ; but if while the
earth moves from E to e, M:irs moves from M to O, his retro*
grade motion will be nearlv represented by the arc aPr. Hence
Mars will have two stationary and one retrograde appearance
What we have here shewn with respect to Mars, is also true of
Jufiiter^ Saturn, and all the superior planets. But Saturn's re-
gressions are more frequent than Jupiter's, and Jupiter's than
those of M irs, because the earth overtakes them, or passes be-
tween them and the sun, oftener than between the sun and Mars ;
but the retrogressions of Mars are greater than those of Jupiter,
and Jupiter's greater than Saturn's, Sec. For more information
on this subject see Keil's Astronomy, lect. 27.
From what we have given above it will be seen that Mars* year
js nearly equal to two of our years ; but that the natural day is only
a little greater than ours. The axis of Mars' diurnal revolution
being nearly perp. to the plane of its orbit, it follows, that the ar-
tificial d ty, exclusive of twilight, is almost constantly equal to the
night, on every p*rt of this planet's surface ; and that therefore
there is little diff between his summer and winter. However,
from the different inclinations of the sun's rays to the horizon, as
on the earth at the equinoxes, the heat constantly diminishes from
his ; quator to the poles ; and hence Dr. Gregory supposes the
fascia, or belts of Mars, to be produced from this phenomenon.
For as the same dt-gree of heat always continues in the same cli-
mate, it is probable that the spots in Mars, like the clouds and
snow on the earth, being produced by heat and cold, are extended,
according to the climate, and make Fascia or belts parallel to tine
OF THE SOLAR SYSTEM. 357
circles of Mars* diurnal motion. Similar phenomena existing in
Jupiter, who, like Mars, has a perpetual equinox, strengthens the
conjecture. (See Gregory's Astr. b. 6. prob. 4.)
In our earth and moon, an observer in Mars will have a phe-
nomenon, which is not seen by us, that of an inferior planet with
a satellite ; though they will never appear to him to be one quar-
ter of a degree from each other To him our earth will appear
about as large as Venus does to us, and its elongation from the
sun, will never appear to him to be more than about 48. Like
Mercury and Venus, it will sometimes be seen by him to pass
over the sun's disk. Venus will be as seldom seen by him as we
see Mercury, and Mercury will never be visible to him, unless as-
sisted with good telescopes or other substitutes. We have re-
marked before (pa. ^77 that, as we are now better acquainted
with the planet's motions and phenomena, than in any preceding
period, these phenomena represented on the globe, or with aa
orrery, would be very interesting. And from what is shewn, in
parts 2 and 3, the learner will be able to pursue this entertaining
subject at his leisure.
CHAP. VI.
OF THE MEW PLANETS,
CERES, PALLAS, JUNO AND VESTA.
HERSCHEL, in : 780, having enriched our system with anew
planet, it seemed at that time, that in this regard, all was then dis-
covered, and that the number of the planets was fixed to seven ;
but recent discoveries have shewn that they are not thus limited,
and that we are, as yet, far from being acquainted with their num-
ber. We have spoken of the new planets discovered, since this
time, in pa. 47, note to def. 119, but here it becomes necessary to
speak more particularly of them.
The celebrated M. Hazzi, astronomer royal at Palermo in Sici-
ly, on the tst of January, 1801, augmented the number of the
planets al^eidy known in our system, by adding another, to which
he gave he name ot Ceres,* called Ceres Fernundea, in honour of
Ferdinand IV. king of the Two Sicilies. Her orbit he found to
be situated between the orbits of Mars and Jupiter, at the distance,
according to some, of about 94 millions of leagues, or according to
others of 2| times that of the earth, from the sun. Her periodic
* Ceres and the names of the other newly discovered planets, Pallas,
Juno, and Vesta, were given these planets in allusion to the heathen names
given the other primary planets, to preserve an uniformity and similarity,
with the names in the ancient system. The planet Herschcl or Uranus be-
ing exterior or superior to Jupiter, has received a title of greater antiquity,
but these being- interior or inferior, have received titles which indicate a
more recent daie. For their mythological explication see Lempriere's Clas-
sical Dictionary, or JLittkton's Latin Dictionary.
358 OF THE SOLAR SYSTEM.
revolution is, according to some, 4 years 7 months, and according
to others, 4 years 8 months; but the elements of her theory is ay
yet very imperfectly known. She is found not to he confined with-
in the ancient limits of the zodiac ; she is invisible to the naked
eye, being of the 8th magnitude, and cannot therefore be seen
-without a goed telescope. Her diameter, according to Dr. Hers-
chel, is about 1 62 miles.
Dr. Others^ of Bremen, discovered a 9th planet on the 28th of
March, 1 802, to which he gave the name of Pallas. It is of the
7th magnitude, and was then situated near the northern wing of
the constellation Virgo. Her orbit is about equally placed between
Mars and Jupiter, and stated to be at nearly the same distance
from the sun as Ceres, The theory of the phenomena of this
planet is still less known than that of Ceres, and hence the account
of her dist. period, magnitude, Sec must be very imperfect. Her
periodic revolution is reckoned to be about 1 month more than the
planet of Piazzi, and her diameter about 95 miles.
These discoveries surprized all the astronomers, in pointing out
to them new planets which till then had escaped their researches,
and attended with phenomena, which they had never before ob-
served. Two planets placed at nearly equal distances from the
central body (the sun) is a phenomenon entirely new, and which
may give place to very extraordinary results, altogether unforeseen
or unexpected. It is true that these planets move in different
Jilanes, and that the eccentricity of their orbits is not the same ;
but after all it may possibly take place, that their approach may be
too near ; and, if so, it will then be curious to observe the effects
which would result from this too near proximity.
Some German astronomers, having considered the relative dis-
tances of the planets from the sun, concluded that there was want-
ing another planet between Mars and Jupiter, and hence they en-
deavoured to find it ^ut ; but their wishes are more than gratified
in the discovt-ry, not only of one or two, but even of four. How-
ever some, before the discovery of the two latter, to account for
the phenomenon of the two planets being equally dist. from the
sun, asserted that they were but one planet divided into pieces, &c,
Mr. Harding of Lilienthal in the duchy of Bremen, on Septem-
ber 1, 1804, discovered a 10th planet which he called Juno. This
new planet is also found to be at nearly the same distance from
the sun as the former two, and it is not yet decided with certainty,
which of the three is the nearest or the most remote from the sun.
As she appears like a star of the 8th magnitude, she is not there-
fore visible to the naked eye.
On the 29th March, 1807, Dr. Olbers discovered another new
planet, which he called Vesta, now the 1 1th in our System, in the
order of discovery : it is very remarkable that this planet is found
to be apparently at the same distance from the sun as the three al-
ready mentioned. At the time that it was discovered, that is on
the 29th March, 8h. 2 1 ; , its rt. ascen. was 1 84P 8', and declination
OF THE SOLAR SYSTEM. 35$
1 1 47'. In size this planet appears like a star of the 5th magni-
tude. If the phenomena of two planets, nearly at the same dist.
from the sun appeared strange to astronomers, that of four must
appear still more extraordinary ; however it is more than probable
that since the creation of the world, they have like the comets per-
formed their motions in their respective orbits, without clashing
with each other, or producing any of those strange phenomena, re-
sulting from their too near proximity ; and it is equally probable,
that the wisdom of the Creator has regulated their motions in
such a manner, as to prevent any of these accidents taking place>
while the Solar System exists.
CHAP. VII.
OF JUPITER,
AND HIS FOUR SATELLITES.
JUPITER is the next planet, in order, in our System, and als
the largest of all the planets, so that notwithstanding his great dis-
tance from the sun and earth, he appears to the naked eye almost
as large as Venus, though not so bright. Jupiter when in opposi-
tion to the sun, appears larger and more luminous than at other
times, being then much nearer to the earth, than a little before or
after his conjunction.* Jupiter will be a morning star when his
longitude is less than that of the sun, and will therefore appear in
tihe east before the sun rises ; but when his longitude is greater
than the sun's, he will be an evening star, and will be in the west
after sun set. Jupiter's periodic or sidereal revolution from ivest
to east, is, according to Laplace, 4332 602208 days or 117y. 31d.
14h. 27' lo"77. Vince makes his periodic revolution lly. 3l5d.
14h. 27' I0"8.f
Jupiter before his opposition, is subject to the same apparent
inequalities as Mars, as we have before remarked, and when he is
about 1 1 5 1 2' distant from it, his motion becomes retrograde, his
velocity increases until the opposition, and after which it dimin-
* Jupiter, when in conjunction with the sun, rises, sets, and comes to the
meridian with the sun ; but when in opposition, he rises when the sun sets,
sets when the sun rises, and comes to the meridian at midnight.
f Here there are two days difference between Vince and the other astron-
omers C except those "who copy tumj this might be accounted for allowing
only 365 days for the year, in reducing the days from Laplace ,- whereas there
are, at least, two leap years ; but Vince y where he makes the same calcula-
tion for Satwn's sidereal revolution, takes 365 days for the year without any
;dlowance for bissextile. Calculating from Jupiter's mean motion according
to Delambre (tab. 119) which is at the rate of 30 20' SL"? in it year, we have
30 20' 3/'7 : 360 :: 365d. : 4330d. Uh. 39 f 49"-2 ; but this is the mean *ro-
pical revolution which is considerably less than the sidereal; and hence Vine*
must have here fallen into an error. Keith in his Treatise on the niches,
makes the same mistake.
360 OF THE SOL AH SYSTEM.
ishes, until the planet in his approach towards the sun, is again
only 1 15 12' distant from him The duration of this retrograde
motion is 121 days, according to Laplace* and the arc of retrogra-
dation 9 54' ; there are, however, perceptible differences in the
duration and extent of the regressions of Jupiter.
The semimajor axis of Jupiter's orbit, or his mean distance from
the sun, is, according to Laplace 5.202592, the earth's being 1, the
proportion of the eccentricity of this axis for the beginning of 175O
is 048877, and the secular increase of this proportion 0.000 1 4345.
Vince makes the relative mean dist. of Jupiter 52v> 79, and the ec-
centricity of his orbit 250 ! 3,3, the mean dist. of the earth from the
sun being 100000. The mean longitude of Jupiter, reckoned from
the mean vernal equinox at the epoch of the 31st of December,
1749, at noon, mean time at Paris, is reckoned according to La-
place (4 201) 3 42 1 2y"i24 ; longitude of the perihelion at the
beginning of 1750 (P50;2) 10 2i'3"883; its sidereal and di-
rect secular motion (203(>"25) '0' 57"80L Vince makes !he
place of the aphelion for the beginning of '750, 6s. 10 21' 4",
and its secular motion in longitude i 34' 33". DeLambre in his
tables, makes the mean long of Jupiter for the beginning of 181 1,
mean time at Paris, Is 25 44' 33"7, of his afihenon 6s. 11 18'
45", and of his node 3s 8 30' 40". He makes Jupiter's mean
annual motion Is 20' 31 "7, that of his aphelion 57", and of his
node 36"; and the secular motion of the aphelion 1 34' 33", and
of the nodes 59' 30" The mean motion of Jupiter for a month is
therefore 2 34' 37"2, of the aphel. 5", and of the nodes 3" For
a day the mean mot. is 4' 59" 3, for an hour i2"5, and for a mi-
nute 0"2. The inclination of Jupiter's orbit to the plane of the
ecliptic, at the beginning of 1750, according to Laplace, was
(1 4636) 1 19' 2"064, and its secular variation to the true eclip-
tic ( 67"40) 2 I "8376 decreasing. Vince makes his inclination
1 18' 56", According to Laplace, the longitude of the ascending
node, on the ecliptic, at the beginning of 1750, was (i08 8062)
97 55' 32"o88, and its sidereal and secular mot. on the true eclip-
tic ( 4509"5) 24' 21 "07 8 decreasing. Vince makes the long,
of the nodes for the beg. of 1750, 3s 7 55' 32", and its xecular
motion 59' 30". The greatest equation for Jupiter, according to
Vince, is 5 30' 38"3, according to Delambre 5 30' 37"7. (See
tab 124 of Delambre, where its secular var. is also given for every
degree of the mean anomaly.) Lap'ace remarks that the path of
Jupiter occasionally deviates from the ecliptic (3 or 4; 2 42'
or 3 36'.
Jupiter is observed to have several obscure belts or stripes on
his surface, which are parallel among themselves and also to his
equator, and therefore nearly parallel to the ecliptic ; there are
likewise other spots, the motion of which has demonstrated the
rotation of this planet from west to east upon an axis nearly pel p.
to the plane of the ecliptic, and in a period, according to Laplace
of 0.4 1 377 days, or 9h, 55' 49"7. Jlf. Casbini, from a remarkable
OF THE SOLAR SYSTEM. 361
spot which he observed in 1665, found the time of rotation to be
9h. 56'. In Oct. 1 69 1, he observed two bright spots almost as
broad as the belts ; at the end of the month he saw two more,
from which he found the rotation of the planet to be performed ia
9h. 51' ; he also found that some spots near the equator revolved
in 9h. 50', and in general he found that the nearer the spots were
to the equator the quicker they revolved ; he also observed, that
several spots which at first were round, grew long by degrees, in
a direction parallel to the belts ; and divided themselves into two
or three spots. These variations in some of the spots, and the
sensible difference in the period of rotation of others, induce
the opinion that they are not attached to Jupiter ; they appear to
be clouds which are transported by the winds with various veloci-
ties, in an atmosphere extremely agitated.* M. Miraldi from,
several observations of the spot observed by Cassini in 1665, found
the time of rotation to be 9h. 56', and concluded that the spots
had a dependence upon the contiguous belt, the spot never ap-
pearing without the belt, though the belt appeared without the
spot. It continued to appear and disappear until 1694, and then
disappeared until 1708 ; he therefore concluded that the spot was
some effusion from the belt upon a fixed place of Jupiter's body,
as it always happened in the same place. Dr Herschd found the
rotation of different spots to vary, and that the time of the rotation
of the same spot diminished. The spot in 1788 revolved as fol-
lows ; from Feb. 25, to March 2, it revolved in 9h 55' 20";
from March 2 to 14, in 9h. 54' 58" ; from April 7 to '2, in 9h.
51' 35". He observes that this is agreeable to the theory of equi-
noctial winds, as the spot may require some time before it can
acquire the velocity of the wind ; and he further remarks, that if
Jupiter's spots were observed in different parts of its revolution to
be accelerated and retarded, it would amount almost to a demon-
stration of its monsoons, and their periodical changes. M tichroe-
ter makes the time of rotation 9h. 55' 36" 6 ; he found the same
variations as Herschel.
According to Lafilace the apparent diameter of Jupiter, in his
opposition, or when it is greatest, is (149") 48" 276, and his
mean diameter, in the direction of his equator, is (120") 38" 88.
From the great magnitude of Jupiter, and his quick revolution on
his axis, it is found that he is flatter towards the poles than at the
* The belts of this planet are also subject to verj great variations, both
as to number and figure ; eight have been sometimes seen at once, and
at other times only one ; they are sometimes found to continue three
months without any change, and sometimes a new belt has been formed
in one or two hours. Large spots have been also seen in these belts, and
when a bell, disappears, the contiguous spots disappear likewise. Hence,
frim their being subject to such sudden changes, it is very probable that
they do not adhere to the body of Jupiter, but are produced andexist in his
atmosphere. If this be the case, those that are produced in one or two
hours,, must require an agitation or velocity in the air, much greater than
we experience in the greatest hurricanes.
Xx
362 OF THE SOLAR SYSTEM.
equator, Lafilace makes the polar diameter : equatorial diaaa. tt
13 : 14. Mr. Pound makes this proportion as 12 : 13. Mr,
'Short as 13 : 14. Dr. Bradley as 12.5 : 13 5. Sir I. Newton
makes the ratio from theory as 91 : IOJ (prop 18. b. 3. prin.J
and on supposition that Jupiter is more dense towards the equator,
lie makes the ratio from 12 : 13 or from 13 to 14. The diame-
ters as observed by Mr. Pound vary from 11 : 12 to 13^ : 14
(see the principia prob. 19 b. 3.) Newton makes the greatest
diam. 37" and least 33" 25'" ; but after allowing 3" for refrac-
tion, he makes them 40'" and 36" 25"' respectively.
Jupiter's mean dist. from the sun, as calculated from his peri-
odic time, is 483342701.3 miles,* his hourly velocity in his orbit
is 29206 miles,f taking his mean diameter 38" 8 or 39" nearly,
his real diameter will be 73687 miles,! and his magnitude 807
times that of the earth. The light and heat which he receives
from the sun is about -^ of that which the earth receives. H
It is a remarkable result which the nicety of modern observa-
tions has determined, and which may be collected from what is
said above, that, while the eccentricities, inclinations to a fixed
* The sidereal revolution of Jupiter being lly. 317d. 14h. 27' 11* near-
ly = 374336831", the square of which is 140128064043122561, this divided
by 995916894538801, the square of the earth's sidereal revolution (see
pa. 350) gives 140.702567, the cube root of which is nearly 5.207, the
relative dist. of Jupiter from the sun, that of the earth being- 1 ; hence
23464.5 X 5.207 = 122179.6515, Jupiter's dist. from the sun in semidi-
ameters of the earth, which multiplied by 3956, gives 483342701.334
miles, the mean dist. of Jupiter from the sun.
t 483342701.3 X 2 X 3.1416 = 3036938860.8 miles, the circumference
of Jupiter's orbit ; hence 4332d. 14h. 27' ll r : lh. r: 3036938860.8 :
29206 miles, the hourly velocity of Jupiter.
t 483342701 92825562 (pa. 255 and 258) = 390517139 miles, the
dist. of the earth from Jupiter. Now by the rule of three, inversely,
390517139 : 39" :: 92825562 : 164"073, the apparent diam. of Jupiter at
a dist from the earth equal to that of the sun ; hence 32^ : 862299 :r.
164"073 : 73687.4 miles, the diameter of Jupiter. The diam. may be als
determined by trig, in the same manner as that of Mercury, pa. 263, or
thus, in the fig. pa. 250, let M represent the earth, AB Jupiter ; then the
angle AMC = 1S"5, and MC = 122179.623464.5 =sx 98715 nearly, Ju-
piter's dist. from the earth in semidiameters of the earth; then cos. 19^5
(log. = 10) : sine 19"4 (log. =r 5.9754667) :: 98715 (log. = 4.9943831}
: 9.3293 (log. = 0.9698498) the semidiameter of Jupiter in semidiam.
of the earth ; hence 9.3293 X 2 = 18.6586, Jupiter's diam. which multi-
plied by 3956, gives 73813.4, Jupiter's diameter in miles.
The cube of the diameter of Jupiter, divided by the cube of the
earth's diameter = ? TJJj^- = (log. 2.9074785) 806.7. Or taking 73813
we get 812.3 nearly. A more accurate method for determining the mag,
and dist of Jupiter will be given in treating of his satellites.
|| The relative mean dist of Jupiter from the sun is 5.207, that of the
earth being 1 ; hence, the effects of light and heat being reciprocally as
the distances from the centre whence they are propagated, we have
= 52072 27.1 nearly.
OF THE SOLAR SYSTEM. 363
plane, the position of the nodes and perihelia of the planetary or-
bits, are subject to small variations, which, as Laplace remarks, ap-
pear; up to the present time, to have increased proportional to the
rimes, their greater axis, half of which is their mean distances from
the sun, appear to be always the same These variations only be-
coming sensible through the lapse of ages, have been therefore
called secular inequalities. We shall speak more fully of these
after the laws of gravity, Sec. Small inequalities have been like-
wise remarked which disturb the periods of the planets, as we
have shewn with respect to the sun (pa. 24-6) but these inequali-
ties are principally sensible in Jufiiter and Saturn, from their mu*
tual actions and respective situations ; these particular causes are
found to alter at length the elements of their orbits, as we shall af-
fcerwards shew.
For the comparative astronomy as regards Jupiter, or the phe-
nomena that would appear to an eye in Jupiter, see Dr. Gre-
gory's Astronomy, b. 6. prob- 5.
OF THE SATELLITES OF JUPITER.
GALLILEO, the inventor of the telescope, was the first who dis-
covered that there are four small stars or satellites invisible to the
naked eye, which constantly accompany Jupiter, and perform their
revolutions round him. He called them Medicea sidera or Mtdi-
eean stars, in honour of the family of the Medici, his patrons.
This discovery, which he made on January 8, 1610, was a very
important one in its consequences ; the eclipses of these satellites
furnishing one of the best methods of determining the longitudes
of places on the earth. From these eclipses Roemer was led to
the discovery of the progressive motion of light, from which Dr.
Bradley was enabled to account for the apparent motion of the fix-
ed stars, called the aberration.
The relative situations of these satellites to each other vary every
moment ; they are observed to oscilate on each side of Jupiter, and
it is from the extent of these elongations, that they are classed ;
that being called the first satellite whose oscilation is the least, and
so on in order. They are frequently concealed from our view or
eclipsed by the shadow of Jupiter, while performing their revolu-
tions from west to east ; and when they move from east to west,
they are observed to pass over his disk, and project a shadow which
then describes a chord of this disk From these phenomena we
discover that Jupiter and his satellites are ofiake bodies enlighten-
ed by the sun ; and also that they revolve round Jupiter in the
same direction that Jupiter revolves round the sun. The three
first satellites are always eclipsed when they are in opposition to
the sun ; they are often found to disappear when at some distance
from the planet's disk, and the duration of their eclipses is differ-
ent at different times ; the third and fourth sometimr s re-appear
on the same side of the disk, and the fourth sometimes passes
364 OF THE SOLAR SYSTEM.
through its opposition without being at all eclipsed. These dis-
appearances are perfectly similar to eclipses of the moon, and are
evidently produced by the conical shadow of Jupiter, which, rela-
tively to the sun, is projected behind him ; for the satellites are
found to disappear in opposition, or on that side of Jupiter where
the shadow is projected ; they are eclipsed nearest to the disk
when the planet is nearest his opposition ; and the duration of their
eclipses exactly corresponds to the time they should employ in
traversing the shadow of Jupiter in his various positions. From,
the same phenomena it appears, that the planes of iheir orbits do
not coincide with that of Jupiter's orbit ; as in that case they would
always puss through the centre of Jupiter's shadow, and therefore,
at every opposition of the sun, an eclipse would take place, and of
the same, or very nearly the same duration By comparing eclip-
ses at long intervals, observed near the planets' opposition, we have
the most accurate methods of determining their motions. It is
thus discovered that the orbits of the satellites of Jupiter are near-
ly circular aud uniform ; for this hypothesis very nearly corres-
ponds with those eclipses which take place while the planet is in
the same position with respect to the sun When the planes of
the orbits pass through the eye, the satellites will appear to de-
scribe straight, lines, passing through the centre of Jupiter's disk ;
when this is not the case, they will appear to describe ellipses of
which Jupiter is the centre.
The following is, in substance, the method given by Vince and
other astronomers for determining the periodic times and distance*
of Jupiter* & satellites The mean time of their synodic revolutions,
or of their revolutions relatively to the sun, is thus found ; let the
passage of a satellite over the body of Jupiter be observed, when
Jupiter is in opposition, and let the time be marked when it ap-
pears to be exactly in conjunction with the centre of Jupiter, this
will also be the time of conjunction with the sun. Let the same
observation be repeated after a considerable interval of time, Ju-
piter being in opposition, and divide this interval by the number of
conjunctions with the sun during this time, and the icsult will be
the satellites synodic revolution This is the revolution on which
the eciipses depend, and is therefore the most important to be
considered ; but on account of the equation of Jupiter's orbit, this
will not give \hs-inean time of a synodic revolution, unless Jupiter
were at the same point of his orbit at both observations ; when,
this is not the case, we must proceed in the following manner.
OF THE SOLAR SYSTEM.
365
Let AIPR be the orbit of Jupi-
ter, S the sun in one focus, about
which the motion may be consi-
dered as uniform, the eccentricity
of the orbit, or SF being small.
(Ward's Theory, pa. 317) Let
Jupiter be in A his aphelion, in
opposition to the earth at T ; and
L a satellite in conjunction ; let I
be the place of Jupiter at his next
opposition with the earth at D, and
the satellite in conjunction at G ;
then if the satellite had been in O,
it would have been in conjunction
with F, or in mean conjunction ; therefore before it comes to its
mean conjunction, it mustdescribe the angle FIS ; this angle is the
equation of the orbit, according to Ward's or the simjile elliptic hy-
fiot/ieftiS) which, as the eccentricity is small, is here used. The
angle FIS therefore measures the difference between the mean
synodic revolutions with respect to F and the synodic revolution
with respect to the sun S. Let n =a the number of the satellites
revolutions as respects the sun ; then n x 360 SIF = the re-
volutions as respects F ; hence n X 360 SIF : 360 :: the
time between the two oppositions : to the time of a mean synodic
revolution about the sun.
The difference between the times of any two successive revolu-
tions, with regard to S and F respectively, is as the variation of the
equation of the orbit, or of the angle FIS ; for the satellite being
at O at the mean conj. and at G when in conj. with the sun, it is
evident that if FIS continued the same, the time of a rev. in respect
to S, would equal the time in respect to F, or that of the mean sy-
nodic re vol. When Jup. is at A, this equation vanishes, and the
conjunctions at F and S happen ai the same time. When Jup.
comes to I, the mean conj. at O takes place after the true conj. at
G by the time of describing FIS. This astronomers call ihejftrst
inequality ; and this inequality of the times of the intervals of the
true conjunctions, affect the times of the eclipses of the satellites.
As it however seldom happens, that a conjunction of the satel-
lite takes place when Jupiter is in opposition, the following method
must be used to find the time of a mean revolution, when he is
not in opposition. Let the earth be in H when the satellite is in
Z, in conjunction with Jup. at R, and again in V when the satel-
lite is in C,in conj. with Jup. at I ; let RH, IV, be produced until
they meet in M ; then Jupiter's motion round the earth, in this
interval, is the same as if the earth was fixed at M. Now the
diff between the true and mean mot. of Jup. is RFI RMI =
FIM + FRM (32 Eucl 1) which shews the excess of the num-
ber of mean revolutions in respect to F, above the same number
of apparent revolutions in respect to the earth ; hence n x 360
366 OF THE SOLAR SYSTEM.
FIM FRM : 360 :: the time between the observations c
the time of a mean synodic revolution of the satellite If C and Z
be at the other side of O and Y, the angles FIM, FRM must be
addtd to n x 360 ; if C be on one side and Z at the other, one
must be added and the other subtracted according to the circum-
stances.
From the great brightness of Jupiter, it is difficult to determine
accurately the time when the satellite is in conjunction with Ju-
piter's centre, in its passage over his disk, and hence it is deter-
mined by observing the satellites entrance upon the disk, and its
going off; but as this cannot be so accurately determined as the
times of immersion into, and emersion from the shadow of Jupiter,
the eclipses will therefore determine the time of conjunction more
accurately.
Let I be the centre of Jupiter's shadow FG ; Nst the orbit of a
satellite, and N its node upon the orbit of Jupiter, let lv be drawn
perp. to IN, and 1C to Nr ; and when the satellite comes to v, it
is in conj.* with the sun. The time of this conjunction is found
thus ; the immersion at s and
emersion at t of the 2d, 3d and 4th
satellites, mav be sometimes ob-
served, the middle of the time be-
tween which will give the middle
of the eclipse at c ; and hence NI
and the L. N being given, cv may
be found, and therefore the time of conjunction at -v. If both the
immersion and emersion cannot be observed, let the time of either
be observed, and after a long interval, let the time of the same be
again observed, when an eclipse happens in the same situation
with respect to the node, as nearly as possible ; from the interval
of these times, the time of a revolution will be obtained.
M. Cassini) by these methods, found the times of the four sa-
tellites mean synodic revolutions to be as follow, viz. First id.
18h 28' 36", Second 3d. -3h. 17' 54", Third 7d. 3h. 59' 36",
and Fourth 16d I8h. 5' 1" .
H nee it follows that 247 revolutions of the first satellite are
performed in 437d. 3h 44' ; 123 revolutions of the 2d in 43 7d.
3h. 41' ; 61 revolutions of the 3d in 4o7d. 3h 35' ; and 26 re-
volutions of the 4th in 43,5d. 1 *h. 13' It appears therefore that
after an interval of 437 days, the three first satellites return to
their relative situations within 9'.
The synodic revolutions of the satellites and the mean motion
of Jupiter being given, their sidereal, or periodic revolutions may
be easily found thus ; let x be the mean angle described by Ju-
piter during a synodic revolution of the satellite, then in the re-
turn of the satel. to the mean conj. it will have to describe this an-
,* A satellite is said to be in conjunction, both when it is between the
sun and Jupiter, anil when it is opposite to the sun ; the latter is called su
nerior, the former inferior conjunction.
OF THE SOLAR SYSTEM, 367
gle to complete its periodic revolution ; hence we have this pro,-
portion 360 -f- x : 360 :: time of a synodic revolution : time
f a periodic revolution. The periodic revolutions of the satellites
are therefore as follow ; the sidereal revolution of the First is Id.
18h 27' 33" ; of the Second 3d. 13h. 13' 42", of the Third 7d.
3h. 42' 43'; and of the Fourth 16d. 16h. 32' 8". Mwton
(Prin. b. 3. phen. 1.) makes them nearly the same. Lajilace gives
their sidereal revolutions as follow ; the First 1.769137787069931
days = .d. 18h, 27' 38" 5; the Second 3551181016734509
days = 3d I3h. 13' 42"; the Third 7.15455^807541524 days
= 7d. 3h 42' 33" 3, and the Fourth 16.689019396008634 days
= 16d. 16h. 32' 11" 27.
At the beginning of V700 the mean longitudes of the satellites
were as follow : First 77 15' 51" 084, of the Second 75 13'
37" 948, of the Third 164 12' 16" 38, and the Fourth "227 50'
20" 58.
The distances of the satellites being compared with the duration
f their revolution, the same proportion has been observed as in
the primary planets, and hence the dist. of one being obtained, the
distances of the others may be therefore found, the squares of the
periodic times being as the cubes of their mean distances from Jupi-
ter. The greatest heliocentric elongation of the 4th satel. from
Jupiter's centre, was taken by M. Puund with a micrometer in a
telescope 15 feet long, and at the mean distance of Jupiter from
the earth, was found about 8' 1 6", that of the 2d taken with a
micrometer in a telescope 123 feet, was found = 4' 42", and from
the periodic times the others were found 2' 56" 47'" and l' 51"
6'" respectively, and from Newton's determination Jupiter's diam.
at its mean dist. being taken 371", the distances of the satellites
are found to be 5.965, 9.494, 15,141 and 2663 semidiameters of
Jupiter, respectively. (See b. 3, prin. phen. 1.) JVetuton also re-
marks that with the 123 feet telescope, Jupiter's diameter reduc-
ed to the earth's mean distance proved always less than 40", never
less than 38", and generally 39". Laplace, taking the diameter
of Jupiter's equator (i20" 37) 39" nearly, finds the mean dis-
tances of the satellites from his centre, the mean distance of Ju-
piter from the being taken 1, as follow ; lat 5.6973, 2d. 9.065898,
3d. 14.461628, and 4th 25.436. The distances may be also thus
found. When a satellite passes over the middle of Jupiter's disk,
let the whole time of its passage be observed, then, the time of a
revolution : time of passage over the disk :: 360 : the arc of its
orbit corresponding to the time of its passage over the disk ; hence,
sine of half this arch : rad. :: Jupiter's semidiameter : the distance
of the satellite. See the distances thus determined by M. Casyinl f
also by Borelli and Townley given by Newton. (Pa. 207 Mode's
trans.)
By knowing the greatest elongations of the satellites in minutes
and seconds, their distances from the centre of Jupiter, compared
with Jupiter's mean distance from the earth is found by saying-,
368 OF THE SOLAR SYSTEM.
sine of the satellite's greatest elongation : radius : distance of the
satellite from Juf liter : the; mean distance of Jufiiter from the earth.
The distances of Jupiter and the sun from the earth may be al-
so compared with each other, by knowing the position of the satel-
lites as seen from Jupiter's centre, which is easily determined
from their periodic time;*, on supposition that they revolve in cir-
cular orbits round him. Let the total duration of an eclipse of the
third satellite, for example, be observed ; and at the middle of
the eclipse the satellite is nearly in opposition to the sun, as seen
from Jupiter's centre. Now its position in the heavens, as observ-
ed from Jupiter's centre, is the same as this centre seen from the
sun ; and from the periodic time of the sun, or from direct obser-
vation, the position of the earth as seen from the sun's centre is
given ; hence if a triangle be conceived to be formed by the right
lines which join the centres of the sun, the earth, and Jupiter,
the rectilinear distance from Jupiter to the earth, and also to the
sun, in parts of the distance of the sun and earth, will be given,
at the instant of the middle of the eclipse. By this method it is
found that the distance of Jupiter, when his apparent diameter is
38" 88, is at least five times that of the sun from us. At this dis-
tance the diameter of the earth would not appear under an angle of
3" 56, and hence the magnitude of Jupiter is at least 1000 times
greater than the earth's, from these data.
The apparent diameters of the satellites being insensible, their
magnitudes cannot, therefore, be exactly measured The attempt
to appreciate it has been made by observing the time they take to
penetrate the shadow of the planet ; but the various powers of tele-
scopes and other circumstances, have produced a great discordance
in these observations.
Caasmi suspected that the satellites had a rotary motion on their
axis) as in their passage over Jupiter's disk, they were sometimes
visible, at other times not : he therefore conjectured that the 1 - had
spots on one side and not on the other, and that they were rendered
visible in their passage when the spots were next tht- earth At
different times they likewise appear of different magnitudes and of
different brightness. The 4th appears generally the smallest, but
sometimes it appears the largest ; and the diameter of its shadow
on Jupiter, appears sometimes greater than the satellite The od
also appears to vary its magnitude, and the same is observed to
take place with regard to the other two. From similar circum-
stances also Mr. Pound concluded that they revolved on theit axis.
The comparative brightness of the satellites ought to afford in-
formation concerning their rotary motion. Dr Her*chrt has ob-
served, that they surpass each other alternately in brilliance, a cir-
cumstance that enables us to judge of their respective ight. The
relation of the maximum and minimum of their light with their
mutual positions, has persuaded him that they revolve upon
themselves like our moon, in the period equal to the duration of
their respective revolutions round Jupiter. (Phil, trans. 1797.)
OF THE SOLAR SYSTEM.
369
M&raldi had already found the same result for the 4th satellite,
From the returns of the same spot observed on its disk, in its pas*-
sage over the planet.
The edifises of Jufiiter's satellites deserve to be particularly con-
sidered, not only for themselves, but also as they serve to settle
the longitude of places, and to explain one of the most interesting
results in modern philosophy ; that is the motion or propagation
of light in a determined time, as we have before noticed.
Let S be the sun, EF the orbit of the
earth, I Jupiter, abc the orbit of one of his
satellites. When the earth is at E before
Jupiter's opposition, the spectator will ob-
serve the immersion at a ; but if the satellite
be the 1st, from its nearness to Jupiter, the
emersion is never visible, the satellite being
then always behind the body of Jupiter ; the
oiher three satellites may have both their im-
mersions and emersions visible ; this will,
however, depend on the earth's position.
When the earth is advanced to F after oppo-
sition, the emersion of the 1st is then seen,
but the immersion can never be seen in this
position ; but both the emersion and immer-
sion of the other three may be visible at this
time. Let Elr be drawn ; then denoting
the centre of the shadow, or the middle point between a and b by
5, sr, the distance of the centre of the shadow from the centre of
Jupiter, referred to the orbit of the satellite, is measured at Jupi-
ter by sr, or by the angle sir, or EIS. The satellite may be hid-
den behind Jupiter at r without being eclipsed, which is called an
occultation. When the earth is at , the conjunction of the satel-
lite happens later at the earth than at the sun ; but sooner when
the earth is at *'.
The diameter of Jupiter's shadow, at the distance of any of the
satellites, is best found by observing the time of an eclipse, when
it happens at the node, at which time the satellite passes through
the centre of the shadow ; for the time of a synodical revolution :
the time the satellite is passing through the centre of the shadow
:: 360" : the diameter of the shadow in degrees. But when in
the nodes, the immersion and emersion of the first and second
satellite cannot be seen. Hence astronomers compare the immer-
sions some days before the opposition of Jupiter with the erner*
sions some days after, and then the number of synodical revolu-
tions being known, the time of the transit through the shadow, and
thence the corresponding degrees are found. But from the
eccentricity of some of the orbits, the times of the central transit
must vary. Example. The second satellite is sometimes found
to pass through the centre of the shadow in 2h. 50', and some-
times to be 2h. 54' in passing ; this indicates an eccentricity.
Yy
370 OF THE SOLAR SYSTEM.
The duration of the eclipses being very unequal, shews that the
orbits are inclined to the orbit of Jupiter, sometimes the fourth
satellite passes the opposition without being eclipsed, as noticed
before. The duration of the eclipses depends on the situation of
the nodes wi Ji respect to the sun, as in a lunar eclipse ; when the
line of the nodes passes through the sun, the satellite will pass
through the centre of the shadow ; but as Jupiter revolves round
the sun, the line of the nodes will be carried out of conjunction
with the sun, and the times of the eclipse will be shortened as the
satellite only describes a chord of the section of the shadow.
The diifiticity of the orbit of the first satellite, as Lafila.ce re-
marks, is insensible, its plane nearly coincides with the equator of
Jupiter, the inclination of which, to the planet's orbit is (4 4444)
4 nearly. The ellipticity of the orbit of the second satellite is
equally insensible. Its inclination to Jupiter's orbit varies, as well
as the position of its nodes. These variations according to Lafllace,
may be represented nearly by supposing the orbit of satellite inclin-
ed about (5 !82") 27' 58" 968 to the equator of Jupiter, and giv-
ing its nodes a retrograde motion on this plane, the period of which
is about )(> Julian years. A slight ellipticity is observable in
the orbit of the third satellite. The extremity of its greater axis
nearest to Jupiter, and which is called fienjove, has a direct motion,
and the eccentricity of its orbit is subject to perceptible alterations.
The equation of the centre, towards the end of the last century,
was at its maximum nearly = (2o61"; 14' 22" 184, then di-
minishing, about 1775, it was at its minimum = (759") 4' 5" 916*
The variations in the inclination of its orbit, and the position of its
nodes, may be nearly represented by supposing the orbit inclined
about (2244") 12' 7" 056 to the equator of Jupiter, and the peri-
od of the retrograde motion of its nodes =137 years. The orbit
of the 4th satellite has a very sensible ellipucity, its perigee has a
direct motion of about (7852") 42' 24" 048. Its inclination te>
the orbit of Jupiter is (27i") I' 28" IS 8, in consequence of which
this satellite passes behind Jupiter relatively to the sun without
being eclipsed. The cause of these irregularities, &c. is explain-
ed in ch. 6. b. 4, Lafilace's Astronomy.
The mean motion of the satellites are such, that the motion of
the first satellite plus that of the third, is nearly equal three times
that of the second. Hence the same proportion evidently exists
between their mean synodical motions.
The mean longitudes, whether synodical or sidereal, of the
three first satellites seen from Jupiter's centre is such, that the
motion of the first satellite, minus three times that of the second
plus, twice that of the third is nearly, or exactly, equal the semi-
circumference.
The periods and the laws of the principal inequalities in these
satellites are the same. The inequality of the first advances OF
retards its eclipses (233") l' 15" 492 in time, at its maximum.
This is found to disappear when the two fircit satellites sejsn from
OF THE SOLAR SYSTEM. 371
the centre of the planet are in opposition to the sun at the same
time ; it afterwards increases, and is the greatest possible when
the first satellite at its opposition is 45 more advanced than the
second. It is again nothing when the satellite is 90 more advanc-
ed ; after this it takes a contrary sign and retards the eclipses, it
augments to 135 distance between the satellites where it is at its
negative max. It then diminishes and disappears, when 180
distance. In the second half of its circumference it follows the
same laws as in the first Hence in the first satellite there is found
an inequality of (5258") 28' 23" 592 at its max. and proportion-
al to the sine of twice the excess of the mean longitude of the first
satellite above that of the second, an excess = the difference of
the mean synodic longitudes of the two satellites. The period of
this inequality is not four days, and yet it transforms itself into a
period of 437d. 18h. in the eclipses of the first satellite.
Let the two first satellites be supposed to set out together from
their mean oppositions to the sun. At every circumference which
the first satellite describes, in consequence of its mean synodic
motion, it will be in the mean opposition. If we now imagine a
fictitious star, whose angular motion may be equal to the excess
of the mean synodic motion of the first satellite above twice that
of the second, then twice the difference of the mean synodic mo-
tions of the two satellites will be, in the eclipses of the first, equal
to the multiple of its circumference, plus the motion of the ficti-
tious star. The sine of this last motion will be then proportional
to the inequality of the first satellite in its eclipses, and may repre-
sent it. Its period is equal to the duration of the motion of the
fictitious star, which from the mean synodic motions is 437.75
days.
The inequality of the second satellite follows a law similar to
that of the first, with this difference, that it has always a contrary
sign. It accelerates or retards the eclipses (1059") 5' 43" 116
in time,* at its maximum. It disappears when the two first satel-
lites are at the same time in opposition to the sun. It then retards
more and more the eclipses of the second satellite, until the two
satellites are 90 distance from each other, at the instant of these
phenomena. This retardation diminishes and becomes nothing
when the satellites are distant 180 ; beyond which term the eclip-
ses advance in the same manner as they were before retarded.
From these observations it has been concluded, that there is an in-
equality of (11923") 1 4' 23" 052 at its maximum in the mo-
tion of the second satellite proportional (but affected with a con-
trary sign) to the sine of the excess of the mean longitude of the
first satellite above that of the second, which is equal the difference
of the mean synodic motions of the two satellites. If we conceive
* These quantities being 1 principally taken from Laplace, his measures
are therefore retained, and the corresponding- measures used in this conn- /
try given ; but in his seconds of time, he probably retains the late French
division of the calendar (see the note page 308) if so, 1059"= 13' 3V97C.
872 OF THE SOLAR SYSTEM.
the two satellites to set out together as before, the second satellite
will be in its mean opposition at each circumference, it will de-
scribe in consequence of its mean synodical revolution. If as be-
fore, a star be taken whose angular motion may be equal the ex-
cess of the mean synodical motion of the first satellite above twice
that of the second, then the difference of the synodical motion of
the two satellites will be, in the eclipses of the second, equal to a
multiple of its circumference, plus the motion of the fictitious
star The inequality of the second satellite in its eclipses, will
then be proportionable to the sine of the motion of this imaginary
star. Hence the period and law of this irregularity is the same as
that of the first satellite
The influence of the first on the second satellite is very proba-
ble ; but if the third satellite produced in the motion of the second
an inequality similar to that which the second seems to produce in
the motion of the first, that is proportional to the sine of double
the difference of the mean longitude of the second and third satel-
lites, this new inequality would confound itself with that of the
first satellite ; for, from the relation which the longitudes of the three
first satellites have to each other, as before remarked, the differ-
ence of the mean longitudes of the two first, is equal to half the
circumference, plus twice the difference of the longitude of the
second and third satellites ; so that the sine of the first difference
is the same as the sine of twice the second difference with a con-
trary sign The inequality produced by the third satellite in the
motion of ?he second, would therefore have the same sign, and fol-
low the same law, as the inequality observed in this motion ; hence
it is probable that this inequality is the result of two inequalities
depending on the first and third satellites. If by the succession of
ages, says Laplace, the relation between the mean longitudes of
these three satellites should cease to exist, these two inequalities,
at present confounded, would separate, and their respective values
might be known But according to observation this relation should
subsist for a long period. Lafilace shews (in the fourth book of
his astronomy) that it is rigorously exact.
Finally, the inequality relating to the third satellite in its eclip-
ses, compared with the respective positions of the second and third
satellites, offers the same proportion as the inequality of the two
first satellites. Hence in the motion of the third satellite there
exists an inequality proportional to the sine of the excess of the
mean longitude of the second satellite above that of the third,
which at its maximum is (827") 4' 27 "948.
If we suppose a star, whose angular motion may be equal to
the excess of the mean synoclical motion of the second satellite,
above twice the mean synodical motion of the third, the inequal-
ity of the third satellite will be. in its eclipses, proportional to the
sine of the motion of this fictitious star. Now in consequence of
the proportion which exists between the mean longitudes, the sine
of this motion is, exclusive of the sign, the same with the motion of
OF THE SOLAR SYSTEM. 373
the first fictitious star which we have considered- The inequality
of the third satellite in its eclipses, has the same petiods, and fol-
lows the same laws, as those of the two first satellites
These are the principal inequalities in the motions of the three
first satellites of Jupitei?. Dr. Bradley^ first remarked them, but
Warirrntin, whose tables of these satellites are used in calculating
the Nautical Almanac, has. since investigated them with the great-
est accuracy.
From these tables of Wargentin the configurations, or the firo-
portional distances, of the satellites of Jupiter and their eclipses
are calculated in the Nautical Almanac.
The eclipses of Jupiter's satellites, besides their bein useful at
sea (see pa 53) are, as Dr Maskelyne remarks, observed by as-
tronomers on land, as well in order to provide materials for im-
proving the theories and tables of their motions, as for the sake of
comparison with the corresponding observations, which may be
made by persons in different parts of the globe, whereby the lon-
gitude of such places will be accurately ascertained. He also re-
marks, that it is to be lamented that persons who visit distant coun-
tries, are not more diligent to multiply observations of this kind ;
for want of which, the observations made by astronomers in estab-
lished observatories lose half their use, and the improvement of
geography is retarded.* The eclipses set down in the Ejihemeris,
pa. 3 of the month, will serve to shew the times when these ob-
servations should be attended to ; having first from the difference
of longitude in time, if he be under any meridian different from
Greenwich, found the apparent time at which the eclipse will hap-
pen at his meridian nearly, that at Greenwich being given. (See
prob. 6, page 58.) He must also have his watch or clock, previous-
ly, well regulated, by some of the methods given in parts first or
second, to know the mean time exactly at which the observation is
to be made Equal altitudes of the sun or stars is, perhaps, the
best method. f
* As so useful a service can be rendered to the public, and to science in
general, Irom multiplied observations of this and a similar nature, it is tru-
ly to be. regretted that public observatories are" not erected in some of our
principal cities in the United States, and that there are to be found only
one solitary observatory near Boston, which is of any credit to the coun-
try. It is, however, to be hoped, that a point of such extensive utility and
importance to a country, will not be much longer overlooked by an enlight-
ened public ; and that New-York, which is so well situated for, and calcu-
lated to support a public undertaking of this nature, will not be backward
in setting- the example to her sister states.
j In prob. 22, pa. 83, we have shewn how a watch or clock may be regu-
lated, and how its rate of going may be ascertained ; but as this point is
extremely useful in determining the longitude, &c. we shall here insert the
following 1 observations collected from Vines' s astronomy, ibr finding the
rate of going, and from thence the mean rate, Sec. Suppose, for instance,
that the rate oi a watch for thirty days be examined, and that on some of
those days it gains and on others loses ; if the quantities winch it has gain-
ed be added, and found for example to amount to 17", and the quantities
374 OP THE SOLAR SYSTEM.
Dr. Maskelyne further remarks, that, the observer, being in a
place whose longitude is well known, should be settled at his teles-
cope three minutes before the expected time of an immersion or
emersion of the three first satellites ; and ten minutes before that
of the fourth. If the longitude of the place be very uncertain, he
Tnust begin to look out for the eclipse proportionally sooner. Thus,
if the longitude be uncertain to 3, answering to I2 f of time, he
ought to fix himself at his telescope 12' sooner than is mentioned
above, 8cc. However, when he has observed one eclipse of any
satellite, and thereby found the error of the tables, he may allow
the same correction to the calculations of the R/ihemeris for seve-
ral months, which will advertise him nearly of the time of expect-
ing the eclipses of the same satellite, and dispense with his attend-
ing so long. (See pa. 178.)
The immersions or emersions of any satellite being carefully
observed in any place according to mean time (the eclipses of the
satellites being now computed to mean time, in the Nautical Al-
manac, instead of afifiarent time as formerly) the longitude from
Greenwich is found immediately, by taking the diff rence of the
-observed time from the corresponding time shewn in the Efihcm-
eris, which must be turned into degrees, Sec. and will be east or
west from Greenwich^ as the time observed is more or less than
that of the J&fihemeris.
Dr. Maskelyne also observes that a corresponding observation of
an eclipse of a satellite of Jupiter, made under a well known me-
ridian, is to be preferred to the calculations of the Efihemeris for
which it has lost to 13*, then the difference 4" is the mean rate of gaining 1
for 30 days ; hence 0" 133 is the mean daily rate of gaining-. Or the daily-
rate may be thus obtained ; take the difference between what the watch
was too fast or too slow on the first and last days of observation, if it be
too fast or too slow on each clay ; but their sum if too fast on one day, and
too slow on the other ; and divide by the number of days between the ob-
servations. (See Waters method of finding the longitude at sea.) To find
the time at the place of trial at any future period by this watch, the time
gained or lost by the watch must be obtained ; then df 133 X the number
of days from the end of the trial, being 1 the quantity gained according 1 to
the above mean rate of gaining 1 , and the true time affected by the error at
the end of the trial, is supposed to be obtained. This will, however, only
diminish the probable error of the watch ; as the temperatures of the air,
and the imperfection of the workmanship, will cause some change in the rate
of the watches going. Hence when the watch is used to determine the longi-
tude at sea, the observer when he goes ashore, if time permits, should compare
his watch, for several days, with the mean time deduced from the sun's or
a star's altitude, to determine its rate of going- more correctly. And when-
ever he comes to a place whose longitude is known, the watch may be cor-
rected and set to Greenwich time. Without these or similar means of ad-
justing- the watch, its error in long voyages may be very considerable, and
therefore the longitude deduced from it proportionably erroneous ; but in
short voyages, and in carrying on the longitude from one known place to
another, or in keeping the longitude from that which is deduced from a lu-
nar observation until another is obtained, the watch is undoubtedly very
useful ; and benc'^ in navigation and geography it may be rendered of great
scrvicfc.
t)F THE SOLAR SYSTEM. 375
comparing with an observation made in a meridian whose longi-
tude is required ; but if no corresponding observation can be ob-
tained, as is frequently the case, it will be best to find what cor-
rections the calculations of the Ephemeris required, by the nearest
observations to the given time that can be obtained ; which cor'
rections applied to the calculation of the given eclipse in the*
Ejihemeris) renders it almost equivalent to an actual observation.
In the actual making the observations, the observers should be
furnished with the same kind of telescopes, should make allowance
for the different states of the air, and remove themselves from all
warmth and light, for a little time, before making the observation,
that the eye may be reduced to a proper state ; and this precau-
tion is also to be attended to, when the difference between the ap-
parent and true times of immersion and emersion is to be settled.
If two telescopes show the disappearance or appearance of the
satellite, at the same distance of time from the immersion or emer-
sion, the difference of the times will be the same as the difference
of the true times of their immersions and emersions, and will
therefore shew the difference of longitude exactly ; but if the ob-
served time at one place be compared with the computed time at
another, then allowance must be made for the difference between
the apparent and true times of immersion and emersion, in order
to obtain the true time when the observation was made, to compare
with the true time from computation at the other place. This
difference may be found by observing an eclipse at any place whose
longitude is known, and comparing it with the time by computa-
tion. At an immersion when the satellite enters the shadow, it
grows fainter and fainter, until at last the quantity of light is so
small, that it becomes invisible, even before it is immersed in the
shadow ; hence the instant that it becomes invisible, will depend on
the quantity of light which the telescope receives, and its magni-
fying power. Therefore the instant of the disappearance of a
satellite will be later, the better the telescope is, and its emersion
will appear sooner.
The apparent position of Jupiter's satellites with respect to each
other, and to Jupiter, or their configurations, are given in pa. 12
of the month in the Nautical Almanac, at such an hour of the
evening or night, as they are most likely to be observed, and serve
to distinguish the satellites from one another. Jupiter is distin-
guished by the mark O, and the satellites by points with figures
annexed, the 1 signifying the 1st satellite, 2 the 2d, Sec. When
the satellite is approaching towards Jupiter, the figure is put be-
tween Jupiter and the point, but when receding from him, the fi-
gure is put on the other side of the point. The satellites are iu
the superior parts of their orbits, that is above the orbit of Jupiter
or furthest from the earth, when they are marked to the right
hand, or west of Jupiter approaching him ; or to the left hand or
east of Jupiter receding from him ; but are in the inferior parts of
their orbits or nearest to. the earth when thev are marked to the
376 OF THE SOLAR SYSTEM.
right hand or west of Jupiter receding from him, or to the left or
east of Jupiter approaching him. The sateiiites that are above
the orb of Jupiter has nonh latitude, those below south latitude
The cypher 0, sometimes annexed to the figure of the satellite
towards the margin, signifies that it is invisible on the face of Ju-
piter, and the black mark 0, signifies that it is invisible, being
eclipsed in Jupiter's shadow, or behind Jupiter, eclipsed by his
body.
The following exhibits the configurations of the satellites of Ju-
piter at 4 o'clock in the morning, Greenwich time (or ./ 56" af-
ter eleven o'clock at night, in New-York) on the following days
in September, 1813, viz 17, 19, 25, 26, and 28, as given pt; 12
of the month in the Nautical Almanac, an explanation of which will
render that page intelligible to young students, for any other year
and month.
'
<}
O' >2
4 <J
S o c f 4
c i\J 1 :
,3.
1C- v
- /
;>26i * 3 .
O' 1
20 J
28,3.0
* 2 o - 1
\
On the 1 7th day of the month, given above, the second and
third satellites distant from Jupiter as represented, wii; be eclip-
sed at 4 o'clock in the morning, at Greenwich, or at 3' 56" after
1 1 at night, in New-York ; the first satellite is to the left hand of
Jupiter, and in north latitude, the fourth satellite to the right and
in south latitude.
On the 19th the third satellite at 4 o'clock, at Greenwich, will
be to the left of Jupiter in north latitude, or higher than Jupiter ;
the first and second will be in conjunction, or appear as one, on
the right hand of Jupiter, and the fourth will be in south latitude
further from Jupiter.
On the 25th the fourth satellite will appear to the left of Jupiter
above his orbit, or in north latitude, and approaching towards Jupi-
ter, the second and third will be also to the left, in south latitude,
approaching towards Jupiter, the first will be eclipsed to the ri^ht
hand of Jupiter.
On the 26th the fourth will be above Jupiter's orbit, the third
below it, both to the left ; the first will be near the body of Jupi-
ter, in south latitude and receding from him ; the second will ap-
pear like a bright spot on the disk of Jupiter.
On the 28th the third is invisible on the disk of Jupiter, the
fourth in north, and the second in south latitude, both to the lett ;
the first will appear to the right of Jupiter above his orbit, and re^
ceding from Jupiter.
OF THE SOLAR SYSTEM. 377
CHAP. VIII.
OF SATURN,
HIS SATELLITES AND
SATURN is the next planet in order, in the solar system, aftej'
Jupiter ; he shines with a pale, feeble light, being the most re-
mote from the sun of any of the planets that are visible without a
telescope. When viewed through a good telescope, the singular-
ity of his appearance engages the attention of the young astron-
omer ; being surrounded by a ring) the only phenomenon bf the
kind observed in the Solar System.
As the periods of the superior planets depends on their ofifiosi-
tions to the sun, we shall here shew how to find these oppositions,
and from thence the periodic times, this article being omiited in
the preceding chapters.
To determine the time of opposition ; Let the time when the
planet is nearly in that situation be observed, the time at which it
passes the meridian (note to prob. 8} and also its right ascension
(prob. 1. pa, 192.) let its meridian altitude be also found, and
likewise the meridian altitude of the sun ; and let the observations
be repeated for several days. From the observed meridian alti-
tudes let the declinations be found (see the note pa 140) and from
the right ascensions and declinations compute the latitudes and
longitndes of the planet (note to prob 3, pa. 195) and the longi-
tudes of the sun. Then let a day be taken when the difference of
their longitudes is 180, and reduce the sun's longitude on that
day, found from observation when it passes the meridian, to the
longitude found at the time (/) the planet passed, by finding from
observation or computation at what rate the longitude then in-
creases. Now as the planet is retrograde in opposition, the differ-
ence between the longitudes of the planets and sun increases by
the sum of their motions. Hence the following rule : As sum (S)
of their daily motions in longitude : the difference (D) between
180 and the difference of their longitudes reduced to the same
time (t)* (subtracting the sun's longitude from the planet's to ob-
tain the difference reckoned from the sun according to the order
of the signs) :: 24h. : the internal between that time (t) and the
time of opposition. This interval added to, or subtracted from that
time (t) according as the difference of their longitudes was great-
er or less than 180, gives the time of opposition. If this be re-
peated for several days, and the mean of the whole taken, the time
will be obtained more accurately. If the time of opposition* found
from observation, be compared with the time by computation from
* The diflf. shewing 1 how far the planet is from oppos. The proportion
is evident from this principle, that the sun approaches the star by spaces
proportionable to the times ; hence the spaces S and tt must kc r\s the.
time 24h. and the time (t} to opposition.
