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SIR JOHN Fºw.”HERSCHEL, BART. K.H.
M.A. D.C.L. F.R.S.L. & E. HoN. M.R.I.A. F.R.A.S. F.G.S. M.O.U.P.S.
Correspondent or Honorary Member of the Imperial, Royal, and National, Academies of Sciences
of Berlin, Brussels, Copenhagen, Göttingen, Haarlem, Massachusetts (U. S.), Modena,
Naples, Paris, Petersburg, Stockholm, Turin, Vienna, and Washington (U.S.);
the Italian and Helvetic Societies;
the Academies, Institutes,"&e., of Albany (U. S.), Bologna, Catania, Djon, L ºsanne,
Nantes, Padua, Palermo, Rome, Venice, Utrecht, and Wilna;
the Philomathic Society of Paris; Asiatic Society of Bengal; South African Lit. and Phil. Society
Literary and Historical Society of Quebec; Historical Society of New York;
Royal Medico-Chirurgical Soc., and Inst. of Civil Engineers, London
Geographical Soc. of Berlin; Astronomical and
Meteorological Soc. of British Guiana;
&c. &c. &c.
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WITH NUMER 0 US PLATES AND W 00I)-CUTS,
NEW YORK :
SHELDON & COMPANY, PUBLISHERS,
498 & 500 BROADWAY.
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N 0 T E
TO
THE F O U R T H EDITION.
*
SEVERAL alterations and additions are made in this Edition, be-
sides what have been introduced into the Third, to bring it up
to the actual state of astronomical discovery. The elements of
four new planets (Parthenope, Egeria, Victoria, and Irene) have
been added, and improved elements of Iris, Metis, Hebe, and
Hygeia, substituted for the provisional elements before given.
The remarkable discovery of an additional ring of Saturn, and
the curious researches of M. Peters on the proper motion of
Sirius, with several minor features, are noticed. Where such
additions are introduced in the text, they are indicated by being
enclosed in brackets [ ]. ,'
J. F. W HERSCHEL.
London, Aug 5 1861.
(v)
PR EFA C E.
THE work here offered to the Public is based upon and may be
considered as an extension, and, it is hoped, an improvement of
a treatise on the same subject, forming Part 43, of the Cabinet
Cyclopædia, published in the year 1833. Its object and general
character are sufficiently stated in the introductory chapter of
that volume, here reprinted with little alteration; but an oppor-
tunity having been afforded me by the Proprietors, preparatory
to its re-appearance in a form of more pretension, I have gladly
availed myself of it, not only to correct some errors which, to
my regret, subsisted in the former volume, but to remodel it alto-
gether (though in complete accordance with its original design as
a work of ea planation); to introduce much new matter in the
earlier portions of it; to re-write, upon a far more matured and
comprehensive plan, the part relating to the lunar and planetary
perturbations, and to bring the subjects of sidereal and nebular
astronomy to the level of the present state of our knowledge in
those departments.
The chief novelty in the volume, as it now stands, will be found
in the manner in which the subject of Perturbations is treated.
It is not — it cannot be made elementary, in the sense in which
that word is understood in these days of light reading. The
chapters devoted to it must, therefore, be considered as addressed
to a class of readers in possession of somewhat more mathematical
knowledge than those who will find the rest of the work readily
(vii)
viii P. R. E. F. A C E .
and easily accessible; to readers desirous of preparing them-
selves, by the possession of a sort of carte du pays, for a cam-
paign in the most difficult, but at the same time the most attract-
ive and the most remunerative of all the applications of modern
geometry. More especially they may be considered as addressed
to students in that university, where the “Principia” of Newton
is not, nor ever will be, put aside as an obsolete book, behind the
age; and where the grand, though rude outlines of the lunar
theory, as delivered in the eleventh section of that immortal
work, are studied less for the sake of the theory itself than for
the spirit of far-reaching thought, superior to and disencumbered
of technical aids, which distinguishes that beyond any other pro-
duction of the human intellect.
In delivering a rational as distinguished from a technical expo-
sition of this subject, however, the course pursued by Newton in
the section of the Principia alluded to, has by no means been
servilely followed. As regards the perturbations of the nodes
and inclinations, indeed, nothing equally luminous can ever be
substituted for his explanation. But as respects the other dis-
turbances, the point of view chosen by Newton has been aban-
doned for another, which it is somewhat difficult to perceive why
he did not, himself, select. By a different resolution of the dis-
turbing forces from that adopted by him, and by the aid of a few
obvious conclusions from the laws of elliptic motion which would
have found their place, naturally and consecutively, as corollaries
of the seventeenth proposition of his first book (a proposition
which seems almost to have been prepared with a special view to
this application), the momentary change of place of the upper
focus of the disturbed ellipse is brought distinctly under inspec-
tion; and a clearness of conception introduced into the pertur-
bations of the excentricities, perihelia, and epochs, which the
author does not think it presumption to believe can be obtained
by no other method, and which certainly is not obtained by that
from which it is a departure. It would be out of keeping with
the rest of the work to have introduced into this part of it any
algebraic investigations; else it would have been easy to show
that the mode of procedure here followed leads direct, and by
P. R. E. F. A C E . IX
steps (for the subject) of the most elementary character, to the
general formulae for these perturbations, delivered by Laplace
in the Mécanique Céleste."
The reader will find one class of the lunar and planetary in-
equalities handled in a very different manner from that in which
their explanation is usually presented. It comprehends those
which are characterized as incident on the epoch, the principal
among them being the annual and secular equations of the moon,
and that very delicate and obscure part of the perturbational
theory (so little satisfactory in the manner in which it emerges
from the analytical treatment of the subject), the constant or
permanent effect of the disturbing force in altering the disturbed
orbit. I will venture to hope that what is here stated will tend
to remove some rather generally diffused misapprehensions as
to the true bearings of Newton's explanation of the annual
Equation.”
If proof were wanted of the inexhaustible fertility of astro-
aomical science in points of novelty and interest, it would suffice
to adduce the addition to the list of members of our system of
no less than eight new planets and satellites during the prepara-
tion of these sheets for the press. Among them is one whose
discovery must ever be regarded as one of the noblest triumphs
of theory. In the account here given of this discovery, I trust
to have expressed myself with complete impartiality; and in the
exposition of the perturbative action on Uranus, by which the
Axistence and situation of the disturbing planet became revealed
to us, I have endeavoured, in pursuance of the general plan of
this work, rather to exhibit a rational view of the dynamical
action, than to convey the slightest idea of the conduct of those
amasterpieces of analytical skill which the researches of Messrs.
Leverrier and Adams exhibit.
To the latter of these eminent geometers, as well as to my
Aucºllent and esteemed friend the Astronomer Royal, I have to
* Livre ii. chap. viii. art. 67.
* Principia, lib. i. prop. 66, cor. 6.
X PR E FA C E.
return ny best thanks for communications which would have
effectually relieved some doubts I at one period entertained, had
I not succeeded in the interim in getting clèar of them, as to the
compatibility of my views on the subject of the annual equation
already alluded to, with the tenor of Newton's account of it. To
my valued friend, Professor De Morgan, I am indebted for some
most ingenious suggestions on the subject of the mistakes com-
mitted in the early working of the Julian reformation of the
calendar, of which I should have availed myself, had it not ap-
peared preferable, on mature consideration, to present the sub-
ject in its simplest form, avoiding altogether entering into mi-
nutiae of chronological discussion.
J. F. W. HERSCII.E.L.
Collingwood, April 12, 1849.
0 0 NT ENTS.
PR2RACE © tº º º g º O & © tº º ſº º & © tº e º ºs e º e º O e g g g g g º e º ºs e e º 'º e º e g º e º º e º e e g º e s tº gº e º e o e º & © e º º ºs e º 'º e º 'º e º 'º º Page vii—x
INTRODUCTION......... & © º º º 'º º G & º 'º º ſº tº dº e º e º e e g c & e g º e º º e º is tº e º e s tº e º 'º e º º e g º º e º 'º e º O & © tº C tº L is e º ºs º º tº e º 'º tº º e 17
PART I.
CHAPTER I.
General notions. Apparent and real motions. Shape and size of the Earth.
The horizon and its dip. The atmosphere. Refraction. Twilight. Appear-
ances resulting from diurnal motion. From change of station in general.
Parallactic motions. Terrestrial parallax. That of the stars insensible.
First step towards forming an idea of the distance of the stars. Copernican
view of the Earth’s motion. Relative motion. Motions partly real, partly
apparent. Geocentric astronomy, or ideal reference of phaenomena to the
Earth's centre as a common conventional station................................... 24
CHAPTER II.
Terminology and elementary geometrical conceptions and relations. Termino-
logy relating to the globe of the Earth—to the celestial sphere. Celestial
perspective............................................................................. ....... 62
CHAPTER III.
Of the nature of astronomical instruments and observations in general. Of
sidereal and solar time. Of the measurements of time. Clocks, chronome-
ters. Of astronomical measurements. Principle of telescopic sights to
increase the accuracy of pointing. Simplest application of this principle.
The transit instrument. Of the measurement of angular intervals. Methods.
of increasing the accuracy of reading. The vernier. The microscope. Of
the mural circle. The Meridian circle. Fixation of polar and horizontal
points. The level, plumb-line, artificial horizon. Principle of collimation.
Collimators of Rittenhouse, Kater, and Benzenberg. Of compound instru-
ments with co-ordinate circles. The equatorial, altitude, and azimuth instru-
ment. Theodolite. Of the sextant and reflecting circle. Principle of repe-
tition. Of micrometers. Parallel wire micrometer. Principle of the dupli-
cation of images. The heliometer. Double refracting eye-piece. Variable
prisin micrometer. Of the position micrometer.................................. 76
s (xi)
XII - CONTENTS.
CHAPTER IV.
O F. G E O G. R. A. P. H. Y.
Of the figure of the Earth. Its exact dimensions. Its form that of equilibrium
modified by centrifugal force. Variation of gravity on its surface. Statical
and dynamical measures of gravity. The pendulum. Gravity to a spheroid.
Other effects of the Earth’s rotation. Trade winds. Determination of geo-
graphical positions—of latitudes—of longitudes. Conduct of a trigonometri-
cal survey. Of maps. Projections of the sphere. Measurement of heights
by the barometer.................. tº gº º º e º gº tº dº º º º ſº tº ºi º º tº g g g g tº e º e º 'º º e º 'º tº e º ºs e º e º ſº e g º º & ſº e º e º tº º e º e ſº 118
CHAPTER. W.
O F U R A N O G. R. A. P. H. Y,
Construction of celestial maps and globes by observations of right ascension and
declination. Celestial objects distinguished into fixed and erratic. Of the
constellations. Natural regions in the heavens. The Milky Way. The Zo-
diac. Of the ecliptic. Celestial latitudes and longitudes. Precession of the
equinoxes. Nutation. Aberration. Refraction. Parallax. Summary view
of the uranographical corrections................. º e º 'º e º ºs e e º 'º g º ºs e º 'º e g g g g º ºs º is e º is e º e º e º e 161
CHAPTER WI.
of T H E S UN’s M O TI O N.
Apparent motion of the sun not uniform. Its apparent diameter also variable.
Variation of its distance concluded. Its apparent orbit an ellipse about the
focus. Law of the angular velocity. Equable description of areas. Parallax
of the Sun. Its distance and magnitude. Copernican explanation of the
Sun's apparent motion. Parallelism of the Earth's axis. The seasons. Heat
received from the Sun in different parts of the orbit. Mean and true longi-
tudes of the Sun. Equation of the centre. Sidereal, tropical, and anoma-
listic years. Physical constitution of the Sun. Its spots. Faculae. Probable
nature and causo of the spots. Atmosphere of the Sun. Its supposed clouds.
Temperature at its surface. Its expenditure of heat. Terrestrial effects of
solar radiation...... ........................................... .............. tº º e º ſº º ſº tº ... 185
CHAPTER VII.
Of the Moon. Its sidereal period. Its apparent diameter. Its parallax, dis-
tance, and real diameter. First approximation to its orbit. An ellipse about
the Earth in the focus. Its excentricity and inclination. Motion of its nodes
and apsides. Of occultations and solar eclipses generally. Limits within
which they are possible. They prove the Moon to be an opaque solid. Its
light derived from the Sun. Its phases. Synodic revolution or lunar month.
Of eclipses more particularly. Their phenomena. Their periodical requr-
CONTENTS. XIII
rence Physical constitution of the Moon. Its mountains and other super-
ficial features. Indications of former volcanic activity. Its atmosphere.
Climate. Radiation of heat from its surface. Rotation on its own axis.
Libration. Appearance of the Earth from it...................................... 218
CHAPTER VIII.
Of terrestrial gravity. Of the law of universal gravitation. Paths of projec-
tiles, apparent, real. The Moon retained in her orbit by gravity. Its law of
diminution. Laws of elliptic motion. Orbit of the Earth round the Sun in
accordance with these laws. Masses of the Earth and Sun compared.
Density of the Sun. Force of gravity at its surface. Disturbing effect of the
Sun on the Moon's motion......... tº e º e e º e º 'o tº º e º 'º e º e tº e º º ºs e e e e e º ºs s is e © tº e g º e e g º C º 'º e tº tº gº tº e º E tº gº 233
CHAPTER IX.
O F T H E S O L A. R. S. Y S T E M .
Apparent motions of the planets. Their stations and retrogradations. The Sun
their natural centre of motion. Inferior planets. Their phases, periods, etc.
Dimensions and form of their orbits. Transits across the Sun. Superior
planets. Their distances, periods, etc. Kepler's laws and their interpreta-
tion. Elliptic elements of a planet's orbit. Its heliocentric and geocentric
place. Empirical law of planetary distances; violated in the case of Nep-
tune. The ultra-zodiacal planets. Physical peculiarities observable in each
of the planets.......... © º ſº dº tº º º Lº º & º & & © tº e º 'º . . . . . . . . . . . . . tº e º 'º e s tº a º º tº º is tº $ g º ºs e º e s e o e º a s e º 'º e e º e º ſº 242
CHAPTER X.
O F T H E S A. T E L L IT E S .
Of the Moon, as a satellite of the Earth. General proximity of satellites to
their primaries, and consequent subordination of their motions. Masses of
the primaries concluded from the periods of their satellites. Maintenance of
Kepler's laws in the secondary systems. Of Jupiter's satellites. Their
eclipses, etc. Velocity of light discovered by their means. Satellites of
Saturn—of Uranus—of Neptune......................... º, e º e º e e tº e º ſº tº e º e g tº tº e s tº a tº e g º ºs e e 282
CHAPTER XI.
OF COMETS.
Great number of recorded comets. The number of those unrecorded probably
much greater. General description of a comet. Comets without tails, or with
more than one. Their extreme tenuity. Their probable structure. Motions
conformable to the law of gravity. Actual dimensions of comets. Periodical
return of several. Halley’s comet. Other ancient comets probably periodic.
2
xiv. CONTENTS.
Encke's comet–Biela’s—Faye's—Lexell's—De Vico's—Brorsen’s—Peter's.
Great comet of 1843. Its probable identity with several older comets. Great
interest at present attached to cometary astronomy, and its reasons. Re-
marks on cometary orbits in general ......... ........................................ 295
PART II.
OF THE PLANETARY PERTURBATIONS.
CHAPTER, XII.
Subject propounded. Problem of three bodies. Superposition of small motions.
Estimation of the disturbing force. Its geometrical representation. Nume-
rical estimation in particular cases. Resolution into rectangular components.
Radial, transversal, and orthogonal disturbing forces. Normal and tangential.
Their characteristic effects. Effects of the orthogonal force. Motion of the
nodes. Conditions of their advance and recess. Cases of an exterior planet
disturbed by an interior. The reverse case. In every case the node ºf the
disturbed orbit recedes on the plane of the disturbing on an average. Com-
bined effect of many such disturbances. Motion of the Moon's nodes.
Change of inclination. Conditions of its increase and diminution. Average
effect in a whole revolution. Compensation in a complete revolution of the
nodes. Lagrange's theorem of the stability of the inclinations of the plane-
tary orbits. Change of obliquity of the ecliptic. Precession of the equinoxes
explained. Nutation. Principle of forced vibrations.......................... 826
CHAPTER XIII.
THEORY OF THE AxES, PERIHELIA, AND ExCENTRICITIES.
Variation of elements in general. Distinction between periodic and secular
variations. Geometrical expression of tangential and normal forces. Varia-
tion of the Major Axis produced only by the tangential force. Lagrange’s
theorem of the conservation of the mean distances and periods. Theory of
the Perihelia and Excentricities. Geometrical representation of their mo-
mentary variations. Estimation of the disturbing forces in nearly circular
orbits. Application to the case of the Moon. Theory of the lunar apsides
and excentricity. Experimental illustration. Application of the foregoing
principles to the planetary theory. Compensation in orbits very nearly cir-
cular. Effects of ellipticity. General results. Lagrange's theorem of the
stability of the excentricities ............................ tº e º e º & e º e º e º º e s e º 'º e º e º e º e º e e º e 354
CHAPTER XIV.
Of the inequalities independent of the excentricities. The Moon's variation and
parallactic inequality. Analogous planetary inequalities. Three cases of
CONTENTS. XV.
planetary perturbation distinguished. Of inequalities dependent on the excen-
tricities. Long inequality of Jupiter and Saturn. Law of reciprocity between
the periodical variations of the elements of both planets. Long inequality of
the Earth and Venus. Variation of the epoch. Inequalities incident on the
epoch affecting the mean motion. Interpretation of the constant part of these
inequalities. Annual equation of the Moon. Her secular acceleration. Lunar
inequalities due to the action of Venus. Effect of the spheroidal figure of the
Earth and other planets on the motions of their satellites. Of the tides.
Masses of disturbing bodies deducible from the perturbations they produce.
Mass of the Moon, and of Jupiter’s satellites, how ascertained. Perturbations
of Uranus resulting in the discovery of Neptune................................. 887
PART III.
OF SIDEREAL ASTRONOMY.
CHAPTER XV.
Of the fixed stars. Their classification by magnitudes. Photometric scale of
magnitudes. Conventional or vulgar scale. Photometric comparison of stars.
Distribution of stars over the heavens. Of the Milky Way or galaxy. Its
supposed form that of a flat stratum partially subdivided. Its visible course
among the constellations. Its internal structure. Its apparently indefinite
extent in certain directions. Of the distance of the fixed stars. Their
annual parallax. Parallactic unit of sidereal distance. Effect of parallax
analogous to that of aberration. How distinguished from it. Detection of
parallax by meridional observations. Henderson’s application to a Centauri.
By differential observations. Discoveries of Bessel and Struve. List of stars
in which parallax has been detected. Of the real magnitudes of the stars.
Comparison of their lights with that of the Sun................................. . 439
CHAPTER XVI.
Variable and periodical stars. List of those already known. Irregularities in
their periods and lustre when brightest. Irregular and temporary stars.
Ancient Chinese records of several. Missing stars. Double stars. Their
classification. Specimens of each class. Binary systems. Revolution round
each other. Describe elliptic orbits under the Newtonian law of gravity.
IElements of orbits of several. Actual dimensions of their orbits. Coloured
double stars. Phaenomenon of complementary colours. Sanguine stars.
Proper motion of the stars. Partly accounted for by a real motion of the Sun.
Situation of the solar apex. Agreement of Southern and northern stars in
giving the same result. Principles on which the investigation of the solar
motion depends. Absolute velocity of the Sun's motion. Supposed revolution
of the whole sidereal system round a common centre. Systematic parallax
and aberration. Effect of the motion of light in altering the apparent period
of a binary star ............................................... tº tº º tº e º 'º e º $ & © tº e º 'º e º e º e º g ..... 467
XVI - CONTENTS.
CHAPTER XVII.
OF CIUSTERS OF STARS AND NEBULAF.
Of clustering groups of stars. Globular clusters. Their stability dynamically
possible. List of the most remarkable. Classification of nebulae and clusters.
Their distribution over the heavens. Irregular clusters. Resolvability of
nebulae. Theory of the formation of clusters by nebulous subsidence. Of
elliptic nebulae. That of Andromeda. Annular and planetary nebulae.
Double nebulae. Nebulous stars. Connection of nebulae with double stars.
Insulated nebulae of forms not wholly irregular. Of amorphous nebulae.
Their law of distribution marks them as outliers of the galaxy. Nebulae and
nebulous group of Orion—of Argo—of Sagittarius—of Cygnus. The Magel-
lanic clouds. Singular nebula in the greater of them. The zodiacal light.
Shooting stars ......... e e s e s e o e s so e o e e s = e s e e s - e. e º e s a se e s • * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 498
PART IV.
O F T H E A C C O U N T OF T H M E.
CHAPTER XVIII.
Natural units of time. Relation of the sidereal to the solar day affected by
precession. Incommensurability of the day and year. Its inconvenience.
How obviated. The Julian Calendar. Irregularities at its first introduction.
Reformed by Augustus. Gregorian reformation. Solar and lunar cycles.
Indiction. Julian period. Table of chronological eras. Rules for calculating
- the days elapsed between given dates. Equinoctial time..................... . 523
APPENDIX.
I. Lists of Northern and Southern Stars, with their approximate Magni-
tudes, on the Vulgar and Photometric Scales ............................... 541
II. Synoptic Table of the Elements of the Planetary System ......... tº e º tº e g g g e 543
III. Synoptic Table of the Elements of the Orbits of the Satellites, so far
as they are known .......................................................... ........ 545
IV. Elements of Periodical Comets at their last appearance............ ........ 548
INDEX e e s e º see e s e º e s e is e a e e s see e s c e s is e s s a se e s & © e º e a se s is e is e e s e s e º 0 & 0 & 0 & 0 & © e & 6 & 8 & 9 & 0 & 0 & G & © e s g c e s is e g º ºs 549
0 UT LINES
OF
A S T R O N 0 M Y.
**Nºvºv-º-º-º- *ſºn
IN T R O DU C TI O N.
(1.) EvKRY student who enters upon a scientific pursuit, especially if
at a somewhat advanced period of life, will find not only that he has
much to learn, but much also to unlearn. Familiar objects and events
are far from presenting themselves to our senses in that aspect and with
those connections under which science requires them to be viewed, and
which constitute their rational explanation. There is, therefore, every
reason to expect that those objects and relations which, taken together,
constitute the subject he is about to enter upon will have been previously
apprehended by him, at least imperfectly, because much has hitherto
escaped his notice which is essential to its right understanding: and not
only so, but too often also erroneously, owing to mistaken analogies, and
the general prevalence of vulgar errors. As a first preparation, therefore,
for the course he is about to commence, he must loosen his hold on all
crude and hastily adopted notions, and must strengthen himself, by some-
thing of an effort and a resolve, for the unprejudiced admission of any
conclusion which shall appear to be supported by careful observation and
logical argument, even should it prove of a nature adverse to notions he
may have previously formed for himself, or taken up, without examina-
tion, on the credit of others. Such an effort is, in fact, a commencement
of that intellectual discipline which forms one of the most important ends
of all science. It is the first movement of approach towards that state of
mental purity which alone can fit us for a full and steady perception of
moral beauty as well as physical adaptation. It is the “euphrasy and
rue” with which we must “purge our sight” before we can receive and
contemplate as they are the lineaments of truth and nature.
2 (17)
18 oUTLINES OF ASTRONOMY.
(2.) There is no science which, more than astronomy, stands in need
of such a preparation, or draws more largely on that intellectual liberality
which is ready to adopt whatever is demonstrated, or concede whatever is
rendered highly probable, however new and uncommon the points of view
may be in which objects the most familiar may thereby become placed.
Almost all its conclusions stand in open and striking contradiction with
those of Superficial and vulgar observation, and with what appears to
every one, until he has understood and weighed the proofs to the con-
trary, the most positive evidence of his senses. Thus, the earth on which
he stands, and which has served for ages as the unshaken foundation of
the firmest structures, either of art or nature, is divested by the astro-
nomer of its attribute of fixity, and conceived by him as turning swiftly
on its centre, and at the same time moving onwards through space with
great rapidity. The sun and the moon, which appear to untaught eyes
round bodies of no very considerable size, become enlarged in his imagi-
nation into vast globes, – the one approaching in magnitude to the earth
itself, the other immensely surpassing it. The planets, which appear
only as stars somewhat brighter than the rest, are to him spacious, elabo-
rate, and habitable worlds; several of them much greater and far more
curiously furnished than the earth he inhabits, as there are also others
less so; and the stars themselves, properly so called, which to ordinary
apprehension present only lucid sparks or brilliant atoms, are to him suns
of various and transcendent glory — effulgent centres of life and light to
myriads of unseen worlds. So that when, after dilating his thoughts to
comprehend the grandeur of those ideas his calculations have called up,
and exhausting his imagination and the powers of his language to devise
similes and metaphors illustrative of the immensity of the scale on which
his universe is constructed, he shrinks back to his native sphere; he finds
it, in comparison, a mere point; so lost—even in the minute system to
which it belongs—as to be invisible and unsuspected from some of its
principal and remoter members.
(3.) There is hardly any thing which sets in a stronger light the inhe-
rent power of truth over the mind of man, when opposed by no motives
of interest or passion, than the perfect readiness with which all these con-
clusions are assented to as soon as their evidence is clearly apprehended,
and the tenacious hold they acquire over our belief when once admitted.
In the conduct, therefore, of this volume, I shall take it for granted that
the reader is more desirous to learn the system which it is its object to
teach as it now stands, than to raise or revive objections against it; and
that, in short, he comes to the task with a willing mind; an assumption
which will not only save the trouble of piling argument on argument to
INTRODUCTION. - 19
convince the sceptical, but will greatly facilitate his actual progress; inas-
much as he will find it at once easier and more satisfactory to pursue from
the outset a straight and definite path, than to be constantly stepping
aside, involving himself in perplexities and circuits, which, after all, can
only terminate in finding himself compelled to adopt the same road.
(4.) The method, therefore, we propose to follow in this work is neither
strictly the analytic nor the synthetic, but rather such a combination of
both, with a leaning to the latter, as may best suit with a didactic com-
position. Its object is not to convince or refute opponents, nor to inquire,
under the semblance of an assumed ignorance, for principles of which we
are all the time in full possession — but simply to teach what is known.
The moderate limit of a single volume, to which it will be confined, and
the necessity of being on every point, within that limit, rather diffuse and
copious in explanation, as well as the eminently matured and ascertained
character of the science itself, render this course both practicable and
eligible. Practicable, because there is now no danger of any revolution
in astronomy, like those which are daily changing the features of the less
advanced sciences, supervening, to destroy all our hypotheses, and throw
our statements into confusion. Eligible, because the space to be bestowed,
either in combating refuted systems, or in leading the reader forward by
slow and measured steps from the known to the unknown, may be more
advantageously devoted to such explanatory illustrations as will impress
on him a familiar and, as it were, a practical sense of the sequence of
phenomena, and the manner in which they are produced. We shall not,
then, reject the analytic course where it leads more easily and directly to
our objects, or in any way fetter ourselves by a rigid adherence to method.
Writing only to be understood, and to communicate as much information
in as little space as possible, consistently with its distinct and effectual
communication, no sacrifice can be afforded to system, to form, or to
affectation. ſº
(5.) We shall take for granted, from the outset, the Copernican system
of the world; relying on the easy, obvious, and natural explanation it
affords of all the phenomena as they come to be described, to itnpress the
student with a sense of its truth, without either the formality of demon-
stration or the superfluous tedium of eulogy, calling to mind that impor-
tant remark of Bacon : —“Theoriarum vires, arcta et quasi se mutuo
Sustinente partium adaptatione, quâ, quasi in orbem cohaerent, firmantur';”
* “The confirmation of theories relies on the compact adaptation of their parts, by
which, like those of an arch or dome, they mutually sustain each other, and form a
coherent whole.” This is what Dr. Whewell expressively terms the consilienee of
inductions.
20 OUTLINES OF ASTRONOMY.
not failing, however, to point out to the reader, as occasion offers, the
contrast which its superior simplicity offers to the complication of other
hypotheses.
(6.) The preliminary knowledge which it is desirable that the student
should possess, in order for the more advantageous perusal of the following
pages, consists in the familiar practice of decimal and sexagesimal arith-
metic; some moderate acquaintance with geometry and trigonometry,
both plane and spherical; the elementary principles of mechanics; and
enough of optics to understand the construction and use of the telescope,
and some other of the simpler instruments. Of course, the more of such
knowledge he brings to the perusal, the easier will be his progress, and
the more complete the information gained; but we shall endeavour in
every case, as far as it can be done without a sacrifice of clearness, and of
that useful brevity which consists in the absence of prolixity and epi-
sode, to render what we have to say as independent of other books as
possible.
(7.) After all, I must distinctly caution such of my readers as may
commence and terminate their astronomical studies with the present work
(though of such, – at least in the latter predicament, — I trust the num-
ber will be few), that its utmost pretension is to place them on the
threshold of this particular wing of the temple of Science, or rather on
an eminence exterior to it, whence they may obtain something like a
general notion of its structure; or, at most, to give those who may wish
to enter a ground-plan of its accesses, and put them in possession of the
pass-word. Admission to its sanctuary, and to the privileges and feelings
of a votary, is only to be gained by one means, – sound and sufficient
knowledge of mathematics, the great instrument of all eacact inquiry,
without which no man can ever make such advances in this or any other
of the higher departments of science as can entitle him to form an inde-
pendent opinion on any subject of discussion within their range. It is
not without an effort that those who possess this knowledge can commu-
nicate on such subjects with those who do not, and adapt their language
and their illustrations to the necessities of such an intercourse. Proposi-
tions which to the one are almost identical, are theorems of import and
difficulty to the other; nor is their evidence presented in the same way to
the mind of each. In teaching such propositions, under such circum-
stances, the appeal has to be made, not to the pure and abstract reason,
but to the sense of analogy—to practice and experience: principles and
modes of action have to be established not by direct argument from
acknowledged axioms, but by continually recurring to the sources from
which the axioms themselves have been drawn; viz. examples; that is to
INTRODUCTION. 21
say, by bringing forward and dwelling on simple and familiar instances in
which the same principles and the same or similar modes of action take
place: thus erecting, as it were, in each particular case, a separate induc-
tion, and constructing at each step a little body of science to meet its
exigencies. The difference is that of pioneering a road through an un-
traversed country and advancing at ease along a broad and beaten high-
way; that is to say, if we are determined to make ourselves distinctly
understood, and will appeal to reason at all. As for the method of asser-
tion, or a direct demand on the faith of the student (though in some com-
plex cases indispensable, where illustrative explanation would defeat its
own end by becoming tedious and burdensome to both parties), it is one
which I shall neither willingly adopt nor would recommend to others.
(8.) On the other hand, although it is something new to abandon the
road of mathematical demonstration in the treatment of subjects suscepti-
ble of it, and to teach any considerable branch of science entirely or chiefly
by the way of illustration and familiar parallels, it is yet not impossible
that those who are already well acquainted with our subject, and whose
knowledge has been acquired by that confessedly higher practice which is
incompatible with the avowed objects of the present work, may yet find their
account in its perusal,— for this reason, that it is always of advantage to
present any given body of knowledge to the mind in as great a variety of
different lights as possible. It is a property of illustrations of this kind
to strike no two minds in the same manner, or with the same force;
because no two minds are stored with the same images, or have acquired
their notions of them by similar habits. Accordingly, it may very well
happen, that a proposition, even to one best acquainted with it, may be
placed not merely in a new and uncommon, but in a more impressive and
satisfactory light by such a course—some obscurity may be dissipated,
some inward misgivings cleared up, or even some links supplied which
may lead to the perception of connections and deductions altogether
unknown before. And the probability of this is increased when, as in
the present instance, the illustrations chosen have not been studiously
selected from books, but are such as have presented themselves freely to
the author's mind as being most in harmony with his own views; by
which, of course, he means to lay no claim to originality in all or any of
them beyond what they may really possess. º
(9.) Besides, there are cases in the application of mechanical principles
with which the mathematical student is but too familiar, where, when the
data are before him, and the numerical and geometrical relations of his
problems all clear to his conception,-when his forces are estimated and
his lines measured, – nay, when even he has followed up the application
22 ouTLINES OF ASTRONOMY.
of his technical processes, and fairly arrived at his conclusion, — there is
still something wanting in his mind—not in the evidence, for he has
examined each link, and finds the chain complete—not in the principles,
for those he well knows are too firmly established to be shaken — but pre-
cisely in the mode of action. He has followed out a train of reasoning
by logical and technical rules, but the signs he has employed are not
pictures of nature, or have lost their original meaning as such to his
mind: he has not seen, as it were, the process of mature passing under
his eye in an instant of time, and presented as a consecutive whole to his
imagination. A familiar parallel, or an illustration drawn from some
artificial or natural process, of which he has that direct and individual
impression which gives it a reality and associates it with a name, will, in
almost every such case, supply in a moment this deficient feature, will
convert all his symbols into real pictures, and infuse an animated meaning
into what was before a lifeless succession of words and signs. I cannot,
indeed, always promise myself to attain this degree of vividness of illus-
tration, nor are the points to be elucidated themselves always capable of
being so paraphrased (if I may use the expression) by any single in-
stance adducible in the ordinary course of experience; but the object will
at least be kept in view; and, as I am very conscious of having, in making
such attempts, gained for myself much clearer views of several of the
more concealed effects of planetary perturbation than I had acquired by
their mathematical investigation in detail, it may reasonably be hoped
that the endeavour will not always be unattended with a similar success
in others. • *.
(10.) From what has been said, it will be evident that our aim is not
to offer to the public a technical treatise, in which the student of practical
or theoretical astronomy shall find consigned the minute description of
methods of observation, or the formulae he requires prepared to his hand,
or their demonstrations drawn out in detail. In all these the present
work will be found meagre, and quite inadequate to his wants. Its aim
is entirely different; being to present in each case the mere ultimate
rationale of facts, arguments, and processes; and, in all cases of mathe-
matical application, avoiding whatever would tend to encumber its pages
with algebraic or geometrical symbols, to place under his inspection that
central thread of common sense on which the pearls of analytical research
are invariably strung; but which, by the attention the latter claim for
themselves, is often concealed from the eye of the gazer, and not always
disposed in the straightest and most convenient form to follow by those
who string them. This is no fault of those who have conducted the
inquiries to which we allude. The contention of mind for which they call
INTRODUCTION. 23
is enormous; and it may, perhaps, be owing to their experience of how
little can be accomplished in carrying such processes on to their conclu-
sion, by mere ordinary clearness of head; and how necessary it often is
to pay more attention to the purely mathematical conditions which ensure
success, - the hooks-and-eyes of their equations and series, – than to
those which enchain causes with their effects, and both with the human
reason, — that we must attribute something of that indistinctness of view
which is often complained of as a grievance by the earnest student, and
still more commonly ascribed ironically to the native cloudiness of an
atmosphere too sublime for vulgar comprehension. We think we shall
render good service to both classes of readers, by dissipating, so far as lies
in our power, that accidental obscurity, and by showing ordinary untu-
tored comprehension clearly what it can, and what it cannot, hope to
attain,
,”
24 oUTLINEs of ASTRONOMY.
sº
CHAPTER I.
GENTERAT, NOTIONS. — APPARENT AND REAL MOTIONS. - SHAPE AND
SIZE OF THE EARTH. — THE HORIZON AND ITS DIP. - THE ATMO-
SPHERE. — REFRACTION.—TWILIGHT. — APPEARANCES RESULTING
FROM DITURNAL MOTION. — FROM CHANGE OF STATION IN GENERAL.
— PARALIACTIC MOTIONS. — TERRESTRIAL PARALLAX. - THAT OF
THE STARS INSENSIBLE. — FIRST STEP TOWARDS FORMING AN IDEA
OF THE DISTANCE OF THE STARS. — COPERNICAN VIEW OF • THE
EARTH's MOTION. — RELATIVE MOTION. — MOTIONS PARTLY REAL,
PARTLY APPARENT.—GEOCENTRIC ASTRONOMY, OR IDEAL REFERENCE
OF PHAENOMENA TO THE EARTH's CENTRE As A COMMON CONVEN-
TIONAL STATION.
(11.) THE magnitudes, distances, arrangement, and motions of the
great bodies which make up the visible universe, their constitution and
physical condition, so far as they can be known to us, with their mutual
influences and actions on each other, so far as they can be traced by the
effects produced, and established by legitimate reasoning, form the assem-
blage of objects to which the intention of the astronomer is directed.
The term astronomy' itself, which denotes the law or rule of the astra (by
which the ancients understood not only the stars properly so called, but
the sun, the moon, and all the visible constituents of the heavens), suffi-
ciently indicates this; and, although the term astrology, which denotes the
reason, theory, or interpretation of the stars,” has become degraded in its
application, and confined to superstitious and delusive attempts to divine
future events by their dependence on pretended planetary influences, the
same meaning originally attached itself to that epithet. -
(12.) But, besides the stars and other celestial bodies, the earth itself,
regarded as an individual body, is one principal object of the astronomer's
consideration, and indeed, the chief of all. It derives its importance, in
* Agrip, a star; vonos, a law , or vegetv, to tend, as a shepherd, his flock; so that
agroovopos means “shepherd of the stars.” The two two etymologies are, however,
coincident,
* Aoyos, reason, or a word, the vehicle of reason; the interpreter of thought.
GENERAL NOTIONS. 25
a practical as well as theoretical sense, not only from its proximity, and
its relation to us as animated beings, who draw from it the supply of all
our wants, but as the station from which we see all the rest, and as the
only one among them to which we can, in the first instance, refer for any
determinate marks and measures by which to recognize their changes of
situation, or with which to compare their distances.
(13.) To the reader who now for the first time takes up a book on
astronomy, it will no doubt seem strange to class the earth with the
heavenly bodies, and to assume any community of nature among things
apparently so different. For what, in fact, can be more apparently differ-
ent than the vast and seemingly immeasurable extent of the earth, and the
stars? The earth is a dark and opaque, while the celestial bodies are
brilliant. We perceive in it no motion, while in them we observe a con-
tinual change of place, as we view them at different hours of the day or
night, or at different seasons of the year. The ancients, accordingly, one
or two of the more enlightened of them only excepted, admitted no such
community of nature; and, by thus placing the heavenly bodies and their
movements without the pale of analogy and experience, effectually inter-
cepted the progress of all reasoning from what passes here below, to what
is going on in the regions where they exist and move. Under such con-
ventions, astronomy, as a science of cause and effect, could not exist, but
must be limited to a mere registry of appearances, unconnected with any
attempt to account for them on reasonable principles, however successful
to a certain extent might be the attempt to follow out their order of
sequence, and to establish empirical laws expressive of this order. To
get rid of this prejudice, therefore, is the first step towards acquiring a
knowledge of what is really the case; and the student has made his first
effort towards the acquisition of sound knowledge, when he has learnt to
familiarize himself with the idea that the earth, after all, may be nothing
but a great star. How correct such an idea may be, and with what limi-
tations and modifications it is to be admitted, we shall see presently.
(14.) It is evident, that, to form any just notions of the arrangement,
in space, of a number of objects which we cannot approach and examine,
but of which all the information we can gain is by sitting still and watch-
ing their evolutions, it must be very important for us to know, in the first
instance, whether what we call sitting still is really such : whether the
station from which we view them, with ourselves, and all objects which
immediately surround us, be not itself in motion, unperceived by us; and
if so, of what nature that motion is. The apparent places of a number
of objects, and their apparent arrangement with respect to each other,
will of course be materially dependent on the situation of the spectator
26 OUTLINES OF ASTRONOMY.
among them; and if this situation be liable to change, unknown to the
spectator himself, an appearance of change in the respective situations of
the objects will arise, without the reality. If, then, such be actually the
case, it will follow that all the movements we think we perceive among
the stars will not be real movements, but that some part, at least, of
whatever changes of relative place we perceive among them must be
merely apparent, the results of the shifting of our own point of view;
and that, if we would ever arrive at a knowledge of their real motions, it
can only be by first investigating our own, and making due allowance for
its effects. Thus, the question whether the earth is in motion or at rest,
and if in motion, what that motion is, is no idle inquiry, but one on which
depends our only chance of arriving at true conclusions respecting the
constitution of the universe. , *
(15.) Nor let it be thought strange that we should speak of a motion
existing in the earth, unperceived by its inhabitants; we must remember
that it is of the earth as a whole, with all that it holds within its substance
or Sustains on its surface, that we are speaking; of a motion common to
the Solid mass beneath, to the ocean which flows around it, the air that
rests upon it, and the clouds which float above it in the air. Such a
motion, which should displace no terrestrial object from its relative situa-
tion among others, interfere with no natural processes, and produce no
Sensations of shocks or jerks, might, it is very evident, subsist undetected
by us. There is no peculiar sensation which advertises us that we are in
motion. We perceive jerks, or shocks, it is true, because these are sud-
den changes of motion, produced, as the laws of mechanics teach us, by
sudden and powerful forces acting during short times; and these forces,
applied to our bodies, are what we feel. When, for example, we are
carried along in a carriage with the blinds down, or with our eyes closed
(to keep us from seeing external objects), we perceive a tremor arising from
inequalities in the road, over which the carriage is successively lifted and
let fall, but we have no sense of progress. As the road is smoother, our
Sense of motion is diminished, though our rate of travelling is accelerated.
Railway travelling, especially by night, or in a tunnel, has familiarized
every one with this remark. Those who have made aeronautic voyages
testify that with closed eyes, and under the influence of a steady breeze
communicating no oscillatory or revolving motion to the car, the sensation
is that of perfect rest, however rapid the transfer from place to place.
(16.) But it is on shipboard, where a great system is maintained in
motion, and where we are surrounded with a multitude of objects which
participate with ourselves and each other in the common progress of the
whole mass, that we feel most satisfactorily the identity of sensation
APPARENT AND REAL MOTIONS. 27
between a state of motion and one of rest. In the cabin of a large and
heavy vessel, going smoothly before the wind in still water, or drawn along
a canal, not the smallest indication acquaints us with the way it is making.
We read, sit, walk, and perform every customary action as if we were on
land. If we throw a ball into the air, it falls back into our hand; or if
we drop it, it alights at our feet. Insects buzz around us as in the free
air; and smoke ascends in the same manner as it would do in an apart-
ment on shore. If, indeed, we come on deck, the case is, in some respects,
different; the air, not being carried along with us, drifts away smoke and
other light bodies — such as feathers abandoned to it — apparently, in
the opposite direction to that of the ship's progress; but, in reality, they
remain at rest, and we leave them behind in the air. Still, the illusion,
so far as massive objects and our own movements are concerned, remains
complete; and when we look at the shore, we then perceive the effect of
our own motion transferred, in a contrary direction, to external objects—
external, that is, to the system of which we form a part.
“Provehimur, portu, terraque urbesque recedunt.
(17.) In order, however, to conceive the earth as in motion, we must
form to ourselves a conception of its shape and size. Now, an object
cannot have shape and size unless it is limited on all sides by some defi-
nite outline, so as to admit of our imagining it, at least, disconnected from
other bodies, and existing insulated in space. The first rude notion we
form of the earth is that of a flat surface, of indefinite extent in all direc-
tions from the spot where we stand, above which are the air and sky;
below, to an indefinite profundity, solid matter. This is a prejudice to be
got rid of, like that of the earth's immobility; — but it is one much
easier to rid ourselves of, inasmuch as it originates only in our own mental
inactivity, in not questioning ourselves where we will place a limit to a
thing we have been accustomed from infancy to regard as immensely
large; and does not, like that, originate in the testimony of our senses
unduly interpreted. On the contrary, the direct testimony of our senses
lies the other way. When we see the sun set in the evening in the west,
and rise again in the east, as we cannot doubt that it is the same sun we
see after a temporary absence, we must do violence to all our notions of
solid matter, to suppose it to have made its way through the substance of
the earth. It must, therefore, have gone under it, and that not by a mere
subterraneous channel; for if we notice the points where it sets and rises
for many successive days, or for a whole year, we shall find them con-
stantly shifting, round a very large extent of the horizon; and, besides,
the moon and stars also set and rise again in all points of the visible
28 OUTLINES OF ASTRONOMY.
horizon. The conclusion is plain : the earth cannot extend indefinitely
in depth downwards, nor indefinitely in surface laterally; it must have not
only bounds in a horizontal direction, but also an under side round which
the sun, moon, and stars can pass; and that side must, at least, be so far
like what we see, that it must have a sky and sunshine, and a day when
it is night to us, and vice versä ; where, in short,
—“redit a nobis Aurora, diemgue reducit.
Nosque ubi primus equis oriens afflavit anhelis,
Illic sera rubens accendit lumina Vesper. Georg.
(18.) As soon as we have familiarized ourselves with the conception of
an earth without foundations or fixed supports — existing insulated in
space from contact of every thing external, it becomes easy to imagine it
in motion — or, rather, difficult to imagine it otherwise: for, since there
is nothing to retain it in one place, should any causes of motion exist, or
any forces act upon it, it must obey their impulse. Let us next see what
obvious circumstances there are to help us to a knowledge of the shape
of the earth.
(19.) Let us first examine what we can actually see of its shape. Now,
it is not on land (unless, indeed, on uncommonly level and extensive
plains), that we can see any thing of the general figure of the earth; —
the hills, trees, and other objects which roughen its surface, and break
and elevate the line of the horizon, though obviously bearing a most mi-
nute proportion to the whole earth, are yet too considerable with respect
to ourselves and to that small portion of it which we can see at a single
view, to allow of our forming any judgment of the form of the whole, from
that of a part so disfigured. But with the surface of the sea or any vastly
extended level plain, the case is otherwise. If we sail out of sight of
land, whether we stand on the deck of the ship or climb the mast, we see
the surface of the sea — not losing itself in distance and mist, but termi-
nated by a sharp, clear, well-defined line or offing as it is called, which
runs all round us in a circle, having our station for its centre. That this
line is really a circle, we conclude, first, from the perfect apparent similar-
ity of all its parts; and, secondly, from the fact of all its parts appearing
at the same distance from us, and that, evidently, a moderate one; and
thirdly, from this, that its apparent diameter, measured with an instru-
ment called the dip sector, is the same (except under some singular atmos.
pheric circumstances, which produce a temporary distortion of the outline),
in whatever direction the measure is taken,_ properties which belong only
to the circle among geometrical figures. If we ascend a high eminence
THE HORIZON AND ITS DIP. 29
on a plain (for instance, one of the Egyptian pyramids,) the same holds
good.
(20.) Masts of ships, however, and the edifices erected by man, are
trifling eminences compared to what nature itself affords; Ætna, Tene-
riffe, Mowna Roa, are eminences from which no contemptible aliquot part
of the whole earth’s surface can be seen; but from these again—in those
few and rare occasions when the transparency of the air will permit the
real boundary of the horizon, the true sea-line, to be seen — the very
same appearances are witnessed, but with this remarkable addition, viz.,
that the angular diameter of the visible area, as measured by the dip sec-
tor, is materially less than at a lower level; or, in other words, that the
apparent size of the earth has sensibly diminished as we have receded
from its surface, while yet the absolute quantity of it seen at once has been
increased. º e
(21.) The same appearances are observed universally, in every part of
the earth’s surface visited by man. Now, the figure of a body which,
however seen, appears always circular, can be no other than a sphere or
globe.
(22.) A diagram will elucidate this. Suppose the earth to be repre-
sented by the sphere LHNQ, whose centre is C, and let A, G, M be sta-
tions at different elevations above various points of its surface, represented
Fig. 1
_-T Z ~~~
2. ^e
º SS
/ ^
Z \
Z \
M \
/ \
|
EX: M |Y
i F-
\ N ſ
* f
º ºx: * - */
\ 2–~~~ = ~ *** * * T--> A
NX-r ~S (Q
2 º
2
7. O
T
O
Ö
C

30 OUTLINES OF ASTRONOMY.
by a, g, m respectively. From each of them (as from M) let a line be
drawn, as MNn, a tangent to the surface at N, then will this line represent
the visual ray along which the spectator at M will see the visible horizon;
and as this tangent sweeps round M, and comes successively into the posi-
tions MOo, MPp, MQq, the point of contact N will mark out on the sur-
face the circle NOPQ. The area of the spherical surface comprehended
within this circle is the portion of the earth's surface visible to a spectator
at M, and the angle NMQ included between the two extreme visual rays
is the measure of its apparent angular diameter. Leaving, at present, out
of consideration the effect of refraction in the air below MI, of which more
hereafter, and which always tends, in some degree, to increase that angle,
or render it more obtuse, this is the angle measured by the dip sector.
Now, it is evident, 1st, that as the point M is more elevated above m, the
point immediately below it on the sphere, the visible area, i. e. the spher-
ical segment or slice NOPQ, increases; 2dly, that the distance of the vis-
ible horizon' or boundary of our view from the eye, viz. the line MN,
increases; and, 3dly, that the angle NMQ becomes less obluse, or, in other
words, the apparent angular diameter of the earth diminishes, being no-
where so great as 180°, or two right angles, but falling short of it by
some sensible quantity, and that more and more the higher we ascend.
The figure exhibits three states or stages of elevation, with the horizon,
&c., corresponding to each, a glance at which will explain our meaning;
or, limiting ourselves to the larger and more distinct, MNOPQ, let the
reader imagine nVM, MQq to be the two legs of a ruler jointed at M, and
kept extended by the globe Nm.0 between them. It is clear, that as the
joint M is urged home towards the surface, the legs will open, and the ruler
will become more nearly straight, but will not attain perfect straightness
till M is brought fairly up to contact with the surface at m, in which case
its whole length will become a tangent to the sphere at m, as is the line
£y.
(23.) This explains what is meant by the dip of the horizon. Mm,
which is perpendicular to the general surface of the sphere at m, is also
the direction in which a plumb-line” would hang; for it is an observed
fact, that in all situations, in every part of the earth, the direction of a
plumb-line is exactly perpendicular to the surface of still water; and,
moreover, that it is also exactly perpendicular to a line or surface truly
adjusted by a spirit-level.” Suppose, then, that at our station M we were
to adjust a line (a wooden ruler for instance) by a spirit-level, with perfect
exactness; then, if we suppose the direction of this line indefinitely pro-
** Outów, to terminate.
*See these instruments described in Chap. III.
THE HORIZON AND ITS DIP. 31
longed both ways, as XMY, the line so drawn will be at right angles to
M m, and therefore parallel to a my, the tangent to the sphere at m. A
spectator placed at M will therefore see not only all the vault of the sky
above this line, as XZY, but also that portion or zone of it which lies
between XN and YQ; in other words, his sky will be more than a hemi-
sphere by the zone YQXN. It is the angular breadth of this redundant
zone—the angle YMC), by which the visible horizon appears depressed
below the direction of a spirit-level—that is called the dip of the horizon.
It is a correction of constant use in nautical astronomy.
(24.) From the foregoing explanations it appears, 1st, That the general
figure of the earth (so far as it can be gathered from this kind of observa-
tion) is that of a sphere or globe. In this we also include that of the sea,
which, wherever it extends, covers and fills in those inequalities and local
irregularities which exist on land, but which can of course only be regarded
as trifling deviations from the general outline of the whole mass, as we
consider an orange not the less round for the roughness on its rind. 2dly,
That the appearance of a visible horizon, or sea-offing, is a consequence
of the curvature of the surface, and does not arise from the inability of
the eye to follow objects to a greater distance, or from atmospheric indis-
tinctness. It will be worth while to pursue the general notion thus
acquired into some of its consequences, by which its consistency with
observations of a different kind, and on a larger scale, will be put to the
test, and a clear conception be formed of the manner in which the parts
of the earth are related to each other, and held together as a whole.
(25.) In the first place, then, every one who has passed a little while
at the sea-side is aware that objects may be seen perfectly well beyond the
offing or visible horizon—but not the whole of them. We only see their
upper parts. Their bases where they rest on, or rise out of the water, are
hid from view by the spherical surface of the sea, which protrudes between
them and ourselves. Suppose a ship, for instance, to sail directly away
from our station; — at first, when the distance of the ship is small, a
spectator, S, situated at some certain height above the sea, sees the whole
of the ship, even to the water line where it rests on the sea, as at A.
As it recedes it diminishes, it is true, in apparent size, but still the whole
is seen down to the water line, till it reaches the visible horizon at B.
but as soon as it has passed this distance, not only does the visible por-
tion still continue to diminish in apparent size, but the hull begins to
disappear bodily, as if sunk below the surface. When it has reached a
certain distance, as at C, its hull has entirely vanished, but the masts
and sails remain, presenting the appearance c. But if, in this state of
things, the spectator quickly ascends to a higher station, T, whose visibie
*
82 OUTLINES OF ASTRONOMY.
horizon is at D, the hull comes again in sight; and, when he descends
again he loses it. The ship still receding, the lower sails seem to sink
Fig. 2.
below the water, as at d, and at length the whole disappears: while yet
the distinctness with which the last portion of the sail d is seen is such
as to satisfy us that were it not for the interposed segment of the sea,
ABCDE, the distance TE is not so great as to have prevented an equally
perfect view of the whole.
(26.) The history of ačronautic adventure affords a curious illustration
of the same principle. The late Mr. Sadler, the celebrated ačronaut,
ascended on one occasion in a balloon from Dublin, and was wafted across
the Irish Channel, when, on his approach to the Welsh coast, the balloon
descended nearly to the surface of the sea. By this time the sun was
set, and the shades of evening began to close in. He threw out nearly
all his ballast, and suddenly sprang upwards to a great height, and by so
doing brought his horizon to dip below the sun, producing the whole
phenomenon of a western sunrise. Subsequently descending in Wales,
he of course witnessed a second sunset on the same evening.
(27.) If we could measure the heights and exact distance of two sta-
tions which could barely be discerned from each other over the edge of
the horizon, we could ascertain the actual size of the earth itself: and, in
fact, were it not for the effect of refraction, by which we are enabled to
see in some small degree round the interposed segment (as will be here-
after explained), this would be a tolerably good method of ascertaining
it. Suppose A and B to be two eminences, whose perpendicular heights
A a and Bb (which, for simplicity, we will suppose to be exactly equal)
are known, as well as their exact horizontal interval a Dºb, by measure-
ment; then it is clear that D, the visible horizon of both, will lie just
half-way between them, and if we suppose a D b to be the sphere of the
earth, and C its centre in the figure CD b B, we know D b, the length of
the arch of the circle between D and b, +viz. balf the measured interval,

SIZE OF THE EARTH. 33
and b B, the excess of its secant above its radius—which is the height of
B,-data which, by the solution of an easy geometrical problem, enable
- Fig. 3.
A T} Tº

us to find the length of the radius DC. If, as is really the case, we sup-
pose both the heights and distance of the stations inconsiderable in com-
parison with the size of the earth, the solution alluded to is contained in
the following proposition : — -
The earth’s diameter bears the same proportion to the distance of the
visible horizon from the eye as that distance does to the height of the
eye above the sea level.
When the stations are unequal in height, the problem is a little more
complicated. -
(28.) Although, as we have observed, the effect of refraction prevents
this from being an exact method of ascertaining the dimensions of the
earth, yet it will suffice to afford such an approximation to it as shall be
of use in the present stage of the reader's knowledge, and help him to
many just conceptions, on which account we shall exemplify its applica-
tion in numbers. Now, it appears by observation, that two points, each
ten feet above the surface, cease to be visible from each other over still
water, and in average atmospheric circumstances, at a distance of about 8
miles. But 10 feet is the 528th part of a mile, so that half their distance,
or 4 miles, is to the height of each as 4 x 528 or 2112:1, and therefore
in the same proportion to 4 miles is the length of the earth's diameter.
It must, therefore, be equal to 4 × 2112 = 8.448, or, in round numbers,
about 8000 miles, which is not very far from the truth.
(29.) Such is the first rough result of an attempt to ascertain the
earth’s magnitude; and it will not be amiss, if we take advantage of it
to compare it with objects we have been accustomed to consider as of vast
size, so as to interpose a few steps between it and our ordinary ideas of
dimension. We have before likened the inequalities on the earth’s sur-
face, arising from mountains, valleys, buildings, &c. to the roughnesses
on the rind of an Orange, compared with its general mass. The compari-
3
34 OljTLINES OF ASTRONOMY.
son is quite free from exaggeration. The highest mountain known hardly
exceeds five miles in perpendicular elevation: this is only one 1600th
part of the earth's diameter; consequently, on a globe of sixteen inches
in diameter, such a mountain would be represented by a protuberance of
no more than one hundredth part of an inch, which is about the thickness
of ordinary drawing-paper. Now, as there is no entire continent, or even
any very extensive tract of land, known, whose general elevation above
the sea is any thing like half this quantity, it follows, that if we would
construct a correct model of our earth, with its seas, continents, and
mountains, on a globe sixteen inches in diameter, the whole of the land,
with the exception of a few prominent points and ridges, must be com-
prised on it within the thickness of thin writing-paper; and the highest
hills would be represented by the smallest visible grains of sand.
(30.) The deepest mine existing does not penetrate half a mile below
the surface: a scratch, or pin-hole, duly representing it, on the surface of
such a globe as our model, would be imperceptible without a magnifier.
(31.) The greatest depth of sea, probably, does not very much exceed
the greatest elevation of the continents; and would, of course, be repre-
sented by an excavation, in about the same proportion, into the substance
of the globe: so that the ocean comes to be conceived as a mere film of
liquid, such as, on our model, would be left by a brush dipped in colour,
and drawn over those parts intended to represent the sea : only, in so con-
ceiving it, we must bear in mind that the resemblance extends no farther
than to proportion in point of quantity. The mechanical laws which
would regulate the distribution and movements of such a film, and its
adhesion to the surface, are altogether different from those which govern
the phenomena of the sea.
(32.) Lastly, the greatest extent of the earth's surface which has
ever been seen at once by man, was that exposed to the view of MM.
Biot and Gay-Lussac, in their celebrated ačronautic expedition to the
enormous height of 25,000 feet, or rather less than five miles. To esti-
mate the proportion of the area visible from this elevation to the whole
earth’s surface, we must have recourse to the geometry of the sphere,
which informs us that the convex surface of a spherical segment is to the
whole surface of the sphere to which it belongs as the versed sine or thick-
ness of the segment is to the diameter of the sphere; and further, that
this thickness, in the case we are considering, is almost exactly equal to
the perpendicular elevation of the point of sight above the surface. The
proportion, therefore, of the visible area, in this case, to the whole earth's
surface, is that of five miles to 8000, or 1 to 1600. The portion visible
from AEtna, the Peak of Teneriffe, or Mowna Roa, is about one 4000th.
THE ATMOSPHERE. 35
(33.) When we ascend to any very considerable elevation above the
surface of the earth, either in a balloon, or on mountains, we are made
aware, by many uneasy sensations, of an insufficient supply of air. The
barometer, an instrument which informs us of the weight of air incum-
bent on a given horizontal surface, confirms this impression, and affords
a direct measure of the rate of diminution of the quantity of air which a
given space includes as we recede from the surface. From its indications
we learn, that when we have ascended to the height of 1000 feet, we have
left below us about one-thirtieth of the whole mass of the atmosphere:—
that at 10,600 feet of perpendicular elevation (which is rather less than
that of the summit of Ætna") we have ascended through about one-third;
and at 18,000 feet (which is nearly that of Cotopaxi) through one-half
the material, or, at least, the ponderable body of air incumbent on the
earth's surface. From the progression of these numbers, as well as a
priori, from the nature of the air itself, which is compressible, i.e. capa-
ble of being condensed or crowded into a smaller space in proportion to
the incumbent pressure, it is easy to see that, although by rising still
higher, we should continually get above more and more of the air, and so
relieve ourselves more and more from the pressure with which it weighs
upon us, yet the amount of this additional relief, or the ponderable quan-
tity of air surmounted, would be by no means in proportion to the addi-
tional height ascended, but in a constantly decreasing ratio. An easy
calculation, however, founded on our experimental knowledge of the pro-
perties of air, and the mechanical laws which regulate its dilatation and
compression, is sufficient to show that, at an altitude above the surface of
the earth not exceeding the hundredth part of its diameter, the tenuity,
or rarefaction, of the air must be so excessive, that not only animal life
could not subsist, or combustion be maintained in it, but that the most
delicate means we possess of ascertaining the existence of any air at all
would fail to afford the slightest perceptible indications of its presence.
(34.) Laying out of consideration, therefore, at present, all nice ques-
tions as to the probable existence of a definite limit to the atmosphere,
beyond which there is, absolutely and rigorously speaking, no air, it is
clear, that, for all practical purposes, we may speak of those regions which
are more distant above the earth's surface than the hundredth part of its
diameter as void of air, and of course of clouds (which are nothing but
visible vapours, diffused and floating in the air, sustained by it, and ren-
dering it turbid as mud does water). It seems probable, from many indi-
*The height of Ætna above the Mediterranean (as it results from a barometrical
measurement of my own, mad - in July, 1824, under very favourable circumstances) is
10,872 English feet.—Author.
36 OTITLINES OF ASTRONOMY.
cations, that the greatest height at which visible clouds ever eaſist does not
exceed ten miles; at which height the density of the air is about an eighth
part of what it is at the level of the sea.
(35.) We are thus led to regard the atmosphere of air, with the clouds
it supports, as constituting a coating of equable or nearly equable thick-
ness, enveloping our globe on all sides; or rather as an aérial ocean, of
which the surface of the sea and land constitutes the bed, and whose infe-
rior portions or strata, within a few miles of the earth, contain by far the
greater part of the whole mass, the density diminishing with extreme
rapidity as we recede upwards, till, within a very moderate distance (such
as would be represented by the sixth of an inch on the model we have
before spoken of, and which is not more in proportion to the globe on
which it rests, than the downy skin of a peach in comparison with the
fruit within it), all sensible trace of the existence of air disappears.
(36.) Arguments, however, are not wanting to render it, if not abso-
lutely certain, at least in the highest degree probable, that the surface of .
the aërial, like that of the aqueous Ocean, has a real and definite limit, as
above hinted at ; beyond which there is positively no air, and above which
a fresh quantity of air, could it be added from without, or carried aloft
from below, instead of dilating itself indefinitely upwards, would, after a
certain very enormous but still finite enlargement of volume, sink and
merge, as water poured into the sea, and distribute itself among the mass
beneath. With the truth of this conclusion, however, astronomy has
little concern; all the effects of the atmosphere in modifying astronomical
phenomena being the same, whether it be supposed of definite extent or
not.
(37.) Moreover, whichever idea we adopt, within those limits in which
it possesses any appreciable density its constitution is the same over all
points of the earth's surface; that is to say, on the great scale, and leaving
out of consideration temporary and local causes of derangement, such as
winds, and great fluctuations, of the nature of waves, which prevail in it
to an immense extent. In other words, the law of diminution of the
air’s density as we recede upwards from the level of the sea is the same
in every column into which we may conceive it divided, or from whatever
point of the surface we may set out. It may therefore be considered as
consisting of successively superposed strata or layers, each of the form of
a spherical shell, concentric with the general surface of the sea and land,
and each of which is rarer, or specifically lighter, than that immediately
beneath it; and denser, or specifically heavier, than that immediately
above it. This, at least, is the kind of distribution which alone would be
consistent with the laws of the equilibrium of fluids. Inasmuch, however,
THE ATMOSPHERE. \ 37
as the atmosphere is not in perfect equilibrium, being always kept in a
state of circulation, owing to the excess of heat in its equatorial regions
over that at the poles, some slight deviation from the rigorous expression
of this law takes place, and in peculiar localities there is reason to believe
that even considerable permanent depressions of the contours of these
strata, below their general or spherical level, subsist. But these are points
of consideration rather for the meteorologist than the astronomer. It
must be observed, moreover, that with this distribution of its strata the
inequalities of mountains and valleys have little concern. These exercise
hardly more influence in modifying their general spherical figure than the
inequalities at the bottom of the sea interfere with the general sphericity
of its surface. They would exercise absolutely none were it not for their
effect in giving another than horizontal direction to the currents of air
constituting winds, as shoals in the ocean throw up the currents which
sweep over them towards the surface, and so in some small degree tend to
disturb the perfect level of that surface.
(38.) It is the power which air possesses, in common with all trans-
parent media, of refracting the rays of light, or bending them out of
their straight course, which renders a knowledge of the constitution of the
atmosphere important to the astronomer. Owing to this property, objects
seen obliquely through it appear otherwise situated than they would to the
same spectator, had the atmosphere no existence. It thus produces a
false impression respecting their places, which must be rectified by ascer-
taining the amount and direction of the displacement so apparently pro-
duced on each, before we can come at a knowledge of the true directions
in which they are situated from us at any assigned moment.
(39.) Suppose a spectator placed at A, any point of the earth's surface
K A k; and let L l, M m, N n, represent the successive strata or layers,
of decreasing density, into which we may conceive the atmosphere to be
divided, and which are spherical surfaces concentric with K k, the earth's
surface. Let S represent a star, or other heavenly body, beyond the ut-
most limit of the atmosphere. Then, if the air were away, the spectator
would see it in the direction of the straight line A S. But, in reality,
when the ray of light S A reaches the atmosphere, suppose at d, it will,
by the laws of optics, begin to bend downwards, and take a more inclined
direction, as d c. This bending will at first be imperceptible, owing to
the extreme tenuity of the uppermost strata; but as it advances down-
wards, the strata continually increasing in density, it will continually
undergo greater and greater refraction in the same direction; and thus,
instead of pursuing the straight line S d A, it will describe a curve S d c
b a, continually more and more concave downwards, and will reach the
38 OuTLINES OF ASTIRONOMY.
earth, not at A, but at a certain point a, nearer to S. This ray, conse.
quently, will not reach the spectator's eye. The ray by which he will see
the star is, therefore, not S d A, but another ray which, had there been
no atmosphere, would have struck the earth at K, a point behind the
spectator; but which, being bent by the air into the curve S D C B A,
actually strikes on A. Now, it is a law of optics, that an object is seen
in the direction which the visual ray has at the instant of arriving at the
eye, without regard to what may have been otherwise its course between
the object and the eye. Hence the star S will be seen, not in the direc-
tion A S, but in that of A s, a tangent to the curve S D C B A, at A.
Fig. 4.
y
C
But because the curve described by the refracted ray is concave down-
wards, the tangent. A s will lie above A S, the unrefracted ray: conse-
quently the object S will appear more elevated above the horizon A H,
when seen through the refracting atmosphere, than it would appear were
there no such atmosphere. Since, however, the disposition of the strata
is the same in all directions around A, the visual ray will not be made to
deviate laterally, but will remain constantly in the same vertical plane,
S A C", passing through the eye, the object, and the earth’s centre.
(40.) The effect of the air's refraction, then, is to raise all the heavenly
bodies higher above the horizon in appearance than they are in reality.
Any such body, situated actually in the true horizon, will appear above it,
or will have some certain apparent altitude (as it is called). Nay, even
some of those actually below the horizon, and which would therefore be
invisible but for the effect of refraction, are, by that effect, raised above it
and brought into sight. Thus, the sun, when situated at P below the true
*

REFRACTION. 39
horizon, A H, of the spectator, becomes visible to him, as if it stood at
p, by the refracted ray P q r t A, to which A p is a tangent.
(41.) The exact estimation of the amount of atmospheric refraction, or
the strict determination of the angle S A s, by which a celestial object at
any assigned altitude, H A S, is raised in appearance above its true place,
is, unfortunately, a very difficult subject of physical inquiry, and one on
which geometers (from whom alone we can look for any information on
the subject) are not yet entirely agreed. The difficulty arises from this,
that the density of any stratum of air (on which its refracting power
depends) is affected not merely by the superincumbent pressure, but also
by its temperature or degree of heat. Now, although we know that as
we recede from the earth's surface the temperature of the air is constantly
diminishing, yet the law, or amount of this diminution at different heights,
is not yet fully ascertained. Moreover, the refracting power of air is per-
ceptibly affected by its moisture; and this, too, is not the same in every
part of an aérial column; neither are we acquainted with the laws of its
distribution. The consequence of our ignorance on these points is to
introduce a correspons . g degree of uncertainty into the determination of
the amount of refraction, which affects, to a certain appreciable extent,
our knowledge of several of the most important data of astronomy.
The uncertainty thus induced, is, however, confined within such very
narrow limits as to be no cause of embarrassment, except in the most
delicate inquiries, and to call for no further allusion in a treatise like the
present.
(42.) A “Table of Refraction,” as it is called, or a statement of the
amount of apparent displacement arising from this cause, at all altitudes,
or in every situation of a heavenly body, from the horizon to the zenith,
or point of the sky vertically above the spectator, and, under all the
circumstances in which astronomical observations are usually performed
which may influence the result, is one of the most important and indis-
pensable of all astronomical tables, since it is only by the use of such a
table we are enabled to get rid of an illusion which must otherwise pervert
all our notions respecting the celestial motions. Such have been, accord.
ingly, constructed with great care, and are to be found in every collection
of astronomical tables. Our design, in the present treatise, will not
admit of the introduction of tables; and we must, therefore, content our-
selves here, and in similar cases, with referring the reader to works
especially destined to furnish these useful aids to calculation. It is, how-
* From an Arabic word of this signification. See this term technically defined in
Chap. II.
40 OUTLINES OF ASTRONOMY.
ever, desirable that he should bear in mind the following general notions
of its amount, and law of variations.
(43.) 1st. In the zenith there is no refraction. A celestial object,
situated vertically over-head, is seen in its true direction, as if there were
no atmosphere, at least if the air be tranquil.
2dly. In descending from the zenith to the horizon, the refraction con-
tinually increases. Objects near the horizon appear more elevated by it
above their true directions than those at a high altitude.
3dly. The rate of its increase is nearly in proportion to the tangent of
the apparent angular distance of the object from the zenith. But this
rule, which is not far from the truth, at moderate zenith distances, ceases
to give correct results in the vicinity of the horizon, where the law
becomes much more complicated in its expression. -
4thly. The average amount of refraction, for an object half-way between
the Zenith and the horizon, or at an apparent altitude of 45°, is about 1'
(more exactly 57"), a quantity hardly sensible to the naked eye; but at
the visible horizon it amounts to no less a quantity than 33", which is
rather more than the greatest apparent diameter of either the sun or the
moon. Hence it follows, that when we see the lower edge of the sun or
moon just apparently resting on the horizon, its whole disk is in reality
below it, and would be entirely out of sight and concealed by the con-
vexity of the earth, but for the bending round it, which the rays of light
have undergone in their passage through the air, as alluded to in art. 40.
5thly. That when the barometer is higher than its average or mean
state, the amount of refraction is greater than its mean amount; when
lower, less; and,
6thly. That in one and the same state of the barometer the refrac-
tion is greater, the colder the air. The variation, owing to these two
causes, from its mean amount (at temp. 55°, pressure 30 inches), are
about one 420th part of that amount for each degree of the thermometer
of Fahrenheit, and one 300th for each tenth of an inch in the height of
the barometer.
(44.) It follows from this, that one obvious effect of refraction must be
to shorten the duration of night and darkness, by actually prolonging the
stay of the sun and moon above the horizon. But even after they are set,
the influence of the atmosphere still continues to send us a portion of their
light; not, indeed, by direct transmission, but by reflection upon the wa-
pours, and minute solid particles which float in it, and, perhaps, also on
the actual material atoms of the air itself. To understand how this takes
place, we must recollect, that it is not only by the direct light of a lumi-
nous object that we see, but that whatever portion of its light which
TWILIGHT. 41
would not otherwise reach our eyes is intercepted in its course, and thrown
back, or laterally, upon us, becomes to us a means of illumination. Such
reflective obstacles always exist floating in the air. The whole course of a
sun-beam penetrating through the chink of a window-shutter into a dark
room is visible as a bright line in the air; and even if it be stifled, or let
out through an opposite crevice, the light scattered through the apartment
from this source is sufficient to prevent entire darkness in the room. The
luminous lines occasionally seen in the air, in a sky full of partially bro-
ken clouds, which the vulgar term “the sun drawing water,” are simi-
larly caused. They are sunbeams, through apertures in the clouds, par-
tially intercepted and reflected on the dust and vapours of the air below.
Thus it is with those solar rays which, after the sun is itself concealed by
the convexity of the earth, continue to traverse the higher regions of the
atmosphere above our heads, and pass through and out of it, without di-
rectly striking on the earth at all. Some portion of them is intercepted
and reflected by the floating particles above mentioned, and thrown back,
or laterally, so as to reach us, and afford us that secondary illumination,
Fig. 5.
which is twilight. The course of such rays will be immediately understood
from the annexed figure, in which A B C D is the earth; A a point on its
surface, where the sun S is in the act of setting; its last lower ray S A M
just grazing the surface at A, while its superior rays SN, SO, traverse
the atmosphere above A without striking the earth, leaving it finally at
the points PQR, after being more or less bent in passing through it, the
lower most, the higher less, and that which, like SR 0, merely grazes
the exterior limit of the atmosphere, not at all. Let us consider several
points, A, B, C, D, each more remote than the last from A, and each more

42 OUTLINES OF ASTRONOMY.
deeply involved in the earth's shadow, which ºccupies the whole space
from A beneath the line A. M. Now, A just receives the sun's last direct
ray, and, besides, is illuminated by the whole reflective atmosphere PQ
R.T. It therefore receives twilight from the whole sky. The point B,
to which the sun has set, receives no direct solar light, nor any, direct or
reflected, from all that part of its visible atmosphere which is below APM;
but from the lenticular portion P R x, which is traversed by the sun's rays,
and which lies above the visible horizon B R of B, it receives a twilight,
which is strongest at It, the point immediately below which the sun is,
and fades away gradually towards P, as the luminous part of the atmo-
sphere thins off. At C, only the last or thinnest portion, PQ z of the
lenticular segment, thus illuminated, lies above the horizon, C Q, of that
place; here, then, the twilight is feeble, and confined to a small space in
and near the horizon, which the sun has quitted, while at D the twilight
has ceased altogether.
(45.) When the sun is above the horizon, it illuminates the atmosphere
and clouds, and these again disperse and scatter a portion of its light in
all directions, so as to send some of its rays to every exposed point, from
every point of the sky. The generally diffused light, therefore, which we
enjoy in the daytime, is a phenomenon originating in the very same causes
as the twilight. Were it not for the reflective and scattering power of the
atmosphere, no objects would be visible to us out of direct sunshine; every
..shadow of a passing cloud would be pitchy darkness; the stars would be
visible all day, and every apartment, into which the sun had not direct
admission, would be involved in nocturnal obscurity. This scattering action
of the atmosphere on the solar light, it should be observed, is increased
by the irregularity of temperature caused by the same luminary in its
different parts, which, during the daytime, throws it into a constant state
of undulation, and, by thus bringing together masses of air of very un-
equal temperatures, produces partial reflections and refractions at their
common boundaries, by which some portion of the light is turned aside
from the direct course, and diverted to the purposes of general illumina-
tion.
(46.) From the explanation we have given, in arts. 39 and 40, of the
nature of atmospheric refraction, and the mode in which it is produced in
the progress of a ray of light through successive strata, or layers of the
atmosphere, it will be evident, that whenever a ray passes obliquely from
a higher level to a lower one, or vice versá, its course is not rectilinear,
but concave downwards; and of course any object seen by means of such
a ray, must appear deviated from its true place, whether that object he,
like the celestial bodies, entirely beyond the atmosphere, or, like the sum-
TWILIGHT. 43
mits of mountains seen from the plains, or other terrestrial stations at
different levels seen from each other, immersed in it. Every difference
of level, accompanied, as it must be, with a difference of density in the
aërial strata, must also have, corresponding to it, a certain amount of re-
fraction; less, indeed, than what would be produced by the whole atmo-
sphere, but still often of very appreciable, and even considerable, amount.
This refraction between terrestrial stations is termed terrestrial refraction,
to distinguish it from that total effect which is only produced on celestial
objects, or such as are beyond the atmosphere, and which is called celes-
tial or astronomical refraction.
(47.) Another effect of refraction is to distort the visible forms and
proportions of objects seen near the horizon. The sun, for instance, which
at a considerable altitude always appears round, assumes, as it approaches
the horizon, a flattened or oval outline; its horizontal diameter being vis-
ibly greater than that in a vertical direction. When very near the horizon,
this flattening is evidently more considerable on the lower side than on the
upper; so that the apparent form is neither circular nor elliptic, but a spe-
cies of oval, which deviates more from a circle below than above. This
singular effect, which any one may notice in a fine sunset, arises from the
rapid rate at which the refraction increases in approaching the horizon.
Were every visible point in the sun's circumference equally raised by re-
fraction, it would still appear circular, though displaced; but the lower
portions being more raised than the upper, the vertical diameter is thereby
shortened, while the two extremities of its horizontal diameter are equally
raised, and in parallel directions, so that its apparent length remains the
same. The dilated size (generally) of the sun or moon, when seen near
the horizon, beyond what they appear to have when high up in the sky,
has nothing to do with refraction. It is an illusion of the judgment,
arising from the terrestrial objects interposed, or placed in close compar-
ison with them. In that situation we view and judge of them as we do
of terrestrial objects—in detail, and with an acquired habit of attention
to parts. Aloft we have no associations to guide us, and their insulation
in the expanse of sky leads us rather to undervalue than to over-rate their
apparent magnifudes. Actual measurement with a proper instrument cor-
rects our error, without, however, dispelling our illusion. By this we
learn, that the sun, when just on the horizon, subtends at our eyes almost
exactly the same, and the moon a materially less angle, than when seen
at a great altitude in the sky, owing to its greater distance from us in the
former situation as compared with the latter, as will be explained farther
Ołł. -
(48.) After what has been said of the small extent ºf the atmosphere
44 OUTLINES OF ASTRONOMY.
in comparison with the mass of the earth, we shall have little hesitation
in admitting those luminaries which people and adorn the sky, and which,
while they obviously form no part of the earth, and receive no support
from it, are yet not borne along at random like clouds upon the air, nor
drifted by the winds, to be external to our atmosphere. As such we have
considered them while speaking of their refractions—as existing in the
immensity of space beyond, and situated, perhaps, for any thing we can
perceive to the contrary, at enormous distances from us and from each
other.
(49.) Could a spectator exist unsustained by the earth, or any solid
support, he would see around him at one view the whole contents of space
— the visible constituents of the universe: and, in the absence of any
means of judging of their distances from him, would refer them, in the
directions in which they were seen from his station, to the concave sur-
face of an imaginary sphere, having his eye for a centre, and its surface
at some vast indeterminate distance. Perhaps he might judge those
which appear to him large and bright, to be nearer to him than the
smaller and less brilliant; but, independent of other means of judging,
he would have no warrant for this opinion, any more than for the idea
that all were equidistant from him, and really arranged on such a
spherical surface. Nevertheless, there would be no impropriety in his
referring their places, geometrically speaking, to those points of such a
purely imaginary sphere, which their respective visual rays intersect; and
there would be much advantage in so doing, as by that means their ap-
pearance and relative situation could be accurately measured, recorded,
and mapped down. The objects in a landscape are at every variety of
distance from the eye, yet we lay them all down in a picture on one
plane, and at one distance, in their actual apparent proportions, and the
likeness is not taxed with incorrectness, though a man in the foreground
should be represented larger than a mountain in the distance. So it is
to a spectator of the heavenly bodies pictured, projected, or mapped down
on that imaginary sphere we call the sky or heaven. Thus, we may
easily conceive that the moon, which appears to us as large as the sun,
though less bright, may owe that apparent equality to its greater prox-
imity, and may be really much less; while both the moon and sun may
only appear larger and brighter than the stars, on account of the remote-
ness of the latter.
(50.) A spectator on the earth's surface is prevented, by the great
mass on which he stands, from seeing into all that portion of space which
is below him, or to see which he must look in any degree downwards
It is true that, if his place of observation be at a great elevation, the dip
CHANGE OF HORIZON IN TRAVELLING. 45
of the horizon will bring within the scope of vision a little more than a
hemisphere, and refraction, wherever he may be situated, will enable him
to look, as it were, a little round the corner; but the Zone thus added to
his visual range can hardly ever, unless in very extraordinary circum-
stances, exceed a couple of degrees in breadth, and is always ill seen on
account of the vapours near the horizon. Unless, then, by a change of
his geographical situation, he should shift his horizon (which is always a
plane passing through his eye, and touching the spherical convexity of
the earth); or unless, by some movements proper to the heavenly bodies,
they should of themselves come above his horizon; or, lastly, unless, by
some rotation of the earth itself on its centre, the point of its surface
which he occupies should be carried round, and presented towards a dif-
ferent region of space; he would never obtain a sight of almost one half
the objects external to our atmosphere. But if any of these cases be
supposed, more, or all, may come into view according to the circum-
stances. -
(51.) A traveller, for example, shifting his locality on our globe, will
obtain a view of celestial objects invisible from his original station, in a
way which may be not inaptly illustrated by comparing him to a person
standing in a park close to a large tree. The massive obstacle presented
by its trunk cuts off his view of all those parts of the landscape which it
Fig. 6.
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occupies as an object; but by walking round it a complete successive view
of the whole panorama may be obtained. Just in the same way, if we
set off from any station, as London, and travel southwards, we shall not
fail to notice that many celestial objects which are never seen from
London come successively into view, as if rising up above the horizon,
night after night, from the south, although it is in reality our horizon,

46 OTTLINES OF ASTRONOMY.
which, travelling with us southwards round the sphere, sinks in succes-
sion beneath them. The novelty and splendour of fresh constellations
thus gradually brought into view in the clear calm nights of tropical
climates, in long voyages to the south, is dwelt upon by all who have
enjoyed this spectacle, and never fails to impress itself on the recollection
among the most delightful and interesting of the associations connected
with extensive travel. A glance at the accompanying figure, exhibiting
three suggessive stations of a traveller, A, B, C, with the horizon corre-
sponding to each, will place this process in clearer evidence than any
description. .
(52.) Again: suppose the earth itself to have a motion of rotation on
its centre. It is evident that a spectator at rest (as it appears to him)
on any part of it will, unperceived by himself, be carried round with it:
unperceived, we say, because his horizon will constantly contain, and be
limited by, the same terrestrial objects. He will have the same land-
scape constantly before his eyes, in which all the familiar objects in it,
which serve him for landmarks and directions, retain, with respect to
himself or to each other, the same invariable situations. The perfect
smoothness and equality of the motion of so vast a mass, in which every
object he sees around him participates alike, will (art. 15) prevent his
entertaining any suspicion of his actual change of place. Yet, with
respect to external objects, – that is to say, all celestial ones which do
Inot participate in the supposed rotation of the earth, – his horizon will
have been all the while shifting in its relation to them, precisely as in
the case of our traveller in the foregoing article. Recurring to the figure
of that article, it is evidently the same thing, so far as their visibility is
concerned, whether he has been carried by the earth's rotation succes-
sively into the situations A, B, C; or whether, the earth remaining at
rest, he has transferred himself personally along its surface to those
stations. Our spectator in the park will obtain precisely the same view
of the landscape, whether he walk round the tree, or whether we suppose
it sawed off, and made to turn on an upright pivot, while he stands on a
projecting step attached to it, and allows himself to be carried round by
its motion. The only difference will be in his view of the tree itself, of
which, in the former case, he will see every part, but, in the latter, only
that portion of it which remains constantly opposite to him, and imme-
diately under his eye.
(53.) By such a rotation of the earth, then, as we have supposed, the
horizon of a stationary spectator will be constantly depressing itself below
those objects which lie in that region of space towards which the rotation
is carrying him, and elevating itself above those in the opposite quarter,
DIURNAL ROTATION OF THE EARTH. 47
admitting into view the former, and successively hiding the latter. As
the horizon of every such spectator, however, appears to him motionless,
all such changes will be referred by him to a motion in the objects them-
selves so successively disclosed and concealed. In place of his horizon
approaching the stars, therefore, he will judge the stars to approach his
horizon; and when it passes over and hides any of them, he will consider
them as having sunk below it, or set; while those it has just disclosed,
and from which it is receding, will seem to be rising above it.
(54.) If we suppose this rotation of the earth to continue in one and
the same direction, — that is to say, to be performed round one and the
same aris, till it has completed an entire revolution, and come back to the
position from which it set out when the spectator began his observations,
—it is manifest that every thing will then be in precisely the same rela-
tive position as at the outset: all the heavenly bodies will appear to occupy
the same places in the concave of the sky which they did at that instant,
except such as may have actually moved in the interim; and if the rota.
tion still continue, the same phenomena of their successive rising and
setting, and return to the same places, will continue to be repeated in the
same order, and (if the velocity of rotation be uniform) in equal intervals
of time, ad infinitum.
(55.) Now, in this we have a lively picture of that grand phenomenon,
the most important beyond all comparison which nature presents, the daily
rising and setting of the sun and stars, their progress through the vault
of the heavens, and their return to the same apparent places at the same
hours of the day and night. The accomplishment of this restoration in
the regular interval of twenty-four hours is the first instance we encounter
of that great law of periodicity,' which, as we shall see, pervades all
astronomy; by which expression we understand the continual reproduction
of the same phenomena, in the same order, at equal intervals of time.
(56.) A free rotation of the earth round its centre, if it exist and be
performed in consonance with the same mechanical laws which obtain in
the motions of masses of matter under our immediate control, and within
our ordinary experience, must be such as to satisfy two essential conditions.
It must be invariable in its direction with respect to the sphere itself, and
uniform in its velocity. The rotation must be performed round an axis
or diameter of the sphere, whose poles or extremities, where it meets the
surface, correspond always to the same points on the sphere. Modes of
rotation of a solid body under the influence of external agency are con-
ceivable, in which the poles of the imaginary line or axis about which it
* IIepiodos, a going round, a circulation or revolution.
48 OUTLINES OF ASTRONOMY.
is at any moment revolving shall hold no fixed places on the surface, but
shift upon it every moment. Such changes, however, are inconsistent
with the idea of a rotation of a body of regular figure about its axis of
symmetry, performed in free space, and without resistance or obstruction
from any surrounding medium, or disturbing influences. The complete
absence of such obstructions draws with it, of necessity, the strict fulfil-
ment of the two conditions above mentioned.
(57.) Now, these conditions are in perfect accordance with what we
observe, and what recorded observation teaches us, in respect of the diur-
nal motions of the heavenly bodies. We have no reason to believe, from
history, that any sensible change has taken place since the earliest ages in
the interval of time elapsing between two successive returns of the same
star to the same point of the sky; or, rather, it is demonstrable from
astronomical records that no such change has taken place. And with
respect to the other condition, — the permanence of the aris of rotation,
—the appearances which any alteration in that respect must produce,
would be marked, as we shall presently show, by a corresponding change
of a very obvious kind in the apparent motions of the stars; which, again,
history decidedly declares them not to have undergºne.
(58.) But, before we proceed to examine more in detail how the hypo-
thesis of the rotation of the earth about an axis accords with the phe-
nomena which the diurnal motion of the heavenly bodies offers to our
notice, it will be proper to describe, with precision, in what that diurnal
motion consists, and how far it is participated in by them all; or whether
any of them form exceptions, wholly or partially, to the common analogy
of the rest. We will, therefore, suppose the reader to station himself, on
a clear evening, just after sunset, when the first stars begin to appear, in
some open situation whence a good general view of the heavens can be
obtained. He will then perceive, above and around him, as it were, a
vast concave hemispherical vault, beset with stars of various magnitudes,
of which the brightest only will first catch his attention in the twilight;
and more and more will appear as the darkness increases, till the whole
sky is over-spangled with them. When he has awhile admired the calm
magnificence of this glorious spectacle, the theme of so much song, and
of so much thought, a spectacle which no one can view without emotion,
and without a longing desire to know something of its nature and purport,
— let him fix his attention more particularly on a few of the most bril-
iiant stars, such as he cannot fail to recognize again without mistake after
looking away from them for some time, and let him refer their apparent
situations to some surrounding objects, as buildings, trees, &c., selecting
purposely such as are in different quarters of his horizon. On comparing
APPARENT DIURNAU, MOTION. 49
them again with their respective points of reference, after a moderate
interval, as the night advances, he will not fail to perceive that they have
changed their places, and advanced, as by a general movement, in a west-
ward direction; those towards the eastern quarter appearing to rise or
recede from the horizon, while those which lie towards the west will be
seen to approach it; and, if watched long enough, will, for the most part,
finally sink beneath it, and disappear; while others, in the eastern quarter,
will be seen to rise as if out of the earth, and, joining in the general
procession, will take their course with the rest towards the opposite
quarter. •
(59.) If he persist for a considerable time in watching their motions,
on the same or on several successive nights, he will perceive that each star
appears to describe, as far as its course lies above the horizon, a circle in
the sky; that the circles so described are not of the same magnitude for
all the stars; and that those described by different stars differ greatly in
respect of the parts of them which lie above the horizon. Some, which
lie towards the quarter of the horizon which is denominated the SouTH,"
only remain for a short time above it, and disappear, after describing in
sight only the small upper segment of their diurnal circle; others, which
rise between the south and east, describe larger segments of their circles
above the horizon, remain proportionally longer in sight, and set precisely
as far to the westward of south as they rose to the eastward; while such
as rise exactly in the east remain just twelve hours visible, describe a
semicircle, and set exactly in the west. With those, again, which rise
between the east and north, the same law obtains; at least, as far as
regards the time of their remaining above the horizon, and the proportion
of the visible segment of their diurnal circles to their whole circum-
ferences. Both go on increasing; they remain in view more than twelve
hours, and their visible diurnal arcs are more than semicircles. But the
magnitudes of the circles themselves diminish, as we go from the east,
northward; the greatest of all the circles being described by those which
rise exactly in the east point. Carrying his eye farther northwards, he
will notice, at length, stars which, in their diurnal motion, just graze the
horizon at its north point, or only dip below it for a moment; while others
never reach it at all, but continue always above it, revolving in entire
circles round on E POINT called the POLE, which appears to be the common
centre of all their motions, and which alone, in the whole heavens, may
be considered immoveable. Not that this point is marked by any star
It is a purely imaginary centre; but there is near it one considerably
* We suppose our observer to be stationed in some northern latitude; some where
in Europe, for example.
50 ouTLINES OF ASTRONOMY.
bright star, called the Pole Star, which is easily recognized by the very
small circle it describes; so small, indeed, that, without paying particular
attention, and referring its position very nicely to some fixed mark, it may
easily be supposed at rest, and be, itself, mistaken for the common centre
about which all the others in that region describe their circles; or it may
be known by its configuration with a very splendid and remarkable con-
stellation or group of stars, called by astronomers the GREAT BEAR.
(60.) He will further ºbserve, that the apparent relative situations of
all the stars among one another, is not changed by their diurnal motion.
In whatever parts of their circles they are observed, or at whatever hour
of the night, they form with each other the same identical groups or con-
figurations, to which the name of CONSTELLATIONS has been given. It
is true, that, in different parts of their course, these groups stand dif-
ferently with respect to the horizon; and those towards the north, when
in the course of their diurnal movement they pass alternately above and
below that common centre of motion described in the last article, become
actually inverted with respect to the horizon, while, on the other hand,
they always turn the same points towards the pole. In short, he will
perceive that the whole assemblage of stars visible at once, or in succes-
sion, in the heavens, may be regarded as one great constellation, which
seems to revolve with a uniform motion, as if it formed one coherent
mass; or as if it were attached to the internal surface of a vast hollow
sphere, having the earth, or rather the spectator, in its centre, and turning
round an axis inclined to his horizon, so as to pass through that fixed
point or pole already mentioned.
(61.) Lastly, he will notice, if he have patience to outwatch a long
winter's night, commencing at the earliest moment when the stars appear,
and continuing till morning twilight, that those stars which he observed
setting in the west have again risen in the east, while those which were
rising when he first began to notice them have completed their course, and
are now set; and that thus the hemisphere, or a great part of it, which
was then above, is now beneath him, and its place supplied by that which
was at first under his feet, which he will thus discover to be no less
copiously furnished with stars than the other, and bespangled with groups
no less permanent and distinctly recognizable. Thus he will learn that
the great constellation that we have above spoken of as revolving round
the pole is co-extensive with the whole surface of the sphere, being in
reality nothing less than a universe of luminaries surrounding the earth
on all sides, and brought in succession before his view, and referred (each
luminary according to its own visual ray or direction from his eye) to the
imaginary spherical surface, of which he himself occº” -- *** **** -
APPARENT DIURNAL MOTION. * 51
(See art. 49.) There is always, therefore (he would justly argue), a star-
bespangled canopy over his head, by day as well as by night, only that
the glare of daylight (which he perceives gradually to efface the stars as
the morning twilight comes on) prevents them from being seen. And
such is really the case. The stars actually continue visible through teles-
copes in the day-time; and, in proportion to the power of the instrument,
not only the largest and brightest of them, but even those of inferior
lustre, such as scarcely strike the eye at night as at all conspicuous, are
readily found and followed even at noonday,+ unless in that part of the
sky which is very near the sun, - by those who possess the means of
pointing a telescope accurately to the proper places. Indeed, from the
bottoms of deep narrow pits, such as a well, or the shaft of a mine, such
bright stars as pass the zenith may even be discerned by the naked eye;
and we have ourselves heard it stated by a celebrated optician, that the
earliest circumstance which drew his attention to astronomy was the
regular appearance, at a certain hour, for several successive days, of a
considerable star, through the shaft of a chimney. Venus in our climate,
and even Jupiter in the clearer skies of tropical countries, are often
visible, without any artificial aid, to the naked eye of one who knows
nearly where to look for them. During total eclipses of the sun, the
larger stars also appear in their proper situations.
(62.) But to return to our incipient astronomer, whom we left contem-
plating the sphere of the heavens, as completed in imagination beneath
his feet, and as rising up from thence in its diurnal course. There is one
portion or segment of this sphere of which he will not thus obtain a view.
As there is a segment towards the north, adjacent to the pole above his
horizon, in which the stars never set, so there is a corresponding segment,
about which the smaller circles of the more southern stars are described,
in which they never rise. The stars which border upon the extreme
circumference of this segment just graze the southern point of his hori-
zon, and show themselves for a few moments above it, precisely as those
near the circumference of the northern segment graze his northern hori-
zon, and dip for a moment below it, to re-appear immediately. Every
point in a spherical surface has, of course, another diametrically opposite
to it; and as the spectator's horizon divides his sphere into two hemi-
spheres—a superior and inferior—there must of necessity exist a de-
pressed pole to the south, corresponding to the elevated one to the north,
and a portion surrounding it, perpetually beneath, as there is another
surrounding the north pole, perpetually above it.
“Hic vertex nobis semper sublimis; at illum *
Sub edibus nox atra videt, manesque profundi.”—VIRGIL.
52 * OUTLINES OF ASTRONOMY.
One pole rides high, one, plunged beneath the main,
Seeks the deep night, and Pluto's dusky reign.
(63.) To get sight of this segment, he must travel southwards. In so
doing, a new set of phenomena come forward. In proportion as he
advances to the south, some of those constellations which, at his original
station, barely grazed the northern horizon, will be observed to sink below
it and set; at first remaining hid only for a very short time, but gradually
for a longer part of the twenty-four hours. They will continue, however,
to circulate about the same point — that is, holding the same invariable
position with respect to them in the concave of the heavens among the
stars; but this point itself will become gradually depressed with respect
to the spectator's horizon. The axis, in short, about which the diurnal
motion is performed, will appear to have become continually less and less
inclined to the horizon; and by the same degrees as the northern pole is
depressed the southern will rise, and constellations surrounding it will
come into view; at first momentarily, but by degrees for longer and longer
times in each diurnal revolution — realizing, in short, what we have
already stated in art. 51.
(64.) If he travel continually southwards, he will at length reach a
line on the earth's surface, called the equator, at any point of which,
indifferently, if he take up his station and recommence his observations,
he will find that he has both the centres of diurnal motion in his horizon,
occupying opposite points, the northern pole having been depressed, and
the southern raised; so that, in this geographical position, the diurnal
rotation of the heavens will appear to him to be performed about a hori-
zontal axis, every star describing half its diurnal circle above and half
beneath his horizon, remaining alternately visible for twelve hours, and
concealed during the same interval. In this situation, no part of the
heavens is concealed from his successive view. In a night cf twelve hours
(supposing such a continuance of darkness possible at the equator) the
whole sphere will have passed in review over him—the whole hemisphere
with which he began his night's observation will have been carried down
beneath him, and the entire opposite one brought up from below.
(65.) If he pass the equator, and travel still farther southwards, the
southern poles of the heavens will become elevated above his horizon, and
the northern will sink below it; and the more, the farther he advances
southwards; and when arrived at a station as far to the south of the
equator as that from which he started was to the north, he will find the
whole phenomena of the heavens reversed. The stars which at his origi-
nal station described their whole diurnal circles above his horizon, and
never set, now describe them entirely below it, and never rise, but remain
PARALLACTIC MOTION. 53
constantly invisibli to him; and vice versá, those stars which at his former
station he never Saw, he will now never cease to see.
(66.) Finally, if, instead of advancing southwards from his first station,
he travel northwards, he will observe the northern pole of the heavens to
become more elevated above'his horizon, and the southern more depressed
below it. In consequence, his hemisphere will present a less variety of
stars, because a greater proportion of the whole surface of the heavens
remains constantly visible or constantly invisible: the circle described by
each star, too, becomes more nearly parallel to the horizon; and, in short,
every appearance leads to suppose that could he travel far enough to the
north, he would at length attain a point vertically under the northern pole
of the heavens, at which none of the stars would either rise or set, but
each would circulate round the horizon in circles parallel to it. Many
endeavours have been made to reach this point, which is called the north
pole of the earth, but hitherto without success; a barrier of almost insur-
mountable difficulty being presented by the increasing rigour of the
climate : but a very near approach to it has been made; and the pheno-
mena of those regions, though not precisely such as we have described as
what must subsist at the pole itself, have proved to be in exact correspon-
dence with its near proximity. A similar remark applies to the south
pole of the earth, which, however, is more unapproachable, or, at least,
has been less nearly approached, than the north. -
(67.) The above is an account of the phenomena of the diurnal motion
of the stars, as modified by different geographical situations, not grounded
on any speculation, but actually observed and recorded by travellers and
voyagers. It is, however, in complete accordance with the hypothesis of
a rotation of the earth round a fixed axis. In order to show this, how-
ever, it will be necessary to premise a few observations on parallactic
motion in general, and on the appearances presented by an assemblage of
remote objects, when viewed from different parts of a small and circum-
scribed station.
(68.) It has been shown (art. 16) that a spectator in smooth motion,
and surrounded by, and forming part of, a great system partaking of
the same motion, is unconscious of his own movement, and transfers it in
idea to objects external and unconnected, in a contrary direction; those
which he leaves behind appearing to recede from, and those which he
advances towards to approach, him Not only, however, do external
objects at rest appear in motion generally, with respect to ourselves when
we are in motion among them, but they appear to move one among the
other—they shift their relative apparent places. Let any one tra 7elling
rapidly along a high road fix his eye steadily on any object, but at the
54 OUTLINES OF ASTRONOMY.
same time not entirely withdraw his attention from the general landscape,
—he will see, or think he sees, the whole landscape thrown into rotation,
and moving round that object as a centre; all objects between it and
himself appearing to move backwards, or the contrary way to his own
motion; and all beyond it, forwards, or in the direction in which he
moves: but let him withdraw his eye from that object, and fix it on
another, — a nearer one, for instance, — immediately the appearance of
rotation shifts also, and the apparent centre about which this illusive
circulation is performed is transferred to the new object, which, for the
moment, appears to rest. This apparent change of situation of objects
with respect to one another, arising from a motion of the spectator, is
called a parallactic motion. To see the reason of it we must consider
that the position of every object is referred by us to the surface of an
imaginary sphere of an indefinite radius, having 'our eye for its centre :
Fig. 7.

and, as we advance in any direction, A B, carrying this imaginary sphere
along with us, the visual rays A P, A Q, by which objects are referred to
its surface (at C, for instance), shift their positions with respect to the
line in which we move, A B, which serves as an axis or line of reference,
and assume new positions, B P p, B Q q, revolving round their respective
objects as centres. Their intersections, therefore, p, q, with our visuai
sphere, will appear to recede on its surface, but with different degrees of
angular velocity in proportion to their proximity; the same distance of
advance A B subtending a greater angle, A P B = c P p, at the near
object P than at the remote one Q.
(69.) A consequence of the familiar appearance we have adduced in
illustration of these principles is worth noticing, as we shall have occa-
sion to refer to it hereafter. We observe that every object nearer to us
than that on which our eye is fixed appears to recede, and those farther
from us to advance in relation to one another. If then we did not know,
or could not judge by any other appearances, which of two objects were y_
nearer to us, this apparent advance or recess of one of them, when the
eye is kept steadily fixed on the other, would furnish a criterion. In a
PARALLACTIC MOTION. 55
dark night, for instance, when all intermediate objects are unseen, the
apparent relative movement of two lights which we are assured are them-
selves fixed, will decide as to their relative proximities. That which seems
to advance with us and gain upon the other, or leave it behind it, is the
farthest from us.
(70.) The apparent angular motion of an object, arising from a change
of our point of view, is called in general parallaz, and it is always ex-
pressed by the angle APB subtended at the object P (see fig, of art. 68)
by a line joining the two points of view A B under consideration. For
it is evident that the difference of angular position of P, with respect to
the invariable direction ABD, when viewed from A and from B, is the
difference of the two angles DBP and DAP; now, DBP being the exte-
rior angle of the triangle ABP, is equal to the sum of the interior and
opposite, DBP = DAP + APB, whence DBP—DAP = APB.
(71.) It follows from what has been said that the amount of parallactic
motion arising from any given change of our point of view is, caeteris
paribus, less, as the distance of an object viewed is greater; and when
that distance is extremely great in comparison with the change in our
point of view, the parallax becomes insensible; or, in other words, objects
do not appear to vary in situation at all. It is on this principle, that in
alpine regions visited for the first time we are surprised and confounded
at the little progress we appear to make by a considerable change of
place. An hour's walk, for instance, produces but a small parallactic
change in the relative situations of the vast and distant masses which
surround us. Whether we walk round a circle of a hundred yards in
diameter, or merely turn ourselves round in its centre, the distant pano-
rama presents almost exactly the same aspect, —we hardly seem to have
changed our point of view.
(72.) Whatever notion, in other respects, we may form of the stars, it
is quite clear they must be immensely distant. Were it not so, the appa-
rent angular interval between any two of them seen over-head would be
much greater than when seen near the horizon, and the constellations,
instead of preserving the same appearances and dimensions during their
whole diurnal course, would appear to enlarge as they rise higher in the
sky, as we see a small cloud in the horizon swell into a great over-
shadowing canopy when drifted by the wind across our zenith, or as may
be seen in the annexed figure, where ab, A B, a b, are three different
positions of the same stars, as they would, if near the earth, be seen
from a spectator S, under the visual angles a Sb, ASB. No such change
of apparent dimension, however, is observed. The nicest measurements
of the apparent angular distance of any two stars inter se, taken in any
56 . OljTLINES OF ASTRONOMIY.
parts of their diurnal course, (after allowing for the unequal effects of
refraction, or when taken at such times that this cause of distortion shall
act equally on both,) manifest not the slightest perceptible variation.
Not only this, but at whatever point of the earth's surface the measure-
ment is performed, the results are absolutely identical. No instruments
ever yet invented by man are delicate enough to indicate, by an increase
or diminution of the angle subtended, that one point of the earth is nearer
to or further from the stars than another. *
(73.) The necessary conclusion from this is, that the dimensions of
the earth, large as it is, are comparatively nothing, absolutely impercep-
tible, when compared with the interval which separates the stars from the
earth. If an observer walk round a circle not more than a few yards in
diameter, and from different points in its circumference measure with a
sextant or other more exact instrument adapted for the purpose, the
angles PAQ, PBQ, PCQ, subtended at those stations by two well-defined
points in his visible horizon, PQ, he will at once be advertised, by the
difference of the results, of his change of distance from them arising from
his change of place, although that difference may be so small as to pro-
duce no change in their general aspect to his unassisted sight. This is
one of the innumerable instances where accurate measurement obtained
by instrumental means places us in a totally different situation in respect
to matters of fact, and conclusions thence deducible, from what we should
hold, were we to rely in all cases on the mere judgment of the eye. To
so great a nicety have such observations been carried by the aid of an
instrument called a theodolite, that a circle of the diameter above men-
tioned may thus be rendered sensible, may thus be detected to have a
size, and an ascertainable place, by reference to objects distant by fully
100,000 times its own dimensions. Observations, differing, it is true,
somewhat in method, but identical in principle, and executed with quite
as much exactness, have been applied to the stars, and with a result such
as has been already stated. Hence it follows, incontrovertibly, that the

-* THE DISTANCE OF THE STARS IMMENSE. , 57
Fig. 9.
distance of the stars from the earth cannot be so small as 100,000 of the
earth’s diameters. It is, indeed, incomparably greater; for we shall here-
after find it fully demonstrated that the distance just named, immense as
it may appear, is yet much underrated.
(74.) From such a distance, to a spectator with our faculties, and
furnished with our instruments, the earth would be imperceptible; and,
reciprocally, an object of the earth's size, placed at the distance of the
stars, would be equally undiscernible. If, therefore, at the point on which
a spectator stands, we draw a plane touching the globe, and prolong it in
imagination till it attain the region of the stars, and through the centre
of the earth conceive another plane parallel to the former, and co-extensive
with it, to pass; these, although separated throughout their whole extent
by the same interval, viz., a semi-diameter of the earth, will yet, on ac-
count of the vast distance at which that interval is seen, be confounded
together, and undistinguishable from each other in the region of the stars,
when viewed by a spectator on the earth. The zone they there include
will be of evanescent breadth to his eye, and will only mark out a great
circle in the heavens, one and the same for both the stations. This great
circle, when spoken of as a circle of the sphere, is called the celestial hori-
zon or simply the horizon, and the two planes just described are also spoken
of as the sensible and the rational horizon of the observer's station.
(75.) From what has been said (art. 73) of the distance of the stars,
it follows, that if we suppose a spectator at the centre of the earth to have
his view bounded by the rational horizon, in exactly the same manner as
that of a corresponding spectator on the surface is by his sensible horizon,
the two observers will see the same stars in the same relative situations,
each beholding that entire hemisphere of the heavens which is above the
celestial horizon, corresponding to their common Zenith. Now, so far ae

58 OUTLINES OF ASTRONOMY.
appearances go, it is clearly the same thing whether the heavens, that is,
all space, with its contents, revolve round a spectator at rest in the earth's
centre, or whether that spectator simply turn round in the opposite direc-
tion in his place, and view them in succession. The aspect of the heavens,
at every instant, as referred to his horizon (which must be supposed to
turn with him), will be the same in both suppositions. And since, as has
been shown, appearances are also, so far as the stars are concerned, the
same to a spectator on the surface as to one at the centre, it follows that,
whether we suppose the heavens to revolve without the earth, or the earth
within the heavens, in the opposite direction, the diurnal phenomenon, to
all its inhabitants, will be no way different.
(76.) The Copernican astronomy adopts the latter as the true explana-
tion of these phenomena, avoiding thereby the necessity of otherwise re-
sorting to the cumbrous mechanism of a solid but invisible sphere, to
which the stars must be supposed attached, in order that they may be
carried round the earth without derangement of their relative situations
inter se. Such a contrivance would, indeed, suffice to explain the diurnal
revolution of the stars, so as to “save appearances;” but the movements
of the sum and moon, as well as those of the planets, are incompatible with
such a supposition, as will appear when we come to treat of these bodies.
On the other hand, that a spherical mass of moderate dimensions (or,
rather, when compared with the surrounding and visible universe, of eva-
nescent magnitude), held by no tie, and free to move and to revolve, should
do so, in conformity with those general laws which, so far as we know,
regulate the motions of all material bodies, is so far from being a postulate
difficult to be conceded, that the wonder would rather be should the fact
prove otherwise. As a postulate, therefore, we shall henceforth regard it;
and as, in the progress of our work, analogies offer themselves in its sup-
port from what we observe of other celestial bodies, we shall not fail to
point them out to the reader's notice.
(77.) The earth's rotation on its axis so admitted, explaining, as it evi-
dently does, the apparent motion of the stars in a completely satisfactory
manner, prepares us for the further admission of its motion, bodily, in
space, should such a motiën enable us to explain, in a manner equally so,
the apparently complex and enigmatical motions of the Sun, moon, and
planets. The Copernican astronomy adopts this idea in its full extent,
ascribing to the earth, in addition to its motion of rotation about an axis,
also one of translation or transference through space, in such a course or
orbit, and so regulated in direction and celerity, as, taken in conjunction
with the motions of the other bodies of the universe, shall render a ration-
*.
RELATIVE MOTION. 59
al account of the appearances they successively present,- that is to say,
an account of which the several parts, postulates, propositions, deductions,
intelligibly cohere, without contradicting each other or the nature of things
as concluded from experience. In this view of the Copernican doctrine
it is rather a geometrical conception than a physical theory, inasmuch as it
simply assumes the requisite motions, without attempting to explain their
mechanical origin, or assign them any dependence on physical causes. The
Newtonian theory of gravitation supplies this deficiency, and, by showing
that all the motions required by the Copernican conception must, and that
no others can, result from a single, intelligible, and very simple dynamical
law, has given a degree of certainty to this conception, as a matter of fact,
which attaches to no other creation of the human mind.
(78.) To understand this conception in its further developments, the
reader must bear steadily in mind the distinction between relative and ab-
solute motion. Nothing is easier to perceive than that, if a spectator at
rest view a certain number of moving objects, they will group and arrange
themselves to his eye, at each successive moment, in a very different way from
what they would do were he in active motion among them,--if he formed
one of them, for instance, and joined in their dance. This is evident from
what has been said before of parallactic motion; but it will be asked, How
is such a spectator to disentangle from each other the two parts of the
apparent motions of these external objects, – that which arises from the
effect of his own change of place, and which is therefore only apparent
(or, as a German metaphysician would say, subjective—having reference
only to him as perceiving it), — and that which is real (or objective—hav- .
ing a positive existence, whether perceived by him or not)? By what
rule is he to ascertain, from the appearances presented to him while him-
self in motion, what would be the appearances were he at rest? It by
no means follows, indeed, that he would even then at once obtain a clear con-
ception of all the motions of all the objects. The appearances so presented
to him would have still something subjective about them. They would be
still appearances, not geometrical realities. They would still have a refe-
rence to the point of view, which might be very unfavourably situated
(as, indeed, is the case in our system) for affording a clear notion of the
real movement of each object. No geometrical figure, or curve, is seen
by the eye as it is conceived by the mind to exist in reality. The laws
of perspective interfere and alter the apparent directions and foreshorten
the dimensions of its several parts. If the spectator be unfavourably
situated, as, for instance, nearly in the plane of the figure (which is the
case we have to deal with), they may do so to such an extent, as to make
60 OUTLINES OF ASTRONOMY.
a considerable effort of imagination necessary to pass from the sensible to
the real form.
(79.) Still, preparatory to this ultimate step, it is first necessary that
the spectator should free or clear the appearances from the disturbing
influence of his own change of place. And this he can always do by the
following general rule or proposition: —
The relative motion of two bodies is the same as if either of them
were at rest, and all its motion communicated to the other in an opposite
direction."
Hence, if two bodies move alike, they will, when seen from each other
(without reference to other near bodies, but only to the starry sphere),
appear at rest. Hence, also, if the absolute motions of two bodies be
uniform and rectilinear, their relative motion is so also.
(80.) The stars are so distant, that as we have seen it is absolutely
indifferent from what point of the earth's surface we view them. Their
configurations inter se are identically the same. It is otherwise with the
sun, moon, and planets, which are near enough (especially the moon) to
be parallactically displaced by change of station from place to place on
one globe. In order that astronomers residing on different points of the
earth's surface should be able to compare their observations with effect, it
is necessary that they should clearly understand and take account of this
effect of the difference of their stations on the appearance of the outward
universe as seen from each. As an exterior object seen from one would
appear to have shifted its place were the spectator suddenly transported to
the other, so two spectators, viewing it from the two stations at the same
instant, do not see it in the same direction. Hence arises a necessity for
the adoption of a conventional centre of reference, or imaginary station
of observation common to all the world, to which each observer, wherever
situated, may refer (or, as it is called, reduce) his observations, by calcu-
lating and allowing for the effect of his local position with respect to that
common centre (supposing him to possess the necessary data). If there
were only two observers, in fixed stations, one might agree to refer his
observations to the other station; but, as every locality on the globe may
be a station of observation, it is far more convenient and natural to fix
* This proposition is equivalent to the following, which precisely meets the case pro-
posed, but requires somewhat more thought for its clear apprehension than can perhaps
be expected from a beginner: — -
PROP.—If two bodies, A and B, be in motion independently of each other, the motion
which B seen from A would appear to have if A were at rest is the same with that which
it would appear to have, A being in motion, if, in addition to its own motion, a motwon
equal to A's and in the same direction were communicated to it.
FELATIVE MOTION. 61
upon a point equally related to all, as the common point of reference;
and this can be no other than the centre of the globe itself. The paral-
lactic change of apparent place which would arise in an object, could any
observer suddenly transport himself to the centre of the earth, is evidently
the angle C S P, subtended or the object S by that radius C P of the
earth which joins the centre and the place P of observation.
Fig. 10.

62 OUTLINES OF ASTRONOMY.
CHAPTER II.
TERMINOLOGY AND ELEMENTARY GEOMETRICAL CONCEPTIONS AND
RELATIONS. — TERMINOLOGY RELATING TO THE GLOBE OF THE
EARTH – TO THE CELESTIAL SPHERE. — CELESTIAL PERSPECTIVE.
(81.) SEVERAL of the terms in use among astronomers have been ex-
plained in the preceding chapter, and others used anticipatively. But the
technical language of every subject requires to be formally stated, both
for consistency of usage and definiteness of conception. We shall there-
fore proceed, in the first place, to define a number of terms in perpetual
use, having relation to the globe of the earth and the celestial sphere.
DEFINITION 1. The azis of the earth is that diameter about which
it revolves, with a uniform motion, from west to east; performing one
revolution in the interval which elapses between any star leaving a cer-
tain point in the heavens, and returning to the same point again.
(83.) DEF. 2. The poles of the earth are the points where its axis
meets its surface. The North Pole is that nearest to Europe; the South
Pole that most remote from it.
(84.) DEF. 3. The earth's equator is a great circle on its surface,
equidistant from its poles, dividing it into two hemispheres—a northern
and a southern; in the midst of which are situated the respective poles
of the earth of those names. The plane of the equator is, therefore, a
plane perpendicular to the earth’s axis, and passing through its centre.
(85.) DEF. 4. The terrestrial meridian of a station on the earth's
surface, is a great circle of the globe passing through both poles and
through the plane. The plane of the meridian is the plane in which
that circle lies.
(86.) DEF. 5. The sensible and the rational horizon of any station
have been already defined in art. 74.
(87.) DEF. 6. A meridian line is the line of intersection of the
plane of the meridian of any station with the plane of the sensible
horizon, and therefore marks the north and south points of the horizon,
or the directions in which a spectator must set out if he would travel
directly towards the north or south pole,
TERMINOLOGY. 63
(88.) DEF. 7. The latitude of a place on the earth's surface is its
angular distance from the equator, measured on its own terrestrial meri-
dian : it is reckoned in degrees, minutes, and seconds, from 0 up to 90°,
and northwards or southwards according to the hemisphere the place lies
in. Thus, the observatory at Greenwich is situated in 51°28' 40" north
latitude. This definition of latitude, it will be observed, is to be con-
sidered as only temporary. A more exact knowledge of the physical
structure and figure of the earth, and a better acquaintance with the
ificeties of astronomy, will render some modification of its terms, or a
different manner of considering it, necessary.
(89.) DEF. 8. Parallels of latitude are small circles on the earth's
surface parallel to the equator. Every point in such a circle has the
same latitude. Thus, Greenwich is said to be situated in the parallel of
519 28' 40".
(90.) DEF. 9. The longitude of a place on the earth’s surface is the
inclination of its meridian to that of some fixed station referred to as a
point to reckon from. English astronomers and geographers use the ob-
servatory at Greenwich for this station; foreigners, the principal observa-
tories of their respective nations. Some geographers have adopted the
island of Ferro. Hereafter, when we speak of longitude, we reckon from
Greenwich. The longitude of a place is, therefore, measured by the arc
of the equator intercepted between the meridian of the place and that of
Greenwich; or, which is the same thing, by the spherical angle at the
pole included between these meridians.
(91.) As latitude is reckoned north or south, so longitude is usually
said to be reckoned west or east. It would add greatly, however, to sys-
tematic regularity, and tend much to avoid confusion and ambiguity in
computations, were this mode of expression abandoned, and longitudes
reckoned invariably westward from their origin round the whole circle
from 0 to 360°. Thus, the longitude of Paris is, in common parlance,
either 2° 20' 22" east, or 357° 39'38" west of Greenwich. But, in the
sense in which we shall henceforth use and recommend others to use the
term, the latter is its proper designation. Longitude is also reckoned in
time at the rate of 24h. for 360°, or 15° per hour. In this system the
longitude of Paris is 23 h. 50 m. 38; s."
(92.) Knowing the longitude and latitude of a place, it may be laid
down on an artificial globe; and thus a map of the earth may be con-
* To distinguish minutes and seconds of time from those of angular measure we
shall invariably adhere to the distinct system of notation here adopted (°'”, and h.m.
s.) Great confusion sometimes arises from the practice of using the same marks
for both.
64. OUTLINES OF ASTRONOMY.
structed. Maps of particular countries are detached portions of this
general map, extended into planes; or rather, they are representations
on planes of such portions, executed according to certain conventional
systems of rules, called projections, the object of which is either to
distort as little as possible the outlines of countries from what they are
on the globe—or to establish easy means of ascertaining, by inspection or
graphical measurement, the latitudes and longitudes of places which
occur in them, without referring to the globe or to books—or for other
peculiar uses. See Chap. IV. AP.
(93.) DEF. 10. The Tropics are two parallels of latitude, one on the
north and the other on the south side of the equator, over every point
of which respectively, the sun in its diurnal course passes vertically on
the 21st of March and the 21st of September in every year. Their
latitudes are about 23°28' respectively, north and south.
(94.) DEF. 11. The Arctic and Antarctic circles are two small circles
or parallels of latitude as distant from the north and south poles as the
tropics are from the equator, that is to say, about 23°28'; their latitudes,
therefore, are about 66° 32'. We say about, for the places of these
circles and of the tropics are continually shifting on the earth's surface,
though with extreme slowness, as will be explained in its proper place.
(95.) DEF, 12. The sphere of the heavens or of the stars is an ima-
ginary spherical surface of infinite radius, having the eye of any specta-
tor for its centre, and which may be conceived as a ground on which the
stars, planets, &c., the visible contents of the universe, are seen projected
as in a vast picture."
(96.) DEF. 13. The poles of the celestial sphere are the points of that
imaginary sphere towards which the earth's axis is directed.
(97.) DEF. 14. The celestial equator, or, as it is often called by as-
1 The ideal sphere without us, to which we refer the places of objects, and whicn
we carry along with us wherever we go, is no doubt intimately connected by associa‘
tion, if not entirely dependent on that obscure perception of sensation in the retinae of
our eyes, of which, even when closed and unexcited, we cannot entirely divest them.
We have a real spherical surface within our eyes, the seat of sensation and vision,
corresponding, point for point, to the external sphere. On this the stars, &c. are really
mapped down, as we have supposed them in the text to be, on the imaginary concave
of the heavens. When the whole surface of the retina is excited by light, habit leads
us to associate it with the idea of a real surface existing without us. Thus we become
impressed with the notion of a sky and a heaven, but the concave surface of the retina
itself is the true seat of all visible angular dimension and angular motion. The sub-
stitution of the retina for the heavens would be awkward and inconvenient in language,
but it may always be mentally made. (See Schiller's pretty enigma on the eye, in his
Turandot.) -
TERMINOLOGY. 65
tronomers, the equinoctial, is a great circle of the celestial sphere, marked
out by the indefinite extension of the plane of the terrestrial equator.
(98.) DEF. 15. The celestial horizon of any place is a great circle of
the sphere marked out by the indefinite extension of the plane of any
spectator's sensible or (which comes to the same thing as will presently
be shown,) his rational horizon, as in the case of the equator. *
(99.) DEF. 16. The zenith and nadir' of a spectator are the two
points of the sphere of the heavens, vertically over his head, and verti-
cally under his feet, or the poles of the celestial horizon; that is to say,
points 90° distant from every point in it.
(100.) DEF. 17. Vertical circles of the sphere are great circles passing
through the zenith and nadir, or great circles perpendicular to the horizon.
On these are measured the altitudes of objects above the horizon — the
complements to which are their zenith distances.
(101.) DEF, 18. The celestial meridian of a spectator is the great circle
marked out on the sphere by the prolongation of the plane of his terres-
trial meridian. If the earth be supposed at rest, this is a fixed circle, and
all the stars are carried across it in their diurnal courses from east to west.
If the stars rest and the earth rotate, the spectator's meridian, like his
horizon (art. 52), sweeps daily across the stars from west to east. When-
ever in future we speak of the meridian of a spectator or observer, we
intend the celestial meridian, which being a circle passing through the
poles of the heavens and the zenith of the observer, is necessarily a verti-
cal circle, and passes through the north and south points of the horizon.
(102.) DEF. 19. The prime vertical is a vertical circle perpendicular to
the meridian, and which therefore passes through the east and west points
of the horizon.
(103.) DEF. 20. Azimuth is the angular distance of a celestial object
from the north or south point of the horizon (according as it is the north
or south pole which is elevated), when the object is referred to the horizon
by a vertical circle; or it is the angle comprised between two vertical
planes — one passing through the elevated pole, the other through the
object. Azimuth may be reckoned eastwards, or westwards, from the
north or south point, and is usually so reckoned only to 180° either way.
But to avoid confusion, and to preserve continuity of interpretation when
algebraic symbols are used (a point of essential importance, hitherto too
little insisted on), we shall always reckon azimuth from the points of the
horizon most remote from the elevated pole, westward (so as to agree in
general directions with the apparent diurnal motion of the stars), and
| From Arabic words. Nadir corresponds evidently to the German nieder, (down
whence our nether.' -
5
56 OUTLINES OF ASTRONOMY.
carry its reckoning from 0° to 360° if always reckoned positive, consider-
ing the eastward reckoning as negative.
(104.) DEF. 21. The altitude of a heavenly body is its apparent angular
elevation above the horizon. It is the complement to 90°, therefore, of
its zenith distance. The altitude and azimuth of an object being known,
its place in the visible heavens is determined.
(105.) DEF. 22. The declination of a heavenly body is its angular
distance from the equinoctial or celestial equator, or the complement to
90° of its angular distance from the nearest pole, which latter distance is
called its Polar distance. Declinations are reckoned plus or minus,
according as the object is situated in the northern or southern celestial
hemisphere. Polar distances are always reckoned from the North Pole,
from 0° up to 180°, by which all doubt or ambiguity of expression with
respect to sign is avoided.
(106.) DEF. 23. Hour circles of the sphere; or circles of declination,
are great circles passing through the poles, and of course perpendicular to
the equinoctial. The hour circle, passing through any particular heavenly
body, serves to refer it to a point in the equinoctial, as a vertical circle
does to a point in the horizon.
(107.) DEF. 24. The hour angle of a heavenly body is the angle at
the pole included between the hour circle passing through the body, and
the celestial meridian of the place of observation. We shall always
reckon it positively from the upper culmination (art. 125) westwards, or
in conformity with the apparent diurnal motion, completely round the
circle from 0° to 360°. Hour angles, generally, are angles included at
the pole between different hour circles.
(108.) DEF. 25. The right ascension of a heavenly body is the are of
the equinoctial included between a certain point in that circle called the
Vernal Equinox, and the point in the same circle to which it is referred
by the circle of declination passing through it. Or it is the angle included
between two hour circles, one of which passes through the vernal equinox
(and is called the equinoctial colure), the other through the body. How
the place of this initial point on the equinoctial is determined, will be
explained further on. -
(109.) The right ascensions of celestial objects are always reckoned
eastwards from the equinox, and are estimated either in degrees, minutes,
and seconds, as in the case of terrestrial longitudes, from 0° to 360°,
which completes the circle; or, in time, in hours, minutes, and seconds,
from 0h. to 24h. The apparent diurnal motion of the heavens being
contrary to the real motion of the earth, this is in conformity with the
westward reckoning of longitudes. (Art. 91.)
TERMINOLOGY. 67
(110.) Sidereal time is reckoned by the diurnal motion of the stars,
or rather of that point in the equinoctial from which right ascensions are
reckoned. This point may be considered as a star, though no star is, in
fact, there; and, moreover, the point itself is liable to a certain slow
variation, — so slow, however, as not to affect, perceptibly, the interval,
of any two of its successive returns to the meridian. This interval is
called a sidereal day, and is divided into 24 sidereal hours, and these again
into minutes and seconds. A clock which marks sidereal time, i. e. which
goes at such a rate as always to show 0h. 0m. 0s, when the equinox comes on
the meridian, is called a sidereal clock, and is an indispensable piece of furni-
ture in every observatory. Hence the hour angle of an object reduced to
time at the rate of 15° per hour, expresses the interval of sidereal time
by which (if its reckoning be positive) it has past the meridian; or, if
negative, the time it wants of arriving at the meridian of the place of
observation. So also the right ascension of an object, if converted into
time at the same rate (since 360° being described uniformly in 24 hours,
15° must be so described in 1 hour), will express the interval of sidereal
time which elapses from the passage of the vernal equinox across the
meridian to that of the object next subsequent.
(111.) As a globe or maps may be made of the whole or particular
regions of the surface of the earth, so also a globe, or general map of the
heavens, as well as charts of particular parts, may be constructed, and the
stars laid down in their proper situations relative to each other, and to
the poles of the heavens and the celestial equator. Such a representa-
tion, once made, will exhibit a true appearance of the stars as they
present themselves in succession to every spectator on the surface, or as
they may be conceived to be seen at once by one at the centre of the
globe. It is, therefore, independent of all geographical localities. There
will occur in such a representation neither zenith, nadir, nor horizon —
neither east nor west points; and although great circles may be drawn on
it from pole to pole, corresponding to terrestrial meridians, they can no
longer, in this point of view, be regarded as the celestial meridians of
fixed points on the earth's surface, since, in the course of one diurnal
revolution, every point in it passes beneath each of them. It is on
account of this change of conception, and with a view to establish a com-
plete distinction between the two branches of Geography and Uranogra-
phy,' that astronomers have adopted different terms, (viz. declination and
right ascension) to represent those arcs in the heavens which correspond
to latitudes and longitudes on the earth. It is for this reason that they
*Tn the earth; ypapew, to describe or represent; ovpavos, the heaven.
68 OUTLINES OF ASTRONOMY.
term the equator of the heavens the equinoctial; that what are meridians
on the earth are called hour circles in the heavens, and the angles they
include between them at the poles are called hour angles. All this is
convenient and intelligible; and had they been content with this nomen-
clature, no confusion could ever have arisen. Unluckily, the early
astronomers have employed also the words latitude and longitude in their
uranography, in speaking of arcs of circles not corresponding to those
meant by the same words on the earth, but having reference to the motion
of the sun and planets among the stars. It is now too late to remedy
this confusion, which is ingrafted into every existing work on astronomy:
we can only regret, and warn the reader of it, that he may be on his
guard when, at a more advanced period of our work, we shall have occa-
sion to define and use the terms in their celestial sense, at the same time
urgently recommending to future writers the adoption of others in their
places. - -
(112.) It remains to illustrate these descriptions by reference to a
figure. Let C be the centre of the earth, N C S its axis; then are N
Fig. 11.
Z,
and S its poles; E Q its equator; A B the parallel of latitude of the
station A on its surface; A P parallel to S C N, the direction in which
an observer at A will see the elevated pole of the heavens; and A. Z, the
prolongation of the terrestrial radius C A, that of his zenith. N A E S
will be his meridian; N G. S that of some fixed station, as Greenwich;
and G. E, or the spherical angle G. N. E., his longitude, and E A his lati-
tude. Moreover, if n s be a plane touching the surface in A, this will

TERMINOLOGY. 69
be his sensible horizon: n A s marked on that plane by its intersection
with his meridian will be his meridian line, and n and s the north and
south points of his horizon. *
(113.) Again, neglecting the size of the earth, or conceiving him
stationed at its centre, and referring every thing to his rational horizon;
let the annexed figure represent the sphere of the heavens; C the specta-
tor; Z his zenith; and N his nadir : then will H. A. O, a great circle of
the sphere, whose poles are Z N, be his celestial horizon; P p the
elevated and depressed Poles of the heavens; H P the altitude of the
pole, and H P Z E O his meridian ; E T Q, a great circle perpendicular
to P p, will be the equinoctial; and if r represent the equinox, ºr T will
be the right ascension, T S the declination, and P S the polar distance
of any star or object S, referred to the equinoctial by the hour circle P
S T p, and B S D will be the diurnal circle it will appear to describe
Fig. 12
IR Z. E
2 : T
PA: ;
# e
g *72 op
H C, lo
A
I) f p
º
Q -
N

about the pole. Again, if we refer it to the horizon by the vertical circle
Z S M, O M will be its azimuth, M S its altitude, and Z S its zenith
distance. H and O are the north and south, e w the east and west points
of his horizon, or of the heavens. Moreover, if H h, O o, be small
circles, or parallels of declination, touching the horizon in its north and
south points, H h will be the circle of perpetual apparition, between
which and the elevated pole the stars never set; O o that of perpetual
occultation, between which and the depressed pole they never rise. In
all the zone of the heavens between H h and O o, they rise and set; any
one of them, as S, remaining above the horizon in that part of its diurnal
circle represented by a BA, and below it throughout all the part repre-
sented by A D a. It will exercise the reader to construct this figure for
t
70 OluTLINES OF ASTRONOMY.
several different elevations of the pole, and for a variety of positions of
the star S in each. , w -
(114.) Celestial perspective is that branch of the general science of
perspective which teaches us to conclude, from a knowledge of the real
situation and forms of objects, lines, angles, motions, &c. with respect to
the spectator, their apparent aspects, as seen by him projected on the
imaginary concave of the heavens; and, vice versä, from the apparent
configurations and movements of objects so seen projected, to conclude,
so far as they can be thence concluded, their real geometrical relations to
each other and to the spectator. It agrees with ordinary perspective when
only a small visual area is contemplated, because the concave ground of
the celestial sphere, for a small extent, may be regarded as a plane sur-
face, on which objects are seen projected or depicted as in common per-
spective. But when large amplitudes of the visual area are considered,
or when the whole contents of space are regarded as projected on the
whole interior surface of the sphere, it becomes necessary to use a different
phraseology, and to resort to a different form of conception. In common
perspective there is a single “point of sight,” or “centre of the picture,”
the visual line from the eye to which is perpendicular to the “plane Cf
the picture,” and all straight lines are represented by straight lines. In
celestial perspective, every point to which the view is for the moment
directed, is equally entitled to be considered as the “centre of the pic-
ture,” every portion of the surface of the sphere being similarly related
to the eye. Moreover, every straight line (supposed to be indefinitely
prolonged) is projected into a semicircle of the sphere, that, namely, in
which a plane passing through the line and the eye cuts its surface. And
every system of parallel straight lines, in whatever direction, is projected
into a system of semicircles of the sphere, meeting in two common apexes,
or vanishing points, diametrically opposite to each other, one of which
corresponds to the vanishing point of parallels in ordinary perspective;
the other, in such perspective has no existence. In other words, every
point in the sphere to which the eye is directed may be regarded as one
ºf the vanishing points, or one apex of a system of straight lines, parallel
to that radius of the sphere which passes through it, or to the direction
of the line of sight, seen in perspective from the earth, and the points
diametrically opposite, or that from which he is looking, as the other.
And any great circle of the sphere may similarly be regarded as the
vanishing circle of a system of planes, parallel to its own.
(115.) A familiar illustration of this is often to be had by attending to
the lines of light seen in the air, when the sun’s rays are darted through
apertures in clouds, the sun itself being at the time obscured behind thcro
\
CELESTIAL PERSPECTIVE. 71
These lines which, marking the course of rays emanating from a point
almost infinitely distant, are to be considered as parallel straight lines, are
thrown into great circles of the sphere, having two apexes or points of
common intersection — one in the place where the sun itself (if not
obscured) would be seen. The other diametrically opposite. The first
only is most commonly suggested when the spectator's view is towards the
sun. But in mountainous countries, the phenomenon of Sunbeams con-
verging towards a point diametrically opposite to the sun, and as much
depressed below the horizon as the sun is elevated above it, is not unfre-
quently noticed, the back of the spectator being turned to the sun's place.
Occasionally, but much more rarely, the whole course of such a system
of sunbeams, stretching in semicircles across the hemisphere from horizon
to horizon (the sun being near setting), may be seen.' Thus again, the
streamers of the Aurora Borealis, which are doubtless electrical rays,
parallel, or nearly parallel to each other, and to the dipping needle, usually
appear to diverge from the point towards which the needle, freely sus-
pended, would dip northwards (i. e. about 70° below the horizon and 23°
west of north from London), and in their upward progress pursue the
course of great circles till they again converge (in appearance) towards
the point diametrically opposite (i.e. 70° above the horizon, and 23° to the
eastward of south), forming a sort of canopy over-head, having that point
for its centre. So also in the phenomenon of shooting stars, the lines of
direction which they appear to take on certain remarkable occasions of
periodical recurrence, are observed, if prolonged backwards, apparently to
meet nearly in one point of the sphere; a certain indication of a general
near approach to parallelism in the real directions of their motions on
those occasions. On which subject more hereafter.
(116.) In relation to this idea of celestial perspective, we may conceive
the north and south poles of the sphere as the two vanishing points of a
system of lines parallel to the axis of the earth; and the zenith and nadir
of those of a system of perpendiculars to its surface at the place of
observation, &c. It will be shown that the direction of a plumb-line, at
every place is perpendicular to the surface of still water at that place
* It is in such cases only that we conceive them as circles, the ordinary conventions
of plane perspective becoming untenable. The author had the good fortune to witness
on one occasion the phenomenon described in the text under circumstances of more
than usual grandeur. Approaching Lyons from the south on Sept. 30, 1826, about 5% h.
P. M., the sun was seen nearly setting behind broken masses of stormy cloud, from
whose apertures streamed forth beams of rose-coloured light, traceable all across the
hemisphere almost to their opposite point of convergence behind the snowy precipices
of Mont Blanc, conspicuously visible at nearly 100 miles to the eastward. The im-
pression produced was that of another but feebler sun about to rise from behind the
mountain, and darting forth precursory beams to meet those of the real one opposite.
72 OlſTLINES OF ASTRONOMY.
which is the true horizon, and though mathematically speaking no two
plumb-lines are exactly parallel (since they converge to the earth’s centre),
yet over very small tracts, such as the area of a building—in one and
the same town, &c., the difference from exact parallelism is so small that
it may be practically disregarded." To a spectator looking upwards such
a system of plumb-lines will appear to converge to his zenith; downwards,
to his nadir.
(117.) So also the celestial equator, or the equinoctial, must be con-
ceived as the vanishing circle of a system of planes parallel to the earth’s
equator, or perpendicular to its axis. The celestial horizon of any spec-
tator is in like manner the vanishing circle of all planes parallel to his
true horizon, of which planes his rational horizon (passing through the
earth’s centre) is one, and his sensible horizon (the tangent plane of his
station) another. -
(118.) Owing, however, to the absence of all the ordinary indications
of distance which influence our judgment in respect of terrestrial objects,
owing to the want of determinate figure and magnitude in the stars and
planets as commonly seen — the projection of the celestial bodies on the
ground of the heavenly concave is not usually regarded in this its true
light, of a perspective representation or picture, and it even requires an
effort of imagination to conceive them in their true relations, as at vastly
different distances, one behind the other, and forming with one another
lines of junction violently foreshortened, and including angles altogether
differing from those which their projected representations appear to make.
To do so at all with effect presupposes a knowledge of their actual situa-
tions in space, which it is the business of astronomy to arrive at by appro-
priate considerations. But the connections which subsist among the
several parts of the picture, the purely geometrical relations among the
angles and sides of the spherical triangles of which it consists, constitute,
under the name of Uranometry,” a preliminary and subordinate branch of
the general science, with which it is necessary to be familiar before any
further progress can be made. Some of the most elementary and fre-
quently occurring of these relations we proceed to explain. And first, as
immediate consequences of the above definitions, the following propositions
will be borne in mind.
(119.) The altitude of the elevated pole is equal to the latitude of the
spectator's geographical station.
For it appears, see fig. art. 112, that the angle P A Z between the
*An interval of a mile corresponds to a convergence of plumb-lines amounting to
somewhat less space than a minute.
* ovpaves, the heavens; perpetv, to measure: the measurement of the heavens.
ELEMENTARY RELATIONS. 73
pole and the zenith is equal to N C A, and the angles Z. A n and N C E
being right angles, we have P A n=A C E. Now the former of these
is the elevation of the pole as seen from E, the latter is the angle at the
earth’s centre subtended by the arc E A, or the latitude of the place.
(120.) Hence to a spectator at the north pole of the earth, the north
pole of the heavens is in his zenith. As he travels southward it becomes
less and less elevated till he reaches the equator, when both poles are in
his horizon — south of the equator the north pole becomes depressed
below, while the south rises above his horizon, and continues to do so till
the south pole of the globe is reached, when that of the heavens will be
in the zenith. *.
(121.) The same stars, in their diurnal revolution, come to the meridian,
successively, of every place on the globe once in twenty-four sidereal hours.
And, since the diurnal rotation is uniform, the interval, in sidereal time,
which elapses between the same star coming upon the meridians of two
different places is measured by the difference of longitudes of the places.
(122.) Vice versá — the interval elapsing between two different stars
coming on the meridian of one and the same place, expressed in sidereal
time, is the measure of the difference of right ascensions of the stars.
(123.) The equinoctial intersects the horizon in the east and west
points, and the meridian is a point whose altitude is equal to the co-lati-
tude of the place. Thus, at Greenwich, of which the latitude is 51° 28′
40", the altitude of the intersection of the equinoctial and meridian is
38° 31' 20". The north and south poles of the heavens are the poles of
the equinoctial. The east and west points of the horizon of a spectator
are the poles of his celestial meridian. The north and south points of his
horizon are the poles of his prime vertical, and his zenith and nadir are
the poles of his horizon. g *
(124.) All the heavenly bodies culminate (i.e. come to their greatest
altitudes) on the meridian ; which is, therefore, the best situation to ob-
serve them, being least confused by the inequalities and vapours of the
atmosphere, as well as least displaced by refraction.
(125.) All celestial objects within the circle of perpetual apparition
come twice on the meridian, above the horizon, in every diurnal revolu-
tion; once above and once below the pole. These are called their upper
and lower culm nations.
(126.) The problems of uranometry, as we have described it, consist
in the solution of a variety of spherical triangles, both right and oblique
angled, according to the rules, and by the formulae of spherical trigonom-
etry, which we suppose known to the reader, or for which he will consult
appropriate treatises. We shall only here observe generally, that in all
74 OUTLINES OF ASTRONOMY.
problems in which spherical geometry is concerned, the student will find
it a useful practical maxim rather to consider the poles of the great circles
which the question before him refers to than the circles themselves. To
use, for example, in the relations he has to consider, polar distances rather
than declinations, Zenith distances rather than altitudes, &c. Bearing
this in mind, there are few problems in uranometry which will offer any
difficulty. The following are the combinations which most commonly
occur for solution when the place of one celestial object only on the sphere
is concerned.
(127.) In the triangle ZPS, Z is the zenith, P the elevated pole, and
S the star, sun, or other celestial object. In this triangle occur, 1st, PZ,
which being the complement of PH (the altitude of the pole), is ob-
viously the complement of the latitude (or the co-latitude, as it is called)
of the place; 2d, PS, the polar distance, or the complement of the decli-
nation (co-declination) of the star; 3d, ZS, the zenith distance or co-alti-
tude of the star. If P S be greater than 90°, the object is situated on
the side of the equinoctial opposite to that of the elevated pole. If 2S
be so, the object is below the horizon.
Fig. 13.
In the same triangle the angles are, 1st, ZPS the lower angle; 2d, PZS
(the supplement of S ZO, which latter is the azimuth of the star or other
heavenly body), 3d, P S Z, an angle which, from the infrequency of any
practical reference to it, has not acquired a name."
*In the practical discussion of the measures of double stars and other objects by the
aid of the position micrometer, this angle is sometimes required to be known; and
when so required, it will be not inconveniently referred to as “the angle of position
ot the zenith.”

ELEMENTARY RELATIONS. 75
The following five astronomical magnitudes, then, occur among the sides
of this most useful triangle: viz., 1st, The co-latitude of the place of
observation; 2d, the polar distance; 3d, the zenith distance; 4th, the
hour angle; and 5th, the sub-azimuth (supplement of azimuth) of a given
celestial object; and by its solution therefore may all problems be resolved,
in which three of these magnitudes are directly or indirectly given, and
the other two required to be found.
(128.) For example, suppose the time of rising or setting of the sun
or of a star were required, having given its right ascension and polar dis-
tance. The star rises when apparently on the horizon, or really about
34' below it (owing to refraction), so that, at the moment of its apparent
rising, its zenith distance is 90° 34'−Z S. Its polar distance PS being also
given, and the co-latitude ZP of the place, we have given the three sides
of the triangle, to find the hour angle Z PS, which, being known, is to
be added to or subtracted from the star's right ascension, to give the side-
real time of setting or rising, which, if we please, may be converted into
solar time by the proper rules and tables. v.
(129.) As another example of the use of the same triangle, we may
propose to find the local sidereal time, and the latitude of the place of
observation, by observing equal altitudes of the same star east and west
of the meridian, and noting the interval of the observations in sidereal
time. -
The hour angles corresponding to equal altitudes of a fixed star being
equal, the hour angle east or west will be measured by half the observed
interval of the observations. In our triangle, then, we have given this
hour angle Z PS, the polar distance P S of the star, and Z S, its co-
altitude at the moment of observation. Hence we may find P Z, the
co-latitude of the place. Moreover, the hour angle of the star being
known, and also its right ascension, the point of the equinoctial is known,
which is on the meridian at the moment of observation; and, therefore,
the local sidereal time at that moment. This is a very useful observation
for determining the latitude and time at an unknown station.
76 OUTLINES OF ASTRONOMY.
CHAPTER III.1
OF THE NATURE OF ASTRONOMICAL INSTRUMENTS AND OBSERVATIONS
IN GENERAL.-OF SIDEREAL AND SOLAR, TIME. –OF THE MEASURE-
MENTS OF TIME. — CLOCKS, CHRONOMETERS. — OF ASTRONOMICAL
MEASUREMENTS.— PRINCIPLE OF TELESCOPIC, SIGHTS TO INCREASE
THE ACCURACY OF POINTING.. — SIMPLEST APPLICATION OF THIS
PRINCIPLE.- THE TRANSIT INSTRUMENT.-- OF THE MEASUREMENT
OF ANGULAR INTERVALS. — METHODS OF INCREASING THE ACCU-
RACY OF READING.—THE VERNIER.—THE MICRoscoPE.—of THE
MURAI, CIRCLE. – THE MERIDIAN CIRCLE. — FIXATION OF POLAR
AND HORIZONTAL POINTS. — THE LEVEL, PLUMB-LINE, ARTIFICIAL
HORIZON.—PRINCIPLE OF COLLIMATION.—COLT IMATOR'S OF RITTEN-
HOUSE, KATER, AND BENZENBERG. — OF COMPOUND INSTRUMENTs
WITH CO-ORDINATE CIRCLES. — THE EQUATORIAL, ALTITUDE, AND
AZIMUTH INSTRUMENTS. — THEODOLITE. — OF THE SEXTANT AND
REFLECTING CIRCLE. — PRINCIPLE OF REPETITION. — OF MICROME-
TERS. — PARALLEL WIRE MICROMETER. — PRINCIPLE OF THE DU-
PLICATION OF IMAGES.— THE EIELIOMETER.— DOTBLE REFRACTING
EYE-PIECE. — WARIABLE PRISM MICROMETER. – OF THE POSITION
MICROMETER.
(130.) OUR first chapters have been devoted to the acquisition chiefly
of preliminary notions respecting the globe we inhabit, its relation to the
celestial objects which surround it, and the physical circumstances under
which all astronomical observations must be made, as well as to provide
ourselves with a stock of technical words and elementary ideas of most
frequent and familiar use in the sequel. We might now proceed to a
more exact and detailed statement of the facts and theories of astronomy;
but, in order to do this with full effect, it will be desirable that the
reader be made acquainted with the principal means which astronomers
* The student who is anxious to become acquainted with the chief subject matter
of this work, may defer the reading of that part of this chapter which is devoted to
the description of particular instruments, or content himself with a cursory perusal
of it until farther advanced, when it will be necessary to return to it.
NATURE OF ASTRONOMICAL INSTRUMENTS. 77
possess, of determining, with the degree of nicety their theories require,
the data on which they ground their conclusions; in other words, of as-
certaining by measurement the apparent and real magnitudes with which
they are conversant. It is only when in possession of this knowledge
that he can fully appreciate either the truth of the theories themselves,
or the degree of reliance to be placed on any of their conclusions ante- .
cedent to trial: since it is only by knowing what amount of error can
certainly be perceived and distinctly measured, that he can satisfy himself
whether any theory offers so close an approximation, in its numerical
results, to actual phenomena, as will justify him in receiving it as a true
representation of nature. -
(131.) Astronomical instrument-making may be justly regarded as the
most refined of the mechanical arts, and that in which the nearest ap-
proach to geometrical precision is required, and has been attained. It
may be thought an easy thing, by one unacquainted with the niceties re-
quired, to turn a circle in metal, to divide its circumference into 360
equal parts, and these again into smaller subdivisions,—to place it accu-
rately on its centre, and to adjust it in a given position; but practically
it is found to be one of the most difficult. Nor will this appear extra-
ordinary, when it is considered that, owing to the application of telescopes
to the purposes of angular measurement, every imperfection of structure
of division becomes magnified by the whole optical power of that instru-
ment; and that thus, not only direct errors of workmanship, arising from
unsteadiness of hand or imperfection of tools, but those inaccuracies
which originate in far more uncontrollable causes, such as the unequal
expansion and contraction of metallic masses, by a change of temperature,
and their unavoidable flexure or bending by their own weight, become
perceptible and measurable. An angle of one minute occupies, on the
circumference of a circle of 10 inches in radius, only about 340th part
of an inch, a quantity too small to be certainly dealt with without the
use of magnifying glasses; yet one minute is a gross quantity in the
astronomical measurement of an angle. With the instruments now em-
ployed in observatories, a single second, or the 60th part of a minute, is
rendered a distinctly visible and appreciable quantity. Now, the arc of
a circle, subtended by one second, is less than the 200,000th part of the
radius, so that on a circle of 6 feet in diameter it would occupy no greater
linear extent than gºoth part of an inch; a quantity requiring a power-
ful microscope to be discerned at all. Let any one figure to himself,
therefore, the difficulty of placing on the circumference of a metallic
circle of such dimensions (supposing the difficulty of its construction sur-
mounted), 360 marks, dots, or cognizable divisions, which shall all be
78 OUTLINES OF ASTRONOMY.
true to their places within such narrow limits; to say nothing of the sub-
division of the degrees so marked off into minutes, and of these again
into seconds. Such a work has probably baffled, and will probably for
ever continue to baffle, the utmost stretch of human skill and industry;
nor, if executed, could it endure. The ever-varying fluctuations of heat
and cold have a tendency to produce not merely temporary and transient,
but permanent, uncompensated changes of form in all considerable masses
of those metals which alone are applicable to such uses; and their own
weight, however symmetrically formed, must always be unequally sus-
tained, since it is impossible to apply the sustaining power to every part
separately : even could this be done, at all events force must be used to
move and to fix them; which can never be done without producing tem-
porary and risking permanent change of form. It is true, by dividing
them on their centres, and in the identical places they are destined to
occupy, and by a thousand ingenious and delicate contrivances, wonders
have been accomplished in this department of art, and a degree of perfec.
tion has been given, not merely to chefs d'oeuvre, but to instruments of
moderate prices and dimensions, and in ordinary use, which, on due con-
sideration, must appear very surprising. But though we are entitled to
look for wonders at the hands of scientific artists, we are not to expect
niracles. The demands of the astronomer will always surpass the power
of the artist; and it must, therefore, be constantly the aim of the former
to make himself, as far as possible, independent of the imperfections inci-
dent to every work the latter can place in his hands. He must, therefore,
endeavour so to combine his observations, so to choose his opportunities,
and so to familiarize himself with all the causes which may produce in-
strumental derangement, and with all the peculiarities of structure and
material of each instrument he possesses, as not to allow himself to be
misled by their errors, but to extract from their indications, as far as pos-
sible, all that is true, and reject all that is erroneous. It is in this that
the art of the practical astronomer consists, –an art of itself of a curious
and intricate nature, and of which we can here only notice some of the
leading and general features.
(132.) The great aim of the practical astronomer being numerical
correctness in the results of instrumental measurement, his constant care
and vigilance must be directed to the detection and compensation of errors,
either by annihilating, or by taking account of, and allowing for them.
Now, if we examine the sources from which errors may arise in any
instrumental determination, we shall find them chiefly reducible to three
principal heads:–
(133.) 1st, External or incidental causes of error; comprehending
INSTRUMENTAL AND OTHER SOURCES OF ERROR. 79
such as depend on external, uncontrollable circumstances: such as, fluc-
tuations of weather, which disturb the amount of refraction from its tabu-
lated value, and, being reducible to no fixed law, induce uncertainty to
the extent of their own possible magnitude; such as, by varying the tem-
perature of the air, vary also the form and position of the instruments
used, by altering the relative magnitudes and the tension of their parts;
and others of the like nature.
(134.) 2dly, Errors of observation: such as arise, for example, from
inexpertness, defective vision, slowness in seizing the exact instant of
occurrence of a phenomenon, or precipitancy in anticipating it, &c.;
from atmospheric indistinctness; insufficient optical power in the instru-
ment, and the like. Under this head may also be classed all errors arising
from momentary instrumental derangement, slips in clamping, looseness
of screws, &c.
(135.) 3dly, The third, and by far the most numerous class of errors
to which astronomical measurements are liable, arise from causes which
may be deemed instrumental, and which may be subdivided into two
principal classes. The first comprehends those which arise from an instru-
ment not being what it professes to be, which is error of workmanship.
Thus, if a pivot or axis, instead of being, as it ought, exactly cylindrical,
be slightly flattened, or elliptical,—if it be not exactly (as it is intended
it should) concentric with the circle it carries;–if this circle (so called)
be in reality not exactly circular, or not in one plane;—if its divisions,
intended to be precisely equidistant, should be placed in reality at unequal
intervals, and a hundred other things of the same sort. These are not
mere speculative sources of error, but practical annoyances, which every
observer has to contend with.
(136.) The other subdivision of instrumental errors comprehends such
as arise from an instrument not being placed in the position it ought to
have; and from those of its parts, which are made purposely moveable,
not being properly disposed inter se. These are errors of adjustment.
Some are unavoidable, as they arise from a general unsteadiness of the
soil or building in which the instruments are placed; which, though too
minute to be noticed in any other way, become appreciable in delicate
astronomical observations; others, again, are consequences of imperfect
workmanship, as where an instrument once well adjusted will not remain
so, but keeps deviating and shifting. But the most important of this class
of errors arise from the non-existence of natural indications, other than
those afforded by astronomical observations themselves, whether an instru
ment has or has not the exact position, with respect to the horizon and its
80 OluTLINES OF ASTRONOMY.
cardinal points, the axis of the earth, or to other principal astronomical
lines and circles, which it ought to have to fulfil properly its objects.
(137.) Now, with respect to the first two classes of error, it must be
observed, that, in so far as they cannot be reduced to known laws, and
thereby become subjects of calculation and due allowance, they actually
vitiate, to their full extent, the results of any observations in which they
subsist. Being, however, in their nature casual and accidental, their
effects necessarily lie sometimes one way, sometimes the other; sometimes
diminishing, sometimes tending to increase the results. Hence, by greatly
multiplying observations, under varied circumstances, by avoiding unfa-
sourable, and taking advantage of favourable circumstances of weather,
or othcrwise using opportunity to advantage—and finally, by taking the
imean or average of the results obtained, this class of errors may be so
far subdued, by setting them to destroy one another, as no longer sensibly
to vitiate any theoretical or practical conclusion. This is the great and
indeed only resource against such errors, not merely to the astronomer,
but to the investigator of numerical results in every department of
physical research.
(138.) With regard to errors of adjustment and workmanship, not
only the possibility, but the certainty of their existence, in every ima-
ginable form, in all instruments, must be contemplated. Human hands
or machines never formed a circle, drew a straight line, or erected a per-
pendicular, nor ever placed an instrument in perfect adjustment, unless
accidentally; and then only during an instant of time. This does not
prevent, however, that a great approximation to all these desiderata
should be attained. But it is the peculiarity of astronomical observation
to be the ultimate means of deteetion of all mechanical defects which
elude by their minuteness every other mode of detection. What the eye
cannot discern nor the touch perceive, a course of astronomical observa-
tions will make distinctly evident. The imperfect products of man's
hands are here tested by being brought into comparison under very great
magnifying powers (corresponding in effect to a great increase in acute-
ness of perception) with the perfect workmanship of nature; and there.
is none which will bear the trial. Now, it may seem like arguing in a
vicious circle, to deduce theoretical conclusions and laws from observation,
and then to turn round upon the instruments with which those observa-
tions were made, accuse them of imperfection, and attempt to detect and
rectify their errors by means of the very laws and theories which they
have helped us to a knowledge of. A little consideration, however, will
suffice to show that such a course of proceeding is perfectly legitimate.
(139.) The steps by which we arrive at the laws of natural phenomena,
INSTRUMENTAL sources OF ERROR. 81
and especially those which depend for their verification on numerical
determinations, are necessarily successive. Gross results and palpable
laws are arrived at by rude observation with coarse instruments, or
without any instruments at all, and are expressed in language which is
not to be considered as absolute, but is to be interpreted with a degree of
latitude commensurate to the imperfection of the observations themselves.
These results are corrected and refined by nicer scrutiny, and with more
delicate means. The first rude expressions of the laws which embody
them are perceived to be inexact. The language used in their expression
is corrected, its terms more rigidly defined, or fresh terms introduced,
until the new state of language and terminology is brought to fit the
improved state of knowledge of facts. In the progress of this scrutiny
subordinate laws are brought into view which still further modify both
the verbal statement and numerical results of those which first offered
themselves to our notice; and when these are traced out and reduced to
certainty, others, again, subordinate to them, make their appearance, and
become subjects of further inquiry. Now, it invariably happens (and
the reason is evident) that the first glimpse we catch of such subordinate
laws — the first form in which they are dimly shadowed out to our minds
—is that of errors. We perceive a discordance between what we expect,
and what we find. The first occurrence of such a discordance we attri-
bute to accident. It happens again and again; and we begin to suspect
our instruments. We then inquire, to what amount of error their deter-
minations can, by possibility, be liable. If their limit of possible error
exceed the observed deviation, we at once condemn the instrument, and
set about improving its construction or adjustments. Still the same
deviations occur, and, so far from being palliated, are more marked and
better defined than before. We are now sure that we are on the traces
of a law of nature, and we pursue it till we have reduced it to a definite
statement, and verified it by repeated observation, under every variety
of circumstances.
(140.) Now, in the course of this inquiry, it will not fail to happen
that other discordances will strike us. Taught by experience, we suspect
the existence of some natural law, before unknown; we tabulate (i. e.
draw out in order) the results of our observations; and we perceive, in
this synoptic statement of them, distinct indications of a regular progres-
sion. Again we improve or vary our instruments, and we now lose sight
of this supposed new law of nature altogether, or find it replaced by some
other, of a totally different character. Thus we are led to suspect an
instrumental cause for what we have noticed. We examine, therefore,
the theory of our instrument; we suppose defects in its structure, and, by
6
82 OUTLINES OF ASTRONOMY.
the aid of geometry, we trace their influence in introducing actual errors
into its indications. These errors have their laws, which, so long as we
have no knowledge of causes to guide us, may be confounded with laws
of nature, as they are mixed up with them in their effects. They are not
fortuitous, like errors of observation, but, as they arise from sources
inherent in the instrument, and unchangeable while it and its adjustments
remain unchanged, they are reducible to fixed and ascertainable forms;
each particular defect, whether of structure or adjustment, producing its
own appropriate form of error. When these are thoroughly investigated,
we recognize among them one which coincides in its nature and progression
with that of our observed discordances. The mystery is at once solved.
We have detected, by direct observation, an instrumental defect.
(141.) It is, therefore, a chief requisite for the practical astronomer to
make himself completely familiar with the theory of his instruments. By
this alone is he enabled at once to decide what effect on his observations any
given imperfection of structure or adjustment will produce in any given
circumstances under which an observation can be made. This alone also
can place him in a condition to derive available and practical means of
destroying and eliminating altogether the influence of such imperfections,
by so arranging his observations, that it shall affect their results in oppo-
site ways, and that its influence shall thus disappear from their mean,
which is one of the chief modes by which precision is attained in practical
astronomy. Suppose, for example, the principle of an instrument required
that a circle should be concentric with the axis on which it is made to
turn. As this is a condition which no workmanship can exactly fulfil, it
becomes necessary to inquire what errors will be produced in observations
made and registered on the faith of such an instrument, by any assigned
deviation in this respect; that is to say, what would be the disagreement
between observations made with it and with one absolutely perfect, could
such be obtained. Now, simple geometrical considerations suffice to show
—1st. that if the axis be excentric by a given fraction (say one thou-
sandth part) of the radius of the circle, all angles read off on that part
of the circle towards which the excentricity lies, will appear by that frac-
tional amount too small, and all on the opposite side too large. And,
2dly, that whatever be the amount of the excentricity, and on whatever
part of the circle any proposed angle is measured, the effect of the error
in question on the result of observations depending on the graduation of
its circumference (or limb, as it is technically called) will be completely
annihilated by the very easy method of always reading off the divisions
on two diametrically opposite points of the circle, and taking a mean; for
the effect of excentricity is always to increase the arc representing the
ILLUSTRATION BY EXAMPLES. — REFRACTION. 83
angle in question on one side of the circle, by just the same quantity by
which it diminishes that on the other. Again, suppose that the proper
use of the instrument required that this axis should be exactly parallel to
that of the earth. As it never can be placed or remain so, it becomes a
question, what amount of error will arise, in its use, from any assigned
deviation, whether in a horizontal or vertical plane, from this precise posi-
tion. Such inquiries constitute the theory of instrumental errors; a
theory of the utmost importance to practice, and one of which a complete
knowledge will enable an observer, with moderate instrumental means,
often to attain a degree of precision which might seem to belong only to
the most refined and costly. This theory, as will readily be apprehended,
turns almost entirely on considerations of pure geometry, and those for
the most part not difficult. In the present work, however, we have no
further concern with it. The Astronomical instruments we propose briefly
to describe in this chapter will be considered as perfect both in construc-
tion and adjustment." . *
(142.) As the above remarks are very essential to a right understand-
ing of the philosophy of our subject and the spirit of astronomical
methods, we shall elucidate them by taking one or two special cases.
Observant persons, before the invention of astronomical instruments, had
already concluded the apparent diurnal motions of the stars to be per-
formed in circles about fixed poles in the heavens, as shown in the fore-
going chapter. In drawing this conclusion, however, refraction was
entirely overlooked, or, if forced on their notice by its great magnitude
in the immediate neighbourhood of the horizon, was regarded as a local
irregularity, and, as such, neglected, or slurred over. As soon, however,
as the diurnal paths of the stars were attempted to be traced by instru-
ments, even of the coarsest kind, it became evident that the motion of
exact circles described about one and the same pole would not represent
the phenomena correctly, but that, owing to some cause or other, the
apparent diurnal orbit of every star is distorted from a circular into an
oval form, its lower segment being flatter than its upper; and the devia-
tion being greater the nearer the star approached the horizon, the effect
being the same as if the circle had been squeezed upwards from below,
and the lower parts more than the higher. For such an effect, as it was
soon found to arise from no casual or instrumental cause, it became neces-
sary to seek a natural one; and refraction readily occurred, to solve the
* The principle on which the chief adjustments of two or three of the most useful
and common instruments, such as the transit, the equatorial, and the sextant, are per
formed, are, however, noticed, for the convenience of readers who may use such in
struments without going farther into the arcana of practica. astronomy.
84 O'DTLINES OF ASTRONOMY.
difficulty. In fact, it is a case precisely analogous to what we have
already noticed (art. 47), of the apparent distortion of the sun near the
horizon, only on a larger scale, and traced up to great ºr altitudes. This
new law once established, it became necessary to modify the expression of
that anciently received, by inserting in it a salvo for the effect of refrac-
tion, or by making a distinction between the apparent diurnal orbits, as
affected by refraction, and the true ones cleared of that effect. This dis-
tinction between the apparent and the true—between the uncorrected
and corrected—between the rough and obvious, and the refined and ulti-
mate —is of perpetual occurrence in every part of astronomy.
(143.) Again. The first impression produced by a view of the diurnal
movement of the heavens is that all the heavenly bodies perform this
revolution in one common period, viz. a day, or 24 hours. But no sooner
do we come to examine the matter instrumentally, i. e. by noting, by
time-keepers, their successive arrivals on the meridian, than we find dif-
ferences which cannot be accounted for by any error of observation. All
the stars, it is true, occupy the same interval of time between their suc-
cessive appulses to the meridian, or to any vertical circle; but this is a
very different one from that occupied by the sun. It is palpably shorter;
being, in fact, only 23h 56' 4.09", instead of 24 hours, such hours as
our common clocks mark. Here, then, we have already two different
days, a sidereal and a solar; and if, instead of the sun, we observe the
moon, we find a third, much longer than either, — a lunar day, whose
average duration is 24' 54" of our ordinary time, which last is solar time,
being of necessity conformable to the Sun's successive re-appearances, on
which all the business of life depends.
(144.) Now, all the stars are found to be unanimous in giving the
same exact duration of 23* 56' 4.09", for the sidereal day; which, there-
fore, we cannot hesitate to receive as the period in which the earth makes
one revolution on its axis. We are, therefore, compelled to look on the
sun and moon as exceptions to the general law; as having a different
nature, or at least a different relation to us, from the stars; and as having
motions, real or apparent, of their own, independent of the rotation of
the earth on its axis. Thus a great and most important distinction is
disclosed to us. §
(145.) To establish these facts, almost no apparatus is required. An
observer need only station himself to the north of some well-defined ver-
tical object, as the angle of a building, and, placing his eye exactly at a
certain fixed point (such as a small hole in a plate of metal nailed to some
immoveable support), notice the successive disappearances of any star be-
SIDEREAL AND SOLAR TIME. 85
hind the building, by a watch." When he observes the sun, he must
shade his eye with a dark-coloured or smoked glass, and notice the moments
when its western and eastern edges successively come up to the wall, from
which, by taking half the interval, he will ascertain (what he cannot di-
rectly observe) the moment of disappearance of its centre.
(146.) When, in pursuing and establishing this general fact, we are
led to attend more nicely to the times of the daily arrival of the sun on
the meridian, irregularities (such they first seem to be) begin to make their
appearance. The intervals between two successive arrivals are not the same
at all times of the year. They are sometimes greater, sometimes less, than
24 hours, as shown by the clock; that is to say, the solar day is not always
of the same length. About the 21st of December, for example, it is half
a minute longer, and about the same day of September nearly as much
shorter, than its average duration. And thus a distinction is again press-
ed upon our notice between the actual solar day, which is never two days
in succession alike, and the mean Solar day of 24 hours, which is an ave-
rage of all the Solar days throughout the year. Here, then, a new source
of inquiry opens to us. The Sun’s apparent motion is not only not the
same with the stars, but it is not (as the latter is) uniform. It is subject
to fluctuations, whose laws become matter of investigation. But to pur-
sue these laws, we require micer means of observation than what we have
described, and are obliged to call in to our aid an instrument called the
transit instrument, especially destined for such observations, and to attend
minutely to all the causes of irregularity in the going of clocks and watches
which may affect our reckoning of time. Thus we become involved by
degrees in more and more delicate instrumental inquiries; and we speed-
ily find that, in proportion as we ascertain the amount and law of one
great or leading fluctuation, or inequality, as it is called, of the sun’s
diurnal motion, we bring into view others continually smaller and smaller,
which were before obscured, or mixed up with errors of observation and
instrumental imperfections. In short, we may not inaptly compare the
mean length of the solar day to the mean or average height of water in
a harbour, or the general level of the sea unagitated by tide or waves.
The great annual fluctuation above noticed may be compared to the daily
* This is an excellent practical method of ascertaining the rate of a clock or watch,
being exceeding accurate if a few precautions are attended to ; the chief of which is,
to take care that that part of the edge behind which the star (a bright one, not a planet)
disappears shall be quite smooth; as otherwise variable refraction may transfer the
point of disappearance from a protuberance to a notch, and thus vary the moment of
observation unduly. This is easily secured, by nailing up a smooth-edged board.
The verticality of its edge should be insured by the use of a plumb-line.
86 OuTLINES OF ASTRONOMY.
variations of level produced by the tides, which are nothing but enormous
waves extending over the whole ocean, while the smaller subordinate ine-
qualities may be assimilated to waves ordinarily so called, on which, when
large, we perceive lesser undulations to ride, and on these, again, minuter
ripplings, to the series of whose subordination we can perceive no end.
(147.) With the causes of these irregularities in the solar motion we
have no concern at present; their explanation belongs to a more advanced
part of our subject; but the distinction between the solar and sidereal
days, as it pervades every part of astronomy, requires to be early intro-
duced, and never lost sight of. It is, as already observed, the mean or
average length of the solar day, which is used in the civil reckoning of
time. It commences at midnight, but astronomers, even when they use .
mean solar time, depart from the civil reckoning, commencing their day
at noon, and reckoning the hours from 0 round to 24. Thus, 11 o'clock
in the forenoon of the second of January, in the civil reckoning of time,
corresponds to January 1 day 23 hours in the astronomical reckoning;
and one o’clock in the afternoon of the former, to January 2 days 1 hour
of the latter reckoning. This usage has its advantages and disadvantages,
but the latter seem to preponderate; and it would be well if, in conse-
quence, it could be broken through, and the civil reckoning substituted.
Uniformity in nomenclature and modes of reckoning in all matters relat-
ing to time, space, weight, measure, &c., is of such vast and paramount
importance in every relation of life as to outweigh every consideration of
technical convenience or custom."
(148.) Both astronomers and civilians, however, who inhabit different
points of the earth's surface, differ from each other in their reckoning of
time; as it is obvious they must, if we consider that, when it is noon at
one place, it is midnight at a place diametrically opposite; sunrise at
another; and sunset, again, at a fourth. Hence arises considerable in-
convenience, especially as respects places differing very widely in situation,
and which may even in some critical cases involve the mistake of a whole
day. To obviate this inconvenience, there has lately been introduced a
system of reckoning time by mean solar days and parts of a day counted
from a fixed instant, common to all the world, and determined by no local
* The only disadvantage to astronomers of using the civil reckoning is this- that
their observations being chiefly carried on during the night, the day of their date will,
in this reckoning, always have to be changed at midnight, and the former and latter
portion of every night's observations will belong to two differently numbered civil days
of the month. There is no denying this to be an inconvenience. Habit, however,
would alleviate it; and some inconveniences must be cheerfully submitted to by all who
resolve to act on general principles. All other classes of men, whose occupation ex-
extends to the night as well as day, submit to it, and find their advantage in doing so.
MEASUREMENT OF TIME. 87
circumstance, such as noon or midnight, but by the motion of the sun
among the stars. Time, so reckoned, is called equinoctial time; and is
numerically the same, at the same instant, in every part of the globe.
Its origin will be explained more fully at a more advanced stage of our
work.
(149.) Time is an essential element in astronomical observation, in a
twofold point of view:— 1st, As the representative of angular motion.
The earth's diurnal motion being uniform, every star describes its diurnal
circle uniformly ; and the time elapsing between the passage of the stars
in succession across the meridian of any observer becomes, therefore, a
direct measure of their differences of right ascension. 2dly, As the
fundamental element (or natural independent variable, to use the lan-
guage of geometers) in all dynamical theories. The great object of as-
tronomy is the determination of the laws of the celestial motions, and
their reference to their proximate or remote causes. Now, the statement
of the law of any observed motion in a celestial object can be no other
than a proposition declaring what has been, is, and will be, the real or
apparent situation of that object at any time, past, present, or future. To
compare such laws, therefore, with observation, we must possess a register
of the observed situations of the object in question, and of the times when
they were observed.
(150.) The measurement of time is performed by clocks, chronometers,
clepsydras, and hour-glasses. The two former are alone used in modern
astronomy. The hour-glass is a coarse and rude contrivance for measur-
ing, or rather counting out, fixed portions of time, and is entirely disused.
The clepsydra, which measured time by the gradual emptying of a large
vessel of water through a determinate orifice, is susceptible of considera-
ble exactness, and was the only dependence of astronomers before the
invention of clocks and watches. At present it is abandoned, owing to
the greater convenience and exactness of the latter instruments. In one
case only has the revival of its use been proposed; viz. for the accurate
measurement of very small portions of time, by the flowing out of mer-
cury from a small orifice in the bottom of a vessel, kept constantly full
to a fixed height. The stream is intercepted at the moment of noting
any event, and directed aside into a receiver, into which it continues to
run, till the moment of noting any other event, when the intercepting
cause is suddenly removed, the stream flows in its original course, and
ceases to run into the receiver. The weight of mercury received, com.
pared with the weight received in an interval of time observed by the
clock, gives the interval between the events observed. This ingenious
and simple method of resolving, with all possible precision, a problem
88 OUTLINES OF ASTRONOMY.
of much importance in many physical inquiries, is due to the late Captain
Kater.
(151.) The pendulum clock, however, and the balance watch, with
those improvements and refinements in its structure which constitute it
emphatically a chronometer,' are the instruments on which the astronomer
depends for his knowledge of the lapse of time. These instruments are
now brought to such perfection, that an habitual irregularity in the rate
of going, to the extent of a single second in twenty-four hours in two
consecutive days, is not tolerated in one of good character; so that any
interval of time less than twenty-four hours may be certainly ascertained
within a few tenths of a second, by their use. In proportion as intervals
are longer, the risk of error, as well as the amount of error risked, be-
comes greater, because the abcidental errors of many days may accumu-
late; and causes producing a slow progressive change in the rate of going
may subsist unperceived. It is not safe, therefore, to trust the determi-
nation of time to clocks, or watches, for many days in succession, without
checking them, and ascertaining their errors by reference to natural events
which we know to happen, day after day, at equal intervals. But if this
be done, the longest intervals may be fixed with the same precision as the
shortest; since, in fact, it is then only the times intervening between the
first and the last moments of such long intervals, and such of those
periodically recurring events adopted for our points of reckoning, as occur
within twenty-four hours respectively of either, that we measure by arti-
ficial means. The whole days are counted out for us by nature; the frac-
tional parts only, at either end, are measured by our clocks. To keep the
reckoning of the integer days correct, so that none shall be lost or counted
twice, is the object of the calendar. Chronology marks out the order of
succession of events, and refers them to their proper years and days;
while chronometry, grounding its determinations on the precise observa-
tion of such regularly periodical events as can be conveniently and exactly
subdivided, enables us to fix the moments in which phenomena occur, with
the last degree of precision.
(152.) In the culmination or transit (i. e. the passage across the
meridian of an observer,) of every star in the heavens, he is furnished
with such a regularly periodical natural event as we allude to. Accord-
ingly, it is to the transits of the brightest and most conveniently situated
fixed stars that astronomers resort to ascertain their exact time, or, which
comes to the same thing, to determine the exact amount of error of their
clocks.
(153.) Before we describe the instrument destined for the purpose of
*Xpovos, time; perpetv, to measure.
PRINCIPLE OF TELESCOPIC, SIGHTS. 89
observing such culminations, however, or those intended for the measure-
ment of angular intervals in the sphere, it is requisite to place clearly
before the reader the principle on which the telescope is applied in astro-
nomy to the precise determination of a direction in space, — that, namely,
of the visual ray by which we see a star or any other distant object.
(154.) The telescope most commonly used in astronomy for these pur-
poses is the refracting telescope, which consists of an object-glass (either
single, or as is now almost universal, double, forming what is called in
optics, an acromatic combination) A.; a tube A B, into which the brass
cell of the object-glass is firmly screwed, and an eye-lens C, for which is
Fig. 14.
A- IB ID
E (W Jº —T-H
r–F–Ü T_F:- Q
JE

often substituted a combination of glasses designed to increase the magni-
fying power of the telescope, or otherwise give more distinctness of vision
according to optical principles which we have no occasion here to refer to.
This also is fitted into a cell, which is screwed firmly into the end B of
the tube, so that object-glass, tube, and eye-glass may be considered as
forming one piece, invariable in the relative position of its parts.
(155) The line PQ joining the centres of the object and eye-glasses
and produced, is called the aaºis, or line of collimation of the telescope.
And it is evident, that the situation of this line holds a fixed relation to
the tube and its appendages, so long as the object and eye-glasses maintain
their fixity in this respect.
(156.) Whatever distant object E, this line is directed to, an inverted
picture or image of that object F is formed (according to the principles of
optics), in the focus of the object-glass, and may there be viewed as if it
were a real object, through the eye-lens C, which (if of short focus) ena-
bles us to magnify it just as such a lens would magnify a material object
in the same place.
(157.) Now as this image is formed and viewed in the air, being itself
immaterial and impalpable—nothing prevents our placing in that very
place F in the axis of the telescope, a real, substantial object of very
definite form and delicate make, such as a fine metallic point, as of a
needle — or better still, a cross formed by two very fine threads (spider-
lines), thin metallic wires, or lines drawn on glass intersecting each other
at right angles—and whose intersection is all but a mathematical point.
If such a point, wire, or cross be carefully placed and firmly fixed in the
Nº.
90 OUTLINES OF ASTRONOMY.
exact focus F, both of the object and eye-glass, it will be seen through the
latter at the same time, and occupying the same precise place as the image
of the distant star E. The magnifying power of the lens renders percep-
tible the smallest deviation from perfect coincidence, which, should it exist,
is a proof, that the axis QP is not directed rigorously towards E. In that
case, a fine motion (by means of a screw duly applied), communicated to
the telescope, will be necessary to vary the direction of the axis till the
coincidence is rendered perfect. So precise is this mode of pointing found
in practice, that the axis of a telescope may be directed towards a star
or other definite celestial object without an error of more than a few tenths
of a second of angular measure.
(158.) This application of the telescope may be considered as completely
annihilating that part of the error of observation which might otherwise
arise from an erroneous estimation of the direction in which an object lies
from the observer's eye, or from the centre of the instrument. It is, in fact,
the grand source of all the precision of modern astronomy, without which all
other refinements in instrumental workmanship would be thrown away;
the errors capable of being committed in pointing to an object, without
such assistance, being far greater than what could arise from any but the
very coarsest graduation." In fact, the telescope thus applied becomes,
with respect to angular, what the microscope is with respect to linear
dimension. By concentrating attention on its smallest parts, and magni-
fying into palpable intervals the minutest differences, it enables us not
only to scrutinise the form and structure of the objects to which it is
* The honour of this capital improvement has been successfully vindicated by Der-
ham (Phil. Trans. xxx. 603) to our young, talented, and unfortunate countryman Gas-
coigne, from his correspondence with Crabtree and Horrockes, in his (Derham’s)
possession. The passages cited by Derham from these letters leave no doubt that, so
early as 1640, Gascoigne had applied telescopes to his quadrants and sextants, with
threads in the common focus of the glasses ; and had even carried the invention so far
as to illuminate the field of view by artificial light, which he found “very helpful when
the moon appeareth not, or it is not otherwise light enough.” These inventions were
freely communicated by him to Crabtree, and through him to his friend Horrockes, the
pride and boast of British astronomy; both of whom expressed their unbounded admi-
ration of this and many other of his delicate and admirable improvements in the art
of observation. Gascoigne, however, perished, at the age of twenty-three, at the
battle of Marston Moor; and the premature and sudden death of Horrockes, at a yet
earlier age, will account for the temporary oblivion of the invention. It was revived,
or re-invented, in 1667, by Picard and Auzout (Lalande, Astron. 2310), after which its
use became universal. Morin, even earlier than Gascoigne (in 1635), had proposed to
substitute the telescope for plain sights; but it is the thread or wire stretched in the
focus with which the image of a star can be brought to exact coincidence, which
gives the telescope its advantage in practice; and the idea of this does not seem to have
occurred to Morin. See Lalande, wbri suprā.
THE TRANSIT INSTRUMENT. 91
pointed, but to refer their apparent places, with all but geometrical pre-
cision, to the parts of any scale with which we propose to compare them.
(159.) We now return to our subject, the determination of time by the
transits or culminations of celestial objects. The instrument with which
such culminations are observed is called a transit instrument. It consists
of a telescope firmly fastened on a horizontal axis directed to the east and
west points of the horizon, or at right angles to the plane of the meridian
of the place of observation. The extremities of the axis are formed into
cylindrical pivots of exactly equal diameters, which rest in notches formed
in metallic supports, bedded (in the case of large instruments) on strong
pieces of stone, and susceptible of nice adjustment by screws, both in a
vertical and horizontal direction. By the former adjustment, the axis can
Fig. 15.
be rendered preciscly horizontal, by levelling it with a level made to rest
on the pivots. By the latter adjustment the axis is brought precisely into
the east and west directions, the criterion of which is furnished by the
observations themselves made with the instrument, in a manner presently
to be explained, or by a well-defined object, called a meridian-mark, origi-
mally determined by such observations, and then, for convenience of ready
reference, permanently established, at a great distance, exactly in a meri-
dian line passing through the central point of the whole instrument. It
is evident, from this description, that, if the axis, or line of collimation
of the telescope be once well adjusted at right angles to the axis of the
transit, it will never quit the plane of the meridian, when the instrument
is turned round on its axis of rotation. ---
(160.) In the focus of the eye-piece, and at right angles to the length
of the telescope, is placed, not a single cross, as in our general explanation
in art. 157, but a system of one horizontal and several equidistant vertical
threads or wires, (five or seven are more usually employed,) as represented
in the annexed figure, which always appear in the field of view, when
properly illuminated, by day by the light of the sky, by night by that of
a lamp introduced by a contrivance not necessary here to explain. The

92 OUTLINES OF ASTRONOMY.

\|
place of this system of wires may be altered by adjusting screws, giving
it a lateral (horizontal) motion; and it is by this means brought to such
a position, that the middle one of the vertical wires shall intersect the line
of collimation of the telescope, where it is arrested and permanently
fastened." In this situation it is evident that the middle thread will be a
visible representation of that portion of the celestial meridian to which
the telescope is pointed; and when a star is seen to cross this wire in
the telescope, it is in the act of culminating, or passing the celestial meri-
dian. The instant of this event is noted by the clock or chronometer,
which forms an indispensable accompaniment of the transit instrument.
For greater precision, the moment of its crossing all the vertical threads
is noted, and a mean taken, which (since the threads are equidistant)
would give exactly the same result, were all the observations perfect, and
will, of course, tend to subdivide and destroy their errors in an average of
the whole in the contrary case. -
(161.) For the mode of executing the adjustments, and allowing for
the errors unavoidable in the use of this simple and elegant instrument,
the reader must consult works especially devoted to this department of
practical astronomy.” We shall here only mention one important verifica-
tion of its correctness, which consists in reversing the ends of the axis, or
turning it east for west. If this be done, and it continue to give the
same results, and intersect the same point on the meridian mark, we may
be sure that the line of collimation of the telescope is truly at right angles
to the axis, and describes strictly a plane, i. e. marks out in the heavens
a great circle. In good transit observations, an error of two or three
tenths of a second of time in the moment of a star's culmination is the
utmost which need be apprehended, exclusive of the error of the clock: in
* There is no way of bringing the true optic axis of the object-glass to coincide
exactly with the line of collimation, but, so long as the object-glass does not shift or
shake in its cell, any line holding an invariable position with respect to that aris, may
pe taken for the conventional or astronomical axis with equal effect.
* See Dr. Pearson’s Treatise on Practical Astronomy. Also Bianchi Sopra lo
Stromente de’ Passagi. Ephem. di Milano, 1824.
MEASUREMENT OF ANGUIAR, INTERVALS. 93
other words, a clock may be compared with the earth's diurnal motio 1 by
a single observation, without risk of greater error. By multiplying
observations, of course, a yet greater degree of precision may be obtained.
(162.) The plane described by the line of collimation of a transit ought
to be that of the meridian of the place of observation. To ascertain
whether it is so or not, celestial observation must be resorted to. Now, as
the meridian is a great circle passing through the pole, it necessarily
bisects the diurnal circles described by all the stars, all which describe the
two semicircles so arising in equal intervals of 12 sidereal hours each.
Hence, if we choose a star whose whole diurnal circle is above the horizon,
or which never sets, and observe the moments of its upper and lower tram-
sits across the middle wire of the telescope, if we find the two semidiurnal
portions east and west of the plane described by the telescope to be
described in precisely equal times, we may be sure that plane is the meridian.
(163.) The angular intervals measured by means of the transit instru-
ment and clock are arcs of the equinoctial, intercepted between circles of
declination passing through the objects observed; and their measurement,
in this case, is performed by no artificial graduation of circles, but by the
help of the earth's diurnal motion, which carries equal arcs of the equi-
noctial across the meridian, in equal times, at the rate of 15° per sidereal
hour. In all other cases, when we would measure angular intervals, it is
necessary to have recourse to circles, or portions of circles, constructed of
metal or other firm and durable material, and mechanically subdivided into
equal parts, such as degrees, minutes, &c. The simplest and most obvious
mode in which the measurement of the angular interval between two
directions in space can be performed is as follows. Let A B C D be a
circle, divided into 360 degrees, (numbered in order from any point 0° in
the circumference, round to the same point again,) and connected with its
centre by spokes or rays, a, y, z, firmly united to its circumference or

94 OUTLINES OF ASTRONOMY.
limb At the centre let a circular hole be pierced, in which shall move
a pivot exactly fitting it, garrying a tube, whose axis, a b, is exactly
parallel to the plane of the circle, or perpendicular to the pivot; and also
two arms, m, n, at right angles to it, and forming one piece with the tube
and the axis; so that the motion of the axis on the centre shall carry the
tube and arms smoothly round the circle, to be arrested and fixed at any
point we please, by a contrivance called a clamp. Suppose, now, we
would measure the angular interval between two fixed objects, S, T. The
plane of the circle must first be adjusted so as to pass through them both,
and immoveably fixed and maintained in that position. This done, let the
awis a b of the tube be directed to one of them, S, and clamped. Then
will a mark on the arm m point either exactly to some one of the divisions
on the limb, or between two of them adjacent. In the former case, the
division must be noted as the reading of the arm m. In the latter, the
fractional part of one whole interval between the consecutive divisions by
which the mark on m surpasses the last inferior division must be estimated
or measured by some mechanical or optical means. (See art. 165.) The
division and fractional part thus noted, and reduced into degrees, minutes,
and seconds, is to be set down as the reading of the limb corresponding
to that position of the tube a b, where it points to the object S. The
same must then be dome for the object T; the tube pointed to it, and the
limb “read off,” the position of the circle remaining meanwhile unaltered.
It is manifest, then, that, if the lesser of these readings be subtracted
from the greater, their difference will be the angular interval between S
and T, as seen from the centre of the circle, at whatever point of the limb
the commencement of the graduations or the point 0° be situated.
(164.) The very same result will be obtained, if, instead of making the
Fig. 18.
tube moveable upon the circle, we connect it invariably with the latter,
and make both revolve together on an axis concentric with the circle, and
forming one piece with it, working in a hollow formed to receive and fit
It in some fixed support. Such a combination is represented in section in
the annexed sketch. T is the tube or sight, fastened, at pp, on the circle
AB, whose axis, D, works in the solid metallic centring E, from which

METHODS OF READING OFF. 95
originates an arm, F, carrying at its extremity an index, or other proper
mark, to point out and read off the exact division of the circle at B, the
point close to it. It is evident that, as the telescope and circle revolve
through any angle, the part of the limb of the latter, which by such
revolution is carried past the index F, will measure the angle described.
This is the most usual mode of applying divided circles in astronomy.
(165.) The index F may either be a simple pointer, like a clock hand
(fig. a); or a vernier (fig. b); or, lastly, a compound microscope (fig.
c), represented in section in fig, d, and furnished with a cross in the com-
mon focus of its object and eye-glass, moveable by a fine-threaded screw,
by which the intersection of the cross may be brought to exact coincidence
with the image of the nearest of the divisions of the circle formed in the
focus of the object lens upon the very same principle with that explained,
art. 157 for the pointing of the telescope, only that here the fiducial cross
is made moveable; and by the turns and parts of a turn of the screw
required for this purpose the distance of that division from the original or
zero point of the microscope may be estimated. This simple but delicate
contrivance gives to the reading off of a circle a degree of accuracy only
limited by the power of the microscope, and the perfection with which a
screw can be executed, and places the subdivision of angles on the same
footing of optical certainty which is introduced into their measurement by
the use of the telescope.
(166.) The exactness of the result thus obtained must depend, 1st, on
the precision with which the tube a b can be pointed to the objects; 2dly,
on the accuracy of graduation of the limb; 3dly, on the accuracy with
which the subdivision of the intervals between any two consecutive gradu-
ations can be performed. The mode of accomplishing the latter object
with any required exactness has been explained in the last article. With
regard to the graduation of the limb, being merely of a mechanical nature,

96 OUTLINES OF ASTRONOMY.
we shall pass it without remark, further than this, that, in the present
state of instrument-making, the amount of error from this source of inac-
curacy is reduced within very narrow limits indeed." With regard to the
first, it must be obvious that, if the sights a b be nothing more than
simple crosses, or pin-holes at the ends of a hollow tube, or an eye-hole
at one end, and a cross at the other, no greater nicety in pointing can be
expected than what simple vision with the naked eye can command. But
if, in place of these simple but coarse contrivances, the tube itself be con-
verted into a telescope, having an object-glass at b, an eye-piece at a, and
a fiducial cross in their common focus, as explained in art. 157; and if
the motion of the tube on the limb of the circle be arrested when the
object is brought just into coincidence with the intersectional point of
that cross, it is evident that a greater degree of exactness may be attained
in the pointing of the tube than by the unassisted eye, in proportion to
the magnifying power and distinctness of the telescope used.
(167.) The simplest mode in which the measurement of an angular in-
terval can be executed, is what we have just described; but, in strictness,
this mode is applicable only to terrestrial angles, such as those occupied
on the Sensible horizon by the objects which surround our station, —be-
cause these only remain stationary during the interval while the telescope
is shifted on the limb from one object to the other. But the diurnal
motion of the heavens, by destroying this essential condition, renders the
direct measurement of angular distance from object to object by this means
impossible. The same objection, however, does not apply if we seek only
to determine the interval between the diurnal circles described by any
two celestial objects. Suppose every star, in its diurnal revolution, were
to leave behind it a visible trace in the heavens, – a fine line of light, for
instance, — then a telescope once pointed to a star, so as to have its image
brought to coincidence with the intersection of the wires, would constantly
remain pointed to some portion or other of this line, which would there-
fore continue to appear in its field as a luminous line, permanently inter-
secting the same point, till the star came round again. From one such
line to another the telescope might be shifted, at leisure, without error;
and then the angular interval between the two diurnal circles, in the plane
of the telescope’s rotation, might be measured. Now, though we cannot
see the path of a star in the heavens, we can wait till the star itself crosses
the field of view, and seize the moment of its passage to place the inter-
section of its wires so that the star shall traverse it; by which, when the
* In the great Ertel circle at Pulkova, the probable amount of the accidental error
of division is stated by M. Struve not to exceed 0"-264. Desc. de l'Obs, centrale de
Pulkova, p. 147.
OF THE MURAL CIRCLE. 97
telescope is well clamped, we equally well secure the position of its diurnal
circle as if we continued to see it ever so long. The reading off of the
limb may then be performed at leisure; and when another star comes
round into the plane of the circle, we may unclamp the telescope, and a
similar observation will enable us to assign the place of its diurnal circle
on the limb : and the observations may be repeated alternately, every day,
as the stars pass, till we are satisfied with their result.
(168.) This is the principle of the mural circle, which is nothing more
than such a circle as we have described in art. 163, firmly supported, in
the plane of the meridian, on a long and powerful horizontal axis. This
axis is let into a massive pier, or wall, of stone (whence the name of the
instrument), and so secured by screws as to be capable of adjustment
both in a vertical and horizontal direction; so that, like the axis of the
transit, it can be maintained in the exact direction of the east and west
points of the horizon, the plane of the circle being consequently truly
meridional.
(169.) The meridian, being at right angles to all the diurnal circles
described by the stars, its arc intercepted between any two of them will
measure the least distance between these circles, and will be equal to the
difference of the declinations, as also to the difference of the meridian
altitudes of the objects—at least when corrected for refraction. These
differences, then, are the angular intervals directly measured by the mural
circle. But from these, supposing the law and amount of refraction
known, it is easy to conclude, not their differences only, but the quantities
themselves, as we shall now explain.
(170.) The declination of a heavenly body is the complement of its
distance from the pole. The pole, being a point in the meridian, might
be directly observed on the limb of the circle, if any star stood exactly
therein; and thence the polar distances, and, of course, the declinations
of all the rest, might be at once determined. But this not being the
case, a bright star as near the pole as can be found is selected, and observed
in its upper and lower culminations; that is, when it passes the meridian
above and below the pole. Now, as its distance from the pole remains
the same, the difference of reading off the circle in the two cases is, of
course (when corrected for refraction), equal to twice the polar distance
of the star; the arc intercepted on the limb of the circle being, in this
case, equal to the angular diameter of the star's diurnal circle. In the
annexed diagram, H PO represents the celestial meridian, P the pole,
BR, A Q, CD the diurnal circles of stars which arrive on the meridian at
B, A, and C in their upper and at R, Q, D in their lower culminations,
of which D and Q happen above the horizon H. O. P is the pole; and if
7
98 Olu TLINES OF ASTRONOMY.
IR
/
we suppose hpo to be the mural circle, having S for its centre, b &cpd
will be the points on its circumference corresponding to B A CPD in the
heavens. Now the arcs ba, b c, b d, and cal are given immediately by ob-
servation; and since C P = PD, we have also cf. = p d, and each of them
= }c d, consequently the place of the polar point, as it is called, upon
the limb of the circle becomes known, and the arcs p b, p a, p c, which
represent on the circle the polar distances required, become also known.
(171.) The situation of the pole star, which is a very brilliant one, is
eminently favourable for this purpose, being only about a degree and an half
from the pole; it is, therefore, the star usually and almost solely chosen
for this important purpose; the more especially because, both its culmina-
tions taking place at great and not very different altitudes, the refractions
by which they are affected are of small amount, and differ but slightly
from each other, so that their correction is easily and safely applied. The
brightness of the pole star, too, allows it to be easily observed in the day-
time. In consequence of these peculiarities, this star is one of constant
resort with astronomers for the adjustment and verification of instruments
of almost every description. In the case of the transit, for instance, it
furnishes an excellent object for the application of the method of testing
the meridional situation of the instrument described in art. 162, in fact,
the most advantageous of any for that purpose, owing to its being the
most remote from the zenith, at its upper culmination, of all bright stars
observable both above and below the pole.
(172.) The place of the polar point on the limb of the mural circle
once determined, becomes an origin, or zero point, from which the polar
distances of all objects, referred to other points on the same limb, reckon.
It matters not whether the actual commencement 0° of the graduations
stand there, or not; sinee it is only by the differences of the readings
that the arcs on the limb are determined; and hence a great advantage is

OF THE MERIDIAN CIRCLE. 99
obtained in the power of commencing anew a fresh series of observations,
in which a different part of the circumference of the circle shall be
employed, and different graduations brought into use, by which inequalities
of division may be detected and neutralized. This is accomplished prac-
tically by detaching the telescope from its old bearings on the circle, and
fixing it afresh, by screws or clamps, on a different part of the circum-
ference.
(173.) A point on the limb of the mural circle, not less important
than the polar point, is the horizontal point, which, being once known,
becomes in like manner an origin, or zero point, from which altitudes are
reckoned. The principle of its determination is ultimately nearly the
same with that of the polar point. As no star exists in the celestial
horizon, the observer must seek to determine two points on the limb,
the one of which shall be precisely as far below the horizontal point as
the other is above it. For this purpose, a star is observed at its culmina-
nation on one night, by pointing the telescope directly to it, and the next,
by pointing to the image of the same star reflected in the still, unruffled
surface of a fluid at perfect rest. Mercury, as the most reflective fluid'
known, is generally chosen for that use. As the surface of a fluid at rest
is necessarily horizontal, and as the angle of reflection, by the laws of optics,
is equal to that of incidence, this image will be just as much depressed
below the horizon as a star itself is above it (allowing for the difference
of refraction at the moment of observation). The arc intercepted on the
limb of the circle between the star and its reflected image thus consecu-
tively observed, when corrected for refraction, is the double altitude of
the star, and its point of bisection the horizontal point. The reflecting
surface of a fluid so used for the determination of the altitudes of objects
is called an artificial horizon.'
(174.) The mural circle is, in fact, at the same time, a transit instru-
ment; and, if furnished with a proper system of vertical wires in the
focus of its telescope, may be used as such. As the axis, however, is
only supported at one end, it has not the strength and permanence neces.
sary for the more delicate purposes of a transit; nor can it be verified, as
a transit may, by the reversal of the two ends of its axis, east or west.
* By a peculiar and delicate manipulation and management of the setting bisection
and reading off of the circle, aided by the use of a moveable horizontal micrometic
wire in the focus of the object-glass, it is found practicable to observe a slow moving -
star (as the pole star) on one and the same night, both by reflection and direct vision, * - .
sufficiently near to either culmination to give the horizontal point, without .isking the : > -
change of refraction in wenty-four hours; so that this source of error is thus com. &
pletely eliminated.
100 OUTLINES OF ASTRONOMY.
&
Nothing, however, prevents a divided circle being permanently fastened
on the axis of a transit instrument, either near to one of its extremities,
or close to the telescope, so as to revolve with it, the reading off being
performed by one or more microscopes fixed on one of its piers. Such an
instrument is called a TRANSIT CIRCLE, or a MERIDIAN CIRCLE, and serves
for the simultaneous determination of the right ascensions and polar dis-
tances of objects observed with it; the time of transit being noted by the
clock, and the circle being read off by the lateral microscopes. There is
much advantage, when extensive catalogues of small stars have to be
formed, in this simultaneous determination of both their celestial co-ordi-
nates: to which may be added the facility of applying to the meridian
circle a telescope of any length and optical power. The construction of
the mural circle renders this highly inconvenient, and indeed impracticable
beyond very moderate limits.
(175.) The determination of the horizontal point on the limb of an
instrument is of such essential importance in astronomy, that the student
should be made acquainted with every means employed for this purpose.
These are, the artificial horizon, the plumb-line, the level, and the colli-
mator. The artificial horizon has been already explained. The plumb-
line is a fine thread or wire, to which is suspended a weight, whose oscil-
lations are impeded and quickly reduced to rest by plunging it in water.
The direction ultimately assumed by such a line, admitting its perfect
flexibility, is that of gravity, or perpendicular to the surface of still
water. Its application to the purposes of astronomy is, however, so deli-
cate, and difficult, and liable to error, unless extraordinary precautions are
taken in its use, that it is at present almost universally abandoned, for
the more convenient, and equally exact instrument the level.
(176.) The level is a glass tube nearly filled with a liquid, (spirit of
wine, or sulphuric ether, being thus now generally used, on account of
Fig. 21.
| 11111111 ill 111111 ill, it ill, it
tº .. their extreme mobility, and not being liable to freeze) the bubble in
. . ." which, when the tube is placed horizontally, would rest indifferently in
any part if the tube could be mathematically straight. But that being


DETERMINATION OF THE HORIZONTAL POINT. 101
impossible to execute, and every tube having some slight curvature; if
the convex side be placed upwards the bubble will occupy the higher part,
as in the figure (where the curvature is purposely exaggerated). Suppose
such a tube, as A B, firmly fastened on a straight bar, C D, and marked
at a b, two points distant by the length of the bubble; then, if the
instrument be so placed that the bubble shall occupy this interval, it is
clear that C D can have no other than one definite inclination to the hori-
zon; because, were it ever so little moved one way or other, the bubble
would shift its place, and run towards the elevated side. Suppose, now,
that we would ascertain whether any given line PQ be horizontal; let
the base of the level C D be set upon it, and note the points a b, between
which the bubble is exactly contained; then turn the level end for end,
so that C shall rest on Q, and D on P. If then the bubble continue to
occupy the same place between a and b, it is evident that P Q can be no
otherwise than horizontal. If not, the side towards which the bubble
runs is highest, and must be lowered. Astronomical levels are furnished
with a divided scale, by which the places of the ends of the bubble can
be nicely marked; and it is said that they can be executed with such
delicacy, as to indicate a single second of angular deviation from exact
horizontality. In such levels accident is not trusted to to give the requi-
site curvature. They are ground and polished internally by peculiar
mechanical processes of great delicacy.
(177.) The mode in which a level may be applied to find the horizontal
point on the limb of a vertical divided circle may be thus explained; let
A B be a telescope firmly fixed to such a circle, DEF, and moveable in
one with it on a horizontal axis C, which must be like that of a transit,
susceptible of reversal (see art. 161) and with which the circle is insep-
arably connected. Direct the telescope on some distant well-defined object
S, and bisect it by its horizontal wire, and in this position clamp it fast.
Let L be a level fastened at right angles to an arm, L E F, furnished with
a microscope, or vernier at F, and, if we please, another at E. Let this
arm be fitted by grinding on the axis C, but capable of moving smoothly
on it without carrying it round, and also of being clamped fast on it, so as
to prevent it from moving until required. While the telescope is kept
fixed on the object S, let the level be set so as to bring its bubble to the
marks a b, and clamp it there. Then will the arm L C F have some cer-
tain determinate inclination (no matter what) to the horizon. In this
position let the circle be read off at F, and then let the whole apparatus
be reversed by turning its horizontal axis end for end, without unclamping
the level arm from the axis. This done, by the motion of the whole in-
strument (level and all) on its axis, restore the level to its horizontal posi-
- t
102 OUTLINES OF ASTRONOMY.
tion with the bubble at a b. Then we are sure that the telescope has
now the same inclination to the horizon the other way, that it had when
pointed to S, and the reading off at F will not have been changed. Now
unclamp the level, and keeping it nearly horizontal, turn round the circle
on the axis, so as to carry back the telescope through the zenith to S, and
in that position clamp the circle and telescope fast. Then it is evident
that an angle equal to twice the zenith distance of S has been moved over
by the axis of the telescope from its last position. Lastly, without un-
clamping the telescope and circle, let the level be once more rectified.
Then will the arm L E F once more assume the same definite position
with respect to the horizon; and, consequently, if the circle be again read
off, the difference between this and the previous reading must measure the
arc of its circumference which has passed under the point F, which may
be considered as having all the while retained an invariable position.
This difference, then, will be the double zenith distance of S, and its half
will be the zenith distance simply, the complement of which is its altitude.
Thus the altitude corresponding to a given reading of the limb becomes
known, or, in other words, the horizontal point on the limb is ascertained.
Circuitous as this process may appear, there is no other mode of employ-
ing the level for this purpose which does not in the end come to the same
thing. Most commonly, however, the level is used as a mere fiducial re-
ference, to preserve a horizontal point once well determined by other
means, which is done by adjusting it so as to stand level when the tele-
scope is truly horizontal, and thus leaving it, depending on the permanence
of its adjustment.
(178.) The last, but probably not the least exact, as it certainly is, in

DETERMINATION OF THE HORIZONTAL POINT. 103
innumerable cases, the most convenient means of ascertaining the horizon-
tal point, is that afforded by the floating collimator, an invention of Cap-
tain Kater, but of which the optical principle was first employed by Rit-
tenhouse, in 1785, for the purpose of fixing a definite direction in space
by the emergence of parallel rays from a material object placed in the
focus of a fixed lens. This elegant instrument is nothing more than a
small telescope furnished with a cross-wire in its focus, and fastened hori-
zontally, or as nearly so as may be, on a flat iron float, which is made to
swim on mercury, and which, of course, will, when left to itself, assume
always one and the same invariable inclination to the horizon. If the
cross-wires of the collimator be illuminated by a lamp, being in the focus
of its object-glass, the rays from them will issue parallel, and will there-
fore be in a fit state to be brought to a focus by the object-glass of any
other telescope, in which they will form an image as if they came from
a celestial object in their direction, i. e. at an altitude equal to their decli-
nation. Thus the intersection of the cross of the collimator may be ob-
served as if it were a star, and that, however near the two telescopes are
to each other. By transferring then, the collimator still floating on a ves-
sel of mercury from the one side to the other of a circle, we are furnished
with two quasi-celestial objects, at precisely equal altitudes, on opposite
sides of the centre; and if these be observed in succession with the tele-
scope of the circle, bringing its cross to bisect the image of the cross of the
collimator (for which end the wires of the latter cross are purposely set
45° inclined to the horizon), the difference of the readings on its limb
will be twice the zenith distance of either; whence, as in the last article,
the horizontal or zenith point is immediately determined. Another, and,
in many respects, preferable form of the floating collimator, in which the
telescope is vertical, and whereby the zenith point is directly ascertained,
is described in the Phil. Trans. 1828, p, 257, by the same author.
(179.) By far the neatest and most delicate application of the princi-
ple of collimation of Rittenhouse, however, is suggested by Benzenberg,
which affords at once, and by a single observation, an exact knowledge
of the nadir point of an astronomical circle. In this combination, the

"... sº 3 * * *.
...] !, ºxº-. 4 ºr
*::: ; º” * , .
104 oUTLINEs of ASTRONº.
Fig. 24.
*
* ,
telescope of the circle is its own collimator. The object observed is the ,
central intersectional cross of the wires in its own focus reflected in mere
cury. A strong illumination being thrown upon the system of wires,
(art. 160) by a lateral lamp, the telescope of the instrument is directed
vertically downwards towards the surface of the mercury, as in the figure
annexed. The rays diverging from the wires issue in parallel pencils
from the object-glass, are incident on the mercury, and are thence re-
flected back (without losing their parallel character) to the object-glass,
which is therefore enabled to collect them again in its focus. Thus is
formed a reflected image of the system of cross-wires, which, when
brought by the slow motion of the telescope to exact coincidence (inter-
section upon intersection) with the real system as seen in the eye-piece
of the instrument, indicates the precise and rigorous verticality of the
optical axis of the telescope when directed to the nadir point.
(180.) The transit and mural circle are essentially meridian instru-
ments, being used only to observe the stars at the moment of their
meridian passage. Independent of this being the most favourable mo-
ment for seeing them, it is that in which their diurnal motion is parallel
to the horizon. It is therefore easier at this time than it could be at any
other, to place the telescope exactly in their true direction; since their
apparent course in the field of view being parallel to the horizontal thread
of the system of wires therein, they may, by giving a fine motion to the
t


COMPOUND INSTRUMENTS WITH CO-ORDINATE CIRCLES. I05
telescope, be brought to exact coincidence with it, and time may be
allowed to examine and correct this coincidence, if not at first accurately
hit, which is the case in no other situation. Generally speaking, all
angular magnitudes which it is of importance to ascertain exactly, should,
if possible, be observed at their maxima or minima of increase or dimi-
nution; because at these points they remain not perceptibly changed
during a time long enough to complete, and even, in many cases, to
repeat and verify, our observations in a careful and leisurely manner.
The angle which, in the case before us, is in this predicament, is the
altitude of the star, which attains its maximum or minimum on the
meridian, and which is measured on the limb of the mural circle.
(181.) The purposes of astronomy, however, require that an observer
should possess the means of observing any object not directly on the
meridian, but at any point of its diurnal course, or wherever it may
present itself in the heavens. Now, a point in the sphere is determined
by reference to two great circles at right angles to each other; or of two
circles, one of which passes through the pole of the other. These, in
the language of geometry, are co-ordinates by which its situation is
ascertained: for instance,— on the earth, a place is known if we know
its longitude and latitude; — in the starry heavens, if we know its right
ascension and declination;–in the visible hemisphere, if we know its
azimuth and altitude, &c.
(182.) To observe an object at any point of its diurnal course, we must
possess the means of directing a telescope to it; which, therefore, must be
capable of motion in two planes at right angles to each other; and the
amount of its angular motion in each must be measured on two circles
co-ordinate to each other, whose planes must be parallel to those in which
the telescope moves. The practical accomplishment of this condition is
effected by making the axis of one of the circles penetrate that of the
other at right angles. The pierced axis turns on fixed supports, while
the other has no connection with any external support, but is sustained
entirely by that which it penetrates, which is strengthened and enlarged
at the point of penetration to receive it. The annexed figure exhibits
the simplest form of such a combination, though very far indeed from
the best in point of mechanism. The two circles are read off by ver-
niers, or microscopes; the one attached to the fixed support which
carries the principal axis, the other to an arm projecting from that axis.
Both circles also are susceptible of being clamped, the clamps being
attached to the same ultimate bearing with which the apparatus for
reading off is connected.
(183.) It is manifest that such a combination, however its principal
106 OUTLINES OF ASTRONOMY.
axis be pointed (provided that its direction be invariable,) will enable us
to ascertain the situation of any object with respect to the observer's
station, by angles reckoned upon two great circles in the visible hemi-
sphere, one of which has for its poles the prolongations of the principal
axis or the vanishing points of a system of lines parallel to it, and the
other passes always through these poles: for the former great circle is
the vanishing line of all planes parallel to the circle A B, while the
latter, in any position of the instrument, is the vanishing line of all the
planes parallel to the circle G. H.; and these two planes being, by the
construction of the instrument, at right angles, the great circles, which
are their vanishing lines, must be so too. Now, if two great circles of
a sphere be at right angles to each other, the one will always pass
through the other’s poles.
(184.) There are, however, but two positions in which such an appa-
ratus can be mounted so as to be of any practical utility in astronomy.
The first is, when the principal axis C D is parallel to the earth's axis,
and therefore points to the poles of the heavens which are the vanishing
points of all lines in this system of parallels; and when, of course, the
plane of the circle A B is parallel to the earth's equator, and therefore
has the equinoctial for its vanishing circle, and measures, by its arcs read
off, hour angles, or differences of right ascension. In this case, the great
circles in the heavens, corresponding to the various positions, which the
circle G H can be made to assume, by the rotation of the instrument
round its axis C D, are all hour-circles; and the arcs read off on this
circle will be declinations, or polar distances, or their differences.

THE EQUATORIAL INSTRUMENT. 107
(185.) In this position the apparatus assumes the name of an equato-
rial, or, as it was formerly called, a parallactic instrument. It is a most
convenient instrument for all such observations as require an object to be
kept long in view, because, being once set upon the object, it can be fol-
lowed as long as we please by a single motion, i.e. by merely turning the
whole apparatus round on its polar axis. For since, when the telescope
is set on a star, the angle between its direction and that of the polar axis is
equal to the polar distance of the star, it follows, that when turned about
its axis, without altering the position of the telescope on the circle G. H.,
the point to which it is directed will always lie in the small circle of the
heavens coincident with the star's diurnal path. In many observations
this is an inestimable advantage, and one which belongs to no other instru-
ment. The equatorial is also used for determining the place of an un-
known by comparison with that of a known object, in a manner to be
described in the fifth chapter. The adjustments of the equatorial are
somewhat complicated and difficult. They are best performed in this
manner: — 1st, Follow the pole star round its whole diurnal course, by
which it will become evident whether the polar axis is directed above or
below, to the right or to the left, of the true pole,_and correct it accord-
ingly (without any attempt, during this process, to correct the errors, if
any, in the position of the declination axis). 2dly, after the polar axis is
thus brought into adjustment, place the plane of the declination circle in
or near the meridian; and, having there secured it, observe the transits
of several known stars of widely different declinations. If the intervals
between these transits correspond to the known differences of right ascen-
sions of the stars, we may be sure that the telescope describes a true
meridian, and that, therefore, the declination axis is truly perpendicular
to the polar one;—if not, the deviation of the intervals from this law
will indicate the direction and amount of the deviation of the axis in
question, and enable us to correct it."
(186.) A very great improvement has, within a few years of the present
time, been introduced into the construction of the equatorial instrument.
It consists in applying a clockwork movement to turn the whole instru-
ment round upon its polar axis, and so to follow the diurnal motion of
any celestial object, without the necessity of the observer's manual inter-
vention. The driving power is the descent of a weight which communi-
* See Littrow on the Adjustment of the Equatorial (Mem. Ast. Soc. vol. ii. p. 45),
where formulae are given for ascertaining the amount and direction of all the misad-
justments simultaneously. But the practical observer, who wishes to avoid bewilder
ing hiriself by doing two things at once, had better proceed as recommended in the
text.
*
*
108 OUTLINES OF ASTRONOMY.
cates motion to a train of wheelwork, and thus, ultimately, to the polar
axis, while, at the same time, its too swift descent is controlled and regu-
lated to the exact and uniform rate required to give that axis one turn in
24 hours, by connecting it with a regulating clock, or (which is found
preferable in practice) by exhausting all the superfluous energy of the
driving power, by causing it to overcome a regulated friction. Artists
have thus succeeded in obtaining a perfectly smooth, uniform, and regula-
ble motion, which, when so applied, serves to retain any object on which the
telescope may be set, commodiously, in the centre of the field of view for
whole hours in succession, leaving the attention of the observer undis-
tracted by having a mechanical movement to direct, and with both his
hands at liberty.
(187.) The other position in which such a compound apparatus as we
have described in art. 182 may be advantageously mounted, is that in
which the principal axis occupies a vertical position, and the one circle, A
B, consequently corresponds to the celestial horizon, and the other, G H,
to a vertical circle of the heavens. The angles measured on the former
are therefore azimuths, or differences of azimuth, and those of the latter
Zenith distances, or altitudes, according as the graduation commences from
the upper point of its limb, or from one 90° distant from it. It is there-
fore known by the name of an azimuth and altitude instrument. The
vertical position of its principal axis is secured either by a plumb-line
suspended from the upper end, which, however it be turned round, should
continue always to intersect one and the same fiducial mark near its lower
extremity, or by a level fixed directly across it, whose bubble ought not
to shift its place, on moving the instrument in azimuth. The north or
south point on the horizontal circle is ascertained by bringing the vertical
circle to coincide with the plane of the meridian, by the same criterion by
which the azimuthal adjustment of the transit is performed (art. 162),
and noting, in this position, the reading off of the lower circle; or by the
following process.
(188.) Let a bright star be observed at a considerable distance to the
east of the meridian, by bringing it on the cross wires of the telescope.
In this position let the horizontal circle be read off, and the telescope
securely clamped on the vertical one. When the star has passed the
meridian, and is in the descending point of its daily course, let it be fol-
lowed by moving the whole instrument round to the west, without, how-
ever, unclamping the telescope, until it comes into the field of view; and
until, by continuing the horizontal motion, the star and the cross of the
wires come once more to coincide. In this position it is evident the star
must have the same precise altitude above the western horizon, that it had
ALTITUDE AND AZIMUT I INSTRUMENT. 109
at the moment of the first observation above the eastern. At this point
let the motion be arrested, and the horizontal circle be again read off.
The difference of the readings will be the azimuthal arc described in the
interval. Now, it is evident that when the altitudes of any star are equal
on either side of the meridian, its azimuths, whether reckoned both from
the north or both from the south point of the horizon, must also be equal,
— consequently the north or south point of the horizon must bisect the
azimuthal arc thus determined, and will therefore become known.
(189.) This method of determining the north and south points of a
horizontal circle is called the “method of equal altitudes,” and is of great
and constant use in practical astronomy. If we note, at the moments of
the two observations, the time, by a clock or chronometer, the instant
half-way between them will be the moment of the star's meridian passage,
which may thus be determined without a transit; and, vice versä, the
error of a clock or chronometer may by this process be discovered. For
this last purpose, it is not necessary that our instrument should be pro-
vided with a horizontal circle at all. Any means by which altitudes can
be measured will enable us to determine the moments when the same star
arrives at equal altitudes in the eastern and western halves of its diurnal
course; and, these once known, the instant of meridian passage and the
error of the clock become also known.
(190.) Thus also a meridian line may be drawn and a meridian mark
erected. For the readings on the north and South points on the limb of
the horizontal circle being known, the vertical circle may be brought ex-
actly into the plane of the meridian, by setting it to that precise reading.
This done, let the telescope be depressed to the north horizon, and let the
point intersected there by its cross-wires be noted, and a mark erected
there, and let the same be done for the south horizon. The line joining
these points is a meridian line, passing through the centre of the hori-
zontal circle. The marks may be made secure and permanent if required.
(191.) One of the chief purposes to which the altitude and azimuth
circle is applicable is the investigation of the amount and laws of refrac-
tion. For, by following with it a circumpolar star which passes the
zenith, and another which grazes the horizon, through their whole diurnal
course, the exact apparent form of their diurnal orbits, or the ovals into
which their circles are distorted by refraction, can be traced; and their
deviation from circles, being at every moment given by the nature of the
observation in the direction in which the refraction itself takes place (i. e.
in altitude), is made a matter of direct observation.
(192.) The zenith sector and the theodolite are peculiar modifications
of the altitude and azimuth instrument. The former is adapted for the
110 OTTLINES OF ASTRONOMY.
very exact observation of stars in or near the zenith, by giving a great
length to the vertical axis, and suppressing all the circumference of the
vertical circle, except a few degrees of its lower part, by which a great
length of radius, and a consequent proportional enlargement of the divi-
sions of its arc, is obtained. The latter is especially devoted to the mea-
sures of horizontal angles between terrestrial objects, in which the telescope
never requires to be elevated more than a few degrees, and in which,
therefore, the vertical circle is either dispensed with, or executed on a
smaller scale, and with less delicacy; while, on the other hand, great care
is bestowed on securing the exact perpendicularity of the plane of the
telescope's motion, by resting its horizontal axis on two supports like the
piers of a transit-instrument, which themselves are firmly bedded on the
spokes of the horizontal circle, and turn with it.
(193.) The next instrument we shall describe is one by whose aid the
angular distance of any two objects may be measured, or the altitude of
a single one determined, either by measuring its distance from the visible
horizon (such as the sea-offing, allowing for its dip), or from its own reflec-
tion on the surface of mercury. It is the sextant, or quadrant, commonly
called Hadley’s, from its reputed inventor, though the priority of inven-
tion belongs undoubtedly to Newton, whose claims to the gratitude of the
navigator are thus doubled, by his having furnished at once the only
theory by which his vessel can be securely guided, and the only instru-
ment which has ever been found to avail, in applying that theory to its
nautical uses."
(194.) The principle of this instrument is the optical property of re-
flected rays, thus announced :-‘‘The angle between the first and last
directions of a ray which has suffered two reflections in one plane is equal to
twice the inclination of the reflecting surfaces to each other. Let A B be
the limb or graduated arc, of a portion of a circle 60° in extent, but
divided into 120 equal parts. On the radius C B let a silvered plane glass
D be fixed, at right angles to the plane of the circle, and on the moveable
radius CE let another such silvered glass, C, be fixed. The glass D is
permanently fixed parallel to A C, and only one half of it is silvered, the
other half allowing objects to be seen through it. The glass C is wholly
silvered, and its plane is parallel to the length of the moveable radius CE,
* Newton communicated it to Dr. Halley, who suppressed it. The description of
the instrument was found, after the death of Halley, among his papers, in Newton's
own handwriting, by his executor, who communicated the papers to the Royal Society,
twenty-five years after Newton's death, and eleven after the publication of Hadley's
invention, which might be, and probably was, independent of any knowledge of New-
ton's, though Hutton insinuates the contrary.
THE SEXTANT. 111
at the extremity E of which a vernier is placed to read off the divisions
of the limb. On the radius A C is set a telescope F, through which any
object, Q, may be seen by direct rays which pass through the unsilvered
portion of the glass D, while another object, P, is seen through the same
telescope, by rays, which, after reflection at C, have been thrown upon the
silvered part of D, and are thence directed by a second refleetion into the
telescope. The two images so formed will both be seen in the field of
view at once, and by moving the radius C E will (if the reflectors be truly
perpendicular to the plane of the circle) meet and pass over, without oblit-
erating each other. The motion, however, is arrested when they meet,
and at this point the angle included between the direction C P of one ob-
ject, and FQ of the other, is twice the angle E CA included between the
fixed and moveable radii CA, CE. Now, the graduations of the limb
being purposely made only half as distant as would correspond to degrees,
the arc, A E, when read off, as if the graduations were whole degrees, will,
in fact, read double its real amount, and therefore the numbers so read off
will express, not the angle E CA, but its double, the angle subtended by
the objects.
(195.) To determine the exact distances between the stars by direct
observation is comparatively of little service; but in nautical astronomy
the measurement of their distances from the moon, and of their altitudes,
is of essential importance; and as the sextant requires no fixed support,
but can be held in the hand, and used on ship-board, the utility of the
instrument becomes at once obvious. For altitudes at sea, as no level,
plumb-line, or artificial horizon can be used, the sea-offing affords the only
resource; and the image of the star observed, seen by reflection, is brought
to coincide with the boundary of the sea seen by direct rays. Thus the
altitude above the sea-line is found; and this corrected for the dip of the
horizon (art. 23) gives the true altitude of the star. On land, an artifi-

112 Oly'ſ LINES OF ASTRONOMY.
cial horizon may be used (art. 173), and the consideration of dip is ren-
dered unnecessary.
(196.) The adjustments of the sextant are simple. They consist in
fixing the two reflectors, the one on the revolving radius CE, the other
on the fixed one C B, so as to have their planes perpendicular to the
plane of the circle, and parallel to each other, when the reading of the
instrument is zero. This adjustment in the latter respect is of little
moment, as its effect is to produce a constant error, whose amount is
readily ascertained by bringing the two images of one and the same star
or other distant object to coincidence; when the instrument ought to read
Zero, and if it does not, the angle which it does read is the zero correction
and must be subtracted from all angles measured with the sextant. The
former adjustments are essential to be maintained, and are performed by
small screws, by whose aid either or both the glasses may be tilted a little
one way or another until the direct and reflected images of a vertical line (a
plumb-line) can be brought to coincidence over their whole eactent, so as
to form a single unbroken straight line, whatever be the position of the
moveable arm, in the middle of the field of view of the telescope, whose
axis is carefully adjusted by the optician to parallelism with the plane of
the limb. In practice it is usual to leave only the reflector D on the fixed
radius adjustable, that on the moveable being set to great nicety by the
maker. In this case the best way of making the adjustment is to view a
pair of lines crossing each other at right angles (one being horizontal, the
other vertical) through the telescope of the instrument, holding the plane
of its limb vertical, - then having brought the horizontal line and its
reflected image to coincidence by the motion of the radius, the two
images of the vertical arm must be brought to coincidence by tilting one
way or other the fixed reflector D by means of an adjusting screw, with
which every sextant is provided for that purpose. When both lines coin-
cide in the centre of the field, the adjustment is correct.
(197.) The reflecting circle is an instrument destined for the same uses
as the sextant, but more complete, the circle being entire, and the divi-
sions carried all roñnd. It is usually furnished with three verniers, so as
to admit of three distinct readings off, by the average of which the error
of graduation and of reading is reduced. This is altogether a very refined
and elegant instrument. * -
(198.) We must not conclude this part of our subject without mention
of the “principle of repetition;” an invention of Borda, by which the
error of graduation may be diminished to any degree, and, practically
speaking, annihilated. Let PQ be two objects which we may suppose
fixed, for purposes of mere explanation, and let K L be a telescope move-
PRINCIPLE OF REPETITION. 113
Fig. 27.
g
able on O, the common axis of two circles, A M L and a be, of which the
former, A M L, is absolutely fixed in the plane of the objects, and carries
the graduations, and the latter is freely moveable on the axis. The tele-
scope is attached permanently to the latter circle, and moves with it. An
arm O a A carries the index, or vernier, which reads off the graduated
limb of the fixed circle. This arm is provided with two clamps, by which
it can be temporarily connected with either circle, and detached at
pleasure. Suppose, now, the telescope directed to P. Clamp the index
arm O A to the inner circle, and unclamp it from the outer, and read off.
Then carry the telescope round to the other object Q. In so doing, the
inner circle, and the index-arm which is clamped to it, will also be carried
round, over an arc A B, on the graduated limb of the outer, equal to the
angle PO Q. Now clamp the index to the outer circle, and unclamp the
inner, and read off: the difference of readings will of course measure the
angle PO Q; but the result will be liable to two sources of error—that
of graduation and that of observation, both which it is our object to get
rid of. To this end transfer the telescope back to P, without unclamping
the arm from the outer circle; then, having made the bisection of P,
clamp the arm to b, and unclamp it from B, and again transfer the tele-
scope to Q, by which the arm will now be carried with it to C, over a
second arc, B C, equal to the angle PO Q. Now again read off; then
will the difference between this reading and the original one measure
twice the angle PO Q, affected with both errors of observation, but only
with the same error of graduation as before. Ilet this process be re-
peated as often as we please (suppose ten times); then will the final arc
A B C D read off on the circle be ten times the required angle, affected
by the joint errors of all the ten observations, but only by the same con-
stant error of graduation, which depends on the initial and final readings
off alone. Now the errors of observation, when numerous, tend to

8
114 OluTLINES OF ASTRONOMY.
balance and destroy one another ; so that, if sufficiently multiplied, their
influence will disappear from the result. There remains, then, only the
constant error of graduation, which comes to be divided in the final result
by the number of observations, and is therefore diminished in its influence
to one tenth of its possible amount, or to less if need be. The abstract
beauty and advantage of this principle seem to be counterbalanced in
practice by some unknown cause, which, probably, must be sought for in
imperfect clamping. w
(199.) Micrometers are instruments (as the name imports') for measur-
ing, with great precision, small angles, not exceeding a few minutes, or at
most a whole degree. They are very various in construction and principle,
nearly all, however, depending on the exceeding delicacy with which space
can be subdivided by the turns and parts of a turn of fine screws. Thus
—in the parallel wire micrometer, two parallel threads (spider's lines are
generally used) stretched on sliding frames, one or both moveable by
tº
screws in a direction perpendicular to that of the threads, are placed in
the common focus of the object and eye-glasses of a telescope, and brought
by the motion of the screws exactly to cover the two extremities of the
image of any small object seen in the telescope, as the diameter of a
planet, &c., the angular distance between which it is required to measure.
This done, the threads are closed up by turning one of the screws till they
exactly cover each other, and the number of turns and parts of a turn
required gives the interval of the threads, which must be converted into
angular measure, either by actual calculation from the linear measure of
the threads of the screw and the focal length of the object-glass, or
experimentally, by measuring the image of a known object placed at a
known distance (as a foot-rule at a hundred yards, &c.) and therefore sub-
tending a known angle.
(200.) The duplication of the image of an object by optical means
furnishes a valuable and fertile resource in micrometry. Suppose by any
optical contrivance the single image A of any object can be converted into
two, exactly equal and similar, A B, at a distance from one another,
r
N
* Mukpos, small; perpetv, to measure.






MICROMETERS, 115
Fig. 29.
OOO Q
dependent (by some mechanical movement) on the will of the observer,
and in any required direction from one another. As these can, therefore,
be made to approach to or recede from each other at pleasure, they may
be brought in the first place to approach till they touch one another on
one side, as at A C, and then being made by continuing the motion to
cross and touch on the opposite side, as A D, it is evident that the
quantity of movement required to produce the change from one contact
to the other, if uniform, will measure the double diameter of the object A.
(201.) Innumerable optical combinations may be devised to operate
such duplication. The chief and most important (from its recent appli-
cations,) is the heliometer, in which the image is divided by bisecting the
object-glass of the telescope, and making its two halves, set in separate
brass frames, slide laterally on each other, as A B, the motion being
Fig. 30.
A.
produced and measured by a screw. Each half, by the laws of optics,
forms its own image (somewhat blurred, it is true, by diffraction,”) in its
own axis; and thus two equal and similar images are formed side by side
in the focus of the eye-piece, which may be made to approach and recede
by the motion of the screw, and thus afford the means of measurement
as above described.
(202.) Double refraction through crystallized media affords another
* This might be cured, though at an expense of light, by limiting each half to a
circular space by diaphragms, as represented by the dotted lines.

116 OljTLINES OF ASTRONOMY.
means of accomplishing the same end. Without going it to the intrica-
cies of this difficult branch of optics, it will suffice to state that objects
viewed through certain crystals (as Iceland spar, or quartz) appear
double, two images equally distinct being formed, whose angular distance
from each other varies from nothing (or perfect coincidence,) up to a
certain limit, according to the direction with respect to a certain fixed
line in the crystal, called its optical axis. Suppose, then, to take the
simplest case, that the eye-lens of a telescope, instead of glass, were
formed of such a crystal (say of quartz, which may be worked as well or
better than glass,) and of a spherical form, so as to offer no difference
when turned about on its centre, other than the inclination of its optical
axis to the visual ray. Then when that axis coincides with the line of
collimation of the object-glass, one image only will be seen, but when
made to revolve on an axis perpendicular to that line, two will arise,
opening gradually out from each other, and thus originating the desired
duplication. In this contrivance, the angular amount of the rotation of
the sphere affords the necessary datum for determining the separation of
the images.
(203.) Of all methods which have been proposed, however, the
simplest and most unobjectionable would appear to be the following. It
Fig. 31.
is well known to every optical student, that two prisms of glass, a flint
and a crown, may be opposed to each other, so as to produce a colourless
deflection of parallel rays. An object seen through such a compound or
achromatic prism, will be seen simply deviated in direction, but in no
way otherwise altered or distorted. Let such a prism be constructed
with its surfaces so nearly parallel that the total deviation produced in
traversing them shall not exceed a small amount (say 5'.) Let this be
cut in half, and from each half let a circular disc be formed, and cemented

*MICROMETERS. - 117
on a circular plate of parallel glass, or otherwise sustained, close to and
concentric with the other by a framework of metal so light as to inter-
cept but a small portion of the light which passes on the outside (as in
the annexed figure,) where the dotted lines represent the radii sustaining
one, and the undotted those carrying the other disc. The whole must
be so mounted as to allow one disc to revolve in its own plane behind
the other, fixed, and to allow the amount of rotation to be read off. It
is evident, then, that when the deviations produced by the two discs
conspire, a total deviation of 10' will be effected on all the light which
has passed through them; that when they oppose each other, the rays
will emerge undeviated, and that in intermediate positions a deviation
varying from 0 to 10', and calculable from the angular rotation of the
one disc on the other, will arise. Now, let this combination be applied
at such a point of the cone of rays, between the object-glass and its
focus, that the discs shall occupy exactly half the area of its section.
Then will half the light of the object lens pass undeviated—the other
half deviated, as above described; and thus a duplication of image,
variable and measureable (as required for micrometric measurement) will
occur. If the object-glass be not very large, the most convenient point
of its application will be externally before it, in which case the diameter
of the discs will be to that of the object-glass as 707: 1000; or (allow-
ing for the spokes) about as 7 to 10.
(204.) The Position Micrometer is simply a straight thread or wire
which is carried round by a smooth revolving motion, in the common focus
of the object and eye-glasses, in a plane perpendicular to the axis of the
telescope. It serves to determine the situation with respect to some fixed
line in the field of view, of the line joining any two objects or points of
an object seen in that field—as two stars, for instance, near enough to be
seen at once. For this purpose the moveable thread is placed so as to
cover both of them, or stand, as may best be judged, parallel to their line
of junction. And its angle, with the fixed one, is then read off upon a
small divided circle exterior to the instrument. When such a micrometer
is applied (as it most commonly is) to an equatorially mounted telescope,
the zero of its position corresponds to a direction of the wire, such as, pro-
longed, will represent a circle of declination in the heavens—and the
“angles of position ” so read off are reckoned invariably from one point,
and in one direction, viz., north, following, south, preceding ; so that 0°
position corresponds to the situation of an object exactly north of that as-
sumed as a centre of reference,—90° to a situation exactly eastward or
following ; 180° exactly south; and 270° exactly west, or preceding in
the crder of diurnal movement.
118 OUTLINES OF ASTRONOMY,
CHAPTER IV.
O F. G E O G. R. A. P. H. Y.
OF THE FIGURE OF THE EARTH.-ITS EXACT DIMENSIONS.—ITS FORM-
THAT OF EQUILIBRIUM MODIFIED BY CENTRIFUGAL FORCE. — VARIA-
TION OF GRAVITY ON ITS SURFACE. – STATICAL AND DYNAMICAL
MEASURES OF GRAVITY. — THE PENDULUM. – GRAVITY TO A SPHE-
ROID.—OTHER EFFECTS OF THE EARTH's ROTATION.—TRADE WINDs.
— DETERMINATION OF GEOGRAPHICAL POSITIONS. — OF LATITUDES.
— OF LONGITUDES. — CONDUCT OF A TRIGONOMETRICAL SURVEY. —
OF MAPS. – PROJECTIONS OF THE SPHERE. — MEASUREMENT OF
HEIGHTS BY THE BAROMETER.
(205.) GEOGRAPHY is not only the most important of the practical
branches of knowledge to which astronomy is applied, but it is also,
theoretically speaking, an essential part of the latter science. The earth
being the general station from which we view the heavens, a knowledge
of the local situation of particular stations on its surface is of great con-
sequence, when we come to inquire the distances of the nearer heavenly
bodies from us, as concluded from observations of their parallax as well
as on all other occasions, where a difference of locality can be supposed to
influence astronomical results. We propose, therefore, in this chapter, to
explain the principles, by which astronomical observation is applied to
geographical determinations, and to give at the same time an outline of
geography so far as it is to be considered a part of astronomy.
(206.) Geography, as the word imports, is a delineation or description
of the earth. In its widest sense, this comprehends not only the delinea-
tion of the form of its continents and seas, its rivers and mountains, but
their physical condition, climates, and products, and their appropriation
by communities of men. With physical and political geography, how-
ever, we have no concern here. Astronomical geography has for its
objects the exact knowledge of the form and dimensions of the earth, the
parts of its surface occupied by sea and land, and the configuration of the
surface of the latter, regarded as protuberant above the ocean, and broken
THE FIGURE OF THE EARTH. 119
into the various forms of mountain, table land, ar.d valley; neither should
the form of the bed of the ocean, regarded as a continuation of the sur-
face of the land beneath the water, be left out of consideration: we
know, it is true, very little of it; but this is an ignorance rather to be
lamented, and, if possible, remedied, than acquiesced in, inasmuch as
there are many very important branches of inquiry which would be
greatly advanced by a better acquaintance with it.
(207.) With regard to the figure of the earth as a whole, we have
already shown that, speaking loosely, it may be regarded as spherical; but
the reader who has duly appreciated the remarks in art. 22 will not be at
a loss to perceive that this result, concluded from observations not suscep-
tible of much exactness, and embracing very small portions of the surface
at once, can only be regarded as a first approximation, and may require te
be materially modified by entering into minutiae before neglected, or by
increasing the delicacy of our observations, or by including in their extent
larger areas of its surface. For instance, if it should turn out (as it will),
on minuter inquiry, that the true figure is somewhat elliptical, or flattened,
in the manner of an orange, having the diameter which coincides with the
axis about shoth part shorter than the diameter of its equatorial circle;
—this is so trifling a deviation from the spherical form that, if a model
of such proportions were turned in wood, and laid before us on a table,
the nicest eye or hand would not detect the flattening, since the difference
of diameters, in a globe of fifteen inches, would amount only to ºth of
an inch. In all common parlance, and for all ordinary purposes, then, it
would still be called a globe; while, nevertheless, by careful measurement,
the difference would not fail to be noticed; and, speaking strictly, it would
be termed, not a globe, but an oblate ellipsoid, or spheroid, which is the
name appropriated by geometers to the form above described.
(208.) The sections of such a figure by a plane are not circles, but
ellipses; so that, on such a shaped earth, the horizon of a spectator would
nowhere (except at the poles) be exactly circular, bnt somewhat elliptical.
It is easy to demonstrate, however, that its deviation from the circular
form, arising from so slight an “ellipticity’ as above supposed, would be
quite imperceptible, not only to our eye-sight, but to the test of the dip-
sector; so that by that mode of observation we should never be led to
notice so small a deviation from perfect sphericity. How we are led to
this conclusion, as a practical result, will appear, when we have explained
the means of determining with accuracy the dimensions of the whole, or
any part of the earth.
(209.) As we cannot grasp the earth, nor recede from it far enough te
view it at once as a whole, and compare it with a known standard of mea-
w
120 OljTLINES OF ASTRONOMY.
sure in any degree commensurate to its own size, but can only creep about
upon it, and apply our diminutive measures to comparatively small parts
of its vast surface in succession, it becomes necessary to supply, by geo-
metrical reasoning, the defect of our physical powers, and from a delicate
and careful measurement of such small parts to conclude the form and
dimensions of the whole mass. This would present little difficulty, if we
were sure the earth was strictly a sphere, for the proportion of the cir-
cumference of a circle to its diameter being known (viz. that of 3.1415926
to 1:0000000), we have only to ascertain the length of the entire circum-
ference of any great circle, such as a meridian, in miles, feet, or any other
standard units, to know the diameter in units of the same kind. Now,
the circumference of the whole circle is known as soon as we know the
exact length of any aliquot part of it, such as 1° or sºoth part; and this,
being not more than about seventy miles in length, is not beyond the
limits of very exact measurement, and could, in fact, be measured (if we
knew its exact termination at each extremity) within a very few feet, or,
indeed, inches, by methods presently to be particularized.
(210.) Supposing, then, we were to begin measuring with all due nicety
from any station, in the exact direction of a meridian and go measuring
on, till by some indication we were informed that we had accomplished an
exact degree from the point we set out from, our problem would then be
at once resolved. It only remains, therefore, to inquire by what indica-
tions we can be sure, 1st, that we have advanced an eacact degree ; and,
2dly, that we have been measuring in the exact direction of a great circle.
(211.) Now the earth has no landmarks on it to indicate degrees, nor
traces inscribed on its surface to guide us in such a course. The compass,
though it affords a tolerable guide to the mariner or the traveller, is far
too uncertain in its indications, and too little known in its laws, to be of
any use in such an operation. We must, therefore, look outwards, and
refer our situation on the surface of our globe to natural marks, eacternal
to it, and which are of equal permanence and stability with the earth
itself. Such marks are afforded by the stars. By observations of their
meridian altitudes, performed at any station, and from their known polar
distances, we conclude the height of the pole; and since the altitude of
the pole is equal to the latitude of the place (art. 119) the same obser-
vations give the latitudes of any stations where we may establish the
requisite instruments. When our latitude then, is found to have dimin-
ished a degree, we know that, provided we have kept to the meridian, we
have described one three hundred and sixtieth part of the earth’s circum-
ference.
(212.) The direction of the meridian may be secured at every instant
FIGURE OF THE EARTH. 121
by the observations described in art. 162, 188; and although local diffi-
culties may oblige us to deviate in our measurement from this exact direc-
tion, yet if we keep a strict account of the amount of this deviation, a
very simple calculation will enable us to reduce our observed measure to
its meridional value.
(213.) Such is the principle of that most important geographical ope-
ration, the measurement of an arc of the meridian. In its detail, however,
a somewhat modified course must be followed. An observatory cannot be
mounted and dismounted at every step; so that we cannot identify and
measure an exact degree neither more nor less. But this is of no conse-
quence, provided we know with equal precision how much, more or less,
we have measured. In place, then, of measuring this precise aliquot
part, we take the more convenient method of measuring from one good
observing station to another, about a degree, or two or three degrees, as
the case may be, or indeed any determinate angular interval apart, and
determining by astronomical observation the precise difference of latitudes
between the stations.
(214.) Again, it is of great consequence to avoid in this operation
every source of uncertainty, because an error committed in the length of
a single degree will be multiplied 360 times in the circumference, and
nearly 115 times in the diameter of the earth concluded from it. Any
error which may affect the astronomical determination of a star's altitude
will be especially influential. Now, there is still too much uncertainty
and fluctuation in the amount of refraction at moderate altitudes, not to
make it especially desirable to avoid this source of error. To effect this,
we take care to select for observation, at the extreme stations, some star
which passes through or near the zeniths of both. The amount of refrac-
tion, within a few degrees of the zenith, is very small, and its fluctuations
and uncertainty, in point of quantity, so excessively minute as to be
utterly inappreciable. Now, it is the same thing whether we observe the
pole to be raised or depressed a degree, or the zenith distance of a star
when on a meridian to have changed by the same quantity (fig, art. 128).
If at one station we observe any star to pass through the zenith, and at
the other to pass one degree south or north of the zenith, we are sure that
the geographical latitudes, or the altitudes of the pole at the two stations,
must differ by the same amount.
(215.) Granting that the terminal points of one degree can be
ascertained, its length may be measured by the methods which will be
presently described, as we have before remarked, to within a very few feet.
Now, the error which may be committed in fixing each of these terminal
points cannot exceed that which may be committed in the observation of
122 OUTLINES OF ASTRONOMY.
the zenith distance of a star properly situated for the purpose in question.
This error, with proper care, can hardly exceed half a second. Supposing
we grant the possibility of ten feet of error in the length of each degree
in a measured arc of five degrees, and of half a second in each of the
Zenith distances of one star, observed at the northern and southern sta-
tions, and, lastly, suppose all these errors to conspire, so as to tend all of
them to give a result greater, or all less, than the truth, it will appear,
by a very easy proportion, that the whole amount of error which would
be thus entailed on an estimate of the earth's diameter, as concluded
from such a measure, would not exceed 1147 yards, or about two thirds
of a mile, and this is ample allowance.
(216.) This, however, supposes that the form of the earth is that of a
perfect sphere, and, in consequence, the lengths of its degrees in all parts
precisely equal. But, when we come to compare the measures of meri-
dional arcs made in various parts of the globe, the results obtained,
although they agree sufficiently to show that the supposition of a spherical
figure is not very remote from the truth, yet exhibit discordances far
greater than what we have shown to be attributable to error of observation,
and which render it evident that the hypothesis, in strictness of its word-
ing, is untenable. The following table exhibits the lengths of arcs of the
meridian (astronomically determined as above described), expressed in
British standard feet, as resulting from actual measurement made with all
possible care and precision, by commissioners of various nations, men of
the first eminence, supplied by their respective governments with the best
instruments, and furnished with every facility which could tend to ensure
a successful result of their important labours. The lengths of the degrees
in the last column are derived from the numbers set down in the two
preceding ones by simple proportion, a method not quite exact when the
arcs are large, but sufficiently so for our purpose.
FIGURE OF THE EARTH. . 123
Mean
M .#. of
º Measured the Degree
Country. sº #. mºre. *:::::: lD1 ****
tude in
Feet.
Sweden,* A B - - |-|-66° 20' 10"-0 | 1937' 1976 593.277 365744
Sweden, A *- - |-|-66 19 37 0 57 30 4 35.1832 367086
Russia, A º - |-|-58 17 37 3 35 5-2 || 13097.42 || 365368
Russia, B & - |-|-56 3 55-5 8 2 28.9 || 29374.39 || 365291
Prussia, B - - - - 54 58 26-0 || 1 30 29-0 || 551073 || 365420
Denmark, B te - |-|- 54 8 13.7 1 31 53-3 5591 21 365087
Hanover, A B - - |-|-52 32 16-6 2 0 57.4 736425 || 365300
England, A º - | + 52 35 45 3 57 13-1 || 1442953 || 364971
England, B wº - + 52 2 19:4 2 50 23-5 | 1036409 || 364951
France, A * - || +46 52 2 8 20 0.3 || 3040605 || 364872
France, A B º - |-|-44 51 2.5 12 22 12.7 4509832 || 36.4572
Rome, A º - +42 59 — 2 9 47 7879 19 364262
America, A º - || +39 12 — 1 28 45.0 538 100 || 363786
India, A B g- - |-|- 16 8 21.5 | 15 57 407 || 5794598 || 363044
India, A B sº - || + 12 32 20:8 1 34 56.4 574318 362956
Peru, A B º - — 1 31 0.4 3 7 3.5 | 1131050 36.3626
Cape of Good Hope, A —33 -18 30 1 13 17.5 445506 || 364713
Cape of Good Hope, B | —35 43 20:0 3 34 34-7 || 1301993 || 364060
It is evident from a mere inspection of the second and fifth columns of
this table, that the measured length of a degree increases with the lati-
tude, being greatest near the poles, and least near the equator. Let us
now consider what interpretation is to be put upon this conclusion, as
regards the form of the earth.
(217.) Suppose we held in our hands a model of the earth smoothly
turned in wood, it would be, as already observed, so nearly spherical, that
neither by the eye nor the touch, unassisted by instruments, could we
detect any deviation from that form. Suppose, too, we were debarred
from measuring directly across from surface to surface in different direc-
tions with any instrument, by which we might at once ascertain whether
* The astronomers by whom these measurements were executed were as fol
iows : —
Sweden, A B–Svanberg. France, A B — Delambre, Mechain.
Sweden, A–Maupertuis. Rome — Boscovich.
Russia, A–Struve. America — Mason and Dixon.
Russia, B – Struve, Tenner. India, 1st — Lambton.
Prussia — Bessel, Bayer. India, 2d — Lambton, Everest.
Denmark – Schumacher. Peru — Lacondamine, Bouguer.
Hanover — Gauss. Cape of Good Hope, A–Lacaille.
England — Roy, Kater. Cape of Good Hope, B — Maclear.
France, A–Lacaille, Cassini. —-Astr. Nachr. 574
124 OUTLINES OF ASTRONOMY.
Fig. 32.
Fº

one diameter were longer than another; how, then, we may ask, are we
to ascertain whether it is a true sphere or not? It is clear that we have
no resource, but to endeavour to discover, by some nicer means than
simple inspection or feeling, whether the convexity of its surface is the
same in every part; and if not, where it is greatest, and where least.
Suppose, then, a thin plate of metal to be cut into a concavity at its edge,
so as exactly to fit the surface at A: let this now be removed from A,
and applied successively to several other parts of the surface, taking care
to keep its plane always on a great circle of the globe, as here represented.
If, then, we find any position, B, in which the light can enter in the
middle between the globe and plate, or any other, C, where the latter tilts
by pressure, or admits the light under its edges, we are sure that the cur-
vature of the surface at B is less, and at C greater, than at A.
(218.) What we here do by the application of a metal plate of deter-
minate length and curvature, we do on the earth by the measurement of
a degree of variation in the altitude of the pole. Curvature of a surface
is nothing but the continual deflection of its tangent from one fixed direc-
tion as we advance along it. When, in the same measured distance of
advance we find the tangent (which answers to our horizon) to have
shifted its position with respect to a fixed direction in space, (such as the
axis of the heavens, or the line joining the earth's centre and some given
star,) more in one part of the earth's meridian than in another, we con-
clude, of necessity, that the curvature of the surface at the former spot is
greater than at the latter; and vice versá, when, in order to produce the
same change of horizon with respect to the pole (suppose 1°) we require
to travel over a longer measured space at one point than at another, we
assign to that point a less curvature. Hence we conclude that the curva-
ture of a meridional section of the earth is sensibly greater at the equa-
vor than towards the poles; or, in other words, that the earth is not
spherical, but flattened at the poles, or, which comes to the same, protu-
berant at the equator.
FIGURE OF THE EARTH, 125
(219.) Let NABI) EF represent a meridional section of the earth, C
its centre, and N A, B D, G E, arcs of a meridian, each corresponding to
one degree of difference of latitude, or to one degree of variation in the
meridian altitude of a star, as referred to the horizon of a spectator travel-
ling along the meridian. Let n N, a A, b B, d D, g G, elº, be the respec-
tive directions of the plumb-line at the stations N, A, B, D, G, E, of
which we will suppose N to be at the pole and E at the equator; then
will the tangents to the surface at these points respectively be perpen-
dicular to these directions; and, consequently, if each pair, viz. n N and
a A, b B and d D, g G and eB, be prolonged till they intersect each other
(at the points ac, y, z), the angles Na: A, By D, Gz E, will each be one
degree, and, therefore, all equal; so that the small curvilinear arcs NA,
BD, GE, may be regarded as arcs of circles of one degree each, described
about ac, ºy, z, as centres. These are what in geometry are called centres
of curvature, and the radii a N or a A, y B or y D, 2 G or z E, represent
Fig. 33.
ill,
radii of curvature, by which the curvatures at those points are deter-
mined and measured. Now, as the arcs of different circles, which sub-
tend equal angles at their respective centres, are in the direct proportion
of their radii, and as the are N A is greater than BD, and that again
than GE, it follows that the radius N a must be greater than By, and
By than Ez. Thus it appears that the mutual intersections of the
plumb-lines will not, as in the sphere, all coincide in one point C, the
centre, but will be arranged along a certain curve, a y z (which will be
rendered more evident by considering a number of intermediate stations).

126 OUTLINES OF ASTRONOMY.
To this curve geometers have given the name of the evolute of the curve
N A B D G E, from whose centres of curvature it is constructed.
(220.) In the flattening of a round figure at two opposite points, and
its protuberance at points rectangularly situated to the former, we recog-
nize the distinguishing feature of the elliptic form. Accordingly, the
next and simplest supposition that we can make respecting the nature of
the meridian, since it is proved not to be a circle, is, that it is an ellipse,
or nearly so, having N S, the axis of the earth, for its shorter, and E F,
the equatorial diameter, for its longer axis; and that the form of the
earth's surface is that which would arise from making such a curve revolve
about its shorter axis N S. This agrees well with the general course of
the increase of the degree in going from the equator to the pole. In the
ellipse, the radius of curvature at E, the extremity of the longer axis is
the least, and at that of the shorter axis, the greatest it admits, and the
form of its evolute agrees with that here represented.' Assuming, then,
that it is an ellipse, the geometrical properties of that curve enable us to
assign the proportion between the lengths of its axes which shall corre-
spond to any proposed rate of variation in its curvature, as well as to fix
upon their absolute lengths, corresponding to any assigned length of the
degree in a given latitude. Without troubling the reader with the inves-
tigation, (which may be found in any work on the conic sections,) it will
be sufficient to state results which have been arrived at by the most sys-
tematic combinations of the measured arcs which have hitherto been made
by geometers. The most recent is that of Bessel", who by a combination
of the ten arcs, marked B in our table, has concluded the dimensions of
the terrestrial spheroid to be as follows:–
Feet. Miles.
Greater or equatorial diameter sº tº - = 41,847,192 = 7925.604
Lesser or polar diameter s º * > - = 41,707,324 – 7899-114
Difference of diameters, or polar compression = 139,768 = 26-471
Proportion of diameters as 299.15 to 298.15.
The other combination whose results we shall state, is that of Mr
Airy”, who concludes as follows:
***.
** Feet. Miles.
Equatorial diameter ge sº gº º - = 41,847,426 = 7925-648
Polar diameter - tº tº - tº º - = 41,707,620 = 7899-170
Polar compression - * tº tº * - - = 139,806 = 26.478
Proportion of diameters as 299-33 to 298.33.
* The dotted lines are the portions of the evolute belonging to the other quadrants.
• Schumacher’s Astronomische Nachrichten, Nos. 333, 334, 335, 438.
* Encyclopædia Metropolitana, “Figure of the 3arth” (1831).
DIMENSIONS OF THE EARTH, 127
These conclusions are based on the consideration of those 13 arcs, to
which the letter A is annexed', and of one other arc of 1° 7' 31":1,
measured in Piedmont by Plana and Carlini, whose discordance with the
rest, owing to local causes hereafter to be explained, arising from the ex-
ceedingly mountainous nature of the country, render the propriety of so
employing it very doubtful. Be that as it may, the strikingly near ac-
cordance of the two sets of dimensions is such as to inspire the greatest
confidence in both. The measurement at the Cape of Good Hope by
Lacaille, also used in this determination, has always been regarded as
unsatisfactory, and has recently been demonstrated by Mr. Maclear to be
erroneous to a considerable extent. The omission of the former, and the
substitution for the latter, of the far preferable result of Mr. Maclear's
second measurement would induce, however, but a trifling change in the
final result. -
(221.) Thus we see that the rough diameter of 8000 miles we have
hitherto used, is rather too great, the excess being about 100 miles, or
sºoth part. As convenient numbers to remember, the reader may bear in
mind, that in our latitude there are just as many thousands of feet in a
degree of the meridian as there are days in the year (365): that, speak-
ing loosely, a degree is about 70 British statute miles, and a second about
100 feet; that the equatorial circumference of the earth is a little less
than 25,000 miles (24,899), and the ellipticity or polar flattening
amounts to one 300th part of the diameter.
(222.) The two sets of results above stated are placed in juxtaposition,
and the particulars given more in detail than may at first sight appear
consonant, either with the general plan of this work, or the state of the
reader's presumed acquaintance with the subject. But it is of importance
that he should early be made to see how, in astronomy, results in admira-
ble concordance emerge from data accumulated from totally different quar-
ters, and how local and accidental irregularities in the data themselves
become neutralized and obliterated by their impartial geometrical treat-
ment. In the cases before us, the modes of calculation followed are
widely different, and in each the mass of figures to be gone through to
arrive at the result, enormous.
(223.) The supposition of an elliptic form of the earth's section through
the axis is recommended by its simplicity, and confirmed by comparing
the numerical results we have just set down with those of actual measure-
ment. When this comparison is executed, discordances, it is true, are
observed, which, although still too great to be referred to error of
* In those which have both A and B, the numbers used by Mr. Airy differ slightlv
from Bessel's, which are those we have preferred.
128 oUTLINEs of ASTRONOMY.
measurement, are yet so small, compared to the errors which would result
from the spherical hypothesis, as completely to justify our regarding the
earth as an ellipsoid, and referring the observed deviations to either local
or, if general, to comparatively small causes.
(224.) Now, it is highly satisfactory to find that the general elliptical
figure thus practically proved to exist, is precisely what ought theoretically
to result from the rotation of the earth on its axis. For, let us suppose
the earth a sphere, at rest, of uniform materials throughout, and exter-
nally covered with an ocean of equal depth in every part. Under such
circumstances it would obviously be in a state of equilibrium ; and the
water on its surface would have no tendency to run one way or the other.
Suppose, now, a quantity of its materials were taken from the polar
regions, and piled up all around the equator, so as to produce that dif-
fe, ence of the polar and equatorial diameters of 26 miles which we know
to exist. It is not less evident that a mountain ridge or equatorial conti.
ment, only, would be thus formed, from which the water would run down
the excavated part at the poles. However solid matter might rest where
it was placed, the liquid part, at least, would not remain there, any more
than if it were thrown on the side of a hill. The consequence therefore,
would be the formation of two great polar seas, hemmed in all round by
equatorial land. Now, this is by no means the case in nature. The
ocean occupies, indifferently, all latitudes, with no more partiality to the
polar than to the equatorial. Since, then, as we see, the water occupies
an elevation above the centre no less than 13 miles greater at the equator
than at the poles, and yet manifests no tendency to leave the former and
run towards the latter, it is evident that it must be retained in that
situation by some adequate power. No such power, however, would exist
in the case we have supposed, which is therefore not conformable to
nature. In other words, the spherical form is not the figure of equili.
brium ; and therefore the earth is either not at rest, or is so internally
constituted as to attract the water to its equatorial regions, and retain it
there. For the latter supposition there is no primá facie probability, nor
any analogy to lead us to such an idea. The former is in accordance with
all the phenomena of the apparent diurnal motion of the heavens; and
therefore, if it will furnish us with the power in question, we can have no
hesitation in adopting it as the true one.
(225.) Now, every body knows that when a weight is whirled round,
It acquires thereby a tendency to recede from the centre of its motion;
which is called the centrifugal force. A stone whirled round in a sling
is a common illustration; but a better, for our present purpose, will be a
pail of water, suspended by a cord, and made to spin round, while the
FIGURE OF THE EARTH. 129
Fig. 33.
JH
cord hangs perpendicularly. The surface of the water, instead of re-
maining horizontal, will become concave, as in the figure. The centri-
fugal force generates a tendency in all the water to leave the axis, and
press towards the circumference; it is, therefore, urged against the pail,
and forced up its sides, till the excess of height, and consequent increase
of pressure downwards, just counterbalances its centrifugal force, and a
state of equilibrium is attained. The experiment is a very easy and
instructive one, and is admirably calculated to show how the form of
equilibrium accommodates itself to varying circumstances. If, for ex-
ample, we allow the rotation to cease by degrees, as it becomes slower we
shall see the concavity of the water regularly diminish; the elevated
outward portion will descend, and the depressed central rise, while all the
time a perfectly smooth surface is maintained, till the rotation is ex-
hausted, when the water resumes its horizontal state.
(226.) Suppose, then, a globe, of the size of the earth, at rest, and
covered with a uniform ocean, were to be set in rotation about a certain
axis, at first very slowly, but by degrees more rapidly, till it turned round
once in twenty-four hours; a centrifugal force would be thus generated,
whose general tendency would be to urge the water at every point of the
surface to recede from the axis. A rotation might, indeed, be conceived
so swift as to flirt the whole ocean from the surface, like water from a
mop. But this would require a far greater velocity than what we now

9
130 OUTLINES OF ASTR3NOMY.
speak of. In the case supposed, the weight of the water would still keep
it on the earth; and the tendency to recede from the axis could only be
satisfied, therefore, by the water leaving the poles, and flowing towards
the equator; there heaping itself up in a ridge, just as the water in our
pail accumulates against the side; and being retained in opposition to its
weight, or natural tendency towards the centre, by the pressure thus
caused. This, however, could not take place without laying dry the
polar portions of the land in the form of immensely protuberant conti-
ments; and the difference of our supposed cases, therefore, is this: — in
the former, a great equatorial continent and polar seas would be formed;
in the latter, protuberant land would appear at the poles, and a Zone of
ocean be disposed around the equator. This would be the first or im-
mediate effect. Let us now see what would afterwards happen, in the
two cases, if things were allowed to take their natural course.
(227.) The sea is constantly beating on the land, grinding it down,
and scattering its worn-off particles and fragments, in the state of mud
and pebbles, over its bed. Geological facts afford abundant proof that the
existing continents have all of them undergone this process, even more
than once, and been entirely torn in fragments, or reduced to powder, and
submerged and reconstructed. Land, in this view of the subject, loses
its attribute of fixity. As a mass it might hold together in opposition to
forces which the water freely obeys; but in its state of successive or
simultaneous degradation, when disseminated through the water, in the
state of sand or mud, it is subject to all the impukses of that fluid. In
the lapse of time, then, the protuberant land in both cases would be des-
troyed, and spread over the bottom of the ocean, filling up the lower parts,
and tending continually to remodel the surface of the solid nucleus, in cor-
respondence with the form of equilibrium in both cases. Thus, after a
sufficient lapse of time, in the case of an earth at rest, the equatorial con-
tinent, thus forcibly constructed, would again be levelled and transferred
to the polar excavations, and the spherical figure be so at length restored.
In that of an earth in rotation, the polar protuberances would gradually
be cut down and disappear, being transferred to the equator (as being
then the deepest sea), till the earth would assume by degrees the form we
observe it to have — that of a flattened or oblate ellipsoid.
(228.) We are far from meaning here to trace the process by which
the earth really assumed its actual form ; all we intend is, to show that
this is the form to which, under the conditions of a rotation on its axis,
it must tend ; and which it would attain, even if originally and (so to
speak) perversely constituted otherwise. gº
(229.) But, further, the dimensions of the earth and the time of its
VARIATION OF TERRESTRIAL GRAVITY. 131
1ctation being known, it is easy thence to calculate the exact amount of
the centrifugal force,' which, at the equator, appears to be g; gth part of
the force or weight by which all bodies, whether solid or liquid, tend to
fall towards the earth. By this fraction of its weight, then, the sea at
the equator is lightened, and thereby rendered susceptible of being sup-
ported on a higher level, or more remote from the the centre than at the
poles, where no such counteracting force exists; and where, in conse-
quence, the water may be considered as specifically heavier. Taking this
principle as a guide, and combining it with the laws of gravity (as devel-
oped by Newton, and as hereafter to be more fully explained), mathemati-
cians have been enabled to investigate, à priori, what would be the figure
of equilibrium of such a body, constituted internally as we have reason
to believe the earth to be; covered wholly or partially with a fluid; and
revolving uniformly in twenty-four hours; and the result of this inquiry
is found to agree very satisfactorily with what experience shows to be the
case. From their investigations it appears that the form of equilibrium
is, in fact, no other than an oblate ellipsoid, of a degree of ellipticity very
nearly identical with what is observed, and which would be no doubt
accurately so, did we know, with precision, the internal constitution and
materials of the earth.
(230.) The confirmation thus incidentally furnished, of the hypothesis
of the earth’s rotation on its axis, cannot fail to strike the reader. A
deviation of its figure from that of a sphere was not contemplated among
the original reasons for adopting that hypothesis, which was assumed
solely on account of the easy explanation it offers of the apparent diurnal
motion of the heavens. Yet we see that, once admitted, it draws with it,
as a necessary consequence, this other remarkable phenomenon, of which
no other satisfactory account could be rendered. Indeed, so direct is their
connection, that the ellipticity of the earth's figure was discovered and
demonstrated by Newton to be a consequence of its rotation, and its amount
actually calculated by him, long before any measurement had suggested
such a conclusion. As we advance with our subject, we shall find
the same simple principle branching out into a whole train of singular
and important consequences, some obvious enough, others which at first
seem entirely unconnected with it, and which, until traced by Newton up
to this their origin, had ranked among the most inscrutable arcana of
astronomy, as well as among its grandest phenomena.
(231.) Of its more obvious consequences, we may here mention one
which falls naturally within our present subject. If the earth really
* Newton's Principia, iii. Prop. 19.
I 32 OUTLINES OF ASTRONOMY.
revolve on its axis, this rotation must generate a centrifugal force (see
art. 225,) the effect of which must of course be to counteract a certain
portion of the weight of every body situated at the equator, as compared
with its weight at the poles, or in any intermediate latitudes. Now, this
is fully confirmed by experience. There is actually observed to exist a
difference in the gravity, or downward tendency, of one and the same
body, when conveyed successively to stations in different latitudes. Ex
periments made with the greatest care, and in every accessible part of the
globe, have fully demonstrated the fact of a regular and progressive
increase in the weights of bodies corresponding to the increase of lati-
tude, and fixed its amount and the law of its progression. From these it
appears, that the extreme amount of this variation of gravity, or the
difference between the equatorial and polar weights of one and the same
mass of matter, is 1 part in 194 of its whole weight, the rate of increase
in travelling from the equator to the pole being as the square of the sine
of the latitude.
(232.) The reader will here naturally inquire, what is meant by
speaking of the same body as having different weights at different sta-
tions; and, how such a fact, if true, can be ascertained. When we
weigh a body by a balance or a steelyard, we do but counteract its weight
by the equal weight of another body under the very same circumstances;
and if both the body weighed and its counterpoise be removed to another
station, their gravity, if changed at all, will be changed equally, so that
they will still continue to counterbalance each other. A difference in the
intensity of gravity could, therefore, never be detected by these means;
nor is it in this sense that we assert that a body weighing 194 pounds at
the equator will weigh 195 at the pole. If counterbalanced in a scale
or steelyard at the former station, an additional pound placed in one or
other scale at the latter would inevitably sink the beam.
(233.) The meaning of the proposition may be thus explained:—
Conceive a weight a suspended at the equator by a string without weight
Fig. 85.

AWARIATION OF TERRESTRIAL GRAVITY. 133
passing over a pulley, A, and conducted (supposing such a thing possi-
ble) over other pulleys, such as B, round the earth's convexity, till the
other end hung down at the pole, and there sustained the weight y. If,
then, the weights a, and y were such as, at any one station, equatorial or
polar, would exactly counterpoise each other on a balance, or when sus-
pended side by side over a single pulley, they would not counterbalance
each other in this supposed situation, but the polar weight y would pre-
ponderate; and to restore the equipoise the weight a must be increased
by Tºith part of its quantity.
(234.) The means by which this variation of gravity may be shown
to exist, and its amount measured, are twofold (like all estimations of
mechanical power,) statical and dynamical. The former consists in
putting the gravity of a weight in equilibrium, not with that of another
weight, but with a natural power of a different kind not liable to be
affected by local situation. Such a power is the elastic force of a spring.
Let A B C be a strong support of brass standing on the foot A E D cast
in one piece with it, into which is let a smooth plate of agate, D, which
can be adjusted to perfect horizontality by a level. At C let a spiral
N
spring G be attached, which carries at its lower end a weight F, polished
and convex below. The length and strength of the spring must be so
adjusted that the weight F shall be sustained by it just to swing clear of
contact with the agate plate in the highest latitude at which it is intended
to use the instrument. Then, if small weights be added cautiously, it
may be made to descend till it just grazes the agate, a contact which can
be made with the utmost imaginable delicacy. Let these weights be
|

134 Olu TLINES OF ASTRONOMY.
noted; the weight F detached; the spring G carefully lifted off its hook,
and secured, for travelling, from rust, strain, or disturbance, and the
whole apparatus conveyed to a station in a lower latitude. It will then
be found, on remounting it, that, although loaded with the same addi-
tional weights as before, the weight F will no longer have power enough
to stretch the spring to the extent required for producing a similar con-
tact. More weights will require to be added; and the additional quan-
tity necessary will, it is evident, measure the difference of gravity
between the two stations, as exerted on the whole quantity of pendent
matter, i. e. the sum of the weight F and half that of the spiral spring
itself. Granting that a spiral spring can be constructed of such strength
and dimensions that a weight of 10,000 grains, including its own, shall
produce an elongation of 10 inches without permanently straining it,'
one additional grain will produce a further extension of roºm oth of an
inch, a quantity which cannot possibly be mistaken in such a contact as
that in question. Thus we should be provided with the means of mea-
suring the power of gravity at any station to within Toºgoth of its whole
quantity.
(235.) The other, or dynamical process, by which the force urging any
given weight to the earth may be determined, consists in ascertaining the
velocity imparted by it to the weight when suffered to fall freely in a given
time, as one second. This velocity cannot, indeed, be directly measured;
but indirectly, the principles of mechanics furnish an easy and certain
means of deducing it, and, consequently, the intensity of gravity, by ob-
serving the oscillations of a pendulum. It is proved from mechanical
principles”, that, if one and the same pendulum be made to oscillate at
different stations, or under the influence of different forces, and the
numbers of oscillations made in the same time in each case be counted,
the intensities of the forces will be to each other as the squares of the
numbers of oscillations made, and thus their proportion becomes known.
For instance, it is found that, under the equator, a pendulum of a certain
form and length makes 86,400 vibrations in a mean solar day; and that,
when transported to London, the same pendulum makes 86,535 vibrations
* Whether the process above described could ever be so far perfected and refined as
to become a substitute for the use of the pendulum must depend on the degree of
permanence and uniformity of action of springs, on the constancy or variability of the
effect of temperature on their elastic force, on the possibility of transporting them,
absolutely unaltered, from place to place, &c. The great advantages, however,
which such an apparatus and mode of observation would possess, in point of conve-
nience, cheapness, portability, and expedition, over the present laborious, tedious, and
expensive process, render the attempt well worth making.
N 'wton's Principia, ii, Prop. 24, Cor. 3.
GRAVITY OF A SPHEROID 135
in the same time. Hence we conclude, that the intensity of the force
urging the pendulum downwards at the equator is to that at London as
(86,400)” to (86,535)”, or as 1 to 1:00315; or, in other words, that a
mass of matter weighing in London 100,000 pounds, exerts the same
pressure on the ground, or the same effort to crush a body placed below
it, that 100,315 of the same pounds transported to the equator would
exert there.
(236.) Experiments of this kind have been made, as above stated, with
the utmost care and minutest precaution to ensure exactness in all acces-
sible latitudes; and their general and final result has been, to give rºi
for the fraction expressing the difference of gravity at the equator and
poles. Now, it will not fail to be noticed by the reader, and will, pro-
bably, occur to him as an objection against the explanation here given of
the fact by the earth’s rotation, that this differs materially from the frac-
tion ºn expressing the centrifugal force at the equator. The difference
by which the former fraction exceeds the latter is gºo, a small quantity
in itself, but still far too large, compared with the others in question, not
to be distinctly accounted for, and not to prove fatal to this explanation
if it will not render a strict account of it.
(237.) The mode in which this difference arises affords a curious and
instructive example of the indirect influence which mechanical causes
often exercise, and of which astronomy furnishes innumerable instances.
The rotation of the earth gives rise to the centrifugal force; the centri-
fugal force produces an ellipticity in the form of the earth itself; and this
very ellipticity of form modifies its power of attraction on bodies placed
at its surface, and thus gives rise to the difference in question. Here,
then, we have the same cause exercising at once a direct and an indirect
influence. The amount of the former is easily calculated, that of the
latter with far more difficulty, by an intricate and profound application of
geometry, whose steps we cannot pretend to trace in a work like the pre-
sent, and can only state its nature and result.
(238.) The weight of a body (considered as undiminished by a centri-
fugal force) is the effect of the earth’s attraction on it. This attraction,
as Newton has demonstrated, consists, not in a tendency of all matter to
any one particular centre, but in a disposition of every particle of matter
in the universe to press towards, and if not opposed to approach to, every
other. The attraction of the earth, then, on a body placed on its surface,
is not a simple but a complex force, resulting from the separate attractions
of all its parts. Now, it is evident, that if the earth were a perfect sphere,
the attraction exerted by it on a body any where placed on its surface,
whether at its equator or pole, must be exactly alike, –for the simple
136 OUTLINES OF ASTRONOMY.
reason of the exact symmetry of the sphere in every direction. It is not
less evident that, the earth being elliptical, and this symmetry or simili-
tude of all its parts not existing, the same result cannot be expected. A
body placed at the equator, and a similar one at the pole of a flattened
ellipsoid, stand in a different geometrical relation to the mass as a whole.
This difference, without entering further into particulars, may be expected
to draw with it a difference in its forces of attraction on the two bodies.
Calculation confirms this idea. It is a question of purely mathematical
investigation, and has been treated with perfect clearness and precision
by Newton, Maclaurin, Clairaut, and many other eminent geometers; and
the result of their investigations is to show that, owing to the elliptic form
of the earth alone, and independent of the centrifugal force, its attraction
ought to increase the weight of a body in going from the equator to the
pole by almost exactly sºoth part; which, together with gºgth due to
the centrifugal force, make up the whole quantity, rhith, observed.
(239.) Another great geographical phenomenon, which owes its exis-
tence to the earth's rotation, is that of the trade-winds. These mighty
currents in our atmosphere, on which so important a part of navigation
depends, arise from, 1st, the unequal exposure of the earth's surface to
the sun's rays, by which it is unequally heated in different latitudes;
and, 2dly, from that general law in the constitution of all fluids, in virtue
of which they occupy a larger bulk, and become specifically lighter when
hot than when cold. These causes, combined with the earth's rotation
from west to east, afford an easy and satisfactory explanation of the mag-
nificent phenomena in question. -
(240.) It is a matter of observed fact, of which we shall give the
explanation farther on, that the sun is constantly vertical over some one
or other part of the earth between two parallels of latitude, called the
tropics, respectively 2349 north, and as much south of the equator; and
that the whole of that zone or belt of the earth's surface included between
the tropics, and equally divided by the equator, is, in consequence of the
great altitude attained by the sun in its diurnal course, maintained at a
much higher temperature than those regions to the north and south
which lie nearer the poles. Now, the heat thus acquired by the earth's
surface is communicated to the incumbent air, which is thereby expanded,
and rendered specifically lighter than the air incumbent on the rest of the
globe. It is therefore, in obedience to the general laws of hydrostatics,
displaced and buoyed up from the surface, and its place occupied by
colder, and therefore heavier air, which glides in, on both sides, along the
surface, from the regions beyond the tropics; while the displaced air, thus
TRADE WINDS. 137
raised above its due level, and unsustained by any lateral pressure, flows
over, as it were, and forms an upper current in the contrary direction, or
towards the poles; which, being cooled in its course, and also sucked
down to supply the deficiency in the extra-tropical regions, keeps up thus
a continual circulation.
(241.) Since the earth revolves about an axis passing through the
poles, the equatorial portion of its surface has the greatest velocity of
rotation, and all other parts less in the proportion of the radii of the
circles of latitude to which they correspond. But as the air, when rela-
tively and apparently at rest on any part of the earth's surface, is only so
because in reality it participates in the motion of rotation proper to that
part, it follows that when a mass of air near the poles is transferred to
the region near the equator by any impulse urging it directly towards that
circle, in every point of its progress towards its new situation it must be
found deficient in rotatory velocity, and therefore unable to keep up with
the speed of the new surface over which it is brought. Hence, the cur-
rents of air which set in towards the equator from the north and south
must, as they glide along the surface, at the same time lag, or hang back,
and drag upon it in the direction opposite to the earth’s rotation, i. e.
from east to west. Thus these currents, which but for the rotation would
be simply northerly and southerly winds, acquire, from this cause, a rela-
tive direction towards the west, and assume the character of permanent
north-easterly and south-easterly winds.
(242.) Were any considerable mass of air to be suddenly transferred
from beyond the topics to the equator, the difference of the rotatory velo-
cities proper to the two situations would be so great as to produce not
merely a wind, but a tempest of the most destructive violence. But this
is not the case: the advance of the air from the north and south is
gradual, and all the while the earth is continually acting on, and by the
friction of its surface accelerating its rotatory velocity. Supposing its
progress towards the equator to cease at any point, this cause would
almost immediately communicate to it the deficient motion of rotation,
after which it would revolve quietly with the earth, and be at relative
rest. We have only to call to mind the comparative thinness of the coat-
ing which the atmosphere forms around the globe (art. 35), and the im-
mense mass of the latter, compared with the former (which it exceeds at
least 100,000,000 times), to appreciate fully the absolute command of
any extensive territory of the earth over the atmosphere immediately
incumbent on it, in point of motion.
(243.) It follows from this, then, that as the winds on both sides ap-
©
138 OUTLINES OF ASTRONOMY.
proach the equator, their easterly tendency must diminish." The lengths
of the diurnal circles increase very slowly in the immediate vicinity
of the equator, and for several degrees on either side of it hardly change
at all. Thus the friction of the surface has more time to act in accelera-
ting the velocity of the air, bringing it towards a state of relative rest,
and diminishing thereby the relative set of the currents from east to west,
which, on the other hand, is feebly, and, at length, not at all reinforced
by the cause which originally produced it. Arrived, then, at the equator,
the trades must be expected to lose their easterly character altogether.
But not only this but the northern and southern currents here meeting
and opposing, will mutually destroy each other, leaving only such pre-
ponderancy as may be due to a difference of local causes acting in the two
hemispheres, – which in some regions around the equator may lie one
way, in some another.
(244.) The result, then, must be the production of two great tropical
belts, in the northern of which a constant north-easterly, and in the
southern a south-easterly, wind must prevail, while the winds in the
equatorial belt, which separates the two former, should be comparatively
calm and free from any steady prevalence of easterly character. All
these consequences are agreeable to observed fact, and the system of ačrial
currents above described constitutes in reality what is understood by the
regular trade winds.
(245.) The constant friction thus produced between the earth and at-
mosphere in the regions near the equator must (it may be objected) by
degrees reduce and at length destroy the rotation of the whole mass.
The laws of dynamics, however, render such a consequence, generally,
impossible; and it is easy to see, in the present case, where and how the
compensation takes place. The heated equatorial air, while it rises and
flows over towards the poles, carries with it the rotatory velocity due to
its equatorial situation into a higher latitude, where the earth’s surface
has less motion. Hence, as it travels northward or southward, it will
gain continually more and more on the surface of the earth in its diurnal
motion, and assume constantly more and more a westerly relative direc-
tion; and when at length it returns to the surface, in its circulation,
which it must do more or less in all the interval between the tropics and
the poles, it will act on it by its friction as a powerful south-west wind in
the northern hemisphere, and a north-west in the Southern, and restore to
it the impulse taken up from it at the equator. We have here the origin
* See Captain Hall’s “Fragments of Voyages and Travels,” 2d series, vol. i. p.
162, where this is very distinctly, and, so far as I am aware, for the first time, reasoned
nut.
©
DETERMINATION OF LATITUDES. 139
of the south-west and westerly gales so prevalent in our latitudes, and of
the almost universal westerly winds in the North Atlantic, which are, in
fact, nothing else than a part of the general system of the re-action of the
trades, and of the process by which the equilibrium of the earth's mo-
tion is maintained under their action."
(246.) In order to construct a map or model of the earth, and obtain
a knowledge of the distribution of sea and land over its surface, the forms
of the outlines of its continents and islands, the courses of its rivers and
mountain chains, and the relative situations, with respect to each other,
of those points which chiefly interest us, as centres of human habitation,
or from other causes, it is necessary to possess the means of determining
correctly the situation of any proposed station on its surface. For this
two elements require to be known, the latitude and longitude, the former
assigning its distance from the poles or the equator, the latter, the meri-
dian on which that distance is to be reckoned. To these, in strictness,
should be added, its height above the sea level; but the consideration of
this had better be deferred, to avoid complicating the subject.
(247.) The latitude of a station on a sphere would be merely the
length of an arc of the meridian, intercepted between the station and the
nearest point of the equator, reduced into degrees. (See art. 88.) But
as the earth is elliptic, this mode of conceiving latitudes becomes inappli-
cable, and we are compelled to resort for our definition of latitude to a
generalization of that property (art. 119,) which affords the readiest
means of determining it by observation, and which has the advantage of
being independent of the figure of the earth, which, after all, is not
exactly an ellipsoid, or any known geometrical solid. The latitude of a
station, then, is the altitude of the elevated pole, and is, therefore, astro-
nomically determined by those methods already explained for ascertaining
* As it is our object merely to illustrate the mode in which the earth's rotation affects
the atmosphere on the great scale, we omit all consideration of local periodical winds,
such as monsoons, &c.
It seems worth inquiry, whether hurricanes in tropical climates may not arise from
portions of the upper currents prematurely diverted downwards before their relative
velocity has been sufficiently reduced by friction on, and gradual mixing with, the
lower strata; and so dashing upon the earth with that tremendous velocity which gives
them their destructive character, and of which hardly any rational account has yet
been given, But it by no means follows that this must always be the case. In
general, a rapid transfer, either way, in latitude, of any mass of air which local or
temporary causes might carry above the immediate reach of the friction of the earth's
surface, would give a fearful exaggeration to its velocity. Wherever such a mass
should strike the earth, a hurricane might arise; and should two such masses encoun
ter in mid air, a tornado of any degree of intensity on record might easily result from
their combination. *
140 OUTLINES OF ASTRONOMY.
\
that important element. In consequence, it will be remembered that, to
make a perfectly correct map of the whole, or any part of the earth’s
surface, equal differences of latitude are not represented by exactly equal
intervals of surface.
(248.) For the purposes of geodesical' measurements and trigonome-
trical surveys, an exceedingly correct determination of the latitudes of the
most important stations is required. For this purpose, therefore, the
zenith sector (an instrument capable of great precision) is most commonly
used to observe stars passing the meridian near the zenith, whose declina-
tions have become known by previous long series of observations at fixed
observatories, and which are therefore called standard or fundamental
stars. Iłecently a method” has been employed with great success, which
Fig. 37.
C
º
º/
__
ID
consists in the use of an instrument similar in every respect to the transit
instrument, but having the plane of motion of the telescope not coinci-
dent with the meridian, but with the prime vertical, so that its axis of
rotation prolonged passes through the north and south points of the
horizon. Let A B C D be the celestial hemisphere projected on the
horizon, P the pole, Z the zenith, A B the meridian, C D the prime
vertical, Q R S part of the diurnal circle of a star passing near the
zenith, whose polar distance P R is but little greater than the co-latitude
of the place, or the arc PZ, between the zenith and pole (art. 112.)
Then the moments of this star's arrival on the prime vertical at Q and S
*Tn, the earth; 3ects (from Čew, to bind,) a joining or connection (of parts.)
* Devised originally by Römer. Revived cr re-invented by Bessel.--A str. Nachr.
No. 40.