Z z
378 OF THE SOLAR SYSTEM.
the tables, the difference will be the error of the tables, which
may serve as means of correcting them.
Examfile. M. de la Lande, on October 24, 1763, observed the
difference between the right ascension of Arics^ and Saturn,
which passed the meridian 12h. 17' 17" apparent time, to be
8 5' 7", the star passing first. Now the apparent right ascen-
sion of the star at that time was 25 24' S3" 6 ; hence the appa-
rent right ascension of Saturn was Is. 3 29' 40" 6 at 12h. 17 P
17" apparent time, or l^h. 1' 37" mean time. On the same
day he found from observation of the meridian altitude of Saturn,
that his declination was 10 35' 20" N His longitude is there*
fore found, from his right ascension and declination = Is. 4 50'
56", and latitude 2 43' 25" S. At the same time the sun's lon-
gitude was found by calculation to be 7s. 1 19' 22", which taken
from Saturn's longitude gives 6s. 3 31' 34" ; hence Saturn was
3 31' 34" beyond oppo-oidon, but being retrograde, will after-
wards come into opposition. From the observations made on sev-
eral days at that time, Saturn's longitude was found to decrease
4' 50" in 24h. and by computation the sun's longitude increased
59' 19" in the same time, the sum of which is 64' 49" ; hence
64' 49'' : 3 31' 34" :: 24h. : 78h. 20' 20", which being add-
ed to October 24, 12h. 1' 37", gives 27d. 18h. 2l' 57" for the
time of opposition. The longitude of Saturn at the time of oppo-
sition may therefore be found by saying 24h. : 78h. 20' 20" ::
4' 50" : 15' 47", the retrograde motion of Saturn in 78h. 20' 20",
which taken from Is. 4 50' 56", leaves Is. 4 35' 9", the lon-
gitude of Saturn at the time of opposition. In like manner the
sun's longitude may be found at the same time, in order to prove
the opposition ; for 24h. : 78h. 20' 20" :: 59' 59" : 3 15' 47",
which added to 7s. 1 19' 22", the sun's longitude at the time oi"
observation gives 7s. 4 n 35' 9" for the sun*s longitude at the time
of opposition, which is exactly opposite that of Saturn. Hence the
latitude of Saturn may be found at the same time, by observing in
like manner the daily variation, or by computation from the tables,
the elements of its motions being known, and the tables construct-
ed : whence it appears, that in the interval between the times of
observation and opposition, the latitude had increased 6", and was
therefore 2 43' 3i", Thus, the times of opposition of alt the sii-
ficrior planets are found.
Vince. remarks, that from the conjunctions* and oppositions of
the planets, their mean motions could be readily determined, if
the place of the aphelia, and eccentricities of their orbits were pre-
viously known ;t for then the equation of the orbit could be found
(pa. 313) and the true reduced to the mean place ; hence the
mean places being determined at two different times, the mean
motion corresponding to the interval between these times will be
* See the method of determining- the conjunctions of the Inferior planets.
pa. 267. See also pa. 260 and pa. 280.
f The method ot finding 1 these is given pa. 2ol, Sec:
OF THE SOLAR SYSTEM. 379
given. The place of the aphelion is however best determined
from, the mean motion To determine, therefore, the mean mo-
tion, independent of the place of the aphelion, such oppositions or
conjunctions must be sought, as take place nearly in the same
points of the heavens ; for the planet being then very nearly in the
same point of its orbit, the equation will be very nearly the same
at each observation, and therefore the comparison between the
true places will be nearly a comparison of their mean places. The
equation must be considered, if it differ much in the two. observa-
tions Now by comparing the modern observations, the time of a
revolution will be nearly obtained ; and then, by comparing the
modern with the ancient observations, the mean motion may be
accurately determined ; for any error, being divided among a great
number of revolutions, will become very small with respeoit to
one revolution. The following example (M Cassini Elem. d* As-
iron, pa. 362 or Vince} will best illustrate this article.
On September 16, 1701, Saturn was in opposition at 2h. when
the sun's place was Virgo 23 2 i' 16", and Saturn therefore in
Pisces 23 2 1' 16", with 2 :) 27' 45" S. latitude On September
10, 1730, the opposition was at 12h. 27', and Saturn in Pisces 17
5S' 57", with 2 U 19' 6" S latitude. On September 23, 1731, the
opposition was at 15h. 5i', in Aries 31' 50", with 2" 3t>' 55" S.
latitude. Now the interval of the two first observations was 29
years (of which 7 were bissextiles) wanting 5d. 13h 33' ; and the
interval of the two last was ly. 1 3d. 3h. 24'. The difference of
she places of Saturn was also, in the two first observations, 5 27'
19", and in the two last 12 36' 53". Hence, from these obser-
vations, Saturn moved over 12 36' 53" in one year ; therefore 12
36' 53" : 527' 19" :: ly. 13d. 3h. 24' :: 163d. I2h. 41', the
time of moving over 5 27' 19" very nearly, Saturn being nearly
in the same part of its orbit, and will therefore move nearly with
the same velocity ; this therefore, added to the interval between
the two first observations (as Saturn, at the 2d observation, wanted
5 27' 19" from being up to his place at the 1st) gives 29 years,
164d. 23h. 8' for the time of one revolution. Hence 29y. 164d.
23h. 8' : 365d. :: 360 : i2 13' 23' 50"', the mean annual mo-
tion of Saturn in a common year of 365 days, that is, on suppo-
sition that it moves uniformly This being divided by 365, gives
2' 0" 28'" for Saturn's mean daily motion.*
* The mean motion thus determined will be sufficiently accurate to de-
termine tlie number of revolutions which the planet must have made, when
\ve compare the modern with the ancient observations, in order to deter-
mine the mean motion more accurately. Delambre in his tables (tab. 142;
makes Saturn's mean mot. in a Julian or common year 12 lo' 36" 8 ;
hence 12 13' 36" 8 : 360 :: ly. : 161y. 19h. 2(/ 18"" 07 the time of Sa-
turn's revolution. He makes Saturn's mot. for a day ^ 0" 6, for an hour
:>", and for a minute G" 1. These tables of Ddambre are calculated from
the theory of Laplace, and examined from a multitude of observations. See
the demonstrations of the principles ia the raernoires of the Academy for
1785 and 1786.
380 OF THE SOLAR SYSTEM.
The most ancient observation which we have of the opposition
of Saturn, was on March 2, in the year 228, before Jesus Christ,
at 10 o'clock in the afternoon, in the meridian of Paris, Saturn be-
ing then in Virgo 8 23', with 2 50' N. latitude. February 26,
1714, at 8h. 15', Saturn was found in opposition in Virgo 7 56'
46", with 2 3' N. latitude. From this time 1 1 days must be
Subtracted (if the observation were made after the year 1800, 12
days should be subtracted, &c. seepage 16) to reduce it to the
same stile as at the i st observation, and therefore this opposition
happened on February 5, at 8h. i 5' ; the difference between these
two places was, therefore, only 26' 14". Also, the opposition in
1715, was on March 1!, at 16h. 55', Saturn being then in Virgo
21 3 ; 14", with 2 25' N latitude. Now between the two first
observations, there were 1945 years ['of which 485 were bissex-
tiles) wanting I4d. 16h. 45', that is, 1943 common years, and 105d.
7h* 15' over. The interval between the times of the two last op-
positions was 378d. 8h. 4o', during which time Saturn had moved
over 13 6' 28"; hence 13 6' 28" : 26' U" :: 378d. 8h. 40'
: 13d. I4h which added to the time of oppos. in 1714, gives the
time when Saturn had the same longitude as at the oppos. in the
year 228 before J. C. This being therefore added to 1943 com.
years, 105d. 7h 15', gives 1943y 118d. 2ih. 15', in which interval
Saturn must have made a certain complete number of revolutions.
Now having found above from the modern observations, that the
time of one revolution n.ust be nearly 29 com. y. 164d. 23h. 8',
it follows that the number of rev. in the above interval was 66 ;
this interval being therefore divided by 66, gives 29y. 162d. 4h.
27' for the time of one revolution. From comparing the opposi-
tions in the years 1714 and 1715, the true mot. of Saturn appears
to be very nearly equal his mean mot. which shews that the op-
positions were observed very near the mean distance, and that
therefore, the mot. of the aphel. could not have caused any con-
siderable error in the determination of the mean mot. Hence the
jnean annual mot. is 12 IS' 35" 14'", and the mean daily mot.
2' 0" 35'". Dr Haliey makes the annual mot. to be 12 13' 21".
M de Lafiia'-e and Ddambre make it 12 13' 36"8. As the revo*
lution here determined is, that in respect to the long, of the planet,
it must be a trofiical revolution ; hence to find the sidereal rev.
we have this proportion, 2' 0" 35 /f/ : 24' 42" 20'" (the preces-
sion in the time of a tropical revol see pa 246 and 305) :: 1 day
: 12d. 7h. 1' 57", which added to 29y 162d. 4h. -J7', gives 29y.
174d 1 1 h. 28' 57", the length of a sidereal year of Saturn. From
more correct observations Vince makes it 29y. 174d. Ih. 51' 1 1"2.
In this manner the periodic times of all the superior planets are
found.
Lafitace makes the sidereal rev. the same as Vince 10759.077213
days, or 29y 174d Ih. 51' 1 1"2. The semimajor axis of his or-
bit, or his mean distance 9 540724, the earth's being 1 ; the pro-
Tjprtion of the eccentricity of half the greater axis for the beginning
OF THE SOLAR SYSTEM. 381
of 1750, 0.056223 ; the secular -variation of this proportion
0.000261553 diminishing; the mean longitude of Saturn at the
beginning of 1750, reckoning from the mean vernal equinox at
the epoch of the 3 1st of December, 1749, at noon, mean time at
Paris (2570438) 231 20' 2i"912 ; long, of the perihelion at the
beginning of 1750 (979466) 88 9' 6^984; the sidereal and se-
cular direct motion of the perihejion (4967"64) 26'49"51536;
the inclination of thf orbit to the ecliptic at the beginning of 1750
(27762) 2 29' 54"888 ; the secular var. of the inclination to the
true ecliptic ( 47"87) 15"5098 decreasing ; longitude of the as-
cending node upon the ecliptic at the beginning of 1750 (1239327)
1 1 1 32' 2l"y48 ; the sidereal and secular mot. of the node on the
true ecliptic ( 5781"54) 31 7 ;3"2i896, retrograde.
According to Vince, the relative mean distance of Saturn from
the sun is 954072, that of the earth being 100000 ; the place of
his aphelion for the beginning of 1750, was 8s. 28 9' 7" ; its
secular motion 1 50' 1" ; the eccentricity of his orbit 53640.42 ;
the greatest equation 6 26' 42" ; the longitude of his node for
the beginning of 1750, 3s. 21 32' 22" ; its secular motion in
respect to the equinox, 55' 30"; and the inclination of his orbit
to the plane of the ecliptic 2 29' 50". Delambre for the begin-
ning of 1800, makes the mean place of Saturn 4s. 3 5' 9" 9, of
\hz aphelion 8s. 29 4' 10", and of the node 3s. 21 56' 40".
For the beginning of 1812, he makes his place 8s, 29 54' 33" 5,
that of his aphelion 8s. 29 17' 23", and of his nodes 3s. 22 2'
58". The annual mean motion of the aphelion, according to De-
lambre, is l' 6", and of the node 32". The greatest equation of
Saturn in his orbit, for 1750, according to the same author, is
6s. 26o 41' 7, and its secular variation 110" 24.
The periodic motion of Saturn in his orbit is from west to east,
and nearly in the plane of the ecliptic ; it is subject to inequalities
similar to those of Jupiter and Mars. Saturn commences and fin-
ishes his retrograde motion when the planet, before and after his
opposition, is about iOS 54' distant from the sun. The duration
of this retrogradation is nearly 1 3 1 days, and the arc of retrogra-
dation about 6 18'. At the moment of his opposition, his diam-
eter is a maximum, and its mean magnitude, according to La-
place, is (54" 4) 17" 6256, or 17" 6 nearly.
From Saturn's periodic revolution, his mean distance from the
sun is found to be 895 35 1645.2 miles ;* and his progressive mo-
tion in his orbit is 21786.5 miles an hour.-\ His real diameter is
* Saturn's per, rev. = 10759d. Hi. 51' 11" nearly, == 929584271", Hit
sq. of which is 864126916890601441, this being- divided by 995916894538801,
the sq. of the seconds in a sidereal year (see pa. 350) gives 867.669705, the
cube root of which is 9.5378 nearly, the relative dist. of Saturn from the.
sun. Hence 9.5378 X 23464.5 =223799.7081 dist. of Saturn in semidmnu
of the earth, which multiplied by 3956, gives 895351645.2436 miles, the
dist. of Saturn from the sun.
j- Saturn's dist. from the sun being multiplied by 2, and then by 3.1416,
fives 5025673457.4 nearly the circumference of his orbit ; hence 10759d,
382 OF THE SOLAR SYSTEM.
67624 miles, * and his magnitude 624.6 times that of the earth.f
The Light and heat which he receives from the sun about -j of
the light and heat which the earth receives |
Cassini and Fato in 1683, suspected that Saturn revolved on
his axis, from having one day observed a bright streak which dis-
appeared the next, when another came into view near the edgt of
his disk ; these streaks are called Belts Cassini considered these
belts as clouds floating in the air ; and having a curvature similar
to the exterior circumference of the ring, he concluded that they
ought to be nearly at the same distance from the planet, and that
therefore the atmosfiherc of Saturn extended to the ring Dr.
Herschel found that the arrangement of the belts always followed
the direction of the ring, so that when the ring opened, the belts
shewed an incurvature answering to it. And during his observa-
tions on June 19, 20 and 21, 1780, he saw the ^same spot in
three different situations He in consequence conjectured that
Saturn revolved about an axis perpendicular to the plane of his
ring. This conjecture receives a greater degree of probability
from the planet being an oblate spheriod, the equatorial diam. or
the diam. in the direction of the ring, being to the diam. perp to
it, or the polar diam. in the proportion of about 11 : 10 accord-
ing to Dr. Herschel ; the measures being taken with a wire mi-
crometer prefixed to his 20 feet reflector. The attraction of the
ring, however, contributes to produce this effect. He afterwards
verified the truth of his conj cture, having determined from di-
rect observation, that Saturn revolves on his axis from west to east
in l"h 16' i/' 4. Phil, trans. 1794. He has also observed^e
belts nearly parallel to Saturn's equator.
For the phenomena that would appear to an observer situated in
Saturn, see Dr. Gregory's Astr b. 6, prob 6. From what has
been delivered these may be easily conceived, and most of them
represented on the globes.
Ih. 51' 11" : lh. or 3600" :: 5625673457.4 : 21786.5 miles the hourly velo-
city of Saturn in his orbit.
* Saturn's dist from the sun, at opposition, is 223799.7 of the earth':;
semidiameters, from which 23464.5, the earth's dist. in semid. being taken,
leaves 200335.2 semidiameters of the earth, Saturn's dist. from the earth.
Now taking his appar. diam. at oppos. 17" 6, we have inversely 200335.2 :
17" 6 :: 23464.5 : 150" 265, the appar. diam. of Saturn at a di'st. from the
earth equal to that of the sun. Hence 32' : 150" 265 :: 864065.5 (sun's
diam.) : 67624 miles the diameter of Saturn. The diam. may be also found
in the same manner as Jupiter's, pa. 362.
676243
t For '/an 3 ^ ( lo - 2 - 7956085 ) 6 24.6.
Saturn being about 9^ times further from the sun than the earth, his
heat and light (being as the square of the dist.) will therefore be 90J times
Ic35; than the earth's.
OF THE SOLAR SYSTEM. 383
OF SATURN'S RING.
SATURN, when viewed through a good telescope, makes a
more remarkable appearance than any of the other planets. Gal-
lileo, in 1610, first discovered his extraordinary shape ; the planet
appearing to him like a large globe between two small ones. In
1612 he was surprized to find only the middle globe ; but after-
wards he discovered again the globes on each side, and found that
their magnitude and form were extremely variable ; sometimes
they appeared round, at other times oblong, sometimes semicir-
cular, then with horns towards the globe in the middle, and by de-
grees growing so long and wide as to encompass it, as it were,
with an oval ring. Huygens in 1656, from the improvements he
had made in grinding glasses, was able to announce the curious
discovery, that these strange phenomena are produced by a large
thin ring, which surrounds the globe of Saturn, and which is every
where separated from it He made the space between the globe
and the ring something greater than the breadth of the ring, and
the greater diameter of the ring (which generally appears elliptic
except when the eye of the spectator is in its plane, when it ap-
pears like a straight line) to that of the globe as 9 : 4. Mr.
Pound made this prop, as 7 : 3. The best description of this
singular phenomenon is that given by Dr. Herschet in the Phil,
trans, for 1790. The following is the substance of his account.
The black disk, or belt, or Saturn's ring, is not in the middle
f its breadth ; nor is the ring subdivided into many such lines, as
some astronomers represent ; but there is one single dark, con-
siderable broad line, belt, or zone, which he has constantly found
on the north side of the ring. As this dark belt is subject to no
change whatever, it is probably owing to some permanent construc-
tion of the ring's surface. This construction cannot be owing to
the shadow of a chain of mountains, since it is visible all round the
ring ; for at the ends of the ring there could be no shade ; and the
same arguments will hold against any supposed caverns. It is
moreover pretty evident, that this dark zone is contained between
two concentric circles, as all the phenomena correspond to the
projection of such a zone.
The matter of the ring is undoubtedly no less solid than the
planet itself ; and it is observed to cast a strong shadow on the
planet. The light of the ring is also generally brighter than that
of the planet ; for the ring appears sufficiently bright when the
telescope affords scarcely light enough for Saturn. Dr. Herschel
next observes the extreme thinness of the ring : he frequently
saw the 1st, 2d, 3d, and 4th satellites pass before and behind it,
in such a manner as to serve as excellent micrometers to measure
its thickness. For an account of these phenomena, consult the.
jihil. trans for 1790 and 1792 ; or Vince's astr. many particulars
will also be found in Dr. Gregory's astr.
From a series of observations upon luminous points of the ring,
he has discovered that it has a rotation about its axis, the time of
384 OF THE SOLAR SYSTEM.
-which is lOh. 32' 15"4. The ring is invisible, with the telescopes
in common use among astronomers, when its plane passes through
the sun, or the earth, or between them ; in the first case the sun
shines only upon its edge, which is too thin to reflect sufficient
light to render it visible ; in the second case, the edge only being
opposed to us, it is not visible, for the same reason ; in the third
case, the dark side of the ring is exposed to us, and therefore the
edge, being the only luminous part which is towards the earth, is
invisible on the same account. Dr. Herschel, with his large tele-
scopes has been, however, able to see it in every situation. He
thinks that the edge of the ring is not flat but spherical, or sphe-
riodical. 3V1. de la Lande thinks that the ring is just visible with
the best telescopes in common use. when the sun is elevated 3'
above its plane, or three days before its plane passes through the
sun ; and when the earth is elevated 2' 20" above the plane, o"one
day from the earth's passing it. The difference of the telescopes
and the state of the atmosphere, will make 10 or 12 days difference
in the time of its becoming invisible.
Dr. Herschd, from his observations on the ring, thinks that he
has sufficient reason to conclude, that Saturn has two concentric
rings, situated in one plane, which is probably not much inclined
to the equator of the planet. The dimensions of the rings and the
space between them, are in the following proportion, as nearly as
they could be ascertained.
Parts. Miles.
Inner diameter of the smaller ring, - - 5900 or 146345
Outside diameter of do. - - - 7510 184393
Inner diam. of the larger ring, - - - 7740 190248
Outside diam of do. - - - 8300 204883
Breadth of the inner ring, 805 20000
Breadth of the outer ring, 280 7200
Breadth of the dark zone, or vacant space
between the rings, 155 2839
Dr. Hcrschd) from the mean of a great many measures of the
diameter of the larger ring, makes it 46" 677 at the mean clist.
of Saturn. Hence his diam. : the earth's diam. :: 25.8914
(according to Vince) : 1 . From the above proportion, therefore,
the diameter of the ring must be 204883 miles;* and the dist.
of the two rings 2839 miles.
From the oblique position of the ring, though circular, it ap-
pears elliptical, and it appears most open when Saturn is 90 from,
the nodes of the ring upon the orbit of Saturn ; or when Saturn's
long, is about 2s. 17, and 8s. 17. In this situation the lesser
* By taking- the appar. diameter of Saturn = 17" 6. we have 17" 6 :
46" 677 :: 67624 miles the diam. of Saturn (see pa. 382) : 179345 miles
the diameter of the ring 1 . If 17" 6 the appar. diam. of Saturn at his oppos.
were reduced to Saturn's mean dist. from the earth, the diameter of the:
ping- would come out greater, as the appar. d,iameter of Saturn would be
diminished.
OF THE SOLAR SYSTEM. 385
axis is very nearly equal to half the greater, when the observations
are reduced to the sun ; and therefore the plane of the ring makes
an angle of about 30 with Saturn's orbit.
In the Mem. de 1'Acad. at Paris, 1787, M. de Laplace sup-
poses that the ring may have many divisions ; but Dr. Herschel
remarks that no observations will justify this supposition. Lafilace
makes the inclination of the ring to the plane of the ecliptic (54 8)
49 19' 12". He remarks that the plane of the ring meeting the
solar orbit at eveiy serni revolution of Saturn, the phenomena of
its disappearance and reappearance return every fifteen years, but
frequently with very different circumstances, two disappearances,
and two reappearances may occur in the same year, but never
more. The incl. of the ring to the ecliptic is measured by the
largest opening which the eclipse presents to us. As the earth is
in the plane of the ring when it disappears or reappears, the posi-
tion of its node may be determined by the appar. situation of Sa-
turn. Lajilace further remarks, that all the disappearances and
appearances from which the same sidereal positions of the nodes
of the ring result, take place when its plane meets the earth.
The others when the same plane meets the sun It may therefore
be known by the situation of Saturn when the ring disappears or
reappears, whether this phenomenon is produced by the sun or
the earth. When the plane passes through the sun, the position
of its nodes gives that of Saturn, as seen from the sun's centre,
and the rectilinear dist. of Saturn from the earth may be deter-
mined as that dist. of Jupiter is by the eclipses of his satellites.
It is thus found that Saturn is about 9- times further from us
than the sun, when his appar. diam. is 17" 6. For more infor-
mation on the phenomena of the ring, and the manner of deter-
mining them, consult Dr. Gregory's Astr. sect. 13, b- 4, and
ch. 6. b. 6. See also Newton's prin. phen. 2. b. 3.
The phenomena of the ring to an eye placed in Saturn make an
interesting and curious part of the comparative Astr. of Saturn,
for which the reader is referred to Gregory's Astr. b. 6. ch. 6.
A learner who understands what is here delivered, will easily con-
ceive these phenomena, and the globe will very much assist in
exhibiting them ; thus, if the equator of the artificial globe be
made to coincide with the horizon, and the globe be turned on its
axis from west to east, its mot. will represent that of Saturn on
his axis, and the wooden horizon will represent the ring, especial-
ly if it be supposed a little more dist. from the globe. This ring
will cause a great variety in the days and nights in Saturn, which,
from its rapid mot. on his axis, are shorter than ours. There is
also a much greater diff. between summer and winter on Saturn's
globe than on the earth, as well on account of the duration on each,
and the sun's great decl. from the equator, as on account of the
meridian darkness in winter, from the interposition of the ring:
'vhich hides the sun.
386 OF THE SOLAR SYSTEM.
OF THE SATELLITES OF SATURN.
SATURN has besides his ring, seven little secondary planets or
satellites, which perform their motions round him from west to
east, in orbits nearly circular. One of them, which till lately
was reckoned the 4th in order, was discovered by Huygcns in' 1655,
with a telescope .00 feet long ; he published tables of its mean
motion in 1659, which were afterwards corrected by Dr. Halley in
1682. M. Cassini, with telescopes 100 and 136 feet long, discov-
ered the 5th in (671, the 3d in 1672, and the 1st and 2d in 1684 ;
he afterwards published tables of their motions, and called them
Sidera Lodoicea, in honour of Louis le Grand, in whose reign and
observatory they were discovered. These tables were afterwards
reformed and corrected by Dr. Halley from Mr. Pound's observa-
tions. Dr. Halley observes that the four innermost satellites des-
cribe orbits very nearly in the plane of the ring, which he says is,
as to sense, parallel to the equator ; and that the orbit of the 5th
is a little inclined to them. The periodic times of the five satel-
lites, and their dist. in semid. of the ring, as determined by Mr.
Pound, with a micrometer fitted to the 123 feet telescope given by
Huygcns to the R. Society, are as follow; first, Id. 2ih. 18' 27"
dist. 2 097 ; second, 2d. I7h. 41' 22" dist 2.686 : third, 4d. 12h.
25' 12" dist. 3752; fourth, I5d. 22h. 41' 12" dist. 8.698 ; and
Jlf'hj 7yd. 7h. 49' dist. 25.348. The distances in semid. of Sa-
turn as given by Pound, are 4.893, 6.286, 8.754, 20.295, and
59.154 respectively. The above distances were deduced from
that of the 4th. which was measured, from the proportion between
the squares of the periodic times and the cubes of their distances,
and found to agree with observation. Cassini, from his own ob~
sel-vations, makes the periods the same except the 5th, which he
makes 79d. 7h. 48'. He makes their dist. in semid. of the ring
as follow ; lf| (or 1|) 2|, 3J, 8, 8, and 23 (or 24 Mwton's
prin. b. 3. phen 2.) their dist. from the periodic times in sem.
of the ring 1.93, 2.47, 3.45, 8, 23.35 ; and their dist. at the mean
dist. of Saturn 43" 5, 56", l' 18", 3', and 8' 42" 5 respec-
tively. Herschel makes the dist. of the 5th 8' 31" 97, which is
probably more exact. Mr. Pound found the dist. of the 4th satel.
3' 7", when it was very near its greatest eastern digression ;
hence at the mean dist. of the earth from Saturn, that distance
becomes 2' 58" 21. Sir Isaac Newton (b. 3. prop. 8, cor. 1.)
makes it 3' 4".
The periodic times for Saturn's satellites are found m the same
manner as for those of Jupiter (pa. 365.) To determine these,
Cassini chose the time when the semiminor axis of the ellipsis
which they describe, were the greatest, as Saturn was then 90
irom their node, because the place of the satellite in its orbit is
then the same as upon the orbit of Saturn ; whereas in every other
case, it would be necessary to apply the reduction to obtain its
place in its orbit. As Saturn and his satellites cannot be seen at
the same time, without difficulty, in the field of view of a teles-
OF THE SOLAR SYSTEM. 387
cope, their distances have sometimes been measured by observing
the time of the passage of Saturn over a wire adjusted as an hour
circle in the field of the telescope, and the interval between the
limes when Saturn and the satellite passed.
, By .comparing the places of the satellites with the ring in dif-
ferent points of their orbits, and the greatest minor axes of the
eclipses which they appear to describe, compared with the major
axes, the first four are found to have the planes of thtir orbits very
nearly in the plane of the ring, and are, therefore, inclined to the
orbit of Saturn about 30 ; but according to Casxini the son, the
orbit of the 5th makes an angle with the ring of about 15. Cos-
sini places the node of the ring, and consequently the nodes of the
first four satellites, from what we have just now remarked, in 5s.
22 upon the orbit of Saturn, and 5s 2> upon the ecliptic.-
Huygens found it equal 5s. 20 30'. M. Maraldi, in 17-6, de-
termined the long, of the node of the ring upon the orbit of Saturn
to be 5s. 19 48' 30", and upon the ecliptic 5s. 16 20'. The
node of the 5th satel. is placed by Cassini in 5s. 5 upon the orbit
of Saturn. M. de la Lande makes it 5s. 27'. From the observa-
tion of M. Bernard at Marseilles, in 1787, it appears that the
node of this satellite is retrograde. Dr. Halley discovered that
the orbit of the 4th satellite was eccentric ; for, from its mean
motion, he found that its observed place was at one time 3 more
forward than by his calculations, and at other observations 2 30'
backward This indicated an eccentricity ; and he placed the line
of the apsides in 10s. 22. Phil, trans. No 145. Or Vince's
astr. from which the principal part of our acct. of the satellites is
extracted.
The re-volutions and mean motions of the satellites are given by
La Lande as follow. In this table the satellites are numbered
from Saturn, as they were before the discovery of the other two by
Dr. Herschel, whose orbits are situated nearer to Saturn than any
of the other five.
flat
I
II
III
IV
V
Diur. mot.
6s. 10 41' 53'
4 11 32 6
2 19 41 25
9 22 34 38
4 32 17
Mot in 365rf.
Period Itevol. \ Synod. Ker-ot.
Id. 21h. 18' 26"222
2 17 44 51,177
4 12 25 11,100
15 22 .41 16,U22
79 7 53 42,772
Id. 21h. 18' 54"778
2 17 45 51,013
4 12 27 55,239
15 23 15 23,153
79 22 3 12,883
Newton in his prin. b. 3. prob. i7, remarks, that the outermost
satellite of Saturn seems to revolve about its axis with a motion
like that of the moon, having the same face continually turned to-
wards Saturn. For in its revolution round Saturn, as often as it
comes to the eastern part of its orbit, it is scarcely visible, and
generally quite disappears ; which he says is probably occasioned
by some spots on that part of its body which is then turned towards
the earth. Newton remarks the same of Jupiter's satellites. Dr.
Herachcl has confirmed this conjecture, by tracing regularly the
periodical change of light through more than ten revolutions,
388 OF THE SOLAR SYSTEM.
which he found, in all appearances, to be cotemporary with the
return of the satellite to the same situation in its orbit. M Ber-
nard, at Marseilles, from his observations in 1787, has further
confirmed this result. Hence this equality in the period of rota-
tion and revolutions appears to be a general law of the motion of
the satellites, and a remarkable instance of analogy in this part ot
the Solar System.
In the phil. trans, for 1789 and 1790, Dr. Herschel gives an
account of the discovery of two other satellites, with some of the
elements of their motions, and tables for calculating them.
The distances of these satellites from the centre of Saturn are
36" 7889, and 28" 6689 ; and their peiiodic times are Id. 8h
53' 8" 9, and 22h. 37' 22" 9. The planes of the orbits of these
satellites lie so near the plane of the ring, that their difference
cannot be perceived.
According to Laplace, if the semidiameter of Saturn seen at
his mean distance from the sun be taken as unity, the distances
of the satellites irom its centre will be as follow : First, 3.080,
Second^ 3.952, Third, 4.893, Fourth, 6.268, Fifth, 8.754, Sixth,
20,295, and Seventh, 59 154 ; and the durations of their sidereal
re-volutions, 0.94271 days = 22h. 37' 30" 144 ; 1*370:24 days ==
Id. 8h. 53' 8" 736 ; 1.8878 days = Id. 21h. 18 f 25" 92 ;
2.73948 days = 2d. 17h. 44' 51" 072 ; 4.5)749 days = 4d. I2h.
25' 11" 136; 159453 days = I5d. 22h. 4l' 13" 92; and
79.3296 days = 79d. 7h 54?' 37" 44 respectively, the satellites
being taken here in order, as their respective orbits are situated
from Saturn. These mean distances of the satellites, as Laplace
remarks, being compared with the durations of thiir revolutions,
the beautiful proportion of Kepler, relative to the planets, and
which we have seen to exist in the satellites of Jupiter, is here
again found to take place.
CHAP. IX.
OF URANUS OR HERSCHEL,*
AND HIS SATELLITES.
THIS is the remotest of the planets belonging to the Solar Sys-
tem, that has hitherto been discovered. From its minuteness it
had escaped the observation of ancient astronomers. Flam&tead ar
the end of the last centuiy, and Mayer and Le Monnicr in this,
had observed it as a small star ; and according to F. de Zach's
account of this planet in the memoirs of the Brussels academy,
* This planet is called by the English the Georgium Sidus, in honour oi
the present King George III. In the Naut. Aim. it is called the Georgian,
By foreigners it is generally called Herschel, in honour of the discoverer,
The royal academy of Prussia and some others call this planet Ouranvs or
Uranus. Laplacz calls it by the same name, but Dehtmbrv in M * v
OF THE SOLAR SYSTEM. 389
1785, there was then in the library of the prince of Orange, four
observations of this planet considered as a star, in a catalogue of
observations written by Tycho Brake. It was not, however, until
1781, that Dr. Herschel discovered its motion, and soon after, by
observing carefully, he was able to ascertain that it was a true
planet.
Its apparent diameter is so small that it can seldom be seen by
the naked eye. When viewed through a telescope of small mag-
nifying power, it appears like a star of the 6th or 7th magnitude.
Laplace makes its apparent diam. about ( 1 2") 3"888. In a clear
night, in the absence of the moon, it may be perceived, by a good
eye, without a telescope ; at the beginning of 1812, its place, as
given in the Naut. Aim. will be long. 7s. 22 2', lat. 18' N. and
decl. 1 8 S. At the beginning of 1 8 1 3, it will be in long. 7s. 26 23',
lat. U'N.anddecl. 19 8' S And on the 31st Dec. 1813,it will be
in long 8s, 36', lat. 1 1' N. and decl. 20 7' S. Like Mars, Ju-
piter and Saturn, its motion is from west to east round the earth.
Accort&ng to Vince, its periodic revolution is performed in 83
years, Isbd. 18h. The place ofhsafihelion for the beginning of
1750, is Us. 16 19' 30", and its annual progressive motion^ ac-
cording to M. de la Grange 3" t7, from the action of Jupiter
and Saturn ; and therefore its motion in long. = 50" 2s (the
precession of the equin.) -f 3" 17 = 53" 42. The longitude of
the nodes in the beginning of 1750 was 2s. 12 47'. The annual
motion of its nodes-) according to La Grange, is 12" 5 from the-
ory ; according to La Lande, who takes a different density for
Venus, it is 20" 40'", which he uses in his tables. The inclina-
tion of its orbit to the ecliptic is 46' 2C' ; . Its distance from the
sun is 1918352, that of the earth being 100000. The eccentricity
of its orbit 90804. Its greatest equation is 5 27' 16".
Laplace makes the sidereal revolution of Uranus 30689 days,
or 84 years 29 days. His mean distance or half the greater axis
of his orbit, 19.18362, that of the earth being 1. The proportion
of the eccentricity of half the greater axis of his orbit, for the be-
ginning of 1750, 0.046683. The secular variation of this pro-
portion 0.000026228, the sign indicating a diminution.
The mean long, at the beginning of 1 750, reckoning from the
mean vernal equinox, at the epoch of the 31st Dec. 1749, at noon,
mean time at Paris was (352 962) 318 33' 53" 64. The lon-
gitude of the perihelion at the beginning of 1750, was (185 1262)
166 36' 48" 888. The sidereal and secular progressive motion
of the perihelion (759" 85) 4' 6" 1914. The inclination of the
orbit to the ecliptic at the beginning of 1750 (0 8599) 46' 26"076.
The secular progressive var. of the incl. to the true ecliptic
,9 38) 3" 039. Longitude of the ascending node upon the
calls it Herschel. It is called Uranus in allusion to the names of the heathen
deities by which the other planets are distinguished, as before remarked ;
thus Uranus was the father of Saturn, Saturn the father of Jupiter, Jupiter
the father of Mars, .c. Herscl&l discovered this planet at Hath in England.,
nn the 13th of March, 1781,
S90 OF THE SOLAR SYSTEM.
ecliptic at the beginning of 1750 (80* 7015) 72 37' 52" 86.
And the sidereal and secular retrograde motion of the node upon
the true ecliptic (10608") 57' 16" 992.
Delambre in his tables (tab. 1 60) gives his mean place for the
beginning of 1 8 1 2, 7s. I 5 4' 9"5, of his afihdion 1 1 s. 17 3 i ' 23",
anri of his node 2s. 12 54' 6". His mean mot. for 365 days 4
17' 44"2, of his aphel. 53", and of his node 16", his mean motion
for an hour is 1 "8. His greatest equat. for 1780, 5 2 1' 2" 7.
Lafilace remarks that his motion, which is nearly in the plane of
the ecliptic, begins to be retrograde when, previous to the opposi-
tion, the planet is (115) 103 30' distant from the sun. The mo-
tion ceases to be retrograde when, after the opposition, the planet
in his approach to the sun is only 103 30' distant from it. The
duration of his retrogradation is about 1 5 1 days, and his arc of re-
trogiadation (4 C ) 3 36'. He further remarks, that if the disk of
Uranus were to be estimated by the slowness of his motion, it
should be on the confines of the planetary system.
From the periodic time of this planet, given above, his distance
from the sun, Sec. may be found as for the other planets. The ratio of
his diam. to that of the earth's is given as 4.32 : 1, hence its mag-
nitude is more than 80 times that of the earth. His hourly vel in
his orbit ; the light and heat on his surface, Sec. may also be found
as for the other planets.
OF THE SATELLITES OF HERSCHEL.
DR. HERSCHEL has discovered six satellites moving round this?
planet, in orbits almost circular and nearly perpendicular to the
plane of the ecliptic. The first two he discovered on Jan. 1 1, 1787,
of which he gives an account in the Phil. Trans, for 1787.
In the Phil. Trans, for 788, he gives a further account of this
discovery, together with their periodic times, distances, and posi-
tions of their orbits, as far as he was then able to ascertain them.
The most convenient method, as Vmce remarks, of determining
the periodic time of a satellite, being, either from Us eclipses, or
from taking its positions in several successive oppositions of the
planets ; but as no eclipses happened since the discovery of the
satellites, and that it would be too tedious to put in practice the
latter method, Hcrschd, therefore, took their situations whenever
he could ascertain them with some degree of precision, and then
reduced them, by computation, to such situations as were necessa-
ry for his purpose. In computing their periods round their prima-
ry, he has taken the synodic revolutions, as the positions of their
orbits, at the times when their situations were taken, were not
sufficiently known, to get very accurate sidereal revolutions. The
mean of several revolutions gave the synodic rev. of the first sa-
tellite 8d. 17h. l 1 19"3, and of the second 13d. 1 Ih. 5' i"5. The
epochs from which their situations may, at any timej be computed;
OF THE SOLAR SYSTEM. 391
are, for the/r/, Oct. 19, 1787, at I9h. 1 1' 28" ; and for the see-
end, at I7h. 22' 40" ; at which times they were 76 43' north, fol-
lowing the planet.
Dr. Herschel has also determined their distances from the plan-
et ; one of which he obtained by observation ; and the other from
the periodic times. While making his observations to discover the
dist. of the second, its orbit seemed elliptical. He found its greatest
longation to be 46"46, and its elong. at the mean dist. of the pri-
mary from the earth 44"23, which will be the true dist. very
nearly, on supposition that the satellites move in circular orbits ;
hence by Kepler's rule, the dist. of the second sat. comes out 33"09.
In this calculation the synodic rev. were used for the sidereal,
which will make but little error.
In determining the inclinations of the orbits and places of their
nodes, Herschel could not determine which part of the orbit was
inclined /o, and which from the earth ; he therefore computed
them on both suppositions, and found that the orbit of the 2d sat.
is inclined to the ecliptic 99 43' 53" 3, or 81 6 ; 4" 4 ; its as-
cending node upon the ecliptic is in 5s. 18, or 8s. 6, and when
the planet comes to the ascending node of this satellite, which hap-
pened about the year i799, and will again take place about the
year 1818, at which time there will be an eclipse of this and the
1st satellite, when they will appear to ascend through the shadow
of the planet, in a direction almost perp. to the ecliptic. M JDe-
lambre makes the ascending node in 5s. 21, or 8s. 9 from Dr.
HerschePs observations. The situation of the orbit of the first
satellite does not materially differ from that of the second. The
light of the satellites is extremely faint ; the 2d is the brightest,
but the difference is small. Here, as in Jupiter's satellites, these
two are called t st and 2d satellites, and are so in the order of dis-
covery, but from the four other satellites which Herschel has dis-
covered to revolve round this planet (Phil, trans. 1798) this oiv
deris changed, and the 1st is now the 2d, and the 2d the 4th.
Most astronomers give their distances from the planet, and
their periods as follow.
I.
JDist. 25"5
II. f III. ) IV.
33' 3b"57 44*2
L I7h.l'19|l0d. 23h. 4'!13d. llh. 5'
V.
8S"4
38d. Ih. 49'
VI.
176*8
107d.16h.4G 1
Lafilace says, that if we take for unity the semidiameter of Ura-
nus, supposed (6") 1"944 seen at the meao dist. of the planet
from the sun, the distances of his satellites will be 13 120, 17.022,
19.845, 22.752, 45.507, 91.008 ; and the durations of their sidereal
revolutions 5.8926 days, 8.70 6 8d. 10.961 Id. 13.4559d. :>8.075d. aud
107.6944 days respectively. These durations, as Laplace remarks,
with the exception of the 2d and 4th, have been concluded from
the greatest observed elongations, and from Kepler's rule, as re-
gards the primary planets, (see pa. 253) a rule which observation
has. confirmed with regard to the 2d and 4th satellites of Herschel,
392 OF THE SOLAR SYSTEM.
so that it should be considered as a general law of the motion of ft
system of bodies round a common focus.
It is a singular circumstance, that the orbits of those satellites
are found to be nearly perp. to the ecliptic, and still more singular,
that they perform their revolutions round Herschel in a retrograde
order, that is contrary to the order of the signs. The first is pro-
bably the cause of the latter ; and if properly examined, might
therefore throw much light on the general cause of the regular
law observed in all the planets, in following the direction of the
sun's motion on his axis ; and also of all the satellites, except those
of Herschel, in performing their motions in the direction of the di-
urnal revolution of their primaries. If the action of the sun in
moving on its axis, carry the planets, or that of the planets the sa-
tellites, it is plain tha* the more oblique their orbits are to the
equator of the body, the less will the effect of the body be upon
that which regards it as its centre.
CHAP. X.
OF THE MATURE AND MOTION OP
COMETS.
BESIDES the primary planets and their satellites already describ-
ed, there are, belonging to our system, other bodies called Comets,
from their hairy appearance ; these appear suddenly in the plan-
etary regions, and again disappear ; they are sufifiosed to move
round the sun in elliptic orbits, like the planets, but very eccen-
tric, so that the Comet is visible but in a small part of it. They
are distinguished from other stars from their bting attended with
a long train of light, always opposite the sun, and which is of a
fainter lustre the further it is from the body. Hence comets are
commonly divided into bearded, tailed, and hairy ; this division,
however, relates not to different comets, but rather to the several
appearances of the same comet. Thus, when the comet is west-
ward of the sun, and moves from it, it is said to be bearded, be-
cause the light precedes it in the manner of a beard ; when the
comet is west of the sun, and therefore sets after him, it is said
to be tailed, because the light or train follows it ; lastly, when the
comet is in opposition to the sun, the train is hidden behind the
body of the cornet, except a small portion that surrounds it like a
border of hair, or coma, whence called hairy, and whence the
comet derives its name.
Like the other stars, the comets participate in the diurnal mo-
tions of the heavens, and thus combined with the smallness of their
parallax, proves that they are not meteors generated in the atmos-
phere, but that they are much higher than the moon, and in the
regions of the planets (JVewton prin. b'3. Lem. 4.) Though
OF THE SOLAR SYSTEM. 393
the opinion prevailed among many of the ancient philosophers that
they were meteors, &c. yet the most ancient and learned of them
supposed comets to be eternal or constant bodies oi the world,
which like planets perform their revolutions in stated times. See
Newton pr. b. 3, or Dr. Gregory's Astr b 5. sect 1 where
their opinions, 8cc. are given. (The fihase observed in the comet
of 1744, of which only half the disk was enlightened, proves that
they are ofiake bodies, which receive their light from the sun.)
Among the moderns Tycho Brake was the first who, after dil-
igently observing the comet of 1577, and finding that it had no
sensible diurnal parallax,* assigned it Us true place in the plan-
etary regions Few comets have approached the earth so near as
to have a diurnal parallax, they however afford sufficient indica-
tions of an annual parallax. This shews that they are not so dis-
tant as the fixed stars f
There have been various theories concerning the nature of co-
mets, which it would be too tedious here to detail \ (they may be
* Kiel in his astr. lect. 17, gives the following 1 simple method of discov-
ering- whether the comet has any sensible parallax. A comet before it dis-
appears moves so slowly, that it seems to be almost without any motion,
and it may be twice observed in this manner, before and after the perihe-
lion ; these places of the comet being selected, then, when it is very high
above the hurizon, take any two stars between which the comet lies in a
right line parallel to the horizon, which may be easily found by extending
the thread before the stars ; next when the comet approaches the horizon,
let the thread be extended again as before, and if the comet is found to be
in the same straight line with the stars as before, it is a proof that it h;:s no
sensible parallax, and must be at an immense distance from us. No error
can arise here from refraction, as it equally affects both the stars and comet.
Kiel also gives the following method ; let the comet be observe^ when it is
near the eastern part of the horizon, and in a right line with two stars that
are both in the same circle, which is perp. to the horizon ; and afterwards
when the stars rise higher, and are not in the same vertical circle as before,
if the comet still appear to be in the same right line with them, it can have
no sensible parallax ; and hence its course must be very hig-h in the hea-
vens. If it should be found more depressed than to appear in the right line
that joins the stars, it must necessarily have a parallax. And if during the
observations, the comet has a proper motion ef his own, this motion must
be allowed for, in proportion to the time between the observations. The
parallax here spoken of is the diurnal. The want of this parallax afforded
an argument of placing the comets higher than the moon ; but their being 1
subject to an annual paral. proves their descent into the planetary regions.
The reason of these methods may be easily understood from considering
the earth's motion, and the nature of a parallax.
\ Hevelius observes, that these motions of the comets are inexplicable,
but on the supposition of the earth's motion round the sun ; which therefore
affords another proof of the truth of this hypothesis. See Newton's prin.
pa. 380, Motte's translation.
$ The principal is that of Sir Isaac J\~eu<ton, He says (pn'n. b. 3,
prop. 41.) that the cornets are solid, compact, fixed, and durable bodies,
like the bodies of the planets ; in a word, they are a kind of planets winch
move in very oblique orbits every way with the greatest freedom ; perse-
vering in their motions even contrary to the planets direction ; and that
their tail is a very thin slender vapour, emitted by the head or nwck-us of
3 B
394 OF THE SOLAR SYSTEM.
found in the firincifiia, in Gregory 9 s ast. b. 5, or in Vince's astr.J
but the truth or falsehood of any one of these theories may be
tried from the following phenomena of comets, collected from
JRee's Cyclopedia (Philadelphia ed.)
first. Those comets which move according to the order of the
signs, do all, a little before they disappear, either advance a little
slower than usual, or else go retrograde, if the earth be between
them and the sun ; and more swiftly if the earth be situated in a
contrary part. On the contrary, those which proceed contrary to
the order of the signs, advance more swiftly than usual, if the earth
be between them and the sun ; and more slowly, or go retrograde,
when the earth is in a contrary part.*
2. As long as their velocity is increased, they move nearly in
great circles : f but towards the end of their course, they deviate
from their circles ; and when the earth advances in one direction,
they advance the contrary way.
3. They move round the sun in ellipses, having one of their
feci in the centre of the sun : and by radii drawn to the sun de-
scribe areas proportional to the times4
the comet heated by the sun. The truth of these principles appear from
their being 1 perfectly conformable to the above phenomena. That the comets
are solid, appears from the heat they are capable of sustaining-, as appears
from that of 1680, whose heat, according- to Neivton, was 2000 times great-
er than that of red hot iron.
* This is evident, as the course of the comets is among the planets, and
must therefore follow the same laws.
f Keil remarks (lect. 17. astr.) that if the distance of the comet be ob-
served every day from two fxed stars, whose longitudes and latitudes are
known, and the places computed from these distances be marked on the
surface of a celestial globe, the course of the comet will thus be found to
be a portion of a great circle, allowance being 1 made for the earth's motion.
From this it is manifest, that the comet moves in a plane passing- through
the eye of the spectator, or more exactly through the sun , for all visible
motion that is made in such a plane, however inclined to the ecliptic, will
always appear to be in the periphery of a great circle. The comet's devia-
tion from a course in a great circle, or the variation in the comet's orbit
with the ecliptic, is only apparent, and does not arise from the real mo-
tion of the comet, but from that of the earth, as was shewn \\\t\\G planets.
whose distances and incl. to the ecliptic vary according to their different;
positions, as seen from the earth ; while they are regular, as seen from th
sun. Newton says that this arises from their parallax. See his prin. b. 3.
Lemma 4.
$ They are supposed, for the ease in calculation, to move ia parabolic
orbits, which, in that part of it near the sun, is sufficiently correct for the
elements of their elliptic orbits, as they are very eccentric. Neivton re-
marks, from considering the curvity of their orbits, that when they disap-
pear, they are much beyond the orbit of Jupiter, and that in their perihelion
they frequently descend below the orbits of Mars and the inferior planets.
{Prin. b. 3. Lemma 4.) He has fully demonstrated, that every body plac-
ed in our planetary system, should be attracted by the sun, with a force
reciprocally proportional to the squares of the distances, which, in conjunc-
tion with the projectile force, would cause the body to move in a conic sec-
tion about the sun placed in the focus, and describe areas proportional to
the times. . He also shews, that if the same comet ever return 4 o oi:<- svs^
OF THE SOLAR SYSTEM. 395
4. The light of their bodies, or nuclei, increases in their recess
from the earth towards the sun ; and on the contrary, decreases in
their recess from the sun.*
5. Their tails appear the largest and brightest immediately af-
ter their transit through the region of the sun, or after their peri-
helion ; because then they are most heated, and must therefore-
emit a greater quantity of vapours f
6. The tails always decline from a just opposition to the sun to-
wards those parts which the bodies or nuclei pass over, in their
progress through their orbits ; because all smoke or vapours emit-
ted from a body in motion, tends upwards in an oblique direction,
and receding from that part towards which the smoking body
proceeds.
7. This declination cceteris paribus, is the smallest, when the-
heads, or nuclei, approach nearest the sun ; and is still less near
the nucleus of the cornet* than towards the extremity of the tail.
Because the vapour ascends with more velocity near the head of
the*comet, than in the higher extremity of the tail ; and also when
the comet is at a less distance from the sun than when at a greater.
See Dr. Gregory's astr, b. 5, prop. 4. cor. 1, &c. In this prop,
and corollaries, many interesting remarks concerning the tails of
comets are given. See also the principia, b. 3, Lemma 4, and
prop. 41.
8. The tails are somewhat brighter and more distinctly defined
in their convex than in their concave part, because the vapour in
the concave part, which goes first, being somewhat nearer and
denser, reflects the light more copiously.
9. The tails always appear broader at their upper extreme than
near the centre of the comet, because the vapour in a free space
is perpetually rarified and dilated, as is also the case with any vir-
tue passing from a centre.
10. The tails are always transparent, and the smallest stars ap-
pear through them, because it consists of thin vapour, 8cc.
Hence the hypothesis of Newton, given in the note to pa. 393,
exactly agrees with the phenomena. Newton, at the end of the
third book of his Principia, fully illustrates this hypothesis, and
gives many other interesting particulars concerning the nature of
comets. Dr. Gregory enters more fully into the investigation of
these particulars. See b. 5 of his Astronomy.
The nuclei, or the heads^ or rather the bodies of comets, when
viewed through a telescope, appear differently, or with different
tern, it must describe an ellipsis, though very eccentric. See Dr. I/alley's
Synopsis of the astronomy of comets, at the end of Dr. Gregory's astronomy.
* As they are in the regions of the planets, their access towards the sun
bears a considerable proportion to their whole distance. See Newton's ob-
servations on the comet of 1680.
f As the heads of the comets are illuminated by the sun, this light being
reflected towards the earth, renders them visible, and shews that they are
not in the region of the fixed stars, for any of the planets which are only
illuminated by the light of the fixed stars, are not visible on the earth.
396 OF THE SOLAR SYSTEM.
phases, from those of the fixed stars or planets. They are subject
to apparen changes, which Newton considered as performed in
their atmosphere ; and this opinion was confirmed by observations
of the comet in i744, Hist. Acad. Sciences, 1744.
Tycho* He-vetius, and some others, give various estimates of the
magnitude of comets, but their estimates are not sufficiently accu-
rate to be depended on ; for it appears that they did not distinguish
between the nucleus and the surrounding atmosphere. Tycho
computes that the true diameter of the comet in 1 577, was in pro-
portion to the earth's diameter as 3 to 14. Hruelius found the
diam. of the nucleus of the comet of 1661, and also that of 1665,
at its commencement, to be less than a I Oth part of the diam. of
the earth ; and that of 1652, from its parallax, and appar. mag. of
its head he computes on the 10th Dec. to be to the diam. of the
earth as 52 to 100. He found the true diam. of the comet of 1 664
to be six times that of the earth, at another time not much more
than a| diameters The diameter of the atmosphere is sometimes
10 or i5 times greater than that of the nucleus. JVamstead ob-
served the comet of 682, with a telescope of 16 feet, and found
with a micrometer the least diam. of its head = 2', but the nucleus
scarcely a tenth part, or about l i" or 12". From comparing the
appar. dist. and mag. of comets, some have been found larger than
the moon, and even equal to some of the primary planets. The
diam of that of 1744, when at the distance of the sun from us,
measured about l'. and therefore its di-.m must be about three
timts the dium of the earth : at another time the diam. of the nu-
cleus was nearly equal that of Jupiter.
Hence Ne A ton shews that more comets are seen in the- hemis-
phere towards the sun, than in that which is opposit*. to it. For as
cornets shine by the reflected light of the sun, they will not become
visible to us until their light, so reflected, is strong enough to affect
Our eyes : and moreover, as comets are principally rendered con*
spicuous from their tails, which they do not emit until heated by
the sun, it is evident that to have the comet or its tail visible, it
must come within a defined distance of the sun. And, as Newton
remarks (Cor 2, lem. 4, b. 3) comets descending into our parts,
neither emit Uils, nor are so well illuminated by the sun, as to dis-
cover themselves to our naked eyes, until they are come nearer to
tis than Jupiter. But the far greater part of that spherical space
which is described about the sun, with so small an interval, lies on
that side of the earth which regards the sun ; and the comets in
that greater part are more strongly illuminated, as being generally
nearer the sun This is more fully elucidated in prop. 5, b. 5,
Gregory's Astr. where it is further proved (Scholium) that by how
much nearer a comet must be to the sun before it becomes visible,
by so much does the number of the comets, seen in the hemis-
phere towards the sun, exceed the number of those which appear
in the opposite hemisphere. If, as Newton remarks, they wrre vi-
sible in the regions far above Saturn, they would appear more frc-
quently in the parts opposite the sun.
OF THE SOLAR SYSTEM. 397
Comets are always surrounded with a very gross dense atmos-
phere, and from the side opposite the sun project a tail, which in-
creases as the comet approaches its perihelion, immediately after
which it is longest and most ruminous. That the tuil depends on
the sun, is evident from these phenomena. The tail is so rare,
that the smallest stars are been through it, am' hence the opinion
of the ancient philosophers was, that $ie tail is a very thin, fiery
vapour arising from the comet, dpian, Cardan, Tycho^ Sndl and
others among the moderns, were of a different opinion, and imagin-
ed that the sun's rays were propagated through the transparent
head of the comet, and refracted as in a lens. But this is contrary
to the laws of Dioptrics, nor does the figure of the tail answer to it ;
and moreover there must be some reflecting substance like dust in
a room, Sec to render the rays visible to an eye placed sideways
from it. Kefiler supposed that the tail was produced by the gross
parts of the comet being carried away by the sun's rays- Hevclius
thought that the thinnest parts of the comet's atmosphere were ra*
rified, and driven towards the parts turned from the sun. De. Cartes
considered the tail as produced by the refraction of light, from, the
nucleus of the comet to the eye of the spectator. If this were the
case, the planets and principal fixed stars ought likewise to have
tails ; nor would the tails, as Dr. Gregory remarks, be free from the
colour of the rainbow, which always accompany refracted light.
(See other opinions, Sec. in Gregory's Ast. prop. 4, b. 5,) Dr. Gre-
gory remarks, that the most obvious opinion to any one that looks
at a comet, is, that the tail has its origin from the head. He shews
this to be the opinion of the principal of the ancient philosophers,
and also that of Neivton, who says, that the tail is nothing else but a
-very thin -vafiour^ ivhich the head or nucleus of the comet emits by its
hf j at. He shews that the atmosphere of comets will furnish vapours
sufficient to form their tails, from this principle, that if the air
should expand itself according to this law, which is confirmed by
experience, viz that the spaces in which it is compressed are reci-
procally proportional to the weights compressing it ; a globe of air
of an inch diameter, if it should become as rare as it would be at
the height of a semidiameter of the earth, would fill all the plane-
tary regions as far as the sphere of Saturn, and far beyond. (See
this demonstrated in prop 3, b 5, Gregory's Ast.) Hence he sup-
poses that when the comet is descending to its perihelion, the va-
pours behind the comet in respect to the sun, being rarified by the
sun's heat, ascend and carry with them the particles of which the
tail is composed, as air rarified by heat carries up the particles of
smoke in a chimney. Since then the coma or atmosphere of a
comet is ten times higher than the surface of the nucleus, reckoning
from its centre ; the tail ascending much higher, must necessarily
be immensely rare ; and hence the stars appear so visibly through
it. The ascent of the vapours will be promoted by their circular
motion round the sun. When the taihis thus formed, like the nu-
cleus, it gravitates towards the sun, and by the projectile force re-
398 OF THE SOLAR SYSTEM.
ceived from the comet, it describes an ellipse about the sun, and
accompanies the comet, Sec. ^ (See the Prin. or Gregory's Astr.)