DETERMINATION OF LATITUDES. 141
will, if the instrument be correctly adjusted, be those of its crossing the
middle wire in the field of view of the telescope (art. 160.) Conse-
quently the interval between these moments will be the time of the star
passing from Q to S, or the measure of the diurnal arc Q R. S, which
corresponds to the angle Q P S at the pole. This angle, therefore, be-
comes known by the mere observation of an interval of time, in which it
is not even necessary to know the error of the clock, and in which, when
the star passes near the zenith, so that the interval in question is small,
even the rate of the clock, or its gain or loss on true sidereal time, may
be neglected. Now the angle Q PS, or its half Q PR, and P Q the
polar distance of the star, being known, PZ the zenith distance of the
pole can be calculated by the resolution of the right angled spherical
triangle PZ Q, and thus the co-latitude (and of course the latitude) of
the place of observation becomes known. The advantages gained by this
mode of observation afe, 1st, that no readings of a divided arc are needed,
so that errors of graduation and reading are avoided: 2dly, that the
are Q R S is very much greater than its versed sine R. Z, so that the
difference R Z between the latitude of the place and the declination of
the star is given by the observation of a magnitude very much greater
than itself, or is, as it were, observed on a greatly enlarged scale. In
consequence, a very minute error is entailed on R Z by the commission
of even a considerable ong in Q R S : 3dly, that in this mode of obser-
vation all the merely instrumental errors which affect the ordinary use of
the transit instrument are either uninfluential or eliminated by simply
reversing the axis.
(249.) To determine the latitude of a station, then, is easy. It is
otherwise with its longitude, whose exact determination is a matter of more
difficulty. The reason is this: — as there are no meridians marked upon
the earth, any more than parallels of latitude, we are obliged in this case,
as in the case of the latitude, to resort to marks external to the earth, i. e.
to the heavenly bodies, for the objects of our measurement; but with this
difference in the two cases — to observers situated at stations on the same
meridian (i. e. differing in latitude) the heavens present different aspects
at all moments. The portions of them which become visible in a com-
plete diurnal rotation are not the same, and stars which are common to
both describe circles differently inclined to their horizons, and differently
divided by them, and attain different altitudes. On the other hand, to
observers situated on the same parallel (i. e. differing only in longitude)
the heavens present the same aspects. Their visible portions are the
same; and the same stars describe circles equally inclined, and similarly
divided by their horizons, and attain the same altitudes. In the former
*
142 OTTLINES OF ASTRONOMY.
case there is, in the latter there is not, any thing in the appearance of the
heavens, watched through a whole diurnal rotation, which indicates a dif-
ference of locality in the observer.
(250.) But not two observers, at different points of the earth's surface,
can have at the same instant the same celestial hemisphere visible. Sup-
pose, to fix our ideas, an observer stationed at a given point of the equator,
and that at the moment when he noticed some bright star to be in his
zenith, and therefore on his meridian, he should be suddenly transported,
in an instant of time, round one quarter of the globe in a westerly direction,
it is evident that he will no longer have the same star vertically above
him : it will now appear to him to be just rising, and he will have to wait
six hours before it again comes to his zenith, i.e. before the earth's rota-
tion from west to east carries him back again to the line joining the star
and the earth's centre from which he set out.
(251.) The difference of the cases, then, may be thus stated, so as to
afford a key to the astronomical solution of the problem of the longitude.
In the case of stations differing only in latitude, the same star comes to
the meridian at the same time, but at different altitudes. In that of
stations differing only in longitude, it comes to the meridian at the same
altitude, but at different times. Supposing, then, that an observer is in
possession of any means by which he can certainly ascertain the time of a
known star’s transit across his meridian, he knows his longitude; or if he
knows the difference between its time of transit across his meridian and
across that of any other station, he knows their difference of longitudes.
For instance, if the same star pass the meridian of a place A at a certain
moment, and that of B exactly one hour of sidereal time, or one twenty-
fourth part of the earth's diurnal period, later, then the difference of lon-
gitude between A and B is one hour of time or 15° of arc, and B is so
much west of A.
(252.) In order to a perfectly clear understanding of the principle on
which the problem of finding the longitude by astronomical observations
is resolved, the reader must learn to distinguish between time, in the
abstract, as common to the whole universe, and therefore reckoned from
an epoch independent of local situation, and local time, which reckons, at
each particular place, from an epoch, or initial instant, determined by local
convenience. Of time reckoned in the former, or abstract manner, we
have an example in what we have before defined as equinoctial time, which
dates from an epoch determined by the sun’s motion among the stars.
Of the latter, or local reckoning, we have instances in every sidereal clock in
an observatory, and in every town clock for common use. Every astrono-
mer regulates, or aims at regulating, his sidereal clock, so that it shall
DETERMINATION OF I. ONGITUDES. 143
indicate 0° 0" 0 ,when a certain point in the heavens, called the equinox,
is on the meridian of his station. This is the epoch of his sidereal time;
which is, therefore, entirely a local reckoning. It gives no information to
say that an event happened at such and such an hour of sidereal time,
unless we particularize the station to which the sidereal time meant apper-
tains. Just so it is with mean or common time. This is also a local
reckoning, having for its epoch mean noon, or the average of all the
times throughout the year, when the sun is on the meridian of that par-
ticular place to which it belongs; and, therefore, in like manner, when
we date any event by mean time, it is necessary to name the place, or
particularize what mean time we intend. On the other hand, a date
by equinoctial time is absolute, and requires no such explanatory addition.
(253.) The astronomer sets and regulates his sidereal clock by observ-
ing the meridian passages of the more conspicuous and well-known stars.
Each of these holds in the heavens a certain determinate and known place
with respect to that imaginary point called the equinox, and by noting the
times of their passage in succession by his clock he knows when the equi-
nox passed. At that moment his clock ought to have marked 0° 0" 0";
and if it did not, he knows and can correct its error, and by the agreement
or disagreement of the errors assigned by each star he can ascertain
whether his clock is correctly regulated to go twenty-four hours in one
diurnal period, and if not, can ascertain and allow for its rate. Thus,
although his clock may not, and indeed cannot, either be set correctly, or
go truly, yet by applying its error and rate (as they are technically
termed), he can correct its indications, and ascertain the exact sidereal
times corresponding to them, and proper to his locality. This indispensa-
ble operation is called getting his local time. For simplicity of explana-
tion, however, we shall suppose the clock a perfect instrument; or, which
comes to the same thing, its error and rate applied at every moment it is
consulted, and included in its indications. -
(254.) Suppose, now, of two observers, at distant stations, A and B,
cach, independently of the other, to set and regulate his clock to the true
sidereal time of his station. It is evident that if one of these clocks
could be taken up without deranging its going, and set down by the side
of the other, they would be found, on comparison, to differ by the exact
difference of their local epochs; that is, by the time occupied by the equi.
nox, or by any star, in passing from the meridian of A to that of B; in
other words, by their difference of longitude, expressed in sidereal hours,
minutes, and seconds. - f
(255.) A pendulum clock cannot be thus taken up and transported
from place to place without derangement, but a chronometer may. Sup
144 OUTLINES OF ASTRONOMY.
pose, then, the observer at B to use a chronometer instead of a clock, he
may, by bodily transfer of the instrument to the other station, procure a
direct comparison of sidereal times, and thus obtain his longitude from A.
And even if he employ a clock, yet by comparing it first with a good
chronometer, and then transferring the latter instrument for comparison
with the other clock, the same end will be accomplished, provided the
going of the chronometer can be depended on.
(256.) Were chronometers perfect, nothing more complete and conve-
nient than this mode of ascertaining differences of longitude could be
desired. An observer, provided with such an instrument, and with a por-
table transit, or some equivalent method of determining the local time at
any given station, might, by journeying from place to place, and observing
the meridian passages of stars at each, (taking care not to alter his chro-
nometer, or let it run down,) ascertain their differences of longitude with
any required precision. In this case, the same time-keeper being used at
every station, if, at one of them, A, it mark true sidereal time, at any
other, B, it will be just so much sidereal time in error as the difference of
longitudes of A and B is equivalent to : in other words, the longitude of
B from A will appear as the error of the time-keeper on the local time of
B. If he travel westward, then his chronometer will appear continually
to gain, although it really goes correctly. Suppose, for instance, he set
out from A, when the equinox was on the meridian, or his chronometer at
0h, and in twenty-four hours (sid. time) had travelled 15° westward to B.
At the moment of arrival there, his chronometer will again point to 0h;
but the equinox will be, not on his new meridian, but on that of A, and
he must wait one hour more for its arrival at that of B. When it does ar-
rive there, then his watch will point not to 0h but to 1", and will therefore
be 1.h fast on the local time of B. If he travel eastward, the reverse will
happen.
(257.) Suppose an observer now to set out from any station as above
described, and constantly travelling westward to make a tour of the globe,
and return to the point he set out from. A singular consequence will
happen: he will have lost a day in his reckoning of time. He will enter
the day of his arrival in his diary, as Monday, for instance, when, in fact,
it is Tuesday. The reason is obvious. Days and nights are caused by the
alternate appearance of the sun and stars, as the rotation of the earth car-
ries the spectator round to view them in succession. So many turns as
he makes absolutely round the centre, so often will he pass through the
earth's shadow, and emerge into light, and so many nights and days will
he experience. But if he travel once round the globe in the direction of
its motion, he will, on his arrival, have really made one turn more round
DETERMINATION OF LONGITUDES. 3' 145
its centre; and if in the opposite direction, one turn less than if he had
remained upon one point of its surface: in the former case, then, he will
have witnessed one alternation of day and night more, in the latter one
less, than if he had trusted to the rotation of the earth alone to carry him
round. As the earth revolves from west to east, it follows that a westward
direction of his journey, by which he counteracts its rotation, will cause
him to lose a day, and an eastward direction, by which he conspires with
it, to gain one. In the former case, all his days will be longer; in the
latter, shorter than those of a stationary observer. This contingency has
actually happened to circumnavigators. Hence, also, it must necessarily
happen that distant settlements, on the same meridian, will differ a day
in their usual reckoning of time, according as they have been colonized by
settlers arriving in an eastward or in a westward direction,--a circumstance
which may produce strange confusion when they come to communicate
with each other. The only mode of correcting the ambiguity, and settling
the disputes which such a difference may give rise to, consists in having
recourse to the equinoctial date, which can never be ambiguous.
(258.) Unfortunately for geography and navigation, the chronometer,
though greatly and indeed wonderfully improved by the skill of modern
artists, is yet far too imperfect an instrument to be relied on implicitly.
However such an instrument may preserve its uniformity of rate for a
few hours, or even days, yet in long absences from home the chances of
error and accident become so multiplied, as to destroy all security of reli-
ance on even the best. To a certain extent this may, indeed, be remedied
by carrying out several, and using them as checks on each other; but,
besides the expense and trouble, this is only a palliation of the evil—
the great and fundamental,—as it is the only one to which the determina-
tion of longitudes by time-keepers is liable. It becomes necessary, there-
fore, to resort to other means of communicating from one station to another
a knowledge of its local time, or of propagating from some principal sta-
tion, as a centre, its local time as a universal standard with which the
local time at any other, however situated, may be at once compared, and
thus the longitudes of all places be referred to the meridian of such cen-
tral point. *
(259.) The simplest and most accurate method by which this object
can be accomplished, when circumstances admit of its adoption, is that by
telegraphic signal. Let A and B be two observatories, or other stations,
provided with accurate means of determining their Tespective local times,
and let us first suppose them visible from each other. Their clocks being
regulated, and their errors and rates ascertained and applied, let a signal
be made at A, of some sudden and definite kind, such as the flash of gun.
10
146 OUTLINES OF ASTRONOMY.
powder, the explosion of a rocket, the sudden extinction of a bright light,
or any other which admits of no mistake, and can be seen at great dis-
tances. The moment of the signal being made must be noted by each
observer at his respective clock or watch, as if it were the transit of a star,
or other astronomical phenomenon, and the error and rate of the clock at
each station being applied, the local time of the signal at each is deter-
mined. Consequently, when the observers communicate their observations
of the signal to each other, since (owing to the almost instantaneous
transmission of light) it must have been seen at the same absolute instant
by both, the difference of their local times, and therefore of their longitudes,
becomes known. For example, at A the signal is observed to happen at
5* 0” 0° sid. time at A, as obtained by applying the error and rate to the
time shown by the clock at A, when the signal was seen there. At B the
same signal was seen at 5' 4" O', sid. time at B, similarly deduced from
the time noted by the clock at B, by applying its error and rate. Conse-
quently, the difference of their local epochs is 4" 0", which is also their
difference of longitudes in time, or 1° 0' 0" in hour angle. .
(260.) The accuracy of the final determination may be increased by
making and observing several signals at stated intervals, each of which
affords a comparison of times, and the mean of all which is, of course,
more to be depended on than the result of any single comparison. By
this means, the error introduced by the comparison of clocks may be re-
garded as altogether destroyed.
(261.) The distances at which signals can be rendered visible must of
course depend on the nature of the interposed country. Over sea the
explosion of rockets may easily be seen at fifty or sixty miles; and in
mountainous countries the flash of gunpowder in an open spoon may be
seen, if a proper station be chosen for its exhibition, at much greater
distances.
(262.) When the direct light of the flash can no longer be perceived,
either owing to the convexity of the interposed segment of the earth, or
to intervening obstacles, the sudden illumination cast on the under surface
of the clouds by the explosion of considerable quantities of powder may
often be observed with success; and in this way signals have been made
at very much greater distances. Whatever means can be devised of exci-
ting in two distant observers the same sensation, whether of sound, light,
or visible motion, at precisely the same instant of time, may be employed
as a longitude signal. Wherever, for instance, an unbroken line of elec-
tro-telegraphic connection has been, or hereafter may be, established, the
means exist of making as complete a comparison of clocks or watches as
if they stood side by side, so that no method more complete for the deter-
DETERMINATION OF LONGITUDES. 147
mination of differences of longitude can be desired. The differences of
longitude between the observatories of New York, Washington, and Phila-
delphia, have been very recently determined in this manner by the astro.
nomers at those observatories.
(263.) Where no such electric communication exists, however, the
interval between observing stations may be increased by causing the
signals to be made not at one of them, but at an intermediate point; for,
provided they are seen by both parties, it is a matter of indifference where
they are exhibited. Still the interval which could be thus embraced
would be very limited, and the method in consequence of little use, but
for the following ingenious contrivance, by which it can be extended to
any distance, and carried over any tract of country, however difficult.
(264.) This contrivance consists in establishing, between the extreme
stations, whose difference of longitude is to be ascertained, and at which
the local times are observed, a chain of intermediate stations, alternately
destined for signals and for observers. Thus, let A and Z be the extreme
stations. At B let a signal station be established, at which rockets, &c.
are fired at stated intervals. At C let an observer be placed, provided
with a chronometer; at D, another signal station; at E, another observer
and chronometer; till the whole line is occupied by stations so arranged,
that the signal at B can be seen from A and C; those at D, from C and
E; and so on. Matters being thus arranged, and the errors and rates of
Fig. 38.
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TD
IE T Z
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the clocks at A and Z ascertained by astronomical observation, let a signal
be made at B, and observed at A and C, and the times noted. Thus the
difference between A's clock and C's chronometer becomes known. After
a short interval (five minutes for instance) let a signal be made at D, and
observed by C and E. Then will the difference between their respective
chronometers be determined; and the difference between the former and
the clock at A being already ascertained, the difference between the clock
A and chronometer E is therefore known. This, however, supposes that
the intermediate chronometer C has kept true sidereal time, or at least a
known rate, in the interval between the signals. Now this interval is
purposely made so very short, that no instrument of any pretensions to
148 OUTLINES OF ASTRONOMY. *
ſharacter can possibly produce an appreciable amount of error in its lapse
by deviations from its usual rate. Thus the time propagated from A to
C may be considered as handed over, without gain or loss (save from error
of observation), to E. Similarly, by the signal made at F, and observed
at E and Z, the time so transmitted to E is forwarded on to Z; and thus
at length the clocks at A and Z are compared. The process may be
repeated as often as is necessary to destroy error by a mean of results;
and when the line of stations is numerous, by keeping up a succes-
sion of signals, so as to allow each observer to note alternately those on
either side, which is easily pre-arranged, many comparisons may be kept
running along the line at once, by which time is saved, and other advan-
tages obtained." In important cases the process is usually repeated on
several nights in succession.
(265.) In place of artificial signals, natural ones, when they occur
sufficiently definite for observation, may be equally employed. In a clear
night the number of those singular meteors, called shooting stars, which
may be observed, is often very great, especially on the 9th and 10th of
August, and some other days, as November 12 and 13; and as they are
sudden in their appearance and disappearance, and from the great height
at which they have been ascertained to take place are visible over exten-
sive regions of the earth’s surface, there is no doubt that they may be
resorted to with advantage, by previous concert and agreement between
distant observers to watch and note them.” Those sudden disturbances
of the magnetic needle, to which the name of magnetic shocks has been
given, have been satisfactorily ascertained to be, very often at least,
simultaneous over whole continents, and in some, perhaps, over the whole
globe. These, if observed at magnetic observatories with precise atten-
tion to astronomical time, may become the means of determining their
differences of longitude with more precision, possibly, than by any other
method, if a sufficient number of remarkable shocks be observed to
ascertain their identity, about which the intervals of time between their
occurrence (exactly alike at both stations) will leave no doubt.
* For a complete account of this method, and the mode of deducing the most advan-
tageous result from a combination of all the observations, see a paper on the difference
of longitudes of Greenwich and Paris, Phil. Trans. 1826; by the Author of this
volume.
* This idea was first suggested by the late Dr. Maskelyne, to whom, however, the
practically useful fact of their periodic recurrence was unknown. Mr. Cooper has thus
employed the meteors of the 10th and 12th August, 1847, to determine the difference
of longitudes of Markree and Mount Eagle, in Ireland. Those of the same epoch have
also been used in Germany for ascertaining the longitudes of several stations, and with
very satisfactory results.
LUNAR METHOD. 149
(266.) Another species of natural signal, visible at once over a whole
terrestrial hemisphere, is afforded by the eclipses of Jupiter's satellites, of
which we shall speak more at large when we come to treat of those bodies.
Every such eclipse is an event which possesses one great advantage in its
applicability to the purpose in question, viz. that the time of its happen-
ing, at any fixed station, such as Greenwich, can be predicted from a long
course of previous recorded observation and calculation thereon founded,
and that this prediction is sufficiently precise and certain, to stand in the
place of a corresponding observation. So that an observer at any other
station wherever, who shall have observed one or more of these eclipses,
and ascertained his local time, instead of waiting for a communication
with Greenwich, to inform him at what moment the eclipse took place
there, may use the predicted Greenwich time instead, and thence, at
once, and on the spot, determine his longitude. This mode of ascertain-
ing longitudes is, however, as will hereafter appear, not susceptible of
great exactness, and should only be resorted to when others cannot be
had. The nature of the observation also is such that it cannot be made
at sea"; so that, however useful to the geographer, it is of no advantage
to navigation. r
(267.) But such phenomena as these are of only occasional occurrence;
and in their intervals, and when cut off from all communication with any
fixed station, it is indispensable to possess some means of determining
longitudes, on which not only the geographer may rely for a knowledge
of the exact position of important stations on land in remote regions, but
on which the navigator can securely stake, at every instant of his adven-
turous course, the lives of himself and comrades, the interests of his
country, and the fortunes of his employers. Such a method is afforded
by LUNAR OBSERVATIONS. Though we have not yet introduced the
reader to the phenomena of the moon's motion, this will not prevent us
from giving here the exposition of the principle of the lunar method; on
the contrary, it will be highly advantageous to do so, since by this course
we shall have to deal with the naked principle, apart from all the peculiar
sources of difficulty with which the lunar theory is encumbered, but
* To accomplish this is still a desideratum. Observing chairs, suspended with stu
dious precaution for ensuring freedem of motion, have been resorted to, under the vain
hope of mitigating the effect of the ship's oscillation. The opposite course seems more
promising, viz. to merely deaden the motion by a somewhat stiff suspension (as by a
coarse and rough cable), and by friction strings attached to weights running through
loops (not pulleys) fixed in the wood-work of the vessel. At least, such means have
been found by the author of singular efficacy in increasing personal comfort in the sus.
pension of a cot.
150 OUTLINES OF ASTRONOMY.
which are, in fact, completely extraneous to the principle of its applica-
tion to the problem of the longitudes, which is quite elementary. -
(268.) If there were in the heavens a clock furnished with a dial-plate
and hands, which always marked Greenwich time, the longitude of any
station would be at once determined, so soon as the local time was known,
by comparing it with this clock. Now, the offices of the dial-plate and
hands of a clock are these : — the former carries a set of marks upon it,
whose position is known; the latter, by passing over and among these
marks, inform us, by the place it holds with respect to them, what it is
o'clock, or what time has elapsed since a certain moment when it stood at
one particular spot.
(269.) In a clock the marks on the dial-plate are uniformly distributed
all around the circumference of a circle, whose centre is that on which the
hands revolve with a uniform motion. But it is clear that we should, with
equal certainty, though with much more trouble, tell what o'clock it were,
if the marks on the dial-plate were unequally distributed,—if the hands
were eaccentric, and their motion not uniform, provided we knew, 1st,
the exact intervals round the circle at which the hour and minute marks
were placed; which would be the case if we had them all registered in a
table, from the results of previous careful measurement: —2dly, if we
knew the exact amount and direction of excentricity of the centre of mo-
tion of the hands; — and 3dly, if we were fully acquainted with all the
mechanism which put the hands in motion, so as to be able to say at every
instant what were their velocity of movement, and so as to be able to cal-
culate, without fear of error, HOW MUCH time should correspond to so
MUCH angular movement.
(270.) The visible surface of the starry heavens is the dial-plate of our
clock, the stars are the fixed marks distributed around its circuit, the moon
is the moveable hand, which, with a motion that, superficially considered,
seems uniform, but which, when carefully examined, is found to be far
otherwise, and which, regulated by mechanical laws of astonishing com-
plexity and intricacy in result, though beautifully simple in principle and
design, performs a monthly circuit among them, passing visibly over and
hiding, or, as it is called, occulting some, and gliding beside and between
others; and whose position among them can, at any moment when it is
visible, be exactly measured by the help of a sextant, just as we might
measure the place of our clock-hand among the marks on its dial-plate
with a pair of compasses, and thence, from the known and calculated laws
of its motion, deduce the time. That the moon does so move among the
stars, while the latter hold constantly, with respect to each other, the same
LUNAR METHOD, & 151
relative position, the notice of a few nights, or even hours, will satisfy the
commencing student, and this is all that at present we require.
(271.) There is only one circumstance wanting to make our analogy
complete. Suppose the hands of our clock, instead of moving quite close
to the dial-plate, were considerably elevated above, or distant in front of
of it. Unless, then, in viewing it, we kept our eye just in the line of
their centre, we should not see them exactly thrown or projected upon
their proper places on the dial. And if we were either unaware of this
cause of optical change of place, this parallaa – or negligent in not
taking it into account—we might make great mistakes in reading the
time, by referring the hand to the wrong mark, or incorrectly appreciating
its distance from the right. On the other hand, if we took care to note,
in every case when we had occasion to observe the time, the exact posi-
tion of the eye, there would be no difficulty in ascertaining and allowing
for the precise influence of this cause of apparent displacement. Now,
this is just what obtains with the apparent motion of the moon among
the stars. The former (as will appear) is comparatively near to the earth
— the latter immensely distant; and in consequence of our not occupy-
ing the centre of the earth, but being carried about on its surface, and
constantly changing place, there arises a parallaa, which displaces the
moon apparently among the stars, and must be allowed for before we can
tell the true place she would occupy if seen from the centre. •
(272.) Such a clock as we have described might, no doubt, be con-
sidered a very bad one; but if it were our only one, and if incalculable
interests were at stake on a perfect knowledge of time, we should justly
regard it as most precious, and think no pains ill bestowed in studying
the laws of its movements, or in facilitating the means of reading it
correctly. Such, in the parallel we are drawing, is the lunar theory,
whose object is to reduce to regularity, the indications of this strangely
irregular-going clock, to enable us to predict, long beforehand, and with
absolute certainty, whereabouts among the stars, at every hour, minute,
and second, in every day of every year, in Greenwich local time, the
moon would be seen from the earth's centre, and will be seen from every
accessible point of its surface; and such is the lunar method of longi-
tudes. The moon’s apparent angular distance from all those principal
and conspicuous stars which lie in its course, as seen from the earth’s
centre, are computed and tabulated with the utmost care and precision in
almanacks published under national control. No sooner does an observer,
in any part of the globe, at sea or on land, measure its actual distance
from any one of those standard stars (whose places in the heavens have
been ascertained for the purpose with the most anxious solicitude,) than
152 OUTLINES OF ASTRONOMY.
he has, in fact, performed that comparison of his local time with the
local times of every observatory in the world, which enables him to as-
certain his difference of longitude from one or all of them.
(273.) The latitudes and longitudes of any number of points on the
earth's surface may be ascertained by the methods above described; and
by thus laying down a sufficient number of principal points, and filling in
the intermediate spaces by local surveys, might maps of countries be
constructed. In practice, however, it is found simpler and easier to
divide each particular nation into a series of great triangles, the angles
of which are stations conspicuously visible from each other. Of these
triangles, the angles only are measured by means of the theodolite, with
the exception of one side only of one triangle, which is called a base,
and which is measured with every refinement which ingenuity can devise
or expense command. This base is of moderate extent, rarely surpassing
six or seven miles, and purposely selected in a perfectly horizontal plane,
otherwise conveniently adapted to the purposes of measurement. Its
length between its two extreme points (which are dots on plates of gold
or platina let into massive blocks of stone, and which are, or at least
ought to be, in all cases preserved with almost religious care, as monu-
mental records of the highest importance,) is then measured, with every
precaution to ensure precision,' and its position with respect to the
meridian, as well as the geographical positions of its extremities, carefully
ascertained.
(274.) The annexed figure represents such a chain of triangles. A B
Fig. 38.
T->\/4%
s + ||
is the base, O, C, stations visible from both its extremities (one of which,
O, we will suppose to be a national observatory, with which it is a prin-
cipal object that the base should be as closely and immediately connected
as possible;) and D, E, F, G, H, K, other stations, remarkable points in
* The greatest possible error in the Irish base of between seven and eight miles,
near Londonderry, is supposed not to exceed two inches.

CONSTRUCTION OF MAPS. 153
the country, by whose connection its whole surface may be covered, as it
were, with a network of triangles. Now, it is evident that the angles of
the triangle A, B, C being observed, and one of its sides, A, B, mea-
sured, the other two sides, A C, B C, may be calculated by the rules of
trigonometry; and thus each of the sides A C and B C becomes in its
turn a base capable of being employed as known sides of other triangles.
For instance, the angles of the triangles A C G and B C F being known
by observation, and their sides A C and B C, we can thence calculate the
lengths A G, C, G, and B F, C F. Again, C G and C F being known
and the included angle G C F, G F may be calculated, and so on. Thus
may all the stations be accurately determined and laid down, and as this
process may be carried on to any extent, a map of the whole country
may be thus constructed, and filled in to any degree of detail we please.
(275.) Now, on this process there are two important remarks to be
made. The first is, that it is necessary to be careful in the selection of
stations, so as to form triangles free from any very great inequality in
their angles. For instance, the triangle KB F would be a very improper
one to determine the situation of F from observations at B and K, because
the angle F being very acute, a small error in the angle K would produce
a great one in the place of F upon the line B F. Such ill-conditioned
triangles, therefore, must be avoided. But if this be attended to, the
accuracy of the determination of the calculated sides will not be much
short of that which would be obtained by actual measurement (were it
practicable); and, therefore, as we recede from the base on all sides as a
centre, it will speedily become practicable to use as bases, the sides of much
larger triangles, such as G F, G, H, H K, &c.; by which means the next
step of the operation will come to be carried on on a much larger scale,
and embrace far greater intervals, than it would have been safe to do (for
the above reason) in the immediate neighbourhood of the base. Thus it
becomes easy to divide the whole face of a country into great triangles of
from 30 to 100 miles in their sides (according to the nature of the ground),
which, being once well determined, may be afterwards, by a second series
of subordinate operations, broken up into smaller ones, and these again
into others of a still minuter order, till the final filling in is brought within
the limits of personal survey and draftsmanship, and till a map is con-
structed, with any required degree of detail.
(276.) The next remark we have to make is, that all the triangles in
question are not, rigorously speaking, plane, but spherical—existing on
the surface of a sphere, or rather, to speak correctly, of an ellipsoid. In
very small triangles, of six or seven miles in the side, this may be
neglected, as the difference is imperceptible; but in the larger ones it
154 OUTLINES (; F ASTRONOMY.
must be taken into consideration. It is evident that, as every object used
for pointing the telescope of a theodolite has some certain elevation, not
only above the soil, but above the level of the sea, and as, moreover,
these elevations differ in every instance, a reduction to the horizon of all
the measured angles would appear to be required. But, in fact, by the
construction of the theodolite (art. 192), which is nothing more than an
altitude and azimuth instrument, this reduction is made in the very act
Fig. gº. ! 2
-/O.”
§ -Ty
%
of reading off the horizontal angles. Let E be the centre of the earth;
A, B, C, the places on its spherical surface, to which three stations, A,
P, Q, in a country are referred by radii E A, E B P, EC Q. If a theo-
dolite be stationed at A, the axis of its horizontal circle will point to E
when truly adjusted, and its plane will be a tangent to the sphere at A,
intersecting the radii E B P, EC Q, at M and N, above the spherical
surface. The telescope of the theodolite, it is true, is pointed in succes-
sion to P, and Q; but the readings off of its azimuth circle give—not
the angle P A Q, between the directions of the telescope, or between the
objects P, Q, as seen from A.; but the azimuthal angle M A N, which is
the measure of the angle A of the spherical triangle B A C. Hence
arises this remarkable circumstance,—that the sum of the three observed
angles of any of the great triangles in geodesical operations is always
found to be rather more than 180°. Were the earth’s surface a plane, it
ought to be exactly 180°; and this eaccess, which is called the spherical
excess, is so far from being a proof of incorrectness in the work, that it is
essential to its accuracy, and offers at the same time another palpable
proof of the earth’s sphericity.
(277.) The true way, then, of conceiving the subject of a triognomet-
rical survey, when the spherical form of the earth is taken into considera-



CONSTRUCTION OF MAPS. 155
tion, is to regard the network of triangles with which the country is
covered, as the bases of an assemblage of pyramids converging to the
centre of the earth. The theodolite gives us the true measures of the
angles included by the planes of these pyramids; and the surface of an
imaginary sphere on the level of the sea intersects them, in an assemblage
of spherical triangles, above whose angles, in the radii prolonged, the real
stations of observation are raised, by the Superficial inequalities of moun-
tain and valley. The operose calculations of spherical trigonometry which
this consideration would seem to render necessary for the reductions of a
survey, are dispensed with in practice by a very simple and easy rule,
called the rule for the spherical eaccess, which is to be found in most works
on trigonometry. If we would take into account the ellipticity of the
earth, it may also be done by appropriate processes of calculation, which,
however, are too abstruse to dwell upon in a work like the present.
(278.) Whatever process of calculation we adopt, the result will be a
reduction to the level of the sea, of all the triangles, and the consequent
determination of the geographical latitude and longitude of every station
observed. Thus we are at length enabled to construct maps of countries;
to lay down the outlines of continents and islands; the courses of rivers;
the places of cities, towns and villages; the direction of mountain ridges,
and the places of their principal summits; and all those details which, as
they belong to physical and statistical, rather than to astronomical geog-
raphy, we need not here dilate on. A few words, however, will be neces-
sary respecting maps, which are used as well in astronomy as in geog-
raphy.
(279.) A map is nothing more than a representation, upon a plane, of
some portion of the surface of a sphere, on which are traced the particu-
lars intended to be expressed, whether they be continuous outlines or
points. Now, as a spherical surface' can by no contrivance be extended
or projected into a plane, without undue enlargement or contraction of
some parts in proportion to others; and as the system adopted in so ex-
tending or projecting it will decide what parts shall be enlarged or rela-
tively contracted, and in what proportions; it follows, that when large
portions of the sphere are to be mapped down, a great difference in their
representations may subsist, according to the system of projection adopted.
(280.) The projections chiefly used in maps, are the orthographic,
stereographic, and Mercator's. In the orthographic projection, every
point of the hemisphere is referred to its diametral plane or base, by a
* We here neglect the ellipticity of the earth, which, for such a purpose as map-
making, is too trifling to have any material influence. -
156 OUTLINES OF ASTRONOMY.
ºf jº
perpendicular let fall on it, so that the representation of the hemisphere
thus mapped on its base, is such as would actually appear to an eye placed
at an infinite distance from it. It is obvious, from the annexed figure,
that in this projection only the central portions are represented of their
true forms, while all the exterior is more and more distorted and crowded
together as we approach the edges of the map. Owing to this cause, the
Orthographic projection, though very good for small portions of the globe,
is of little service for large ones.
(281.) The stereographic projection is in great measure free from this
defect. To understand this projection, we must conceive an eye to be
placed at E, one extremity of a diameter, ECB, of the sphere, and to
view the concave surface of the sphere, every point of which, as P, is
referred to the diametral plane A D F, perpendicular to E B by the
visual line PM E. The stereographic projection of a sphere, then, is a
Fig. 42.
true perspective representation of its concavity on a diametral plane;
and, as such, it possesses some singularly elegant geometrical properties,
of which we shall state one or two of the principal.
(282.) And first, then, all circles on the sphere are represented by

PROJECTIONS OF THE SPEIERE. 157
circles in the projection. Thus the circle X is projected into a. Only
great circles passing through the vertex B are projected into straight lines
traversing the centre C : thus, B P A is projected into C A.
2dly. Every very small triangle, G H K, on the sphere, is represented
by a similar triangle, g h k, in the projection. This is a very valuable
property, as it insures a general similarity of appearance in the map to
the reality in all its parts, and enables us to project at least a hemisphere
in a single map, without any violent distortion of the configurations on
the surface from their real forms. As in the orthographic projection, the
borders of the hemisphere are unduly crowded together; in the stereo-
graphic, their projected dimensions are, on the contrary, somewhat enlarged
in receding from the centre.
(283.) Both these projections may be considered natural ones, inas-
much as they are really perspective representations of the surface on a
plane. Mercator's is entirely an artificial one, representing the sphere as
it cannot be seen from any one point, but as it might be seen by an eye
carried successively over every part of it. In it, the degrees of longitude,
and those of latitude, bear always to each other their due proportion: the
|||st,
60 < 2-} 60

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40 #B 40
20 3. º,’ w 20
or=ſ===T-T- | | O
equator is conceived to be extended out into a straight line, and the meri-
dians are straight lines at right angles to it, as in the figure. Altogether,
the general character of maps on this projection is not very dissimilar to
what would be produced by referring every point in the globe to a circum-
scribing cylinder, by lines drawn from the centre, and then unrolling the
cylinder into a plane. Like the stereographic projection, it gives a true
representation, as to form, of every particular small part, but varies
greatly in point of scale in its different regions; the polar portions in
particular being extravagantly enlarged; and the whole map, even of a
single hemisphere, not being comprisable within any finite limits.
(284.) We shall not of course, enter here into any geographical
details; but one result of maritime discovery on the great scale is, so to
158 OUTLINES OF ASTRONOMY.
speak, massive enough to call for mention as an astronomical feature.
When the continents and seas are laid down on a globe (and since the
discovery of Australia and the recent addition to our antarctic knowledge
of Victoria Land by Sir J. C. Ross, we are sure that no very extensive
tracts of land remain unknown), we find that it is possible so to divide
the globe into two hemispheres, that one shall contain nearly all the land;
the other being almost entirely sea. It is a fact, not a little interesting
to Englishmen, and, combined with our insular station in that great high-
way of nations, the Atlantic, not a little explanatory of our commercial
eminence, that London' occupies nearly the centre of the terrestrial hemi-
sphere. Astronomically speaking, the fact of this divisibility of the
globe into an Oceanic and a terrestrial hemisphere is important, as demon-
strative of a want of absolute equality in the density of the solid mate-
rial of the two hemispheres. Considering the whole mass of land and
water as in a state of equilibrium, it is evident that the half which pro-
trudes must of necessity be buoyant ; not, of course, that we mean to
assert it to be lighter than water, but, as compared with the whole globe,
in a less degree heavier than that fluid. We leave to geologists to draw
from these premises their own conclusions (and we think them obvious
enough) as to the internal constitution of the globe, and the immediate
nature of the forces which sustain its continents at their actual elevation;
but in any future investigations which may have for their object to explain
the local deviations of the intensity of gravity, from what the hypothesis
of an exact elliptic figure would require, this, as a general fact, ought not
to be lost sight of
(285.) Our knowledge of the surface of our globe is incomplete, un-
less it include the heights above the sea level of every part of the land,
and the depression of the bed of the ocean below the surface over all its
extent. The latter object is attainable (with whatever difficulty, and
howsoever slowly) by direct sounding; the former by two distinct methods:
the one consisting in trignometrical measurement of the differences
of level of all the stations of a survey; the other, by the use of the
barometer, the principle of which is, in fact, identical with that of the
sounding line. In both cases we measure the distance of the point whose
level we would know from the surface of an equilibrated ocean: only in
the one case it is an ocean of water; in the other, of air. In the one
“More exactly, Falmouth. The central point of the hemisphere which contains the
maximum of land falls very nearly indeed upon this port. The land in the opposite
hemisphere, with exception of the tapering extremity of South America and the
slender peninsula of Malacca, is wholly insular, and were it not for New Holland
would be quite insignificant in amount.
BAROMETRIC MEASUREMENT OF HEIGHTS. 159
case our sounding is real and tangible; in the other, an imaginary one,
measured by the length of the column of quicksilver the superincumbent
air is capable of counterbalancing.
(286.) Suppose that instead of air, the earth and ocean were covered
with oil, and that human life could subsist under such circumstances.
Let A B C D E be a continent, of which the portion A B C projects
Fig. 44.
G-
above the water, but is covered by the oil, which also floats at an uniform
depth on the whole ocean. Then if we would know the depth of any
point D below the sea-level, we let down a plummet from F. But, if
we would know the height of B above the same level, we have only to
send up a float from B to the surface of the oil; and having done the
same at C, a point at the sea level, the difference of the two float lines
gives the height in question. -
(287.) Now, though the atmosphere differs from oil in not having a
positive surface equally definite, and in not being capable of carrying up
any float adequate to such an use, yet it possesses all the properties of a
fluid really essential to the purpose in view, and this in particular, - that,
over the whole surface of the globe, its strata of equal density supposed
in a state of equilibrium, are parallel to the surface of equilibrium, or to
what would be the surface of the sea, if prolonged under the continents,
and therefore each or any of them has all the characters of a definite
surface to measure from, provided it can be ascertained and identified.
Now, the height at which, at any station B, the mercury in a barometer
is supported, informs us at once how much of the atmosphere is incum-
bent on B, or, in other words, in what stratum of the general atmosphere
(indicated by its density) B is situated: whence we are enabled finally to
conclude, by mechanical reasoning,' at what height above the sea-level
that degree of density is to be found over the whole surface of the globe.
Such is the principle of the application of the barometer to the measure-
ment of heights. For details, the reader is referred to other works.” .
* Newton’s Princip. ii. Prop. 22.
* Biot, Astronomie Physique, vol. iii. For tables, see the work of Biot cited. Also
those of Oltmann, annually published by the French board of longitudes in their
Annuaire; and Mr. Baily’s collection of Astronomical Tables and Formulae.

160 oUTLINES OF ASTRONOMY.
(288.) We will content ourselves here with a general caution against
an implicit dependence on barometric measurements, except as a differ-
ential process, at stations not too remote from each other. They rely in
their application on the assumption of a state of equilibrium in the atmo-
spheric strata over the whole globe—which is very far from being their
actual state (art. 37.) Winds, especially steady and general currents
sweeping over extensive continents, undoubtedly tend to produce some
degree of conformity in the curvature of these strata to the general form
of the land-surface, and therefore to give an undue elevation to the mer-
curial column at some points. On the other hand, the existence of
localities on the earth’s surface where a permanent depression of the
barometer prevails to the astonishing extent of nearly an inch, has been
clearly proved by the observations of Ermann in Siberia and of Ross in
the Antaretic Seas, and is probably a result of the same cause, and may
be conceived as complementary to an undue habitual elevation in other
regions.
(289.) Possessed of a knowledge of the height of stations above the
sea, we may connect all stations at the same altitude by level lines, the
lowest of which will be the outline of the sea-coast; and the rest will
mark out the successive coast-lines which would take place were the sea
to rise by regular and equal accessions of level over the whole world, till
the highest mountains were submerged. The bottoms of valleys, and the
ridge-lines of hills are determined by their property of intersecting all
these level lines at right angles, and being, subject to that condition, the
shortest and longest, that is to say, the steepest, and the most gently
sloping courses respectively which can be pursued from the summit to
the sea. The former constitute the “water courses” of a country; the
latter its lines of “water-shed” by which it is divided into distinct basins
of drainage. Thus originate natural districts of the most ineffaceable
character, on which the distribution, limits, and peculiarities of human
communities are in a great measure dependent. The mean height of the
continent of Europe, or that height which its surface would have were all
inequalities levelled and the mountains spread equally over the plains, is
according to Humboldt 671 English feet; that of Asia, 1137; of North
America, 748; and of South America, 1151.
CONSTRUCTION OF CELESTIAL MAPS. 161
CHAPTER V.
O F U R A N O G. R. A. P. H. Y.
CONSTRUCTION OF CELESTIAL MAPS AND GLOBES BY OBSERVATIONS
OF RIGHT ASCENSION AND DECLINATION. – CELESTIAL OBJECTS DIS-
TINGUISHED INTO FIXED AND ERRATIC. — OF THE CONSTELLATIONS.
— NATURAL REGIONS IN THE HEAVENS.—THE MILKY WAY. — THE
ZODIA.C. — OF THE ECLIPTIC. — CELESTIAL LATITU DES AND LONGI-
TUDES. — PRECESSION OF THE EQUINOXES. — NUTATION.—ABERRA-
TION.—REFRACTION.—PARALLAx—SUMMARY VIEW OF THE URANO-
GRAPHICAL CORRECTIONS.
(290.) THE determination of the relative situations of objects in the
heavens, and the construction of maps and globes which shall truly re-
present their mutual configurations as well as of catalogues which shall
preserve a more precise numerical record of the position of each, is a task
at once simpler and less laborious than that by which the surface of the
earth is mapped and measured. Every star in the great constellation
which appears to revolve above us, constitutes, so to speak, a celestial sta-
tion; and among these stations we may, as upon the earth, triangulate, by
measuring with proper instrument their angular distances from each
other, which, cleared of the effect of refraction, are then in a state for
laying down on charts, as we would the towns and villages of a country:
and this without moving from our place, at least for all the stars which
rise above our horizon.
(291.) Great exactness might, no doubt, be attained by this means,
and excellent celestial charts constructed; but there is a far simpler and
easier, and at the same time, infinitely more accurate course laid open to
us if we take advantage of the earth's rotation on its axis, and by observ-
ing each celestial object as it passes our meridian, refer it separately and
independently to the celestial equator, and thus ascertain its place on the
surface of an imaginary sphere, which may be conceived to revolve with
it, and on which it may be considered as projected.
(292.) The right ascension and declination of a point in the heavens
162 OUTLINES OF ASTRONOMY.
correspond to the longitude and latitude of a station on the earth; and
the place of a star on the celestial sphere is determined, when the former
elements are known, just as that of a town on a map, by knowing the
latter. The great advantages which the method of meridian observation
possesses over that of triangulation from star to star, are, then, 1st, That
in it every star is observed in that point of its diurnal course, when it is
best seen and least displaced by refraction. 2dly, That the instruments
required (the transit and meridian circle) are the simplest and least liable
to error or derangement of any used by astronomers. 3dly, That all the
abservations can be made systematically, in regular succession, and with
equal advantages; there being here no question about advantageous or
disadvantageous triangles, &c. And, lastly, That, by adopting this
course, the very quantities which we should otherwise have to calculate
by long and tedious operations of spherical trigonometry, and which are
essential to the formation of a catalogue, are made the objects of imme-
diate measurement. It is almost needless to state, then, that this is the
course adopted by astronomers, .
(293.) To determine the right ascension of a celestial object, all that
is necessary is to observe the moment of its meridian passage with a
transit instrument, by a clock regulated to exact sidereal time, or reduced
to such by applying its known error and rate. The rate may be obtained
by repeated observations of the same star at its successive meridian pas-
sages. The error, however, requires a knowledge of the equinox, or
initial point from which all right ascensions in the heavens reckon, as
longitudes do on the earth from a first meridian.
(294.) The nature of this point will be explained presently; but for
the purposes of uranography, in so far as they concern only the actual
configurations of the stars inter se, a knowledge of the equinox is not neces-
sary. The choice of the equinox, as a zero point of right ascensions, is
purely artificial, and a matter of convenience; but as on the earth, any
station (as a national observatory) may be chosen for an origin of longi-
tides; so in uranography, any conspicuous star might be selected as an
initial point from which hour angles might be reckoned, and from which,
by merely observing differences or intervals of time, the situation of all
others might be deduced. In practice, these intervals are affected by
certain minute causes of inequality, which must be allowed for, and
which will be explained in their proper places.
(295.) The declinations of celestial objects are obtained, 1st, By ob.
servation of their meridian altitudes, with the mural or meridian circle,
or other proper instruments. This requires a knowledge of the geogra-
phical latitude of the station of observation, which itself is only to be
FIXED AND ERRATIC STARS. 163
obtained by celestial observation. 2dly, And more directly, by observa-
tion of their polar distances on the mural circle, as explained in art. 170,
which is independent of any previous determination of the latitude
of the station; neither, however, in this case, does observation give
directly and immediately the exact declinations. The observations re-
quire to be corrected, first for refraction, and moreover for those minute
causes of inequality which have been just alluded to in the case of right
ascensions.
(296.) In this manner, then, may the places, one among the other, of
all celestial objects be ascertained, and maps and globes constructed.
Now here arises a very important question. How far are these places
permanent? Do these stars and the greater luminaries of heaven pre-
serve for ever one invariable connection and relation of place inter se, as
if they formed part of a solid though invisible firmament; and, like the
great natural land-marks on the earth, preserve immutably the same
distances and bearings each from the other? If so, the most rational
idea we could form of the universe would be that of an earth at absolute
rest in the centre, and a hollow crystalline sphere circulating round it,
and carrying Sun, moon, and stars along in its diurnal motion. If not,
we must dismiss all such notions, and inquire individually into the dis-
tinct history of each object, with a view to discovering the laws of its
peculiar motions, and whether any and what other connection subsists
between them.
(297.) So far is this, however, from being the case, that observations,
even of the most cursory nature, are sufficient to show that some, at least,
of the celestial bodies, and those the most conspicuous, are in a state of
continual change of place among the rest. In the case of the moon,
indeed, the change is so rapid and remarkable, that its alteration of situa-
tion with respect to such bright stars as may happen to be near it may be
noticed any fine night in a few hours; and if noticed on two successive
nights, cannot fail to strike the most careless observer. With the sun,
too, the change of place among the stars is constant and rapid; though,
from the invisibility of stars to the naked eye in the day-time, it is not so
readily recognized, and requires either the use of telescopes and angular
instruments to measure it, or a longer continuance of observation to be
struck with it. Nevertheless, it is only necessary to call to mind its
greater meridian altitude in summer than in winter, and the fact that the
stars which come into view at night (and which are therefore situated in
an hemisphere opposite to that occupied by the sun, and having that
luminary for its centre) vary with the season of the year, to perceive that
a great change must have taken place in that interval in its relative situa-
164 OUTLINES OF ASTRONOMY.
tion with respect to all the stars. Besides the sun and moon, too, there
are several other bodies, called planets, which, for the most part, appear
to the naked eye only as the largest and most brilliant stars, and which
offer the same phenomenon of a constant change of place among the
stars; now approaching, and now receding from, such of them as we may
refer them to as marks; and, some in longer, some in shorter periods,
making, like the sun and moon, the complete tour of the heavens.
(298.) These, however, are exceptions to the general rule. The innu-
merable multitude of the stars which are distributed over the vault of the
heavens form a constellation, which preserves, not only to the eye of the
casual observer, but to the nice examination of the astronomer, a uni-
formity of aspect which, when contrasted with the perpetual change in
the configurations of the Sun, moon, and planets, may well be termed
invariable. It is true, indeed, that, by the refinement of exact measure-
ments prosecuted from age to age, some small changes of apparent place,
attributable to no illusion and to no terrestrial cause, have been detected
in many of them. Such are called, in astronomy, the proper motions of
the stars. But these are so excessively slow, that their accumulated
amount (even in those stars for which they are greatest) has been insuffi-
cient, in the whole duration of astronomical history, to produce any
obvious or material alteration in the appearance of the starry heavens.
(299.) This circumstance, then, establishes a broad distinction of the
heavenly bodies into two great classes;– the fixed, among which (unless
in a course of observations continued for many years) no change of mutual
situation can be detected; and the erratic, or wandering— (which is
implied in the word planet')—including the sun, moon, and planets, as
well as the singular class of bodies termed comets, in whose apparent
places among the stars, and among each other, the observation of a few
days, or even hours, is sufficient to exhibit an indisputable alteration.
(300.) Uranography, then, as it concerns the fixed celestial bodies (or,
as they are usually called, the ficed stars), is reduced to a simple marking
down of their relative places on a globe or on maps; to the insertion on
that globe, in its due place in the great constellation of the stars, of the
pole of the heavens, or the vanishing point of parallels to the earth's
axis; and of the equator and place of the equinox: points and circles
these, which, though artificial, and having reference entirely to our earth,
and therefore subject to all changes (if any) to which the earth's axis may
be liable, are yet so convenient in practice, that they have obtained an
admission (with some other circles and lines), sanctioned by usage, in all
globes and planispheres. The reader, however, will take care to keep
* IIXavnrns, a wanderer.
OF THE CONSTELLATIONS. 165
them separate in his mind, and to familiarize himself with the idea rather
of two or more celestial globes, superposed and fitting on each other, on
one of which — a real one — are inscribed the stars; on the others those
imaginary points, lines, and circles, which astronomers have devised for
their own uses, and to aid their calculations; and to accustom himself to
conceive in the latter or artificial spheres a capability of being shifted in
any manner upon the surface of the other; so that, should experience
demonstrate (as it does) that these artificial points and lines are brought,
by a slow motion of the earth’s axis, or by other secular variations (as
they are called), to coincide, at very distant intervals of times, with dif.
ferent stars, he may not be unprepared for the change, and may have no
confusion to correct in his notions.
(301.) Of course we do not here speak of those uncouth figures and
outlines of men and monsters, which are usually scribbled over celestial
globes and maps, and serve, in a rude and barbarous way, to enable us to
talk of groups of stars, or districts in the heavens, by names which,
though absurd or puerile in their origin, have obtained a currency from
which it would be difficult to dislodge them. In so far as they have really (as
some have) any slight resemblance to the figures called up in imagination
by a view of the more splendid “constellations,” they have a certain con-
venience; but as they are otherwise entirely arbitrary, and correspond to no
natural subdivisions or groupings of the stars, astronomers treat them lightly,
or altogether disregard them,' except for briefly naming remarkable stars, as
a Leonis, 3 Scorpii, &c. &c., by letters of the Greek alphabet attached to
them. The reader will find them on any celestial charts or globes, and may
compare them with the heavens, and there learn for himself their position.
(302.) There are not wanting, however, natural districts in the heavens,
which offer great peculiarities of character, and strike every observer:
such is the milky way, that great luminous band, which stretches, every
evening, all across the sky, from horizon to horizon, and which, when
traced with diligence, and mapped down, is found to form a zone com-
pletely encircling the whole sphere, almost in a great circle, which is neither
an hour circle, nor coincident with any other of our astronomical gram-
mata. It is divided in one part of its course, sending off a kind of
branch, which unites again with the main body, after remaining distinct
for about 150 degrees, within which it suffers an interruption in its con-
* This disregard is neither supercilious nor causeless. The constellations seem to
have been almost purposely named and delineated to cause as much confusion and
inconvenience as possible. Innumerable snakes twine through long and contorted
areas of the heavens, where no memory can follow them ; bears, lions, and fishes,
large and small, northern and southern, confuse all nomenclature, &c. A better sys
tem of constellations might have been a material help as an artificial memory.
166 OUTLINES OF ASTRONOMY.
tinuity. This remarkable belt has maintained, from the earliest ages, the
Same relative situation among the stars; and, when examined through
powerful telescope, is found (wonderful to relate () to consist entirely of
stars scattered by millions, like glittering dust, on the black ground of the
general heavens. It will be described more particularly in the subsequent
portion of this work.
(803.) Another remarkable region in the heavens is the zodiac, not
from any thing peculiar in its own constitution, but from its being the
area within which the apparent motions of the sun, moon, and all the
greater planets are confined. To trace the path of any one of these, it is
only necessary to ascertain, by continued observation, its places at succes-
sive epochs, and entering these upon our map or sphere in sufficient num-
ber to form a series, not too far disjoined, to connect them by lines from
point to point, as we mark out the course of a vessel at sea by mapping
down its place from day to day. Now when this is done, it is found, first,
that the apparent path, or track, of the sun on the surface of the heavens,
is no other than an exact great circle of the sphere which is called the
ecliptic, and which is inclined to the equinoctial at an angle of about 23°
28', intersecting it at two opposite points, called the equinoctial points, or
equinoxes, and which are distinguished from each other by the epithets
vernal and autumnal; the vernal being that at which the sun crosses the
equinoctial from south to north; the autumnal, when it quits the northern
and enters the southern hemisphere. Secondly, that the moon and all
the planets pursue paths which, in like manner, encircle the whole
heavens, but are not, like that of the sun, great circles exactly returning
into themselves and bisecting the sphere, but rather spiral curves of much
complexity, and described with very unequal velocities in their different
parts. They have all, however, this in common, that the general direc-
tion of their motions is the same with that of the sun, viz. from west to
east, that is to say, the contrary to that in which both they and the stars
appear to be carried by the diurnal motion of the heavens; and, more-
over, that they never deviate far from the ecliptic on either side, crossing
and recrossing it at regular and equal intervals of time, and confining
themselves within a zone, or belt (the zodiac already spoken of), extend-
ing (with certain exceptions among the smaller planets) not further than
8° or 9° on either side of the ecliptic.
(304.) It would manifestly be useless to map down on globes or charts
the apparent paths of any of those bodies which never retrace the same
course, and which, therefore, demonstrably, must occupy at some one mo-
ment or other of their history, every point in the area of that zone of the
neavens within which they are circumscribed. The apparent complication
4.
OF THE ECLIPTIC AND ZODIAC. 167
of their movements arise (that of the moon excepted) from our viewing
them from a station which is itself in motion, and would disappear, could
we shift our point of view and observe them from the sun. On the other
hand the apparent motion of the sun is presented to us under its least
involved form, and is studied, from the station we occupy, to the greatest
advantage. So that, independent of the importance of that luminary to
us in other respects, it is by the investigation of the laws of its motions
in the first instance that we must rise to a knowledge of those of all the
other bodies of our system.
(305.) The ecliptic, which is its apparent path among the stars, is tra-
versed by it in the period called the sidereal year, which consists of
365d 6h 9m 9.6°, reckoned in mean solar time or 366' 6' 9" 9-6 reck-
oned in sidereal time. The reason of this difference (and it is this which
constitutes the origin of the difference between solar and sidereal time)
is, that as the sun’s apparent annual motion among the stars is performed
in a contrary direction to the apparent diurnal motion of both sun and
stars, it comes to the same thing as if the diurnal motion of the sun were
so much slower than that of the stars, or as if the Sun lagged behind
them in its daily course. When this has gone on for a whole year, the
sun will have fallen behind the stars by a whole circumference of the
heavens — or, in other words—in a year the Sun will have made fewer
diurnal revolutions, by one, than the stars. So that the same interval of
time which is measured by 366" 6", &c. of sidereal time, will be called
365 days, 6 hours, &c., if reckoned in mean solar time. Thus, then, is
the proportion between the mean solar and sidereal time established,
which, reduced into a decimal fraction, is that of 1:00.273791 to 1. The
measurement of time by these different standards may be compared to
that of space by the standard feet, or ells of two different nations; the
proportions of which, once settled and borne in mind, can never become
a source of error.
(306.) The position of the ecliptic among the stars may, for our pre-
sent purpose, be regarded as invariable. It is true that this is not strictly
the case; and on comparing together its position at present with that
which it held at the most distant epoch at which we possess observations,
we find evidences of a small change, which theory accounts for, and whose
nature will be hereafter explained; but that change is so excessively slow,
that for a great many successive years, or even for whole centuries, this
circle may be regarded, for most ordinary purposes, as holding the same
position in the sidereal heavens.
(307.) The poles of the ecliptic, like those of any other great circle
of the sphere, are opposite points on its surface, equidistant from the
168 Olſ'TLINES OF ASTRONOMY.
ecliptic in every direction. They are of course not coincident with those
of the equinoctial, but removed from it by an angular interval equal to
the inclination of the ecliptic to the equinoctial (23°28'), which is called
the obliquity of the ecliptic. In the next figure, if P p represent the
north and south poles (by which when used without qualification we al-
ways mean the poles of the equinoctial), and EA QW the equinoctial,
W S AW the ecliptic, and K k, its poles — the spherical angle Q W S is
the ºbliquity of the ecliptic, and is equal in angular measure to PK or
S Q. If we suppose the sun’s apparent motion to be in the direction
WSA W, W will be the vernal and A the autumnal equinox. S and W,
the two points at which the ecliptic is most distant from the equinoctial,
are termed solstices, because, when arrived there, the sun ceases to recede
from the equator, and (in that sense, so far as its motion in declination is
concerned) to stand still in the heavens. S, the point where the sun has
the greatest northern declination, is called the summer, and W, that where
it is farthest south, the winter solstice. These epithets obviously have
their origin in the dependence of the seasons on the sun’s declination,
which will be explained in the next chapter. The circle E K P Qkp,
which passes through the poles of the ecliptic and equinoctial, is called
the solstitial colure; and a meridian drawn through the equinoxes, PW
p A, the equinoctial colure.
(308.) Since the ecliptic holds a determinate situation in the starry
heavens, it may be employed, like the equinoctial, to refer the positions
Fig. 45.
of the stars to, by circles drawn through them from its poles, and there-
fore perpendicular to it. Such circles are termed, in astronomy, circles
of latitude—the distance of a star from the ecliptic, reckoned on the circle
of latitude passing through it, is called the latitude of the stars—and the

NONAGESIMAL, ANGLE OF SITUATION. 169
are of the ecliptic intercepted between the vernal equinox and this circle,
its longitude. In the figure, X is a star, PX R a circle of declination
drawn through it, by which it is referred to the equinoctial, and KXT a
circle of latitude referring it to the ecliptic — then, as W R is the right
ascension, and R X the declination, of X, so also is W T its longitude, and
TX its latitude. The use of the terms longitude and latitude, in this
sense, seems to have originated in considering the ecliptic as forming a
kind of natural equator to the heavens, as the terrestrial equator does to
the earth—the former holding an invariable position with respect to the
stars, as the latter does with respect to stations on the earth's surface.
The force of this observation will presently become apparent.
(309.) Knowing the right ascension and declination of an object, we
may find its longitude and latitude, and vice versa. This is a problem
of great use in physical astronomy—the following is its solution: – In
our last figure, E KPQ, the solstitial colure is of course 90° distant from
V, the vernal equinox, which is one of its poles—so that W R (the right
ascension) being given, and also V E, the arc E R, and its measure, the
spherical angle E PR, or KPX, is known. In the spherical triangle
KPX, then, we have given, 1st, The side PK, which, being the distance
of the poles of the ecliptic and equinoctial, is equal to the obliquity of
the ecliptic; 2d., The side PX, the polar distance, or the complement
of the declination R X; and, 3d, the included angle KPX; and there-
fore, by spherical trigonometry, it is easy to find the other side KX, and
the remaining angles. Now K X is the complement of the required lati-
tude XT, and the angle P K X being known, and P K V being a right
angle (because S W is 90°), the angle X K V becomes known. Now
this is no other than the measure of the longitude W T of the object.
The inverse problem is resolved by the same triangle, and by a process
exactly similar.
(310.) It is often of use to know the situation of the ecliptic in the visible
heavens at any instant; that is to say, the points where it cuts the horizon, and
the altitude of its highest point, or, as it is sometimes called, the nomagesimal
point of the ecliptic, as well as the longitude of this point on the ecliptic
itself from the equinox. These, and all questions referable to the same
data and quaesita, are resolved by the spherical triangle Z PE, formed by
the zenith Z (considered as the pole of the horizon), the pole of the
equinoctial P, and the pole of the ecliptic E. The sidereal time being
given, and also the right ascension of the pole of the ecliptic (which is
always the same, viz. 18' 0" 0°), the hour angle Z PE of that point is
known. Then, in this triangle we have given PZ, the colatitude; PE,
the polar distance of the pole of the ecliptic, 23°28', and the angle Z Pl:
170 oUTLINES OF ASTRONOMY.
A
Fig. 46.
from which we may find, 1st, the side ZE, which is easily seen to be
equal to the altitude of the nonagesimal point sought; and 2dly, the angle
PZ E, which is the azimuth of the pole of the ecliptic, and which, there-
fore, being added to and subtracted from 90°, gives the azimuth of the
eastern and western intersections of the ecliptic with the horizon. Lastly,
the longitude of the nonagesimal point may be had, by calculating in the
same triangle the angle PE Z, which is its complement.
(311.) The angle of situation of a star is the angle included between
circles of latitude and of declination passing through it. To determine it
in any proposed case, we must resolve the triangle PS E, in which are
given PS, PE, and the angle SPE, which is the difference between the
star's right ascension and 18 hours; from which it is easy to find the
angle PS E required. This angle is of use in many inquiries in physical
astronomy. It is called in most books on astronomy, the angle of post-
tion, but this expression has become otherwise and more conveniently
appropriated. (See Art. 204.)
(312.) The same course of observations by which the path of the sun
among the fixed stars is traced, and the ecliptic marked out among them,
determines, of course, the place of the equinox V (Fig. art. 308) upon
the starry sphere, at that time—a point of great importance in practical
astronomy, as it is the origin or zero point of right ascension. Now,
when this process is repeated at considerably distant intervals of time, a
very remarkable phenomenon is observed; viz. that the equinox does not
preserve a constant place among the stars, but shifts its position, travel-
ling continually and regularly, although with extreme slowness, back-
wards, along the ecliptic, in the direction WW from east to west, or the
contrary to that in which the sun appears to move in that circle. As
the ecliptic and equinoctial are not very much inclined, this motion of the

PRECESSION OF THE EQUINOXES. 71
equinox from east to west along the former, conspires (speaking generally)
with the diurnal motion, and carries it, with reference to that motion,
continually in advance upon the stars: hence it has acquired the name
of the precession of the equinoxes, because the place of the equinox among
the stars, at every subsequent moment, precedes (with reference to the
diurnal motion) that which it held the moment before. The amount of
this motion by which the equinox travels backward, or retrogrades (as it
is called), on the ecliptic, is 0° 0' 50-10" per annum, an extremely
minute quantity, but which, by its continual accumulation from year to
year, at last makes itself very palpable, and that in a way highly incon-
venient to practical astronomers, by destroying, in the lapse of a moderate
number of years, the arrangement of their catalogues of stars, and making
it necessary to reconstruct them. Since the formation of the earliest cata-
logue on record, the place of the equinox has retrograded already ahout
30°. The period in which it performs a complete tour of the ecliptic, is
25,868 years." t
(313.) The immediate uranographical effect of the precession of the
equinoxes is to produce a uniform increase of longitude in all the heavenly
bodies, whether fixed or erratic. For the vernal equinox being the initial
point of longitudes, as well as of right ascension, a retreat of this point
on the ecliptic tells upon the longitudes of all alike, whether at rest or in
motion, and produces, so far as its amount extends, the appearance of a
motion in longitude common to all, as if the whole heavens had a slow
rotation round the poles of the ecliptic in the long period above mentioned,
similar to what they have in twenty-four hours round those of the equi-
noctial. This increase of longitude, the reader will of course observe and
bear in mind, is, properly speaking, neither a real nor an apparent move-
ment of the stars. It is a purely technical result, arising from the gradual
shifting of the zero point from which longitudes are reckoned. Had a
fixed star been chosen as the origin of longitudes, they would have been
invariable.
(314.) To form a just idea of this curious astronomical phenomenon,
irowever, we must abandon, for a time, the consideration of the ecliptic,
as tending to produce confusion in our ideas; for this reason, that the
stability of the ecliptic itself among the stars is (as already hinted, art.
306) only approximate, and that in consequence its intersection with the
equinoctial is liable to a certain amount of change, arising from its fluc-
tuation, which mixes itself with what is due to the principal uranogra-
phical cause of the phenomenon. This cause will become at once appa-
*Incipiunt magni procedere menses. –VIRGIL, Pollio.
172 OUTLINES OF ASTRONOMY.
rent, if, instead of regarding the equinox, we fix our attention on the pole
of the equinoctial, or the vanishing point of the earth's axis.
(315.) The place of this point among the stars is easily determined at
any epoch, by the most direct of all astronomical observations,—those
with the meridian or mural circle. By this instrument we are enabled to
ascertain at every moment the exact distance of the polar point from any
three or more stars, and therefore to lay it down, by triangulating from
these stars, with unerring precision, on a chart or globe, without the
least reference to the position of the ecliptic, or to any other circle not
naturally connected with it. Now, when this is done with proper dili-
gence and exactness, it results that, although for short intervals of time,
such as a few days, the place of the pole may be regarded as not sensibly
variable, yet in reality it is in a state of constant, although extremely
slow motion; and, what is still more remarkable, this motion is not
uniform, but compounded of one principal uniform, or nearly uniform,
part, and other smaller and subordinate periodical fluctuations: the
former giving rise to the phenomena of precession; the latter to another
distinct phenomenon called mutation. These two phenomena, it is true,
belong, theoretically speaking, to one and the same general head, and are
intimately connected together, forming part of a great and complicated
chain of consequences flowing from the earth's rotation on its axis: but
it will be conducive to clearness at present to consider them separately.
(316.) It is found, then, that in virtue of the uniform part of the
motion of the pole, it describes a circle in the heavens around the pole of
the ecliptic as a centre, keeping constantly at the same distance of 23°
28' from it in a direction from east to west, and with such a velocity, that
the annual angle described by it, in this its imaginary orbit, is 50:10";
so that the whole circle would be described by it in the above-mentioned
period of 25,868 years. It is easy to perceive how such a motion of the
pole will give rise to the retrograde motion of the equinoxes; for in the
figure, art. 308, suppose the pole P in the progress of its motion in the
small circle P O Z round K to come to 0, then, as the situation of the
equinoctial E W Q is determined by that of the pole, this, it is evident,
must cause a displacement of the equinoctial, which will take a new
situation, E U Q, 90° distant in every part from the new position O of
the pole. The point U, therefore, in which the displaced equinoctial will
intersect the ecliptic, i. e. the displaced equinox, will lie on that side of
W, its original position, towards which the motion of the pole is directed,
or to the westward. -
(317.) The precession of the equinoxes thus conceived, consists, then,
in a real but very slow motion of the pole of the heavens among the
THE POLE STAR NOT ALWAYS THE SAME. 173
stars, in a small circle round the pole of the ecliptic. Now this cannot
happen without producing corresponding changes in the apparent diurnal
motion of the sphere, and the aspect which the heavens must present at
very remote periods of history. The pole is nothing more than the
vanishing point of the earth’s axis. As this point, then, has such a
motion as we have described, it necessarily follows that the earth's azis
must have a conical motion, in virtue of which it points successively to
every part of the small circle in question. We may form the best idea
of such a motion by noticing a child’s peg-top, when it spins not upright,
or that amusing toy the te-to-tum, which, when delicately executed, and
nicely balanced, becomes an elegant philosophical instrument, and ex-
hibits, in the most beautiful manner, the whole phenomenon. The
reader will take care not to confound the variation of the position of the
earth's axis in space with a mere shifting of the imaginary line about
which it revolves, in its interior. The whole earth participates in the
motion, and goes along with the axis as if it were really a bar of iron
driven through it. That such is the case is proved by the two great facts:
1st, that the latitudes of places on the earth, or their geographical situa-
tion with respect to the poles, have undergone no perceptible change from
the earliest ages. 2dly, that the sea maintains its level, which could not
be the case if the motion of the axis were not accompanied with a motion
of the whole mass of the earth.'
(318.) The visible effect of precession on the aspect of the heavens
consists in the apparent approach of some stars and constellations to the
pole and recess of others. The bright star of the Lesser Bear, which we
call the pole star, has not always been, nor will always continue to be,
our cynosure: at the time of the construction of the earliest catalogues it
was 12° from the pole—it is now only 1° 24', and will approach yet
nearer, to within half a degree, after which it will again recede, and
slowly give place to others, which will succeed in its companionship to
the pole. After a lapse of about 12,000 years, the star a Lyrae, the
brightest in the northern hemisphere, will occupy the remarkable situa-
tion of a pole star approaching within about 5° of the pole.
(319.) At the date of the erection of the Great Pyramid of Gizeh,
which precedes by 3970 years (say 4000) the present epoch, the longi-
tudes of all the stars were less by 55° 45' than at present. Calculating
* Local changes of the sea level, arising from purely geological causes, are easily
distinguished from that general and systematic alteration which a shifting of the axis
of rotation would give rise to.
174 OUTLINES OF ASTRONOMY.
from this datum' the place of the pole of the heavens among the stars, it
will be found to fall near a Draconis; its distance from that star being 8°
44'25". This being the most conspicuous star in the immediate neigh-
bourhood was therefore the pole star at that epoch. And the latitude of
Gizeh being just 30° north, and consequently the altitude of the north
pole there also 30°, it follows that the star in question must have had at
its lower culmination, at Gizeh, an altitude of 26° 15' 35". Now it is a
remarkable fact, ascertained by the late researches of Col. Wyse, that of
the nine pyramids still existing at Gizeh, six (including all the largest)
have the narrow passages by which alone they can be entered, (all which
open out on the northern faces of their respective pyramids) inclined to
the horizon downwards at angles as follows.
1st, or Pyramid of Cheops .................................... 260 41
2d, or Pyramid of Cephren.................................... 25 55
3d, or Pyramid of Mycerinus ................................. 26 2
4th, ............................................................... 27 0
5th, ............................................................... 27 12
9th, ............................................................... 28 0
Mean - 26 47
Of the two pyramids at Abousseir also, which alone exist in a state of
sufficient preservation to admit of the inclinations of their entrance pas-
sages being determined, one has the angle 27° 5', the other 26°.
(320.) At the bottom of every one of these passages therefore, the then
pole star must have been visible at its lower culmination, a circumstance
which can hardly be supposed to have been unintentional, and was doubt-
less connected (perhaps superstitiously) with the astronomical observation
of that star, of whose proximity to the pole at the epoch of the erection
of these wonderful structures, we are thus furnished with a monumental
record of the most imperishable nature.
(321.) The mutation of the earth’s axis is a small and slow subordinate
gyratory movement, by which, if subsisting alone, the pole would describe
among the stars, in a period of about nineteen years, a minute ellipsis,
having its longer axis equal to 18".5, and its shorter to 13".74; the longer
being directed towards the pole of the ecliptic, and the shorter, of course,
at right angles to it. The consequence of this real motion of the pole is
* On this calculation the diminution of the obliquity of the eliptic in the 4000 years
elapsed has no influence. That diminution arises from a change in the plane of the
earth’s orbit, and has nothing to do with the change in the position of its aris, as re-
ferred to the starry sphere.
NUTATION OF THE EARTH's Axts, 175
an apparent approach and recess of all the stars in the heavens to the
pole in the same period. Since, also, the place of the equinox on
the ecliptic is determined by the place of the pole in the heavens, the
same cause will give rise to a small alternate advance and recess of the
equinoctial points, by which, in the same period, both the longitudes and
the right ascensions of the stars will be also alternately increased and
diminished.
(322.) Both these motions, however, although here considered sepa.
rately, subsist jointly; and since, while in virtue of the nutation, the pole
is describing its little ellipse of 18".5 in diameter, it is carried by the
greater and regularly progressive motion of precession over so much of its
circle round the pole of the ecliptic as corresponds to nineteen years,
that is to say, over an angle of nineteen times 50"-1 round the centre
(which, in a small circle of 23°28' in diameter, corresponds to 6' 20",
as seen from the centre of the sphere): the path which it will pursue in
virtue of the two motions, subsisting jointly, will be neither an ellipse
nor an exact circle, but a gently undulated ring like that in the figure
(where, however, the undulations are much exaggerated). (See fig.
to art. 325.)
(323.) These movements of precession and nutation are common to all
the celestial bodies, both fixed and erratic; and this circumstance makes
it impossible to attribute them to any other cause than a real motion of the
carth's axis such as we have described. Did they only affect the stars,
they might, with equal plausibility, be urged to arise from a real rotation
of the starry heavens, as a solid shell, round an axis passing through the
poles of the ecliptic in 25,868 years, and a real ecliptic gyration of that
axis in nineteen years: but since they also affect the sun, moon, and pla-
nets, which, having motions independent of the general body of the stars,
cannot without extravagance be supposed attached to the celestial concave,"
this idea falls to the ground; and there only remains, then, a real motion
in the earth by which they can be accounted for. It will be shown in a
subsequent chapter that they are necessary consequences of the rotation
of the earth, combined with its elliptical figure, and the unequal attrac-
tion of the sun and moon on its polar and equatorial regions.
(324.) Uranographically considered, as affecting the apparent places of
the stars, they are of the utmost importance in practical astronomy. When
we speak of the right ascension and declination of a celestial object, it
* This argument, cogent as it is, acquires additional and decisive force from the law
of mutation, which is dependent on the position, for the time, of the lunar orbit. If
we attribute it to a real motion of the celestial sphere, we must then maintain that sphere
to be kept in a constant state of tremor by the motion of the moon.
176 * OUTLINES OF ASTRONOMY.
becomes necessary to state what epoch we intend, and whether we mean
the mean right ascension—cleared, that is, of the periodical fluctuation in
its amount, which arises from nutation, or the apparent right ascension,
which, being reckoned from the actual place of the vernal equinox, is
affected by the periodical advance and recess of the equinoctial point pro-
duced by nutation—and so of the other elements. It is the practice of
astronomers to reduce, as it is termed, all their observations, both of right
ascension and declination, to some common and convenient epoch — such
as the beginning of the year for temporary purposes, or of the decade, or
the century for more permanent uses, by subtracting from them the whole
effect of precession in the interval; and, moreover, to divest them of the
influence of nutation by investigating and subducting the amount of
change, both in right ascension and declination, due to the displacement
of the pole from the centre to the circumference of the little ellipse above
mentioned. This last process is technically termed correcting or equating
the observation for nutation; by which latter word is always understood,
in astronomy, the getting rid of a periodical cause of fluctuation, and pre-
senting a result, not as it was observed, but as it would have been observed,
had that cause of fluctuation had no existence.
(325.) For these purposes, in the present case, very convenient
formulae have been derived, and tables constructed. They are, however,
of too technical a character for this work; we shall, however, point out
the manner in which the investigation is conducted. It has been shown
in art. 309 by what means the right ascension and declination of an
object are derived from its longitude and latitude. Referring to the
figure of that article, and supposing the triangle KPX orthographically
projected on the plane of the ecliptic as in the annexed figure: in the
triangle KPX, K P is the obliquity of the ecliptic, KX the co-latitude
(or complement of latitude), and the angle P KX the co-longitude of the
object X. These are the data of our question, of which the second is
constant, and the other two are varied by the effect of precession and
mutation: and their variations (considering the minuteness of the latter
effect generally, and the small number of years in comparison of the
whole period of 25,868, for which we ever require to estimate the effect
of the former,) are of that order which may be regarded as infinitesimal
in geometry, and treated as such without fear of error. The whole ques-
tion, then, is reduced to this:–In a spherical triangle KPX, in which
one side KX is constant, and an angle K, and adjacent side K P vary by
given infinitesimal changes of the position of P: required the changes
thence arising in the other side PX, and the angle KPX. This is a
very simple and easy broblem of spherical geometry, and being resolved,
CORRECTIONS FOR PRECESSION AND NUTATION. I77
Fig. 47.
º
it gives at once the reductions we are seeking; for PX being the polar
distance of the object, and the angle KPX its right ascension plus 90°,
their variations are the very quantities we seek. It only remains, then,
to express in proper form the amount of the precession and mutation in
longitude and latitude, when their amount in right ascension and declina-
tion will immediately be obtained.
(326.) The precession in latitude is zero, since the latitudes of objects
are not changed by it: that in longitude is a quantity proportional to the
time at the rate of 50":10 per annum. With regard to the mutation in
longitude and latitude, these are no other than the abscissa and ordinate
of the little ellipse in which the pole moves. The law of its motion,
however, therein, cannot be understood till the reader has been made
acquainted with the principal features of the moon’s motion on which it
depends.
(327.) Another consequence of what has been shown respecting pre-
cession and mutation is, that sidereal time, as astronomers use it, i. e. as
reckoned from the transit of the equinoctial point, is not a mean or uni-
Jormly flowing quantity, being affected by mutation; and moreover, that
so reckoned, even when cleared of the periodical fluctuatiºn of mutation,
it does not strictly correspond to the earth's diurnal rotation. As the sun
loses one day in the year on the stars, by its direct motion in longitude;
so the equinox gains one day in 25,868 years on them by its retrograda-
tion. We ought, therefore, as carefully to distinguish between mean and
apparent sidereal as between mean and apparent solar time.
(328.) Neither precession nor mutation changes the apparent places of
celestial objects inter se. We see them, so far as these causes go, as they

12
178 OUTLINES OF ASTRONOMY.
are, though from a station more or less unstable, as we see distant land
objects correctly formed, though appearing to rise and fall when viewed
from the heaving deck of a ship in the act of pitching and rolling. But
there is an optical cause, independent of refraction or of perspective, which
displaces them one among the other, and causes us to view the heavens
under an aspect always to a certain slight extent false; and whose in-
fluence must be estimated and allowed for before we can obtain a precise
knowledge of the place of any object. This cause is what is called the
aberration of light; a singular and surprising effect arising from this, that
we occupy a station not at rest but in rapid motion; and that the apparent
directions of the rays of light are not the same to a spectator in motion
as to one at rest. As the estimation of its effect belongs to uranography,
we must explain it here, though, in so doing, we must anticipate some of
the results to be detailed in subsequent chapters.
(329.) Suppose a shower of rain to fall perpendicularly in a dead calm;
a person exposed to the shower, who would stand quite still and upright,
would receive the drops on his hat, which would thus shelter him, but if
he ran forward in any direction they would strike him in the face. The
effect would be the same as if he remained still, and a wind should arise
of the same velocity, and drift them against him. Suppose a ball let fall
Fig. 48.
S}~7S S/Q
from a point A above a horizontal line E F, and that at B were placed to
receive it the open mouth of an inclined hollow tube PQ; if the tube
were held immoveable the ball would strike on its lower side, but if the
tube were carried forward in the direction E F, with a velocity properly
adjusted at every instant to that of the ball, while preserving its inclina-
tion to the horizon, so that when the ball in its natural descent reached
C, the tube should have been carried into the position RS, it is evident
that the ball would, throughout its whole descent, be found in the axis of

ABERRATION OF LIGHT. 170
the tube; and a spectator referring to the tube the motion of the ball,
and carried along with the former, unconscious of its motion, would fancy
that the ball had been moving in the inclined direction R S of the tube's
axis. e
(330.) Our eyes and telescopes are such tubes. In whatever manner
we consider light, whether as an advancing wave in a motionless ether, or
a shower of atoms traversing space, (provided that in both cases we regard
it as absolutely incapable of suffering resistance or corporeal obstruction
from the particles of transparent media traversed by it,') if in the interval
between the rays traversing the object glass of the one or the cornea of
the other (at which moment they acquire that convergence which directs
them to a certain point in ficed space), and their arrival at their focus,
the cross wires of the one or the retina of the other, be slipped aside, the
point of convergence (which remains unchanged) will no longer corres-
pond to the intersection of the wires or the central point of our visual
area. The object then will appear displaced; and the amount of this
displacement is aberration.
(331.) The earth is moving through space with a velocity of about 19
miles per second, in an elliptic path round the sun, and is therefore
changing the direction of its motion at every instant. Light travels with
a velocity of 192,000 miles per second, which, although much greater than
that of the earth, is yet not infinitely so. Time is occupied by it in
traversing any space, and in that time the earth describes a space which is
to the former as 19 to 192,000, or as the tangent of 20".5 to radius.
Suppose now A PS to represent a ray of light from a star at A, and let
the tube PQ be that of a telescope so inclined forward that the focus
formed by its object-glass shall be received upon its cross wire, it is
evident from what has been said, that the inclination of the tube must
be such as to make P S : S Q :: velocity of light: velocity of the earth ::
1 : tan. 20".5; and, therefore, the angle SPQ, or PSR, by which the
axis of the telescope must deviate from the true direction of the star,
must be 20".5.
(332.) A similar reasoning will hold good when the direction of the
earth's motion is not perpendicular to the visual ray. If S B be the true
* This condition is indispensable. Without it we fall into all those difficulties which
M. Doppler has so well pointed out in his paper on Aberration (Abhandlungen der k.
boemischen Gesellschaft der Wissenschaften. Folge W. vol. iii.). If light itself, or
the luminiferous ether, be corporeal, the condition insisted on amounts to a formal sur-
render of the dogma, either of the extension or of the impenetrability of matter; at
least in the sense in which those terms have been hitherto used by metaphysicians.
At the point to which science is arrived, probably few will be found disposed to main
tain either the one of the other.
180 OUTLINES OF ASTRONOMY.
Fig. 48.

IB JYS
direction of the visual ray, and A C the position in which the telescope
requires to be held in the apparent direction, we must still have the pro-
portion B C : B A : : velocity of light: velocity of the earth :: rad.:
sine of 20"-5 (for in such small angles it matters not whether we use the
sines or tangents). But we have, also, by trigonometry, B C : B A ::
sine of B A C : sine of A C B or CBD, which last is the apparent dis-
placement caused by aberration. Thus it appears that the sine of the
aberration, or (since the angle is extremely small) the aberration itself, is
proportional to the sine of the angle made by the earth’s motion in space
with the visual ray, and is therefore a maximum when the line of sight is
perpendicular to the direction of the earth's motion.
(333.) The uranographical effect of aberration, then, is to distort the
aspect of the heavens, causing all the stars to crowd as it were directly
towards that point in the heavens which is the vanishing point of all lines
parallel to that in which the earth is for the time moving. As the earth
moves round the sun in the plane of the ecliptic, this point must lie in
that plane, 90° in advance of the earth’s longitude, or 90° behind the
sun's, and shifts of course continually, describing the circumference of the
ecliptic in a year. It is easy to demonstrate that the effect on each par-
ticular star will be to make it apparently describe a small ellipse in the
heavens, having for its centre the point in which the star would be seen
if the earth were at rest.
(334.) Aberration then affects the apparent right ascensions and decli-
nations of all the stars, and that by quantities easily calculable. The
formulae most convenient for that purpose, and which, systematically
embracing at the same time the corrections for precession and mutation,
enable the observer, with the utmost readiness, to disencumber his obser-
vations of right ascension and declination of their influence, have been
constructed by Prof. Bessel, and tabulated in the appendix to the first
volume of the Transactions of the Astronomical Society, where they will
he found accompanied with an extensive catalogue of the places, for 1830,
ABERRATION OF LIGHT. 181
of the principal fixed stars, one of the most useful and best arranged works
of the kind which has ever appeared.
(335.) When the body from which the visual ray emanates is itself in
motion, an effect arises which is not properly speaking aberration, though
it is usually treated under that head in astronomical books, and indeed
confounded with it, to the production of some confusion in the mind of
the student. The effect in question (which is independent of any theo-
rutical views respecting the nature of light') may be explained as follows.
The ray by which we see any object is not that which it emits at the
moment we look at it, but that which it did emit some time before, viz.,
the time occupied by light in traversing the interval which separates it
from us. The aberration of such a body then arising from the earth’s
velocity must be applied as a correction, not to the line joining the earth's
place at the moment of observation with that occupied by the body at the
same moment, but at that antecedent instant when the ray quitted it.
Hence it is easy to derive the rule given by astronomical writers for the
case of a moving object. From the known laws of its motion and the
earth's, calculate its apparent or relative angular motion in the time
taken by light to traverse its distance from the earth. This is the total
amount of its apparent displacement. Its effect is to displace the body
observed in a direction contrary to its apparent motion in the heavens.
And it is a compound or aggregate effect consisting of two parts, one of
which is the aberration, properly so called, resulting from the composition
of the earth’s motion with that of light, the other being what is not
inaptly termed the Equation of light, being the allowance to be made for
the time occupied by the light in traversing a variable space.
(336.) The complete Reduction, as it is called, of an astronomical
observation consists in applying to the place of the observed heavenly
body as read off on the instruments (supposed perfect and in perfect
adjustment) five distinct and independent corrections, viz. those for
refraction, parallax, aberration, precession, and mutation. Of these the
correction for refraction, enables us to declare what would have been the
* The results of the undulatory and corpuscular theories of light, in the matter of
aberration are, in the main, the same. We say in the main. There is, however, a
minute difference even of numerical results. In the undulatory doctrine, the propa-
gation of light takes place with equal velocity in all directions, whether the luminary
be at rest or in motion. In the corpuscular, with an excess of velocity in the direc-
tion of the motion over that in the contrary equal to twice the velocity of the body’s
motion. In the cases, then, of a body moving with equal velocity directly to and direct
ly from the earth, the aberration will be alike on the undulatory, but different on the
corpuscular hypothesis. The utmost difference which can arise from this cause in our
system cannot amount to above six thousandths of a second.
182 OUTLINES OF ASTRONOMY.
observed place, were there no atmosphere to displace it. That for paral
lax enables us to say from its place observed at the surface of the earth
where it would have been seen if observed from the centre. That for
aberration, where it would have been observed from a motionless, instead
of a moving station: while the corrections for precession and nutation
refer it to fixed and determinate instead of constantly varying celestial
circles. The great importance of these corrections, which pervade all
astronomy, afid have to be applied to every observation before it can be
employed for any practical or theoretical purpose, renders this recapitula-
tion far from superfluous.
(337.) Refraction has been already sufficiently explained, Art. 40, and
it is only, therefore, necessary here to add that in its use as an astronomi-
cal correction its amount must be applied in a contrary sense to that in
which it affects the observation; a remark equally applicable to all other
corrections.
(338.) The general nature of parallax or rather of parallactic motion
has also been explained in Art. 80. But parallax in the uranographical
sense of the word has a more technical meaning. It is understood to
express that optical displacement of a body observed which is due to its
being observed, not from that point which we have fixed upon as a con-
ventional central station (from which we conceive the apparent motion
would be more simple in its laws,) but from some other station remote
from such conventional centre: not from the centre of the earth, for
instance, but from its surface: not from the centre of the sun (which, as
we shall hereafter see, is for some purposes a preferable conventional
station), but from that of the earth. In the former case this optical dis-
placement is called the diurnal or geocentric parallax; in the latter the
annual or heliocentric. In either case parallax is the correction to be
applied to the apparent place of the heavenly body, as actually seen from
the station of observation, to reduce it to its place as it would have been
seen at that instant from the conventional station.
(339.) The diurnal or geocentric parallax at any place of the earth's
surface is easily calculated if we know the distance of the body, and, vice
versä, if we know the diurnal parallax that distance may be calculated.
For supposing S the object, C the centre of the earth, A the station of
observation at its surface, and C A Z the direction of a perpendicular to
the surface at A, then will the object be seen from A in the direction A
S, and its apparent zenith distance will be Z A S; whereas, if seen from
the centre, it will appear in the direction C S, with an angular distance
from the zenith of A equal to Z C S; so that Z A S –Z C S or A S C
is the parallax. Now since by trignometry C S : C A :: sin C A S
SUMMARY OF URANOGRAPHICAL CORRECTIONS. I83
Fig. 50.
2,

T A
= sin Z A S : sin A S C, it follows that the sine of the parallax
Radius of earth iº
TIDistance of body × sin Z. A S.
(340.) The diurnal or geocentric parallax, therefore, at a given place,
and for a given distance of the body observed, is proportional to the 'sine
of its apparent Zenith distance, and is, therefore, the greatest when the
body is observed in the act of rising or setting, in which case its parallax
is called its horizontal parallax, so that at any other zenith distance,
parallax = horizontal parallax × sine of apparent Zenith distance, and
since A C S is always less than Z. A S it appears that the application of
the reduction or correction for parallax always acts in diminution of the
apparent zenith distance or increase of the apparent altitude or distance
from the Nadir, i. e. in a contrary sense to that for refraction.
(341.) In precisely the same manner as the geocentric or diurnal
parallax refers itself to the zenith of the observer for its direction and
quantitative rule, so the heliocentric or annual parallax refers itself for its
law to the point in the heavens diametrically opposite to the place of the
sun as seen from the earth. Applied as a correction, its effect takes place
in a plane passing through the sun, the earth, and the observed body.
Its effect is always to decrease its observed distance from that point or to
increase its angular distance from the sun. And its sine is given by the
relation, Distance of the observed body from the sun : distance of the
earth from the sun :: sine of apparent angular distance of the body from
the sun (or its apparent elongation) : sine of heliocentric parallax."
* This account of the law of heliocentric parallax is in anticipation of what follows in
a subsequent chapter, and will be better understood by the student when somewhat
* -- . .dvanced,
184 - OUTLINES OF : ASTRONOMY.
(342.) On a summary view of the whole of the uranographical correc
tions, they divide themselves into two classes, those which do, and those
which do not, alter the apparent configurations of the heavenly bodies
inter se. The former are real, the latter technical corrections. The real
corrections are refraction, aberration and parallax. The technical are pre-
cession and mutation, unless, indeed, we choose to consider parallax as a
technical correction introduced with a view to simplification by a better
choice of our point of sight.
(343.) The corrections of the first of these classes have one peculiarity
in respect of their law, common to them all, which the student of prac-
tical astronomy will do well to fix in his memory. They all refer them-
selves to definite ape.ces or points of convergence in the sphere. Thus,
refraction in its apparent effect causes all celestial objects to draw together
or converge towards the zenith of the observer: geocentric parallax,
towards his Nadir : heliocentric, towards the place of the sun in the
heavens: aberration towards that point in the celestial sphere which is
the vanishing point of all lines parallel to the direction of the earth's
motion at the moment, or (as will be hereafter explained) towards a point
in the great circle called the ecliptic, 90° behind the sun's place in that
circle. When applied as corrections to an observation, these directions
are of course to be reversed.
(344.) In the quantitative law, too, which this class of corrections
follow, a like agreement takes place, at least as regards the geocentric and
heliocentric parallax and aberration, in all three of which the amount of
the correction (or more strictly its sine) increases in the direct proportion
of the sine of the apparent distance of the observed body from the apex.
appropriate to the particular correction in question. In the case of re-
fraction the law is less simple, agreeing more nearly with the tangent than
the sine of that distance, but agreeing with the others in placing the
maximum at 90° from its apex.
(345.) As respects the order in which these corrections are to be
applied to any observation, it is as follows: 1. Refraction; 2. Aberration;
3. Geocentric Parallax; 4. Heliocentric Parallax; 5. Nutation; 6. Pre-
cession. Such, at least, is the order in theoretical strictness. But as the
amount of aberration and nutation is in all cases a very minute quantity,
it matters not in what order they are applied; so that for practical conve-
nience they are always thrown together with the precession, and applied
after the others. -
of THE SUN’s MOTION. 185.
CHAPTER WI.
o F T H E SU N'S MO TI O N.
APPARENT MOTION OF THE SUN NOT UNIFORM. — ITS APPARENT
DIAMETER ALSO VARIABLE. — VARIATION OF ITS DISTANCE CON-
CLUIDED. — ITS APPARENT ORBIT AN ELLIPSE ABOUT THE FOCUS.
— LAW OF THE ANGULAR, VELOCITY. – EQUABLE DESCRIPTION OF
AREAs. – PARALLAX OF THE SUN. —ITS DISTANCE AND MAGNI-
TUDE. — COPERNICAN EXPLANATION OF THE SUN’s APPARENT
MoTION.— PARALLELISM OF THE EARTH's AXIS. — THE SEASONS.
— HEAT RECEIVED FROM THE SUN IN DIFFERENT PARTS OF THE
ORBIT. — MEAN AND TRUE LONGITUDES OF THE SUN. — EQUATION
oR THE CENTRE.-SIDEREAL, TROPICAL, AND ANOMALISTIC YEARS..
—PHYSICAL CONSTITUTION OF THE SUN. —ITs SPOTS. — FACULAE.
— PROBABLE NATURE AND CAUSE OF THE SPOTS. — ATMOSPHERE
OF THE SUN. — ITS SUPPOSED CLOUDS. — TEMPERATURE AT ITS
suPFACE. —ITs ExPENDITURE OF HEAT. —TERRESTRIAL EFFECTs
OF. 3OLAR RADIATION.
(346.) IN the foregoing chapters, it has been shown that the apparent
path of the sum is a great circle of the sphere, which it performs in a
period of one sidereal year. From this it follows, that the line joining
the earth and sun lies constantly in one plane; and that, therefore,
whatever be the real motion from which this apparent motion arises, it
must be confined to one plane, which is called the plane of the ecliptic.
(347.) We have already seen (art. 146) that the sun’s motion in right
ascension among the stars is not uniform. This is partly accounted for
by the obliquity of the ecliptic, in consequence of which equal variations
in longitude do not correspond to equal changes of right ascension. But
if we observe the place of the sun daily throughout the year, by the
transit and circle, and from these calculate the longitude for each day, it
will still be found that, even in its own proper path, its apparent angular
motion is far from uniform. The change of longitude in twenty-four
mean solar hours averages 0° 59'8":33; but about the 31st of Decem-
186 OUTLINES OF ASTRONOMY.
ber it amounts to 191' 9".9, and about the 1st of July is only 0° 57"
11"-5. Such are the extreme limits, and such the mean value of the
sun's apparent angular velocity in its annual orbit. .
(348.) This variation of its angular velocity is accompanied with a
•orresponding change of its distance from us. The change of distance is
•ecognized by a variation observed to take place in its apparent diameter,
when measured at different seasons of the year, with an instrument
adapted for that purpose, called the heliometer,' or, by calculating from
khe time which its disc takes to traverse the meridian in the transit
instrument. The greatest apparent diameter corresponds to the 1st of
December, or to the greatest angular velocity, and measures 32' 35".6,
the least is 31' 31"-0; and corresponds to the 1st of July; at which
epochs, as we have seen, the angular motion is also at its extreme limit
either way. Now, as we cannot suppose the sun to alter its real size
periodically, the observed change of its apparent size can only arise from
an actual change of distance. And the sines or tangents of such small
wres being proportional to the arcs themselves, its distances from us, at
the above-named epoch, must be in the inverse proportion of the apparent
diameters. It appears, therefore, that the greatest, the mean, and the
least distances of the sun from us are in the respective proportions of the
numbers 1.01679, 1.00000, and 0.98321; and that its apparent angular
velocity diminishes as the distance increases, and vice versä.
(349.) It follows from this, that the real orbit of the sun, as referred
to the earth supposed at rest, is not a circle with the earth in the centre.
The situation of the earth within it is eaccentric, the eaccentricity amount-
*ing to 0.01679 of the mean distance, which may be regarded as our unit
of measure in this inquiry. But besides this, the form of the orbit is
not circular, but elliptic. If from any point O, taken to represent the
earth, we draw a line, O A, in some fixed direction, from which we then
Fig. 51.
T}
set off a series of angles, A. OB, A O C, &c. equal to the observed longi-
tudes of the sun throughout the year, and in these respective directions
“‘HAtos the sun, and perpetv to measure.

FORM OF THE SUN’s APPARENT ORBIT. 187
measure off from O the distances O A, O B, O C, &c. representing the
distances deduced from the observed diameter, and then connect all the
extremities A, B, C, &c. of these lines by a continuous curve, it is evident
this will be a correct representation of the relative orbit of the sun about
the earth. Now, when this is done, a deviation from the circular figure
in the resulting curve becomes apparent; it is found to be evidently longer
than it is broad—that is to say, elliptic, and the point O to occupy, not
the centre, but one of the foci of the ellipse. The graphical process here
described is sufficient to point out the general figure of the curve in ques-
tion; but for the purposes of exact verification, it is necessary to recur to
the properties of the ellipse,' and to express the distance of any one of
its points in terms of the angular situation of that point with respect to
the longer axis, or diameter of the ellipse. This, however, is readily
done; and when numerically calculated, on the supposition of the excen-
tricity, being such as above stated, a perfect coincidence is found to
subsist between the distances thus computed, and those derived from the
measurement of the apparent diameter.
(350.) The mean distance of the earth and sun being taken for unity,
the extremes are 1.01679 and 0.98321. But if we compare, in like
manner, the mean or average angular velocity with the extremes, greatest
and least, we shall find these to be in the proportions of 1.03386, 1:00000,
and 0.96670. The variation of the sun's angular velocity, then, is much
greater in proportion than that of its distance — fully twice as great; and
if we examine its numerical expressions at different periods, comparing
them with the mean value, and also with the corresponding distances, it
will be found, that, by whatever fraction of its mean value the distance
exceeds the mean, the angular velocity will fall short of its mean or ave-
rage quantity by very nearly twice as great a fraction of the latter, and
vice versä. Hence we are led to conclude that the angular velocity is in
the inverse proportion, not of the distance simply, but of its square; so
that, to compare the daily motion in longitude of the sun, at one point,
A, of its path, with that at B, we must state the proportion thus:–
O B" : O A* :: daily motion at A : daily motion at B. And this is
found to be exactly verified in every part of the orbit.
(351.) Hence we deduce another remarkable conclusion—viz. that if
the Sun be supposed really to move around the circumference of this
ellipse, its actual speed cannot be uniform, but must be greatest at its
least distance and less at its greatest. For, were it uniform, the apparent
* See Conic Sections, by the Rev. H. P. Hamilton, or any other of the very
numerous works on this subject. -
188 OUTLINES OF ASTRONOMY.
-
angular velocity would be, of course, inversely proportional to the distance;
simply because the same linear change of place, being produced in the
same time at different distances from the eye, must, by the laws of per-
spective, correspond to apparent angular displacements inversely as those
distances. Since, then, observation indicates a more rapid law of varia-
tion in the angular velocities, it is evident that mere change of distance,
unaccompanied with a change of actual speed, is insufficient to account for
it; and that the increased proximity of the sun to the earth must be
accompanied with an actual increase of its real velocity of motion along
its path.
(352.) This elliptic form of the sun's path, the excentric position of
the earth within it, and the unequal speed with which it is actually
traversed by the sun itself, all tend to render the calculation of its longi-
tude from theory (i. e. from a knowledge of the causes and nature of its
motion) difficult; and indeed impossible, so long as the law of its actual
velocity continues unknown. This law, however, is not immediately
apparent. It does not come forward, as it were, and present itself at once,
like the elliptic form of the orbit, by a direct comparison of angles and
distances, but requires an attentive consideration of the whole series of
observations registered during an entire period. It was not, therefore,
without much painful and laborious calculation, that it was discovered by
Kepler (who was also the first to ascertain the elliptic form of the orbit),
and announced in the following terms: — Let a line be always supposed
to connect the Sun, supposed in motion, with the earth, supposed at rest;
then, as the sun moves along its ellipse, this line (which is called in astro-
nomy the radius vector) will describe or sweep over that portion of the
whole area or surface of the ellipse which is included between its consec-
utive positions: and the motion of the sun will be such that equal areas
are thus swept over by the revolving radius vector in equal times, in what-
ever part of the circumference of the ellipse the sun may be moving.
(353.) From this it necessarily follows, that in unequal times, the areas
described must be proportional to the times. Thus, in the figure of art.
349, the time in which the sun moves from A to B, is to the time in
which it moves from C to D, as the area of the elliptic sector AO B is to
the area of the sector D O C.
(354.) The circumstances of the sun's apparent annual motion may,
therefore, be summed up as follows:—It is performed in an orbit lying in
one plane passing through the earth’s centre, called the plane of the ecliptic,
and whose projeetion on the heavens is the great circle so called. In this
plane, however, the actual path is not circular, but elliptical; having the
earth, not in its centre, but in one focus. The excentricity of this ellipse
DISTANCE OF THE SUN. 189
is 0.01679, in parts of a unit equal to the mean distance, or half the
longer diameter of the ellipse; i. e. about one sixtieth part of that semi-
diameter; and the motion of the sun in its circumference is so regulated,
that equal areas of the ellipse are passed over by the radius vector in
equal times. &
(355.) What we have here stated supposes no knowledge of the sun's
actual distance from the earth, nor, consequently, of the actual dimen-
sions of its orbit, nor of the body of the sun itself. To come to any
conclusions on these points, we must first consider by what means we can
arrive at any knowledge of the distance of an object to which we have no
access. Now, it is obvious, that its parallaa alone can afford us any in-
formation on this subject. Suppose P A B Q to represent the earth, C
its centre, and S the sun, and A, B two situations of a spectator, or,
which comes to the same thing, the stations of two spectators, both ob-
serving the Sun S at the same instant. The spectator A will see it in the
direction A Sa, and will refer it to a point a in the infinitely distant
sphere of the stars, while the spectator B will see it in the direction BS b,
and refer it to b. The angle included between these directions, or the
Fig. 52.
measure of the celestial arc a b, by which it is displaced, is equal to the
angle ASB; and if this angle be known, and the local situations of A
and B, with the part of the earth’s surface A B included between them,
it is evident that the distance C S may be calculated. Now, since ASC
(art. 339) is the parallax of the sun as seen from A, and BSC as seen
from B, the angle ASB, or the total apparent displacement is the sum
of the two parallaxes. Suppose, then, two observers — one in the
northern, the other in the southern hemisphere—at stations on the same
meridian, to observe on the same day the meridian altitudes of the sun's
centre. Having thence derived the apparent zenith distances, and cleared
them of the effects of refraction, if the distance of the sun were equal
to that of the fixed stars, the sum of the zenith distances thus found
would be precisely equal to the sum of the latitudes north and south of
the places of observation. For the sum in question would then be equal

*
f
190 OUTLINES OF ASTRONOMY.
to the angle Z CX, which is the meridional distance of the stations
across the equator. But the effect of parallax being in both cases to in-
crease the apparent zenith distances, their observed sum will be greater
than the sum of the latitudes, by the sum of the two parallaxes, or by the
angle ASB. This angle, then, is obtained by subducting the sum of
the north and south latitudes from that of the zenith distances; and this
once determined, the horizontal parallax is easily found, by dividing the
angle so determined by the sum of the sines of the two latitudes.
(356.) If the two stations be not exactly on the same meridian (a cons
dition very difficult to fulfil), the same process will apply, if we take care
to allow for the change of the sun's actual zenith distance in the interval
of time elapsing between its arrival on the meridians of the stations. This
change is readily ascertained, either from tables of the sun's motion,
grounded on the experience of a long course of observations, or by actual
observation of its meridional altitude on several days before and after that
on which the observations for parallax are taken. Of course, the nearer
the stations are to each other in longitude, the less is this interval of time,
and, consequently, the smaller the amount of this correction; and, there-
fore, the less injurious to the accuracy of the final result is any uncertainty
in the daily change of zenith distance which may arise from imperfection
in the solar tables, or in the observations made to determine it.
(357.) The horizontal parallax of the sun has been concluded from
observations of the nature above described, performed in stations the most
remote from each other in latitude, at which observatories have been in-
stituted. It has also been deduced from other methods of a more refined
nature, and susceptible of much greater exactness, to be hereafter de-
scribed. Its amount so obtained, is about 8"-6. Minute as this quan-
tity is, there can be no doubt that it is a tolerably correct approximation
to the truth; and in conformity with it, we must admit the sun to be
situated at a mean distance from us, of no less than 23984 times the
length of the earth's radius, or about 95000000 miles.
(358.) That at so vast a distance the sun should appear to us of the
size it does, and should so powerfully influence our condition by its heat
and light, requires us to form a very grand conception of its actual mag-
nitude, and of the scale on which those important processes are carried on
within it, by which it is enabled to keep up its liberal and unceasing
supply of these elements. As to its actual magnitude we can be at no
loss, knowing its distance, and the angles under which its diameter appears
to us. An object, placed at the distance of 95000000 miles, and sub-
tending an angle of 32' 3", must have a real diameter of 882000 miles.
Such, then, is the diameter of this stupendous globe. If we compare it
THE EARTH's ANNUAL MOTION. 191
with what we have already ascertained of the dimensions of our own, w8
shall find that in linear magnitude it exceeds the earth in the proportion
1114 to 1, and in bulk in that of 1384472 to 1.
(359.) It is hardly possible to avoid associating our conception of an
object of definite globular figure, and of such enormous dimensions, with
some corresponding attribute of massiveness and material solidity. That
the sun is not a mere phantom, but a body having its own peculiar struc-
ture and economy, our telescopes distinctly inform us. They show us
dark spots on its surface, which slowly change their places and forms, and
by attending to whose situation, at different times, astronomers have ascer-
tained that the sun revolves about an axis nearly perpendicular to the
plane of the ecliptic, performing one rotation in a period of about 25 days,
and in the same direction with the diurnal rotation of the earth, i. e. from
west to east. Here, then, we have an analogy with our own globe; the
slower and more majestic movement only corresponding with the greater
dimensions of the machinery, and impressing us with the prevalence of
similar mechanical laws, and of, at least, such a community of nature as
the existence of inertia and obedience to force may argue. Now, in the
exact proportion in which we invest our idea of this immense bulk with
the attribute of inertia, or weight, it becomes difficult to conceive its circu.
lation round so comparatively small a body as the earth, without, on the
one hand, dragging it along, and displacing it, if bound to it by some in-
visible tie; or, on the other hand, if not so held to it, pursuing its course
alone in space, and leaving the earth behind. If we connect two solid
masses by a rod, and fling them aloft, we see them circulate about a point
between them, which is their common centre of gravity; but if one of
them be greatly more ponderous than the other, this common centre will
be proportionally nearer to that one, and even within its surface; so that
the smaller one will circulate, in fact, about the larger, which will be com-
paratively but little disturbed from its place. -
(360.) Whether the earth move round the sun, the sun round the
earth, or both round their common centre of gravity, will make no dif-
ference, so far as appearances are concerned, provided the stars be sup-
posed sufficiently distant to undergo no sensible apparent parallactic
displacement by the motion so attributed to the earth. Whether they are
so or not must still be a matter of inquiry; and from the absence of any
measurable amount of such displacement, we can conclude nothing but
this, that the scale of the sidereal universe is so great, that the mutual
orbit of the earth and sun may be regarded as an imperceptible point in
comparison with the distance of its nearest members. Admitting, then,
in conformity with the laws of dynamics, that two bodies connected with
192 OUTLINES OF ASTRONOMY.
and revolving about each other in free space do, in fact, revolve about their
common centre of gravity, which remains immoveable by their mutual
action, it becomes a matter of further inquiry, whereabouts between them
this centre is situated. Mechanics teach us that its place will divide their
mutual distance in the inverse ratio of their weights or masses;' and
calculations grounded on phenomena, of which an account will be given
further on, inform us that this ratio, in the case of the sun and earth, is
actually that of 354936 to 1,– the sun being, in that proportion, more
ponderous than the earth. From this it will follow that the common point
about which they both circulate is only 267 miles from the sun's centre,
or about asſooth part of its own diameter.
(361.) Henceforward, then, in conformity with the above statements,
and with the Copernican view of our system, we must learn to look upon
the sun as the comparatively motionless centre about which the earth per-
forms an annual elliptic orbit of the dimensions and excentricity, and with
a velocity, regulated according to the law above assigned; the sun occu-
pying one of the foci of the ellipse, and from that station quietly dissemi-
nating on all sides its light and heat; while the earth travelling round it,
and presenting itself differently to it at different times of the year and
day, passes through the varieties of day and night, summer and winter,
which we enjoy. --
(362.) In this annual motion of the earth, its axis preserves, at all
times, the same direction as if the orbitual movement had no existence;
and is carried round parallel to itself, and pointing always to the same
vanishing point in the sphere of the fixed stars. This it is which gives
rise to the variety of seasons, as we shall now explain. In so doing, we
shall neglect (for a reason which will be presently explained) the ellipticity
of the orbit, and suppose it a circle, with the Sun in the centre.
* Principia, lib. i. lex. iii. cor. 14.

OF THE SEASONS. 193
(363.) Let, then, S represent the sun, and A, B, C, D, four positions
of the earth in its orbit 90° apart, viz. A that which it has on the 21st
of March, or at the time of the vernal equinox; B that of the 21st of
June, or the summer solstice; C that of the 21st of September, or the
autumnal equinox; and D that of the 21st of December, or the winter
solstice. In each of these positions let PQ represent the axis of the
earth, about which its diurnal rotation is performed without interfering
with its annual motion in its orbit. Then, since the sun can only en-
lighten one half of the surface at once, viz. that turned towards it, the
shaded portions of the globe in its several positions will represent the
dark, and the bright, the enlightened halves of the earth's surface in these
positions. Now, 1st, in the position A, the sun is vertically over the
intersection of the equinoctial FE and the ecliptic H. G. It is, therefore,
in the equinox; and in this position the poles PQ, both fall on the ex-
treme confines of the enlightened side. In this position, therefore, it is
day over half the northern and half the southern hemisphere at once;
and as the earth revolves on its axis, every point of its surface describes
half its diurnal course in light, and half in darkness; in other words, the
duration of day and might is here equal over the whole globe: hence the
term equinox. The same holds good at the autumnal equinox on the
position C.
(364.) B is the position of the earth at the time of the northern
summer solstice. Here the north pole P, and a considerable portion of
the earth’s surface in its neighbourhood, as far as B, are situated within
the enlightened half. As the earth turns on its axis in this position,
therefore, the whole of that part remains constantly enlightened; there-
fore, at this point of its orbit, or at this season of the year, it is continual
day at the north pole, and in all that region of the earth which encircles
this pole as far as B-that is, to the distance of 23°28' from the pole,
or within what is called in geography, the arctic circle. On the other
hand, the opposite or south pole Q, with all the region comprised within
the antarctic circle, as far as 23°28' from the south pole, are immersed
at this season in darkness during the entire diurnal rotation, so that it is
here continual night.
(365.) With regard to that portion of the surface comprehended
between the arctic and antarctic circles, it is no less evident that the
nearer any point is to the north pole, the larger will be the portion of
its diurnal course comprised within the bright, and the smaller within
the dark hemisphere; that is to say, the longer will be its day, and the
shorter its night. Every station north of the equator will have a day of
more and a night of less than twelve hours' duration, and vice versä.
13
194 ouTLINES OF ASTRONOMY.
All these phenomena are exactly inverted when the earth comes to the
opposite point D of its orbit.
(366.) Now, the temperature of any part of the earth's surface
depends mainly on its exposure to the sun's rays. Whenever the sun is
above the horizon of any place, that place is receiving heat; when below,
parting with it, by the process called radiation; and the whole quantities
received and parted with in the year (secondary causes apart) must
balance each other at every station, or the equilibrium of temperature
(that is to say, the constancy which is observed to prevail in the annual
averages of temperature as indicated by the thermometer) would not be
supported. Whenever, then, the sun remains more than twelve hours
above the horizon of any place, and less beneath, the general temperature
of that place will be above the average; when the reverse, below. As
the earth, then, moves from A to B, the days growing longer, and the
nights shorter, in the northern hemisphere, the temperature of every part
of that hemisphere increases, and we pass from spring to summer; while,
at the same time, the reverse obtains in the southern hemisphere. As
the earth passes from B to C, the days and nights again approach to
equality—the excess of temperature in the northern hemisphere above
the mean state grows less, as well as its defect in the southern ; and at
the autumnal equinox C, the mean state is once more attained. From
thence to D, and, finally, round again to A, all the same phenomena, it
is obvious, must again occur, but reversed,—it being now winter in the
northern and summer in the southern hemisphere.
(367.) All this is exactly consonant to observed fact. The continual
day within the polar circles in summer, and night in winter, the general
increase of temperature and length of day as the sun approaches the
elevated pole, and the reversal of the seasons in the northern and southern
hemispheres, are all facts too well known to require further comment.
The positions A, C of the earth correspond, as we have said, to the
equinoxes; those at B, D to the solstices. This term must be explained.
If, at any point, X, of the orbit, we draw XP the earth's axis, and X S
to the sun, it is evident that the angle PX S will be the sun's polar
distance. Now, this angle is at its maximum in the position D, and at
its minimum at B; being in the former case=90°,+23° 28–103°28',
and in the latter 90°–23°28'-66° 32'. At these points the sun
ceases to approach to or to recede from the pole, and hence the name
Rolstice.
(368.) The elliptic form of the earth's orbit has but a very trifling
share in producing the variation of temperature corresponding to the
difference of seasons This assertion may at first sight seem incompati-
EQUAL SUPPLY OF HEAT TO BOTH HEMISPHERES. 195
ble with what we know of the laws of the communication of heat from
a luminary placed at a variable distance. Heat, like light, being equally
dispersed from the Sun in all directions, and being spread over the surface
of a sphere continually enlarging as we recede from the centre, must, of
course, diminish in intensity according to the inverse proportion of the
surface of the sphere over which it is spread; that is, in the inverse pro-
portion of the square of the distance. But we have seen (art. 350) that
this is also the proportion in which the angulaš velocity of the earth
about the sun varies. Hence it appears, that the momentary supply of *.
heat received by the earth from the sun varies in the exact proportion of
angular velocity, i.e. of the momentary increase of longitude: and from
this it follows, that equal amounts of heat are received from the sun in
passing over equal angles round it, in whatever part of the ellipse those
angles may be situated. Let, then, S represent the sun; A Q M P the
Fig. 54.
earth's orbit; A its nearest point to the sun, or, as it is called, the peri-
helion of its orbit; M the farthest, or the aphelion; and therefore A S
M the aa is of the ellipse. Now, suppose the orbit divided into two
segments by a straight line PS Q, drawn through the sun, and anyhow
situated as to direction; then, if we suppose the earth to circulate in the
direction P A Q MP, it will have passed over 180° of longitude in
moving from P to Q, and as many in moving from Q to P. It appears,
therefore, from what has been shown, that the supplies of heat received
from the sun will be equal in the two segments, in whatever direction the
line PS Q be drawn. They will, indeed, be described in unequal times;
that in which the perihelion. A lies in a shorter, and the other in a longer,
in proportion to their unequal area: but the greater proximity of the sun
in the smaller segment compensates exactly for its more rapid description,
and thus an equilibrium of heat is, as it were, maintained. Were it not
for this, the excentricity of the orbit would materially influence the tran-
sition of seasons. The fluctuation of distance amounts to nearly goth of

196 OUTLINES OF ASTRONOMY
its mean quantity, and, consequently, the fluctuation in the sun's direct
heating power to double this, or ºth of the whole. Now, the perihelion
of the orbit is situated nearly at the place of the northern winter solstice;
so that, were it not for the compensation we have just described, the effect
would be to exaggerate the difference of summer and winter in the
southern hemisphere, and to moderate it in the northern ; thus producing
a more violent alternation of climate in the one hemisphere, and an
approach to perpetual spring in the other. As it is, however, no guch
inequality subsists, but an equal and impartial distribution of heat and
light is accorded to both.
(369.) This does not prevent, however, the direct impression of the
solar heat in the height of summer, — the glow and ardour of his rays,
under a perfectly clear sky, at noon, in equal latitudes and under equal
circumstances of exposure, from being very materially greater in the
southern hemisphere than in the northern. One fifteenth is too considera-
ble a fraction of the whole intensity of sunshine not to aggravate in a
serious degree the sufferings of those who are exposed to it in thirsty
deserts, without shelter. The accounts of these sufferings in the interior
of Australia, for instance, are of the most frightful kind, and would seem
far to exceed what have ever been undergone by travellers in the northern
deserts of Africa."
(370.) A conclusion of a very remarkable kind, recently drawn by Pro-
fessor Dove from the comparison of the thermometric observations at
different seasons in very remote regions of the globe, may appear on first
sight at variance with what is above stated. That eminent meteorologist
has shown, by taking at all seasons the mean of the temperatures of points
diametrically opposite to each other, that the mean temperature of the
whole earth's surface in June considerably exceeds that in December.
This result, which is at variance with the greater proximity of the sun in
December, is, however, due to a totally different and very powerful cause,
— the greater amount of land in that hemisphere which has its summer
solstice in June (i. e. the northern, see art. 362); and the fact is so
explained by him. The effect of land under sunshine is to throw heat
into the general atmosphere, and so distribute it by the carrying power of
the latter over the whole earth. Water is much less effective in this
respect, the heat penetrating its depths, and being there absorbed; so that
* See the account of Captain Sturt's exploration in Athenaeum, No. 1012. “The
ground was almost a molten surface, and if a match accidentally fell upon it, it imme-
diately ignited.” The author has observed the temperature of the sulface soil in
South Africa as high as 159° Fahrenheit. An ordinary lucifer match does not ignite
when simply pressed upon a smooth surface at 212°, but in the act of withdrawing it,
it takes fire, and the slightest friction upon such a surface of course ignites it.
MEAN TEMPERATURE OF THE EARTH's SURFACE. 197
the surface never acquires a very elevated temperature even under the
equator.
(371.) The great key to simplicity of conception in astronomy, and,
indeed, in all Sciences where motion is concerned, consists in contempla-
ting every movement as referred to points which are either permanently
fixed, or so nearly so, as that their motions shall be too small to interfere
materially with and confuse our motions. In the choice of these primary
points of reference, too, we must endeavour, as far as possible, to select
such as have simple and symmetrical geometrical relations of situation with
respect to the curves described by the moving parts of the system, and
which are thereby fitted to perform the office of natural centres—advan-
tageous stations for the eye of reason and theory. Having learned to
attribute an orbitual motion to the earth, it loses this advantage, which is
transferred to the sun, as the fixed centre about which its orbit is per-
formed. Precisely as, when embarrassed by the earth’s diurnal motion,
we have learned to transfer, in imagination, our station of observation
from its surface to its centre, by the application of the diurnal parallax;
so, when we come to inquire into the movements of the planets, we shall
find ourselves continually embarrassed by the orbitual motion of our point
of view, unless, by the consideration of the annual or heliocentric paral-
law, we consent to refer all our observations on them to the centre of the
sun, or rather to the common centre of gravity of the sun, and the other
bodies which are connected with it in our system. Hence arises the dis-
tinction between the geocentric and heliocentric place of an object. The
former refers its situation in space to an imaginary sphere of infinite
radius, having the centre of the earth for its centre — the latte, to one
concentric with the sun. Thus, when we speak of the heliocentric longi-
tudes and latitudes of objects, we suppose the spectator situated in the sun
and referring them by circles perpendicular to the plane of the ecliptic,
to the great circle marked out in the heavens by the infinite prolongation
of that plane.
(372.) The point in the imaginary concave of an infinite heaven, to
which a spectator in the sun refers the earth, must, of course, be diame-
trically opposite to that to which a spectator on the earth refers the sun’s
centre; consequently the heliocentric latitude of the earth is always
nothing, and its heliocentric longitude always equal to the sun's geocentric
longitude--180°. The heliocentric equinoxes and solstices are, therefore,
the same as the geocentric reversely named; and to conceive them, we
have only to imagine a plane passing through the sun's centre, parallel to
the earth’s equator, and prolonged infinitely on all sides. The line of inter-
198 OUTLINES OF ASTRONOMY.
section of this plane and the plane of the ecliptic is the line of equinoxes,
and the solstices are 90° distant from it.
(373.) The position of the longer axis of the earth’s orbit is a point
of great importance. In the figure (art. 368) let E C L I be the ecliptic,
E the vernal equinox, L the autumnal (i. e. the points to which the earth
is referred from the sun when its heliocentric longitudes are 0° and 180°
respectively). Supposing the earth's motion to be performed in the direc-
tion E C L I, the angle E SA, or the longitude of the perihelion, in the
year 1800 was 99° 30' 5": we say in the year 1800, because, in point of
fact, by the operation of causes hereafter to be explained, its position is
subject to an extremely slow variation of about 12" per annum to the
eastward, and which in the progress of an immensely long period—of no
less than 20984 years—carries the axis A S M of the orbit completely
round the whole circumference of the ecliptic. But this motion must be
disregarded for the present, as well as many other minute deviations, to be
brought into view when they can be better understood.
(374.) Were the earth's orbit a circle, described with a uniform
velocity about the sun placed in its centre, nothing could be easier than
to calculate its position at any time with respect to the line of equinoxes,
or its longitude, for we should only have to reduce to numbers the pro-
portion following; viz. One year : the time elapsed :: 360°: the arc of
longitude passed over. The longitude so calculated is called in astronomy
the mean longitnde of the earth. But since the earth's orbit is neither
circular, nor uniformly described, this rule will not give us the true place
in the orbit of any proposed moment. Nevertheless, as the excentricity
and deviation from a circle are small, the true place will never deviate
very far from that so determined (which for distinction's sake, is called
the mean place), and the former may at all times be calculated from the
latter, by applying to it a correction or equation (as it is termed), whose
amount is never very great, and whose computation is a question of pure
geometry, depending on the equable description of areas by the earth
about the sun. For since, in elliptic motion according to Kepler's law
above stated, areas not angles are described uniformly, the proportion
must now be stated thus;—One year : the time elapsed :: the whole area
of the ellipse : the area of the sector swept over by the radius vector in
that time. This area, therefore, becomes known, and it is then, as above
observed, a problem of pure geometry to ascertain the angle about the sun
(ASP, fig. art. 368), which corresponds to any proposed fractional area of
the whole ellipse supposed to be contained in the sector A. P. S. Suppose
we set out from A the perihelion, then will the angle A S P at first
increase more rapidly than the mean longitude, and will, therefore, during
$
OF THE SUN's MEAN AND TRUE LONGITUDES. 199
the whole semi-revolution from A to M, exceed it in amount; or, ill other
words, the true place will be in advance of the mean: at M, one half the
year will have elapsed, and one half the orbit have been described,
whether it be circular or elliptic. Here, then, the mean and true places
coincide; but in all the other half of the orbit, from M to A, the true
place will fall short of the mean, since at M the angular motion is slowest,
and the true place from this point begins to lag behind the mean – to
make up with it, however, as it approaches A, where it once more over-
takes it.
(375.) The quantity by which the true longitude of the earth differs
from the mean longitude is called the equation of the centre, and is addi-
tive during all the half-year, in which the earth passes from A to M,
beginning at 0° 0' 0", increasing to a maximum, and again diminishing
to zero at M ; after which it becomes subtractive, attains a maximum of
subtractive magnitude between M and A, and again diminishes to 0 at A.
Its maximum, both additive and subtractive, is 1° 55' 33"-3.
(376.) By applying, then, to the earth’s mean longitude, the equation
of the centre corresponding to any given time at which we would ascer-
tain its place, the true longitude becomes known; and since the sun is
always seen from the earth in 180° more longitude than the earth from
the sun, in this way also the sun's true place in the ediptic becomes
known. The calculation of the equation of the centre is performed by a
table constructed for that purpose, to be found in all “Solar Tables.”
(377.) The maximum value of the equation of the centre depends only
on the ellipticity of the orbit, and may be expressed in terms of the ex-
centricity. Vice versá, therefore, if the former quantity can be ascer-
tained by observation, the latter may be derived from it; because, when-
ever the law, or numerical connection, between two quantities is known,
the one can always be determined from the other. Now, by assiduous
observation of the sun's transits over the meridian, we can ascertain, for
every day, its exact right ascension, and thence conclude its longitude
(art. 309). After this, it is easy to assign the angle by which this
observed longitude exceeds or falls short of the mean; and the greatest
amount of this excess or defect which occurs in the whole year, is the
maximum equation of the centre. This, as a means of ascertaining the
excentricity of the orbit, is a far more easy and accurate method than
that of concluding the sun's distance by measuring its apparent diameter.
The results of the two methods coincide, however, perfectly.
(378.) If the ecliptic coincided with the equinoctial, the effect of the
equation of the centre, by disturbing the uniformity of the sun's apparent
motion in longitude, would cause an inequality in its time of coming on
200 - OUTLINES OF ASTRONOMY.
the meridian on successive days. When the sun's centre comes to the
meridian, it is apparent noon, and if its motion in longitude were uni-
form, and the ecliptic coincident with the equinoctial, this would always
coincide with mean noon, or the stroke of 12 on a well-regulated solar
clock. But, independent of the want of uniformity in its motion, the
obliquity of the ecliptic gives rise to another inequality in this respect; in
consequence of which, the sun, even supposing its motion in the ecliptic
uniform, would yet alternately, in its time of attaining the meridian, anti-
cipate and fall short of the mean noon as shown by the clock. For the
right ascension of a celestial object forming a side of a right-angled sphe-
rical triangle, of which its longitude is the hypothenuse, it is clear that
the uniform increase of the latter must necessitate a deviation from uni-
formity in the increase of the former.
(379.) These two causes, then, acting conjointly, produce, in fact, a
very considerable fluctuation in the time as shown per clock, when the sun
really attains the meridian. It amounts, in fact, to upwards of half an
hour; apparent noon sometimes taking place as much as 164 min. before
mean noon, and at others as much as 14% min. after. This difference
between apparent and mean noon is called the equation of time, and is
calculated and inserted in ephemerides for every day of the year, under
that title: or else, which comes to the same thing, the moment, in mean
time, of the Sun's culmination for each day, is set down as an astrono-
mical phaenomenon to be observed. -
(380.) As the sun, in its apparent annual course, is carried along the
ecliptic, its declination is continually varying between the extreme limits
of 23° 27' 30" north, and as much south, which it attains at the sol-
stices. It is consequently always vertical over some part or other of that
zone or belt of the earth's surface which lies between the north and south
parallels of 23° 27' 30". These parallels are called in geography the
tropics; the northern one that of Cancer, and the southern, of Capri-
corn; because the sum, at the respective solstices, is situated in the divi-
sions, or signs of the ecliptic so denominated. Of these signs there are
twelve, each occupying 30° of its circumference. They commence at the
vernal equinox, and are named in order— Aries, Taurus, Gemini, Cancer,
Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces."
They are denoted also by the following symbols:–90, 8, II, gö, Sl. 11, =,
ſil, 1. V5, 3%, 36. Longitude itself is also divided into signs, degrees, and
minutes, &c. Thus 5* 27° 0' corresponds to 177° 0'.
* They may be remembered by the following memorial hexameters : —
Sunt Aries, Taurus, Gemini, Cancer, Leo, Virgo,
Libraque, Scorpius, Arcitemens, Caper, Amphora, Pisces.
SIDEREAL, TROPICAL, AND ANOMALISTIC YEARS. 201
(381.) These Signs are purely technical subdivisions of the ecliptic,
commencing from the actual equinox, and are not to be confounded with
the constellations so called (and sometimes so symbolized). The constel-
lations of the zodiac, as they now stand arranged on the ecliptic, are all a
full “sign” in advance or anticipation of their symbolic cognomens thereon
marked. Thus the constellation Aries actually occupies the sign Taurus
8, the constellation Taurus, the sign Gemini II, and so on, the signs
having retreated' among the stars (together with the equinox their origin),
by the effect of precession. The bright star Spica in the constellation
Virgo (a Virginis), by the observations of Hipparchus, 128 years B. C.,
preceded, or was westward of the autumnal equinox in longitude by 6°.
In 1750 it followed or stood eastward of the same equinox by 20° 21'.
Its place then, as referred to the ecliptic at the former epoch, would be in
longitude 5" 24° 0', or in the 24th degree of the sign Sl, whereas in the
latter epoch it stood in the 21st degree of ſº, the equinox having retreated
by 26° 21' in the interval, 1878 years, elapsed. To avoid this source of
misunderstanding, the use of “signs” and their symbols in the reckoning
of celestial longitudes is now almost entirely abandoned, and the ordinary
reckoning (by degrees, &c. from 0 to 360) adopted in its place, and the
names Aries, Virgo, &c. are becoming restricted to the constellations so
called.”
(382.) When the sun is in either tropic, it enlightens, as we have seen,
the pole on that side the equator, and shines over or beyond it to the
extent of 23° 27' 30". The parallels of latitude, at this distance from
either pole, are called the polar circles, and are distinguished from each
other by the names arctic and antarctic. The regions within these
circles are sometimes termed frigid zones, while the belt between the
tropics is called the torrid zone, and the intermediate belts temperate zones.
These last, however, are merely names given for the sake of naming; as,
in fact, owing to the different distribution of land and sea in the two hemi-
spheres, zones of climate are not co-terminal with zones of latitude.
(383.) Our seasons are determined by the apparent passages of the sun
across the equinoctial, and its alternate arrival in the northern and south-
ern hemisphere. Were the equinox invariable, this would happen at
intervals precisely equal to the duration of the sidereal year; but, in fact,
* Retreated is here used with reference to longitude, not to the apparent diurnal
motion.
* When, however, the place of the sun is spoken of the old usage prevails. Thus,
if we say “the sun is in Aries,” it would be interpreted to mean between 0° and 30°
of longitude. So, also, “the first point of Aries” is still understood to mean the
vernal, and “the first point of Libra,” the autumnal equinox; and so in a few other
CàSBS's
202 OUTLINES OF ASTRONOMY.
owing to the slow conical motion of the earth’s axis described in art. 317,
the equinox retreats on the ecliptic, and meets the advancing sun some-
what before the whole sidereal circuit is completed. The annual retreat
of the equinox is 50"-1, and this are is described by the sun in the eclip-
tic in 20m 1949. By so much shorter, then, is the periodical return of
our seasons than the true sidereal revolution of the earth round the sun.
As the latter period, or sidereal year, is equal to 365' 6' 9" 9*6, it fol-
lows, then, that the former must be only 365° 5' 48* 49*7; and this is
what is meant by the tropical year. -
(384.) We have already mentioned that the longer axis of the ellipse
described by the earth has a slow motion of 11".8 per annum in advance.
From this it results, that when the earth, setting out from the perihelion,
has completed one sidereal period, the perihelion will have moved
forward by 11”.8, which are must be described by the earth before it can
again reach the perihelion. In so doing, it occupies 4* 39*7, and this
must therefore be added to the sidereal period, to give the interval between
two consecutive returns to the perihelion. This interval, then, is 365"
6, 13m 49-3, and is what is called the anomalistic year. All these
periods have their uses in astronomy; but that in which mankind in
general are most interested is the tropical year, on which the return of
the seasons depends, and which we thus perceive to be a compound phe-
nomenon, depending chiefly and directly on the annual revolution of the
earth round the sun, but subordinately also, and indirectly, on its rotation
round its own axis, which is what occasions the precession of the equi-
noxes; thus affording an instructive example of the way in which a
motion, once admitted in any part of our system, may be traced in its
influence on others with which at first sight it could not possibly be sup-
posed to have any thing to do.
(385.) As a rough consideration of the appearance of the earth points
out the general roundness of its form, and more exact inquiry has led us
first to the discovery of its elliptic figure, and, in the further progress of
refinement, to the perception of minuter local deviations from that figure;
so, in investigating the solar motions, the first notion we obtain is that of
an orbit, generally speaking, round, and not far from a circle, which, on
more careful and exact examination, proves to be an ellipse of small excen-
tricity, and described in conformity with certain laws, as above stated.
Still minuter inquiry, however, detects yet smaller deviations again from
this form and from these laws, of which we have a specimen in the slow
motion of the axis of the orbit spoken of in art. 372; and which are
* These numbers, as well as all the other numerical data of our system, are taken
from Mr. Baily's Astronomical Tables and Formulae, unless the contrary is expressed
PHYSICAL CONSTITUTION OF THE SUN, 203
generally comprehended under the name of perturbations and secular in-
equalities. Of these deviations, and their causes, we shall speak here-
after at length. It is the triumph of physical astronomy to have rendered
a complete account of them all, and to have left nothing unexplained,
either in the motions of the sun or in those of any other of the bodies of
our system. But the nature of this explanation cannot be understood till
we have developed the law of gravitation, and carried it into its more
direct consequences. This will be the object of our three following chap-
ters; in which we shall take advantage of the proximity of the moon,
and its immediate connection with and dependence on the earth, to render
it, as it were, a stepping-stone to the general explanation of the planet-
ary movements. We shall conclude this by describing what is known of
the physical constitution of the sun.
(386.) When viewed through powerful telescopes, provided with
coloured glasses, to take off the heat, which would otherwise injure our
eyes, the sun is observed to have frequently large and perfectly black
spots upon it, surrounded with a kind of border, less completely dark,
called a Penº; Aşgº of these are represented at a, b, c, d, in Plate
I. fig. 2, at the § ºffs volume. They are, however, not permanent.
When watched from day to day, or even from hour to hour, they appear
to enlarge or contract, to change their forms, and at length to disappear
altogether, or to break out "anew in parts of the surface where none were
before. In such cases of disappearance, the central dark spot always con-
tracts into a point, and vanishes before the border. Occasionally they
break up, or divide into two or more, and in those cases offer every evi-
dence of that extreme mobility which belongs only to the fluid state, and
of that excessively violent agitation which seems only compatible with the
atmospheric or gaseous state of matter. The scale on which their move-
ments take place is immense. A single second of angular measure, as
seen from the earth, corresponds on the sun's disc to 461 miles; and a
circle of this diameter (containing therefore nearly 167000 square miles)
is the least space which can be distinctly discerned on the sun as a visible
area. Spots have been observed, however, whose linear diameter has
been upwards of 45000 miles;' and even, if some records are to be
trusted, of very much greater extent. That such a spot should close up
in six weeks' time (for they seldom last much longer), its borders must
approach at the rate of more than 1000 miles a day.
(387.) Many other circumstances tend to corroborate this view of the
subject. The part of the sun's disc not occupied by spots is far from
* Mayer, Obs. Mar. 15, 1758. “Ingens macula in sole conspiciebatur, cujus diam
eter= 3; diam. Solis.
204 oUTLINEs of ASTRONOMY.
uniformly bright. Its ground is finely mottled with an appearance of
minute, dark dots, or pores, which, when attentively watched, are found
to be in a constant state of change. There is nothing which represents
so faithfully this appearance as the slow subsidence of some flocculent
chemical precipitates in a transparent fluid, when viewed perpendicularly
from above: so faithfully, indeed, that it is hardly possible not to be im-
pressed with the idea of a luminous medium intermixed, but not con-
founded, with a transparent and non-luminous atmosphere, either floating
as clouds in our air, or pervading it in vast sheets and columns like flame,
or the streamers of our northern lights, directed in lines perpendicular to
the surface.
(388.) Lastly, in the neighbourhood of great spots, or extensive groups
of them, large spaces of the surface are often observed to be covered with
strongly marked curved or branching streaks, more luminous than the
rest, called faculae, and among these, if not already existing, spots fre-
quently break out. They may, perhaps, be regarded with most proba-
bility, as the ridges of immense waves in the luminous regions of the
Sun's atmosphere, indicative of violent agitation in their neighbourhood.
They are most commonly, and best seen, towards the borders of the
visible disc, and their appearance is as represented in Plate I. fig. 1.
(389.) But what are the spots? Many fanciful notions have been
broached on this subject, but only one seems to have any degree of
physical probability, viz. that they are the dark, or at least comparatively
dark, solid body of the sun itself, laid bare to our view by those immense
fluctuations in the luminous regions of its atmosphere, to which it appears
to be subject. Respecting the manner in which this disclosure takes
place, different ideas again have been advocated. Lalande (art. 3240)
suggests, that eminences in the nature of mountains are actually laid
bare, and project above the luminous ocean, appearing black above it,
while their shoaling declivities produce the penumbrae, where the lumi-
nous fluid is less deep. A fatal objection to this theory is the uniform
shade of the penumbra and its sharp termination, both inwards, where it
joins the spot, and outwards, where it borders on the bright surface. A
more probable view has been taken by Sir William Herschel,' who con-
siders the luminous strata of the atmosphere to be sustained far above
the level of the solid body by a transparent elastic medium, carrying on
its upper surface (or, rather, to avoid the former objection, at some con-
siderably lower level within its depth) a cloudy stratum which, being
strongly illuminated from above, reflects a considerable portion of the
light to our eyes, and forms a penumbra, while the solid body shaded by
* Phil. Trans. 1801.
NATURE OF THE SUN'S SPOTS. 205
the clouds, reflects none. (See fig.) The temporary removal of both
the strata, but more of the upper than the lower, he supposes effected by
powerful upward currents of the atmosphere, arising, perhaps, from
spiracles in the body, or from local agitations.
(390.) When the spots are attentively watched, their situation on the
disc of the sun is observed to change. They advance regularly towards
its western limb or border, where they disappear, and are replaced by *
others which enter at the eastern limb, and which, pursuing their respec-
tive courses, in their turn disappear at the western. The apparent
rapidity of this movement is not uniform, as it would be were the spots
dark bodies passing, by an independent motion of their own, between the
earth and the sun; but is swiftest in the middle of their paths across
the disc, and very slow at its borders. This is precisely what would be
the case supposing them to appertain to and make part of the visible
surface of the sun's globe, and to be carried round by a uniform rotation
of that globe on its axis, so that each spot should describe a circle parallel
to the sun's equator, rendered elliptic by the effect of perspective. Their
apparent paths also across the disc conform to this view of their nature,
being, generally speaking, ellipses, much elongated, concentric with the
sun's disc, each having one of its chords for its longer axis, and all these
axes parallel to each other. At two periods of the year only do the
spots appear to describe straight lines, viz. on and near to the 11th of
June and the 12th of December, on which days, therefore, the plane of
the circle, which a spot situated on the sun's equator describes (and con-
sequently, the plane of that equator itself.) passes through the earth.
Hence it is obvious, that the plane of the sun's equator is inclined to that
of the ecliptic, and intersects it in a line which passes through the place
of the earth on these days. The situation of this line, or the line of the