The vapours of the comets^ being thus ratified and dilated, may be
scattered through'lfhe heavens, and mix with the atmosphere of the
planets. Mairan supposes that the tails are formed out of the lu-
minous matter that composes the sun's atmosphere, which is sup-
posed to extend as far as the orbit of the earth, and to furnish mat-
ter for those northern lights, called the Aurora Borealis. He calls
the tail of a comet the .Aurora Borealis of the comet. This hypo-
thesis La Lande combines with that of JVeyvton. He thinks that
part of thevapour which form them, arises out of the atmosphere
rarified by heat., and is pushed forward by the force of the light
streaming from the sun ; and also that a comet passing through the
sun's atmosphere is drenched therein, and carries away some of it.
Euler (Mem. de 1'Acad. de Berlin, torn. 2, pa. 1 17, seq , thinks
that there is great affinity between these tails, the zodiacal light,
and the aurora borealis ; and that the common cause of them all is
the action of the sun's light on the atmosphere of the comets, of
the sun, and of the earth. It may, from thence happen, that the vet-
of the comet in its perihelion may be so great, that the force of the
sun's rays may produce a new tail before the old one, varied from,
the comet's mot. in its orbit and about an axis, can follow ; in which
case the comet might have two or more tails. The possibility of
this is confirmed by the comet of 1744, which was observed to have
several tails while it was in perihelion. Dr Hamilton, in his Phi*
losofihical Essays, urges several objections against the Newtonian
hypothesis. He observes that we have no proof of the existence of
a solar atmosphere ; and if we had, that when the comet is moving
in its perihelion, in a direction at right angles to the direction of its
tail, the vapours which then arise, partaking of the great velocity
of the comet, and being also specifically lighter than the medium
in which they move, must suffer a much greater resistance than
the dense body of the comet, and therefore ought to be left behind?
and would not appear opposite the sun ; and afterwards they ought
to appear towards the sun. Besides, if the splendour of the tails be
owing to the reflection and refraction of the sun's rays, it ought to
diminish the lustre of the stars seen through it, which would have
their light reflected and refracted in like manner, and consequently
their brightness diminished. He concludes that the tail of a comet
is composed of a matter which has not the power of refractity or
reflecting the rays of light ; but that it is a lucid or self shining
substance ; and from its similarity to the Aurora Borealis^ produced
by the same cause, and a proper electrical phenomenon. He con-
jectures that the use of the comets are destined to supply the sun
with fresh fuel, in place of what he loses from the emission of
light. This he conjectured from the proximity of the comet of
1 680 to the sun, and the resistance it must receive from the sun's
atmosphere. Hcyeliua informs us that he observed the comet of
1665 to cast a shadow upon the tail, a dark line appearing in its
OF THE SOLAR SYSTEM. 399
middle. Cassini observed the same phenomenon in the comet of
1680 ; and the same appearance was taken notice of by a curious
observer in the comet of 1744.
The lengths of the tails of comets are 'various?, and depend on
a variety of circumstances. Longomontanus mentions a comet that
in 1688, Dec. 10th. had a tail .which appeared under an angle of
140 ; that of 1680 oh the month of Dec. when it was scarcely
equal in light to the stars of the second mag. emitted a remarka-
ble tail, extending '50, 60, or 70, and more ; the comet of
1744 had a tail extending 16 from its body, and w'hich, allowing
the sun's parallax to be 10", must have been about 2 3, millions of
miles in length. The diameter of its body was equal to that of
Jupiter. The tail of the comet of. 1759, according to M. Pingre,
subtended an angle of 90 ; but the light was very faint. The
length of a comet*s tail may be thus found ;
let S represent the sun, E the earth, C the
comet, CL the tail when directed from the
sun ; then the place of the comet being giv-
en, we have the angle ECL, the side EC,
and the angle GEL, the angle under which
the tail appears ; hence CL the length of
the tail is given. If the tail deviate by an
angle LCM found from observation, we shall
then know the angle ECM, with CE and
the angle CEM, to find CM.
The analogy between the periodic times of the planets and their
distances from the sun, takes place also in the comets ; hence the
comet's mean distance may be found by comparing its period with
the time of the earth's revolution round the sun : thus the period
of the comet which appeared in 1531, 1607, 1682, and 1759,
feeing about 76 years, its mean dist. is found by saying as I 2
(or 1 year) : 76 2 (= 5776) :: 100 3 (or 1000000) : 5776000000
the cube of the comet's mean dist the cube root of which is 1794,
the mean dist. in such parts as that of the earth contains 100. If
the per. dist. of this comet 58 betaken from 3588, double the
mean dist. the aphel. dist. will be given = 3530, which is a lit-
tle more than 35 times the earth's mean dist. from the sun. la
like manner the aphel. dist. of the comet of 1680 is found to be
135 times the earth's mean dist. from the sun, its period being-
supposed 575 years ; so that in the aphel. it is more than 14 times
more distant from the sun than Saturn.
If the tail of the comet be supposed directed from the sun, the
limit of the comet's distance may be easilv ascertained from it.
400 OF THE SOLAR SYSTEM.
Let S be the sun, E the earth, ET the line in
which the head of the comet appears, EW the line
in which the extremity of the tail is observed, and
draw ST parallel to EW, then the comet is nithin
the dist. ET ; for if the comet were at T, the tail
would be directed in a line parallel to EW, and
therefore it could not appear in that line. The an-
gle TEW being given from observation, and there-
fore ETS its equal, together with TES the an-
gular distance of the comet from the sun, and ES
(from the theory of the earth, part 4 c.h, 4. or the
Nautical Aim.) hence ST the limit is given by saying sine ETS :
sine TES :: ES : ST. Or it may be found as 'n the following
example. On Dec 21, i680, the comet's elongation from the
sun was 32 24', and length of the tail 70; hence ST : SE ::
sine 32 24' : sine 70 :: 4 : 7 nearly ; therefore the comet's
dist. from the sun was ^ of the earth's dist from the same
Hence Aeivton found that all comets, while visible, are not fur-
ther from the sun than 3 times the earth's dist. from the sun. But
this computation depends on the goodness of the telescope, and the
mag. of the comet.
With regard to the motion and periods of comets, astronomers
are not agreed. Jheivton, Flamstead, Hallry, Gregory^ and all the
English astronomers, and also Cassini and others of the French,
seem satisfied that they return But de la Hire and others sup-
posed not It would be too tedious to give their respective rea-
sonings on this subject. Dr. Hallcy was the first who predicted
the return of a comet, and found that it was one and the same
comet which appeared in 1682, 1607, 15 i3, 14-56 and -305
Dr. Halley in his Synopsis of the Astronomy of Comets (see Dr.
Gregory's astr ) shews that comets describe ellipses, and not
parabolas or hyperbolas, and thence ventures to foretell the return
of the comet of 168^;, about the end of 1758 or beginning of 17^9 ;
it appeared Dec 14, 1758 He also shewed that it was the same
as the comet of 1607, 15 '3, 1456, and '305 In this computa-
tion he allowed for the action of Jupiter on this comet, which he
found would increase its periodic time about a year M. C'a^auf
computed the effects both of Saturn and Jupiter, and found that
Saturn wouM retard its return, in the last period, 1 days, and
Jupiter 51 1 clays. He, therefore, determined the time when the
comet would come toitsficnhelton to be in \pril > 5, 1759, observ-
ing that he might err a month from neglecting small quantities in
the computation It passed the perihelion on March 13, within
33 days of the time computed If we suppose that the time meant
by Dr. Halley to be that of the comet's passing the periht lion,
and not of its first appearing ; the addition of the 100 duys, from
the action of Saturn, which he did not consider, will bring it very
near the time in which it did pass the perihelion, and will also shew
that Jupiter's effect on its motion, as computed by him, was very
OF THE SOLAR SYSTEM. 401
accurate. He also observed that the action of Jupiter, in the de-
scent of the comet towards its perihelion in 1 682, would tend to
increase the inclination of its orbit, and accordingly the incl. in
1682 was found 22' greater than in 1607.
From the observations of M. Messier upon a comet in 1770, M.
JEdric Prosfieriti) member of the Royal Acad. of Stockholm and
Upsal, shewed that a parabolic orbit would not answer to its mo-
tions, and therefore recommended to astronomers to seek for the
elliptic orbit. This laborious task was undertaken by M. Lexdl,
who has shewn, that an ellipsis in which the periodic time is about
5 years and 7 months, agrees very well with the observations.-
(Phil, trans. 1779J
The ellipsis which the comets describe being very eccentric, as-
tronomers, for the ease in calculation, suppose them to revolve in
parabolic orbits, for those parts of their orbits which are within the
reach of calculation ; on this supposition they can very accurately
find the place of the perihelion of a comet, its dist from the sun,
the inclination of the plane of its orbit to the ecliptic, and the place
of its node ; which are the elements of the comet's orbit. Before
we can, however, determine the orbit of a comet, from observa-
tion, it will be necessary to premise such particulars relative to the
motion of a body in a parabola, as may be requisite for such an la-
ve stigation.
Let APM be a parabola, S its focus, .
A the vertex, P the place of the body ;
draw PQ perp. to AS, PD perp. to the
tang. PT, and SM perp. to AD. Now
by the property of the parabola QD=4^
the latus rectum (Emerson's Conic
Sect. prob. 11, b. 3) hence if AS be
taken = 1, then QD = 2 ; also the T A
angle PSA = 2PDA ; therefore if '
QD be made rad. PQ will be tang, of PDA or PSA ; hence to
the rad. AS, PQ will be twice the tang, of - PSA ; so that if t
be taken = the tang, of (z) half the true anomaly PS \, to the
radius AS =* 1,2?=== PQ. Also AQ x 4 AS = PQ 2 (Emer-
son's con. sect. b. 3. prop. 3, cor. 3) hence AQ = ? 2 ; also the
area AQP = f AQ X QP (Em con. b. 3, pr. 55) =* J. t* x 2( *s
| e* ; and as QS = 1 /*, the area QPS = QS X P Q = *_,s .
hence the area ASP = $1* -f t ; and the area ASM = 4 (AS
being = 1 SM, or ^ the latus rectum.) Let a and b be the times
in which the comet moves from A to M, and from \ to P ; then
the areas described about S being proportional to the times, we
have a : b :: | : -^ 3 + e, therefore at 3 + 5at = 4*. Hence
if a and the true anomaly be given, we have the time b = a 3 -f-
%at. And because a : b :: | : * -f. t , therefore if the true
anomalies, and consequently t, be given ^n different patabolas, the
402
OF THE SOLAR SYSTEM.
times of describing those true anomalies from the perihelion, will
be proportional to the times of describing 90 from the perihelion.
If the times a and b be given, the true anomaly may be found
from resolving the cubic equation t* -f 3t = which may be
4
done thus. Make a right angled triangle, one of whose sides is
expressed by 1, and the other by 7 and find the hypotenuse (h)
then find two mean proportionals between h -f- -p- and h 7- and
their difference will be the value of f.
Take the fluxion of * 3 -f 3t = and we have 3t* i -f- 3t =
a
- (a being considered constant) ; hence } = X - ; but
/= 1 -f t 2 x z, therefore 2z = -, x r~ = -- Xcos. z*xb,
3a 1-fr 2 Sa
the variation of the true anomaly corresponding to any small vari-
ation b of time expressed in decimals of a day, a being expressed
in days.
Let SA be the mean dist. of the earth from the sun = 1 ; then
the area of the circle described with that radius will be 3.14159 ;
also the area A MS -J. Now the velocity in the parabola : vel.
in the circle :: \/ 2 ; 1 ; for let Pfi be an indefinitely small arc
described by the body, S the place of the sun, SN a line drawn
from the focus S perp. to a tangent to the parabola APD at the
point P ; then 1 st. The vel. a in any fioint
P of the parabola, is as the square root of
the parameter to the axis, divided by SN ;
For the vel. is as the arc P/t, or u = fiP :
now/zM being perp. to PS, in the similar
rt. angled triangles /zPM, PSN, SN :
SP :: PM
But the
parameter is as the square of the sectors
described ; hence put R = the parameter,
from the nature of
we have R = /zM 2 x SP 2 , and R 2 =
/*M X SP ; and by substitution, fiP or u
.' RJ _ ^4AS
~SN j01 SN
the parabola. 2dly. The velocity u in any fioint P of the parabola^
is to the velocity Vofa body running through the cir conference of
a circle, 'with a central force tending to its centre, the rad. being
= SP, as 2^ : 1 . For, since u == , u z = r-^- 2 ; or as
SN 2 = SP x SA (from the nature of the parabola) w 2 =
<DF THE SOLAR SYSTEM. 403
A A d A
. ''5 T ** rn 5 now tne circle whose radius is SP being taken
orXAb b.r
as an ellipse, its parameter is = 2SP ; and the vel V being uni-
f - - ^ 2SP xr 2 2SP 2 ,u
form, it is every where as - ; hence V 2 = r- = rrj : there-
ojr ol 01
fore u* : V> :: ^ : Ja :: 2 : 1 ; hence u : V :: 2* : 1* :: fail.
or br
The areas described in the same time will be in the same ratio
as the velocities, because at A the motion in each orbit being per-
pendicular to SA, the areas described will be as the velocities ;
and this being the case in one instance, it must hold always so,
because in each orbit respectively, equal areas are described in
equal times. But the times of describing any two areas are as the
areas directly, and the areas described in the same time inversely ;
3.14159 4 /v/8i
therefore :: I :: the time of the revolution
o \/ 2 \ 3 /
in the circle = 365d. 6h. 9' : the time, of desciibing AM =t
109d. i4h. 46' 20". Now, as the time of describing AM, is in a
given ratio to the time in the circle, which varies as AS|, there-
fore if r = the perihelion dist. in any parabola, we have l| : r%
:: 109d (4h 46' 20" : the time of describing 90 in that para*
bola from the perihelion. Hence the time corresponding to any
true anomaly in that parabola whose perihelion dist = 1, being
given, we know the time corresponding to the same true anomaly
in any other parabola, because the times of describing 90 are as
the times corresponding to the same true anomaly. Hence if n
= the number of days corresponding to any given anomaly in that
parabola, whose perihelion dist is unity, then nr* will be the time
t corresponding to the same anomaly in that whose perihelion dis-
tance is r, which may be readily found thus ; multiply the log. r
by 3 and divide by 2, and to the quoiient add the log. w, and the
sum will be the log. of the time required. Hence also n as ~
therefore if from the log. t we subtract f log. r, the remainder will
be the log. n of the number of days corresponding to the same ano-
maly in the parabola whose perihelion dist = 1 ; hence the ano-
maly will be found from a table, which gives the times corres-
ponding to the true anomaly for 200000 days from the perihelion, in
that parabola whose perihelion dist. is unity. This table may be
constructed by the preceding prob. by taking a = 109.6155, and
assuming b = 1, 2, 3, 4, 8cc. and finding the corresponding values
of t. Dr. Halley first constructed a table of this kind.* De la
* See his synopsis of the astronomy of comets at the end of Dr. Gregory's
Astr. tthere he also gives the elements of 24, from 1337 to 1698. The tbl-
1 owing is the method he makes use of in calculating- his table. Let S be the sun
(fig-, pa. 402) ARP the orbit of a comet, A the perihelion, R the place where
the comet is 90 distant from the perjhel. P any other place. Let the lines
AP, PS, he drawn, and make SII, SQ equal to ?S ; and having 1 drawn the
404 OF THE SOLAR SYSTEM
Caille changed it into a more convenient form, by putting the areas
for the times ; but the most extensive and complete table is that
computed by De.lambre, and inserted in the tables annexed to the
3d ed. of La Landers Ast (tab. pa 204 to 234.) See art. 3118, La
Lande's Ast- The true anomaly in this tab. is calculated for days
nd quarters of a day to 200 days, then for days and half days to 4-00,
afterwards for each day to 700, for 2 days to 1200, for every 5 days
.to 1800, for 10 days to 3000, for I2i days to 4000, for 25 days to
7000, for 50 days to 12000, for 100 days to 24000, for 200 days to
40000, for 250 days to 48000, for 500 days to 100000, and for 1000
days to 200000 days. He also gives a small table for reducing
hours, minutes, and seconds, to decimals of a day. The following
is an example of the use of this table. The perih. dist. of a comet
being given = 0.5835 to find its true anomaly for 49d. 18h. 55' 16"
rt. lines PQ, PH, one of which is tang, and the other perp. to the curve ; let
fall PO perp. to the axis AOQ. . Now any area, as PRAS, being given, the
angle PSA, and the distance PS is required. From the nature of the parabo-
la, QO is always = \ the parameter or latus rectum of the axis (Emerson's
Con. Sect. b. 3, prop. 11) and hence if the param. = 2, then QO = 1. Let
PO = z, then AO = z, and the parabolic segment PRA = li" ~ 3 - But
the triangle APS = z, and hence the area ASPR = z 3 4- 3 ; r = a ;
whence r3 _j_ 3^ __. a . Wherefore solving- this cubic equation, z or the or-
dinate PO will be kiiown. Now let the area ARS be proposed to be divided
into 100 parts, this area is -j^ of the sq. of the param. and therefore 12 =
that sq. = 4. If therefore ihe roots of these equations z 3 4- 3z = 0.04,
0.08, 0.09, &c. be successively extracted, there will be obtained so many va-
lues of z or the ordinate PO respectively, and the area PRS will be divided
into - 00 equal parts. In like manner the calculus is to be continued beyond
the place R. Now the root of this equation (since OQ, = 1) is the tabular
tang-, of the /_ PQO or ASP ; hence the i_ ASP is given. RC, also
the secant of PRO, is a mean prop between QO = 1, and HQ = 2HS. But
if AS =r 1, and latus rectum therefore =4 (as in Halleifs table) then HQ will
be the dist. sought, that is 2PS in the former parabola. He then determines
the place of a comet, by the numbers in his table, thus. The vel. of a comet
in a parabola, being- every where to that of a planet describing 1 a circle about
the sun, at the same distance from the sun, as x /2 : 1 (from what we have
before dem. or Newt on* s prin. b. 1, prop. 16, cor. 7.) If therefore a comet
in its perihelion were supposed to be as far distant from the sun as the earth
is, then the diurnal arc, which the comet would describe, would be to the
diurnal arc of the earth as v/2 : 1. And therefore the time of the annual
rev. : the time in which such a comet would describe the quadrant of its orbit
from the perih. :: 3.14159 : */ \. Hence the comet would describe that
Quadrant in I09d. 14h. 46'; and so the parabolic area (analogous to the area
ASR) being* divided into 100 parts, to each day there would be allotted
0.912280 of those parts, the log-, of which, that is, 9.960128, is to be kept
for constant use. But then the times in which comets, at a gr. or less dist.
would describe similar quadrants, are as the times of the rev. in circles, that
is, in the sesquiplicate ratio of the distances ; whence the diurnal arcs esti-
mated in centessimal parts of the quadrant (which are put for the measures
of the mean mot. like degrees) are in each, in the sesquialtcra proportion of
the distance from the sun in the perihelion. See the application of the above,
in computing the appar. place of a comet, &:c. with examples, in Dr. Halle?/' t,
Synopsis, and which, except in the change in the tables, differs but little
from the method given abow.
OF THE SOLAR SYSTEM.
405
before or after its perihelion. (La Lande, art, 3121.) 1 8h. 5 5' 1 6"
being reduced to decimals of a day, gives ft. 7500 0.03819, and
0.00018 respectively, the sum of which added to 49 days, gives
49.78837 days.
Log. dist. of the perihelion 0.5835 9.7660409
Half the same log. 9.8830204
Log. of the perh. multiplied by - 9.6490613
Log. of the given time 49.78837 - 1.6971279
The remcl. is the log. of 1 1 1 .7034 days 2.0480666
Now 111.5 days correspond 90 38' 57"4? true anomaly; the
difference given in the table between 1 1 1.50 and 1 1 1 .75, is 5' 6"\ ;
hence .25 : .2034* :: 5' 6"l : *' 9" ; therefore the true anomaly
required is 90 43' 6".
Vince has given Delambre's table in pa. 454, vol. 1, of his Astr.
Draw SYperp. to the tang. (fig. pa. 401) then SP : SY :: SY: SA;
whence SP* : SA* :: SP : SY :: rad. : cos. PSY,or|PSA the
true anomaly ; or SP : SA :: rad 2 : cos. square of 1 the true ano-
maly. Hence if SA = 1, and a + x = ^PSA, a x == |/zSA ;
then 1 : SP* :: CO s. (a-\-x) : rad. and S/z* ; i :: ra( j. : cos<
(a x) ; hence S/z 2 : SP* :: cos. (a + or) : cos. (a x"). Hence
SP = SA the square of i the true anomaly, rad. being = 1 ;
therefore from log. SA, subtract twice the log. cos. 5 true anomaly,
the remainder is the log. of the dist. of the comet from the sun.
Now let BD be made perp to AB, take BC
= AB, join AC and produce it to E, from D
let fall DE perp to AE, and produce ED until
it meets AF parallel to BD in F, join AD and
draw DG, CH parallel to B A. Then as the
angle EAF = 45, EFA = 45, (32 Eucl. 1)
hence AE= EF (6 Eucl. l) ; also FG= GD
= AB (34 Eucl. 1) ; hence AF = BD + BA,
and GH = BD BA ; also by similar trian-
gles AF or BD -f BA : CD = GH or BD
BA :: EF or EA : ED :: rad. : tang. DAE ;
but AB : BD :: rad. : tang. BAD, from which subtract 45, and
we have BD -f BA : BD BA :: rad. : tang, of that difference.
If BD = SP 2 " (see last fig.) and BA = S/**, then SP* : Sfl* ::
rad. : tang. B AD = | ~ J * ; hence to get this angle, take half the
difference of the logarithms of SP and S//, and add 10 to the index
(because in the log. tangents, the index of log. tang, of 45, or log,
' The above .2034 (.703450) is given in Delatnbrr-'s tables .7034, t lift-
ex, b^ing taken from ther.
E
G
406
OF THE SOLAR SYSTEM
of rad. = 1, is 10, in place of 0) and this last sum will be the logo
tang, of the angle, from which let 45 be taken, and we have
SP 2 -f- S/* 2 : SP 2 " S/z* :: rad. : tang, of that difference.
Hence if two radii SP, Sfi (fig pa. 402) and the contained angle
fiSP be given, the two anomalies will thence be gi en. For let a be i
of ASP -{- AS/z, and x be i of ASP AS/? ; then i ASP = a -f- or,
and ^ AS/i = a x ; hence S/j* : SP 3 :: cos. a -f x : cos. a a:
:: (Trig.) cos. a X cos. or sine a x sin. a: : cos. a X cos. x -}-
sine a X sine x; therefore SP 2 -f- S/* 2 : SP 2 S/i 2 :: cos. a.
X cos. x : sin. a X sin. x :: cos. a -~ sin a : sin. x -~ cos. .r ::
cos. a : tang. or. Now the ratio of the two first terms is found
from the last art. and as the angle PS/* is given, the value of or will
be given ; hence a is also given, and therefore the sum, arid dif-
ference of ASP, AS/* is given, and hence the angles themselves.
If fi be on the other side of A, then a is given to find x.
Given two distances SP, S/2, from a v
the focus to the curve of a parabola,
and the angle between them tojind the
parabola. With the centres P and /*,
and radii PS, /jS. describe two circu-
lar arcs riut, mvn, to which draw the
tangent ai^T ; draw ST perp to it,
and bisect it in A, and it will be the
vertex of the parabola. For S A be-
ing = AT, and ST to Pw, or S/i to
pi>i the parabola will pass through A
and P, pi &c. from a well known
method of describing the parabola.*
prob. in Newton'k prin b. 1, prop.
Gregory's Astronomy.
See a general solution to this
19. Or prop. 24, b. 5, Dr.
* If one end of a thread equal in length to the longest side t>B of a ruler
or square (similar to that which carpenters use) be fixed at the point S, and
the other end at B the end of the square vB. If the side av of the squares
be moved along- the rig'ht line To, and always coincide with it ; then, the
string being always kept tight, and close to the side of the square, the
point p will describe a parabola. For other methods consult the writers on
conic sections.
If through either of the given points a circle be described with S as a
centre, meeting again with the trajectory, and this point of intersection be
joined to the former by a straight line ; on this line let a perp. be let fall
from S until it meets the parabola, this last point will be the perihelium of
the trajectory ; the dist. of which from the focus will be the latus rectum.
The reason is evident ; for the focus S being found in the axis of the para-
bola, a circle described about the centre S will cut the parabola, if it cuts it
at all, in two points equally distant from the axis, and hence the rt. line joining
the intersections, will be perp. to the axis, &c. If a time be taken who^e in-
terval, from the first obs. (the comet being then supposed in the point c of
its orbit) is to the interval of time between the first and third obs. as thd
urea cAS to the area cSd (c and d being the observed places of the comet,
and A the perihelion) that will be the time of the comet's perih.
OF THE SOLAR SYSTEM.
407
The elements of the orbit of a comet being given, to compute
its place at any time. The elements of a comet's orbit are, 1.
The time when the cemet passes the perihelion, 2 . The place of
the perihelion, 3. The distance of the perihelion from the sun, 4
The place of the ascending node, 5 The inclination of the orbit
to the ecliptic. From these elements the place at any time may
be computed. The example given by M. de la Cattle in his astr.
is the comet of 1739, which passed its perihelion on June 17, at
lOh. 9' 30" meantime; the place of the perihelion was in 3s.
12 38' 40" ; the perihelion distance was 0.67358, the mean
dist. of the earth from the sun being I ; the ascending node was
in Os 27 25' U", and the inclination of the orbit 55 42' 44";
to compute the place seen from the earth on Aug. 17, at 14 ; 20"
mean time
Let APV be the parabolic orbit of the
comet, N the ascending node, P the co-
met's place, T the corresponding place of
the earth ; draw Pv perp to the ecliptic ;
produce SN, Sv, SP, ST to w, w, p, and
/ the sphere of the fixed stars, and de-
scribe the great circles nfi 9 nu^t and/m.
I. The interval of time from the peri-
helion to the given time, is 6 id. 4h. 10'
30" = 61.17.-, whose log. = 1.786567 ;
also the log. of .67358 is 9.828388, f of
which log. (from the nature of logar )
is 9.742582, which sub. from 1.786567,
leaves 2.043985, the log. of 1 10.6587
days, which by the table (see the ex. pa. 405) answers to 3s. 21'
38", the true anomaly PSA at the given time.
II. Subtract 3s. 21' 38" from 3s. 12 38' 40", the place
of the perihelion, the comet being retrograde, and after passing
the perihelion, and the remainder is 12 17' 1" for the heliocen-
tric place p of the comet in its orbit.
III. The longitude of n is 27 25' 14", also np = 27 25' 14"
12 17' 1" = 15 8' 13" ; hence rad. : cos.flnu = 55 42' 44"
:: tang./zrc = 15 8' 13" : tang, un = 8 39' 53", the distance
f the comet from the ascending node measured upon the ecliptic.
IV. Take this value of un from the place of the node, and there
remains 18 45' 21"
comet reduced to the ecliptic.
V. As rad. : sine pn == 15 8'
the true heliocentric place of the
13
sine finu = 55 42' 44
: sine pu = 12 37' 34" the heliocentric latitude, or lat. seen
from the sun, which is south.
VI. The true place T of the earth at the same time is 10s. 24
34' 36" ; hence TST => 35 25' 24" ; therefore TST -f YSw
= TSv = Is. 24 10' 45". Also TS = 1.0115.
VII. By what is shewn in a preceding article (pa. 405) cos,
s.quare of 45 10' 48" ; rad. 3 :: .67358 : SP = 1.3S57.
408 OF THE SOLAR SYSTEM.
VIII. As rad. : cos. PSv = 12 27' 34" :: SP = 1.3557 :
Sv = 1.32377.
IX. In the triangle TS-y, TS, St>, and the included angle TSv
are given ; hence the angle S fi; is found = 77 33' 38", which
being taken from 4s. 24 34' 36" the sun's place, leaves 2s. 7 0'
57}" for the comet's true geocentric longitude.
X. Again, sine 54 10' 45" : sine 77 33' 38V' :: tang. PS^
= 12 27' 34" : tang. PTi; = 14 54' 4", the comet's true geo-
centric latitude.
To determine the orbit of a comet from observation. Sir Isaac
Newton, aided by his theory of the planets, and by those observa-
tions which shewed him that the comets descended from immense
distances into the planetary regions, was the first who solved this
important problem, which he called firoblema longe difficittimum.
The following is the method given by him in his principia. Select
three observations distant from one another by intervals of time
nearly equal. But let the interval of time in which the comet
moves more slowly, be somewhat greater than the other ; so that,
for ex. the diff. of the times may be to the sum of the times, as
the sum of the times to about 600 days. See Newton prop. 41, b. 3.
If such observations be not at hand, a new place of the comet must
be found. (See Lem. 6. b. 3. JVeivton's prin. or Emersorfs differ-
ential method, prob. 10. ex. 17 and 18.)
Let S represent the
sun, T, t, /, three pla- MS
ces of the earth in the
orbis magnus ; TA,
tB, *C, three observed
longitudes of the com-
et ; V the time be-
tween the 1st obs and
the 2d. W the time
between the 2d and 3d.
X the length which in if\ G /
the whole time, V + "
Wthe comet might
describe, with that ve-
locity which it had in
the mean clist. of the earth from the sun (which is found as di-
rected pa. 402, or cor. 3. prop. 40. b. 3 Jirin ) and tV a perp.
upon the chord Tt. In the mean observed long. tB, take the
point B at pleasure, for the place of the comet in the plane of the
ecliptic ; and from thence towards the sun S, draw the line BE,
which may be to the perp. tV as SB X St 2 to the cube of the
hyp. of the rt angled triangle, whose sides are SB, and the tang.
*f the lat. of the comet in the 2d obs. to the radius tB. Through
OF THE SOLAR SYSTEM. 409
the point E (see lemma 7 firin. *) draw the right line AEC, whose
parts AE and EC, terminating in the right lines TA, and C, may
be to each other as the times V and VV ; then A and C will be
nearly the places of the comet in the plane of the ecliptic, in the
1st and 3d observations, if B was its place rightly assumed in the
second.
Upon AC bisected in I, erect the perp. li ; through B draw the
obscure line Bz parallel to AC ; join the obscure line Sz cutting
AC in /, and complete the parallelogram Him. Take Is = 3 1/,
and through the sun S draw the obscure line ax = 3S* 4- 3i7.
Then cancelling the letters A, E, C, I, from the point B towards
,, draw the new obscure line BE, which may be to the former
BE in the duplicate proportion of the dist. BS to the quantity Sm
-f- -^ il. Again, through the point E draw the right line AEC as
before, that is so that AE : EC :: the time V : the time W,
between the observations. Then A and C will be the places of
the comet more accurately.
Upon AC bisected in I, erect the perpendiculars AM, CN, IO,
of which AM, CN may be the tangents of the latitudes in the 1st
and 3d obs. to the radii TA and tC ; join MN cutting IO in O ;
draw the rect. parallelogram Him as before ; in z'A produced
take ID = Sm -f fzV. Then in MN. towards N take MP to X,
found above, hi the subduplicate propor. of the earth's mean dist.
from the sun (or of the semid. of the orbis magnus) to the dist.
OD. If the point P fall on N ; A, B, C, will be three places of the
comet through which its orbit is to be described in the plane of
the ecliptic. But if P do not fall on N ; in the right line AC take
CG = NP, so that the points G and P may be on the same side
of the line NC.
By the same method as the points E, A, C, G, were found
from the assumed point B, from other points b and b assumed at
pleasure, find out the new points e, a, c, g, and e, a, c, g. Then
through G, g and g, draw the circle Cgg- cutting the rt. line fC in
Z ; Z will then be one place of the comet in the plane of the
ecliptic. And in AC, ac, ac, taking AF, af, of, equal respectively
to CG, eg, eg, through the points F, f and/ draw the circle Ff/
cutting the rt. line AT in X ; the point X will be another place
of the comet in the plane of the ecliptic. And at the points X
aad Z, erecting the tangents of the latitudes of the comet to the
* This lemma is as follows. Through a given
point P to draw a rt. line RC, -whose parts PB,
PC, cutoff by tivo rt. lines AB, AC, given in po-
sition, may he to each ether, i?i a given proportion.
From the given point P suppose any rt. line PD
to be drawn to either of the right lines given, as
AB (produced if necessary) and produce PD
towards AC the other given right line to E, so
that PE may be to PD in the given proportion.
(12 Eucl. 6) Draw EC parallel to AD ; also draw
CPB, and by sim. triangles PC : PB :: PE : PD.
3D
410 OF THE SOLAR SYSTEM.
radii TX and /Z, two places of the comet in its orbit will be de-
termined If therefore a parabola be described to the focus S
through those two places, by the method given pa. 406, this para-
bola will be the orbit of the comet See the demonstration of this
construction in the lemmas (7, 8, 11 and 10) given by Newton,
with a further explanation in Emerson's comment on the priii. pa.
105. 106, &c This prob. is more fully explained by Dr. Gregory
in his Astronomy, b 5, prob. 26. This explanation would be pre-
ferred were it not too tedious.
It will be convenient nut to assume the points B, b, #, at ran-
dom, but nearly true If the angle AQt, at which the projection
of the orbit in the plane of the ecliptic cuts the rt. line tB be
rudely known ;* at that angle with Bt draw the obscure line AC,
which may be to | Tt in the reciprocal subduplicate proportion of
SQ to St (See Emerson's Comment, pa. 106) or as \/St to
<\/SQ t And drawing the rt. line SEB so that its parts EB may
be equal to the length Vr, the point B will be determined, which
we are to use for the first time. Then cancelling the rt. line \C,
and drawing anew AC according to the preceding construction,
and moreover finding the length MP ; in tB take the point b by
this rule, that if TA and tC intersect in Y, the dist. Yb may be to
YB in a proportion compounded of the proportion of MP to MN,
and the subduplicate proper, of SB to v ^b.| By the same method
the 3d point b may be found if required. But if this method be
followed, two operations will in general suffice. For if the dist.
Bb happen to be very small after the points F, f, and G, g, are
found, draw the right lines Ff and Gg, and they will cut TA and
tC in the points X and Z. (See this also investigated in Gregory's
ast. b 5, prob 27.)
Newton corrects the comet's trajectory found by the foregoing
method as follows.
Ojicration 1 \ssume that position of the plane of the trajectory
which was determined according to the preceding method, and
select three places of the comet found from very accurate obser-
vations, ana at great distances one from the other- Then sup-
* See the method of determining this in the Schol. prop. 13, b. 5. Greg-
Ast. In what follows, in this chap, there arc also given different methods,
whicn are more convenient in practice.
j- For vel. of a comet at Q. in a parab. : vel. at Q in a circle : ^/2 : 1,
nearly as f : 1. Also vel. at Q in a circle : vel. at t in a circle :: St^ : SQ- ;
therefore ex erjuo, vel. of the corn, in Q : vel. of the earth at t :: nearly as
AC : Tt; hence AC : Tt :: |St* : SQ^ ; or AC : -f Tt :: St* : SQ* '
Wherefore Q is nearly in the chord of the parabola, and B nearly a point of
the comet's orbit.
* If MP = MN, or AG = AC, then Yb : YB :: Yc : YE :: ac : AC ::
vel. in b : vel. in B :: SB^ : Sb^. But if Sb = SB, and MP or AG invari-
able, it will be Yb : YB :: ac or AG : AC when G falls in CY. Therefore
universally Yb : YB :: AG X SB* : AC X Sb :: MP X SB^ MN X
Sb^> to find b truly.
OF THE SOLAR SYSTEM. 4H
pose A to represent the time between the 1st obs. and the 2d, and
B the time between the 2d and 3d It will be convenient that in
one of those times the comet be in its perigeon, or at least not far
from it. From those apparent places find by trigonometry the
three true places of the comet in that assumed plane of the trajec-
tory, then through the places found, and about the centre of the
sun as the focus, describe a conic section by arithm. operations
(priricipia, b. 1, prob. 2 1, or pa. 406.) Let the areas of this figure
which are terminated by radii drawn from the sun to the places
found, be D and E, to wit, D the area between the 1st and 2d
obs. and E the area between the 2d and 3d. Let T represent the
whole time in which the whole area D -}- E should be described
with the vel of the comet found as in prob. 16, b. 1. prin. See
the laws of Gravity, Sec. at the end of this work.
Ofier. 2 Retaining the incl. of the plane of the trajectory to the
plane of the ecliptic, let the longitude of the nodes of the plane of
the trajectory be increased by the addition of 20 ; or 30'. which
call P. Then from the foresaid three observed places of the co-
met, let the three true places be found, as before, in this new
plane, as also the orbit passing through those places, and the two
areas of the same, described between the true observations, \vhich
call d and e, and let t be the whole time in which the whole area,
d + e should be described.
Oper. 3. Retaining the long, of the nodes in the 1st operation,
let the incl. of the plane of the trajectory to the plane of the eclip-
tic be increased by adding 20' or 30' to it, the sum of which call Q.
Then from the foregoing three appar places of the comet, let the
three new places be found in this new phne, as well as the orbit
passing through them, and the two areas of the same described
between the observation, which call d and e, and let t be the whole
time in which the whole area d + e should be described.
Then taking C : 1 :: A : B, and G : 1 :: D : E, also g :
1 :: d : e, and g : 1 :: d : e ; let S be the true time between
the 1 st and 3d. obs. and observing well the signs -j- and , let
such numbers m and n be found as will make 2G 2C = mG
rag -f- n * n ar| d 2T 2S = mT mi -f- nT nt.
If in the 1st oper. I represents the incl. of the plane of the tra-
jectory to the plane of the ecliptic, and K the long, of either node,
then I -f W Q will be the true incl. of the plane of the trajectory to
the plane of the ecliptic ; and K ~j- wP the true longitude of the
node. And lastly, if, in the 1st. 2d. and 3d. oper. the quantities
R, r and r represent the parameters of the trajectory, and the
quantities L, 1 and /, the transverse diam. of the same; then R-f-
mr mR -f nr nR will be the true parameter, 1 divided by
L -f ml mL -\- nl wL will be the true transverse diam. of
the trajectory which the comet describes And the transverse
dUm. being given, the periodic time is given. See the investiga-
tion of these expressions, and of computing them by the rule" of
false, in Emerson's comment, pa, 108, Sec, The same investiga-
412 OF THE SOLAR SYSTEM.
lions, 8cc. arc also given in Gregory's astr. prob. 31, to which the
reader is referred, as it would be rather tedious to insert them
here ; their application will be given in the following part of this
chapter.
Newton remarks, that the periodic times of the rev. of comets,
and the transverse diameters of their orbits, cannot be accurately
enough determined but by comparing comets together, which ap-
pear at different times. If after equal intervals of time, several
comets are found to have described the same orbits, we may thence
conclude, that they are all but one and the same comet revolved
in the same orbit, and then from the times of their revolutions,,
the transverse diameters of their orbits will be given ; and from
those diameters the elliptic orbits themselves will be determined.
For this purpose an extensive table of the elements of the comets
is given in this chap.
M. la Lande gives the following mechanical method of deter-
ming the orbit nearly. Let the dist of the earth from the sun be
divided into equal parts, and let 1 parabolas be described, whose
perihelion distances are 1, 2, 3, &c. of those pans ; and divide
these parabolas into days from the perihelion, answering to the
motion of a body in each. Let S be the
sun, a, #, c the places of the earth at
the times of three observations of the
comet. Then let three geocentric la-
titudes and longitudes of the comet be
found, and set off the elongations Sarf,
S6e, Sc/in longitude. From a, 5,c, ex-
tend three fine threads, am, bn, ofa ver-
tical to ad, be, cf, making angles, with
them equal the geocentric latitudes res-
pectively. Then let any one of the pa-
rabolas be taken, place its focus in S,
apply the edge to the threads, and observe whether it can be made
to touch them all, and whether the intervals of time cut off by the
threads upon the parabola, be equal to the respective intervals of
the observations, or very nearly so ; if this be the case, the true
parabola, or very nearly the true one, is found. But if the parabola
do not agree, let others be tried, until there be one found which
agrees, or very nearly agrees, and the true, or nearly the true pa-
rabola will then be obtained, whose inclination, place of the node,
and perihelion, are to be determined as accurately as possible from
mensuration ; also the projection upon the ecliptic. If none of the
parabolas nearly answer, it shews that the perihelion dist. must be
greater than the dist of the earth from the sun, in which case other
parabolas must be constructed ; but this does not very often happen.
As this method is rather troublesome, when only the elements of
"one comet is required, though useful when there are many, as it
determines the elements very nearly, Vince proposes the following
method by means of one parabola, without dividing it.
OF THE SOLAR SYSTEM.
413
Take a firm board perfectly plane, and fix on paper for the pro-
jection ; let a grove be cut near the edge, and five perpendiculars
be moveable in it, so that they may be fixed at any distances. Let
S represent the sun, and describe any
number of circles about it ; compute
five geocentric latitudes and longi-
tudes of the comet, from which you
will have the five elongations of the
comet at the times of the respective
observations. Draw SA, SB, SC,
SD, SE, making the angles ASB,
BSC, CSD, DSE, equal to the sun's
motion in the intervals of the obser-
vations ; and on any one of the circles , _
make the angles So/, Sbg, Sc/;, Srfi,
S' j , equal the respective elongations in longitude, and fix the five
perpendiculars, so that the edge of each may coincide with /, #,
^, /, k. From the points a, 6, c, d, e, extend threads to the res-
pective perpendiculars, making angles with the plane equal to the
geocentric latitudes of the comet ; then fix the focus of the para-
bola in S and apply its edge to the threads, and if it can be made
to touch them all, it will be the parabola required, corresponding
to the mean distance Sa of the earth, which is here supposed to
revolve in a circle, being sufficiently accurate for the present pur-
pose. If the parabola cannot be made to touch all the threads,
change the points a, 6, c, d, e, to such of the other circles as will
be judged from the present trial, most likely to succeed, and let
the former trial be repeated again ; by a few such repetitions,
such a distance for the earth will be obtained, that the parabola will
touch all the threads, in which position find the inclination, ob-
serve the place of the node, and measure the perihelion distance,
compared with the earth's dist. and the elements of the comet's
rbit will be nearly obtained.
Bosco-uich gives the following method of approximating to the
rbit of a comet.
Let S be the sun,
XZ the orbit of the
earth, supposed to
be a circle ; T the
place of the earth at
the first observa-
tion, and t at the
third ; draw TC, tc
to represent the ob-
served long, of the
comet ; and let L,
/, A, be the longi-
tudes at the first, second, and third observations ; m and n the geo-
centric latitudes of the comet at the first and third observations ;
ar$ t } T, the intervals of time between the first and second, and
414 OF THE SOLAR SYSTEM.
second and third observations. Assume C for the place of the
comet at the first observation, reduced to the ecliptic ; then to
determine the point at the third observation we have T x sin. * I
: t X sin. / L :: TC : tc, and c will be nearly the place re-
quired (see Buscov. Opuscula vol 3, or Sir H Englefield's de-
termination of the orbits of comets, pa, 37) join Cc, and it will
represent the path of the comet on the ecliptic, upon this assump-
tion. Draw CK, ck, perp. to the ecliptic, and make CK : TC ::
tang, m : racl. and ck : tc :: tang n : rad join K>, and it will
represent the orbit of the comet, if the first assumption be true.
Bisect Cc in x, and draw xy parallel to ck, and y will bisect K ;
join yS. Let SX = 1 ; then if v be put for the mean ve locity of
i^
the earth in its orbit, the velocity of the comet at y will be -
hence takings = Tt, let the value of -7^ be found, and
if this be equal to K measured by the scale, the assumed point C
was the true point. But if these quantities be not equal, assume
a new point for C, in doing which the error of the first assump-
tion will be a guide ; if for instance the computed value of K be
greater than the true value, and the lines CK, ck art diverging
from each other and receding from the sun, the point C must be
taken further from T, and how much further, may be conjectured
from the value of the error, and also from hence that the velocity
of the comet diminishes as it recedes from the sun These con-
siderations will lead us to make a second assumption near the
truth Having thus determined the true points C, c, very nearly,
produce cC, K to meet at N, join NS, and it will be the line of
the nodes. Draw O, cz perp to SN, and the angles KK^, kzc,
will measure the inclination of the orbit. From the two distances
SC, Sc, and the included angle CSc, the parabola may be con-
structed, and applied as in the preceding method, from which the
time of passing the perihelion may be found
The following is another method by which the orbit is readily,
and very nearly obtained. Let S be the
sun, T the earth, T, t, t, three places
of the earth at the times of the three
observations ; extend three threads Tfi,
in, tm, in the direction of the comet, as
before directed ; assume a point y for
the place of the comet at the second ob-
servation, and measure St/ ; then if ST
= i, and the velocity of the earth oe v,
the velocity of the comet at y will be
2 2 v -n
. , ^ t ; let v be represented by Tt, it, and upon any straight
2~ X Tt 2- X t?
edge PQ 5 set off cc = ~y^-rr> and ed = ~ 2 ; then apply
OF THE SOLAR SYSTEM. 415
the point e to ?/, and by turning about the edge, try whether you
can make the point C fall in T/fc, and the point d in tm ; if this
cannot be done, the error will be a guide to assume another dis-
tance, and by a few trials the point y, where the points c and d
will fall in T/?, tm. This method is very easy in practice, and
sufficiently accurate to obtain a dist Sz/ from which to begin to
compute, in order to find the orbit more correctly, when the comet
is not too near the sun.
The parabola being determined nearly, let some quantity be as-
sumed as known at the first and second observations, from which let
the place of the comet be computed at those times, and also the time
between ; if that time agree with the observed interval, a parabola
which agrees with the two first observations is obtained ; if the
tinus do not agree, let one of the assumed quantities be altered,
and see how it then agrees : and then by the rule of false, the sup-
position which was altered may be corrected, and a parabola ob-
tained which will agree with the two first observations. (See Dr.
Gregory's Ast. b. 5, prop. 31 and 26.) In like manner by altering
the other assumed quantity, another parabola is obtained, agreeing
with the two first observations Then if these do not agree with
the third observation, a correction must be made by proportion,
and the three observations will be answered.*
As the comets do not however move in parabolas, but in very
eccentric ellipses, it is impossible to find a parabola that will accu-
rately agree to all the data ; it will therefore be sufficient when it
nearly agrees. When great accuracy is required, we must take
into consideration the effect of aberration and fiarallax ; the former
may be computed from the methods given in the following chap.
and the latter by taking the horizontal parallax to the sun's hori-
zontal parallax = 8"75 (see pa. 284) as the distance of the sun to
the distance of the comet, and then finding the parallax in lat. and
long, as directed pa. 331, Stc.t
* For further particulars we must refer to Vtnce's ask vol. pa. 428, &c.
See also " An account of the discoveries concerning- comets, with the way
to find their orbits, and some improvements in constructing- and calculating-
their places ; to which are added new tables fitted for those purposes." By
Thomas Barker, Gent. London, 1757.
f Newton, in determining- nearly the
dist. of a comet, determines also its pa-
raliax. The following- is his method.
Let T QA, TQB, TQC be three ob-
served longitudes of the comet about
the time of its first appearing 1 , and
'Y'QF its last observed long-, before its
disappearing 1 . Draw tliert. line ABC,
whose parts AB, BC intercepted be-
tween the right lines QA and QB, QB
and QC, may be to each other as the
times between the three first observa-
tions respectively. Produce AC to G,
go that AG may be tQ AB as the time
416
OF THE SOLAR SYSTEM.
R
To ascertain the periodic time of a comet, and the axis of its orbit.
If comets, after receding from the lower regions of the Solar
System, to vast distances beyond the orbits of the most distant
planets, return again to the neighbourhood of the sun, the paths
which they describe must be nearly elliptic : if then observations
be made sufficiently accurate to be a basis of the operations, the
requisites of the prob. may be determined in the following man-
ner. Let AKBi be the trajectory
of a comet, AB its major axis, IK
the minor, S, F the two foci, the
former of which represents the
sun's place, C the comet's place,
CS its dist. from the sun, Cc the
space it passes over in a very small
portion of time, DCE a tangent to
the curve in the point C ; SD, FE
perpendiculars, let fall thereon
from the foci ; let SG be drawn
parallel to the tangent, and join
FC. Also let \LB be a circle,
described on the tranverse axis
AB ; APTB a rectangle about the ellipse AIB, and AQRB a
square about the circle ALB. Lastly, let A NO be the elliptic
orbit of any planet, S, f, its foci ; let SC = a, SD =e b y Cc =* e,
the time in which e is described = /; the transverse or greater
axis of the comet's orbit AB = x, that of the planet's orbit /\O =
g, the circumference of the circle AVO, described on the same axis
==/?, the periodic time of the comet = , and that of the planet == n.
between the first and last obs to the time between the 1st and 2d. and join
QG. Now if the comet moved uniformly in a rt. line, and the earth either
-stood still, or was likewise carried forwards in a rt. line by an uniform mo-
tion, the long 1 . 'Y'QG would be the comet's long 1 , at the last obs. Hence the
/^ FQG, which is the difl'. of long-, proceeds from the unequal motions of tbe
comet and the earth. If the eartb and comet move in contrary directions,
this angle is added to TQ^, and accelerates the comet's appar. mot. but if
they move in the Siime direction it is subtracted, and either retards the mo-
tion of the comet or renders it retrograde This angle therefore proceeding 1
from tbe earth's motion, is properly esteemed the comet's parallax ; the small^
increment or decrement that may arise from the unequal mot. of the comet
in its orbit being- neglected; From this parallax the comet's distance is found
thus. Let S represent the sun, acT the earth's orbit, a the place in the 1st
obs. c its place in the 3d obs. Q its place in the last, and Q'V a rt. line
drawn to the beginning- of aries. Join ac, and produce it to g t so that ag
nc :: AG : AC, and g, will be the place at which the earth would have ar-
rived at the time of the last obs. if it continued to move uniformly in ac. If
therefore $<* be drawn parallel to Q^, and the _ at g made' = TQG,
(gF being drawn parallel to QG, meeting- at F, or at any other point) the
j__ <y>{rF will then be equal to the long, of' the comet seen from g, and QF^
will be the parallax which arises from the earth being transferred from the
place g into the place Q ; and therefore F will be the place of the comet in
the plane of the ecliptic. This place F, J\*e?vton found to be commonly
Tower than the orb of Jupiter.
OF THE SOLAR SYSTEM. 417
The space described Cc, the distance SC, and the angle SCD,
are determined frcwn observation. Let the mean dist. of the comet
AH or its equal SK =* f x, and that of the planet Ag or SN = \q ;
and the squares of the periodic times being as the cubes of the
mean distances, we have \ y* : $x 3 :: n 2 : t 2 i hence/ 2
i jr 3 n* nx x
r -r~> an d t = */ -
i? 3 99
It is however necessary to find another expression for the peri-
odic time i, which may be thus found. Cc being a small portion
of the orbit, may be considered as a straight line, and the sector
CSc a rectilineal triangle, whose area = | SD x Cc 3= * be is
given j then as -| be : the area of the ellipse AKBI a A :: f t
X A. Now to determine the area A, the semiconjugate HK
%O
must be found ; in order to which AB = SC -j- FC ; hence FC
SB x -~ a ; and the triangle SDC, FEC, being similar, we have
SC : SD :: FC : FE ; that is, a bx ~ ab
FE ; therefore FG = FE GE = -. Again SC : CD
:: FC : CE ; or a : (** J 9 )* :: xa : *^ x (a* A)
hence DE or SG=CE + CD = ~~ X (a 2 brf +
-f a 2
T ( a * J 2 ) 2 . But FG = ; therefore FS =
+ 4a 6 2 \i , __
- - J ; hence SH = ^ FS
+ **
Moreover, as SK - AH
HK = SK-
= A> x (ajr a 2 )^, and X (d^r o')i = area of the rectJJi-
gle APTB. Let P = the periphery of the circle whose diarn. = x,
then its area will be \ LH X P = \ xP ; hence x* : 1 x :: '^a
: |*P :: AQRB : ALB :: APTB : AIB :: g* : A ^ ; that is
7 2 : ifi!A : ^ X (a* a 2 )^ : ^f x ( a:c _ 2)i AIB ;
but 2 AIB == AIKB = A ** ~ x (ax 2 )i Let this value
xiuy
of A be substituted in the preceding expression, and we get t *?=
tied X ^ "~ aa ^' which being equated with that already given,
3 B
418 OF THE SOLAR SYSTEM.
then x/ -sat' X (ax a 2 ) 2 , from which x is found =r
q q aeq
a fz *2 q
72~~i = AB > the g reafer v xis f thc comet's elliptic
/! p 2 q ae 2 n*
trajectory.
If tliis value of x be substituted in the above equation for f, we
shall get t = 2 ' 2 ^~2\s == the periodic time. Also,be-
n 1
cause the conjugate IK = X (ax a 2 )* = c, we have x =
- ^ ... ~ = rr-^r T~7 whence, by reducdon, we find
c = 2 den X I-TS r s ^ *, the lesser axis of (he comet's orbit.
V P 9 ae 2 n z /
From these equations it is evident, that when the velocity of the
comet is such that f 2 p z 9 = ae z n 2 , the axis x will be infinite,
and therefore the trajectory will be a parabola ; if ae 2 n 2 be greater
than f 2 p 2 y, the direction of the axis will be on the other side of
the curve, which will be an hyperbola; in either of which cases
the comet can never return : but when f 2 p 2 q is greater than
ae 2 n 2 , the comet will describe an ellipsis ; among the ellipses
a f 2 A 2 Q
we may comprise the circle where x = 2a = ^ ^-5
fz p? q ae z n ,
and/ 2 fi* g = 2ae 2 n*, whence e = Cc = ^ x ^ ) , the arc
c \^a'
cf the circle described in 1 day, I hour, Sec. according as the value
of n is given in days, or hours, &c. The solution of the above is
also given in Simpson's Fluxions, art. 240.
If the earth, for example, be the planet which is supposed to
describe the ellipse ANO ; and taking its mean dist. q 1 00000,
or q = 200000, then p = 6283 1 8 ; the periodic time n = 1 year ;
hence, if Cc be the portion of the comet's orbit described in 1 day,
we have /= = 0.0027378. The other expressions
3uo><*5 oo
591826599235 X a
will become as follow: x = S9AM659923s _ , nd t -
4750560000 X fl
(591826599235
To obtain the elliptic orbit of a comet from computation to any
degree of exactness, is extremely difficult ; for when the orbit is
very eccentric, a small error in the observation will change the
computed orbit into a parabola, or hyperbole. And, from the
thickness and inequality of the atmosphere with which the comet
is surrounded, it is impossible to determine with great precision
when either the limb or centre of the comet pass the wire at the
time of observation. This uncertainty in the observations wHl
subject the computed orbit to a great error- Hence it happened
W THE SOLAR SYSTEM.
that Baugucr determined the orbit of the comet in 1729, to be an
hyperbola. Euler determined the same for the comet of 1 744,
but from more accurate observations, he found it to be an ellipse.
The period of the comet in 1 680 appears, fit>m observation, to be
575 years, which Euler by his computation determined to be 1 66
years. The only safe way to get the periods of comets, as Vincc
remarks, is to compare the elements of all those which have been
computed, and where they are found to agree very well, it may be
concluded that they are elements of the same comet, it being so
extremely improbable that the orbits of two different comets should
have the same inclination, the same perihelion distance, and the
places of the perihelion and node the same. Thus, knowing the
periodic time, we get the greater axis of the ellipse ; and the per-
ihelion dist. being known, the lesser axis will be known. When
the elements of the orbits agree, the comets may be the same,
although the periodic times should vary a little ; as that may arise
from the attraction of the bodies in our system, .and which may
also alter all the other elements a little.
It has been already observed, that the comet which appeared in
1759, had its periodic time increased considerably by the attraction
of Jupiter and Saturn, This comet was seen in 1682, 1607, and
1531, all the elements agreeing except a little variation of the pe->
riodic time. Dr. Halley suspected the comet in 1680 to have
been the same which appeared in 1 106, 531, and 44 years before
Christ. He also conjectured that the comet observed by Apian
in 1532, was the same as that observed by Hevelius in 1661 ; if
BO, it ought to have returned in 1790, but it has never been observ-
ed. But M. Mechain having collected all the observations in 1 532,
and calculated the orbit again, found it to be sensibly different
from that determined by Dr. Halley. The comet in 1770, whose
periodic time M. Lexell has found to be 5 years 7 months, has
not been since observed, owing probably to the disturbing force
of Jupiter. From the elements calculated by Lexell) the comet
would be in conj. with Jupiter on Aug 23, i779, and its distance
from Jupiter would be only -^ of its dist. from the sun ; hence
the sun's attraction would be only ^ part of Jupiter's attraction.
What a change should this make in the orbit ! The comet would
not be visible if it returned to its perih. in March, 1776. See
JLexeWs account in Phil, trans. 1779. The elements of the or-
bits of the comets in 1264 and 1556, were so nearly the same,
that it is probably the same comet which appeared at each, time ;
if it be, it ought to appear again about the year 1848.
The number of comets that, from the most accurate accounts,
are stated to have appeared, since the commencement of our aera,
is about 500 ; and before that sera, about 100 others are recorded
to have been seen.
The elements of the comet for 1 770, with the trajectory of its
path, may be found in the transactions of ^he American Phil.
Society, vol. 1.
420 OF TH SOLAR SYSTEM.
In order to obtain the course of a comet, its distance from two
known fixed stars must be observed ; or its alt. taken when in the
same azimuth with any two stars ; from either of these observa-
tions its place may be calculated by spher. trigonometry,* or laid
down on a globe. If several places of the comet be thus found,
and marked on the globes, the great circle passing through them
will be the way of the comet. This great circle may be drawn by
the quad of alt. or the poles may be elevated or depressed, until
all the places marked are, at the same time, found in the horizon ;
for thjen the circle denoted by the hor. on the surface of the
globe will be that required. Hence its intersections with the
ecliptic will be the nodes of the orbit of the comet, and the angle
which the ecliptic makes with the horizon, measured by the alt.
of the nonagesimat degree, will be the incl. of its orbit to the eclip-
tic The hng. latitude, &c. of a comet, may therefore be easily
found on the globes.
Let a circle be described the diam. of which is equal to that ol
the globe (or reduced proportionally to a smaller scale if necessary)
as ABO, whose centre is T, and A a
point in its circumference, represent-
ing its place among the fixed stars, in
which the comet was first observed.
Let the arcs AB, AD he taken equal
to the dist of the place of the comet,
marked on the surface of the globe,
from the place first observed, and let
TB, TD be drawn. Draw the right
line through the point A, so that AE :
EC :: R : S (see the note pa. 409)
that is, by construction, as the time be-
tween the 1st obs. and 2d to the time
between the 2d and 3d. From T let fall TP perp. to AO, and
* If its dist from two known fixed stars be taken, its place may be found
thus : Let S be the comet (see fig. pa. 208) s one of the stars, Z the other,
and P the pole of the equator ; then ZP, sP, are the co. decl. of the stars,
and the angle sPZ the difF. of their rt. ascensions, which are given ; there-
fore the dist. between the stars sZ, and the angle ZsP are given. Now in
the triangle SZs, SZ, Ss, the correct dist. of the comet from each of the
given staVs, and also sZ are given ; hence the angle SsZ, and therefore SsP
are given. Now in the triangle SPs, the dist. S,v and sP, and the /_ SsP
are given, hence SP the co. decl, of the comet, and the angle SP,?, which is
the diff. between the rt. ascen. of the star and comet ; therefore the comet's
rt. ascen. and decl. are given, from which its lat. and long, is found, as
shewn in the note to prob. 3, pa. 195.
When exactness is required, the apparent distances must be first correct-
ed ; thus, the places of the stars and the hour being given, their alt. may be
found ; and as the appar. place of the comet is given, and the hour, its appar
alt. may be also found, for which the refraction for that alt. will be nearly
the refraction for its true alt. and hence from the appar. alt. the true alt.
may be nearly found. If the refraction for this last alt. be again found (see the
table, pa 155) and taken from the appar. alt. the true alt. of the comot w2U
OF THE SOLAR SYSTEM. 421
producing it if necessary as far as the circumference at G ; an arc
equal to AG be ing transferred from the place first observed, to the
way of the comet, described above on the surface of the globe, the
point G will shew the place among the fixed stars, in which the
comet will be in its pengceum.
If the place of the comet can be observed when it has no lati-
tude, the place and time of being in one of its nodes will then be
exactly known ; but as this can seldom be actually observed, these
elements are generally observed by approximation from other
methods. The appar. diam. of the comet must also be often ob-
served ; as by this means a judgment may be formed of its re/a*
live distance at different times. Its degree of motion, its bright-
ness, Sec. must also be regarded ; for when it moves with the
greatest velocity, or appears most bright, it may be inferred that
it is near its fierihetion.
If four stars round the comet be observed, such that the comet
may be in the intersection of the rt. lines which join the two op-
posite, which are easily found, by extending the thread, placed
before the eye, over the stars and comet ; let the thread be ex-
tended in like manner, over those stars found on a globe, and the
point of intersection will shew the place of the comet.*
Although the orbit of a comet may be computed from three ob-
servations, yet from these data the direct solution of the prob. is
impossible We have therefore given several indirect methods to
find the orbit very near the truth, by mechanical and graphical
operations (as did Aewton himself for the comet of 1680, see his
Jirin.) then by computation it may be corrected by what is given,
until a parabola be found to satisfy the observations very nearly.
The result of these methods as given by Clairautt, Fenn and
others, and as pointed out in the preceding part of this chap, is as
follows :
Let the rt. ascension and decl. of the comet be found, and from
thence its long, reduced to the eclip. and its lat corresponding to
each obs. as shewn in the preceding note. Let the sun's long, be
then be obtained very nearly ; the appar. place of the comet found on a good
globe, will be sufficiently exact for trial. Now from the apparent and truq
altitudes, and the appar. distances, the true distances may be found, as
shewn in the note pa. 224.
If the quantity of the comet's parallax be known, which may be.;estimated
from its clist. from the earth, or as shewn in the note pa. 120, it may be al-
lowed for, and also an allowance may be made for the aberration of light.
Whoever wants more information on this subject, besides the works al-
siderations on the situations of the orbits of the planets and comets which
have hitherto been calculated, inserted in the memoirs of the Academy of
Sciences of Berlin ; O. Gregory's treatise on astronomy, 1803 ; De la Lande
Theorie des Cometes, 1759 ; and Astronomic, vol. 3. An account of the
discoveries concerning comets with the way to find their orbits, See., by T/;o-
mav Vttrker. 1727* &c.
422 OF THE SOLAR SYSTEM,
computed at each observation (selecting three best calculated for
that purpose) and the diff. (A. a, a) between the comtt's long,
and that of the sun, corresponding to each obs will be the elonga-
tion of the comet or its dist. from the sun reduced to the ecliptic.
Let the dist (B, b, b} of the earth from the sun at the time of
each obs be computed (as shewn in the theory of the earth, ch. 4,
or n;ore readily by the Nautical Aim.* or Dclambre's tables.)
Then let y and z be the assumed distances of the comet from
the sun reduced to the ecliptic (found as nearly as possible by
some of the foregoing- methods) at the 1st and 2d obs. then the
true distance may be determined as follows :
As the assumed dist. y or z : sine A or a the elong. :: dist. B or
b of the earth at the \st or 2rf. obs. : sine L C or c contained by
the rt. line drawn from the earth and sun to the comet* This
angle C or c being added to the elongation A or a, their sum
will be the supplement of the angle of commutation D or A.\ Thea
(by the proper, pa. 351) sine A or a : sine D or d :: tang, obser-
ved geocentric lat. of the comet corresponding to the 1st or %d obs.
: tang, corresponding heliocentric lat. of the comet E or e.
Each ot the curtate distances y and z divided by the cos of the
conesponding helioc. lat. E and e, will give the true distances of
the comet V. v, from the sun. Now to find the angle contained
by those distances^ add to or subtract from the places of the earth
(according to the comet's pos with respect to the signs) the corres-
ponding ^ 's of com. D, d, the sum. or diff will be the heliocen-
tric long tudes L, 1, of the comet whose diff'. F, is the helioc. motion
of the comet in the plane of the ecliptic. Then as rad. : cos. F ::
cot. greatest of the two hel. lat. : tang. X. Let this arc X be ta-
ken from the compl. of the least hel lat. and the rem. call x.
Then cos. X : cos. x :: sine gr of the two lat. : cos. /_ contained
by the two -vector rays or distances V, v, of the comet.
Now by what is shewn pa. 405, the place of the perihelion may-
be found by this rule : take the log. of the least vector ray from
that of the greatest, add 10 to the characteristic of half the remain-
der, it will be the tang, of an angle, from which 45 being subtr.
the log tang, of the remainder added to log cot. 1 of the mot. of
the comet in its orbit, will be the log tang, of an /_ to which i of
* In the Naut. Aim. pa. 3, the log-, of the dist. of the earth from the sun is
given every 6th day in the month, the earth's mean dist. being 1 1.
f In the triangle ESC, pa. 399, let SC represent y and ES, B, then the
/_ SEC will be the elong-ation of the comet from the sun, and SCE that
found by lines from the earth to the sun and comet, whence from plane trig-.,
the proper, is evident.
* In this fig\ pa. 351, if B represent the plane of the comet reduced to the
ecliptic, then (32 Eucl. i) TBS + STB the elong. = 180 TSB; hence
TSB the commut. = 180 TBS STB.
In the fig*, pa. 351, rad. : cos. PST :: PS : SB the curt. dist. B being-
supposed as before the comet's place, &c. ; hence rad. being- taken = 1, PS
QTJ
tlie comet's true dist. (P being- its true place) = < prv
OF THE SOLAR SYSTEM. 423
the mot. of the comet in its orbit being added, the sum will be \
the greatest true anomaly, and their difference will be % the least
of the two true anomalies. These quantities doubled will be the
two true anomalies, which will be both on the same side of the
perihelion^ when their diff. is the whole motion of the comet ; but
on different sides, when it is their sum, which is equal to the
whole motion of the comet
Let the fierihelion dist, be found by adding twice the log. cos.
f the greatest of the halves of the two true anomalies, to that of
the greatest of the two distances, the sum will be the log. of the
perihelion distance required. See pa. 405.
The time in which the comet describes the two vector rays may
be thus determined. To the constant log. 1.9149328,* add the
log. tang, of % each true anom. add the triple of this same log.
tang, to the constant log. 1.4378 1 1 6, the sum of the two numbers
corresponding to those two sums of logs, will be the exact num-
ber of days corresponding to each true anomaly, in a parabola
whose perihelion dist is 1. (by what is shewn pa. 403.) Find the
log. of the diff. or sum of those two numbers, according as the two
anom. are situated on the same, or on different sides of the perihe-
lion ; to this log. add | of the log. of the perihelion dist. the sum
will be the log. of the time in which the comet describes the
angle contained between the two vector rays j as shewn pa. 406,
which see.
The above is called the \st hypothesis ; the following is the 2c?
supposition of this hypothesis. If the time thus found does not
agree with the observed time, another value of the curt. dist. z is
to be assumed, corresponding to the 2d obs. the val of y corres-
ponding to the 1st being still retained, and the helioc. long, and lat.
from thence deduced, and all the operations indicated in the fore-
going articles being repeated ; another expression will be found
for the interval of time between the two observations. If this time
approaches nearer the observed time, the 2d val. assumed for z is
to be preferred to the 1st ; if not, a 3d val. for z is to be assumed,
and by the increase or decrease of the errors, the value to be as-
sumed, so that the interval of time calculated may agree with the
observed one, will be easily discovered ; and therefore a parabola
will be found which answers the two first observations, or the first
hypothesis.
The parabola answering the two first obs. would be the true orbit
if it also answered the 3d obs. but as this seldom or never happens,
a second hypothesis becomes necessary, in which another parabola
is to be found which answers the two first observations, by increas-
ing or diminishing the curt. dist. y, preserved constant in the 1st
hypot. and preserving it still constant, but varying the 2d assumed
* IJy what is shewn pa. 401, b = at* -f- at, and a = 109.6155, of
'149328 ; also of 0* =
Hence this rule is take*
sdiich or | a ra 82.2116, the log. of which is 1.9149328 ; also of a
27.4038, the log- of which is 1.4378116 as above.
from the vglue of *.
424 OF THE SOLAR SYSTEM.
dist. r until this 2d parab. is obtained. The 3d obs. calculated in
those two parab. will shew which of them approaches nearest the
true orbit sought. To calculate this 3d obs in each hypot. the
time of the comet's passing the perih. the ind. of its orbit, and the
plaee of the nodes of each parabola, is first to be determined
To determine the comet's passage at the perihelion* Find the
number of days corresponding to one of the true anoni. for ex to
that which corresponds to the 1st obs in the parab whose perih.
dist, is 1. ^by Delambre's table, or as shewn pa. 403, &c. where the
method of constructing the table is given) the log. of this number
of days being added to of the log. of the perih. dist. (found above)
will be the log. of the interval of time elapsed between the 1st obs.
and the comet's passing the perih. which is to be added to or sub-
tracted from the time of the obs. according as it was made before
or after the passage of the comet at the perihelion.
To determine the place of the node we have this proportion ; sine
of the 2d arc x : sine 1st arc X :: tang, comet's mot. in the eclip.
: tang, of an angle r ; then rad : sine least lat. :: tang, r :: tang.
dist. from the node (See also pa. 407.) From this dist. and the
helioc long of the comet, found as shewn above, the heliocentric
long, of the node is obtained. With this hel. long, and the dist mea-
sured on the orbit of the comet, the/J/ac<? of the perihelion is deter-
mined. To Jind the dist. measured on the comet's orbit, we have
this proportion ; sine r : rad. :: dist* measured on the ecliptic :
dist. required. To determine the inclination it will be rad. : sine r
:: cos. least lat. : cos. . of incl.
The elements of each parabola being determined, the geocentric
place of the comet answering to the 3d obs. is computed in each
by the following rules : 1st. Let the log. of the diff'. between the
time of the 3d. obs. and the time of the comet's passing the perih.
be taken, from which take f of the log. of the perih. dist. the rem.
will be the log. of the diff. between the time of the 3d obs. and
that of the comet's passage at the perih. of a parab. whose perih.
dist. is 1 . 2d. Let the true anomaly corresponding to this time
b b
be found, by solving the equation t z -f 3t = -^ = 4033'* as
shewn pa. 402,* in which t = tang. the true anom. and b the
time of describing it 3d. When the mot of the comet is direct^
add the true anom. to the place of the perih. but subtract it if the
* The mean proportional required here between h -f- - - and h
to find t (see pa. 402) will be obtained by finding the cube root of
the ratio between the two quantities ; the root thus found will be the ratio
between the four quantities ; hence this ratio multiplied by the least ex-
treme, will give the next term, or one of the mean proportionals, and this
multiplied again by the same ratio, will give the other mean proportional.
The equation may be also solved by any of the known methods for solving
cubic equations.
OF THE SOLAR SYSTEM.
425-
obs. was made before the comet's passing the perih. When the
comet's mot. is retrograde, add the true anom. to the place of the
perih. -if the obs. was made before the passage, at the perih. but
subtract it if the obs. was made after the time of perih. the true
helioc. long-, of the comet in its orbit will be thus obtained. Sec
also pa 407. '4th. The -cliff, between this long, and the long, of
the ascending node will be the argument of the lat. of the comet.
5th. As rad. : cos. inch :: tang, argum. of lat. : tang, of this arg,
measured on the ecliptic, which added to the true place of the
node, gives the hetioc. long, reduced to the ecliptic. 6th. Rad. ;
sine arg. of lat. :: sine incl. of comet's orbit : sine of its hel. lat.
which when the motion of the comet is direct, is north or south
according as the argument of lat. is less or greater than 6 signs ;
but when retrograde, is north or south, according as the arg. of
lat. is greater or less than 6 signs. 7th. Log. cos. hel. Jat. -f log.
perih. dist. log. of twice cos. | the true anom. = log. of the
curt. dist. answering to the 3d. obs. 8th. Log. curt. dist. log.
dist. of the earth from the sun -f- 10 added to the characteristic, or
index = log. tang, of an angle, from which subtract 45, and to
log. tang remd. add log. tang, com pi. of ~ the angle of commuta-
tion, the sum = log. tang, of an arc, which added to this compL
if the curt. dist. of the comet from the sun exceed the earth's dist.
but subtracted if the comet's dist. be less than the earth's, the sum
er diff. will be the angle of elongation ; this angle added to, or
subtracted from the sun's true place or long, according as the
eomet seen from the earth is east or west of the sun, will give the
geocentric long, of the comet. 9th. Sine L commut. : sine elon-
fation :: tang, heiioc. lat. of the comet : tang, of its geocentric lat.
ee these different cases exemplified in pa. 407 and 408, and in what
follows. The long, and lat. thus found, ought to agree with those
observed, if the parabola obtained were really the comet's orbit.
As these rules without sufficient examples may, especially to
beginners, be rather difficult in their application,' the following
example of the comet, which was observed in Europe, about the
beginning of March) 1742, is given.
1742.
Afean Time.
Obs. long,
of the
comet.
Obs. hit.
north of
the cornet.
Long, of the
sun calcu-
lated.
Log. of the
earth's
dist. from
the sun.
Elong, of
the comet
from the
sun.
4thMar.l6h. 9' 50"
28 . . at 13 39
24 Apr. 9 39
9s. 16 G'4/'
2 18 52 45
3 1 5 53
3445'3/"
63 3 55
50 32 50
Ils.l427'44"
8 11 28
I 4 27 16
9.996910 5827'4/V.
9.999840!
0.003092 56 38 17*
1st. Supposition, y = 0879 and z 0.957, of the earth's
mean dist. from the sun, which is taken equal .1 ; then C = 105o
42
61 31', C 4- A
164 9' 52", and c -f a = 118
9' 17" ; hence D = 15 50' 8", and d = 61 50' 43" ; there-
fore E = 12 3i' 42" north, and e = 52 3' 38", and log-. V **
9.954455, and log. v = 0.192159.
31
426 OF THE SOLAR SYSTEM-
The places of the earth at the 1st and 2d. obs. being 5s. 14 27^
44", and 7s. 4 27' 16" respectively, hence D -f 5s. 14 27' 44"
(the comet being east of the earth's place) = L 6s. 17' 52",
and 7s. 4 27' 16" d (the comet being west of the earth) =
1 = 5s. 2 36' 33" ; then L 1 = F = 27 41' 19" the com-
et's mot. in the eclip. Also X = 34 37' 1 1", and x = 42 51'
7", the angle contained by the two vector rays = 45 22' 8", the
comet's motion in its orbit.
Log. v 0.1 92 159 log. V 9.954*455 = 0.237704, half of which,
together with 10 added to its characteristic = 10.1 18852 = tang.
52 44' 38", from which 45 being taken, leaves 7 44' 38" ;
whence log. tang, of 7 44' 38" -f log. cot. of 1 1 20' 32''
(I of 4-50 22' 8") = log. tang, of 34 8' 5" ; therefore 34 8'
54" 1 i 20' 32" = 22" 47' 33", half the least true anomaly ;
and 34 8' 5" + 1 1" 20' 32" = 45 28' 37" half the greatest
true anom. Hence the least true anom. = 45 35' 7", and the
greatest 90 57' 15", and their diff being equal the comet's mot.
they are both therefore on the same side of the perihelion. Hence
log. perih. clist. is found = 9.883835.
To find the time in which the comet described the angle con-
tained by the two vector rays, we have log. 1.914-9328 + log-
0.007233 (log. tang, of 45 28' 37-J") = 1.922166, and log,
1.438112 -j- 0.021699 (triple log. same tang.) =* 1.459512, the
numbers corresponding to which are 83.592 and 28.808 respec-
tively ; hence 112.400 days is the time corresponding to the true
anomaly 90 57' 15" in a parabola, whose perih. dist. is 1. In
like manner the number of days, in the same parab. corresponding
to the true anom. 45 35' 7", is 36.579. Now taking the diff.
of those times = 75.821 days (the two anom. being on the same
side of the perih.) the log- of which is 1.879789 added to 9.825752
(i log. perih. dist.) gives log. 1.705541 corresponding to 50.76^
days, the time required.
Comparing this time with the interval 50 728 * between the two
observations, it is found to exceed it by 0.033, if therefore a vari-
ation of 0.001 be made in the dist. z, in order to discover which
way, and by how much the elements of the corresponding parabola
will be changed.
2d. Sup. y =t 0.879 and z = 956, and repeating the same cal-
culations as in the foregoing sup. we find E = 12 31' 42", e =
52 1' 54|", log. of the dist. V = 9.954455, and v = 0.191424,
the helioc. long. L = 6s. 17' 52", 1 = 5s. 2 43" 11" ; the
mot. of the comet in the eclip. = 27 34' 41", and in its orbit
45 18' 13", the true anomalies 45 32' 5" and 90 15' 16" ?
the corresponding days 36.529 and 112.056, log. perih. dist. =
9.883997, and the reduced time of describing the angle contained
by the two vector rays 50,594 days. Hence increasing z by 0.001,
the time is diminished by 0.168; therefore 0.168 : 0.001 :: 0.033-|
: 0.0002 ; hence z is diminished by 0.0002 to obtain a parabola
answering the required conditions.
OF THE SOLAR SYSTEM. 427
3d. Sufifios. y = 0.879, r = 0.9568, from which the helioc.
lat. E = I2o 31' 42" and e = 52 3' 16J"; log. V = 9.954455
and log. v = 0.192009 ; L = 6s. 17' 52" and 1 = 5s. 2 37'
53" ; comet's mot. in eclip = 27 39' 59", motion in its orbit
45 21' 22"; the true anomalies 45 34' 26" and 90 55 ; 50";
corresponding times 36.5f<8| and 1 12.330 days ; log. perih dist.
= 9.883870 ; reduced time 50.728 days, agreeing with obser-
vation.
Having therefore found a parab. corresponding to the two first
obs. another must be found answering the same obs. by making a
variation in the dist. y preserved constant in the 1st hyp.
Second Hyp. 4th. Sufifws. y = 0.878, and z = 0.957, from
which E = 12 42' 11", e =* 52 3' 38", log. V = 9.954257,
log. v = 0.192159, L = 6s. 31' 54", 1 = 5s. 2 36' 33",
comet's mot. in the eclip. = 27 55' 21", angle contained by the
two vector rays = 45 17' 56", true anomalies 45 44' 56" and
9lo 2' 52", corresponding times 36.743 and 112680 days, log.
perih. dist. = 9 8831 15, reduced time = 50.714, differing 0.014
from the observed interval ; hence diminishing y by 0.001, the
time is diminished 0.048 ; therefore 0.048 : 0.001 :: 0.0 1 4 J- :
0.0003.
5. Sup. y = 0.8783, z == 0.957, hence E 12 39' 2", e =*
52 3' 38", log. V = 9.954316, log. v =0.192159, L = 6s.
27' 40", 1 == 5s. 2 36' 33", comet's mot. = 27 51' 7", an-
gle contained by the vector rays = 45 19' 20", true anomalies
45 41' 45" and 91 I' 5", corresponding times 36.689 and
112.590, log. perih. dist. = 9.883344, reduced time 50.729,
agreeing with observation.
Two parabolas being therefore obtained answering the 1st and
2d observations, the 3d obs. must be calculated in each to find
which of them approaches nearest the real orbit of the comet
hence the place of the perih. the time of the comet's passage at it,
the incl. to the eclip. and the place of the nodes of each parabola,
must be first found.
In the first parab, R = 23 40' Is", comet's dist. from the
ascending node reduced to the eclip. at the 1st obs. 5 2s' 45",
which added to its helioc. long. 4th March, 6s. 17' 52" (its
mot. being retrograde as 6s. 17' 52" is greater than 5s. 2
36' 33" at the 2d obs. and the comet after passing its perih.)
gives 6s. 5 43' 37" for the place of the node. Dist. of the com-
et from its node measured on its orbit = 13 38' 14", which ta^
ken from the place of the node, gives the place of the comet in
its orbit at the 1st obs. and its true anomaly being then 45 34' 28",
therefore these being added to the comet's place in its orbit, give
7s. 7 39' 51" for the place of the perihelion ; | of the log. of
which added to log. of 36. 5 68 days, the time corresponding to
the least true anomaly 45 34' 28", gives 24.486 days for the
interval of the elapsed time between the 1st obs. and that of the
perih. which being taken from March 4d. J6h. 9' 50" or 4th
428 OF THE SOLAR SYSTEM.
March 0.673, the time of the 1st obs. fixes the passage of the
perihelion on the 8th of Feb. at 0.187-. The incl. of the comet's
orbit to the ecliptic is found = 66 56' 14".
In the 2d parab. the same elements are, the ascending node in
6s. 5<> 59' 6", place of the perih. 7s. 7 53' 42", incl. 66 47'
14", time of passing the perih. 8th Feb. O.i5l|.
The geocentric long, for March '^8, a. 0.569 - of the day in each
parabola is thus calculated ; the interval between this time and the
comet's passing the peril), is 48.381 days ; log. perih. distance
9.883870, its triple is 9.651610, its \ = 9.825805, which taken
from log. of 48.381 or 1 684675, leaves log. 1.858870 corres-
ponding to 72.255 days, answering to 73 11' 7" or 2s. 13 ll f
7" anomaly, this subtracted from the place of the perih. 7s. 7 39/
51" (the comet being retrograde) gives 4s. 24 28' 44" for the
true helioc. place of the comet in its orbit, from which 6s, 5 43'
37" the place of the ascending node being subtracted, leaves 10s.
18 45' 7" the argum. oflat. which on the ecliptic is 11s. 11
2 ; 47" ; hence the comet's helioc. long, is 5s. 16 46' 24", and
hel. lat. 37 20' 41" north, the arg. oflat. of the comet, which
is retrograde, being greater than 6s.
The sun's true place March 28th at 13h. 39' is 8 11' 28",
and log. dist. from the earth is 9.999841, hence the true place of
the earth seen from the sun is 6s 8 11' 28", from which 5s.
16 46' 24" being subtr. gives 21 25' 4" the angle of commut.
Log. of the curt. dist. for 3d obs. is 9.974915, which being taken
from 9.99811, leaves 0.024926, to the caracter. of which 10 be-
ing added, gives the sum is 10.024926 log. tang, of 46 38' 42" f,
from which taking 45 the log. tang. remd. 1 38' 42" -f log.
tang. 79 17' 28" (com pi. of -*- Z. of commut) = log. tang.
S* 37' 39", which taken from 79 17' 28" (because the comet's
dist. from the sun is less than the earth's) gives 70 39' 4-9" or
2s. 10 39' 49" the elongation. If the places of the sun, the
earth, and the comet found as above, be marked on the ecliptic of
an artificial globe (or any circle divided into 12 signs) it will be
seen that the comet to an observer on the earth appears east of the
sun. Hence the elongation is to be added to the sun's place to
find the true geocentric long, of the comet, which is 2s. 18 51'
17", which is less than the observed long, by l' 28". In like
manner the comet's geocentric long, in the 2d parabola March 28,
is 2s. 18 4.5 f 14-", which is less than the observed long, by 7' 31" ;
hence neither of the two parabolas is the comet's orbit.
3d. Hypothesis. As the variations of the orbits are, however,
sensibly proportional to those made in the curt, distances ; hence
to obtain the two curt, distances answering to the re<jd. orbit, we
have from the rule of false position those two proportions ; as
6' 3" (diff.* of the two errors 1' 28" and 7' 31") : least of the
* If one of the errors was by excess and the other by defect, the awn of
the errors would be here used. Sec this rule investigated, and different
methods given in prob. 80. b. 1, and Corollaries Emerson's Algebra,
OF THE SOLAR SYSTEM. 429
twoj' 28" :: 0.0007 and 0.0002 (corrections of y and z to ob-
tain "parabolas in the 1st and 2d obs.) : G.000235 and 0.000065,
corrections for those distances to obtain the orbit required.
Now as y, supposed = 0.879 gives an error of I' 28", and
y, supposed = 0.8783, gives an error of - 7' 31", by diminish-
ing ij the error is increased ; hence the true value of y = 0,879 -f-
0.000235 = 0.879235. Reasoning in like manner we find z =
0.956735.
6, Suppos. y 0.879235, and z = 0.956735, whence E =
12 29' 171" ; e = 52 3' 10-|", log. of the vector rays V =
9.954504 and v = 0.191963, L = 6s. 0<> 14' 37" and 1 = 5s. 2
38' 19", true anomalies => 45 32' and 90 54 f 4", the corres-
ponding times 36.528 and 1 12.243 days, log. perih. dist. 9.884049
time of describing the angle contained by the two vector rays
50.729, place of the node 6s. 5 38' 29'', place of the perih. 7s.
7 35' 13", inclination of orbit 66 59' 14", and time of passing
the perih. 8th Feb. at 4h. 48'. From those elements the geo-
centric long, on the 28th March, at 13h. 39', is 2s. 18 53' 18",
and geoc. lat. 63 3' 57" north, agreeing with observation.
The following table of Mr. Lee's taken from Dr. Rees* New
Cyclopedia, calculated on an extensive scale, and with immense
labour, will be found extremely useful in comparing the com-
puted orbits of new comets with those before observed, Sec. The
Elements of the foregoing comet determined as above, exactly
agrees with that given in the following table by la Caille> except
the time of the passage of the perihelion, which in the table is
given Jan. 28th. 4h. 38' 40". The 11 days difference arises from
the old stile being used in the table,
430 OF THE SOLAR SYSTEM.
THE ELEMENTS OF NINETY-SEVEN COMETS.
^
Passage at per. mean tim
at Greenwich.
Long, of th
per. on the orb
of the comet.
Per. dis
earth's be
ing 1.
Long, of th
ascending
node.
/we/, ofth
orbit.
1
Anno V. C.
539 Oct. 20 15h, O f
I0s.l330'
0.3412
Is.28or7s.28
10-f- r-
D
Anno Domini OldStile.
2
837 March
9 19 3
0.5800
6 S .2633' C
10 or 12
11
3
1097 Sept. 21 21 36
11 2 30
0.7385
6 27 30
7330'
D
i
1231 Jan. 30 7 12 40
4 14 48
0.9478
13 30
650
D
i
1264 July 6800
9 21
0.445
5 19
36 30
D
July 17 6 40
9 5 45
0.41081
5 28 45
30 25
1299 Mar. 31 7 28 40
3 20
0.3179
3 17 8
68 57
R
]
1301 Oct. + or
9s. or 10s.
0.457
15 + or
70 -f- or
R
8
1337 June 2 6 25
Is. 759 / 0'
0.40666
2s.242V
32 11
R
June 1 30 40
20
0.6445
g 6 22
32 11
9
1351 Nov. 26 12
2900
1.0000
I)
10
1456 June 8 22
10 1
0.5855
1 18 30
1756
R
11
1472 Feb. 28 22 23
1 15 33 30
0.54273
9 11 46 20
5 20
R
1531 Aug. 24 21 18 30
10 1 39
0.56700
1 19 25
17 56
D
12
1532 Oct. 19 22 12
3 21 7
0.50910
2 20 2f
32 36
D
13
1533 June 16 19 30
4 27 16
20280
4 5 44
35 49
R
14
1556 April 21 20 3
9 8 50
0.46390
5 25 42
>2 6 30
])
15
1577 Oct. 26 18 45
4 9 22
0.18342
25 52
74 32 45
R
16
1580 Nov. 28 13 44 40
3 19 11 5
0.59553
19 7 37
64 51 50
1)
15
3 19 5 50
0.59628
18 57 20
64 40
17
1582 May 7
8s.5or9sll
23 or .04
7s. 5 or21
59 or 61
R
18
1585 Sept. 27 1920
Os. 851' 0"
1.09358
Is. 742' 30"
6 4' C'
1)
19
1590 Jan. 29 3 45
7 6 54 30
0.57661
5 15 30 40
29 40 40
R
20
1593 July 8 13 38 40
5 26 19
0.08911
5 14 15
87 58
D
21
1596 July 31 19 55
7 18 16
0.51293
10 12 12 30
55 12
R
29 15 33 40
7 28 30 50
0.549415
10 15 36 50
52 9 45
R
"10
607 Oct. 16 3 50
10 2 16
0.58680
1 20 21
17 2
R
22
618 Aug. 7 3 2 40
10 18 20
0.51298
9 23 25
21 28
D
23
618 Oct. 29 12 23
2 14
0.37975
2 16 1
37 34
D
2-1
652 Nov. 2 15 40
28 18 40
0.84750
2 28 10
79 28
D
25
661 Jan. 16 23 41
3 25 58 50
0.44851
2 22 30 30
32 35 50
1)
26
664 Nov. 24 11 52
4 10 41 25
.025755
2 21 13 55
21 18 40
R
27
665 Apr. 14 5 15 30
2 11 54 30
0.10649
7 18 2
76 5
i
28
672 Feb 20 8 37
1 16 59 30
.69739
9 27 30 30
83 22 10
)
29
677 Apr. 26 37 30
4 17 37 5
.28059
7 26 49 10
79 3 15
t
SC-
678 Aug. 16 14 3
27 46
.23801
5 11 40
3 4 20
J
SI
1680 Dec. 8012
8 22 40 10
.006030
9 1 57 13
1 22 55
)
7040
8 22 33
9 1 53
1 20 20
7 23 9
8 22 44 25
.006170
9220
1 6 48
7 20 38 39
8 23 26 48
.006565
9 2 59 9
8 39 50
8040
8 27 43
.005920
9 1 53
1 20 20
'10
682 Sept 4 7 39
2 52 50
.58328
1 21 16 30
7 56
I
21 31
1 36
.58250
1 20 48
7 42
32
683 July 2 3 50
2 25 29 30
.56020
5 23 23
3 11
I
33
684 May 29 10 16
7 28 52
.96015
8 28 15
5 48 40
5
34
686 Sept. 6 14 33
2 17 30
.32500
1 ^0 34 40
1 21 40
35
689 Nov. 21 14 55 40
8 23 44 45
.016889
23 45 20
9 17
^
36
698 Oct. 8 16 57
9 51 15
.69129
8 27 44 15
1 46
OF THE SOLAR SYSTEM.
431
o
<
Passage at per. mean time
at Greenwich,
Long, of the
per. on the orb.
of the cornet.
Per. (list,
earth's be-
ing 1.
Long, of the
ascending
node.
Incl. of the
orbit.
37
1699 Jan. 3 8h.22' 19''
7s. 23V 6"
0.74400
10s.2145' 35"
6920* G"
38
1702 Mar. 2 14 12 19
4 18 41 3
0.64590
6 9 25 15
4 30
39
1706 Jan. 19 4 22 39
2 12 29 10
0.42580
13 11 40
55 14 10
4 56 4
2 12 36 25
0.426865
13 11 23
55 14 5
40
1707 Nov. 30 23 29 39
2 19 54 56
0.8597
1 32 46 35
88 36
23 43 6
2 19 58 9
0.85904
1 22 50 29
88 37 40
41
1718 Jan. 3 23 38 39
4 1 30
1.02650
4 8 43
30 20
4 1 14 55
4 1 26 36
1.02565
4 7 55 20
31 12 53
42
1723 Sept. 16 16 10
1 12 15 20
0.998651
14 16
49 59
43
1729 June 14 11 640
10 22 40
4.26140
10 10 32 37
76 58 4
12 6 32 2
10 22 16 53
4.0698
10 10 35 15
77^ 1 58
44
1737 Jan. 19 8 20
10 25 55
0.22282
7 16 22
18 20 45
45
1739 June 6 9 59 40
3 12 38 40
0.67358
27 15 14
55 43 44
46
1742 Jan. 28 4 38 40
7 7 35 13
0.76568
6 5 38 29
66 59 14
4 20 50
7 7 33 44
0.765555
6 5 34 45
67 4 11
27 4 14 39
7 10 49 23
0.7521
6 9 32 7
61 43 44
47
1742 Dec. 30 20 25 40
3 2 41 45
0.83501
2 8 21 15
2 19 33
21 15 16
3 2 58 4
0.838115
2 8 10 48
2 15 50
48
1743 Sept. 9 21 16 18
8 6 33 52
0.52157
5 16 25
45 48 21
49
1744 Feb. 19 8 27
6 17 12 55
0.22206
1 15 45 20
47 8 36
50
1747 Feb. 20 7 10 40
9720
2.19851
4 27 18 50
79 6 20
17 11 44 38
9 10 5 41
2.29388
4 26 58 27
77 56 55
51
1748 Apr. 17 19 25
7 5 50
0.84067
7 22 52 16
85 26 57
52
1748 June 7 1 24 15
9 6 9 24
0.65525
1 4 39 43
56 59 3
JVew Stile.
53
1757 Oct. 21 9 46 40
4 2 49
0.33800
440
12 48
54
1758 June 11 3 17 40
8 27 38
0.21535
7 20 50
68 19
10
1759 Mar. 12 13 31 40
10 3 16
0.58349
1 23 49
17 39
13 50 40
10 3 8 10
58490
1 23 45 35
17 40 14
13 48 16
10 3 16 20
0.58360
i 23 49 21
17 35 20
.55
1759 Nov. 27 2 37
1 23 34 19
0.80139
4 19 39 41
79 6 38
56
1759 Dec. 16 21 3 40
4 18 24 35
0.96599
2 19 50 45
4 51 32
77
1762 May 28 15 17 40
3 15 15
1.0124
11 19 20
85 45
6 51 29
3 14 29 46
0.009856
11 19 2 22
85 3 2
29 18 28
3 15 22 23
1.01415
11 18 55 31
85 22 21
>8
1763 Nov. 1 19 43 18
2 24 51 54
0.49876
11 26 23 26
72 40 40
20 58 14
2 25 1 6
0.49820
11 26 27 0-
72 28
:79
1764 Feb. 12 13 42 16
15 14 52
0.55522
4 4 33
52 53 31
60
1766 Feb. 17 8 40 40
4 23 15 25
0.50533
8 4 10 50
40 50 20
61
1766 Apr. 22 20 46 20
8 2 17 53
0.33274
2 14 22 50
ll" 8 4
*2
1769 Oct. 7 12 20 40
4 24 5 24
0.12376
5 25 43
40 37 33
13 36 53
4 24 11 7
0.12272
5 25 6 33
40 48 49
14 56 39
4 24 16
0.12265
5 25 3
40 50
15 28 16
4 24 10 51
0.1227
5 25 4 41
40 49 33
12 34 9
4 24 U 8
0.1232852
5 25 2 24
40 48 29
15 42 2
4 24 15 53
0.12275
5 25 6 4
40 46 42
12 44 38
4 24 11 32
0.12327
5 25 3 40
40 47 56
63
1770 Aug. 14 044
11 26 26 13
0.676893
4 12 17 3
1 34 30
13 12 55 40
11 26 16 26
0.6743815
4 12
1 33 40
13 12 37 35
11 23 15
0.67435
4 It 54 50
1 .14 31
OF THE SOLAR SYSTEM.
%
Passage at per. mean time
at Greenwich.
Long, of the
per, on the orb.
of the comet.
Per. dist.
earth's be-
ing 1.
Lon^.. of the
ascending
node.
Inch of the
orbit.
64
1770 Nov. 22 5h.38'40"
6s.2822' 4*"
0.52824
3s.l842'10 f/
3125' 55'
65
1771 Apr. 18 22 5 7
3 13 28 13
0.90576
27 51
11 15 20
19 5 1 21
3 14 2 54
0.90340
27 50 27
11 16
66
1772 Feb. 18 20 41 13
3 18. 6 22
1.01815
8 12 43 5
18 59 40
67
1773 Sept. 5 11 9 25
2 15 35 43
1.13390
4 1 15 37
61 25 21
14 33 48
2- 15 10 58
0.10120
4 1 5 30
61 14 7
68
1774 Aug. 15 10 46 15
10 17 22 4
1.4286
6 49 48
83 25
69
1779 Jan. 4 2 2 40
2 27 13 11
0.71312
25 5 51
32 24
2 15 10
2 27 13 40
0.7132
25 3 57
32 25 30
70
1780 Sept. 30 18 3 30
8 6 21 18
0.09925
4 4 9 19
53 48 5
71
1781 July 7 4 32
7 29 11 25
0.775861
2 23 38
81 43 26
72
1781 Nov. 29 12 32 26
16 3 28
0.96101
2 17 22 52
27 13 8
12 33 25
16 3 7
0.960995
2 17 22 55
27 12 4
73
1783 Nov. 15 5 44 3
I 15 24 46
1.5655
1 24 13 50
53 9 9
74
1784 Jan. 21 4 47 40
2 20 44 24
0.70786
1 26 49 21
51 9 12
75
1784 Apr. 9 21 7 46
10 28 54 57
0.650531
2 26 52 9
47 55 8
76
1785 Jan. 27 7 48 44
3 19 51 56
1.143398
8 24 12 15
70 14 12
77
1785 Apr. 8 8 58 52
9 27 29 33
0.427300
2 4 33 36
87 31 54
78
1786 July 7 21 50 52
5 9 25 36
0.41010
6 14 22 40
50 54 28
79
1787 May 10 19 48 40
7 44 9
0.34891
3 16 51 36
48 15 51
80
1788 Nov. 10 7 25 40
3 9 8 27
1.06301
5 7 10 38
12 28 20
81 1788 Nov. 20 9 4 25
23 12 22
0.7669H
11 21 42 15
64 52 32
82
1790 Jan. 15 5 5 39
2 14 32
1.7581
5 26 11 46
31 54 15
83
1790 Jan. 28 7 36 13
3 21 43 37
0.063286
8 27 8 37
56 58 13
84
1790 May 21 5 46 54
9 3 43 27
0.79796
1 3 11 2
63 52 27
85
1792 Jan. 13 13 35 9
1 6 29 42
1.293
6 10 46 15
39 46 55
86
1795 Dec. 14 23 17 53
5 13 37
0.227
11 29 11
24 17
15 15 6 18
5 7 37
0.258
ll 13 23
20 3
14 18 43
5 15 S3
0.215
0170
24 42 u
87
1796 April 2 19 47 51
6 12 44
1.578
17 2
64 55
88
1797 July 9 2 43 31
1 19 34 48
0.52545
10 29 16 35
50 35 50
89
1797 April 4 11 32 21
3 14 59
0.48476
4290
43 52 16
90
1798 Dec. 31 12 58 57
1 4 29 48
0-77968
8 9 30 44
42 23 25
91
1799 Sept. 7 5 49 48
3 39 12
0.839865
3 9 31 59
50 55 37
5 34 4
3 39 10
0.840178
3 9 27 19
50 57 30
4 24 39
3 36
3 9 34
50 52 30
92
1799 Dec. 25 21 30 49
6 10 20 12
0.625810
10 26 49 11
77 1 38
93
1801 Aug. 8 13 22 39
6 3 49
0.2617
1 14 28
21 20
94
1803 Sept. 9 20 33 54
11 2 8
1.0942
10 10 17
57
95
1804 Feb. 13 13 31 7
4 28 44 51
1.07117
5 26 47 58
56 28 40
13 15 30 39
4 28 53 32
1.072277
5 26 49 47
56 44 20
96
1805 Nov. 18 15 39
4 29 28
0.37567
11 15 6 51 115 58 12
37
1805 Dec. 31 6 1 36
3 19 23 40
0.89159
8 10 35 24 116 25 25
OF THE SOLAR SYSTEM. 433
The following were calculated by M. Burckhardt ; 1, 3, 9, 58,
2d. 63, 3d in an elliptic orbit semi major axis 3.1435, period, 65
Apr. 19, in an hyperbolic orbit, 67, Sept. 14. 89, 90, 91, 3d. 93.
Those that follow by M. Pingre, No. 2, 4, 5, July 17, sup-
posed to be the same as No. 14, 6, 8, 2d. 10, period 75| years.
16, 17 (nearly) 21, 2d. 22 (nearly) 31, 35 (nearly) 53, 54 (see
mem. de PAcad. 1757 and 1759) 55, 58 (mem. de I'Acad. 1774)
59 (mem. 1774) 60 (nearly, mem. 1776) 61 (mem. 1778) 63,
6th. cal. in an elliptic orbit, mem. 1771. 63, 1st. in an elliptic
orbit, mean dist 3.08891, period 5.42886 years. 64, 65.
M. Dunthorn calculated the elements of the 5th, 1st supposed
to be the same as No. 14-.
Dr. Haitey calculated the elem. of the following ; 8, 1st nearly
11, 1st 12, 13, 14, 15, the four last also nearly, No. 12 is supp.
the same as No. 25, and No. 14 as 5. 16, 2d. 18, 19, 10,* (see
Newton's Prin) 23, 24, 26 (see the firing 27, 28, 29, 3i, 1st.
2d. in an ellipse, mean dist. 138.2957, period 575 years (see
flrtn.) 10,* 1st 2d in an ellipse, period 75 years. 32, 33, 34 and 36.
The following were calculated by M. Douwes ; 13, 30 (nearly)
41, 2d.
M. de la Caille calculated the following 20 (nearly) 37 (nearly)
38, 39, 1st 40, 1st 41, 1st 43, 1st see Mem. de I'Acad. Roy.
1763. 45, 46, 1st 47, 1st nearly. 50, 1st (mem. 1757J 10,*
1st 56.
M. Euler in his Theoria motuum Pianetarum et Cometarum,
calculates the following, 31, 4th an elliptic orbit. 46, 3d 62, 3d
an elliptic orbit.
The 39, 2d 40, 2d 46, 2d 47, 2d. 52 (nearly) 57, 2d were
calculated by M. Siruyck.
The 42 (see Newton's firing and 44, were calculated very accu-
rately by M. Bradley.
M. Klinkenberg calculated 48. M. Belts calculated 49 very ac-
cur. (Phil, trans, vol. 43.) M. Chezeaux 50, 2d. M. Maraldi
51, 10,* 3d. (mem. de fAcad. 1759) 57, 3d. M de la Lande
10,* 3d 57, 1st (mem. de fAcad. 1762 and 1763) 62, 1st (mem.
1769 and 1765.) M. Prosjierin 62, 2d 86, 3d. M. Lexell 62,
4th an elliptic orbit. (See mem. 1795, pa. 430) 63, 2d an ellip-
tic orbit, semimajor axis 3 14786, period 5.585 years. (Phil,
tram. vol. 69.) Sir Henry Englejield 62, 5th M. Mechain 68,
69, 2d 70, 71, 72, 73 (nearly) 74, 76, 77, 78 (mem. 1786) 80
(mem. 1788) 81, 83, 91, 1st (see Connoissance des Terns, an. 12)
92 (La Landc supposes this to be the same as No. 37) 94- (Con.
des T. an. 14) Chev. d^Angos 69, 2d 75. M. Lcgendre 72, 2d
(see A'ou-uelle Mcthodes, &c.) 96 and 97. P. de Saron 79. (Mem.
1787) 82 (con. de 7') M Zach 86, 1st 91, 2d (con. tie T. an. 12)
M. Bouvard 86, 2d 88. Dr. Others 87 M. Gauss 95, 1st
(con. de T. an. 15) I 2d. con. an. 1808
The preceding article being nearly written, when, from the po-
liteness of Professor Farrar in Cambridge, Massachusetts, the
434 OF THE SOLAR SYSTEM.
author bad been favoured with his observations on the present co-
met (1811) from Sep. 6 to Nov. 12.* From these observation:*
those of Sept. 26, Oct 1 7 and Nov. 10, have been selected for
calculating the elements of this comet*s orbit ; as more nearly
conforming with the directions given by Newon t As these sheets
are immediately wanting for the press, there is therefore little
time to enter into calculations so tedious as the present : the fol-
lowing results have, however, been hastily deduced ; the time
being changed to that of the meridian of Greenwich, allowing
71 7' for cliff, long, and the correct distances! of the comet, for
Sept. 26, -2h. 11' 9" being taken from Arcturus 30 11' 46 /r
from Lyra 60 15'; on Oct. 17, llh. 20' 48" from Arct. 34
2' from Lyra 27 26' 35" ; Nov. 10, 12h. 34' 58", from Lyra
16 30' 5", from Deneb 33 10'.
* The following- being 1 , according- to Mr. Farrar the most correct, are in-
serted here, and the Cambridge times, as given by him, retained. They may
serve as an exercise for learners in making- more accurate calculations, &c.
ARCTURUS. LYRA.
Jlppar. time. Jip/jar. dist. JJpp. time. Jlp. dist.
Sept. 6d. 7h. 52' 14" 47 43 ; 2i/' 8h. 3' 14" 82 11' C'
9 7 59 20 45 38 20 7 54 40 79 32 18
10 8 19 38 44 53 16 7 47 31 78 30 46
14 8 38 18 42 15 8 28 20 74 33 50
18 7 39 23 38 58 45 7 27 2 70 18 50
26 7 26 41 30 8 15 7 32 16 60 17 10
30 7 12 30 49 30 7 19 20 54 38 30
Oct. 2 7 33 12 29 54 45 7 49 40 51 37 45
17 6 36 20 34 4 30 6 41 50 27 27 05
POLAR STAR.
('21)19 8 6 30 49 7 5925 24 13 15
' 21 8 58 10 50 34 15 8 55 21 7 15
23 7 23 30 52 20 30 7 17 30 18 27 45
29 7 3 50 57 40 15 71 20 12 24 45
Nov. 2 7 15 5 61 17 45 775 11 32 37
4 7 23 63 6 45 7 18 30 12 12 45
10 7 58 20 6S 1 7 50 30 16 30 05
118 8 30 68 38 15 82 50 17 21 55
12 7 15 57 18 11 21
The distances were also taken from fleneb a. Cygni (or c&Jlrided as marked
on Cary's Globes) as follow. Nov. lOcl. Sh. 6' 20" ap. dist. 35 1C' 25" Nov
11. 8h. 11' 50" dist. 33 12' 25" and Nov. 12. 7h. 24' 45" dist. 33 14' 55'.
^t would have been better if the distances could have been taken at one in -
stunt by two observers and the altitudes given, as then the trigonometrical
calculations, and allowances would become more simple. But the alt. in
order to allow for refraction, &c. can be found nearly by the Globes.
f The distances corrected above, were not found "from strict calculation,
but when the star and comet were found to have nearly the same alt. a
mean of the alt. was taken, and refraction being- allowed, the dist. was in-
creased as cos. variation of the altitudes from refraction, &c. When the
star and comet were found to be nearly vertical, then the cliff, of their re-
tractions were subtracted. It being too tedious to give the trig 1 , cal. not
having 1 day to perform the oper. Some allowance was made for the com
et's motion, but the aberrations and parallax were omitted, the calculation
not being rigorously exact.
OF THE SOLAR SYSTEM.
435
1811
Mean time at
Greenwich.
obs. long.-
of the
come t
y)bs. hit.
north
of the
comet.
sim'-'i lon^.
calculated.
log. of the
earth's
dist. from
the sun.
comet's elong
from the
sun.
Sept. 26d. 12h. $ 32"
Oct. 17 11 6 18
Nov. 10 12 19 14
5s. 18 2
7 12 8
9 12 30
46 17'
62 40
45 15
6s. 2 46' 57"
6 23 4-1
7 17 44 33
0.000715
9.998334
9.995520
14 47 f 57"\V.
18 27 E.
54 45 27 E.
Now the mean distances of the comet from the sun, by projec-
tion, was found 1.058 corresponding to the obs. of Sept. 26.
And 1.422 corresponding to the obs. of Nov. 10, the earth's
mean dist. being 1 . From which data the elements of the comet
may be found as before directed * *
Mr. Farrar remarks, that with a common night glass, he ob*
served a dark ground of considerable extent immediately surround-
ing the coma, exterior to which a sort of halo, which after making
a curve of about 180, receded in a tangential direction forming
the two branches of the tail, of which the convex, on that next the
sun was somewhat the longest, and both a little incurvated ; the
length of the tail, found by taking the dist. of a star at its extremi-
ty and the comet, was Sept. 10, in the evening 5, breadth 2 ;
Sept. 13, even 7 10'; Sept. 18, 12; Oct 19, the evening
being fine, the tail measured 14i. Nov. 6, the diameter of the
head including the coma, was measured with an object glass mi-
crometer, fitted to a 3 feet Gregorian Reflector, made by Short,
and found 2' 46". The nucleus had very much the colour and
nearly the apparent mag. of Saturn. The exterior light surround-
ing the comet before mentioned, was judged to be five times the
diameter of the head. Mr. Farrar further remarks, that the
curved part of this light seemed very much to resemble in form,
a current of water flowing round a stone or other obstacle placed
in it.
* We had thus far proceeded when the sheets were wanting- for the press,
we may however resume the calculation in some of the following- articles.
Mr. N. Botuditch of Massachusetts, determines the elements of this comet
as follow. Perih. dist. 1.032, time of passing- the perih. Sept. 12d. 31i.
Greenw. time, place of the perih. reckoned on the comet's orbit 2s. 15 14',
long-, of the ascending node, 4s. 20 24', incl. of its orbit to the eclip. 73.
Ijts motion he finds to be retrograde.
Mr. Wood of Richmond says that the mot. is direct, as it moves according-
to the order of the signs since its appearing ,- but it is evident that its mo-
tion may be direct, as seen from the earth, though retrograde as seen from
the sun, from which this motion is estimated in the cal. The heliocentric
longitudes immediately shew whether the comet be direct or not, and not
the geocentric.
436 OF THE FIXED STARS,
CHAP. XI.
OF THE FIXED STARS.
HAVING given as comprehensive an account of the Solar System
as an elementary work of this nature would admit, we shall now
give a short description of those bodies which are situated beyond
the limits of this System, and which are called fixed stars, from
their not having any proper sensible motion of their own, except a
few Of their general division into constellations, &c. we have
already given an account in the first part
From the most accurate observations, the whole diameter of the
earth's orbit, seen from them, is found to dwindle into a point, or
in other words they are found to have no sensible parallax, and
must therefore be immensely distant from us. From this circum-
stance, and that when examined by the most powerful telescopes,
their disks appear but as luminous points, it is with reason inferred,
that their magnitudes must be very great, as otherwise they would
not be visible ; and considering the weakness of reflected light,
there can be no doubt but that they shine with their own light.
They therefore differ from Planets in these two circumstances,
that they do not borrow their light, or reflect it, as the planets do,
and that their apparent diameters are not increased by telescopes.
The smallness of their appar. diameters is also proved from their
rapid disappearance in their occultations with the moon, the time
of which not amounting to \" proves, -as Lafilace remarks, that
their appar. diam. is less than (5") l"62. And as the. smallest
stars are subject to the same motions as the most brilliant, and
constantly preserve the same relative positions, it is therefore prob-
able that all those bodies are of the saine nature and placed more
or less distant from the planetary system. Whether they have
planets revolving around them or not, is a question that can never
be decided ; but as they seem to be of the same nature with the
sun, and of a mag. at least equal to him, analogy would lead us to
suppose that they are destined to perform the same functions, and
are, therefore, probably suns to other systems ; for there are mil-
lions of them that are not at all visible to an inhabitant in our sys-
tem.
The number of the stars visible to the naked eye is about 1 000,
as may be seen by reckoning all those to the 6th mag incl. which
are on a hemisphere of the celestial globe. But from the im-
provements in reflecting telescopes, Dr. Herschel and other astro-
nomers have discovered that their number is great beyond concep*
tion. Every improvement of the telescope discovers stars not vis-
ible before, so that there seems to be no limits to their number or
to the extent of the universe.
We have marked in the table of the constellations, in a separate
column, all those stars that are not single, when viewed with good
telescopes, but as some consist of 3- 4, 01 more stars, we shall give
OF THE FIXED STARS. 437
iin account here of the most remarkable, following the order of
the table, page 28.
In Aries a and & are double.
In Taurus Aldebaran is quadruple, the largest in the Pleiades
double.
In Gemini a or Castor is double, or Pillux, or Hercules con-
sist of several, there are 1 1 marked on it, on Gary's Globe.
In Leo Regulus, 7, and some others are double.
In Virgt y is double, and S quadruple.
In Capricorn a, is double, one is much larger than the other)
the largest is marked 1, the other 2.
In Aquarius is double.
In Ursa Minor a is double, very unequal, the largest white,,
the smallest red.
In Ursa Major Mizar is double, or triple, two are easily dis-
tinguished, one larger than the other, the smallest is called Alcor.
In Draco v 1, v 2 appears single to the naked eye.
/? in Ceplicus is double. In Cassiopeia K or Shedir, and t arc doub.
a small star marked d is triple, a or Capella, an\l & in Auriga are
double. The star marked 2 in Lynx is double. Cor Caroli is
double. In Bootes Arcturus is triple or quadruple, two of the stars
in it appear of the 8th mag. e or Mirach is double, very unequal,
largest reddish, smallest blue, or faint blue, very beautiful , f, i,
and others are double. In Corona Borealis f and a are quadruple,
v is quintuple. In Hercules a, y and others appear double, the
stars marked 70 and 71, also 80 and 81 appear single to the naked
eye. In Lyra appears double, very unequal, the smallest ap-
pears with a power of 277. Dr. Herschel measured the diameter
of this star and found it = /; 3553. # is quadruple, unequal, one
white, three reddish, ^ 1 and $ 2, appear as one, e and others are
double. |5, a-, X, 32, o2 and others, are double, y triple, f* is also
double, unequal, largest white, smallest blue. In Delfihinus (3 and
7 are double In Equuleus 7, <r, , and others are double. In
Pegasus E or Enir and e are double. In Andromeda x and 7 or
Almaach, are double, the latter are unequal, the largest reddish
white, the smallest blue, inclining to green, a beautiful object.