206 OUTLINES OF ASTRONOMY.
nodes of the sun's equator as it is called, is, therefore, defined by the
longitudes of the earth as seen from the sun at those epochs, which are
respectively 80° 21' and 260° 21' (=80°21'4-180°) being, of course,
diametrically opposite in direction.
(391.) The inclination of the sun's axis (that of the plane of its
equator) to the ecliptic is determined by ascertaining the proportion of
the longer and the shorter diameter of the apparent ellipse, described by
any remarkable, well-defined spot; in order to do which, its apparent
place on the sun's disc must be very precisely ascertained by micrometric
measures, repeated from day to day as long as it continues visible (usually
about 12 or 13 days, according to the magnitude of the spots, which
always vanish by the effect of foreshortening before they attain the actual
border of the disc—but the larger spots being traceable closer to the limb
than the smaller.") The reduction of such observations, or the conclu-
sion from them of the element in question, is complicated with the effect
of the earth’s motion in the interval of the observations, and with its
situation in the ecliptic, with respect to the line of nodes. For simplicity,
we will suppose the earth situated as it is on the 10th of March, in a line
at right angles to that of the nodes, i. e. in the heliocentric longitude
170° 21', and to remain there stationary during the whole passage of a
spot across the disc. In this case the axis of rotation of the sun will be
Fig. 56.
N

29 S
situated in a plane passing through the earth and at right angles to the
plane of the ecliptic. Suppose C to represent the sun's centre, P C p
its axis, E C the line of sight, P N Q A p S a section of the sun passing
* The great spot of December, 1719, is stated to have been seen as a notch in the
limb of the sun.
\
;
. OF THE SUN's ROTATION ON ITS AxIS. 20.
through the earth, and Q a spot situated on its equator, and in that plane
and consequently in the middle of its apparent path across the disc. If
the axis of rotation were perpendicular to the ecliptic, as N S, this spot
would be at A, and would be seen projected on C, the centre of the sun.
It is actually at Q, projected upon D, at an apparent distance C D to the
north of the centre, which is the apparent smaller semi-axis of the ellipse
described by the spot, which being known by micrometric measurement,
the value of º or the cosine of Q C N, the inclination of the sun’s
equator becomes known, C N being the apparent semi-diameter of the
sun at the time. At this epoch, moreover, the northern half of the circle
described by the spot is visible (the southern passing behind the body of
the sun,) and the south pole p of the sun is within the visible hemi-
sphere. This is the case in the whole interval from December 11th to
July 12th, during which, the visual ray falls upon the southern side of
the sun's equator. The contrary happens in the other half year, from
July 12th to December 11th, and this is what is understood when we say
that the ascending node (denoted Q) of the sun's equator lies in 80° 21'
longitude—a spot on the equator passing that node being then in the act
of ascending from the southern to the northern side of the plane of the
ecliptic—such being the conventional language of astronomers in speaking
of these matters.
(392.) If the observations are made at other seasons (which, however,
are the less favourable for this purpose the more remote they are from
the epochs here assigned); when, moreover, as in strictness is necessary,
the motion of the earth in the interval of the measures is allowed for (as
for a change of the point of sight); the calculations requisite to deduce
the situation of the axis in space, and the duration of the revolution
around it, become much more intricate, and it would be beyond the scope
of this work to enter into them." According to the best determinations
we possess, the inclination of the sun's equator to the ecliptic is about 7°
20’ (its nodes being as above stated), and the period of rotation 25 days
7 hours 48 minutes.”
(393.) The region of the spots is confined, generally speaking, within
about 25° on either side of the sun's equator; beyond 30° they are very
* See the theory in Leland's Astronomy, art. 3258, and the formulae of computation
in a paper by Petersen Schumacher’s Nachrichten, No. 419.
*Bianchi (Schumacher's Nach. 483), agreeing with Laugier. Lelambre makes it
25°. On 17"; Petersen, 25° 4° 30”. The inclination of the axis is uncertain to half a
degree, and the node to several degrees. The continual changes in the spots them
selves cause this uncertainty.
208 OUTLINES OF ASTRONOMY.
rarely seen; in the polar regions, never. The actual equator of the sun
is also less frequently visited by spots than the adjacent zones on either
side, and a very material difference in their frequency and magnitude
subsists in its northern and southern hemisphere, those on the northern
preponderating in both respects. The zone comprised between the 11th
and 15th degree to the northward of the equator is particularly fertile in
large and durable spots. These circumstances, as well as the frequent
occurrence of a more or less regular arrangement of the spots, when
numerous, in the manner of belts parallel to the equator, point evidently
to physical peculiarities in certain parts of the sun's body more favourable
than in others to the production of the spots, on the one hand; and on
the other, to a general influence of its rotation on its axis as a determining
cause of their distribution and arrangement, and would appear indicative
of a system of movements in the fluids which constitute its luminous
surface bearing no remote analogy to our trade winds—from whatever
cause arising. (See art. 239. et seq.)
(394.) The duration of individual spots is commonly not great; some
are formed and disappear within the limit of a single transit across the
disc—but such are for the most part small and insignificant. Frequently
they make one or two revolutions, being recognized at their reappearance
by their situation with respect to the equator, their configurations inter se,
their size, or other peculiarities, as well as by the interval elapsing be-
tween their disappearance at one limb and reappearance on the other. In
a few rare cases, however, they have been watched round many revolu-
tions. The great spot of 1779 appeared during six months, and one and
the same group was observed in 1840 by Schwabe to return eight
times." It has been surmised, with considerable apparent probability, that
some spots, at least, are generated again and again, at distant intervals of
time, over the same identical points of the sun's body (as hurricanes, for
example, are known to affect given localities on the earth's surface, and to
pursue definite tracks). The uncertainty which still prevails with respect
to the exact duration of its rotation renders it very difficult to obtain con-
vincing evidence of this; mor, indeed, can it be expected, until by bring-
ing together into one connected view the recorded state of the sun's sur-
face during a very long period of time, and comparing together remarka-
ble spots which have appeared on the same parallel, some precise periodic
time shall be found which shall exactly conciliate numerous and well-
characterized appearances. The inquiry is one of singular interest, as
there can be no reasonable doubt that the supply of light and heat
Schum. Nach. No. 418, p. 150. The recent papers of Biela, Capocci, Schwabe,
Pastorff, and Schmidt, in that collection, will be found highly interesting.
OF THE SUN's SPOTs. 209
afforded to our globe stands in intimate connexion with those processes
which are taking place on the solar surface, and to which the spots in
some way or other owe their origin. f
(395.) Above the luminous surface of the sun, and the region in which
the spots reside, there are strong indications of the existence of a gaseous
atmosphere having a somewhat imperfect transparency. When the whole
disc of the sun is seen at once through a telescope magnifying moderately
cnough to allow it, and with a darkening glass such as to suffer it to be
contemplated with perfect comfort, it is very evident that the borders of
the disc are much less luminous than the centre. That this is no illusion
is shown by projecting the sun's image undarkened and moderately mag-
nified, so as to occupy a circle two or three inches in diameter, on a sheet
of white paper, taking care to have it well in focus, when the same ap-
pearance will be observed. This can only arise from the circumferential
rays having undergone the absorptive action of a much greater thickness
of some imperfectly transparent envelope (due to greater obliquity of
their passage through it) than the central. — But a still more convincing
and indeed decisive evidence is offered by the phaenomena attending a
total eclipse of the sun. Such eclipses (as will be shown hereafter) are
produced by the interposition of the dark body of the moon between the
earth and sun, the moon being large enough to cover and surpass, by a
very small breadth, the whole disc of the sun. Now when this takes
place, were there no vaporous atmosphere capable of reflecting any light
about the sun, the sky ought to appear totally dark, since (as will here-
after abundantly appear) there is not the smallest reason for believing the
moon to have any atmosphere capable of doing so. So far, however, is
this from being the case, that a bright ring or corona of light is seen,
fading gradually away, as represented in Pl. I. fig. 3., which (in cases
where the moon is not centrally superposed on the sun) is observed to be
concentric with the latter, not the former body. This corona was beauti-
fully seen in the eclipse of July 7, 1842, and with this most remarkable
addition — witnessed by every spectator in Pavia, Milan, Vienna, and
elsewhere: there distinct and very conspicuous rose-coloured protuberances
(as represented in the figure cited) were seen to project beyond the dark
limb of the moon, likened by some to flames, by others to mountains, but
which their enormous magnitude (för to have been seen at all by the
naked eye their height must have exceeded 40,000 miles), and their faint
degree of illumination, clearly prove to have been cloudy masses of the
most eaccessive tenuity, and which doubtless owed their support, and proba-
bly their existence, to such an atmosphere as we are now speaking of
(396.) That the temperature at the visible surface of the sun cannot
14 p
210 OUTLINES OF ASTRONOMY.
be otherwise than very elevated, much more so than any artificial heat
produced in our furnaces, or by chemical or galvanic processes, we have
indications of several distinct kinds: 1st, From the law of decrease of
radiant heat and light, which, being inversely as the squares of the dis-
tances, it follows, that the heat received on a given area exposed at the
distance of the earth, and on an equal area at the visible surface of the
sun, must be in the proportion of the area of the sky oceupied by the
sun's apparent disc to the whole hemisphere, or as 1 to about 300000.
A far less intensity of solar radiation, collected in the focus of a burning
glass, suffices to dissipate gold and platina in vapour. 2dly, From the
facility with which the calorific rays of the sun traverse glass, a property
which is found to belong to the heat of artificial fires in the direct pro-
portion of their intensity." 3dly, From the fact, that the most vivid
flames disappear, and the most intensely ignited solids appear only as
black spots on the disc of the sun when held between it and the eye.”
From the last remark it follows, that the body of the sun, however dark
it may appear when seen through its spots, may, nevertheless, be in a
state of most intense ignition. It does not, however, follow of necessity
that it must be so. The contrary is at least physically possible. A per-
fectly reflective canopy would effectually defend it from the radiation of
the luminous regions above its atmosphere, and no heat would be con-
ducted downwards through a gaseous medium increasing rapidly in
density. That the penumbral clouds are highly reflective, the fact of
their visibility in such a situation can leave no doubt.
(397.) As the magnitude of the sun has been measured, and (as we
shall hereafter see) its weight, or quantity of ponderable matter, ascer-
tained, so also attempts have been made, and not wholly without success,
from the heat actually communicated by its rays to given surfaces of
material bodies exposed to their vertical action on the earth’s surface, to
estimate the total expenditure of heat by that luminary in a given time.
The result of such experiments has been thus announced. Supposing a
cylinder of ice 45 miles in diameter, to be continually darted into the sun
with the velocity of light, and that the water produced by its fusion were
* By direct measurement with the actinometer, I find that out of 1000 calorific solar
rays, 816 penetrate a sheet of plate glass 0-12 inch thick; and that of 1000 rays which
have passed through one such plate, 859 are capable of passing through another. H.
1827. -
* The ball of ignited quicklime, in Lieutenant Drummond’s oxy-hydrogen lamp,
gives the nearest imitation of the solar splendour which has yet been produced. The
appearance of this against the sun, was, however, as described in an imperfect trial at
which I was present. The experiment ought to be repeated under favourable circum
stances.—Note to the ed. of 1833
TERRESTRIAL EFFECTs of THE SUN's RADIATION. 211
continually carried off, the heat now given off constantly by radiation
would then be wholly expended in its liquefaction, on the one hand, so as
to leave no radiant surplus; while, on the other, the actual temperature,
at its surface would undergo no diminution.
(398.) This immense escape of heat by radiation, we may remark, will
fully explain the constant state of tumultuous agitation in which the fluids
composing the visible surface are maintained, and the continual generation
and filling in of the pores, without having recourse to internal causes.
The mode of action here alluded to is perfectly represented to the eye in
, the disturbed subsidence of a precipitate, as described in art. 387, when
the fluid from which it subsides is warm, and losing heat from its surface.
(399.) The sun's rays are the ultimate source of almost every motion
which takes place on the surface of the earth. By its heat are produced
all winds, and those disturbances in the electric equilibrium of the atmo-
sphere which give rise to the phenomena of lightning, and probably also
to those of terrestrial magnetism and the aurora. By their vivifying
action vegetables are enabled to draw support from inorganic matter, and
become, in their turn the support of animals and of man, and the sources
of those great deposits of dynamical efficiency which are laid up for
human use in our coal strata." By them the waters of the sea are made
to circulate in vapour through the air, and irrigate the land, producing
springs and rivers. By them are produced all disturbances of the
chemical equilibrium of the elements of nature, which, by a series of
compositions and decompositions, give rise to new products, and originate .
a transfer of materials. Even the slow degradation of the solid con-
stituents of the surface, in which its chief geological changes consist, is
almost entirely due on the one hand to the abrasion of wind and rain, and
the alternation of heat and frost; on the other to the continual beating
of the sea waves, agitated by winds, the results of solar radiation. Tidal
action (itself partly due to the sun's agency) exercises here a compara-
tively slight influence. The effect of oceanic currents (mainly originating
in that influence,) though slight in abrasion, is powerful in diffusing and
transporting the matter abraded; and when we consider the immense
transfer of matter so produced, the increase of pressure over large spaces
in the bed of the ocean, and diminution over corresponding portions of
the land, we are not at a loss to perceive how the elastic power of sub-
terranean fires, thus repressed on the one hand and relieved on the other,
may break forth in points where the resistance is barely adequate to their
retention, and thus bring the phenomena of even volcanic activity under
the general law of solar influence.”
* So in the edition of 1833. * So in the edition of 1833,
212 - oUTLINEs of ASTRONOMY.
(400.) The great mystery, however, is to conceive how so enorm. Jus a
conflagration (if such it be) can be kept up. Every discovery in chemi-
cal science here leaves us completely at a loss, or rather, seems to remove
farther the prospect of probable explanation. If conjecture might be
hazarded, we should look rather to the known possibility of an indefinite
generation of heat by friction, or to its excitement by the electric dis-
charge, than to any actual combustion of ponderable fuel, whether solid
or gaseous, for the origin of the solar radiation."
* Electricity traversing excessively rarefied air or vapours, gives out light, and,
doubtless, also heat. May not a continual current of electric matter be constantly
circulating in the sun’s immediate neighbourhood, or traversing the planetary spaces,
and exciting, in the upper regions of its atmosphere, those phenomena of which, on
however diminutive a scale, we have yet an unequivocal manifestation in our aurora
borealis. The possible analogy of the solar light to that of the aurora has been
distinctly insisted on by the late Sir W. Herschel, in his paper already cited. It would
be a highly curious subject of experimental inquiry, how far a mere reduplication of
sheets of flame, at a distance one behind the other (by which their light might he
brought to any required intensity,) would communicate to the heat of the resulting
compound ray the penetrating character which distinguishes the solar calorific rays.
We may also observe, that the tranquillity of the sun's polar, as compared with its
equatortal regions (if its spots be really atmospheric,) cannot be accounted for by its
rotation on its axis only, but must arise from some cause eaternal to the luminous sur-
face of the sun, as we see the belts of Jupiter and Saturn, and our trade-winds arise
from a cause, external to these planets, combining itself with their rotation, which
alone can produce no motions when once the form of equilibrium is attained.
The prismatic analysis of the solar beam exhibits in the spectrum a series of “fixed
fines,” totally unlike those which belong to the light of any known terrestrial flame.
This may hereafter lead us to a clearer insight into its origin. But, before we can
draw any conclusions from such an indication, we must recollect that previous to
reaching us it has undergone the whole absorptive action of our atmosphere, as well
as of the sun's. Of the latter we know nothing, and may conjecture every thing;
but of the blue colour of the former we are sure; and if this be an inherent (i. e. an
absorptive) colour, the air must be expected to act on the spectrum after the analogy
of other coloured media, which often (and especially light blue media) leave unab-
sorbed portions separated by dark intervals. It deserves inquiry, therefore, whether
some or all the fixed lines observed by Wollaston and Fraunhofer may not have their
origin in our own atmosphere. Experiments made on lofty mountains, or the cars of bal-
loons, on the one hand, and on the other with reflected beams which have been made
to traverse several miles of additional air near the surface, would decide this point.
The absorptive effect of the sun's atmosphere, and possibly also of the medium sur-
rounding it (whatever it be) which resists the motions of comets, cannot be thus
eliminated.—Note to the edition of 1833.
OF THE MOON. 213
CHAPTER VII.
OF THE MOON. - ITS SIDEREAL PERIOD. —ITS APPARENT DIAMETER.
— ITS PARALLAX, DISTANCE, AND REAL DIAMETER.—FIRST AP-
PROXIMATION TO ITS ORBIT. — AN ELLIPSE ABOUT THE EARTH IN
THE FOCUS. — ITS EXCENTRICITY AND INCLINATION. — MOTION OF
ITS NODES AND APSIDES. — OF OCCULTATIONS AND SOLAR ECLIPSES
GENERALLY. — LIMITS WITHIN WEIICH THEY ARE POSSIBLE. - THEY
PROVE THE MOON TO BE AN OPAKE SOLID. — ITS LIGHT DERIVED
FROM THE SUN. — ITS PHASES. — SYNODIC REVOLUTION OR LUNAR,
MONTH. — OF ECLIPSES MORE PARTICULARLY. — THEIR. PHENOMENA.
— THEIR PERIODICAL RECURRENCE. — PHYSICAL CONSTITUTION OF
THE MOON. – ITS MOUNTAINS AND OTHER SUPERFICIAL FEATURES.
—INDICATIONS OF FORMER VOLCANIC ACTIVITY..—ITS ATMOSPHERE.
—CLIMATE. –RADIATION OF HEAT FROM ITs SURFACE. – ROTATION
ON ITS OWN AXIS. – LIBRATION. — APPEARANCE OF THE EARTH
FROM IT. -
(401.) THE moon, like the sun, appears to advance among the stars
with a movement contrary to the general diurnal motion of the heavens,
but much more rapid, so as to be very readily perceived (as we have
before observed) by a few hours' cursory attention on any moonlight
night. By this continual advance, which, though sometimes quicker,
sometimes slower, is never intermitted or reversed, it makes the tour of
the heavens in a mean or average period of 27° 7' 48" 11°5, returning,
in that time, to a position among the stars nearly coincident with that it
had before, and which would be exactly so, but for reasons presently to
be stated. -
(402.) The moon, then, like the sun, apparently describes an orbit
round the earth, and this orbit cannot be very different from a circle, be-
cause the apparent angular diameter of the full moon is not liable to any
great extent of variation.
(403.) The distance of the moon from the earth is concluded from its
horizontal parallax, which may be found either directly, by observations
at remote geographical stations, exactly similar to those described in art.
$55, in the case of the sun, or by means of the phaenomena called occul-
214 * -. OUTLINES OF ASTRONOMY.
tations, from which also its apparent diameter is most readily and cor-
rectly found. From such observations it results that the mean or average
distance of the centre of the moon from that of the earth is 59-9643 of
the earth’s equatorial radii, or about 237,000 miles. This distance,
great as it is, is little more than one-fourth of the diameter of the sun’s
body, so that the globe of the sun would nearly twice include the whole
orbit of the moon; a consideration wonderfully calculated to raise our
ideas of that stupendous luminary 1 &
(404.) The distance of the moon’s centre from an observer at any
station on the earth's surface, compared with its apparent angular diameter
as measured from that station, will give its real or linear diameter. N OW,
the former distance is easily calculated when the distance from the earth's
centre is known, and the apparent Zenith distance of the moon also deter-
mined by observation; for if we turn to the figure of art. 339, and suppose
S the moon, A the station, and C the earth's centre, the distance S C, and
the earth’s radius CA, two sides of the triangle A C S are given, and the
angle CAS, which is the supplement of Z A S, the observed zenith dis-
tance, whence it is easy to find A S, the moon's distance from A. From
such observations and calculations it results, that the real diameter of the
moon is 2160 miles, or about 0.2729 of that of the earth, whence it follows
that, the bulk of the latter being considered as 1, that of the former will
be 0.0204, or about al. The difference of the apparent diameter of the
moon, as seen from the earth's centre and from any point of its surface,
is technically called the augmentation of the apparent diameter, and its
maximum occurs when the moon is in the zenith of the spectator. Her
mean angular diameter, as seen from the centre, is 31' 7", and is always
= 0-545 × her horizontal parallax.
(405.) By a series of observations, such as described in art. 403, if
continued during one or more revolutions of the moon, its real distance
may be ascertained at every point of its orbit; and if at the same time its
apparent places in the heavens be observed, and reduced by means of its
parallax to the earth's centre, their angular intervals will become known,
so that the path of the moon may then be laid down on a chart supposed
to represent the plane in which its orbit lies, just as was explained in the
case of the solar ellipse (art. 349.) Now, when this is done, it is found
that, neglecting certain small (though very perceptible) deviations (of
which a satisfactory account will hereafter be rendered), the form of the
apparent orbit, like that of the sun, is elliptic, but considerably more
eccentric, the eccentricity amounting to 0.05484 of the mean distance, or
the major semi-axis of the ellipse, and the earth's centre being situated in
its focus.
OF THE MOON'S MOTION. 215
(406.) The plane in which this orbit lies is not the ecliptic, however,
but is inclined to it at an angle of 5°8'48", which is called the incli-
nation of the lunar orbit, and intersects it in two opposite points, which
are called its nodes—the ascending mode being that in which the moon
passes from the Southern side of the ecliptic to the northern, and the
descending the reverse. The points of the orbit at which the moon is
nearest to, and farthest from, the earth, are called respectively its perigee
and apogee, and the line joining them and the earth of the line of apsides.
(407.) There are, however, several remarkable circumstances which
interrupt the closeness of the analogy, which cannot fail to strike the
reader, between the motion of the moon around the earth, and of the
earth around the sun. In the latter case, the ellipse described remains,
during a great many revolutions, unaltered in its position and dimensions;
or, at least, the changes which it undergoes are not perceptible but in a
course of very nice observations, which have disclosed, it is true, the
existence of “perturbations,” but of so minute an order, that, in ordinary
parlance, and for common purposes, we may leave them unconsidered.
But this cannot be done in the case of the moon. Even in a single revo-
lution, its deviation from a perfect ellipse is very sensible. It does not
return to the same exact position among the stars from which it set out,
thereby indicating a continual change in the plane of its orbit. And, in
effect, if we trace by observation, from month to month, the point where
it traverses the ecliptic, we shall find that the nodes of its orbit are in a
continual state of retreat upon the ecliptic. Suppose O to be the earth,
and A b a d that portion of the plane of the ecliptic which is intersected
by the moon, in its alternate passages through it, from south to north, and
vice versä ; and let A B C D E F be a portion of the moon’s orbit, em-
Fig. 57.
bracing a complete sidereal revolution. Suppose it to set out from the
ascending node, A.; then, if the orbit lay all in one plane, passing through
O, it would have a, the opposite point in the ecliptic, for its descending
node; after passing which, it would again ascend at A. But, in fact, its
real path carries it not to a, but along a certain curve, A B C, to C, a

216 - OUTLINES OF ASTRONOMY.
point in the ecliptic less than 180° distant from A ; so that the angle
A O C, or the arc of longitude described between the ascending and the
descending node, is somewhat less than 180°. It then pursues its course
below the ecliptic, along the curve CD E, and rises again above it, not at
the point c, diametrically opposite to C, but at a point E, less advanced in
longitude. On the whole, then, the arc described in longitude between
two consecutive passages from South to north, through the plane of the
ecliptic, falls short of 360° by the angle A O E.; or, in other words, the
ascending node appears to have retreated in one lunation on the plane of
the ecliptic by that amount. To complete a sidereal revolution, then, it
must still go on to describe an arc, E F, on its orbit, which will no longer,
however, bring it exactly back to A, but to a point somewhat above it, or
having north latitude.
(408.) The actual amount of this retreat of the moon's node is about
3' 10"-64 per diem, on an average, and in a period of 6793:39 mean
solar days, or about 18.6 years, the ascending node is carried round in a
direction contrary to the moon’s motion in its orbit (or from east to west)
over a whole circumference of the ecliptic. Of course, in the middle of
this period the position of the orbit must have been precisely reversed
from what it was at the beginning. Its apparent path, then, will lie
among totally different stars and constellations at different parts of this
period; and this kind of spiral revolution being continually kept up, it
will, at one time or other, cover with its disc every point of the heavens
within that limit of latitude or distance from the ecliptic which its inclina-
tion permits; that is to say, a belt or zone of the heavens, of 10° 18' in
breadth, having the ecliptic for its middle line. Nevertheless, it still
remains true that the actual place of the moon, in consequence of this
motion, deviates in a single revolution very little from what it would be
were the nodes at rest. Supposing the moon to set out from its node A,
its latitude, when it comes to F, having completed a revolution in longi-
tude, will not exceed 8'; which, though small in a single revolution,
accumulates in its effect in a succession of many : it is to account for, and
represent geometrically, this deviation, that the motion of the modes is
devised. -
(409.) The moon’s orbit, then, is not, strictly speaking, an ellipse
returning into itself, by reason of the variation of the plane in which it
lies, and the motion of its nodes. But even laying aside this considera-
tion, the axis of the ellipse is itself constantly changing is direction in
space, as has already been stated of the solar ellipfe, but much more
rapidly; making a complete revolution, in the satno direction with the
moon's own motion, in 3232°5753 mean solar days, or about nine years,
MOTION OF THE NODES AND APSIDES. 217
being about 3° of angular motion in a whole revolution of the moon.
This is a phenomenon known by the name of the revolution of the moon’s
apsides. Its cause will be hereafter explained. Its immediate effect is
to produce a variation in the moon’s distance from the earth, which is not
included in the laws of exact elliptic motion. In a single revolution of
the moon, this variation of distance is trifling; but in the course of many
it becomes considerable, as is easily seen, if we consider that in four years
and a half the position of the axis will be completely reversed, and the
apogee of the moon will occur where the perigee occurred before.
(410.) The best way to form a distinct conception of the moon's motion
is to regard it as describing an ellipse about the earth in the focus, and,
at the same time, to regard this ellipse itself to be in a twofold state of
revolution, 1st, in its own plane, by a continual advance of its axis in
that plane; and 2dly, by a continual tilting motion of the plane itself,
exactly similar to, but much more rapid than, that of the earth's equator
produced by the conical motion of its axis described in art. 317. .
(411.) As the moon is at a very moderate distance from us (astronomi-
cally speaking), and is in fact our nearest neighbour, while the sun and,
stars are in comparison immensely beyond it, it must of necessity happen,
that at one time or other it must pass over and occult or eclipse every star
and planet within the zone above described (and, as seen from the surface
of the earth, even somewhat beyond it, by reason of parallax, which may
throw it apparently nearly a degree either way from its place as seen from
the centre, according to the observer’s station). Nor is the sun itself
exempt from being thus hidden, whenever any part of the moon's dise,
in this her tortuous course, comes to overlap any part of the space occu-
pied in the heavens by that luminary. On these occasions is exhibited
the most striking and impressive of all the occasional phenomena of
astronomy, an eclipse of the sun, in which a greater or less portion, or
even in some rare conjunctures the whole, of its dise is obscured, and, as it
were, obliterated, by the superposition of that of the moon, which appears
upon it as a circularly-terminated black spot, producing a temporary dimi-
nution. of daylight, or even nocturnal darkness, so that the stars appear as
if at midnight. In other cases, when, at the moment that the moon is
centrally superposed on the sun, it so happens that her distance from
the earth is such as to render her angular diameter less than the sun's,
the very singular phenomenon of an annular Solar eclipse takes place,
when the edge of the sun appears for a few minutes as a narrow ring of
light, projecting on all sides beyond the dark circle occupied by the moon
in its centre.
(412.) A solar eclipse can only happen when the sun and moon are in
218 OUTLINES OF ASTRONOMY.
conjuxction, that is to say, have the same, or nearly the same, position in
the heavens, or the same longitude. It appears by art. 409 that this
condition can only be fulfilled at the time of a new moon, though it by no
means follows, that at every conjunction there must be an eclipse of the
sun. If the lunar orbit coincided with the ecliptic, this would be the
case, but as it is inclined to it at an angle of upwards of 5°, it is evident
that the conjunction, or equality, of longitudes, may take place when the
moon is in the part of her orbit too remote from the ecliptic to permit the
Wiscs to meet and overlap. It is easy, however, to assign the limits
within which an eclipse is possible. To this end we must consider, that,
by the effect of parallax, the moon’s apparent edge may be thrown in
any direction, according to a spectator's geographical station, by any
amount not exceeding the horizontal parallax. Now, this comes to the
same (so far as the possibility of an eclipse is concerned) as if the ap-
parent diameter of the moon, seen from the earth's centre, were dilated
by twice its horizontal parallax; for if, when so dilated, it can touch or
overlap the sun, there must be an eclipse at Some part or other of the
earth's surface. If, then, at the moment of the nearest conjunction, the
geocentric distance of the centres of the two luminaries do not exceed the
sum of their semidiameters and of the moon’s horizontal parallax, there
will be an eclipse. This sum is, at its maximum, about 1° 34' 27". In
the spherical triangle SN M, then, in which S is the sun's centre, M the
moon's, S N the ecliptic, M N the moon's orbit, and N the node, we may
Fig. 58.

S\U †
suppose the angle N S M a right angle, S M = 1° 34' 27", and the angle
M N S = 5° 8' 48", the inclination of the orbit. Hence we calculate
SN, which comes out 16° 58'. If, then, at the moment of the new
moon, the moon's node is farther from the sun in longitude than this
limit, there can be no eclipse; if within, there may, and probably will, at
some part or other of the earth. To ascertain precisely whether there
will or not, and, if there be, how great will be the part eclipsed, the solar
and lunar tables must be consulted, the place of the node and the semi-
diameters exactly ascertained, and the local parallax, and apparent aug-
mentation of the moon’s diameter due to the difference of her distance
OCCULTATION OF A STAR BY THE MOON. 219
from the observer and from the centre of the earth (which may amount
to a sixtieth part of her horizontal diameter), determined; after which it
is easy, from the above considerations, to calculate the amount overlapped
of the two discs, and their moment of contact.
(413.) The calculation of the occultation of a star depends on similar
considerations. An occultation is possible, when the moon's course, as
seen from the earth's centre, carries her within a distance from the star
equal to the sum of her semidiameter and horizontal parallax, and it will
happen at any particular spot, when her apparent path, as seen from that
spot, carries her centre within a distance equal to the sum of her aug-
mented semidiameter and actual parallax. The details of these calcula-
tions, which are somewhat troublesome, must be sought elsewhere."
(414.) The phenomenon of a solar eclipse and of an occultation are
highly interesting and instructive in a physical point of view. They
teach us that the moon is an opaque body, terminated by a real and sharply
defined surface intercepting light like a solid. They prove to us, also,
that at those times when we cannot see the moon, she really exists, and
pursues her course, and that when we see her only as a crescent, however
narrow, the whole globular body is there, filling up the deficient outline,
though unseen. For occultations take place indifferently at the dark and
bright, the visible and invisible outline, whichever happens to be towards
the direction in which the moon is moving; with this only difference, that
a star occulted by the bright limb, if the phenomenon be watched with a
telescope, gives notice, by its gradual approach to the visible edge, when
to expect its disappearance, while, if occulted at the dark limb, if the
moon, at least, be more than a few days old, it is, as it were, extinguished
in mid-air, without notice or visible cause for its disappearance, which, as
it happens instantaneously, and without the slightest previous diminution
of its light, is always surprising; and, if the star be a large and bright
one, even startling from its suddenness. The re-appearance of the star,
too, when the moon has passed over it, takes place in those cases when
the bright side of the moon is foremost, not at the concave outline of the
crescent, but at the invisible outline of the complete circle, and is scarcely
less surprising, from its suddenness, than its disappearance in the other
C3Se.
* Woodhouse's Astronomy, vol. i. See also Trans. Ast. Soc. vol. i. p. 325.
* There is an optical illusion of a very strange and unaccountable nature which has
often been remarked in occultations. The star appears to advance actually upon and
within the edge of the disc before it disappears, and that sometimes to a considerable
depth. I have never myself witnessed this singular effect, but it rests on most une-
quivocal testimony. I have called it an optical illusion; but it is barely possible that a
star may shine on such occasions through leep fissures in the substance of the moon.
220 OUTLINES OF ASTRONOMY.
(415.) The existence of the complete circle of the disc, even when
the moon is not full, does not, however, rest only on the evidence of
occultations and eclipses. It may be seen, when the moon is crescent or
waning, a few days before and after the new moon, with the naked eye,
as a pale round body, to which the crescent seems attached, and some-
what projeeting beyond its outline (which is an optical illusion arising
from the greater intensity of its light.) The cause of this appearance
will presently be explained. Meanwhile the fact is sufficient to show
that the moon is not inherently luminous like the sun, but that her light
is of an adventitious nature. And its crescent form, increasing regularly
from a narrow semicircular line to a complete circular disc, corresponds to
the appearance a globe would present, one hemisphere of which was
black, the other white, when differently turned towards the eye, so as to
present a greater or less portion of each. The obvious conclusion from
this is, that the moon is such a globe, one half of which is brightened by
the rays of some luminary sufficiently distant to enlighten, the complete
hemisphere, and sufficiently intense to give it the degree of splendour we
see. Now, the sun alone is competent to such an effect. Its distance
and light suffice; and, moreover, it is invariably observed that, when a
crescent, the bright edge is towards the sun, and that in proportion as
the moon in her monthly course becomes more and more distant from the
sun, the breadth of the crescent increases, and vice versá.
(416.) The sun's distance being 23984 radii of the earth, and the
moon's only 60, the former is nearly 400 times the latter. Lines, there-
fore, drawn from the sun to every part of the moon’s orbit may be
regarded as very nearly parallel." Suppose, now, O to be the earth,
A B C D, &c. Various positions of the moon in its orbit, and S the sun,
at the vast distance above stated; as is shown, then, in the figure, the
hemisphere of the lunar globe turned towards it (on the right) will be
bright, the opposite dark, wherever it may stand in its orbit. Now, in
the position A, when in conjunction with the sun, the dark part is
entirely turned towards O, and the bright from it. In this case, then,
The occultations of close double stars ought to be narrowly watched, to see whether
both individuals are thus projecied, as well as for other purposes connected with their
theory. I will only hint at one, viz. that a double star, too close to be seen divided
with any telescope, may yet be detected to be double by the mode of its disapppear-
ance. Should a considerable star, for instance, instead of undergoing instantaneous
and complete extinction, go out by two distinct steps, following close upon each other;
first losing a portion, then the whole remainder of its light, we may be sure it is a
double star, though we cannot see the individuals separately.—Note to the edit. of 1833.
* The angle subtended by the moon's orbit, as seen from the sun, (in the mean state
of things) is only 17' 12".
PHASES OF THE MOON EXPLAINED. 221
Fig. 59.
O, ºpe
ſºft||| S ſºft
ſºil-- #TT =#||
|
ſ
i
1
* \o
i. - *D7
# =# ºf Y_{.
sº-g . S * {
º
the moon is not seen, it is new moon. When the moon has come to C,
half the bright and half the dark hemisphere are presented to 0, and the
same in the opposite situation G: these are the first and third quarters
"of the moon. Lastly, when at E, the whole bright face is towards the
earth, the whole dark side from it, and it is then seen wholly bright or
full moon. In the intermediate positions B D F H, the portions of the
bright face presented to O will be at first less than half the visible sur-
face, then greater, and finally less again, till it wanishes altogether, as it
comes round again to A. - - -
(417.) These monthly changes of appearance, or phases, as they are
called, arise, then, from the moon, an opaque body, being illuminated on
one side by the sun, and reflecting from it, in all directions, a portion of
the light so received. Nor let it be thought surprising that a solid sub-
stance thus illuminated should appear to shine and again illuminate the
earth. It is no more than a white cloud does standing off upon the clear
blue sky. By day, the moon can hardly be distinguished in brightness
from such a cloud; and, in the dusk of the evening, clouds catching the
last rays of the sun appear with a dazzling splendour, not inferior to the
seeming brightness of the moon at night.' That the earth sends also
such a light to the moon, only probably more powerful by reason of its
greater apparent size”, is agreeable to optical principles, and explains the
(º) s
§ -
S
* The actual illumination of the lunar surface is not much superior to that of weathered
sandstone rock in full sunshine. I have frequently compared the moon setting behind
the grey perpendicular façade of the Table Mountain, illuminated by the sun just risen
in the opposite quarter of the horizon, when it has been scarcely distinguishable in
brightness from the rock in contact with it. The sun and moon being nearly at equal
altitudes and the atmosphere perfectly free from cloud or vapour, its effect is alike on
both luminaries. (H. 1848).
*The apparent diameter of the moon is 32' from the earth; that of the earth seen
from the moon is twice her horizontal parallax, or 1954'. The apparent surfaces
therefore, are as (114)”; (32)”, or as 13 : 1 nearly.








222 OUTLINES OF ASTRONOMY.
appearance of the dark portion of the young or waning moon completing
its crescent (art. 413). For, when the moon is nearly new to the earth,
the latter (so to speak) is nearly full to the former; it then illuminates its
dark half by strong earth-light; and it is a portion of this, reflected back
again, which makes it visible to us in the twilight sky. As the moon
gains age, the earth offers it a less portion of its bright side, and the phe-
nomenon in question dies away.
(418.) The lunar month is determined by the recurrence of its phases:
it reckons from new moon to new moon; that is, from leaving its conjunc-
tion with the Sun to its return to conjunction. If the sun stood still, like
a fixed star, the interval between two conjunctions would be the same as
the period of the moon’s sidereal revolution (art. 401); but, as the sun
apparently advances in the heavens in the same direction with the moon,
only slower, the latter has more than a complete sidereal period to perform
to come up with the sun again, and will require for it a longer time, which,
is the lunar month, or, as it is generally termed in astronomy, a synodical
period. The difference is easily calculated by considering that the super-
fluous are (whatever it be) is described by the sun with the velocity of
0°-98565 per diem, in the same time that the moon describes that arc
plus a complete revolution, with her velocity of 13°-17640 per diem; and,
the times of description being identical, the spaces are to each other in the
proportion of the velocities. Let W and v be the mean angular velocities,
a; the superfluous arc; then W : v :: 1 + æ : ac; and W — v : v :: 1: æ,
whence ac is found, and + = the time of describing ac, or the difference of
the sidereal and synodical periods. From these data a slight knowledge
of arithmetic will suffice to derive the arc in question, and the time of its
description by the moon; which being the excess of the synodic over the
sidereal period, the former will be had, and will appear to be 29" 12"
44m 25.87. - .
(419.) Supposing the position of the nodes of the moon's orbit to
permit it, when the moon stands at A (or at the new moon), it will inter-
cept a part or the whole of the sun's rays, and cause a solar eclipse. On
the other hand, when at E (or at the full moon), the earth O will inter-
cept the rays of the Sun, and cast a shadow on the moon, thereby causing
a lunar eclipse. And this is perfectly consonant to fact, such eclipses
never happening but at the exact time of the full moon. But, what is
still more remarkable, as confirmatory of the position of the earth's sphe-
ricity, this shadow, which we plainly see to enter upon and, as it were,
eat away the disc of the moon, is always terminated by a circular outline,
though, from the greater size of the circle, it is only partially seen at any
gº
SOLAR AND LUNAR. ECLIPSES. 223.
one time. Now, a body which always casts a circular shadow must itself
be spherical.
(420.) Eclipses of the sun are best understood by regarding the sun
and moon as two independent luminaries, each moving according to known
laws, and viewed from the earth: but it is also instructive to consider
eclipses generally as arising from the shadow of one body thrown on ano-
ther by a luminary much larger than either. Suppose then, A B to
represent the sun, and C D a spherical body, whether earth or moon, illu-
minated by it. If we join and prolong A C, B D; since A B is greater
than CD, these lines will meet in a point E, more or less distant from
the body CD, according to its size, and within the space C E D (which
represents a cone, since CD and A B are spheres), there will be a total
shadow. This shadow is called the umbra, and a spectator situated within
it can see no part of the sun's disc. Beyond the umbra are two diverging
Fig. 60.
4 º |
| º
liſi;|\,
lili
T3
spaces (or rather, a portion of a single conical space, having K for its
vertex), where if a spectator be situated, as at M, he will see a portion
only (A O N P) of the sun's surface, the rest (B O N P) being obscured
by the earth. He will, therefore, receive only partial sunshine; and the
more, the nearer he is to the exterior borders of that cone which is called
the penumbra. Beyond this he will see the whole sun, and be in full
illumination. All these circumstances may be perfectly well shown by
holding a small globe up in the sun, and receiving its shadow at different
distances On a sheet of paper.
(421.) In a lunar eclipse (represented in the upper figure), the moon
is seen to enter' the penumbra first, and, by degrees, get involved in the
umbra, the former bordering the latter like a smoky haze. At this period
of the eclipse, and while yet a considerable part of the moon remains
*The actual contact with the penumbra is never seen; the defalcation of light comes
on sove-y gradually that it is not till when already deeply immersed, that it is perceived
to be sensibly darkened.


224 OUTLINES OF ASTRONOMY.
unobscured, the portion involved in the umbra is invisible to the naked
eye, though still perceptible in a telescope, and of a dark grey hue. But
as the eclipse advances, and the enlightened part diminishes in extent, and
grows gradually more and more obscured by the advance of the penumbra,
the eye, relieved from its glare, becomes more sensible to feeble impres-
sions of light and colour; and phenomena of a remarkable and instruc-
tive character begin to be developed. The umbra is seen to be very far
from totally dark: and in its faint illumination it exhibits a gradation of
colour, being bluish, or even (by contrast) somewhat greenish, towards
the borders for a space of about 4' or 5’ of apparent angular breadth
inwards, thence passing, by delicate but rapid gradation, through rose red
to a fiery or copper-coloured glow, like that of dull red-hot iron. As
the eclipse proceeds this glow spreads over the whole surface of the moon,
which then becomes on some occasions so strongly illuminated, as to cast
a very sensible shadow, and allow the spots on its surface to be perfectly
well distinguished through a telescope.
(422.) The cause of these singular, and sometimes very beautiful
appearances, is the refraction of the sun's light in passing through our
atmosphere, which at the same time becomes coloured with the hues of
sunset by the absorption of more or less of the violet and blue rays, as it
passes through strata nearer or more remote from the earth's surface, and
therefore, more or less loaded with vapour. To show this, let A D a be
a section of the cone of the wmbra, and FBh,f of the penumbra, through
their common axis D ES, passing through the centres E S of the earth
and sun, and let K M k be a section of these cones at a distance E M from
E, equal to the radius of the moon’s orbit, or 60 radii of the earth."
Taking this radius for unity, since ES, the distance of the sun, is 23984,
and the semidiameter of the sun 111; such units, we readily calculate
|D E=218, D M-158, for the distances at which the apex of the geome-
trical umbra lies behind the earth and the moon respectively. We also
find for the measure of the angle E D B, 15'46", and therefore D B E=
89° 44' 14", whence also we get M C (the linear semidiameter of the
umbra)=0-725 (or in miles 2864), and the angle. CEM, its apparent
angular semidiameter as seen from E=41' 30". And instituting similar
calculations for the geometrical penumbra we get M K=1:005 (3970
miles), and KEM 57° 36"; and it may be well to remember that the
doubles of these angles, or the mean angular diameters of the umbra and
penumbra, are described by the moon with its mean velocity in 2, 43",
and 3h 47 = respectively, which are therefore the respective durations of
* The figure is unavoidably drawn out of all proportion, and the angles violently
exaggerated. The reader snould endeavour to draw the figure in its true proportions.
PHENOMENA OF A LUNAR ECLIPSE. 225
the total and partial obscuration of any one point of the moon's disc in
traversing centrally the geometrical shadow. -
(423.) Were the earth devoid of atmosphere, the whole of the phe-
momena of a lunar eclipse would consist in these partial or total obscura-
tions. Within the space C c the whole of the light, and within K C and
ck a greater or less portion of it, would be intercepted by the solid body
B b of the earth. The refracting atmosphere, however, extends from
B, b, to a certain unknown, but very small distance BH, bh, which, acting
as a convex lens, of gradually (and very rapidly) decreasing density, dis-
perses all that comparatively small portion of light which falls upon it
over a space bounded externally by Hg, parallel and very nearly coinci-
dent with BF, and internally by a line B 2, the former representing the
extreme exterior ray from the limb a of the sun, the latter, the extreme
interior ray from the limb A. To avoid complication, however, we will
trace only the courses of rays which just graze the surface at B, viz: B2
from the upper border, A, and B v from the lower, a, of the sun. Each
of these rays is bent inwards from its original course by double the
amount of the horizontal refraction (33') i. e. by 1° 6', because, in
passing from B out of the atmosphere, it undergoes a deviation equal to
that at entering, and in the same direction. Instead, therefore, of pur-
suing the courses BD, BF, these rays respectively will occupy the posi-
tions B 2 y, B v, making, with the aforesaid lines, the angles D B b, FB v,
each 19 6'. Now we have found D B E = 89° 44' 14” and therefore

15
226 OUTLINES OF ASTRONOMY.
PBE (=DBE + angular diam. of O) = 90° 17' 17", consequently
the angles EBy and E B v will be respectively 88° 38'14" and 89° 11’
17” from which we conclude E2 = 42-03 and Ev = 88-89, the former
falling short of the moon’s orbit by 17:07, and the latter surpassing it by
28-89 radii of the earth.
(424.) The penumbra, therefore, of rays refracted at B, will be spread
over the space v B y, that at H over g H d, and at the intermediate
points, over similar intermediate spaces, and through this compound of
superposed penumbrae the moon passes during the whole of its path
through the geometrical shadow, never attaining the absolute umbra
B 2 b at all. Without going into detail as to the intensity of the
refracted rays, it is evident that the totality of light so thrown into the
shadow is to that which the earth intercepts, as the area of a circular
section of the atmosphere to that of a diametrical section of the earth
itself, and, therefore, at all events but feeble. And it is still further
enfeebled by actual clouds suspended in that portion of the air which
forms the visible border of the earth's disc as seen from the moon, as
well as by the general want of transparency caused by invisible vapour,
which is especially effective in the lowermost strata, within three or four
miles of the surface, and which will impart to all the rays they transmit,
the ruddy hue of sunset, only of double the depth of tint which we
admire in our glowing sunsets, by reason of the rays having to traverse
twice as great a thickness of atmosphere. This redness will be most
intense at the points ac, y, of the moon’s path through the umbra, and
will thence degrade very rapidly outwardly, over the spaces a c, y C, less
so inwardly, over a y. And at C, c, its hue will be mingled with the
bluish or greenish light which the atmosphere scatters by irregular dis-
persion, or in other words by our twilight (art. 44). Nor will the phe-
nomenon be uniformly conspicuous at all times. Supposing a generally
and deeply clouded state of the atmosphere around the edge of the earth's
disc visible from the moon (i. e. around that great circle of the earth, in
which, at the moment the sun is in the horizon,) little or no refracted
light may reach the moon.' Supposing that circle partly clouded and
partly clear, patches of red light corresponding to the clear portions will
be thrown into the umbra, and may give rise to various and changeable
distributions of light on the eclipsed disc;” while, if entirely clear, the
eclipse will be remarkable for the conspicuousness of the moon during
the whole or a part of its immersion in the umbra.”
* As in the eclipses of June 5, 1620, April 25, 1642. Lalande, Ast. 1769.
* As in the eclipse of Oct. 13, 1837, observed by the author.
* As in that of March 19, 1848, when the moon is described as giving “good light”
during more than an hour after its total immersion, and some persons even doubted
its heing eclipsed. (Notices of R. Ast. Soc. viii. p. 132.)
I, UNAR, ECLIPTIC LIMITS, 227
(425.) Owing to the great size of the earth, the cone of its umbra
always projects far beyond the moon; so that, if, at the time of a lunar
eclipse, the moon’s path be properly directed, it is sure to pass through
the umbra. This is not, however, the case in solar eclipses. It so
happens, from the adjustment of the size and distance of the moon, that
the extremity of her umbra always falls near the earth, but sometimes
attains and sometimes falls short of its surface. In the former case
(represented in the lower figure art. 420) a black spot, surrounded by a
fainter shadow, is formed, beyond which there is no eclipse on any part
of the earth, but within which there may be either a total or partial one,
as the spectator is within the umbra or penumbra. When the apex of
the umbra falls on the surface, the moon at that point will appear, for an
instant, to just cover the sun; but, when it falls short, there will be no
total eclipse on any part of the earth; but a spectator, situated in or near
the prolongation of the axis of the cone, will see the whole of the moon
on the sun, although not large enough to cover it, i. e. he will witness an
annular eclipse.
(426.) Owing to a remarkable enough adjustment of the periods in
which the moon's Synodical revolution, and that of her nodes, are per-
formed, eclipses return after a certain period, very nearly in the same
order and of the same magnitude. For 223 of the moon's mean synodi-
cal revolutions, or lunations, as they are called, will be found to occupy
6585.32 days, and nineteen complete synodical revolutions of the node to
occupy 6585-78. The difference in the mean position of the node, then,
at the beginning and end of 223 lunations, is nearly insensible; so that
a recurrence of all eclipses within that interval must take place. Accord-
ingly, this period of 223 lunations, or eighteen years and ten days, is a
very important one in the calculation of eclipses. It is supposed to have
been known to the Chaldeans, the earliest astronomers, the regular return
of eclipses having been known as a physical fact for ages before their
exact theory was understood. In this period there occur ordinarily 70
eclipses, 29 of the moon and 41 of the sun, visible in some part of the
earth. Seven eclipses of either sun or moon at most, and two at least
(both of the sun,) may occur in a year.
(427.) The commencement, duration, and magnitude of a lunar eclipse
are much more easily calculated than those of a solar, being independent
of the position of the spectator on the earth's surface, and the same as if
viewed from its centre. The common centre of the umbra and penumbra
lies always in the ecliptic, at a point opposite to the sun, and the path
described by the moon in passing through it is its true orbit as it stands
at the moment of the full moon. In this orbit, its position, at every
228 OUTLINES OF ASTRONOMY.
instant, is known from the lunar tables and ephemeris; and all we have,
therefore, to ascertain, is, the moment when the distance between the
moon’s centre and the centre of the shadow is exactly equal to the sum
of the semidiameters of the moon and penumbra, or of the moon and
wmbra, to know when it enters upon and leaves them respectively. No
lunar eclipse can take place, if, at the moment of the full moon, the sun
be at a greater angular distance from the mode of the moon’s orbit than
11° 21', meaning by an eclipse the immersion of any part of the moon in
the umbra, as its contact with the penumbra cannot be observed (see note
to art. 421).
(428.) The dimensions of the shadow, at the place where it crosses the
moon's path, require us to know the distances of the sun and moon at the
time. These are variable; but are calculated and set down, as well as
their semidiameters, for every day, in the ephemeris, so that none of the
data are wanting. The sun's distance is easily calculated from its elliptic
orbit; but the moon’s is a matter of more difficulty, by reason of the pro-
gressive motion of the axis of the lunar orbit. (Art. 409.)
(429.) The physical constitution of the moon is better known to us
than that of any other heavenly body. By the aid of telescopes, we
discern inequalities in its surface which can be no other than mountains
and valleys, for this plain reason, we see the shadows cast by the former
in the exact proportion as to length which they ought to have, when we
take into account the inclination of the sun's rays to that part of the
moon’s surface on which they stand. The convex, outline of the limb
turned towards the sun is always circular, and very nearly smooth; but
the opposite border of the enlightened part, which (were the moon a per-
fect sphere) ought to be an exact and sharply defined ellipse, is always
observed to be extremely ragged, and indented with deep recesses and pro-
minent points. The mountains near this edge cast long black shadows,
as they should evidently do, when we consider that the sun is in the act
of rising or setting to the parts of the moon so circumstanced. But as
the enlightened edge advances beyond them, i. e. as the sun to them gains
altitude, their shadows shorten; and at the full moon, when all the light
falls in our line of sight, no shadows are seen on any part of her surface.
From micrometrical measures of the lengths of the shadows of the more
conspicuous mountains, taken under the most favourable circumstances,
the heights of many of them have been calculated. Messrs. Beer and
Maedler in their elaborate work, entitled “Der Mond,” have given a list
of heights resulting from such measurements, for no less than 1095 lunar
mountains, among which occur all degrees of elevation up to 3569 toises,
(22823 British feet), or about 1400 feet higher than Chimborazo in the
…”
PHYSICAL CONSTITUTION OF THE MOON. 229
Andes. The existence of such mountains is further corroborated by their
appearance, as small points or islands of light beyond the extreme edge
of the enlightened part, which are their tops catching the sun-beams
before the intermediate plain, and which, as the light advances, at length
connect themselves with it, and appear as prominences from the general
edge.
(430.) The generality of the lunar mountains present a striking uni-
formity and singularity of aspect. They are wonderfully numerous,
especially towards the Southern portion of the disc, occupying by far the
larger portion of the surface, and almost universally of an exactly circu-
lar or cup-shaped form, foreshortened, however, into ellipses towards the
limb; but the larger have for the most part flat bottoms within, from
which rises centrally a small, steep, conical hill. They offer, in short, in
its highest perfection, the true volcanic character, as it may be seen in the
crater of Vesuvius, and in a map of the volcanic districts of the Campi
IPhlegraei" or the Puy de Dôme, but with this remarkable peculiarity,
viz.: that the bottoms of many of the craters are very deeply depressed
below the general surface of the moon, the internal depth being often
twice or three times the external height. In some of the principal ones,
decisive marks of volcanic stratification, arising from successive deposits
of ejected matter, and evident indications of lava currents streaming
outwards in all directions, may be clearly traced with powerful telescopes.
(See Pl. W. fig. 2.*) In Lord Rosse's magnificent reflector, the flat
bottom of the crater called Albategnius is seen to be strewed with blocks
not visible in inferior telescopes, while the exterior of another (Aristillus)
is all hatched over with deep gullies radiating towards its centre. What is,
moreover, extremely singular in the geology of the moon is, that, although
nothing having the character of seas can be traced, (for the dusky spots,
which are commonly called seas, when closely examined, present appear-
ances incompatible with the supposition of deep water,) yet there are
large regions perfectly level, and apparently of a decided alluvial cha-
racter.
(431.) The moon has no clouds, nor any other decisive indications of
an atmosphere. Were there any, it could not fail to be perceived in the
occultations of stars and the phaenomena of solar eclipses, as well as in
a great variety of other phaenomena. The moon's diameter, for example,
as measured micrometrically, and as estimated by the interval between
the disappearance and reappearance of a star in an occultation, ought to
differ by twice the horizontal refraction at the moon's surface. No appre-
* See Breislak’s map of the environs of Naples, and Desmarest's of Auvergne.
* From a drawing taken with a reflector of twenty feet focal length (h.)
230 OUTLINES OF ASTRONOMY.
ciable difference being perceived, we are entitled to conclude the non-
existence of any atmosphere dense enough to cause a refraction of 1" i. e.
having one 1980th part of the density of the earth's atmosphere. In a
Solar eclipse, the existence of any sensible refracting atmosphere in the
moon, would enable us to trace the limb of the latter beyond the cusps,
externally to the sun's disc, by a narrow, but brilliant line of light,
extending to some distance along its edge. No such phaenomenon is
seen. Very faint stars ought to be extinguished before occultation, were
any appreciable amount of vapour suspended near the surface of the moon.
But such is not the case; when occulted at the bright edge, indeed, the
light of the moon extinguishes small stars, and even at the dark limb,
the glare in the sky caused by the near presence of the moon, renders
the occultation of very minute stars unobservable. But during the con-
tinuance of a total lunar eclipse, stars of the tenth and eleventh magni-
tude are seen to come up to the limb, and undergo sudden extinction as
well as those of greater brightness." Hence, the climate of the moon
must be very extraordinary; the alternation being that of unmitigated
and burning Sunshine fiercer than an equatorial noon, continued for a
whole fortnight, and the keenest severity of frost, far exceeding that of
our polar winters, for an equal time. Such a disposition of things must
produce a constant transfer of whatever moisture may exist on its surface,
from the point beneath the sun to that opposite, by distillation in vacuo
after the manner of the little instrument called a cryophorus. The con-
sequence must be absolute aridity below the vertical sun, constant accre-
tion of hoar frost in the opposite region, and, perhaps, a narrow Zone of
running water at the borders of the enlightened hemisphere." It is
possible, then, that evaporation on the one hand, and condensation on the
other, may to a certain extent preserve an equilibrium of temperature,
and mitigate the extreme severity of both climates; but this process,
which would imply the continual generation and destruction of an atmo-
sphere of aqueous vapour, must, in conformity with what has been said
above of a lunar atmosphere, be confined within very narrow limits.
(482.) Though the surface of the full moon exposed to us, must neces-
sarily be very much heated,—possibly to a degree much exceeding that of
boiling water, yet we feel no heat from it, and even in the focus of large
reflectors, it fails to affect the thermometer. No doubt, therefore, its heat
(conformably to what has been observed of that of bodies heated below
the point of luminosity) is much more readily absorbed in traversing
transparent media than direct solar heat, and is extinguished in the upper
|
* As observed by myself in the eclipse of Oct. 13, 1837.
* So in ed. of 1833.
CLIMATE AND HEAT OF THE MOON. 231
regions of our atmosphere, never reaching the surface of the earth at all.
Some probability is given to this by the tendency to disappearance
of clouds under the full moon, a meteorological fact, (for as such we
think it fully entitled to rank") for which it is necessary to seek a cause,
and for which no other rational explanation seems to offer. As for any
other influence of the moon on the weather, we have no decisive evidence
in its favour. k
(433.) A circle of one second in diameter, as seen from the earth, on
the surface of the moon, contains about a square mile. Telescopes, there-
fore, must yet be greatly improved, before we could expect to see signs of
inhabitants, as manifested by edifices or by changes on the surface of the
soil. It should, however, be observed, that, owing to the small density
of the materials of the moon, and the comparatively feeble gravitation of
bodies on her surface, muscular force would there go six times as far in
overcoming the weight of materials as on the earth. Owing to the want
of air, however, it seems impossible that any form of life, analogous to
those on earth, can subsist there. No appearance indicating vegetation,
or the slightest variation of surface, which can, in our opinion, fairly be
ascribed to change of season, can any where be discerned.
(484.) The lunar summer and winter arise, in fact, from the rotation
of the moon on its own axis, the period of which rotation is eacactly equal
to its sidereal revolution about the earth, and is performed in a plane 19
30' 11" inclined to the ecliptic, whose ascending node is always precisely
coincident with the descending node of the lunar orbit. So that the axis
of rotation describes a conical surface about the pole of the ecliptic in
one revolution of the node. The remarkable coincidence of the two rota-
tions, that about the axis and that about the earth, which at first sight
would seem perfectly distinct, has been asserted (but we think somewhat
too hastily”) to be a consequence of the general laws to be explained here-
after. Be that as it may, it is the cause why we always see the same face
of the moon, and have no knowledge of the other side.
(435.) The moon's rotation on her axis is uniform; but since her
motion in her orbit (like that of the sun) is not so, we are enabled to
look a few degrees round the equatorial parts of her visible border, on the
eastern or western side, according to circumstances; or, in other words,
the line joining the centres of the earth and moon fluctuates a little in its
position, from its mean or average intersection with her surface, to the
“From my own observation, made quite independently of any knowledge of such
a tendency having been observed by others. Humboldt, however, in his personal nar-
rative, speaks of it as well known to the pilots and seamen of Spanish America: see
note at the end of the chapter (h.)
* See Edinburgh Review, No. 175, p. 192.
232 OUTLINES OF ASTRONOMY.
east or westward. And, moreover, since the axis about which she revolves
is neither exactly perpendicular to her orbit, nor holds an invariable direc-
tion in space, her poles come alternately into view for a small space at the
edges of her disc. These phenomena are known by the name of librations
In consequence of these two distinct kinds of libration, the same identi-
cal point of the moon's surface is not always the centre of her disc, and
we therefore get sight of a zone of a few degrees in breadth on all sides
of the border, beyond an exact hemisphere.
(436.) If there be inhabitants in the moon, the earth must present to
them the extraordinary appearance of a moon of nearly 2° degrees in
diameter, exhibiting phases complementary to those which we see the
moon to do, but immoveably ficed in their sky, (or, at least, changing its
apparent place only by the small amount of the libration,) while the stars
must seem to pass slowly beside and behind it. It will appear clouded
with variable spots, and belted with equatorial and tropical zones corres-
ponding to our trade-winds; and it may be doubted whether, in their per-
petual change, the outlines of our continents and seas can ever be clearly
discerned. During a solar eclipse, the earth's atmosphere will become
visible as a narrow, but bright luminous ring of a ruddy colour, where it
rests on the earth, gradually passing into faint blue, encircling the whole
or part of the dark disc of the earth, the remainder being dark and rugged ,
with clouds.
(437.) The best charts of the lunar surface are those of Cassini, of
Russel (engraved from drawings, made by the aid of a seven feet reflect-
ing telescope,) the seleno-topographical charts of Lohrmann, and the very
elaborate projection of Beer and Maedler accompanying their work
already cited." Madame Witte, a Hanoverian lady, has recently suc-
ceeded in producing from her own observations, aided by Maedlar's
charts, more than one complete model of the whole visible lunar hemi-
sphere, of the most perfect kind, the result of incredible diligence and
assiduity. Single craters have also been modelled on a large scale, both
by her and Mr. Nasmyth. [Still more recently (1851) photography has
been successfully applied to the exact delineation of the lunar surface by
Mr. Whipple, using for the purpose the great Fraunhofer equatorial of
the Observatory at Cambridge, U. S.]
* The representations of Hevelius in his Selenographia, though not without great
merit at the time, and fine specimens of his own engraving, are now become antiquated.
Additional Note on Art. 432.
M. Arago has shown, from a comparison of rain, registered as having fallen during
a long period, that a slight preponderance in respect of quantity falls near the new
Moon over that which falls near the full. This would be a natural and necessary con-
sequence of a preponderance of a cloudless sky about the full, and forms, therefore,
part and parcel of the same meteorological fact.
OF TERRESTIAL GRAVITY. 233
CHAPTER VIII.
OF TERRESTRIAL GRAVITY. — OF THE LAW OF UNIVERSAL GRAVITA-
TION.— PATHS OF PROJECTILES; APPARENT — REAL — THE MOON
RETAINED IN HER, ORBIT BY GRAVITY. —ITS LAW OF DIMINUTION –
LAWS OF ELDIEPTIC MOTION. — ORBIT OF THE EARTH ROUND THE SUN
IN ACCORDANCE WITH THESE IAWS. — MASSES OF THE EARTH AND
SUN COMPARED.—DENSITY OF THE SUN.—FORCE OF GRAVITY AT ITS
SURFACE.-DISTURBING EFFECT OF THE SUN ON THE MOON'S MOTION.
(438.) THE reader has now been made acquainted with the chief phe-
nomena of the motions of the earth in its orbit round the sun, and of the
moon about the earth. —We come next to speak of the physical cause
which maintains and perpetuates these motions, and causes the massive
bodies so revolving to deviate continually from the directions they would
naturally seek to follow, in pursuance of the first law of motion," and
bend their courses into curves concave to their centres.
(439.) Whatever attempts may have been made by metaphysical
writers to reason away the connection of cause and effect, and fritter it
down into the unsatisfactory relation of habitual sequence,” it is certain
that the conception of some more real and intimate connection is quite as
strongly impressed upon the human mind as that of the existence of an
external world,—the vindication of whose reality has (strange to say)
been regarded as an achievement of no common merit in the annals of
this branch of philosophy. It is our own immediate consciousness of
effort, when we exert force to put matter in motion, or to oppose and neu-
tralize force, which gives us this internal conviction of power and causa-
* Princip. Lex. i.
* See Brown “On Cause and Effect,”— a work of great acuteness and subtlety of
reasoning on some points, but in which the whole train of argument is vitiated by one
enjrmous oversight; the omission, namely, of a distinct and immediate personal con-
sciousness of causation in his enumeration of that sequence of events, by which the
volition of the mind is made to terminate in the motion of material objects. I mean
the consciousness of effort, accompanied with intention thereby to accomplish an end,
as a thing entirely distinct from mere desire or volition on the one hand, and from mere
spasmodic contraction of muscles on the other. Brown, 3d edit. Edin. 1818, p. 47.
(Note to edition of 1833.)
234 OUTLINES OF ASTRONOMY.
tion so far as it refers to the material world, and compels us to believe
that whenever we see material objects put in motion from a state of rest,
or deflected from their rectilinear paths and changed in their velocities if
already in motion, it is in consequence of such an EFFORT somehow
exerted, though not accompanied with our consciousness. That such an
effort should be exerted with success through an interposed space, is no
more difficult to conceive, than that our hand should communicate motion
to a stone, with which it is demonstrably not in contact.
(440.) All bodies with which we are acquainted, when raised into the
air and quietly abandoned, descend to the earth's surface in lines perpen-
dicular to it. They are therefore urged thereto by a force or effort, which
it is but reasonable to regard as the direct or indirect result of a conscious-
mess and a will existing somewhere, though beyond our power to trace,
which force we term gravity, and whose tendency or direction, as uni-
versal experience teaches, is towards the earth's centre; or rather, to
speak strictly, with reference to its spheroidal figure, perpendicular to the
surface of still water. But if we cast a body obliquely into the air, this
tendency, though not extinguished or diminished, is materially modified
in its ultimate effect. The upward impetus we give the stone is, it is
true, after a time destroyed, and a downward one communicated to it,
which ultimately brings it to the surface, where it is opposed in its fur-
ther progress, and brought to rest. But all the while it has been conti-
nually deflected or bent aside from its rectilinear progress, and made to
describe a curved line concave to the earth’s centre; and having a highest
point, vertex, or apogee, just as the moon has in its orbit, where the direc-
tion of its motion is perpendicular to the radius.
(441.) When the stone which we fling obliquely upwards meets and is
stopped in its descent by the earth’s surface, its motion is not towards the
centre, but inclined to the earth's radius at the same angle as when it
quitted our hand. As we are sure that, if not stopped by the resistance
of the earth, it would continue to descend, and that obliquely, what pre-
sumption, we may ask, is there that it would ever reach the centre towards
which its motion, in no part of its visible course, was ever directed?
What reason have we to believe that it might not rather circulate round
it, as the moon does round the earth, returning again to the point it set
out from, after completing an elliptic orbit of which the earth’s centre
occupies the lower focus? And if so, is it not reasonable to imagine that
the same force of gravity may (since we know that it is exerted at all
accessible heights above the surface, and even in the highest regions of
the atmosphere) extend as far as 60 radii of the earth, or to the moon?
and may not this be the power, —for some power there must be, - which
GRAVITATION OF THE MOON TO THE EARTH. 235
deflects her at every instant from the tangent of her orbit, and keeps her
in the elliptic path which experience teaches us she actually pursues?
(442.) If a stone be whirled round at the end of a string it will stretch
the string by a centrifugal force, which, if the speed of rotation be suffi-
ciently increased, will at length break the string, and let the stone escape.
However strong the string, it may, by a sufficient rotary velocity of the
stone, be brought to the utmost tension it will bear without breaking; and
if we know what weight it is capable of carrying, the velocity necessary
for this purpose is easily calculated. Suppose, now, a string to connect
the earth’s centre with a weight at its surface, whose strength should be
just sufficient to sustain that weight suspended from it. Let us, however,
for a moment imagine gravity to have no existence, and that the weight
is made to revolve with the limiting velocity which that string can barely
counteract: then will its tension be just equal to the weight of the re-
volving body; and any power which should continually urge the body
towards the centre with a force equal to its weight would perform the
office, and might supply the place of the string, if divided. Divide it
then, and in its place let gravity act, and the body will circulate as before;
its tendency to the centre, or its weight, being just balanced by its centri-
fugal force. Knowing the radius of the earth, we can calculate by the
principles of mechanics the periodical time in which a body so balanced
must circulate to keep it up; and this appears to be 1* 23* 22°.
(443.) If we make the same calculation for a body at the distance of
the moon, supposing its weight or gravity the same as at the earth's
surface, we shall find the period required to be 10° 45′ 30°. The actual
period of the moon's revolution, however, is 27° 7' 43*; and hence it is
clear that the moon’s velocity is not nearly sufficient to sustain it against
such a power, supposing it to revolve in a circle, or neglecting (for the
present) the slight ellipticity of its orbit. In order that a body at the
distance of the moon (or the moon itself) should be capable of keeping
its distance from the earth by the outward effort of its centrifugal force,
while yet its time of revolution should be what the moon’s actually is,
it will appear' that gravity, instead of being as intense as at the surface,
would require to be very nearly 3600 times less energetic; or, in other
words, that its intensity is so enfeebled by the remoteness of the body on
which it acts, as to be capable of producing in it, in the same time, only
smºoth part of the motion which it would impart to the same mass of
matter at the earth's surface.
(444.) The distance of the moon from the earth's centre is a very little
less than sixty times the distance from the centre to the surface, and
* Newton, Princip. b. i., Prop. 4., Cor. 2.
236 OUTLINES OF ASTRONOMY.
3600 : 1 :: 60°: 1*; so that the proportion in which we must admit the
earth's gravity to be enfeebled at the moon's distance, if it be really the
force which retains the moon in her orbit, must be (at least in this par-
ticular instance) that of the squares of the distances at which it is com-
pared. Now, in such a diminution of energy with increase of distance,
there is nothing prima facie inadmissible. Emanations from a centre,
such as light and heat, do really diminish in intensity by increase of dis-
tance, and in this identical proportion; and though we cannot certainly
argue much from this analogy, yet we do see that the power of magnetic
and electric attractions and repulsions is actually enfeebled by distance,
and much more rapidly than in the simple proportion of the increased
distances. The argument, therefore, stands thus:– On the one hand,
Gravity is a real power, of whose agency we have daily experience. We
know that it extends to the greatest accessible heights, and far beyond;
and we see no reason for drawing a line at any particular height, and
there asserting that it must cease entirely; though we have analogies to
lead us to suppose its energy may diminish as we ascend to great heights
from the surface, such as that of the moon. On the other hand we are
sure the moon is urged towards the earth by some power which retains
her in her orbit, and that the intensity of this power is such as would
correspond to a gravity, diminished in the proportion — otherwise not
improbable — of the squares of the distances. If gravity be not that
power, there must exist some other; and, besides this, gravity must
cease at some inferior level, or the nature of the moon must be different
from that of ponderable matter; —for if not, it would be urged by both
powers, and therefore too much urged and forced inwards from her path.
(445.) It is on such an argument that Newton is understood to have
rested, in the first instance, and provisionally, his law of universal gravi-
tation, which may be thus abstractly stated:—“Every particle of matter
in the universe attracts every other particle, with a force directly propor-
tioned to the mass of the attracting particle, and inversely to the square
of the distance between them.” In this abstract and general form,
however, the proposition is not applicable to the case before us. The
earth and moon are not mere particles, but great spherical bodies, and to
such the general law does not immediately apply; and, before we can
make it applicable, it becomes necessary to inquire what will be the force
with which a congeries of particles, constituting a solid mass of any
assigned figure, will attract another such collection of material atoms.
This problem is one purely dynamical, and, in this its general form, is
of extreme difficulty. Fortunately however, for human knowledge, when
the attracting and attracted bodies are spheres, it admits of an easy and
GENERAL LAW OF GRAVITATION. - 237.
direct solution. Newton himself has shown (Princip. b. i. prop. 75)
that, in that case, the attraction is precisely the same as if the whole
matter of each sphere were collected into its centre, and the spheres were
single particles there placed; so that, in this case, the general law applies
in its strict wording. The effect of the trifling deviation of the earth
from a spherical form is of too minute an order to need attention at pre-
sent. It is, however, perceptible, and may be hereafter noticed.
(446.) The next step in the Newtonian argument is one which divests
the law of gravitation of its provisional character, as derived from a loose
and superficial consideration of the lunar orbit as a circle described with an
average or mean velocity, and elevates it to the rank of a general and pri-
mordial relation by proving its applicability to the state of existing nature
in all its circumstances. This step consists in demonstrating, as he has
done" (Princip. i. 17. i. 75.), that, under the influence of such an attract-
ive force mutually urging two spherical gravitating bodies towards each
other, they will each, when moving in each other's neighbourhood, be
deflected into an orbit concave towards the other, and describe, one about
the other regarded as fixed, or both round their common centre of gravity,
curves whose forms are limited to those figures known in geometry by the
general name of conic sections. It will depend, he shows, in any assigned
case, upon the particular circumstances or velocity, distance, and direction,
which of these curves shall be described, - whether an ellipse, a circle, a
parabola, or an hyperbola; but one or other it must be; and any one of
any degree of excentricity it may be, according to the circumstances of
the case; and, in all cases, the point to which the motion is referred,
whether it be the centre of one of the spheres, or their common centre of
gravity, will of necessity be the focus of the conic section described. He
shows, furthermore (Princip. i. 1.), that, in every case, the angular velo-
city with which the line joining their centres moves, must be inversely
proportional to the square of their mutual distance, and that equal areas
of the curves described will be swept over by their line of junction in
equal times.
(447.) All this is in conformity with what we have stated of the solar
*We refer for these fundamental propositions, as a point of duty, to the immortal
work in which they were first propounded. It is impossible for us, in this volume, to
go into these investigations: even did our limits permit, it would be utterly inconsist-
ent with our plan : a general idea, however, of their conduct will be given in the next
chapter. We trust that the careful and attentive study of the Principia in its original
form will never be laid aside, whatever be the improvements of the modern analysis
as respects facility of calculation and expression. From no other quarter can a thorough
and complete comprehension of the mechanism of our system, (so far as the immedi-
ate scope of that work, be anything like so well, and we may add so easily obtained.
238 OUTLINES OF ASTRONOMY.
and lunar movements. Their orbits are ellipses, but of different degrees
of excentricity; and this circumstance already indicates the general appli-
cability of the principles in question.
(448.) But here we have already, by a natural and ready implication
(such is always the progress of generalization), taken a further and most
important step, almost unperceived. We have extended the action of
gravity to the case of the earth and sun, to a distance immensely greater
than that of the moon, and to a body apparently quite of a different
nature from either. Are we justified in this? or, at all events, are there
no modifications introduced by the change of data, if not into the general
expression, at least into the particular interpretation, of the law of gravi-
tation? Now, the moment we come to numbers, an obvious incongruity
strikes us. When we calculate, as above, from the known distance of the
sun (art. 357), and from the period in which the earth circulates about it
(art. 305), what must be the centrifugal force of the latter by which the
sun's attraction is balanced, (and which, therefore, becomes an exact mea-
sure of the sun's attractive energy as exerted on the earth,) we find it to
be immensely greater than would suffice to counteract the earth's attrac-
tion on an equal body at that distance—greater in the high proportion of
35.4936 to 1. It is clear, then, that if the earth be retained in its orbit
about the sun by solar attraction, conformable in its rate of diminution
with the general law, this force must be no less than 354936 times more
intense than what the earth would be capable of exerting, caeteris paribus,
at an equal distance.
(449.) What, then, are we to understand from this result Simply
this, that the sun attracts as a collection of 354936 earths occupying
its place would do, or, in other words, that the sun contains 354936 times
the mass or quantity of ponderable matter that the earth consists of Nor
let this conclusion startle us. We have only to recall what has been
already shown (in art. 358) of the gigantic dimensions of this magnifi-
cent body, to perceive that, in assigning to it so vast a mass, we are not
outstepping a reasonable proportion. In fact, when we come to compare
its mass with its bulk, we find its density' to be less than that of the earth,
being no more than 0.2543. So that it must consist, in reality, of far
lighter materials, especially when we consider the force under which its
central parts must be condensed. This consideration renders it highly
probable that an intense heat prevails in its interior by which its elasticity
* The density of a material body is as the mass directly, and the volume inversely :
hence density of G); density of “B:: r;#34; 1 : 0.2543 : 1.
GRAVITATION AND MASS OF THE SUN. 239
is reinforced, and rendered capable of resisting this almost inconceivable
pressure without collapsing into smaller dimensions.
(450.) This will be more distinctly appreciated, if we estimate, as we
are now prepared to do, the intensity of gravity at the sun's surface.
The attraction of a sphere being the same (art. 445) as if its whole
mass were collected in its centre, will, of course, be proportional to the
mass directly, and the square of the distance inversely; and, in this case,
the distance is the radius of the sphere. Hence we conclude", that the
intensities of solar and terrestrial gravity at the surfaces of the two globes
are in the proportions of 279 to 1. A pound of terrestrial matter at the
sun's surface, then, would exert a pressure equal to what 27.9 such pounds
would do at the earth’s. The efficacy of muscular power to overcome
weight is therefore proportionally nearly 28 times less on the sun than on
the earth. An ordinary man, for example, would not only be unable to
sustain his own weight on the sun, but would be literally crushed to atoms
under the load.”
(451.) Henceforward, then, we must consent to dismiss all idea of the
earth's immobility, and transfer that attribute to the sun, whose ponderous
mass is calculated to exhaust the feeble attractions of such comparative
atoms as the earth and moon, without being perceptibly dragged from its
place. Their centre of gravity lies, as we have already hinted, almost
close to the centre of the solar globe, at an interval quite imperceptible
from our distance; and whether we regard the earth's orbit as being per-
formed about the one or the other centre makes no appreciable difference
in any one phenomenon of astronomy.
(452.) It is in consequence of the mutual gravitation of all the several
parts of matter, which the Newtonian law supposes, that the earth and
moon, while in the act of revolving, monthly, in their mutual orbits about
their common centre of gravity, yet continue to circulate, without parting
company, in a greater annual orbit round the sun. We may conceive this
motion by connecting two unequal balls by a stick, which, at their centre
of gravity, is tied by a long string, and whirled round. Their joint sys-
tem will circulate as one body about the common centre to which the
string is attached, while yet they may go on circulating round each other in
subordinate gyrations, as if the stick were quite free from any such tie,
and merely hurled through the air. If the earth alone, and not the
moon, gravitated to the sun, it would be dragged away, and leave the moon
“Solar gravity: terrestrial : ; ######2 : trººm ; ; 27.9 : 1; the respective radii
of the sun and earth being 440000, and 4000 miles.
*A mass weighing 12 stone or 168lbs. on the earth, would produce a pressure of
4687 lbs. on the sun.
240 OUTLINES OF ASTRONOMY.
behind—and vice versá, but, acting on both, they continue together
under its attraction, just as the loose parts of the earth’s surface continue
to rest upon it. It is, then, in strictness, not the earth or the moon
which describes an ellipse around the sun, but their common centre of
gravity. The effect is to produce a small, but very perceptible, monthly
equation in the sun's apparent motion as seen from the earth, which is
always taken into account in calculating the sun's place. The moon's
actual path in its compound orbit round the earth and sun is an epicycloi-
dal curve intersecting the orbit of the earth twice in every lunar month,
and alternately within and without it. But as there are not more than
twelve such months in the year, and as the total departure of the moon
from it either way does not exceed one 400th part of the radius, this
amounts only to a slight undulation upon the earth’s ellipse, so slight,
indeed, that if drawn in true proportion on a large sheet of paper, no eye
unaided by measurement with compasses would detect it. The real orbit
of the moon is everywhere concave towards the sun.
(453.) Here moreover, i. e. in the attraction of the sun, we have the
key to all those differences from an exact elliptic movement of the moon
in her monthly orbit, which we have already noticed (arts. 407, 409), viz.
to the retrograde revolution of her nodes; to the direct circulation of the
axis of her ellipse; and to all the other deviations from the laws of elliptic
motion at which we have further hinted. If the moon simply revolved
about the earth under the influence of its gravity, none of these pheno-
mena would take place. Its orbit would be a perfect ellipse, returning
into itself, and always lying in one and the same plane. That it is not so,
is a proof that some cause disturbs it, and interferes with the earth's
attraction; and this cause is no other than the sun's attraction—or rather,
that part of it which is not equally exerted on the earth.
(454.) Suppose two stones, side by side, or otherwise situated with re-
spect to each other, to be let fall together; then, as gravity accelerates
them equally, they will retain their relative positions, and fall together as
if they formed one mass. But suppose gravity to be rather more intensely
exerted on one than on the other; then would that one be rather more
accelerated in its fall, and would gradually leave the other; and thus a
relative motion between them would arise from the difference of action,
however slight.
(455.) The sun is about 400 times more remote than the moon; and,
in consequence, while the moon describes her monthly orbit round the
earth, her distance from the sun is alternately 4% oth part greater and as
much less than the earth's. Small as this is, it is yet sufficient to produce
a perceptible excess of attractive tendency of the moon towards the sun,
SOLAR DISTUBANCE OF THE MOON'S SURFACE. 241
Fig. 62.
NOM
E

*
above that of the earth when in the nearer point of her orbit, M, and a
corresponding defect on the opposite part, N.; and, in the intermediate
positions, not only will a difference of forces subsist, but a difference of
directions also; since however small the lunar orbit M N, it is not a point,
and, therefore, the lines drawn from the sun S to its several parts cannot
be regarded as strictly parallel. If, as we have already seen, the force of
the sun were equally exerted, and in parallel directions on both, no disturb-
ance of their relative situations would take place; but from the non-veri-
fication of these conditions arises a disturbing force, oblique to the line
joining the moon and earth, which in some situations acts to accelerate, in
others to retard, her elliptic orbitual motion; in some to draw the earth
from the moon, in others the moon from the earth. Again, the lunar
orbit, though very nearly, is yet not quite coincident with the plane of the
ecliptic; and hence the action of the sun, which is very nearly parallel to
the last-mentioned plane, tends to draw her somewhat out of the plane of
her orbit, and does actually do so—producing the revolution of her nodes,
and other phenomena less striking. We are not yet prepared to go into
the subject of these perturbations, as they are called; but they are intro-
duced to the reader's notice as early as possible, for the purpose of re-
assuring his mind, should doubts have arisen as to the logical correctness
of our argument, in consequence of our temporary neglect of them while
working our way upward to the law of gravity from a general considera-
tion of the moon's orbit.
16
242 GlyTLINES OF ASTRONOMY.
CHAPTER IX.
O R T H E S O L A R S Y S T E M.
APPARENT MOTIONS OF THE PLANETS. — THEIR STATIONS AND RE-
TROGRADATIONS. — THE SUN THEIR NATURAL CENTRE OF MOTION.
— INFERIOR PLANETS. — THEIR PHASEs, PERIODs, ETC. — DIMEN-
SIONS AND FORM OF THEIR, ORBITS. — TRANSITS ACROSS THE SUN.
— SUPERIOR PLANETS. – THEIR DISTANCEs, PERIODs, ETC. — KEP-
LER'S LAWS AND THEIR INTERPRETATION. —ELLIPTIC ELEMENTS
of A PLANET's ORBIT. —ITs HELIOCENTRIC AND GEOCENTRIC
PLACE. – EMPIRICAL LAW OF PLANETARY DISTANCEs;— VIoIATED
IN THE CASE OF NEPTUNE. – THE ULTRA-ZODIACAL, PLANETS. —
PHYSICAL PECULIARITIES OBSERVABLE IN EACH OF THE, PLANETS.
(456.) THE sun and moon are not the only celestial objects which
appear to have a motion independent of that by which the great con-
stellation of the heavens is daily carried round the earth. Among the
stars there are several, - and those among the brightest and most con-
spicuous, – which, when attentively watched from night to night, are
found to change their relative situations among the rest; some rapidly,
others much more slowly. These are called planets. Four of them —
Venus, Mars, Jupiter, and Saturn – are remarkably large and brilliant;
another, Mercury, is also visible to the naked eye as a large star, but, for
a reason which will presently appear, is seldom conspicuous; a sixth,
Uranus, is barely discernible without a telescope; and nine others—Nep-
tune, Ceres, Pallas, Westa, Juno, Astraea, Hebe, Iris, Flora— are never
visible to the naked eye. Besides these fifteen, others yet undiscovered
may exist;" and it is extremely probable that such is the case, the mul-
titude of telescopic stars being so great that only a small fraction of their
number has been sufficiently noticed to ascertain whether they retain the
same places or not, and the ten last-mentioned planets having all been
discovered within little more than half a century from the present time.
* While this sheet is passing through the press, a sixteenth, not yet named, has
been added to the list, by the observations of Mr. Graham, astronomical assistant to
E. Cooper, Esq. at his observatory at Markree, Sligo, Ireland.
APPARENT MOTIONS OF THE PLANETS. 243
(457.) The apparent motions of the planets are much more irregular
than those of the Sun or moon. Generally speaking, and comparing their
places at distant times, they all advance, though with very different ave-
rage or mean velocities, in the same direction as those luminaries, i. e. in
opposition to the apparent diurnal motion, or from west to east: all of
them make the entire tour of the heavens, though under very different
circumstances; and all of them, with the exception of the eight teles-
copic planets, Ceres, Pallas, Juno, Westa, Astraea, Hebe, Iris, and Flora
(which may therefore be termed ultra-zodiacal)—are confined in their
visible paths within very narrow limits on either side the ecliptic, and
perform their movements within that zone of the heavens we have called,
above, the Zodiac (art. 303.) -
(458.) The obvious conclusion from this is, that whatever be, other-
wise, the nature and law of their motions, they are performed nearly in
the plane of the ecliptic—that plane, namely, in which our own motion
about the sun is performed. Hence it follows, that we see their evolu-
tions, not in plan, but in section; their real angular movements and
linear distances being all foreshortened and confounded undistinguishably,
while only their deviations from the ecliptic appear of their natural mag-
nitude, undiminished by the effect of perspective.
(459.) The apparent motions of the sun and moon, though not uni-
form, do not deviate very greatly from uniformity; a moderate accelera-
tion and retardation, accountable for by the ellipticity of their orbits,
being all that is remarked. But the case is widely different with the
planets: sometimes they advance rapidly; then relax in their apparent
speed—come to a momentary stop; and then actually reverse their
motion, and run back upon their former course, with a rapidity at first
increasing, then diminishing, till the reversed or retrograde motion ceases
altogether. Another station, or moment of apparent rest or indecision,
now takes place; after which the movement is again reversed, and
resumes its original direct character. On the whole, however, the amount
of direct motion more than compensates the retrograde; and by the
Fig. 63.
Tº -
r TRE9 C
N-Es=L

S
excess of the former over the latter, the gradual advance of the planet
from west to east is maintained. Thus, supposing the Zodiac to pe
unfolded into a plane surface, (or represented as in Mercator's projection,
244 OUTLINES OF ASTRONOMY.
art. 283, taking the ecliptic E C for its ground line,) the track of a planet
when mapped down by observation from day to day, will offer the appear-
ance PQRS, &c.; the motion from P to Q being direct, at Q stationary,
from Q to R retrograde, at R again stationary, from R to S direct, and
SO Oſl.
(460.) In the midst of the irregularity and fluctuation of this motion,
one remarkable feature of uniformity is observed. Whenever the planet
crosses the ecliptic, as at N in the figure, it is said (like the moon) to be
in its node; and as the earth necessarily lies in the plane of the ecliptic,
the planet cannot be apparently or uranographically situated in the
celestial circle so called, without being really and locally situated in that
plane. The visible passage of a planet through its node, then, is a phe-
nomenon indicative of a circumstance in its real motion quite independent
of the station from which we view it. Now, it is easy to ascertain, by
observation, when a planet passes from the north to the south side of the
ecliptic: we have only to convert its right ascensions and declinations into
longitudes and latitudes, and the change from north to south latitude on
two successive days will advertise us on what day the transition took
place; while a simple proportion, grounded on the observed state of its
motion in latitude in the interval, will suffice to fix the precise hour and
minute of its arrival on the ecliptic. Now, this being done for several
transitions from side to side of the ecliptic, and their dates thereby fixed,
we find, universally, that the interval of time elapsing between the suc-
cessive passages of each planet through the same node (whether it be the
ascending or the descending) is always alike, whether the planet at the
moment of such passage be direct or retrograde, swift or slow, in its
apparent movement.
(461.) Here, then, we have a circumstance which, while it shows that
the motions of the planets are in fact subject to certain laws and fixed
periods, may lead us very naturally to suspect that the apparent irregu-
larities and complexities of their movements may be owing to our not
seeing them from their natural centre (art. 338, 371), and from our mixing
up with their own proper motions movements of a parallactic kind, due to
our own change of place, in virtue of the orbitual motion of the earth
about the sun.
(462.) If we abandon the earth as a centre of the planetary motions, it
cannot admit of a moment’s hesitation where we should place that centre
with the greatest probability of truth. It must surely be the sun which
is entitled to the first trial, as a station to which to refer to them. If it
be not connected with them by any physical relation, it at least possesses
the advantage, which the earth does not, of comparative immobility. But
THE SUN THE CENTRE OF OUR SYSTEM. 245
after what has been shown in art. 449, of the immense mass of that lumi-
nary, and of the office it performs to us as a quiescent centre of our orbi-
tual motion, nothing can be more natural than to suppose it may perform
the same to other globes which, like the earth, may be revolving round it;
and these globes may be visible to us by its light reflected from them, as
the moon is. Now there are many facts which give a strong support to
the idea that the planets are in this predicament.
(463.) In the first place, the planets really are great globes, of a size
commensurate with the earth, and several of them much greater. When
examined through powerful telescopes, they are seen to be round bodies,
of sensible and even of considerable apparent diameter, and offering dis-
tinct and characteristic peculiarities, which show them to be solid masses,
each possessing its individual structure and mechanism; and that, in one
instance at least, an exceedingly artificial and complex one. (See the
representations of Mars, Jupiter, and Saturn, in Plate III.) That their
distances from us are great, much greater than that of the moon, and some
of them even greater than that of the sun, we infer, 1st, from their being
occulted by the moon, and 2dly, from the smallness of their diurnal
parallax, which, even for the nearest of them, when most favourably
situated, does not exceed a few seconds, and for the remote ones is almost
imperceptible. From the comparison of the diurnal parallax of a celestial
body, with its apparent semidiameter, we can at once estimate its real size.
For the parallax is, in fact, nothing else than the apparent semidiameter
of the earth as seen from the body in question (art. 339 et seq.); and, the
intervening distance being the same, the real diameters must be to each
other in the proportion of the apparent ones. Without going into parti-
culars, it will suffice to state it as a general result of that comparison, that
the planets are all of them incomparably smaller than the sun, but some
of them as large as the earth, and others much greater.
(464.) The next fact respecting them is, that their distances from us,
as estimated from the measurement of their angular diameters, are in a
continual state of change, periodically increasing and decreasing within
certain limits, but by no means corresponding with the supposition of
regular circular or elliptic orbits described by them about the earth as a
centre or focus, but maintaining a constant and obvious relation to their
apparent angular distances or elongations from the sun. For example;
the apparent diameter of Mars is greatest when in opposition (as it is
called) to the sun, i. e. when in the opposite part of the ecliptic, or when
it comes on the meridian at midnight, — being then about 18", -but
diminishes rapidly from that to about 4", which is its apparent diameter
when in conjunction, or when seen in nearly the same direction as that
246 OUTLINES OF ASTRONOMY.
luminary. This, and facts of a similar character, observed with respect
to the apparent diameters of the other planets, clearly point out the sun
as having more than an accidental relation to their movements.
(465.) Lastly, certain of the planets, (Mercury, Venus, and Mars,)
when viewed through telescopes, exhibit the appearance of phases like
those of the moon. This proves that they are opaque bodies, shining
only by reflected light, which can be no other than that of the sun's;
not only because there is no other source of light external to them suffi-
ciently powerful, but because the appearance and succession of the phases
themselves are (like their visible diameters) intimately connected with
their elongations from the Sun, as will presently be shown.
(466.) Accordingly it is found, that, when we refer the planetary move-
ments to the sun as a centre, all that apparent irregularity which they
offer when viewed from the earth disappears at once, and resolves itself
into one simple and general law, of which the earth’s motion, as ex-
plained in a former chapter, is only a particular case. In order to show
how this happens, let us take the case of a single planet, which we will
suppose to revolve round the sun, in a plane nearly, but not quite, coin-
cident with the ecliptic, but passing through the Sun, and of course inter-
secting the ecliptic in a fixed line, which is the line of the planet's nodes.
This line must of course divide its orbit into two segments; and it is
evident that, so long as the circumstances of the planet's motion remain
otherwise unchanged, the times of describing these segments must remain
the same. The interval, then, between the planet’s quitting either node,
and returning to the same node again, must be that in which it describes
one complete revolution round the sun, or its periodic time; and thus we
are furnished with a direct method of ascertaining the periodic time of
each planet.
(467.) We have said (art. 457) that the planets make the entire tour
of the heavens under very different circumstances. This must be ex-
plained. Two of them — Mercury and Venus—perform this circuit
evidently as attendants upon the sun, from whose vicinity they never
depart beyond a certain limit. They are seen sometimes to the east,
sometimes to the west of it. In the former case they appear conspicuous
over the western horizon, just after sunset, and are called evening stars:
Venus, especially, appears occasionally in this situation with a dazzling
lustre; and in favourable circumstances may be observed to cast a pretty
strong shadow." When they happen to be to the west of the sun, they
* It must be thrown upon a white ground. An open window in a white-washed room
is the best exposure. In this situation I have observed not only the shadow, but the
diffracted fringes edging its outline. — H. Note to the edition of 1833. Venus may
often be seen with the naked eye in the daytime.
INFERIOR PLANETS. 247
rise before that luminary in the morning, and appear over the eastern
horizon as morning stars: they do not, however, attain the same elonga-
tion from the Sun. Mercury never attains a greater angular distance from
it than about 29°, while Venus extends her excursions on either side to
about 47°. When they have receded from the sun, eastward, to their
respective distances, they remain for a time, as it were, immoveable with
respect to it, and are carried along with it in the ecliptic with a motion
equal to its own; but presently they begin to approach it, or, which comes
to the same, their motion in longitude diminishes, and the sun gains upon
them. As this approach goes on, their continuance above the horizon
after sunset becomes daily shorter, till at length they set before the dark-
ness has become sufficient to allow of their being seen. For a time, then,
they are not seen at all, unless on very rare occasions, when they are to
be observed passing across the sun's disc as Small, round, well-defined
black spots, totally different in appearance from the solar spots (art. 386.)
These phenomena are emphatically called transits of the respective
planets across the sun, and take place when the earth happens to be
passing the line of their nodes while they are in that part of their orbits,
just as in the account we have given (art. 412) of a solar eclipse. After
having thus continued invisible for a time, however, they begin to appear
on the other side of the sun, at first showing themselves only for a few
minutes before sunrise, and gradually longer and longer as they recede
from him. At this time their motion in longitude is rapidly retrograde.
Before they attain their greatest elongation, however, they become station-
ary in the heavens; but their recess from the sun is still maintained by
the advance of that luminary along the ecliptic, which continues to leave
them behind, until, having reversed their , motion, and become again
direct, they acquire sufficient speed to commence overtaking him — at
which moment they have their greatest western elongation : and thus is a
kind of oscillatory movement kept up, while the general advance along
the ecliptic goes on.
(468.) Suppose PQ to be the ecliptic, and A B D the orbit of one of
these planets, (for instance, Mercury,) seen almost edgewise by an eye
situated very nearly in its plane; S, the sun, its centre; and A, B, D, S,
successive positions of the planet, of which B and S are in the nodes.
If, then, the sun S stood apparently still in the ecliptic, the planets would
simply appear to oscillate backwards and forwards from A to D, alter-
nately passing before and behind the sun; and, if the eye happened to
lie exactly in the plane of the orbit, transiting his disc in the former
case, and being covered by it in the latter. But as the sun is not so
stationary, but apparently carried along the ecliptic PQ, let it be supposed.
248 OUTLINES OF ASTRONOMY.
Fig. 64.
to move over the spaces ST, TU, UW, while the planet in each case exe-
cutes one quarter of its period. Then will its orbit be apparently carried
along with the sun, into the successive positions represented in the figure;
and while its real motion round the sun brings it into the respective points,
B, D, S, A, its apparent movement in the heavens will seem to have been
along the wavy or zigzag line A N H K. In this, its motion in longitude
will have been direct in the parts A N, N H, and retrograde in the parts
H n K; while at the turns of the zigzag, as at H, it will have been sta-
tionary.
- (469.) The only two planets — Mercury and Venus—whose evolu-
tions are such as above described, are called inferior planets; their points
of farthest recess from the Sun are called (as above) their greatest eastern
and western elongations; and their points of nearest approach to it, their
tnferior and superior conjunctions,—the former when the planet passes
between the earth and the sun, the latter when behind the sun.
(470.) In art. 467 we have traced the apparent path of an inferior
planet, by considering its orbit in section, or as viewed from a point in
the plane of the ecliptic. Let us now contemplate it in plan, or as viewed
from a station above that plane, and projected on it. Suppose then, S to
represent the sun, a b c d the orbit of Mercury, and A B C D a part of
that of the earth—the direction of the circulation being the same in
Fig. 65.
, both, viz. that of the arrow. When the planet stands at a, let the earth
be situated at A, in the direction of a tangent, a A, to its orbit; then it
is evident that it will appear at its greatest elongation from the sun, -


INFERIOR PLANETS. 249
the angle a A S, which measures their apparent interval as seen from A,
being then greater than in any other situation of a upon its own circle.
(471.) Now, this angle being known by observation, we are hereby
furnished with a ready means of ascertaining, at least approximately, the
distance of the planet from the sun, or the radius of its orbit, supposed a
circle. For the triangle SA a is right-angled at a, and consequently we
have S a S.A. : : sin. S A a radius, by which proportion the radii S a,
S.A of the two orbits are directly compared. If the orbits were both
exact circles, this would of course be a perfectly rigorous mode of proceed.
ing: but (as is proved by the inequality of the resulting values of S a
obtained at different times) this is not the case; and it becomes necessary
to admit an excentricity of position, and a deviation from the exact circu-
lar form in both orbits, to account for this difference. Neglecting, how-
ever, at present this inequality, a mean or average value of S a may, at
least, be obtained from the frequent repetition of this process in all vari-
eties of situation of the two bodies. The calculations being performed,
it is concluded that the mean distance of Mercury from the sun is
about 36000000 miles; and that of Venus, similarly derived, about
68000000; the radius of the earth’s orbit being 95000000.
(472.) The sidereal periods of the planets may be obtained (as before
observed), with a considerable approach to accuracy, by observing their
passages through the nodes of their orbits; and indeed, when a certain
very minute motion of these nodes and the apsides of their orbits (similar
to that of the moon’s nodes and apsides, but incomparably slower) is
allowed for, with a precision only limited by the imperfection of the
appropriate observations. By such observations, so corrected, it appears
that the sidereal period of Mercury is 87° 23' 15" 43.9°; and that of
Venus, 2244 16” 49- 8.0°. These periods, however, are widely different
from the intervals at which the successive appearances of the two planets
at their eastern and western elongations from the sun are observed to
happen. Mercury is seen at its greatest splendour as an evening star, at
average intervals of about 116, and Venus at intervals of about 584 days.
The difference between the sidereal and synodical revolutions (art. 418)
accounts for this. Referring again to the figure of art. 470, if the sun
stood still at A, while the planet advanced in its orbit, the lapse of a
sidereal period, which should bring it round again to a, would also produce
a similar elongation from the sun. But, meanwhile, the earth has
advanced in its orbit in the same direction towards E, and therefore the
next greatest elongation on the same side of the sun will happen — not
in the position a A of the two bodies, but in some more advanced posi-
tion, e E. The determination of this position depends on a calculation
250 OUTLINES OF ASTRONOMY.
exactly similar to what has been explained in the article referred to ; and
we need, therefore, only state the resulting synodical revolutions of the
two planets, which come out respectively 115,877", and 583.920".
(473.) In this interval, the planet will have described a whole revolu-
tion plus the arc a ce, and the earth only the arc ACE of its orbit.
During its lapse, the inferior conjunction will happen when the earth has
a certain intermediate situation, B, and the planet has reached b, a point
between the sun and earth. The greatest elongation on the opposite side
of the sun will happen when the earth has come to C, and the planet to
c, where the line of junction C c is a tangent to the interior circle on the
opposite side from M. Lastly, the superior conjunction will happen
when the earth arrives at D, and the planet at d in the same line pro-
longed on the other side of the sun. The intervals at which these phe-
nomena happen may easily be computed from a knowledge of the synodi-
cal periods and the radii of the orbits.
(474.) The circumferences of circles are in the proportion of their
radii. If, then, we calculate the circumferences of the orbits of Mercury
and Venus, and the earth, and compare them with the times in which
their revolutions are performed, we shall find that the actual velocities
with which they move in their orbits differ greatly; that of Mercury
being about 109360 miles per hour, of Venus 80000, and of the earth
68040. From this it follows, that at the inferior conjunction, or at b,
either planet is moving in the same direction as the earth, but with a
greater velocity; it will, therefore, leave the earth behind it; and the
apparent motion of the planet viewed from the earth, will be as if the
planet stood still, and the earth moved in a contrary direction from what
it really does. In this situation, then, the apparent motion of the planet
must be contrary to the apparent motion of the sun; and, therefore,
retrograde. On the other hand, at the superior conjunction, the real
motion of the planet being in the opposite direction to that of the earth,
the relative motion will be the same as if the planet stood still, and the
earth advanced with their united velocities in its own proper direction.
In this situation, then, the apparent motion will be direct. Both these
results are in accordance with observed fact.
(475.) The stationary points may be determined by the following con-
sideration. At a or c, the points of greatest elongation, the motion of
the planet is directly to or from the earth, or along their line of junction,
while that of the earth is nearly perpendicular to it. Here, then, the
apparent motion must be direct. At b, the inferior conjunction, we have
seen that it must be retrograde, owing to the planet's motion (which is
there, as well as the earth's, perpendicular to the line of junction) sur-
INEERIOR PLANETS. 251
passing the earth's. Hence, the stationary points ought to lie, as it is
found by observation they do, between a and b, or c and b, viz. in such a
position that the obliquity of the planet's motion with respect to the line
of junction shall just compensate for the excess of its velocity, and cause
an equal advance of each extremity of that line, by the motion of the
planet at one end, and of the earth at the other: so that, for an instant
of time, the whole line shall move parallel to itself. The question thus
proposed is purely geometrical, and its solution on the supposition of
circular orbits is easy. Let E e and P p represent small arcs of the
Fig. 66. &-
S
R. F-e
orbits of the earth and planet described contemporaneously, at the moment
when the latter appears stationary, about S, the sun. Produce p P and
e E, tangents at P and E, to meet at R, and prolong EP backwards to
Q, join ep. Then since PE, p e are parallel, we have by similar trian-
gles P p : E e :: P R : R E, and since, putting v and W for the respec-
tive velocities of the planet and the earth, P p : E e :: v : V ; therefore
v : V :: P R : R E::: sin. P E R : sin. E P R
:: cos. S E P : cos. S P Q
:: cos. S E P : cos. (S E P4-E SP)
because the angles S E R and S P R are right angles. Moreover, if r
and R be the radii of the respective orbits, we have also
r: R. : : sin. S E P : sin. (S E P4-E SP)
from which two relations it is easy to deduce the values of the two angles
S E P and E S P; the former of which is the apparent elongation of the

252. OUTLINES OF ASTRONOMY.
planet from the sun,' the latter the difference of heliocentric longitudes
of the earth and planet.
(476.) When we regard the orbits as other than circles (which they
really are), the problem becomes somewhat complex—too much so to be
here entered upon. It will suffice to state the results which experience
verifies, and which assigns the stationary points of Mercury at from 15°
to 20° of elongation from the Sun, according to circumstances; and of
Venus, at an elongation never varying much from 29°. The former con-
tinues to retrograde during about 22 days; the latter, about 42.
(477.) We have said that some of the planets exhibit phases like the
moon. This is the case with both Mercury and Venus; and is readily
explained by a consideration of their orbits, such as we have above sup-
posed them. In fact, it requires little more than mere inspection of the
figure annexed, to show, that to a spectator situated on the earth E, an
inferior planet, illuminated by the sun, and therefore bright on the side
next to him, and dark on that turned from him, will appear full at the
superior conjunction A; gibbous (i. e. more than half full, like the moon
between the first and second quarter) between that point and the points
B C of its greatest elongation; half-mooned at these points; and crescent-
shaped, or horned, between these and the inferior conjunction D. As it
approaches this point, the crescent ought to thin off till it vanishes alto-
gether, rendering the planet invisible, unless in those cases where it
transits the sun's disc, and appears on it as a black spot. All these phe-
nomena are exactly conformable to observation. -
(478.) The variation in brightness of Venus in different parts of its
apparent orbit is very remarkable. This arises from two causes: 1st, the
V -
1 If |-n and--n, SEP=p, ESP=}, the equations to be resolved are sin,
1 +m m.
m+n.
(**)=m sin. $, and cos. (4 +})=n cos, p, which give cos. ip-

TRANSITS OF VENUS AND MERCURY. 253
varying j:oportion of its visible illuminated area to its whole disc; and,
2dly, the varying angular diameter, or whole apparent magnitude of the
disc itself. As it approaches its inferior conjunction from its greater
elongation, the half-moon becomes a crescent, which thins off; but this is
more than compensated, for some time, by the increasing apparent magni-
tude, in consequence of its diminishing distance. Thus the total light
received from it goes on increasing, till at length it attains a maximum,
which takes place when the planet's elongation is about 40°.
(479.) The transits of Venus are of very rare occurrence, taking place
alternately at the very unequal but regularly recurring intervals of 8, 122,
8, 105, 8, 122, &c., years in succession, and always in June or December.
As astronomical phaenomena, they are extremely important; since they
afford the best and most exact means we possess of ascertaining the sun’s
distance, or its parallax. Without going into the niceties of calculation
of this problem, which, owing to the great multitude of circumstances to
be attended to, are extremely intricate, we shall here explain its principle,
which, in the abstract, is very simple and obvious. Let E be the earth,
V Venus, and S the sun, and CD the portion of Venus's relative orbit
which she describes while in the act of transiting the sun's disc. Suppose
A B two spectators at opposite extremities of that diameter of the earth
Fig. 67.
which is perpendicular to the ecliptic, and, to avoid complicating the case,
let us lay out of consideration the earth's rotation, and suppose A, B, to
retain that situation during the whole time of the transit. Then, at any
moment when the spectator at A sees the centre of Venus projected at a
on the sun’s disc, he at B will see it projected at b. If them one or other
spectator could suddenly transport himself from A to B, he would see
Venus suddenly displaced on the disc from a to b; and if he had any
means of noting accurately the place of the points on the disc, either by
micrometrical measures from its edge, or by other means, he might ascer-
tain the angular measure of a b as seen from the earth. Now, since
A W a, BW b, are straight lines, and therefore make equal angles on each
side W, a b will be to A B as the distance of Venus from the sun is to its
distance from the earth, or as 68 to 27, or nearly as 23 to 1; a b there-

254 OUTLINES OF ASTRONOMY.
fore occupies on the sun's disc a space 24 times as great as the earth's
diameter; and its angular measure is therefore equal to about 2% times
the earth's apparent diameter at the distance of the sun, or (which is the
same thing) to five times the sun's horizontal parallax (art. 339). Any
error, therefore, which may be committed in measuring a b, will entail
only one fifth of that error on the horizontal parallax concluded from it.
(480.) The thing to be ascertained, therefore, is, in fact, neither more
nor less than the breadth of the zone PQRS, p q r s, included between
the extreme apparent paths of the centre of Venus across the sun’s disc,
from its entry on one side to its quitting it on the other. The whole
business of the observers at A, B, therefore, resolves itself into this;–to
ascertain, with all possible care and precision, each at his own station, this
path, where it enters, where it quits, and what segment of the sun's disc
it cuts off. Now, one of the most exact ways in which (conjoined with
careful micrometric measures) this can be done, is by noting the time occu-
pied in the whole transit; for the relative angular motion of Venus being,
in fact, very precisely known from the tables of her motion, and the appa-
rent path being very nearly a straight line, these times give us a measure
(on a very enlarged scale) of the lengths of the chords of the segments
cut off; and the sun's diameter being known also with great precision,
their versed sines, and therefore their difference, or the breadth of the
zone required, becomes known. To obtain these times correctly, each
observer must ascertain the instants of ingress and egress of the centre.
To do this, he must note, 1st, the instant when the first visible impression
or notch on the edge of the disc at P is produced, or the first external
contact; 2dly, when the planet is just wholly immersed, and the broken
edge of the dise just closes again at Q, or the first internal contact; and,
Jastly, he must make the same observations at the egress at R, S. The
mean of the internal end external contacts, corrected for the curvature of
the sun's limb in the intervals of the respective points of contact, internal
and external, gives the entry and egress of the planet's centre.
(481.) The modifications introduced into this process by the earth's
rotation on its axis, and by other geographical stations of the observers
thereon than here supposed, are similar in their principles to those which
enter into the calculation of a solar eclipse, or the occultation of a star by
the moon, only more refined. Any consideration of them, however, here,
would lead us too far; but in the view we have taken of the subject, it
affords an admirable example of the way in which minute elements in
astronomy may become magnified in their effects, and, by being made sub-
ject to measurement on a greatly enlarged scale, or by substituting the
measure of time for space, may be ascertained with a degree of precision
SUPERIOR PLANETS. 255
-*
adequate to every purpose, by only watching favourable opportunities, and
taking advantage of nicely adjusted combinations of circumstance. So im-
portant has this observation appeared to astronomers, that at the last transit
of Venus, in 1769, expeditions were fitted out, on the most extensive scale,
by the British, French, Russian, and other governments, to the remotest
corners of the globe, for the express purpose of performing it. The cele-
brated expedition of Captain Cook to Otaheite was one of them. The gene-
ral result of all the observations made on this most memorable occasion gives
8°5776 for the sun's horizontal parallax. The two next occurrences of
this phaenomenon will happen on Dec. 8, 1874, and Dec. 6, 1882.
(482.) The orbit of Mercury is very elliptical, the excentricity being
nearly one fourth of the mean distance. This appears from the inequality
of the greatest elongations from the sun, as observed at different times,
and which vary between the limits 16°12' and 28° 48', and, from exact
measures of such elongations, it is not difficult to show that the orbit of
Venus also is slightly excentric, and that both these planets, in fact,
describe ellipses, having the sun in their common focus.
(483.) Transits of Mercury over the sun's disc occasionally occur, as
in the case of Vénus, but more frequently; those at the ascending node
in November, at the descending in May. The intervals (considering
each node separately) are usually either 13 or 7 years, and in the order
13, 13, 13, 7, &c.; but owing to the considerable inclination of the orbit
of Mercury to the ecliptic, this cannot be taken as an exact expression
of the said recurrence, and it requires a period of at least 217 years to
bring round the transits in regular order. One will occur in the present
year (1848,) the next in 1861. They are of much less astronomical
importance than that of Venus, on account of the proximity of Mercury
to the sun, which affords a much less favourable combination for the
determination of the sun’s parallax.
(484.) Let us now consider the superior planets, or those whose orbits
enclose on all sides that of the earth. That they do so is proved by
several circumstances:–1st, They are not, like the inferior planets, con-
fined to certain limits of elongation from the sun, but appear at all dis.
tances from it, even in the opposite quarter of the heavens, or, as it is
called, in opposition; which could not happen, did not the earth at such
times place itself between them and the sun : 2dly, They never appear
horned, like Venus or Mercury, nor even semilunar. Those, on the
contrary, which, from the minuteness of their parallax, we conclude to be
the most distant from us, viz. Jupiter, Saturn, Uranus, and Neptune,
never appear otherwise than round; a sufficient proof, of itself, that we
see them always in a direction not very remote from that in which the
256 OUTLINES OF ASTRONOMY.
sun's rays illuminate them; and that, therefore, we occupy a station which
is never very widely removed from the centre of their orbits, or, in other
words, that the earth's orbit is entirely enclosed within theirs, and of
comparatively small diameter. Only one of them, Mars, exhibits any
perceptible phase, and in its deficiency from a circular outline, never
surpasses a moderately gibbous appearance,—the enlightened portion of
the disc being never less than seven-eighths of the whole. To understand
this, we need only cast our eyes on the annexed figure, in which E is the
earth, at its apparent greatest elongation from the sun S, as seen from
Mars, M. In this position, the angle S M E, included between the lines
Fig. 69.
S M and E M, is at its maximum; and therefore, in this state of things,
a spectator on the earth is enabled to see a greater portion of the dark
hemisphere of Mars than in any other situation. The extent of the
phase, then, or greatest observable degree of gibbosity, affords a measure
—a sure, although a coarse and rude one — of the angle S M E, and
therefore of the proportion of the distance S M, of Mars, to SE, that of
the earth from the sun, by which it appears that the diameter of the
orbit of Mars cannot be less than 14 times that of the earth's. The
phases of Jupiter, Saturn, Uranus, and Neptune, being imperceptible, it
follows that their orbits must include not only that of the earth, but of
Mars also. -
(485.) All the superior planets are retrograde in their apparent
motions when in opposition, and for some time before and after; but they
differ greatly from each other, both in the extent of their arc of retrogra-
dation, in the duration of their retrograde movement, and in its rapidity
when swiftest. It is more extensive and rapid in the case of Mars than

SUPERIOR PLANETS. 257
of Jupiter, of Jupiter than of Saturn, of that planet than of Uranus,
and of Uranus again than Neptune. The angular velocity with which a
planet appears to retrograde is easily ascertained by observing its apparent
place in the heavens from day to day; and from such observations, made
about the time of opposition, it is easy to conclude the relative magni-
tudes of their orbits, as compared with the earth's, supposing their
periodical times known. For, from these, their mean angular velocities
are known also, being inversely as the times. Suppose, then, E e to be a
Fig. 70.
_-f 7// 3/
S-E- --- X
IE TVI
very small portion of the earth’s orbit, and M. m. a corresponding portion
of that of a superior planet, described on the day of opposition, about
the sun S, on which day the three bodies lie in one straight line SEM X.
Then the angles E S e and M S m are given. Now, if em be joined and
prolonged to meet S M continued in X, the angle e X E, which is equal
to the alternate angle X e Y, is evidently the retrogradation of Mars on
that day, and is, therefore, also given. E e, therefore, and the angle
E X e, being given in the right-angled triangle E e X, the side E X is
easily calculated, and thus S X becomes known. Consequently, in the
triangle S m X, we have given the side S X and the two angles m S X,
and m X S, whence the other sides, S m, m X, are easily determined.
Now, S m is no other than the radius of the orbit of the superior planet
required, which in this calculation is supposed circular, as well as that of
the earth; a supposition not exact, but sufficiently so to afford a satisfac-
tory approximation to the dimensions of its orbit, and which, if the
process be often repeated, in every variety of situation at which the
opposition can occur, will ultimately afford an average or mean value of
its diameter fully to be depended upon. *
(486.) To apply this principle, however, to practice, it is necessary to
know the periodic times of the several planets. These may be obtained
directly, as has been already stated, by observing the intervals of their
passages through the ecliptic; but, owing to the very small inclination of
the orbits of some of them to its plane, they cross it so obliquely that
the precise moment of their arrival on it is not ascertainable, unless by
very nice observations. A better method consists in determining, from
the observations of several successive days, the exact moments of their
arriving in opposition with the sum, the criterion of which is a difference
of longitudes between the sun and planet of exactly 180°. The Interval
17
258 OTOTLINES OF ASTRONOMY.
between successive oppositions thus obtained is nearly one synodical pe.
riod; and would be exactly so, were the planet's orbit and that of the
earth both circles, and uniformly described; but as that is found not to
be the case (and the criterion is, the inequality of successive synodical
revolutions so observed), the average of a great number, taken in all va-
rieties of situation in which the oppositions occur, will be freed from the
elliptic inequality, and may be taken as a mean synodical period. From
this, by the considerations and by the process of calculation, indicated
(art. 418) the sidereal periods are readily obtained. The accuracy of this
determination will, of course, be greatly increased by embracing a long
interval between the extreme observations employed. In point of fact,
that interval extends to nearly 2000 years in the cases of the planets
known to the ancients, who have recorded their observations of them in
a manner sufficiently careful to be made use of. Their periods may, there-
fore, be regarded as ascertained with the utmost exactness. Their nume-
rical values will be found stated, as well as the mean distances, and all
the other elements of the planetary orbits, in the synoptic table at the
cnd of the volume, to which (to avoid repetition) the reader is once for
all referred. *.
(487.) In casting our eyes down the list of the planetary distances, and
comparing them with the periodic times, we cannot but be struck with a
certain correspondence. The greater the distance, or the larger the orbit,
evidently the longer the period. The order of the planets, beginning
from the sun, is the same, whether we arrange them according to their
distances, or to the time they occupy in completing their revolutions; and
is as follows: — Mercury, Venus, Earth, Mars, the ultra-Zodiacal pla-
nets, or, as they are sometimes also called, Asteroids,-Jupiter, Saturn, Ura-
nus, and Neptune. Nevertheless, when we come to examine the numbers
expressing them, we find that the relation between the two series is not
that of simple proportional increase. The periods increase more than in
proportion to the distances. Thus, the period of Mercury is about 88 days,
and that of the earth 365 — being in proportion as 1 to 4:15, while their
distances are in the less proportion of 1 to 2:56; and a similar remark
holds good in every instance. Still, the ratio of increase of the times is
not so rapid as that of the squares of the distances. The square of 2:56
is 6:5536, which is considerably greater than 4:15. An intermediate
rate of increase, between the simple proportion of the distances and that
of their squares is therefore clearly pointed out by the sequence of the
numbers; but it required no ordinary penetration in the illustrious Kep-
ler, backed by uncommon perseverance and industry, at a period when
the data themselves were involved in obscurity, and when the pro-
KEPLER’s THIRD LAW. 259
eesses of trigonometry and of numerical calculation were encumbered
with difficulties, of which the more recent invention of logarithmic tables
has happily left us no conception, to perceive and demonstrate the real
law of their connection. This connection is expressed in the following
proposition : —“The squares of the periodic times of any two planets are
to each other, in the same proportion as the cubes of their mean distances
from the sun.” Take, for example, the Earth and Mars,' whose periods
are in the proportion of 3652564 to 6869796, and whose distance from
the sun is that of 100000 to 152369; and it will be found, by any one
who will take the trouble to go through the calculation, that—
(3652564) : (6869796) :: (100000)*: (152369)”.
(488.) Of all the laws to which induction from pure observation has
ever conducted man, this third law (as it is called) of Kepler may justly
be regarded as the most remarkable, and the most pregnant with
important consequences. When we contemplate the constituents of the
planetary system from the point of view which this relation affords us, it
is no longer mere analogy which strikes us — no longer a general resem-
blance among them, as individuals independent of each other, and circula-
ting about the sun, each according to its own peculiar nature, and con-
nected with it by its own peculiar tie. The resemblance is now perceived
to be a true family likeness; they are bound up in one chain—inter-
woven in one web of mutual relation and harmonious agreement—suh-
jected to one pervading influence, which extends from the centre to the
farthest limits of that great system, of which all of them, the earth
included, must henceforth be regarded as members.
(489.) The laws of elliptic motion about the sun as a focus, and of the
equable description of areas by lines joining the sun and planets, were
originally established by Kepler, from a consideration of the observed
motions of Mars; and were by him extended, analogically, to all the other
planets. However precarious such an extension might then have ap-
peared, modern astronomy has completely verified it as a matter of fact,
by the general coincidence of its results with entire series of observations
of the apparent places of the planets. These are found to accord satis.
factorily with the assumption of a particular ellipse for each planet, whose
magnitude, degree of excentricity, and situation in space, are numerically
assigned in the synoptic table before referred to. It is true, that when
observations are carried to a high degree of precision, and when each
planet is traced through many successive revolutions, and its history car-
* The expression of this law of Kepler requires a slight modification when we come
to the extreme nicety of numerical calculation, for the greater planets, due to the
influence of their masses. This correction is imperceptible for the Earth and Mars.
260 oUTLINES OF ASTRONOMY.
ried back, by the aid of calculations founded on these data, for many cen-
turies, we learn to regard the laws of Kepler as only first approacimations
to the much more complicated ones which actually prevail; and that to
bring remote observations into rigorous and mathematical accordance
with each other, and at the same time to retain the extremely convenient
nomenclature and relations of the ELLIPTIC SYSTEM, it becomes necessary
to modify, to a certain extent, our verbal expression of the laws, and to
regard the numerical data or elliptic elements of the planetary orbits as
not absolutely permanent, but subject to a series of extremely slow and
almost imperceptible changes. These changes may be neglected when we
consider only a few revolutions; but going on from century to century,
and continually accumulating, they at length produce material departures
in the orbits from their original state. Their explanation will form the
subject of a subsequent chapter; but for the present we must lay them
out of consideration, as of an order too minute to affect the general con-
clusions with which we are now concerned. By what means astronomers
are enabled to compare the results of the elliptic theory with observation,
and thus satisfy themselves of its accordance with nature, will be ex-
plained presently.
(490.) It will first, however, be proper to point out what particular
theoretical conclusion is involved in each of the three laws of Kepler,
considered as satisfactorily established,—what indication each of them,
separately, affords of the mechanical forces prevalent in our system, and
the mode in which its parts are connected,—and how, when thus con-
sidered, they constitute the basis on which the Newtonian explanation of
the mechanism of the heavens is mainly supported. To begin with the
first law, that of the equable description of areas.-Since the planets move
in curvilinear paths, they must (if they be bodies obeying the laws of
dynamics) be deflected from their otherwise natural rectilinear progress
by force. And from this law, taken as a matter of observed fact, it fol-
lows, that the direction of such force, at every point of the orbit of each
planet, always passes through the sun. No matter from what ultimate
cause the power which is called gravitation originates, be it a virtue
lodged in the sun as its receptacle, or be it pressure from without, or the
resultant of many pressures or solicitations of unknown fluids, magnetic
or electric ethers, or impulses, still, when finally brought under our con-
templation, and summed up into a single resultant energy—its direction
is, from every point on all sides, towards the sun's centre. As an abstract
dynamical proposition, the reader will find it demonstrated by Newton, in
the first proposition of the Principia, with an elementary simplicity to
which we really could add nothing but obscurity by amplification, that
INTERPRETATION OF KEPLER'S LAWS. 261
.*
any body, urged towards a certain central point by a force continually
directed thereto, and thereby deflected into a curvilinear path, will describe
about that centre equal areas in equal times; and vice versä, that such
equable description of areas is itself the essential criterion of a continual
direction of the acting force towards the centre to which this character
belongs. The first law of Kepler, then, gives us no information as to the
nature or intensity of the force urging the planets to the sun; the only
conclusion it involves is, that it does so urge them. It is a property of
orbitual rotation under the influence of central forces generally, and, as
such, we daily see it exemplified in a thousand familiar instances. A
simple experimental illustration of it is to tie a bullet to a thin string,
and, having whirled it round with a moderate velocity in a vertical plane,
to draw the end of the string through a small ring, or allow it to coil
itself round the finger, or round a cylindrical rod held very firmly in a
horizontal position. The bullet will then approach the centre of motion
in a spiral line; and the increase not only of its angular but of its linear
velocity, and the rapid diminution of its periodic time when near the
centre, will express, more clearly than any words, the compensation by
which its uniform description of areas is maintained under a constantly
diminishing distance. If the motion be reversed, and the thread allowed
to uncoil, beginning with a rapid impulse, the velocity will diminish by
the same degrees as it before increased. The increasing rapidity of a
dancer's pirouette, as he draws in his limbs and straightens his whole per-
son, so as to bring every part of his frame as near as possible to the axis
of his motion, is another instance where the connection of the observed
effect with the central force exerted, though equally real, is much less
obvious.
(491.) The second law of Kepler, or that which asserts that the planets
describe ellipses about the sun as their focus, involves, as a consequence,
the law of solar gravitation (so be it allowed to call the force, whatever it
be, which urges them towards the sun) as exerted on each individual
planet, apart from all connection with the rest. A straight line, dynamic-
ally speaking, is the only path which can be pursued by a body absolutely
free, and under the action of no external force. All deflection into a
curve is evidence of the exertion of a force ; and the greater the deflection
in equal times, the more intense the force. Deflection from a straight
line is only another word for curvature of path ; and as a circle is char-
acterized by the uniformity of its curvatures in all its parts—so is every
other curve (as an ellipse) characterized by the particular law which regu-
lates the increase and diminution of its curvature as we advance along its
circumference. The deflecting force, then, which continually bends a
262 OUTLINES OF ASTRONOMY.
moving body into a curve, may be ascertained, provided its direction, in
the first place, and, secondly, the law of curvature of the curve itself, be
known. Both these enter as elements into the expression of the force.
A body may describe, for instance, an ellipse, under a great variety of
dispositions of the acting forces: it may glide along it, for example, as a
bead upon a polished wire, bent into an elliptic form; in which case the
acting force is always perpendicular to the wire, and the velocity is uni-
form. In this case the force is directed to no fixed centre, and there is
no equable description of areas at all. Or it may describe it as we may
see done, if we suspend a ball by a very long string, and, drawing it a
little aside from the perpendicular, throw it round with a gentle impulse.
In this case the acting force is directed to the centre of the ellipse, about
which areas are described equally, and to which a force proportional to
the distance (the decomposed result of terrestrial gravity) perpetually
urges it.* This is at once a very easy experiment, and a very instructive
one, and we shall again refer to it. In the case before us, of an ellipse
described by the action of a force directed to the focus, the steps of the
investigation of the law of force are these : 1st, The law of the areas de-
termines the actual velocity of the revolving body at every point, or the
space really run over by it in a given minute portion of time; 2dly, The
law of curvature of the ellipse determines the linear amount of deflection
from the tangent in the direction of the focus, which corresponds to that
space so run over; 3dly, and lastly, The laws of accelerated motion de-
clare that the intensity of the acting force causing such deflection in its
own direction, is measured by or proportional to the amount of that de-
flection, and may therefore be calculated in any particular position, or
generally expressed by geometrical or algebraic symbols, as a law inde-
pendent of particular positions, when that deflection is so calculated or
expressed. We have here the spirit of the process by which Newton has
resolved this interesting problem. For its geometrical detail, we must
refer to the 3d section of his Principia. We know of no artificial mode
of imitating this species of elliptic motion; though a rude approximation
to it—enough, however, to give a conception of the alternate approach
and recess of the revolving body to and from the focus, and the variation
of its velocity—may be had by suspending a small steel bead to a fine and
very long silk fibre, and setting it to revolve in a small orbit round the
pole of a powerful cylindrical magnet, held upright, and vertically under
the point of suspension.
* If the suspended body be a vessel full of fine sand, having a small hole at its bottom
the elliptic trace of its orbit will be left in a sand streak on a table placed below it
'This meat illustration is due, to the best of my knowledge, to Mr. Babbage.
INTERPRETATION OF KEPLER'S LAWS. 263
(492.) The third law of Kepler, which connects the distances and
periods of the planets by a general rule, bears with it, as its theoretical
interpretation, this important consequence, viz. that it is one and the same
force, modified only by distance from the sun, which retains all the planets
in their orbits about it. That the attraction of the sun (if such it be)
is exerted upon all the bodies of our system indifferently, without regard
to the peculiar materials of which they may consist, in the exact pro-
portion of their inertiae, or quantities of matter; that it is not, therefore,
of the nature of the elective attractions of chemistry or of magnetic
action, which is powerless on other substances than iron and some one or
two more, but is of a more universal character, and extends equally to all
the material constituents of our system, and (as we shall hereafter see
abundant reason to admit) to those of other systems than our own. This
law, important and general as it is, results, as the simplest of corollaries,
from the relations established by Newton in the section of the Principia
referred to (Prop. xv.), from which proposition it results, that if the earth
were taken from its actual orbit, and launched anew in space at the place,
in the direction, and with the velocity of any of the other planets, it
would describe the very same orbit, and in the same period, which that
planet actually does, a minute correction of the period only excepted,
arising from the difference between the mass of the earth and that of the
planet. Small as the planets are compared to the sun, some off them are
not, as the earth is, mere atoms in the comparison. The strict wording
of Kepler's law, as Newton has proved in his fifty-ninth proposition, is
applicable only to the case of planets whose proportion to the central
body is absolutely inappreciable. When this is not the case, the periodic
time is shortened in the proportion of the square root of the number ex-
pressing the sun's mass or inertiae, to that of the sum of the numbers
expressing the masses of the Sun and planet; and in general, whatever
be the masses of two bodies revolving round each other under the influ-
ence of the Newtonian law of gravity, the square of their periodic time
will be expressed by a fraction whose numerator is the cube of their mean
distance, i. e. the greater semi-axis of their elliptic orbit, and whose de-
nominator is the sum of their masses. When one of the masses is in-
comparably greater than the other, this resolves into Kepler's law; but
when this is not the case, the proposition thus generalized stands in lieu
of that law. In the system of the sun and planets, however, the numerical
correction thus introduced into the results of Kepler's law is too small to
be of any importance, the mass of the largest of the planets (Jupiter)
being much less than a thousandth part of that of the sun. We shall
264 oUTLINES OF ASTRONOMY.
presently, however, perceive all the importance of this generalization,
when we come to speak of the satellites. -
(493.) It will first, however, be proper to explain by what process of
calculation the expression of a planet's elliptic orbit by its elements can be
compared with observation, and how we can satisfy ourselves that the
numerical data contained in a table of such elements for the whole system
does really exhibit a true picture of it, and afford the means of deter-
mining its state at every instant of time, by the mere application of Kep-
ler's laws. Now, for each planet, it is necessary for this purpose to know,
1st, the magnitude and form of its ellipse; 2dly, the situation of this
ellipse in space, with respect to the ecliptic, and to a fixed line drawn
therein; 3dly, the local situation of the planet in its ellipse at some known
epoch, and its periodic time or mean angular velocity, or, as it is called,
its mean motion.
(494.) The magnitude and form of an ellipse are determined by its
greatest length and least breadth, or its two principal axes; but for astro-
nomical uses it is preferable to use the semi-axis major (or half the greatest
length), and the excentricity or distance of the focus from the centre,
which last is usually estimated in parts of the former. Thus, an ellipse,
whose length is 10 and breadth 8 parts of any scale, has for its major
semi-axis 5, and for its excentricity 3 such parts; but when estimated in
parts of the semi-axis, regarded as a unit, the excentricity is expressed by
the fraction #.
(495.) The ecliptic is the plane to which an inhabitant of the earth
most naturally refers the rest of the solar system, as a sort of ground-
plane; and the axis of its orbit might be taken for a line of departure in
that plane or origin of angular reckoning. Were the axis fixed, this
would be the best possible origin of longitudes; but as it has a motion
(though an excessively slow one), there is, in fact, no advantage in reck-
oning from the axis more than from the line of the equinoxes, and astro-
nomers therefore prefer the latter, taking account of its variation by
the effect of precession, and restoring it, by calculation at every in-
stant, to a fixed position. Now, to determine the situation of the ellipse
described by a planet with respect to this plane, three elements require
to be known: — 1st, the inclination of the plane of the planet's orbit
to the plane of the ecliptic; 2dly, the line in which these two planes
intersect each other, which of necessity passes through the sun, and
whose position with respect to the line of the equinoxes is therefore
given by stating its longitude. This line is called the line of the
nodes. When the planet is in this line, in the act of passing from the
south to the north side of the ecliptic, it is in its ascending node, and
ELEMENTS OF A PLANET'S ORBIT. 265
its longitude at that moment is the element called the longitude of the
mode. These two data determine the situation of the plane of the orbit;
and there only remains, for the complete determination of the situation
of the planet's ellipse, to know how it is placed in that plane, which
(since its focus is necessarily in the sun) is ascertained by stating the
longitude of its perihelion, or the place which the extremity of the axis
nearest the sun occupies, when orthographically projected on the ecliptic.
(496.) The dimensions and situation of the planet's orbit thus deter-
mined, it only remains, for a complete acquaintance with its history, to
determine the circumstances of its motion in the orbit so precisely fixed.
Now, for this purpose, all that is needed is to know the moment of time
when it is either at the perihelion, or at any other precisely determined
point of its orbit, and its whole period; for these being known, the law
of the areas determines the place at every other instant. This moment is
called (when the perihelion is the point chosen) the perihelion passage,
or, when some point of the orbit is fixed upon, without special reference
to the perihelion, the epoch.
(497.) Thus, then, we have seven particulars or elements, which must
be numerically stated, before we can reduce to calculation the state of the
system at any given moment. But, these known, it is easy to ascertain
the apparent positions of each planet, as it would be seen from the sun, or
is seen from the earth at any time. The former is called the heliocentric,
the latter the geocentric, place of the planet.
(498.) To commence with the heliocentric places. Let S represent
the sun; P A N the orbit of the planet, being an ellipse, having the S in
its focus, and A for its perihelion; and let p a NT represent the projection
of the orbit on the plane of the ecliptic, intersecting the line of equinoxes
Sºr in T, which, therefore, is the origin of longitudes. Then will SN be
the line of nodes; and if we suppose B to lie on the south, and A on the
north side of the ecliptic, and the direction of the planet's motion to be
from B to A, N will be the ascending node, and the angle T S N the lon-
gitude of the mode. In like manner, if P be the place of the planet at
any time, and if it and the perihelion A be projected on the ecliptic, upon
the points p, q, the angles T Sp, TS a, will be the respective heliocentric

266 OUTLINES OF ASTRONOMY.
longitudes of the planet and of the perihelion, the former of which—is to
be determined, and the latter is one of the given elements. Lastly, the
angle pSP is the heliocentric latitude of the planet, which is also required
to be known. t w
(499.) Now, the time being given, and also the moment of the planet's
passing the perihelion, the interval, or the time of describing the portion
A P of the orbit, is given, and the periodical time, and the whole area of
the ellipse being known, the law of proportionality of areas to the times
of their description gives the magnitude of the area A SP. From this
it is a problem of pure geometry to determine the corresponding angle
A SP, which is called the planet's true anomaly. This problem is of
the kind called transcendental, and has been resolved by a great variety
of processes, some more, some less intricate. It offers, however, no
peculiar difficulty, and is practically resolved with great facility by the
help of tables constructed for the purpose, adapted to the case of each
particular planet." -
(500.) The true anomaly thus obtained, the planet's angular distance
from the node, or the angle N SP, is to be found. Now, the longitudes of
the perihelion and node being respectively r a and r N, which are given,
their difference a N is also given, and the angle N of the spherical right-
angled triangle A N a, being the inclination of the plane of the orbit to
the ecliptic, is known. Hence we calculate the arc N A, or the angle
N S.A., which, added to A SP, gives the angle N S P required. And
from this, regarded as the measure of the arc N P, forming the hypothe-
muse of the right-angled spherical triangle P N p, whose angle N, as
before, is known, it is easy to obtain the other two sides, N p and P p.
The latter, being the measure of the angle p SP, expresses the planet's
heliocentric latitude; the former measures the angle N S p, or the planet's
distance in longitude from its node, which, added to the known angle
r SN, the longitude of the node, gives the heliocentric longitude. This
* It will readily be understood, that, except in the case of uniform circular motion,
an equable description of areas about any centre is incompatible with an equable de-
scription of angles. The object of the problem in the text is to pass from the area
supposed known, to the angle, supposed unknown : in other words, to derive the true
amount of angular motion from the perihelion, or the true anomaly from what is tech-
nically called the mean anomaly, that is, the mean angular motion which would have
been performed had the motion in angle been uniform instead of the motion in area.
It happens fortunately, that this is the simplest of all problems of the transcendental
kind, and can be resolved, in the most difficult case, by the rule of “false position,”
or trial and error, in a very few minutes. Nay, it may even be resolved instantly on
inspection by a simple and easily constructed piece of mechanism, of which the reader
may see a description in the Cambridge Philosophical Transactions, vol. iv. p. 425, by
the author of this work.
GEOCENTRIC PLACE OF A PLANET. 267
process, however circuitous it may appear, when once well understood may
be gone through numerically by the aid of the usual logarithmic and tri-
gonometrical tables, in little more time than it will have taken the reader
to peruse its description.
(501.) The geocentric differs from the heliocentric place of a planet by
reason of that parallactic change of apparent situation which arises from
the earth’s motion in its orbit. Were the planet's distances as vast as
those of the stars, the earth's orbitual motion would be insensible when
viewed from them, and they would always appear to us to hold the same
relative situations among the fixed stars, as if viewed from the Sun, i. e.
they would then be seen in their heliocentric places. The difference, then,
between the heliocentric and geocentric places of a planet is, in fact, the
same thing with its parallax, arising from the earth's removal from the
centre of the system and its annual motion. It follows from this, that
the first step towards a knowledge of its amount, and the consequent
determination of the apparent place of each planet, as referred from the
earth to the sphere of the fixed stars, must be to ascertain the proportion
of its linear distances from the earth and from the sun, as compared with
the earth's distance from the sun, and the angular positions of all three
with respect to each other.
(502.) Suppose, therefore, S to represent the sun, E the earth, and P
the planet; S T the line of equinoxes, T E the earth's orbit, and P p
a perpendicular let fall from the planet on the ecliptic. Then will the
angle S P E (according to the general notion of parallax conveyed in art.
69) represent the parallax of the planet arising from the change of station
Fig. 72.
|
º
amº
f ſº
iſſiºn
from S to E; EP will be the apparent direction of the planet seen from
E; and if S Q be drawn parallel to Ep, the angle T S Q will be the geo-
centric longitude of the planet, while T S E represents the heliocentric
longitude of the earth, T S p that of the planet. The former of these,
TS E, is given by the solar table; the latter, T S p, is found by the pro-
cess above described (art. 500). Moreover, S P is the radius vector of
the planet's orbit, and SE that of the earth's, both of which are determined









268 OUTLINES OF ASTRONOMY.
from the known dimensions of their respective ellipses, and the places of
the bodies in them at the assigned time. Lastly, the angle PS p is the
planet's heliocentric latitude.
(503.) Our object, then, is, from all these data, to determine the angle
ºr S Q, and PE p, which is the geocentric latitude. The process, then,
will stand as follows:—1st, In the triangle S P p, right-angled at p, given
SP, and the angle PS p (the planet's radius vector and heliocentric lati-
tude), find S p and P p ; 2dly, In the triangle S E p, given S p (just
found), S E (the earth's radius vector), and the angle E S p (the difference
of heliocentric longitudes of the earth and planet), find the angle S p E,
and the side Ep. The former being equal to the alternate angle p S Q,
is the parallactic removal of the planet in longitude, which, added to Y S p,
gives its geocentric longitude. The latter, E. p (which is called the curtate
distance of the planet from the earth), gives at once the geocentric lati-
tude, by means of the right-angled triangle PE p, of which Ep and P p
are known sides, and the angle PE p is the geocentric latitude sought.
(504.) The calculations required for these purposes are nothing but
the most ordinary processes of plane trigonometry; and, though some-
what tedious, are neither intricate nor difficult. When executed, how-
ever, they afford us the means of comparing the places of the planets
actually observed with the elliptic theory, with the utmost exactness, and
thus putting it to the severest trial; and it is upon the testimony of such
computations, so brought into comparison with observed facts, that we
declare that theory to be a true representation of nature.
(505.) The planets Mercury, Venus, Mars, Jupiter, and Saturn, have
been known from the earliest ages in which astronomy has been culti-
vated. Uranus was discovered by Sir W. Herschel in 1781, March 13th,
in the course of a review of the heavens, in which every star visible in a
telescope of a certain power was brought under close examination, when
the new planet was immediately detected by its disc, under a high magni-
fying power. It has since been ascertained to have been observed on
many previous occasions, with telescopes of insufficient power to show its
disc, and even entered in catalogues as a star; and some of the observa-
tions which have been so recorded have been used to improve and extend
our knowledge of its orbit. The discovery of the ultra-zodiacal planets
dates from the first day of 1801, when Ceres was discovered by Piazzi, at
Palermo; a discovery speedily followed by those of Juno by professor
Harding, of Göttingen, in 1804; and of Pallas and Vesta, by Dr. Olbers,
of Bremen, in 1802 and 1807 respectively. It is extremely remarkable
that this important addition to our system had been in some sort surmised
as a thing not unlikely, on the ground that the interval between the orbit
ORDER AND DISCOVERY OF THE PLANETS. 269
of Mercury and the other planetary orbits, go on doubling as we recede
from the sun, or nearly so. Thus, the interval between the orbits of the
Earth and Mercury is nearly twice that between those of Venus and Mer-
cury; that between the orbits of Mars and Mercury nearly twice that
between the Earth and Mercury: and so on. The interval between the
orbits of Jupiter and Mercury, however, is much too great, and would
form an exception to this law, which is, however, again resumed in the
case of the three planets next in order of remoteness, Jupiter, Saturn, and
Uranus. It was therefore thrown out, by the late professor Bode, of Ber-
lin,' as a possible surmise, that a planet not then yet discovered might
exist between Mars and Jupiter: and it may easily be imagined what
was the astonishment of astronomers on finding not only one, but four
planets, differing greatly in all the other elements of their orbits, but
agreeing very nearly, both inter se, and with the above stated empirical
law, in respect of their mean distances from the sun. No account, à pri-
ori or from theory, was to be given of this singular progression, which is
not, like Kepler’s laws, strictly exact in numerical verification; but the
circumstances we have just mentioned tended to create a strong belief that
it was something beyond a mere accidental coincidence, and bore reference
to the essential structure of the planetary system. It was even conjec-
tured that the ultra-zodiacal planets are fragments of some greater planet
which formerly circulated in that interval, but which has been blown to
atoms by an explosion; an idea countenanced by the exceeding minute-
ness of these bodies which present discs; and it was argued that in that
case innumerable more such fragments must exist and might come to be
hereafter discovered. Whatever may be thought of such a speculation as
a physical hypothesis, this conclusion has been verified to a considerable
extent as a matter of fact by subsequent discovery, the result of a careful
and minute examination and mapping down of the smaller stars in and
near the zodiac, undertaken with that express object. Zodiacal charts of
this kind, the product of the zeal and industry of many astronomers, have
been constructed, in which every star down to the ninth or tenth magni-
tude is inserted, and these stars being compared with the actual stars of
the heavens, the intrusion of any stranger within their limits cannot fail
to be noticed when the comparison is systematically conducted. The dis-
covery of Astraea, and that of Hebe by Professor Hencke, date respec-
tively from December 8th, 1845, and July 1st, 1847; those of Iris and
* The empirical law itself, as we have above stated it, is ascribed by Voron, not to
Bode (who would appear, however, at all events, to have first drawn attention to this
interpretation of its interruption), but to Professor Titius of Wittemberg. (Voiron,
Supplemen' to Bailly.)
270 OUTLINES OF ASTRONOMY.
Flora, by Mr. Hind, from August 13th and October 18th, 1847; of Me
tis, by Mr. Graham, from April 25, 1848; and of Hygeia, by M. De
Gasparis, April 12th, 1849.
(506.) The discovery of Neptune marks in a signal manner the matu-
rity of astronomical science. The proof, or at least the urgent presump-
tion of the existence of such a planet, as a means of accounting (by its
attraction) for certain small irregularities observed in the motions of
Uranus, was afforded almost simultaneously by the independent researches
of two geometers, Messrs. Adams of Cambridge and Leverrier of Paris,
who were enabled, from theory alone, to calculate whereabouts it ought
to appear in the heavens, if visible, the places thus independently calcu-
lated agreeing surprisingly. Within a single degree of the place assigned
by M. Leverrier's calculations, and by him communicated to Dr. Galle of
the Royal Observatory at Berlin, it was actually found by that astronomer
on the very first night after the receipt of that communication, on turning
a telescope on the spot, and comparing the stars in its immediate neigh-
bourhood with those previously laid down in one of the zodiacal charts
already alluded to." This remarkable verification of an indication so
extraordinary took place on the 23d of September, 1846.”
(507.) The mean distance of Neptune from the sun, however, so far
from falling in with the supposed law of planetary distances above men-
tioned, offers a decided case of discordance. The interval between its
orbit and that of Mercury, instead of being nearly double the interval
between those of Uranus and Mercury, does not, in fact, exceed the latter
interval by much more than half its amount. This remarkable exception
may serve to make us cautious in the too ready admission of empirical
laws of this nature to the rank of fundamental truths, though, as in the
present instance, they may prove useful auxiliaries, and serve as stepping
stones, affording a temporary footing in the path to great discoveries. The
force of this remark will be more apparent when we come to explain more
* Constructed by Dr. Bremiker, of Berlin. On reading the history of this noble
discovery, we are ready to exclaim with Schiller—
“Mit dem Genius steht die Natur in evigem Bunde,
Was der Eine verspricht liestet die Andre gewiss.”
* Professor Challis, of the Cambridge Observatory, directing the Northumberland
telescope of that Institution to the place assigned by Mr. Adams's calculations and its
vicinity, on the 4th and 12th of August 1846, saw the planet on both those days, and
noted its place (among those of other stars) for re-observation. He, however, post-
poned the comparison of the places observed, and, not possessing Dr. Bremiker's
chart (which would have at once indicated the presence of an unmapped star,)
remained in ignorance of the planet’s existence as a visible object till its announce.
ment as such by Dr. Galle.
PHYSICAL DESCRIPTION OF THE PLANETS. 271
$my
particularly the nature of the theoretical views which led to the discovery
of Neptune itself.
(508.) We shall devote the rest of this chapter to an account of the
physical peculiarities and probable condition of the several planets, so far
as the former are known by observation, or the latter rest on probable
grounds of conjecture. In this, three features principally strike us as
necessarily productive of extraordinary diversity in the provisions by
which, if they be, like our earth, inhabited, animal life must be supported.
These are, first, the difference in their respective supplies of light and
heat from the sun; secondly, the difference in the intensities of the
gravitating forces which must subsist at their surfaces, or the different
ratios which, on their several globes, the inertiae of bodies must bear to
their weights; and, thirdly, the difference in the nature of the materials
of which, from what we know of their mean density, we have every
reason to believe they consist. The intensity of solar radiation is nearly
seven times greater on Mercury than on the Earth, and on Uranus 330
times less; the proportion between the two extremes being that of upwards
of 2000 to 1. Let any one figure to himself the condition of our globe,
were the sun to be septupled, to say nothing of the greater ratio ! or were
it diminished to a seventh, or to a 300th of its actual power I Again,
the intensity of gravity, or its efficacy in counteracting muscular power
and repressing animal activity, on Jupiter, is nearly two and a half times
that on the earth, on Mars not more than one-half, on the Moon one-
sixth, and on the smaller planets probably not more than one-twentieth ;
giving a scale of which the extremes are in the proportion of sixty to one.
Lastly, the density of Saturn hardly exceeds one-eighth of the mean
density of the Earth, so that it must consist of materials not much heavier
than cork. Now, under the various combinations of elements so important
to life as these, what immense diversity must we not admit in the condi-
tions of that great problem, the maintenance of animal and intellectual
existence and happiness, which seems, so far as we can judge by what we
see around us in our own planet, and by the way in which every corner
of it is crowded with living beings, to form an unceasing and worthy
object for the exercise of the Benevolence and Wisdom which preside
over all !
(509.) Quitting, however, the region of mere speculation, we will now
show what information the telescope affords us of the actual condition of
the several planets within its reach. Of Mercury we can see little more
than that it is round, and exhibits phases. It is too small, and too much
lost in the constant neighbourhood of the Sun, to allow us to make out
272 - OUTLINES OF ASTRONOMY.
more of its nature. The real diameter of Mercury is about 3200 miles:
its apparent diameter varies from 5" to 12". Nor does Venus offer any
remarkable peculiarities: although its real diameter is 7800 miles, and
although it occasionally attains the considerable apparent diameter of 61”,
which is larger than that of any other planet, it is yet the most difficult
of them all to define with telescopes. The intense lustre of its illumin-
ated part dazzles the sight, and exaggerates every imperfection of the tele-
scope; yet we see clearly that its surface is not mottled over with permanent
spots like the Moon; we notice in it neither mountains nor shadows, but
a uniform brightness, in which sometimes we may indeed fancy, or per-
haps more than fancy, brighter or obscurer portions, but can seldom or
never rest fully satisfied of the fact. It is from some observations of this
kind that both Venus and Mercury have been concluded to revolve on
their axes in about the same time as the Earth, though in the case of
Venus, Bianchini and other more recent observers have contended for a
period of twenty-four times that length. The most natural conclusion,
from the very rare appearance and want of permanence in the spots, is,
that we do not see, as in the Moon, the real surface of these planets, but
only their atmospheres, much loaded with clouds, and which may serve
to mitigate the otherwise intense glare of their sunshine.
(510.) The case is very different with Mars. In this planet we fre-
quently discern, with perfect distinctness, the outlines of what may be
continents and seas. (See Plate III, fig.1., which represents Mars in its
gibbous state, as seen on the 16th of August, 1830, in the 20-feet
reflector at Slough.) Of these, the former are distinguished by that
ruddy colour which characterizes the light of this planet (which always
appears red and fiery), and indicates, no doubt, an ochrey tinge in the
general soil, like what the red sandstone districts on the Earth may pos-
sibly offer to the inhabitants of Mars, only more decided. Contrasted
with this (by a general law in optics), the seas, as we may call them,
appear greenish." These spots, however, are not always to be seen
equally distinct, but, when seen, they offer the appearance of forms con-
siderably definite and highly characteristic,” brought successively into
view by the rotation of the planet, from the assiduous observation of
* I have noticed the phaenomena described in the text on many occasions, but never
more distinct than on the occasion when the drawing was made from which the figure
in Plate III. is engraved. — Author.
2 The reader will find many of these forms represented in Schumacher's Astrono-
mische Nachrichten, No. 191, 434, and in the chart in No. 349, by Messrs. Beer and
VI: idler.
JUPITER. 273.
which it has even been found practicable to construct a rude chart of the
surface of the planet. The variety in the spots may arise from the planet
not being destitute of atmosphere and clouds; and what adds greatly io
the probability of this is the appearance of brilliant white spots at its
poles, – one of which appears in our figure, — which have been conjec-
tured, with some probability, to be snow; as they disappear when they
have been long exposed to the sun, and are greatest when just emerging
ſrom the long night of their polar winter, the snow line then extending
to about six degrees (reckoned on a meridian of the planet) from the pole.
By watching the spots during a whole night, and on successive nights, it
is found that Mars has a rotation on an axis inclined about 30° 18' to the
ecliptic, and in a period of 24° 37" 23” in the same direction as the
Earth's, or from west to east. The greatest and least apparent diameters
of Mars are 4" and 18", and its real diameter about 4100 miles.
(511.) We now come to a much more magnificent planet, Jupiter, the
largest of them all, being in diameter no less than 87,000 miles, and in
bulk exceeding that of the Earth nearly 1300 times. It is, moreover,
dignified by the attendance of four moons, satellites, or Secondary planets,
as they are called, which constantly accompany and revolve about it, as
the Moon does round the Earth, and in the same direction, forming with
their principal, or primary, a beautiful miniature system, entirely analo-
gous to that greater one of which their central body is itself a member,
obeying the same laws, and exemplifying, in the most striking and in-
structive manner, the prevalence of the gravitating power as the ruling
principle of their motions: of these, however, we shall speak more at
large in the next chapter.
(512.) The disc of Jupiter is always observed to be crossed in one
certain direction by dark bands or belts presenting the appearance, in
Plate III. fig. 2., which represents this planet as seen on the 23d of Sep-
tember, 1832, in the 20-feet reflector at Slough. These belts are, how-
ever, by no means alike at all times; they vary in breadth and in situa-
tion on the disc (though never in their general direction). They have
even been seen broken up and distributed over the whole face of the
planet; but this phaenomenon is extremely rare. Branches running out
from them, and subdivisions, as represented in the figure, as well as evi-
dent dark spots, are by no means uncommon; and from these, attentively
watched, it is concluded that this planet revolves in the surprisingly short
period of 9° 55 ° 50' (sid. time), on an axis perpendicular to the direction
* Beer and Mädler, Astr. Nachr. 349.
18
274 OTTLINES OF ASTRONOMY.
of the belts. Now, it is very remarkable, and forms a most satisfactory
comment on the reasoning by which the spheroidal figure of the Earth
has been deduced from its diurnal rotation, that the outline of Jupiter's
disc is evidently not circular, but elliptic, being considerably flattened in
the direction of its axis of rotation. This appearance is no optical illu-
sion, but is authenticated by micrometrical measures, which assign 107 to
100 for the proportion of the equatorial and polar diameters. And to
confirm in the strongest manner, the truth of those principles on which
our former conclusions have been founded, and fully to authorize their
extension to this remote system, it appears, on calculation, that this is
really the degree of oblateness which corresponds, on those principles, to
the dimensions of Jupiter, and to the time of his rotation.
(513.) The parallelism of the belts to the equator of Jupiter, their
occasional variations, and the appearances of spots seen upon them, render
it extremely probable that they subsist in the atmosphere of the planet,
forming tracts of comparatively clear sky, determined by currents analo-
gous to our trade-winds, but of a much more steady and decided character,
as might indeed be expected from the immense velocity of its rotation.
That it is the comparatively darker body of the planet which appears in
the belts is evident from this, -that they do not come up in all their
strength to the edge of the disc, but fade away gradually before they
reach it. (See Plate III. fig. 2.) The apparent diameter of Jupiter
varies from 30" to 46”.
(514.) A still more wonderful, and, as it may be termed, elaborately
artificial mechanism, is displayed in Saturn, the next in order of remote-
ness to Jupiter, to which it is not much inferior in magnitude, being about
79,000 miles in diameter, nearly 1000 times exceeding the earth in bulk,
and subtending an apparent angular diameter at the earth, of about 18"
at its mean distance This stupendous globe, besides being attended by
no less than ºf tº or moons, is surrounded by two broad, flat,
extremely thin rings, concentric with the planet and with each other;
both lying in one plane, and separated by a very narrow interval from
each other throughout their whole circumference, as they are from the
planet by a much wider. The dimensions of this extraordinary appendage
are as follows':— w
1 These dimensions are calculated from Prof. Struve's micrometric measures, Mem.
Art. Soc. iii. 301, with the exception of the thickness of the ring, which is concluded
from its total disappearance in 1833, in a telescope which would certainly have shown,
as a visible object, a line of light one-twentieth of a second in breadth. The interval
of the rings here stated is possibly somewhat too small.
SATURN. 275
*g Miles.
Exterior diameter of exterior ring . . . . . . . . . . . . . . . . . . . 40’095 = 176,418
Interior ditto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35:289 = 155,272
Exterior diameter of interior ring . . . . . . . . . . . . . . . . . . . . 34°475 = 151,690
Interior ditto . . . . . $ 9 & 0 e º ºs e e s tº e e s e º ºs e g º e s tº e º 'º e º 'º º ºs º º 26-668 – 117,339
Equatorial diameter of the body . . . . . . . . . . . . . . . . . . . . . 17.991 = 79,160
Interval between the planet and interior ring . . . . . . . . . 4.339 = 19,090
Interval of the rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O'408 = 1,791
Thickness of the rings not exceeding. . . . . . . . . . . . . . . . = 250
The figure (fig. 3, Plate III.) represents Saturn surrounded by its rings,
and having its body striped with dark belts, somewhat similar, but broader
and less strongly marked than those of Jupiter, and owing, doubtless, to
a similar cause." That the ring is a solid opake substance is shown by its
throwing its shadow on the body of the planet, on the side nearest the
sun, and on the other side receiving that of the body, as shown in the
figure. From the parallelism of the belts with the plane of the ring it
may be conjectured that the axis of rotation of the planet is perpendicular
to that plane; and this conjecture is confirmed by the occasional appearance
of extensive dusky spots on its surface, which when watched, like the
spots on Mars or Jupiter, indicate a rotation in 10" 29* 17° about an axis
so situated.
(515.) The axis of rotation, like that of the earth, preserves its paral-
lelism to itself during the motion of the planet in its orbit; and the same
is also the case with the ring, whose plane is constantly inclined at the
same, or very nearly the same, angle to that of the orbit, and, therefore,
to the ecliptic, viz. 28° 11'; and intersects the latter plane in a line,
which makes at present” an angle with the line of equinoxes of 167° 31'.
So that the nodes of the ring lie in 167° 31' and 347° 31' of longitude.
Whenever, then, the planet happens to be situated in one or other of these
longitudes, as at C, the plane of the ring passes through the sun, which
then illuminates only the edge of it. And if the earth at that moment
be in F, it will see the ring edgeways, the planet being in opposition, and
therefore most favourably situated (caeteris paribus) for observation.
Under these circumstances the ring, if seen at all, can only appear as a
very narrow straight line of light projecting on either side of the body as
a prolongation of its diameter. In fact, it is quite invisible in any but
telescopes of extraordinary power.” This remarkable phenomenon takes
* The equatorial bright belt is generally well seen. The subdivision of the dark one
by two narrow bright bands is seldom so distinct as represented in the plate.
* According to Bessel, the longitude of the node of the ring increases 46”-452 per
annum. In 1800 it was 166° 53' 8"9.
*Its disappearance was complete when observed with a reflector eighteen inches in
aperture and twenty feet in focal length, on the 29th of April, 1833, by the author.
276 g OUTLINES OF ASTRONOMY.
Fig. 73.
place at intervals of fifteen years nearly (being a semi-period of Saturn in
its orbit). One disappearance at least must take place whenever Saturn
passes either node of its orbit; but three must frequently happen, and
two are possible. To show this, suppose S to be the sun, A B C D part
of Saturn's orbit situated so as to include the node of the ring (at C);
E F G H the earth's orbit: S C the line of the node; EB, G D parallel
to SC touching the earth's orbit in E G ; and let the direction of motion
of both bodies be that indicated by the arrow. Then since the ring pre-
serves its parallelism, its plane can nowhere intersect the earth's orbit, and
therefore no disappearance can take place, unless the planet be between B
and D: and, on the other hand, a disappearance is possible (if the earth
be rightly situated) during the whole time of the description of the arc
B.D. Now, since S B or SD, the distance of Saturn from the Sun, is to
SE or S G, that of the Earth, as 9:54 to 1, the angle CSD or CS B =
6° 1', and the whole angle BSD = 12° 2', which is described by Saturn
(on an average) in 359:46 days, wanting only 5.8 days of a complete
year. The Earth then describes very nearly an entire revolution within
the limits of time when a disappearance is possible; and since, in either
half of its orbit E F G or G H F, it may equally encounter the plane
of the ring, one such encounter at least is unavoidable within the time
specified.
(516.) Let G a be the are of the Earth's orbit described from G in
5.8 days. Then if, at the moment of Saturn's arrival at B, the Earth be
at a, it will encounter the plane of the ring advancing parallel to itself
and to B E to meet it, somewhere in the quadrant H E, as at M, after
which it will be behind that plane (with reference to the direction of
Saturn's motion) through all the are M EFG up to G, where it will
*

DISAPPEARANCE OF SATURN's RING. 277
again overtake it at the very moment of the planet quitting the arc B. D.
In this state of things there will be two disappearances. If, when Saturn
is at B, the Earth be anywhere in the arc a HE, it is equally evident
that it will meet and pass through the advancing plane of the ring some-
where in the quadrant H E, that it will again overtake and pass through
it somewhere in the semicircle E FG, and again meet it in, some point
of the quadrant G. H., so that three disappearances will take place. So,
also, if the Earth be at E when Saturn is at B, the motion of the Earth
being at that instant directly towards B, the plane of the ring will for a
short time leave it behind; but the ground so lost being rapidly regained
as the earth’s motion becomes oblique to the line of junction, it will soon
overtake and pass through the plane in the early part of the quadrant
E F, and passing on through G before Saturn arrives at D, will meet the
plane again in the quadrant G. H. The same will continue up to a certain
point b, at which, if the Earth be initially situated, there will be but two
disappearances—the plane of the ring there overtaking the Earth for an
instant, and being immediately again left behind by it, to be again en-
countered by it in G. H. Finally, if the initial place of the Earth (when
Saturn is at B) be in the are b F a, there will be but one passage through
the plane of the ring, viz., in the semicircle G H E, the Earth being in
advance of that plane throughout the whole of b G.
(517.) The appearances will moreover be varied according as the Earth
passes from the enlightened to the unenlightened side of•the ring, or vice
versá. If C be the ascending node of the ring, and if the under side of
the paper be supposed south and the upper north of the ecliptic, then,
when the Earth meets the plane of the ring in the quadrant H E, it
passes from the bright to the dark side : where it overtakes it in the
quadrant E F, the contrary. Vice versä, when it overtakes it in FG,
the transition is from the bright to the dark side, and the contrary where
it meets it in G. H. On the other hand, when the Earth is overtaken by
the ring-plane in the interval E b, the change is from the bright to the
dark side. When the dark side is exposed to sight, the aspect of the
planet is very singular. It appears as a bright round disc, with its belts,
&c., but crossed equatorially by a narrow and perfect black line. This
can never of course happen when the planet is more than 6° 1' from the
node of the ring. Generally, the northern side is enlightened and visible
when the heliocentric longitude of Saturn is between 173° 32' and
341° 30', and the southern when between 353° 32' and 161° 30'. The
greatest opening of the ring occurs when the planet is situated at 90° dis-
tance from the node of the ring, or in longitudes 77° 31' and 257° 31',
278 OuTLINES OF ASTRONOMY.
and at these points the longer diameter of its apparent ellipse is almost
exactly double the shorter.
(518.) It will naturally be asked how so stupendous an arch, if com-
posed of solid and ponderous materials, can be sustained without collaps-
ing and falling in upon the planet! The answer to this is to be found in
a swift rotation of the ring in its own plane, which observation has de-
tected, owing to some portion of the ring being a little less bright than
others, and assigned its period at 10" 32° 15', which, from what we know
of its dimensions, and of the force of gravity in the Saturnian system, is
very nearly the periodic time of a satellite revolving at the same distance
as the middle of its breadth. It is the centrifugal force, then, arising
from this rotation, which sustains it; and, although no observation nice
enough to exhibit a difference of periods between the outer and inner
rings have hitherto been made, it is more than probable that such a diffe-
rence does subsist as to place each independently of the other in a similar
state of equilibrium.
(519.) Although the rings are, as we have said, very nearly concentric
with the body of Saturn, yet micrometrical measurements of extreme
delicacy have demonstrated that the coincident is not mathematically ex-
act, but that the centre of gravity of the rings oscillates round that of the
body describing a very minute orbit, probably under laws of much com-
plexity. Trifling as this remark may appear, it is of the utmost import-
ance to the stability of the system of the rings. Supposing them mathe-
matically perfect in their circular form, and exactly concentric with the
planet, it is demonstrable that they would form a system in a state of un-
stable equilibrium, which the slightest external power would subvert—
not by causing a rupture in the substance of the rings—but by precipi-
tating them, unbroken, on the surface of the planet. For the attraction
of such a ring or rings on a point or sphere excentrically within them, is
not the same in all directions, but tends to draw the point or sphere
towards the nearest part of the ring, or away from the centre. Hence,
supposing the body to become, from any cause, ever so little excentric to
the ring, the tendency of their mutual gravity is not to correct but to
increase this excentricity, and to bring the nearest parts of them together.
Now, external powers, capable of producing such excentricity, exist in the
attractions of the satellites, as will be shown in Chap. XII. ; and in order
that the system may be stable, and possess within itself a power of resist-
ing the first inroads of such a tendency, while yet nascent and feeble, and
opposing them by an opposite or maintaining power, it has been shown
that it is sufficient to admit the rings to be loaded in some part of their
circumference, either by some minute inequality of thickness, or by some
EQUILIBRIUM OF SATURN’s RINGS. 279
portions being denser than others. Such a load would give to the whole
ring to which it was attached somewhat of the character of a heavy and
sluggish satellite maintaining itself in an orbit with a certain energy suffi-
cient to overcome minute causes of disturbance, and establish an average
bearing on its centre. But even without supposing the existence of any
such load, of which, after all, we have no proof-and granting, in its
full extent, the general instability of the equilibrium, we think we per-
ceive, in the rapid periodicity of all the causes of disturbance, a sufficient
guarantee of its preservation. However homely be the illustration, we
can conceive nothing more apt, in every way, to give a general conception
of this maintenance of equilibrium under a constant tendency to subver-
sion, than the mode in which a practised hand will sustain a long pole in
a perpendicular position resting on the finger by a continual and almost
imperceptible variation of the point of support. Be that, however, as it
may, the observed oscillation of the centres of the rings about that of the
planet is in itself the evidence of a perpetual contest between conserva-
tive and destructive powers — both extremely feeble, but so antagonizing
one another as to prevent the latter from ever acquiring an uncontrollable
ascendancy, and rushing to a catastrophe.
(520.) This is also the place to observe, that as the smallest difference
of velocity between the body and the rings must infallibly precipitate the
latter on the former, never more to separate, (for they would, once in
contact, have attained a position of stable equilibrium, and be held toge-
ther ever after by an immense force;) it follows, either that their motions
in their common orbit round the sun must have been adjusted to each
other by an external power, with the minutest precision, or that the rings
must have been formed about the planet while subject to their common
orbitual motion, and under the full and free influence of all the acting
forces.
(521.) Several astronomers have suspected, and even consider them-
selves to have certainly observed, the rings of Saturn to be occasionally,
at least, streaked with more or less numerous dark lines parallel to the
decided black interval which separates the two rings, which latter being
permanent, and seen equally and in the same part of the breadth on both
sides of the ring, cannot be doubted to be a real separation."
(522.) [The exterior ring of Saturn is described by many observers as
rather less luminous than the interior, and the inner portion of this latter
* The passage of Saturn across any considerable star would afford an admirable
opportunity of testing the reality of such fissures, as it would flash in succession
through them. The opportunity of watching for such occultations – when Saturn
traverses the Milky- Way, for instance—should not be neglected.
280 OUTLINES OF ASTRONOMY.
than its outer. On the night of Nov. 11, 1850, however, Mr. G. B
Bond, of the Harvard Observatory (Cambridge, U. S.,) using the great
Fraunhofer equatorial of that institution, became aware of a line of
demarcation between these two portions so definite, and an extension
inwards of the dusky border to such an extent (one fifth, by measure-
ment, of the joint breadth of the two old rings,) as to justify him in
considering it as a newly-discovered ring. On the nights of the 25th
and 29th of the same month, and without knowledge of Mr. Bond's
observations, Mr. Dawes, at his observatory at Wateringbury, by the aid
of an exquisite achromatic by Merz, of 6% inches aperture, observed the
very same fact, and even more distinctly, so as to be sure of a decidedly
darker interval between the old and new rings, and even to subdivide the
latter into two of unequal degrees of obscurity, separated by a line more
obscure than either.] º
(523.) Of Uranus we see nothing but a small round uniformly illumi-
nated disc, without rings, belts, or discernible spots. Its apparent dia-
meter is about 4", from which it never varies much, owing to the small-
ness of our orbit in comparison of its own. Its real diameter is about
35,000 miles, and its bulk 82 times that of the earth. It is attended by
satellites—four at least, probably five or six—whose orbits (as will be seen
in the next chapter) offer remarkable peculiarities.
(524.) The discovery of Neptune is so recent, and its situation in the
ecliptic at present so little favourable for seeing it with perfect distinctness,
that nothing very positive can be stated as to its physical appearance. To
two observers it has afforded strong suspicion of being surrounded with a
ring very highly inclined. And from the observations of Mr. Lassell,
1M. Otto Struve, and Mr. Bond, it appears to be attended certainly by one,
and very probably by two satellites—though the existence of the second
can hardly yet be considered as quite demonstrated.
(525.) If the immense distance of Neptune precludes all hope of
coming at much knowledge of its physical state, the minuteness of the
ultra-zodiacal planets is no less a bar into any inquiry into theirs. One
of them, Pallas, has been said to have somewhat of a nebulous or hazy
appearance, indicative of an extensive and vaporous atmosphere, little
repressed and condensed by the inadequate gravity of so small a mass.
It is probable, however, that the appearance in question has originated
in some imperfection in the telescope employed or other temporary causes
of illusion. In Westa and Pallas only have sensible discs been hitherto
observed, and those only with very high magnifying powers. Westa was
once seen by Schroeter with the naked eye. No doubt the most remark-
able of their peculiarities must lie in this condition of their state. A
GENERAL VIEW OF THE SOLAR SYSTEM. 281
man placed on one of them would spring with ease 60 feet high, and
sustain no greater shock in his descent than he does on the earth from
leaping a yard. On such planets giants might exist; and those enormous
animals, which on earth require the buoyant power of water to counteract
their weight, might there be denizens of the land. But of such specula-
tions there is no end.
(526.) We shall close this chapter with an illustration calculated to
convey to the minds of our readers a general impression of the relative
magnitudes and distances of the parts of our system. Choose any well
levelled field or bowling-green. On it place a globe, two feet in diameter;
this will represent the Sun; Mercury will be represented by a grain of
mustard seed, on the circumference of a circle 164 feet in diameter for its
orbit; Venus a pea, on a circle 284 feet in diameter; the Earth also a
pea, on a circle of 430 feet; Mars a rather large pin's head, on a circle
of 654 feet; Juno, Ceres, Westa, and Pallas, grains of sand, in orbits of
from 1000 to 1200 feet; Jupiter a moderate-sized orange, in a circle nearly
half a mile across, Saturn a small orange, on a circle of four-fifths of a
mile; Uranus a full-sized cherry, or small plum, upon the circumference
of a circle more than a mile and a half, and Neptune a good-sized plum
on a circle about two miles and a half in diameter. As to getting correct
notions on this subject by drawing circles on paper, or, still worse, from
those very childish toys called orreries, it is out of the question. To imi-
tate the motions of the planets, in the above-mentioned orbits, Mercury
must describe its own diameter in 41 seconds; Venus in 4" 14*; the
Earth, in 7 minutes; Mars, in 4° 48'; Jupiter, 2* 56"; Saturn, in 3"
13*; Uranus, in 2° 16"; and Neptune in 3° 30°.
282 OUTLINES OF ASTRONOMY.
CHAPTER X.
O F T H E S A. T E L L IT E S.
OF THE MOON, AS A SATELLITE OF THE EARTH. — GENERAL PROXIMITY
OF SATELLITES TO THEIR PRIMARIES, AND CONSEQUENT SUBORDINA-
TION OF THEIR, MOTIONS. — MASSES OF THE PRIMARIES CONCLUDED
FROM THE PERIODS OF THEIR, SATELLITES. — MAINTENANCE OF KEP-
LER's LAWS IN THE SECONDARY SYSTEMS. – OF JUPITER’s SATEL-
IITEs. – THEIR ECLIPSEs, ETC. — VELOCITY OF LIGHT DISCOVERED
BY THEIR, MEANS. — SATELLITES OF SATURN — OF URANU'S - OF
NEPTUNE.
(527.) IN the annual circuit of the earth about the sun, it is constantly
attended by its satellite, the moon, which revolves round it, or rather
both round their common centre of gravity; while this centre, strictly
speaking, and not either of the two bodies thus connected, moves in an
elliptic orbit, undisturbed by their mutual action, just as the centre of
gravity of a large and small stone tied together and flung into the air
describes a parabola as if it were a real material substance under the
earth's attraction, while the stones circulate round it or round each other,
as we choose to conceive the matter.
(528.) If we trace, therefore, the real curve actually described by
either the moon’s or the earth's centres, in virtue of this compound mo-
tion, it will appear to be, not an exact ellipse, but an undulated curve,
like that represented in the figure to article 324, only that the number
of undulations in a whole revolution is but 13, and their actual deviation
from the general ellipse, which serves them as a central line, is compara-
tively very much smaller—so much so, indeed, that every part of the
curve described by either the earth or moon is concave towards the sun.
The excursions of the earth on either side of the ellipse, indeed, are so
very small as to be hardly appreciable. In fact, the centre of gravity of
the earth and moon lies always within the surface of the earth, so that
the monthly orbit described by the earth's centre about the common
centre of gravity is comprehended within a space less than the size of the
earth itself. The effect is, nevertheless, sensible, in producing an appa-
OF THE MOON AS A SATELLITE. . 283
rent monthly displacement of the sun in longitude, of a parallactic kind,
which is called the menstrual equation; whose greatest amount is, how-
ever, less than the sun's horizontal parallax, or than 8-6".
(529.) The moon, as we have seen, is about 60 radii of the earth dis-
tant from the centre of the latter. Its proximity, therefore, to its centre
of attraction, thus estimated, is much greater than that of the planets to
the sun; of which Mercury, the nearest, is 84, and Uranus 2026 solar
radii from its centre. It is owing to this proximity that the moon
remains attached to the earth as a satellite. Were it much farther, the
feebleness of its gravity towards the earth would be inadequate to produce
that alternate acceleration and retardation in its motion about the sun,
which divests it of the character of an independent planet, and keeps its
movements subordinate to those of the earth. The one would outrun, or
be left behind the other, in their revolutions round the sun (by reason of
Kepler's third law,) according to the relative dimensions of their helio-
centric orbits, after which the whole influence of the earth would be
confined to producing some considerable periodical disturbance in the
moon's motion, as it passed or was passed by it in each synodical revolu-
tion. -
(530.) At the distance at which the moon really is from us, its gravity
towards the earth is actually less than towards the sun. That this is the
case, appears sufficiently from what we have already stated, that the
moon's real path, even when between the earth and sun, is concave
towards the latter. But it will appear still more clearly if, from the
known periodic times' in which the earth completes its annual and the
moon its monthly orbit, and from the dimensions of those orbits, we
calculate the amount of deflection, in either, from their tangents, in equal
very minute portions of time, as one second. These are the versed sines
of the arcs described in that time in the two orbits, and these are the
measures of the acting forces which produce those deflections. If we
execute the numerical calculation in the case before us, we shall find
2.233; 1 for the proportion in which the intensity of the force which
retains the earth in its orbit round the sun actually exceeds that by which
the moon is retained in its orbit about the earth.
* R and r radii of two orbits (supposed circular,) P and p the periodic times; then
the arcs in question (A and a) are to each other as F to #; and since the versed sines
are as the squares of the arcs directly and the radii inversely, these are to each other
8S } w; and in this ratio are the forces acting on the revolving bodies in either
CâSë.
284 OUTLINES OF ASTRONOMY.
(531.) Now the sun is 399 times more remote from the earth than the
moon is. And, as gravity increases as the squares of the distances de-
crease, it must follow that at equal distances, the intensity of solar would
exceed that of terrestrial gravity in the above proportion, augmented in
the further ratio of the square of 400 to 1; that is, in the proportion of
355000 to 1; and therefore, if we grant that the intensity of the gravi-
tating energy is commensurate with the mass or inertia of the attracting
body, we are compelled to admit the mass of the earth to be no more
than 3 sºooo of that of the sun.
(532.) The argument is, in fact, nothing more than a recapitulation of
what has been adduced in Chap. VIII. (art. 448.) But it is here re-
introduced, in order to show how the mass of a planet which is attended
by one or more satellites can be as it were weighed against the sun, pro-
vided we have learned from observation the dimensions of the orbits
described by the planet about the sun, and by the satellites about the
planet, and also the periods in which these orbits are respectively
described. It is by this method that the masses of Jupiter, Saturn,
Uranus, and Neptune have been ascertained. (See Synoptic Table.)
(583.) Jupiter, as already stated, is attended by four satellites, Saturn
by seven; Uranus, certainly by four, and perhaps by six; and Neptune
by two or more. These, with their respective primaries (as the central
planets are called,) form in each case miniature systems entirely analogous,
in the general laws by which their motions are governed, to the great
system in which the sun acts the part of the primary, and the planets of
its satellites. In each of these systems the laws of Kepler are obeyed,
in the sense, that is to say, in which they are obeyed in the planetary
system — approximately, and without prejudice to the effects of mutual
perturbation, of extraneous interference, if any, and of that small but
not imperceptible correction which arises from the elliptic form of the
central body. Their orbits are circles or ellipses of very moderate excen-
tricity, the primary occupying one focus. About this they describe areas
very nearly proportional to the times; and the squares of the periodical
times of all the satellites belonging to each planet are in proportion to
each other as the cubes of their distances. The tables at the end of the
volume exhibit a synoptic view of the distances and periods in these
several systems, so far as they are at present known; and to all of them
it will be observed that the same remark respecting their proximity to
their primaries holds good, as in the case of the moon, with a similar
reason for such close connection.
(534.) Of these systems, however, the only one which has been studied
SATELLITES OF JUPITER. 285
with attention to all its details, is that of Jupiter; partly on account of
the conspicuous brilliancy of its four attendants, which are large enough
to offer visible and measurable discs in telescopes of great power; but
more for the sake of their eclipses, which, as they happen very frequently,
and are easily observed, afford signals of considerable use for the determi-
nation of terrestrial longitudes (art. 266). This method, indeed, until
thrown into the back ground by the greater facility and exactness now
attainable by lunar observations (art. 267) was the best, or rather the
only one, which could be relied on for great distances and long intervals.
(535.) The satellites of Jupiter revolve from west to east (following the
analogy of the planets and moon,) in planes very nearly, although not ex-
actly, coincident with that of the equator of the planet, or parallel to its
belts. This latter plane is inclined 3° 5' 30" to the orbit of the planet,
and is therefore but little different from the plane of the ecliptic. Accord-
ingly, we see their orbits projected very nearly into straight lines, in which
they appear to oscillate to and fro, someti.' ... passing before Jupiter, and
casting shadows on his disc, (which are very visible in good telescopes,
like small round ink spots, the circular form of which is very evident,)
and sometimes disappearing behind the body, or being eclipsed in its
shadow at a distance from it. It is by these eclipses that we are furnished
with accurate data for the construction of tables of the satellites' motions,
as well as with signals for determining differences of longitude.
(536.) The eclipses of the satellites, in their general conception, are
perfectly analogous to those of the moon, but in their detail they differ in
several particulars. Owing to the much greater distance of Jupiter from
the sun, and its greater magnitude, the cone of its shadow or umbra
(art. 420) is greatly more elongated, and of far greater dimension, than
that of the earth. The satellites are, moreover, much less in proportion
to their primary, their orbits less inclined to its ecliptic, and (compara-
tively to the diameter of the planet) of smaller dimensions, than is the
case with the moon. Owing to these causes, the three interior satellites
of Jupiter pass through the shadow, and are totally eclipsed, every revo-
lution; and the fourth, though, from the greater inclination of its orbit, it
sometimes escapes eclipse, and may occasionally graze as it were the border
of the shadow, and suffer partial eclipse, yet does so comparatively seldom,
and, ordinarily speaking, its eclipses happen, like those of the rest, each
revolution.
(537.) These eclipses, moreover, are not seen, as is the case with those
of the moon, from the centre of their motion, but from a remote station,
and one whose situation with respect to the line of shadow is variable.
286 . OUTLINES OF ASTRONOMY.
This, of course, makes no difference in the times of the eclipses, but a
very great one in their visibility, and in their apparent situations with
respect to the planet at the moments of their entering and quitting the
shadow.
(538.) Suppose S to be the sun, E the earth in its orbit E F G. K. J .
Jupiter, and a b the orbit of one of its satellites. The cone of the shadow,
then, will have its vertex at X, a point far beyond the orbits of all the
satellites; and the penumbra, owing to the great distance of the sun, and
the consequent smallness of the angle (about 6' only) its disc subtends at
Jupiter, will hardly extend, within the limits of the satellites' orbits, to
any perceptible distance beyond the shadow, -for which reason it is not
represented in the figure. A satellite revolving from west to east (in the
direction of the arrows) will be eclipsed when it enters the shadow at a,
but not suddenly, because, like the moon, it has a considerable diameter
seen from the planet; so that the time elapsing from the first perceptible
loss of light to its total extinction will be that which it occupies in de-
scribing about Jupiter an angle equal to its apparent diameter as seen from
the centre of the planet, or rather somewhat more, by reason of the
Fig. 74.
penumbra; and the same remark applies to its emergence at b. Now,
owing to the difference of telescopes and of eyes, it is not possible to assign
the precise moment of incipient obscuration, or of total extinction at a, nor
that of the first glimpse of light falling on the satellite at b, or the complete
recovery of its light. The observation of an eclipse, then, in which only
the immersion, or only the emersion, is seen, is incomplete, and inadequate
to afford any precise information, theoretical or practical. But, if both
the immersion and emersion can be observed with the same telescope, and
by the same person, the interval of the times will give the duration, and
their mean the exact middle of the eclipse, when the satellite is in the
fine S J X, i. e. the true moment of its opposition to the sun. Such ob-
servations, and such only, are of use for determining the periods and other

ECLIPSES OF JUPITER’s SATELLITES. 287
particulars of the motions of the satellites, and for affording data of any
material use for the calculation of terrestrial longitudes. The intervals
of the eclipses, it will be observed, give the synodic periods of the satel-
lites’ revolutions; from which their sidereal periods must be concluded by
the method in art. 418.
(539.) It is evident, from a mere inspection of our figure, that the
eclipses take place to the west of the planet, when the earth is situated
to the west of the line S J, i.e. before the opposition of Jupiter; and to
the east, when in the other half of its orbit, or after the opposition.
When the earth approaches the opposition, the visual line becomes more
and more nearly coincident with the direction of the shadow, and the ap-
parent place where the eclipses happen will be continually nearer and
nearer to the body of the planet. When the earth comes to F, a point
determined by drawing b F to touch the body of the planet, the emersions
will cease to be visible, and will thenceforth, up to the time of the oppo-
sition, happen behind the disc of the planet. Similarly, from the oppo-
sition till the time when the earth arrives at I, a point determined by
drawing a I tangent to the eastern limb of Jupiter, the immersions will
be concealed from our view. When the earth arrives at G (or H) the
immersion (or emersion) will happen at the very edge of the visible dise,
and when between G and H (a very small space), the satellites will pass
uneclipsed behind the limb of the planet.
(540.) Both the satellites and their shadows are frequently observed to
transit or pass across the disc of the planet. When a satellite comes to
m, its shadow will be thrown on Jupiter, and will appear to move across
it as a black spot till the satellite comes to n. But the satellite itself will
not appear to enter on the disc till it comes up to the line drawn from E
to the eastern edge of the disc, and will not leave it till it attains a similar
line drawn to the western edge. It appears then that the shadow will
precede the satellite in its progress over the disc before the opposition of
Jupiter, and vice versâ. In these transits of the satellites, which, with
very powerful telescopes, may be observed with great precision, it fre-
quently happens that the satellite itself is discernible on the disc as a
bright spot if projected on a dark belt; but occasionally also as a dark
spot of smaller dimensions than the shadow. This curious fact (observed
by Schroeter and Harding) has led to a conclusion that certain of the
satellites have occasionally on their own bodies, or in their atmospheres,
obscure spots of great extent. We say of great extent; for the satellites
of Jupiter, small as they appear to us, are really bodies of considerable
size, as the following comparative table will show : '--
* Struve, Mem. Art. Soc. iii. 301.
288 OUTLINES OF ASTRONOMY.

Mean apparent Mean apparent
diameter as seen | diameter as seen | Diameter in miles. Mass.”
from the Earth. from Jupiter.
Jupiter........... 387-327 87.000 1.0000000
1st satellite ..... 1-017 33' 11tt 2508 0-0000173
2d —........ 0.911 17 35 20.68 0.0000232
3d —......... 1'488 18 0 3377 0-0000885
4th —......... I-273 8 46 2890 0-0000427
From which it follows, that the first satellite appears to a spectator on
Jupiter, as large as our moon to us; the second and third nearly equal to
each other, and of somewhat more than half the apparent diameter of the
first, and the fourth about one quarter of that diameter. So seen, they
will frequently, of course, eclipse one another, and cause eclipses of the
Sun (the latter visible, however, only over a very small portion of the
planet), and their motions and aspects with respect to each other must
offer a perpetual variety and singular and pleasing interest to the inhabi-
tants of their primary. &
(541.) Besides the eclipses and the transits of the satellites across the
disc, they may also disappear to us when not eclipsed, by passing behind
the body of the planet. Thus, when the earth is at E, the immersion of
the satellite will be seen at a, and its emersion at b, both to the west of
the planet, after which the satellite, still continuing its course in the di-
rection ab, will pass behind the body, and again emerge on the opposite
side, after an interval of occultation greater or less according to the dis-
tance of the satellite. This interval (on account of the great distance of
the earth compared with the radii of the orbits of the satellites) varies but
little in the case of each satellite, being nearly equal to the time which
the satellite requires to describe an arc of its orbit, equal to the angular
diameter of Jupiter as seen from its centre, which time, for the several
satellites, is as follows: viz., for the first, 2h 20m; £or the second, 2* 56"; for
the third, 3° 43"; and for the fourth, 4h 56"; the corresponding diameters
of the planet as seen from these respective satellites being, 19° 49'; 12°
25'; 7° 47'; and 4° 25'.” Before the opposition of Jupiter, these occul-
tations of the satellites happen after the eclipses: after the opposition
(when, for instance, the earth is in the situation K), the occultations take
place before the eclipses. It is to be observed that owing to the proximity
of the orbits of the first and second satellites to the planet, both the im-
mersion and emersion of either of them can never be obs3rved in any
* Laplace, Mec. Cel. liv. viii. § 27.
* These data are taken approximately from Mr. Woodhouse's paper in the swpplement
.n the Nautical Almanack for 1835. -
ECLIPSEs of JUPITER's SATELLITEs. 289
single eclipse, the immersion being concealed by the body, if the plane:
be past its opposition, the emersion if not yet arrived at it. So also of
the occultation. The commencement of the occultation, or the passage
of the satellite behind the disc, takes place while obscured by the shadow,
before opposition, and its re-emergence after. All these particulars will
be easily apparent on mere inspection of the figure (art. 536). It is only
during the short time that the earth is in the arc G H (i.e. between the
sm and Jupiter, that the cone of the shadow converging (while that of
the visual rays diverges) behind the planet, permits their occultations to
be completely observed both at ingress and egress, unobscured, the eclipses
being then invisible.
(542.) An extremely singular relation subsists between the mean
angular velocities or mean motions (as they are termed) of the three first
satellites of Jupiter. If the mean angular velocity of the first satellite
be added to twice that of the third, the sum will equal three times that
of the second. From this relation it follows, that if from the mean lon-
gitude of the first added to twice that of the third, be subducted three
times that of the second, the remainder will always be the same, or con-
stant, and observation informs us that this constant is 180°, or two right
angles; so that, the situations of any two of them being given, that
of the third may be found. It has been attempted to account for this
remarkable fact, on the theory of gravity by their mutual action; and
Laplace has demonstrated, that if this relation were at any one epoch ap-
proximately true, the mutual attractions of the satellites would, in process
of time, render it exactly so. One curious consequence is, that these
three satellites cannot be all eclipsed at once; for, in consequence of the
last-mentioned relation, when the second and third lie in the same direc-
tion from the centre, the first must lie on the opposite; and therefore,
when at such a conjuncture the first is eclipsed, the other two must lie
between the sun and planet, throwing its shadow on the disc, and vice
versdº.
(543.) Although, however, for the above mentioned reason, the satel-
lites cannot be all eclipsed at once, yet it may happen, and occasionally
does so, that all are either eclipsed, occulted, or projected on the body, in
which case they are, generally speaking, equally invisible, since it requires
an excellent telescope to discern a satellite on the body, except in peculiar
circumstances. Instances of the actual observations of Jupiter thus
denuded of its usual attendance and offering the appearance of a solitary
disc, though rare, have been more than once recorded. The first occasion
in which this was noticed was by Molyneux, on November 2d., (old style)
19
290 OUTLINES OF ASTRONOMY.
1681." A similar observation is recorded by Sir W. Herschel as made
by him on May 22d, 1802. The phaenomenon has also been observed
by Mr. Wallis, on April 15th, 1826; (in which case the deprivation con-
tinued two whole hours;) and last y by Mr. H. Griesbach, on September
27th, 1843.
(544.) The discovery of Jupiter's satellites, one of the first fruits of
the invention of the telescope, and of Galileo's early and happy idea of
directing its new-found powers to the examination of the heavens, forms
one of the most memorable epochs in the history of astronomy. The
first astronomical solution of the great problem of “the longitude”—prac-
tically the most important for the interests of mankind which has ever
been brought under the dominion of strict scientific principles, dates
immediately from their discovery. The final and conclusive establish-
ment of the Copernican system of astronomy may also be considered as
referable to the discovery and study of this exquisite miniature system,
in which the laws of the planetary motions, as ascertained by Kepler,
and especially that which connects their periods and distances, were
speedily traced, and found to be satisfactorily maintained. And (as if to
accumulate historical interest on this point) it is to the observation of
their eclipses that we owe the grand discovery of the aberration of light,
and the consequent determination of the enormous velocity of that won-
derful element. This we must explain now at large.
(545.) The earth's orbit being concentric with that of Jupiter and
interior to it (see fig. art. 536), their mutual distance is continually
7arying, the variation extending from the sum to the difference of the
radii of the two orbits; and the difference of the greater and least dis-
tances being equal to a diameter of the earth’s orbit. Now, it was
observed by Roemer, (a Danish astronomer, in 1675,) on comparing to-
gether observations of eclipses of the satellites during many successive
years, that the eclipses at and about the opposition of Jupiter (or its
nearest point to the earth) took place too soon—sooner, that is, than, by
calculation from an average, he expected them; whereas those which hap-
pened when the earth was in the part of its orbit most remote from
Jupiter were always too late. Connecting the observed error in their
computed times with the variation of distance, he concluded, that, to
make the calculation on an average period correspond with fact, an allow-
ance in respect of time behoved to be made proportional to the excess or
defect of Jupiter's distance from the earth above or below its average
amount, and such that a difference of distance of one diameter of the
earth's orbit should correspond to 16m 26*6 of time allowed. Specu-
Molyneux, Optics, p. 271.
SATELLITES OF SATURN. 291
er
lating on the probable physical cause, he was naturally led to think of a
gradual instead of an instantaneous propagation of light. This explained
every particular of the observed phenomenon, but the velocity required
(192000 miles per second) was so great as to startle many, and, at all
events, to require confirmation. This has been afforded since, and of the
most unequivocal kind, by Bradley's discovery of the aberration of light
(art. 329.) The velocity of light deduced from this last phaenomenon
differs by less than one-eightieth of its amount from that calculated from
the eclipses, and even this difference will no doubt be destroyed by nicer .
and more rigorously reduced observations. .
(546.) The orbits of Jupiter's satellites are but little excentric, those
of the two interior, indeed, have no perceptible excentricity. Their
mutual action produces in them perturbations analogous to those of the
planets about the sun, and which have been diligently investigated by
Laplace and others. By assiduous observation it has been ascertained
that they are subject to marked fluctuations in respect of brightness, and
that these fluctuations happen periodically, according to their position
with respect to the sun. From this it has been concluded, apparently
with reason, that they turn on their axes, like our moon, in periods equal
to their respective sidereal revolutions about their primary.
(547.) The satellites of Saturn have been much less studied than those
of Jupiter, being far more difficult to observe. The most distant has its
orbit materially inclined (no less than 12° 14')' to the plane of the ring,
with which the orbits of all the rest nearly coincide. Nor is this the only
circumstance which separates it by a marked difference of character from
the system of the six interior ones, and renders it in some sort an anoma-
lous member of the Saturnian system. Its distance from the planet's centre
exceeds in the proportion of nearly three to one that of the most distant
of all the rest, being no less than 64 times the radius of the globe of
Saturn, a distance from the primary to which our own moon (at 60 radii)
offers the only parallel. Its variation of light also in different parts of its
orbit is very much greater than the case of any other secondary planet.
Dominic Cassini indeed (its first discoverer, A. D. 1671) found it to disap-
pear for nearly half its revolution, when to the east of Saturn, and though
the more powerful telescopes now in use enable us to follow it round the
whole of its circuit, its diminution of light is so great in the eastern half
of its orbit as to render it somewhat difficult to perceive. From this cir-
cumstance (viz. from the defalcation of light occurring constantly on the
same side of Saturn as seen from the earth, the visual ray from which is
never very oblique to the direction in which the sun's light falls on it) it
* Lalande, Astron. Art. 3075.
292 OUTLINES OF ASTRONOMY.
is presumed with much certainty that this satellite revolves on its axis in
the exact time of rotation about the primary; as we know to be the case
with the moon, and as there is considerable ground for believing to be so
with all secondaries.
(548.) The next satellite in order proceeding inwards (the first in order
of discovery') is by far the largest and most conspicuous of all, and is
probably not much inferior to Mars in size. It is the only one of the
number whose theory and perturbations have been at all inquired into *
farther than to verify Kepler's law of the periodic times, which holds
good, mutatis mutandis, and under the requisite reservations, in this, as
in the system of Jupiter. The three next satellites still proceeding
inwards* are very minute, and require pretty powerful telescopes to see
them; while the two interior satellites which just skirt the edge of the
ring * can only be seen with telescopes of extraordinary power and per-
fection, and under the most favourable atmospheric circumstances. At
the epoch of their discovery they were seen to thread, like beads, the
almost infinitely thin fibre of light to which the ring then seen edge-
ways, was reduced, and for a short time to advance off it at either end,
speedily to return, and hastening to their habitual concealment behind
the body.”
(549.) Owing to the obliquity of the ring and of the orbits of the
satellites to Saturn's ecliptic, there are no eclipses, occultations, or
* By Huyghens, March 25, 1655.
* By Bessel, Astr. Nachr. Nos. 193, 214.
* Discovered by Dominic Cassini in 1672 and 1684.
* Discovered by Sir William Herschel in 1789.
* Considerable confusion prevails in the nomenclature of the Saturnian system,
owing to the order of discovery not coinciding with that of distances, Astronomers
have not yet agreed whether to call the two interior satellites the 6th and 7th (reckon-
ing inward) and the older ones the 1st, 2d, 3d, 4th, and 5th, reckoning outward; or to
commence with the innermost and reckon outwards from 1 to 7. This confusion has
been attempted to be obviated by a mythological nomenclature, in consonance with
that at length completely established for the primary planets. Taking the names of
the Titanian divinities, the following pentameters afford an easy artificial memory,
commencing with the most distant.
Iapetus, Titan ; Rhea, Dione, Tethys; (pron. Têthys)
Enceladus, Mimas
It is worth remarking that Simon Marius, who disputed the priority of the discovery
of Jupiter's satellites with Galileo, proposed for them mythological names, viz: –Io
Europa, Ganymede, and Callisto. The revival of these names would savour of a prefe-
rence of Marius's claim, which, even if an absolute priority were conceded (which it is
not), would still leave Galileo's general claim to the use of the telescope as a means of
astronomical discovery intact. But in the case of Jupiter's satellites there exists no
confusion to rectify. They are constantly referred to by their numerical designations
in cvery almanack.
SATELLITES OF URANUS. 298.
transits of these bodies or their shadows across the disc of their primary
(the interior ones excepted), until near the time when the ring is seen
edgewise, and when they do take place, their observation is attended with
too much difficulty to be of any practical use, like the eclipses of Jupiter's
satellites, for the determination of longitudes, for which reason they
have been hitherto little attended to by astronomers. .
(550.) A remarkable relation subsists between the periodic times of the
two interior satellites of Saturn, and those of the two next in order of
distance; viz. that the period of the third (Tethys) is double that of the
first (Mimas), and that of the fourth (Dione) double that of the second
(Enceladus). The coincidence is exact in either case to about one-800th
part of the larger period.
(551.) The satellites of Uranus require very powerful and perfect tele-
scopes for their observation. Two are, however, much more conspicuous
than the rest, and their periods and distances from the planet have been
ascertained with tolerable certainty. They are the second and fourth of
those set down in the synoptic table. Of the remaining four, whose ex-
istence, though announced with considerable confidence by their original
discoverer, could hardly be regarded as fully demonstrated, two only have
been hitherto reobserved; viz. the first of our table, interior to the two
larger ones, by the independent observations of Mr. Lassell,' and M.
Otto Struve,” and the fourth, intermediate between the larger ones, by
the former of these astronomers. The remaining two, if future observa-
tion should satisfactorily establish their real existence, will probably be
found to revolve in orbits exterior to all these.
(552.) The orbits of these satellites offer remarkable, and, indeed,
quite unexpected and unexampled peculiarities. Contrary to the un-
broken analogy of the whole planetary system — whether of primaries or
secondaries—the planes of their orbits are nearly perpendicular to the
ecliptic, being inclined no less than 78° 58' to that plane, and in these
orbits their motions are retrograde; that is to say, their positions, when
projected on the ecliptic, instead of advancing from west to east round the
centre of their primary, as is the case with every other planet and satel-
lite, move in the opposite direction. Their orbits are nearly or quite
circular, and they do not appear to have any sensible, or, at least, any
rapid motion of nodes, or to have undergone any material change of incli-
nation, in the course, at least, of half a revolution of their primary round
the sun. When the earth is in the plane of their orbits, or nearly so,
their apparent paths are straight lines or very elongated ellipses, in which
* September 14th to November 9th, 1847.
* October 8th to December 10th, 1847.
N
: 94 OUTLINES OF ASTRONOMY.
case they become invisible, their feeble light being effaced by the superior
light of the planet, long before they come up to its disc, so that the ob-
servation of any eclipses or occultations they may undergo is quite out
of the question, with our present telescopes.
(553.) If the observation of the satellites of Uranus be difficult, those
of Neptune, owing to the immense distance of that planet, may be readily
imagined to offer still greater difficulties. Of the existence of one, dis.
covered by Mr. Lassell," there can remain no doubt, it having also been
observed by other astronomers, both in Europe and America. Accord-
ing to M. Otto Struve” its orbit is inclined to the ecliptic at the considera.
ble angle of 35°; but whether, as in the case of the satellites of Uranus,
the direction of its motion be retrograde, it is not possible to say, until it
shall have been longer observed.
* On July 8th, 1847.
* Astron. Nachr. No. 629, from his own observations, Sept Ambor 11th to Decem-
ber 20th, 1847.
OF COMETS. 295
CHAPTER XI.
O F C O M ET. S.
GREAT NUMBER OF RECORDED COMETS. — T HE NUMBER OF THOSE
UNRECORDED PROBABLY MUCH GREATER. -- GENERAL DESCRIPTION
OF A COMET. — COMETS WITHOUT TAILS, OR WITH MORE THAN
ONE. — THEIR EXTREME TENUITY. — THEIR PROBABIE STRUCTURE.
— MOTIONS CONFORMABLE TO THE LAW OF GRAVITY. — ACTUAL
DIMENSIONS OF COMETS. — PERIODICAL RETURN OF SEVERAL. —
HALLEY'S COMET. — OTHER ANCIENT COMETS PROBABLY PERIODIC.
—ENCKE's CoMET. —BIELA’s. – FAYE'S.–LEXELL’s. – DE VICO's.
—BRORSEN’s.—PETERs’s.—GREAT COMET OF 1843.−ITS PROBABLE
IDENTITY WITH SEVERAL OLDER, COMETS. — GREAT INTEREST AT
PRESENT ATTACHED TO COMETARY ASTRONOMY, AND ITS REASONS.
—REMARKs on compTARY ORBITS IN GENERAL.
(554.) THE extraordinary aspect of comets, their rapid and seemingly
irregular motions, the unexpected manner in which they often burst upon
us, and the imposing magnitudes which they occasionally assume, have in
all ages rendered them objects of astonishment, not unmixed with super-
stitious dread to the uninstructed, and an enigma to those most conversant
with the wonders of creation and the operations of natural causes. Even
now, that we have ceased to regard their movements as irregular, or as
governed by other laws than those which retain the planets in their orbits,
their intimate nature, and the offices they perform in the economy of our
system, are as much unknown as ever. No distinct and satisfactory
account has yet been rendered of those immensely voluminous append.
ages which they bear about with them, and which are known by the name
of their tails, (though improperly, since they often precede them in their
motions,) any more than of several other singularities which they
present. -
(555.) The number of comets which have been astronomically observed,
Br of which notices have been recorded in history, is very great, amount-
296. OUTLINES OF A$$. RONOMY.
ing to several hundreds;" and when we consider that in the earlier ages
of astronomy, and indeed in more recent times, before the invention of
the telescope, only large and conspicuous ones were noticed; and that,
since due attention has been paid to the subject, scarcely a year has passed
without the observation of one or two of these bodies, and that sometimes
two and even three have appeared at once; it will be easily supposed that
their actual number must be at least many thousands. Multitudes,
indeed, must escape all observation, by reason of their paths traversing
only that part of the heavens which is above the horizon in the daytime.
Comets so circumstanced can only become visible by the rare coincidence
of a total eclipse of the Sun, - a coincidence which happened, as related
by Seneca, sixty-two years before Christ, when a large comet was actually
observed very near the sun. Several, however, stand on record as having
been bright enough to be seen with the naked eye in the daytime, even
at noon and in bright sunshine. Such were the comets of 1402, 1532,
and 1843, and that of 43 B.C. which appeared during the games cele-
brated by Augustus in honour of Venus shortly after the death of Caesar,
and which the flattery of poets declared to be the soul of that hero taking
its place among the divinities.
(556.) That feelings of awe and astonishment should be excited by the
sudden and unexpected appearance of a great comet, is no way surprising;
being, in fact, according to the accounts we have of such events, one of
the most imposing of all natural phenomena. Comets consist for the most
part of a large and more or less splendid, but ill-defined nebulous mass
of light, called the head, which is usually much brighter towards its
centre, and offers the appearance of a vivid nucleus, like a star or planet.
From the head, and in a direction opposite to that in which the sun is
situated from the comet, appear to diverge two streams of light, which
grow broader and more diffused at a distance from the head, and which
most commonly close in and unite at a little distance behind it, but some-
times continue distinct for a great part of their course; producing an effect
like that of the trains left by some bright meteors, or like the diverging
fire of a sky-rocket (only without sparks or perceptible motion.) This is
* See catalogues in the Almagest of Riccioli; Pingré's Cometographie; Delambre's
Astron. vol. iii.; Astronomische Abhandlungen, No. 1, (which contains the elements
of all the orbits of comets which have been computed to the time of its publication.
1823;) also a catalogue, by the Rev. T. J. Hussey. Lond. & Ed. Phil. Mag. vol. ii.
No. 9, et seq. In a list cited by Lalande from the 1st vol. of the Tables de Berlin,
700 comets are enumerated. See also notices of the Astronomical Society and Astron.
Nachr. passim. A great many of the more ancient comets are recorded in the Chinese
Annals, and in some cases with sufficient precision to allow of the calculation of
rudely approximate orbits from their motions so described.
EXTREME TENUITY OF COMETS. 297
the tail. This magnificent appendage attains occasionally an immense
apparent length. Aristotle relates of the tail of the comet of 371 B. C.,
that it occupied a third of the hemisphere, or 60°; that of A. D. 1618 is
stated to have been attended by a train no less than 104° in length. The
comet of 1680, the most celebrated of modern times, and on many ac-
counts the most remarkable of all, with a head not exceeding in bright-
ness a star of the second magnitude, covered with its tail an extent of
more than 70° of the heavens, or, as some accounts state, 90°; that of
the comet of 1769 extended 97°, and that of the last great comet (1843)
was estimated at about 65° when longest. The figure (fig. 2, Plate II.)
is a representation of the comet of 1819 —by no means one of the most
considerable, but which was, however, very conspicuous to the naked eye.
(557.) The tail is, however, by no means an invariable appendage of
comets. Many of the brightest have been observed to have short and
feeble tails, and a few great comets have been entirely without them.
Those of 1585 and 1763 offered no vestige of a tail; and Cassini describes
the comets of 1665 and 1682 as being as round' and as well defined as
Jupiter. On the other hand, instances are not wanting of comets fur-
nished with many tails or streams of diverging light. That of 1744 had
no less than six, spread out like an immense fan, extending to a distance
of nearly 30° in length. The small comet of 1823 had two, making an
angle of about 160°, the brighter turned as usual from the sun, the fainter
towards it, or nearly so. The tails of comets, too, are often somewhat
curved, bending, in general, towards the region which the comet has left,
as if moving somewhat more slowly, or as if resisted in their course.
(558.) The smaller comets, such as are visible only in telescopes, or
with difficulty by the naked eye, and which are by far the most numerous,
offer very frequently no appearance of a tail, and appear only as round
or somewhat oval vaporous masses, more dense towards the centre, where,
however, they appear to have no distinct nucleus, or anything which seems
entitled to be considered as a solid body. Stars of the smallest magni-
tudes remain distinctly visible, though covered by what appears to be the
densest portion of their substance; although the same stars would be
completely obliterated by a moderate fog, extending only a few yards from
the surface of the earth. And since it is an observed fact, that even those
* This description however applies to the “disc” of the head of these comets as seen
in a telescope. Cassini's expressions are, “aussi rond, aussi net, et aussi clair que
Jupiter,” (where it is to be observed that the latter epithet must by no means be trans-
lated bright). To understand this passage fully, the reader must refer to the descrip-
tion given further on, of the “disc” of Halley’s comet, after its perihelior, passage
in 1835-6.
298 OUTLINES OF ASTRONOMY.
larger comets which have presented the appearance of a nucleus have yet
exhibited no phases, though we cannot doubt that they shine by the re-
flected solar light, it follows that even these can only be regarded as great
masses of thin vapour, susceptible of being penetrated through their
whole substance by the sunbeams, and reflecting them alike from their
interior parts and from their surfaces. Nor will any one regard this ex-
planation as forced, or feel disposed to resort to a phosphorescent quality
in the comet itself, to account for the phenomena in question, when we
consider, (what will hereafter be shown) the enormous magnitude of the
space thus illuminated, and the extremely small mass which there is
ground to attribute to these bodies. It will then be evident that the most
unsubstantial clouds which float in the highest regions of our atmosphere,
and seem at sunset to be drenched in light, and to glow throughout their
whole depth as if in actual ignition, without any shadow or dark side,
must be looked upon as dense and massive bodies compared with the filmy
mnd all but spiritual texture of a comet. Accordingly, whenever powerful
telescopes have been turned on these bodies, they have not failed to dispel
the illusion which attributes solidity to that more condensed part of the
head, which appears to the naked eye as a nucleus; though it is true that
in some, a very minute stellar point has been seen, indicating the existence
of a solid body.
(559.) It is in all probability to the feeble coercion of the elastic power
of their gaseous parts, by the gravitation of so small a central mass, that
we must attribute this extraordinary development of the atmospheres of
comets If the earth, retaining its present size, were reduced, by any in-
terna, enange (as by hollowing out its central parts) to one thousandth part
of its actual mass, its coercive power over the atmosphere would be dimi-
nished in the same proportion, and in consequence the latter would expand
to a thousand times its actual bulk; and indeed much more, owing to the
still farther diminution of gravity, by the recess of the upper parts from
the centre." An atmosphere, however, free to expand equally in all di-
rections, would envelope the nucleus spherically, so that it becomes neces-
sary to admit the action of other causes to account for its enormous exten-
sion in the direction of the tail,- a subject to which we shall presently
take occasion to recur. - -> -
(560.) That the luminous part of a comet is something in the nature of
* Newton has calculated (Princ. III. p. 512,) that a globe of air of ordinary density
at the earth's surface, of one inch in diameter, if reduced to the density due to the
altitude above the surface of one radius of the earth, would occupy a sphere exceeding
in radius the orbit of Saturn. The tail of a great comet then, for aught wa can tell,
may consist of only a very few pounds or even ounces of matter.
MOTIONS OF COMETS. 299
a smoke, fog, or cloud, suspended in a transparent atmosphere, is evident
from a fact which has been often noticed, viz. –that the portion of the tail
where it comes up, and surrounds the head, is yet separate from it by an
interval less luminous, as if sustained and kept off from contact by a
transparent stratum, as we often see one layer of clouds over another with
a considerable clear space between. These and most of the other facts ob-
served in the history of comets, appear to indicate that the structure of a
comet, as seen in section in the direction of its length, must be that of a
hollow envelope, of a parabolic form, enclosing near its vertex the nucleus
and head, something as represented in the annexed figure. This would
account for the apparent division of the tail into two principal lateral
Fig. 75.
branches, the envelope being oblique to the line of sight at its borders,
and therefore a greater depth of illuminated matter being there exposed
to the eye. In all probability, however, they admit great varieties of
structure, and among them may very possibly be bodies of widely diffe-
rent physical constitution, and there is no doubt that one and the same
comet at different epochs undergoes great changes, both in the disposition
of its materials and in their physical state.
(561.) We come now to speak of the motions of comets. These are
apparently most irregular and capricious. Sometimes they remain in
sight only for a few days, at others for many months; some move with
extreme slowness, others with extraordinary velocity; while not unfre-
quently, the two extremes of apparent speed are exhibited by the same
comet in different parts of its course. The comet of 1472 described an
arc of the heavens of 40° of a great circle' in a single day. Some
pursue a direct, some a retrograde, and others a tortuous and very irregu-
lar course; nor do they confine themselves, like the planets, within any
certain region of the heavens, but traverse indifferently every part. Their
variations in apparent size, during the time they continue visible, are no
less remarkable than those of their velocity; sometimes they make their
first appearance as faint and slow moving objects, with little or no tail;
* 120° in extent in the former editions. But this was the arc described in longitude.
and the comet at the time referred to had great north latitude.

300 OluTLINES OF ASTRONOMY.
but by degrees accelerate, enlarge, and throw out from them this appen-
dage, which increases in length and brightness till (as always happens in
such cases) they approach the sun, and are lost in his beams. After a
time they again emerge, on the other side, receding from the sun with a
velocity at first rapid, but gradually decaying. It is for the most part
after thus passing the sun, that they shine forth in all their splendour,
and that their tails acquire their greatest length and developement; thus
indicating plainly the action of the sun's rays as the exciting cause of that
extraordinary emanation. As they continue to recede from the Sun, their
motion diminishes and the tail dies away, or is absorbed into the head,
which itself grows continually feebler, and is at length altogether lost
sight of, in by far the greater number of cases never to be seen more.
(562.) Without the clue furnished by the theory of gravitation, the
enigma of these seemingly irregular and capricious movements might
have remained for ever unresolved. But Newton, having demonstrated
the possibility of any conic section whatever being described about the
sun, by a body revolving under the dominion of that law, immediately
perceived the applicability of the general proposition to the case of come-
tary orbits; and the great comet of 1680, one of the most remarkable on
record, both for the immense length of its tail and for the excessive close-
ness of its approach to the sun (within one-sixth of the diameter of that
luminary), afforded him an excellent opportunity for the trial of his
theory. The success of the attempt was complete. He ascertained that
this comet described about the sun as its focus an elliptic orbit of so great
an excentricity as to be undistinguishable from a parabola, (which is the
extreme, or limiting form of the ellipse when the axis becomes infinite,)
and that in this orbit the areas described about the sun were, as in the
planetary ellipses, proportional to the times. The representation of the
apparent motions of this comet by such an orbit, throughout its whole
observed course, was found to be as satisfactory as those of the motions
of the planets in their nearly circular paths. From that time it became
a received truth, that the motions of comets are regulated by the same
general laws as those of the planets—the difference of the cases consisting
only in the extravagant elongation of their ellipses, and in the absence
of any limit to the inclinations of their planes to that of the ecliptic—or
any general coincidence in the direction of their motions from west to
east, rather than from east to west, like what is observed among the
planets. -
(563.) It is a problem of pure geometry, from the general laws of
elliptic or parabolic motion, to find the situation and dimensions of the
ellipse or parabola which shall represent the motion of any given comet.
MOTIONS OF COMETS. 301
In general, three complete observations of its right ascension and declina-
tion, with the times at which they were made, suffice for the solution of
this problem, (which is, however, by no means an easy one,) and for the
determination of the elements of the orbit, These consist, mutatis mu-
tandis, of the same data as are required for the computation of the mo-
tion of a planet; (that is to say, the longitude of the perihelion, that of
the ascending node, the inclination to the ecliptic, the semiaxis, excen-
tricity, and time of perihelion passage, as also whether the motion is
direct or retrograde;) and, once determined, it becomes very easy to com-
pare them with the whole observed course of the comet, by a process
exactly similar to that of art. 502, and thus at once to ascertain their
correctness, and to put to the severest trial the truth of those general
laws on which all such calculations are founded. -
(564.) For the most part, it is found that the motions of comets may
be sufficiently well represented by parabolic orbits, – that is to say,
ellipses whose axes are of infinite length, or, at least, so very long that
no appreciable error in the calculation of their motions, during all the
time they continue visible, would be incurred by supposing them actually
infinite. The parabola is that conic section which is the limit between
the ellipse on the one hand, which returns into itself, and the hyperbola
on the other, which runs out to infinity. A comet, therefore, which
should describe an elliptic path, however long its axis, must have visited
the sun before, and must again return (unless disturbed) in some deter-
minate period, – but should its orbit be of the hyperbolic character, when
once it had passed its perihelion, it could never more return within the
sphere of our observation, but must run off to visit other systems, or be
lost in the immensity of space. A very few comets have been ascertained
to move in hyperbolas', but many more in ellipses. These latter, in so
far as their orbits can remain unaltered by the attractions of the planets,
must be regarded as permanent members of our system.
(565.) We must now say a few words on the actual dimensions of
comets. The calculation of the diameters of their heads, and the lengths
and breadths of their tails, offers not the slightest difficulty when once
the elements of their orbits are known, for by these we know their real
distances from the earth at any time, and the true direction of the tail,
which we see only foreshortened. Now calculations instituted on these
principles lead to the surprising fact, that the comets are by far the most
voluminous bodies in our system. The following are the dimensions of
some of those which have been made the subjects of such inquiry.
* For example, that of 1723, calculated by Burckhardt; that of 1771, by both Burck
hardt and Encke ; and the second comet of 1818, by Rosenberg and Schwabe.
302 OljTLINES OF ASTRONOMY.
(566.) The tail of the great comet of 1680, immediately after its peri-
helion passage, was found by Newton to have no less than 20000000 of
leagues in length, and to have occupied only two days in its emission
from the comet's body a decisive proof this of its being darted forth by
some active force, the origin of which, to judge from the direction of the
tail, must be sought in the sun itself. Its greatest length amounted to
41000000 leagues, a length much exceeding the whole interval between
the sun and earth. The tail of the comet of 1769 extended 16000000
leagues, and that of the great comet of 1811, 36000000. The portion
of the head of this last, comprised within the transparent atmospheric en-
velope which separated it from the tail, was 180000 leagues in diameter.
It is hardly conceivable, that matter once projected to such enormous dis-
tances should ever be collected again by the feeble attraction of such a
body as a comet—a consideration which accounts for the surmised pro-
gressive diminution of the tails of such as have been frequently observed.
(567.) The most remarkable of those comets which have been ascer-
tained to move in elliptic orbits is that of Halley, so called from the cele-
brated Edmund Halley, who, on calculating its elements from its perihelion
passage in 1682, when it appeared in great splendour, with a tail 30° in
length, was led to conclude its identity with the great comets of 1531 and
1607, whose elements he had also ascertained. The intervals of these
successive apparitions being 75 and 76 years, Halley was encouraged to
predict its reappearance about the year 1759. So remarkable a predic-
tion could not fail to attract the attention of all astronomers, and, as the
time approached, it became extremely interesting to know whether the
attractions of the larger planets might not materially interfere with its
orbitual motion. The computation of their influence from the Newtonian
law of gravity, a most difficult and intricate piece of calculation, was
undertaken and accomplished by Clairaut, who found that the action of
Saturn would retard its return by 100 days, and that of Jupiter by no
less than 518, making in all 618 days, by which the expected return
would happen later than on the supposition of its retaining an unaltered
period, and that, in short, the time of the expected perihelion passage
would take place within a month, one way or other, of the middle of
April, 1759–It actually happened on the 12th of March in that year,
Its next return was calculated by several eminent geometers', and fixed
successively for the 4th, the 7th, the 11th, and the 26th of November,
1835; the two latter determinations appearing entitled to the higher de-
gree of confidence, owing partly to the more complete discussion bestowed
on the observations of 1682 and 1759, and partly to the continually im-
* Damoiseau, Pontecoulant, Rosenberger, and Lehmann.
HALLEY'S COMET. 303
proving state of our knowledge of the methods of estimating the dis-
turbing effect of the several planets. The last of these predictions, that
of M. Lehmann, was published on the 25th of July. On the 5th of
August the comet first became visible in the clear atmosphere of Rome
as an exceedingly faint telescopic nebula, within a degree of its place as
predicted by M. Rosenberger for that day. On or about the 20th of
August it became generally visible, and, pursuing very nearly its calcu-
lated path among the stars, passed its perihelion on the 16th of November;
after which, its course carrying it south, it ceased to be visible in Europe,
though it continued to be conspicuously so in the southern hemisphere
throughout February, March, and April, 1836, disappearing finally on the
5th of May. -
(568.) Although the appearance of this celebrated comet at its last
apparition was not such as might be reasonably considered likely to excite
lively sensations of terror, even in superstitious ages, yet, having been an
object of the most diligent attention in all parts of the world to astrono-
mers, furnished with telescopes very far surpassing in power those which
had been applied to it at its former appearance in 1759, and indeed to any
of the greater comets on record, the opportunity thus afforded of studying
its physical structure, and the extraordinary phaenomena which it pre-
sented when so examined, have rendered this a memorable epoch in cometic
history. Its first appearance, while yet very remote from the sun, was
that of a small round or somewhat oval nebula, quite destitute of tail, and
having a minute point of more concentrated light excentrically situated
within it. It was not before the 2d of October that the tail began to be
developed, and thenceforward increased pretty rapidly, being already 4°
or 5° long on the 5th. It attained its greatest apparent length (about
20°) on the 15th of October. From that time, though not yet arrived
at its perihelion, it decreased with such rapidity, that already on the 29th
it was only 3°, and on November the 5th 23° in length. There is every
reason to believe that before the perihelion, the tail had altogether dis-
appeared, as, though it continued to be observed at Pulkowa up to the
very day of its perihelion passage, no mention whatever is made of any
tail being then seen.
(569.) By far the most striking phaenomena, however, observed in this
part of its career, were those which, commencing simultaneously with the
growth of the tail, connected themselves evidently with the production of
that appendage and its projection from the head. On the 2d of October
(the veryday of the first observed commencement of the tail) the nucleus,
which had been faint and small, was observed suddenly to have become
much brighter, and to be in the act of throwing out a jet or stream of
304 OljTLINES OF ASTRONOMY.
light from its anterior part, or that turned towards the sun. This ejection
after ceasing awhile was resumed, and with much greater apparent vio-
lence, on the 8th, and continued, with occasional intermittences, so long
as the tail itself continued visible. Both the form of this luminous ejec-
tion, and the direction in which it issued from the nucleus, meanwhile
underwent singular and capricious alterations, the different phases suc-
ceeding each other with such rapidity that on no two successive nights
were the appearances alike. At one time the emitted jet was single, and
confined within narrow limits of divergence from the nucleus. At others
it presented a fan-shaped or swallow-tailed form, analogous to that of a
gas-flame issuing from a flattened orifice: while at others again two, three,
or even more jets were darted forth in different directions." (See figures
a, b, c, d, plate I., fig. 4, which represent, highly magnified, the appear-
ances of the nucleus with its jets of light, on the 8th, 9th, 10th, and 12th
of October, and in which the direction of the anterior portion of the head,
or that fronting the Sun, is supposed alike in all, viz. towards the upper
part of the engraving. In these representations the head itself is omitted,
the scale of the figures not permitting its introduction: e represents the
nucleus and head as seen October 9th on a less scale.) The direction of
the principal jet was observed meanwhile to oscillate to and fro on either
side of a line directed to the sun in the manner of a compass-needle when
thrown into vibration and oscillating about a mean position, the change
of direction being conspicuous even from hour to hour. These jets,
though very bright at their point of emanation from the nucleus, faded
rapidly away, and became diffused as they expanded into the coma, at the
same time curving backwards as streams of steam or smoke would do, if
thrown out from narrow orifices, more or less obliquely in opposition to a
powerful wind, against which they were unable to make way, and ulti-
mately yielding to its force, so as to be drifted back and confounded in a
vaporous train, following the general direction of the current.”
(570.) Reflecting on these phaenomena, and carefully considering the
evidence afforded by the numerous and elaborately executed drawings
which have been placed on record by observers, it seems impossible to
avoid the following conclusions. 1st. That the matter of the nucleus of
* See the exquisite lithographic representations of these phenomena by Bessel.
Astron. Nachi, No. 302, and the fine series by Schwabe in No. 297 of that collec-
tion, as also the magnificent drawings of Struve, from which our figures a, b, c, d, alo
copies. - - -
2 On this point Schwabe's and Bessel's drawings are very express and unequiv-
ocal. Struve's attention seems to have been more especially directed to the scrutiny
of the nucleus.
HALLEY'S COMET. 305
a comet is powerfully excited and dilated into a vaporous state by the
action of the sun's rays, escaping in streams and jets at those points of its
surface which oppose the least resistance, and in all probability throwing
that surface or the nucleus itself into irregular motions by its reaction in
the act of so escaping, and thus altering its direction.
2dly. That this process chiefly takes place in that portion of the
nucleus which is turned towards the sun; the vapour escaping chiefly in
that direction.
3dly. That when so emitted, it is prevented from proceeding in the
direction originally impressed upon it, by some force directed from the
sun, drifting it back and carrying it out to vast distances behind the
nucleus, forming the tail or so much of the tail as can be considered as
consisting of material substance.
4thly. That this force, whatever its nature, acts unequally on the ma-
terials of the comet, the greater portion remaining unvaporized, and a
considerable part of the vapour actually produced, remaining in its neigh-
bourhood, forming the head and coma. *
5thly. That the force thus acting on the materials of the tail cannot
possibly be identical with the ordinary gravitation of matter, being centri-
fugal or repulsive, as respects the sun, and of an energy very far exceeding
the gravitating force towards that luminary. This will be evident if we
consider the enormous velocity with which the matter of the tail is carried
backwards, in opposition both to the motion which it had as part of the
nucleus, and to that which it acquired in the act of its emission, both
which motions have to be destroyed in the first instance, before any move-
ment in the contrary direction can be impressed.
6thly. That unless the matter of the tail thus repelled from the sun be
retained by a peculiar and highly energetic attraction to the nucleus, dif-
fering from and exceptional to the ordinary power of gravitation, it must
leave the nucleus altogether; being in effect carried far beyond the coer-
cive power of so feeble a gravitating force as would correspond to the
minute mass of the nucleus; and it is therefore very conceivable that a
comet may lose, at every approach to the sun, a portion of that peculiar
matter, whatever it be, on which the production of its tail depends, the
remainder being of course less excitable by the solar action, and more
inpassive to his rays, and therefore, pro tanto, more nearly approximating
to the nature of the planetary bodies.
(571.) After the perihelion passage, the comet was lost sight of for
upwards of two months, and at its reappearance (on the 24th of January
1836) presented itself under quite a different aspect, having in the in-
terval evidently undergone some great physical change which had operated
20
306 OljTLINES OF ASTRONOMY.
an entire transformation in its appearance. It no longer presented any
vestige of tail, but appeared to the naked eye as a hazy star of about the
fourth or fifth magnitude, and in powerful telescopes as a small, round,
well-defined disc, rather more than 2' in diameter, surrounded with a
nebulous chevelure or coma of much greater extent. Within the disc,
and somewhat excentrically situated, a minute but bright nucleus appeared,
from which extended towards the posterior edge of the disc (or that remote
from the sun) a short vivid luminous ray. (See fig. 4 of pl. I.) As the
comet receded from the sun, the coma speedily disappeared, as if absorbed
into the disc, which, on the other hand, increased continually in dimen-
sions, and that with such rapidity, that in the week elapsed from January
25th to February 1st, (calculating from micrometrical measures, and from
the known distance of the comet from the earth on those days) the actual
volume or real solid content of the illuminated space had dilated in the
ratio of upwards of 40 to 1. And so it continued to swell out with un-
diminished rapidity, until from this cause alone it ceased to be visible, the
illumination becoming fainter as the magnitude increased; till at length
the outline became undistinguishable from simple want of light to trace
it. While this increase of dimension proceeded, the form of the disc
passed, by gradual and successive additions to its length in the direction
opposite to the sun, to that of a paraboloid, as represented in g, fig. 4,
plate I., the anterior curved portion preserving its planetary sharpness,
but the base being faint and ill-defined. It is evident that had this pro.
cess continued with sufficient light to render the result visible, a tail would
have been ultimately reproduced; but the increase of dimension being
accompanied with diminution of brightness, a short, imperfect, and as it
were rudimentary tail only was formed, visible as such for a few nights to
the naked eye, or in a low magnifying telescope, and that only when the
comet itself had begun to fade away by reason of its increasing distance.
(572.) While the parabolic envelope was thus continually dilating and
growing fainter, the nucleus underwent little change, but the ray proceed-
ing from it increased in length and comparative brightness, preserving all
the time its direction along the axis of the paraboloid, and offering none
of those irregular and capricious phaenomena which characterized the jets
of light emitted anteriorly, previous to the perihelion. If the office of
those jets was to feed the tail, the converse office of conducting back its
successively condensing matter to the nucleus would seem to be that of
the ray now in question. By degrees this also faded, and the last appear-
ance presented by the comet was that which it offered at its first appear-
ance in August; viz. that of a small round nebula with a bright point in
or near the centre. -
OTHER PERIODICAL COMETS. 307.
(573.) Besides the comet of Halley, several other of the great comets
recorded in history have been surmised with more or less probability to
return periodically, and therefore to move in elongated ellipses around the
sun. Such is the great comet of 1680, whose period is estimated at 575
years, and which is considered, with the highest appearance of probability,
to be identical with a magnificent comet observed at Constantinople and
in Palestine, and referred by contemporary historians, both European and
Chinese, to the year A. D. 1105; with that of A. D. 575, which was seen
at noon-day close to the sun; with the comet of 43 B.C., already spoken
of as having appeared after the death of Caesar, and which was also
observed in the day-time; and finally with two other comets, mention of
which occurs in the Sibylline Oracles, and in a passage of Homer, and
which are referred, as well as the obscurity of chronology and the indica-
tions themselves will allow, to the years 618 and 1194 B. C. It is to the
assumed near approach of this comet to the earth about the time of the
Deluge, that Whiston ascribed that overwhelming tide wave to whose
agency his wild fancy ascribed that great catastrophe-a speculation, it is
needless to remark, purely visionary.
(574.) Another great comet, whose return in the year actually current
(1848) has been considered by more than one eminent authority in this
department of astronomy' highly probable, is that of 1556, to the terror
of whose aspect some historians have attributed the abdication of the
Emperor Charles W. This comet is supposed to be identical with that
of 1264, mentioned by many historians as a great comet, and observed
also in China, – the conclusion in this case resting upon the coincidence
of elements calculated on the observations, such as they are, which have
been recorded. On the subject of this coincidence Mr. Hind has recently
entered into many elaborate calculations, the result of which is strongly
in favour of the supposed identity. This probability is farther increased
by the fact of a comet with a tail of 40° and a head bright enough to be
visible after sunrise having appeared in A. D. 975; and of two others
having been recorded by the Chinese annalists in A. D. 395 and 104. It
is true that if these be the same, the mean period would be somewhat
short of 292 years. But the effect of planetary perturbation might
reconcile even greater differences, and though up to the time of our
writing no such comet has yet been observed, at least another year must
elapse before its return can be pronounced hopeless.
(575.) In 1661, 1532, 1402, 1145, 891, and 243 great comets
appeared—that of 1402 being bright enough to be seen at noon-day. A
period of 129 years would conciliate all these appearances, and should
* Pingré Cometographie, i. 411. Lalande, Astr. 3185.
308 OUTLINES OF ASTRONOMY.
have brought back the comet in 1789 or 1790 (other circumstances
agreeing.) That no such comet was observed about that time is no proof
that it did not return, since, owing to the situation of its orbit, had the
perihelion passage taken place in July it might have escaped observation.
Mechain, indeed, from an elaborate discussion of the observations of 1532
and 1661, came to the conclusion that these comets were not the same;
but the elements assigned by Olbers to the earlier of them, differ so
widely from those of Mechain for the same comet on the one hand, and
agree so well with those of the last named astronomer for the other,'
that we are perhaps justified in regarding the question as not yet set at
rest.
(576.) We come now, however, to a class of comets of short period,
respecting whose return there is no doubt, inasmuch as two at least of
them have been identified as having performed successive revolutions
round the sun; have had their return predicted already several times;
and have on each occasion scrupulously kept to their appointments. The
first of these is the comet of Encke, so called from Professor Encke of
Berlin, who first ascertained its periodical return. It revolves in an
ellipse of great excentricity (though not comparable to that of Halley's,.)
the plane of which is inclined at an angle of about 13° 22' to the plane
of the ecliptic, and in the short period of 1211 days, or about 34 years.
This remarkable discovery was made on the occasion of its fourth recorded
appearance, in 1819. From an ellipse then calculated by Encke, its
return in 1822 was predicted by him, and observed at Paramatta, in New
South Wales, by M. Rümker, being invisible in Europe: since which it
has been re-predicted and re-observed in all the principal observatories,
both in the northern and southern hemispheres, as a phenomenon of
regular occurrence.
(577.) On comparing the intervals between the successive perihelion
passages of this comet, after allowing in the most careful and exact manner
for all the disturbances due to the actions of the planets, a very singular
fact has come to light, viz. that the periods are continually diminishing,
or, in other words, the mean distance from the sun, or the major axis of
the ellipse, dwindling by slow and regular degrees at the rate of about
04:11 per revolution. This is evidently the effect which would be pro-
duced by a resistance experienced by the comet from a very rare ethereal
medium pervading the regions in which it moves; for such resistance, by
diminishing its actual velocity, would diminish also its centrifugal force,
and thus give the sun more power over it to draw it nearer. Accordingly
this is the solution proposed by Encke, and at present generally received
* See Schumacher's Catal. Astron, Abhandl. i.
DOUBLE COMET OF BIELA. . º 309
It will, therefore, probably fall ultimately into the sun, should it not first
be dissipated altogether, a thing no way improbable, when the lightness
of its materials is considered.
(578.) By measuring the apparent magnitude of this comet at different
distances from the sun, and thence, from a knowledge of its actual dis-
tance from the earth at the time, concluding its real volume, it has been
ascertained to contract in bulk as it approaches to, and to expand as it
recedes from, that luminary. M. Walz, who was the first to notice this
fact, accounts for it by supposing it to undergo a real compression or con-
densation of volume arising from the pressure of an aethereal medium which
he conceives to grow more dense in the sun's neighbourhood. But such
an hypothesis is evidently inadmissible, since it would require us to assume
the exterior of the comet to be in the nature of a skin or bag impervious
to the compressing medium. The phenomenon is analogous to the increase
of dimension above described as observed in the comet of Halley when in
the act of receding from the sun, and is doubtless referable to a similar
cause, viz. the alternate conversion of evaporable matter into the states of
visible cloud and invisible gas by the alternating action of cold and heat.
This comet has no tail, but offers to the view only a small ill-defined
nucleus, excentrically situated within a more or less elongated oval mass
of vapours, being nearest to that vertex which is towards the sun.
(579.) Another comet of short period is that of Biela, so called from
M. Biela, of Josephstadt, who first arrived at this interesting conclusion
on the occasion of its appearance in 1826. It is considered to be identi-
cal with comets which appeared in 1772, 1805, &c., and describes its very
excentric ellipse about the sun in 2410 days or about 63 years; and in a
plane inclined 12° 34' to the ecliptic. It appeared again according to the
prediction in 1832, and in 1846. Its orbit, by a remarkable coincidence,
very nearly intersects that of the earth; and had the latter at the time of
its passage in 1832, been a month in advance of its actual place, it would
have passed through the comet,_a singular rencontre, perhaps not un-
attended with danger."
}
* Should calculation establish the fact of a resistance experienced also by this comet,
the subject of periodical comets will assume an extraordinary degree of interest. It
cannot be doubted that many more will be discovered, and by their resistance questions
will come to be decided, such as the following:—What is the law of density of the re-
sisting medium which surrounds the sun ? Is it at rest or in motion ? If the latter, in
what direction does it move 2 Circularly round the sun, or traversing space : If cir-
cularly, in what plane It is obvious that a circular or vorticose motion of the ether
would accelerate some comels and retard others, according as their revolution was, reia-
tive to such motion, direct or retrograde. Supposing the neighbourhood of the sun to
be filled with a mattſal fluid, it is not conceivable that the circulation of the planets in
310 oUTLINES OF ASTRONOMY.
(580.) This comet is small and hardly visible to the naked eye even
when brightest. Nevertheless, as if to make up for its seeming insig-ſifi-
cance by the interest attaching to it in a physical point of view, it exhi-
bited at its last appearance, in 1846, a phaenomenon which struck every
astronomer with amazement, as a thing without previous example in the
history of our system." It was actually seen to separate itself into two dis-
tinct comets, which, after thus parting company, continued to journey along
amicably through an arc of upwards of 70° of their apparent orbit, keeping
all the while within the same field of view of the telescope pointed towards
them. The first indication of something unusual being about to take
place, might be, perhaps, referred to the 19th of December, 1845, when
the comet appeared pear-shaped, the nebulosity being unduly elongated in
the north following direction.” But on the 13th of January, at Wash.
ington in America, and on the 15th and subsequently in every part of
Europe, it was distinctly seen to have become double; a very small and
faint cometic body, having a nucleus of its own, being observed appended
to it, at a distance of about 2' (in arc) from its centre, and in a direction
forming an angle of about 328° with the meridian, running northwards
from the principal or original comet (see art. 204). From this time the
separation of the two comets went on progressively, though slowly. On
the 30th of January, the apparent distance of the nucleus had increased
to 3', on the 7th of February to 4', and on the 13th to 5', and so on,
until on the 5th of March the two comets were separated by an interval
of 9' 19'', the apparent direction of the line of junction all the while
varying but little with respect to the parallel.”
(581.) During this separation very remarkable changes were observed
to be going on both in the original comet and its companion. Both had
it for ages should not have impressed upon it some degree of rotation in their own
direction. And this may preserve them from the ext:eme effects of accumulated
resistance.—Author.
* Perhaps not quite so. To say nothing of a singular surmise of Kepler, that two
great comets seen at once in 1618, might be a single comet separated into two, the fol-
lowing passage of Helvelius cited by M. Littrow (Nachr. 564) does really seem to
refer to some phaenomenon bearing at least a certain analogy to it. “In ipso disco,”
he says (Cometographia, p. 326). “quatuor vel quinque corpuscula quædam sive nu-
cleos reliquo corpore aliquanto densiores ostendebat.”
* According to Mr. Hind's observation. But there can be little doubt that by a mis-
take of the most common occurrence, when no measure of the position is taken, north
following is an error of entry or printing for north preceding (n f for n p). In fact, an
elongation from north following to south preceding would agree with the regular direc-
tion of the tail and would occasion no remark.
* By far the greater portion of this increase of apparent distance was due to the
comet's increased proximity to the earth. The real increase reduced to a distance = 1
of the comet was at the rate of about 3" per diem.
DOUBLE COMET OF BIELA. 31]
nuclei, both had short tails, parallel in direction, and nearly perpendicular
to the line of junction, but whereas at its first observation on January
13th, the new comet was extremely small and faint in comparison with
the old, the difference both in point of light and apparent magnitude di.
minished. On the 10th of February, they were nearly equal, although
the day before the moonlight had effaced the new one, leaving the other
bright enough to be well observed. On the 14th and 16th, however, the
new comet had gained a decided superiority of light over the old, pre-
senting at the same time a sharp and starlike nucleus, compared by Lieut.
Maury to a diamond spark. But this state of things was not to continue.
Already, on the 18th, the old comet had regained its superiority, being
nearly twice as bright as its companion, and offering an unusually bright
and starlike nucleus. From this period the new companion began to fade
away, but continued visible up to the 15th of March. On the 24th the
comet was again single, and on the 22d of April both had disappeared.
(582.) While this singular interchange of light was going forwards,
indications of some sort of communication between the comets were exhi-
bited. The new or companion comet, besides its tail, extending in a
direction parallel to that of the other, threw out a faint are of light, which
extended as a kind of bridge from the one to the other; and after the
restoration of the original comet to its former preeminence, it, on its part,
threw forth additional rays, so as to present (on the 22d and 23d Febru-
ary) the appearance of a comet with three faint tails, forming angles of
about 120° with each other, one of which extended towards its com-
panion."
(583.) Professor Plantamour, director of the observatory of Geneva,
having investigated the orbits of both these comets as separate and inde-
pendent bodies, from the extensive and careful series of observations
made upon them, has arrived at the conclusion that the increase of dis-
tance between the two nuclei, at least during the interval from February
10th to March 22d, was simply apparent, being due to the variation of
distance from the earth, and to the angle under which their line of junc-
tion presented itself to the visual ray; the real distance during all that
interval (neglecting small fractions) having been on an average about
thirty-nine times the semi-diameter of the earth, or less than two-thirds
the distance of the moon from its centre. From this it would appear,
that already, at this distance, the two bodies had ceased to exercise any
* These last-mentioned particulars rest on the testimony of Lieutenant Maury of
Washington, who had the advantage of using a nine-inch object-glass of Munich.
manufacture. It does not appear that any large telescope was turned upon it in Eu
rope on the dates in question
312 OUTLINES OF ASTRONOMY.
perceptible amount of perturbative gravitation on each other; as, indeed,
from the probable minuteness of cometary masses we might reasonably
expect. Calculating upon the elements assigned by him', we find 16*4
for the interval of their next perihelion passages. And it will be, there-
fore, necessary at their next reappearance, to look out for each comet as a
separate and independent body, computing its place from these elements
as if the other had no existence. Nevertheless, as it is still perfectly
possible that some link of connection may subsist between them, (if, in-
deed, by some unknown process the companion has not been actually
reabsorbed,) it will not be advisable to rely on this calculation to the
neglect of a most vigilant search throughout the whole neighbourhood of
the more conspicuous one, lest the opportunity should be lost of pursuing
to its conclusion the history of this strange occurrence.
(584.) A third comet of short period has still more recently been added
to our list by M. Faye, of the observatory of Paris, who detected it on
the 22d of November 1843. A very few observations sufficed to show
that no parabola would satisfy the cónditions of its motion, and that to
represent them completely, it was necessary to assign to it an elliptic orbit
of very moderate excentricity. The calculations of M. Nicolai, subse-
quently revised and slightly corrected by M. Leverrier, have shown that
an almost perfect representation of its motions during the whole period
of its visibility would be afforded by assuming it to revolve in a period of
2717*68 (or somewhat less than 73 years) in an ellipse whose excen-
tricity is 0-55596, and inclination to the ecliptic 11° 22' 31"; and taking
this for a basis of further calculation, and by means of these data and the
other elements of the orbit estimating the effect of planetary perturbation
during the revolution now in progress, he has fixed its next return to the
perihelion for the 3d of April 1851, with a probable error one way or
other not exceeding one or two days.
(585.) The effect of planetary perturbation on the motion of comets
has been more than once alluded to in what has been above said. With-
out going minutely into this part of the subject, which will be better un-
derstood after the perusal of a subsequent chapter, it must be obvious,
that as the orbits of comets are very excentric, and inclined in all sorts of
1 Original Comet. Companion.
Perihelion passage, 1846, Feb. 11:00476 . . . . . . . . . . . . 11:07.1 11 Geneva M.T.
Long. semiaxis major . . . . . . . . . . . . 0°5471002 . . . . . . . . . . . 0.5451271
Perihelion distance . . . . . . . . . . . . . . 9932701 1 . . . . . . . . . . . 9'9326965
Angle of excentricity or whose
sine=e . . . . . . . . . . . . . . . . . . . . . 49° 12' 2/.5 . . . . . . . . 499 6' 14”-4
Inclination . . . . . . . . . . . . . . . . . . . . 12 34 53 '3 . . . . . . . . 12 34 14 °3
Node 38 . . . . . .. . . . . . . . . . . . . . . 245 54 38 ‘8 . . . . . . . 245 56 1 °7
Perihelion . . . . . . . . . . . . . . . . . . . 109 2 20 °1 . . . . . . . 109 2 39 '6
Mean equinox of 1846, 0.
COMETS OF LEXELL AND DE WICO. 313
angles to the ecliptic, they must in many instances, if not actually inter.
sect, at least pass very near to the orbits of some of the planets. We
have already seen, for instance, that the orbit of Biela's comet so nearly
intersects that of the earth, that an actual collision is not impossible, and
indeed (supposing neither orbit variable) must in all likelihood happen in
the lapse of some millions of years. Neither are instances wanting of
comets having actually approached the earth within comparatively short
distances, as that of 1770, which on the 1st of July of that year was
witin little more than seven times the moon’s distance. The same comet
in 1767 passed Jupiter at a distance only one 58th of the radius of that
planet's orbit, and it has been rendered extremely probable that it is to
the disturbance its former orbit underwent during that appulse that we
owe its appearance within our own range of vision. This exceedingly re-
markable comet was found by Lexell to describe an elliptic orbit with an
excentricity of 0-7858, with a periodic time of about five years and a
half, and in a plane only 1° 34' inclined to the ecliptic, having passed its
perihelion on the 13th of August 1770. Its return of course was eagerly
expected, but in vain, for the comet has never been seen since. Its ob-
servation on its first return in 1776 was rendered impossible by the rela-
tive situations of the perihelion and of the earth at the time, and before
another revolution could be accomplished (as has since been ascertained,)
viz: about the 23d of August 1779, by a singular coincidence it again
approached Jupiter within one 491st part of its distance from the sun,
being nearer to that planet by one-fifth than its fourth satellite. No
wonder, therefore, that the planet's attraction (which at that distance
would exceed that of the sun in the proportion of at least 200 to 1)
should completely alter the orbit and deflect it into a curve, not one of
whose elements would have the least resemblance to those of the ellipse
of Lexell. It is worthy of notice that by this rencontre with the system
of Jupiter's satellites, none of their motions suffered any perceptible
derangement, — a sufficient proof of the smallness of its mass. Jupiter
indeed, seems, by some strange fatality, to be constantly in the way of
comets, and to serve as a perpetual stumbling-block to them.
(586.) On the 22nd of August, 1844, Signor De Vico, director of the
observatory of the Collegio Romano, discovered a comet, the motions of
which, a very few observations sufficed to show, deviated remarkably from
a parabolic orbit. It passed its perihelion on the 2nd of September, and
continued to be observed until the 7th of December. Elliptic elements
of this comet, agreeing remarkably well with each other, were accordingly
calculated by several astronomers; from which it appears that the period
of revolution is about 1990 days, or 53 (5.4357) years, which (supposing
314 OUTLINES OF ASTRONOMY.
its orbit undisturbed in the interim) would bring it back to the perihelion
on or about the 13th of January, 1850. As the assemblage and com-
parison of these elements thus computed independently, will serve better,
perhaps, than any other example, to afford the student an idea of the
degree of arithmetical certainty capable of being attained in this branch
of astronomy, difficult and complex as the calculations themselves are, and
liable to error as individual observations of a body so ill-defined as the
smaller comets are for the most part; we shall present them in a tabular
form, as on the next page: the elements being as usual; the time of peri-
helion passage, longitude of the perihelion, that of the ascending node,
the inclination to the ecliptic, semiaxis and excentricity of the orbit, and
the periodic time.
This comet, when brightest, was visible to the naked eye, and had a
small tail. It is especially interesting to astronomers from the circum-
stance of its having been rendered exceedingly probable by the researches
of M. Leverrier, that it is identical with one which appeared in 1678 with
some of its elements considerably changed by perturbation. This comet
is further remarkable, from having been concluded by Messrs. Laugier
and Mauvais, to be identical with the comet of 1585 observed by Tycho
Brahe, and possibly also with those of 1743, 1766, and 1819.
(587.) Eiliptic elements have in like manner been assigned to the
comet discovered by M. Brorsen, on the 26th of February, 1846, which,
like that last mentioned, speedily after its discovery began to show evident
symptoms of deviation from a parabola. These elements, with the names
of their respective calculators, are as follow. The dates are for February
1846, Greenwich time.

| -
| Computed by Brunnow. Hind. Y. Yº..."
|
| Perihelion passage............…. 25d. 37794 254. 33109 25d. 02227
| Long. of Perihelion................. 116 o 28' 34”-0 || 1160 28, 1777-8 || 1160 23: 527.9
Long of 0............................. 102 39 36° 5 || 102 45 20° 9 || 103 31 25- 7
Inclination............................. 30 55 6' 5 30 49 3' 6 30 30 30- 2
Semiaxis................................ 3-15021 3-12292 2-87052
Excentricity........................... 0.79363 0-79771 0-77313
* (days)........................ 2042 2016 1776
COMET OF DE WICO.
315

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316 OUTLINES OF ASTRONOMY.
This comet is faint, and presents nothing remarkable in its appearance.
Its chief interest arises from the great similarity of its parabolic elements
to those of the comet of 1532, the place of the perihelion and node, and
the inclination of the orbit, being almost identical.
(588.) Elliptic elements have also been calculated by M. D'Arrest, for
a comet discovered by M. Peters, on the 26th of June, 1846, which go
to assign it a place among the comets of short period, viz. 5804*3, or
very nearly 16 years. The excentricity of the orbit is 0-75672, its semi-
axis 6.32066, and the inclination of its plane to that of the ecliptic 31°2'
14". This comet passed its perihelion on the 1st of June, 1846.
(589.) By far the most remarkable comet, however, which has been
seen during the present century, is that which appeared in the spring of
1843, and whose tail became visible in the twilight of the 17th of March
in England as a great beam of nebulous light, extending from a point
above the western horizon, through the stars of Eridanus and Lepus,
under the belt of Orion. This situation was low and unfavourable; and
it was not till the 19th that the head was seen, and then only as a faint
and ill-defined nebula, very rapidly fading on subsequent nights. In
more southern latitudes, however, not only the tail was seen, as a mag-
nificent train of light extending 50° or 60° in length; but the head and
nucleus appeared with extraordinary splendour, exciting in every country
where it was seen the greatest astonishment and admiration. Indeed, all
descriptions agree in representing it as a stupendous spectacle, such as in
superstitious ages would not fail to have carried terror into every bosom.
In tropical latitudes in the northern hemisphere, the tail appeared on the
3d of March, and in Van Diemen's Land, so early as the 1st, the comet
having passed its perihelion on the 27th of February. Already on the
3d the head was so far disengaged from the immediate vicinity of the sun,
as to appear for a short time above the horizon after sunset. On this day
when viewed through a 46-inch achromatic telescope it presented a plan.
etary disc, from which rays emerged in the direction of the tail. The tail
was double, consisting of two principal lateral streamers, making a very
small angle with each other, and divided by a comparatively dark line, of
the estimated length of 25°, prolonged, however, on the north side by a
divergent streamer, making an angle of 5° or 6° with the general direc-
tion of the axis, and traceable às far as 65° from the head. A similar
though fainter lateral prolongation appeared on the south side. A fine
drawing of it of this date by C. P. Smyth, Esq., of the Royal Observatory,
C. G. H., represents it as highly symmetrical, and gives the idea of a
vivid cone of light, with a dark axis, and nearly rectilinear sides, inclosed
in a fainter cone, the sides of which curve slightly outwards. The light
GREAT COMET OF 1843. 317
of the nucleus at this period is compared to that of a star of the first or
second magnitude; and on the 11th, of the third; from which time it
degraded in light so rapidly, that on the 19th it was invisible to the naked
eye, the tail all the while continuing brilliantly visible, though much
more so at a distance from the nucleus, with which, indeed, its connexion
was not then obvious to the unassisted sight — a singular feature in the
history of this body. The tail, subsequent to the 3d, was, generally
speaking, a single straight or slightly curved broad band of light, but on
the 11th it is recorded by Mr. Clerihew, who observed it at Calcutta, to
have shot forth a lateral tail nearly twice as long as the regular one, but
fainter, and making an angle of about 18° with its direction on the
southern side. The projection of this ray (which was not seen either
before or after the day in question) to so enormous a length, (nearly
100°) in a single day conveys an impression of the intensity of the forces
acting to produce such a velocity of material transfer through space, such
as no other natural phaenomenon is capable of exciting. It is clear that
if we have to deal here with matter, such as we conceive it, viz. possessing
inertia—at all, it must be under the dominion of forces incomparably
more energetic than gravitation.
(590.) There is abundant evidence of the comet in question having
been seen in full daylight, and in the Sun's immediate vicinity. It was
so seen on the 28th of February, the day after its perihelion passage, by
every person on board the H. E. I. C. S. Owen Glendower, then off the
Cape, as a short, dagger-like object, close to the sun, a little before sunset.
On the same day, at 3*6" P. M., and consequently in full sunshine, the
distance of the nucleus from the sun was actually measured with a sex-
tant by Mr. Clarke, of Portland, United States, the distance, centre from
centre, being then only 3° 50'43". He describes it in the following
terms: “The nucleus, and also every part of the tail, were as well defined
as the moon on a clear day. The nucleus and tail bore the same appear
ance, and resembled a perfectly pure white cloud, without any variation,
except a slight change near the head, just sufficient to distinguish the
nucleus from the tail at that point.” The denseness of the nucleus was
so considerable, that Mr. Clarke had no doubt it might have been visible
upon the sun's disc, had it passed between that and the observer. The
length of the visible tail resulting from these measures was 59', or not far
from double the apparent diameter of the sun; and as we shall presently
see that on the day in question the distance from the earth of the sun and
comet must have been very nearly equal, this gives us about 1700000
miles for the linear dimensions of this, the densest portion of that ap.
3.18 ouTLINES OF ASTRONOMY.
pendage, making no allowance for the foreshortening, which at that time
was very considerable.
(591.) The elements of this comet are among the most remarkable of
any recorded. They have been calculated by several eminent astronomers,
among whose results we shall specify only those which agree best; the
earlier attempts to compute its path having been rendered uncertain by
the difficulty attending exact observations of it in the first part of its
visible career. The following are those which seem entitled to most
confidence:

Encke. Plantamour. Rnorre. Nicolai. Peters.
Perihel. pass., 1843,
Feb... mean time at
Greenwich............ 27d.45096 27 d. 42935 27d.39638 27d.43023 27d.41319
Long, of perihel....... 2799 2. 30” 27So IS’ 3” 27SO 28' 25" | 27SO 36' 33" | 2790 59' 7"
Long of $3.............. 4 15 5 0 51 4 1 48 3 1 37 55 3 55 17
Inclination ............. 35 12 3S 35 8 56 35 35 29 35 36 29 35 15 42
Perihel. dist............. 0-00522 ()'ſ)(\581 0-00579 0-00558 0-00428
Motion .................... Retrograde. Retrograde. Retrograde. Retrograde. Retrograde.
(592.) What renders these elements so remarkable is the smallness of
the perihelion distance. Of all comets which have been recorded this has
made the nearest approach to the Sun. The Sun's radius being the sine
of his apparent semi-diameter (16' 1" .5) to a radius equal to the earth’s
mean distance=1, is represented on that scale by 0-00466, which falls
short of 0.00534, the perihelion distance found by taking a mean of all
the foregoing results, by only 0.00067, or about one seventh of its whole
magnitude. The comet, therefore, approached the luminous surface of
the sun within about the seventh part of the Sun’s radius ! It is worth
while to consider what is implied in such a fact. In the first place, the
intensity both of the light and radiant heat of the sun at different dis-
tances from that luminary increase proportionally to the spherical area of
the portion of the visible hemisphere covered by the Sun's disc. This
disc, in the case of the earth, at its mean distance has an angular dia-
meter of 32' 3". At our comet in perihelio the apparent angular dia-
meter of the sun was no less than 121° 32'. The ratio of the spherical
surfaces thus occupied (as appears from spherical geometry) is that of the
squares of the sines of the fourth parts of these angles to each other, or
that of 1:47042. And in this proportion are to each other the amounts
of light and heat thrown by the sun on an equal area of exposed surface
on our earth and at the comet in equal instants of time. Let any one
imagine the effect of so fierce a glare as that of 47000 suns such as we
experience the warmth of, on the materials of which the earth's surface
is composed. To form some practical idea of it we may compare it with
GREAT COMET OF 1843. 319
what is recorded of Parker's great lens, whose diameter was 32% inches
and focal length six feet eight inches. The effect of this, supposing all
the light and heat transmitted, and the focal concentration perfect, (both
conditions very imperfectly satisfied,) would be to enlarge the sun’s
effective angular diameter to 23° 26', which, compared on the same
principle with a sun of 32' in diameter, would give a multiplier of only
1015 instead of 47000. The heat to which the comet was subjected
therefore surpassed that in the focus of the lens in question, on the
lowest calculation, in the proportion of 243 to 1. Yet that lens melted
cornelian, agate, and rock crystall
(593.) To this extremity of heat however the comet was exposed but
for a short time. Its actual velocity in perihelio was no less than 366
miles per second, and the whole of that segment of its orbit above (i. e.
north of) the plane of the ecliptic, and in which, as will appear from a
consideration of the elements, the perihelion was situated, was described
in little more than two hours; such being the whole duration of the time
from the ascending to the descending node, or in which the comet had
north latitude. Arrived at the descending node, its distance from the
sun would be already doubled, and the radiation reduced to one fourth
of its maximum amount. The comet of 1680, whose perihelion distance
was 0.0062, and which therefore approached the sun's surface within one
third part of his radius (more than double the distance of the comet now
in question) was computed by Newton to have been subjected to an
intensity of heat 2000 times that of red-hot iron, a term of comparison
indeed of a very vague description, and which modern thermotics do not
recognize as affording a legitimate measure of radiant heat."
(594.) Although some of the observations of this comet were vague
and inaccurate, yet there seem good grounds for believing that its whole
course cannot be reconciled with a parabolic orbit, and that it really
describes an ellipse. Previous to any calculation, it was remarked that
in the year 1668 the tail of an immense comet was seen in Lisbon, at
Bologna, in Brazil, and elsewhere, occupying nearly the same situation
among the stars, and at the same season of the year, viz. on the 5th of
March and the following days. Its brightness was such that its reflected
* A transit of this comet over the sun's disc must probably have taken place shortly
after its passage through its decending node. It is greatly to be regretted that so
interesting a phaenomenon should have passed unobserved. Whether it be possible
that some offset of its tail, darted off so late as the 7th of March, when the comet
was already far south of the ecliptic, should have crossed that plane and been seen
near the Pleiades, may be doubted. Certain it is, that on the evening of that day, a
decidedly cometic ray was seen in the immediate neighbourhood of those stars by Mr.
Nasmyth. (Ast Soc. Notices, vol. v. p. 270.)
320 OUTLINES OF ASTRONOMY.
trace was easily distinguished on the sea. The head, when it at length
came in sight, was comparatively faint and scarce discernible. No precise
observations were made of this comet, but the singular coincidence of
situation, season of the year, and physical resemblance, excited a strong
suspicion of the identity of the two bodies, implying a period of 175
years within a day or two more or less. This suspicion has been con-
verted almost into a certainty by a careful examination of what is recorded
of the older comet. Locating on a celestial chart the situation of the
head, concluded from the direction and appearance of the tail, when only
that was seen, and its visible place, when mentioned, according to the
descriptions given, it has been found practicable to derive a rough orbit
from the course thus laid down: and this agrees in all its features so well
with that of the modern comet as nearly to remove all doubt on the
subject. Comets, moreover, are recorded to have been seen in A. D. 268,
442–3, 791, 968, 1143, 1317, 1494, which may have been returns of
this, since the period above-mentioned would bring round its appearance
to the years 268, 443, 618, 793, 968, 1143, 1318, and 1493, and a
certain latitude must always be allowed for unknown perturbations.
(595.) But this is not the only comet on record whose identity with
the comet of '43 has been maintained. In 1689 a comet bearing a con-
siderable resemblance to it was observed from the 8th to the 23d of De-
cember, and from the few and rudely observed places recorded, its elements
had been calculated by Pingré, one of the most diligent inquirers into this
part of astronomy." From these it appears that the perihelion distance
of that comet was very remarkably small, and a sufficient though indeed
rough coincidence in the places of the perihelion and node tended to
corroborate the suspicion. But the inclination (69°) assigned to it by
Pingré appeared conclusive against it. On recomputing the elements,
however, from his data, Professor Pierce has assigned to that comet an
inclination widely differing from Pingré's, viz. 30°4'4, and quite within
reasonable limits of resemblance. But how does this agree with the
longer period of 175 years before assigned 7 To reconcile this we must
*Author of the “Cométographie,” a work indispensable to all who would studv
this interesting department of the science.
* United States Gazette, May 29, 1843. Considering that all the observations lie
near the descending node of the orbit, the proximity of the comet at that time to the
sun, and the loose nature of the recorded observations, no doubt almost any given in-
clination might be deduced from them. The true test in such cases is not to ascend
from the old incorrect data to elements, but to descend from known and certain ele-
ments to the older data, and ascertain whether the recorded phaenomena can be repre-
sented by them (perturbations included) within fair limits of interpretation. Such is
the course pursued by Clausen.
GREAT come T OF 1843. 321
suppose that these 175 years comprise at least eight returns of the comet,
and that in effect a mean period of 215.875 must be allowed for its return.
Now it is worth remarking that this period calculated backwards from
1843-156 will bring us upon a series of years remarkable for the appear-
ance of great comets, many of which, as well as the imperfect descriptions
we have of their appearance and situation in the heavens, offer at least no
obvious contradiction to the supposition of their identity with this. Be-
sides those already mentioned as indicated by the period of 175 years, we
may specify as probable or possible intermediate returns, those of the
comets of 1733 7", 1689 above-mentioned, 1559 7, 1537°, 1515°, 1471,
1426, 1405–6, 1383, 1861, 1340", 1296, 1274, 1230°, 1208, 1098, 1056,
1034, 1012°, 990?", 925?, 858??, 684%, 552, 530°, 421, 245 or 247",
180", 158. Should this view of the subject be the true one, we may ex-
pect its return about the end of 1864 or beginning of 1865, in which
event it will be observable in the Southern Hemisphere both before and
after its perihelion passage”.
(596.) M. Clausen, from the assemblage of all the observations of this
comet known to him, has calculated elliptic elements which give the extra-
ordinarily short period of 6-38 years. And in effect it has been suggested
that a still further subdivision of the period of 21.875 into three of
7.292 years, would reconcile this with other remarkable comets. This
seems going too far; but at all events the possibility of representing its
motions by so short an ellipse will easily reconcile us to the admission of
a period of 21 years. That it should only be visible in certain apparitions,
and not in others, is sufficiently explained by the situation of its orbit.
(597.) We have been somewhat diffuse on the subject of this comet,
* P. Passage 1733.781. This great southern comet of May 17th seems too early in
the year.
* P. P. 1536'906. In January 1537, a comet was seen in Pisces.
* P. P. 1515-031. A comet predicted the death of Ferdinand the Catholic. He died
Jan. 23, 1515.
* P. P. 1340.031. Evidently a southern comet, and a very probable appearance.
* P. P. 1230-656, was perhaps a return of Halley’s.
* P. P. 1011:906. In 1012, a very great comet in the southern part of the heavens.
“Son éclat blessait les yeux.” (Pingré Cométographie, from whom all these recorded
appearances are taken.)
* P. P. 990’031. “Comète fort épouvantable,” some year between 989 and 998.
* P. P. 383-781. In 684, appeared two or three comets. Dates begin to be obscure.
* Two distinct comets (one probably the comet of Caesar and 1680) appeared in 530
and 531, the former observed in China, the latter in Europe.
* P. P. 246-281 ; both southern comets of the Chinese annals. The year of one or
other may be wrong. -
* P. P. 180°656. Nov. 6, A. D. 180. A southern comet of the Chinese annals.
* Clausen, Astron. Nachr. No. 485.
21
322 OljTLINES OF ASTRONOMY.
for the sake of showing the degree and kind of interest which attaches to
cometic astronomy in the present state of the science. In fact there is no
branch of astronomy more replete with interest, and we may add more
eagerly pursued at present, inasmuch as the hold which exact calculation
gives us on it may be regarded as completely established; so that whatever
may be concluded as to the motions of any comet which shall hencefor-
ward come to be observed, will be concluded on sure grounds and with
numerical precision; while the improvements which have been introduced
into the calculation of cometary perturbation, and the daily increasing
familiarity of numerous astronomers with computations of this nature,
enable us to trace their past and future history with a certainty, which at
the commencement of the present century could hardly have been looked
upon as attainable. Every comet newly discovered is at once subjected to
the ordeal of a most rigorous inquiry. Its elements, roughly calculated
within a few days of its appearance, are gradually approximated to as ob-
servations accumulate, by a multitude of ardent and expert computists.
On the least indication of a deviation from a parabolic orbit, its elliptic
elements become a subject of universal and lively interest and discussion.
Old records are ransacked, with all the advantage of improved data and
methods, so as to rescue from oblivion the orbits of ancient comets which
present any similarity to that of the new visitor. The disturbances under-
gone in the interval by the action of the planets are investigated, and the
past, thus brought into unbroken connexion with the present, is made to
afford substantial ground for prediction of the future. A great impulse,
meanwhile, has been given of late years to the discovery of comets, by
the establishment in 1840', by his late Majesty the King of Denmark,
of a prize medal, to be awarded for every such discovery, to the first ob-
server, (the influence of which may be most unequivocally traced in the
great number of these bodies which every successive year sees added to
our list,) and by the circulation of notices, by special letter”, of every such
discovery (accompanied, when possible, by an ephemeris), to all observers
who have shown that they take an interest in the inquiry, so as to ensure
the full and complete observation of the new comet, so long as it remains
within the reach of our telescopes.
(598.) It is by no means merely as a subject of antiquarian interest, or
on account of the brilliant spectacle which comets occasionally afford, that
astronomers attach a high degree of importance to all that regards them.
Apart even from the singularity and mystery which appertains to their
* See the announcement of this institution in Astron. Nachr. No. 400.
* By Prof. Schumacher, Director of the Royal Observatory of Altona.
INTEREST ATTACHIED TO COMETARY ASTRONOMY. 323
physical constitution, they have become, through the medium of exact
calculation, unexpected instruments of inquiry into points connected with
the planetary system itself, of no small importance. We have seen that
the movements of the comet of Encke, thus minutely and perseveringly
traced by the eminent astronomer whose name is used to distinguish it,
has afforded ground for believing in the presence of a resisting medium
filling the whole of our system. Similar inquiries, prosecuted in the
cases of other periodical comets, will extend, confirm, or modify our con-
clusions on this head. The perturbations, too, which comets experience
in passing near any of the planets, may afford, and have afforded, infor-
mation as to the magnitude of the disturbing masses, which could not well
be otherwise obtained. Thus the approach of this comet to the planet
Mercury in 1838 afforded an estimation of the mass of that planet, the
more precious, by reason of the great uncertainty under which all previous
determinations of that element laboured. Its approach to the same planet
in the present year (1848) will be still nearer. On the 22d of November
their mutual distance will be only fifteen times the moon's distance from
the earth.
(599.) It is, however, in a physical point of view that these bodies offer
the greatest stimulus to our curiosity. There is, beyond question, some
profound secret and mystery of nature concerned in the phaenomenon of
their tails. Perhaps it is not too much to hope that future observation,
borrowing every aid from rational speculation, grounded on the progress of
physical science generally, (especially those branches of it which relate to
the aethereal or imponderable elements), may ere long enable us to pene-
trate this mystery, and to declare whether it is really matter, in the ordi-
mary acceptation of the term, which is projected from their heads with
such extravagant velocity, and if not impelled, at least directed in its
course by a reference to the sun, as its point of avoidance. In no respect
is the question as to the materiality of the tail more forcibly pressed on us
for consideration, than in that of the enormous sweep which it makes
round the sun in perihelio, in the manner of a straight and rigid rod, in
defiance of the law of gravitation, nay, even of the received laws of mo-
tion, extending (as we have seen in the comets of 1680 and 1843) from
near the sun's surface to the earth's orbit, yet whirled round unbroken;
in the latter case through an angle of 180° in little more than two hours.
It seems utterly incredible that in such a case it is one and the same
material object which is thus brandished. If there could be conceived
such a thing as a negative shadow, a momentary impression made upon
the luminiferous aether behind the comet, this would represent in some
degree the conception such a phaenomenon irresistibly calls up. But this
324 OUTLINES OF ASTRONOMY.
is not all. Even such an extraordinary excitement of the aether, conceive
it as we will, will afford no account of the projection of lateral streamers;
of the effusion of light from the nucleus of a comet towards the sun; and
its subsequent rejection; of the irregular and capricious mode in which
that effusion has been seen to take place; none of the clear indications of
alternate evaporation and condensation going on in the immense regions of
space occupied by the tail and coma, – none, in short, of innumerable
other facts which link themselves with almost equally irresistible cogency
to our ordinary notions of matter and force.
(600.) The great number of comets which appear to move in parabolic
orbits, or orbits at least undistinguishable from parabolas during their de-
scription of that comparatively small part within the range of their visi-
bility to us, has given rise to an impression that they are bodies extraneous
to our system, wandering through space, and merely yielding a local and
temporary obedience to its laws during their sojourn. What truth there
may be in this view, we may never have satisfactory grounds for deciding.
On such an hypothesis, our elliptic comets owe their permanent denizen-
ship within the sphere of the sun's predominant attraction to the action
of one or other of the planets near which they may have passed, in such
a manner as to diminish their velocity, and render it compatible with
elliptic motion." A similar cause acting the other way, might with equal
probability, give rise to a hyperbolic motion. But whereas in the former
case, the comet would remain in the system, and might make an indefinite
number of revolutions, in the latter it would return no more. This may
possibly be the cause of the exceedingly rare occurrence of a hyperbolic
comet as compared with elliptic ones. *
(601.) All the planets without exception, and almost all the satellites,
circulate in one direction. Retrograde comets, however, are of very com-
mon occurrence, which certainly would go to assign them an exterior or
at least an independent origin. Laplace, from a consideration of all the
cometary orbits known in the earlier part of the present century, con-
cluded that the mean or average situation of the planes of all the cometary
orbits, with respect to the ecliptic, was so nearly that of perpendicularity,
as to afford no presumption of any cause biassing their directions in this
respect. Yet we think it worth noticing, that among the comets which
are as yet known to describe elliptic orbits, not one whose inclination is
under 17° is retrograde; and that out of thirty-six comets which have
had elliptic elements assigned to them, whether of great or small excen-
tricities, and without any limit of inclination, only five are retrograded,
“The velocity in an ellipse is always less than in a parabola, at equal distances from
the sun; in an hyperbola always greater.
GENERAL REMARKS. 325
and of these only two, viz. Halley's and the great comet of 1843, can be
regarded as satisfactorily made out. Finally, of the 125 comets whose
elements are given in the collection of Schumacher and Olbers, up to
1823, the number of retrograde comets under 10° of inclination is only
2 out of 9, and under 20°, 7 out of 23. A plane of motion, therefore,
nearly coincident with the ecliptic, and a periodical return, are circum-
stances eminently favourable to direct revolution in the cometary as they
are decisive among the planetary orbits. [Here also we may notice a very
curious remark of Mr. Hind, (Ast. Nachr. No. 724,) respecting periodic
comets, viz., that, so far as at present known, they divide themselves for
the most part into two families, – the one having periods of about 75
years, corresponding to a mean distance about that of Uranus; the other
corresponding more nearly with those of the asteroids, and with a mean
distance between those small planets and Jupiter. The former group
consists of four members, Halley's comet revolving in 76 years, one dis-
covered by Olbers in 74, De Vico's 4th comet in 73, and Brorsen's 3d in
75, respectively. Examples of the latter group are to be seen in the
table, p. 552, at the end of this volume. It may be added, also, that
one or two of the asteroids are described as having a faint nebulous enve.
lope about them, indicating somewhat of a cometic nature.]
326 OUTLINES OF ASTRONOMY.
P A R T II.
OF THE LUNAR AND PLANETARY PERTURBATIONS.
“Magnus ab integro saclorum nascitur ordo.”—VIRG. Pollio.
CHAPTER XII.
SUBJECT PROPOUNDED.—PROBLEM OF THREE BODIES.–SUPERPOSITION
OE SMALL MOTIONS. – ESTIMATION OF THE DISTURBING FORCE. —
ITS GEOMETRICAL REPRESENTATION. – NUMERICAL ESTIMATION IN
PARTICULAR CASEs. – RESOLUTION INTO RECTANGULAR COMPO-
NENTs. – RADIAL, TRANSVERSAL, AND ORTHOGONAL DISTURBING
FORCES. — NORMAL AND TANGENTIAL. – THEIR CHARACTERISTIC
EFFECTS.—EFFECTS OF THE ORTHOGONAL FORCE.-MOTION OF THE
NODES.— CONDITIONS OF THEIR, ADVANCE AND RECESS.— CASES OF
AN EXTERIOR PLANET DISTURBED BY AN INTERIOR.—THE REVERSE
CASE.— IN EVERY CASE THE NODE OF THE DISTURBED ORBIT RE-
CEDES ON THE PLANE OF THE DISTURBING ON AN AVER.A.G.E. –
COMBINED EFFECT OF MANY SUCH DISTURBANCES.—MOTION OF THE
MOON'S NODES. — CHANGE OF INCLINATION. — CONDITIONS OF ITS
INCREASE AND DIMINUTION. — AVERAGE EFFECT IN A WHOLE RE-
VOLUTION. — COMPENSATION IN A COMPLETE REVOLUTION OF THE
NODES.—LAGRANGE's THEOREM of THE STABILITY OF THE INCLI-
NATIONS OF THE PLANETARY ORBITS.— CHANGE OF OBLIQUITY OF
THE ECLIPTIC. — PRECESSION OF THE EQUINOXES EXPLAINED. —
NUTATION. – PRINCIPLE OF FORCED VIIBRATIONS.
(602.) IN the progress of this work, we have more than once called the
reader's attention to the existence of inequalities in the lunar and plane-
tary motions not included in the expression of Kepler's laws, but in some
sort supplementary to them, and of an order so far subordinate to those
leading features of the celestial movements, as to require, for their detec-
ion, nicer observations, and longer continued comparison between facts
PERTURBATIONS. 327
and theories, than suffice for the establishment and verification of the
elliptic theory. These inequalitics are known, in physical astronomy, by
the name of perturbations. They arise, in the case of the primary
planets, from the mutual gravitations of these planets towards each other,
which derange their elliptic motions round the sun; and in that of the
Secondaries, partly from the mutual gravitation of the secondaries of the
same system similarly deranging their elliptic motions round their common
primary, and partly from the unequal attraction of the Sun and planets on
them and on their primary. These perturbations, although Small, and, in
most instances, inscrisible in short intervals of time, yet, when accumu-
lated, as some of them may become, in the lapse of ages, alter very
greatly the original elliptic relations, so as to render the same elements of
the planetary orbits, which at one epoch represented perfectly well their
movements, inadequate and unsatisfactory after long intervals of time.
(603.) When Newton first reasoned his way from the broad features
of the celestial motions, up to the law of universal gravitation, as affect-
ing all matter, and rendering every particle in the universe subject to the
influence of every other, he was not unaware of the modifications which
this generalization would induce upon the results of a more partial and
limited application of the same law to the revolutions of the planets about
the sun, and the satellites about their primaries, as their only centres of
attraction. So far from it, his extraordinary sagacity enabled him to per-
ceive very distinctly how several of the most important of the lunar
inequalitics take their origin, in this more general way of conceiving the
agency of the attractive power, especially the retrograde motion of the
nodes, and the direct revolution of the apsides of her orbit. And if he
did not extend his investigations to the mutual perturbations of the planets,
it was not for want of perceiving that such perturbations must exist, and
might go the length of producing great derangements from the actual state
of the system, but was owing to the then undeveloped state of the prac-
tical part of astronomy, which had not yet attained the precision requisite
to make such an attempt inviting, or indeed feasible. What Newton left
undone, however, his successors have accomplished; and, at this day, it
is hardly too much to assert that there is not a single perturbation, great
or small, which observation has become precise enough clearly to detect
and place in evidence, which has not been traced up to its origin in the
mutual gravitation of the parts of our system, and minutely accounted
for, in its numerical amount and value, by strict calculation on Newton's
principles.
(604.) Calculations of this nature require a very high analysis for their
successful performance, such as is far beyond the scope and object of this
328 OUTLINES OF ASTRONOMY.
work to attempt exhibiting. The reader who would master them must
prepare himself for the undertaking by an extensive course of prepara-
tory study, and must ascend by steps which we must not here even digress
to point out. It will be our object, in this chapter, however, to give some
general insight into the nature and manner of operation of the acting forces,
and to point out what are the circumstances which, in some cases, give them
a high degree of efficiency—a sort of purchase on the balance of the sys-
tem; while, in others, with no less amount of intensity, their effective
agency in producing extensive and lasting changes is compensated or ren-
dered abortive; as well as to explain the nature of those admirable results
respecting the stability of our system, to which the researches of geome-
ters have conducted them; and which, under the form of mathematical
theorems of great simplicity and elegance, involve the history of the past
and future state of the planetary orbits during ages, of which, contem-
plating the subject in this point of view, we neither perceive the begin-
ning nor the end.
(605.) Were there no other bodies in the universe but the sun and one
planet, the latter would describe an exact ellipse about the former (or both
round their common centre of gravity), and continue to perform its revo-
lutions in one and the same orbit for ever; but the moment we add to
our combination a third body, the attraction of this will draw both the
former bodies out of their mutual orbits, and, by acting on them un-
equally, will disturb their relation to each other, and put an end to the
rigorous and mathematical exactness of their elliptic motions, not only
about a fixed point in space, but about one another. From this way of
propounding the subject, we see that it is not the whole attraction of the
newly-introduced body which produces perturbation, but the difference of
its attractions on the two originally present.
(606.) Compared to the sun, all the planets are of extreme minuteness;
the mass of Jupiter, the greatest of them all, being not more than about
one 1100th part that of the sun. Their attractions on each other, there-
fore, are all very feeble, compared with the presiding central power, and
the effects of their disturbing forces are proportionally minute. In the
case of the secondaries, the chief agent by which their motions are
deranged is the Sun itself, whose mass is indeed great, but whose disturb-
ing influence is immensely diminished by their near proximity to their
primaries, compared to their distances from the sun, which renders the
difference of attractions on both extremely small, compared to the whole
amount. In this case the greatest part of the Sun's attraction, viz. that
which is common to both, is exerted to retain both primary and secondary
in their common Orbit about itself, and prevent their parting company.
SUPERPOSITION OF SMALL MOTIONS. 329
Only the small overplus of force on one as compared with the other acts
as a disturbing power. The mean value of this overplus, in the case of
the moon disturbed by the sun, is calculated by Newton to amount to
no higher a fraction than gasomo of gravity at the earth's surface, or +44;
of the principal force which retains the moon in its orbit.
(607.) From this extreme minuteness of the intensities of the disturb-
ing, compared to the principal forces, and the consequent smallness of
their momentary effects, it happens that we can estimate each of these
effects separately, as if the others did not take place, without fear of
inducing error in our conclusions beyond the limits necessarily incident
to a first approximation. It is a principle in mechanics, immediately
flowing from the primary relations between forces and the motions they
produce, that when a number of very minute forces act at once on a
system, their joint effect is the sum or aggregate of their separate effects,
at least within such limits, that the original relation of the parts of the
system shall not have been materially changed by their action. Such
effects supervening on the greater movements due to the action of the
primary forces may be compared to the small ripplings caused by a
thousand varying breezes on the broad and regular swell of a deep and
rolling ocean, which run on as if the surface were a plane, and cross in all
directions without interfering, each as if the other had no existence. It
is only when their effects become accumulated in lapse of time, so as to
alter the primary relations or data of the system, that it becomes neces-
sary to have especial regard to the changes correspondingly introduced
into the estimation of their momentary efficiency, by which the rate of the
subsequent changes is affected, and periods or cycles of immense length
take their origin. From this consideration arise some of the most curious
theories of physical astronomy.
(608.) Hence it is evident, that in estimating the disturbing influence
of several bodies forming a system, in which one has a remarkable pre-
ponderance over all the rest, we need not embarrass ourselves with combi-
nations of the disturbing powers one among another, unless where
immensely long periods are concerned; such as consist of many hundreds
of revolutions of the bodies in question about their common centre. So
that, in effect, so far as we propose to go into its consideration, the
problem of the investigation of the perturbations of a system, however
numerous, constituted as ours is, reduces itself to that of a system of three
bodies: a predominant central body, a disturbing, and a disturbed; the
two latter of which may exchange denominations, according as the motions
of the one or the other are the subject of inquiry.
(609.) Both the intensity and direction of the disturbing force are
330 OUTLINES OF ASTRONOMY.
continually varying, according to the relative situation of the disturbing
and disturbed body with respect to the sun. If the attraction of the dis-
turbing body M, on the central body S, and the disturbed body P, (by
which designations, for brevity, we shall hereafter indicate them,) were
equal, and acted in parallel lines, whatever might otherwise be its law of
variation, there would be no deviation caused in the elliptic motion of P
about S, or of each about the other. The case would be strictly that of
art. 454; the attraction of M, so circumstanced, being at every moment
exactly analogous in its effects to terrestrial gravity, which acts in parallel
lines, and is equally intense on all bodies, great and small. But this is
not the case of nature. Whatever is stated in the subsequent article to
that last cited, of the disturbing effect of the Sun and moon, is, mutatis
mutandis, applicable to every case of perturbation; and it must be now
our business to enter, somewhat more in detail, into the general heads of
the subject there merely hinted at.
(610.) To obtain clear ideas of the manner in which the disturbing
force produces its various effects, we must ascertain at any given moment,
and in any relative situations of the three bodies, its direction and inten-
sity as compared with the gravitation of P towards S, in virtue of which
latter force alone P would describe an ellipse about S regarded as fixed,
or rather P and S about their common centre of gravity in virtue of their
mutual gravitation to each other. In the treatment of the problem of
three bodies, it is convenient, and tends to clearness of apprehension, to
regard one of them as fixed, and refer the motions of the others to it as
to a relative centre. In the case of two planets disturbing each other's
motions, the sun is naturally chosen as this fixed centre; but in that of
satellites disturbing each other, or disturbed by the sun, the centre of
their primary is taken as their point of reference, and the sun itself is
regarded in the light of a very distant and massive satellite revolving
about the primary in a relative orbit, equal and similar to that which the
primary describes absolutely round the sun. Thus the generality of our
language is preserved, and when, referring to any particular central body,
we speak of an exterior and an interior planet, we include the cases in
which the former is the sun and the latter a satellite; as, for example, in
the Lunar theory. It is a principle in dynamics, that the relative motions
of a system of bodies inter se are no way altered by impressing on all of
them a common motion or motions, or a common force or forces accelera-
ting or retarding them all equally in common directions, i. e. in parallel
lines. Suppose, therefore, we apply to all the three bodies, S, P, and M,
alike, forces equal to those with which M and P attract S, but in opposite
directions. Then will the relative motions both of M and P about S be
ESTIMATION OF THE DISTURBING FORCE. 331
unaltered; but S, being now urged by equal and opposite forces to and
from both M and P, will remain at rest. Let us now consider how either
of the other bodies, as P, stands affected by these newly-introduced forces,
in addition to those which before acted on it. It is clear that now P will
be simultaneously acted on by four forces; firstly, the attraction of S in
the direction PS; Secondly, an additional force, in the same direction,
equal to its attraction on S; thirdly, the attraction of M in the direction
PM; and fourthly, a force parallel to MS, and equal to M's attraction
on S. Of these, the two first, following the same law of the inverse
square of the distance SP, may be regarded as one force, precisely as if
the sum of the masses of S and P were collected in S; and in virtue of
their joint action, P will describe an ellipse about S, except in so far as
that elliptic motion is disturbed by the other two forces. Thus we see
that in this view of the subject the relative disturbing force acting on P
is no longer the mere single attraction of M, but a force resulting from
the composition of that attraction with M’s attraction on S transferred to
P in a contrary direction.
(611.) Let CPA be part of the relative orbit of the disturbed, and MB
of the disturbing body, their planes intersecting in the line of nodes SAB,
Fig. 76.
N
D
and having to each other the inclination expressed by the spherical angle
P Aa. In MP, produced if required, take M N : M S :: M S : M P.
Then, if S M' be taken to represent, in quantity and direction, the accele-
* The reader will be careful to observe the order of the letters, where forces are
represented by lines. M S represents a force acting from M towards S. S M from S
towards M. ---

332 OUTLINES OF ASTRONOMY.
rative attraction of M on S, M S will represent in quantity and direction
the new force applied to P, parallel to that line, and N M will represent
on the same scale the accelerative attraction of M on P. Consequently,
the disturbing force acting on P will be the resultant of two forces applied
at P, represented respectively by N M and M S, which by the laws of
dynamics are equivalent to a single force represented in quantity and
direction by NS, but having P for its point of application.
(6.12.) The line N S, is easily calculated by trigonometry, when the
relative situations and real distances of the bodies are known; and the
force expressed by that line is directly comparable with the attractive
forces of S on P by the following proportions, in which M, S, represent
the masses of those bodies which are supposed to be known, and to which,
at equal distances, their attractions are proportional :—
Disturbing force : M’s attraction on S :: N S : S M ;
M’s attraction on S : S's attraction on M :: M : S.;
S’s attraction on M : S's attraction on P :: S P : S M*: by com-
pounding which proportions we collect as follows:–
Disturbing force : S's attraction on P :: M. N. S. S P : S. S M*.
A few numerical examples are subjoined, exhibiting the results of this
calculation in particular cases, chosen so as to exemplify its application
under very various circumstances, throughout the planetary system. In
each case the numbers set down express the proportion in which the central
force retaining the disturbed body in its elliptic orbit exceeds the disturb-
ing force, to the nearest whole number. The calculation is made for three
positions of the disturbing body—viz. at its greatest, its least, and its mean
distance from the disturbed.