In Musca Borealis the star marked 33 is double, 41 is triple, 3d
mag. In Perseus is triple, p, H 31, and others, are double. -
In Serftens 5, 5 and others are double. In Ophicicus a has two
small stars of the 6th mag. nearly touching it, A and others are dou-
ble. In Cetusy Menkar is double, one of the 2d, and the other of
the 6th mag. Mira, * and others are double, Mira is changeable
when greatest of the 2d, when smallest in vis. period 344d. In
Orion, Betelgeux, and Rigel, , , *j. T, e and others, are double,
/, v 1 and v 2 contain each 6 or 7 small stars, cr has 6 or two triple
;tars. In Canis Minor Procyon has two stars of the 9th mag. very
near its body, the star marked 31 is double. In Hydra Cor Hydra
is triple, the star marked 2 is double and variable. In Cor-vus $ is
double. In Ccntaurus K is double, the 1st is of the 1st and the 2d
438 OF THE FIXED STARS.
of the 4th mag. In Plscis Australia Fomalhaut has a sjtar of the
6th mag near its edge, e is double, the 1st of the 3d, the 2d of tire
5th mag. In the Shift Argo Canopus is double, the 1st is of the 1st,
the 2d of the 6th mag. ft in the milky way has on its south side an
innumerable multitude of stars, and in its body 9 or 10. The
southern constellations have not been examined with the same care
as the northern, which is the reason that so few of them are marked
double, &c. Dr. Herschel has given a catalogue of the double
stars in the Phil, trans. 1782 and 1785.
Many of the stars observed by ancient astronomers, do not ap-
pear at present, and others are at present observed which are not
found in their catalogues, It was the appearance of a new star
about 120 years before J C. that caused Hififiarchus first to under-
take making a catalogue. There is however no account where
this star appeared. A second is said to have appeared in the year
130 ; a 3d in 389 ; a 4th in the 9th century, in 15 of Scorpio ; a
5th in 945, and a 6th in 1264 ; the accounts we have of these are
however imperfect. Cornelius Gemma on Nov 8, J 572, observed,
in the chair of Cassiofieia, the 1st of which we have any regular ac-
count. It exceeded Sirius in brightness, and was seen at midday.
It first appeared larger than Jupiter, then gradually decayed, and
in 1 6 months vanished. Some suppose that it was this which ap-
peared in 945 and 1264?.
David FabriciuS) on Aug. 13, 1596, discovered a new star in
the neck of the whale in 25 45' of Aries, lat 15 s4' S. In Oct.
the same year it disappeared. It was again seen in 1637 ; the pe-
riodic time between its greatest brightness is determined to be 333
days. Its greatest brightness is that of the 2d mag. and least that
of a star of the 6th But its greatest splendour and also its period,
are found to be variable.
William Jansenius, in 1 600, discovered a changeable star in the
neck of the Swan. Kejiler, who wrote a treatise on it, fixes its place
in ym 16 18', with 55 30' or 32' S. lat Ricdolus observed it in
1616, 1621, 1624, and 1629 ; and says that it was invisible in 1640
and 1650. Cassini observed it in 1655, from which it increased to
1660, then grew less, and at the end of 1660 disappeared. In
Nov. 1666, it again appeared, and disappeared in 1681. In 1715,
it appeared of the 6th mag.
p. Anthelme on June 20, 1 670, discovered another changeable
star near the swan's head In Oct. it disap. and appeared again
on March 17, 1761, and disap. Sep. 11. In March, 1672, it ap-
peared again, disap in the same month, and was not observed
since. Its long, was XX 1 52' 26", lat. 47 25' 22" N. The
days are here as in the new stile.
In 1686 Kirchius observed % in the swan to be a changeable
star ; and from 20 years obs. it was found that the period of the
return of the same phases is 405 days ; its magnitude is, however,
subject to some irregularity. In 1604 Kefiler discovered a new
star in the right foot of Serpentarius, which exceeded even Jupi-
OF THE FIXED STARS. 439
ter in mag. Near the horizon it appeared white, in every other
posit it continually varied its colour into some of the colours of
the rainbow, it disappeared in 1605, and was not seen since. Its
longitude was / 17 40', lat. 1 56' N. it had no parallax.
The stars j3 and y in Virgo were found by M>mtanari to be
wanting. They were visible in 1664, but were wanting in 1668.
He found 6 in the serpent visible from the time he observed until
1695-. ]/ in Leo disappeared, and was again seen in 1667. # in
the head of Medusa also varied its mag.
Cassini discovered one new star of the 4th and two of the 5th
mag. in Cassiopeia ; he afterwards discovered Jive new stars in
the same constel three of which disappeared He discovered
two new stars in Eridanus, one of the 4th and the other of the 5th
mag. He observed that e in the Little Bear disappeared : that
in Andromeda which had disappeared, had again appeared in 1695 :
that in place of there are two stars more northerly, and that | is
diminished.
Maraldi says that * which was of the 3d mag. in 1 67 1, was of the
6th in 1676, Dr Hattey found it again of the 3d in 1692, it was
almost imperceptible, but in 1693 and 1694 it was of the 4th mag.
In 1704 he discovered a new star in Hydra, in a rt. line with ?r
and y. The period of its changes is about 2 years. J. Goodricke
has found the periodic variation of Algol in Perseus to be about
3d. 21h. He has also discovered that & Lyra; completes all its
phases in 12d. I9h. Cejihei, according to him performs the
periodic variation of its phases in 5d. 8h. 37 J'. E. Pigott has
discovered that Antinoui is a variable star, and fixes the period of
its variation at 7d. 4h. 38'. For a further detail, consult Vince's
Astr. or Phil, trans. 1785, and Herschel's remarks and method
f observing these changes, Phil, trans. 1796. In the Phil, trans.
for 1783, in a paper on the proper motion of the Solar System^ he
has given a large collection of stars which were formerly seen, but
are now lost ; also a catalogue of variable stars and of new stars.
These variations in those stars, considered as fixed, have afford-
ed ample scope for conjecture. Maupertnis supposes the varia-
tions to arise from their quick motion round their axis, which he
thinks may reduce them to very oblate spheroids, like a mill stone,
and that when the flat side is presented to the earth, they become
nearly invisible. Laplace remarks, that the extensive spots which
these fixed stars present to us periodically in turning round their
axis, nearly in the same manner as the last satellite of Saturn, and
the interposition of large opake bodies which circulate round them*
are sufficient to explain their periodic variations ; and further re-
marks, that as to those stars which suddenly appear with a very
brilliant light, and then vanish, it may be supposed that this takes
place by means of great conflagrations on their surface, occasioned
by extraordinary causes. As light takes a considerable time to
pass from us to the fixed stars, it may have considerable effect in
Changing the apparent place of those that become invisible, wheu
440
OF THE FIXED STARS,
S'
they again reappear. And as the nature of the aberration of the
stars is not only useful to be known in this inquiry, but also in cor-
recting their apparent places to obtain the true, we shall here col-
lect, principally from Vince's Astronomy, what is most practical and
important on this subject.
The situation of any object in the heavens, is determined by the
position of the axis of a telescope annexed to the instrument with
which we measure ; for the telescope is so placed, that the rays
of light from the object may pass through it in the direction of its
axis or length, and then the index shews the angular dist. requir-
ed. Now if light take a determined time in passing from one ob-
ject to another, when a ray from any distant object as a star de-
scends down the axis, the position of the telescope must be differ-
ent from what it would have been, if light had been instantaneous,
and therefore the place to which the telescope is directed, will be
different from the true place of the object. For let S' be a fixed
star, VF the direction of the earth's mo-
tion, S'F the direction of a particle of light
entering the axis ac of a telescope at a,
and moving through aF while the earth
moves from c to F, then if the telescope
continue parallel to itself, the light will
descend in the axis. For let the axis nm,
Fw, continue parallel to ac, then consid-
ering each mot. as uniform (the rotary
inot of the earth being here neglected,
as producing no sensible effect) the spa-
ces described in the same time will con-
tinue in the same proportion ; but cF :
F :: en : av, and cF, aF are, by suppos.
described in the same time ; en, av, are therefore described in
the same time ; hence when the telescope comes into the situa-
tion nm, the particle of light will be in the axis at -v, and this is
true for every instant, the position of the telescope remaining the
same ; hence when the telescope is at F, the place of the star, as
determined by it, will be *' and the angle S'F./ will measure the
change made in the appar. position of the star from the motion of
the earth, combined with the progressive motion of light, or the
aberration. If we therefore take FS : F* :: the vel of light :
the earth's vel. and join S/, and complete the paral. F/Ss, the L.
FSr will represent the aberr. S will be the true place of the star,
and s the place determined by the instrument.
A similar change will be produced in the place of the star, seen
with the naked eye. For if a ray of light fall upon the eye in
motion, its relative mot in respect to the eye, will be the same as
if equal motions were impressed in the same direction upon each,
at the moment of contact ; as equal motions in the same direction,
impressed upon two bodies, will not affect their relative motions,
and therefore the effect of one upon the other will not be allowed.
/*
c t
-V
OF THE FIXED STARS. 44}
Let VF be a tangent to the earth's orbit at F, which will represent
the direction of the earth's mot. at F, S a star, join SF and produce
it to G, and make FG : n :: vel. of light : vel. of the earth in its
orbit ; complete the paral. and draw the diag. Fit Now as FG
represents the mot. of light* and wF that of the earth in its orbit,
let a mot. Fra, equal and oppos. to wF, be conceived to be impress-
ed upon the eye at F, and upon the ray of light, then the eye will
be at rest, and the ray, by the two motions FG, F, will describe
the diagonal FH ; this is therefore the rel mot of the ray in res-
pect to the eye itself. The object will therefore appear in the di-
rection HF, and the angle GFH = FS* will measure the diff. be-
tween its appar. and true place, as before. The place, therefore
determined by the instrument, is properly called its apparent place.
Now sin FS? : sine F*S :: F; : FS :: vel. of the earth : vel of
light ; hence sine of aberration =e sin. F^S X vel. of the earth
vel. of light. If we therefore consider the vel. of the earth and light
constant, sine aber. or the aber itself, as it never exceeds 20", va-
ries as sine F*S, and is therefore greatest when F*S = 90 ; taking
5 = sin. F*S, it will be rad. : s :: 20" : s x 20" the aberration.
By obs. JL FS* = 20" ; hence when F/S = 90, vel. of the
earth : vel. of light :: sin. 20" : rad. :: ] : 10314. The aber.
S'*' lies from the true place of the star, in a direction paral. to the
direction of the earth's motion, and towards the same part.
While the earth makes one revolution in its orbit, the curve de-
scribed by the appar. place of a fixed star, parallel to the ecliptic,
is a circle. ForletAFBQ
be the earth's orbit, K the S'
focus in which the sun is,
H the other focus, on AB
the greater axis let a circle
be described in the same
plane ; to the point F draw
the tangent yFZ, and kij
HZ perp. to it, then the
points y and Z will be al*
ways in the circumference
of the circle ( Vince's conic
sect. prop. 5, el. or Emer-
son's, b. 1, prob. 20 ) Let S be the true place of the star, out of
the plane of the ecliptic, and therefore elevated above the plane
AFBQ, and let *F be to FS as vel. of the earth to vel of light ; com-
plete the paral. F*Ss, and by what is shewn in the first part of this
art. s will be the star's appar. place Let FL be drawn perp. to AB,
and let WsVar be the curve described by the point s ; draw WS V
parallel to FL. Now from a well known principle in physics, the
vel of the earth varies as Ky, or as HZ (Knee's con sect. pr. 6 el.)
but /F, or Sa, represents the earth's vel. hence Ss varies as HZ.
AncJ as S, SV are paral. to Fy, FL, the angle sSV = yFL =
ZHL ; for LFZ added to each makes two rt. angles, the angles at
3 H
442 OF THE FIXED STARS.
L and Z being right angles Hence as S* varies as HZ, and *SV
ass. ZHA, the figures described by the points * and Z must be
similar; but Z descril.es a circle in the time of one rev of the
earth, .s must therefore describe a circle about S in the same time.
As S* 's always paral. to *F in the plane of the ecliptic, the circle
W*Vor is therefore paral to the ecliptic. Also as S and H are
two points similarly situated in VVV ancl AB, it appears that the
true place of the star divides the diam (which although in a dif-
ferent plane, may be considered as perp. to the greater axis of the
earth's orbit) in the same ratio as the focus divides the greater'
axis. But as the earth's orbit is nearly a circle, S may be con-
sidered in the centre of the c T .rcle without any sensible error.
The earth's orbit AFBQ being considered circular, and therefore
coinciding with AZB, draw C ' paral. to S* or vF, *' will then be
the point in that circle corresponding to s in the circle W.s V ; and
as FA'= 90 the appur. place ol the star in the circle of aber. is al-
ways 90 before the earth's place in its orbit, and hence the appar.
angular vel of the star and earth, about their respective centres,
are always equal. Moreover S' being at an indefinitely great dist.
the true place S being supposed not altered from the earth's mot.
or the stai to have no parallax, and FS being considered as always
parallel to itself, it will be always directed to S'. Hence also the
appar place of the sun being oppos to the earth's, the star's appar.
place, in the circle of aberration, is 90 behind that of the sun
A small portion of the heavens being considered as perp to a
line joining the earth and star, the circle
anbm, paral to the ecliptic, described by
the appar. place of the star, projected
on that plane, wM (from the principles
of orthographic project- see Emerson's
tracts) be an ellipse ; hence the star's
appar. path will be an ellipse, and the
trans, will be to the conjugate as rad. to
sine of the star's Int. For let CE be the
ptane of the ecliptic, P its pole, PE a
secondary toil, PC perp to EC, C, the
place of the eye, and let ab be pural to CE, then ab will be the
diam. of the circle anbm of aberration, which is seen most oblique-
ly, and therefore that diam which is projected into the lesser axis
of the c llipst- ; let mn be perp to ab, and it will be seen directly,
being perp to a line drawn from it to the eye, and will therefore
be the greater axis ; let Ca, Cbd, he drawn, and ab will be the pro-
jection of ab ; and as ad may be considered as a straight line, it
will be, mn or ab the greater axis : ad the lesser axis : rad. :
sine abd, or tLCd the star's lat. Hence the circle is projected on
a plane making an angle wiih it equal to the comp. of the star's
lat. for bad is the comp of abd* or of the star's lat.
The lesser axis da coinciding with a secondary to the ecliptic, is
therefore perp. to it ; and the greater axis mn is parallel to it, its
OF THE FIXED STARS.
443
posit, not being altered by projection Hence in the pole of the
ecliptic the sine of the lat. being rad the ellipse becomes a circle ;
and in the plane of the ecliptic sine star's lat being = 0, the les-
ser axis vanishes, and the ellipse becomes a straight line, or rather
a very small circular arc.
To find the aberration in Latitude and Longitude ; let ABCD
be the earth's orbit, supposed
a circle with the sun in the
centre at x, let P be in a line
drawn from x perp. to ABCD
so as to represent the pole of
the ecliptic ; let S be the star's
true place, and let a/icg be the
circle of aber paral to the
ecliptic, and abed the ellipse
into which it is projected ; let
T T be an arc of the ecliptic,
and draw the secondary PSG
to it, and, by one of the fore-
going articles, it will coincide
\vith the lesser axis bd, into
which the diam fig is pro-
jected ; let GC^A be drawn,
and it will be parallel to fig, and B.rD, perp. to AC, is paral. to the
greater axis ac ; then, when the earth is at A, the star is in conj.
and when the earth is at C, it is in oppos. Now the star's place
in the circle of aber. being always 90 before the earth in its or-
bit, as shewn before, when the earth is at A, B, C, D, the appar.
places of the star in the circle will be at a,/z, c, y, and in the ellipse
at , b, c, d. To find the place of the star in the circle when the
earth is at any point E, take the angle /*S = E.rB, and s will be
the corresponding place of the star in the circle ; to find the pro-
jected place in the ellipse, draw s~u perp to Sc, and -vt perp. to Sc
in the plane of the ellipse, then t will be the appar place of the
star in the ellipse ; let at be joined, and it will be perp to -vt (the
project, of the circle into the ellipse being in lines perp to the
ellipse) draw the secondary Pz^K, which, as to sense, will coin-
cide with we, unless when the star is very near P , hence, except
in this case, the rules here given will be sufficiently accurate.
Now as cyS is paral. to the eclip S and v have the same lat.
hence vt is the. aberration in Lat. and G being the true and K the
appar. place of the star in the cclip OK is the aberration in lung.
To determine these quantities, let m and n be the sine and cos. of
*S'.-, or CorE, the clist of the earth from syzygies, rad being = 1 ;
and .is _ s-vt = star's co. lat as we have before shewn, -uxt = the
la, tor the sine, and cos. of which let i> and iv be taken, let r = Sa
or Sor ; then in the rt angled triangle S.vv, 1 : ra :: r : &v = rm ;
In nee in the triangle vf*, > : v :: rm : t~v =s r-vm the aber in lat.
Also in the triangle S&v, 1 : n :: r : iS =* rn, Now, as the simi-
444
OF THE FIXED STARS.
lar arches of circles contain the same number of degrees w : 1 fc
rn : GK = the aber. in long. When the earth is in syzygies
m = o ; and hence there is no aber. in lat. and n being then greatest,
there is the greatest aber. in long, if the earth be at A or" the star
in conj. the star's appar. piace is at , and reduced to the ecliptic
at H ; GH is therefore the aber which diminishes the star's long,
the order of the signs being TGT ; but when the earth is at C,
or the star in oppos. the appar. place c reduced to the eclip. is at F,
and the aber. GF increases the long, the long, is therefore the
greatest when the star is in oppos and least when in conj. When
the earth is in quadratures at D or B, then n = o, and m is great-
est ; hence there is no aber in long, but the greatest in lat. when
the earth is at D, the appar. place of the star is at d, and the lat.
is there increased ; but when at B the appar. place of the star is
at , and the lat. is diminished ; hence the lat. is least in quadra-
ture before oppos. and greatest in quad, after oppos. From the
mean of a great number of observations, Dr. Bradley determined
the value of r to be 20".*
To find the aber. in rt. as- P
cen. and dccl. Let AEL be
the equator, fi its pole ; ACL
the ecliptic, P its pole ; S the
star's true filace, and * its ap-
par. place in, the ellipse ;
draw the great circles Psa,
P*, /zSw, fiSv, and Si>, st,
perp. to P6, ii-v. Now, as
we have before shewn, sv =*= . . v .
i-vm ; and Si; = rn ; hence A J -<
rvm (vs) : rn (Si>) :: rad. : tang. Ssv = n ~ -vm = cot. earth's dist.
syzy divided by sine of the star's lat. = cos. star's lat. X cot. earth's
dist. from syzy. Thus L. Ssv is immediately computed ; in like
manner Psfi the angle of posit, is computed from the three sides of
that triangle being given, the angle Ssfi is given, being the sum or
* Example 1. What is the greatest aber. in lat. and long, of $ Ursa Mi-
noris, whose lat. the beginning of 1812 will be 74 55' 28"? Here m = 1, v
= sine 74 55' 28"= .9655836, which mult, by 20"= 19"31 nearly, the great-
est aber. in latitude. For the greatest aber. in long, n = 1, w = .2600927.
which divided into 20" gives 76"9 nearly, the gr. aber. in lat. and long.
2- When the earth is 30 from syzyg-ies, what is the aber. in lat. and long,
of the same star ?
Here m sine 30 = .5, hence 19"31 X .5 = 5"65 nearly, the aber. in.
lat. If the earth were 30 past conj. or before oppos. the lat. is diminished :
but increased if the earth be 30 after oppos. or before conj. Also 11 = cos.
30 = .866 ; hence 70"9 X -866 = 66"59, the aber. in long. If the earth be
30 from conj. the long, is diminished : but increased if 30 from oppos.
3. For the sun, m =. o, n = 1, and -w = 1 ; hence the sun has no aber.
in lat. and the aber in long. = r = 20" corstantly. This aberration an-
swers to the sun's mean mot. in 8' 7" 30'", whiclTis therefore the time in
which light-moves from the sun to the earth at its mean dist. Hence the sun
appears 20 more backward than hi? trur nlace.
OF THE FIXED STARS. 445
iliff. of Ss-v and Psfi. Let a the sine, and b = the cos. of Ssv ; c
r= sine and d = cos. of Ssfi. z = cos. star's decl. then (sv, st, being
the co sines of SAT;, Sst, to rad. *S) b, : d :: si; (== rvm) : st =
rv7tix- =20"x ~um X the aber. in declination ; and (as Si;, St,
b b
are the sines of Ssu, Sstj rad. being 58) a : c :: Sv (=* rra) : Sf
= ; hence, from the property of similar arches, v (St -r- cos.
decl.) = 20" x the aber. in rt. ascension. Or the correct lat.
az
and long, being given, the corresponding correct rt ascen. and decl.
may be found, and hence the aber in rt. as. and decl. Before we
conclude this chap, we shall collect a few remarks on the nature
of the nebulous afifiearances observed in the heavens^ which from
the improvements in telescopes, have lately become so interesting.
The greater part of the fixed stars are collected into clusters, of
which it requires a large magnifying power, with a great quantity
of light, to be able to distinguish the stars separately. With a tele-
scope of small magnifying power, and light, these clusters appear
like small whitish spots, and thence were called Nebula ; the Milky
Way is a continuation of such nebulae or perhaps the one in which
we are situated, as Herschel observes. Allowing an observer (says
Herschel) the use of a common telescope, he begins to suspect
that all the milkiness of the bright path which surrounds the sphere
may be owing to stars. He perceives a few clusters of them in va-
rious parts of the heavens, and finds also that there are a kind of
nebulous patches : still his views are not extended to reach so far
us the end of the stratum in which himself is situated, so that he
looks upon these patches as belonging to that system which to him
seems to comprehend every celestial object. He now increases his
power of vision ; and applying himself to a close observation, finds
that the milky way is in reality no other than a collection of very
nmall stars. He perceives that those objects which had been called
ne.bulx, are evidently nothing but clusters of stars. Their number
increases upon him, and when he resolves one nebula into stars, be
discovers ten new ones which he cannot resolve, he then forms the
idea of immense strata of fixed stars, of clusters of stars, and of ne-
buliK ; till going on with such interesting observations, he now per-
ceives, that all these appearances must naturally arise, from the
confined situation in which he is placed. Confincdii may justly be
called, though in no less a space than what appeared before to be
the whole region of the fixed stars, but which now has assumed the
shape of a crooked, branching nebula ; not indeed one of the least,
but, perhaps, very far from being the most considerable of those
numberless clusters that enter into the construction of the heavens.
There are some nebul&, however, which do not receive their
light from stars, For in 1656, Huygens discovered a nebula in the
middle of Orion's sword ; it contains only seven stars, the other
part being a bright spot upon a dark ground, and appears like an
446 OF THE FIXED STARS.
opening into brighter regions beyond. Simon Marius in 161 2, dis-
covered a nebula in the Girdle of Andromeda. Dr. Hallfy^ when
observing the southern starsj discovered one in the Centaur^ and
in 1714 another in Hercules, in rt ascen. above 248f and decl.
37 N. this is visible to the nakc d eye when the sky is clear and
the moon is absent M. Cassini discovered one between the Great
Dog and Argo. M de !a Cailic gives an account of 42.
Meatier and Mechain* in the con. dr Temfift for I 783 and 1 784.. has
given a catalogue of 103 nebulae ; but Dr. Herschei, from his own
observations, has given a catalogue of 2000 nebula and clusters.
Some of them form a round, compact system ; others are more ir-
regular, of various forms ; some are long and narrow.; others are
hollow in the middle as that in. the constcl Telescofiium, and others
are thicker in the middle or more condensed towards the centre ^
the globular systems of stars are of the latter kind. On Gary's large
globes there are described, including clusters, clusters and nebulae,
and nebulae, exclusive of the milky way, at least 31 1. And as their
figti'es and places are marked on the globes, the learner will have
no difficulty in rinding their places in the heavens.
That the stars should be thus accidentally disposed, is a suppo-
sition too improbable to be admitted. Dr Hcrsc/iel* therefore,
supposes that they are thus brought together by their mutual at-
tractions, and that the gradual condensation towards the t entre is a
proof of a central power of that kind. He also observes thut there
are additional circumstances in the appearance of extended clus-
ters and nebulas, that favour he idea of a power lodged in the
brightest part He supposes the milky way to be a nebula of
which our sun is one of its component pans See his account in
the Phil trans. 1786 and 1789, or in Low's or the Philadelphia
ed. of the Encyclopedia, art astr
Dr Herschei has discovered other phenomena in the heavens
which he calls Nebulous sar.\; that is stars surrounded with a faint
luminous atmosphere of a consideiable extent He has given an
account of seventeen of these stars, one of which he describes thus.
" Nov 13, i790, A most singular phenomenon ; a star of the 8th
mag with a faint luminous atmosphere of a circular form, and of
about 3' diameter. The star is perfectly in the centre, and the at-
mosphere is so diluted, faint, and equal throughout, that there can
be no surmise of its consisting of stars , nor can there be a doubt
of the evident connection between the atmosphere and ihe star.
Another star not much less in brightness, and in the same field of
view with the above, was perfectly free of any such appearance "
Herschei therefore draws the following conclusions : that the cen-
tral star must be immensely greater than those which give the ne-
bulous appearance, if this consist of stars very remote and connect-
ed with the stur which it surrounds ; or thut if the central star be
not larger than common, the other luminous points must be ex-
tremely small and compressed to form the nebulosity. Hence ac-
cording to the former supposition, as the central point must far ex-
ceed the standard of what we call a star, there must exist a central
OF THE FIXED STARS. 447
body which is not a star. That this may be, and is very probably the
case, we have shewn in the note pa 296 If" the latter supposi-
tion be granted, there must exist a shining fluid surrounding a star
of a nature entirely unknown to us. Dr. Herschel adopts the latter
opinion, and says, that the existence of this shining matter does not
seem to be so essentially connected with the central points, that it
might not exist without them. The great resemblance there is be-
tween the chevelure, or the beams or hairy appearance, of these
stars, and the diffused nebulosity about the constellation Orz'cw,
which occupies a space of more than t>0 sq. degrees, renders it ex-
tremely probable that they are of the same nature. This being ad-
mitted, the separate existence of the luminous matter is fully prov-
ed This is also extremely probable from what is shewn in the
note pa. '296 ; moreover light reflected from the star could not be
visible at this dist. and besides the outward parts are nearly as bright
as those next the star. Her&chel further observes in coniir of this
supposition, that a cluster of stars will not account for the milkiness
or soft tint of the light of those nebulae, as a self luminous fluid.
There is a telescopic milky way extending in rt. ascen. from 5h.
15' 8" to 5h. 39' '", and in polar dist. from 87 46' to 98 10'.
Dr Herschel thinks that this is better accounted for by a luminous
matter, than from a collection of stars. He observes that some
may account for those nebulous stars, from a star being accidental-
ly placed nearer, which appears in the centre of a collection of
stars placed at an immense distance- but he is of opinion that this
milky appearance does not at all favour the suppos that it is pro-
duced by a great number of stars. But as Vince remarks, when
we reflect that nothing but a solid body is self luminous, or at least,
that a fixed luminary must immediately depend upon such, as the
flame of a candle upon the candle itself, it is extremely difficult to
admit this suppos. Our knowledge of the nature of these phenom-
ena must however be very imperfect, as we are. as yet, but imper*
xectly acquainted with the nature of light and its various modes of
existence. See Dr. JFferschel's account in the Phil, trans. 1791.
The distance of the fixed stars are great beyond conception ; for
at the dist of the nearest, the whole diameter of the earth's orbit
does not amount to, or at most much exceed, a single second. If
thi- angle, or their parallax could be accurately determined their
disk might be found in the same manner as that of the .superior
planets For other methods see Dr. Gregory's ast. sect. 9, b 3.
Dr Bradley estimates the dist of the nearest at 80000 times that
of the sun, and of y Draconis 400000 times the earth's mean dist.
from the sun, its parallax not amounting to 1 ". How great then
must he the dist of the nebulous stars ! Dr Herschel remarks that
a nebula whose light is perfectly milky, cannot be supposed at less
than 6000 or 8000 times the dist. of Sirius, considered the nearest
of the fixed stars ; so that a ray of light, which traverses the im-
mense space between the earth and the sun in 8' 7"iO'" (pa 444)
would take 36000 or 38000 years, to arrive from one of these ne-
bula to us 3 according to the distances which Herschel assumes.
448 OF SOLAR AND LUNAR ECLIPSES.
CHAP. XII.
OF SOLAR AND LUNAR ECLIPSES.
AN eclipse of the moon is evidently caused by the interposition of
some opake body which deprives it of the light of the sun ; and it
is equally evident that this opake body is the earth, as an eclipse of
the moon never happens but at the full moon or oppositions, at
which lime the earth is between her and the sun ; and projects
behind it relatively to the sun, a conical shadow, the axis of which
is the straight line which joins the centres of the sun and the
earth, and terminates in a point where the diameters of these two
bodies are the same. Hence the cone of the terrestrial shadow is
at least three times the length of the moon's dist. from the earth,
and its breadth at the points where it is crossed by the moon is more
than double her diameter. Hence there would be a lunar eclipse
at every oppos. if the plane of the moon's orbit coincided with the
ecliptic. But from the incl of these planes, the moon in oppos.
is often elevated above or depressed below the lunar shadow, and
does not enter it but when she is near the nodes. If the whole of
the disk be immersed in the shadow, the eclipse is total; if only
a portion of the disk be obscured, it is partial.
In calculating an eclipse of the moon, the first thing to be found
is the time of the mean ofifios. or the time when the oppos. would
have taken place were the motions uniform. To obtain which,
from the table of Efiacts (see Delambre's or Burg's tables as pub-
lished by Vince) take out the epact for the year and month, and
take this sum from 29d. 12h. 44' 3", one synodic rev. of the
moon, or two if necessary, so that the rem. be less than a rev. this
rem. will be the time of mean conjunction. If 14d 18h. 22' l /f 5,
half a revolution be added to the time of mean conj. the sum will
be the time of the next mean oppos. or if it be subtracted, the rem.
will be the time of the preceding mean oppos If it be bissextile,
one day is to be taken from the sum of the epacts in Jan. and Feb.
before the above subtr. is made. When the day of the mean conj.
is 0, it denotes the last day of the preceding month.*
To determine whether an eclipse may happen at oppos. let the
earth's mean long, at the time of mean oppos. be found, and also
the long, of the moon's node ; then according to M Cassini, if
the diff. between the mean long, of the earth and the moon's node
be less than 7 30', there will be an eclipse, if this diff. be great-
er than 14 30', there will be no eclipse ; but between 7 30' and
* Ex. To find the times of the mean new and ful moons in Feb. 1813.
Here 27d. 16h. 17' 18" 4. Id. llh. 15' 57" (the epact for Feb.) = 29d. 3b.
33' 15'/, which subt.from 29d. 12h. 44' 3", leaves Od. 9h. !(/ 48" for mean
ne-w moon on January 31, the day being- = 0. 14d 18h. 22' 1"5 added to
this, gives 15d. 3h. 32' 4S"5 the time of mean full moon. (See pa. 19.)
OF SOLAR AND LUNAR ECLIPSES. 449
j 4 30', there may or may not be an eclipse. M. Delambre
makes these limits 7 47' and 13 21.*
The new and full moons for a month before and after the time
at which the sun comes to the place of the nodes of the lunar or-
bit, being thus examined, no eclipse will be omitted. Or if the
eclipses for the preceding 1 8 years be given, and to the times of
the middle of these eclipses 18y. lOd. 7h. 43^' or 18y. lid. 7h.
431' (see note pa. 176) be added, the times at which the return
of the eclipses may be expected will be given.
Next compute by the tables the true long, of the sun and
moon, and the moon's true lat. for the time of mean oppos. and
also their horary motions in long, the diflf. (a*) of the horary mot.
is the moon's relative hor. mot. in respect to the sun, or the mot.
with which the moon approaches to, or recedes from the sun ; let
the moon's hor. mot. in lat. be also found ; and suppose the moon
is at the dist. (m) from oppos. at the time (t) of mean oppos. then
as the moon's access or recess from the sun may be supposed uni-
form d : m :: I hour : the time (w) between t and oppos. which
added to or subtr. from /, according as the time is before or after
the moon's oppos. gives the time of the ecliptic oppos.
To find the moon's place in oppos. let n be the moon's hor. mot.
in long, then Ih. : iv :: n : the increase of the moon's long, in
the time w, which applied to the moon's long, at mean oppos.
gives the moon's true long, at the ecliptic oppos. The opposite
point to this is the surSs true place or long. Let the moon's true
lat. at the time of oppos. be also found, by this propor. Ih. : w ::
the hor. mot. in lat. : mot. in lat. in the time w, which applied to
the moon's lat. at the time of the mean oppos. gives the true lat.
at the time of the true oppos.f In like manner the true time of
the ecliptic conjunction may be computed, and the places of the^
sun and moon for that time, when a solar eclipse is calculated.
From the sun's hor. mot. in long, and the moon's in long, and
lat. the incl. of the relative orbit, and the horary mot. on it> may
be thus found ; let LM be the moon's hor.
mot. in long. SM that of the sun ; let Ma
perp. to LM = the moon's hor. mot. in lat.
take S6 = and parallel to aM, and join La,
L6, then La is the moorfs true orbit, and L6
her relative orbit in respect to the sun. Hence
* To find whether there will be an eclipse at the full moon on Feb. 15,
1813. Sun's mean long, at 15d. 3h. 32' 49" 5 = 10s. 15 14' 19" 6, hence
the earth's mean long-. = 4s. 15 14' 19" 6, and long 1 , of the moon's node
= 4s. i9 23' 28" 6, the dilF. of which is 4 9' 9", hence there must he an
eclipse, because this diflT. is less than the limit given above. The ecliptic
limit being- found, as will be shewn afterwards, to which if the greatest dill'
of the true and mean places be applied, the above limit will be obtained.
j- Vince directs that for greater certainty the places of the sun and moon
may be computed again from the tables, and if they be not exactly in oppos.
which may happen not to be the cuse, as the moon's long-, does not increase
uniformly, the operation may be repeated. This accuracy is however gen r
crally unnecessary in eclipses, unless where very great accuracy is required',*
31
450
OF SOLAR AND LUNAR ECLIPSES,
LS (diff. hot*, mot. in long.) : Sb (moon's hor. mot. in lat.) :: rad.
: tang. 6LS the incl. of the rel orbit, and cos. 6LS : rad. :: LS
: L6, the hor. mot. in the relative orbit. The moon's hor. paral-
lax, \\Qvsemidiam. and the semidiam of the sun, the hor. parallax
of which may be here taken = 9", must also be found from the.
tables at the time of oppos.
Tojind the semidiam. of the earth's shadow at the moon seen from
the earth. Let AB be the sun's
diam. TR the diam. of the earth,
O and C the centres ; let AT, BR, ]
be produced to meet at I, and draw
OCI ; let FGH be the diam. of the
earth's shadow at the dist. of the moon, and join OT, CF. Now
the L FCG = CFA CIA (32 Eucl. 1) but CIA == OTA
TOC ; hence FfG = CFA OTA -f TOG ; that is the angle
under which the semidiam. of the earth's shadow, at tlie raoon^ ap-
pear -Sj is equal the sum of the horizontal parallaxes of the sun and
moon less the sun's appar. semidiam. From the earth's atmos-
phere, the shadow, in lunar eclipses, is found to be a little greater
than this rule gives it. According to M. Cassini the augmen. is
20" ; according to M. Monnier 30", and to M. de la Hire 60".
Mayer makes the correction ^ of the semidiam. of the shadow.
Some computers always add 50", but this must be subject to un-
certainty.
The L. CIT (= OTA TOC) being known, we have sin.
TIC : cos. TIC :: TC : CI the length 'of the earth's shadow.
If the sun's mean semid. or the L ATO be taken = 16' 3 f/ , his
hor. parallax TOC = a", we have CIT = 15' 54" ; hence sin.
15' 54" : cos. 15' 54", or 1 : 216,2 :: TC : CI = 216,2 TC.
The different eclipses which may happen of the moon is thus
explained by differ-
ent authors. Let M
CL represent the &\ M
plane of the ecliptic,
OR the moon's or-
bit cutting the eclip-
tic in the node N ;
and let SH repre-
sent a section of the earth's shadow, at the dist. of the moon from
the earth, and M the moon when she passes nearest the centre of
the earth's shadow. Hence if the oppos. happen as in pos. I, the
moon will touch the earth's shadow, without entering it, and hence
there will be no eclipse. In posit. II, a part of the moon will pass
through the earth's shadow, and there will be a partial eclipse.
In posit. Ill, the whole of the moon passes through the earth's
shadow, and there is a total eclipse. In posit. IV, the moon's
centre passes through that of the shadow, and there is a total and
central eclipse. Hence it is evident, that whether an eclipse will
happen at the time of oppos. or not, depends upon the moon's
OF SOLAR AND LUNAR ECLIPSES.
451
dist. from the node at that time ; or the dist. of the earth's shad-
ow, or of the earth, from the node In lunar eclipses the angle at
N may be taken = 5 17', and in posit. I the value of EV, according
to Fince, is about 1 3' 30" ; hence sin. 5 17' : rad. :: sin. 1 3'
SO" : sine QN 11 34' ; when EN is therefore greater than
QN at oppos. there can be no eclipse. This quantity 1 1 34' is
called the edi/itic limit.
Let ArBA be that half of the earth's shadow which the moon passes
through, NL the moon's
relative orbit ; let Cmr be
drawn perp. to NL. and
let z be the centre of the
moon at the beginning of
the eclipse, m at the mid-
dle, x at the end ; also
let AB be the ecliptic,
B
and Cn perp. to it. Now in the rt. angled triangle Cnm, Cn the
moon's lat. at the time of the eclip. conj. is given (as shewn in the
beginning) and the L. Cnm, the comp. of the angle which the
moon's rel. orbit makes with the ecliptic ; hence rad : cos. Cnm
:: Cn : rim, which is called the reduction ; and rad. : sin. Cnm
:: Cn : Cm. The moon's horary motion (A) in her rel. orbit be-
ing known, the time of describing mn is thus found ; h : mn r.
Hi. : the time of descr. mn t The time of the ecliptic conj. at n
being known, we therefore know the time of the middle of the
eclifise at m. Again, in the rt. angled tri. Cwz, O?, and Cr, the
sum of the semidiameters of the earth's shadow and the moon are
given : hence mz is given. (47 Eucl. 1.) For mz =* v/Cz 2 Cm*
= (Cz -f Cm x Cz C?w)^, and therefore log. mz = J x log.
(Cz -f Cm + log. Cz Cm.) The moon's lior. mot. being there-
fore known, we know the time of describing Z7n, which subtracted
from the time at m y gives the time of the beginning, and added,
the time of the end of the eclipse. The magn. of the eclipse at
the middle is represented by tr ; which is the greatest dist. of the
moon within the earth's shadow, and is measured in terms of the
moon's diam. conceived to be divided into 12 parts called Digits^
or Paris drjicient ; to find which the diff. between Cm and Cr
gives rar, which added to mf, or if m fall without the shadow, the
diff. between mr and mt^ and we have tr ; hence to find the digits
eclijised we have mt : tr :: 6 digits 360' (the digits being usually
divided into 60 equal parts, and these parts called minutes) : the,
digits eclijised. If the moon's lat- be N. the ufifier semicircle is
used ; if S, the lower. And in the fig. if the moon at n have N.
or S. lat. increasing, the L. Cn?n is to be set off to the right ;
otherwise to the left of Cn.
If the earth had no atmosphere, the moon would be invisible
when totally eclipsed ; but from the refraction of the atmosphere,
some rays will fall on the moon's surface, upon which account the
452 OF SOLAR AND LUNAR ECLIPSES
moon will be visible at that time, and appear of a red, dusky co-
lour. The earth's umbra in general, at a certain dist. is divided
by a kind of penumbra from this refraction. And hence in some
total eclipses the moon will be more visible than in otuers.
An eclip.se of the sun is caused by the interposition of the moon
between the sun and the spectator, or by the shadow of the moon
falling on the earth at the place of the observer ; for it is only in
the conjunction of the sun and moon that we can observe a solar
eclipse. Let the sun and moon be observed in the same straight
line with the eye of the observer, he will then see the sun eclips-
ed, arid if the appar. diameter of the moon be greater than that of
the sun, the eclipse will be total; but if less, aluminous ring will
be seen, formed by that part of the sun's disk which extends be-
yond the disk of the moon, in which case the eclipse will be an-
nular. If the moon be not in the rt. line which joins the centre of
the sun and the observer, the moon may then conceal only a part
of the solar disk, and hence the eclipse will be partial, Thus the
circumstances of a solar eclipse is subject to great variety, as well
from the difference in the distances of the sun and moon, and the
proximity of the moon to the node, as from the elevation of the
moon above the horizon, which changes the angle under v uich
her appar. diam. is seen, and which by the effect of the lunar par-
allax, may so augment cr diminish the appar. distances of the sun
and moon, that an eclipse of the sun which is visible to one ob-
server, may be totally invisible to another. The length of a solar
eclipse is also affected by the earth's rotation about its axis. M. du
Sejour determines that an eclipse can never be annular longer
than 12' 24". nor total longer than 7' 58".
A total eclipse of the sun is thus described by Laplace. We
often see the shadow of a cloud transported by the winds, rapidly
pass over the hills and valleys, depriving those spectators which
it reaches of the light of the sun, which others are enjoying ; this
is the exact image of a total eclipse of the sun ; a profound night,
which under favourable circumstances may last five minutes, ac-
companies these eclipses ; the sudden disapparition of the sun t
with the sudden darkness that succeeds, fills all animals with
dread ; the stars which had been effaced by the light of day, shew
themselves in their full lustre, and the heavens resemble the
most profound night. Dr. Halley in his remarks on the total
eclipse of the sun which happened on April 22, 1715, says, that a
few seconds before the sun was totally obscured, he observed round
the moon a luminous ring, about a digit, or perhaps a tenth part
of the moon's diam. in breadth : that it was of a pale whiteness, or
rather pearl colour, and seemed a little tinged with the colour of
the iris, and to be concentric with the moon ; and hence he con-
cluded that it was the moon's atmosphere. But, says he, the great
height of it far exceeding that of the earth's atmosphere ; and the
observations of some one who found the breadth of the ring to in-
crease on the west side of the moon as the emersion approached.
OF SOLAR AND LUNAR ECLIPSES.
453
together with the contrary sentiments of those whose judgment I
shall always revere, make me less confident, especially in a mat-
ter to which I paid not all the attention requisite. Lafilace there-
fore concludes, that it must be the solar atmosphere, its extent
not agreeing with that of the moon, as we are assured from the
eclipses of \he sun and stars that the lunar atmosphere is nearly
insensible, if any.
The different eclipses of the sun may be thus explained. Let
CL be the orbit of
the earth, OR the Q-
line described by
the centres of the
moon's umbra and
penu. at the earth ;
N the moon's node,
SF the earth, E its
centre ; fin the
moon's penumbra,
u the umbra. Then in pos. I, there will be no eclipse, as no
part of the earth enters into the penumbra. In pos. II, the pen-
umbra fin falls upon the earth, but the umbra u does not ; there
will, therefore, be no total eclipse, but there will be a partial
eclipse where the penumbra falls. In pos. Ill there will be both
a partial and total eclipse, as the umbra and penumbra both fall
on the earth ; where the umbra w falls, the eclipse will be total;
where only the penumbra falls, it will be but partial, and where
neither falls, there will be no eclipse. Now we may find the
ecliptic limit thus ; the" /_ N may be taken = 5 17', and in pos.
I, the value of Ew (u being the centre of the umbra) is about 1
34' 27"; hence sin. 5 17' : rad. :: sine 1 34?' 27" : sine EN
17^ 2 1' 27" the ecliptic limit; hence if the earth be within
this dist of the node at the time of conjunction, there will be no
eclipse.
We may calculate a solar eclipse, or rather an eclipse of the
earth, without respect to any particular place, in the same manner
as a lunar eclipse, that is, the times when the moon's umbra or
penumbra first touches and leaves the earth ; but to obtain the
times of the beginning, middle, and end, at any particular place, is
attended with more difficulty, as the apparent place of the moon,
as seen from thence must be determined, and hence the parallax
in lat- and long, must be computed, which renders the calculation
of a solar eclipse extremely long and tedious.
To calculate the eclipse of the sun for any particular place, the
first oper is to determine that there will be an eclipse somewhere
upon the earth, or that the earth at the time of conj. is not further
than 17 21' 27" from the node ; the true long, of the sun and
moon must be computed, by the astrom. tables, and the moon's
true lat. at the time of mean conj. (determined as shewn before.)
The horary mot, of the sun and moon in long, and the moon's her
454 OF SOL AH AND LUNAR ECLIPSES.
mot. in lat. must also be found in the same manner as the time of
the ecliptic oppos. was computed. At the time of the ecliptic conj.
let the sun and moon 'slat, be computed (as shewn in the beginning
of this ch.) and also the moon's lat. let the horizontal parallax of
the moon be also found, from the tables of the moon's mot. from
which let the sun's horiz. parallax be subtracted, the rem. is the
horiz. parallax of the moon from the sun.
Let the moon's parallax in lat. and long, from the sun, be com-
puted (pa. 331) at the time of the ecliptic conj to the lat. of the
given place, and the moon's horiz. parallax from the sun ;* the
paral. in lat. applied to the true lat. gives the apparent lat. (L) of
the moon from the sun, and the paral. in long, shews the appar.
diff. (D) of the long, of the sun and moon.
Let S be the sun, C'E the w
ecliptic, according to the or-
der of the signs ; let SM =
D, and MN be perp. to MS
and = L ; then N is the
moon's appar. place, and SN
\
(D
= moons
appar. dist. from the sun If the moon be east of the nonages deg.
the parallel increases the long, if ivest^ it diminishes it (see pa. 331)
hence if the true long, of the sun and moon be equal, in the former
case, the appar. place will be from S towards E, in the latter to-
wards C.
Find the true long, of the sun and moon, and also the moon's true
lat from their horary motions, for some time, as an hour after
the true conj. if the moon be to the west of the nonagesimal deg.
or before, if east ; and let the parallax in lat. and long, from the
sun be found ; the parallax in lat. being applied to the true lat.
gives the appar. lat. (/) of the moon from the sun The appar.
dist. (d} of the moon from the sun in long, is also found by taking
the cliff between the sun and moon's true long, and applying the
parallax in long. Now from S let SP be taken = rf, and draw PQ
perp. to EC and equal to /, Q will then be the moon's appar. place
Ih. from the true conj. and SQ = (d 2 -f / 2 ) = the moon's appar.
dist. from the sun ; hence the it. line NQ being drawn, will repre-
sent the relative afifiar. path of the moon,f and its value will also
represent the rel. horary mot. of the moon in the appar. orbit, the
rel. mot. in long being = MP.
The appar. hor. mot. (r) in long, of the moon from the sun is
found from the din , between the moon's appar. dist. in long, from
the sun at the time of the ecliptic conj. and at the interval of an
* The horizontal parallax of the moon from the sun is here used, instead
of the moon's horiz. paral. in order to determine \vliat effect the parallax
lias in varying their appar. relative situations.
j The small portion of this path here considered is taken as a straight
line, it being 1 in general very nearly so.
OF SOLAR AND LUNAR ECLIPSES. x 455
hour ; and the moon's appar dist. in long, from the sun at the true
time of the ecliptic conj is = the diff. (D) between the true long,
at the ecliptic conj. and the moon's appar. long, hence, r : D ::
Ih. : the time from the true to the appar. conj the time of the ap-
par. conj. is therefore given. To find whether this time be accu-
rate, let the true longitudes of the sun and moon (from their hora-
ry motions) be computed, and also the moon's paral in long, from
the sun, which applied to the true long, gives the appar. long, if
this be the same as the sun's long, the time of appar. conj. was
rightly determined ; if they do not agree, the true time must be
found from thence as before. For the true lime of appar. conj.
let the moon's true lat. from its horary mot. and her paral. in lat.
be found, from which her appar. lat. at the time of the appar. conj.
is obtained. Let SA be drawn perp to CE and equal to this appar*
lat. then as the point A will not probably fall in NQ, let us suppose
it to fall in QN, to which let SB be perp and draw NR parallel to
PM. Then NR (= PM) and QR (= QP MN) being given,
we have by trig. NR : RQ :: rad. : tang. QNR, or ASB ; and
sin. QNR : rad. :: QR : QN. The time of describing NQ in
the appar. orbit, being = the time from M to P in long. QN is
therefore the hor. mot. in the orbit. Moreover, rad. : sin. ASB ::
AS : AB and rad. : cos. ASB :: AS : SB.
The moon appears at A at the appar. conj. the time of which is
known from the preceding article ; when the moon appears at B, she
is then at her nearest dist. from the sun, and the time corresponding
is therefore that of the greatest obscur. or the time of the middle of
the eclipse.* Now the quantity of the eclipse ; its beginning, and
end, are thus found. The mot. being considered uniform, it will
be, QN : AB :: time of describing NQ : time of describing AB,
which added to or subtracted from the time at A (according as the
appar. lat. is decreasing or increasing) will give the time of the
greatest obscuration.
To find the digits eclipsed ; take BS from the sum of the appar.
semid. of the sun and moon, and the rem. will shew how much of
the sun is covered by the moon, or the parts deficient ; hence sun's
semid. : parts deficient :: 6 digits : the digits eclipsed: If SB
be less than the diff. of the semid. of the sun and moon, and the
moon's semid. greater than the sun's, the eclipse will be total ; if
the moon's semid. be less, the eclipse will be annular, the edge of
the sun appearing like a ring round the moon's disk ; but if B and
S coincide, the eclipse will be central.
Let QN be produced if necessary, and let S V, S W = the sum
of the appar. semid. of the sun and moon, at the beginning and end
of the eclipse respectively ; then BV = (SV 2 SB 3 )3, and BW
= (SW 2 SB 2 )i j to find the times of describing those we have
NQ r BV :: Ih. : the time of describing BV ; and NQ : BW ::
Ih. : the time of descr. BW, which times respectively subtr. from
* This is provided there be &u eclipsp, which will always be the case,
when SB is less than the sum of the appar. semidiamelers of the sun ami moon.
456 OF SOLAR AND LUNAR ECLIPSES.
and added to the time of the greatest obscur. will give the times of
the beginning and end nearly. A different method must however
be adopted where accuracy is required ; for in supposing VVV to
be a straight line, there will arise errors too considerable to be neg-
lected. It will however serve as a rule to assume the beg and end.
It therefore follows that the time of the greatest obscur. at B is
not necessarily equidistant from the beg and end. If the eclipse
be total, let SV, S W, be taken equal the diff. of the semid. of the
sun and moon, then BV = B W = (SW 2 SB 2 )i, from whence
the times of describing BV, BW, may be found as before ; these
times may be considered as equal, and if applied to the time of the
middle of the eclipse, or gr. obscur. will give the beg. and end of
total darkness.
To determine the time of the beg. and end of the eclipse more
accurately, we must proceed thus. From the horary motions, and
computed parallaxes, let the appar. lat. VD of the moon be found
at the estimated time of the beg and also her appar. long. DS from
the sun, and we get SV = (SD 2 -f DV*M ; if this be equal to the
appar. semid. of the moon, added to that of the sun (which sum
call S) the estimated time is the beginning ; if it be not equal, let
another time be assumed (as the error directs) at a small interval
from it, before if SV be Less than S, but after if greater ; let the
moon's appar. lat. mv, and appar. long. Sm from the sun, be again
computed for that time, and we find Sv = (Sm 2 -f .v 2 )$, which
if not = S, say, diff. of Sv and SV : diff. St/ and SL (~=~ S) :: the
assumed interval, or time of mot. through Vz> : the time through
-i>L, which added to or subtr. from the time at -y, according as Si>
is greater or less than SL, will give the time at the beginning.* In
the same manner the end of the eclipse may be computed.
* The reason of this oper. is, that as Vr, vL are very small, they will be
very nearly prop, to the diff'. of SV, St; } and Sr, SL. But the var. of the ap-
par. dist. of the sun and moon, not being- exactly proportional to the var. of
the diff'. of the appar. long-, and lat. where great accuracy is required, the
time of the beg-, thus found (if not correct) may be corrected by assuming- it
for a 3d time and proceeding* as before. This correction is however never
necessary, unless where extreme accuracy is required in order to deduce
some consequences from it. But the time thus found is to be considered as
accurate, only so far as the tables of the sun and moon can be depended on
for their accuracy ; the lunar tables of Mr. Burg and solar tables of JJelam-
6re, republished and corrected by Vince, are, as before remarked, by far the
most correct. If however there remains any error in the tables, and some small
errors are unavoidable, accurate observations of an eclipse compared with
the computed time, furnish the best means of correcting- the lunar tables.
The above directions principally collected from Vince will, together with
g-ood tables, which the young- astronomer should always be furnished with,
be found sufficient in calculations of this nature. For examples, &c and
more information, this author may be consulted. See also Rees Cyclopedia,
the Philadelphia ed. of the Encyc. art. Astr. Lead better, E-winy, Ferguson^
and other practical writers on this subject. Dr. Gregory treats this subject
at larg-e in B. 4. of his Astronomy, and also Kiel in his Astr. Lectures 11,
12, 13 and 14, where besides the calcul. he gives various graphical ir.eth.-
ods, for computing both solar and lunar eclipses, &c.
OF SOLAR AND LUNAR ECLIPSES. 457
*The duration of an eclipse of the sun can never exceed 2 hours ;
nor of the moon, from the first touching the earth until her leaving
it, cannot exceed 5^ hours. The moon cannot remain in the earth's
umbra longer than 3| h, in any eclipse, nor be totally eclipsed for a
longer period than 1| hours. (Emerson's ast sec. 7, pa. 347 &c 339.)
If a conj. of the sun or moon happen at, or very near the node>
there will then be a great solar eclipse ; but in this case, at the
preceding oppos. the earth was not within the lunar ecliptic limits,
and next oppos it will be beyond it ; hence it may happen that at
each node there may be but one solar eclipse, and therefore in a
year there may happen but two ; and this is the least that can
happen in a year, as there must be one conj. in the time in which
the earth passes through the solar eclip. limits, and hence there
must be one solar eclipse at each node. If there be an oppos im-
mediately before the earth enters the ecliptic limit, the next may
not happen until the earth is beyond the limit on the other side of
the node ; hence there may not be a lunar eclipse at the node,
and not therefore in the course of a year There can be at most
but three lunar eclipses in a year ; for when there is a lunar eclipse^
as soon as the sun gets within the lunar eclip lim. it will be out
of this lim. before the next oppos. and hence there can be but one
lunar eclipse at each node ; but as the moon's nodes have a retro-
grade mot. of about 19-^ in a year (see ch. 4) the earth may come
again within the lunar eclip. lim. at the same node in the course of a
year. There may happen at each node two eclipses of the sun and
one of the moon ; for when a lunar eclipse happen at, or near the
node, a conj. may take place before and after, while the earth is
within the solar ecliptic limits ; the eclipses of the sun in this
case will be small, and that of the moon large. Hence when the
eclipses do not happen a second time at either node, there may be
six eclipses in a year, four of which will be of the sun, and two of
the moon. But if, as in the last case, an eclipse happen at the
same node a second time in a year, there may be six eclipses,
three of the sun and three of the moon. These six may take place
during 12 lunations or 354 days, or 1 1 days less than a common
year ; hence if an eclipse of the sun should happen before Jan. 1 1,
and the last cases should also take place, there may be seven eclip-
ses in a year, five of the sun and two of the moon j but there can
be no more ; the mean number is however four, and seven can
seldom happen.
As the earth describes 19J- in about 19J days, hence the mid-
dle of the seasons of the eclipses is about 1 9 days sooner each year
than the preceding. The solar ecliptic limits being greater than
the lunar, in the ratio of 17 21' 27" to 11 34' (as shewn before)
or nearly of 3 : 2, there will therefore be more solar than lunar
eclipses in about the same proportion ; but, as a lunar eclipse is
visible to a whole hemisphere at once, and a solar only to a part,
there is greater probability of seeing a lunar than a solar eclipse,
and hence more lunar than solar eclipses are seen at any place^
3 K
458 OF THE TIDES.
CHAP. XIII.
OF THE TIDES.*
A TIDE is that motion of the waters in the seas and rivera, t>y
which they are found to rise and fall in regular succession. Newton
shews (Prop. 66, b. I, cor. 9 and 20 ; and prop. 24, b. 3) that the
waters of the sea ought to rise and fall twice every day, as well lu-
nar as solar, from the combined attraction of the sun and moon.