Ratio at the Ratio at the Ratio at the
Disturbing body. Disturbed body. greatest histance IſleáIl * least alºne
The Sun........... The Moon......... 90 179 89
Jupiter............ Saturn.............. 354 312 128
Jupiter............ The Earth ........ 95683 147575 53268
Venus ............. The Earth ........ 255.208 21.0245 26833
Neptune .......... Uranus............. 57420 56592 5519
Mercury .......... Neptune............ 526 526 526
Jupiter ............ Ceres............... 6433 6937 1033
Saturn ............ Jupiter............. 2024.8 21579 3065
(613.) If the orbit of the disturbing body be circular, S M is invariable.
In this case, N S will continue to represent the disturbing force on the
same invariable scale, whatever may be the configuration of the three
bodies with respect to each other. If the orbit of M be but little elliptic,
ESTIMATION OF THE DISTURBING FORCE. 333
the same will be nearly the case. In what follows throughout this chapter,
except where the contrary is expressly mentioned, we shall neglect the
excentricity of the disturbing orbit.
(614.) If P be nearer to M than S is, M N is greater than MP, and
N lies in M P prolonged, and therefore on the opposite side of the plane
of P's orbit from that on which M is situated. The force N S therefore
urges P towards MI's plane, and towards a point X, situated between S
and M, in the line S M. If the distance M P be equal to M S as when
P is situated, suppose, at D or E, M N is also equal to M P or M S, so
that N coincides with P, and therefore X with S, the disturbing forces being
in these cases directed towards the central body. But if M P be greater
than MS, M N is less than MP, and N lies between M and P, or on the
same side of the plane of P's orbit that M is situated on. The force N
S, therefore, applied at P, urges P towards the contrary side of that plane
towards a point in the line M S produced, so that X now shifts to the
farther side of S. In all cases, the disturbing force is wholly effective in
the plane M P S, in which the three bodies lie.
Fig. 77.
It is very important for the student to fix distinctly and bear constantly
in his mind these relations of the disturbing agency considered as a single
wnresolved force, since their recollection will preserve him from many
mistakes in conceiving the mutual actions of the planets, &c. on each
other. For example, in the figures here referred to, that of Art. 611
corresponds to the case of a nearer disturbed by a more distant body, as
the earth by Jupiter, or the moon by the Sun; and that of the present
article to the converse case: as, for instance, of Mars disturbed by the
earth. Now, in this latter class of cases, whenever M P is greater than

334 OUTLINES OF ASTRONOMY.
M S, or SP, greater than 2 S M, N lies on the same side of the plane
of P's orbit with M, so that N S, the disturbing force, contrary to what
might at first be supposed, always urges the disturbed planet out of the
plane of its orbit towards the opposite side to that on which the disturbing
planet lies. It will tend greatly to give clearness and definiteness to his
ideas on the subject, if he will trace out on various suppositions as to the
relative magnitude of the disturbing and disturbed orbits (supposed to lie
in one plane) the form of the oval about M considered as a fixed point, in
which the point N lies when P makes a complete revolution round S.
(615.) Although it is necessary for obtaining in the first instance a
clear conception of the action of the disturbing force, to consider it in this
way as a single force having a definite direction in space and a determinate
intensity, yet as that direction is continually varying with the position of
N S, both with respect to the radii SP, S M, the distance PM, and the
direction of P's motion, it would be impossible, by so considering it, to
attain clear views of its dynamical effect after any considerable lapse of
time, and it therefore becomes necessary to resolve it into other equivalent
forces acting in such directions as shall admit of distinct and separate
consideration. Now this may be done in several different modes. First,
we may resolve it into three forces acting in fixed directions in space
rectangular to one another, and by estimating its effect in each of these
three directions separately, conclude the total or joint effect. This is the
mode of procedure which affords the readiest and most advantageous
handle to the problem of perturbations when taken up in all its generality,
and is accordingly that resorted to by geometers of the modern School in
all their profound researches on the subject. Another mode consists in
resolving it also into three rectangular components, not, however, in fixed
directions, but in variable ones, viz. in the directions of the lines N Q,
Q L, and LS, of which L S is in the direction of the radius vector SP,
Q L in a direction perpendicular to it, and in the plane in which S P and
a tangent to P's orbit at P both lie; and lastly, N Q in a direction per-
pendicular to the plane in which P is at the instant moving about S.
The first of these resolved portions we may term the radial component
of the disturbing force, or simply the radial disturbing force; the second the
transversal; and the third the orthogonal." When the disturbed orbit is
one of small excentricity, the transversal component acts nearly in the
direction of the tangent to P's orbit at P, and is therefore confounded with
that resolved component which we shall presently describe (art. 618) under
* This is a term coined for the occasion. The want of some appellation for this com-
ponent of the disturbing force is often felt.
RESOLUTION OF THE DISTURBING FORCE. 335
the name of the tangential force. This is the mode of resolving the
disturbing force followed by Newton and his immediate successors.
(616.) The immediate actions of these components of the disturbing
force are evidently independent of each other, being rectangular in their
directions; and they affect the movement of the disturbed body in modes
perfectly distinct and characteristic. Thus, the radial component, being
directed to or from the central body, has no tendency to disturb either
the plane of P's orbit, or the equable description of areas by P about S,
since the law of areas proportional to the times is not a character of the
force of gravity only, but holds good equally, whatever be the force which
retains a body in an orbit, provided only its direction is always towards a
fixed centre.' Inasmuch, however, as its law of variation is not conform-
able to the simple law of gravity, it alters the elliptic form of P's orbit,
by directly affecting both its curvature and velocity at every point. In
virtue, therefore, of the action of this disturbing force, the orbit deviates
from the elliptic form by the approach or recess of P to or from S, so that
the effect of the perturbations produced by this part of the disturbing
, force falls wholly on the radius vector of the disturbed orbit.
(617.) The transversal disturbing force represented by QL, on the
other hand, has no direct action to draw P to or from S. Its whole effi-
ciency is directed to accelerate or retard P's motion in a direction at right
angles to SP. Now the area momentarily described by P about S, is,
caeteris paribus, directly as the velocity of P in a direction perpendicular
to S.P. Whatever force, therefore, increases this transverse velocity of
P, accelerates the description of areas, and vice versá. With the area
ASP is directly connected, by the nature of the ellipse, the angle ASP
described or to be described by P from a fixed line in the plane of the
orbit, so that any change in the rate of description of areas ultimately
resolves itself into a change in the amount of angular motion about S,
and gives rise to a departure from the elliptic laws. Hence arise what
are called in the perturbational theory equations (i.e. changes or fluctua-
tions to and fro about an average quantity) of the mean motion of the
disturbed body. -
(618.) There is yet another mode of resolving the disturbing force into
rectangular components, which, though not so well adapted to the compu-
tation of results, in reducing to numerical calculation the motions of the
disturbed body, is fitted to afford a clearer insight into the nature of the
modifications which the form, magnitude, and situation of its orbit un-
dergo in virtue of its action, and which we shall therefore employ in
preference. It consists in estimating the components of the disturbing
* Newton, i. 1.
336 OUTLINES OF ASTRONOMY.
force, which lie in the plane of the orbit, not in the direction we have
termed radial and transversal, i. e. in that of the radius vector PS and
perpendicular to it, but in the direction of a tangent to the orbit at P,
and in that of a normal to the curve, and at right angles to the tangent,
for which reason these components may be called the tangential and
normal disturbing forces. When the orbit of the disturbed body is cir-
cular, or nearly so, this mode of resolution coincides with or differs but
little from the former, but, when the ellipticity is considerable, these
directions may deviate from the radial and transversal directions to any
extent. As in the Newtonian mode of resolution, the effect of the one
component falls wholly upon the approach and recess of the body P to
the central body S, and of the other wholly on the rate of description of
areas by P round S, so in this which we are now considering, the direct
effect of the one component (the normal) falls wholly on the curvature of the
orbit at the point of its action, increasing that curvature when the normal
force acts inwards, or towards the concavity of the orbit, and diminishing
it when in the opposite direction; while, on the other hand, the tangential
component is directly effective on the velocity of the disturbed body, in-
creasing or diminishing it according as its direction conspires with or
opposes its motion. It is evident enough that where the object is to trace
simply the changes produced by the disturbing force, in angle and distance
from the central body, the former mode of resolution must have the
advantage in perspicuity of view and applicability to calculation. It is
less obvious, but will abundantly appear in the sequel that the latter offers
peculiar advantages in exhibiting to the eye and the reason the momen-
tary influence of the disturbing force on the elements of the orbit itself.
(519.) Neither of the last mentioned pairs of the resolved portions of
the disturbing force tends to draw P out of the plane of its orbit P S A.
But the remaining or orthogonal portion N Q acts directly and solely to
produce that effect. In consequence, under the influence of this force, P
must quit that plane, and (the same cause continuing in action) must
describe a curve of double curvature as it is called, no two consecutive
portions of which lie in the same plane passing through S. The effect
of this is to produce a continual variation in those elements of the orbit
of P on which the situation of its plane in space depends; i.e. on its
inclination to a fixed plane, and the position in such a plane of the node
or line of its intersection therewith. As this, among all the various
effects of perturbation, is that which is at once the most simple in its
conception, and the easiest to follow into its remoter consequences, we
shall begin with its explanation. * -
(620.) Suppose that up to P (Art. 611, 614,) the body were describing
*
EFFECTS OF THE ORTHOGONAL FORCE. 337
an undisturbed orbit C P. Then at P it would be moving in the direction
of a tangent P R to the ellipse PA, which prolonged will intersect the
plane of M's orbit somewhere in the line of nodes, as at R. Now, at P,
let the disturbing force parallel to N Q act momentarily on P; then P
will be deflected in the direction of that force, and instead of the arc Pºp,
which it would have described in the next instant if undisturbed, will
describe the are P g lying in the state of things represented in Art. 611,
below, and in Art. 614, above, P p with reference to the plane P S A.
Thus, by this action of the disturbing force, the plane of P's orbit will
have shifted its position in space from PS p (an elementary portion of
the old orbit) to P S q, one of the new. Now the lines of nodes S A B
in the former is determined by prolonging P p into the tangent PR,
intersecting the plane M S B in R, and joining S R. And in like manner,
if we prolong P q into the tangent P r, meeting the same plane in r, and
join Sr, this will be the new line of nodes. Thus we see that, under the
circumstances expressed in the former figure, the momentary action of the
orthogonal disturbing force will have caused the line of nodes to retro-
grade upon the plane of the orbit of the disturbing body, and under
those represented in the latter to advance. And it is evident that the
action of the other resolved portions of the disturbing force will not in
the least interfere with this result, for neither of them tends either to
carry P out of its former plane of motion, or to prevent its quitting it,
Their influence would merely go to transfer the points of intersection of
the tangents P p or P q from R or r to R' or '', points nearer to or far-
ther from S than R r, but in the same lines.
(621.) Supposing, now, MI to lie to the left instead of the right side
of the line of nodes in fig. 1., P retaining its situation, and M P being
less than M1 S, so that X shall still lie between M and S. In this situation
of things (or configuration, as it is termed of the three bodies with
respect to each other,) N will lie below the plane A SP, and the disturb-
ing force will tend to raise the body P above the plane, the resolved
orthogonal portion N Q in this case acting upwards. The disturbed arc
P q will therefore lie above Pºp, and when prolonged to meet the plane
M S B, will intersect it in a point in advance of R; so that in this con-
figuration the node will advance upon the plane of the orbit of M, pro-
vided always that the latter orbit remains fixed, or, at least, does not itself
shift its position in such a direction as to defeat this result.
(622.) Generally speaking, the node of the disturbed orbit will recede
upon any plane which we may consider as fixed, whenever the action of
the orthogonal disturbing force tends to bring the disturbed body nearer
to that plane: and vice versá. This will be evident on a mere inspection
22
338 OUTLINES OF ASTRONOMY.
of the annexed figure, in which C A represents a semicircle of the projec-
tion of the fixed plane as seen from S on the sphere of the heavens, and
CPA that of the plane of P's undisturbed orbit, the motion of P being
in the direction of the arrow, from C the ascending, to A the descend-
ing node. It is at once seen, by prolonging P q, P q' into arcs of great
Fig. 78. \
P" A. 7° TEI 7" C Ps
circles, Pr, Pr, (forwards or backwards, as the case may be) to meet
C A, that the node will have retrograded through the are A r, or C r,
whenever P q lies between C P A and C A, or when the perturbing force
carries P towards the fixed plane, but will have advanced through A r" or
Cr' when P g’ lies above C PA, or when the disturbing impulse has
lifted P above its old orbit or away from the fixed plane, and this with-
out any reference to whether the undisturbed orbitual motion o P at the
moment is carrying it towards the plane C A or from it, as in the two
cases represented in the figure.
(623.) Let us now consider the mutual disturbance of two bodies M
and P, in the various configurations in which they may be presented to
each other and to their common central body. And first, let us take the
case, as the simplest, where the disturbed orbit is exterior to that of the
disturbing body (as in fig, art. 614), and the distance between the orbits
greater-than the semiaxis of the smaller. First, let both planets lie on
the same side of the line of nodes. Then (as in art. 620) the direc-
tion of the whole disturbing force, and therefore also that of its ortho-
gonal component, will be towards the opposite side of the plane of P's
orbit from that on which M lies. Its effect therefore will be, to draw P
out of its plane in a direction from the plane of M's orbit, so that in this
state of things the node will advance on the latter plane, however P and
M may be situated in these semicircumferences of their respective orbits.
Suppose, next, M transferred to the opposite side of the line of nodes,
then will the direction of its action on P, with respect to the plane of P’s
orbit, be reversed, and P in quitting that plane will now approach to
instead of receding from the plane of M's orbit, so that its node will now
recede on that plane.
(624.) Thus, while M and Prevolve about S, and in the course of many
revolutions of each are presented to each other and to S in all possible
eonfigurations, the node of P's orbit will always advance on M’s when

MOTION OF THE NODES. 339
both bodies are on the same side of the line of nodes, and recede when
on the opposite. They will, therefore, on an average, advance and recede
during equal times (supposing the orbits nearly circular). And, there-
fore, if their advance were at each instant of its duration equally rapid
with their recess at each corresponding instant during that phase of the
movement, they would merely oscillate to and fro about a mean position,
without any permanent motion in either direction. But this is not the
case. The rapidity of their recess in every position favonrable to recess
is greater than that of their advance in the corresponding opposite posi-
tion. To show this, let us consider any two configurations in which M’s
Fig. 79.
phases are diametrically opposite, so that the triangles P S M, P S M',
shall lie in one plane, having any inclination to P's orbit, according to the
situation of P. Produce PS, and draw M m, M'm' perpendicular to it,
which will therefore be equal. Take M N : M S :: M S : M Pº, and
Mſ N' : M S :: M'S*: M'P', then, if the orbits be nearly circles, and
therefore M S = M' S, N'M' will be less than M. N.; and therefore
(since PM' is greater than PM) P N' : P M' in a greater ratio than
PN : P M ; and consequently, by similar triangles, drawing N m, N' n
perpendicular to PS, N' n' : M'm in a greater ratio than N n : M m, and
therefore N' n' is greater than N m. Now the plane P M M' intersects
P’s orbit in PS, and being inclined to that orbit at the same angle
through its whole extent, if from N and N' perpendiculars be conceived
let fall on that orbit, these will be to each other in the proportion of N n,
N' n'; and therefore the perpendicular from N' will be greater than that
from N. Now since by art. 611 N' S and N S represent in quantity and
direction the total disturbing forces of M’ and M on P respectively, there-
fore these perpendiculars express (art. 615) the orthogonal disturbing

340 OUTLINES OF ASTRONOMY.
forces, the former of which tends (as above shown) to make the nodes
recede, and the latter to advance; and therefore the preponderance in
every such pair of situations of M is in favour of a retrograde motion. I
(625.) Let us next consider the case where the distance between the
orbits is less than the semiaxis of the interior, or in which the least dis.
tance of M from P is less than M. S. Take any situation of P with
Fig. 80.
respect to the line of nodes AC. Then two points d and e, distant by
less than 120°, can be taken on the orbit of M equidistant from P with
S. Suppose M to occupy successively every possible situation in its orbit,
P retaining its place; — then, if it were not for the existence of the are
de, in which the relations of art. 624 are reversed, it would appear by
the reasoning of that article that the motion of the node is direct when
M occupies any part of the semiorbit FM B, and retrograde when it is in
the opposite, but that the retrograde motion on the whole would predom-
inate. Much more then will it predominate when there exists an arc
dMe within which if M be placed, its action will produce a retrograde
instead of a direct motion.
(626.) This supposes that the arc de lies wholly in the semicirclo
Fallb. But suppose it to lie, as in the annexed figure, partly within and
partly without that circle. The greater part d B necessarily lies within
Fig. 81.
it, and not only so, but within that portion, the point of M’s orbit nearest
to P, in which, therefore, the retrograding force has its maximum, is sit-
uated. Although, therefore, in the portion Be, it is true, the retrograde
tendency otherwise general over the whole of that semicircle (Art. 624)


MOTION OF THE NODES. 341
will be reversed, yet the effect of this will be much more than counter-
balanced by the more energetic and more prolonged retrograde action over
d R; and, therefore, in this case also, on the average of every possible
situation of M, the motion of the node will be retrograde.
(627.) Let us lastly consider an interior planet disturbed by an exte-
rior. Take MD and M E (fig. of art. 611.) each equal to MS. Then
first, when P is between D and the node A, being nearer than S to M,
the disturbing force acts towards M's orbit on the side on which M lies,
and the node recedes. It also recedes when (M. retaining the same situa-
tion) P is in any part of the are EC from E to the other node, because
in that situation the direction of the disturbing force, it is true, is re-
versed, but that portion of P's orbit being also reversely situated with
respect to the plane of M's, P is still urged towards the latter plane, but
on the side opposite to M. Thus, (M holding its place) whenever P is
anywhere in D A or EC, the node recedes. On the other hand, it ad-
vances whenever P is between A and E or between C and D, because, in
these arcs, only one of the two determining elements (viz. the direction
of the disturbing force with respect to the plane of P's orbit; and the
situation of the one plane with respect to the other as to above and be-
low) has undergone reversal. Now first, whenever M is anywhere but in
the line of nodes, the sum of the arcs DA and EC exceeds a semicircle,
and that the more, the nearer M is to a position at right angles to the
line of nodes. Secondly, the arcs favourable to the recess of the node
comprehend those situations in which the Orthogonal disturbing force is
most powerful, and vice versá. This is evident, because as P approaches
D or E, this component decreases, and vanishes at those points (612.)
The movement of the node itself also vanishes when P comes to the
node, for although in this position the disturbing orthogonal force neither
vanishes nor changes its direction, yet, since at the instant of P's passing
the node (A) the recess of the node is changed into an advance, it must
necessarily at that point be stationary." Owing to both these causes,
therefore, (that the mode recedes during a longer time than it advances,
and that a more energetic force acting in its recess causes it to recede
more rapidly,) the retrograde motion will preponderate on the whole in
each complete synodic revolution of P. And it is evident that the rea-
soning of this and the foregoing articles, is no way vitiated by a moderate
amount of excentricity in either orbit.
It would seem, at first sight, as if a change per saltum took place here, but the
continuity of the node's motion will be apparent from an inspection of the annexed
figure, where bad is a portion of P's disturbed path near the node A, concave towards
the plane (; A. The momentary place of the moving node is determined by the inter
342 OTTLINES OF ASTRONOMY.
i.
(628.) It is therefore a general proposition, that on the average of each
complete synodic revolution, the node of every disturbed planet recedes
upon the orbit of the disturbing one, or in other words, that in every pair
of orbits, the node of each recedes upon the other, and of course upon any
intermediate plane which we may regard as fixed. On a plane not inter-
mediate between them, however, the node of one orbit will advance, and
that of the other will recede. Suppose for instance, CA C to be a plane
Fig. 83.
intermediate between PP and M M the two orbits. If pp and mm be
the new positions of the orbits, the node of P on M will have receded
from A to 5, that of M on P from A to 4, that of P and M on C C re-
spectively from A to 1 and from A to 2. But if F A F be a plane not
intermediate, the node of M on that plane has receded from A to 6, but
that of P will have advanced from A to 7. If the fixed plane have not
a common intersection with those of both orbits, it is equally easy to see
that the node of the disturbed orbit may either recede on both that plane
and the disturbing orbit, or advance on the one and recede on the other,
according to the relative situation of the planes.
(629.) This is the case with the planetary orbits. They do not all
section of the tangent be with AG, which as 5 passes through a to d, recedes from A
Fig. 82.
©
A. / →
(2.
d
to a, rests there for an instant, and then advances again.

MOTION OF THE NODES. 343
intersect each other in a common node. Although perfectly true, there-
fore, that the node of any one planet would recede on the orbit of any and
each other by the individual action of that other, yet, when all act to-
gether, recess on one plane may be equivalent to advance on another, so
that the motion of the node of any one orbit on a given plane, arising from
their joint action, taking into account the different situations of all the
planes, becomes a curiously complicated phaenomenon whose law cannot
be very easily expressed in words, though reducible to strict numerical
statement, being, in fact, a mere geometrical result of what is above
shown. -
(630.) The nodes of all the planetary orbits on the true ecliptic, as a
matter of fact, are retrograde, though they are not all so on a fixed plane,
such as we may conceive to exist in the planetary system, and to be a
plane of reference unaffected by their mutual disturbances. It is, how-
ever, to the ecliptic, that we are under the necessity of referring their
movements from our station in the system; and if we would transfer our
ideas to a fixed plane, it becomes necessary to take account of the varia-
tion of the ecliptic itself, produced by the joint action of all the planets.
(631.) Owing to the smallness of the masses of the planets, and their
great distances from each other, the revolutions of their nodes are exces-
sively slow, being in every case less than a single degree per century, and
in most cases not amounting to half that quantity. It is otherwise with
the moon, and that owing to two distinct reasons. First, that the disturb-
ing force itself arising from the sun’s action, (as appears from the table
given in art. 612,) bears a much larger proportion to the earth's central
attraction on the moon than in the case of any planet disturbed by any
other. And secondly, because the synodic revolution of the moon,
within which the average is struck (and always on the side of recess), is
only 293 days, a period much shorter than that of any of the planets,
and vastly so than that of several among them. All this is agreeable to
what has already been stated (art. 407,408,) respecting the motion of the
moon's nodes, and it is hardly necessary to mention that, when calculated,
as it has been, d priori, from an exact estimation of all the acting forces,
the result is found to coincide with perfect precision with that immediately
derived from observation, so that not a doubt can subsist as to this being
the real process by which so remarkable an effect is produced.
(632.) So far as the physical condition of each planet is concerned, it
is evident that the position of their nodes can be of little importance. It
is otherwise with the mutual inclinations of their orbits with respect to
each other, and to the equator of each. A variation in the position of the
ecliptic, for instance, by which its pole should shift its distance from the
344 OUTLINES OF ASTRONOMY.
pole of the equator, would disturb our seasons. Should the plane of the
earth's orbis, for instance, ever be so changed as to bring the ecliptic to
coincide with the equator, we should have perpetual spring over all the
world; and on the other hand, should it coincide with a meridian, the
extremes of summer and winter would become intolerable. The inquiry,
then, of the variations of inclination of the planetary orbits inter se, is
one of much higher practical interest than those of their nodes.
(633) Referring to the figures of art. 610, et seq., it is evident that
the plane S P q, in which the disturbed body moves during an instant of
time from its quitting P, is differently inclined to the orbit of M, or to a
fixed plane, from the original or undisturbed plane PSp. The difference
of absolute position of these two planes in space is the angle between the
planes P S R and P S r, and is therefore calculable by spherical trigono-
metry, when the angle R S r or the momentary recess of the node is
known, and also the inclination of the planes of the orbits to each other.
We perceive, then, that between the momentary change of inclination,
and the momentary recess of the node, there exists an intimate relation,
and that the research of the one is in fact bound up in that of the other.
This may be, perhaps, made clearer, by considering the orbit of P to be
not merely an imaginary line, but an actual circle or elliptic hoop of some
rigid material, without inertia, of which, as on a wire, the body P may
slide as a bead. It is evident that the position of this hoop will be deter-
mined at any instant, by its inclination to the ground plane to which it
is referred, and by the place of its intersection therewith, or node. It
will also be determined by the momentary direction of P's motion, which
(having no inertia) it must obey; and any change by which P should, in
the next instant, alter its orbit, would be equivalent to a shifting, bodily,
of the whole hoop, changing at once its inclination and nodes.
(634.) One immediate conclusion from what has been pointed out
above, is that where the orbits, as in the case of the planetary system and
the moon, are slightly inclined to one another, the momentary variations
of the inclination are of an order much inferior in magnitude to those in
the place of the node. This is evident on a mere inspection of our figure,
the angle R P r being, by reason of the small inclination of the planes
SPR and R S r, necessarily much smaller than the angle R S r. In pro-
portion as the planes of the orbits are brought to coincidence, a very tri-
fling angular movement of Pp about P S as an axis will make a great
variation in the situation of the point r, where its prolongation intersects
the ground plane.
(635.) Referring to the figure of art. 622, we perceive that although
the motion of the node is retrograde whenever the momentary disturbed
CHANGE OF INCLINATION. 345
arc PQ lies between the planes C A and C G A of the two orbits, and
vice versä, indifferently whether P be in the act of receding from the
plane C A, as in the quadrant C G, or of approaching to it, as in G A,
yet the same identity as to the character of the change does not subsist
in respect of the inclination. The inclination of the disturbed orbit (i. e.
of its momentary element) PQ or PQ', is measured by the spherical angle
Pr H or Pr' H. Now in the quadrant C G, Pr H is less, and Pr' H
greater than PC H; but in G A, the converse. Hence this rule:—1st,
If the disturbing force urge P towards the plane of M's orbit, and the
undisturbed motion of P carry it also towards that plane; and 2dly, if the
disturbing force urge P from that plane, while P's undisturbed motion also
carries it from it, in either case the inclination momentarily increases; but
if, 3dly, the disturbing force act to, and P's motion carry it from — or if
the force act from, and the motion carry it to, that plane, the inclination
momentarily diminishes. Or (including all the cases under one alternative)
if the action of the disturbing force and the undisturbed motion of P with
reference to the plane of M's orbit be of the same character, the inclina-
tion increases; if of contrary characters, it diminishes.
(636.) To pass from the momentary changes which take place in the
relations of nature to the accumulated effects produced in considerable
lapses of time by the continued action of the same causes, under circum-
stances varied by these very effects, is the business of the integral calculus.
Without going into any calculations, however, it will be easy for us to
demonstrate from the principles above laid down, the leading features of
this part of the planetary theory, viz. the periodic nature of the change
of the inclinations of two orbits to each other, the re-establishment of their
original values, and the consequent oscillation of each plane about a certain
mean position. As in explaining the motion of the nodes, we will com-
mence, as the simplest case, with that of an exterior planet disturbed by
an interior one at less than half its distance from the central body. Let
A C A' be the great circle of the heavens into which M's orbit seen from
S is projected, extended into a straight line, and Ag Ch A' the corre.
sponding projection of the orbit of P so seen. Let M occupy some fixed
situation, suppose in the semicircle A C, and let P describe a complete
revolution from A through g C h to A'. Then while it is between A and
g or in its first quadrant, its motion is from the plane of M's orbit, and
at the same time the orthogonal force acts from that plane: the inclina-
tion, therefore, (art. 635) increases. In the second quadrant the motion
of P is to, but the force continues to act from, the plane, and the inclina-
tion again decreases. A similar alternation takes place in its course
through the quadrants Ch and h A. Thus the plane of P's orbit oscil-
346 OUTLINES OF ASTRONOMY.
Fig. 84.

lates to and fro about its mean position twice in each revolution of P.
During this process if M held a fixed position at G, the forces being
symmetrically alike on either side, the extent of these oscillations would be
exactly equal, and the inclination at the end of one revolution of P would
revert precisely to its original value. But if M be elsewhere, this will
not be the case, and in a single revolution of P, only a partial compensa-
tion will be operated, and an overplus on the side, suppose of diminution,
will remain outstanding. But when M comes to M', a point equidistant
from G on the other side, this effect will be precisely reversed (supposing
the orbits circular). On the average of both situations, therefore, the
effect will be the same as if M were divided into two equal portions, one
placed at M and the other at M', which will annihilate the preponderance
in question and effect a perfect restoration. And on an average of all
possible situations of M, the effect will in like manner be the same as if
its mass were distributed over the whole circumference of its orbit, forming
a ring, each portion of which will exactly destroy the effect of that simi-
larly situated on the opposite side of the line of nodes.
(673.) The reasoning is precisely similar for the more complicated
cases of arts. (625) and (627.) Suppose that owing either to the
proximity of the two orbits, (in the case of an exterior disturbed planet)
or to the disturbed orbit being interior to the disturbing one, there were
a larger or less portion, d e, of P's orbit in which these relations were
reversed. Let M be the position of M' corresponding to de, then taking
G. M'-G M, there will be a similar portion d'e' bearing precisely the
same reversed relation to M', and therefore, the actions of M' M, will
equally neutralize each other in this as in the former state of things.
(638.) To operate a complete and rigorous compensation, however, it
is necessary that M should be presented to P in every possible configura-
tion, not only with respect to Pitself, but to the line of nodes, to the
position of which line the whole reasoning bears reference. In the case
of the moon for example, the disturbed body (the moon) revolves in
274-322, the disturbing (the sun) in 365°-256, and the line of nodes in
67934-391, numbers in proportion to each other about as 1 to 13 and 249
respectively. Now in 13 revolutions of P, and one of M, if the node
remained fixed, P would have been presented to M so nearly in every
CHANGE OF INCLINATION. 347
configuration as to operate an almost exact compensation But in 1
revolution of M, or 13 of P, the node itself has shifted º, or about ºn
of a revolution, in a direction opposite to the revolutions of M and P, so
that although P has been brought back to the same configuration with
respect to M, both are ſº of a revolution in advance of the same configu.
ration as respects the node. The compensation, therefore, will not be
exact, and to make it so, this process must be gone through 19 times, at
the end of which both the bodies will be restored to the same relative
position, not only with respect to each other, but to the node. The
fractional parts of entire revolutions, which in this explanation have been
neglected, are evidently no farther influential than as rendering the com-
pensation thus operated in a revolution of the node slightly inexact, and
thus giving rise to a compound period of greater duration, at the end of
which a compensation almost mathematically rigorous, will have been
effected.
(639.) It is clear then, that if the orbits be circles, the lapse of a very
moderate number of revolutions of the bodies will very nearly, and that
of a revolution of the node almost exactly, bring about a perfect restora-
tion of the inclinations. If, however, we suppose the orbits excentric, it
is no less evident, owing to the want of symmetry in the distribution of
the forces, that a perfect compensation will not be effected either in one
or in any number of revolutions of P and M, independent of the motion
of the node itself, as there will always be some configuration more favour-
able to either an increase of inclination than its opposite is unfavourable.
Thus will arise a change of inclination which, were the nodes and apsides
of the orbits fixed, would be always progressive in one direction until
the planes were brought to coincidence. But, 1st, half a revolution of
the nodes would of itself reverse the direction of this progression by
making the position in question favour the opposite movement of inclina-
tion; aud, 2dly, the planetary apsides are themselves in motion with
unequal velocities, and thus the configuration whose influence destroys
the balance, is, itself, always shifting its place on the orbits. The varia-
tions of inclination dependent on the excentricities are therefore, like those
independent of them, periodical, and being, moreover, of an order more
minute (by reason of the smallness of the excentricities) than the latter,
it is evident that the total variation of the planetary inclinations must
fluctuate within very narrow limits. Geometers have accordingly demon-
strated by an accurate analysis of all the circumstances, and an exact
estimation of the acting forces, that such is the case; and this is what
is meant by asserting the stability of the planetary system as to the
mutual inclinations of its orbits. By the researches of Lagrange (of
348 OUTLINES OF ASTRONOMY.
whose analytical conduct it is impossible here to give any idea,) the
following elegant theorem has been demonstrated:— -
“If the mass of every planet be multiplied by the square root of the
major axis of its orbit, and the product by the square of the tangent of
*ts inclination to a fixed plane, the sum of all these products will be con-
stantly the same under the influence of their mutual attraction.” If the
present situation of the plane of the ecliptic be taken for that fixed plane
(the ecliptic itself being variable like the other orbits), it is found that
this sum is actually very small: it must, therefore, always remain so.
This remarkable theorem alone, then, would guarantee the stability of the
orbits of the greater planets; but from what has above been shown of the
teadency of each planet to work out a compensation on every other, it is
evident that the minor ones are not excluded from this beneficial arrange-
ment.
(640.) Meanwhile, there is no doubt that the plane of the ecliptic does
actually vary by the actions of the planets. The amount of this variation
is about 48" per century, and has long been recognized by astronomers,
by an increase of the latitudes of all the stars in certain situations, and
their diminution in the opposite regions. Its effect is to bring the ecliptic
by so much per annum nearer to coincidence with the equator; but from
what we have above seen, this diminution of the obliquity of the ecliptic
will not go on beyond certain very moderate limits, after which (although
in an immense period of ages, being a compound cycle resulting from the
joint action of all the planets,) it will again increase, and thus oscillate
backward and forward about a mean position, the extent of its deviation
to one side and the other being less than 1% 21'.
(641.) One effect of this variation of the plane of the ecliptic,-that
which causes its nodes on a fixed plane to change,_is mixed up with the
precession of the equinoxes, and undistinguishable from it, except in
theory. This last-mentioned phaenomenon is, however, due to another
cause, analogous, it is true, in a general point of view, to those above
considered, but singularly modified by the circumstances under which it
is produced. We shall endeavour to render these modifications intelli-
gible, as far as they can be made so without the intervention of analytical
formulae.
(642.) The precession of the equinoxes, as we have shown in art. 312,
consists in a continual retrogradation of the node of the earth's equator on
the ecliptic; and is, therefore, obviously an effect so far analogous to the
general phaenomenon of the retrogradation of the nodes of the orbits on
each other. The immense distance of the planets, however, compared
with the size of the earth, and the smallness of their masses compared to
PRECESSION OF THE EQUINOXES. 349
that of the sun, puts their action out of the question in the inquiry of its
cause, and we must, therefore, look to the massive though distant sun,
and to our near though minute neighbour, the moon, for its explanation.
This will, accordingly, be found in their disturbing action on the redun-
dant matter accumulated on the equator of the earth, by which its figure
is rendered spheroidal, combined with the earth's rotation on its axis. It
is to the sagacity of Newton that we owe the discovery of this singular
mode of action.
(643.) Suppose in our figure (art. 611,) that instead of one body, P,
revolving round S, there were a succession of particles not coherent, but
forming a kind of fluid ring, free to change its form by any force applied.
Then, while this ring revolved round S in its own plane, under the dis-
turbing influence of the distant body M, (which now represents the moon
or the sun, as P does one of the particles of the earth's equator,) two
things would happen: 1st, its figure would be bent out of a plane into an
undulated form, those parts of it within the arcs D A and E C being ren-
dered more inclined to the plane of M's orbit, and those within the arcs
A E, CD, less so than they would otherwise be; 2dly, the nodes of this
ring, regarded as a whole, without respect to its change of figure, would
retreat upon that plane.
(644.) But suppose this ring, instead of consisting of discrete mole
cules free to move independently, to be rigid and incapable of such flexure,
like the hoop we have supposed in art. 633, but having inertia, then it is
evident that the effort of those parts of it which tend to become more
inclined will act through the medium of the ring itself (as a mechanical
engine or lever) to counteract the effort of those which have at the same
tnstant a contrary tendency. In so far only, then, as there exists an excess
on the one or the other side will the inclination change, an average being
struck at every moment of the ring's motion; just as was shown to
happen in the view we have taken of the inclinations, in every complete
revolution of a single disturbed body, under the influence of a fixed dis-
turbing one.
(645.) Meanwhile, however, the nodes of the rigid ring will retrograde,
the general or average tendency of the nodes of every molecule being to
do so. Here, as in the other case, a struggle will take place by the coun-
teracting efforts of the molecules contrarily disposed, propagated through
the solid substance of the ring; and thus at every instant of time, an
average will be struck, which being identical in its nature with that
effected in the complete revolution of a single disturbed body, will, in
every case, be in favour of a recess of the node, save only when the dis.
350 OUTLINES OF ASTRONOMY.
turbing body, be it sun or moon, is situated in the plane of the earth's
equator.
(646.) This reasoning is evidently independent of any consideration
of the cause which maintains the rotation of the ring; whether the par-
ticles be small satellites retained in circular orbits under the equilibrated
action of attractive and centrifugal forces, or whether they be small masses
conceived as attached to a set of imaginary spokes, as of a wheel, center-
ing in S, and free only to shift their planes by a motion of those spokes
perpendicular to the plane of the wheel. This makes no difference in
the general effect; though the different velocities of rotation, which may
be impressed on such a system, may and will have a very great influence
both on the absolute and relative magnitudes of the two effects in ques-
tion — the motion of the nodes and change of inclination. This will be
easily understood, if we suppose the ring without a rotatory motion, in which
extreme case it is obvious that so long as M remained fixed there would
take place no recess of nodes at all, but only a tendency of the ring to tilt
its plane round a diameter perpendicular to the position of M, bringing it
towards the line S M.
(647.) The motion of such a ring, then, as we have been considering,
would imitate, so far as the recess of the nodes goes, the precession of the
equinoxes, only that its nodes would retrograde far more rapidly than the
observed precession, which is excessively slow. But now conceive this
ring to be loaded with a spherical mass enormously heavier than itself,
placed concentrically within it, and cohering firmly to it, but indifferent,
or very nearly so, to any such cause of motion; and suppose, moreover,
that, instead of one such ring there are a vast multitude heaped together
around the equator of such a globe, so as to form an elliptical protube-
rance, enveloping it like a shell on all sides, but whose mass, taken toge-
ther, should form but a very minute fraction of the whole spheroid. We
have now before us a tolerable representation of the case of nature;' and
it is evident that the rings, having to drag round with them in their nodal
* That a perfect sphere would be so inert and indifferent as to a revolution of the
nodes of its equator under the influence of a distant attracting body appears from this,
— that the direction of the resultant attraction of such a body, or of that single force
which, opposed, would neutralize and destroy its whole action, is necessarily in a line
passing through the centre of the sphere, and, therefore, can have no tendency to turn
the sphere one way or other. It may be objected by the reader, that the whole sphere
may be conceived as consisting of rings parallel to its equator, of every possible dia-
meter, and that, therefore, its nodes should retrograde even without a protuberant
equatcr. The inference is incorrect, but our limits will not allow us to go into an ex-
position of the fallacy. We should, however, caution him, generally, that no dyna-
mica, subject is open to more mistakes of this kind, which nothing but the closest
attention, in every varied point of view, will detect.
PRECESSION AND NUTATION EXPLAINED. 351
revolution this great inert mass, will have their velocity of retrogradation
proportionally diminished. Thus, then, it is easy to conceive how a mo-
tion similar to the precession of the equinoxes, and, like it, characterized
by extreme slowness, will arise from the causes in action.
(648.) Now a recess of the node of the earth's equator, upon a given
plane, corresponds to a conical motion of its axis round a perpendicular to
that plane. But in the case before us, that plane is not the ecliptic, but
the moon's orbit for the time being; and it may be asked how we are to
reconcile this with what is stated in art. 317 respecting the nature of the
motion in question. To this we reply, that the nodes of the lunar orbit,
being in a state of continual and rapid retrogradation, while its inclination
is prescrved nearly invariable, the point in the sphere of the heavens round
which the pole of the earth's equator revolves (with that extreme slow-
ness characteristic of the precession) is itself in a state of continual circu-
lation round the pole of the ecliptic, with that much more rapid motion
which belongs to the lunar node. A glance at the annexed figure will
Fig. 85.
C
--|->'6
B C d.
explain this better than words. P is the pole of the ecliptic, A the pole
of the moon's orbit, moving round the small circle A B C D in 19 years;
a the pole of the earth's equator, which at each moment of its progress
has a direction perpendicular to the varying position of the line A a, and
a velocity depending on the varying intensity of the acting causes during
the period of the nodes. This velocity however being extremely small,
when A comes to B, C, D, E, the line A a will have taken up the posi-
tions B b, C c, D d, E e, and the earth's pole a will thus, in one tropical
revolution of the node, have arrived at e, having described not an exactly
circular are a e, but a single undulation of a wave-shape or epicycloidal
curve, a b c de, with a velocity alternately greater and less than its mean

352 OUTLINES OF ASTRONOMY.
motion, and this will be repeated in every succeeding revolution of the
node. -
(649.) Now this is precisely the kind of motion which, as we have
seen in art. 325, the pole of the earth's equator really has round the pole
of the ecliptic, in consequence of the joint effects of precession and nuta-
tion, which are thus uranographically represented. If we superadd to the
effect of lunar precession that of the solar, which alone would cause the
pole to describe a circle uniformly about P, this will only affect the undu-
lations of our waved curve, by extending them in length, but will produce
no effect on the depth of the waves, or the excursions of the earth's axis
to and from the pole of the ecliptic. Thus we see that the two phenomena
of nutation and precession are intimately connected, or rather both of
them essential constituent parts of one and the same phenomenon. It is
hardly necessary to state that a rigorous analysis of this great problem, by
an exact estimation of all the acting forces and summation of their dy-
namical effects, leads to the precise value of the co-efficients of precession
and mutation, which observation assigns to them. The solar and lunar
portions of the precession of the equinoxes, that is to say, those portions
which are uniform, are to each other in the proportion of about 2 to 5.
(650.) In the nutation of the earth's axis we have an example (the
first of its kind which has occurred to us), of a periodical movement in
one part of the system, giving rise to a motion having the same precise
period in another. The motion of the moon’s modes is here, we see,
represented, though under a very different form, yet in the same exact
periodic time, by a movement of a peculiar oscillatory kind impressed on
the solid mass of the earth. We must not let the opportunity pass of
generalizing the principle involved in this result, as it is one which we
shall find again and again exemplified in every part of physical astronomy,
nay, in every department of natural science. It may be stated as “the
principle of forced oscillations, or of forced vibrations,” and thus gene-
rally announced :-
If one part of any system connected either by material ties, or by the
mutual attractions of its members, be continually maintained by any
cause, whether inherent in the constitution of the system or external to it,
in a state of regular periodic motion, that motion will be propagated
throughout the whole systems, and will give rise, in every member of it
and in every part of each member, to periodic movements eacecuted in
equal period, with that to which they owe their origin, though not neces-
sarily synchronous with them in their maacima and minima.' -
* See a demonstration of this theorem for the forced vibrations of systems connected
by material ties of imperfect elasticity, in my treatise on Sound, Encyc. Metrop. art.
323. The demonstration is easily extended and generalized to take in other systems.
PRINCIPLE OF FORCED VIBRATIONS. 353
The system may be favourably or unfavourably constituted for such a
transfer of periodic movements, or favourably in some of its parts and
unfavourably in others; and accordingly as it is the one or the other, the
derivative oscillation (as it may be termed) will be imperceptible in one
case, of appreciable magnitude in another, and even more perceptible in
its visible effects than the original cause in a third; of this last kind we have
an instance in the moon’s acceleration, to be hereafter noticed.
(651.) It so happens that our situation on the earth, and the delicacy
which our observations have attained, enable us to make it as it were an
instrument to feel these forced vibrations,— these derivative motions,
communicated from various quarters, especially from our near neighbour,
the moon, much in the same way as we detect, by the trembling of a
board beneath us, the secret transfer of motion by which the sound of an
organ-pipe is dispersed through the air, and carried down into the earth.
Accordingly, the monthly revolution of the moon, and the annual motion
of the sun, produce, each of them, small mutations in the earth's axis,
whose periods are respectively half a month and half a year, each of
which, in this view of the subject, is to be regarded as one portion of a
period consisting of two equal and similar parts. But the most remark-
able instance, by far, of this propagation of periods, and one of high
importance to mankind, is that of the tides, which are forced oscillations,
excited by the rotation of the earth in an ocean disturbed from its figure
by the varying attractions of the sun and moon, each revolving in its own
orbit, and propagating its own period into the joint phenomenon. The
explanation of the tides, however, belongs more properly to that part of
the general subject of perturbations which treats of the action of the
radial component of the disturbing force, and is therefore postponed to a
subsequent chapter. -
354 OUTLINES OF ASTRONOMY.
CHAPTER XIII.
THEORY OF THE AXES, PERIHELIA, AND ExCENTRICITIES.
VARIATION OF ELEMENTS IN GENERAL. — DISTINCTION BETWEEN PE-
RIODIC AND SECULAR VARIATIONS. — GEOMETRICAI, EXPRESSION OF
TANGENTIAL AND NORMAL FORCES. — VARIATION OF THE MAJOR.
AxIS PRODUCED ONLY BY THE TANGENTIAL FORCE. —LAGRANGE's
THEOREM OF THE CONSERVATION OF THE MEAN DISTANCES AND
PERIODS. — THEORY OF THE PERIIIELIA AND EXCENTRICITIES. —
GEOMETRICAT, REPRESENTATION OF THEIR. MOMENTARY WARIA-
TIONS. — ESTIMATION OF THE DISTURBING FORCES IN NEARLY
CIRCULAR ORBITs. – APPLICATION TO THE CASE OF THE MOON.—
THEORY OF THE LUNAR, APSIDES AND EXCENTRIC1TY. – EXIPERI-
MENTAL, ILT, USTRATION. — APPLICATION OF THE FOREGOING PRIN-
CIPLES TO THE PLANETARY THEORY. — COMPENSATION IN ORBITS
VERY NEARLY CIRCUILAR. — EFFECTS OF ELI, IPTICITY. — GENERAL
RESULTs. – LAGRANGE's THEOREM OF THE STABILITY OF THE
FXCENTRICITIES. -
(652.) IN the foregoing chapter we have sufficiently explained the action
of the orthogonal component of the disturbing force, and traced it to its
results in a continual displacement of the plane of the disturbed orbit, in
virtue of which the nodes of that plane alternately advance and recede
upon the plane of the disturbing body’s orbit, with a general preponde-
rance on the side of advance, so as after the lapse of a long period to
cause the nodes to make a complete revolution and come round to their
former situation. At the same time the inclination of the plane of the
disturbed motion continually changes, alternately increasing and diminish-
ing; the increase and diminution, however, compensating each other,
nearly in single revolutions of the disturbed and disturbing bodies, more
exactly in many, and with perfect accuracy in long periods, such as those
of a complete revolution of the nodes and apsides. In the present and
following chapters we shall endeavour to trace the effects of the other
components of the disturbing force, — those which act in the plane (for
WARIATION OF ELEMENTS. 355
the time being) of the disturbed orbit, and which tend to derange the
elliptic form of the orbit, and the laws of elliptic motion in that plane.
The small inclination, generally speaking, of the orbits of the planets and
satellites to each other, permits us to separate these effects in theory one
from the other, and thereby greatly to simplify their consideration. Ac-
cordingly, in what follows, we shall throughout neglect the mutual incli-
nation of the orbits of the disturbed and disturbing bodies, and regard all
the forces as acting and all the motions as performed in one plane.
(653.) In considering the changes induced by the mutual action of two
bodies, in different aspects with respect to each other, on the magnitudes
and forms of their orbits, and in their positions therein, it will be proper
in the first instance to explain the conventions under which geometers
and astronomers have alike agreed to use the language and laws of the
elliptic system, and to continue to apply them to disturbed orbits, although
those orbits so disturbed are no longer, in mathematical strictness, ellipses,
or any known curves. This they do, partly on account of the convenience
of conception and calculation which attaches to this system, but much
more for this reason, — that it is found, and may be demonstrated from
the dynamical relations of the case, that the departure of each planet from
its ellipse, as determined at any epoch, is capable of being truly repre-
sented, by supposing the ellipse itself to be slowly variable, to change its
magnitude and excentricity, and to shift its position and the plane in
which it lies according to certain laws, while the planet all the time con-
tinues to move in this ellipse, just as it would do if the ellipse remained
invariable and the disturbing forces had no existence. By this way of
considering the subject, the whole effect of the disturbing forces is regarded
as thrown upon the orbit, while the relations of the planet to that orbit
remain unchanged. This course of procedure, indeed, is the most natural,
and is in some sort forced upon us by the extreme slowness with which
the variation of the elements, at least where the planets only are con-
cerned, develop themselves. For instance, the fraction expressing the
excentricity of the earth's orbit changes no more than 0.00004 in its
amount in a century; and the place of its perihelion, as referred to the
sphere of the heavens, by only 19' 39" in the same time. For several
years, therefore, it would be next to impossible to distinguish between an
ellipse so varied and one that had not varied at all; and in a single revo-
lution, the difference between the original ellipse and the curve really
represented by the varying one, is so excessively minute, that, if accu.
rately drawn on a table, six feet in diameter, the nicest examination, with
microscopes, continued along the whole outlines of the two curves, would
hardly detect any perceptible interval between them. Not to call a mo-
356 oUTLINES OF ASTRONOMY.
tion so minutely conforming itself to an elliptic curve, elliptic, would be
affectation, even granting the existence of trivial departures, alternately on
one side or on the other; though on the other hand, to neglect a varia-
tion, which continues to accumulate from age to age, till it forces itself on
our notice, would be wilful blindness.
(654.) Geometers, then, have agreed, in each single revolution, or for
any moderate interval of time, to regard the motion of each planet as
elliptic, and performed according to Kepler's laws, with a reserve in
favour of those very small and transient fluctuations which take place
within that time, but at the same time to regard all the elements of each
ellipse as in a continual, though extremely slow, state of change; and, in
tracing the effects of perturbation on the system, they take account prin-
cipally, or entirely, of this change of the elements, as that upon which any
material change in the great features of the system will ultimately depend.
(655.) And here we encounter the distinction between what are termed
secular variations, and such as are rapidly periodic, and are compensated
in short intervals. In our exposition of the variation of the inclination
of a disturbed orbit (art. 636,) for instance, we showed that, in each
single revolution of the disturbed body, the plane of its motion underwent
fluctuations to and fro in its inclination to that of the disturbing body,
which nearly compensated each other; leaving, however, a portion out-
standing, which again is nearly compensated by the revolution of the dis-
turbing body, yet still leaving outstanding and uncompensated a minute
portion of the change which requires a whole revolution of the node to
compensate and bring it back to an average or mean value. Now, the
two first compensations which are operated by the planets going through
the succession of configurations with each other, and therefore in compara-
tively short periods, are called periodic variations; and the deviations thus
compensated are called inequalities depending on configurations; while
the last, which is operated by a period of the node (one of the elements,)
has nothing to do with the configurations of the individual planets, requires
a very long period of time for its consummation, and is, therefore, distin-
guished from the former by the term secular variation.
(656.) It is true, that, to afford an exact representation of the motions
of a disturbed body, whether planet or satellite, both periodical and
secular variations, with their corresponding inequalities, require to be
expressed; and, indeed, the former even more than the latter; seeing that
the secular inequalities are, in fact, nothing but what remains after the
mutual destruction of a much larger amount (as it very often is) of peri-
odical. Bnt these are in their nature transient and temporary: they
disappear in short periods, and leave no trace. The planet is temporarily
GEOMETRICAL ExPRESSION OF FORCE. 357
drawn from its orbit (its slowly varying orbit,) but forthwith returns to
it, to deviate presently as much the other way, while the varied orbit
accommodates and adjusts itself to the average of these excursions on
either side of it; and thus continues to present, for a succession of indefi-
nite ages, a kind of medium picture of all that the planet has been doing
in their lapse, in which the expression and the character is preserved; but
the individual features are merged and lost. These periodic inequalities,
however, are, as we have observed, by no means neglected, but it is more
convenient to take account of them by a separate process, independent
of the secular variations of the elements.
(657.) In order to avoid complication, while endeavouring to give the
reader an insight into both kinds of variations, we shall henceforward
conceive all the orbits to lie in one plane, and confine our attention to the
case of two only, that of the disturbed and disturbing body, a view of the
subject which (as we have seen) comprehends the case of the moon dis-
turbed by the sun, since any one of the bodies may be regarded as fixed
at pleasure, provided we conceive all its motions transferred in a contrary
Fig. 86.
direction to each of the others. Let therefore A PB be the undisturbed
elliptic orbit of a planet P; M a disturbing body, join MP, and supposing
M K= M S take M N : M K :: M K* : M Pº. Then if SN be joined,
NS will represent the disturbing force of M or P, on the same scale that
S M represents M’s attraction on S. Suppose ZP Y a tangent at P, SY
perpendicular to it, and NT, NL perpendicular respectively to SY and
PS produced. Then will NT represent the tangential, TS the normal,
NL the transversal, and L S the radial components of the disturbing
force. In circular orbits or orbits only slightly elliptic, the directions

858 - OUTLINES OF ASTRONOMY.
PSL and SY are nearly coincident, and the former pair of forces will
differ but slightly from the latter. We shall here, however, take the
general case, and proceed to investigate in an elliptic orbit of any degree
of excentricity the momentary changes produced by the action of the dis-
turbing force in those elements on which the magnitude, situation, and
form of the orbit depend (i. e. the length and position of the major axis
and the excentricity,) in the same way as in the last chapter we deter-
mined the momentary changes of the inclination and node similarly pro-
duced by the orthogonal force.
(658.) We shall begin with the momentary variation in the length of
the axis, an element of the first importance, as on it depends (art. 487)
thé periodic time and mean angular motion of the planet, as well as the
average supply of light and heat it receives in a given time from the sun,
any permanent or constantly progressive change in which would alter
most materially the conditions of existence of living beings on its surface.
Now it is a property of elliptic motion performed under the influence of
gravity, and in conformity with Kepler's laws, that if the velocity with
which a planet moves at any point of its orbit be given, and also the
distance of that point from the sun, the major axis of the orbit is thereby
also given. It is no matter in what direction the planet may be moving
at that moment. This will influence the excentricity and the position of
its ellipse, but not its length. This property of elliptic motion has been
demonstrated by Newton, and is one of the most obvious and elementary
conclusions from his theory. Let us now consider a planet describing an
indefinitely small arc of its orbit about the sun, under the joint influence
of its attraction, and the disturbing power of another planet. This are
will have some certain curvature and direction, and, therefore, may be
considered as an arc of a certain ellipse described about the sun as a
focus, for this plain reason, that whatever be the curvature and direction
of the are in question, an ellipse may always be assigned, whose focus
shall be in the sun, and which shall coincide with it throughout the whole
interval (supposed indefinitely small) between its extreme points. This
is a matter of pure geometry. It does not follow, however, that the
ellipse thus instantaneously determined will have the same elements as
that similarly determined from the arc described in either the previous or
the subsequent instant. If the disturbing force did not exist, this would
be the case; but, by its action, a variation of the element from instant to
instant is produced, and the ellipse so determined is in a continual state
of change. Now when the planet has reached the end of the small arc
under consideration, the question whether it will in the next instant
describe an are of an ellipse having the same or a varied axis will depend,
WARIATION OF THE MAJOR AXIS. 359
not on the new direction impressed upon it by the acting forces, for the
axis, as we have seen, is independent of that direction,-not on its change
of distance from the sun, while describing the former arc, - for the ele-
ments of that arc are accommodated to it, so that one and the same axis
must belong to its beginning and its end. The question, in short, whether
in the next arc it shall take up a new major axis or go on with the old
one will depend solely on this—whether its velocity has or has not under-
gone a change by the action of the disturbing force. For the central
force residing in the focus can impress on it no such change of velocity
as to be incompatible with the permanence of its ellipse, seeing that it is
by the action of that force that the velocity is maintained in that due
proportion to the distance which elliptic motion, as such, requires.
(659.) Thus we see that the momentary variation of the major axis
depends on notii:g but the momentary deviation from the law of elliptic
velocity produced by the disturbing force, without the least regard te the
direction in which that extraneous velocity is impressed, or the distance
from the sun at which the planet may be situated, at the moment of its
impression. Nay, we may even go farther, for, as this holds good at every
instant of its motion, it will follow that after the lapse of any time, how-
ever great, the total amount of change which the axis may have under-
gone will be determined only by the total deviation produced by the action
of the disturbing force in the velocity of the disturbed body from that
which it would have had in its undisturbed ellipse, at the same distance
from the centre, and that therefore the total amount of change produced
in the axis in any lapse of time may be estimated, if we know at every
instant the efficacy of the disturbing force to alter the velocity of the
body's motion, and that without any regard to the alterations which the
action of that force may have produced in the other elements of the
motion in the same time.
(660.) Now it is not the whole disturbing force which is effective in
changing P's velocity, but only its tangential component. The normal
component tends merely to alter the curvature of the orbit or to deflect it
into conformity with a circle of curvature of greater or lesser radius, as
the case may be, and in no way to alter the velocity. Hence it appears
that the variation of the length of the axis is due entirely to the tangen-
tial force, and is quite independent on the normal. Now it is easily shown
that as the velocity increases, the axis increases (the distance remaining
unaltered') though not in the same exact proportion. Hence it follows
* If a be the semiaxis, r the radius vector, and v the velocity of P in any point of an
e tº a - 2 1 ſº e - &
ellipse, a is given by the relation v=F-3, the units of velocity and force being pro-
perly assumed.
360 OUTLINES OF ASTRONOMY.
that if the tangential disturbing force conspires with the motion of P, its
momentary action increases the axis of the disturbed orbit, whatever be
the situation of P in its orbit, and vice versä.
(661.) Let A S B (fig, art. 657) be the major axis of the ellipse A P
b, and on the opposite side of A B take two points P' and M', similarly
situated with respect to the axis with P and M on their side. Then if at
P' and M' bodies equal to P and M be placed, the forces exerted by M’
on P' and S will be equal to those exerted by M on P and S, and there-
fore the tangential disturbing force of M' on P' exerted in the direction
P’ Z' (suppose) will equal that exerted by M on P in the direction P. Z.
P" therefore (supposing it to revolve in the same direction round S as P)
will be retarded (or accelerated, as the case may be) by precisely the same
force by which P is accelerated (or retarded), so that the variation in the
axis of the respective orbits of P and P' will be equal in amount, but con-
trary in character. Suppose now M's orbit to be circular. Then (if the
periodic times of M and P be not commensurate, so that a moderate
number of revolutions may bring them back to the same precise relative
positions) it will necessarily happen, that in the course of a very great
number of revolutions of both bodies, P will have been presented to M
on one side of the axis, at some one moment, in the same manner as at
some other moment on the other. Whatever variation may have been
effected in its axis in the one situation will have been reversed in that
symmetrically opposite, and the ultimate result, on a general average of
an infinite number of revolutions, will be a complete and exact compen-
sation of the variations in one direction by those in the direction opposite.
(662.) Suppose, next, P's orbit to be circular. If now M’s orbit were
so also, it is evident that in one complete synodic revolution, an exact
restoration of the axis to its original length would take place, because the
tangential forces would be symmetrically equal and opposite during each
alternate quarter revolution. But let M, during a synodic revolution,
have receded somewhat from S, then will its disturbing power have become
gradually weaker, so that, in a synodic revolution, the tangential force in
each quadrant, though reversed in direction being inferior in power, an
exact compensation will not have been effected, but there will be left an
outstanding uncompensated portion, the excess of the stronger over the
feebler effects. But now suppose M to approach by the same gradations
as it before receded. It is clear that this result will be reversed; since
... the uncompensated stronger actions will all lie in the opposite direction.
Now suppose M’s orbit to be elliptic. Then during its recess from S, or
in the half revolution from its perihelion to its aphelion, a continual un-
compensated variation will go on accumulating in one direction. But,
VARIATION OF THE MAJOR AXIS. 361
from what has been said, it is clear that this will be destroyed, during M’s.
approach to S in the other half of its orbit, so that here again, on the
average of a multitude of revolutions during which P has been presented
to M in every situation for every distance of M from S, the restoration
will be effected.
(663.) If neither P's nor M's orbit be circular, and if moreover the
directions of their axes be different, this reasoning, drawn from the sym-
metry of their relations to each other, does not apply, and it becomes
necessary to take a more general view of the matter. Among the funda-
mental relations of dynamics, relations which presuppose no particular
law of force like that of gravitation, but which express in general terms
the results of the action of force on matter during time, to produce or
change velocity, is one usually cited as the “Principle of the conserva-
tion of the vis viva,” which applies directly to the case before us. This
principle (or rather this theorem) declares that if a body subjected at
every instant of its motion to the action of forces directed to fixed centres
(no matter how numerous), and having their intensity dependent only on
the distances from their respective centres of action, travel from one point
of space to another, the velocity which it has on its arrival at the latter
point will differ from that which it had on setting out from the former, by
a quantity depending only on the different relative situations of these two
points in space, without the least reference to the form of the curve in
which it may have moved in passing from one point to the other, whether
that curve have been described freely under the simple influence of the
central forces, or the body have been compelled to glide upon it, as a bead
upon a smooth wire. Among the forces thus acting may be included any
constant forces, acting in parallel directions, which may be regarded as
directed to fixed centres infinitely distant. It follows from this theorem,
that, if the body return to the point P, from which it set out, its velocity
of arrival will be the same with that of its departure; a conclusion which
(for the purpose we have in view) sets us free from the necessity of enter-
ing into any consideration of the laws of the disturbing force, the change
which its action may have induced in the form of the orbit of P, or the
successive steps by which velocity generated at one point of its interme
diate path is destroyed at another, by the reversed action of the tangen-
tial force. Now to apply this theorem to the case in question, let M be
supposed to retain a fixed position during one whole revolution of P.
P then is acted on, during that revolution, by three forces: 1st. by the
central attraction of S directed always to S; 2nd. by that to M, always
directed to M ; 3rd. by a force equal to M’s attraction on S; but in the
dircetion MS, which therefore is a constant force, acting always in parallel
362 OuTLINES OF ASTRONOMY.
directions. On completing its revolution, then, P's velocity, and therefore
the major axis of its orbit, will be found unaltered, at least neglecting
that excessively minute difference which will result from the non-arrival
after a revolution at the exact point of its departure by reason of the per-
turbations in the orbit produced in the interim by the disturbing force,
which for the present we may neglect.
(664.) Now suppose M to revolve, and it will appear, by a reasoning
precisely similar to that of art. 662, that whatever uncompensated varia-
tion of the velocity arises in successive revolutions of P during M’s recess
from S will be destroyed by contrary uncompensated variations arising
during its approach. Or, more simply and generally thus: whatever M's
situation may be, for every place which P can have, there must exist some
other place of P (as P'), in which the action of M shall be precisely
reversed. Now if the periods be incommensurable, in an indefinite
number of revolutions of both bodies, for every possible combination of
situations (M, P) there will occur, at some time or other, the combination
(M, P’) which neutralizes the effect of the other, when carried to the
general account; so that ultimately, and when very long periods of time
are embraced, a complete compensation will be found to be worked out.
(665.) This supposes, however, that in such long periods the orbit of
M is not so altered as to render the occurrence of the compensating situ-
ation (M, P’) impossible. This would be the case if M's orbit were to
dilate or contract indefinitely by a variation in its axis. But the same
reasoning which applies to P, applies also to M. P. retaining a fixed
situation, M’s velocity, and therefore the axis of its orbit, would be ex-
actly restored at the end of a revolution of M ; so that for every position
PM there exists a compensating position PM’. Thus M’s orbit is main-
tained of the same magnitude, and the possibility of the occurrence of
the compensating situation (M, P’) is secured.
(666.) To demonstrate as a rigorous mathematical truth the complete
and absolute ultimate compensation of the variations in question, it would
be requisite to show that the minute outstanding changes due to the non-
arrivals of P and M at the same eacact points at the end of each revolu-
tion, cannot accumulate in the course of infinite ages in one direction.
Now it will appear in the subsequent part of this chapter, that the effect
of perturbation on the excentricities and apsides of the orbits is to cause
the former to undergo only periodical variations, and the latter to revolve
and take up in succession every possible situation. Hence in the course
of infinite ages, the points of arrival of P and M at fixed lines of direc-
tion, SP, SM, in successive revolutions, though at one time they will
approach S, at another will recede from it, fluctuating to and fro about
WARIATION OF THE MAJOR AXIS. 363
mean points from which they never greatly depart. And if the arrival.
of either of them at P, at a point nearer S, at the end of a complete
revolution, cause an excess of velocity, its arrival at a more distant point
will cause a deficiency, and thus, as the fluctuations of distance to and fro
ultimately balance each other, so will also the excesses and defects of
velocity, though in periods of enormous length, being no less than that
of a complete revolution of P's apsides for the one cause of inequality,
and of a complete restoration of its excentricity for the other.
(667.) The dynamical proposition on which this reasoning is based is
general, and applies equally well to cases wherein the forces act in one
plane, or are directed to centres anywhere situated in space. Hence, if
we take into consideration the inclination of P's orbit to that of M, the
same reasoning will apply. Only that in this case, upon a complete revo-
lution of P, the variation of inclination and the motion of the nodes of
B's orbit will prevent its returning to a point in the exact plane of its
original orbit, as that of the excentricity and perihelion prevent its arrival
at the same exact distance from S. But since it has been shown that the
inclination fluctuates round a mean state from which it never departs
much, and since the node revolves and makes a complete circuit, it is
obvious that in a complete period of the latter the points of arrival of P
at the same longitude will deviate as often and by the same quantities
above as below its original point of departure from exact coincidence;
and, therefore, that on the average of an infinite number of revolutions,
the effect of this cause of non-compensation will also be destroyed.
(668.) It is evident, also, that the dynamical proposition in question.
being general, and applying equally to any number of fixed centres, as
well as to any distribution of them in space, the conclusion would be pre-
cisely the same whatever be the number of disturbing bodies, only that
the periods of compensation would become more intricately involved.
We are, therefore, conducted to this most remarkable and important con-
clusion, viz. that the major axes of the planetary (and lunar) orbits, and,
consequently, also their mean motions and periodic times, are subject to
none but periodical changes; that the iength of the year, for example, in
the lapse of infinite ages, has no preponderating tendency either to increase
or diminution,-that the planets will neither recede idefinitely from the
sun, nor fall into it, but continue, so far as their mutual perturbations at
least are concerned, to revolve for ever in orbits of very nearly the same
dimensions as at present.
(669.) This theorem (the Magna Charta of our system), the discovery
of which is due to Lagrange, is justly regarded as the most important, as
a single result, of any which have hitherto rewarded the researches of
364 OlſtLINES OF ASTRONOMY.
(mathematicians in this application of their science; and it is especially
worthy of remark, and follows evidently from the view here taken of it,
that it would not be true but for the influence of the perturbing forces on
other elements of the orbit, viz. the perihelion and excentricity, and the
inclination and nodes; since we have seen that the revolution of the ap-
sides and nodes, and the periodical increase and diminution of the ex-
centricities and inclinations, are both essential towards operating that final
and complete compensation which gives it a character of mathematical
exactness. We have here an instance of a perturbation of one kind
operating on a perturbation of another to annihilate an effect which would
otherwise accumulate to the destruction of the system. It must, however,
be borne in mind, that it is the smallness of the excentricities of the more
influential planets, which gives this theorem its practical importance, and
distinguishes it from a mere barren speculative result. Within the limits
of ultimate restoration, it is this alone which keeps the periodical fluctua-
tions of the axis to and fro about a mean value within moderate and
reasonable limits. Although the earth might not fall into the sun, or re-
cede from it beyond the present limits of our system, any considerable
increase or diminution of its mean distance, to the extent, for instance, of
a tenth of its actual amount, would not fail to subvert the conditions on
which the existence of the present race of animated beings depends.
Constituted as our system is, however, changes to anything like this ex-
tent are utterly precluded. The greatest departure from the mean value
of the axis of any planetary orbit yet recognized by theory or observation
(that of the orbit of Saturn disturbed by Jupiter), does not amount to a
thousandth part of its length." The effects of these fluctuations, how-
ever, are very sensible, and manifest themselves in alternate accelerations
and retardations in the angular motions of the disturbed about the central
body, which cause it alternately to outrun and to lag behind its elliptic
place in its orbit, giving rise to what are called equations in its motion,
some of the chief instances of which will be hereafter specified when we
come to trace more particularly in detail the effects of the tangential force
in various configurations of the disturbed and disturbing bodies, and to
explain the consequences of a near approach to commensurability in their
periodic times. An exact commensurability in this respect, such, for in-
stance, as would bring both planets round to the same configuration in two
or three revolutions of one of them, would appear at first sight to destroy
one of the essential elements of our demonstration. But even supposing
* Greater deviations will probably be found to exist in the orbits of the small extra-
tropical planets. But these are too insignificant members of our system to need special
notice in a work of this nature. - -
DISPLACEMENT OF THE UPPER FOCUS. 365
such an exact adjustment to subsist at any epoch, it could not remain per-
manent, since by a remarkable property of perturbations of this class,
which geometers have demonstrated, but the reasons of which we cannot
stop to explain, any change produced on the axis of the disturbed planet's
orbit is necessarily accompanied by a change in the contrary direction in
that of the disturbing, so that the periods would recede from commensu-
rability by the mere effect of their mutual action. Cases are not wanting
in the planetary system of a certain approach to commensurability, and in
one very remarkable case (that of Uranus and Neptune) of a considerably
near one, not near enough, however, in the smallest degree to affect the
validity of the argument, but only to give rise to inequalities of very long
periods, of which more presently."
(670.) The variation of the length of the axis of the disturbed orbit
is due solely to the action of the tangential disturbing force. It is other-
wise with that of its excentricity and of the position of its axis, or, which
is the same thing, the longitude of its perillelion. Both the normal and
tangential components of the disturbing force affect these elements. We
shall, however, consider separately the influence of each, and, commencing,
Fig. 87.
as the simplest case, with that of the tangential force;—let P be the
place of the disturbed planet in its elliptic orbit A PB, whose axis at the
moment is A S B and focus S. Suppose Y PZ to be a tangent to this
orbit at P. Then, if we suppose A B = 2a, the other focus of the
ellipse, H, will be found by making the angle Z P H = Y P S or Y PH
= 180° — Y PZ, or S P H = 180°–2 YPS, and taking P H = 2 a.
— SP. This is evident from the nature of the ellipse, in which lines
drawn from any point to the two foci make equal angles with the tangent,
*41 revolutions of Neptune are nearly equal to 81 of Uranus, giving rise to an ine.
quality, having 6805 years for its period.
ºw

366 ouTLINES OF ASTRONOMY.
and have their sum equal to the major axis. Suppose, now, the tangen-
tial force to act on P and to increase its velocity. It will therefore increase
the axis, so that the new value assumed by a (viz. a') will be greater than
a. But the tangential force does not alter the angle of tangency, so that
to find the new position (H') of the upper focus, we must measure off
along the same line PH, a distance PH' (= 2a, – SP) greater than
P H. Do this then, and join S H' and produce it. Then will A'B' be
the new position of the axis, and # SH' the new excentricity. Hence we
conclude, 1st, that the new position of the perihelion A' will deviate from
the old one A towards the same side of the axis A B on which P is when
the tangential force acts to increase the velocity, whether P be moving
from perihelion to aphelion, or the contrary. 2dly, That on the same sup-
position as to the action of the tangential force, the excentricity increases
when P is between the perihelion and the perpendicular to the axis FH G.
drawn through the upper focus, and diminishes when between the aphelion
and the same perpendicular. 3dly, That for a given change of velocity,
i. e. for a given value of the tangential force, the momentary variation in
the place of the perihelion is a maximum when P is at F or G, from
which situation of P to the perihelion or aphelion, it decreases to nothing,
the perihelion being stationary when P is at A or B. 4thly, That the
variation of the excentricity due to this cause is complementary in its law
of increase and decrease to that of the perihelion, being a maximum for a
given tangential force when P is at A or B, and vanishing when at G or
F. And lastly, that where the tangential force acts to diminish the velo-
city, all these results are reversed. If the orbit be very nearly circular'
the points F, G, will be so situated that, although not at opposite extremi
ties of a diameter, the times of describing A F, FB, B G, and G. A will
be all equal, and each of course one quarter of the whole periodic time
of P.
(671.) Let us now consider the effects of the normal component of the
disturbing force upon the same elements. The direct effect of this force
is to increase or diminish the curvature of the orbit at the point P of its
action, without producing any change on the velocity, so that the length
of the axis remains unaltered by its action. Now, an increase of curva-
ture at P is synonymous with a decrease in the angle of tangency S P Y
when P is approaching towards S, and with an increase in that angle
when receding from S. Suppose the former case, and while P approaches
S (or is moving from aphelion to perihelion), let the normal force act
inwards or towards the concavity of the ellipse. Then will the tangent
PY by the action of that force have taken up the position PY'. To find
* So nearly that the cube of the excentricity may be neglected.
DISPLACEMENT OF THE UPPER, FOCUS, 367
-- Z'
the corresponding position H' taken up by the focus of the orbit so dis.
turbed, we must make the angle S P H'-180°–2 S P Y', or, which
comes to the same, draw PH' on the side of PH opposite to S, making
the angle H PH'-twice the angle of deflection Y PY’ and in PH' take
PH'-P H. Joining, then, S H' and producing it, A'S H'M' will be
the new position of the axis, A' the new perihelion, and 4 SH' the new
excentricity. Hence we conclude, 1st, that the normal force acting
Žnwards, and P moving towards the perihelion, the new direction SA’
of the perihelion is in advance (with reference to the direction of P's
revolution) of the old–or the apsides advance—when P is anywhere
situated between F and A (since when at F the point H' falls upon HM
between H and M.) When P is at F the apsides are stationary, but
when P is anywhere between M and F the apsides retrograde, H' in this
case lying on the opposite side of the axis. 2dly, That the same direc-
tions of the normal force and of P's motion being supposed, the excentri-
city increases while P moves through the whole semiellipse from aphelion
to perihelion—the rate of its increase being a maximum when P is at F,
and nothing at the aphelion and perihelion. 3dly, That these effects are
reversed in the opposite half of the orbit, A G M, in which P passes from
perihelion to aphelion or recedes from S. 4thly, That they are also
reversed by a reversal of the direction of the normal force, outwards, in
place of inwards. 5thly, That here also the variations of the excentricity
and perihelion are complementary to each other; the one variation being
most rapid when the other vanishes, and vice versá. 6thly, And lastly,
that the changes in the situation of the focus H produced by the actions
of the tangential and normal components of the disturbing force are at
right angles to each other in every situation of P, and therefore where
the tangential force is most efficacious (in proportion to its intensity) in
varying either the one or the other of the elements in question, the
normal is least so, and vice versä.

368 OUTLINES OF ASTRONOMY.
(672.) To determine the momentary effect of the whole disturbing
force, then, we have only to resolve it into its tangential and normal
components, and estimating by these principles separately the effects of
either constituent on both elements, add or substract the results according
as they conspire or oppose each other. Or we may at once make the
angle H PH" equal to twice the angle of deflection produced by the
normal force, and lay off PH"=P H+ twice the variation of a produced
in the same moment of time by the tangential force, and H” will be the
new focus. . The momentary velocity generated by the tangential force is
calculable from a knowledge of that force by the ordinary principles of
dynamics; and from this, the variation of the axis is easily derived."
The momentary velocity generated by the normal force in its own direc-
tion is in like manner caculable from a knowledge of that force, and
dividing this by the linear velocity of P at that instant, we deduce the
angular velocity of the tangent about P, or the momentary variation of
the angle of tangency S P Y, corresponding.
(673.) The following résumé of these several results in a tabular form
includes every variety of case according as P is approaching to or receding
from S; as it is situated in the are F A G of its orbit about the perihe.
lion, or in the remoter arc GM F about the aphelion, as the tangential
force accelerates or retards the disturbed body, or as the normal acts in-
words or outwards with reference to the concavity of the orbit.

EFFECTS OF THE TANGENTIAL DISTURBING FORCE.
Direction of P's mo- Situation of P in Action of Tangential sy
tion. Orbit. Force. Effect on Eigments.
Approaching S. Anywhere. Accelerating P. Apsides recede.
Ditto. Ditto. Retarding P. advance.
Receding from S. Ditto. Accelerating P. advance.
Ditto. Ditto. Retarding P. recede.
Indifferent. About Aphelion. Accelerating P. Excentr, decreases.
Ditto. Ditto. Retarding P. increases.
Ditto. About Perihelion. Accelerating P. increases.
Ditto. Ditto. Retarding P. decreases.
1 2 1 2 1 I
* + = -—vº, and 7 = --—v” ... +–- = wº—v'*=(v4-v') (v–v') or when infi.
0. r 0. 7". 0. O.
p
— 0.
nutesimal variations only are considered
=2v (v'—v) or a’—a =2a”v (v'— v)
from which it appears that the variation of the axis arising from a given variation of
velocity is independent of r, or is the same at whatever distance from S the change
takes place, and that cateris-paribus it is greater for a given change of velocity (or for
a given tangential force) in the direct ratio of the velocity itself.
EFFECTS ON THE APSIDES AND EXCENTRICITIES.
369
EFFECTS OF THE NORMAL DISTURBING FORCE.

* | |
e e y ituation o - ion of Normal t
Directºr S DOO- Si º P in Actio Fº Effect on Elements.
Iridifferent. About Aphelion. Inwards. Apsides recede.
Ditto. Ditto. Outwards. advance.
Ditto. About Perihelion. Inwards. advance.
Ditto. Ditto. Outwards. recede.
Approaching S. Anywhere. Inwards. Excentr. increases.
Ditto. Ditto. Outwards. decreases.
Receding from S. Ditto. Inwards. decreases.
Ditto. Ditto. Outwards. increases.
f ſ
(674.) From the momentary changes in the elements of the disturbed
orbit corresponding to successive situations of P and M, to conclude the
total amount of change produced in any given time is the business of the
integral calculus, and lies far beyond the scope of the present work.
Without its aid, however, and by general consideration of the periodical
recurrence of configurations of the same character, we have been able to
demonstrate many of the most interesting conclusions to which geometers
have been conducted, examples of which have already been given in the
reasoning by which the permanence of the axes, the periodicity of the
inclinations, and the revolutions of the nodes of the planetary orbits have
been demonstrated. We shall now proceed to apply similar considerations
to the motion of the apsides, and the variations of the excentricities. To
this end we must first trace the changes induced on the disturbing forces
themselves with the varying positions of the bodies, and here as in treat-
ing of the inclinations we shall suppose, unless the contrary is expressly
indicated, both orbits to be very nearly circular, without which limitation
the complication of the subject would become too embarrassing for the
reader to follow, and defeat the end of explanation.
(675.) On this supposition the directions of S P and SY, the perpen-
dicular on the tangent at P, may be regarded as coincident, and the
normal and radial disturbing forces become nearly identical in quantity,
also the tangential and transversal, by the near coincidence of the points
T and L (fig, art. 687). So far then as the intensity of the forces is con-
cerned, it will make very little difference in which way the forces are re-
solved, nor will it at all materially affect our conclusions as to the effects
of the normal and tangential forces, if in estimating their quantitative
values, we take advantage of the simplification introduced into their nu-
merical expression by the neglect of the angle PSY, i.e. by the substi-
tution for them of the radial and transversal components. The character
24
370 OUTLINES OF ASTRONOMY.
Fig. 89.
D;
of these effects depends (art. 670, 671,) on the direction in which the
forces act, which we shall suppose normal and tangential as before, and it
is only on the estimation of their quantitative effects that the error in-
duced by the neglect of this angle can fall. In the lunar orbit this angle
never exceeds 3° 10', and its influence on the quantitative estimation of
the acting forces may therefore be safely neglected in a first approxima-
tion. Now M N being found by the proportion M P : M S : : M S :
MN, N P (= M N — M. P) is also known, and therefore N L = NP.
sin N P S = NP. sin (ASP-- S M P) and I, S = PL–PS = NP.
cos N P S –PS = NP. cos (ASP-SM P)—S P become known,
which express respectively the tangential and normal forces on the same
scale that S M represents M’s attraction on S." Suppose. P to revolve in
the direction E A D B. Then, by drawing the figure in various situations
of P throughout the whole circle, the reader will easily satisfy himself —
1st. That the tangential force accelerates P, as it moves from E towards
A, and from D towards B, but retards it as it passes from A to D, and
from B to E. 2d. That the tangential force vanishes at the four points
A, D, E, B, and attains a maximum at some intermediate points.
3dly. That the normal force is directed outwards at the syzygies A, B,
and inwards at the points D, E, at which points respectively its outward
and inward intensities attain their maxima. Lastly, that this force wa-
* M s— R; S P = r ; M P = f; A S P = 0; A M P = M ; M N = #: ; N P =
*#4-(R—f) (1+}+}. ; whence we have N L = (R—f). sin (0-H. M).
(i+}+% ; L S = (R—f). co, ºr M). (-|-}+})–: When R and f,
owing to the great distance of M, are nearly equal, we have R —f = P v,;-
nearly, and the angle M may be neglected; so that we have N P = 3 PW.

DISTURBING FORCES IN CIRCULAR ORBITS. 371
nishes at points intermediate between A D, DB, B E, and E A, which
points, when M is considerably remote, are situated nearer to the quadra-
ture than the syzygies.
(676.) In the lunar theory, to which we shall now proceed to apply
these principles, both the geometrical representation and the algebraic
expression of the disturbing forces admit of great simplification. Owing
to the great distance of the sun M, at whose centre the radius of the
moon's orbit never subtends an angle of more than about 8', N P may be
regarded as parallel to A B. And D S E becomes a straight line, coinci-
dent with the line of quadratures, so that W P becomes the cosine of
A S P to radius SP, and N L = N P. sin A SP; L P = NP. cos
A SP. Moreover, in this case (see the note on the last article) N P =
3 P W = 3 SP. cos A SP; and consequently N L = 3 SP. cos A SP.
sin A S P = 3 SP. sin 2 A SP, and L S = S P (3. cos ASP”—-1)
= , S P (1+3. cos 2 A SP) which vanishes when cos A S P = }, or at
64° 14' from the syzygy. Suppose through every point of P's orbit
there be drawn S Q = 3 S P. cos A S Pº, then will Q trace out a certain
looped oval, as in the figure, cutting the orbit in four points 64° 14' from
A and B respectively, and PQ will always represent in quantity and di-
rection the normal force àcting at P.
(677.) It is important to remark here, because upon this the whole
lunar theory and especially that of the motion of the apsides hinges, that
all the acting disturbing forces, at equal angles of elongation A.S.P of the
moon from the sun, are caeteris paribus proportional to SP, the moon’s
distance from the earth, and are therefore greater when the moon is near
its apogee than when near its perigee; the extreme proportion being that
of about 28:25. This premised, let us first consider the effect of the
normal force in displacing the lunar apsides. This we shall best be ena.
bled to do by examining separately those cases in which the effects are

372 OUTLINES OF ASTRONOMY.
most strongly contrasted; viz, when the major axis of the moon's orbit is
directed towards the sun, and when at right angles to that direction.
First, then, let the line of apsides be directed to the sun as in the an-
nexed figure, where A is the perigee, and take the arcs A a, Alb, Bc, Bd
Fig. 91.
=_9_3:
each =64° 14'. Then while P is between a and b the normal force act-
ing outwards, and the moon being near its perigee, by art. 671, the
apsides will recede, but when between c and d, the force there acting out-
wards, but the moon being near its apogee, they will advance. The ra-
pidity of these movements will be respectively at its maxima at A and B,
not only because the disturbing forces are then most intense, but also
because (see art. 671) they act most advantageously at those points to
displace the axis. Proceeding from A and B towards the neutral points
abcd, the rapidity of their recess and advance diminishes, and is nothing
(or the apsides are stationary) when P is at either of these points. From
b to D, or rather to a point some little beyond D (art. 671) the force acts
inwards, and the moon is still near perigee, so that in this arc of the orbit
the apsides advance. But the rate of advance is feeble, because in the
early part of that are the normal force is small, and as P approaches D
and the force gains power, it acts disadvantageously to move the axis, its
effect vanishing altogether when it arrives beyond D at the extremity of
the perpendicular to the upper focus of the lunar ellipse. Theuce up to
c this feeble advance is reversed and converted into a recess, the force still
acting inwards, but the moon now being near its apogee. And so also
for the arcs d'E, Ea. In the figure these changes are indicated by + +
for rapid advance,——for rapid recess, + and — for feeble advance and
recess, and 0 for the stationary points. Now if the forces were equal on
the sides of + and — it is evident that there would be an exact counter-
balance of advance and recess on the average of a whole revolution. But
this is not the case. The force in apogee is greater than that in perigee



MOTION OF THE LUNAR, APSIDES. 873
in the proportion of 28:25, while in the quadratures about D and E
they are equal. Therefore, while the feeble movements + and — in the
neighbourhood of these points destroy each other almost exactly, there
will necessarily remain a considerable balance in favour of advance, in
this situation of the line of apsides. -
(678.) Next, suppose the apogee to lie at A, and the perigee at B. In
this case it is evident that, so far as the direction of the motions of the
apsides is concerned, all the conclusions of the foregoing reasoning will
be reversed by the substitution of the word perigee for apogee, and vice
versä ; and all the signs in the figure referred to will be changed. But
now the most powerful forces act on the side of A, that is to say, still on
the side of advance, this condition also being reversed. In either situa-
tion of the orbit, then, the apsides advance.
(679.) (Case 3.) Suppose, now, the major axis to have the situation
DE, and the perigee to be on the side of D. Here, in the are b c of P's
motion the normal force acts inwards, and the moon is near perigee, con-
sequently the apsides advance, but with a moderate rapidity, the maxi-
mum of the inward normal force being only half that of the outward.
In the arcs A b and c B the moon is still near perigee, and the force acts
outwards, but though powerfully towards A and B, yet at a constantly
increasing disadvantage (art. 671.) Therefore in these arcs the apsides
recede, but moderately. In a A and B d (being towards apogee) they
again advance, still with a moderate velocity. Lastly, throughout the are
da, being about apogee with an inward force, they recede. Here as
before, if the perigee and apogee forces were equal, the advance and recess
would counterbalance; but as in fact the apogee forces preponderate, there
will be a balance on the entire revolution in favour of recess. The same
reasoning of course holds good if the perigee be towards E. But now,
between these cases and those in the foregoing articles, there is this dif-
ference, viz. that in this the dominant effect results from the inward action
of the normal force in quadratures, while in the others it results from its
outward, and doubly powerful action in syzygies. The recess of the ap-
sides in their quadratures arising from the action of the normal force will
therefore be less than their advance in their syzygies; and not only on
this account, but also because of the much less extent of the arcs b c and
da on which the balance is mainly struck in this case, than of a b and
c d, the corresponding most influential arcs in the other.
(680.) In intermediate situations of the line of apsides, the effect will
be intermediate, and there will of course be a situation of them in which
on an average of a whole revolution, they are stationary. This situation
it is easy to see will be nearer to the line of quadratures than of syzygies,
t
Gº .
374 OluTLINES OF ASTRONOMY.
and the preponderance of advance will be maintained over a much more
considerable arc than that of recess, among the possible situations which
they can hold. On every account, therefore, the action of the normal
force causes the lunar apsides to progress in a complete revolution of M
or in a synodical year, during which the motion of the sun round the
earth (as we consider the earth at rest) brings the line of syzygies into all
situations with respect to that of apsides.
(681.) Let us next consider the action of the tangential force. And
as before (Case 1.), supposing the perigree of the moon at A, and the
direction of her revolution to be A D B E, the tangential force retards
her motion through the quadrant A D, in which she recedes from S, there-
fore by art. 670 the apsides recede. Through D B the force accelerates,
while the moon still recedes, therefore they advance. Through B E the
force retards, and the moon approaches, therefore they continue to advance,
and finally throughout the quadrant E A the force accelerates, and the
moon approaches, therefore they recede. In virtue therefore of this force,
the apsides recede, during the description of the arc E A D, and advance
during D B E, but the force being in this case as in that of the normal
force more powerful at apogee, the latter will preponderate, and the apsides
will advance on an average of a whole revolution.
(682.) (Case 2.) The perigee being towards B, we have to substitute
in the foregoing reasoning approach to S, for recess from it, and vice versä,
the accelerations and retardations remaining as before. Therefore the re-
sults, as far as direction is concerned, will be reversed in each quadrant,
the apsides advance during E A D and recede during DBE. But the
situation of the apogee being also reversed, the predominance remains on
the side of E A D, that is, of advance. -
(683.) (Case 3.) Apsides in quadratures, perigee near D.—Over qua-
drant AD, approach and retardation, therefore advance of apsides. Over
D B recess and acceleration, therefore again advance; over B E recess
and retardation with recess of apsides, and lastly over E A approach and
acceleration, producing their continued recess. Total result: advance
during the half revolution A D B, and recess during B E A, the acting
forces being more powerful in the latter, whence of course a preponderant
recess. The same result when the perigee is at E.
(684.) So far the analogy of reasoning between the action of the tan-
gential and normal forces is perfect. But from this point they diverge.
It is not here as before. The recess of the apsides in quadratures does
not now arise from the predominance of feeble over feebler forces, while
that in syzygies results from that of powerful over powerful ones. The
maximum accelerating action of the tangential force is equal to its maxi-
MOTION OF THE LUNAR APSIDES. 375
mum retarding, while the inward action of the normal at its maximum is
only half the maximum of its outward. Neither is there that difference
in the extent of the arcs over which the balance is struck in this, as in
the other case, the action of the tangential force being inward and outward
alternately over equal arcs, each a complete quadrant. Whereas, there-
fore, in tracing the action of the normal force, we found reason to con-
clude it much more effective to produce progress of the apsides in their
syzygy, than in their quadrature situations, we can draw no such conclu-
sion in that of the tangential forces: there being, as regards that force, a
complete symmetry, in the four quadrants, while in regard of the normal
force the symmetry is only a half-symmetry having relation to two semi-
circles.
(685.) Taking the average of many revolutions of the sun about the
earth, in which it shall present itself in every possible variety of situations
to the line of apsides, we see that the effect of the normal force is to pro-
duce a rapid advance in the syzygy of the apsides, and a less rapid recess
in their quadrature, and on the whole, therefore, a moderately rapid gene-
ral advance, while that of the tangential is to produce an equally rapid
advance in syzygy, and recess in quadrature. Tirectly, therefore, the
tangential force would appear to have no ultimate influence in causing
either increase or diminution in the mean motion of the apsides resulting
from the action of the normal force. It does so, however, indirectly,
conspiring in that respect with, and greatly increasing, an indirect
action of the normal force in a manner which we shall now proceed to
explain.
(686.) The sun moving uniformly, or nearly so, in the same direction
as P, the line of apsides when in or near the syzygy, in advancing follows
the sun, and therefore remains materially longer in the neighbourhood of
syzygy than if it rested. On the other hand, when the apsides are in
quadrature they recede, and, moving therefore contrary to the sun's
motion, remain a shorter time in that neighbourhood, than if they rested.
Thus the advance, already preponderant, is made to preponderate more
by its longer continuance, and the recess, already deficient, is rendered
still more so by the shortening of its duration." Whatever cause, then,
increases directly the rapidity of both advance and recess, though it may
do both equally, aids in this indirect process, and it is thus that the tan-
gential force becomes effective through the medium of the progress already
produced, in doing and aiding the normal force to do that which alone it
would be unable to effect. Thus we have perturbation exaggerating
perturbation, and thus we see what is meant by geometers, when they
* Newton, Princ. i. 66. Cor. 8.
376 f . Olſ'TLINES OF ASTRONOMY.
declare that a considerable part of the motion of the lunar apsides is due
to the square of the disturbing force, or, in other words, arises out of a
second approximation in which the influence of the first in altering the
data of the problem is taken into account.
(687.) The curious and complicated effect of perturbation, described in
the last article, has given more trouble to geometers than any other part
of the lunar theory. Newton himself had succeeded in tracing that part
of the motion of the apogee which is due to the direct action of the radial
force; but finding the amount only half what observation assigns, he
appears to have abandoned the subject in despair. Nor, when resumed
by his successors, did the inquiry, for a very long period, assume a more
promising aspect. On the contrary, Newton's result appeared to be even
minutely verified, and the elaborate investigations which were lavished
upon the subject without success, began to excite strong doubts whether
this feature of the lunar motions could be explained at all by the New-
tonian law of gravitation. The doubt was removed, however, almost in
the instant of its origin, by the same geometer, Clairaut, who first gave it
currency, and who gloriously repaired the error of his momentary hesita-
tion, by demonstrating the exact coincidence between theory and observa-
tion, when the effect of the tangential force is properly taken into the
account. The lunar apogee circulates, in 3232°575343, or about 9%
years.
(688.) Let us now proceed to investigate the influence of the disturbing
forces so resolved on the excentricity of the lunar orbit, and the foregoing
articles having sufficiently familiarized the reader with our mode of fol-
Fig. 92.
lowing out the changes in different situations of the orbit, we shall take
at once a more general situation, and suppose the line of apsides in any
position with respect to the sun, such as Z Y, the perigee being at Z, a
point between the lower syzygy and the quadrature next following it, the
direction of P's motion as all along supposed being A D B E. Then

VARIATION OF THE MOON'S EXCENTRICITY. 377
(comrºencing with the normal force) the momentary change of excentri-
city will vanish at a, b, c, d, by the vanishing of that force, and at Z and
Y by the effect of situation in the orbit annulling its action (art. 671).
In the arcs Z b and Y d therefore the change of excentricity will be small,
the acting force nowhere attaining either a great magnitude or an advan-
tageous situation within their limits. And the force within these two
arcs having the same character as to inward and outward, but being oppos
sitely influential by reason of the approach of P to S in one of them and
its recess in the other, it is evident that, so far as these arcs are concerned,
a very near compensation of effects will take place, and though the apo-
geal arc Y d will be somewhat more influential, this will tell for little
upon the average of a revolution. º -
(689.) The arcs b D c and d E a are each much less than a quadrant
in extent, and the force acting inwards throughout them (which at its
maximum in D and E is only half the outward force at A, B) degrades
very rapidly in intensity towards either syzygy (see art. 676). Hence
whether Z be between b c or b A, the effects of the force in these arcs
will not produce very extensive changes on the excentricity, and the
changes which it does produce will (for the reason already given) be op-
posed to each other. Although, then, the are a d be farther from perigee
than b c, and therefore the force in it is greater, yet the predominance of
effect here will not be very marked, and will moreover be partially neu-
tralized by the small predominance of an opposite character in Y d over
Z b. On the other hand, the arcs a Z, c Y are both larger in extent than
either of the others, and the seats of action of forces doubly powerful.
Their influence, therefore, will be of most importance, and their prepon-
derance one over the other (being opposite in their tendencies), will decide
the question whether, on an average of the revolution, the excentricity
shall increase or diminish. It is clear that the decision must be in favour
of c Y, the apogeal arc, and, since in this the force is outwards and the
moon receding from the earth, an increase of the excentricity will arise
from its influence. A similar reasoning will, evidently, lead to the same
conclusion were the apogee and perigee to change places, for the directions
of P's motion as to approach and recess to S will be indeed reversed, but
at the same time the dominant forces will have changed sides, and the
arc a A Z will now give the character to the result. But when Z lies
between A and E, as the reader may easily satisfy himself, the case will
be altogether different, and the reverse conclusion will obtain. Hence the
changes of excentricity emergent on the average of single revolutions
from the action of the normal force will be as represented by the signs
+ and — in the figure above annexed.
378 OUTLINES OF ASTRONOMY.
(690.) Let us next consider the effect of the tangential force. This
retards P in the quadrants A D, B.E, and accelerates it in the alternate
Fig. 93.
ID

Z
\ M
IB A •
ones. In the whole quadrant A D, therefore, the effect is of one charac.
ter, the perigee being less than 90° from every point in it, and in the
whole quadrant B E it is of the opposite, the apogee being so situated
(art. 670). Moreover, in the middle of each quadrant, the tangential
force is at its maximum. Now, in the other quadrants, E A and D B,
the change from perigeal to apogeal vicinity takes place, and the tangen-
tial force, however powerful, has its effect annulled by situation (art. 670),
and this happens more or less nearly about the points where the force is a
maximum. These quadrants, then, are far less influential on the total
result, so that the character of that result will be decided by the predo-
minance of one or other of the former quadrants, and will lean to that
which has the apogee in it. Now in the quadrant B E the force retards
the moon, and the moon is in apogee. Therefore the excentricity in-
creases. In this situation therefore of the apogee, such is the average
result of a complete revolution of the moon. Here again also, if the
perigee and apogee change places, so will also the character of all the par-
tial influences, arc for arc. But the quadrant A D will now preponderate
instead of D E, so that under this double reversal of conditions the result
will be identical. Lastly, if the line of apsides be in A E, BD, it may
be shown in like manner that the excentricity will diminish on the average
of a revolution.
(691.) Thus it appears, that in varying the excentricity, precisely as in
moving the line of apsides, the direct effect of the taugential force con-
spires with that of the normal, and tends to increase the extent of the
deviations to and fro on either side of a mean value which the varying
situation of the sun with respect to the line of apsides gives rise to, having
for their period of restoration a synodical revolution of the sun and apse.
Supposing the sun and apsis to start together, the Sun of course will
outrun the apsis (whose period is nine years,) and in the lapse of about
EXPERIMENTAL, ILLUSTRATION. 379
(; +a;) part of a year will have gained on it 90°, during al which inter.
val the apse will have been in the quadrant A E of our figure, and the
excentricity continually decreasing. The decrease will then cease, but
the excentricity itself will be a minimum, the Sun being now at right
angles to the line of apsides. Thence it will increase to a maximum
when the sun has gained another 90°, and again attained the line of
apsides, and so on alternately. The actual effect on the numerical value
of the lunar excentricity is very considerable, the greatest and least excen-
tricities being in the ratio of 3 to 2."
(692.) The motion of the apsides of the lunar orbit may be illustrated
by a very pretty mechanical experiment, which is otherwise instructive in
giving an idea of the mode in which orbitual motion is carried on under
the action of central forces variable according to the situation of the
revolving body. Let a leaden weight be suspended by a brass or iron
wire to a hook in the under side of a firm beam, so as to allow of its free
motion on all sides of the vertical, and so that when in a state of rest it
shall just clear the floor of the room, or a table placed ten or twelve feet
beneath the hook. The point of support should be well secured from
wagging to and fro by the oscillation of the weight, which should be
sufficient to keep the wire as tightly stretched as it will bear, with the
certainty of not breaking. Now, let a very small motion be communi-
cated to the weight, not by merely withdrawing it from the vertical and
letting it fall, but by giving it a slight impulse sideways. It will be seen
to describe a regular ellipse about the point of rest as its centre. If the
weight be heavy, and carry attached to it a pencil, whose point lies
exactly in the direction of the string, the ellipse may be transferred to
paper lightly stretched and gently pressed against it. In these circum-
stances, the situation of the major and minor axes of the ellipse will
remain for a long time very nearly the same, though the resistance of the
air and the stiffness of the wire will gradually diminish its dimensions and
excentricity. But if the impulse communicated to the weight be con-
siderable, so as to carry it out to a great angle (15° or 20° from the
vertical,) this permanence of situation of the ellipse will no longer subsist.
Its axis will be seen to shift its position at every revolution of the weight,
advancing in the same direction with the weight's motion, by an uniform
and regular progression, which at length will entirely reverse its situation,
bringing the direction of the longest excursions to coincide with that in
which the shortest were previously made; and so on, round the whole
circle; and, in a word, imitating to the eye, very completely, the motion
of the apsides of the moon's orbit.
* Airy, Gravitation, p. 106.
380 OUTLINES OF ASTRONOMY.
(693.) Now, if we inquire into the cause of this progression of the
apsides, it will not be difficult of detection. When a weight is suspended
by a wire, and drawn aside from the vertical, it is urged to the lowest
point (or rather in a direction at every instant perpendicular to the wire)
by a foree which varies as the sine of the deviation of the wire from the
perpendicular. Now, the sines of very small arcs are nearly in the pro-
portion of the arcs themselves; and the more nearly, as the arcs are
smaller. If, therefore, the deviations from the vertical be so small that
we may neglect the curvature of the spherical surface in which the weight
moves, and regard the curve described as coincident with its projection on
a horizontal plane, it will be then moving under the same circumstances
as if it were a revolving body attracted to a centre by a force varying
directly as the distance; and, in this case, the curve described would be
an ellipse, having its centre of attraction not in the focus, but in the
centre', and the apsides of this ellipse would remain fixed. But if the
excursions of the weight from the vertical be considerable, the force urging
it towards the centre will deviate in its law from the simple ratio of the
distances; being as the sine, while the distances are as the arc. Now the
sine, though it continues to increase as the arc increases, yet does not in-
crease so fast. So soon as the arc has any sensible extent, the sine begins
to fall somewhat short of the magnitude which an exact numerical propor-
tionality would require; and therefore the force urging the weight towards
Fig. 94.
its centre or point of rest at great distances falls, in like proportion, some-
what short of that which would keep the body in its precise elliptic orbit.
It will no longer, therefore, have, at those greater distances, the same
command over the weight, in proportion to its speed, which would enable
it to deflect it from its rectilinear tangential course into an ellipse. The
true path which it describes will be less curved in the remoter parts than
is consistent with the elliptic figure, as in the annexed cut; and, therefore,
it will not so soon. have its motion brought to be again at right angles to
* Newton, Princip. i. 47.

APPLICATION OF THE PLANETARY THEORY. 381
the radius. It will require a longer continued action of the central force
to do this; and before it is accomplished, more than a quadrant of its
revolution must be passed over in angular motion round the centre. But
this is only stating at length, and in a more circuitous manner, that fact
which is more briefly and summarily expressed by saying that the apsides
of its orbit are progressive. Nothing beyond a familiar illustration is of
course intended in what is above said. The case is not an exact parallel
with that of the lunar orbit, the disturbing force being simply radial,
whereas in the lunar orbit a transversal force is also concerned, and even
were it otherwise, only a confused and indistinct view of aspidal motion
can be obtained from this kind of consideration of the curvature of the
disturbed path. If we would obtain a clear one, the two foci of the in-
stantaneous ellipse must be found from the laws of elliptic motion per-
formed under the influence of a force directly as the distance, and the
radial disturbing force being decomposed into its tangential and normal
components, the momentary influence of either in altering their positions
and consequently the directions and lengths of the axis of the ellipse
must be ascertained. The student will find it neither a difficult nor an
uninstructive exercise to work out the case from these principles, which
we cannot afford the space to do.
(694.) The theory of the motion of the planetary apsides and the
variation of their excentricities is in one point of view much more simple,
but in another mučh more complicated than that of the lunar. It is
simpler, because owing to the exceeding minuteness of the changes ope-
rated in the course of a single revolution, the angular position of the
bodies with respect to the line of apsides is very little altered by the
motion of the apsides themselves. The line of apsides neither follows
up the motion of the disturbing body in its state of advance, nor vice
versá, in any degree capable of prolonging materially their advancing or
shortening materially their receding phase. Hence no second approxima-
tion of the kind explained (in art. 686), by which the motion of the lunar
apsides is so powerfully modified as to be actually doubled in amount, is
at all required in the planetary theory. On the other hand, the latter
theory is rendered more complicated than the former, at least in the cases
of planets whose periodic times are to each other in a ratio much less than
13 to 1, by the consideration that the disturbing body shifts its position
with respect to the line, of apsides by a much greater angular quantity in
a revolution of the disturbed body than in the case of the moon. In that
case we were at liberty to suppose (for the sake of explanation), without
any very egregious error, that the sun held nearly a fixed position during
a single lunation. But in the case of planets whose times of revolution
382 OUTLINES OF ASTRONOMY.
are in a much lower ratio this cannot be permitted. In the case of Jupiter
disturbed by Saturn for example, in one sidereal revolution of Jupiter,
Saturn has advanced in its orbit with respect to the line of apsides of
Jupiter by more than 140°, a change of direction which entirely alters
the conditions under which the disturbing forces act. And in the case
of an exterior disturbed by an interior planet, the situation of the latter
with respect to the line of the apsides varies even more rapidly than the
situation of the exterior or disturbed planet with respect to the central
body. To such cases then the reasoning which we have applied to the
lunar perturbatisms becomes totally inapplicable; and when we take into
consideration also the excentricity of the orbit of the disturbing body,
which in the most important cases is exceedingly influential, the subject
becomes far too complicated for verbal explanation, and can only be suc-
cessfully followed out with the help of algebraic expression and the appli-
cation of the integral calculus To Mercury, Venus, and the earth indeed,
as disturbed by Jupiter, and planets superior to Jupiter, this objection to
the reasoning in question does not apply; and in each of these cases
therefore we are entitled to conclude that the apsides are kept in a state
of progression by the action of all the larger planets of our system.
Under certain conditions of distance, excentricity, and relative situation
of the axes of the orbits of the disturbed and disturbing planets, it is
perfectly possible that the reverse may happen, an instance of which is
afforded by Venus, whose apsides recede under the combined action of the
earth and Mercury more rapidly than they advance under the joint actions
of all the other planets. Nay, it is even possible under certain conditions
that the line of apsides of the disturbed planet, instead of revolving always
in one direction, may librate to and fro within assignable limits, and in a
definite and regularly recurring period of time.
(695.) Under any conditions, however, as to these particulars, the
view we have above taken of the subject enables us to assign at every
instant, and in every configuration of the two planets, the momentary
effect of each upon the perihelion and excentricity of the other. In the
simplest case, that in which the two orbits are so nearly circular, that
the relative situation of their perihelia shall produce no appreciable differ-
ence in the intensities of the disturbing forces, it is very easy to show
that whatever temporary oscillations to and fro in the positions of the line
of apsides, and whatever temporary increase or diminution in the excen-
tricity of either planet may take place, the final effect on the average of
a great multitude of revolutions, presenting them to each other in all
possible configurations, must be nil, for both elements.
(696) To show this, all that is necessary is to cast our eyes on the
*
EFFECTS GRP RLLIPTICITY. 383
synoptic table in art. 673. If M, the disturbing body, be supposed to be
successively placed in two diametrically opposite situations in its orbit, the
aphelion of P will stand related to M in one of these situations precisely
as its perihelion in the other. Now the orbits being so nearly circles as
supposed, the distribution of the disturbing forces, whether normal or
tangential, is symmetrical relative to their common diameter passing
through M, or to the line of syzygies. Hence it follows that the half of
P's orbit “about perihelion” (art. 673) will stand related to all the acting
forces in the one situation of M, precisely as the half “about aphelion”
does in the other: and also, that the half of the orbit in which P “ap.
proaches S,” stands related to them in the one situation precisely as the
half in which it “recedes from S'' in the other. Whether as regards,
therefore, the normal or tangential force, the conditions of advance or
recess of apsides, and of increase or diminution of excentricities, are
reversed in the two supposed cases. Hence it appears that whatever
situation be assigned to M, and whatever influence it may exert on P in
that situation, that influence will be annihilated in situations of M and
P, diametrically opposite to those supposed, and thus, on a general
average, the effect on both apsides and excentricities is reduced to
nothing.
(697.) If the orbits, however, be excentric, the symmetry above in-
sisted on in the distribution of the forces does not exist. But, in the first
place, it is evident that if the excentricities be moderate, (as in the planet-
ary orbits,) by far the larger part of the effects of the disturbing forces
destroys itself in the manner described in the last article, and that it is
only a residual portion, viz. that which arises from the greater proximity
of the orbits at one place than at another, which can tend to produce per-
manent or secular effects. The precise estimation of these effects is too
complicated an affair for us to enter upon; but we may at least give some
idea of the process by which they are produced, and the order in which
they arise. In so doing, it is necessary to distinguish between the effects
of the normal and tangential forces. The effects of the former are greatest
at the point of conjunction of the planets, because the normal force itself
is there always at its maximum; and although, where the conjunction
takes place at 90° from the line of apsides, its effect to move the apsides
is nullified by situation, and when in that line its effect on the excentri-
cities is similarly nullified, yet, in the situations rectangular to these, it
acts to its greatest advantage. On the other hand, the tangential force
vanishes at conjunction, whatever be the place of conjunction with respect
to the line of apsides, and where it is at its maximum its effect is still
liable to be annulled by situation. Thus it appears that the normal
384 OUTLINES OF ASTRONOMY,
force is most influential, and mainly determines the character of the ge-
neral effect. It is, therefore, at conjunction that the most influential
effect is produced, and therefore, on the long average, those conjunctions
which happen about the place where the orbits are nearest will determine
the general character of the effect. Now, the nearest points of approach
of two ellipses, which have a common focus, may be variously situated
with respect to the perihelion of either. It may be at the perihelion or
the aphelion of the disturbed orbit, or in any intermediate position. Sup-
pose it to be at the perihelion. Then, if the disturbed orbit be interior
to the disturbing, the force acts outwards, and therefore the apsides re-
cede : if exterior, the force acts inwards, and they advance. In neither
case does the excentricity change. If the conjunction take place at the
aphelion of the disturbed orbit, the effects will be reversed: if interme-
diate, the apsides will be less, and the excentricity more affected.
(698.) Supposing only two planets, this process would go on till the
apsides and excentricities had so far changed as to alter the point of
nearest approach of the orbits, so as either to accelerate or retard and
perhaps reverse the motion of the apsides, and give to the variation of the
excentricity a corresponding periodical character. But there are many
planets, all disturbing one another. And this gives rise to variations in
the points of nearest approach of all the orbits, taken two and two toge-
ther, of a very complex nature.
(699.) It cannot fail to have been remarked, by any one who has fol-
lowed attentively the above reasonings, that a close analogy subsists between
two sets of relations; viz. that between the inclinations and nodes on the
one hand, and between the excentricity and apsides on the other. In fact,
the strict geometrical theories of the two cases present a close analogy,
and lead to final results of the very same nature. What the variation of
excentricity is to the motion of the perihelion, the change of inclination
is to the motion of the node. In either case the period of the one is also
the period of the other; and while the perihelia describe considerable
angles by an oscillatory motion to and fro, or circulate in immense periods
of time round the entire circle, the excentricities increase and decrease by
comparatively small changes, and are at length restored to their original
magnitudes. In the lunar orbit, as the rapid rotation of the nodes pre-
vents the change of inclination from accumulating to any material
amount, so the still more rapid revolution of its apogee effects a speedy
compensation in the fluctuations of its excentricity, and never suffers them
to go to any material extent; while the same causes, by presenting in
quick succession the lunar orbit in every possible situation to all the dis-
turbing forces, whether of the sun, the planets, or the protuberant matter
*OMPOUND CYCLES OF EXCENTRICITIES, ETC. 385
at the earth's equator, prevent any secular accumulation of small changes,
by which, in the lapse of ages, its ellipticity might be materially increased
or diminished. Accordingly, observation shows the mean excentricity of
the moon’s orbit to be the same now as in the earliest ages of astronomy.
(700.) The movements of the perihelia, and variations of excentricity
of the planetary orbits, are interlaced and complicated together in the
same manner and nearly by the same laws as the variations of their nodes
and inclinations. Each acts upon every other, and every such mutual
action generates its own peculiar period of circulation or compensation;
and every such period, in pursuance of the principles of art. 650, is
thence propagated throughout the system. Thus arise cycles upon cycles,
of whose compound duration some notion may be formed, when we con-
sider what is the length of one such period in the case of the two prin-
cipal planets — Jupiter and Saturn. Neglecting the action of the rest,
the effect of their mutual attraction would be to produce a secular varia-
tion in the excentricity of Saturn's orbit, from 0-08409, its mazimum,
to 0-01345, its minimum value: while that of Jupiter would vary be-
tween the narrow limits, 0.06036 and 0.02606: the greatest excentricity
of Jupiter corresponding to the least of Saturn, and vice versá. The
period in which these changes are gone through, would be 70414 years.
After this example, it will be easily conceived that many millions of years
will require to elapse before a complete fulfilment of the joint cycle which
shall restore the whole system to its original state as far as the excentri-
cities of its orbits are concerned.
(701.) The place of the perihelion of a planet's orbit is of little con-
sequence to its well-being; but its excentricity is most important, as upon
this (the axes of the orbits being permanent) depends the mean tempera-
ture of its surface, and the extreme variations to which its seasons may
be liable. For it may be easily shown that the mean annual amount
of light and heat reeeived by a planet from the sun is, caeteris paribus,
as the minor axis of the ellipse described by it. Any variation, there.
fore, in the excentricity, by changing the minor axis, will alter the mean
temperature of the surface. How such a change will also influence the
extremes of temperature appears from art. 368. Now it may naturally
be inquired whether (in the vast cycle above spoken of, in which, at some
period or other, conspiring changes may accumulate on the orbit of one
planet from several quarters,) it may not happen that the excentricity of
any one planet— as the earth—may become exorbitantly great, so as to
subvert those relations which render it habitable to man, or to give rise to
great changes, at least, in the physical comfort of his state. To this the
researches of geometers have enabled us to answer in the negative. A
25
386. OUTLINES OF ASTRONOMY.
relation has been demonstrated by Lagrange between the masses, axes of
the orbits, and excentricities of each planet, similar to what we have al-
ready stated with respect to their inclinations, viz. that if the mass of
each planet be multiplied by the square root of the axis of its orbit, and
the product by the square of its excentricity, the sum of all such products
throughout the system is invariable; and as, in point of fact, this sum is
extremely small, so it will always remain. Now, since the axes of the
orbits are liable to no secular changes, this is equivalent to saying that no
one orbit shall increase its excentricity, unless at the expense of a common
fund, the whole amount of which is, and must for ever remain, extremely
minute."
* There is nothing in this relation, however, taken per se, to secure the smaller pla-
nets—Mercury, Mars, Juno, Ceres, &c.—from a catastrophe, could they accumulate
on themselves, or any one of them, the whole amount of this excentricity fund. But
that can never be : Jupiter and Saturn will always retain the lion's share of it. A
similar remark applies to the inclination fund of art. 639. These funds, be it observed,
can never get into debt. Every term of them is essentially positive.
COMPOUND MOTION OF THE UPPER FOCUS. 387
CHAPTER XIV.
OF THE INEQUALITIES INDEPENDENT OF THE EXCENTRICITIES.—THE
Moon’s VARIATION AND PARALLACTIC INEQUALITY. — ANALOGOUS
PLANETARY INEQUALITIES.—THREE CASES OF PLANETARY PERTUR-
BATION IDISTINGUISHED. — OF INEQUALITIES DEPENDENT ON THE
EXCENTRICITIES. — LONG INEQUALITY OF JUPITER AND SATURN.—
LAW OF RECIPROCITY BETWEEN THE PERIODICAL VARIATIONS OF
THE ELEMENTS OF BOTH PLANETS.—LONG INEQUALITY OF THE EARTFI
AND VENUS.—VARIATION OF THE EPOCII.-INEQUALITIES INCIDENT
oN THE EPoCH AFFECTING THE MEAN MOTION.—INTERPRETATION OF
THE CONSTANT PART OF THESE INEQUALITIES. — ANNUAL EQUATION -
OF THE MOON. – HER SECULAR ACCELERATION.— LUNAR INEQUALI-
TIES DUE TO THE ACTION OF VENUS. — EFFECT OF THE SPHEROIDAL
FIGURE OF THE EARTH AND OTHER PLANETS ON THE MOTIONS OF
THEIR, SATELLITES.—OF THE TIDES.—MASSES OF DISTURBING BODIE8
I) EDUCIBLE FROM THE PERTURBATIONS THEY PRODUCE. – MASS OF
THE MOON, AND OF JUPITER’s SATELLITES, HOW ASCERTAINED. –
PERTURBATIONS OF URANU'S RESULTIFG IN THE DISCOVERY OF
NEPTUNE. .
(702.) To calculate the actual place of a planet or the moon, in longi-
tude and latitude at any assigned time, it is not enough to know the
changes produced by perturbation in the elements of its orbit, still less to
know the secular changes so produced, which are only the outstanding or
uncompensated portions of much greater changes induced in short periods
of configuration. We must be enabled to estimate the actual effect on its
longitude of those periodical accelerations and retardations in the rate of
its mean angular motion, and on its latitude of those deviations above and
below the mean plane of its orbit, which result from the continued action
of the perturbative forces, not as compensated in long periods, but as in
the act of their generation and destruction in short ones. In this chapter
we purpose to give an account of some of the most prominent of the
equations or inequalities thence arising, several of which are of high his-
torical interest, as having become known by observation previous to the
Q.
388 OUTLINES OF ASTRONOMY.
discovery of their theoretical causes, and as having, by their successive
explanations from the theory of gravitation, removed what were in some
instances regarded as formidable objections against that theory, and afforded
in all most satisfactory and triumphant verifications of it.
(703.) We shall begin with those which compensate themselves in a
synodic revolution of the disturbed and disturbing body, and which are
independent of any permanent excentricity of either orbit, going through
their changes and effecting their compensation in orbits slightly elliptic,
almost precisely as if they were circular. These inequalities result, in
fact, from a circulation of the true upper focus of the disturbed ellipse
about its mean place in a curve whose form and magnitude the principles
laid down in the last chapter enable us to assign in any proposed case.
If the disturbed orbit be circular, this mean place coincides with its cen.
tre: if elliptic, with the situation of its upper focus, as determined from
the principles laid down in the last chapter.
(704) To understand the nature of this circulation, we must consider
the joint action of the two elements of the disturbing force. Suppose H
to be the place of the upper focus, corresponding to any situation P of the
..Fig. 95.
disturbed body, and let P Pº be an infinitesimal element of its orbit, de-
scribed in an instant of time. Then supposing no disturbing force to act,
P P will be a portion of an ellipse, having H for its focus, equally
whether the point P or P be regarded. But now let the disturbing
forces act during the instant of describing P P'. Then the focus H will
shift its position to H' to find which point we must recollect, 1st. What is
demonstrated (in art. 671), viz. that the effect of the normal force is to
vary the position of the line P' H so as to make the angle H P H' equal
to double the variation of the angle of tangency due to the action of that
tº

“WARIATION’ OF THE MOON EXPLAINED. 389
*
force, without altering the distance PH: so that in virtue of the normal
force alone, H would move to a point h, along the line H Q, drawn from
H to a point Q, 90° in advance of P, (because S H being exceedingly
small, the angle PH Q may be taken as a right angle when P S Q is so,)
H approaching Q if the normal force act outwards, but receding from Q
if inwards. And similarly the effect of the tangential force (art. 670) is
to vary the position of H in the direction H P or PH, according as the
force retards or accelerates P's motion. . To find H’ then from H draw
H. P., H Q, to P and to a point of P's orbit 90° in advance of P. On
H Q take H h, the motion of the focus due to the normal force, and on
H P take H k the motion due to the tangential force; complete the
parallelogram H. H', and its diagonal H H' will be the element of the
true path of H in virtue of the joint action of both forces.
(705.) The most conspicuous case in the planetary system to which the
above reasoning is applicable, is that of the moon disturbed by the sun.
The inequality thus arising is known by the name of the moon’s varia-
tion, and was discovered so early as about the year 975 by the Arabian
astronomer Aboul Wefa." Its magnitude (or the extent of fluctuation to
and fro in the moon’s longitude which it produces) is considerable, being
no less than 184', and it is otherwise interesting as being the first iné.
quality produced by perturbation, which Newton succeeded in explaining
by the theory of gravity. A good general idea of its nature may be
formed by considering the direct action of the disturbing forces on the moon,
supposed to move in a circular orbit. In such an orbit undisturbed, the
velocity would be uniform; but the tangential force acting to accelerate
her motion through the quadrants preceding her conjunction and oppo.
sition, and to retard it through the alternate quadrants, it is evident that
the velocity will have two maxima and two minima, the former at the
syzygies, the latter at the quadratures. Hence at the syzygies the velocity
will exceed that which corresponds to a circular orbit, and at quadratures
will fall short of it. The true orbit will therefore be less curved or more
flattened than a circle in syzygies, and more curved (i. e. protuberant be-
yond a circle) in quadratures. This would be the case even were the
normal force not to act. But the action of that force increases the effect
in question, for at the syzygies, and as far as 64° 14' on either side of
them, it acts outwards, or in counteraction of the earth's attraction, and
thereby prevents the orbit from being so much curved as it otherwise
would be; while at quadratures, and for 25° 46' on either side of them,
it acts inwards, aiding the earth’s attraction, and rendering that portion
* Sedillot, Nouvelles Recherches pour servir à l’Histoire de l'Astronomie chez .es
Arabes.
390 * OUTLINES OF ASTRONOMY.
of the orbit more curved than it otherwise would be. Thus the joint
action of both forces distorts the orbit from a circle into a flattened or
elliptic form, having the longer axis in quadratures, and the shorter in
syzygies; and in this orbit the moon moves faster than with her mean
velocity at syzygy (i. e. where she is nearest the earth) and slower at
quadratures where farthest. Her angular motion about the earth is there-
fore for both reasons greater in the former than in the latter situation.
Hence at syzygy her true longitude seen from the earth will be in the act
of gaining on her mean,—in quadratures of losing, and at some interme-
diate points (not very remote from the octants) will neither be gaining
nor losing. But at these points, having been gaining or losing through
the whole previous 90° the amount of gain or loss will have attained its
maximum. Consequently at the octants the true longitude will deviate
most from the mean in excess and defect, and the inequality in question
is therefore nil at syzygies and quadratures, and attains its maxima in
advance or retardation at the octants, which is agreeable to observation.
(706.) Let us, however, now see what account can be rendered of this
inequality by the simultaneous variations of the axis and excentricity as
above explained. The tangential force, as will be recollected, is nil at
syzygies and quadratures, and a maximum at the octants, accelerative in
the quadrants E A and D B, and retarding in A D and B E. In the two
former then the axis is in process of lengthening; in the two latter, short-
ening. On the other hand the normal force vanishes at (a, b, d, e) 64°
14 from the syzygies. It acts outwards over e A a, b B d, and inwards
over a D b and d Ee. In virtue of the tangential force, then, the point
H moves towards P when P is in A D, B.E, and from it when in D B,
E A, the motion being nil when at A, B, D, E, and most rapid when at
the octant D, at which points, therefore, (so far as this force is concerned,)
the focus H would have its mean situation. And in virtue of the normal
focus, the motion of H in the direction H Q will be at its maximum of
rapidity towards Q at A, or B, from Q at D or E, and nil, at a, b, d, e. It
will assist us in following out these indications to obtain a notion of the
form of the curve really described by H, if we trace separately the paths
which H would pursue in virtue of either motion separately, since its true
motion will necessarily result from the superposition of these partial mo-
tions, because at every instant they are at right angles to each other, and
therefore cannot interfere. First, then, it is evident, from what we have
said of the tangential force, that when P is at A, H is for an instant at
rest, but that as Premoves from A towards D, H continually approaches
P along their line of junction HP, which is, therefore, at each instant a
tangent to the path of H. When P is in the octant, H is at its mean
“WARIATION” OF THE Moon EXPLAINED. 391
Fig. 96.
E
A.
distance from P (equal to PS), and is then in the act of approaching P
most rapidly. From thence to the quadrature D the movement of H to-
wards P decreases in rapidity till the quadrature is attained, when H rests
for an instant, and then begins to reverse its motion, and travel from P
at the same rate of progress as before towards it. Thus it is clear that,
in virtue of the tangential force alone, H would describe a four-cusped
curve a, d, b, e, its direction of motion round S in this curve being oppo-
site to that of P, so that A and a, D and d, B and b, E and e, shall be
corresponding points.
(707.) Next as regards the normal force. When the moon is at A the
motion of H is towards D, and is at its maximum of rapidity, but slackens
as P proceeds towards D and as Q proceeds towards B. To the curve
described, H Q will be always a tangent, and since at the neutral point of
the normal force (or when P is 64° 14' from A, and Q 64° 14' from D),
the motion of H is for an instant nil and is then reversed, the curve will
have a cusp at l corresponding, and H will then begin to travel along the
arc l m, while P describes the corresponding arc from neutral point to
neutral point through D. Arrived at the neutral point between D and B,
the motion of H along Q H will be again arrested and reversed, giving
rise to another cusp at m, and so on. Thus, in virtue of the normal force
acting alone, the path of H would be the four-cusped, elongated curve
lm no, described with a motion round S the reverse of P's, and having
a, d, b, e for points corresponding to A, B, D, E, places of P.
(708.) Nothing is now easier than to superpose these motions Sup-

392 OUTLINES OF ASTRONOMY.
*
Fig. 97.
B
A.
posing Hi, H, to be the points in either curve corresponding to P, we
have nothing to do but to set from off S, Sh equal and parallel to S H,
in the one curve and from h, h H equal and parallel to SH, in the other.
Let this be done for every corresponding point in the two curves, and
there results an oval curve a b d e, having for its semiaxis Sa-S al-HSa;;
and Sd=Sd,--S d, And this will be the true path of the upper focus,
the points a, d, b, e, corresponding to A, D, B, E, places of P. And from
this it follows, 1st, that at A, B, the syzygies, the moon is in perigee in
her momentary ellipse, the lower focus being nearer than the upper.
2dly, That in quadratures D, E, the moon is in apogee in her then mo-
mentary ellipse, the upper focus being then nearer than the lower. 3dly,
Fig. 98.
That H revolves in the oval ad be the contrary way to P in its orbit,
making a complete revolution from syzygy to syzygy in one synodic revo-
tution of the moon.
(709.) Taking 1 for the moon's mean distance from the earth, suppose
p


“WARIATION ?’ OF THE MOON EXPLAINED. 393
we represent Sal or Sd, (for they are equal) by 2a, Sa, by 2b, and Sd, by
2c, then will the semiaxes of the oval a d be, Sa and Sd be respectively
2a+2b and 2d-H2c, so that the excentricities of the momentary ellipses
at A and D will be respectively a + b and a +c. The total amount of the
effect of the tangential force on the axis, in passing from syzygy to qua-
rature, will evidently be equal to the length of the curvilinear arc a, d,
(fig. art. 708), which is necessarily less than San-i-Sal, or 4a. There-
fore the total effect on the semiaacis or distance of the moon is less than
2a, and the excess and defect of the greatest and least values of this dis-
tance thus varied above and below the mean value SA = 1 (which call a.)
will be less than a. The moon then is moving at A in the periyee of an
ellipse whose semiaxis is 1+a and excentricity a + b, so that its actual
distance from the earth there is 1+a – a – b, which (because a is less
than a) is less than 1 — b. Again, at D it is moving in apogee of an
ellipse whose semiaxis is 1–0 and excentricity a + c, so that its distance
then from the earth is 1 — a +a+c, which (a being greater than a) is
greater than 1+ c, the latter distance exceeding the former by 2a–2a+
b+c.
(710.) Let us next consider the corresponding changes induced upon
the angular velocity. Now it is a law of elliptic motion that at different
points of different ellipses, each differing very little from a circle, the an-
gular velocities are to each other as the square roots of the semiaxes
directly, and as the squares of the distances inversely. In this case the
semiaxes at A and D are to each other as 1+a to 1—a, or as 1 : 1–2a,
so that their square roots are to each other as 1 : 1–0. Again, the dis-
tances being to each other as 1+a – a – b : 1–0 +a+ c, the inverse
ratio of their squares (since a, a, b, c, are all very small quantities) is that
of 1–2a+2a+2c : 1+2a–2a–2b, or as 1:1–4a–4a–2b–2c.
The angular velocities then are to each other in a ratio compounded of these
two proportions, that is in the ratio of
1 : 1+30 – 4a–2b–2c,
which is evidently that of a greater to a less quantity. It is obvious also,
from the constitution of the second term of this ratio, that the normal
force is far more influential in producing this result than the tangential.
(711.) In the foregoing reasoning the sun has been regarded as fixed
Let us now suppose it in motion (in a circular orbit), then it is evident that
at equal angles of elongation (of P from M seen from S), equal disturb-
ing forces, both tangential and normal, will act: only the syzygies and
quadratures, as well as the neutral points of the normal force, instead of
being points fixed in longitude on the orbit of the moon, will advance on
that orbit with a uniform angular motion equal to the angular motion of
*
394 OUTLINES OF ASTRONOMY.
the sun. The cuspidated curves a, d, b, e, and a, d, b, e, fig, art. 708, will,
therefore, no longer be re-entering curves; but each will have its cusps
screwed round as it were in the direction of the sun's motion, so as to in-
crease the angles between them in the ratio of the synodical to the side-
real revolution of the moon (art. 418). And if, in like manner, the mo-
tions in these two curves, thus separately described by H, be compounded,
the resulting curve, though still (loosely speaking) a species of oval, will
not return into itself, but will make successive spiroidal convolutions about
S, its farthest and nearest points being in the same ratio more than 90°
asunder. And to this movement that of the moon herself will conform,
describing a species of elliptic spiroid, having its least distances always in
the line of syzygies and its greatest in that of quadratures. It is evident,
also, that, owing to the longer continued action of both forces, i.e. owing
to the greater arc over which their intensities increase and decrease by
equal steps, the branches of each curve between the cusps will be longer,
and the cusps themselves will be more remote from S, and in the same
degree will the dimensions of the resulting oval be enlarged, and with
them the amount of the inequality in the moon's motion which they
represent.
(712.) In the above reasoning the sum's distance is supposed so great,
that the disturbing forces in the semi-orbit nearer to it shall not sensibly
differ from those in the more remote. The sun, however, is actually
nearer to the moon in conjunction than in opposition by about one two-
hundredth part of its whole distanceſ and this suffices to give rise to a
very sensible inequality (called the parallactie inequality) in the lunar
motions, amounting to about 2' in its effect on the moon’s longitude, and
having for its period one synodical revolution or one lunation. As this
inequality, though subordinate in the case of the moon to the great ine-
quality of the variation with which it stands in connexion, becomes a
prominent feature in the system of inequalities corresponding to it in the
planetary perturbations (by reason of the very great variations of their
distances from conjunction to opposition,) it will be necessary to indicate
what modifications this consideration will introduce into the forms of our
focus curves, and of their superposed oval. Recurring then to our figures
in art. 706, 707, and supposing the moon to set out from E, and the
upper focus, in each curve from e, it is evident that the intercuspidal arcs
ea, a d, in the one, and ep, p a l, ld, in the other, being described under
the influence of more powerful forces, will be greater than the arcs d b,
be, and d m, m b m, n e corresponding in the other half revolution. The
two extremities of these curves then, the initial and terminal places of e
in each, will not meet, and the same conclusion will hold respecting those
*VARIATION" OF THE MOON EXPLAINED. 39
5
Fig. 99.
of the compound oval in which the focus really revolves, which will,
therefore, be as in the annexed figure. Thus, at the end of a complete
lunation, the focus will have shifted its place from e to fin a line parallel
to the line of quadratures. The next revolution, and the next, the same
thing would happen. Meanwhile, however, the sun has advanced in its
orbit, and the line of quadratures has changed its situation by an equal
angular motion. In consequence, the next terminal situation (g) of the
forces will not lie in the line ef prolonged, but in a line parallel to the
new situation of the line of quadratures, and this process continuing, will
evidently give rise to a movement of circulation of the point e, round a
mean situation in an annual period; and this, it is evident, is equivalent
to an annual circulation of the central point of the compound oval itself,
in a small orbit about its mean position S. Thus we see that no perma-
ment and indefinite increase of excentricity can arise from this cause;
which would be the case, however, but for the annual motion of the sum.
(713.) Inequalities precisely similar in principle to the variation and
parallactic inequality of the moon, though greatly modified by the different
relations of the dimensions of the orbits, prevail in all cases where planet
disturbs planet. To what extent this modification is carried will be evi-
dent, if we cast our eyes on the examples given in art. 612, where it will
be seen that the disturbing force in conjunction often exceeds that in
opposition in a very high ratio, (being in the case of Neptune disturbing
Uranus more than ten times as great.) The effect will be, that the orbit
described by the centre of the compound oval about S, will be much
greater relatively to the dimensions of that oval itself, than in the case of
the moon. Bearing in mind the nature and import of this modification,
we may proceed to inquire, apart from it, into the number and distribution
of the undulations in the contour of the oval itself, arising from the alter-

396 OUTLINES OF ASTRONOMY.
nations of direction plus and minus of the disturbing forces in the course
of a synodic revolution. But first it should be mentioned that, in the
case of an exterior disturbed by an interior planet, the disturbing body's
angular motion exceeds that of the disturbed. Hence P, though advan-
cing in its orbit, recedes relatively to the line of syzygies, or, which comes
to the same thing, the neutral points of either force overtake it in succes-
sion, and each, as it comes up to it, gives rise to a cusp in the corresponding
focus curve. The angles between the successive cusps will therefore be
to the angles between the corresponding neutral points for a fixed position
of M, in the same constant ratio of the synodic to the sidereal period of P,
which however is now a ratio of less inequality. These angles then will
be contracted in amplitude, and, for the same reason as before, the excur-
sions of the focus will be diminished, and the more so the shorter the
synodic revolution.
(714.) Since the cusps of either curve recur, in successive synodic
revolutions in the same order, and at the same angular distances from
each other, and from the line of conjunction, the same will be true of all
the corresponding points in the curve resulting from their superposition.
In that curve, every cusp, of either constituent, will give rise to a con-
vexity, and every intercuspidal arc to a relative concavity. It is evident
then that the compound curve or true path of the focus so resulting, but
for the cause above mentioned, would return into itself, whenever the
periodic times of the disturbing and disturbed bodies are commensurate,
because in that case the synodic period will also be commensurate
with either, and the arc of longitude intercepted between the sidereal
Fig. 100.
place of any one conjunction, and that next following it, will be an ali-
quot part of 360°. In all other cases it would be a non-reentering, more or
less undulating and more or less regular, spiroid, according to the number
of cusps in each of the constituent curves (that is to say, according to
the number of neutral points or changes of direction from inwards to

ANALOGOUS PLANETARY DISTURBANCES. 397
outwards, or from accelerating to retarding, and vice versa, of the normal
and tangential forces,) in a complete synodic revolution, and their distri-
bution over the circumference.
(715.) With regard to these changes, it is necessary to distinguish
three cases, in which the perturbations of planet by planet are very dis-
tinct in character. 1st. When the disturbing planet is exterior. In this
case there are four neutral points of either force. Those of the tangen-
tial force occur at the syzygies, and at the points of the disturbed orbit
(which we shall call points of equidistance), equidistant from the sun
and the disturbing planet (at which points, as we have shown (art. 614),
the total disturbing force is always directed inwards towards the sun.)
Those of the normal force occur at points intermediate between these last
mentioned points, and the syzygies, which, if the disturbing planet be
very distant, hold nearly the situation they do in the lunar theory, i. e.
considerably nearer the quadratures than the syzygies. In proportion as the
distance of the disturbing planet diminishes, two of these points, viz. those
nearest the syzygy, approach to each other, and to the syzygy, and in the
extreme case, when the dimensions of the orbits are equal, coincide with it.
(716.) The second case is that in which the disturbing planet is inte-
rior to the disturbed, but at a distance from the sun greater than half that
of the latter. In this case there are four neutral points of the tangential
force, and only two of the normal. Those of the tangential force occur
at the syzygies, and at the points of equidistance. The force retards the
disturbed body from conjunction to the first such points after conjunction,
accelerates it thence to the opposition, thence again retards it to the next
point of equidistance, and finally again accelerates it up to the conjunc-
tion. As the disturbing orbit contracts in dimension, the points of equi,
distance approach; their distance from syzygy from 60° (the extreme
case) diminishing to nothing, when they coincide with each other, and
with the conjunction. In the case of Saturn disturbed by Jupiter, that
distance is only 23° 33'. The neutral points of the normal force lie
somewhat beyond the quadratures, on the side of the opposition, and do
not undergo any very material change of situation with the contraction
of the disturbing orbit.
(717) The third case is that in which the diameter of the disturbing
interior orbit is less than half that of the disturbed. In this case there
are only two points of evanescence for either force. Those of the tan-
gential force are the syzygies. The disturbed planet is accelerated through-
out the whole semi-revolution from conjunction to opposition, and retarded
from opposition to conjunction, the maxima of acceleration and retardation
occurring not far from quadrature. The neutral points of the norm il
398 OUTLINES OF ASTRONOMY.
force are situated nearly as in the last case; that is to say, beyond the
quadratures towards the opposition. All these varieties the student will
easily trace out by simply drawing the figures, and resolving the forces in
a series of cases, beginning with a very large and ending with a very
small diameter of the disturbing orbit. It will greatly aid him in im-
pressing on his imagination the general relations of the subject, if he
construct, as he proceeds, for each case, the elegant and symmetrical ovals
Fig. 101.
D
sº -N 'ſ 77ty
is sº A.
E.
in which the points N and L (fig. art. 675,) always lie, for a fixed posi.
tion of M, and of which the annexed figure expresses the forms they
respectively assume in the third case now under consideration. The
second only differs from this, in having the common vertex m, of both
ovals, outside of the disturbed orbit A P, while in the case of an exterior
disturbing planet, the oval m L assumes a four-lobed form; its lobes
respectively touching the oval m N in its vertices, and cutting the orbit
A P in the points of equidistance and of tangency, (i. e. where M P S is .
a right angle) as in this figure. e'



THREE CASES DISTINGUISHED. 399
(718.) It would be easy now, bearing these features in mind, to trace
in any proposed case the form of the spiroid curve, described, as above
explained, by the upper focus. It will suffice, however, for our present
purpose, to remark, 1st, That between every two successive conjunctions
of P and M, the same general form, the same subordinate undulations,
and the same terminal displacement of the upper focus, are continually
repeated. 2dly, That the motion of the focus in this curve is retrograde
whenever the disturbing planet is exterior, and that in consequence the
apsides of the momentary ellipse also recede, with a mean velocity such
as, but for that displacement, would bring them round at the each con-
junction to the same relative situation with respect to the line of syzygies.
3dly, That in consequence of this retrograde movement of the apse, the
disturbed planet, apart from that consideration, would be twice in peri-
helio and twice in aphelio in its momentary ellipse in each synodic revo-
lution, just as in the case of the moon disturbed by the sun – and that in
consequence of this and of the undulating movement of the focus Hit-
self, an inequality will arise, analogous, mutatis mutandis in each case, to
the moon's variation; under which term we comprehend (not exactly in
conformity to its strict technical meaning in the lunar theory) not only
the principal inequality thus arising, but all its subordinate fluctuations.
And on this the parallactic inequality thus violently exaggerated is
superposed. - -
(719.) We come now to the class of inequalities which depend for
their existence on an appreciable amount of permanent excentricity in
the orbit of one or of both the disturbing and disturbed planets, in con-
sequence of which all their conjunctions do not take place at equal
distances either from the central body or from each other, and therefore
that symmetry in every synodic revolution on which depends the exact
restoration of both the axis and excentricity to their original values at the
completion of each such revolution no longer subsists. In passing from
conjunction to conjunction, then, there will no longer be effected a com-
plete restoration of the upper focus to the same relative situation, or of
the axis to the same length, which they respectively had at the outset.
At the same time it is not less evident that the differences in both re-
spects are only what remain outstanding, after the compensation of by far
the greater part of the deviations to and fro from a mean state, which
occur in the course of the revolution; and that they amount to but small
fractions of the total excursions of the focus from its first position, or of
the increase and decrease in the length of the axis effected by the direct
action of the tangential force, — so small, indeed, that, unless owing to
peculiar adjustments they be enabled to accumulate again and again at
400 OUTLINES OF ASTRONOMY.
successive conjunctions in the same direction, they would be altogether
undeserving of any especial notice in a work of this nature. Such ad-
justments, however, would evidently exist if the periodic times of the
planets were exactly commensurable; since in that case all the possible
conjunctions which could ever happen (the elements not being materially
changed) would take place at fixed points in longitude, the intermediate
points being never visited by a conjunction. Now, of the conjunctions
thus distributed, their relations to the lines of symmetry in the orbits
being all dissimilar, some one must be more influential than the rest on
each of the elements (not necessarily the same upon all). Consequently,
in a complete cycle of conjunctions, wherein each has been visited in its
turn, the influence of that one on the element to which it stands so espe-
cially related, will preponderate over the counteracting and compensating
influence of the rest, and thus, although in such a cycle as above specified
a further and much more exact compensation will have been effected in
its value than in a single revolution, still that compensation will not be
complete, but a portion of the effect (be it to increase or to diminish the
excentricity or the axis, or to cause the apse to advance or to recede,)
will remain outstanding. In the next cycle of the same kind this will be
repeated, and the result will be of the same character, and so on, till at
length a sensible and ultimately a large amount of change shall have
taken place, and in fact until the axis (and with it the mean motion) shall
have so altered as to destroy the commensurability of periods, and the ap-
sides have so shifted as to alter the place of the most influential conjunction.
(720.) Now, although it is true that the mean motions of no two
planets are exactly commensurate, yet cases are not wanting in which
there exists an approach to this adjustment. For instance, in the case of
Jupiter and Saturn, a cycle composed of five periods of Jupiter and two
of Saturn, although it does not excactly bring about the same configura-
tion, does so pretty nearly. Five periods of Jupiter are 21663 days, and
two periods of Saturn, 21519 days. The difference is only 146 days, in
which Jupiter describes, on an average, 12°, and Saturn about 5°; so
that after the lapse of the former interval they will only be 7° from a
conjunction in the same parts of their orbits as before. If we calculate
the time which will exactly bring about, on the average, three conjunc-
tions of the two planets, we shall find it to be 21760 days, their synodical
period being 7253.4 days. In this interval Saturn will have described
8°6' in excess of two sidereal revolutions, and Jupiter the same angle in
excess of five. Every third conjunction, then, will take place 8° 6' in
advance of the preceding, which is near enough to establish, not, it is
true, an identity with, but still a great approach to the case in question.
LONG EQUATION OF JUPITER AND SATURN. 401
The excess of action, for several such triple conjunctions (7 or 8) in suc-
cession, will lie the same way, and at each of them the elements of P's
orbit and its angular motion will be similarly influenced, so as to accu-
mulate the effect upon its longitude; thus giving rise to an irregularity
of considerable magnitude and very long period, which is well known to
astronomers by the name of the great inequality of Jupiter and Saturn.
(721.) The are 8°6' is contained 444 times in the whole circumference
of 360°; and accordingly, if we trace round this particular conjunction,
we shall find it will return to the same point of the orbit in so many
times 21760 days, or in 2648 years. But the conjunction we are now
considuring is only one out of three. The other two will happen at
points of the orbit about 123° and 246° distant, and these points also will
advance by the same are of 8° 6' in 21760 days. Consequently the
period of 2648 years will bring them all round, and in that interval each
of them will pass through that point of the two orbits from which we
commenced: hence a conjunction (one or other of the three) will happen
at that point once in one third of this period, or in 883 years; and this
is, therefore, the cycle in which the “great inequality” would undergo
its full compensation, did the elements of the orbits continue all that
time invariable. Their variation, however, is considerable in so long an
interval; and, owing to this cause, the period itself is prolonged to about
918 years.
(722.) We have selected this inequality as the most remarkable in-
stance of this kind of action on account of its magnitude, the length of
its period, and its high historical interest. It had long been remarked
by astronomers, that on comparing together modern with ancient observa-
tions of Jupiter and Saturn, the mean motions of these planets did not
appear to be uniform. The period of Saturn, for instance, appeared to
have been lengthening throughout the whole of the seventeenth century,
and that of Jupiter shortening—that is to say, the one planet was con-
stantly lagging behind, and the other getting in advance of its calculated
place. On the other hand, in the eighteenth century, a process precisely
the reverse seemed to be going on. It is true the whole retardations and
accelerations observed were not very great; but, as their influence went on
accumulating, they produced, at length, material differences between the
observed and calculated places of both these planets, which as they could
not then be accounted for by any theory, excited a high degree of atten-
tion, and were even, at one time, too hastily regarded as almost subversive
of the Newtonian doctrine of gravity. For a long while this diffrence
baffled every endeavour to account for it; till at length Laplace pointed
26
402 OUTLINES OF ASTRONOMY.
out its cause in the near commensurability of the mean motions, as above
shown, and succeeded in calculating its period and amount.
(723.) The inequality in question amounts, at its maximum, to an al-
ternate gain and loss of about 0°49' in the longitude of Saturn, and a
corresponding loss and gain of about 0° 21' in that of Jupiter. That an
acceleration in the one planet must necessarily be accompanied by a re-
tardation in the other, might appear at first sight self-evident, if we con-
sider, that action and reaction being equal, and in contrary directions,
whatever momentum Jupiter communicates to Saturn in the direction
PM, the same momentum must Saturn communicate to Jupiter in the
direction MP. The one, therefore, it might seem to be plausibly argued,
will be dragged forward, whenever the other is pulled back in its orbit.
The inference is correct, so far as the general and final result goes; but
the reasoning by which it would, on the first glance, appear to be thus
summarily established is fallacious, or at least incomplete. It is perfectly
true that whatever momentum Jupiter communicates directly to Saturn,
Saturn communicates an equal momentum to Jupiter in an opposite linear
direction. But it is not with the absolute motions of the two planets in
space that we are now concerned, but with the relative motion of each
separately, with respect to the sun regarded as at rest. The perturbative
forces (the forces which disturb these relative motions) do not act along
the line of junction of the planets (art. 614.) In the reasoning thus
objected to, the attraction of each on the sun has been left out of the
account', and it remains to be shown that these attractions neutralize and
destroy each other's effects in considerable periods of time, as bearing
upon the result in question. Suppose then that we for a moment abandon
the point of view, in which we have hitherto all along considered the
subject, and regard the sun as free to move, and liable to be displaced by
the attractions of the two planets. Then will the movements of all be
performed about the common centre of gravity, just as they would have
been about the sun's centre regarded as immoveable, the sun all the while
circulating in a small orbit (with a motion compounded of the two elliptic
motions it would have in virtue of their separate attractions) about the
same centre. Now in this case M still disturbs P, and P, M, but the
whole disturbing force now acts along their line of junction, and since it
remains true that whatever momentum M generates in P, P will generate
the same in M in a contrary direction; it will also be strictly true that, so
* We are here reading a sort of recantation. In the edition of 1833 the remarkable
result in question is sought to be established by this vicious reasoning. The mistake
is a very natural one, and is so apt to haunt the ideas of beginners in this department
of physics, that it is worth while expressly to warn them against it.
LAW OF RECIPROCITY. 403
far as a disturbance of their elliptic motions about the common centre of
gravity of the system is alone regarded, whatever disturbance of velocity
is generated in the one, a contrary disturbance of velocity (only in the
inverse ratio of the masses and modified, though never contradicted, by
the directions in which they are respectively moving), will be generated
in the other. Now when we are considering only inequalities of long
period comprehending many complete revolutions of both planets, and
which arise from changes in the axes of the orbits, affecting their mean
motions, it matters not whether we suppose these motions performed
about the ommon centre of gravity, or about the sun, which never de-
parts from that centre to any material extent (the mass of the sun being
such in comparison with that of the planets, that that centre always lies
within his surface.) The mean motion therefore, regarded as the average
angular velocity during a revolution, is the same whether estimated by
reference to the sun's centre, or to the centre of gravity, or, in other
words, the relative mean motion referred to the sun is identical with the
absolute mean motion referred to the centre of gravity.
(724.) This reasaning applies equally to every case of mutual disturb-
ance resulting in a long inequality such as may arise from a slow and
long-continued periodical increase and diminution of the axes, and geom-
eters have accordingly demonstrated as a consequence from it, that the
proportion in which such inequalities affect the longitudes of the two
planets concerned, or the maxima of the excesses and defects of their
longitudes above and below their elliptic values, thence arising, in each,
are to each other in the inverse ratio of their masses multiplied by the
square roots of the major axes of their orbits, and this result is confirmed
by observation, and will be found verified in the instance immediately in
question as nearly as the uncertainty still subsisting as to the masses of
the two planets will permit.
(725.) The inequality in question, as has been observed in general,
(art. 718,) would be much greater, were it not for the partial compensa-
tion which is operated in it in every triple conjunction of the planets.
Suppose PQR to be Saturn's orbit, and p q r Jupiter's; and suppose a
conjunction to take place at Pp, on the line S.A.; a second at 123° dis-
tance, on the line S B; a third at 246° distance, on S C ; and the next at
368°, on SD. This last-mentioned conjunction, taking place nearly in
the situation of the first, will produce nearly a repetition of the first effect
in retarding or accelerating the planets; but the other two, being in the
most remote situations possible from the first, will happen under entirely
different circumstances as to the position of the perihelia of the orbits.
Now, we have seen that a presentation of the one planet to the other in
404 OUTLINES OF ASTRONOMY.
conjunction, in a variety of situations, tends to produce compensation;
and, in fact, the greatest possible amount of compensation which can be
produced by only three conjunctions is when they are thus equally dis
tributed round the centre. Hence we see that it is not the whole amount
of perturbation which is thus accumulated in each triple conjunction, but
only that small part which is left uncompensated by the intermediate
ones. The reader, who possesses already some acquaintance with the
subject, will not be at a loss to perceive how this consideration is, in fact,
equivalent to that part of the geometrical investigation of this inequality
which leads us to seek its expression in terms of the third order, or in-
volving the cubes and products of three dimensions of the excentricities
and inclinations; and how the continual accumulation of small quantities,
during long periods, corresponds to what geometers intend when they
speak of small terms receiving great accessions of magnitude by the intro-
duction of large coefficients in the process of integration.
(726.) Similar considerations apply to every case of approximate com-
mensurability which can take place among the mean motions of any two
planets. Such, for instance, is that which obtains between the mean
motion of the earth and Venus, 13 times the period of Venus being very
nearly equal to 8 times that of the earth. This gives rise to an extremely
near coincidence of every fifth conjunction, in the same parts of each orbit
(within 346th part of a circumference,) and therefore to a correspondingly
extensive accuinulation of the resulting uncompensated perturbation.
But, on the other hand, the part of the perturbation thus accumulated is
only that which remains outstanding after passing the equalizing ordeal
of five conjunctions equally distributed round the circle; or, in the lan-
guage of geometers, is dependent on powers and products of the excen-
tricities and inclinations of the fifth order. It is, therefore, extremely
minute, and the whole resulting inequality, according to the elaborate
calculations of Mr. Airy, to whom it owes its detection, amounts to no

LONG INEQUALITIES OF ELEMENTS. 405
more than a few seconds at its maximum, while its period is no less than
240 years. This example will serve to show to what minuteness these
inquiries have been carried to the planetary theory.
(727.) That variations of long period arising in the way above described
are necessarily accompanied by similarly periodical displacements of the
upper focus, equivalent in their effect to periodical fluctuations in the
magnitude of the excentricity, and in the position of the line of apsides,
is evident from what has been already said respecting the motion of the
upper focus under the influence of the disturbing forces. In the case of
circular orbits the mean place of H coincides with S the centre of the sun,
but if the orbits have any independent ellipticity, this coincidence will no
longer exist—and the mean place of the upper focus will come to be
inferred from the average of all the situations which it actually holds
during an entire revolution. Now the fixity of this point depends on the
equality of each of the branches of the cuspidated curves, and consequent
equality of excursion of the focus in each particular direction, in every
successive situation of the line of conjunction. But if there be some
one line of conjunction in which these excursions are greater in any one
particular direction than in another, the mean place of the focus will be
displaced, and if this process be repeated, that mean place will continue
to deviate more and more from its original position, and thus will arise a
circulation of the mean place of the focus for a revolution about another
mean situation, the average of all the former mean places during a com-
plete cycle of conjunctions. Supposing S to be the sun, O the situation
the upper focus would have, had these inequalities no existence, and H. K.
the path of the upper focus, which it pursues about O by reason of them,
then it is evident that in the course of a complete cycle of the inequality
in question, the excentricity will have fluctuated between the extreme
limits SJ and S I and the direction of the longer axis between the
extreme position S H and S K, and that if we suppose if h k to be the
corresponding mean places of the focus, i.j will be the extent of the fluctu-
ation of the mean excentricity, and the angle h s k, that of the longitude
of the perigee.
(728.) The periods then in which these fluctuations go through their
phases are neeessarily equal in duration with that of the inequality in
longitude, with which they stand in connexion. But it by no means
follows that their maxima all coincide. The variation of the axis to which
that of the mean motion corresponds, depends on the tangential force only
whose maximum is not at conjunction or opposition, but at points remote
from either, while the excentricity depends both on the normal and tan-
gential forces, the maximum of the former of which is at the conjunction
g
406 OUTLINES OF ASTRONOMY.
Fig. 104.
That particular conjunction therefore, which is most influential on the
axis, is not so on the excentricity, so that it can by no means be concluded
that either the maximum value of the axis coincides with the maximum,
or the minimum of the excentricity, or with the greatest excursion to or
fro of the line of apsides from its mean situation, all that can be safely
asserted is, that as either the axis or the excentricity of the one orbit
varies, that of the other will vary in the opposite direction.
(729.) The primary elements of the lunar and planetary orbits, which
may be regarded as variable, are the longitude of the node, the inclina-
tion, the axis, excentricity, longitude of the perihelion, and epoch (art.
496). In the foregoing articles we have shown in what manner each of
the first five of these elements is made to vary, by the direct action of
the perturbing forces. It remains to explain in what manner the last
comes to be affected by them. And here it is necessary, in the first in-
stance, to remove some degree of obscurity which may be thought to hang
about the sense in which the term itself is to be understood in speaking of
an orbit, every other element of which is regarded as in a continual state
of variation. Supposing, then, that we were to reverse the process of
calculation described in arts. 499 and 500 by which a planet's heliocentric
longitude in an elliptic orbit is computed for a given time; and setting
out with a heliocentric longitude ascertained by observation, all the other
elements being known, we were to calculate either what mean longitude
the planet had at a given epochal time, or, which would come to the
same thing, at what moment of time (thenceforward to be assumed as
an epoch) it had a given mean longitude. It is evident that by this
means the epoch, if not otherwise known, would become known, whether
we consider it as the moment of time corresponding to a convenient mean
longitude, or as the mean longitude corresponding to a convenient time.
The latter way of considering it has some advantages in respect of general
convenience, and astronomers are in agreement in employing, as an ele.

\ +
WARHATION OF THE EPOCH. 407
ment under the title “Epoch of the mean longitude,” the mean longitude
of the planet so computed for a fixed date; as, for instance, the commence
ment of the year 1800, mean time at a given place. Supposing now all
elements of the orbit invariable, if we were to go through this reverse
process, and thus ascertain the epoch (so defined) from any number of
different perfectly correct heliocentric longitudes, it is clear we should
always come to the same result. One and the same “epoch” would come
out from all the calculations.
(730.) Considering then the “epoch” in this light, as merely a result
of this reversed process of calculation, and not as the direct result of an
observation instituted for the purpose at the precise epochal moment of
time, (which would be, generally speaking, impracticable,) it might be
conceived subject to variation in two distinct ways, viz. dependently and
independently, ºpendently it must vary, as a necessary consequence of
the variation of the other elements; because, if setting out from one and
the same observed heliocentric longitude of the planet, we calculate back
to the epoch with two different sets of intermediate elements, the one set
consisting of those which it had immediately before its arrival at that
longitude, the other that which it takes up immediately after (i. e. with
an unvaried and varied system), we cannot (unless by singular accident of
mutual counteraction) arrive at the same result; and the difference of the
results is evidently the variation of the epoch. On the other hand, how-
ever, it cannot vary independently; for since this is the only mode in
which the unvaried and varied epochs can become known, and as both
result from direct processes of calculation involving only given data, the
results can only differ by reason of the difference of those data. Or we
may argue thus. The change in the path of the planet, and its place in
that path so changed, at any future time (supposing it to undergo no fur.
ther variation), are entirely owing to the change in its velocity and direc-
tion, produced by the disturbing forces at the point of disturbance; now
these latter changes (as we have above seen) are completely represented by
the momentary change in the situation of the upper focus, taken in com-
bination with the momentary variation in the plane of the orbit; and these
therefore express the total effect of the disturbing forces. There is, there-
fore, no direct and specific action on the epoch as an independent variable.
It is simply left to accommodate itself to the altered state of things in the
mode already indicated.
(731.) Nevertheless, should the effects of perturbation by inducing
changes on these other elements affect the mean longitude of the planet
in any other way than can be considered as properly taken account of, by
the varied periodic time due to a change of axis, such effects must be re-
408 OUTLINES OF ASTRONOMY.
garded as incident on the epoch. This is the case with a very curious
class of perturbations which we are now to consider, and which have their
origin in an alteration of the average distance at which the disturbed body
is found at every instant of a complete revolution, distinct from, and not
brought about by the variation of the major semi-axis, or momentary
“mean distance” which is an imaginary magnitude, to be carefully distin-
guished from the averge of the actual distances now contemplated. Per-
turbations of this class (like the moon’s variation, with which they are
intimately connected) are independent on the excentricity of the disturbed
orbit; for which reason we shall simplify our treatment of this part of the
subject, by supposing that orbit to have no permanent excentricity, the
upper focus in its successive displacements merely revolving about a mean
position coincident with the lower. We shall also suppose M very dis-
tant, as in the lunar theory.
(732.) Referring to what is said in arts. 706 and 707, and to the figures
accompanying those articles, and considering first the effect of the tangential
force, we see that besides the effect of that force in changing the length
of the axis, and consequently the periodic time, it causes the upper focus
II to describe, in each revolution of P, a four-cusped curve, a, b, d, e,
about S, all whose intercuspidal arcs are similar and equal. This supposes
M fixed, and at an invariable distance,—suppositions which simplify the
relations of the subject, and (as we shall afterwards show) do not affect
the general nature of the conclusions to be drawn. In virtue, then, of
the excentricity thus given rise to, P will be at the perigee of its momen-
tary ellipse at syzygies and in its apogee at quadratures. Apart, therefore,
from the change arising from the variation of axis, the distance of P
from S will be less at syzygies, and greater at quadratures, than in the
original circle. But the average of all the distances during a whole revo-
lution will be unaltered; because the distances of a, d, b, e, from S being
equal, and the arcs symmetrical, the approach in and about perigee will
be equal to the recess in and about the apogee. And, in like manner, the
effect of the changes going on in the length of the axis itself, on the
average in question, is nil, because the alternate increases and decreases
of that length balance each other in a complete revolution. Thus we see
that the tangential force is excluded from all influence in producing the
class of perturbations now under consideration.
(733.) It is otherwise with respect to the normal force. In virtue of
the action of that force the upper focus describes, in each revolution of P,
the four-cusped curve (fig. art. 707), whose intercuspidal arcs are alter-
nately of very unequal extent, arising, as we have seen, from the longer
duration and greater energy of the outward than of the inward action of
INEQUALITIES INCIDENT ON THE EPOCH. 409
the disturbing force. Although, therefore, in perigee at syzygies and in
apogee at quadratures, the apogeal recess is much greater than the perigeal
approach, inasmuch as S d greatly exceeds S a. On the average of a
whole revolution, then, the recesses will preponderate, and the average
distance will therefore be greater in the disturbed than in the undisturbed
orbit. And it is manifest that this conclusion is quite independent of any
change in the length of the axis, which the normal force has no power to
produce.
(734.) But neither does the normal force operate any change of linear
velocity in the disturbed body. When carried out, therefore, by the
effect of that force to a greater distance from S, the angular velocity of its
motion round S will be diminished: and contrariwise when brought nearer.
The average of all the momentary angular motions, therefore, will de-
crease with the increase in that of the momentary distances; and in a
higher ratio, since the angular velocity, under an equable description of
areas, is inversely as the square of the distance, and the disturbing force,
being (in the case supposed) directed to or from the centre, does not dis-
turb that equable description (art. 616). Consequently, on the average
of a whole revolution, the angular motion is slower, and therefore the time
of completing a revolution, and returning to the same longitude, longer
than in the undisturbed orbit, and that independent of and without any
reference to the length of the momentary axis, and the “periodic time”
or “mean motion” dependent thereon. We leave to the reader to follow
out (as is easy to do) the same train of reasoning in the cases of planetary
perturbation, when M is not very remote, and when it is interior to the
disturbed orbit. In the latter case the preponderant effect changes from
a retardation of angular velocity to an acceleration, and the dilatation of
the average dimensions of P's orbit to a contraction. -
(735.) The above is an accurate analysis, according to strict dynamical
principles, of an effect which, speaking roughly, may be assimilated to an
alteration cf M's gravitation towards S by the mean preponderant amount
of the outward and inward action of the normal forces constantly exerted
—nearly as would be the case if the mass of the disturbing body were
formed into a ring of uniform thickness, concentric with S, and of such
diameter as to exert an action on P everywhere equal to such mean pre-
ponderant force, and in the same direction as to inwards or outwards. For
it is clear that the action of such a ring on P, will be the difference of its
attractions on the two points P and S, of which the latter occupies its
centre, the former is excentric. Now the attraction of a ring on its
centre is manifestly equal in all directions, and therefore, estimated in
any one direction, is zero. On the other hand, on a point P out of its
410 OljTLINES OF ASTRONOMY.
centre, if within the ring, the resulting attraction will always be outwards,
towards the nearest point of the ring, or directly from the centre.' But
if P lie without the ring, the resulting force will act always inwards,
urging P towards its centre. Hence it appears that the mean effect of
the radial force of the ring will be different in its direction, according as
the orbit of the disturbing body is exterior or interior to that of the dis-
turbed. In the former case it will act in diminution, in the latter in
augmentation of the central gravity.
(736.) Regarding, still, only the mean effect, as produced in a great
number of revolutions of both bodies, it is evident that such an increase
of central force will be accompanied with a diminution of périodic time
and distance of a body revolving with a stated velocity, and vice versá.
This, as we have shown, is the first and most obvious effect of the radial
part of the disturbing force, when exactly analyzed. It alters permanently,
and by a certain mean amount, the distances and times of revolution of
all the bodies composing the planetary system, from what they would be,
did each planet circulate about the sun uninfluenged by the attraction of
the rest; the angular motion of the interior boºies of the system being
thus rendered less, and those of the exterior greater, than on that suppo-
sition. The latter effect, indeed, might be at once concluded from this
obvious consideration, — that all the planets revolving interiorly to any
* As this is a proposition which the equilibrium of Saturn's ring renders not merely
speculative or illustrative, it will be well to demonstrate it; which may be done very
simply, and without the aid of any calculus. Conceive a spherical shell, and a point
within it: every line passing through the point, and terminating both ways in the shell,
will, of course, be equally inclined to its surface at either end, being a chord of a sphe-
rical surface, and therefore symmetrically related to all its parts. Now, conceive a
small double cone, or pyramid, having its apex at the point, and formed by the conical
motion of such a line round the point. Then will the two portions of the spherical
shell, which form the bases of both the cones, or pyramids, be similar and equally in-
clined to their axes. Therefore their areas will be to each other as the squares of their
distances from the common apex. Therefore their attractions on it will be equal, be-
cause the attraction is as the attracting matter directly, and the square of its distance
inversely. Now, these attractions act in opposite directions, and therefore counteract
each other. Therefore the point is in equilibrium between them ; and as the same is
true of every such pair of areas into which the spherical shell can be broken up, there-
fore the point will be in equilibrium however situated within such a spherical shell.
Now take a ring, and treat it similarly, breaking its circumference up into pairs of ele-
ments, the bases of triangles formed by lines passing through the attracted point.
Here the attracting elements being lines, not surfaces, are in the simple ratio of the
distances, not the duplicate, as they should be to maintain the equilibrium. Therefore
it will not be maintained, but the nearest elements will have the superiority, and the
point will, on the whole, be urged towards the nearest part of the 4ng. The same is
true of every linear ring, and is therefore true of any assemblagº of concentric ones
fºrming a flat antiulus, like the ring of Saturn.
INEQUALITIES INCIDENT ON THE EPOCH. 411
orbit may be considered as adding to the general aggregate of the attract-
ing matter within, which is not the less efficient for being distributed over
space, and maintained in a state of circulation.
(737.) This effect, however, is one which we have no means of mea-
suring, or even of detecting, otherwise than by calculation. For our
knowledge of the periods of the planets is drawn from observations made
on them in their actual state, and therefore under the influence of this,
which may be regarded as a sort of constant part of the perturbative
action. Their observed mean motions are therefore affected by the whole
amount of its influence; and we have no means of distinguishing this by
observation from the direct effect of the sun's attraction, with which it is
blended. Our knowledge, however, of the masses of the planets assures
us that it is extremely small; and this, in fact, is all which it is at all
important to us to know, in the theory of their motions.
(738.) The action of the sun upon the moon, in like manner, tends, by
its mean influence during many successive revolutions of both bodies, to
increase permanently the moon's distance and periodic time. But this
general average is not established, either in the case of the moon or
planets, without a series of subordinate fluctuations, which we have pur-
posely neglected to take account of in the above reasoning, and which
obviously tend, in the average of a great multitude of revolutions, to
neutralize each other. In the lunar theory, however, some of these sub-
ordinate fluctuations are very sensible to observation. The most conspi-
cuous of these is the moon’s annual equation; so called because it consists
in an alternate increase and decrease in her longitude, corresponding with
|
*g.
Fig. 105.

\\ –
A.
the earth's situation in its annual orbit; i. e. to its angular distance from
the perihelion, and therefore having a year instead of a month, or aliquot
part of a month, for its period. To understand the mode of its produc-
tion, let us suppose the sun, still holding a fixed position in longitude, to
approach gradually nearer to the earth. Then will all its disturbing forces
be gradually increased in a very high ratio compared with the diminution
of the distance (being inversely as its cube; so that its effects of every
kind are three times greater in respect of any change ºf distance, than
412 OUTLINES OF ASTRONOMY.
they would be by the simple law of proportionality). Hence, it is obvious
that the focus H (art. 707) in the act of describing each intercuspidal are
of the curve a, d, b, e, will be continually carried out farther and farther
from S; and the curve, instead of returning into itself at the end of each
revolution, will open out into a sort of cuspidated spiral, as in the figure
annexed. Retracing now the reasoning of art. 733, as adapted to this
state of things, it will be seen that so long as this dilatation goes on, so
long will the difference between M's recess from S in aphelio and its
approach in perihelio (which is equal to the difference of consecutive long
and short semidiameters of this curve) also continue to increase, and with
it the average of the distances of M from S in a whole revolution, and
consequently also the time of performing such a revolution. The reverse
process will go on as the sun again recedes. Thus it appears that, as the
sun approaches the earth, the mean angular motion of the moon on the
average of a whole revolution will diminish, and the duration of each
lunation will therefore exceed that of the foregoing, and vice versá.
(739.) The moon’s orbit being supposed circular, the sun's orbitual
motion will have no other effect than to keep the moon longer under the
influence of every gradation of the disturbing force, than would have been
the case had his situation in longitude remained unaltered (art. 711.) The
effects, therefore, will take place only on an increased scale in the propor-
tion of the increased time; i. e. in the proportion of the synodic to the
sidereal revolution of the moon. Observation confirms these results, and
assigns to the inequality in question a maximum value of between 10' and
11', by which the moon is at one time in advance of, and at another be-
hind, its mean place, in consequence of this perturbation.
(740.) To this class of inequalities we must refer one of great import-
ance, and extending over an immense period of time, known by the name
of the secular acceleration of the moon's mean motion. It had been
observed by Dr. Halley, on comparing together the records of the most
ancient lunar eclipses of the Chaldean astronomers with those of modern
times, that the period of the moon’s revolution at present is sensibly
shorter than at that remote epoch; and this result was confirmed by a
further comparison of both sets of observations with those of the Arabian
astronomers of the eighth and ninth centuries. It appeared, from these
comparisons, that the rate at which the moon's mean motion, increases is
about 11 seconds per century, - a quantity small in itself, but becoming
considerable by its accumulation during a succession of ages. This re-
markable fact, like the great equation of Jupiter and Saturn, had been
long the subject of toilsome investigation to geometers. Indeed, so
difficult did it appear to render any exact account of, that while some
ANNUAL INEQUALITY OF THE MOON. 413
were on the point of again declaring the theory of gravity inadequate to
its explanation, others were for rejecting altogether the evidence on which
it rested, although quite as satisfactory as that on which most historical
events are credited. It was in this dilemma that Laplace once more
stepped in to rescue physical astronomy from its reproach, by pointing out
the real cause of the phaenomenon in question, which, when so explained,
is one of the most curious and instructive in the whole range of our sub-
ject, — one which leads our speculations farther into the past and future,
and points to longer vistas in the dim perspective of changes which our
system has undergone and is yet to undergo, than any other which obser-
vation assisted by theory has developed.
(741.) The year is not an exact number of lunations. It consists of
twelve and a fraction. Supposing then the sun and moon to set out from
conjunction together; at the twelfth conjunction subsequent the sun will
not have returned precisely to the same point of its annual orbit, but will
fall somewhat short of it, and at the thirteenth will have overpassed” it.
Hence in twelve lunations the gain of longitude during the first half year
will be somewhat under and in thirteen somewhat over-compensated. In
twenty-six it will be nearly twice as much over-compensated, in thirty-nine
not quite so nearly three times as much, and so on, until, after a certain
number of such multiples of a lunation have elapsed, the sun will be
found half a revolution in advance, and in place of receding farther at the
expiration of the next, it will have begun to approach. From this time
every succeeding cycle will destroy some portion of that over-compensa-
tion, until a complete revolution of the sun in excess shall be accom-
plished. Thus arises a subordinate or rather supplementary inequality,
having for its period as many years as is necessary to multiply the defi-
cient arc into a whole revolution, at the end of which time a much more
exact compensation will have been operated, and so on. Thus after a
moderate number of years an almost perfect compensation will be effected,
and if we extend our views to centuries we may consider it as quite so.
Such at least would be the case if the solar ellipse were invariable. But
that ellipse is kept in a continual but excessively slow state of change by
the action of the planets on the earth. Its axis, it is true, remains unal-
tered; but its excentricity is, and has been since the earliest ages, dimin-
ishing; and this diminution will continue (there is little reason to doubt)
till the excentricity is annihilated altogether, and the earth’s orbit becomes
a perfect circle; after which it will again open out into an ellipse, the
excentricity will again increase, attain a certain moderate amount, and
then again decrease. The time required for these evolutions, though
scalculable, has not been calculated, further than to satisfy us that it is
414 Olj TLINES OF ASTRONOMY.
not to be reckoned by hundreds or by thousands of years. It is a period,
in short, in which the whole history of astronomy and of the human race
occupies but as it were a point, during which all its changes are to be
regarded as uniform. Now, it is by this variation in the excentricity of
the earth's orbit that the secular acceleration of the moon is caused. The
compensation above spoken of (even after the lapse of centuries) will now,
we see, be only imperfectly effected, owing to this slow shifting of one
of the essential data. The steps of restoration are no longer identical
with, nor equal to, those of change. The struggle up hill is not main-
tained on equal terms with the downward tendency. The ground is all
the while slowly sliding beneath the feet of the antagonists. During the
whole time that the earth's excentricity is diminishing, a preponderance
is given to the reaction over the action; and it is not till that diminution
shall cease, that the tables will be turned, and the process of ultimate
restoration will commence. Meanwhile, a minute, outstanding, and un-
compensated effect in favour of acceleration is left at each recurrence, or
near recurrence, of the same configurations of the sun, the moon, and the
solar perigee. These accumulate, and at length affect the moon’s longi.
tude to an extent not to be overlooked.
(742.) The phaenomenon, of which we have now given an account, is
another and very striking example of the propagation of a periodic change
from one part of a system to another. The planets, with one exception,
have no direct appreciable action on the lunar motions as referred to the
earth. Their masses are too small, and their distances too great, for their
difference of action on the moon and earth ever to become sensible. Yet
their effect on the earth’s orbit is thus, we see, propagated through the
sun to that of the moon; and, what is very remarkable, the transmitted
effect thus indirectly produced on the angle described by the moon round
the earth is more sensible to observation than that directly produced by
them on the angle described by the earth round the sun.
(743.) Referring to the reasoning of art. 738, we shall perceive that
if, owing to any other cause than its elliptic motion, the sun's distance
from the earth be subject to a periodical increase and decrease, that varia-
tion will give rise to a lunar inequality of equal period analogous to the
annual equation. It thus happens that very minute changes impressed
on the orbit of the earth, by the direct action of the planets, (provided
their periods, though not properly speaking secular, be of considerable
length,) may make themselves sensible in the lunar motions. The longi-
tude of that satellite, as observed from the earth, is, in fact, singularly
sensible to this kind of reflected action, which illustrates in a striking .
manner the principle of forced vibrations laid down in art. 650. The
INDIRECT ACTION OF VENUS ON THE MOON. 415
reason of this will be readily apprehended, if we consider that however
trifling the increase of her longitude which would arise in a single revolu-
tion, from a minute and almost infinitesimal increase of her mean angu-
lar velocity, that increase is not only repeated in each subsequent revolu-
tion, but is reinforced during each by a similar fresh accession of angular
motion generated in its lapse. This process goes on so long as the angu-
lar motion continues to increase, and only begins to be reversed when
lapse of time, bringing round a contrary action on the angular motion,
shall have destroyed the excess of velocity previously gained, and begun
to operate a retardation. In this respect, the advance gained by the moon
on her undisturbed place may be assimilated, during its increase, to the
space described from rest under the action of a continually accelerating
force. The velocity gained in each instant is not only effective in carry.
ing the body forward during each subsequent instant, but new velocities
are every instant generated, and go on adding their cumulative effects to
those before produced.
(744.) The distance of the earth from the sun, like that of the moon
from the earth, may be affected in its average value estimated over long
periods embracing many revolutions, in two modes, conformably to the
theory above delivered. 1st, it may vary by a variation in the length of
the axis major of its orbit, arising from the direct action of some tangen-
tial disturbing force on its velocity, and thereby producing a change of
mean motion and periodic time in virtue of the Keplerian law of periods,
which declares that the periodic times are in the sesquiplicate ratio of the
mean distances. Or, 2dly, it may vary by reason of that peculiar action
on the average of actual distances during a revolution, which arises from
variations of excentricity and perihelion only, and which produces that
sort of change in the mean motion which we have characterized as inci-
dent on the epoch. The change of mean motion thus arising, has nothing
whatever to do with any variation of the major axis. It does not depend
on the change of distance by the Keplerian law of periods, but by that
of areas. The altered mean motion is not sub-sesquiplicate to the altered
axis of the ellipse, which in fact does not alter at all, but is sub-dupli.
cate to the altered average of distances in a revolution; a distinction
which must be carefully borne in mind by every one who will clearly un-
derstand either the subject itself, or the force of Newton's explanation
of it in the 6th Corollary of his celebrated 66th Proposition. In which-
ever mode, however, an alteration in the mean motion is effected, if we
accommodate the general sense of our language to the specialities of the
case, it remains true that every change in the mean motion is accompa-
nied with a corresponding change in the mean distance. -
416 OUTLINES OF ASTRONOMY.
(745.) Now we have seen (art. 726), that Venus produces in the earth
a perturbation in lengitude, of so long a period (240 years), that it can-
not well be regarded without violence to ordinary language, otherwise than
as an equation of the mean motion. Of course, therefore, it follows that
during that half of this long period of time, in which the earth’s motion
is retarded, the distance between the Sun and earth is on the increase, and
vice varsá. Minute as is the equation in question, and consequent altera-
tion of solar distance, and almost inconceivably minute as is the effect
produced on the moon’s mean angular velocity in a single lunation, yet
the great number of lunations (1484), during which the effect goes on
accumulating in one direction, causes the moon, at the moment when that
accumulation has attained its maximum to be very sensibly in advance of
its undisturbed place (viz. by 23" of longitude), and after 1484 more
lunations, as much in arrear. The calculations by which this curious
result has been established, formidable from their length and intricacy,
are due to the industry, as the discovery of its origin is to the Sagacity,
of Professor Hansen.
(746.) The action of Venus, just explained, is indirect, being as it
were a sort of reflection of its influence on the earth’s orbit. But a very
remarkable instance of its influence, in actually perturbing the moon’s mo-
tions by its direct attraction, has been pointed out, and the inequality due
to it computed by the same eminent geometer." As the details of his
processes have not yet appeared, we can here only explain, in general
terms, the principle on which the result in question depends, and the
nature of the peculiar adjustment of the mean angular velocities of the
earth and Venus which render it effective. The disturbing forces of
Venus on the moon are capable of being represented or expressed (as is
indeed generally the case with all the forces concerned in producing pla-
netary disturbance) by the substitution for them of a series of other forces,
each having a period or cycle within which it attains a maximum in one
direction, decreases to nothing, reverses its action, attains a maximum in
the opposite direction, again decreases to nothing, again reverses its action,
and reattains its former magnitude, and so on. These cycles differ for
each particular constituent or term, as it is called, of the total forces con-
sidered as so broken up into partial ones, and, generally speaking, every
combination which can be formed by subtracting a multiple of the mean
motion of one of the bodies concerned from a multiple of that of the
other, and when there are three bodies disturbing one another, every such
triple combination becomes, under the technical name of an argument,
the cyclical representative of a force acting in the manner and according
- * Astronomische Nachrichten, No. 597.
DIRECT ACTION OF VENUs on THE MOON. 417
to the law described. Each of these periodically acting forces produces
its perturbative effect, according to the law of the superposition of small
motions, as if the others had no existence. And if it happen, as in an
immense majority of cases it does, that the cycle of any particular one of
these partial forces has no relation to the periodic time of the disturbed
body, so as to bring it to the same, or very nearly the same point of its
orbit, or to any situation favourable to any particular form of disturbance,
over and over again when the force is at its maximum; that force will, in
a few revolutions, neutralize its own effect, and nothing but fluctuations
of brief duration can result from its action. The contrary will evidently
be the case, if the cycle of the force coincide so nearly with the cycle of
the moon’s anomalistic revolution, as to bring round the maximum of the
force acting in one and the same direction (whether tangential or normal)
either accurately, or very nearly indeed to some definite point, as, for ex-
ample, the apogee of her orbit. Whatever the effect produced by such a
force on the angular motion of the moon, if it be not exactly compen-
sated in one cycle of its action, it will go on accumulating, being repeated
over and over again under circumstances very nearly the same, for many
successive revolutions, until at length, owing to the want of precise accu-
racy in the adjustment of that cycle to the anomalistic period, the maxi-
mum of the force (in the same phase of its action) is brought to coincide
with a point in the orbit (as the perigee), determinative of an opposite
effect, and thus, at length, a compensation will be worked out; in a time,
however, so much the longer as the difference between the cycle of the
force and the moon’s anomalistic period is less.
(747.) Now, in fact, in the case of Venus disturbing the moon, there
exists a cyclical combination of this kind. Of course the disturbing force
of Venus on the moon varies with her distance from the earth, and this
distance again depends on her configuration with respect to the earth and
the sun, taking into account the ellipticity of both their orbits. Among
the combinations which take their rise from this latter consideration, and
which, as may easily be supposed, are of great complexity, there is a term
(an exceedingly minute one), whose argument or cycle is determined by
subtracting 16 times the mean motion of the earth from 18 times that of
Venus, The difference is so very nearly the mean motion of the moon
in her anomalistic revolution, that whereas the latter revolution is com-
pleted in 27° 13' 18- 32-3s, the cycle of the force is completed in 27
13h 7m 35.6°, differing from the other by no more than 10m 56.7°, or
about one 3625th part of a complete period of the moon from apogee to
apogee. During half of this very long interval (that is to say, during
about 1864 years), the perturbations produced by a force of this character,
27
418 OUTLINES OF ASTRONOMY.
go on increasing and accumulating, and are destroyed in another equal in-
terval. Although therefore excessively minute in their actual effect on
the angular motion, this minuteness is compensated by the number of re-
peated acts of accumulation, and by the length of time during which they
continue to act on the longitude. Accordingly M. Hansen has found the
total amount of fluctuation to and fro, or the value of the equation of the
moon's longitude so arising, to be 27'4". It is exceedingly interesting to
observe that the two equations considered in these latter paragraphs,
account satisfactorily for the only remaining material differences between
theory and observation in the modern history of this hitherto rebellious
satellite. We have not thought it necessary (indeed it would have required
a treatise on the subject) to go into a special account of the almost innu-
merable other lunar inequalities which have been computed and tabulated,
and which are necessary to be taken into account in every computation
of her place from the tables. Many of them are of very much larger
amount than these. We ought not, however, to pass unnoticed, that the
parallactic inequality, already explained (art. 712), is interesting, as afford-
ing a measure of the sun's distance. For this equation originates, as there
shown, in the fact that the disturbing forces are not precisely alike in the
two halves of the moon's orbit nearest to and most remote from the sun,
all their values being greater in the former half. As a knowledge of the
relative dimensions of the solar and lunar orbits enables us to calculate à
priori, the amount of this inequality, so a knowledge of that amount
deduced by the comparison of a great number of observed places of the
moon with tables in which every inequality but this should be included,
would enable us conversely to ascertain the ratio of the distances in ques-
tion. Owing to the smallness of the inequality, this is not a very accu-
rate mode of obtaining an element of so much importance in astronomy
as the sun’s distance, but were it larger (i. e. were the moon's orbit con-
siderably larger than it actually is), this would be, perhaps, the most
exact method of any by which it could be concluded.
(748.) The greatest of all the lunar inequalities, produced by pertur-
bation, is that called the evection. It arises directly from the variation
of the excentricity of her orbit, and from the fluctuation to and fro in the
general progress of the line of apsides, caused by the different situation
of the sun, with respect to that line (arts. 685, 691). Owing to these
causes the moon is alternately in advance, and in arrear of her elliptic
place by about 1° 20' 20". This equation was known to the ancients,
having been discovered by Ptolemy, by the comparison of a long series
of observations, handed down to him from the earliest ages of astronomy.
The mode in which the effects of these several sources of inequality be-
EFFECT OF THE EARTH'S SPHEROIDAL FIGURE. 419
come grouped together under one principal argument, common to them
all, belongs, for its explanation, rather to works specially treating of the
lunar theory than to a treatise of this kind.
(749.) Some small perturbations are produced in the lunar orbit by
the protuberant matter of the earth's equator. The attraction of a sphere
is the same as if all its matter were condensed into a point in its centre;
but that is not the case with a spheroid. The attraction of such a mass
is neither exactly directed to its centre, nor does it exactly follow the law
of the inverse squares of the distances. Hence will arise a series of
perturbations, extremely small in amount, but still perceptible in the
lunar motions, by which the node and the apogee will be affected. A
more remarkable consequence of this cause, however, is a small nutation
of the lunar orbit, exactly analogous to that which the moon causes in
the plane of the earth's equator, by its action on the same elliptic protu-
berance. And, in general, it may be observed, that in the systems of
planets which have satellites, the elliptic figure of the primary has a ten-
dency to bring the orbits of the satellites to coincide with its equator, La
tendency which, though small in the case of the earth, yet in that of Jupiter,
whose ellipticity is very considerable, and of Saturn especially, where the
ellipticity of the body is reinforced by the attraction of the rings, becomes
predominant over every external and internal, cause of disturbance, and
produces and maintains an almost exact coincidence of the planes in
question. Such, at least, is the case with the nearer satellites. The
more distant are comparatively less affected by this cause, the difference
of attractions between a sphere and spheroid diminishing with great ra-
pidity as the distance increases. Thus, while the orbits of all the interior
satellites of Saturn lie almost exactly in the plane of the ring and equator
of the planet, that of the external satellite, whose distance from Saturn
is between sixty and seventy diameters of the planet, is inclined to that
plane considerably. On the other hand, this considerable distance, while
it permits the satellite to retain its actual inclination, prevents (by parity
of reasoning) the ring and equator of the planet from being perceptibly
disturbed by its attraction, or being subjected to any appreciable move-
ments analogous to our nutation and precession. If such exist, they
must be much slower than those of the earth; the mass of this satellite
being, as far as can be judged by its apparent size, a much smaller frac-
tion of that of Saturn than the moon is of the earth; while the solar
precession, by reason of the immense distance of the sun, must be quite
imperceptible.
(750.) The subject of the tides, though rather belonging to terrestrial
physics than properly to astronomy, is yet so directly connected with the
420 OUTLINES OF ASTRONOMY.
theory of the lunar perturbations, that we cannot omit some explanatºry
notice of it, especially since many persons find a strange difficulty in con-
ceiving the manner in which they are produced. That the sun, or moon,
should by its attraction heap up the waters of the ocean under it, seems
to them very natural. That it should at the same time heap them up on
the opposite side seems, on the contrary, palpably absurd. The error of
this class of objectors is of the same kind with that noticed in art. 723,
and consists in disregarding the attraction of the disturbing body on the
mass of the earth, and looking on it as wholly effective on the superficial
water. Were the earth indeed absolutely fixed, held in its place by an
external force, and the water left free to move, no doubt the effect of the
disturbing power would be to produce a single accumulation vertically
under the disturbing body. But it is not by its whole attraction, but by
the difference of its attractions on the superficial water at both sides, and
on the central mass, that the waters are raised : just as in the theory of
the moon, the difference of the sun's attractions on the moon and on the
earth (regarded as moveable and as obeying that amount of attraction
which is due to its situation) gives rise to a relative tendency in the moon
to recede from the earth in conjunction and opposition, and to approach
it in quadratures. Referring to the figure of art. 675, instead of sup-
posing A D B C to represent the moon’s orbit, let it be supposed to repre-
sent a section of the (comparatively) thin film of water reposing on the
globe of the earth, in a great circle, the plane of which passes through
the disturbing body M, which we shall suppose to be the moon. The
disturbing force on a particle at P will then (exactly as in the lunar
theory) be represented in amount and direction by NS, on the same scale
on which S M represents the moon’s whole attraction on a particle situ-
ated at S. This force, applied at P, will urge it in the direction P X
parallel to N S; and therefore, when compounded with the direct force of
Fig. 106.
D
P
S Q.

gravity which (neglecting as of no account in this theory the spheroidal
OF THE TIDES. 421
form of the earth) urges P towards S, will be equivalent to a single force
deviating from the direction PS towards X. Suppose PT to be the di-
rection of this force, which, it is easy to see, will be directed towards a
point in D S produced, at an extremely small distance below S, because
of the excessive minuteness of the disturbing force compared to gravity."
Then if this be done at every point of the quadrant A D, it will be evi-
dent that the direction PT of the resultant force will be always that of a
tangent to the small cuspidated curve a d at T, to which tangent the sur-
face of the ocean at P must everywhere be perpendicular, by reason of
that law of hydrostatics which requires the direction of gravity to be
everywhere perpendicular to the surface of a fluid in equilibrio. The
form of the curve DPA, to which the surface of the ocean will tend to
conform itself, so as to place itself everywhere in equilibrio under two
acting forces, will be that which always has PT for its radius of curva-
ture. It will therefore be slightly less curved at D, and more so at A,
being in fact no other than an ellipse, having S for its centre, d a for its
evolute, and SA, SD for its longer and shorter semi-axes respectively; so
that the whole surface (supposing it covered with water) will tend to as-
sume, as its form of equilibrium, that of an oblongated ellipsoid, having
its longer axis directed towards the disturbing body, and its shorter of
course at right angles to that direction. The difference of the longer and
shorter semi-axes of this ellipsoid due to the moon's attraction would be
about 58 inches: that of the ellipsoid, similarly formed in virtue of the
sun, about 23 times less, or about 23 inches.
(751.) Let us suppose the moon only to act, and to have no orbitual
motion; then if the earth also had no diurnal motion, the ellipsoid of
equilibrium would be quietly formed, and all would be thenceforward
tranquil. There is never time, however, for this spheroid to be fully
formed. Before the waters can take their level, the moon has advanced
in her orbit, both diurnal and monthly, (for in this theory it will answer
the purpose of clearness better, if we suppose the earth's diurnal motion
transferred to the sun and moon in the contrary direction,) the vertex of
the spheroid has shifted on the earth's surface, and the ocean has to seek
a new bearing. The effect is to produce an immensely broad and exces-
sively flat wave (not a circulating current), which follows, or endeavours
* According to Newton’s calculation, the maximum disturbing force of the sun on the
water does not exceed one 25736400th part of its gravity. That of the moon will
therefore be to this fraction as the cube of the sun's distance to that of the moon’s di-
rectly, and as the mass of the sun to that of the moon inversely, i.e. as (400)*X 0.012517
: 354936, which, reduced to numbers, gives, for the moon's maximum of power to dis-
turb the waters, about one 11400000th of gravity, or somewhat less than 2% times the
sun's. *.
422 OUTLINES OF ASTRONOMY.
to follow, the apparent motions of the moon, and must, in fact, by the
principle of forced vibrations, imitate, by equal though not by synchronous
periods, all the periodical inequalities of that motion. When the higher
or lower parts of this wave strike our coasts, they experience what we
call high and low water. .
(752.) The sun also produces precisely such a wave, whose vertex tends
to follow the apparent motion of the sun in the heavens, and also to imi-
tate its periodic inequalities. This solar wave co-exists with the lunar—
is sometimes superposed on it, sometimes transverse to it, so as to partly
neutralize it, according to the monthly synodical configuration of the two
luminaries. This alternate mutual reinforcement and destruction of the
solar and lunar tides cause what are called the spring and neap tides—the
former being their sum, the latter their difference. Although the real
amount of either tide is, at present, hardly within the reach of exact cal-
culation, yet their proportion at any one place is probably not very remote
from that of the ellipticities which would belong to their respective sphe-
roids, could an equilibrium be attained. Now these ellipticities, for the
solar and lunar spheroids, are respectively about two and five feet; so that
the average spring tide will be to the neap as 7 to 3, or thereabouts.
(753.) Another effect of the combination of the solar and lunar tides
is what is called the priming and lagging of the tides. If the moon
alone existed, and moved in the plane of the equator, the tide-day (i. e.
the interval between two successive arrivals at the same place of the same
vertex of the tide-wave) would be the lunar day (art. 143), formed by
the combination of the moon’s sidereal period and that of the earth’s
diurnal motion. Similarly, did the sun alone exist, and move always on
the equator, the tide-day would be the mean solar day. The actual tide-
day, then, or the interval of the occurrence of two successive mazima of
their superposed waves, will vary as the separate waves approach to or re-
cede from coincidence; because, when the vertices of two waves do not
coincide, their joint height has its maximum at a point intermediate be-
tween them. This variation from uniformity in the lengths of successive
tide-days is particularly to be remarked about the time of the new and
full moon. -
(754.) Quite different in its origin is that deviation of the time of high
and low water at any port or harbour, from the culmination of the lumi-
naries, or of the theoretical maximum of their superposed spheroids, which
is called the “establishment” of that port. If the water were without
inertia, and free from obstruction, either owing to the friction of the bed
of the sea, the narrowness of channels along which the wave has to travel
before reaching the port, their length, &c., &c., the times above distin-
OF THE TIDES. 423
guished would be identical. Bnt all these causes tend to create a dif.
ference, and to make that difference not alike at all ports. The observa-
tion of the establishments of harbours is a point of great maritime im-
portance; nor is it of less consequence, theoretically speaking, to a
knowledge of the true distribution of the tide-waves over the globe. In
making such observations, care must be taken not to confound the time
of “slack water,” when the current caused by the tide ceases to flow
visibly one way or the other, and that of high or low water, when the
level of the surface ceases to rise or fall. These are totally distinct phae-
nomena, and depend on entirely different causes, though it is true they
may sometimes coincide in point of time. They are, it is feared, too often
mistaken one for the other by practical men; a circumstance which,
whenever it occurs, must produce the greatest confusion in any attempt to
reduce the systern ºf the tides to distinct and intelligible laws.
(755.) The declination of the sun and moon materially affects the tides
at any particular spot. As the vertex of the tide-wave tends to place
itself vertically under the luminary which produces it, when this vertical
changes its point of incidence on the surface, the tide-wave must tend to
shift accordingly, and thus, by monthly and annual periods, must tend to
increase and diminish alternately the principal tides. The period of the
moon’s nodés is thus introduced into this subject; her excursions in de-
clination in one part of that period being 29°, and in another only 17°,
on either side the equator.
(756.) Geometry demonstrates that the efficacy of a luminary in raising
tides is inversely proportional to the cube of its distance. The sun and
moon, however, by reason of the ellipticity of their orbits, are alternately
nearer to and farther from the earth than their mean distances. In con-
sequence of this, the efficacy of the sun will fluctuate between the ex-
tremes 19 and 21, taking 20 for its mean value, and that of the moon
between 43 and 59. Taking into account this cause of difference, the
highest spring tide will be to the lowest meap as 59 +21 to 43—19, or
as 80 to 24, or 10 to 3. Of all the causes of differences in the height
of tides however, local situation is the most influential. In some places
the tide-wave, rushing up a narrow channel, is suddenly raised to an ex-
traordinary height. At Annapolis, for instance, in the Bay of Fundy, it
is said to rise 120 feet. Even at Bristol the difference of high and low
water occasionally amounts to 50 feet.
(757.) It is by means of the perturbations of the planets, as ascertained
by observation and compared with theory, that we arrive at a knowledge.
of the masses of those planets which having no satellites, offer no other
hold upon them for this purpose. Every planet produces an amount of
424 OUTLINES OF ASTRONOMY.
perturbation in the motions of every other, proportioned to its mass, and
to the degree of advantage or purchase which its situation in the system
gives it over their movements. The latter is a subject of exact calcula-
tion; the former is unknown, otherwise than by observation of its effects.
In the determination, however, of the masses of the planets by this
means, theory lends the greatest assistance to observation, by pointing out
the combinations most favourable for eliciting this knowledge from the
confused mass of superposed inequalities which affect every observed place
of a planet; by pointing out the laws of each inequality in its periodical
rise and decay; and by showing how every particular inequality depends
for its magnitude on the mass producing it. It is thus that the mass of
Jupiter itself (employed by Laplace in his investigations, and interwoven
with all the planetary tables) has been ascertained, by observations of the
derangements produced by it in the motions of the ultra-zodiacal planets,
to have been insufficiently determined, or rather considerably mistaken,
by relying too much on observations of its satellites, made long ago by
Pound and others, with inadequate instrumental means. The same con-
clusion has been arrived at, and nearly the same mass obtained, by means
of the perturbations produced by Jupiter on Encke's comet. The error
was one of great importance; the mass of Jupiter being by far the most
influential element in the planetary system, after that of the sun. It is
satisfactory, then, to have ascertained, as Mr. Airy has done, the cause of
the error; to have traced it up to its source, in insufficient micrometric
measurements of the greatest elongations of the satellites; and to have
found it disappear when measures, taken with more care and with infinitely
superior instruments, are substituted for those before employed.
(758.) In the same way that the perturbations of the planets lead us
to a knowledge of their masses, as compared with that of the sun, so the
perturbations of the satellites of Jupiter have led, and those of Saturn's
attendants will no doubt hereafter lead, to a knowledge of the proportion
their masses bear to their respective primaries. The system of Jupiter's
satellites has been elaborately treated by Laplace; and it is from his
theory, compared with innumerable observations of their eclipses, that the
masses assigned to them, in art. 540 have been fixed. Few results of
theory are more surprising than to see these minute atoms weighed in the
same balance, which we have applied to the ponderous mass of the sun,
which exceeds the least of them in the enormous proportion of 65,000,000
to 1.
(759.) The mass of the moon is concluded, 1st, from the proportion of
the lunar to the solar tide, as observed at various stations, the effects being
separated from each other by a long series of observations of the relative
MASS OF THE MOON DISCOWERED. º 425
Tºrr
heights of spring and neap tides which, we have seen, (art. 752,) depends
on the proportional influence of the two luminaries. 2dly, from the
phaenomenon of mutation, which, being the result of the moon's attraction
alone, affords a means of calculating her mass, independent of any know-
ledge of the sun's. Both methods agree in assigning to our satellite a
mass about one seventy-fifth that of the earth.'
(760.) Not only, however, has a knowledge of the perturbations pro-
duced on other bodies of our system enabled us to estimate the mass of a
disturbing body already known to exist, and to produce disturbance. It
has done much more, and enabled geometers to satisfy themselves of the
existence, and even to indicate the situation of a planet previously un-
known, with such precision, as to lead to its immediate discovery on the
very first occasion of pointing a telescope to the place indicated. We
have already (art. 506,) had occasion to mention in general terms this
great discovery; but its importance, and its connexion with the subject
before us, calls for a more specific notice of the circumstances attending it.
When the regular observation of Uranus, consequent on its discovery in
1781, had afforded some certain knowledge of the elements of its orbit, it
became possible to calculate backwards into time past, with a view to
ascertain whether certain stars of about the same apparent magnitude,
observed by Flamsteed, and since reported as missing, might not possibly
be this planet. No less than six ancient observations of it as a supposed
star were thus found to have been recorded by that astronomer, — one in
1690, one in 1712, and four in 1715. On further inquiry, it was also
ascertained to have been observed by Bradley in 1753, by Mayer in 1756,
and no less than twelve times by Le Monnier, in 1750, 1764, 1768, 1769,
and 1771, all the time without the least suspicion of its planetary nature.
The observations, however, so made, being all circumstantially registered,
and made with instruments the best that their respective dates admitted,
were quite available for correcting the elements of the orbit, which, as
will be easily understood, is done with so much the greater precision the
larger the arc of the ellipse embraced by the extreme observations em-
ployed. It was, therefore, reasonably hoped and expected, that, by
making use of the data thus afforded, and duly allowing for the perturba-
tions produced since 1690, by Saturn, Jupiter, and the inferior planets,
elliptic elements would be obtained, which, taken in conjunction with
those perturbations, would represent not only all the observations up to
the time of executing the calculations, but also all future observations, in
as satisfactory a manner as those of any of the other planets are actually
represented. This expectation, however, proved delusive. M. Bouvard,
* Laplace, Expos. du Syst. du Monde, pp. 285,300.
426 OUTLINES OF ASTRONOMY.
one of the most expert and laborious calculators of whom astronomy has
had to boast, and to whose zeal and indefatigable industry we owe the
tables of Jupiter and Saturn in actual use, having undertaken the task of
constructing similar tables for Uranus, found it impossible to reconcile
the ancient observations above mentioned with those made from 1781 to
1820, so as to represent both series by means of the same ellipse and the
same system of perturbations. He therefore rejected altogether the ancient
series, and grounded his computations solely on the modern, although
evidently not without serious misgivings as to the grounds of such a pro-
ceeding, and “leaving it to future time to determine whether the difficulty
of reconciling the two series arose from inaccuracy in the older observa-
tions, or whether it depend on some extraneous and unperceived influence
which may have acted on the planet.” .
(761.) But neither did the tables so calculated continue to represent,
with due precision, observations subsequently made. The “error of the
tables” after attaining a certain amount, by which the true longitude of
Uranus was in advance of the computed, and which advance was steadily
maintained from about the year 1795 to 1822, began, about the latter
epoch, rapidly to diminish, till, in 1830–31, the tabular and observed
longitudes agreed. But, far from remaining in accordance, the planet,
still losing ground, fell, and continued to fall behind its calculated place,
and that with such rapidity as to make it evident that the existing tables
could no longer be received as representing, with any tolerable precision,
the true laws of its motion.
(762.) The reader will easily understand the nature and progression of
these discordancies by casting his eye on fig. 1, Plate A, in which the
horizontal line or abscissa is divided into equal parts, each representing
50° of heliocentric longitude in the motion of Uranus round the sun, and
in which the distances between the horizontal lines represent each 100"
of error in longitude. The result of each year's observation of Uranus
(or of the mean of all the observations obtained during that year) in lon.
gitude, is represented by a black dot placed above or below the point of
the abscissa, corresponding to the mean of the observed longitudes for the
year above, if the observed longitude be in excess of the calculated, below
if it fall short of it, and on the line if they agree; and at a distance from
the line corresponding to their difference on the scale above mentioned."
* The points are laid down from M. Leverrier's comparison of the whole series of
observations of Uranus, with an ephemeris of his own calculation, founded on a com-
plete and searching revision of the tables of Bouvard, and a rigorous computation of
the perturbations caused by all the known planets capable of exercising any influence
on it. The differences of longitude are geocentric ; but for our present purpose it
matters not in the least whether we consider the errors in heliocentric or in geocentric
longitude,
PERTURBATION OF URANUs, 427
Thus in Flamsteed's earliest observations in 1690, the dot so marked is
placed above the line at 65".9 above the line, the observed longitude being
so much greater than the calculated.
(763.) If, neglecting the individual points, we draw a curve (indicated
in the figure by a fine, unbroken line,) through their general course, we
shall at once perceive a certain regularity in its undulations. It presents
two great elevations above, and one nearly as great intermediate depres-
sion below the medial line or abscissa. And it is evident that these un-
dulations would be very much reduced, and the errors in consequence
greatly palliated, if each dot were removed in the vertical direction
through a distance and in the direction indicated by the corresponding
point of the curve A, B, C, D, E, F, G, H, intersecting the abscissa at
points 180° distant, and making equal excursions on either side. Thus
the point a for 1750 being removed upwards or in the direction to-
wards b through a distance equal to c b, would be brought almost to
precise coincidence with the point e in the abscissa. Now, this is a
clear indication that a very large part of the differences in question are
due, not to perturbation, but simply to error in the elements of Uranus,
which have been assumed as the basis of calculation. For such ex-
cesses and defects of longitude alternating over arcs of 180° are pre-
cisely what would arise from error in the excentricity, or in the place
of the perihelion, or in both. In ellipses slightly excentric, the true lon-
gitude alternately exceeds and falls short of the mean during 180° for
each deviation, and the greater the excentricity, the greater these alternate
fluctuations to and fro. If then the excentricity of a planet's orbit be
assumed erroneously (suppose too great) the observed longitudes will ex-
hibit a less amount of such fluctuation above and below the mean than
the computed, and the difference of thc two, instead of being, as it ought
to be, always nil, will be alternately + and — over arcs of 180°. If then
a difference be observed following such a law, it may arise from erro-
neously assumed excentricity, provided always the longitudes at which
they agree (supposed to differ by 180°) be coincident with those of the
perihelion and aphelion; for in elliptic motion nearly circular, these are
the points where the mean and true longitudes agree, so that any fluctua-
tion of the nature observed, if this condition be not satisfied, cannot arise
from error of excentricity. Now the longitude of the perihelion of
Uranus in the elements employed by Bouvard is (neglecting fractions of
a degree) 168°, and of the aphelion 348°. These points, then, in our
figure, fall at x and a, respectively, that is to say, nearly half way between
A C, CE, EG, &c. It is evident, therefore, that it is not to error or
excentricity that the fluctuation in question is mainly due.
428 OUTLINES OF ASTRONOMY.
•
(764.) Let us now consider the effect of an erroneous assumption of
the place of the perihelion. Suppose in fig. 2, Plate A, o a to represent
the longitude of a planet, and a y the excess of its true above its mean
longitude, due to ellipticity. Then if R be the place of the perihelion,
and P, or T, the aphelion in longitude, y will always lie in a certain un-
dulating curve PQRST, above' PT, between R and T, and below it
between P and R. Now suppose the place of the perihelion shifted for-
ward to r, or the whole curve shifted bodily forward into the situation
p q r s t, then at the same longitude o ac, the excess of the true above the
mean longitude will be a y only; in other words, this excess will have
diminished by the quantity y y' below its former amount. Take therefore
in o N (fig. 3, Pl. A.) oſ = oa and y?/ always = y/ in fig. 2, and
having thus constructed the curve K L MNO, the ordinate y y' will
always represent the effect of the supposed change of perihelion. It is
evident (the excentricity being always supposed small), that this curve
will consist also of alternate superior and inferior waves of 180° each in
amplitude, and the points L, N of its intersection with the axis, will
occur at longitudes corresponding to X, Y, intermediate between the
maxima Q, g; and S, s of the original curves, that is to say (if these in-
tervals Q q, Ss, or R r, to which both are equal, be very small,) very
nearly at 90° from the perihelion and aphelion. Now this agrees with
the conditions of the case in hand, and we are therefore authorised to
conclude that the major portion of the errors in question has arisen from
error in the place of the perihelion of Uranus itself, and not from pertur-
bation, and that to correct this portion, the perihelion must be shifted
somewhat forward. As to the amount of this shifting, our only object
being explanation, it will not be necessary here to inquire into it. It will
suffice that it must be such as shall make the curve A B C D E F G as
nearly as possible similar, equal, and opposite to the curve traced out by
the dots on the other side. And this being done, we may next proceed
to lay down a curve of the residual differences between observation and
theory in the mode indicated in art. (763.)
(765.) This being done, by laying off at each point of the line of
longitudes an ordinate equal to the difference of the ordinates of the two
curves in fig. 1, when on opposite, and their sum when on the same side
of the abscissa, the result will be as indicated by the dots in fig. 4. And
here it is at once seen that a still farther reduction of the differences under
consideration would result, if, instead of taking the line A B for the line
of longitudes, a line, a b, slightly inclined to it, were substituted, in which
case the whole of the differences between observation and theory, from
* The curves, figs. 2. 3, are inverted in the engraving.
PERTURBATIONS OF URANUS BY NEPTUNE. 429
1712 to 1800, would be annihilated, or at least so far reduced as hardly
to exceed the ordinary errors of observation; and as respects the observa-
tion of 1690, the still outstanding difference of about 35” would not be
more than might be attributed to a not very careful observation at so early
an epoch. Now the assumption of such a new line of longitudes as the
correct one, is in effect equivalent to the admission of a slight amount of
error in the periodic time and epoch of Uranus; for it is evident that by
reckoning from the inclined instead of the horizontal line, we in effect
alter all the apparent outstanding errors by an amount proportional to the
time before or after the date at which the two lines intersect (viz. about
1789). As to the direction in which this correction should be made, it is
obvious by inspection of the course of the dots, that if we reckon from
A B, or any line parallel to it, the observed planet on the long run keeps
falling more and more behind the calculated one; i. e. its assigned mean
angular velocity by the tables is too great, and must be diminished, or its
periodic time requires to be increased.
(766.) Let this increase of period be made, and in correspondence with
that change let the longitudes be reckoned on a b, and the residual diffe-
rences from that line instead of A B, and we shall have then done all that
can be done in the way of reducing and palliating these differences, and
that, with such success, that up to the year 1804 it might have been safely
asserted that positively no ground whatever existed for suspecting any
disturbing influence. But with this epoch an action appears to have
commenced, and gone on increasing, producing an acceleration of the
motion in longitude, in consequence of which Uranus continually gains on
its elliptic place, and continued to do so till 1822, when it ceased to gain,
and the excess of longitude was at its maximum, after which it began
rapidly to lose ground, and has continued to do so up to the present time.
It is perfectly clear, then, that in this interval some extraneous cause must
have come into action which was not so before, or not in sufficient power
to manifest itself by any marked effect, and that that cause must have
ceased to act, or rather begun to reverse its action, in or about the year
1822, the reverse action being even more energetic than the direct.
(767.) Such is the phaenomenon in the simplest form we are now able
to present it. Of the various hypotheses formed to account for it, during
the progress of its developement, none seemed to have any degree of ra-
tional probability but that of the existence of an exterior, and hitherto
undiscovered, planet, disturbing, according to the received laws of plan-
etary disturbance, the motion of Uranus by its attraction, or rather super-
posing its disturbance on those produced by Jupiter and Saturn, the only
two of the old planets which exercise any sensible disturbing action on
430 OljTLINES OF ASTRONOMY.
that planet. Accordingly, this was the explanation which naturally, and
almost of necessity, suggested itself to those conversant with the plane-
tary perturbations who considered the subject with any degree of atten-
tion. The idea, however, of setting out from the observed anomalous
deviations, and employing them as data to ascertain the distance and situ-
ation of the unknown body, or, in other words, to resolve the inverse
problem of perturbations, “given the disturbances to find the orbit, and
place in that orbit of the disturbing planet,” appears to have occurred
only to two mathematicians, Mr. Adams in England and M. Leverrier in
France, with sufficient distinctness and hopefulness of success to induce them
to attempt its solution. Both succeeded, and their solutions, arrived at with
perfect independence, and by each in entire ignorance of the other's at-
tempt, were found to agree in a surprising manner when the nature and
difficulty of the problem is considered; the calculations of M. Leverrier
assigning for the heliocentric longitude of the disturbing planet for the
23d Sept. 1846, 326° 0', and those of Mr. Adams (brought to the same
date) 329°19', differing only 3°19'; the plane of its orbit deviating very
slightly, if it all, from that of the ecliptic.
(768.) On the day above mentioned—a day for ever memorable in the
annals of astronomy— Dr. Galle, one of the astronomers of the Royal
Observatory at Berlin, received a letter from M. Leverrier, announcing to
him the result he had arrived at, and requesting him to look for the dis-
turbing planet in or near the place assigned by his calculation. He did
so, and on that very night actually found it. A star of the eighth mag-
nitude was seen by him and by M. Encke in a situation where no star
was marked as existing in Dr. Bremiker's chart, then recently published
by the Berlin Academy. The next night it was found to have moved
from its place, and was therefore assuredly a planet. Subsequent obser-
vations and calculations have fully demonstrated this planet, to which the
name of Neptune has been assigned, to be really that body to whose dis-
turbing attraction, according to the Newtonian law of gravity, the observed
anomalies in the motion of Uranus were owing. The geocentric longi-
tude determined by Dr. Galle from this observation was 325°53', which,
converted into heliocentric, gives 326° 52', differing 0° 52' from M. Le-
verrier's place, 2° 27' from that of Mr. Adams, and only 47' from a
mean of the two calculations.
(769.) It would be quite beyond the scope of this work, and far in
advance of the amount of mathematical knowledge we have assumed our
readers to possess, to attempt giving more than a superficial idea of the
course followed by these geometers in their arduous investigations. Suf.
fice it to say, that it consisted in regarding, as unknown quantities, to be
PERTURBATI Ns of URANUS BY NEPTUNE. 431
determined, the mass, and all the elements of the unknown planet (sup-
posed to revolve in the same plane and the same direction with Uranus),
except its major semiaxis. This was assumed in the first instance (in
conformity with “Bode's law,” art. (505), and certainly at the time with
a high primâ facie probability,) to be double that of Uranus, or 38.364
radii of the Earth's orbit. Without some assumption as to the value of
this element, owing to the peculiar form of the analytical expression of
the perturbations, the analytical investigation would have presented diffi-
culties apparently insuperable. But besides these, it was also necessary
to regard as unknown, or at least as liable to corrections of unknown
magnitude of the same order as the perturbations, all the elements of
Uranus itself, a circumstance whose necessity will be easily understood,
when we consider that the received elements could only be regarded as
provisional, and must certainly be erroneous, the places from which they
were obtained being affected by at least some pºrtions of the very pertur-
bations in question. This consideration, though indispensable, added
vastly both to the complication and the labour of the inquiry. The axis
(and therefore the mean motion) of the one orbit, then, being known very
nearly, and that of the other thus hypothetically assumed, it became prac.
ticable to express in terms, partly algebraic, partly numerical, the amount
of perturbation at any instant, by the aid of general expressions delivered
by Laplace in his “Mécanique Céleste” and elsewhere. These, then,
together with the corrections due to the altered elements of Uranus itself,
being applied to the tabular longitudes, furnished, when compared with
those observed, a series of equations, in which the elements and mass of
Neptune, and the corrections of those of Uranus entered as the unknown
quantities, and by whose resolution (no slight effort of analytical skill) all
their values were at length obtained. The calculations were then repeated,
reducing at the same time the value of the assumed distance of the new
planet, the discordances between the given and calculated results indicating
it to have been assumed too large when the results were found to agree
better, and the solutions to be, in fact, more satisfactory. Thus, at length,
elements were arrived at for the orbit of the unknown planet, as below.

Leverrier. Adams.
Epoch of Elements......................... Jan. 1, 1847. Oct. 6, 1846.
Mean longitude in Epoch................. 3180 47' 4// 323 o 27
Semiaxis Major.............................. 36-1539 37-2474
Excentricity .......................... * * * * * * * 0-107610 0-120615
Longitude of Perihelion................... 2840 45' 8" 2990 11/
Mass (the Sun being 1).................... 0-0001 07.27 000015603 |
—"
432 OljTLINES OF ASTRONOMY.
The elements of M. Leverrier were obtained from a consideration of the
observations up to the year 1845, those of Mr. Adams only as far as
1840. On subsequently taking into account, however, those of the five
years up to 1845, the latter was led to conclude that the semiaxis ought
to be still much further diminished, and that a mean distance of 33-33
(being to that of Uranus as 1: 0-574) would probably satisfy all the obser-
vations very nearly." • *
(770.) On the actual discovery of the planet, it was, of course, assidu-
ously observed, and it was soon ascertained that a mean distance, even less
than Mr. Adams's last presumed value, agreed better with its motion ;
and no sooner were elements obtained from direct observation, sufficiently
approximate to trace back its path in the heavens for a considerable inter-
val of time, than it was ascertained to have been observed as a star by
Lalande on the 8th and 10th of May, 1795, the latter of the two obser-
vations, however, having been rejected by him as faulty, by reason of its
non-agreement with the former (a consequence of the motion of the
planet in the interval.) From these observations, combined with those
since accumulated, the elements calculated by Prof. Walker, U. S., result
as follows:—
Epoch of Elements * *
Mean longitude at Epoch
Semiaxis major - º
Excentricity
Longitude of the Perihelion
Ascending Node ºf
Jan. 1, 1847, M. noon, Greenwich.
32So 32' 44// 2
30'0367
O 00S71946
470 12' 6'-50
130° 4' 20".81
Inclination & sº 1° 46'58".97
Periodic time tºs * > 164-6181 tropical year.
Mean annual Motion tº 29-18688
(771.) The great disagreement between these elements and those assigned
either by M. Leverrier or Mr. Adams will not fail to be remarked; and it
will naturally be asked how it has come to pass, that elements so widely
different from the truth should afford anything like a satisfactory represen-
tation of the perturbation in question, and that the true situation of the
planet in the heavens should have been so well, and indeed accurately,
pointed out by them. As to the latter point, any one may satisfy himself
by half an hour's calculation that both sets of elements do really place the
planet, on the day of its discovery, not only in the longitudes assigned in
art. 763, i. e. extremely near its apparent place, but also at a distance from
the sun very much more approximately correct than the mean distances or
semiaxes of the respective orbits. Thus the radius vector of Neptune,
* In a letter to the Astronomer Royal, dated Sept. 2, 1846,-i, e. three weeks previous
to the optical discovery of the planet. *
PERTURBATIONS OF URANUS BY NEPTUNE. 433
calculated from M. Leverrier's elements for the day in question, instead
of 36:1539 (the mean distance) comes out almost exactly 33; and indeed,
if we consider that the excentricity assigned by those elements gives for
the perihelion distance 32-2634, the longitude assigned to the perihelion
brings the whole arc of the orbit (more than 83°), described in the in-
terval from 1806 to 1847 to, lie within 42° one way or the other of the
perihelion, and therefore, during the whole of that interval, the hypothe.
tical planet would be moving within limits of distance from the sun, 32.6
and 33-0. The following comparative tables of the relative situations of
Uranus, the real and hypothetical planet, will exhibit more clearly than
any lengthened statement, the near imitation of the motion of the former
by the latter within that interval. The longitudes are heliocentric."
}

Uranus. Neptune. Leverrier. Adams.
A. D.
Long. Long. Rad. Wec. Long. Rad. Wec. Long. Rad. Wec.
1805-0 1979.8 235 o'9 30-3 241 o'2 33-1 2469-5 34-2
1810-0 220.9 24.7-0 30-3 25I-1 32-8 255-9 33-7
181 5-0 2.ſ 3-2 258.0 30-3 261-2 32°5 265-5 33-3
1820-0 264-7 268.8 30-2 271-4 32°4 275'4 33-1
1821 - 0 269-0 271-0 30-2 273-5 32-3 277.4 33-0
1822.0 273.3 273-2 30-2 275-6 32-3 279.5 33-0
IS23-0 277-6 275-3 30-2 277-6 32-3 281-5 32.9
1824' 0 281-8 277.4 30-2 279.7 32.3 283.6 32.9
1825'0 2S5.8 279-6 30.2 281-8 32-3 285-6 32-8
1830'0 306.1 290.5 30-1 292.1 32.3 296-0 32.8
I 835'0 326-0 301 -4 30:1 302-5 32°4. 306.3 32-8
1840-0 345-7 312.2 30-1 31 2-6 32-6 . 316.3 32-9
1845'0 365-3 . 323-1 30.0 322.6 32.9 326-0 33-1
1847:0 373-3 3.27.6 30-0 326-5 33-1 329-3 33-2
(772.) From this comparison it will be seen that Uranus arrived at its
conjunction with Neptune at or immediately before the commencement of
1822, with the calculated planet of Leverrier at the beginning of the fol-
lowing year 1823, and with that of Adams about the end of 1824. Both
the theoretical planets, and especially that of M. Leverrier, therefore,
during the whole of the above interval of time, so far as the directions
of their attractive forces on Uranus are concerned, would act nearly on it
as the true planet must have done. As regards the intensity of the rela-
tive disturbing forces, if we estimate these by the principles of art. (612)
at the epochs of conjunction, and for the commencement of 1805 and
* The calculations are carried only to tenths of degrees, as quite sufficient for the
object in view.
434 OUTLINES OF ASTRONOMY.
1845, we find for the respective denominators of the fractions of the sun’s
attraction on Uranus regarded as unity, which express the total disturbing
force, NS, in each case, as below:
1805. Conjunction: 1845.
& & y —-1-- 7508 32390
Neptune with { Pierce º In a SS Tº 83 0 27540 f
Struve's mass Triºs 20244 5519 23810
Leverrier's theoretical Planet, mass I 20837 51.93 19935
T; 33.3
The masses here assigned to Neptune are those respectively deduced by
Prof. Peirce and M. Struve from observations of the satellite discovered
by Mr. Lassell made with the large telescopes of Fraunhofer in the obser-
vatories of Cambridge, U. S. and Pulkova respectively. These it will be
perceived differ very considerably, as might reasonably be expected in the
results of micrometrical measurements of such difficulty, and it is not pos-
sible at present to say to which the preference ought to be given. As
compared with the mass assigned by M. Struve, an agreement on the
whole more satisfactory could not have been looked for within the interval
in tºdiately in question.
(773.) Subject then to this uncertainty as to the real mass of Neptune,
the theoretical planet of Leverrier must be considered as representing with
quite as much fidelity as could possibly be expected in a research of such
exceeding delicacy, the particulars of its motion and perturbative action
during the forty years elapsed from 1805 to 1845, an interval which (as
is obvious from the rapid diminution of the forces on either side of the
conjunction indicated by the numbers here set down) comprises all the
most influential range of its action. This will, however, be placed in full
evidence by the construction of curves representing the normal and tan-
gential forces on the principles laid down (as far as the normal constituent
is concerned) in art 717, one slight change only being made, which, for
the purpose in view, conduces greatly to clearness of conception. The
force L S (in the figure of that article) being supposed applied at P in the
direction L s, we here construct the curve of the normal force by erecting
at P (fig. 5, Plate A) P W always perpendicular to the disturbed orbit,
A P; at P, measured from P in the same direction that S lies from L, and
equal in length to L S. P W will then always represent both the direc-
tion and magnitude of the normal force acting at P. And in like man-
ner, if we take always PZ on the tangent to the disturbed orbit at P,
equal to N L of the former figure, and measured in the same direction
from P that L is from N, PZ will represent both in magnitude and direc-
tion the tangential force acting at P. Thus will be traced out the two
curious ovals represented in our figure of their proper forms and propor-
tions for the case in question. That expressing the normal force is formed
PERTURBATIONS OF URANUS BY NEPTUNE. 435
of four lobes, having a common point in S, viz., S W m XS a Sn Sö SW,
and that expressing the tangential, A Z cf B e d'Y A Z, consisting of four
mutually intersecting loops, surrounding and touching the disturbed orbit
in four points, A B c d. The normal force acts outward over all that part
of the orbit, both in conjunction and opposition, corresponding to the por-
tions of the lobes m, n, exterior to the disturbed orbit, and inwards in
every other part. The figure sets in a clear light the great disproportion
between the energy of this force near the conjunction, and in any other
configuration of the planets; its exceedingly rapid degradation as P ap-
proaches the point of neutrality (whose situation is 35° 5' on either side
of the conjunction, an arc described synodically by Uranus in 165-72);
and the comparatively short duration and consequent inefficacy to produce
any great amount of perturbation, of the more intense part of its inward
action in the small portions of the orbit corresponding to the lobes a, b,
in which the line representing the inward force exceeds the radius of the
circle. It exhibits, too, with no less distinctness, the gradual develope-
ment, and rapid degradation and extinction of the tangential force from
its neutral points, c, d, on either side up to the conjunction, where its
action is reversed, being accelerative over the arc d'A, and retardative
over A c, each of which arcs has an amplitude of 71° 20', and is de-
scribed by Uranus synodically in 347.00. The insignificance of the tan-
gential force in the configurations remote from conjunction throughout the
arc c B d is also obviously expressed by the small comparative develope-
ment of the loops e, f.
(774.) Let us now consider how the action of these forces results in
the production of that peculiar character of perturbation which is exhi-
bited in our curve, fig. 4, Plate A. It is at once evident that the increase
of the longitude from 1800 to 1822, the cessation of that increase in
1822, and its conversion into a decrease during the subsequent interval is
in complete accordance with the growth, rapid decay, extinction at conjunc-
tion, and subsequent reproduction in a reversed sense of the tangential
force: so that we cannot hesitate in attributing the greater part of the
perturbation expressed by the swell and subsidence of the curve between
the years 1800 and 1845,-all that part, indeed, which is symmetrical on
either side of 1822 — to the action of the tangential force.
(775.) But it will be asked, – has then the normal force (which, on
the plain showing of fig. 5, is nearly twice as powerful as the tangential,
and which does not reverse its action, like the latter force, at the point of
junction, but, on the contrary, is there most energetic,) no influence in
producing the observed effects? We answer, very little, within the period
to which observation had extended up to 1845. The effect of the .an.
-)
436 oUTLINES OF ASTRONOMY.
gential force on the longitude is direct and immediate (art. 660), that of
the normal indirect, consequential, and cumulative with the progress of
time (art. 734.) The effect of the tangential force on the mean motion
takes place through the medium of the change it produces on the axis,
and is transient: the reversed action after conjunction (supposing the
orbits circular), exactly destroying all the previous effect, and leaving the
mean motion on the whole unaffected. In the passage through the con-
junction, then, the tangential forge produces a sudden and powerful accel-
eration, succeeded by an equally powerful and equally sudden retardation,
which done, its action is completed, and no trace remains in the subse-
quent motion of the planet that it ever existed, for its action on the peri-
helion and excentricity is in like manner also nullified by its reversal of
direction. But with the normal force the case is far otherwise. Its im-
mediate effect on the angular motion is nil. It is not till it has acted
long enough to produce a perceptible change in the distance of the dis-
turbed planet from the sun that the angular velocity begins to be sensibly
affected, and it is not till its whole outward action has been exerted (i. e.
over the whole interval from neutral point to neutral point) that its maxi-
mum effect in lifting the disturbed planet away from the sun has been
produced, and the full amount of diminution in angular velocity it is
capable of causing has been developed. This continues to act in pro-
ducing a retardation in longitude long after the normal force itself has
reversed its action, and from a powerful outward force has become a feeble
inward motion. A certain portion of this perturbation is incident on the
epoch in the mode described in art. (731.) et seq., and permanently dis.
turbs the mean motion from what it would have been, had Neptune no
existence. The rest of its effect is compensated in a single synodic
revolution, not by the reversal of the action of the force (for that reversed
action is far too feeble for this purpose), but by the effect of the perma-
nent alteration produced in the excentricity, which (the axis being un-
changed) compensates by increased proximity in one part of the revolu-
tion, for increased distance in the other. Sufficient time has not yet
elapsed since the conjunction to bring out into full evidence the influence
of this force. Still its commencement is quite unequivocally marked in
the more rapid descent of our curve fig. 4, subsequent to the conjunction
than ascent previous to that epoch, which indicates the commencement of
a series of undulations in its future course of an elliptic character, con-
sequent on the altered excentricity and perihelion (the total and ultimate
effect of this constituent of the disturbing force) which will be maintained
till within about 20 years from the next conjunction, with the exception,
perhaps, of some trifling inequalities about the time of the opposition;
PERTURBATIONS OF URANUS BY NEPTUNE. 437
similar in character, but far inferior in magnitude to those now under
discussion.
(776.) Posterity will hardly credit that, with a full knowledge of all
the circumstances attending this great discovery — of the calculations of
Leverrier and Adams – of the communication of its predicted place to
Dr. Galle — and of the new planet being actually found by him in that
place, in the remarkable manner above commemorated; not only have
doubts been expressed as to the validity of the calculations of those
geometers, and the legitimacy of their conclusions, but these doubts have
been carried so far as to lead the objectors to attribute the acknowledged
fact of a planet previously unknown occupying that precise place in the
heavens at that precise time, to sheer accidentſ' What share accident
may have had in the successful issue of the calculations, we presume the
reader, after what has been said, will have little difficulty in satisfying
himself. As regards the time when the discovery was made, much has
also been attributed to fortunate coincidence. The following considera-
tions will, we apprehend, completely dissipate this idea, if still lingering
in the mind of any one at all conversant with the subject. The period
of Uranus being 84-0140 years, and that of Neptune 1646181, their
synodic revolution (art. 418.), or the interval between two successive con-
junctions, is 171.58 years. The late conjunction having taken place
about the beginning of 1822; that next preceding must have happened
in 1649, or more than 40 years before the first recorded observation of
* These doubts seem to have originated partly in the great disagreement between the
predicted and real elements of Neptune, partly in the near (possibly precise) commen-
surability of the mean motions of Neptune and Uranus. We conceive them however
to be founded in a total misconception of the nature of the problem, which was not,
from such obviously uncertain indications as the observed discordances could give, to
determine as astronomical quantities the axis, excentricity and mass of the disturbing
planet; but practically to discover where to look for it: when, if once found, these
elements would be far better ascertained. To do this, any avis, excentricity, perihe-
lion, and mass, however wide of the truth, which would represent, even roughly the
amount, but with tolerable correctness the direction of the disturbing force during the
very moderate interval when the departures from theory were really considerable,
would equally serve their purposes; and with an excentricity, mass, and perihelion dis
posable, it is obvious that any assumption of the axis between the limits 30 and 38
may, even with a much wider inferior limit, would serve the purpose. In his attempt
to assign an inferior limit to the axis, and in the value so assigned, M. Leverrier, it
must be admitted, was not successful. Mr. Adams, on the other hand, influenced ly
no considerations of the kind which appear to have weighed with his brother geometer,
fixed ultimately (as we have seen) on an axis not very egregiously wrong. Still it were
to be wished, for the satisfaction of all parties, that some one would undertake the
problem de novo, employing formulae not liable to the passage through infinity, which,
technically speaking, hampers, or may be supposed to hamper the continuous applica
tion of the usual perturbational formulae when cases of commensurability occur.
438 OUTLINES OF ASTRONOMY.
Uranus in 1690, to say nothing of its discovery as a planet. In 1690,
then, it must have been effectually out of reach of any perturbative influ-
ence worth considering, and so it remained during the whole interval from
thence to 1800. From that time the effect of perturbation began to be-
come sensible, about 1805 prominent, and in 1820 had nearly reached its
maximum. At this epoch an alarm was sounded. The maximum was
not attained,—the event, so important to astronomy, was still in progress
of developement, — when the fact (any thing rather than a striking one)
was noticed, and made matter of complaint. But the time for discussing
its cause with any prospect of success was not yet come. Every thing
turns upon the precise determination of the epoch of the maximum,
when the perturbing and perturbed planet were in conjunction, and upon
the law of increase and diminution of the perturbation itself on either
side of that point. Now it is always difficult to assign the time of the
occurrence of a maximum by observations liable to errors bearing a ratio
far from inconsiderable to the whole quantity observed. Until the lapse
of some years from 1822 it would have been impossible to have fixed that
epoch with any certainty, and as respects the law of degradation and total
arc of longitude oyer which the sensible perturbations extend, we are
hardly yet arrived at a period when this can be said to be completely de-
terminable from observation alone. In all this we see nothing of acci-
dent, unless it be accidental that an event which must have happened
between 1781 and 1953, actually happened in 1822; and that we live in
an age when astronomy has reached that perfection, and its cultivators
exercise that vigilance which neither permit such an event, nor its scientific
importance, to pass unnoticed. The blossom had been watched with in-
terest in its developement, and the fruit was gathered in the very moment
of maturity."
*
* The student who may wish to see the perturbations of Uranus produced by Nep-
tune, as computed from a knowledge of the elements and mass of that planet, such
as we now know to be pretty near the truth, will find them stated at length from the
calculations of Mr. Walker, (of Washington, U. S.) in the “Proceedings of the Amer-
icar, Academy of Arts and Sciences,” vol. i. p. 334, et seq. On examining the com-
parisons of the results of Mr. Walker's formulae with those of Mr. Adam's theory in
p. 342, he will perhaps be surprised at the enormous difference between the actions of
Neptune and Mr. A £am’s “hypothetical planet” on the longitude of Uranus. This
is easily explained. Mr. Adam's perturbations are deviations from Bouvard’s orbit of
Uranus, as it stood immediately previous to the late conjunction. Mr. Walker's are
the deviations from a mean or undisturbed orbit freed from the influence of the long
inequality resulting from the near commensurability of the motions.
©F SIDEREAL ASTRONOMY. 439
P A R T III.
O F SID E R E A L A S T R O N O M Y,
CHAPTER XV.
OF THE FIXED STARS. — THEIR CLASSIFICATION BY MAGNITUDES.-
PHOTOMETP (‘ & CALE OF MAGNITUDES.–CONVENTIONAL OR VUILGAR
SCALE.-H. H. OTO METRIC COMPARISON OF STAR.S.– DISTRIBUTION OF
STARS OVER THE HEAVENS.—OF THE MILKY WAY OR GALAXY.-
ITS SUPPOSED FORM THAT OF A FLAT STRATUM PARTIALLY SUB-
DIVIDED.—ITS VISIBLE COURSE AMONG THE CONSTELLATIONS.–ITS
INTERNAL STRUCTURE. — ITS APPARENTLY INDEFINITE EXTENT IN
CERTAIN IDIRECTIONS.—OF THE DISTANCE OF THE FIXED STARS.--
THEIR ANNUAL PARALLAX. — PARALLACTIG UNIT OF SIDEREAL
DISTANCE.-EFFECT OF PARALLAX ANALOGOUS TO THAT OF ABER-
RATION.—HOW DISTINGUISHED FROM IT.-DETECTION OF PARAL-
LAx BY MERIDIONAL OBSERVATIONS.—HENDERSON'S APPLICATION
TO o CENTAURI.-BY DIFFERENTIAL OBSERVATIONS.—DISCOVERIES
OF BESSEL AND STRUWE. – LIST OF STARS IN WEHIJH PARAI, LAX
HAS BEEN DETECTED.—OF THE REAL MAGNITUDES OF THE STARS.--
COMPARISON OF THEIR LIGHTS WITH THAT OF THE SUN.
(777.) BESIDEs the bodies we have described in the foregoing chap-
ters, the heavens present us with an innumerable multitude of other
objects, which are called generally by the name of stars. Though com-
prehending individuals differing from each other, not merely in brightness,
but in many other essential points, they all agree in one attribute, a
high degree of permanence as to apparent relative situation. This has
procured them the title of “fixed stars;” an expression which is to be
understood in a comparative and not an absolute sense, it being certain
that many, and probable that all, are in a state of motion, although too
slow to be perceptible unless by means of very delicate observations, con-
tinued during a long series of years.
(778.) Astronomers are in the habit of distinguishing the stars into
440 OUTLINES OF ASTRONOMY.
classes, according to their apparent brightness. These are termed magni-
tudes. The brightest stars are said to be of the first magnitude; those
which fall so far short of the first degree of brightness as to make a
strongly marked distinction are classed in the second; and so on down to
the sixth or seventh, which comprise the smallest stars visible to the
naked eye, in the clearest and darkest night. Beyond these, however,
telescopes continue the range of visibility, and magnitndes from the 8th
down to the 16th are familiar to those who are in the practice of using
powerful instruments; nor does there seem the least reason to assign a
limit to this progression; every increase in the dimensions and power of
instruments, which successive improvements in optical science have
attained, having brought into view multitudes innumerable of objects
invisible before; so that, for any thing experience has hitherto taught us,
the number of the stars may be really infinite, in the only sense in which
we can assign a meaning to the word.
(779.) This classification into magnitudes, however, it must be ob-
served, is entirely arbitrary. Of a multitude of bright objects, differing
probably, intrinsically, both in size and in splendour, and arranged at
unequal distances from us, one must of necessity appear the brightest, one
next below it, and so on. An order of succession (relative, of course, to
our local situation among them) must exist, and it is a matter of absolute
indifference, where, in that infinite progression downwards, from the one
brightest to the invisible, we choose to draw our lines of demarcation. All
this is a matter of pure convention. Usage, however, has established
such a convention; and though it is impossible to determine exactly, or
à priori, where one magnitude ends and the next begins, and although
different observers have differed in their magnitudes, yet, on the whole,
astronomers have restricted their first magnitude to about 23 or 24 prin-
cipal stars; their second to 50 or 60 next inferior; their third to about
200 yet smaller, and so on; the numbers increasing very rapidly as we
descend in the scale of brightness, the whole number of stars already regis-
tered down to the seventh magnitude, inclusive, amounting to from 12000
tu 15000.
(780.) As we do not see the actual disc of a star, but judge only of its
brightness by the total impression made upon the eye, the apparent “mag.
nitude” of any star will, it is evident, depend, 1st, on the star's distance
from us; 2d, on the absolute magnitude of its illuminated surface; 3d,
on the intrinsic brightness of that surface. Now, as we know nothing or
next to nothing, of any of these data, and have every reason for believing
that each of them may differ in different individuals, in the proportion of
many millions to one, it is clear that we are not to expect much satisfac-
MAGNITUDES OF THE STARS. 441
tion in any conclusions we may draw from numerical statements of the
number of individuals, which have been arranged in our artificial classes
antecedent to any general or definite principle of arrangement. In fact,
astronomers have not yet agreed upon any principle by which the magni-
tudes may be photometrically classed a priori, whether for example a
scale of brightnesses decreasing in geometrical progression should be
adopted, each term being one-half of the preceding, or one-third, or any
other ratio, or whether it would not be preferable to adopt a scale decreas-
ing as the squares of the terms of an harmonic progression, i.e. according
to the series 1, #, #, #3, #, &c. The former would be a purely photo-
metric scale, and would have the apparent advantage, that the light of a
star of any magnitude would bear a fixed proportion to that of the mag-
nitude next above it, an advantage, however, merely apparent, as it is cer-
tain, from many optical facts, that the unaided eye forms very different
judgments of the proportions existing between bright lights, and those
between feeble ones. The latter scale involves a physical idea, that of
supposing the scale of magnitudes to correspond to the appearance of a
first magnitude standard star, removed successively to twice, three times, &c.,
its original or standard distance. Such a scale, which would make the
nominal magnitude a sort of index to the presumable or average distance,
on the supposition of an equality among the real lights of the stars,
would facilitate the expression of speculative ideas on the constitution of
the sidereal heavens. On the other hand, it would at first sight appear
to make too small a difference between the lights in the lower magnitudes.
For example, on this principle of nomenclature, the light of a star of the
seventh magnitude would be thirty-six 49ths of that of one of the sixth,
and of the tenth 81 hundredths of the ninth, while between the first and
the second the proportion would be that of four to one. So far, however,
from this being really objectionable, it falls in well with the general tenor
of the optical facts already alluded to, inasmuch as the eye (in the ab-
sence of disturbing causes) does actually discriminate with greater preci-
sion between the relative intensities of feeble lights than of bright ones,
so that the fraction #ff, for instance, expresses quite as great a step down-
wards (physiologically speaking) from the sixth magnitude, as 3 does from
the first. As the choice, therefore, so far as we can see, lies between these
two scales, in drawing the lines of demarcation between what may be
termed the photometrical magnitudes of the stars, we have no hesitation
in adopting, and recommending others to adopt, the latter system in pre-
ference to the former. -
(781.) The conventional magnitudes actually in use among astrono-
mers, so far as their usage is consistent with itself, conforms moreover
442 OUTLINES OF ASTRON 9 MY.
*-
very much more nearly to this than to the geometrical progression. It has
been shown," by direct photometric measurement of the light of a consi-
derable number of stars, from the first to the fourth magnitude, that if M
be the number expressing the magnitude of a star on the above system,
and m the number expressing the magnitude of the same star in the loose
and irregular language at present conventionally or rather provisionally
adopted, so far as it can be collected from the conflicting authorities of
different observers, the difference between these numbers, or M-m, is
the same in all the higher parts of the scale, and is less than half a mag-
nitude (0m. 414). The standard star assumed as the unit of magnitude
in the measurements referred to, is the bright southern star a Centauri, a
star somewhat superior to Arcturus in lustre. If we take the distance of
this star for unity, it follows that when removed to the distances 1:414,
2:414, 3:414, &c., its apparent lustre would equal those of average stars
of the 1st, 2d, 3d, &c. magnitudes, as ordinarily reckoned, respectively.
(782.) The difference of lustre between stars of two consecutive mag-
nitudes is so considerable as to allow of many intermediate gradations
being perfectly well distinguished. Hardly any two stars of the first, or
of the second magnitude, would be judged by an eye practised in such
comparisons to be exactly equal in brightness. Hence, the necessity, if
anything like accuracy be aimed at, of subdividing the magnitudes and
admitting fractions into our nomenclature of brightness. When this ne-
cessity first began to be felt, a simple bisection of the interval was recog-
nized, and the intermediate degree of brightness was thus designated, viz.
1.2 m, 2.3 m, and so on. At present it is not unfrequent to find the in-
terval trisected thus: 1 m, 1.2 m, 2.1 m, 2 m, &c. where the expression
1.2 m denotes a magnitude intermediate between the first and second, but
nearer 1 than 2; while 2.1 m designates a magnitude also intermediate,
but nearer 2 than 1. This may suffice for common parlance, but as this
department of astronomy progresses towards exactness, a decimal subdi-
vision will of necessity supersede these rude forms of expression, and the
magnitude will be expressed by an integer number, followed by a decimal
fraction; as, for instance, 2.51, which expresses the magnitude of y Gemi-
norum on the vulgar or conventional scale of magnitudes, by which we at
once perceive that its place is almost exactly half way between the 2d and
3d average magnitudes, and that its light is to that of an average first
magnitude star in that scale (of which a Orionis in its usual or normal
state” may be taken as a typical specimen) as 1”: (2:51)*, and to that of
* See “Results of observations made at the Cape of Good Hope, &c. &c.” p. 371.
By the Author. -
* In the interval from 1836 to 1839 this star underwent considerable and remarkable
fluctuations of brightness.
PHOTOMETRIC SCALE OF MAGNITUDES. 443
a Centauri as 1*: (2924)", making its place in the photometric scale (so
defined) 2.924. Lists of stars, northern and southern, comprehending
those of the vulgar first, second and third magnitudes, with their magni-
tudes decimally expressed in both systems, will be found at the end of
this work. The light of a star of the sixth magnitude may be roughly
stated as about the hundredth part of one of the first. Sirius would make
between three and four hundred stars of that magnitude.
(783.) The exact photometrical determination of the comparative in-
tensities of light of the stars is attended with many and great difficulties,
arising partly from their differences of colour; partly from the circum-
stance that no invariable standard of artificial light has yet been disco-
vered; partly from the physiological cause above alluded to, by which the
eye is incapacitated from judging correctly of the proportion of two lights,
and can only decide (and that with not very great precision) as to their
equality or inequality; and partly from other physiological causes. The
least objectionable mode hitherto proposed would appear to be the follow-
ing. A natural standard of comparison is in the first instance selected,
brighter than any of the stars, so as to allow of being equalized with any
of them by a reduction of its light optically effected, and at the same time
either invariable, or at least only so variable that its changes can be ex-
actly calculated and reduced to numerical estimation. Such a standard is
offered by the planet Jupiter, which, being much brighter than any star,
subject to no phases, and variable in light only by the variation of its dis-
tance, from the sun, and which, moreover, comes in succession above the
horizon at a convenient altitude, simultaneously with all the fixed stars,
and, in the absence of the moon, twilight, and other disturbing causes
(which fatally affect all observations of this nature), combines all the re-
quisite conditions. Let us suppose, now, that Jupiter being at A and the
star to be compared with it at B, a glass prism, C, is so placed that the
light of the planet deflected by total internal reflexion at its base, shall
emerge parallel to B E, the direction of the star's visual ray. After re-
flexion let it be received on a lens, D, in whose focus, F, it will form a
small, bright, star-like image, capable of being viewed by an eye placed
at E, so far out of the axis of the cone of diverging rays as to admit of
seeing at the same time, and with the same eye, and so comparing this
image with the star seen directly. By bringing the eye nearer to or
further from the focus, F, the apparent brightness of the focal point will
be varied in the inverse ratio of the square of the distance, E F, and
therefore may be equalized, as well as the eye can judge of such equali-
ties, with the star. If this be done for two stars several times alternately,
and a mean of the results taken, by measuring EF, their relative bright.
444 OUTLINES OF ASTRONOMY.
Fig. 107. & /
ness will be obtained: that of Jupiter, the temporary standard of com.
parison, being altogether eliminated from the result.
(784.) A moderate number of well selected stars being thus photometri-
cally determined by repeated and careful measurements, so as to afford an
ascertained and graduated scale of brightness among the stars themselves,
the intermediate steps or grades of magnitude may be filled up, by insert-
ing between them, according to the judgment of the eye, other stars,
forming an ascending or descending sequence, each member of such a
sequence being brighter than that below, and less bright than that
above it; and thus at length, a scale of numerical magnitudes will
become established, complete in all its members, from Sirius, the brightest
of the stars, down to the least visible magnitude." It were much to be
wished that this branch of astronomy, which at present can hardly be said
to be advanced beyond its infancy, were perseveringly and systematically
cultivated. It is by no means a subject of mere barren curiosity, as will
abundantly appear when we come to speak of the phaenomena of variable
stars, and being moreover, one in which amateurs of the science may
easily chalk out for themselves a useful and available path, may naturally
be expected to receive large and interesting accessions at their hands.
(785.) If the comparison of the apparent magnitudes of the stars with
their numbers leads to no immediately obvious conclusion, it is otherwise
when we view them in connexion with their local distribution over the
heavens. If indeed we confine ourselves to the three or four brightest
classes, we shall find them distributed with a considerable approach to
* For the method of combining and treating such sequences, where accumulated
in considerable numbers, so as to eliminate from their results the influence of erroneous
judgment, atmospheric circumstances, &c., which often give rise to contradictory
arrangements in the order of stars differing but little in magnitude, as well as for an
account of a series of photometric comparisons (in which, however, not Jupiter, but
the moon was used as an intermediate standard), see the work above cited, note on p.
353 (Results of Observations, &c.)

GENERAL FORM OF THE GALAXY. 445
impartiality over the sphere: a marked preference however being observa-
ble, especially in the southern hemisphere, to a zone or belt, following the
direction of a great circle passing through s Orionis and a Crucis. But
if we take in the whole amount visible to the naked eye, we shall perceive
a great increase of number as we approach the borders of the Milky Way.
And when we come to telescopic magnitudes, we find them crowded
beyond imagination, along the extent of that circle, and of the branches
which it sends off from it; so that in fact its whole light is composed of
nothing but stars of every magnitude, from such as are visible to the
naked eye down to the smallest point of light perceptible with the best
telescopes.
(786.) These phaenomena agree with the supposition that the stars of
our firmament, instead of being scattered in all directions indifferently
through space, form a stratum of which the thickness is small, in com-
parison with its length and breadth; and in which the earth occupies a
place somewhere about the middle of its thickness, and near the point
where it subdivides into two principal laminae, inclined at a small angle to
each other (art. 302). For it is certain that, to an eye so situated, the
apparent density of the stars, supposing them pretty equally scattered
through the space they occupy, would be least in a direction of the visual
ray (as SA), perpendicular to the lamina, and greatest in that of its
breadth, as SB, S C, S D; increasing rapidly in passing from one to the
other direction, just as we see a slight haze in the atmosphere thickening
into a decided fog-bank near the horizon, by the rapid increase of the
mere length of the visual ray. Such is the view of the construction of
the starry firmament taken by Sir William Herschel, whose powerful
Fig. 108.
telescopes first effected a complete analysis of this wonderful zone, and
demonstrated the fact of its entirely consisting of stars.' So crowded are
they in some parts of it, that by counting the stars in a single field of his
* Thomas Wright of Durham (Theory of the Universe, London, 1750) appears so
early as 1734 to have entertained the same general view as to the constitution of the
Milky Way and starry firmament, founded, quite in the spirit of just astronomical
speculation, on a partial resolution of a portion of it with a “one-foot reflector” (a
reflector one foot in focal length). See an account of this rare work by M. de Morgan
in Phil. Mag. Ser. 3. xxxii. p. 241. et seq.

446 OUTLINES OF ASTRONOMY.
telescope, he was led to conclude that 50,000 had passed under his review
in a zone two degrees in breadth, during a single hour's observation. In
that part of the milky way which is situated in 10h 30m R A and between
the 147th and 150th degree of N P D, upwards of 5000 stars have been
reckoned to exist in a square degree. The immense distances at which
the remoter regions must be situated will sufficiently account for the vast
predominance of small magnitudes which are observed in it.
(787.) The course of the Milky Way as traced through the heavens
by the unaided eye, neglecting occasional deviations and following the line
of its greatest brightness as well as its varying breadth and intensity will
permit, conforms nearly to that of a great circle inclined at an angle of
about 63° to the equinoctial, and cutting that circle in R A 0h 47m and
12h 47m, so that its northern and southern poles respectively are situated
in R. A. 12h 47m N P D 63° and R. A. Oh 47m NPD 117°. Through-
out the region where it is so remarkably subdivided (art. 186), this great
circle holds an intermediate situation between the two great streams; with
a nearer approximation however to the brighter and continuous stream,
than to the fainter and interrupted one. If we trace its course in order
of right ascension, we find it traversing the constellation Cassiopeia, its
brightest part passing about two degrees to the north of the star 8 of that
constellation, i. e. in about 62° of north declination, or 28° N P D.
Passing thence between y and s Cassiopeiae it sends off a branch to the
south-preceding side, towards a Persei, very conspicuous as far as that
star, prolonged faintly towards s of the same constellation, and possibly
traceable towards the Hyades and Pleiades as remote outliers. The main
stream however (which is here very faint), passes on through Auriga, over
the three remarkable stars, s, &, n, of that constellation preceding Capella,
called the Hoedi, preceding Capella, between the feet of Gemini and the
horns of the Bull (where it intersects the ecliptic nearly in the Solstitial
Colure) and thence over the club of Orion to the neck of Monoceros,
intersecting the equinoctial in R. A. 6h 54m. Up to this point, from
the offset in Perseus, its light is feeble and indefinite, but thenceforward
it receives a gradual accession of brightness, and where it passes through
the shoulder of Monoceros and over the head of Canis Major it presents
abroad, moderately bright, very uniform, and to the naked eye, starless
stream up to the point where it enters the prow of the ship Argo, nearly
on the southern tropic." Here it again subdivides (about the star m
• In reading this description a celestial globe will be a necessary companion. It may
be thought needless to detail the course of the Milky Way verbally, since it is mapped
down on all celestial charts and globes. But in the generality of them, indeed in all
whicn nave come to our knowledge, this is done so very loosely and incorrectly, as by
no means to dispense with a verbal description.
COURSE OF THE WIA LACTEA TRACED. 447
Puppis), sending off a narrow and winding branch on the preceding side
as far as y Argås, where it terminates abruptly. The main stream pur-
sues its southward course to the 123d parallel of N P D, where it diffuses
itself broadly and again subdivides, opening out into a wide fan-like ex-
panse nearly 20° in breadth formed of interlacing branches, all which
terminate abruptly, in a line drawn nearly through A and y Argås.
(788.) At this place the continuity of the Milky Way is interrupted
by a wide gap, and where it recommences on the opposite side it is by a
somewhat similar fan-shaped assemblage of branches which converge upon
the bright star n Argås. Thence it crosses the hind feet of the Centaur,
forming a curious and sharply defined semicircular concavity of small
radius, and enters the Cross by a very bright neck or isthmus of not more
than 3 or 4 degrees in breadth, being the narrowest portion of the Milky
Way. After this it immediately expands into a broad and bright mass,
enclosing the stars a and 3 Crucis, and 6 Centauri, and extending almost
up to a of the latter constellation. In the midst of this bright mass, sur-
rounded by it on all sides, and occupying about half its breadth, occurs a
singular dark pear-shaped vacancy, so conspicuous and remarkable as to
attract the notice of the most superficial gazer, and to have acquired among
the early southern navigators the uncouth but expressive appellation of
the coal-sack. In this vacancy which is about 8° in length, and 5° broad,
only one very small star visible to the naked eye occurs, though it is far
from devoid of telescopic stars, so that its striking blackness is simply due
to the effect of contrast with the brilliant ground with which it is on all
sides surrounded. This is the place of nearest approach of the Milky
Way to the South Pole. Throughout all this region its brightness is very
striking, and when compared with that of its more northern course already
traced, conveys strongly the impression of greater proximity, and would
almost lead to a belief that our situation as spectators is separated on all
sides by a considerable interval from the dense body of stars composing
the Galaxy, which in this view of the subject would come to be considered
as a flat ring of immense and irregular breadth and thickness, within
which we are excentrically situated, nearer to the southern than to the
northern part of its circuit.
(789.) At a Centauri, the Milky Way again subdivides', sending off a
great branch of nearly half its breadth, but which thins off rapidly, at an
angle of about 20° with its general direction, towards the preceding side,
to n and d Lupi, beyond which it loses itself in a narrow and faint stream-
let. The main stream passes on increasing in breadth to y Normae, where
it makes an abrupt elbow and again subdivides into one principal and con
* All the maps and globes place this subdivision at 3 Centauri, but erroneously.
*.
448 OUTLINES OF ASTRONOMY.
t
tinuous stream of very irregular breadth and brightness on the following
side, and a complicated system of interlaced streaks and masses on the
preceding, which covers the tail of Scorpio, and terminates in a vast and
faint effusion over the whole extensive region occupied by the preceding
leg of Ophiuchus, extending northwards to the parallel of 103° N PD,
beyond which it cannot be traced; a wide interval of 14°, free from all
appearance of nebulous light, separating it from the great branch on the
north side of the equinoctial of which it is usually represented as a con-
tinuation.
(790.) Returning to the point of separation of this great branch from
the main stream, let us now pursue the course of the latter. Making an
abrupt bend to the following side, it passes over the stars Arae, 9 and a
Scorpii, and y Tubi to y Sagittarii, where it suddenly collects into a vivid
oval mass about 6° in length and 4° in breadth, so excessively rich in
stars that a very moderate calculation makes their number exceed 100,000.
Northward of this mass, this stream crosses the ecliptic in longitude about
276°, and proceeding along the bow of Sagittarius into Antimous has its
course rippled by three deep concavities, separated from each other by
remarkable protuberances, of which the larger and brighter (situated
between Flamstead's stars 3 and 6 Aquilae) forms the most conspicuous
patch in the southern portion of the Milky Way visible in our latitudes.
(791.) Crossing the equinoctial at the 19th hour of right ascension, it
next runs in an irregular, patchy, and winding stream through Aquila,
Sagitta and Vulpecula up to Cygnus; at s of which constellation its con-
tinuity is interrupted, and a very confused and irregular region commences,
marked by a broad dark vacuity, not unlike the southern “coal-sack,”
occupying the space between s, a, and y Cygni, which serves as a kind of
centre for the divergence of three great streams; one, which we have
already traced; a second, the continuation of the first (across the interval)
from a northward, between Lacerta and the head of Cepheus to the point
in Cassiopeia whence we-set out, and a third branching off from y Cygni,
very vivid and conspicuous, running off in a southern direction through 3
Cygni, and s Aquilae almost to the equinoctial, where it loses itself in a
region thinly sprinkled with stars, where in some maps the modern con-
stellation Taurus Poniatovii is placed. This is the branch which, if con-
tinued across the equinoctial, might be supposed to unite with the great
southern effusion in Ophiuchus already noticed (art. 789). A considerable
offset, or protuberant appendage, is also thrown off by the northern stream
from the head of Cepheus directly towards the pole, occupying the greater
part of the quartile formed by a, 3, v, and 6 of that constellation.
COURSE OF THE WIA LACTEA TRACED. . 449
(792.) We have been somewhat circumstantial in describing the course
and principal features of the Via Lactea, not only because there does not
occur anywhere (so far as we know) any correct account of it, but chiefly
by reason of its high interest in sidereal astronomy, and that the reader
may perceive how very difficult it must necessarily be to form any just
conception of the real, solid form, as it exists in space, of an object so
complicated, and which we see from a point of view so unfavourable.
The difficulty is of the same kind which we experience when we set our-
selves to conceive the real shape of an auroral arch or of the clouds, but
far greater in degree, because we know the laws which regulate the forma.
tion of the latter, and limit them to certain conditions of altitude — be-
cause their motion presents them to us in various aspects, but chiefly
because we contemplate them from a station considerably below their
general plane, so as to allow of our mapping out some kind of ground-
plan of their shape. All these aids are wanting when we attempt to map
and model out the Galaxy, and beyond the obvious conclusion that its
form must be, generally speaking, flat, and of a thickness small in com-
parison with its area in length and breadth, the laws of perspective afford
us little further assistance in the inquiry. Probability may, it is true,
here and there enlighten us as to certain features. Thus when we see, as
in the coal-sack, a sharply defined oval space free from stars, insulated in
the midst of a uniform band of not much more than twice its breadth, it
would seem much less probable that a conical or tubular hollow traverses
the whole of a starry stratum, continuously extended from the eye out-
wards, than that a distant mass of comparatively moderate thickness
should be simply perforated from side to side, or that an oval vacuity
should be seen foreshortened in a distant foreshortened area, not really
exceeding two or three times its own breadth. Neither can we without
obvious improbability refuse to admit that the long lateral offsets which at
so many places quit the main stream and run out to great distances, are
either planes seen edgeways, or the convexities of curved surfaces viewed
tangentially, rather than cylindrical or columnar excrescences bristling up
obliquely from the general level. And in the same spirit of probable
surmise we may account for the intricate reticulations above described as
existing in the region of Scorpio, rather by the accidental crossing of
streaks thus originating, at very different distances, or by a cellular struc.
ture of the mass, than by real intersections, Those cirrous clouds which
are often seen in windy weather, convey no unapt impression either of the
kind of appearance in question, or of the structure it suggests. It is to
other indications however, and chiefly to the telescopic examination of its
intimate constitution, and to the law of the distribution of stars, not only
29
450 † OUTLINES OF ASTRONOMY.
within its bosom, but generally over the heavens, that we must look for
more definite knowledge respecting its true form and extent.
(793.) It is on observations of this latter class, and not on merely
speculative or conjectural views, that the generalization in Art. 786, which
refers the phaenomena of the starry firmament to the system of the Ga-
laxy as their embodying fact, is brought to depend. The process of
“gauging” the heavens was devised by Sir W. Herschel for this purpose.
It consisted in simply counting the stars of all magnitudes which occur in
single fields of view, of 15' in diameter, visible through a reflecting tele-
scope of 18 inches aperture, and 20 feet focal length, with a magnifying
power of 180°: the points of observation being very numerous and taken
indiscriminately in every part of the surface of the sphere visible in our
latitudes. On a comparison of many hundred such “gauges” or local
enumerations it appears that the density of star-light (or the number of
stars existing on an average of several such enumerations in any one im-
mediate neighbourhood) is least in the pole of the Galactic circle,' and
increases on all sides, with the Galactic polar distance (and that nearly
equally in all directions) down to the Milky Way itself, where it attains
its maximum. The progressive rate of increase in proceeding from the
pole is at first slow, but becomes more and more rapid as we approach the
plane of that circle according to a law of which the following numbers,
deduced by M. Struve from a careful analysis of all the gauges in ques-
tion, will afford a correct idea.
Galactic 2 North Polar Distance. *º. in a
0o - 4-15
159 4-68
300 6-52
459 10.36
600 17.68
750 30-30
900 122:00
From which it appears that the mean density of the stars in the galactic
circle exceeds in a ratio of very nearly 30 to 1 that in its pole, and in a
proportion of more than 4 to 1 that in a direction 15° inclined to its
plane. -
* From yaxa, yakakros, milk; meaning the great circle spoken of in Art. 787, to
which the course of the Via Lactea most nearly conforms. Every subject has its tech-
nical or conventional terms, by whose use circumlocution is avoided, and ideas ren-
dered definite. This circle is to sidereal what the invariable ecliptic is to planetary
astronomy — a plane of ultimate reference, the ground-plane of the sidereal system.
* Etudes d’Astronomic Stellaire, p, 71. - *
LAW OF DISTRIBUTION OF THE STARS. 451
(794.) These numbers fully bear out the statement in Art. 786, and
even draw closer the resemblance by which that statement is there illus-
trated. For the rapidly increasing density of a fog-bank as the visual ray
is depressed towards the plane of the horizon is a consequence not only of
the mere increase in length of the foggy space traversed, but also of an
actual increase of density in the fog itself in its lower strata. Now this
very conclusion follows from a comparison inter se of the numbers above
sct down, as M. Struve has clearly shown from a mathematical analysis
of the empirical formula, which faithfully represents their law of progres-
sion, and of which he states the result in the following table, expressing
the densities of the stars at the respective distances, 1, 2, 3, &c., from the
galactic plane, taking the mean density of the stars in that plane itself
for unity.