In their conjunction or oppos. their forces being joined, or acting
on the earth in the same straight line, will produce the greatest
flood and ebb, and these tides are called sfiring tides, In the qua-
dratures, or about the time of the first and last quarters of the
moon, the sun's action will tend to raise the waters which the moon
depresses, and depress those which she raises, and hence from the
diff. of their actions the least tide will follow : these are called neaji
tides. And as there is an oppos and conj. with the sun once in every
lunation, there will be two spring, and two neap tides, in that period.
A mean lunation or synodic rev. is 29d. 12h. 44' 2"8 (pa. 325) the
mean retardation of the tides, or of the moon's coming to the merid.
in 24h. is therefore 48' 45"7, hence the interval between two suc-
cessive tides is I2h. 25 ; 14"2, and the daily retardation of high
water is 50' 28"4, at a medium. But this retardation is consider-
ably altered from the variation in the respective dist. of the sun and
moon, and their different declinations, as also the change of the lat.
of the place. (See Prin. b. 3, prop. 26, or McKay's pr. Nav pa. 18.)
Newton remarks that there will also arise some variation from the
force of reciprocation, which the waters retain after being put in
motion, &c.
Laplace makes the mean interval of the return of the tides, be-
tween two consecutive returns of the moon to the same meridian,
equal 1.035050 days, or id. Oh. 50' 28"32. The mean value of a
total tide (or half the sum of the heights of two successive high
tides above the level of the intermediate low tide) at Brest is 5.888
metres =* 19.318528 feet at its maximum about the syzygies, and
* We have sometimes in the preceding parts of this work referred to
this chap, as if given after the laws of motion, &c. this was our intention, as a
subject so interesting 1 merited a full investigation, and that this investiga-
tion could not be entered into without previously laying down the principles
of gravity, &c. on which it depends. But our time at present not being-
sufficient to discuss a subject, the most intricate in physical astronomy, and
the work already swelled beyond our intended plan, we have only inserted
extracts from Newton, Laplace and others, sufficient to give the learner a
comprehensive idea of this phenomenon. Convinced that the knowledge
which only touches at the surface can be attended with no real utility, and
can only nourish that vanity, too common in the present age, of appearing
learned in matters of which we know nothing, we have without deviating-
from the simplicity of an elementary introduction, all along joined the the
ory with the practical part, and entered as deep into each subject, as the
nature of a contracted School book would permit.
OF THE TIDES. 459
sJ.789 me. = 9.150709 feet, from whence Lafilace concludes that
the mean lunar tide, which corresponds to the constant part of the
parallax of the moon, is three times less than the mean solar tide,
or in other words, that the action of the moon to elevate the waters
of the ocean, is three times as great as that of the sun* The height
of the tides, all other circumstances remaining the same, augment
and diminish with the lunar parallax, but in a greater ratio ; so that
if this paral. increase -fa, the total tide will increase | in the syzy-
gies, and \ in the quadratures ; and as the tide is nearly twice as
great in the first as in the second case, its increase in the two cases
is the same.
The greatest var. in the moon's diam. being about T T T of the
whole, the corresponding var of the total tide in the syz. is 5 3 V of
its mean height, or about 2.997 feet : thus the entire effect of the
hange of dist. between the earth and moon is (1.766 me.) 3.794
feet nearly. The var. in the sun's dist. influences the tides in a
much less degree.
The decl. of the sun and moon diminish the total tides of the
syz. At Brest the dimin. is (08 me.) 2.62 feet nearly, less in the
solstices than at the equinoxes ; and in the quadratures, they are
less by the same quantity in the equinoxes than at the solst The
greatest tide at Brest follows the syz. about 1 days, or is the 3d
afte syz. and the dimin of the total tides that are near it, is propor-
tional to the squares of the times elapsed from that instant to the
time of the intermediate low tide, to which the total tide is referred,
it is (0.1064 me.) 0.349 feet nearly, when this interval is a lunar
day. The following var. from the decl. of the sun and moon are
also found at Brest. In the syz of the sum. solstice the morning
tides the 1 st and 2d day alter it, are less than the evening tides by
(0.183 me.) or .6 feet nearly. They are gr. by the same quantity
in the syz. of the winter solstice. In the quadratures of the au-
tumnal equinox, the morning tides the 1 st and 2d day after, exceed
the evening tides by (0.131 me ) 0.429 feet, and are less by the
same quantity in the quadr. of the ver. equinox.
* The power of a celestial body to raise a particle of water placed be-
tween it and the centre of the earth, is equal the difT". of its action on the
centre and on the particle ; let b be put for the mass of the body, 7-the se-
mid of the earth, d = the dist. between the body and the earth, then the
above diiT. = br~- d* = relatively to the sun, the one hundred and seventy-
ninth part of the force of gravity acting 1 on the moon mult, by the propor-
tion of the terrestrial rad. to the moon's dist. this force of gravity is nearly
= sum of the masses of the earth and moon div. by the sq. of the lunar dist.
hence the power of the sun to raise the waters of the sea is 89 times less
than the sum of the masses of the earth and moon mult, by r and div. by
the cube of the lunar dist. this force, as shewn above., being- only f that of
the moon, which is equal to double its mass mult, by r and div by the cube
of its dist. thus the mass of the moon is to the sum of the masses of the earth
and moon, as 3 : 179 ; from whence it follows, that this mass is very near-
ly TIT f that of the earth, its volume being only -jij/jy^ of the earth's, its
density is 0.8401, that of the earth being 1 ; and the weight which on the
earth's surface is 1, would on the moon's surface be reduced to 0.2291.
bF THE TIDES.
The interval of the tides offer other phenomena. At Brest the
high tide the moment of syz. follows midday or midnight at
0. 14822d. or 3h 33' ^6"^, this is called the hour ofthe fiort^ and
differs in different harbours At quadr. the high tide in Brest
follows midnight or midday at 0.35464 days, or 8h. 30' 40" 9.
The tide of the syz. advances or retards (264") 3' 48", for every
hour that it precedes or follows the syz and the tide of the quadr.
advances or retards (4 16") 5' 59"4 for every hour before or after
the quadr. In the syz also, every minute of increase or dimin.
in the moon's appar. semid advances or retards the hour of high
water (354") 5' 6" 6 ; this phenomenon is three times less at the
quadr.
From the sun and moon's decl. the time of high water advances
about (2') 2' 52"8 in the syz of the solstices, and is equally retard-
ed in the syz of the equinoxes, but in the quadr. oftheequin.
high water advances (8') 11' 3 i"2, and is equally retarded in the
quadr. of the solstices.
The daily retardation of the tides varies also with the phases of
the moon i it is a minimum at the syz, when the total tides is at
their max. and is only 0.2705d. or 38' 57"l2, when the tides
are at their min it is then greatest, and amounts to 0.05 207d or
Ih 14' 58"8, Thus the diff. 20642d. (035464 0.14822) of
the times of high water at the syz. and quadr. increases, for the
tides which follow in the same manner these two phases, and be-
come nearly a quarter of a day relatively to the max. and min. of
the tides
Every minute of incr. or dimin in the moon's appar. semid aug-
ments this daily retardation (258") 3' 42"9 about the syz and is
three times less in quadr. From the var. of the sun and moon's
decl it varies likewise in the syz. of the solstices about (155") 2'
1 3 "9 greater than in its mean state, and equally less in the syz. of
the equinoxes ; on the contrary in the quadr of the equin. it ex-
ceeds the mean by (543") 7' 4'/'l, and is surpassed by this quan-
tity in the quadr of the solstices Hence the var. in the heights
and intervals of the tides have verv diff. periods ; some of half a
day, and a day, others of half a month, a month, half a year, and of
a year ; others again vary with the rev. of the nodes and the perigee
of the lunar orbit, as they vary the decl and dist. from the earth.
All these phenomena appear to have been the same in the new as
in t<ht full moon.
These phenomena equally take place in all the harbours along
the sea sfyore ; but local circumstances, without changing the laws
of the tides,, have a considerable influence in changing the heights
of the tides and the hour of high water for a given port.
JLaptace gives the following method of determining the time of
high wafer on any day Considering each of otir ports as the ex-
tremity of a canal at whose embouchure (its mouth or entrance) the
partial tides happen at the moment of the passage 01 the sun and
moon over the meridian, and employ a day and a halt to arrive at
OF THE TIDES. 461
ij;s extremity (supposed eastward of its embouchure) by a certain
number of hours, called the fundamental hour of the port, and may
be easily computed from the hour of the establishment of the port,
by considering this as the hour of the full tide, when it coincides
with the syz. The daily retardation of the tides being (2705") 38'
57" 1, it will be for 4 days = (395 ,") 56' 53"o, which quantity is
to be added to the hour of the establishment to have the funda-
mental hour Now if the hours of the tides at the embouchure be
augmented by (15 hours) 36h. plus the fundamental hours, we
shall have the hours of the corresponding tides in our ports. We
shall now make a few remarks on the cause of those interesting
phenomena.
As the action of gravity decreases as the sq. of the distance in-
creases, the waters that are on the side next the sun or moon, will
be more attracted, by them respectively, than the central parts of
the earth, and the central parts than the surface on the opposite
side ; therefore the distances between the centre of the earth and
the surface of the water under the Zenith and Nadir, by the laws
of attraction, will be increased ; for that part of the surface which
is nearest to the sun or moon, will move with greater vel. towards
those bodies, and that part that is more dist. with less vel. than
the centre ; from which it is evident, that high water must taka
place nearly at the same instant at opposite parts of the earth ; and
from the earth's mot. on its axis in 24h. there will be two tides of
jiriod and two of ebb in that time agreeable to experience. It ap-
pears from the foregoing explanation, that the fig. of the earth
caused by the tides, would be an oblate spheriod, having its longer
axis passing through the moon, on suppos. that the whole surface
was fluid.*
Laplace thus determines the law by which the waters rise and
fall. Let a vertical circle be conceived whose circumference rep-
resents half a day, and whose diam is equal to the whole tide or
the diff. between the height of high and low water, and let the arc
of this circum. from the lowest point express the time elapsed
since low water, the -versed sines of these arcs will express the
heights of the water corresponding to these, so that the ocean in
rising, covers, in equal times, equal arcs of this circumference. |
This law is exactly observed in the middle of the ocean, but in
our harbours local circumstances produce some deviation. The
sea is also found to employ a longer time to fall than rise ; this
cliff, is found at Brest about (10') 15' 7"2. The greater the
extent of the surface of the water, the more perceptible will be
the phenomena of the tides, the motion which is communicated to
a part of a fluid being communicated to the whole ; hence such
remarkable effects are produced in the ocean, and the waters com-
municating with it, which are insensible in lakes and small seas.
If we imagine at the bottom of tne sea a curved canal, terminated
* See Simpson's Fluxions, art. 403.
j- See Emerson's Fluxions, prob. 25, where the. jne^hod, of determining
the height of t|ie tides is investigated.
462 OF THE TIDES.
at one of its extremities by a vertical tube rising above the sur-
face of the water, and which if prolonged would pass through the
centre of the sun or moon, the water would rise in this tube by
the direct 'action of the sun or moon, which diminishes the gravity
of its particles, and particularly by the pressure of the panicles en-
closed in the canal, all of which make an effort to unite under the
sun and moon, and from the integral of all their efforts arise the
elev. of the water in the tube above its natural level. If we con-
ceive a similar canal communicating with the sea and extending
into the land, the undulations caused by the tides at its entrance,
will be propagated through its whole length, in the interval of half
a day ; but the hoin s will be retarded, in proportion as the points
are lu'ther from the entrance of the canal.* This reasoning may
be applied to rivers, whose surfaces rise and fall by similar waves,
notwithstanding their contrary motion. t
The solar and lunar tides do not happen at the same instant,
their periods being different ; hence the lunar tide will retard
upon a solar tide by the excels ol half a lunar day above half a solar
dc-y, that is (1752") 25' l3"7. See Laplace** astr. b. 4, ch. 10.
In further confirmation of the theory, it is found, that in Brest the
solar tide follows the passage of the sun (18358") 4h 24' 2 "3,
and the lunar the passage of the moon (13101") 3h 8 ; 39"2 ;
the greater (l' 26"4) tide following the syz. being nearly \\ days :
that (100") l' 2< ." * variat in the moon's semid. answers to half
a metre or 1.6405 feet of variat. in the total tide when the moon
is in the equinoctial : that (!'; l' 26"4 retards the tide as given
in the preceding theory (354") 5 ; 6"6 very nearly : that (!')
1' 26"4 var. in the moon's semid produces a var. of (258")
S' 42"y agreeable to obs The above is on suppos. that the sun
and moon move in the equinoctial ; the phenomena resulting from
their change ofdecl are also found to correspond with what is
given in the preceding theory. See this subject further detailed
in Lafilace, b 4, ch. 104
* See this investigated in prob. 20, Emerson's Flux, in the Schol. of which
lie shews, that it' a pendulum be made whose length is the breadth of a
wave from top to top, then in the time that it will perform one vibration, the
waves will advance forwards a space equal their breadth. See also New-
ton's prin. b. 2, prop. 46, 47, &c. Their velocity, according to Newton, is
in the subduplicate ratio of their breadths, li. 2, prop. 45.
f The action of the sun and moon is usually found separate, (~Principia>
b. 3, prop. 36 and 37) which by the composition of forces (see the next
chap.) are combined, and from the resulting force result the tides, which
are observed in our ports.
+ We have, according- to the remark pa. 308, changed Laplace's measures
of time, in all this ch. allowing- 10 hours to the day, 100' to an hour, 100"
to a minute, &c. And we are further confirmed in this opinion, not only
from the result agreeing- with the known established hour in other ports,
but also from Laplace's augmenting the. hours of the tides at embouchure by
15 hours to find the time of high water, as this corresponds to 36 hours* of
our measures, or to 1 flays, agreeable to theory. We have before remark-
ed, that this was not among the least considerable errors in Mr. Pond\
translation of Laplace's Astr.
OF THE GENERAL LAWS OF MOTION, &c. 4G3
CHAP. XIV.
OF THE GENERAL LAWS OF
MOTION, FORCES, GRAVITY, &c.
THESE laws being necessary in understanding those in the solar
system, which we have given in the preceding chapters, and not
only the foundation of Physical Astronomy, but the very basis of
all Natural Philosophy, we have therefore given them a place in
this chapter But, as in the short compass of a single chap, it can-
not be expected that we should enter into the detail of these laws,
we shall therefore only insert what is necessary to give a general
knowledge of the motions of bodies, particularly those in the Solar
System, ^first premising the rules of reasoning in philosophy as
delivered in the 3d. b. of the Principia.
Rules of reasoning in Philosophy.*
1 . We are to admit no more causes of natural things, than such
as are both true and sufficient to explain their appearances.
2. Therefore to the same effects we must as far as possible as-
sign the same causes.
3. The qualities which are found in all bodies upon which ex-
periments can be made, and which can neither be increased or di-
minished, are to be esteemed as belonging to all bodies.
4. In experimental philosophy, propositions collected from phe-
nomena by general induction, are to be admitted as accurately or
* In rule 3, Newton lays it down as a principle, that the properties of
matter cannot be known otherwise than by the senses, from which we know
that innumerable objects exist around us, and act upon us. Their powers,
properties, causes, &c. is an interesting subject for our contemplation, and
what is properly called philosophy. These objects may be divided into two
general classes ; the first is of those which have a self moving 1 power, and
several properties similar to those of our minds. The second is of those
which never move of themselves, without the action of some external or in-
visible object. The former is called Mind or Spirit, and the latter .Body
or Matter. The properties of matter are extension, figure, solidity, motion,
divisibility, gravity and vis inertia. These properties are therefore the foun-
dation of all Nat. Phil. Extension i's considered with regard to length,
breadth and depth, and Figure the boundary of extension. Solidity or impen-
etrability is that property of matter by which it fills space, or excludes any
other portion of matter from that space which it occupies. Motion is the
change of place. Divisibility of matter the capacity of being separated into
parts. Gravity is the force or tendency of a body to a centre, and Vis ln~
trtice the innate force by which it resists any change. Motion is also abso-
lute or relative ; absolute- when it is compared with a body at rest, and re-
lative when compared with others in mot. The rate of this motion is called
the Velocity of a body, causing the body to pass over a certain space in a
given time, and whatever produces or changes this motion is termed Force ,~
in moving bodies this is called their Momentum. The Density of a body is
the proportional weight or quantity of matter in it, and is proportional to the
specific Gravity which is the proportion of the weights of different bodies pi'
ftjuaJ. magnitude.
464 OF THE GENERAL LAWS OF MOTION, 8cc.
nearly true, until from other phenomena they are rendered more
accurate, or liable to exceptions.
Axioms, Laws, b'c.
5. Every body perseveres in its state of rest, or of its uniform
mot in a right line, unless compelled to change that state by other
forces impressed on it.
6. The alteration of motion, or the motion generated or des'
troyed, in any body, is proportional to the force applied ; and is
made in the direction of that straight line in which the force acts.
7. To every action, there is always opposed an equal reaction ;
or the mutual actions of two bodies upon each other, are always
equal and directed to contrary points.
8. The mass) or quantity of matter > or the -volume in all bodies^ is
in the compound ratio of their magnitudes and densities.
If b be put for the body, m its mag and d its density, 6 is as*
jnd. For if the bodies be equal, the mass is as the density ; and
if the densities be equal, the mass is as the mag but when neither
are equal, the mass is therefore in the compound ratio of the mag*
and density.
9. The volumes are als as the densities and cubes of the diam.
the magnitudes being in this proportion
10. The masses are also as the mag and specific gravities, the
density being as the spec gravity.
1 1 . The quantities of motion in moving bodies are in the comfiound
ratio of the masses and velocities. That is m is as b*v, where m repre-
sents the momentum and v the vel. For it the velocities be equal,
the quantities of mot will be as the quantities of matter, and if the
masses be equal, the momentum will be as the velocities, hence, Sec.
12 The momentum generated by any momentary force is as the
force. For every effect is proportional to its adequate cause.
13. In uniform motions the sfiaces arc as the "velocities and times
of description. For the vel. being the same, the spaces are as the
times, and the times being equal, the spaces are as the velocities ;
hence, &c.
14. From this art. it appears that in uniform motions the time
is as the space directly and vel. reciprocally, or as the space divi-
ded by the vel. and that the vel. is as the space directly and time
reciprocally, Sec. From art. 11, 12 and 13, many other general
proportions may be deduced (See Emerson's Mechanics, 4to.
page 8, or Mutton's Mathematics, vol. 2, pa 134.
15. The quantity of motion generated by a constant and uniform
force, is in the compound ratio uf the force and time of acting. For
if the time be divided into very small parts, the momentum in
each is the same (art. 12} the whole momentum will therefore
be as the sum of all the parts or the whole time ; but the momen-
* That is whatever variation is made in b, a proportional change will fcc
made in mil
OF THE GENERAL LAWS OF MOTION, &c. 465
turn for each time is also as the force (art. 12) hence the whole
momentum is in the compound ratio of the force and time.
1 6. From the preceding art. it is evident that the mot. lost or
destroyed in any time is also in the same proportion. And that
the vel. gener. or destroyed in any time, is as the force and lime
directly, and the body or mass reciprocally ; hence if the body or
force be given, the vel. will be as the time, Sec.*
17. The spaces passed over by bodies, urged by constant and uni-
form forces, are in the compound ratio of the forces and squares of
the times of acting directly, and the body or mass reciprocally. For
let b be put for the body, s the space passed over with the vel. v
in the time t. Then art. 16, -\ -u is the vel. at ~ t the middle of
the time ; hence the increase of vel. being uniform, * will be de-
scribed in the same lime t by the uniform vel. \ v, s is therefore
as ^ tv, or s = J tv ; but art. i 6, v is as ft , hence * or tv
is as ft* -r- b.
18. If b and/ be given, s is at ** ; hence art. 16, s is as tv, or
as v 2 . s or \ tv is the space actually described, that is half the
space that would be uniformly described with the last or greatest
vel. in the same time t From these general proportions, a table
of all the particular relations of uniformly accelerated forces may
be easily formed. See Hutton's Math pa. 136, vol. 2, or Emer-
son's Mechanics.
19. What is given in art. 17 and 18, hold equally true, for the
spaces passed over by bodies freely descending by their own gravi-
ty, this force being considered uniform at all places at. or at equal
distances from, the earth's surface It is found that in the lat of
London a body falls 16 T ^ feet in 1", and that (art. 18) at the end
of this time it has acquired a vel that would carry it over 32 feet
in \" ; hence if g = 16 T \- feet the space passed through in 1",
and 2g- the vel. generated in that time ; then art 16 and 17 we
have \" : t 2 ' ?g '- 2gt = v, and 1 : I 2 :: g : gt* = s the
space ; from which proportions if the different values of s, v, g, t,
be found in general, we shull obtain the general equations for the
descents of gravity, &c. Hence art. 16 and 18, if the time be as
the numbers 1, 2, 3, 4, 5, See. the velocities will be as the same,
and the spaces as their squares 1, 4, 9, 16, 25. &c. and the spaces
for each time the difference of these squares, or 1, 3, 5, 7, 9, &c.
20. These relations between the times, velocities, and spaces,
HI ay be represented by a rt angled triangle thus. If one side of
the triangle (/) represent the time, and the other side (v} perp. to it,
the vel. gained at the end of this time. Then if t be divided into
any number of equal parts, and through these points lines be drawn
parallel to the base to meet the hyp these will represent the ve-
locities in each corresponding interval of time, and by similar tri-
angles will be proportional to the former Also the area of the
triangle being = \ tv will therefore represent the space s passed
* In art. 8 and 9 vchime was inadvertently used for the mass, the vol. is
properly the size or mug-.
3 L
466 OF THE GENERAL LAWS OF MOTION, Sec.
over in the time t ; and the smaller triangles, for the same reason?
will represent the different spaces passed over in the corresponding
intervals of time, and their other sides the velocities. Now these
areas or spaces being as the squares of their sides, shews that is
as t z or T' 2 , as in art. ! 8.
21. The relations, Sec given in art. 20, may be more naturally
represented by the abscissas and ordinates of a parabola For if the
ordinates represent the respective times from the beginning, or the
velocities which are proportional to them, then the corresponding
abscissas drawn parallel to the axis of the parabola, will represent
the spaces described by a falling body in those times ; the abscis-
sas which represent the spaces, being as the squares of the ordi-
nates which represent the trme, by a well known property of the
parabola.
22. As motions are destroyed in the same manner as produced,
and by the same forces acting in contrary directions ; hence 1 . A
body thrown directly upwards, will lose equal velocities in equal
times. 2. If the body be projected upwards with the same vel.
acquired in falling, it will lose all its motion in the same time in
which it fell, and will have the same vel. in any point of the same
line both in ascending and descending. 3. The respective heights
ascended to, will be as the squares of the velocities with which they
wer,e projected, or as the squares of the times, until they lose all
their motion These properties are accurately true in Vacuo,
but near the earth's surface the resistance of the air, particularly
in very swift motions, has considerable effect in changing the ve-
locities, See.
23. If a body at A be acted upon by any two similar forces, so that
they would separately cause the body to pass over the spaces AB, AC
iri an equal time ; then if both forces act together, they ivill cause the
body to move, in the same time, through AD the diagonal of the pa-
rallelogram ABCD. Let cd, bd, be drawn parallel to AB, AC res-
pectively ; let t be the time in Ab or Ac, and
T the time in AB or AC ; then if fhe forces be . b_
impulsive or momentary, it will be Ab or cd : **
AB or CD :: t : T, and bd or Ac : BD or Ac ::
t : T ; therefore by equality (1 1 Eucl. 5) Ab :
bd :: AB : BD ; the parallelograms Abdc,
ABDC are therefore similar, and consequent-
ly (26 Eucl. 6) they are about the same diam.
hence d is in the diagonal AD And this may be shewn of any-
other point d, the path of the body is therefore in ArfD, the diag-
onal of the parallelogram.
If the forces be uniformly accelerated or retarded, the spaces
will then be as the squares of the times (art. 18) in which case A6
or cd : AB or CD :: t* : T 2 , and bd or Ac : BD or AC :: t 2 : T 2 ,
hence Ab : bd :: AB : BD as before.
24 From the preceding art. it appears, 1. That the diag. AD
by both forces, is described in the same time as AB, AC, by the
OF THE GENERAL LAWS OF MOTION, &c. 467
single forces impressed in these directions. 2. That the forces in
the directions AB, AC, AD, are as these lines respectively. 3. That
the single force AD is equal to the two AB, AC, and compounded
of them ; so that any single force as AD, may be resolved into
two or more forces by describing any parallelogram whose diag.-is
AD, and each of these may be resolved in like manner. This is
called the resolution of forces. 4. That any two or more forces
may be compounded into one, by reducing any two of them to one,
as in art. 23, and this again with the other ; the force resulting
will be that compounded of the whole, and hence this is called the
composition of forces. 5. And hence the effect of any given force
as AD in any other direction AB, CD, or AC, BD may be found,
being as those sides respectively 6. Hence also if the two forces
act in the same line, in the same or contrary directions, their sum
or diff. will be the resulting force, which will always act in the di-
rection of the greater, 7 From the same principle it will appear,
that if an elastic body impinge on a firm plane, the angle of inci-
dence will be found equal the angle of reflection, by resolving the
forces before and after the stroke, action and reaction being equal
and contrary.
25. The forces of bodies acting on others, may be found from
the same principle Thus the perp. force of AD or the body at
D, on CD, is as BD or AC, that is as sine Z- ADC the angfe of
incidence ; AD which represents the force being radius. For the
force AD may be resolved into AC, CD, the latter of which does
not act on the plane, being parallel to it.
From this art. it appears, 1 That the action of any force and in
any direction, is always perp. to the surface acted on. 2. That if
the plane acted on be not fixed, it will move after the stroke, in a
direction perpendicular to its surface.
26. If one of the forces AD be uniform^ and the other AH uni-
formly accelerated, as the force of gravity, the motion resulting from
these tivo forces will be in the curve AEFG of a parabola. For
If the body be projected from A in the direction
AD with an uniform vel. then art. 14, it would
describe the spaces AB, BC, CD, supposed equal,
in equal times, when not acted on by any other
force. Let BE, CF, DG, be drawn perp. to the
horizon, so as to represent the spaces the body
would fall through by the accelerated force, or "
force of gravity, in the same time that by the uniform force it de-
scribed AB, BC, CD ; hence by the composition of motion the
body will be in the points E, F, G respectively at the end of those
times ; therefore the real path of the body will be in the curve
A, E, F, G. But the spaces in AD are as the times art. 13, and the
spaces BF, CF, DG, as the squares of the times art. 18 ; hence
AB, BC, &c. are as BE 2 , CF 2 , &c. which is the property of the
parabola; therefore the projectile will move in the curve of the
parabola. This demonstration holds whether AD be parallel to
ihe horizon or in an oblique direction.
468 6F THE GENERAL LAWS OF MOTION, Sec.
27 If three forces acting together in the same plane^ keep owe
another in equilibria, they will be proportional to the three sides DE,
C, CD of a triangle which are drawn parallel to the directions of
the forces \D, DB, CD
Let AD, BD be produced, and CF, CE drawn
parallel to them ; then by suppos th, force in
CD is equal to the two in AD, BD ; but the
force in CD is also equal to the two represented
by ED, Ch or FD ; hence if CD represent the
force C, ED, FD will represent the forces A
and B ; therefore the three forces A, B, C, are
proportional to the three lines DE, CE, DC,
parallel to the directions in which they act.
From this it follows, 1. That the three forces, v/hen in equili-
brio, are proportional to the sines of the angles of the triangle
formed by their lines of direction. 2 That these three forces are
also proportional to the three sides of any other triangle drawn
perp to their lines of direction, or forming any angle with them ;
for this triangle will be similar to that whose sides are parallel to
the lines of direction. 3. That if any number of forces acting
against one another, be kept in equilibrio by these actions, they
may be all reduced to two equal and opposite ones. For any two
of the forces may, by composition, be reduced to one acting in the
same plane ; and this last force and any other may likewise be re-
duced to one force acting in the plane of these ; and so on until
they are all reduced to two equal and opposite forces.
28 //' a heavy body or weight W, be sustained on an inclined
plane AB, by a power P, acting in a direction WP parallel to the
plane , then if AC represent the weight VV , BC will represent the
power P, and the base AC the pressure (p) against the plane.
Let < D be drawn perp to AB ; then the
weight W, or force o1 gravity acting perp.
to \C or parallel to BC, the power P acting
parallel to DB, and the pressure p acting
perp to AB or paral. to DC ; will be to one
another as BC, BD, CD respectively (art.
27 that is, from similar triangles as AB,
BC, AC, or as W, P, p
Hence W, P, /z, are as rad. sine and cos BAC the plane's ele-
vation ; or as AC, CD, AD perp. to their directions ; hence
also the relative weight down BA equal \V x BC -f- AB
29 When the pow.er P acts in any other direction as W/z, let
CFrf be perp. to W/? ; then W, P, p are as AC, CF, AF perp.
to the direction of those forces.
30. If a heavy body descend freely down an inclined plane AB,
its velocity in any time, is to the vel. of a body falling perpendicular'
ly, in the same time y as BC the height of the plane > to AB its length.
For the force ot gravity in the directions BA and BC is constant,
and art. ii8, as those sides ; but art. 17, the velocities are as the
OF THE GENERAL LAWS OF MOTION, &c. 469
forces, that is, in the same time, as the force on BA to that on
BC, or as BC : BA.
3 1 . From the preceding art it follows, 1 . That, as the mot.
down an inclined plane is uniformly accelerated, being produced by
a constant force, the laws which are given for accelerated forces in
general, hold equally for motions on inclined planes ; that is, vel.
are as the times ; the spaces as the sq. of the times or velocities ; if
the body be thrown upwards on the incl plane, it will lose its mot.
and ascend to the same height in the same time, &c 2. That the
space descended on the incl. plane, is to the perp. descent in the
same time as CB : AB, or as sine of the plane's incl. to rad 3. That
the velocities on inclined planes are as the sines of their elevations.
4. That if CD be perp. to AB, BD and BC will be described in the
same time ; for by sim. triangles BC : BD :: BA : BC. 5, That
in a rt. angled triangle, whose hyp. is perp. to the horizon, a body
will descend down any of its sides in the same time ; hence if a
circle be described on the hyp. the time of descending down any
of its chords, drawn from either extremity of the hyp. or its perp.
diam will be all equal, and also to the time of falling freely down
the perp. diam.
32. The time of descending down the incl. filane BA is to that of
falling through its height BC as B \ : BC. Let the time of de-
scribing BD or BC, which are equal (4 art. 31) be called , and
that of describing BA, T ; then t* : T :: ED : B \, the forces
being constant ; but BD : BC ;: BC : BA ; hence BD : BA ::
BC 2 : BA 2 ; therefore by equality t* : T a :: BC 2 : BA 2 , or t :
T :: BC : BA.
Cor. Hence it follows, that the times of descending down dif-
ferent planes of the same height, are as the lengths of the planes.
33 A body acquires the same vel. in descending- down an inclined
filane BA, as in falling fierfi. through its height BC. For let F
be the force of gravity in BC, /the force on AB, t the time of
falling through BC, and T of descending down AB ; then art. 28,
F : / :: BA : BC, and art. 32, t : T :: BC : BA ; hence by
comp F : /T :: I : 1 ; butFr, /T, areas the velocities, art.
18, therefore the velocities are equal.
34. From the preceding art. it follows, 1. That the velocities
acquired by bodies descending on any planes, from the same
height to the same horizontal line are equal. 2 That if the veloci-
ties be equal at any two equal altitudes D, E, they will be equal
at all other equal altitudes A, C. 3. That the velocities acquired
in descending down any planes, are as the sq. roots of the heights.
35. If a body descend from the same height through any number
of contiguous jilanes AB, BC, CD ; it will at last acquire the same
velocity as a body falling fierjiendicularly from the same height ; the
Tjel. being sufi/wstd not altered in passing from one jilacc to another .
470 OF THE GENERAL LAWS OF MOTION, See.
Let the planes DC, CB, be produced to
meet the horizontal line EG in F and G ;
then (l art. 34, the vel at B is the same
whether the body descends through AB or
FB ; and in C, for the same reason, the vel is
the same whether the body descends through
ABC, FC or GC ; hence at D it will ac-
quire the same vel. in des. through the
planes AB, BC, CD as in des. through GD, that is, an. 33, as in
falling through ED.
36 The velocity acquired by a body descending along any curve
surface is the same, an ifitftlifierfi. through the same height. This
is evident from the last art. by supposing the lines, AB. BC, &c.
indefinitely small, in which case they will form a curve.
37. Hence also it appears, 1. That the velocities acquired by
bodies descending down any planes, or curve s, or falling perp.
from the same height, are the same. 2. That if the velocities be
equal at any alt. they will be equal at any other alt 3. That the
vel. are as the sq. roots of the perp heights 4. That a body af-
ter its descent through any curve will acquire a vel. that will carry
it through the same height in any ether curve, and in any direc-
tion, or by being retained in the curve by a string, and vibrating
like a fiendulum.* 5 That the velocities will be equal, at equal
altitudes, and also the times of ascending and descending will be
the same, if the curves be of equal altitudes.
38. The times in which bodies descend through similar parts of
simitar curves, in similar positions, are as the sg. roots of their
lengths. Let ABCD, abed, in the foregoing figure, be two simi-
lar curves, and let any corresponding parts as AB, ab, be taken,
these will be proportional to the whole ; and as they are similarly
situated, they will be parallel to each other ; hence the times of
describing these corresponding parallels are as the square roots of
their lengths, art. 31, that is, as \/AD : Vad. In the same
manner it is proved that the times of describing any other two
* A simple pendulum consists of a ball or any other heavy body suspended
by a fine string or thread, moveable about a fixed centre. If the pendulum
te moved from its vertical situation, and then let fall, the ball, from its
gravity, in descending, will describe a circular arc, in the lowest point of
wlaich, or where the pend. regains its vertical position, it will have that vel.
that it would acquire by falling perpendicularly the length of the pendu-
lum (1 art 37) and this vel. will be sufficient to cause the ball to ascend
through an equal arc to the same height (4 art. 37) from whence it fell ;
having there lost all its mot. it will again fall, by its own gravity, in the
same manner as before, and will thus perform continual vibrations. Hence
if the mot. of a pendulum suffered no resistance from the air, or from the
friction at the centre of motion, its vibrations would never cease. But from
these obstructions the vel. of the ball is a little diminished in every vibration,
and hence the arcs described must become continually shorter, until at
length they Vanish with the mot. of the pendulum. To" prevent this taking
place in clocks, there is a mechanical contrivance culled a maintaining- power
See Helsham's Lectures, lecture 10,
OF THE GENERAL LAWS OF MOTION, &c. 471
similar parts, are as \/AD : \/ad ; hence by compos, the whole
times of describing are in the same ratio.
39. Hence it appears, 1. That the times of descent in curves are
as the sq. roots of their axes, or as ^/KD : v/Erf. the axes of sim-
ilar curves being as the lengths of the similar parts. 2. That as
the vibrations of pendulums are similar to the descent of bodies in
curves ; therefore, the times of the -vibration offiendulums in simi-
lar arcs of any curves, are as the square roots of the lengths of the
pendulums. 3. That the -velocity of a pendulum at its lowest point
is as the chord of the arch it descends through See art. 37.
4. That pendulums of the same length vibrate in the same time.
40. When a fiendulum -vibrates in a cycloid,* the time of one "vi-
bration, is to the time a body falls through half the length of the
pendulum, as the circumference of a circle to its diameter. Let
ABC be the cycloid, DB its
axis, or the diameter of the
generating semicircle DB,
OB or 2DB the length of the
pendulum, or radius of curva-
ture at B (see Simpson's Flux-
ions, art. 72.) Let the ball de-
scend from F, and in vibrating
describe the arc FB/; let FB
be divided into innumerable
parts, and let Ga be one of
those parts ; draw FEL, GM,
ab, perp. to DB ; on LB describe the semicircle LMB whose
centre is Q ; draw Mm parallel to DB, and also the chords BE,
BH, EH and rad. QM. Now the triangles BEH, BHI are simi-
lar ; hence BI : BH :: BH : BE (6 Eucl. 4) or BH 2 = BI X
BE, or BH = \/ BI x BE. Also in the sim. tria. Mmt>, MQrc,
Mm : M6 :: Mn : MQ, and by the nature of the cycloid H/r as
Ga. If another body descend down the chord EB, it will acquire
the same velocity as the ball in the cycloid falling from F, art. 37.
* If a circle BED (supposed complete) be rolled on a rt. line AC, until
the fixed point B, which at first touched the line at A, arrives at C ; the
point B will then describe the curve ABC, which is called a cycloid. See its
properties investigated in Emerson's properties of curve lines at the end of
his Conic Sections, sect. 3, some of which are the following-. 1. The rt. line
AD ss the cir. DEB. 2. Any rt. line FE paral. to AD => the arc EB. 3. If
xy be drawn paral. to AD, the tang-, xz is parallel to the chord. 4. The
length of the arc Bx is double the chord Tty. 5. The length of the semicy-
cloid BA = 2DB the diameter of the generating- circle, &c.
The contrivance by which, a pendulum is made to vibrate in. the curve of
a cycloid is the following-. Let the semicycloids OA, OC be described each
= ABC, their vertices being- at A and C. If then OA, OC be supposed to
be two plates of some breadth, and the pendulum OB to vibrate between
these plates, the upper part of the string- will constantly apply itself to that
plate towards which the body moves, and will thus describe the cycloid
ABC. Here AO or OC is called the evolute, and OB becomes the radius of
Curvature of the cycloid. Jffny^ens is the author of this contrivance
472 OF THE GENERAL LAWS OF MOTION, &c.
Hence lc and Ga are passed over with the same vel and therefore
the time in passing them will be as their lengths, or as HA : Ic ?
or by sim tri. as BH (or BI X BE) : BI, or as v/ BE : v/BL
or sim tri. as v/BL : \/En.' That is, the time in Ga : time in
lc :: x/BL : v/Brc Again vel at I : vel. at B :: v/EK -x/EB
(3 art. 34) or \/L : \/LB. Now the uniform vel. for EB is
equal half the vel. at the point B, and the time of describing
any space with a uniform motion, being directly as the space and
reciprocally as the velocity ; hence the time in lc : EB :: lc -*-
%/Lw : KB -~ 4VLB :: (sim. tri.) nr -~ */Ln : LB -r- Jv'LB
:: nr or Mw : 2 (BL x Ln)i. That is time in lc : time in EB
:: Mn : 2 (BL x Ln~)%. But it was shewn that time in Ga :
time in lc :: \/BL : v/Bw ; hence by comp time in Ga :
time in EB :: Mn : 2 (BL X Ln)i or 2wM (35 Eucl 3.) And
by sim. tri. Mn : 2QM or BL :: Mm : 2nH ; hence time in
Ga : time in EB :: Mm : BL. Therefore the sum of all the
times in all the Ga's : time in EB or in DB :: sum of all the
Mm's : BL ; that is time in Fa : time in DB :: L6 : LB, and
time FB : time in DB':: LMB : LB, or time FB/ : time in
DB :: 2LMB : LB, or as 3.U16 : i Q R D.
41. Hence all the vibrations of a pendulum in a cycloid, whether
great or small, are performed in the same time If/z = 3.1416,
/ =t length of the pend and s the space fallen by a heavy body in
1" ; then \/s : \/\l :: 1" : (/ ~ 2.v)i the time of falling through
\l ; therefore 1 : fi :: (I -v- 2*)i : ft X (/ -r- 2*)^, the time of
one vibration.
42. The time of vibration in the small arc of a circle is nearly
equal to that in th< cycloid, as both arcs nearly coincide at B ;
hence fi X \r'l - 2s is the time of vibr. in a small circular arc,
where / is the radius. If & or / be given, the rest is therefore
given. The easiest way is to find by experiment the length of the
pendulum vibrating seconds ; this in the lat. of London is found
a= 39^ inches; hence we have fi X f >9^ -~ 2s)| = i", from
which s = J-/* 8 /, is found = 193 07 in or 16^3 feet.
43. From art 4 1 and 42, if n the number of vibrations per-
formed in the time t ; then the lengths of pendulums describing
similar arcs being as the squares of the times, we have S9 : i 2
:: / : / -f- 39| = the sq of the time of one vibration ; hence t
divided by (I ~- 39)* = > fr m which t 2 x 39^ = rc 2 /, and
therefore n z : t* :: 3^| : /. Thus to find the length of a half-
seconds pendulum, then 1 : -J :: 39| : 9 inches, &c
The reverse of this prop being also true, that is / : 39| :: t
: n 2 , the number of vibrations made by a pendulum of a given
length may from thence be found.
44. The lengths of pendulums vibrating hi the same time, in dif-
ferent places on the earth, will be as the forces of gravity. For the
vel. being as the force of gravity (the quantity of matter being
the same in both pendulums) the space is as the vel. or as the
gravity. Now as pendulums of the same length will vibrate in the
OF THE GENERAL LAWS OF MOTION, Sec. 473
same time (4 art. 39) and the lengths of pendulums are as the
spaces fallen through in equal times, that is as the forces of
gravity.
45 By a similar reasoning it appears, 1 . That the time in which
pendulums of the same length will vibrate, by different forces of
gravity, are reciprocally as the sq. roots of those forces respective-
ly. 2 That the lengths of pendulums in different places are as
the forces of gravity, and the sq. of the times of vibr. 3. That
times of vibr. are as the sq. roots of the lengths of the pend. di-
rectly, and the sq. roots of the gravitating forces reciprocally.
4. That the forces of gravity in different places are as the lengths
of pend. directly, and the sq. of the times of vibr. reciprocally.*
46. If a body revolving in a circle, be retained in if, by a centri-
petal force tending to the centre of the circle ; then its periodic time,
or the time of one rev. will be fit X ("2r -*- s)^ and the vel. or
space if describes in the time t will be v/2r*. Where r = rad. AC,
a the space fallen through in the time t, by the force at A ; and
ft = 3.1416.
Let AB be a tang, at A, AF an indefi-
nitely small arc ; let FB, FD, be drawn perp.
to AB, AC, respectively. Let the body de-
scend through AD or BF in the time 1 ;
then AF will be described in the time 1
The whole circum. = 2/zr, and AF =
(2r x ADH. Now art. 18, si : t :: ADi
: t v/AD -f- * = time of moving through
AD or AF ; and AF : t (AD ~ )i ::
circum. AFEA : the time of one rev by subst. = pt X (2r -T- &*)%.
And by uniform mot. time of descr. AF : AF or v/2r x AD :: / :
v/2rs = vel. of the body (b) or the space descr. in the time t.
47. Vel, of b = vel. acquired in descending through -|r, by the
force (f) at A uniformly continued For s\ : 2v (the vel.) ::
v/^r : \/2r = vel. acquired in falling through \r.
48 Hence the arc described by b in any time is a mean pro-
portional between \r and s. For 2r : \/2rs :: x/2r* : s,
49. Hence if AFE be any curve, and AC, or r its radius of
curvature in any point A, and its centre of force be any other
point (S) then vel. in A = \X2r*. For this is the vel. in the
circle, and therefore in the curve which coincides with it.
50. The periodic times (P) of several bodies revolving in circles
round the same or different centres, are as the sg. roots of their
* In these articles the rod of the pendulum, or the thread, is supposed
very fine, or of no weight; and that the ball is very small, or has its matter
united in a point. Hence as this cannot be so, the length of the pend. is
nearly its di&t. from the point of suspension to the centre of the ball, or
rather to the centre of oscillation of the pend. See Emerson's Tracts, sect.
2, prop, 28. The Methods of finding the centres of (iravity, Percussion,
Oscillation, &.c. with a further detail of the properties of the pend. are g-ivpu
In Emerson's Mechanics.
3 M
474 OF THE GENERAL LAWS OF MOTION, fcc.
radir directly, and the sg. roots of the cent, forces recifirocally. For
(art 45) P = fit X (2r -*- *)i, and s is as/, the force ; henc- P
=3 /if x (2r -*-/) ; but 2, /z, and t are given, therefore P is as
51. From the preceding art. it appears, 1. That the periodic
times are as r -~ t>. For t; = (2r,<?):2 = v'2r/", hence x> 2 = 2>/;
and P 2 = fi 2 t* x 2r -*-/, therefore P 2 v 2 = A 2 r 2 X 4r 2 , from
which P = 2/r -~ x> ; whence P is as r -=- v. 2. That the peri-
odic times are as v -4-/ For v 3 = -2rs = <2rf, and r = v 2 -+- 2/,
also r -r- -i> is as v -s- / ; but P is as r -~ v, that is as T> -r- /I
3. That i) and/ are as r. For if P be given, then r -T-/, r -f- v,
and v -~-f are each given. 4. That if P be as \/r, v will be as
\/r, and the centripetal forces equal For (art. 49 and 50, cor. .)
taking \/ r for P, we have \/r is as (r --/) that is as r ~ v.
Hence J is as 1 -~ \/f, or as \/r ~- v, and \/r is as iy ; the v//
is therefore a given quantity. 5. That if P be as r the velocities
will be equal, and/ as 1 -5- r. For taking r for P, we have r is as
(r -f-/) or as r -~- t> ; hence \/r is as 1 ~ \/f* and 1 is as
1 -s- ^, therefore T is a given or constant quantity (that is V is as^)
and r is as 1 -~~f 6. Tnat if the periodic times be in the sesqui-
plicate ratio of the radii, or if P be as r, then -v will be as C/r,
and/as 1 -f- r 2 . For taking r 3 for P, r| is as (r -T-/)2, or as
r -f- -v ; and r is as 1 ~ v//J or r* is as 1 /, also \/r is as
1 ~ v. 7. That if P be as r n , then v will be as 1 -T- r w ~ 1 , and
/as r 2n "- 1 . For taking r n for P, then r n is as (r -v-/)i, or as
r-f-u ; whence r n is as r~-/, and r n ' ~ is as 1 /. Also r n ~ is
as l-7-i>.
52. 7%<? velocities of several bodies revolving in circles round the
same or different centres are as the radii directly , and fieriodic times
recifirocally ; or v is as r -f- /. For (art 46) v = 2rs = \/2r^ and
P is as v -^-f (2 art. 51) and Pf is as v ; also / is v -~- P ; hence
v = y^ 2 ?^ = (2r X v -f- P), and v 2 = 2rv -v- P, and v =2r ~ P,
that is T; is as r -j- P.
53. From the preceding art. it follows, 1. That v is as P/*. 2.
That v 2 is as r/i For t> = \/%rf. 3. That the velocities being equal,
P is as r, and r as 1 -~-f. For r H- P is a given ratio, r being given ;
and as ^/rfis given, r 4 is as 1 /. 4. That if v be as r, the pe-
riodic times will be the same, and /as r. For then v or r is as
r _i_ p 5 and 1 is as 1 -f- P. \lso r = v / 2r/, hence r is as y! 5.
That if -y be as 1 ~ r, then/ will be as 1 -5- r 3 , and P as r 2 . For
taking I r for v, then (cor. 2) 1 -j- r = v/2r/J or 1 -r- r =2r/";
whence/is as -f- *" 3 i also 1 -4- r is as r -=- P, and P is as r 2 .
54. '/%<? centripetal forces are as the radii directly, and the squares
of the fieriodic times recifirocally. For (art. 46) P = fit X
* When any quantity is divided into another, the reciprocal of that quan-
tity may be taken. The reciprocal of any quantity is 1 divided by that
quantity. Thus the reciprocal of a is 1 -7- a,
OF THE GENERAL LAWS OF MOTION, &c. 475
(2r-r- *H =/tf(2r -*-/)!, and P 2 = fl 2 t X 2r /; also/* 2 /
= 2/z 2 * 2 r . hence/ = 2/* 2 / 2 r -7- P 2 is as r ~ P 2
55. From the lastart.it follows, 1 That/ is as v "t- P For
(art. 52) T; is r -r- P, and / is as r -f- P 2 , that is as v + P. 2.
That f is as 7> -T- r. For /is as ^ -r- P, and /P is as v j
but (I art. 44) P is r -~ v ; hence /P is as /r -5- v ; therefore
fr -T- vis as T>, and /is as v* -*- r. 3. That, the centripetal for-
ces being equal, v will be as P, and r as P 2 or as T> 2 . 4. That if
/be as r, the periodic times will be equal. For if/is as r -j- pa,
and f-~ r is as 1 ~ P 2 ; and if/-*- r be a given ratio, 1 -r- P*
will be given, as also P. 5. That ifj be as 1 -7- r 2 ; Mew P 2 // fo
<zs r 3 , and T> as 1 -7- >/* For taking 1 -j- r 2 for/ then I ~- r 2
is as r -r- P 2 , and r 3 -~ P 2 a given quantity. Also 1 -r- r 3 is as
v 2 -T- r, and 1 -5- r as 7> 2 , or (1 -- r)A as v,
56. 77* e* racfcY are directly as the centripetal forces and the squares
of the periodic times. For (art 46 or 54) P 2 = /z 2 t 2 X 2r -~ f\
and P 2 /= 2/z 2 2 r ; hence P 2 / is as r, 2,/z, and t being given.
57. Hence it follows, 1. That r is vP. For P/ 1 is as v (1 art.
53) and P 2 / is as r ; hence Pv is as r. 2. That r is v 2 /.
For (2 art. 51) P is as v -r-/; but r is as Pt;, by the preceding ;
hence r is as ~v z -?-/. 3. That the radii being equal, /is as t> 2 -5-
P 2 , and that v is as 1 ~- P. For in this case f 2 ~-/, P 2 /, and
Pv are all given ; and/is as i> 2 , or /"is as 1 ~- P 2 ; hence v is as
1 -f- P.
JVW<?. The converse of all these articles, &c. are also true ; and what is
shewn of cent, forces is equally true of centrifugal forces, they being- equal
and contrary. Moreover whatever is demonstrated in these articles, con-
cerning the forces, velocities, and periodic times of bodies moving 1 in circles,
hold equally true in ellipses, taking" the mean distances, or half the trans-
verse axis, instead of the radii. The truth of this is fully shewn in Emer-
son's Centripetal forces, where the different cases of bodies revolving in the
conic sections are investigated. The reader is therefore referred to this, or
to Newton's prin. for more information on this subject. In Gregory's Ast.
the Laws of Centripetal and Centrifugal forces are fully discussed. But for
some recent improvements, consult Laplace's System of the World, B. 3, or
his Celestial Mechanics. Simpson, Emerson, Me Laurin, &c. in their res-
pective treatises on Fluxions, have given the analytic investigation of these
laws. See also pa. 402.
58. 772<? quantities of matter (M) in all attracting bodies^ having
others revolving about them in circles^ are as the cubes of their dia~
tances directly, and the squares of the periodic times reciprocally.
For as observations have fully proved that the squares of the pe-
riodic times are as the cubes of the distances, both of the planets
and satellites from their respective centres Hence (6 art. 51) /
is as 1 -~ r 2 ; and the attractive forces at the same dist being as
the quantity of matter M, hence the absolute force of M is as
M +- r 2 , and (art 54) since / is as r -T- P 2 , if we take M ~ r 2 in
place of/ then M -T- r 2 is as r -4- P 2 . and therefore V isas r 3 -~P 2 .
Cor. 1. Hence M ~- r* which is the force of the attracting bod>-
C at A) may be substituted for/in any of the foregoing articles,
476 OF THE GENERAL LAWS OF MOTION, &c.
Cor. f 2. Hence also the attracting force of any body is as the qium*>
tity of matter directly, and the sq. of the dist. reciprocally.
JVote. If the mean dist. be taken for the radii of the circles, the same
properties hold also in ellipses, &.c.
59 The densities (^) of central attracting bodies are recihro*
caliy as the cubes of the parallaxes of the bodies revolving round them
(as seen from those central bodies) and recifirocally as the squares
of the periodic tijnes. For D X d 3 (the cube of the diam ) is as
the quantity of matter (art. 9) that is as m 3 (cube of the mean
dist.) or r 3 -f- P 2 (art. 58) hence D is as r* (or m 3 ) ~ d 3 P 2 .
But d -f- r or m, is as the angle of the parallax (a) therefore D
is as 1 -T- a 3 P 2 .
Cor. Hence D is as 1 ~ d 3 P 2 *
* From this and the foregoing- article the masses and densities of the plan-
ets are thus found. The result in art. 58, being- applied to Jupiter and his
4th satellite, we have the ang-le subtended by mean rad. of the orbit of this
satel. at .Tup. mean dist. from the sun = (1530"d6 according 1 to Laplace J
8' 15"99 nearly, this ang-le at the earth's mean dist. from the sun = (7964"75)
15' o"57the rad. of the circle = (636619 ff 8) 57 17' 44"8. Hence the mean
vad. of the orbit of the satcl. : the mean rad of the earth's :: 43' G"57 : 57
17' 44"S. The sidereal rev. of the 4th satel. is 16.689 days, and of the earth
365.2564 days ; from which the mass of Jupiter is found = TSTfl.tnr tnat ^
the sun being 1. Laplace remarks that the denom. of this frac. must be
augmented by 1, as the force which retains Jup. in his relative orbit round
the sun is the sum of the attractions of the sun and of Jupiter. The mass
of Jup. will then be T<> ST.^T- The mass of Saturn determined in the same
manner is found to be Tj-yV-Zo^ anc ^ f Herschel T^TO*' The mass of the
earth may be found in the same manner. Laplace gives a different method
in B. 4, ch. 2, Ast. from which he determines that the masses of the sun
and earth are as 1479560.5 and 4.48855, from whence it follows that the
mass of the earth is -jT-gV^TT taking the sun's parallax (27"2) S"8126. If the
sun's parallax differ any thing from this, the value of the earth's mass should
vary as the cube of this parallax compared with that of 8*8. Laplace deter-
mines the masses of Venus and Jlfars from the secular diminution of the obL
of the ecliptic supposed (154"3) 50" nearly, and from the acceleration of the
Moon's mean motion fixing it at (34" r 36) 11"13 nearly, for the 1st century
comm. with 1700. The mass of Venus is thus found = jTSTa'T' ana * tn at
of Mars T^^V^f H C has found the mass of Mercury from his mag.
supposing the densities of this planet and the earth's inversely as their mean
dist. from the sun, = ^TTJ-y^fTy
The densities of bodies being proportional to their masses divided by their
volumes or mag. that is by the cubes of their radii when spherical. Their
densities are therefore as their masses divided by the cubes of their radii ;
but to obtain greater accuracy that radius of a planet must be taken (as La-
place remarks) which corresponds to that parallel the square of the sine of
whose latitude is -p and which is equal to -5- of the sum of the radius ofthoj
pole added to twice the radius of the equator. It is thus that Laplace de-
termines the densities of the Earth, Jup. Saturn and Herschel to be 3.9393,
0.8601, 0.4951 and 1.1376 respectively, the sun's mean density being taken 1
As the force of gravity at the equator of the planets is as their masses di-
vided by the squares of their diameter, supposing them spher. and deprived
of their rotary motion. Now the equatorial diam. of Jup. is (62^"26) 5' 32"9 ;
OF THE GENERAL LAWS OF MOTION, &c. 477
60. The areas which a revolving 'body describes, by radii draiv?i
to a fixed centre of force, are firofiortional to the times of descrip-
tion ; and are all in the same immoveable plane. Let S be the
centre of force ; and let the time be di-
vided into very small equal parts. In
the first part of that time let the body
describe the line AB, then (art. 5) if
no other force be impressed the body
will proceed towards c, and describe
Be a* AB in the 2d part of time, so
that the area ASB = BSc (38 Eucl 1)
But when the body comes to B, suppose
that a centripetal force act by a single,
but strong impulse, then the body will describe the diagonal BC
of a parallelogram whose sides Be, Br or cC parallel to SB repre-
sent those forces (art. 27) hence the triangle SBC =: SBc =s= also
SB A, all of which are in the same plane. In like manner it may
be shewn that if the centripetal force act successively in C, D, E,
&c. and make the body in each single particle of time describe the
lines BC, CD, DE, EF, &c. they will all be in the same plane,
and the triangles CSrf, CSD, DS<?, DSE, ES/, ESF, &c. are all
equal. Hence in equal times, equal areas are described in ono
immoveable plane ; and by comp. any sum of those areas are as
the times of description. Now let the number of these triangles
be increased, and their breadth diminished ad injinitum ; the cen-
tripetal force will then act constantly, and the figure ABCD, &c.
will be a curve.
61. From the precedingart.it appears, 1. That if a body de-
scribe areas proportional to the times about any fixed point, it is
urged by a centripetal force directed to that point. For a body
cannot describe areas proportional to the times about two fixed
centres. 2. That the vel. of a body, in any point of a curve, is
reciprocally proportional to the perp. on the tang, at that point.
The base of any of the triangles (which represents the vel.) being
reciprocally as the perp. 3. That the angular velocity at the cen-
tre offeree is reciprocally as the square of the dist. from that cen-
tre. For ABCD, &c. being considered as a curve, the small tri-
angle SFE = SAB, they are descr. in equal times, and area of
the 1st = SF X FQ 2, and of the 2d SB X Efi -r- 2 ; hence
SF X FQ = SB X B/J. But Z.FSE : ASB :: FQ :: ab (Sa be-
ing = SF) :: SF X FQ : SF or Sa X ab :: SB X B/* : Sa X a*
:: area SBA : area Scd :: SB 2 : Sa 2 or SF Z . See Newion'*
firm. B. 1. sect. 2.
and of the earth (54"5) 17"6, at the earth's mean dist. from the sun. Hence
if the weight of a body at the terrestrial equator be 1, this body transported
to the equator of Jup. would weigh 2.509. But this must be diminished ^
from the effects of the centifugal force at Jupiter's eq. this eilcci may be
determined by the preceding- articles. The same body at the sun's eq.
would weigh 27.65, and heavy bodies \vould fall 100 metres the 1st second
of their descent, according 1 to Laplace. The second here must, no doubty
be that adopted in the French measures.
478 OF THE GENERAL LAWS OF MOTION, Sec.
62. If two bodies A, B, revolve about each other , they 'will both
revolve about their centre of gravity. Let C be the centre of gravity
of A B acting on each other by
any centr. forces. Let AD be the
direction of A's mot draw BE
paral to AD for B's direction.
Let the time of descr YD, BF
be very short, so that AD : BF ::
AC : 'BC, C will then be the
centre of grav. of D and F, the
tri, ACD, BCF being sim. (see
the note pa 253 ) Hence AC : ^ - E
CB :: DC CF. Let Aa, B6, be the spaces through which A and
B will advance towards each other in the same time by their mu-
tual attractions, these spaces will be reciprocally as the bodies, or
directly as the dist. from C the centre of gravity ; that is, Aa : B
:: AC : BC Complete the paral Ac, Bo 1 ; c, and d will then be
the corresponding places of the bodies, instead of D, F. (art 23.)
Now as AC : BC :: Aa : B, by div AC : BC :: aC : bC ; but
AC : BC :: AD : BF :: ac : bd. Hence aC : bC :: ac : bd >
the tri cCa, and dCb are therefore similar, whence Cc : Cd :: ac :
bd :: AC BC :: B : A. Therefore C is still the centre of grav.
of the bodies, at c and d.
If Bo? and Ac be now produced until df= B</, and ci = Ac, and
if ck, dg, be thr spaces drawn through from their mutual attr. and
cr'j ft/, be compl. then it will appear, in like manner, that C is the
centre of grav of the bodies at h and e, and also i and/*; and Ac,
cz, Sec and Bo*, df, 8cc are therefore the paths of A and B round C.
If B be at rest while A moves towards G ; then C will move
uniformly along CH paral to AG Hence if the space the bodies
move in, moves in the direction CH with the vel. of the centre of
grav this centre will then be at rest in that space, and B will
move in the direction BF paral. to CH or AD, which comes to
the same as the former case. Hence the bodies will always move
round the centre of grav. which is either at rest, or moves uni-
formly in a rt. line.
In the same manner it may be proved, that if A and B repel
each other, they will also constantly move round their centre of
gravity
6 From the preceding art it appears, 1. That in estimating
the motions of a system of bodies among themselves, their mo-
tions round their common centre of grav. should be taken 2.
Th^t the directions of the bodies in oppos points of their orbits
are always parallel to each other For as AD : DC :: BF : Fd,
AD, DC is therefore paral to BF, Fd ; hence DAc =* HBrf ;
Be/ is therefore paral to Ac. For the same reason c^is paral. to
', Sec 3. That two bodies actim- on each other by any forces,
describe sim. figures about their centre of gravity. For Ac, Btf,
Sec. are parallel, and always proportional to AC, BC. 4. That if
OF THE GENERAL LAWS OF MOTION, See. 479
the forces be directly as the chst the bodies will describe concen-
tric ellipses round the centre of grav 5 . That if the forces be
reciprocally as the squares of the dist the bodies will describe
similar ellipses, or sonif conic sections, about each other, having
the centre of gravity in the focus of both, &c.*
64. If a body bt projected from A, in a given direction AD, and
be attracted to two fixed centres S, T not in the same plane with
AD, the revolving triangle SAT formed by lines drawn from the
two fixed centres to the body, will describe equal solids in equal
times, about ST the line joining the fixed centres. Let -the time
* Our limits would not permit us to enter into an investigation of any of the
properties of the centre of grav. however interesting-: the following- observa-
tions of Laplace deserve, however, to be mentioned. He remarks, that when
a body receives an impulsion in the direction passing- through the centre of
grav. all its parts move with equal vel. That if the direction pass on one side
of this point, the different parts of the body acquire unequal velocities, from
which results a mot. of rotation of the body about its centre of grav. togeth-
er with its progessive mot. He then remarks that this is the case with the
earth and the planets. He makes the dist. of the prim, impulse from the
centre of grav. of the earth = T ^Q of its rad. supposing it homogeneous.
Sir Isaac JVeiuton makes a similar suppos. in accounting for the centrifugal
forces of the planets. But neither he, or any other person since his time,
has ever shewn whence proceeded, or what was the cause of this primitive
impulse, for it is certain that there is no effect without a cause. It is in-
finitely more probable that both the motion of rotation, and also that in
the orbits of the planets, depend upon the sun, and are regulated by him.
For as we have remarked before, we find no other active principle in mat-
ter, or emanating from it, capable of producing such an effect, except light,
which modified, reflected, refracted, and varied in a thousand different man-
ners, is very probably the cause of gravity, electricity, &c. and innumerable
phenomena of which no rational account can be given, from our igno-
rance of the nature of this subtle fluid. If its action produces the rotary
motion of the planets, it could not be from the situation of their centres of
grav. For if the centre of motion be the same as the centre of the body,
and the centre of gravity a little distant from it, let the line joining these
centres be perp. to the direction of the rays of light, &c. from the sun; then
the unequal action on the body may produce a rotary mot. but when this
line is parallel to the direction of the rays, they have no effect on turning
the body ; and when the line passes this paral. posit, the action of the rays
will tend to turn the body in a contrary direction - r and hence their effect
would be to produce a vibratory mot. which with a little resistance would
subside, and the same side remain turned towards the sun This may be
the case with the secondary planets with respect to their primaries, whose
action on them is greater than that of the sun's. If the unequal action of
the sun's rays on the planets, from a rarification of their atmospheres, ex-
ceeded the former, a rotary motion would ensue. This may be the case
with the primary planets. For if we suppose the former case, it may be
asked why has not the secondary planets a rotary mot. Why does not the
change of the sun's decl change that in the primaries ? Until these ques-
tions be solved, Laplace's obs. can have no force.
480 OF THE GENERAL LAWS OF MOTION, Sec.
be divided into infinitely small equal
parts, then in those equal times it
is evident that the lines AB, BC,
CD, &c. will be described, and
hence that the solids STAB,
STBC, STCD, Sec. described in
the same equal times, would be
equal, if the focus at S and T did
not act on the moving body Let
S and T be now supposed to act at
the end of each interval of time,
let T act at B in the direction BT, in which direction the body is
always drawn by T, so that instead of being at C, it is drawn in the
direction CF ; and by the force S, for the same reason, it is drawn
from C in a direction CE parallel to SB. Hence from both for-
ces the body at the end of the time must be somewhere in the
plane ECF paral. to SB T as at I.* Now as pyramids upon the
same base and of equal altitudes are equal (Emerson's Geom. b.
6, prop '7) the pyramid STBI = STBC, being between the
paral, planes ECF, SBT, and therefore of equal altitude ; hence
pyr. STBI = STAB.
If BI be now continued to K so that IK = BI, the body in the
next part of time would advance to K describing the pyr. STIK =
STBI, but at the end of the time the body being drawn by the
forces S, T parall to their directions KL, KN, it will be found
somewhere in the plane LKN, as suppose at O. Then the solid
STIO = STIK = STBI = pyr. STAB
If IO be now produced making OP = IO the body attracted by
S, T at P will descr. another equal pyr. Hence equal pyramids
will be described in equal times : and therefore the whole described
as the times of description.
It may be here remarked, 1. That the orbit becomes a curve
when the number of the lines AB, BI, IO, &c. is increased, and
their may: dim. ad infinitum. 2. That any lines AB, BI, IO, Sec.
are tangents at A, I, O, &c. being corresponding points in the
curve. 3. That the curve is not in the same plane except when
the forces on each side of it are equal.
65. If the body T revolves
above S at a great dist. in the
orbit TV, and M a leaser body
about T which is near M. Then
if F = the centr. force of S
ufion T, the disturbing force of
S ufion M = F X SMI -T- ST,
M
MI being paral. and IV fierfi. to ST. And F X MT -4- ST
be the increase of the centr. force from M towards T. Let
* This point may be found by resolving 1 the three forces in the directions
BC, CF, CE, as shewn art. 27. And any number of forces may be resolved
Jn like manner.
OF THE GENERAL LAWS OF MOTION, &c. 481
ST = r, MT = , MI = a-, g = the force of gravity, s the"
space fallen by this force in the time 1. h = the space descended
by the force F in the time 1. P = the period, time of T about S,
and t = the period, time of M about T, /'= the centr. force of T
at M, fi =3.1416. Now as attraction is reciprocally as the sq. of
the dist. then the force of S at T : its force at M :: I -t- r* :
I -7- SM 2 :: 1 -7- r 2 :: 1 -7- (r a-) 2 :: r : r -f 2o? -f 3.r 3 *-
r -f 4o? s -7- r 2 , &c. And force of S at T : diff. of the forces ::
r : 1x -f 3o? 2 H- r, &c. that is r : 2.r -f- So? 2 ~ r :: F : 2F.r
-r- r -f- SFo? 2 -r- r 2 nearly, the cliff, of the forces, or the single
force by which M is drawn from its orbit in the direction IM or
MS. Let the forces MS be divided into the two MT, TS, which
substituted for it and proceeding as before, we have force of S at T
: force at M :: 1 -j- r 2 : 1 -f- (r a?) 2 . And force of S on M
in direct. MS : force on M in direct. TS :: r a? : r :: 1 -4- r
: 1 -f- (r x}. Hence from equality of propor. force of S at
T : force on M in direct. TS :: 1 -7- r 3 : 1 ~- (r .r) 3 ::
1 -7- r 3 : 1 divided by r 3 3r ? o? -f 3rx* a? 3 :: r : r -f- So? +
6.r 2 r, &c. And force at T : diff. of the forces :: r : 3x +
6^2 -f. r , &c or r : 3o? -f 6o? 2 -r- r :: F :: 3Fo? -5- r + 6Fo? 2 -f- r 3
= the disturbing force of P paral. to TS. Also x : a :: increase
of the disturbing force in direct MI (Fa? -7- r -f 3 Fa? 2 -f- r*, Sec.)
: Fa -7- r -f 3Faa? -7- r 2 , &c. the addition of the centripetal force
in direction MT. For in the former disturbing force 2Fa? -f- r
-f 3F.r 2 -4- r 2 , there was a diminution of centr. force at T, as
appears from the next art.
66. From the preceding art. it appears, 1. That the simple
disturbing force at M towards S = 2 Fa? -f- r nearly (rejecting the
other terms of the series as inconsiderable) and dimin. of centr.
force of M towards T = Ft> ~- r. Also accelerating force of M
in the arc MA = Fz -7- r. z and i> being the sine and versed
sine of 2MQ. For TI = y, and let IK be perp. to MT ; then
sim. trian. a : x :: x : MK = a? 2 -~ a :: force MI (2 Fa? ~ r
nearly) : force in direct. KM or TM = zF a? 2 -7- ar = Fv ~- r.
Also a : y :: a? : IK :: force MI : force in direct. IK or MA
= 2Fya? -7- ar = (Trig. Emerson's, prop. 2. Schol.) Fz ~ r.
2. That if M be the moon, and S the sun, the disturbing force at
M =/? -5- 59.574, q being sine dist. from the quadrature Q.
For (art. 54) F =f( r -r- P 2 a, and SFo? -7- r, &c. = Sft 2 * -7-
P 2 a (nearly) = 3/? -7- 178.724 (because & ~- P 2 = 1 ~ 178.724
see pa. 308 or 325, and 304 or 350, and x -~- a = q -r- 1) *=f<j
-7- 59.6 nearly. 3. That if M be a body in the equator the dis-
turbing force of the sun at M = gg -f- 13067671. For M being
the moon, the force is then fq -f- 59.6; but g = 60.3 x 60. 3/
(see pa. 308) ory = g -r- 3636 nearly ; hence the force becomes
gg _:_ 59.6 x 3636, and at the earth = gg ~- 59.6 x 60. 3 3 = qg
-f- 13067671 nearly. 4. That the disturbing force of the moon on
the equinoctial qg ~ 3595802. For the general perturbating
force was nearly 3 Fa? -7- r, where F must be the centr. force at
the moon. But the centr. force of the earth at the dist. of the
3 N
482 OF THE GENERAL LAWS OF MOTION, &c.
moon = 1 -7- 60. 3 *g. And the moon being 49.2 times less than
the earth (see pa. 327 or 344*) the centr. force of the moon at the
same dist =ssg-r- 49.2 X 60 3 2 , which being substitute d for F,
then the force of the moon in the equator = 3x -~ r mult, by g -*-
49 2 X 60.3 2 = 3gx divided by 60 3a X 49.2 x 60. 3 2 = yg- divid-
ed by 20 1 X 49.2 X 60.3 2 = 359 ;> 802 nearly. 5. That the rfzs-
turbing force of the sun to that of the moon ufion the < quator is
as 1 : 3.6 nearly. For I -j- 1306767! and 1 3595802 is- near-
ly in this proportion. 6. That if D be the apparent diam. and d
the density of the perturbating body, then the disturbing force
will be as dQ z x For that force = 3F.r -H r, or as F.T-T- r. If
the diam =* b, its quantity of matter = m 5 then F is as cfd 3 ~r 2 ,
and the disturbing force is therelcae db*oc -j- r 3 , or as dD^x. 7.
That the centrifugal force of M (at the equator) : perturbating
force MT :: P* : / 2 , t being here the time of one rev. ot the
earth on its axis. For t : 2/wz (circum.) :: \" : 2/?a -i- t = arc
descr in l" ; and versed sine = 4/z 2 a 2 -r- 2ar 2 = 2/? 2 a -r- t =
the ascent or descent from the earth's centrifugal force. But for-
ces are as the effects produc d ; hence * : g ' "2fi 2 a ~~ t z
(ascent) : Vfi^ag -i- t z s the centrifugal force itself Now as the
perturb force = Fa -f- r = ahg -T- rs, we have centrif force :
perturb force :: Zfi 2 ag -f- t z s : ahg -4- rs :: 2/z 2 ~- t* : h r
:: 2/i 2 r : t*h :: 2// 2 r ^- h : / 2 . But 2/z 2 r -i- h = P 2 ; for
*/2rA : \" :: 3jir : P = 2/zr ~ x/2rA ; hence P 2 = 4^ a r 2 ~
Irh = 2/i 2 r -r- A 8 That the body M is therefore accelerated
from the quadr. Q, Z, to the syzygies A, B ; and retarded from
syzy. to the quadr. And also that the vel. and area descr. in the
syz are greater than in the quadr
67. The linear error gen. in M in any time, is as the disturb,
force and sq. of the time. And the angular error, seen from T, is
as the force and sq of the time directly, and the dist. TM recipro-
cally. For the mot. genr. in any portion of time is as the force,
and in any other time as the force and sq. of the time ; this mot.
is the linear error of M, being carried out of its proper orbit by
the force SFx -T- r. This error as seen from T is as the angle
under which it appears, that is as the linear error divided by the
dist TM ; and hence is as the force and sq. of the time divided
by the dist*
68. From the preceding art. it appears, 1. That the linear er-
ror generated in one rev. of M, is as the distu. force and sq. of the
period, time 3Fa 2 ~- r. And the angular error in one rev. is as
the force and sq of the period, time divided by the dist. 2. That
the mean error of M in any given time will be as the force and
period, time at -~- r And the mean angular error is as the
force and period, time divided by the dist. For let l = given
time ; then t : Fat* ~- r (whole error) :: 1 : Fa -^ r error in
the time 1. 3. That the mean linear error in any given time is
as at -f- P 2 . And the mean angular error as t -f- P 2 . For (art 54)
F is as r ~- P 2 ; hence FaJ ~- r is as at -i- P 2 . And ang. error
as t ~ P 2 . 4, In any given time the linear error is as at -~ r 3 ,
OF THE GENERAL LAWS OF MOTION, &c. 483
Kor (art. 58) P 2 is as r3, hence at -r- P 3 is as at ~ r 3 . 5. That
the linear error in a given time is as Fa| -f- r. And the angular
error as FaJ -r- r. For t 2 is as a 3 , hence Fa -f- r is as Pa| ~ r.
6. That universally the angular errors in the whole rev. of any sa-
tellites are as t 2 -^- P 2 . And the mean ang errors as t -f- P 2 . For
(1) ang. error is as Fa -~ r X f 2 -f- > that is as t 2 P 2 , because F
is as r -f- P*. The latter case has been already proved. (3.)
69. To determine the disturbing force of Jupiter or Saturn, ufion
the earth in its orbit ; that of the sun ufion the moon being given.
Let the quant, of matter in the sun and Jup. be as m : I . P, p, t the
periodic times of the earth, Jup. and the moon, r, , the distan-
ces of the earth and Jup. from the sun, a the moon's dist. from
the earth, F, f, the centr. forces of the sun and Jup. Now
(art. 65) the distur. force of S the sun on M the moon is 3F.r -r- r,
or as Fa ~- r ; but if S be Jup M the earth, and T the sun ; the
force is then/r -r- b. That is the sun's distur. force on the moon
: Jup distur. force on the earth :: Fa ~- r : fr -r- b :: Ybr :
fr 2 .^ But (2 art. 58) F = m ~ r*, and F = I -r- b 2 ; hence the
sun's force on the moon : Jupiter's on the earth :: abm ~- r 2 :
r 2 -*- b 2 :: ab*m : r 4 :: aP 2 w : rP 2 (art. 65.) But the sun's
disturbing force on the moon is given, and therefore that of Jup.
If for P and w, Saturn's periodic time and quantity of matter be
substituted, his disturbing force will be known.
70. From the preceding art. it appears, 1. That the angular er-
rors generated in the moon by the sun are to those gen. by Jup. in
the same time, as p 3 rm : P 3 . For (2 art. 68) these errors are as
the forces and periodic times divided by the distances. Hence the
sun's to that of Jup is as afi 2 mt -=- a. : rP 2 X P ~ r :: p 2 tm : r 2 .
2. That the error in the moon from the sun's action, is to the
earth's by Jupiter's as 1 1466 : 1 ; and to that gener. by Saturn as
222600 : 1. For let p = 4332.6 days, t = 27.32 days nearly, m =
1067.1 nearly. P = 365.25 ; then p z tm -j- P* = 1 1466. And
takingp = K)759.07, and m = 3359.4 for Saturn ; then p 2 /m -j-
P 3 SB 222600. 3. That the force of Saturn to that of Jup. to dis-
turb the earth is as 1 : 19.4. 4. That the secular motion of the
nodes of the earth's orbit by Jupiter's action is 10' 9 "2, and by
Saturn's 3 1"4. For the annual mot. of the moon's nodes = 19 19'
43" (pa. 323) == 69583", which divided by 11466 gives 6"0686,
and mult by 100= 606"86 ; this being increased in the ratio of
cos. of incl. of Jup. orbit 1 19' to that of the moon's 5 8' 48" (pa.
324) gives 10' 9"2, which divided by i9.4 gives 31"4> for Saturn.
5. That the secular mot. of the earth's afihel. by the action of Jup.
is 21' 16" in consequentia, and by Saturn 65"77. For the an-
nual mot of the moon's apogee = 40 39' 5'.'" (pa 323) = 14G390"
div by 1 1466 = 12"76, and rnult. by 100 gives 1276" = 2l' 16".
This div. by 19.4 gives 65"77.
71. If a planet P (or the moon) fierfirm its mot. round an im-
movable centre C in the orbit NTnl, whose pla.ie is inclined to that
of the ecliptic NKnjflwd i acted P?J by a force per p. to AB and par al.
R
484 OF THE GENERAL LAWS OF MOTION, &c.
(o the eclifitic, and always directed from the fdane AB to either
side. To find the mot. of the nodes N, fl, and the ~uar. of the or-
bit's ind. PNE. Let NTBn be half
the orbit above the eclip. NE, Nn
the lines of the nodes, T, ?, the tro-
pics. Let CM be drawn perp. to
the eclip. and CO perp. to the plan-
et's orbit. Let the circle GORX
be clescr. round M as a pole ; then
GC will be the axis of the orbit, and
MC of the eclip. Through T, C,
draw the plane GMC, and another
through N, C, M, cutting the cir.
GRF in X and V ; then GF is perp.
to XV, and GRF paral. to NBra.
Here GNX represents the upper
surface. Let the planet at P de-
scribe the space P2 in any short in-
terval of time, and let Ps be the
space it would be drawn in the same
time by the force acting from the plane MAB. Compl. the para!.
P3, then P3 will be the direction from both forces. Now as P#
is paral. to the eclip. the point 3 is below the plane of the orbit,
and the plane CP2 will be moved in the pos. CP3, about CP ;
hence GC will be moved perp. to CP. The pole G will there-
fore, be moved to some point between F and V. Hence the mot.
of G will be known for all the places of P in its orbit. For about
A, G moves perp. to MQ ; at N perp. to MX, or in the direct.
GM ; at T paral. to MV, or in the curve RO ; about B it moves
perp. from MQ ; hence in the passage of P from A to B, the
pole of its orbit G describes the curve G 1234. But in the other
half of the orbit Brz^A, G will return back at 4, and describe a sim.
curve 4567, the force being directed the contrary way from the
plane ABMR. Hence when P has made 1 rev. G will be at 7.
In this posit, of the nodes 7 will be within the circle ; for the
points G, 4, being equidistant from QR ; 4, 7 will be also equi-
dist. If the force and plane ABRQ revolve round CM in the
direct. ANTrc ; then when the ascend, node N is as far on the
other side of A, as at a, the pole G will be as far on the other
side of O, as at 6, and being equally dist. from RQ on the same
side, the curves (12467) will approach OV there, by the same
degrees as they receded from it at GO. Hence G will, by de-
grees, be brought to the circle again. Thus in every two corresp.
points on each side of O, the forces and their effects balance one
another, and G will be at the same dist. from the circle GOV.
Hence after half a rev. of the plane AB to the nodes, the Z.GCM,
or incl. of orbit, becomes the same as at first. And as G moves
forward or backward in the circle, the mot. of the nodes N, n :
v/iil be forward or backward.
OF THE GENERAL LAWS OF MOTION, &c. 485
72. From this art. it appears, 1. That in the pos. of the nodes
at N and /*, the inclination of the orbit will be dimin. every rev. of
P, but at a increased. For then 4 and 7 approach M and recede
at b. 2. That the incl. decreases when P is in AT or B, and
increases in TB, *A. For G moves to 3, while P moves through
AT. At 3 it is nearest M ; from 3 to 4, G recedes from M
while P moves through TB. The same will happen in the other
half of the orbit. 3. That when P is in AN and Bn, the nodes
move forward, but backward in NB, n\. For while G describes
Gl, its mot. is forward, that is from G towards Q ; at 1 it is stat.
P being in N. G moves backwards or towards O through 1 234 ;
and then P is in NB. 4. That in general the nodes are always
regressive except when P is between a node and its quadr. and
then they are progr. wherever they are situated. 5 That the
nodes move faster when P is in T and t. For then G is at 3 and 6.
6. That the incl. varies most when P is at N and n. For G is
then at 1 and 5. 7. That therefore theincr. or deer, of the incl.
may be easily found, the place of P, and diff. situations of N, n
being given. 8. That hence the forces being given, the mot. of G
may be delin. on the surface GXFV ; and the incl. and place f
the node at any time found.
For more information consult Emerson's Centr. forces, where
this subject is fully investigated.
73. Tojind the secular -uariat. of incl. of the earth's orbit^ from
the action ofJufi. and the same for Saturn. Let TSS X? be the
eclip. NGn the orbit of Jup. N Jup.
ascending node, E, I, Q the poles of
the eclip. Jup. orbit, and the equat.
respectively. ECD a circle paral. to
NG, and F;nQ paral. to the eclip.
Q moves regularly along the circle
Q/F, from the preces. of the equin. and
as Jup. has no force to alter this re-
gular mot. his force being only exert-
ed on the whole body of the earth,
and therefore in altering its orbit and
the pole E of the ecliptic, which
therefore moves in the circle ECD.
Hence the orbit of Jup. must be considered as fixed, and there-
fore the pole I and circle ECD ; in which circle the mot. of
E must therefore be computed. The precess. of the equin.
in 100 years = 1 23' 45" (pa. 244 and 305.) The secular
motion of the nodes of Jupiter is 10' 9"2* (4 art. 70) = 609"2
Jup. asc. node N in 1812 (Z- QEN) 55 8 31' 15". Inci. of
Jup. orbit (1812) 1 18' 49". Hence making Z-QEa = 1 23'
45" and EIC = 10' 9"2. Make Co perp. to Ea, then Eo is the
See pa. 360, where the true secular var. is given.
486 OF THE GENER \L LAWS OF MOTION, fcc.
decrease of EQ or Ea, and is the same as that of the incl. of the
eclip. and equinoctial, (art 71.)
In the AEIC, the 2L.EIC being very small, we have rad : a. 1C
1 18' 40" :: Z.EIC 10' 9"2 :' EC = 13"94 nearly. Again
AcEN or <zEC = 8 3 ' \5" -f l 18' 47" = 9 50' 1" ; *&ien
in the small rt A'd A ECo rad. : EC :: cos. oEC : Eo =
13"73 nearly, the secular decrease of the equin. by the action of
Jup.
The same comput. being applied to Saturn, then EIC = 31 "4,
1C = 2o 29' 45", oEC *= 2lo 32' -f 1 23' 45" = 22 55' 45" ;
hence the deer, by Saturn will be 1"26 nearly, and therefore by
both the deer, will be i 5" nearly.
74 From the preceding art. it appears, 1. That the incl. will
decrease until E and a be at their nearest dist in the two circles,
which will be about 6600 years, after which it will incr. again.
It has been decreasing for more than 8000 years. For the diam.
of the circles ED and FQ being nearly as I : 19.4. And the
AIEQ = 8 1 28' 45", and the diff of the mot of E and Q being
1 13' 36", it will decrease nearly as many centuries as 81 28' 45"
-4- 1 13' 36", which is 66. Also suppl 98 31' 15" ~- 1 13' 36",
gives 80 centuries it has been decreasing. The decrease for every
century is not the same. For at its max. or min. it is very slow,
and is at a stand for a long time. 2. That the incl. can never be
less than about 21, or greater than about 26. For the nearest
and great dist. of the two circles EQ, FQ amount but to these.
Laplace makes these limits (3) 2 42'.
To obtain these results more accurately, more terms of the
series should be made use of; the above results are too small, ' ut
are however sufficient to give the learner an idea of this subject,
the most interesting in modern astronomy For more information
the Celestial Mechanics of Laplace, or his Astr vol 2 may be
consulted. See also Newton's prin. where different methods of
performing this and the preceding articles, relative to the moon,
are given. The above calculation being according to Em.crsorfs
method. Dr. Gregory's Astr. may also be consulted.
DECLINATION OF THE SUN, &c.
487
Declination of the Sun for the years 1312, 1816, 1820, 1824,
BEING LEAP YEARS.
1
Jan.
S.
Feb.
S.
Mar.
S&N.
April
J\fay.
June.
July.
Jug.
Sept.
Oc*.
S.
Nov.
S.
Dec.
B,
N.
N.
N.
N.
N.
N&S.
1
23 5'
1720'
731 /
43tf
15 7'
22 5'
23 8'
lb S
8 16'
314'il* 2i;'
2i51'
2|23
17 3
7 8
4 59
15 25
22 13
23 3
17 46
7 54
3 37
14 48
22
3
22 55
16 46
6 45
5 22
15 42
22 20
22 59
17 31
7 3~
4
15 7
& 9
4
22 49
16 28
6 22
5 45
16
22 2?
2 54
17 15
7 10
4 24
15 26
22 ir
5
22 43
16 10
5 59
6 7
16 17
22 34
22 48
16 5i
6 48
4 47
15 44
22 25
6
22 36
15 52
5 36
6 30
16 34
22 40
22 42
16 42
6 26
5 10
16 3
22 3
7
22 29
15 34
5 13
6 53
16 51
22 46
S2 30
16 26
6 3
5 33
16 20
^2 39
8
22 22
15 15
4 49
7 15
17 7
22 52
22 2i-
16 9
5 41
5 56
16 38
22 45
9
22 14
14 56
4 26
7 38
17 23
22 57
^2 22
lo 51
5 Ib
6 U
17 55
22 51
10
22 5
14 37
4 2
8
17 39
23 2
22 lJ
15 34
4 5-1
642
17 12
22 57
11
21 56
14 17
3 o9
8 22
17 5J
23 6
^2 /
15 16
4 3,
7 4
17 2t
^3 3
12
21 47
13 58
3 15
8 44
18 10
23 10
21 5<J
14 58
4 I
7 27
17 45
23 r
13
21 37
13 38
3 52
9 6
18 25
23 14
21 51
14 40
3 4i
7 49
18 1
23 11
14
21 27
13 18
2 28
9 27
18 39
23 17
21 42
14 22
3 2o
8 12
18 17
23 15
15
21 17
12 57
2 4
9 49
18 54
23 20
21 52
14 3
3 C
8 34
18 33
23 18
16
21 6
12 37
1 41
10 10
19 8
23 22
21 23
13 44
2 37
8 56
1848
23 21
17
20 54
12 16
1 17
10 31
19 21
23 24
2i 13
13 2o
2 14
9 18
19 3
23 23
18
20 43
11 55
53
10 52
19 35
23 26
21 2
13 6
1 50
9 40
19 17
23 25
19
20 31
11 34
29
11 13
19 48
23 27
20 5i
12 46
1 2:
10 2
19 31
23 26
20
20 18
11 U
6
11 34
20
23 27
20 40
12 26
1 4
10 24
19 45
23 27
21
20 5
10 51
N 18
11 54
20 13
23 28
20 29
12 7
40
10 45
19 58
23 28
2O
6
19 52
10 29
42
12 14
20 25
23 28
20 17
11 46
17
11 6
20 11
23 28
23
19 38
10 8
1 5
12 34
20 36
23 27
20 5
11 26
S 6
11 28
20 24
23 27
24
19 24
9 45
1 29
12 54
20 48
23 26
19 53
11 6
30
11 49l 20 36
23 26
25
19 10
9 24
1 52
13 14
20 58
23 25
19 40
10 45
53
12 9
20 48
23 25
26
18 55
9 1
2 16
13 33
21 9
23 23
19 27
10 24
1 17 12 30
21 u
23 23
27
18 40
8 39
2 39
13 52
21 19
23 21
19 13
10 3
i 40 12 50
21 11
23 20
28
18 25
8 16
3 3
14 11
21 29
23 18
19
9 42
2 4 13 11
21 21
23 17
29
18 9
r 54
3 26
14 30
21 38
23 15
18 46
9 21
2 27 13 31
21 32
23 14
SO
10 53
3 49
14 48
21 48123 11
18 31
8 59
2 50113 50
21 4'*,
23 10
31
10 37
21 56f
18 16
8 38
'14 10
23 6
Change of the Sun's decl. for fieriods of four years.
U
4
"8
12
16
20
JANUARY.
FEBRUARY.
MARCH.
to -* >- 1 Periods
o o> to co *^ J of years.
Day^
Days.
Days.
1
Lp.p
vO
1
7
131 19
25
1
7
13
It
25
( +
1.4
2.1
2.8
3.6
OM
0'.2
4
(X.3
6
0'.4
7
O f .4
8
v/.r
1.0
1.5
2.C
2.5
1/5
1.1
1.6
I c -'~
2.7
(j'.C
1.2
1.7
2.3
2.9
O'.c
1 .2
1.9
2 5
3.1
G'.7
1.3
2.0
2.6
3.3
0'.7
1.3
2.0
2.6
3.3
0'.7
1 .4
2.1
2.7
3.5
0'.7'0'.7
1 .4 1 .4
2.1:2/.
2 .8,2 .8
3 ,5|3 .6
.4
.6
.7
.7
.9
1.1
.9
1.2
1.5
1.1
1.4
1.8
1.3
1.7
2.1
In the above tables of declination, the declination is given for the noon of
each day, under the meridian of Greenwich, for four successive years. S and
N shew when the decl. is north or south. These tables are principally cal-
culated for the years 1810, 1811, 1812 and 1813.
The tablt; of the change of the sun's decl. is intended to reduce this decl.
to a subsequent period. The variation of decl. is given opposite the years,
and under the 1st, 7th, Sec. days of the month, which is to be added to or sub-
tracted from the d^cl. in the table, according as the sine over it is -f- or ,
488
DECLINATION OF THE SUN, Sec.
Declination of the Sun for the years 1813, 1817, 1821, 1825, We.
BEING THE FIRST AFTER LEAP YEAR.
i
Jan.
Feb.
Mar.
Jlpril
May.
June.
July.
Jug-.
Sept.
Oct.
'JYov.
Dec.
t='
S.
S.
S&N.
N.
N.
N.
N.
N.
NkS.
S.
S.
S.
1
23 1'
17 7
737'
430'
15 5'
22 3'
23 '
18 5'
821'
3 8'
1425'
2149'
2
22 56
16 50
7 14
4 53
15 21
22 11
23 5
17 50
7 59
3 31
14 44
21 58
3
22 51
16 32
6 51
5 16
15 38
22 18
23
17 35
7 37
3 55
15 3
22 7
4
22 45
16 14
6 28
5 39
15 56
22 26
22 55
17 19
7 16
4 18
15 22
22 15
5
22 38
15 56
6 5
6 2
16 13
22 33
22 50
17 3
6 53
4 41
15 40
22 23
6
22 31
15 38
5 41
6 25
16 30
22 39
22 44
16 46
6 31
5 4
15 58
22 30
7
22 24
15 19
5 18
6 47
16 47
22 45
22 38
16 30
6 8
5 27
16 16
22 37
8
22 16
15
4 55
7 10
17 3
22 51
22 31
16 13
5 46
5 50
16 34
22 44
9
22 7
14 41
4 51
7 32
17 20
22 56
22 24
15 56
5 23
6 13
16 51
22 50
10J21 59
14 22
4 8
7 55
17 35
23 1
22 17
15 38
5 1
6 36
17 8
22 56
11I21 49
14 2
3 44
8 17
17 51
23 5
22 S
15 21
4 38
6 59
17 25
23 1
12
21 40
13 43
3 21
8 39
18 6
23 10
22 1
15 3
4 15
7 21
17 41
23 6
13
21 30
13 22
2 57
9
18 21
23 13
21 53
14 45
3 52
7 44
17 58
23 10
14 21 19
13 2
2 34
9 22
18 36
23 16
21 44
14 26
3 29
8 6
18 13
23 14
15 21 9
12 42
2 10
9 44
18 50
23 19
21 35
14 8
3 6
8 29
18 29
23 17
16)20 57
12 21
1 46
10 5
19 4
23 22
21 25
13 49
2 43
8 51
18 44
23 20
17
>20 46
12
1 23
10 26
19 18
23 24
21 15
13 30
2 19
9 13
18 59
23 23
18
20 34
11 39
59
10 47
19 32
23 25
21 5
13 11
1 56
9 35
19 14
23 25
19
20 21
11 18
35
11 8
19 45
23 26
20 54
12 51
1 33
9 57
19 28
23 26
20
20 8
10 56
12
11 29
19 57
23 27
20 43
12 31
1 9
10 18
19 42
23 27
21
19 55
10 35
N 12
11 49
20 10
23 28
20 32
12 12
46
10 40
19 55
23 28
22
19 42
10 13
36
12 9
20 22
23 28
20 20
11 52
23
11 1
20 8
23 28
23
19 28
9 51
59
12 30
20 34
23 27
20 8
11 31
S 1
11 22
20 21
23 27
24
19 13
9 29
1 23
12 49
20 45
23 26
19 56
11 11
24
11 44
20 33
23 26
25
18 59
9 7
1 47
13 9
20 56
23 25
19 43
10 50
48
12 4
20 45
23 25
26
18 44
8 44
2 10
13 29
21 7
23 23
19 30
10 29
1 11
12 23
20 57
23 23
27
18 28
8 22
2 34
13 48
21 17
23 21
19 17
10 8
1 34
12 46
21 8
23 21
28
18 13
7 59
2 57
14 7
21 27
23 19
19 3
9 47
1 58
13 6
21 19
23 18
29
17 57
3 21
14 26
21 36
23 16
18 49
9 26
2 21
13 26
21 29
23 15
30
17 41
3 44
14 44
21 46
23 12
18 35
8 5
2 45
L3 46
21 39
23 11
31
17 24
4 7
21 54
18 43)
8 43
14 5
25 7
Change of the Sun's dec I. for periods of four years.
8 2
o a
T ^
APRIL.
MAY.
JUKE.
I!
Days.
Days.
Days.
11 I
13! 19
25
1
7
13
19
25
1
7
13
19
25
^^
+ i +
+! +
+
+
+
+
-f
+
+
-f
-f
f
+
4
8
0'.7|0'.7
I All A
0'.7
1.3
0'.6
1.3
V.6
1.2
0'.6
1.1
0'.5
1.0
0'.5
.9
0'.4
.8
0'.3
.7
0'.3
.5
V-2
A
OM
.2
O'.O
.0
OM
.1
4
8
12
2.1:2.1
2.0
1.9
1.8
1.7
1.6
1.4
1.2
1.0
.8
.5
.3
.1
is 12
16
2.8J2.7
2.6
2.5
2.4
2.3
2.1
1.9
1.6
1.3
1.0
.7
.4
.1
.3
16
20
3 .513 .4
3.3)
3.2
3.0
2.8
2.6
2.3
2.0
1.6
1.3
.9
.5
.1
.3
20
Thus if the sun's declination for the 1st of May, 1824, be required. The given
year being leap year, or 12 years after 1812. Hence
The sun's declination 1st of May, 1812, is 15 7' N.
Equation or change for 12 years, is -j- 1.7
Sun's declination, 1st of May, 1824, 15 8.7
Again, required the sun's decl. at noon, for the 30th of September, 1833 ?
Here the given year is the 1st after leap year, and is 20 years after 1815
Hence
DECLINATION OF THE SUN,
489
Declination of the Sun for the years 1810, i814, 1818, 1822,
BEING THE SECOND AFTER LEAP YEAR.
\~
Jan.
Feb.
Mar.
April.
May.
June.
July.
.lug.
Sept,
Oct.
wVou
Dec.
1
" S.
S.
S&N.
IS.
N.
N.
N.
N.
is &s-.
S.
S.
S.
1
23 3'
17 il'
742'
424 /
1458'
9/0
1'
2310,
18 9'
827'
3 2'
1420 /
2i47'
2
22 58
16
54
7 20
4 47
15
16
22
9
23 6
17 54
8
5
3 25
14
39
21 56
3
22 53
16
3/
6
5 10
15
34
22
17
23 2
17 38
7 43
3 48
14
58
22 5
4
22 47
Lu
19
6 34
5 33
15
52
22 24
22 57
17
23
7 21
4 12
15
i;
22 13
5
22 40
16
1
6
11
5 56
16
9
2 31
2'4 5*
17
7
6 59
4 35
15
36
22 21
6
22 33
15
43
5 47
6 19
16
26
22 38
22 4b
16 51
6 37
4 58
15
54
-22 28
7
22 26
15
24
5 24
6 42
16
43
22 44
22 40
16 34
6 14
5 21
16
12
22 35
8
22 18
15
5
5
1
7 4
17
22 50
22 34
i6
17
5 52
5 44
16
30
22 42
9
22 10
14
46
4 38
7 27
17
16
22 55
22 27
16
5
o9
6 7
16
47
22 48
10
22 1
14 27
4
15
7 4.'
17
32
23
22 20
15
43
5
6
6 30
17
4
22 54
11J21 52
14 8
3 51
8 11
17
47
23
4
22 12
15 25
4 44
6 53
17
21
23
12 21 43
13
48
3 27
80 .->
oo
18
2
23
8
22 4
15 8
4
Jl
7 16
17
37
23 5
1321 33
lo
28
3
4
8 55
18
17
23 12
21 56
14 50
3 58
7 38
17
53
23 9
1421 23
13
8
2 40
9 17
18
32
23
16
21 48
14 31
3 35
8 1
18
9
23 13
15121 12
12
47
2
16
9 38
J8
47
23
!<>
21 39
14 13
3
12
8 23
18
25
23 17
1621
12 26
1
52
10
19
I
23
21
21 29
13 54
2 49
8 45
18
41
23 20
17UO 48
12 5
1 29
10 21
19
15
23
23
21 19
13 35
2 25
9 8
18
56
23 22
1820 3b
11 44
1
6
10 42
19 29
23
25
21 9
13 16
2
2
9 30
19
10
23 24
19! 20 24
11 23
42
11 3
19 42
23
26
20 58
12 56
1 39
9 51
19
24
23 26
20! 20 11
11 2
18
11 24
19 55
23
27
20 47
12 37
1
15
10 13
19
38
23 27
21 1 :> 58
10 40
N
5
11 44
20 7
23
27
20 36
lie
17
52
10 35
19
52
23 28
22 19 45
10 ly
29
12 J
20 19
23
28
20 24
11 57
29
10 56
20
6
23 28
23
19 31
9 57
52
12 25
20 31
23
28
20 12
11 37
5
11 17
20
19
23 27
24
19 17
9 35
1
16
12 45
20 42
23
27
19 59
11
17
S
18
11 38
20
31
23 27
.25
19 2
9 12
1
40
13 4
20 53
23
26
19 46
10 56
42
11 59
20
42
23 26
26
18 47
8 50
2
4
13 24
21 4
23
24
19 33
10 35
1
5
12 20
20
54
23 24
27
la 3^
8 28
2
28
13 43
21 14
23
22
19 20
10 14
1 29
12 41
21
5
23 122
28
l8 17
8 5
2
51
14 2
21 24
23
20
19 6
9 53
1 52
13 1
21
16
23 19
29
18 1
3
15
14 21
21 34
23
17
18 52
9 32
2
15
13 21
21
27
23 16
30
17 45
3
38
14 40
21 43
23
14
18 38
9 10
2 39
13 41
21
37
23 13
31
17 28
4
1
21 52
18 24
8 49
14 1
25 9
Change of the Sun's decl. for periods of four years.
% i
JULY.
AUGUST.
SEPTEMBER.
82'
% B
Days.
Days.
Days.
e 1
*>
1
7
13
19)25
1
7
13
19 (
25
1
7
13119
25
~
<5
4
OM
o'.2 ex.:
5 0'.4 O f .4
(X.5
O'o
O 7 .
5 O'.o C
' ?'
0'.7
0'.7
O'.70'. ( (
X.7
4
8
.3
.4
.(
> .7 .9
1.0
1*1
1.'
21.3 1
.3
1.4
1 ,4|l .41 .4
.4
8
12
.4
.7
,c
) 1 .1 1 .3
1.5
1.6
I.
31.92
.0
2.0
2.1
2.1 2. J .
'. .1
12
16
.6
.9
I/
> 1 .4 1 .7
2.0
1 .2
2.
42.52
.6
2.7
2.8
2.82 .9 ,.
9
16
20
.7
1.1
I
51.82.2
2.5
2.7
3.
3 3.2C
,3
3.43.5
3 .
5 3 .61:
f.6
20
The sun's declination 30th of Sept. 1813, is 2 45' S.
Equation or variation for 20 years, is -f- 3.6
Sun's declination 30th Sept. 1833, 2 48'.6
The correction may be found independent of the table thus; From the given
year take as many times 4 as will reduce it to one of the years lo which the
table is adopted, and take out the decl. answering- the given time, as before ;
find also the decl. for the following day, and multiply the difference between
them by ith of the difference between the given and tabular years, and the
3 O
490
DECLINATION OF THE SUN, Sec,
Declination of the Sun for ike years 1811, 1815, 1819, 1823, &c.
BEING THE THIRD AFTER LEAP YEAH.
i
Jan.
Feb.
Mar.
April.
N.
May.
June.
July.
Jlug.
Sept.
Oc*
'Wav.
Dec.
S.
S.
S&N.
Nv
N.
N.
N.
N&S.
S.
S.
S.
1
3 4'
17 16'
748'
418'1453'
2159'
23ll'
1813'
852'
256'
14 15'
44>
2
2 59
16 59
7 25
4 4215 11
22 7
23 7
17 58
8 11
3 1.
14 34
21 53
3
2 53
16 41
7 3
5 5;15 29
22 15
23 2
17 4~
7 49
3 43
14 55
22 2
4
2 48
16 24
6 40
5 28(15 47
22 22
22 58
17 27
7 2?
4 6
15 11,
22 11
5
2 41
16 6
6 17
5 50:16 4
22 29
22 5
17 11
7 4
4 29
15 3i
22 19
6
2 35
15 47
5 53
6 1316 21
22 36
22 47
16 55
6 42
4 52
15 4:
22 27
7
2 27
15 ft
5 JO
6 3616 38
22 4^
J2 4i
16 38
6 .LO
5 15
16 ?
2? 34
8
2 20
15 10
5 7
6 58' -16 55
22 48
32 35
16 21
5 5.
5 38
16 25
2 41
9
2 12
14 51
4 43
7 21J17 U
22 53
2? 28
16 4
5 3
6 1
16 42
22 47
10
22 3
14 32
4 20
7 43 ' 17 27
22 5^
22 21
L> 4.
5 1'-
6 24
16 59
22 53
1J
21 54
14 13
3 57
8 5 17 43
23 3
22 13
15 30
4 *9
', 47
17 If
2V 58
12
21 45
13 53
3 33
8 27 17 58
23 7
22 5
15 l~
4 2o
7 10
17 33
$J3 3
13
21 35
15 33
3 9
8 4918 IH
23 11
21 57
14 54
4 4
7 3*.
17 45
25 8
14
21 25
13 13
2 46
9 1118 28
23 15
21 48
14 36
4 4i
7 55
18 5
2 12
15
21 14
12 52
2 22
9 33
18 43
23 18
21' 3;'
14 17
3 17
8 17
18 21
23 16
16
?1 3
12 32
1 59
9 54
18 57
23 21
21 30
IS 58
2 54
8 40il8 36
23 19
17
20 52
12 11
1 35
10 15
19 11
23 23
21 20
13 39
2 Si
9 2!l8 51
23 21
18
20 40
11 50
1 1
10 36
19 25
23 25
2! 10
13 20
2 8
9 24| 19 6
25 23
19
20 28
11 29
47
10 57
19 38
23 26
21
13 1
1 45
9 46
19 21
25 25
20
C-0 15
11 7
24
11 18
19 51
23 27
2o 4:
1-:; 41
i 21
10 7
IP 35
23 27
21
20 2
10 46
11 39
20 3
23 28
20 38
12 22
58
I., 29|19 48
23 27
2?
19 4:
10 24
N 24
11 59
20 16
23 28
JO 26
12 2
3:
10 50
20 2
25 28
2
10 3j
iO 2
47
12 i>
20 28
23 27
20 14
11 41
11
11 12 20 14
23 28
2
19 21
9 40
1 11
12 3
20 3'
23 2r
20 2
11 21
S 12
11 33 20 2:
23 37
2
19 6
9 IS
1 So
12 5-;
20 50
25 2*
19 50
11 .
36
11 54120 3C
23 26
2
1H 51
8 5
1 58
15 1
21 1
23-24
19 3,
10 4i-
59 12 15
20 51
2 24
2
18 36
8 33
2 2
13 38
21 12
23 2J
19 24
10 V.
1 23 12 33
21 2
23 22
2
18 21
8 11
24-
13 57
21 22
23 20
19 10
9 58
I 46 12 55
21 13
23 20
2
18 5
3 9
14 .6
21 3-2
23 1-'
18 56
9 37
2 9 13 16
21 24
25 17
3
17 4S
3 3
14 35
21 41
23 14
18 42
9 1-
2 33 13 30
21 34
23 13
3
17 3$
3 55
21 iO
18 28
8 54
13 5.>
23 9
1 1
OCTOBER.
N V E M B E K
DECEMBER.
Co &3
o C3
' ^
Days.
Days
Days
^
1
7
13 19
25
1
IP
19
25
1
7
13
19
25
^
_|_
_|_
_|_ _{_
_[_
-{-
~H ~f~
+
-f-
_|_
_|.
+
_j_
4
O'./
o'.r
o'.r o'.e
O r .6
0'.5
0'.5
0'2
or
O'.O
O'O
4
8
1.4
1.4
1.31.3
1.2
1.1
1.0! .9
.8
.7
.5
.4
.2
.0
.1
8
12
2.1
2.0
2 .01 .9
1.8
1.6
lvtfl.3
1.2
1.0
.7
.5
.3
.1
.1
12
16
20
2 JB|2 .7
3 .5J3 .4
2 .6;2 .5
3.3J3.2
2 .4
s!o
2.2
2.8
2. Oil. 8 1.5
2 .5 ! 2 .212 .0
1.3
1.7
1.0
1.3
.7
.9
.4
.1
.2
.2
.3
16
20
product divided by 33 will give the correction ; which is to be added when
the decl. is increasing- or to be subtracted when decreasing-. If the given lime
be anterior to that in the tables, the correction is to be applied in a contrary
manner. Thus, if the sun's decl for the 1st of May, 1824, be required. Here
the given year is 12 years after 1812, the sun's decl. on the 1st of May, 1812,
is 15 7', and for the 2d, 15 C 2*', the difference is 18'. Now 18 X 3 (because
12 -T- 4 = 3) gives 54 ; hence 54 -i- 33 = 1.6 nearly, the correction, which
added to 15 7', because the decl. is increasing 1 , gives 15 8'. 6, the declination
for the 1st of May, 1824.
A TABLE
OP THE
Latitudes and Longitudes
OF SOME OF THE
PRINCIPAL PLACES IN THE WORLD,
Collected from the most authentic Tables, Mafo, Astronomical
Observations^ &c.
THE longitude, or difference of meridians, is reckoned from the
meridian of Greenwich observatory, \vhichis 5' 37" east of Lon-
don, 2 19' according to Mayer (or 2 19' 42" according to
Delambre) west of Paris observatory, 9 5 >', according to Mayer,
west of Gottingen observatory, 17 33' 45" east of the town of
Ferro, 75 14' 22" east of Philadelphia, 74 l' ast of New-
York, 71 3' 37" east of Boston, and 77 14' east of Wash-
ington City, generally reckoned 77 9'.
There is nothing in which authors disagree more than in the
lat and long, of places, as they generally copy each other, or the
mistakes of those who have actually made observations, and whose
errors must have arisen from the imperfection in their instru-
ments, or want of the necessary knowledge in using them.
The Tide-Table annexed to the following latitude and longitude
of places, gives the distance of the moon from the meridian,
when it is high water at those respective places. As there are
two tides in the day, it is evident that this table will only give the
moon's distance from the meridian at one of them : if this time be
however taken from 12, the dist. of the moon from the meridian at
the other will be given. Thus, if when the moon is full, high water
at New- York be at 9 o'clock, P. M. it is evident that the next tide
will be about 9 o'clock in the morning, when the moon will be
about 3 hours distant from the meridian ; and this is the reason
that some make the time of hi^h water in New- York 9 hours,
while others make it 3. The same observation will hold for any
other place.
492 LATITUDES AND LONGITUDES OF PLACES.
Names of Places.
Country or Sea.
Lat.
Long.
H. -water
Fee:
A.
Acre,
Syria,
32 42 N.
35 10 E.
Abbeville,
France,
50 7'N.
149'E.
10h.30'
Aberdeen,
Scotland,
57 9 N.
2 8 W
45
Abo,
Finland,
60 27 N.
22 13 E.
Acapulco,
Mexico,
17 10 N.
101 26 W
Achen,
Sumatra, I.
5 22 N.
95 35 E.
Adrianople,
Turkey,
41 12 N.
26 28 E.
Adventure Bay,
Van Dieman's, L
43 22 S.
147 30 E.
4 36
Agra,
Hindoostan,
26 43 N.
78 45 E,
Air,
Scotland,
54 25 N.
4 26 W
10 30
Aix,
France,
43 32 N.
5 26 E.
Albany,
N. America,
42 39 N.
73 46 W
3 24
Aleppo,
Syria,
36 11 N.
37 10 E.
Alexandretta,
Syria,
36 35 N.
36 15 E.
Alexandria,
Egypt,
31 12 N.
29 55 E.
Alexandria,
Virginia,
38 45 N.
77 16W
Algiers,
Africa,
36 4-9 N.
2 12 E.
Alicant,
Spain,
38 21 N.
SOW
Altorf,
Switzerland,
46 53 N.
8 37 E.
Am boy,
N. Jersey,
40 33 N.
74 20 W
8 9
Amboyna, I.
Moluccas,
3 36 S.
128 15 E.
Amiens,
France,
49 54 N.
2 18 E.
Amsterdam,
Holland,
52 22 N.
4 51 E.
3
r
Ancona,
Italy,
43 38 N.
13 30 E.
Angers,
France,
47 28 N.
34 W
Angra,
Tercera I. Azores
38 39 N.
27 12W
11 45
st
Annapolis,
Maryland,
39 2 N.
76 45 W
10
Antigua, I. St>
T 1 9 *
Carib. Sea,
17 4 N.
62 9 W
John s town, 5
Antioch,
Syria,
35 55 N.
36 15 E.
Antrim,
Ireland,
54 58 N.
6 27W
Antwerp,
Netherlands,
51 13 N.
4 24 E.
6 45
Archangel,
Kussia,
64 34 N.
38 55 E.
6
Arica,
Peru,
18 27 S.
71 13W
Ascension, I.
S. Atlant. Ocean,
7 56 S.
14 21 W
Astracan,
Russia,
46 21 N.
48 3 E.
Athens,
Turkey Eur.
38 5 N.
23 52 E.
Ausburg,
Germany,
48 19 N
10 56 E.
Augusta,
Georgia, U. S.
33 20 N.
81 4W
St. Augustine,
E. Florida,
29 58 N.
81 40W
4 30
Ava,
East India,
21 56 N.
95 15 E.
Avignon,
France,
43 57 N.
4 48 E.
Avranches,
France,
48 41 N.
1 22 W
6 30
Auxerre,
France,
47 48 N.
3 34 E.
;
LATITUDES AND LONGITUDES OF PLACES. 493
Names of Places.
Country or Sea.
Lat.
Long.
//. water
Feet
B.
Babel mandebstr,
Red Sea,
12 50 ; N.
43 45'E.
Oh. 0'
Babylon, (anc.)
Syria,
33 N
42 46 E.
Badajoz,
Spain,
38 46 N.
6 45W.
Bagdad,
Syria,
33 20 N.
44 23 E.
Balasore,
Hindoostan,
20 21 N-
86 45 E.
10
Balbec,
Syria,
33 50 N
36 20 E.
Baltimore,
Ireland,
51 16N.
9 30 VV.
4 30
Baltimore,
Maryland,
39 20 N.
76 43W.
Banca, I. S. end,
Indian Ocean,
3 15 S.
107 10 E.
Bantam point,
Java I.
5 50 S.
106 9 E.
Bantiy,
Ireland,
51 27 N.
9 46W.
5 15
Barbuda I.
Atlantic,
17 49 N.
62 OW.
Barcelona,
Spain,
41 26 N.
2 12 E.
Basil or Basle,
Switzerland,
47 34 N.
7 35 E.
Basse Terre,
Guadaloupe,
15 59 N.
61 54W.
Bassora or Basra,
Turkey, A.
30 25 N.
47 30 E.
Bastia,
Corsica,
42 42 N.
9 25 E.
Batavia,
Java I.
6 11 S.
106 52 E.
Bayonne,
France,
43 29 N.
1 29W-
3 30
15
Beachy head,
England,
50 44 N.
16 E.
10 30
Belfast,
Ireland,
54 43 N.
5 57W.
10
Belgrade,
Turkey, E.
45 ON.
21 20 E.
Bencoolen,
Sumatra,
3 49 S.
102 3 E.
Bennington,
Vermont, U. S.
42 50 N.
73 6W.
Bergen,
Norway,
60 23 N.
5 12 E
1 30
Bergen-op Zoom
Holland,
51 30 N.
4 17 E.
I 30
Berlin,
Germany,
52 32 N.
13 23 E.
Bermudas I. N.
Atlantic,
32 35 N.
64 28 VV.
6 30
*1
Berne,
Switzerland,
46 57 N.
7 26 E
Berwick,
Scotland,
55 47 N.
2 5W.
2
Bethlehem,
Pennsylvania,
40 37 N.
75 25W.
Bilboa,
Spain,
43 26 N.
2 47 \V.
3 45
15
Bologna,
Italy,
44 30 N.
11 21 E.
Bologne,
France,
50 43 N.
1 36 E.
10 45
Bombay I.
India, E.
18 56 N.
72 54 E.
Boston,
Massachusetts,
42 23iN
71 OW.
11 9
Botany Bay,
N. Holland,
34 S.
151 20 E.
8
Bourbon, I. N.
Ind. Ocean,
20 51 S.
55 30 E.
Bourdeaux,
France,
44 50 N.
35 W.
7 14
15
Braganza,
Portugal,
41 53 N.
7 3VV.
Breda,
Netherlands,
51 36 N.
4 46 E.
Bremen,
Germany,
53 5 N.
8 49 E.
5 45
Breslaw,
Silesia,
51 5N.
17 6 E.
Brest,
France,
48 23 N.
4 30 E.
3 45
\9
494 LATITUDES AND LONG1TODES OF PLACES.
J\f(imeft of Places.
Country or Sea.
Lat.
Long.
H tvater
Bristol,
England,
5l28'N
235'W
6h.45'
Bruges,
Netherlands,
51 3 N
3 13 E
Brunswick,
Germany,
52 25 N
10 3i E
Brunswick,
Dist. Maine,
43 52 N
69 59 VV
Brunswick,
New-Jersey,
39 39 N
74 18VV
Brussels,
Netherlands,
50 51 N
4 21 E
Breda,
Hungary,
47 30 N
19 E
Buenos Ayres,
S. America,
34 35 S
58 34 W
Bukarest,
Turkey,
44 27 N
26 8 E
Burlington,
New- Jersey,
40 5 N
75 6W
9 14
Burgos,
Spain,
42 20 N
3 30W.
C
Cabello port,
S. America,
10 31 N
67 32W.
Cadiz,
Spain,
36 3 ! N
6 17W.
2 30
Caernarvon,
Wales,
53 6 N
4 30W.
7
Cagiiari,
Sardinia I.
39 25 v
9 38 E.
Cairo,
Egypt,
30 3 N
31 17 E.
Caithness point,
Scotland,
58 46 N.
3 22W.
9
Calais,
France,
50 57^ N
1 50 E.
11 45
Calcutta, F. VV.
Bengal,
22 35 N
88 28 E.
3 5
Callao,
Peru,
12 2 N
76 58W.
6 30
Calmar,
Sweden,
56 41 N.
16 25 E,
Cambray,
Motherlands,
50 It N.
SUE.
Cambodia,
East India,
13 1 N.
105 E.
Cambridge,
England,
52 13 N.
5 E.
Cambridge,
Massachusetts,
42 23N
71 7W.
Canary, I. N. E.
Canary, Is.
28 13 N.
15 39W.
3
Cancli,
Ceylon,
7 45 N.
80 46 E.
Candia,
^andy, I.
35 19 N.
25 18 E.
Canton,
China,
23 7 N"
113 16 E.
Cape Canso,
Siova Scotia,
45 16 N
60 55W
8 30
Can tin,
Morocco,
32 44 N.
9 10VV.
1
Clear,
re land,
51 18 N.
9 30W.
4 30
Ortegal,
Spain,
43 46 N
7 39 W
3
Finisterre,
Spain,
42 53 N.
9 18W.
3 15
St. Vincent,
3 ortugal,
37 2 N.
9 2W.
3
Bajodor,
Africa,
26 13 N.
14 27W.
Blanco,
do.
20 55 N.
17 10W.
9 45
Verd,
do.
14 47 N.
17 33W.
7 45
Sieru Leon,
do.
8 30
13 9VV.
8 15
Mount,
do.
6 46
11 48 W.
Palmas,
do.
4 30
7 41W.
Good Hope. ~>
do.
34 29 S.
18 23 E,
3
Town. 5
do.
33 5o S.
18 23 E.
2 30
Comorin,
iindoostan,
8 4N.
77 34 E.
LATITUDES AND LONGITUDES OF PLACES. 495
.Mimes of Places.
C<iirtKt.ry or Sea.
Lat.
Long-.
//. water
Feet
Cape Lapotka,
iamskatcha,
51 O'N.
15642' E.
Race,
Newfoundland,
46 40 N.
53 44 W.
Sable,
Nova Scotia,
43 24 N.
65 39 W.
8h.l5'
Cod, (light)
Massachusetts,
42 5 N.
70 14 W.
1 1 30
H
Charles,
Virginia,
37 12 N.
76 9 W.
7
Hatteras,
N. Carolina,
35 12 N.
75 5 \V.
11
Francois (new)
do.
19 46 N.
72 18 \V.
6 Q
3
Horn,
S. America,
55 58 S.
67 26 W.
Blanco,
Patagonia,
47 20 N.
64 42 VV.
Farewell,
Greenland,
59 38 N.
42 42 W.
Florida,
America,
25 47 N.
80 35 \V.
7 30
Capricorn,
N. Holland,
23 27 ^.
151 6 E.
8
Diggs,
Labradore,
62 41 N.
78 51 W.
Henry,
Virginia,
36 57 N.
76 19 W.
10 54
4
Lahogue,
France,
49 45 N.
1 57 W.
8 30
18
May,
N ew-Jersey,
39 4 N.
74 54 \V.
8 9
Cardigan,
Wales,
52 2 N.
4 45 W.
7 15
Carthage ruins,
Tunis,
36 35 N.
10 10 K.
Carthagena,
Spam,
37 37 N.
1 1 W
8
Carthagena,
Terra Fir ma,
10 26 N.
75 21 VV.
2
10
Casan,
Siberia,
55 44 N.
49 28 E.
Cayenne,
Cayenne, I. S. A
4 56 N.
52 16 E.
4
6
Charleston,
South Carolina,
32 50 N.
80 1 VV.
7 54
6
Charlestown,
Massachusetts,
42 22 N.
71 1 VV.
Christiana,
Norway,
59 55 N
10 48 E.
Christiunsand,
Norway,
58 10 N.
8 2 E.
Chri-stianstat,
Sweden,
56 5 N.
14 2 E.
Christmas sound,
Terra del Fuego,
55 22 S.
70 3 W.
2 30
St. Christophers,
West Indies,
17 15 N.
62 43 W.
Cologne,
Germany,
50 55 ;N.
6 55 E.
Columbia,
South Carolina,
33 58 N.
81 5 W.
Conception,
S. America,
36 43 S.
73 6 W.
3
Constance,
Germany,
47 37 N.
9 13 E.
Constantinople,
Turkey,
41 1 N.
28 55 E.
Copenhagen,
Denmark,
55 41 N.
12 35 E.
Corinth,
Turkey,
37 54 N.
22 55 E.
Cork,
Ireland,
51 54 N.
8 28
4 45
Corvo,
Azores,
39 43 N.
31 5VV.
Cracow,
Poland,
50 11 N
19 50 E
St. Cruz, I.
Atlantic,
17 49 N.
64 53 W.
Curacoa, I. 7
W. Indies,
12 16
69 7 W
north point, >
Cusco,
Peru,
12 25 S
73 35 W.
496 LATITUDES AND LONGITUDES OF PLACES.
JVames of Places.
Country or _Sca.
Lat.
Lonff.
H.ivater
Feet
D.
Damascus,
Syria,
33 16'N.
3620'E.
Dantzick,
Poland,
54 22 N.
18 38 E.
Dardanelles,
Turkey,
30 10 N.
26 26 E.
St. David's head,
Wales,
51 55 N.
5 27W.
6h. 0'
36
Delhi,
Hindoostan,
28 37 N.
77 40 E.
Deseada, I.
W. Indies,
16 36 N.
61 10W.
Detroit,
United States,
42 31 N.
83 12W.
Deventer,
United Prov.
52 17 N.
6 13 E.
Dieppe,
France,
49 56 N.
1 4 E.
10 30
Dijon,
Burgundy,
47 19 N.
5 1 E.
Dingle Bay,
Ireland,
51 55 N.
10 49W'
4
St. Domingo,
Hispaniola,
18 20 N.
69 46 W.
Dort,
Hohand,
51 47 N.
4 35 E.
3
Douay,
Flanders,
50 22 N.
3 5 E.
Douglas,
I. of Man,
54 7 N.
4 38W.
10 30
Dover,
England,
51 8 N.
1 19 E.
11 45
16
Dresden,
Germany,
51 3 N-
13 41 E.
Drontheim,
Norway,
63 26 N-
10 22 E.
Dublin,
Ireland,
53 22 N.
6 17W.
9
Dublin, obs
Do.
53 23 N.
6 20W
Dunbar,
Scotland,
56 1 N.
2 33 W.
3 30
Dungarvon,
Ireland,
52 N.
6 50W,
4 30
Dungeness,
England,
50 52 N.
59 E.
9 45
Dunkirk,
France,
51 2 N.
2 22 E.
Dun nose,
I. of Wight,
50 37 N.
1 HW.
9 45
E.
East Cape,
New Zealand,
37 44 S.
178 58W.
Eddystone light,
England,
50 8 N.
4 24W.
5 30
18
Edenton,
N. Carolina,
36 6 N.
76 50W.
Edinburgh,
Scotland,
55 57 N.
3 12W.
4 30
13
Embden,
Germany,
53 23 N.
7 10 E.
15
Ephesus,
Natolia,
38 N.
27 53 E.
Erzerum,
Natolia,
39 56 N.
41 10 E.
Eustatia I.
West-Indies,
17 30 N.
63 2W.
Exeter,
England,
50 44 N.
3 34W.
10 30
F.
Fair Island,
Orkney Is.
59 30 N.
1 46W-
10
Fal mouth,
England,
50 8 N.
5 2W-
5 30
18
False Cape,
Delaware,
38 38 N.
75 9W-
Fayal Town,
Azores,
38 32 N.
28 41W-
2 20
Fayetteville,
N. Carolina,
35 11 N.
78 50 VV-
Ferrara,
Italy,
44 50 N.
11 36 .
Ferro (Town)
Canaries,
27 47 N.
17 46W.
3
LATITUDES AND LONGITUDES OF PLACES. 497
Jfcme* of places.
Country or Sea,
Lat.
Long.
ff water \Feet
Ferrol,
Spain,
4329'N.
815'W.
3h. 0'
15
Fez,
Africa,
33 31 N.
5 OW.
Florence,
Italy,
43 46 N.
11 3 E.
Flores,
Azores,
39 26 N.
31 11W.
flushing,
United Prov.
51 27 N.
3 34 E.
45
N. Foreland,
England,
51 23 N.
1 27 E.
10 20
JFort Royal,
Martinico,
14 36 N.
61 10W.
7 30
2*
France,!, of S.W.
Indian Ocean,
20 27 N.
57 15 E.
30
3
Francfort on ?
the Main, 3
Germany,
50 8 N.
8 35 E.
Frankfort,
Kentucky,
38 4 N.
85 12 W.
Fribourg,
Switzerland,
46 48 N.
7 8 E.
Fuego I.
Cape Verd Is.
14 57 N.
24 22 W.
Funchal,
Madeira,
32 38 N.
16 56W.
4
11
Gal way,
Ireland,
53 10 N.
10 1W.
4
Geneva,
Switzerland,
46 12 N.
6 8 E.
Genoa,
Italy,
44 25 N.
8 50 E.
Georgetown,
Columbia dist.
38 55 N.
77 14
Georgetown,
S. Carolina,
33 32 N.
79 3W.
Fort St. George,
or Madras,
13 5 N.
80 25 E.
St. George's town
Bermudas I.
32 22 N.
64 33W.
5 30
St. George's?
Isle, W. 5
Azores,
28 53 N.
28 10W.
Ghent,
Netherlands,
51 3 N.
3 43 E.
Gibraltar,
Spain,
36 5 N.
5 4W.
Glasgow,
Scotland,
55 52 N.
4 15W.
3
18
Gluckstad)
Holstein,
53 48 N.
9 27 E.
Goa,
Malabar,
15 28 N.
73 59 E.
Gondar,
Abyssinia,
12 34 N.
37 28 E.
1 30
Gottenburg,
Sweden,
57 42 N.
1 1 57 E.
Gottingen (ob.)
Germany,
51 32 N.
9 54 E.
Gvavesend,
England,
51 28 N.
20 E.
Greenwich (ob.)
England,
51 28^N.
2 40
Groningen,
United Prov.
53 10 N.
6 22 E.
Guadaloupe,
West-Indies,
15 59 N.
61 59 W.
Guernsey,
British ch.
49 30 N
2 52W.
1 30
Hnerlem,
Holland,
52 22 N.
4 36 E.
9
Haeue,
Holland,
52 4 N.
4 17 E.
8 15
Halifax,
Nova Scotia,
44 44 N.
63 36W.
7 30
Hamburg,
Germany,
53 34 N.
9 54 h.
6 15
Hinghoo,
China,
30 25 N.
120 12 E.
Hanover,
Germany,
52 22 N.
9 45 E.
Hu'*tford,
Connecticut,
41 50 N.
72 35W.
11 14
Havannah,
Cuba I.
23 12 N.
82 18 W,
3 P
498 LATITUDES AND LONGITUDES OF PLACES.
Names of Places.
Country or Sea.
Lat.
Long.
H. -water
Havre de Grace,
France,
490 29'N.
0* 6' E.
9h. 0'
St. Helena, 7
James town, 5
Atlantic,
15 55 S.
5 49^W.
2 15
Hervey's I.
Society Isles,
19 \7 S.
158 56 W.
Holla,
Iceland,
65 45 N.
19 44 N.
Holyhead,
Wales,
53 23 N.
4 45 W.
9 45
Hull,
England,
53 48 N.
33 W
6
I.
Jackson TPort)
New Holland,
33 52 S.
151 14 E.
8 15
Jickutskoi,
Siberia,
62 2 N.
129 44 E.
Jaffa,
Siberia,
32 5 N
35 10 E.
St. Jago,
Cuba I.
19 55 N.
75 35 W.
Jassay,
Moldavia,
47 8 N.
27 30 E.
Java head,
Java I.
6 49 S
105 7 E.
Ice Cape,
Nova Zembla,
75 30 N.| 67 30 E.
Jeddo,
Japan Is.
36 30 N.
140 E.
Jersey I. St. 7
Aubins, 5
Eng. Channel,
49 13 N.,
2 12 W.
30
Jerusalem,
Syria,
31 45 N.
35 20 W.
Ingolstadt,
Germany,
48 46 N.
11 25 E.
Inverness,
Scotland,
57 36 N.
4 15 W.
11 50
St. John's,
Newfoundland,
47 32 N.; 52 26 W.
6
St John's,
Antigua,
17 4 N. 62 9 W
St. Joseph's,
California,
23 4 N. 109 42 W.
St. Julian (Port)
Ispahan,
Patagonia,
Persia,
49 10 S. 68 44 W.
32 25 .N. 52 50 E.
4 45
Isthmus of Corinth joins the Morea to Greece.
of Darien joins North and South- America.
of Suez joins Africa to Asia.
Ivica I.
Mediterranean,
35 50 N.| 1 30 E-.
St. Juan,
Porto Rico I.
18 30 N.
66 29 VV.
K.
Kamtschatka,
Siberia,
56 30 N.
161 E.
Kiel,
Holstein,
54 20 N.I 10 18 E.
Kilkenny,
Ireland,
52 37 N.
7 15 W.
Kingston,
Jamaica I.
18 15 N.
76 44 W.
Kinsale,
Ireland,
51 32 N,
8 38 W
4 45
Kiow,
Russia,
50 27 N
30 27 E.
Koningsberg,
Prussia,
54 43 N.
21 35 E.
L.
Laguna, TeneO
riffe I. 3
Canaries, '
28 29 N.
16 27 W
3
Lancaster,
England,
54 4 N.
2 50 E.
11 15
L \TITUDES AND LONGITUDES OF PLACES. 499
Names of Places.
Country ov Sea.
Lat.
Long,
/' -water
Feet
Lancaster,
Pennsylvania,
40 3'N
7 60 20' W.
Lands End,
England,
50 6 N.
5 54 W.
4h30'
Leghorn,
Italy,
43 33 N.
10 16 E.
Leuwarden,
United Prov.
53 9 N.
5 55 E.
Leipsic,
Germany,
51 22 N.
12 20 E.
Lexington,
Kentucky, U. S.
37 59 N.
84 46 W.
Leyden,
United Prov.
52 8 N
4 28 E.
Liege,
Netherlands,
50 39 N.
5 31 E.
Lima,
Peru,
12 2 S.
76 5 W.
6 30
2
Limburg,
Netherlands,
50 40 N.
5 5 7 E.
Limerick,
Ireland,
52 33 N.
8 4 2 W.
4 30
Limoges,
France,
45 50
1 15 E.
Lisbon,
Portugal,
38 42 N.
9 9 W.
3 30
Liverpool,
England,
53 22 N.
2 57 W.
11 15
Lizard,
England,
49 57 N.
5 13 W.
5 30
20
London(stpauPs)
England,
51 31 N.
6W.
3
Londonderry,
Ireland,
54 59 N.
7 15 W.
6
Louisbourg,
Cape Breton I.
45 54 N.
59 59 W.
7 15
Louisville,
Georgia, U. S.
32 54 N.
82 44 W.
Louvain,
Netherlands,
50 53 N.
4 41 E.
Lubec,
Germany.
53 51 N.
10 41 E.
Lucerne,
Switzerland,
47 3 N.
8 18 E.
St. Lucia I.
West-Indies,
13 24 N.
60 51 W.
Lunden,
Sweden,
55 42 N.
13 12 E.
Luxemburg,
Netherlands,
49 37 N.
6 11 E.
Lyons,
France,
45 46 N.
4 49 E.
5 50
M.
Macao,
China,
22 13 N.
113 35 V-
5 50
3$
Macassar,
Celebes I.
5 9 S.
119 49 E.
Madeira I. 7
Funchal, J
Atlantic,
32 38 N.
16 56 W.
4
11
Madras,
India,
13 5 N.
80 25 E.
Madrid,
Spain,
40 25 N.
3 38 W.
Mahon (Port)
Minorca I.
39 52 N.
3 48 E.
Majorca I.
Mediterranean,
39 35 N. 2 30 E.
Malacca,
E. India,
2 12 N. 102 9 E.
St. Maloes,
France,
48 39 N.j 2 2 W.
6
45
Malta I.
Mediterranean,
35 54 N. 14.28 E.
0<
Manilla,
Philippine Is.
14 36 N
120 52 E.
Mantua,
Italy,
45 8 N,
10 52 E.
Marigalante,
W. India,
15 55 N.
61 11 W.
Marietta,
Ohio, U.S.
39 8 N
81 38 W.
Marseilles,
France,
43 18 N.
5 22 E.
500 LATITUDES AND LONGITUDES OF PLACES.
Names of Places.
Coitntry or Sea.
Lat.
Long.
H. -water
Fee
Martha's vine-1
yard I. Ed- V
Massachusetts,
4122' N.
7026'W.
gar's town, J
Martinico I.
Fort Royal, 5
W. Indies,
14 36 N.
61 10W.
Mecca,
Arabia,
21 45 N.
40 15 E.
Mexico,
N. America,
19 54 N.
100 7 W .
St. Michael's I.
Azores,
37 47 N.
25 42 W.
Milan,
Italy,
45 28 N.
9 14 I,.
Milford,
Wales,
51 45 N.
5 21 W.
5h.l5'
Minorca (Port 7
Mahon) 5
Mediterranean,
39 51 N.
3 54 E.
Mocha,
Arabia,
13 44 N.
44 E.
Modena,
Italy,
44 47 N.
10 55 E.
Mf>ns,
Netherlands,
50 27 N.
3 57 E.
M'm pelier,
France,
43 37 N.
3 52 E.
Montreal,
Canada,
45 33 N.
73 18 W.
Montserat, N. E.
West-Indies,
16 49 N.
62 27 W.
Morocco,
Burbai y,
31 N.
7 4W.
Moscow,
Russia,
55 45 N.
37 46 E.
Munich,
Germany,
48 8 N.
11 35 E.
N.
Namur,
Netherlands,
50 28 ^
4 51 E.
Nancy,
France,
48 42 N.
6 10 E.
Naiifcrasaki,
Japan,
32 45 >.
130 15 E.
6
Nankin,
China,
32 5 N.
118 46 E
Nantes,
France,
47 13 N.
1 34 W
3 45
Nan tucket >
Nantucket I.
41 18 N.
70 10 W.
3
6
Naples,
Italy,
40 50 N,
14 17 E.
9 30
Newhern,
N. Carolina,
35 17 N.
77 18 W.
Newburyport,
Massachusetts,
42 47 N.
70 52 W.
11 30
10
Newcastle,
England,
55 3 -N.
1 30 W.
5 15
New-Haven,
Connecticut,
41 18 N.
72 53 W.
10 44
8
Newport,
Rhode-Isiand,
41 25 N.
71 15 W.
7 37
5
New^Orleans,
Louisiana,
29 58 N.
90 6V\
New-York,
New-York,
40 42|N.
74 1 W.
8 54
5
N. York light h.
New-York,
40 28 N.
74 W .
7 30
Niagara,
New-York,
43 16 N.
79 W,
Nice,
Italy,
43 43 N,
7 15 E.
Nieuport,
Flanders,
51 8 N.
2 45 E.
Nootka,
N. America,
49 36 N.
126 43 W.
20
Norfolk,
Virginia,
36 55 N
76 22 W.
North Catpe,
Lapland,
71 10 N.
25-49 E.
3 40
LATITUDES AND LONGITUDES OF PLACES. 501
Names of Places.
Country or Sea.
Lot.
Long.
H water
Peer
o.
Ochotsk,
Russia,
5920' N.
143013'N.
Odense,
Fun en I.
55 24 N.
10 11 E.
Ohitatoo I.
Society I.
9 55 $.
139 6W.
2h.30'
Olmutz,
Moravia,
49 37 N.
17 5 E.
St. Omer,
Netherlands,
50 45 N.
2 15 E.
Oporto,
Portugal,
41 10 N.
8 27W.
3 15
L'Orient, (port)
France,
47 45 N.
3 22 E.
3 30
Ostend,
Netherlands,
51 16 N.
2 56 E.
Otaheite,
S. Pacific Ocean,
17 20 S.
149 30 E.
Oviedo,
Spain,
43 18 N.
5 50W.
Owhyhee,S.point
S. Pacific Ocean,
18 54 N.
155 48W.
3 45
24
P.
Padua,
Italy,
45 23 N.
11 53 E.
Palermo,
Sicily I.
38 7 N.
13 35 E.
Palmyra,
Arabia,
33 58 N.
38 42 E.
Panama,
Mexico,
8 58 N.
80 15VV,
3
6|
Paris ;obs.)
France,
48 50 N.
219'^2"E
Parma,
Italy,
44 47 N.
10 21 E.
Pegu,
East India,
17 55 N.
96 45 E
Pekin (obs.)
China,
39 54 N.
116 27 E.
Pensacola,
W. Florida,
30 30 N.
87 10W.
Perth Amboy,
New-Jersey,
40 33 N.
74 20W.
8 9
Petersburg,
Russia,
59 56 N.
30 18 E.
Philadelphia,
Pennsylvania,
39 57 N.
75 14VV.
1 54
Pico 1.
Azores,
38 27 N.
28 28 VV.
Pittsburg,
Pennsylvania,
40 26 N.
80 OW.
Placentia,
Newfoundland,
47 26 N.
53 SOW.
9
6*
Plymouth,
England,
50 22 N.
4 12W.
6
Plymouth,
Massachusetts,
41 57 N.
70 40 VV.
10 40
6*-
Poitiers,
France,
46 35 N.
21 E.
Pondicherry,
East India,
11 56 N.
79 52 E.
Port au Prince,
St. Domingo I.
18 34 N.
72 28W.
Portland,
Dist. Maine,
43 39 N.
70 28W.
10 45
9
Portland (light)
England,
50 31 N.
2 27 W.
7 30
Porto Bello,
Terra Firma,
9 33 N.
79 50 VV.
8
Port Royal,
Jamaica,
18 N.
76 45VV.
Portsmouth,
England,
50 47 N.
1 6W
11 15
Portsmouth,
New Hampshire
43 4 N.
70 46W.
11 15
10
Potosi,
Peru,
20 S.
66 15W.
Prague,
Bohemia,
50 6 N
14 24 E.
Presburg,
Hungary,
48 8 N.
17 10 E.
Providence,
Rhode I. U. S.
41 47 N.
71 22W.
8 11
502 LATITUDES AND LONGITUDES OF PLACES.
Names of Places.
Country or Sea.
Lat.
Long.
H. ivat
Fe
Q-
Quebec,
Canada,
46048' N
71o e'W
7h.30
Quimper,
France,
47 58 N
4 7W
2 30
Quito,
Peru,
13 S
78 10W
R.
Ramsay,
I. of Man,
54 17 N
4 26W
10 30
Revel,
Russia,
59 27 N
24 39 E
Rheims,
France,
49 15 N
4 2 E
Rhodes,
Rhodes I.
35 27 N
28 45 E
Richmond,
Virginia,
37 35 N
77 43 E
Riga,
Russia,
56 55 N
24 E
Rio Janeiro,
Brazil,
22 54 S
42 44 W
4 30
Rochelle,
France,
46 9 N
1 10W
3 45
1C
Rochester,
England,
51 26 N
30 E
45
Rome (St. Pet.)
Italy,
41 54 N.
12 28 E
Rotterdam,
Jnited Prov.
51 56 N
4 28 h
3 45
7
Rouen,
? rance,
49 27 N.
1 5W
2 45
Rugen I.
Baltic,
54 32 N.
14 30 E
S.
Saba I.
W. Indies,
17 39 N.
63 17W.
Sagan,
Silesia,
51 36 N.
15 13 E.
Salem,
Massachusetts,
42 29 N.
70 52W
11 30
12
Salonica,
Turkey,
40 41 N.
23 7 E.
St. Salvador,
Brazil,
2 58 N.
39 OW
Samarcand,
W Tartary,
39 35 N.
64 20 E.
Samos 1.
Archipelago,
37 46 N.
27 13 E.
Sancta Cruz,
feneriffe I.
38 39 N.
16 22 W.
Sancta Fee,
SJew Mexico,
36 34 N.
104 SOW.
Saragossa,
Spain,
41 43 N.
50W
Saratov,
Russia,
1 35 N.
46 E.
Savannah,
Georgia, U. S.
2 4 N.
81 11W.
7 45
Scanderoon,
Syria,
6 35 N.
36 14 E.
Scaff-house*
Switzerland,
7 42 N.
8 37 E.
Sigov
Africa,
4 N.
2 15W.
30
Senegal (Fort)
Africa,
5 53 N.
16 31W.
30
Sion,
Switzerland,
6 14 N.
7 22 E.
Serin gatapam,
lindoostan,
2 22 N.
76 50 E.
Siam,
S. India,
4 18 N.
00 49W.
Si^an,
China,
4 16 N.
09 E.
Sinope,
^atolia,
2 2 N.
35 E,
Smyrna,
^atolia,
8 28 N.
27 7 E.
Stans,
witzerland,
6 57 N.
8 22 E.
Stockholm,
weden,
9 21 N.
18 4 E.
Strasburg,
France,
8 35 N.
7 45 E.
LATITUDES AND LONGITUDES OF PLACES. 503
JVawie* of Places.
Country or Sea.
Lut.
Long.
H. -water
Suez,
Egypt,
29 so'N.
3327' E
Surrinam,
S. America,
6 30 N.
55 30 W
Syracuse,
Sicily I.
36 53 N.
15 17 E.
T.
Tamarin town,
Socotra,
12 30 N.
52 9 E.
9h. 0'
Tanjore,
Tavira,
Hirmoostan,
Portugal,
10 46 N.
37 8 N.
79 48 E.
7 40 E.
1 30
Teflis,
Ptrsia,
42 6 N.
45 15 E.
Teneriffe Peak,
Canary I.
28 15 N.
16 45W.
3
Tercera I.
Azores,
38 39 N.
27 12W
11 45
Texel I.
United prov.
53 10 N
4 59 E.
7 30
Tobolsk,
Siberia,
58 12 N.
68 19 E
Toledo,
Spain,
39 50 N
3 20 W.
Tornea,
Lapland,
65 51 N.
24 14 E.
Toulon,
France,
43 7 N
5 55 E.
3 14
Toulouse,
France,
43 46 N
1 26 E
Tours,
France,
47 24 N.
42 E.
Trent,
Germany,
46 5 N.
11 6 E.
Trenton,
New- Jersey,
40 13 N,
74 50 W.
Trincomale,
Ceylon, I.
8 33 N
81 21 E.
6
Tripoly,
Barbary,
32 54 N.
13 20 E.
Troyes,
France,
48 18 N.
4 5 E
Tunis,
Barbary,
36 16 N.
10 40 N
Turin,
Italy,
45 5 N.
7 39 E.
U.
Upsal,
Sweden,
59 52 N.
17 43 E.
Uraniburg,
Denmark,
55 54 N.
12 51 E.
Ushant I.
Coast of France,
48 28 N.
5 4W.
4 30
Utrecht,
United prov.
52 5 N.
5 9 E.
V.
Valencia,
Spain,
39 25 N.
25W.
Venice,
Italy,
45 27 N.
12 4 E.
10 30
Vera Cruz,
Mexico,
19 10 N.
97 20 W.
Vernon, (mount)
Virginia,
38 46 N.
77 11W.
Verona,
Italy,
45 26 N.
11 IE.
Versailles,
France,
48 48 N.
2 7 E.
Vienna (obs.)
Austria,
48 12 N.
16 22 E.
W.
Wardhuys,
Lapland,
70 23 N.
31 7 E.
Warsaw,
Poland,
52 16 N.
21 3 E.
Washington city,
N. America,
38 53 N.
77 13W-
Waterford,
Ireland,
52 12 N.
7 6W.
4 45
Wells,
England,
51 12 N.
2 45 W.
6
Wexford,
Ireland,
52 20 N.
6 24 W.
8 30
Weymouth,
England,
52 40 N.
2 34W.
7 20
504 LATITUDES AND LONGITUDES OF PLACES.
Names of Places.
Country or Sea.
Lat.
Long.
ff. -water
Williamsburg,
Virginia,
37014' N.
7649'W.
llh.10'
"Wilmington,
N. Carolina,
34 11 N.
78 5 W.
Wilna,
Poland,
54 42 N.
25 27 E.
Wyburg,
Russia,
60 55 N.
30 20 E,
Y.
Yarmouth,
England,
52 55 N.
1 40 E.
9
York,
England,
53 58 N.
1 7W.
York Town,
Virginia,
37 14 N.
76 36 VV.
Z.
2115,
Switzerland,
47 10 N.
8 31 E.
Zurich,
Switzerland,
47 22 N.
8 32 E.
Zutphen,
United Prov.
52 12 N.
6 15 E.
Feet
A table of the mean rt. ascen. and decl. of the firincifial Fixed Stars
adapted to the beginning of 1800.*
Names of Stars.
y Pegasi Algenib,
ce. Cassiopeiae, Schedar,
a Urs. Min. Pole Star,
6 Androm, Mirach,
8 Cassiopeiae,
# Arietis,
7 Androm. Almaach,
Arietis,
a. Ceti, Menkar,
& Persei, Algol, var.
Persei,
i Pleiadum, Alcyone,
y Tauri,
Tauri, Aldebaran,
& Eridani,
a Aurigse, Cafidla,
Orionis, Rigel,
& Tauri,
f) Orionis, Bcllatrix,
<l Orionis,
1 Orionis,
f Orionis,
a Columbae,
Columbae,
* The above is from the latest observations of La Lande, but fitted to the
v'ear 1800, the latest British globes being adapted to that year.
May.
Rt. Jlscen.
an.var.
Declination.
Jin. var*
2
023' 16*
45"9
144'23"N.
+20"2
3
7 18 38
49 7
55 26 20N.
-f-19 9
2.3
13 8 46
173 2
88 14.26N.
+ 19 4
2
14 38 35
49 9
34 33 25N.
-f-19 4
3
18 12 44
57
59 11 28N.
+ 18 9
3.4
25 54 12
49 4
19 49 27N.
4-18
2
27 55 24-
54 1
41 21 46N.
+ 17 7
2.3
28 58 54
50 2
22 30 39N.
+ 17 5
2
42 57 33
46 6
3 17 54N.
4-14 7
2.5
43 48 15
57 5
40 8 19N.
+ 14 5
2
47 31 45
62 9
49 8 13N.
+ 13 7
3
53 55 19
53
23 28 38N.
+ 11 9
3
62 6 22
50 8
i5 8 2N.
+ 9 5
1
66 6 53
51 2
16 5 43N.
4-80
3
74 30 21
43 7
5 21 18S.
54
1
75 29 3
66
45 6 39N.
+ 46
1
76 13 59
43
8 26 30S.
49
2
78 33 53
56 6
28 25 SON.
+ 39
2
78 35 23
45 1
2 35 33S
40
2
80 26 54
45 5
27 27S.
35
2
81 31 3
45 4
1 20 28S.
30
2
82 40 6
45 1
2 3 33S.
26
2
83 6 11
32 2
34 11 17S.
24
3
85 58 46
32 1
35 51 6S.
1 8
A TABLE, &c.
505
d table of the mean rt. ascen. and decl. of the firincifial Fixed Stars
adapted to the beginning of 1800.
J\*iimes of Stars.
a Orionis, Betelguese,
Ciinis Majoris,
$ Canis Majoris,
a Canis Maj. Sinus,
Canis Majoris,
3 Canis Majoris,
Canis Majoris,
Geminorum, Castor
& Canis Mill. Procyon
Geminorum, Pollux,
N"ivis
Ursae Majoris,
a 2 Cancri, .Icuhens,
Hylae, Alfihard^
P Leonis,
a> Leonis, Reffulus,
@ Ursge Majoris,
a Urs. Maj. Dub he,
Leonis, Dcneb,
$ Vir^inis,
y Ursrc Majoris,
^ Ursae Majoris,
y Corvi, 4l^orab^
B Ursse Maj. Allot h,
a. Vir$. -Wca Pzrg-.
Ursse Majoris,
) Ursse Maj. Benetnach
* Oraconis,
a Bo Otis, ArcturuS)
y Bootis, Se$inus,
a. 2 Librae, Zubenelch.
# Ui^a Min. Kochab.
@ Librae, Zubenelg.
& Coron. bor. Alfihacca^
a, Serpentis,
Serpentis,
S:-,orpii,
$ Scorpii,
3 Ophiuchi,
at Scorpii, AntareS)
$ Herculis,
Herculis, Ras Afgethi
X Scorpii, Lcsath,
a, Ophiuchi, nas Alhagus
y Draconis, Rastabcn^
Mag
Rt. Ascen,
an.var
Declination.
An. var.
86 8 12
48"5
7-21' 29''N
+ I "5
2.3
93 9 3
34
29 58 56S
+ l
2.3
93 28 22
39 *
17 53 2S
+ 1 2
99 5 4
39 5
16 26 54S
+ 45
102 41 28
35 I
28 42 19S
+ 43
2.
105 3 5o
.36 2
26 5 2S
+ 5 i
109 2 41
35 2
28 55 16S
+ 65
1.
110 27 13
.57 6
32 18 46N
7
1.
1 12 12 20
47
5 43 34N
67
2.
H3 15 5
55
28 29 50N
80
119 8 2<
31 5
i9 26 45S.
+ 98
131 21 45
63 4
49 49 2N
1 3 5
3
131 52 58
49 2
12 37 24 v
13 4
2
139 26 23
43 9
7 47 SOS.
+ 15
3
145 20 26
51 6
C 26 56 24N
16 5
1
149 25 38
47 9
12 56 2oN
17 2
2
162 25 4
55 4
o7 27 7N.
19 1
2.1
l6'J 48 57
57 2
6J 49 40N.
.9 3
2.1
. 74 42 4 ;
45 9
i5 41 25N.
20
3
175 4 8
46 8
2 53 34N.
2o 3
2
175 48 41
48 1
54 48 25N.
20 2
3
81 21 49
45 3
58 8 4 IN.
20 2
3
81 23 6
45 7
16 25 44S
+ 20
3
91 17 5.
40 4
57 2 55.S.
( 9 7
1
198 40 7
46 9
10 6 34S.
+ 19 I
2.3
198 57 43
36 8
55 58 29N
19
2
204 54 45
35 2
50 19 IN.
18 2
3
209 24 45
21- 6
65 20 8N.
17 3
1
211 38 8
40 7
20 13 54N
19
3
216 15
36 4
39 10 22N.
16 I
2.3
219 57 34
49 2
15 17 4S.
+ 15 5
3
22 51 58
4- 8
74 58 27N.
14 9
2.3
2^6 33 58
47 9
8 38 2S.
+ 13 r
2.3
31 33 24
37 8
27 23 50N.
12 5
2.3
33 36 24
43 9
7 3 59N.
11 8
3
34 14 27
41 3
16 3 30N.
ii r
3
37 9 56
52 8
22 2 18 S.
h 10 9
2
-36 27 28
51 9
19 14 39 S.
[-10 4
3
241 58 10
46 7
3 9 58 S.
-97
1
44 17 3
54 6
25 51 16 S.
- 8 S
3
244 24 27
38 3
21 56 9N.
83
3
256 3
40 8
14 37 54N.
46
2J260 3(5
60 5
36 56 31 S.
4-34
2 '261 24 50
41 .1
12 43 7N. 3 1
3 267 59 31
20 4
5131 N. 7
3Q
506
A TABLE, &c.
A table of the mean rt. ascen. and ded. of the firincipal fixed Star,')
adajited to the beginning of 1800.
/Vrt;M6's of Stars.
:ifay.
Rt. -Iscen.
an.var.
Declination.
Jin. rar.
s Sagitarii,
3
27243'27"
59''5
3427'4l''N.
1
a Lyrx, Vega,
1
277 32 29
30 4
38 36 19N.
+ 29
& Lyrx,
2.3
280 40 28
33
33 8 35N.
+ 36
er Sagitarii,
2.3
280 42 52
55 7
26 31 41 S.
37
y Lyrx,
3
282 51 56
33 5
32 25 SON.
+ 4 5
5 Aquilx,
3
288 51 9
45 2
2 43 44N.
+ 66
Q Cygni, Albirco,
3
290 39 50
36 3
17 32 56N.
+ 7 1
Aquibe, Altair,
1.2
295 15 20
43 8
8 21 8N.
+ 89
& Aquilx,
3
296 22 16
44 5
5 55 ION.
+ 86
a 1 Capricorni,
3.4-
301 38 16
49 9
13 6 44 S.
10 5
a 2 Capricorni,
3
301 44 11
49 9
13 9 3S.
10 8
# Capricorni,
3
302 26 23
50 6
15 23 56 S.
10 9
Delphini,
3
307 35 13
41 6
15 12 59N,
+ 11 "4
i 'ygni, Deneb,
2
308 39 13
30 5
44 34 18N.
+ 12 6
a Cephei, Alderamin,
3
318 26 51
21 5
61 44 29N.
+ 14 9
& Aquarii,
3
320 15 17
47 3
6 26 31 S.
15 6
Pegasi,
3
323 35 22
44 5
8 57 57N.
+ 16 2
$ Capricorni,
3
323 59 43
49 8
17 1 32 S.
16 3
a Aquarii,
3
328 52 36
46
1 17 6S.
17 3
y Aquarii,
3
332 49 45
46 3
2 23 19 S.
18
v Pise. \us. Fomalhaut
1
341 38 34
49 9
30 4-0 SOS.
19
& Pegasi, S cheat,
2
343 31 22
42 9
27 7N.
+ 19 4
a. Pegasi, Markab,
2
343 42 5
44 4
14? 8 4N.
+ 19 2
u, Anclrom. Alfihcratz,
2.3
359 30 8
45 9
27 59 14N.
+ 20
8 Cassiopx,
2.3
359 38 41
46 7
58 2 48N.
+ 19 9
INDEX.
THE NUMBERS INDICATE THE PAGES.
A.
ABERRATION, what 49. How discovered, &c. 30 1.
Altitude of an object, what 9. Meridian 9. Quadrant of 9. How found
with artificial horiz. 50. Of a star, how found 203.
Amphiscii, who and why so called 4.
Amplitude, what 9. How found 145, Sic.
Analemma, what 63.
Angle of position, what 24. How found 135. Differing from the bearing
of places 135.
Anomaly, true, mean, &c. what 48. How found 313.
Anomalistic year, what and how found 305.
Antseci who, their seasons, hours, &,c. 5. How found 57.
Antipodes, who, &c. 5. How found 57.
Aphelion, what 48.
Apogee, what 48.
Apparent place of a body differs from its true 25.
Apsides, line of the 48.
Apsis of an orbit, what 48.
Arch, diurnal and nocturnal, what 25.
Argument, what 49. Of latitude, how found 350.
Ascension, right, oblique, &c. what 23. How found 145, 192, 194
Ascensional difference, what 24. How found 145.
Aspect of the stars or planets, what, &.c. 48.
Atmosphere, what 25.
Axis of a sphere, of the earth, and of the artificial globe 2.
Azimuth, what 9. Sun's, how found 148. Of a star, how found 203.
B.
Bayer's Characters, what, remarks on, &c. 45.
C.
Calendar, reformat, of, method of calculating, &c. 16.
Centre of gravity, what, of a system of bodies, how found 253.
Ceres, a new planet, when and by whom discovered 47. Remarks on 357.
Circle great, lesser, &c. 2- Of reflection, use of, &c. 50.
Circles of perpetual apparition and occultation, what 25.
Climates, what, &c. 22. How found 115. Their breadth, &c. 117.
Clocks, &.c. how regulated 83, 373.
Clusters, what, their number, &.c. 27, &c. 46.
Colures equinoctial and solsticial, mark the seasons, &c. 7.
Comets appar. paths of 242. Their nature, motion, periods, tails, &c. from
392 to 435.
Commutation, angle of 259, 351.
Compass mariner's, variation of, &c. 9.
Conjunction, &c. what 48. Of the inferior planets, how found 267.
Constellations, what, tables of, description, of origin of, 8cc, 27 to 45.
Course, or a ship's way, what 24.
Crepusculum or twilight, what, probable cause of, 8cc. 24.
Culminating point, what 24.
Curtate distance of a planet from the sun or earth, what 49. How found 351
Cycle, what 17- Of the sun and moon 17, 18. Of indiction 19.
508 INDEX.
D.
Day* a true and mean solar, Natural or Astronomical, artificial, &c. what 11,
Civil, sidereal, &c. what 12. Length of the longest in any latitude how
found 77.
Declination, what 23. Of the sun, how found 62, 63, 64, 74, 166, 171. Of
a star 192.
Degree, length of, Sec. 3, 118, 119, Sec.
Degrees, reduced to t.me, and the contrary 51,
Pt-seension oblique, what 23. How found 145, &c,
Diaii.q;, principles ofj &c. 183 to 191.
Digit, what 49.
D;sk, what 49.
Dp, &c. how calculated 158, 159, 160.
Distance of places on the earth, how found 118. Of stars, Sec. from each
other ^24 Of planets from the sun, how found 259, 352.
Dogdays, how found, &e- 233.
Dominical Letter, what, and how found 18.
Duration, what 10.
E.
Earth, diurnal motion of the, and poles 2. Daily mean mot. of 15. Jta
pK.cc, how found 64. Its surface irregular 120, 121. Its mean distance
from the sun 256. Figure 28- to 288. Magnitude, how found 288, kc.
Diurna. mot 293. Probable cause of its diur. mot. 296- Annual mot*
297 Its perihelion, apogee, &.c 305. Its mot. theory of 312, Sec. Its
orbit, secular variation of its inch from the action ol Jupiter and Saturn
485, Sec
Easter Sunday, how found 20.
Eccentricity of a planet's orbit, what 48. How found 318.
Eclipse, solar and iu;ar, where visible 173, 174. \Vhen likely to happen 17&
Eclipses, useful in finding- the long. 5o. Ol Jupiter's satellites, where visi-
ble 1/7. Solar and lunar, how determined 448 to 457-
Ecliptic, what, and why so called, signs of, &c. 6. Cardinal points of 8.
Ellipse, how described 253.
Elongation of a planet from the sun 49. How found 351, 352.
Emersion of a satellite, what, &c. 178.
Epctct, what, how found, &c. 18, 19.
Equation of the centre, what 49 How found 313.
Equation of time, causes of 84. Calculation of 85. Tables of 86, 87, 88.
Equator, what, and why so called 3.
Equinoctial, what 3. Points of what 7-
Equinox, vern 1 and autumnal, what 7.
Equinoxes, precession of 15. Illustrated 243. Its quantity determined 305,
G,
Galaxy. Via Lactea, or Milkyway 45.
Globe, what, terrestrial, its spherical figure, proofs of, 8cc. 1.
H.
Harvest moon, phenomenon of the 179.
Herschel, his periods, &c. 388, 389, &c- His satellites 390 to 392.
Heteroscii, who, and why so called 4,
Hig-h water, how found 240, 460.
Horizon, what, sensible, rational, circles on, wooden, &c. 7, 8.
Hour, six o'clock line, what 24 Of the day, how found 144, 150, 151, 152,
172. Of the night, how found 204, 206, 207, 208, Sic. From the moon's
shining on a dial 214.
Hours, what, and how divided 11. Babylonish, Italian, Jewish, Planetary,
&c. what 12.
INDEX,
I.
Immersion of a satellite, what, &c. 178.
Incl MUUOM, recunation and dec!- of a plane, how found 191.
Inclined planes, motion of b dies on 468 to 470.
Index or hour circle, what, &.c. 6.
Juno, a new planet, when and by whom discovered 46. Remarks on 357.
Jupiter, his per.ods 359 ; retr. mot. mean dist. eccentricity, mean long,
perihelion, incl of orbit, gr. equation, belts, &c. mot. on his axis, &c 36X>
and 362 ; diameter 361 ; light and heat on, &c. 362. His satellites 363 ;
their mean mot.ons, &c. 364 ; their irnmer. emersions and revolutions
366 ; their long, dist &c. 367 ; rotary motion 368 ; their eclipses 369 ;
inequalities 370, &c. their configurations 375, &c. His disturbing 1 force
on the earth, &c. 483,
K.
Kepler's laws 253, 475, 477, &c.
L.
Latitude, of a place, what, diff. of, &c. 3. How found 50 to 57, also 155,
159, &c. to 172. Or a star, how found 195. Of a place 214 to 222. Geo*,
centric and heliocentric, what 48. How found 351.
Laws of motion, forces, gravity, &.c. from 463 to 486.
Light or heat, &c. decreases as the square of the dist. 270.
Longitude, of a place, what 4. How round 50 to 57 and 266. Of a star,
how found 19-5. Difference of, what 3. How found 56, 57, 284. Geo-
centric, how found 352.
Lunv.ere cendree, what 337.
M.
Magnetic needle used in finding- the latitude 54.
Mars, his periodic and rotary motions 347. Mean dist. 348, 549. Apparent
diameter, parallax, &c. 349. Velocity in his orbit, light and heat, mean
longitude, incl. of orbit, place of his node, aphelion, magnitude, Sec. 350.
Fasciae or belts of 356.
Measures, French, &c. 132, 133.
Medium Cseli, what, and how found 210.
Mercator's sailing explained, &c. 134.
Mercury,' his direct, retr. and stat. motion explained 257 and 266. His dist.
from, and period round, the sun 258. Elongation, &c, 259. Eccentricity
and mean mot. 260. Aphelion, nodes, and their secular variations 262,
263, &c. Diam. how found, magnitude, &c. 262, 263.
Meridian, what, why so called, how drawn, brazen or universal, what,
first what, &c. 5, 6. Alt. of the sun, when to be observed 90. Line,
how drawn 181.
Meridional parts, how found 134.
Month, astronomical, periodical, synodic, civil, solar, &c. 13, 14.
Months, their names, origin of, &c. 13, 14.
Moon, age of, how found 20. Time of coming to the meridian or southing-,
how found 20, 213, 238. New, full, &c. 20, 21. Situation of her orbit,
how found 226. Where vertical 237. Distance from the earth, how
found 250 Her mean mot. 320. Her sidereal revol. 321. Eccentricity,
gr. equation of her orbit, &c. 322, 324. Mean mot. of her apogee, place
of the nodes, how found, their mean mot. inclination of her orbit, &c. 323.
Evection, variation, annual equation, &c. 325. Her revol. table of 325.
Appar. diameter 326. Her spots 337- Libration 338. Libr. in lat. and
long 1 339. Atmosphere 340. Refraction 342. Height of her mountains,
how determined 343. Magnitude 344. Her disturbing force, &c. 481, &c.
Motion of bodies in general 463, &c. In circular orbits, &c. 473.
N.
Nadir, what 9.
Nebula*, what, their number, &c. 27, 28, 29, 46, 44 x
510 INDEX.
Nebulous stars, &c. 46, 447.
Night, what, variable in different latitudes 12.
Nodes, what, ascending-, descending, line of, &c. 47, 48. How found, &c. 264 ;
Nonagesimal degree, what, and how found 210.
Noon, mean or true 1 1.
Numbers of the months, what, and how found 19.
Nutation, what, &c. 243.
O.
Occultation of a star or planet, what 49.
Opposition, &c. what 48 Time of, for the sup. planets, how found 377,
Orbit of a planet, &c. what 47. How found 260, 261.
P.
Pallas, a new planet, when, and by whom discovered 47. Remarks on 357.
Parallax, what, &c. 25, 2?6. How found, &c. 279, 328, &c. In lat. and
long. 331.
Parallels of latitude, what 4.
Pendulum, properties of, &c 470 to 473.
Periceci, who, their seasons, hours, &c. 5. How found 57, 58.
Perigee, what 48.
Perihelion, what 48.
Penscii, who, and why so called 4.
Perpetual darkness, observations on 73, 298.
Place of a body, mean and true 49.
Plane of a circle, what 2.
Planets why so called, primary, secondary, opake 47. Time of passing 1
the meridian 201. Above the horiz. after sun setting, or any hour, how
found 235, 236. Heliocentric place of, and dist. from the sun 365, 348,
352, 353. Phases of 268. Mot. of, general remarks on, &c. 305. Mean
long, how found 351. Nodes determined 353. Appearances of superior
355. Masses and densities of 476.
Polar circles, what 4.
Polar distance, what 9.
Pole s of a great circle 2. Of the earth, arctic, antartic, celestial, &c. 2.
Prosthapheresis, what 49. How found 313, 318.
R.
Refraction, what, &c. ancients not unacquainted with, horizontal greatest;,
Sec. 25, 26.
Rhumb, or a rhumb line, what 24.
S,
Sailing parallel, properties of, &c. 128.
Satellites, what, &c. 47. Of Jupiter 363. Of Herschel 390.
Saturn, his periods 380. His mean long, perihelion, incl. of his orbit, long\
of his nodes, eccentricity, gr. equat. diam. mag. mean dist. &c. 381.
Rotary mot. 382. His ring 383, &c. His satellites 386. His disturbing
force on the earth, &c. how found 483.
Seasons, anticipation of the not owing to the precess. of the equinoxes 15.
Alteration of, &c. 89, 90, &c.
Solstitial points, what 7.
Sphere diam. and circum. of 2, 3. Right, parallel, oblique, &c. 21, 22.
Stars, poetical rising and setting of, 15, 27. Fixed, classes of, mag. tel-
escopic, &c. 26. Unformed, what 27. Number of in Orion, Pleiades, &.c,
46. Rising, setting, coming to the meridian, &.c. how found 199. Near
the moon's path 226. Achronical rising and setting of 226, 234. t'os-
mical 230, 234. Heliacal 233, 234. Immense dist. of 302, 447. Their
appearances, mot &c. 436. Aberration 440. General remarks on 445, &c.
Stile, new, old, &c. 16.
INDEX. 511
Sun, what, &c. 47. Place of the, how found 62, 152. Rising and setting
68, 69, 145. Merid. alt. when to be observed 90. Alt. how found 140,
141, 143, &c. 148, 151, 168. His magnitude 250, 255. Spots 251.
Rotary mot. 251. Appar. diameter 253. Real diameter 255. His at-
mosphere 256.
Syzygies, what, and why so called 48.
T.
Tides happen always when the moon is in the same position 20. Their
phenomena 458 to 462.
Time what, how divided, &c. 10. Its physical essence unknown 10. Equa-
tion of what, causes of, &c. 10. How found 83.
Tropics what, and why so called 4.
Twilight, how found, cause of, &c. 107.
V.
Variation chart used in finding the longitude 53.
Variation of the compass, how found 147, 149.
Venus, how long she rises before the sun or sets after him 236. Position
of, when brightest 269. Her periods 270. Elongation 270. Dist. from
the sun, eccentricity, aphel. gr. equat. mean mot. diameter, &c. 271.
Incl. of her orbit, her nodes, variation of, mag. brightness, mountains, &c.
273. Rotary mot. 274. Phenomena 276, 277, &c. Conjunctions and
transit 281, &c.
Vesta, a new planet, Sec. 47. Remarks on 357.
Universe, a general idea of 296.
Up and down, how understood 5.
Week what, the most ancient collection of days, &c. 13. Days of the, calk
ed after the heathen names of the planets 14.
Y.
Year what, how divided, variation of, its seasons, &c. 13. Solar or tropical,
sidereal, civil, lunar, civil solar, Julian, Gregorian, &c. 15, 16, 17. Move-
able feasts of, how regulated 21. Sidereal and tropical length of, how
found 246, 303.
Z.
Zenith, what, distance 8, 9.
Zodiac, what, why so called 27.
Zone, what, why so called, torrid, temperate, frigid, &c. 4. Torrid limits
of, how found by the ancients 66.
INDEX TO THE TABLES.
TABLE of the Climates, &c. 22.
Table of the constellations, number of stars, See. as described on the new
British globes 28, 29.
Table of the length of the longest day in almost every degree of latitude 76.
Tables of the equations of time 86, 87, 88.
Tables for finding the length of a degree 118, 119, 129, &c.
Table for finding how many miles make a degree of longitude in any lati-
tude 127.
Tables of ancient and modern measures 129 to 152.
Tables of the dip or degression of the horizon, &c. 155, 158.
Table of refractions, &c. 155.
Table of the sun's parallax, in altitude 155.
Table of the sun's semidiameter in minutes, &c. 155.
Table of the dip when the land intervenes 158.
Table shewing when the harvest moon is least and most beneficial 181.
512 INDEX.
Table of the hour arches for a horizontal dial, lat. of N. York, 187.
Table of the hour arches for an erect south dial. lat. '-0 43', 189.
Table of the lat. and Ion y. of the nine principal fixed stars used in deter-
mining the longitude 196,
Tables for finding- high water 241.
Tables of the sun's declination, &c. 487", &c.
Table of the latitudes and longitudes of places, time of high water, &c
491 to 504.
Table of the rt. ascension and decl. of the principal fixed stars 504, Sec.
FINIS.
0. A. M. Z). G.
ERRATA.
PAGE 1, line 3, for Definition, razd Description. p.2, 1. 37, for 23d, r. 25th,
p. 3, 1. 40, omit has. p. 4, 1. 43, for decrease, r. decreases. p. 7, 1. 1, for
follows, r. follow. p. U, 1. 16, for spctce, r. interval, &c. p. 14, 1. 25, for
their, r. these. p. 14, 1 41, for follows, r. follow. p. 15, 1. 30, for 20 24',
r. 20 1 24". p. 15, 1. 51, for anticipation*, >-. anticipation p. 23, 1. 42, for in-
creases, r. increase p 32, 1. 6, for who, r. which. p 52, 1 27, after equal-
ly r as. p. 54, 1. 1, for places those, r those places. p. 54, 1. 51, for in, r.
on p. 56, 1. 38, for 75 8' 4.7', r. 75 (V 2/' p. 56. 1 47, for 30 U', r . 30'
11" p. 57, 1. 28, 30, 33 for 75 1 7 2 .", r. 77 14' ?2". p. 57, I. 29, 30, for
1 13' 22", r. 3 13' 2J". p. 57, 1. 33, for 14 52 f S", r. 12 52 f 8". p. 75, 1 6,
for brass meridian, r. meridian. p. Ill, 1. 45, for towards the eastward, r.
eastward. p. 128, 1. 33, for position, r. portion. p. 140, 1. 25, for place r.
plane. p. 140, 1. 32, for where, r. when. p. 147, 1 24, for sun, r. sum. p.
151, 1. 4, for 13th, r. 32d. p. 158, 1. 18, for by r. from. p. 18 >, 1. 42, for
place, r. plane. p. 188, 1. 50, 51, for set method above, r. above method.
p 197, 1. 37, after 16 &, place : p. 230, 1. 44, after similar, r. manner.
p. 251, 1. 29, for ecliptical, r. elliptical. 1 39, for 29 36', r. 3 '36r. p. 255,
1. 25, for foregoing- fig-, r fig-, pa. 250. p. 261, 1. 2, for near distance, r. mean
distance. p. 266, 1. 21, for Venus, r. the planet p. 270, last line, for chap.
7, r.chap. 8. p. 289, 1. 29, omit act. p. 353, 1. 15, for ST, * S*3. p. >67,
1. 37, after the r. sun. p. 373, 1. 26, for he, r. the observer. p. 392, 1. 41,
for thus, r. this. p. 393, 1. 25, for the r. a. p. 399, 1. 7, for on r. in.
CALIFORNIA TTT> RARY
Wallace, J
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A new treatise on th
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