*...* | Density of Stars. | *.*.* | Density of stars.
0.00 1.00000 0-50, 0.08646
0-05 0-48568 0-60 0-05510
0.10 0-33288 0-70 0-03079
0-20 0-23895 0-80 0-01414.
0-30 0-17980 0°866 0.00532
0-40 0-13021
The unit of distance, of which the first column of this table expresses
fractional parts, is the distance at which such a telescope is capable of
rendering just visible a star of average magnitude, or, as it is termed, its
space-penetrating power. As we ascend therefore from the galactic plane
into this kind of stellar atmosphere, we perceive that the density of its
parallel strata decreases with great rapidity. At an altitude above that
plane equal to only one-twentieth of the telescopic limit, it has already
diminished to one-half, and at an altitude of 0-866, to hardly more than
one-two-hundredth of its amount in that plane. So far as we can perceive
there is no flaw in this reasoning, if only it be granted, 1st, that the level
planes are continuous, and of equal density throughout; and, 2dly, tha.
an absolute and definite limit is set to telescopic vision, beyond which, if
stars eacist, they elude our sight, and are to us as if they existed not : a
postulate whose probability the reader will be in a better condition to esti-
mate, when in possession of some other particulars respecting the consti-
tution of the Galaxy to be described presently.
(795.) A similar course of observation followed out in the southern
hemisphere, leads independently to the same conclusion as to the law of
the visible distribution of stars over the southern galactic hemisphere, or
that half of the celestial surface which has the south galactic pole for its
centre. A system of gauges, extending over the whole surface of that
452 OUTLINES OF ASTRONOMY.
hemisphere taken with the same telescope, field of view and magnifying
power employed in Sir William Herschel's gauges, has afforded the ave-
rage numbers of stars per field of 15' in diameter, within the areas of
zones encircling that pole at intervals of 15°, set down in the following
table.
Zones of Galactic South Average Number of Stars
Polar Distance. per Field of 15'.
0° to 15° 6-05
15 to 30 6-62
30 to 45 9.08
45 to 60 - 18-49
60 to 75 26.29
75 to 90 59.06
(796.) These numbers are not directly comparable with those of M.
Struve, given in Art. 793, because the latter corresponds to the limiting
polar distances, while these are the averages for the included zones. That
eminent astronomer, however, has given a table of the average gauges ap-
propriate to each degree of north galactic polar distance,' from which it is
easy to calculate averages for the whole extent of each zone. How near
a parallel the results of this calculation for the northern hemisphere ex-
hibit with those above stated for the southern, will be seen by the follow-
ing table.
Zones of Galactic North Average Number of Stars per Field
Polar Distance. of 15' from M. Struve's Table.
09 to 15° 4.32
15 to 30 5:42
30 to 45 8.21
45 to 60 13-61
60 to 75 24.09
75 to 90 53.43
It would appear from this that, with an almost exactly similar law of ap-
parent density in the two hemispheres, the southern were somewhat richer
in stars than the northern, which may, and not improbably does, arise
from our situation not being precisely in the middle of its thickness, but
somewhat nearer to its northern surface.
(797.) When examined with powerful telescopes, the constitution of
this wonderful zone is found to be no less various than its aspect to the
naked eye is irregular. In some regions the stars of which it is wholly
composed are scattered with remarkable uniformity over immense tracts,
while in others the irregularity of their distribution is quite as striking,
* Etudes d'Astronomie Stellaire, p. 34.
TELESCOPIC CONSTITUTION OF THE GALAXY. 453 .
exhibiting a rapid succession of closely clustering rich patches separated
by comparatively poor intervals, and indeed in some instances by spaces
absolutely dark and completely void of any star, even of the smallest
telescopic magnitude. In some places not more than 40 or 50 stars on
an average occur in a “gauge” field of 15', while in others a similar
average gives a result of 400 or 500. Nor is less variety observable
in the character of its different regions in respect of the magnitudes of
the stars they exhibit, and the proportional numbers of the larger and
smaller magnitudes associated together, than in respect of their aggregate
numbers. In some, for instance, extremely minute stars, though never
altogether wanting, occur in numbers so moderate as to lead us irresistibly
to the conclusion that in these regions we see fairly through the starry
stratum, since it is impossible otherwise (supposing their light not inter-
cepted) that the numbers of the smaller magnitudes should not go on
continually increasing ad infinitum. In such cases moreover the ground
of the heavens, as seen between the stars, is for the most part perfectly
dark, which again would not be the case, if innumerable multitudes of
stars, too minute to be individually discernible, existed beyond. In
other regions we are presented with the phenomenon of an almost uni-
form degree of brightness of the individual stars, accompanied with a
very even distribution of them over the ground of the heavens, both the
larger and smaller magnitudes being strikingly deficient. In such cases
it is equally impossible not to perceive that we are looking through a sheet
of stars nearly of a size, and of no great thickness compared with the dis.
tance which separates them from us. Were it otherwise we should be
driven to suppose the more distant stars uniformly the larger, so as to
compensate by their greater intrinsic brightness for their greater distance,
a supposition contrary to all probability. In others again, and that not
unfrequently, we are presented with a double phaenomenon of the same
kind, viz. a tissue as it were of large stars spread over another of very
small ones, the intermediate magnitudes being wanting. The conclusion
here seems equally evident that in such cases we look through two side-
real sheets separated by a starless interval.
(798.) Throughout by far the larger portion of the extent of the Milky
Way in both hemispheres, the general blackness of the ground of the
heavens on which its stars are projected, and the absence of that innu-
merable multitude and excessive crowding of the smallest visible magni-
tudes, and of glare produced by the aggregate light of multitudes too
small to affect the eye singly, which the contrary supposition would appear
to necessitate, must, we think, be considered unequivocal indications that
its dimensions in directions where these conditions obtain, are not only not
454 OljTLINES OF ASTRONOMY.
infinite, but that the space-penetrating power of our telescopes suffices
fairly to pierce through and beyond it. It is but right, however, to warm
our readers that this conclusion has been controverted, and that by an
authority not lightly to be put aside, on the ground of certain views taken
by Olbers as to a defect of perfect transparency in the celestial spaces, in
virtue of which the light of the more distant stars is enfeebled more than
in proportion to their distance. The extinction of light thus originating,
proceeding in geometrical progression while the distance increases in
arithmetical, a limit, it is argued, is placed to the space-penetrating powers
of telescopes, far within that which distance alone apart from such obscu-
ration would assign. It would lead us too far aside of the objects of a
treatise of this nature to enter upon any discussion of the grounds (partly
metaphysical) on which these views rely. It must suffice here to observe
that the objection alluded to, if applicable to any, is equally so to every
part of the galaxy. We are not at liberty to argue that at one part of its
circumference, our view is limited by this sort of cosmical veil which
extinguishes the smaller magnitudes, cuts off the nebulous light of distant
masses, and closes our view in impenetrable darkness; while at another
we are compelled by the clearest evidence telescopes can afford to believe
that star-strown vistas lie open, exhausting their powers and stretching
out beyond their utmost reach, as is proved by that very phaenomenon
which the existence of such a veil would render impossible, viz. infinite
increase of number and diminution of magnitude, terminating in complete
irresolvable nebulosity. Such is, in effect, the spectacle afforded by a very
large portion of the Milky Way in that interesting region near its point
of bifurcation in Scorpio (arts. 789, 792,) where, through the hollows
and deep recesses of its complicated structure we behold what has all the
appearance of a wide and indefinitely prolonged area strewed over with
discontinuous masses and clouds of stars which the telescope at length
refuses to analyse." Whatever other conclusions we may draw, this must
any how be regarded as the direction of the greatest linear extension of
the ground-plan of the galaxy. And it would appear to follow, also, as a
not less obvious consequence, that in those regions where that zone is
clearly resolved into stars well separated and seen projected on a black
ground, and where by consequence it is certain if the foregoing views be
• It would be doing great injustice to the illustrious astronomer of Pulkova (whose
opinion, if we here seem to controvert, it is with the utmost possible deference and
respect) not to mention that at the time of his writing the remarkable essay already
more than once cited, in which the views in question are delivered, he could not have
been aware of the important facts alluded to in the text, the work in which they are
described being then unpublished.
DISTANCE OF THE FIXED STARS, 455
correct that we look out beyond them into space, the smallest visible stars
appear as such, not by reason of excessive distance, but of a real inferiority
of size or brightness.
(799.) When we speak of the comparative remoteness of certain
regions of the starry heavens beyond others, and of our own situation in
them, the question immediately arises, what is the distance of the nearest
fixed star 7 What is the scale on which our visible firmament is con-
structed 7 And what proportion do its dimensions bear to those of our
wn immediate system 7 To these questions astronomy has at length
been enabled to afford an answer.
(800.) The diameter of the carth has served us for the base of a tri-
angle, in the trigonometrical survey of our system (art. 274,) by which to
calculate the distance of the sun; but the extreme minuteness of the sun’s
parallax (art. 3, 7,) renders the calculation from this “ill-conditioned”
triangle (art. 275,) so delicate, that nothing but the fortunate combination
of favourable circumstances, afforded by the transits of Venus (art. 479,)
could render its results even tolerably worthy of reliance. But the earth's
diameter is too small a base for direct triangulation to the verge even of
our own system (art. 526,) and we are, therefore obliged, to substitute
the annual parallax for the diurnal, or, which comes to the same thing,
to ground our calculation on the relative velocities of the earth and planets
in their orbits (art. 486,) when we would push our triangulation to that
extent. It might be naturally enough expected, that by this enlargement
of our base to the vast diameter of the earth's orbit, the next step in our
survey (art. 275,) would be made at a great advantage;—that our change
of station, from side to side of it, would produce a considerable and easily
measurable amount of annual parallax in the stars, and that by its means
we should come to a knowledge of their distance. But, after exhausting
every refinement of observation, astronomers were, up to a very late period,
unable to come to any positive and coincident conclusion upon this head;
and the amount of such parallax, even for the nearest fixed star examined
with the requisite attention, remained mixed up with, and concealed
among, the errors incidental to all astronomical determinations. The
nature of these errors has been explained in the earlier part of this work,
and we need not remind the reader of the difficulties which must necessa-
rily attend the attempt to disentangle an element not exceeding a few
tenths of a second, or at most a whole second, from the host of uncertain-
ties entailed on the results of observations by them : none of them indi-
vidually perhaps of greater magnitude, but embarrassing by their number
and fluctuating amount. Nevertheless, by successive refinements in
instrument making, and by constantly progressive approximation to the
456 OUTLINES OF ASTRONOMY.
exact knowledge of the Uranographical corrections, that assurance had
been obtained, even in the earlier years of the present century, viz. that
no star visible in northern latitudes, to which attention had been directed,
manifested an amount of parallax exceeding a single second of are. It is
worth while to pause for a moment to consider what conclusions would
follow from the admission of a parallax to this amount.
(801.) Radius is to the sine of 1" as 206265 to 1. In this proportion
then at least must the distance of the fixed stars from the sun exceed that
of the sun from the earth. Again, the latter distance, as we have already
seem (art. 357,) exceeds the earth's radius in the proportion of 23984 to
1. Taking therefore the earth's radius for unity, a parallax of 1" sup-
poses a distance of 4947059760 or nearly five thousand millions of such
units: and lastly, to descend to ordinary standards, since the earth's radius
may be taken at 4000 of our miles, we find 19788239040000 or about
twenty billions of miles for our resulting distance.
(802.) In such numbers the imagination is lost. The only mode we
have of conceiving such intervals at all is by the time which it would
require for light to traverse them. Light, as we know (art. 545,) travels
at the rate of 192000 miles per second, traversing a semidiameter of the
earth’s orbit in 8" 13'-3. It would, therefore, occupy 206265 times this
interval, or 3 years and 83 days to traverse the distance in question. Now
as this is an inferior limit which it is already ascertained that even the
brightest and therefore (in the absence of all other indications) the dis-
tance of those innumerable stars of the smaller magnitudes which the
telescope discloses to us! What for the dimensions of the galaxy in
whose remoter regions, as we have seen, the united lustre of myriads
of stars is perceptible only in powerful telescopes as a feeble nebulous
gleam
(803.) The space-penetrating power of a telescope, or the comparative
distance to which a given star would require to be removed, in order that
it may appear of the same brightness in the telescope as before to the
naked eye, may be calculated from the aperture of the telescope compared
with that of the pupil of the eye, and from its reflecting or transmitting
power, i, e, the proportion of the incident light it conveys to the observer's
eye. Thus it has been computed that the space-penetrating power of such
a reflector as that used in the star-gauges above referred to is expressed
by the number 75. A star, then, of the sixth magnitude, removed to
75 times its distance, would still be perceptible as a star with that instru-
ment, and admitting such a star to have 100th part of the light of a
standard star of the first magnitude, it will follow that such a standard
star, if removed to 750 times its distance, would excite in the eye, when
DISTANCE OF THE FIXED STARS. 457
viewed through the gauging telescope, the same impression as a star of
the sixth magnitude does to the naked eye. Among the infinite multitude
of such stars in the remoter regions of the galaxy, it is but fair to con-
clude that innumerable individuals equal in intrinsic brightness to those
which immediately surround us, must exist. The light of such stars,
then, must have occupied upwards of 2000 years in travelling over the
distance which separates them from our own system. It follows, then,
that when we observe the places and note the appearance of such stars, we
are only reading their history of two thousand years' anterior date, thus
wonderfully recorded. We cannot escape this conclusion, but by adopting
as an alternative an intrinsic inferiority of light in all the smaller stars
of the galaxy. We shall be better able to estimate the probability of this
alternative when we shall have made acquaintance with other sidereal
systems, whose existence the telescope discloses to us, and whose analogy
will satisfy us that the view of the subject here taken is in perfect har-
mony with the general tenor of astronomical facts.
(804.) Hitherto we have spoken of a parallax of 1" as a mere limit,
below which that of any star yet examined, assuredly, or at least very
probably, falls, and it is not without a certain convenience to regard this
amount of parallax as a sort of unit of reference, which, connected in the
reader's recollection with a parallactic unit of distance from our system
of 20 billions of miles, and with a 33 years’ journey of light, may save
him the trouble of such calculations, and ourselves the necessity of cover-
ing our pages with such enormous numbers, when speaking of stars whose
parallax has actually been ascertained with some approach to certainty,
either by direct meridian observation, or by more refined and delicate
methods. These we shall proceed to explain, after first pointing out the
theoretical peculiarities which enable us to separate and disentangle its
effects from those of the Uranographical corrections, and from other causes
of error, which, being periodical in their nature, add greatly to the diffi-
culty of the subject. The effects of precession and proper motion (see
art. 852,) which are uniformly progressive from year to year, and that of
mutation, which runs through its period in nineteen years, it is obvious
enough, separate themselves at once by these characters from that of pa-
rallax; and, being known with very great precision, and being certainly
independent, as regards their causes, of any individual peculiarity in the
stars affected by them, whatever small uncertainty may remain respecting
the numerical elements which enter into their computation (or, in mathe.
matical language, their co-efficients), can give rise to no embarrassment.
With regard to aberration the case is materially different. This correc.
tion affects the place of a star by a fluctuation, annual in its period, and
458 . OUTLINES OF ASTRONOMY.
therefore, so far agreeing with parallax. It is also very similar in the
law of its variation at different seasons of the year, parallax having for
its apex (see art. 343, 344,) the apparent place of the sun in the ecliptic,
and aberration a point in the same great circle 90° behind that place, so
that in fact the formulae of calculation (the coefficients excepted) are the
same for both, substituting only for the sun's longitude in the expression
for the one, that longitude diminished by 90° for the other. Moreover, in
the absence of absolute certainty respecting the nature of the propagation
of light, astronomers have hitherto considered it necessary to assume, at
least as a possibility, that the velocity of light may be to some slight
amount dependent on individual peculiarities in the body emitting it.'
(805.) If we suppose a line drawn from the star to the earth at all
seasons of the year, it is evident that this line will sweep over the surface
of an exceedingly acute, oblique cone, having for its axis the line joining
the sun and star, and for its base the earth’s annual orbit, which, for the
present purpose, we may suppose circular. The star will therefore appear
to describe each year about its mean place, regarded as fixed, and in virtue
of parallax alone, a minute ellipse, the section of this cone by the surface
of the celestial sphere, perpendicular to the visual ray. But there is also
another way in which the same fact may be represented. The apparent
orbit of the star about its mean place as a centre, will be precisely that
which it would appear to describe, if seen from the sun, supposing it.
really revolved about that place in a circle exactly equal to the earth's
annual orbit, in a plane parallel to the ecliptic. This is evident from the
equality and parallelism of the lines and directions concerned. Now the
effect of aberration (disregarding the slight variation of the earth’s ve-
locity in different parts of its orbit) is precisely similar in law, and differs
only in amount, and in its bearing reference to a direction 90° different
in longitude. Suppose, in order to fix our ideas, the maximum of parallax
to be 1" and that of aberration 20-5", and let A B, a b, be two circles
imagined to be described separately, as above, by the star about its mean place
S, in virtue of these two causes respectively, SSP being a line parallel to that
of the line of equinoxes. Then, if in virtue of parallax alone, the star
would be found at a in the smaller orbit, it would, in virtue of aberration
alone, be found at A, in the larger, the angle a SA being a right angle.
Trawing then A C equal and parallel to S a, and joining S C, it will in
* In the actual state of astronomy and photology this necessity can hardly be consi-
dered as still existing, and it is desirable, therefore, that the practice of astronomers
of introducing an unknown correction for the constant of aberration into their ‘ equa-
tions of condition” for the determination of parallax, should be disused, since it actu-
allv tends to introduce error into the final result. º
PARALLAX OF THE FIXED STARS. 459
virtue of both simultaneously be found in C, i. e. in the circumference of
a circle whose radius is SC, and at a point in that circle in advance of A,
the aberrational place, by the angle A S C. Now, since S.A.: A C : :
20.5:1, we find for the angle A S C2°47'35", and for the length of
Fig. 109.
the radius SC of the circle representing the compound motion 20".524.
The difference (0"-024) between this and S C, the radius of the aberra-
tion circle, is quite imperceptible, and even supposing a quantity so mi-
nute to be capable of detection by a prolonged series of observations, it
would remain a question whether it were produced by parallax or by a
specific difference of aberration from the general average 20".5 in the star
itself. It is therefore to the difference of 2°48' between the angular
situation of the displaced star in this hypothetical orbit, i. e. in the
arguments (as they are called) of the joint correction (PSC) and that of
aberration alone (PSA), that we have to look for the resolution of the
problem of parallax. The reader may easily figure to himself the deli-
cacy of an inquiry which turns wholly (even when stripped of all its other
difficulties) on the precise determination of a quantity of this nature, and
of such very moderate magnitude.
(806.) But these other difficulties themselves are of no trifling order.
All astronomical instruments are affected by differences of temperature,
Not only do the materials of which they are composed expand and con
tract, but the masonry and solid piers on which they are erected, nay even
the very soil on which these are founded, participate in the general change
from summer warmth to winter cold. Hence arise slow oscillatory move-
ments of exceedingly minute amount, which levels and plumb-lines afford

460 OUTLINES OF ASTRONOMY.
but very inadequate means of detecting, and which being also annual in
their period (after rejecting whatever is merely casual and momentary)
mix themselves intimately with the matter of our inquiry. Refraction
too, besides its casual variations from night to night, which a long series
of observations would eliminate, depends for its theoretical expression on
the constitution of the strata of our atmosphere, and the law of the dis-
tribution of heat and moisture at different elevations, which cannot be
unaffected by difference of season. No wonder then that mere meridional
observations should, almost up to the present time, have proved insufficient,
except in one very remarkable instance, to afford unquestionable evidence,
and satisfactory quantitative measurement of the parallel of any fixed
Star.
(807.) The instance referred to is that of a Centauri, one of the brightest
and for many other reasons, one of the most remarkable of the southern
stars. From a series of observations of this star, made at the Royal
Observatory of the Cape of Good Hope in the years 1832 and 1833, by
Professor Henderson, with the mural circle of that establishment, a paral-
lax to the amount of an entire second was concluded on his reduction of
the observations in question after his return to England. Subsequent
observations by Mr. Maclear, partly with the same, and partly with a new
and far more efficiently constructed instrument of the same description
made in the years 1839 and 1840, have fully confirmed the reality of the
parallax indicated by Professor Henderson's observations, though with a
slight diminution in its concluded amount, which comes out equal to
0".9128 or about 4%ths of a second; bright stars in its immediate neigh-
bourhood being unaffected by a similar periodical displacement, and thus
affording satisfactory proof that the displacement indicated in the case
of the star in question is not merely a result of annual variations of
temperature. As it is impossible at present to answer for so minute a
quantity as that by which this result differs from an exact second, we may
consider the distance of this star as approximately expressed by the paral
lactic unit of distance referred to in art. 804.
(808.) A short time previous to the publication' of this important
result, the detection of a sensible and measurable amount of parallax in the
star N° 61 Cygni of Flamsteed’s catalogue of stars was announced by the
celebrated astronomer of Königsberg, the late M. Bessel.” This is a small
and inconspicuous star, hardly exceeding the sixth magnitude, but which
had been pointed out for especial observation by the remarkable circum-
* Professor Henderson's paper was read before the Astronomical Society of London,
Jan. 3, 1839. It bears date Dec. 24, 1838.
* Astronomische Nachrichten, Nos. 365, 366. Dec. 13, 1838.
PARALLAX OF THE FIXED STABS. 461
stance of its being affected by a proper motion (see art. 852) i. e. a regular
and continually progressive annual displacement among the surrounding
stars to the extent of more than 5" per annum, a quantity so very much
exceeding the average of similar minute annual displacements which many
other stars exhibit, as to lead to a suspicion of its being actually nearer to
our system. It is not a little remarkable that a similar presumption of
proximity exists also in the case of a Centauri, whose unusually large
proper motion of nearly 4" per annum is stated by Professor Henderson
to have been the motive which induced him to subject his observations of
that star to that severe discussion which led to the detection of its parallax.
M. Bessel's observations of 61 Cygni were commenced in August 1837,
immediately on the establishment at the Königsberg observatory of a
magnificent heliometer, the workmanship of the celebrated optician Fraun-
hofer, of Munich, an instrument especially fitted for the system of obser-
vation adopted; which being totally different from that of direct meri-
dional observation, more refined in its coliception, and susceptible of far
greater accuracy in its practical application, we must now explain.
(809.) Parallax, proper motion, and specific aberration (denoting by
the latter phrase that part of the aberration of a star's light which may
be supposed to arise from its individual peculiarities, and which we have
every reason to believe at all events an exceedingly minute fraction of the
whole,) are the only uranographical corrections which do not necessarily
affect alike the apparent places of two stars situated in, or very nearly in,
the same visual line. Supposing then two stars at an immense distance,
the one behind the other, but otherwise so situated as to appear very
nearly along the same visual line, they will constitute what is called a star
optically double, to distinguish it from a star physically double, of which
more hereafter. Aberration (that which is common to all stars), preces-
sion, nutation, nay, even refraction, and instrumental causes of apparent
displacement, will affect them alike, or so very nearly alike (if the minute
difference of their apparent places be taken into account) as to admit of
the difference being neglected, or very accurately allowed for, by an easy
calculation. If then, instead of attempting to determine by observation
the place of the nearer of two very unequal stars (which will probably be
the larger) by direct observation of its right ascension and polar distance,
we content ourselves with referring its place to that of its remoter and
smaller companion by differential observation, i.e. by measuring only its
difference of situation from the latter, we are at once relieved of the
necessity of making these corrections, and from all uncertainty as to their
influence on the result. And for the very same reason, errors of adjust-
ment (art. 136), of graduation, and a host of instrumental errors, which
462 oUTLINES OF ASTRONOMY.
would for this delicate purpose fatally affect the absolute determination of
either star's place, are harmless when only the difference of their places,
each equally affected by such causes, is required to be known.
(810.) Throwing aside therefore the consideration of all these errors
and corrections, and disregarding for the present the minute effect of
Fig. 110.
A
C )
specific aberration and the uniformly progressive effect of proper motion,
let us trace the effect of the differences of the parallaxes of two stars thus
juxtaposed, or their apparent relative distance and position at various
seasons of the year. Now the parallax being inversely as the distance,
the dimensions of the small ellipses apparently described (art. 805) by
each star on the concave surface of the heavens by parallactic displacement
will differ, the nearer star describing the larger ellipse. But both stars
lying very nearly in the same direction from the sun, these ellipses will be
similar and similarly situated. Suppose S and s to be the positions of the
two stars as seen from the sun, and let A B C D, a b c d, be their paral-
lactic ellipses; then, since they will be at all times similarly situated in
these ellipses, when the one star is seen at A, the other will be seen at a.
When the earth has made a quarter of a revolution in its orbit, their
apparent places will be B b ; when another quarter, C c ; and when
another, D d. If, then, we measure carefully, with micrometers adapted
for the purpose, their apparent situation with respect to each other, at
different times of the year, we should perceive a periodical change, both
in the direction of the line joining them, and in the distance between
their centres. For the lines A a and C c cannot be parallel, nor the lines
Bºb and D dequal, unless the ellipses be of equal dimensions, i.e. unless
the two stars have the same parallax, or are equidistant from the earth.
(811.) Now, micrometers, properly mounted, enable us to measure very

PARALLAX OF THE FIXED STARs. 463
exactly both the distance between two objects which can be seen together
in the same field of a telescope, and the position of the line joining them
with respect to the horizon, or the meridian, or any other determinate
direction in the heavens. The double image micrometer, and especially
the heliometer (art. 200, 201) is peculiarly adapted for this purpose. The
images of the two stars formed side by side, or in the same line, prolonged,
however momentarily displaced by temporary refraction or instrumental
tremor, move together, preserving their relative situation, the judgment
of which is no way disturbed by such irregular movements. The helio-
meter also, taking in a greater range than ordinary micrometers, enables
us to compare one large star with more than one adjacent small one, and
to select such of the latter among many near it, as shall be most favour-
ably situated for the detection of any motion in the large one, not partici-
pated in by its neighbours. -
(812.) The star examined by Bessel has two such neighbours, both very
minute, and therefore probably very distant, most favourably situated, the
one (s) at a distance of 7' 42", the other (s) at 11'46" from the large
star, and so situated, that their directions from that star make nearly a
right angle with each other. The effect of parallax therefore would
necessarily cause the two distances S s and Ss to vary, so as to attain
their maximum and minimum values alternately at three-monthly inter-
vals, and this is what was actually observed to take place, the one distance
being always on the increase or decrease when the other was stationary *
(the uniform effect of proper motion being understood of course to be
always duly accounted for). This alternation, though so small in amount
as to indicate, as a final result, a parallax, or rather a difference of paral-
laxes between the large and small stars of hardly more than one third of
a second, was maintained with such regularity as to leave no room for
reasonable doubt as to its cause, and having been confirmed by the further
continuance of these observations, and quite recently by the exact coinci-
dence between the result thus obtained, and that deduced by M. Peters
from observations of the same star at the observatory of Pulkova', is con-
sidered on all hands as fully established. The parallax of this star finally
resulting from Bessel's observation is 0".848, so that its distance from our
system is very nearly three parallactic units. (Art. 804.)
(813.) The bright star a Lyrae has also near it, at only 43" distance
(and therefore within the reach of the parallel wire or ordinary double
image micrometer) a very minute star, which has been subjected since
1835 to a severe and assiduotis scrutiny by M. Struve, on the same prin-
ciple of differential observation. He has thus established the existence
*With the great vertical circle by Ertel.
464 OUTLINES OF ASTRONOMY.
of a measurable amount of parallax in the large star, less indeed than
that of 61 Cygni (being only about 3 of a second), but yet sufficient
(such was the delicacy of his measurements) to justify this excellent
observer in announcing the result as at least highly probable, on the
strength of only five nights’ observation, in 1835 and 1836. This pro-
bability, the continuation of the measures to the end of 1838 and the
corroborative, though not in this case precisely coincident, result of Mr.
Peters’s investigations have converted into a certainty. M. Struve has
the merit of being the first to bring into practical application this method
of observation, which, though proposed for the purpose, and its great ad-
vantages pointed out by Sir William Herschel so early as 1781", remained
long unproductive of any result, owing partly to the imperfection of
micrometers for the measurement of distance, and partly to a reason
which we shall presently have occasion to refer to.
(814.) If the component individuals S, s (fig. art. 810,) be (as is often
the case) very close to each other, the parallactic variation of their angle
of position, or the extreme angle included between the lines A a, C c,
may be very considerable, even for a small amount of difference of paral-
laxes between the large and small stars. For instance in the case of two
adjacent stars 15" asunder, and otherwise favourably situated for observa-
tion, an annual fluctuation to and fro in the apparent direction of their
line of junction to the extent of half a degree (a quantity which could
not escape notice in the means of numerous and careful measurements)
would correspond to a difference of parallax of only 3 of a second. A
difference of 1" between two stars apparently situated at 5" distance
might cause an oscillation in that line to the extent of no less than 11°,
and if nearer one proportionally still greater. This mode of observation
has not yet been put in practice, but seems to offer great advantages.”
(815.) The following is a list of stars to which parallax has been up
to the present time more or less probably assigned :
a Centauri - q : º 0.913 (Henderson.)
61 Cygni * * * 0.348 (Bessel.)
a Lyrae gº ºf 0.261 (Struve.)
Sirius sº gº 0.230 (Henderson.)
1830 Groombridge” 0.226 (Peters.)
Ursae Majoris * 0.133 ditto.
Arcturus 0-127 ditto.
Polaris tº 0.067 ditto.
Capella dº tº- gº & *E. 0.046 ditto.
• It has been referred. even to Galileo. But the general explanation of Parallax in
the Systema Cosmicum, Dial. iii. p. 271 (Leyden edit. 1699) to which the reference
applies, does not touch any of the peculiar features of the case, or meet any of its
difficulties.
* See Phil. Trans. 1826. p. 266, et seq. and 1827, for a list of stars well adapted for
such observation, with the times of the year most favourable.—The list in Phil. Trans.
1826, is incorrect.
* Groombridge's catalogue of circumpolar stars.
PARALLAX OF THE FIXED STARS. 465
Although the extreme minuteness of the last four of these results de-
prives them of much numerical reliance, it is at least certain that the
parallaxes by no means follow the order of magnitudes, and this is farther.
shown by the fact that a Cygni, one of M. Peters's stars, shows absolutely
no indications of any measurable parallax whatever. -
(816.) From the distance of the stars we are naturally led to the con-
sideration of their real magnitudes. But here a difficulty arises, which,
so far as we can judge of what optical instruments are capable of effect-
ing, must always remain insuperable. Telescopes afford us only negative
information as to the apparent angular diameter of any star. The round,
well-defined, planetary discs which good telescopes show when turned upon
any of the brighter stars are phaenomena of diffraction, dependent, though
at present somewhat enigmatically, on the mutual interference of the rays
of light. They are consequently, so far as this inquiry is concerned,
mere optical illusions, and have therefore been termed spurious discs.
The proof of this is that telescopes of different apertures and magnifying
powers, when applied for the purpose of measuring their angular diame-
ters, give different results, the greater aperture (even with the same mag-
nifying power) giving the smaller disc. That the true disc of even a
large and bright star can have but a very minute angular measure, ap-
pears from the fact that in the occultation of such a star by the moon, its
extinction is absolutely instantaneous, not the smallest trace of gradual
diminution of light being perceptible. The apparent or spurious disc
also remains perfectly round and of its full size up to the instant of dis-
appearance, which could not be the case were it a real object. If our sun
were removed to the distance expressed by our parallactic unit (art. 804),
its apparent diameter of 32' 3" would be reduced to only 0"-0093, or less
than the hundredth of a second, a quantity which we have not the smallest
reason to hope any practical improvement in telescopes will ever show as
an object having distinguishable form.
t817.) There remains therefore only the indication which the quantity
of light they send to us may afford. But here again another difficulty
besets us. The light of the sun is so immensely superior in intensity to
that of any star, that it is impracticable to obtain any direct comparison
between them. But by using the moon as an intermediate term of com.
parison it may be done, not indeed with much precision, but sufficiently
well to satisfy in some degree our curiosity on the subject. Now a Cen-
tauri has been directly compared with the moon by the method explained
in Art. 783. By a mean of eleven such comparisons made in various
states of the moon, duly reduced and making the proper allowance on
photometric principles for the moon's light lost by transmission through
30
466 OUTLINES OF ASTRONOMY.
the lens and prism, it appears that the mean quantity of light sent to the
earth by a full moon exceeds that sent by a Centauri in the proportion of
27408 to 1. Now Wollaston, by a method apparently unobjectionable,
found' the proportion of the sun's light to that of the full moon to be
that of 801072 to 1. Combining these results, we find the hight sent us
by the sun to be that sent by a Centauri as 21,955,000,000, or about
twenty-two thousand millions to 1. Hence from the parallax assigned,
above to that star, it is easy to conclude that its intrinsic splendour, as
compared with that of our sun at equal distances, is 2.3247, that of the
sun being unity.”
(818.) The light of Sirius is four times that of a Centauri and its pa-
rallax only 0"-230 (Art. 230). This in effect ascribes to it an intrinsic
splendour equal to 63-02 times that of our sun.”
* Wollaston, Phil. Trans. 1829, p. 27. g
* Results of Astronomical Observations at the Cape of Good Hope, &c. Art. 278, p.
363. If only the results obtained near the quadratures of the moon (which is the sit-
uation most favourable to exactness) be used, the resulting value of the intrinsic light
of the star (the sun being unity) is 4°1586. On the other hand, if only those procured
near the full moon (the worst time for observation) be employed, the result is 1-4017.
Discordances of this kind will startle no one conversant with Photometry. That a
Centauri really emits more light than our sun must, we conceive, be regarded as an
established fact. To those who may refer to the work cited it is necessary to mention
that the quantity there designated by M, expresses, on the scale there adopted, 500
times the actual illuminating power of the moon at the time of observation, that of the
mean full moon being unity.
* See the work above cited, p. 367.-Wollaston makes the light of Sirius one 20,000.
millionth of the sun's. Steinheil by a very uncertain method found G) = (3286500)^x
Arcturus. -
PERIODICAL STARS. 467
CHAPTER XVI.
wARIABLE AND PERIODICAL STARS.—LIST OF THOSE ALREADY KNOWN.
— IRREGULARITIES IN THEIR PERIODS AND LUSTRE WHEN BRIGHT-
EST.-IRREGULAR AND TEMPORARY STARS.— ANCIENT CHINESE RE-
CORDS OF SEVERAL. — MISSING STARS. — DOUBLE STARS. — THEIR
CLASSIFICATION.— SPECIMIENS OF EACH CLASS.—BINARY SYSTEMs.
— REVOLUTION ROUND EACH OTHER, - DESCRIBE ELLIPTIC ORBITS
UNIDER THE NIEWTONIAN LAW OF GRAVITY..— ELEMENTS OF ORBITS
OF SEVERAL.- ACTUAL DIMENSIONS OF THEIR, ORBITS.— COLOURED
DOTUBLE STARs. – PHAENOMENON OF COMPLEMENTARY COLOURS. -
SANGUINE STARS. —DROPER MOTION OF THE STARs. –PARTLY AC-
COUNTED FOR BY A REAL MOTION OF THE SUN. — SITUATION OF
THE soilAR APEX. — AGREEMENT OF SOUTHERN AND NORTHERN
STARS IN GIVING THE SAME RESULT.—PRINCIPLES ON WEHICH THE
INVESTIGATION OF THE so.AR MOTION DEPENDS. — ABSOLUTE VE-
LOCITY OF THE SUN’s MOTION.— SUPPOSED REVOLUTION OF THE
WHOLE SIDEREAL SYSTEM ROUND A comMon CENTRE.- SYSTEMA-
TIC PARALLAX AND AIBERRATION. — EFFECT OF THE MOTION OF
LIGHT IN ALTERING THE APIPARENT PERIOD OF A BINARY STAR.
(819.) Now, for what purpose are we to suppose such magnificent bodies
scattered through the abyss of space? Surely not to illuminate our
nights, which an additional moon of the thousandth part of the size of
our own would do much better, nor to sparkle as a pageant void of mean-
ing and reality, and bewilder us among vain conjectures. Useful, it is
true, they are to man as points of exact and permanent reference; but he
must have studied astronomy to little purpose, who can suppose man to be
the only object of his Creator's care, or who does not see in the vast and
wonderful apparatus around us provision for other races of animated
beings. The planets, as we have seen, derive their light from the sun;
but that cannot be the case with the stars. These doubtless, then, are
themselves suns, and may, perhaps, each in its sphere, be the presiding
centre round which other planets, or bodies of which we can form no con-
ception from any analogy offered by our own system, may be circulating,
468 OljTLINES OF ASTRONOMY.
(820.) Analogies, however, more than conjectural, are not wanting to
indicate a correspondence between the dynamical laws which prevail in the
remote regions of the stars and those which govern the motions of our own
system. Wherever we can trace the law of periodicity — the regular re-
currence of the same phaenomena in the same times—we are strongly
impressed with the idea of rotatory or orbitual motion. Among the stars
are several which, though no way distinguishable from others by any appa-
rent change of place, nor by any difference of appearance in telescopes,
yet undergo a more or less regular periodical increase and diminution of
lustre, involving in one or two cases a complete extinction and revival.
These are called periodical stars. The longest known and one of the most
remarkable is the star Omicron, in the constellation Cetus (sometimes
called Mira Ceti), which was first noticed as variable by Fabricius in 1596.
It appears about twelve times in eleven years, or more exactly in a period
of 331° 15' 7"; remains at its greatest brightness about a fortnight, being
then on some occasions equal to a large star of the second magnitude;
decreases during about three months, till it becomes completely invisible
to the naked eye, in which state it remains about five months: and con-
tinues increasing during the remainder of its period. Such is the general
course of its phases. It does not always however return to the same
degree of brightness, nor increase and diminish by the same gradations,
neither are the successive intervals of its maxima equal. From the recent
observations and inquiries into its history by M. Argelander, the mean
period above assigned would appear to be subject to a cyclical fluctua-
tion embracing eighty-eight such periods, and having the effect of
gradually lengthening and shortening alternately those intervals to the
extent of twenty-five days one way and the other." The irregularities in
the degree of brightness attained at the maximum are probably also peri-
odical. Hevelius relates” that during the four years between October
1672 and December 1676 it did not appear at all. It was unusually
bright on October 5, 1839 (the epoch of its maximum for that year ac-
cording to M. Argelander's observations) when it exceeded a Ceti and
equall?: 3 Aurigae in lustre.
(821.) Another very remarkable periodical star is that called Algol, or
6 Persei. It is usually visible as a star of the second magnitude, and
such it continues for the space of 2* 134", when it suddenly begins to di-
minish in splendour, and in about 33 hours is reduced to the fourth mag-
nitude, at which it continues about 15". It then begins again to increase,
and in 33 hours more is restored to its usual brightness, going through all
its changes in 24 20° 48* 58°-5. This remarkable law of variation cer-
* Astronom. Nachr. No. 624. * Lalande's Astronomy, Art 794.
PERIODICAL STARS. 469
tainly appears strongly to suggest the revolution round it of some opaque
body, which, when interposed between us and Algol, cuts off a large por-
tion of its light; and this is accordingly the view taken of the matter by
Goodricke, to whom we owe the discovery of this remarkable fact,' in the
year 1782; since which time the same phaenomena have continued to be
observed, but with this remarkable additional point of interest, viz. that
the more recent observations, as compared with the earlier ones, indicate
a diminution in the periodic time. The latest observations of Argelander,
Heis, and Schmidt, even go to prove that this diminution is not uniformly
progressive, but is actually proceeding with accelerated rapidity, which
however will probably not continue, but, like other cyclical combinations
in astronomy, will by degrees relax, and then be changed into an increase,
according to laws of periodicity which, as well as their causes, remain to
be discovered. The first minimum of this star in the year 1844 occurred
on Jan. 3, at 4° 14* Greenwich mean time.”
(822.) The star 8 in the constellation Cepheus is also subject to peri-
odical variations, which, from the epoch of its first observation by Good-
ricke in 1784 to the present time, have been continued with perfect regu-
larity. Its period from minimum to minimum is 5' 8" 47" 39°5, the
first or epochal minimum for 1849 falling on Jan. 2, 3, 13m 37° M. T. at
Greenwich. The extent of its variation is from the fifth to between the
third and fourth magnitudes. Its increase is more rapid than its diminu-
tion, the interval between the minimum and maximum of its light being
only 1° 14'', while that from the maximum to the minimum is 3' 19°.
(823.) The periodical star 3 Lyrae, discovered by Goodricke also in
1784, has a period which has been usually stated at from 649 to 6' 11",
and there is no doubt that in about this interval of time its light under-
goes a remarkable diminution and recovery. The more accurate observa-
tions of M. Argelander however have led him to conclude” the true period
to be 12" 21h 53m 10°, and that in this period a double maximum and
minimum takes place, the two maxima being nearly equal and both about
* The same discovery appears to have been made nearly about the same time by
Palitzch, a farmer of Prolitz, near Dresden,_a peasant by station, an astronomer by
nature,—who, from his familiar acquaintance with the aspect of the heavens, had been
led to notice among so many thousand stars this one as distinguished from the rest by its
variation, and had ascertained its period. The same Palitzch was also the first to re-
discover the predicted comet of Halley in 1759, which he saw nearly a month before
any of the astronomers, who, armed with their telescopes, were anxiously watching
its return. These anecdotes carry us back to the era of the Chaldean shepherds.
2 Ast. Nach. No. 472. ,”
* Astron. Nachr. No. 264. See also the valuable papers by this excellent astron-
omer in A. N. Nos. 417, 455, &c.
;
470 OTITLINES OF ASTRONOMY.
the 3.4 magnitude, but the minima, considerably unequal, viz. 4:3 and
4.5m. In addition to this curious subdivision of the whole interval of
change into two semi-periods, we are presented in the case of this star
with another instance of slow alteration of period, which has all the ap-
pearance of being itself periodical. From the epoch of its discovery in
1784 to the year 1840 the period was continually lengthening, but more
and more slowly, till at the last-mentioned epoch it ceased to increase, and
has since been slowly on the decrease. As an epoch for the least or ab-
solute minimum of this star, M. Argelander's calculations enable us to
assign 1846 January 340h 9m 53° G. M. T.
(824.) Another periodical star whose changes have been carefully ob-
served is n Aquilae or Antinoi, first pointed out by Pigott in 1784 (a year
fertile in such discoveries) as belonging to that class. Its period is 7" 4"
13m 53°, the first minimum for 1849 occurring on Jan. 2, at 19, 22m 55°
G. M. T. It occupies fifty-seven hours in its increase from 5m to 4:3m,
and 115 hours in its decrease. +
(825.) These are all the variable stars which have been observed with
sufficient care and for a sufficient length of time to enable us to speak
with precision as to their periods, epochs, and phases of brightness. But
the number of those whose period is approximately or roughly known is
considerable, and of those whose change is certain, though its period and
limits are as yet unknown, still more so. The following table includes
the principal among them, though each year adds to their number :—
F-
; Star. Period. Change of Mag. Discovered by
d. dec. from to
8 Persei (Algol)........................ 2-8673 2 4. Goodricke, 1782.
A Tauri................................... 4 4 5-4 Baxendell, 1848.
Cephei................................. 5-3664 3°4. 5 Goodricke, 1784.
m Aquilae ................................. 7-1763 3°4. 4°5 || Pigott, 1784.
# Cancri R. A. (1800) =
8h 32m'5 N. P. D. 70° 15'......... 9.015 7-8 10 Hind, 1848.
Geminorum........................... 10-2 # 4-3 4-5 Schmidt, 1847.
Lyrae ................................... 12:9,119 3°4. 4-5 | Goodricke, 1784.
a Herculis .............................. 63 + 3 4 Herschel, 1796.
59 B. Scuti R. A. 1801 = |
18h 37m; N. P. D. = 95° 57*.....] 71-200 5 0 Pigott, 1795.
e Aurigae........... ...................... 250 + 3 4. Heis, 1846.
o Ceti (Mira)............................ 331-63 2 0 Fabricius, 1596.
# Serpentis R. A. 1828 =
15h 46m 45°; P. D. 74° 20' 30" | 335-i- 7 ? 0 Harding, 1826.
X Cygni...............::::::::... ......... 396-87.5 6 II Kirch, 1687.
J Hydrae (B. A. C. 4501.)............ 494-E 4 10 Maraldi, 1704.
# Cephei (B. A. C. 7582.)............ 5 or 6 years 3 6 Herschel, 1782.
34 Cygni (B. A. C. 6990.)............ 18 years —- 6 0 Janson, 1600.
* Leonis (B. A. C. 3345.)............ Many years | 6 0 Roch, 1782.
< Sagittarii.............................. Ditto 3 6 Halley, 1676.
tº Leonis...... • e º e º e e s e a e º se e s tº a º e s e e º e º a Ditto 6 0 Montanari, 1667.
WARIAJ3LE AND PERIODICAL STARS. 471

Star. Period. Change of Mag. Discovered by
d. dec. from to
m Cygni........................... * * * * * * * Many years || 4-5 5-6 | Herschel, jr., 1842?
* Virginis R. A. (1840) =
12h 3m ; N. P. D. 82°8'........... 145 days 6-7 0 Harding, 1814.
* Coronae Bor. (B. A. C. 5236).... 10% months | 6 0 Pigott, 1795.
7 Arietis (B. A. C. 581.) ............. 5 years? 6 8 Piazzi, 1798.
m Argūs .................................. Irregular 1 4 Burchell, 1827.
a Orionis ................ e - e. e. e. e. e. e. g. tº e º 'º - e. e. Ditto 1 1-2 | Herschel, jr., 1836.
a Ursae Majoris........................ Some years I-2 2 Ditto, 1846.
m Ursae Majoris...... e e º e º º e º sº e º ºs º º e º e e Ditto 1.2 2 Ditto, 1846.
8 Ursae Minoris........................ 2 or 3 years?| 2 2-3 || Struve, 1838.
a Cassiopeiae............................ 225 days? 2 2:3 IIerschel, jr., 1838.
a Hydra?................................. 29 or 30 days?| 2:3 3 Ditto, 1837.
* R. A. (1847) = 22, 58m 57-9 N.
P. D. = 80° 17' 30"................ Unknown 82 0 Hind, 1848.
# R. A. (1848) = 7h 33m 552 N.
P. D. = 66° 11'56"................ Ditto 9 0 Ditto, 1848.
* R. A. (1848) = 7h 40m 10°-3 N.
P. D. = 03° 3'," "................ Ditto 9 0 Ditto, 1848.
Near * R. A. ..." .. 1, 0-4 (1848.)
N. P. D. 100°42' 40".............. Ditto 7-8 0 Riimker.
* R. A. (1848) 14h 44m 394-6 N. P.
D. 101° 45' 25"..................... IDitto 8 9-10 || Schumacher.
& Ursae Majoris.......... tº e º e º 'º e º g º º e º 'º Many years 2? 2-3 || Matter of general
- remark.
N. B. In the above list the letters B. A. C. indicate the catalogue of the British Asso-
ciation, B. the catalogue of Bode. Numbers before the name of the constellation (as
34 Cygni) denote Flamsteed's stars. Since this table was drawn up, four additional
stars, variable from the 8th or 9th magnitude to 0, have been communicated to us by
Mr. Hind, whose places are as follows: (1.) R. A. 1 38" 24"; N. P. D. 81° 9' 39";
(2.) 4h 50m 42", 82°6' 36” (1846); (3) Sh 43* 8, 86°11' (1800); (4) 22, 12m 9, 82°
59' 24” (1800.) Mr. Hind remarks that about several variable stars some degree of
haziness is perceptible at their minimum. Have they clouds revolving round them as
planetary or cometary attendants? He also draws attention to the fact that the red
colour predominates among variable stars generally. The double star, No. 2718 of
Struve's Catalogue, R. A. 20° 34", P. D. 77°54', is stated by the author to be variable.
Captain Smyth (Celestial Cycle, i. 274) mentions also 3 Leonis and 18 Leonis as
variable, the former from 6" to 0, P = 78 days, the latter from 5m to 10m, P = 311” 23°,
but without citing any authority. Piazzi sets down 96 and 97 Virginis and 38 Herculis
as variable stars. [The blood-red star, 4" 51* 50.9°, 102° 2'4” (1850), discovered by
Mr. Hind, is stated by Schmidt (Ast. Nachr. 760) to have been seen by him 6" in Jan.
1850, and to have totally disappeared in Dec. 1850 and Jan. 1851.]
(826.) Irregularities similar to those which have been noticed in the
case of o Ceti, in respect of the maxima and minima of brightness attained
un successive periods, have been also observed in several others of the stars
in the foregoing list. 2 Cygni, for example, is stated by Cassini to have
been scarcely visible throughout the years 1699, 1700, 1701, at those
times when it was expected to be most conspicuous. No. 59 Scuti is
sometimes visible to the naked eye at its minimum, and sometimes not so,
and its maximum is also very irregular. Pigott’s variable star in Corona.
472 OUTLINES OF ASTRONOMY.
is stated by M. Argelander to vary for the most part so little that the
unaided eye can hardly decide on its maxima and minima, while yet after
the lapse of whole years of these slight fluctuations, they suddenly become
so great that the star completely vanishes. The variations of a Orionis,
which were most striking and unequivocal in the years 1836–1840,
within the years since elapsed became much less conspicuous. They
seem now (Jan. 1849) to have recommenced.
(827.) These irregularities prepare us for other phaenomena of stellar
variation, which have hitherto been reduced to no law of periodicity, and
must be looked upon, in relation to our ignorance and inexperience, as
altogether casual; or, if periodic, of periods too long to have occurred
more than once within the limits of recorded observation. The phaeno-
mena we allude to are those of Temporary Stars, which have appeared,
from time to time, in different parts of the heavens, blazing forth with
extraordinary lustre; and after remaining awhile apparently immoveable,
have died away, and left no trace. Such is the star which, suddenly ap-
pearing some time about the year 125 B.C., and which was visible in the
day-time, is said to have attracted the attention of Hipparchus, and led
him to draw up a catalogue of stars, the earliest on record. Such, too,
was the star which appeared, A. D. 389, near a Aquilae, remaining for
three weeks as bright as Venus, and disappearing entirely. In the years
945, 1264, and 1572, brilliant stars appeared in the region of the heavens
between Cepheus and Cassiopeia; and, from the imperfect account we
have of the places of the two earlier, as compared with that of the last,
which was well determined, as well as from the tolerably near coincidence
of the intervals of their appearance, we may suspect them, with Good-
ricke, to be one and the same star, with a period of 312 or perhaps of
156 years. The appearance of the star of 1572 was so sudden, that
Tycho Brahe, a celebrated Danish astronomer, returning one evening (the
11th of November) from his laboratory to his dwelling-house, was sur-
prised to find a group of country people gazing at a star, which he was
sure did not exist half an hour before. This was the star in question.
It was then as bright as Sirius, and continued to increase till it surpassed
Jupiter when brightest, and was visible at mid-day. It began to diminish
in December of the same year, and in March, 1574, had entirely disap-
peared. So, also, on the 10th of October, 1604, a star of this kind, and
not less brilliant, burst forth in the constellation of Serpentarius, which
continued visible till October, 1605.
(828) Similar phaenomena, though of a less splendid character, have
taken place more recently, as in the case of the star of the third magni-
tude discovered in 1670, by Anthelm, in the head of the Swan; which,
IRREGULAR AND TEMPORARY STARS. | 473.
after becoming completely invisible, re-appeared, and, after undergoing
one or two singular fluctuations of light, during two years, at last died
away entirely, and has not since been seen. ,
(829.) On the night of the 28th of April, 1848, Mr. Hind observed a
star of the fifth magnitude or 5:4 (very conspicuous to the naked eye) in
a part of the constellation Ophiuchus (R.A. 16' 51" 1-5. N.P.D. 102°
39'14"), where, from perfect familiarity with that region, he was certain
that up to the fifth of that month no star so bright as 9-10 m. previously
existed. Neither has any record been discovered of a star being there
observed at any previous time. From the time of its discovery it con-
tinued to diminish, without any alteration of place, and before the
advance of the season rendered further observation impracticable, was
nearly extinct. Its colour was ruddy, and was thought by many
observers to undergo remarkable changes, an effect probably of its low
situation. s
(830.) The alterations of brightness in the southern star n Argås, which
have been recorded, are very singular and surprising. In the time of Halley
(1677) it appeared as a star of the fourth magnitude. Lacaille, in 1751,
observed it of the second. In the interval from 1811 to 1815, it was again
of the fourth; and again from 1822 to 1826 of the second. On the 1st
of February, 1827, it was noticed by Mr. Burchell to have increased to
the first magnitude, and to equal a Crucis. Thence again it receded to
the second ; and so continued until the end of 1837. All at once in the
beginning of 1838 it suddenly increased in lustre so as to surpass all the
stars of the first magnitude except Sirius, Canopus, and a Centauri, which
last star it nearly equalled. Thence it again diminished, but this time
not below the first magnitude until April, 1843, when it had again
increased so as to surpass Canopus, and nearly equal Sirius in splendour.
“A strange field of speculation,” it has been remarked, “is opened by
this phaenomenon. The temporary stars heretofore recorded have all
become totally extinct. Variable stars, so far as they have been carefully
attended to, have exhibited periodical alternations, in some degree at
least regular, of splendour and comparative obscurity. But here we have
a star fitfully variable to an astonishing extent, and whose fluctuations are
spread over centuries, apparently in no settled period, and with no regu-
larity of progression. What origin can we ascribe to these sudden flashes
and relapses? What conclusions are we to draw as to the comfort or
habitability of a system depending for its supply of light and heat on so
uncertain a source?” Speculations of this kind can hardly be termed
visionary, when we consider that, from what has before been said, we are
compelled to admit a community of nature between the fixed stars and our
474 OUTLINES OF ASTRONOMY.
own sun; and when we reflect that geology testifies to the fact of exten-
sive changes having taken place at epochs of the most remote antiquity in
the climate and temperature of our globe; changes difficult to reconcile
with the operation of secondary causes, such as a different distribution of
sea and land, but which would find an easy and natural explanation in a
slow variation of the supply of light and heat afforded primarily by the Sun
itself. *
(831.) The Chinese annals of Ma-touan-lin,’ in which stand officially
recorded, though rudely, remarkable astronomical phaenomena, supply a
long list of “strange stars,” among which, though the greater part are
evidently comets, some may be recognized as belonging in all probability
to the class of Temporary Stars as above characterized. Such is that
which is recorded to have appeared in A. D. 173, between a and 3 Cen-
tauri, which (no doubt, scintillating from its low situation) exhibited
“the five colours,” and remained visible from December in that year till
July in the next. And another which these annals assign to A. D. 1011,
and which would seem to be identical with a star elsewhere referred to
A. D. 1012, “which was of extraordinary brilliancy, and remained visible
in the Southern part of the heavens during three months,” a situation.
agreeing with the Chinese record, which places it low in Sagittarius.
Among several less unequivocal is one referred to B. c. 134, in Scorpio,
which may possibly have been Hipparchus's star. None of the stars of
A. D. 389, 945, 1264, and 1572, however, are noticed in these records
It is worthy of especial notice, that all the stars of this kind on record,
of which the places are distinctly indicated, have occurred, without excep
tion, in or close upon the borders of the Milky Way, and that only within
the following semicircle, the preceding having offered no example of the
kind.
(832.) On a careful re-examination of the heavens, and a comparison
of catalogues, many stars are now found to be missing; and although
there is no doubt that these losses have arisen in the great majority of
instances from mistaken entries, and in some from planets having been
mistaken for stars, yet in some it is equally certain that there is no mis-
take in the observation or entry, and that the star has really been observed,
and as really has disappeared from the heavens. The whole subject of
variable stars is a branch of practical astronomy which has been too little
followed up, and it is precisely that in which amateurs of the science, and
* Translated by M. Edward Biot, Connoissance des Temps, 1846.
* Hind. Notices of the Astronomical Society, viii. 156, citing Hepidamnus. He places
the Chinese star of 173 B.C. between a and 3 Canis Minoris, but M. Biot distinctly says
a 3 pied oriental du Centaure.
DOUBLE STARS. 475
especially voyagers at sea, provided with only good eyes, or moderate in-
struments, might employ their time to excellent advantage. It holds out
a sure promise of rich discovery, and is one in which astronomers in
established observatories are almost of necessity precluded from taking a
part, by the nature of the observations required. Catalogues of the com-
parative brightness of the stars in each constellation have been constructed
by Sir Wm. Herschel, with the express object of facilitating these re-
searches, and the reader will find them, and a full account of his method
of comparison, in the Phil. Trans. 1796, and subsequent years.
(833.) We come now to a class of phaenomena of quite a different
character, and which give us a real and positive insight into the nature
of at least some among the stars, and enable us unhesitatingly to declare
them subject to the same dynamical laws, and obedient to the same power
of gravitation which governs our own system. Many of the stars, when
examined with telescopes, are found to be double, i. e. to consist of two
(in some cases three or more)individuals placed near together. This might
be attributed to accidental proximity, did it occur only in a few instances;
but the frequency of this companionship, the extreme closeness, and, in
many cases, the near equality of the stars so conjoined, would alone lead
to a strong suspicion of a more near and intimate relation than mere
casual juxtaposition. The bright star, Castor, for example, when much
magnified, is found to consist of two stars of nearly the third magnitude,
within 5" of each other. Stars of this magnitude, however, are not so
common in the heavens as to render it otherwise than excessively impro-
bable that, if scattered at random, they would fall so near. But this im-
probability becomes immensely increased by a consideration of the fact, that
this is only one out of a great many similar instances. Mitchell, in 1767,
applying the rules for the calculation of probabilities to the case of the
six brightest stars in the group called the Pleiades, found the odds to be
500000 to 1 against their proximity being the mere result of a random
scattering of 1500 stars (which he supposed to be the total number of
stars of that magnitude in the celestial sphere') over the heavens.
Speculating further on this, as an indication of physical connexion rather
than fortuitous assemblage, he was led to surmise the possibility, (since
converted into a certainty, but at that time, antecedent to any observation)
of the existence of compound stars revolving about one another, or rather
about their common centre of gravity. M. Struve, pursuing the same
train of thought as applied specially to the cases of double and triple
* This number is considerably too small, and in consequence, Mitchell's odds in
this case materially overrated. But enough will remain, if this be rectified, fully to
bear out his argument. Phil. Trans. vol. 57.
476 oUTLINES OF ASTRONOMY.
combinations of stars, and grounding his computations on a more perfect
enumeration of the stars visible down to the 7th magnitude, in the part
of the heavens visible at Dorpat, calculates that the odds are 9570 to 1
against any two stars, from the 1st to the 7th magnitude, inclusive, out
of the whole possible number of binary combinations then visible, falling,
(if fortuitously scattered) within 4" of each other. Now, the number
of instances of such binary combinations actually observed at the date
of this calculation was already 91, and many more have since been
added to the list. Again, he calculates that the odds against any
such stars fortuitously scattered, falling within 32” of a third, so as to
constitute a triple star, is not less than 173524 to 1. Now, four such
combinations occur in the heavens; viz. 6 Orionis, o Orionis, 11 Monoce-
rotis, and & Cancri. The conclusion of a physical connexion of some kind
or other is therefore unavoidable.
(834.) Presumptive evidence of another kind is furnished by the fol-
lowing consideration. Both a Centauri and 61 Cygni are “Double Stars.”
Both consist of two individuals, nearly equal, and separated from each
other by an interval of about a quarter of a minute. In the case of 61
Cygni, the stars exceeding the 7th magnitude, there is already a primä
facie probability of 9578 to 1, against their apparent proximity. The
two stars of a Centauri are both at least of the 2d magnitude, of which
altogether not more than about 50 or 60 exist in the whole heavens.
But, waiving this consideration, both these stars, as we have already seen,
have a proper motion, so considerable that, supposing the constituent in-
dividuals unconnected, one would speedily leave the other behind. Yet,
at the earliest dates at which they were respectively observed, these stars
were not perceived to be double, and it is only to the employment of tele-
scopes magnifying at least 8 or 10 times, that we owe the knowledge we
now possess of their being so. With such a telescope, Lacaille, in 1751,
was barely able to perceive the separation of the two constituents of a Cen-
tauri, whereas, had one of them only been affected with the observed
proper motion, they should then have been 6' asunder. In these cases,
then, some physical connexion may be regarded as proved by this fact
alone.
(835.) Sir William Herschel has enumerated upwards of 500 double
stars, of which the individuals are less than 32” asunder. M. Struve,
prosecuting the inquiry with instruments more conveniently mounted for
the purpose, and wrought to an astonishing pitch of optical perfection,
has added more than five times that number. And other observers have
extended still further the catalogue of “Double Stars,” without exhaust-
ing the fertility of the heavens. Among these are a great many, in
DOUBLE STARS, 477
which the distance between the component individuals does not exceed a
single second. They are divided into classes by M. Struve (the first living
authority in this department of Astronomy) according to the proximity
of their component individuals. The first class eomprises those only in
which the distance does not exceed 1"; the second those in which it ex-
ceeds 1", but falls short of 2"; the 3d class extends from 2" to 4" dis-
tance; the 4th from 4" to 8"; the 5th from 8" to 12"; the 6th from 12"
to 16"; the 7th from 16" to 24"; the 8th from 24" to 32". Each class
he again subdivides into two sub-classes of which the one under the ap-
pellation of conspicuous double stars (duplices lucidae) comprehends those
in which both individuals exceed the 83 magnitude, that is to say, are
separately bright enough to be easily seen in any moderately good tele-
scope. All others, in which one or both the constituents are below this
limit of easy visibility, are collected into another sub-class, which he
terms residuary (Duplices reliquae). This arrangement is so far conve-
nient, that after a little practice in the use of telescopes as applied to
such objects, it is easy to judge what optical power will probably suffice
to resolve a star of any proposed class and either sub-class, or would at
least be so if the second or residuary sub-class were further sub-divided
by placing in a third sub-class “delicate” double stars, or those in which
the companion star is so very minute as to require a high degree of optical
power to perceive it, of which instances will presently be given.
(836.) The following may be taken as specimens of each class. They
are all taken from among the lucid, or conspicuous stars, and to such of
our readers as may be in possession of telescopes, and may be disposed to
try them on such objects, will afford him a ready test of their degree of
e'ficiency. e
Class I., 0" to 1".
y Coronae Bor. m Coronae. 7 Ophiuchi. Atlas Pleiadum.
y Centauri. m Herculis. * Draconis. 4 Aquarii.
y Lupi.. X Cassiopeiae. * Ursae Majoris. 42 Comas.
e Arietis. X Ophiuchi. .x Aquila. 52 Arietis.
& Herculis. tr Lupi. to Leonis. 66 Piscium.
Class II., 1" to 2".
y Circini. & Bootis. & Ursae Majoris, 2 Camelopardi.
6 Cygni. t Cassiopeiae. tr Aquilaº. 32 Orionis.
& Chamaeleontis. * 2 Cancri. a Coronae Bor. 52 Orionis.
Class III., 2" to 4".
a Piscium. y Virginis. & Aquarii. p. Draconis.
3 Hydrae. & Serpentis. § Orionis. A Canis.
y Ceti. e Bootis. * Leonis. p Herculis.
y Leonis. e Draconis. t Trianguli. a Cassiopeiae.
y Coronae Aus. & Hydrae. * Leporis. 44 Bootis.
478 OUTLINES OF ASTRONOMY.
Class IV., 4" to 8”.
a Crucis. 6 Phoenicis. § Cephei. * Eridani, ,
a Herculis. k Cephei. * Bootis. 70 Ophiuchi.
a Geminorum. A Orionis. p Capricorni. 12 Eridani.
6 Geminorum. A Cygni. v Argüs. 32 Eridani.
§ Coronae Bor. § Bootis. o Aurigae. 95 Herculis.
- Class W., 8" to 12".
6 Orionis. & Antliae. t Orionis.
y Arietis. m Cassiopeiae. f Eridani.
y Delphini. 6 Eridani. 2 Canum Wen.
Class WI., 12" to 16”.
a Centauri. - y Volantis. k Bootis.
6 Cephei. n Lupi: . . 8 Monocerotis.
3 Scorpii. & Ursae Major. 61 Cygni.
Class VII., 16" to 24".
a Canum Ven. 6 Serpentis. \ 24 Comae.
e Normae. k Coronae Aus. 41 Draconis.
§ Piscium. X Fauri. 61 Ophiuchi.
Class VIII., 24” to 32".
3 Herculis. x Herculis. X Cygni.
m Lyrae. k Cephei. 23 Orionis.
* Cancri. Al Draconis.
(837.) Among the most remarkable triple, quadruple, or multiple stars
(for such also occur) may be enumerated,
a Andromedae. 6 Orionis. & Scorpii.
e Lyrae. p. Lupi: 11 Monocerotis.
§ Cancri. u Bootis. 12 Lyncis.
Of these q. Andromedae, w Bootis, and a Lupi, appear in telescopes, even
of considerable optical power, only as ordinary double stars; and it is only
when excellent instruments are used that their smaller companions are
subdivided and found to be in fact, extremely close double stars. Lyrae
offers the remarkable combination of a double-double star. Viewed with
Fig 111.
a telescope of low power it appears as a coarse and easily divided double
star, but on increasing the magnifying power, each individual is perceived

OF BINARY STARS. •. 479
to be beautifully and closely double, the one pair being about 23", the
other about 3" asunder. Each of the stars & Cancri, Ś Scorpii, 11 Mono-
cerotis, and 12 Lyncis consists of a principal star, closely double, and a
smaller and more distant attendant, while 0 Orionis presents the phae-
nomenon of four brilliant principal stars, of the respective 4th, 6th, 7th,
and 8th magnitudes, forming a trapezium, the longest diagonal of which
is 21”.4, and accompanied by two excessively minute and very close com-
panions (as in the annexed figure), to perceive both which is one of the
severest tests which can be applied to a telescope.
(838.) Of the “delicate” sub-class of double stars, or those consisting
of very large and conspicuous principal stars, accompanied by very minute
companions, the following specimens may suffice:
a 2 Cancri. a Polaris. k Circini. q, Virginis.
a 2 Capricorni. 6 Aquarii. k Geminorum. X Eridani.
a Indi. y Hydrae. p Persei. 16 Aurigae.
a Lyrae. Ursae Majoris. 7 Bootis. 94 Ceti.
(839.) To the amateur of Astronomy the double stars offer a subject
of very pleasing interest, as tests of the performance of his telescopes,
and by reason of the finely contrasted colours which many of them ex-
hibit, of which more hereafter. But it is the high degree of physical
interest which attaches to them, which assigns them a conspicuous place
in modern Astronomy, and justifies the minute attention and unwearied
diligence bestowed on the measurement of their angles of position and
distances, and the continual enlargement of our catalogues of them by
the discovery of new ones. It was, as we have seen, under an impression
that such combinations, if diligently observed, might afford a measure
of parallax through the periodical variations it might be expected to pro-
duce in the relative situation of the small attendant star, that Sir W.
Herschel was induced (between the years 1779 and 1784) to form the
first extensive catalogues of them, under the scrutiny of higher magni-
fying powers than had ever previously been applied to such purposes. In
the pursuit of this object, the end to which it was instituted as a means
was necessarily laid aside for a time, until the accumulation of more
abundant materials should have afforded a choice of stars favourably cir-
cumstanced for systematic observation. Epochal measures however, of
each star, were secured, and, on resuming the subject, his attention was
altogether diverted from the original object of the inquiry by phaenomena
of a very unexpected character, which at once engrossed his whole atten-
tion. Instead of finding, as he expected, that annual fluctuation to and
fro of one star of a double star with respect to the other, that alternate
annual increase and decrease of their distance and angle of position, which
the parallax of the earth's annual motion would produce,—he observed,
480 OUTLINES OF ASTRONOMY.
in many instances, a regular progressive change; in some cases bearing
chiefly on their distance,—in others on their position, and advancing
steadily in one direction, so as clearly to indicate either a real motion of
the stars themselves, or a general rectilinear motion of the sun and whole'
solar system, producing a parallax of a higher order than would arise from
the earth’s orbitual motion, and which might be called systematic
parallax.
(840.) Supposing the two stars, and also the sun, in motion independ-
ently of each other, it is clear that for the interval of several years, these
motions must be regarded as rectilinear and uniform. Hence, a very
slight acquaintance with geometry will suffice to show that the apparent
motion of one star of a double star, referred to the other as a centre, and
mapped down, as it were, on a plane in which that other shall be taken
for a fixed or zero point, can be no other than a right line. This, at
least, must be the case if the stars be independent of each other; but it
will be otherwise if they have a physical connexion, such as, for instance,
real proximity and mutual gravitation would establish. In that case, they
would describe orbits round each other, and round their common centre
of gravity; and therefore the apparent path of either, referred to the other
as fixed, instead of being a portion of a straight line, would be bent into
a curve concave towards that other. The observed motions, however, were
so slow, that many years’ observation was required to ascertain this point;
and it was not, therefore, until the year 1803, twenty-five years from the
commencement of the inquiry, that any thing like a positive conclusion
could be come to respecting the rectilinear or orbitual character of the
observed changes of position.
(84.1.) In that, and the subsequent year, it was distinctly announced
by him, in two papers, which will be found in the Transactions of the
Royal Society for those years', that there exist sidereal systems, composed
of two stars revolving about each other in regular orbits, and constituting
what may be termed binary stars, to distinguish them from double stars
generally so called, in which these physically connected stars are con
founded, perhaps, with others only optically double, or casually juxta-
posed in the heavens at different distances from the eye; whereas the in-
dividuals of a binary star are, of course, equidistant from the eye, or, at
least, cannot differ more in distance than the semi-diameter of the orbit
they describe about each other, which is quite insignificant compared with
the immense distance between them and the earth. Between fifty and
sixty instances of changes, to a greater or less amount, in the angles of
* The announcement was in fact made in 1802, but unaccompanied by the observa-
tions establishing the fact.
OF BINARY STARS. 481
position of double stars, are adduced in the memoirs above mentioned;
many of which are too decided, and too regularly progressive, to allow of
their nature being misconceived. In particular, among the more con-
spicuous stars, Castor, y Virginis, ; Ursae, 70 Ophiuchi, 6 and n Coronae,
# Bootis, n Cassiopeiae, y Leonis, & Herculis, 6 Cygni, u Bootis, s 4 and s
5 Lyrae, A. Ophiuchi, tº Draconis, and & Aquarii, are enumerated as among
the most remarkable instances of the observed motion; and to some of
them even periodic times of revolution are assigned; approximative only,
of course, and rather to be regarded as rough guesses than as results of
any exact calculation, for which the data were at the time quite inade-
quate. For instance, the revolution of Castor is set down at 334 years,
that of y Virginis at 708, and that of y Leonis at 1200 years.
(842.) Subsequent observation has fully confirmed these results. Of
all the stars above named, there is not one which is not found to be fully
entitled to be regarded as binary; and, in fact, this list comprises nearly
all the most considerable objects of that description which have yet been
detected, though (as attention has been closely drawn to the subject, and
observations have multiplied) it has, of late, received large accessions.
Upwards of a hundred double stars, certainly known to possess this cha-
racter, were enumerated by M. Mädler in 1841,' and more are emerging
into notice with every fresh mass of observations which come before the
public. They require excellent telescopes for their effective observation,
being for the most part so close as to necessitate the use of very high
magnifiers (such as would be considered extremely powerful microscopes
if employed to examine objects within our reach), to perceive an interval
between the individuals which compose them.
(843.) It may easily be supposed, that phaenomena of this kind would
not pass without attempts to connect them with dynamical theories. From
their first discovery, they were naturally referred to the agency of some
power, like that of gravitation, connecting the stars thus demonstrated to
be in a state of circulation about each other; and the extension of the
Newtonian law of gravitation to these remote systems was a step so ob-
vious, and so well warranted by our experience of its all-sufficient agency
in our own, as to have been expressly or tacitly made by every one who
has given the subject any share of his attention. We owe, however, the
first distinct system of calculation, by which the elliptic elements of the
orbit of a binary star could be deduced from observations of its angle of
position and distance at different epochs, to M. Savary, who showed” that
the motions of one of the most remarkable among them (; Ursae) were
* Dorpat Observations, vol. ix. 1840 and 1841. * Connoiss. des Temps, 1830.
31
482 OUTLINES OF ASTRONOMY.
explicable, within the limits allowable for error of observation, on the
supposition of an elliptic orbit described in the short period of 58% years.
A different process of computation conducted Professor Encke' to an
elliptic orbit for 70 Ophiuchi, described in a period of seventy-four years.
M. Mädler has especially signalized himself in this line of inquiry (see
note). Several orbits have also been calculated by Mr. Hind and Cap-
tain Smyth, and the author of these pages has himself attempted to con-
tribute his mite to these interesting investigations. The following may
be stated as the chief results which have been hitherto obtained in this
branch of astronomy: —
* Berlin Ephem. 1832.
The elements Nos. 1, 2, 3, 4 c, 5, 6 c, 7, 11 b, 12 a, are extracted from M. Mād-
ler’s synoptic view of the history of double stars in vol. ix. of the Dorpat Observations:
4 a, from the Connoiss. des Temps, 1830: 4 by, 6 b, and 11 a, from vol. v. Trans
Astron. Soc. Lond. : 6 a., from Berlin Ephemeris, 1832: No. 8, from Trans. Astron.
Soc. vol. vi. : No. 9, 11 c, 12 b, and 13 from Notices of the Astronomical Society,
vol. vii. p. 22, and viii. p. 159, and No. 10 from the author’s “Results of Astronomica,
Observations, &c. at the Cape of Good Hope,” p. 297. The X prefixed to No. 7,
denotes the number of the star in M. Struve's Dorpat Catalogue (Catalogus Novus
Stellarum Duplicium, &c. Dorpat, 1827), which contains the places for 1826 of 3112
of these objects.
The “position of the node” in col. 4, expresses the angle of position (see Art. 204)
of the line of intersection of the plane of the orbit, with the plane of the heavens on
which it is seen projected. The “inclination” in col. 6 is the inclination of these two
planes to one another. Col. 5 shows the angle actually included in the plane of the
orbit, between the line of nodes (defined as above) and the line c' apsides. The ele-
ments assigned in this table to a Leonis, ; Bootis, and Castor must be considered as
very doubtful, and the same may perhaps be said of those ascribed to u 2 Bootis, which
rest on too small an arc of the orbit, and that too imperfectly a served, to afford a
secure basis of calculation.
Period in
Perihelion.
Star's Name. *. Excentricity. Pºlº Of º Inclination. “Y. #." | By whom computed.
1. Herculis.................... 1//~189 0°44.454 39 o 26/ 262o 4? 500 53? 31'468 1829-50 | Mädler.
2. m Coronae B....... * * * * * * * * * * c 1-088 0-33760 24 18 261 21 71 8 43-246 1815-23 Ditto.
3. § Cancri ........ * * * * * * * * * * tº g g ſº 1°292 0.23486 1 28 266 0 63 17 58-910 1853-37 Ditto.
4. a. § Ursae Majoris............ 3-857 0°41640 95 22 131 38 50 40 58.262 1817-25 Savary.
4. b. Ditto. ............ 3-278 0.37770 97 47 134 22 56 6 60-720 1816-73 Herschel, junior.
4. c. Ditto. ............ 2.417 0°4.1350 98 52 130 48 54 56 61°464 1816°44. Mädler.
4. d. Ditto. tº e º ºs e º º is tº e s a 2°439 0.43148 || 95 50 128 57 52 49 61-576 1816-86 Villarceaux.
5. to Leonis...................... 0.857 0-64.338 || 135 II 185 27 46 33 82°533 1849-76 Ditto.
6. a. p. Ophiuchi................... 4’328 0.43007 | 1.47 12 125 22 46 25 73-862 I806-88 Encke.
6. b. Ditto ...... • * * * g º e je e º e º º 4-392 0-466.70 || 137 2 145 46 48 5 80°340 1807-06 Herschel, junior.
6. c. Ditto ............ tº e º ſº, º is is 4-192 0.44380 | 126 55 142 53 64. 51 92-870 1812-73 Mädler.
7. 2: 3062......................... 1-255 0.44958 15 3 137 27 35 31 94-765 1837-41 Ditto.
8. * Bootis .................. ....ſ 12°560 0.59374 || 359 59 100 59 80 5 117-140 I779-88 IHerschel, junior.
9. 6 Cygni ........ & g g º e º e tº e º is ſº tº º 1°811 0-60667 24 54 243 24 46 23 178-700 1862-87 Hind.
10. y Virginis.................... 3-580 0-87952 5 33 313 45 23 36 I 82°120 1836-43 Herschel, junior.
11. a. Castor..... ſº º e º is a tº $ tº gº tº s & © tº tº º 8-086 0-75820 58 6 97 29 70 3 25.2°660 1855-83 Ditto.
11. b. Ditto ............... tº t tº gº tº sº tº 7-008 0.797.25 23 5 87 37 70 58 232°124 1913-90 Mädler.
11. c. Ditto ..... * @ & © tº e º e º e g º ºs e º 'º e 6:300 0-24050 11 24 356 22 43 14 632°270 1699.26 Hind.
12. a. o. Coronae B.................. 3.918 0-69978 25 7 64. 38 29 29 608°450 1826-60 Mädler.
12. b. Ditto ........ & e º ſº tº e º º e tº 5-194. 0-72560 21 3 69 24 25 39 736-880 1826°48 Hind.
13. P. 2 Bootis ................... 3.218 0-84010 | 1.17 21 103 17 46 57 649.720 1852-50 Ditto.
14. a Centauri...................] 15:500 0.95000 86 7 291 22 47 56 77.000 I851°50 Jacob.
15. & Herculis.................... 1°254 0-44821 34 21 104 55 43 43 36-357 1830°48 Willarceaux.
§

484 OUTLINES OF ASTRONOMY.
(844.) Of the stars in the above list, that which has been most assidu-
ously watched, and has offered phaenomenon of the greatest interest, is
y Virginis. It is a star of the vulgar 3rd magnitude (3.08 = Photom.
3:494), and its component individuals are very nearly equal, and as it
would seem in some slight degree variable, since, according to the obser-
vations of M. Struve, the one is alternately a little greater and a little
less than the other, and occasionally exactly equal to it. It has been
known to consist of two stars since the beginning of the eighteenth cen-
tury; the distance being then between six and seven seconds, so that any
tolerably good telescope would resolve it. When observed by Herschel
in 1780, it was 5"-66, and continued to decrease gradually and regularly
till at length, in 1836, the two stars had approached so closely as to appear
perfectly round and single under the highest magnifying power which
could be applied to most excellent instruments — the great refractor at
Pulkowa alone, with a magnifying power of 1000, continuing to indicate,
by the wedge-shaped form of the disc of the star its composite nature.
By estimating the ratio of its length to its breadth and measuring the
former, M. Struve concludes that, at this epoch (1836.41), the distance
of the two stars, centre from centre, might be stated at 0"-22. From that
time the star again opened, and at present (1849) the individuals are more
than 2" asunder. This very remarkable diminution and subsequent in-
crease of distance has been accompanied by a corresponding and equally
remarkable increase and subsequent diminution of relative angular motion.
Thus, in the year 1783 the apparent angular motion hardly amounted to
half a degree per annum, while in 1830 it had increased to 5°, in 1834
to 20°, in 1835 to 40°, and about the middle of 1836 to upwards of 70°
per annum, or at the rate of a degree in five days. This is in entire con-
fermity with the principles of dynamics, which establish a necessary con-
nexion between the angular velocity and the distance, as well in the
apparent as in the real orbit of one body revolving about another under
the influence of mutual attraction; the former varying inversely as the
square of the latter, whatever be the curve described and whatever the
law of the attractive force. It fortunately happens that Bradley, in 1718,
had noticed and recorded in the margin of one of his observation books,
the apparent direction of the line of junction of the two stars, as seen on
the meridian in his transit telescope, viz., parallel te the line joining two
conspicuous stars a and 8 of the same constellation, as seen by the naked
eye. This note, rescued from oblivion by the late Professor Rigaud, has
proved of singular service in the verification of the elements above
assigned to the orbit, which represent the whole series of recorded obser-
vations that date up to the end of 1846 (comprising an angular movement
ORBITS OF BINARY STARS. 485
of nearly nine-tenths of a complete circuit), both in angle and distance,
with a degree of exactness fully equal to that of observation itself. No
doubt can, therefore, remain as to the prevalence in this remote system of
the Newtonian law of gravitation.
(845.) The observations of ; Ursae Majoris are equally well repre-
sented by M. Mädler's elements (4 c of our table,) thus fully justifying
the assumption of the Newtonian law as that which regulates the motions
of their binary systems. And even should it be the case, as M. Mädler
appears to consider, that in one instance at least (that of p Ophiuchi,)
deviations from elliptic motion, too considerable to arise from mere error
of observation, exist (a position we are by no means prepared to grant,)"
we should rather be disposed to look for the cause of such deviations in
perturbations arising (as Bessel has suggested) from the large or central
star itself being actually a close and hitherto unrecognized double star
than in any defect of generality in the Newtonian law.
(846.) If the great length of the periods of some of these bodies be
remarkable, the shortness of those of others is hardly less so. & Herculis
has already completed two revolutions since the epoch of its first discovery,
exhibiting in its course the extraordinary spectacle of a sidereal occulta-
tion, the small star having twice been completely hidden behind the large
one. n Coronae, & Cancri, and # Ursae have each performed more than
one entire circuit, and 70 Ophiuchi and y Virginis have accomplished by
far the larger portion of one in angular motion. If any doubt, therefore,
could remain as to the reality of their orbitual motions, or any idea of
explaining them by mere parallactic changes, or by any other hypothesis
than the agency of centripetal force, these facts must suffice for their com-
plete dissipation. We have the same evidence, indeed, of their rotations
about each other, that we have of those of Uranus and Neptune about the
sun ; and the correspondence between their calculated and observed places
in such very elongated ellipses, must be admitted to carry with it proof of
the prevalence of the Newtonian law of gravity in their systems, of the
very samé nature and cogency as that of the calculated and observed places
of comets round the central body of our own.
(847.) But it is not with the revolutions of bodies of a planetary or
‘p Ophiuchi belongs to the class of very unequal double stars, the magnitudes of the
individuals being 4 and 7. Such stars present difficulties in the exact measurement of
their angles of position which even yet continue to embarrass the observer, though,
owing to later improvements in the art of executing such measurements, their influ-
ence is confined within much narrower limits than in the earlier history of the subject,
in simply placing a fine single wire parallel to the line of junction of two such stars it
is easily possible to commit an error of 3° or 4°. By placing them between two paralle
•hick wires such errors are in great measure obviated.
486 oUTLINEs of ASTRONOMY.
cometary nature round a solar centre that we are now concerned; it is
with that of sun round sun — each, perhaps, at least in some binary sys-
tems where the individuals are very remote and their period of revolution
very long, accompanied with its train of planets and their satellites, closely
shrouded from our view by the splendour of their respective suns, and
crowded into a space bearing hardly a greater proportion to the enormous
interval which separates them, than the distances of the satellites of our
planets from their primaries bear to their distances from the sun itself.
A less distinctly characterized subordination would be incompatible with
the stability of their systems, and with the planetary nature of their orbits.
Unless closely nestled under the protecting wing of their immediate supe-
rior, the sweep of their other sun in its perihelion passage round their
own might carry them off, or whirl them into orbits utterly incompatible
with the conditions necessary for the existence of their inhabitants. It
must be confessed, that we have here a strangely wide and novel field
for speculative excursions, and one which it is not easy to avoid luxu-
riating in.
(848.) The discovery of the parallaxes of a Centauri and 61 Cygni,
both which are above enumerated among the “conspicuous” double stars
of the 6th class (a distinction fully merited in the case of the former by
the brilliancy of both its constituents), enables us to speak with an ap-
proach to certainty as to the absolute dimensions of both their orbits, and
thence to form a probable opinion as to the general scale on which these
astonishing systems are constructed. The distance of the two stars of 61
Cygni subtends at the earth an angle which, since the earliest micro-
metrical measures in 1781, has varied hardly half a second from a mean
value 15".5. On the other hand, the angle of position has altered since
the same epoch by nearly 50°, so that it would appear probable that the
true form of the orbit is not far from circular, its situation at right angles
to the visual line, and its periodic time probably not short of 500 years.
Now, as the ascertained parallax of this star is 0":348, which is, there-
fore, the angle the radius of the earth's orbit would subtend if equally
1emote, it follows that the mean distance between the stars is to that
radius, as 15" 5:0":348, or as 44:54:1. The orbit described by these
two stars about each other undoubtedly, therefore, greatly exceeds in
dimensions that described by Neptune about the sun. Moreover, suppo-
sing the period to be five centuries (and the distance being actually on the
increase, it can hardly be less) the general propositions laid down by
Newton', taken in conjunction with Kepler's third law, enable us to calcu-
late the sum of the masses of the two stars, which, on these data we find
‘Principia, l. i. Prop. 57, 58, 59.
COLOURED DOUBLE STARS. 487
to be 0.353, the mass of our sun being 1. The sun, therefore, is neither
vastly greater nor vastly less than the stars composing 61 Cygni.
(849.) The data in the case of a Centauri are more uncertain. Since
the year 1822, the distance has been steadily and pretty rapidly decreasing
at the rate of about half a second per annum, and that with very little
change in the angle of position. Hence, it follows evidently that the
plane of its orbit passes nearly through the earth, and (the distance about
the middle of 1834 having been 173") it is very probable that either an
occultation, like that observed in . Herculis, on a close appulse of the
two stars, will take place about the year 1867. As the observations we
possess afford no sufficient grounds for a satisfactory calculation of elliptic
elements' we must be content to assume what, at all events, they fully
justify, viz., that the major semiaxis must exceed 12", and is very pro-
bably consideral.'ſ greater. Now this with a parallax of 0".913 would
give for the real value of the semiaxis 13:15 radii of the earth's orbit, as
a minimum. The real dimensions of their ellipse, therefore, cannot be so
small as the orbit of Saturn; in all probability exceeds that of Uranus;
and may possibly be much greater than either.
(850.) The parallel between these two double stars is a remarkable one.
Owing no doubt to their comparative proximity to our system, their appa-
rent proper motions are both unusually great, and for the same reason
probably rather than owing to unusually large dimensions, their orbits
appear to us under what, for binary double stars, we must call unusually
large angles. Each consists, moreover, of stars, not very unequal in
brightness, and in each both the stars are of a high yellow approaching
to orange colour, the smaller individual, in each case, being also of a
deeper tint. Whatever the diversity, therefore, which may obtain among
other sidereal objects, these would appear to belong to the same family or
genus.”
(851.) Many of the double stars exhibit the curious and beautiful
phaenomenon of contrasted or complementary colours.” In such instances,
* Elements have been recently computed by Captain Jacob, for which see the table,
p. 483.
*Similar combinations are very numerous. Many remarkable instances occur among
the double stars catalogued by the author in the 2nd, 3rd, 4th, 6th and 9th volumes of
Trans. Roy. Ast. Soc. and in the volume of Southern observations already cited. See
Nos. 121, 375, 1066, 1907, 2030, 2146, 2244, 2772, 3853, 3395, 3998, 4000, 4055, 4196,
4210,4615, 4649, 4765, 5003, 5012, of these catalogues. The fine binary star, B. A. C.
No. 4923, has its constituents 15" apart, the one 6m. yellow, the other, 7m. orange.
* “ — other suns, perhaps,
With their attendant moons, thou wilt descry,
Communicating male and female light,
488 OUTLINES OF ASTRONOMY.
the larger star is usually of a ruddy or orange hue, while the smaller one
appears blue or green, probably in virtue of that general law of optics,
which provides, that when the retina is under the influence of excitement
by any bright, coloured light; feebler lights, which seen alone would pro-
duce no sensation but of whiteness, shall for the time appear coloured with
the tint complementary to that of the brighter. Thus a yellow colour
predominating in the light of the orighter star, that of the less bright one
in the same field of view will appear blue ; while, if the tint of the
brighter star verge to crimson, that of the other will exhibit a tendency to
green—or even appear as a vivid green, under favourable circumstances.
The former contrast is beautifully exhibited by , Cancri—the latter by y
Andromedae', both fine double stars. If, however, the coloured star be
much the less bright of the two, it will not materially affect the other.
Thus, for instance, n Cassiopeiae exhibits the beautiful combination of a
large white star, and a small one of a rich ruddy purple. It is by no
means, however, intended to say, that in all such cases one of the colours
is a mere effect of contrast, and it may be easier suggested in words, than
conceived in imagination, what variety of illumination two sums—a red
and a green, or a yellow and a blue one—must afford a planet circulating
about either; and what charming contrasts and “grateful vicissitudes,”
— a red and a green day, for instance, alternating with a white one and
with darkness, might arise from the presence or absence of one or other,
or both, above the horizon. Insulated stars of a red colour, almost as
deep, as that of blood,” occur in many parts of the heavens, but no green
or blue star (of any decided hue) has, we believe, ever been noticed un-
associated with a companion brighter than itself. Many of the red stars
are variable. -
(Which two great sexes animate the world.)
Stored in each orb, perhaps, with some that live.”
Paradise Lost, viii. 148.
* The small star of y Andromedae is close double. Both its individuals are green: a
similar combination, with even more decided colours, is presented by the double star,
h. 881.
* The following are the R. ascensions and N. P. distances for 1830, of some of the
most remarkable of these sanguine or ruby stars:— **.
R. A. N. P. D. B. A. N. P. D. R. A. N. P. D.
h. m. s. O / / / h. m. s. O / / / h. m. s. O / WP
4 40 53 61 46 21 9 48 3 | 130 47 12 20 7 8 111 50 11
4 51 51 102 2 4 10 52 10 107 24 40 21 37 18 3] 59 47 e
5 38 29 136 32 15 12 37 31 148 45 47 21 37 20 52 54 47
9 27 56 152 2 48 16 29 44 122 2 0
Of these No. 5 (in order of right ascension) is in the same field of view with a Hydrae,
and No. 9 with 3 Crucis.
No. 2 (in the same order) is variable.
PROPER MOTIONS OF THE STARS. 489
(852.) Another very interesting subject of inquiry, in the physical
history of the stars, is their proper motion. It was first noticed by
Halley, that three principal stars, Sirius, Arcturus, and Aldebaran, are
placed by Ptolemy, on the strength of observations made by Hipparchus,
130 years B. C., in latitudes respectively 20', 22", and 33' more northerly
than he actually found them in 1717." Making due allowance for the
diminution of obliquity of the ecliptic in the interval (1847 years) they
ought to have stood, if really fixed, respectively 10", 14', and 0' more
southerly. As the circumstances of the statement exclude the supposi-
tion of error of transcription in the MSS., we are necessitated to admit a
southward motion in latitude in these stars to the very considerable extent,
respectively, of 37", 42", and 33', and this is corroborated by an observa-
tion of Aldebaran at Athens, in the year A. D. 509, which star, on the
11th of March in that year, was seen immediately after its emergence
from occultation by the moon, in such a position as it could not have had
if the occultation were not nearly central. Now, from the knowledge we
have of the lunar motions, this could not have been the case had Alde-
baran at that time so much southern latitude as at present. A priori, it
might be expected that apparent motions of some kind or other should
be detected among so great a multitude of individuals scattered through
space, and with nothing to keep them fixed. Their mutual attractions
even, however inconceivably enfeebled by distance, and counteracted by
opposing attractions from opposite quarters, must in the lapse of count-
less ages produce some movements—some change of internal arrangement
—resulting from the difference of the opposing actions. And it is a fact,
that such apparent motions are really proved to exist by the exact obser
vations of modern astronomy. Thus, as we have seen, the two stars of
61 Cygni have remained constantly at the same, or very nearly the same,
distance, of 15", for at least fifty years past, although they have shifted
their local situation in the heavens, in this interval of time, through no
less than 4'23", the annual proper motion of each star being 5'-3; by
which quantity (exceeding a third of their interval) this system is every
year carried bodily along in some unknown path, by a motion which, for
many centuries, must be regarded as uniform and rectilinear. Among
stars not double, and no way differing from the rest in any other obvious
particular, s Indi* and u Cassiopeiae are to be remarked as having the
greatest proper motions of any yet ascertained, amounting respectively to
7"-74 and 3”.74 of annual displacement. And a great many others have
* Phil. Trans. 1717, vol. xxx. fo. 736.
* D'Arrest. Astr. Nachr., No. 618.
490 OUTLINES OF ASTRONOMY.
been observed to be thus constantly carried away from their places by
smaller, but not less unequivocal motions.”
(853.) Motions which require whole centuries to accumulate before
they produce changes of arrangement, such as the naked eye can detect,
though quite sufficient to destroy that idea of mathematical fixity which
precludes speculation, are yet too trifling, as far as practical applications
go, to induce a change of language, and lead us to speak of the stars in
common parlance as otherwise than fixed. Small as they are, however,
astronomers, once assured of their reality, have not been wanting in at-
tempts to explain and reduce them to general laws. No one, who reflects
with due attention on the subject, will be inclined to deny the high proba-
bility, nay certainty, that the sun as well as the stars must have a proper
motion in some direction; and the inevitable consequence of such a mo-
tion, if unparticipated by the rest, must be a slow average apparent ten-
dency of all the stars to the vanishing point of lines parallel to that
direction, and to the region which he is leaving, however greatly indi-
vidual stars might differ from such average by reason of their own pecu-
liar proper motion. This is the necessary effect of perspective; and it is
certain that it must be detected by observation, if we knew accurately the
apparent proper motions of all the stars, and if we were sure that they
were independent, i. e. that the whole firmament, or at least all that part
which we see in our own neighbourhood, were not drifting along together,
by a general set as it were, in one direction, the result of unknown pro-
cesses and slow internal changes going on in the sidereal stratum to
which our system belongs, as we see motes sailing in a current of air,
and keeping nearly the same relative situation with respect to one another.
(854.) It was on this assumption, tacitly made indeed, but necessarily
implied in every step of his reasoning, that Sir William Herschel, in
1783, on a consideration of the apparent proper motions of such stars as
could at that period be considered as tolerably (though still imperfectly)
ascertained, arrived at the conclusion that a relative motion of the sun,
among the fixed stars in the direction of a point or parallactic apex, situ-
ated near 2. Herculis, that is to say, in R. A. 17* 22n=260° 34', N. P. D.
63° 43' (1790), would account for the chief observed apparent motions,
leaving, however, some still outstanding and not explicable by this cause;
and in the same year Prevost, taking nearly the same view of the subject,
arrived at a conclusion as to the solar apex (or point of the sphere towards
which the Sun relatively advances), agreeing nearly in polar distance with
The reader may consult “a list of 314 stars having, or supposed to have, a proper
motion of not less than about 0”5 of a great circle” (per annum) by the late F. Baily,
Esq. Trans. Ast. Soc, v. p. 158.
l
MOTION OF THE SUN IN SPACE. 491
the foregoing, but differing from it about 27° in right ascension. Since
that time methods of calculation have been improved and concinnated,
our knowledge of the proper motions of the stars has been rendered more
precise, and a greater number of cases of such motions have been re-
corded. The subject has been resumed by several eminent astronomers
and mathematicians: viz. 1st, by M. Argelander, who, from the conside-
ration of the proper motions of 21 stars exceeding 1" per annum in arc,
has placed the solar apex in R. A. 256°25', N. P. D. 51°23'; from those
of 50 stars between 0".5 and 1"-0, in 255° 10' 51° 26'; and from those
of 319 stars having motions between 0°-1 and 0".5 per annum, in 261°
11' 59° 2': 2ndly, by M. Luhndahl, whose calculations, founded on the
proper motions of 147 stars, give 252°53', 75° 34'; and 3rdly, by M.
Otto Struve, whose result 261° 22', 62° 24', emerges from a very elabo-
rate discussion of the proper motions of 392 stars. All these places are
for A. D. 1790. -
(855), The most probable mean of the results obtained by these three
astronomers, is (for the same epoch) R. A. =259°9', N. P. D. 55° 23'.
Their researches, however, extending only to stars visible in European
observatories, it became a point of high interest to ascertain how far the
stars of the southern hemisphere not so visible, treated independently on
the same system of procedure, would corroborate or controvert their con-
clusion. The observations of Lacaille, at the Cape of Good Hope, in
1751 and 1752, compared with those of Mr. Johnson at St. Helena, in
1829–33, and of Henderson at the Cape in 1830 and 1831, have afforded
the means of deciding this question. The task has very recently been
executed in a masterly manner by Mr. Galloway, in a paper published in
the Philosophical Transactions for 1847 (to which we may also refer the
reader for a more particular account of the history of the subject than our
limits allow us to give.) On comparing the records, Mr. Galloway finds
eighty-one southern stars not employed in the previous investigations above
referred to, whose proper motions in the intervals elapsed appear consider-
able enough to assure us that they have not originated in error of the
earlier observations. Subjecting these to the same process of computation
he concludes for the place of the solar apex, for 1790, as follows: viz.
R. A. 260° 1', N. P. D. 55° 37', a result so nearly identical with that
afforded by the northern hemisphere, as to afford a full conviction of its
near approach to truth, and what may fairly be considered a demonstration
of the physical cause assigned. -
(856.) Of the mathematical conduct of this inquiry the nature of this
work precludes our giving any account; but as the philosophical principle
on which it is based has been misconceived, it is necessary to say a few
492 OUTLINES OF ASTRONOMY.
words in explanation of it. Almost all the greatest discoveries in astron-
omy have resulted from the consideration of what we have elsewhere
termed RESIDUAL PHAENOMENA', of a quantitative or numerical kind,
that is to say, of such portions of the numerical or quantitative results of
observation as remain outstanding and unaccounted for after subducting
and allowing for all that would result from the strict application of known
principles. It was thus that the grand discovery of the precession of the
equinoxes resulted as a residual phaenomenon, from the imperfect explana-
tion of the return of the seasons by the return of the sun to the same
apparent place among the fixed stars. Thus, also, aberration and mutation
resulted as residual phaenomena from that portion of the changes of the
apparent places of the fixed stars which was left unaccounted for by pre-
cession. And thus again the apparent proper motions of the stars are
the observed residues of their apparent movements outstanding and unac-
counted for by strict calculation of the effects of precession, nutation, and
aberration. The nearest approach which human theories can make to
perfection is to diminish this residue, this caput mortuum of observation,
as it may be considered, as much as practicable, and, if possible, to reduce
it to nothing, either by showing that something has been neglected in our
estimation of known causes, or by reasoning upon it as a new fact, and on
the principle of the inductive philosophy ascending from the effect to its
cause or causes. On the suggestion of any new cause hitherto unresorted
to for its explanation, our first object must of course be to decide whether
such a cause would produce such a result in kind: the next, to assign to
it such an intensity as shall account for the greatest possible amount of the
residual matter in hand. The proper motion of the sun being suggested
as such a cause, we have two things disposable—its direction and velocity,
both which it is evident, if they ever became known to us at all, can only
be so by the consideration of the very phaenomena in question. Our
object, of course, is to account, if possible, for the whole of the observed
proper motions by the proper assumption of these elements. If this be
impracticable, what remains unaccounted for is a residue of a more recon-
dite kind, but which, so long as it is unaccounted for, we must regard as
purely casual, seeing that, for anything we can perceive to the contrary,
it might with equal probability be one way as the other. The theory of
chances, therefore, necessitates (as it does in all such cases) the application
of a general mathematical process, known as “the method of least squares,”
which leads, as a matter of strict geometrical conclusion, to the values of
the elements sought, which, under all the circumstances, are the most
probable.
* Discourse on the Study of Natural Philosophy. Cab. Cyclopædia, No 14.
PROPER MOTION OF THE SUN. 493
(857.) This is the process resorted to by all the geometers we have
enumerated in the foregoing articles (arts. 854, 855). It gives not only
the direction in space, but also the velocity of the solar motion, estimated
on a scale conformable to that in which the velocity of the sidereal motions
to be explained are given; i. e. in seconds of arc as subtended at the
average distance of the stars concerned, by its annual motion in space.
But here a consideration occurs which tends materially to complicate the
problem, and to introduce into its solution an element depending on sup-
positions more or less arbitrary. The distance of the stars being, except
in two or three instances, unknown, we are compelled either to restrict our
inquiry to these, which are too few to ground any result on, or to make
some supposition as to the relative distances of the several stars umployed.
In this we have nothing but general probability to guide us, and two
courses only present themselves, either, 1st, To class the distances of the
stars according to their magnitudes, or apparent brightnesses, and to
institute separate and independent calculations for each class, including
stars assumed to be equidistant, or nearly so: or, 2dly, To class them
according to the observed amount of their apparent proper motions, on
the presumption that those which appear to move fastest are really nearest
to us. The former is the course pursued by M. Otto Struve, the latter
by M. Argelander. With regard to this latter principle of classification,
however, two considerations interfere with its applicability, viz. 1st that
we see the real motion of the stars foreshortened by the effect of perspec-
tive; and 2dly, that that portion of the total apparent proper motion
which arises from the real motion of the sun depends, not simply on the
absolute distance of the star from the sun, but also on its angular apparent
distance from the solar apex, being, casteris paribus, as the sine of that
angle. To execute such a classification correctly, therefore, we ought to
know both these particulars for each star. The first is evidently out of
our reach. We are therefore, for that very reason, compelled to regard it
as casual, and to assume that on the average of a great number of stars it
would be uninfluential on the result. But the second cannot be so sum-
marily disposed of. By the aid of an approximate knowledge of the solar
apex, it is true, approximate values may be found of the simply apparent
portions of the proper motions, supposing all the stars equidistant, and
these being subducted from the total observed motions, the residues might
afford ground for the classification in question." This, however, would be
* M. Argelander's classes, however, are constructed without reference to this con-
sideration, on the sole basis of the total apparent amount proper motion, and are, there-
fore, pro tanto, questionable. It is the more satisfactory then to find so considerable
an agreement among his partial results as actually obtains.
494 - OUTLINES OF ASTRONOMY.
a long, and to a certain extent precarious system of procedure. On the
other hand, the classification by apparent brightness is open to no such
difficulties, since we are fully justified in assuming that, on a general
average, the brighter stars are the nearer, and that the exceptions to this
rule are casual in that sense of the word which it always bears in such
inquiries, expressing solely our ignorance of any ground for assuming a
bias one way or other on either side of a determinate numerical rule. In
Mr. Galloway's discussion of the southern stars the consideration of dis.
tance is waived altogether, which is equivalent to an admission of complete.
ignorance on this point, as well as respecting the real directions and
velocities of the individual motions. -
(858.) The velocity of the solar motion which results from M. Otto
Struve's calculations is such as would carry it over an angular subtense
of 0°-3392 if seen at right angles from the average distance of a star of
the first magnitude. If we take, with M. Struve, senior, the parallax of
such a star as probably equal to 0"-209,' we shall at once be enabled to
compare this annual motion with the radius of the earth's orbit, the result
being 1623 of such units. The sun then advances through space (rela-
tively, at least, among the stars,) carrying with it the whole planetary
and cometary system with a velocity of 1.623 radii of the earth's orbit,
or 154,185,000 miles per annum, or 422,000 miles (that is to say,
nearly its own semi-diameter) per diem : in other words, with a velocity
a very little greater than one-fourth of the earth's annual motion in its
orbit.
(859.) Another generation of astronomers, perhaps many, must pass
away before we are in a condition to decide from a more precise and
extensive knowledge of the proper motions of the stars than we at present
possess, how far the direction and velocity above assigned to the solar
motion deviates from exactness, whether it continue uniform, and whether
it show any sign of deflection from rectilinearity; so as to hold out a
prospect of one day being enabled to trace out an arc of the solar orbit,
and to indicate the direction in which the preponderant gravitation of the
sidereal firmanent is urging the central body of our system. An analogy
for such deviation from uniformity would seem to present itself in the
alleged existence of a similar deviation in the proper motions of Sirius
and Procyon, both which stars are considered to have varied sensibly in
this respect within the limits of authentic and dependable observation.
Such, indeed, would appear to be the amount of evidence for this as a
matter of fact as to have given rise to a speculation on the probable circu-
lation of these stars round opaque (and therefore invisible) bodies at no
* Etudes d’Astronomie Stellaire, p. 107.
SPECULATIONS ON A CENTRAI, SUN. 495
great distances from them respectively, in the manner of binary stars:
[and it has been recently shown by M. Peters (Ast. Nachr. 748,) that,
in the case of Sirius, such a circulation, performed in a period of 50-093
years in an ellipse whose excentricity is 0.7994, the perihelion passage
taking place at the epoch A. D. 1701431, would reconcile in a remark-
able manner the observed anomalies, and reduce the residual motion to
uniformity.]
(860.) The whole of the reasoning upon which the determination of
the solar motion in space rests, is based upon the entire exclusion of any
law either derived from observation or assumed in theory, affecting the
amount and direction of the real motions both of the sun and stars. . It
supposes an absolute non-recognition, in those motions, of any general
directive cause, such as, for example, a common circulation of all about a
common centre. Any such limitation introduced into the conditions of
the problem of the solar motion would alter in toto both its nature and
the form of its solution. Suppose, for instance, that, conformably to the
speculations of several astronomers, the whole system of the Milky Way,
including our sun, and the stars, our more immediate neighbours, which
constitute our sidereal firmament, should have a general movement of rota-
tion in the plane of the galactic circle (any other would be exceedingly
improbable, indeed hardly reconcilable with dynamical principles,) being
held together in opposition to the centrifugal force thus generated by the
mutual gravitation of its constituent stars. Except we at the same time
admitted that the scale on which this movement proceeds is so enormous
that all the stars whose proper motions we include in our calculations go
together in a body, so far as that movement is concerned (as forming too
small an integrant portion of the whole to differ sensibly in their relation
to its central point;) we stand precluded from drawing any conclusion
whatever, not only respectiug the absolute motion of the sun, but respect.
ing even its relative movement among those stars, until we have established
some law, or at all events framed some hypothesis having the provisional
force of a law, connecting the whole, or a part of the motion of each indi-
vidual with its situation in space.
(861.) Speculations of this kind have not been wanting in astronomy,
and recently an attempt has been made by M. Mädler to assign the local
centre in space, round which the sun and stars revolve, which he places
in the group of the Pleiades, a situation in itself improbable, lying as it
does no less than 26° out of the plane of the galactic circle, out of which
plane it is almost inconceivable that any general circulation can take place.
In the present defective state of our knowledge respecting the proper
motion of the smaller stars, especially in right ascension, (an element for
496 OUTLINES OF ASTRONOMY.
the most part far less exactly ascertainable than the polar distance, or at
least which has been hitherto far less accurately ascertained,) we cannot
but regard all attempts of the kind as to a certain extent premature,
though by no means to be discouraged as forerunners of something more
decisive. The question, as a matter of fact, whether a rotation of the
galaxy in its own plane exist or not, might be at once resolved by the
assiduous observation, both in R. A. and polar distance, of a considerable
number of stars of the Milky Way, judiciously selected for the purpose,
and including all magnitudes, down to the smallest distinctly identifiable,
and capable of being observed with normal accuracy: and we would re-
commend the inquiry to the special attention of the directors of permanent
observatories, provided with adequate instrumental means, in both hemi-
spheres. Thirty or forty years of observation, perseveringly directed to
the object in view, could not fail to settle the question."
(862.) The solar motion through space, if real and not simply relative,
must give rise to uranographical corrections analogous to parallax and
aberration. The solar or systematic parallax is no other than that part
of the proper motion of each star which is simply apparent, arising from
the sun's motion, and until the distances of the stars be known, must
remain inextricably mixed up with the other or real portion. The syste-
matic aberration, amounting at its maximum (for stars 90° from the solar
apex) to about 5", displaces all the stars in great circles diverging from
that apex through angles proportional to the sines of their respective dis-
tances from it. This displacement, however, is permanent, and therefore
uncognizable by any phaenomenon, so long as the solar motion remains
invariable; but should it, in the course of ages, alter its direction and
velocity, both the direction and amount of the displacement in question
would alter with it. The change, however, would become mixed up with
other changes in the apparent proper motions of the stars, and it would
seem hopeless to attempt disentangling them.
(863.) A singular, and at first sight paradoxical effect of the progres-
sive movement of light, combined with the proper motion of the stars, is
that it alters the apparent periodic time in which the individuals of a
binary star circulate about each other.” To make this apparent, suppose
them to circulate round each other in a plane perpendicular to the visual
* An examination of the proper motions of the stars of the B. Assoc. Catal. in the
portion of the Milky Way nearest either pole (where the motion should be almost
wholly in R A) indicates no distinct symptom of such a rotation. If the question be
taken up fundamentally, it will involve a redetermination from the recorded proper
motions, both of the precession of the equinoxes and the change of obliquity of the
acliptic. -
* Astronomische Nachrichten, No. 520.
SPECULATIONS ON A CENTRAL SUN. 497
ray in a period of 10,000 days. Then if both the sun and the centre of
gravity of the binary system remained fixed in space, the relative apparent
situation of the stars would be exactly restored to its former state after
the lapse of this interval, and if the angle of position were 0° at first,
after 10,000 days it would again be so. But now suppose that the centre
of gravity of the star were in the act of receding in a direct line from
the sun, with a velocity of one-tenth part of the radius of the earth's
orbit per diem. Then at the expiration of 10,000 days it would be more
remote from us by 1000 such radii, a space which light would require 57
days to traverse. Although really, therefore, the stars would have arrived
at the position 0° at the exact expiration of 10,000 days, it would require
57 days more for the notice of that fact to reach our system. In other
words, the period would appear to us to be 10,057 days, since we could
only conclude the period to be completed, when to us, as observers, the
original angle of position was again restored. A contrary motion would
produce a contrary effect.
32
498 OUTLINES OF ASTRONOMY.
CHAPTER XVII.
OF C L U S T E R S OF ST. A R S A N D N E B U L AE .
OF CIUSTERING GROUPS OF STARS. — GLOBULAR CLUSTERS. — THEIB.
STABILITY DYNAMICALLY POSSIBLE. – LIST OF THE MOST REMARK-
A.B.I.E. — CLASSIFICATION OF NEBULAF AND CLUSTERS. — THEIR
I) ISTRIBUTION OVER THE HEAVENS. – IRREGULAR CLUSTER.S. -
RESOLVABILITY OF NEBUILAC. — THEORY OF THE FORMATION OF
CLU STERS BY NEBUILOUS SUBSIDENCE. — OF ELLIPTIC NEBULAF.
— THAT OF ANDROMEDA. – ANNULAR AND PLANETARY NEBUILAE.
— DOUBLE NEBUL.AE. — NEBULOljS STARS. — CONNEXION OF NEBUT, AE
WITH DOUBLE STARS. — INSULATED NEBULAE, OF FORMS NOT
WHOLLY IRREGULAR. — OF AMORPHOUS NEBULAE. — THEIR LAW OF
DISTRIBUTION MARKS THEM AS OlyTLIERS OF THE GALAXY. —
NEBULAE, AND NEBULOUS GROUP OF ORION.— OF ARGO. — OF SA-
{{ITTARIUS. — OF CYGNU.S. — THE MAGELLANIC CLOUDS. - SINGUIAR
*NEBULA IN THE GREATER, OF THEM. — THE ZODIACAL LIGHT. —
RHOOTING STARS.
(864.) WHEN we cast our eyes over the concave of the heavens in a clear
night, we do not fail to observe that here and there are groups of stars
which secm to be compressed together in a more condensed manner than
in the neighbouring parts, forming bright patches and clusters, which
attract attention, as if they were there brought together by some general
cause other than casual distribution. There is a group, called the Pleiades,
in which six or seven stars may be noticed, if the eye be directed full
upon it; and many more if the eye be turned carelessly aside, while the
attention is kept directed' upon the group. Telescopes show fifty or sixty
large stars thus crowded together in a very moderate space, comparatively
• It is a very remarkable fact, that the centre of the visual area is far less sensible
to feeble impressions of light, than the exterior portions of the retina. Few persons
are aware of the extent to which this comparative insensibility extends, previous to
trial. To estimate it, let the reader look alternately full at a star of the fifth magni-
tude, and beside it; or choose two equally bright, and about 3° or 4° apart, and look
full at one of them, the probability is he will see only the other. The fact accounts for
the multitude of stars with which we are impressed by a general view of the heavens;
their paucity when we come to count them.
CLUSTERS OF STARS AND NEBULAE. 499
insulated from the rest of the heavens. The constellation called Coma
Berenices is another such group, more diffused, and consisting on the
whole of larger stars.
(865.) In the constellation Cancer, there is a somewhat similar, but
less definite, luminous spot, called Praesepe, or the bee-hive, which a very
moderate telescope, – an ordinary night-glass for instance, — resolves en-
tirely into stars. In the sword-handle of Perseus, also, is another such
spot, crowded with stars, which requires rather a better telescope to resolve
into individuals, separated from each other. These are called clusters of
stars; and, whatever be their nature, it is certain that other laws of ag-
gregation subsist in these spots, than those which have determined the
scattering of stars over the general surface of the sky. This conclusion
is still more strongly pressed upon us, when we come to bring very
powerful telescopes to bear on these and similar spots. There are a
great number of objects which have been mistaken for comets, and, in
fact, have very much the appearance of comets without tails: small round,
or oval nebulous specks, which telescopes of moderate power only show
as such. Messier has given, in the Connois, des Temps for 1784, a list
of the places of 103 objects of this sort; which all those who search for
comets ought to be familiar with, to avoid being misled by their similarity
of appearance. That they are not, however, comets, their fixity suffi-
ciently proves; and when we come to examine them with instruments of
great power, — such as reflectors of eighteen inches, two feet, or more in
aperture, any such idea is completely destroyed. They are then, for the
most part, perceived to consist entirely of stars crowded together so as to
occupy almost a definite outline, and to run up to a blaze of light in the
centre, where their condensation is usually the greatest. (See fig. 1,
pl. II., which represents (somewhat rudely) the thirteenth nebula of
Messier's list (described by him as nébuleuse sans étoiles), as seen in a
reflector of 18 inches aperture and 20 feet focal length.) Many of them,
indeed, are of an exactly round figure, and convey the complete idea of a
globular space filled full of stars, insulated in the heavens, and constitut-
ing in itself a family or society apart from the rest, and subject only to
its own internal laws. It would be a vain task to attempt to count the
stars in one of these globular clusters. They are not to be reckoned by
hundreds; and on a rough calculation, grounded on the apparent intervals
between them at the borders, and the angular diameter of the whole group,
it would appear that many clusters of this description must contain at
least five thousand stars, compacted and wedged together in a round space,
whose angular diameter does not exceed eight or ten minutes; that is to
say, in an area not more than a tenth part of that covered by the moon.
500 OUTLINES OF ASTRONOMY.
(866.) Perhaps it may be thought to savour of the gigantesque to look
upon the individuals of such a group as suns like our own, and their mu-
tual distances as equal to those which separate our sun from the nearest
fixed star: yet, when we consider that their united lustre affects the eye
with a less impression of light than a star of the fourth magnitude, (for
the largest of these clusters is barely visible to the naked eye,) the idea
we are thus compelled to form of their distance from us may prepare us
for almost any estimate of their dimensions. At all events, we can
hardly look upon a group thus insulated, thus in seipso totus, teres, afque
Tofundus, as not forming a system of a peculiar and definite character.
Their round figure clearly indicates the existence of some general bond
of union in the nature of an attractive force; and, in many of them, there
is an evident acceleration in the rate of condensation as we approach the
centre, which is not referable to a merely uniform distribution of equidis-
tant stars through a globular space, but marks an intrinsic density in their
state of aggregation, greater in the centre than at the surface of the mass.
It is difficult to form any conception of the dynamical state of such a
system. On the one hand, without a rotatory motion and a centrifugal
force, it is hardly possible not to regard them as in a state of progressive
collapse. On the other, granting such a motion and such a force, we find
it no less difficult to reconcile the apparent sphericity of their form with
a rotation of the whole system round any single axis, without which in-
ternal collisions might at first sight appear to be inevitable. If we sup-
pose a globular space filled with equal stars, uniformly dispersed through
it, and very numerous, each of them attracting every other with a force
inversely as the square of the distance, the resultant force by which any
one of them (those at the surface alone excepted) will be urged, in virtue
of their joint attractions, will be directed towards the common centre of
the sphere, and will be directly as the distance therefrom. This follows
from what Newton has proved of the internal attraction of a homogeneous
sphere. (See also note on Art. 735.) Now, under such a law of force,
each particular star would describe a perfect ellipse about the common
centre of gravity as its centre, and that, in whatever plane and whatever
direction it might revolve. The condition, therefore, of a rotation of the
cluster, as a mass, about a single axis would be unnecessary. Each
ellipse, whatever might be the proportion of its axis, or the inclination
of its plane to the others, would be invariable in every particular, and all
would be described in one common period, so that at the end of every
such period, or annus magnus of the system, every star of the cluster
(except the superficial ones) would be exactly re-established in its original
position, thence to set out afresh, and run the same unvarying round for
CLASSIFICATION OF NEBULAE. 501
an indefinite succession of ages. Supposing their motions, therefore, to
be so adjusted at any one moment as that the orbits should not intersect
each other, and so that the magnitude of each star, and the sphere of its
more intense attraction, should bear but a small proportion to the distance
separating the individuals, such a system, it is obvious, might subsist, and
realize, in great measure, that abstract and ideal harmony, which Newton,
in the 89th Proposition of the First Book of the Principia, has shown
to characterize a law of force directly as the distance."
(867.) The following are the places, for 1830, of the principal of these
remarkable objects, as specimens of their class : —

R. A. N. P. D. R. A. N. P. D. R. A. N. P. D.
S. O / h. m.
h. m. S. O A h. m. 8. O f
0 16 25 163 2 1 5 9 56. 87 16 17 26 51 143 34
9 8 33 154 10 15 34, 56 127 13 17 28 42 93 8
12 47 41 159 57 16 6 55 112 33 11, 26 4. 114, 2
13 4. 30 70 55 16 23 2 102 40 18 55 49 150 14 .
13 16 38 136 35 16 35 37 53 13 21 21 43 '78 34
!
13 34 10 60 46 16 50 24 || 119 51 21 24 40 91 34 |
Of these, by far the most conspicuous and remarkable is a Centauri the
fifth of the list in order of Right Ascension. It is visible to the naked
eye as a dim round cometic object about equal to a star 4.5 m., though
probably if concentered in a single point, the impression on the eye would
be much greater. Viewed in a powerful telescope it appears as a globe
of fully 20' in diameter, very gradually increasing in brightness to the
centre, and composed of innumerable stars of the 13th and 15th magni-
tudes (the former probably being two or more of the latter closely juxta-
posed). The 11th in order of the list (R. A. 16' 35") is also visible to
the naked eye in very fine nights, between n and & Herculis, and is a
superb object in a large telescope. Both were discovered by Halley, the
former in 1677, and the latter in 1714.
(868.) It is to Sir William Herschel that we owe the most complete
analysis of the great variety of those objects which are generally classed.
under the common head of Nebulae, but which have been separated by
him into — 1st. Clusters of stars, in which the stars are clearly distin-
guishable; and these, again, into globular and irregular clusters; 2d.
Resolvable nebulae, or such as excite a suspicion that they consist of stars,
and which any increase of the optical power of the telescope may be ex-
pected to resolve into distinct stars; 3d. Nebulae, properly so called, in
which there is no appearance whatever of stars; which, again, have been
subdivided into subordinate uses, according to their brightness and size;
* See also Quarterly Review, No. 94, p. 540.
502 OUTLINES OF ASTRONOMY.
4th. Planetary nebulae; 5th. Stellar nebulae; and, 6th. Nebulous stars.
The great power of his telescope disclosed the existence of an immense
number of these objects before unknown, and showed them to be distri-
buted over the heavens, not by any means uniformly, but with a marked
preference to a certain district, extending over the northern pole of the
galactic circle, and occupying the constellations Leo, Leo Minor, the body,
tail, and hind legs of Ursa Major, Canes Wenatici, Coma Berenices, the
preceding leg of Bootes, and the head, wing, and shoulders of Virgo. In
this region, occupying about one-eighth of the whole surface of the sphere,
one-third of the entire nebulous contents of the heavens are congregated.
On the other hand, they are very sparingly scattered over the constella-
tions Aries, Taurus, the head and shoulders of Orion, Auriga, Perseus,
Camelopardalus, Draco, Hercules, the northern part of Serpentarius, the
tail of Serpens, that of Aquila, and the whole of Lyra. The hours 3, 4,
5, and 16, 17, 18, of right ascension in the northern hemisphere are sin-
gularly poor, and, on the other hand, the hours 10, 11, and 12 (but espe-
cially 12), extraordinarily rich in these objects. In the southern hemi-
sphere a much greater uniformity of distribution prevails, and with
exception of two very remarkable centres of accumulation, called the
Magellanic clouds (of which more presently), there is no very decided
tendency to their assemblage in any particular region.
(869.) Clusters of stars are either globular, such as we have already
described, or of irregular figure. These latter are, generally speaking,
less rich in stars, and especially less condensed towards the centre. They
are also less definite in outline; so that it is often not easy to say where
they terminate, or whether they are to be regarded otherwise than as
merely richer parts of the heavens than those around them.” Many,
indeed, the greater proportion of them, are situated in or close on the
borders of the Milky Way. In some of them the stars are nearly all of
a size, in others extremely different; and it is no uncommon thing to find
a very red star much brighter than the rest, occupying a conspicuous
situation in them. Sir William Herschel regards these as globular clus-
ters in a less advanced state of condensation, conceiving all such groups
as approaching, by their mutual attraction, to the globular figure, and
assembling themselves together from all the surrounding region, under
laws of which we have, it is true, no other proof than the observance of
a gradation by which their characters shade into one another, so that it is
impossible to say where one species ends and the other begins. Among
the most beautiful objects of this class is that which surrounds the star
* Crucis, set down as a nebula by Lacaille. It occupies an area of about
ºne 48th part of a square degree, and consists of about 110 stars from the
RESOLWABLE NEBULAF. - 503
7th magnitude downwards, eight of the more conspicuous of which are
coloured with various shades of red, green, and blue, so as to give to the
whole the appearance of a rich piece of jewellery.
(870.) Resolvable nebulae can, of course, only be considered as clusters
either too remote, or consisting of stars intrinsically too faint to affect us
by their individual light, unless where two or three happen to be close
enough to make a joint impression, and give the idea of a point brighter
than the rest. They are almost universally round or oval—their loose
appendages, and irregularities of form, being as it were extinguished by
the distance, and the only general figure of the more condensed parts being
discernible. It is under the appearance of objects of this character that
all the greater globular clusters exhibit themselves in telescopes of insuffi-
cent optical power to show them well; and the conclusion is obvious, that
those which the most powerful can barely render resolvable, and even
those which, with such powers as are usually applied, show no sign of
being composed of stars, would be completely resolved by a further in-
crease of optical power. In fact, this probability has almost been con-
verted into a certainty by the magnificent reflecting telescope constructed
by Lord Rosse, of six feet in aperture, which has resolved or rendered
resolvable multitudes of nebulae which had resisted all inferior powers.
The sublimity of the spectacle afforded by that instrument of some of the
larger globular and other clusters enumerated in the list given in Art. 867,
is declared by all who have witnessed it to be such as no words can express.
(871.) Although, therefore, nebulae do exist, which even in this power-
ful telescope appear as nebulae, without any sign of resolution, it may
very reasonably be doubted whether there be really any essential physical
distinction between nebulae and clusters of stars, at least in the nature of
the matter of which they consist, and whether the distinction between
such nebulae as are easily resolved, barely resolvable with excellent tele-
scopes, and altogether irresolvable with the best, be any thing else than
one of degree, arising merely from the excessive minuteness and multitude
of the stars, of which the latter, as compared with the former, consist.
The first impression which Halley, and other early discoverers of nebulous
objects received from their peculiar aspect, so different from the keen,
concentrated light of mere stars, was that of a phosphorescent vapour (like
the matter of a comet's tail) or a gaseous and (so to speak) elementary form
of luminous sidereal matter." Admitting the existence of such a medium,
dispersed in some cases irregularly through vast regions in space, in others
confined to narrower and more definite limits, Sir W. Herschel was led to
speculate on its gradual subsidence and condensation by the effect of its
* Halley, Phil. Trans., xxix. p. 390,
504 OUTLINES OF ASTRONOMY.
own gravity, into more or less regular spherical or spheroidal forms,
denser (as they must in that case be) towards the centre. Assuming that
in the progress of this subsidence, local centres of condensation, subordi-
nate to the general tendency, would not be wanting, he conceived that in
this way solid nuclei might arise, whose local gravitation still further
condensing, and so absorbing the nebulous matter, each in its immediate
neighbourhood, might ultimately become stars, and the whole nebulae
finally take on the state of a cluster of stars. Among the multitude of
nebulae revealed by his telescopes, every stage of this process might be
considered as displayed to our eyes, and in every modification of form to
which the general principle might be conceived to apply. The more or
less advanced state of a nebula towards its segregation into discrete stars
and of these stars themselves towards a denser state of aggregation round
a central nucleus, would thus be in some sort an indication of age.
Neither is there any variety of aspect which nebulae offer, which stands at
all in contradiction to this view. Even though we should feel ourselves
compelled to reject the idea of a gaseous or vaporous “nebulous matter,”
it loses little or none of its force. Subsidence, and the central aggrega-
tion consequent on subsidence, may go on quite as well among a multi-
tude of discrete bodies under the influence of mutual attraction, and
feeble or partially opposing projectile motions, as among the particles of a
gaseous fluid.
(872.) The “nebular hypothesis,” as it has been termed, and the
theory of sidereal aggregation stand, in fact, quite independent of each
other, the one as a physical conception of processes which may yet, for
aught we know, have formed part of that mysterious chain of causes and
effects antecedent to the existence of separate self-luminous solid bodies;
the other, as an application of dynamical principles to cases of a very
complicated nature no doubt, but in which the possibility or impossibility,
at least, of certain general results may be determined on perfectly legiti-
mate principles. Among a crowd of solid bodies of whatever size, ani-
mated by independent and partially opposing impulses, motions opposite
to each other must produce collision, destruction of velocity, and subsi-
dence or near approach towards the centre of preponderant attraction;
while those which conspire, or which remain outstanding after such con-
flicts, must ultimately give rise to circulation of a permanent character.
Whatever we may think of such collisions as events, there is nothing in
this conception contrary to sound mechanical principles. It will be recol-
lected that the appearance of central condensation among a multitude of
separate bodies in motion, by no means implies permanent proximity to
the centre in each ; any more than the habitually crowded State of a
THEORY OF THE FORMATION OF CLUSTERS. 505
market-place, to which a large proportion of the inhabitants of a town
frequeñtly or occasionally resort, implies the permanent residence of each
individual within its area. It is a fact that clusters thus centrally crowded
do exist, and therefore the conditions of their existence must be dynami-
cally possible, and in what has been said we may at least perceive some
glimpses of the manner in which they are so. The actual intervals be-
tween the stars, even in the most crowded parts of a resolved nebula, to
be seen at all by us, must be enormous. Ages, which to us may well
appear indefinite, may easily be conceived to pass without a single instance
of collision, in the nature of a catastrophe. Such may have gradually
become rarer as the system has emerged from what must be considered its
chaotic state, till at length, in the fulness of time, and under the pre-
arranging guidance of that DESIGN which pervades universal nature, each
individual may have taken up such a course as to annul the possibility of
further destructive interference. -
(873.) But to return from the regions of speculation to the description
of facts. Next in regularity of form to the globular clusters, whose con-
sideration has led us into this digression, are elliptic nebulae, more or less
elongated. And of these it may be generally remarked, as a fact un-
doubtedly connected in some very intimate manner with the dynamical
conditions of their subsistence, that such nebulae are, for the most part,
beyond comparison more difficult of resolution than those of globular form.
They are of all degrees of excentricity, from moderately oval forms to
ellipses so elongated as to be almost linear, which are, no doubt, edge-
views of very flat ellipsoids. In all of them the density increases towards
the centre, and as a general law it may be remarked that, so far as we
can judge from their telescopic appearance, their internal strata approach
more nearly to the spherical form than their external. Their resolva-
bility, too, is greater in the central parts, whether owing to a real supe-
riority of size in the central stars or to the greater frequency of cases of
close juxta-position of individuals, so that two or three united appear as
one. In some the condensation is slight and gradual, in others great and
sudden ; so sudden, indeed, as to offer the appearance of a dull and
blotted star, standing in the midst of a faint, nearly equable elliptic nebu-
losity, of which two remarkable specimens occur in R. A. 12* 10* 33°,
N. P. D. 41° 46', and in 13, 27m 28°, 119° 0' (1830).
(874.) The largest and finest specimens of elliptic nebulae which the
heavens afford are that in the girdle of Andromeda (near the star v of
that constellation) and that discovered in 1783, by Miss Carolina Herschel,
in R. A. 0h 39" 12", N. P. D. 116° 13'. The nebula in Andromeda
(Plate II. fig. 3.) is visible to the naked eye, and is continually mistaken
506 OljTLINES OF ASTRONOMY.
for a comet by those unacquainted with the heavens. Simon Marius,
who noticed it in 1612 (though it appears also to have been seen and
described as oval, in 995), describes its appearance as that of a candle
shining through horn, and the resemblance is not inapt. Its form, as seen
through ordinary telescopes, is a pretty long oval, increasing by insensible
gradations of brightness, at first very gradually, but at last more rapidly,
up to a central point, which, though very much brighter than the rest, is
decidedly not a star, but nebula of the same general character with the
rest in a state of extreme condensation. Casual stars are scattered over
it, but with a reflector of 18 inches in diameter, there is nothing to excite
any suspicion of its consisting of stars. Examined with instruments of
superior defining power, however, the evidence of its resolvability into
stars, may be regarded as decisive. Mr. G. P. Bond, assistant at the
observatory of Cambridge, U. S., describes and figures it as extending
nearly 23° in length, and upwards of a degree in breadth (so as to include
two other smaller adjacent nebulae), of a form, generally speaking, oval,
but with a considerably protuberant irregularity at its north following ex-
tremity, very suddenly condensed at the nucleus almost to the semblance
of a star, and though not itself clearly resolved, yet thickly sown over
with visible minute stars, so numerous as to allow of 200 being counted
within a field of 20' diameter in the richest parts. But the most remark-
able feature in his description is that of two perfectly straight, narrow,
and comparatively or totally obscure streaks which run nearly the whole
length of one side of the nebula, and (though slightly divergent from
each other) nearly parallel to its longer axis. These streaks (which
obviously indicate a stratified structure in the nebula, if, indeed, they do
not originate in the interposition of imperfectly transparent matter between
us and it) are not seen on a general and cursory view of the nebula; they
require attention to distinguish them,' and this circumstance must be borne
in mind when inspecting the very extraordinary engraving which illustrates
Mr. Bond's account. The figure given in our Plate II. fig. 3, is from a
rather hasty sketch, and makes no pretensions to exactness. A similar,
but much more strongly marked case of parallel arrangement than that
noticed by Mr. Bond in this, is one in which the two semi-ovals of an
elliptically formed nebula appear cut asunder and separated by a broad
obscure band parallel to the larger axis of the nebula, in the midst of
which a faint streak of light parallel to the sides of the cut appears, is
seen in the southern hemisphere in R. A. 13h 15" 31°, N. P. D. 132°8'
• Account of the nebula in Andromeda, by G. P. Bond, Assistant at the Cambridge
Observatory, U. S. Trans. American Acad., vol. iii. p. 80.
PLANETARY NEBULAE. - T 507
(1830) The nebulae in 12, 27m 3, 63° 5', and 12° 31' 11", 100° 40'
present analogous features. -
(875.) Annular nebulae also exist, but are among the rarest objects in
the heavens. The most conspicuous of this class is to be found almost
exactly half way between 3 and y Lyrae, and may be seen with a telescope
of moderate power. It is small and particularly well defined, so as to
have more the appearance of a flat oval solid ring than of a nebula. The
axes of the ellipse are to each other in the proportion of about 4 to 5, and
the opening occupies about half or rather more than half the diameter.
The central vacuity is not quite dark, but is filled in with faint nebula,
like a gauze stretched over a hoop. The powerful telescopes of Lord
Rosse resolve this object into excessively minute stars, and show filaments
of stars adhering to its edges." -
(876.) PLANETARY NEBUL.A. are very extraordinary objects. They
have, as their name imports, a near, in Some instances, a perfect resem-
blance to planets, presenting discs round, or slightly oval, in some quite
sharply terminated, in others a little hazy or softened at the borders.
Their light is in some perfectly equable, in others mottled and of a very
peculiar texture, as if curdled. They are comparatively rare objects, not
above four or five and twenty having been hitherto observed, and of these
nearly three-fourths are situated in the southern hemisphere. Being very
interesting objects, we subjoin a list of the most remarkable.” Among
these may be more particularly specified the sixth in order, situated in the
Cross. Its light is about equal to that of a star of the 6-7 magnitude,
its diameter about 12", its disc circular or very slightly elliptic, and with
a clear, sharp, well-defined outline, having exactly the appearance of a
planet with the exception only of its colour, which is a fine and full blue
verging somewhat upon green. And it is not a little remarkable that this
phaenomenon of a blue colour, which is so rare among stars (except when
in the immediate proximity of yellow stars) occurs, though less strikingly,
in three other objects of this class, viz. in No. 4, whose colour is sky-blue,
* The places of the annular nebulae, at present known (for 1830) are,
R. A. N. P. D. R. A. N. P. D.
1. 17h 10m 39s 1289 18' 3. 18h 47m 13s 57o 11?
2. 17 19 - 2 113 4. 20 9 33 59 57
* Places for 1830 of twelve of the most remarkable planetary nebulae.
| --
R. A. N. P. D. R. A. N. P. D. R. A. N. P. D.
h. m. S. O / h. m. s. O / h. m. S. O /
1. 7 34 2 104 20 5. 11 4 49 34 4 9. 19 34 21 104 33
2. 9 16 39 147 35 6. 11 41 56 146 14 10. 19 40 19 39 54
3. 9 59 52 129 36 7. 15 5 18 135 1 11. 20 54 53 102 2
4. 10 16 36 107 47 8. 19 10 9 83 46 12. 23 17 44 48 24 . .
508 OUTLINES OF ASTRONOMY.
and in Nos. 11 and 12, where the tint, though paler, is still evident.
Nos. 2, 7, 9, and 12, are also exceedingly characteristic objects of this
class. Nos. 3, 5, and 11 (the latter in the parallel of v Aquarii, and
about 5" preceding that star), are considerably elliptic, and (respectively)
about 38", 30" and 15" in diameter. On the disc of No 3, and very
nearly in the centre of the ellipse, is a star 9", and the texture of its light,
being velvety, or as if formed of fine dust, clearly indicates its resolvability
into stars. The largest of these objects is No. 5, situated somewhat south
of the parallel of 3 Ursae Majoris and about 12" following that star. Its
apparent diameter is 2' 40", which, supposing it placed at a distance from
us not more than that of 61 Cygni, would imply a linear one seven times
greater than that of the orbit of Neptune. The light of this stupendous
globe is perfectly equable (except just at the edge where it is slightly
softened), and of considerable brightness. Such an appearance would not
be presented by a globnlar space uniformly filled with stars or luminous
matter, which structure would necessarily give rise to an apparent increase
of brightness towards the centre in proportion to the thickness traversed
by the visual ray. We might, therefore, be induced to conclude its real
constitution to be either that of a hollow spherical shell or of a flat disc,
presented to us (by a highly improbable coincidence) in a plane precisely
perpendicular to the visual ray.
(877.) Whatever idea we may form of the real nature of such a body,
or of the planetary nebulae in general, which all agree in the absence of
central condensation, it is evident that the intrinsic splendour of their
surfaces, if continuous, must be almost infinitely less than that of the
sun. A circular portion of the sun's disc, subtending an angle of 1’,
would give a light equal to that of 780 full moons; while among all the
objects in question there is not one which can be seen with the naked eye.
M. Arago has surmised that they may possibly be envelopes shining by
reflected light, from a solar body placed in their centre, invisible to us by
the effect of its excessive distance; removing, or attempting to remove
the apparent paradox of such an explanation, by the optical principle that
an illuminated surface is equally bright at all distances, and, therefore, if
large enough to subtend a measurable angle, can be equally well seen,
whereas the eentral body, subtending no such angle, has its effect on our
sight diminished in the inverse ratio of the square of its distance." The
* With due deference to so high an authority we must demur to the conclusion.
Even supposing the envelope to reflect and scatter (equally in all directions) all the
light of the central sun, the portion of the light so scattered which would fall to our
share, could not exceed that which that sun itself would send to us by direct radiation,
£ut this, ea hypothesi, is too small to affect the eye with any luminous perception, much
DOTUBLE NEBULAE. 509
assiduous application of the immense optical powers recently brought to
bear on the heavens, will probably remove some portion of the mystery
which at present hangs about these enigmatical objects.
(878.) Double nebulae occasionally occur—and when such is the case,
the constituents most commonly belong to the class of spherical nebulae,
and are in some instances undoubtedly globular clusters. All the varieties
of double stars, in fact, as to distance, position, and relative brightness,
have their counterparts in double nebulae; besides which the varieties of
form and gradation of light in the latter afford room for combinations
peculiar to this class of objects. Though the conclusive evidence of ob-
served relative motion be yet wanting, and though from the vast scale on
which such systems are constructed, and the probable extreme slowness
of the angular motion, it may continue for ages to be so, yet it is impos-
sible, when we cast our eyes upon such objects, or on the figures which
have been given of them,' to doubt their physical connexion. The argu-
ment drawn from the comparative rarity of the objects in proportion to
the whole extent of the heavens, so cogent in the case of the double stars,
is infinitely more so in that of the double nebulae. Nothing more magni-
ficent can be presented to our consideration, than such combinations.
Their stupendous scale, the multitude of individuals they involve, the
perfect symmetry and regularity which many of them present, the utter
disregard of complication in thus heaping together system upon system,
and construction upon construction, leave us lost in wonder and admira-
tion at the evidence they afford of infinite power and unfathomable
design.
(879.) Nebulae of regular forms often stand in marked and symmetrical
relation to stars, both single and double. Thus we are occasionally pre-
sented with the beautiful and striking phaenomenon of a sharp and bril-
liant star concentrically surrounded by a perfectly circular disc or atmo-
sphere of faint light, in some cases dying away insensibly on all sides, in
others almost suddenly terminated. These are Nebulous Stars. Fine
examples of this kind are the 45th and 69th nebulae of Sir Wm. Her-
schel's fourth class” (R. A. Th 19m 8°, N. P. D. 68°45', and 3° 58° 36',
less then could it do so if spread over a surface many million times exceeding in angular
area the apparent disc of the central sun itself. (See Annuaire du Bureau des Longi-
tudes, 1842, p. 409, 410, 411.) M. Arago is expressly contending for reflected light,
If the envelope be self-luminous, his reasoning is perfectly sound.
1 Phil. Trans., 1833. Plate vii.
2 The classes here referred to are not the species described in Art. 868, but lists of
nebulae, eight in number, arranged according to brightness, size, density of clustering,
&c., in one or other of which all nebulae were originally classed by him. Class I.
contains “Bright nebulae;” II. “Faint do.;” III. “Very faint do. ;” IV. “Planetary
510 O'DTLINES OF ASTRONOMY.
59° 40'), in which stars of the 8th magnitude are surrounded by photo-
spheres of the kind above described respectively of 12” and 25" in dia-
meter. Among stars of larger magnitudes, 55 Andromedae and 8 Canum
Wenaticorum may be named as exhibiting the same phaenomenon with
more brilliancy, but perhaps with less perfect regularity.
(880.) The connexion of nebulae with double stars is in many instances
extremely remarkable. Thus in R. A. 18h 7m 1", N. P. D. 109° 56',
occurs an elliptic nebula having its longer axis about 50", in which, sym- .
metrically placed, and rather nearer the vertices than the foci of the ellipse,
are the equal individuals of a double star, each of the 10th magnitude.
In a similar combination noticed by M. Struve (in R. A. 18' 25", N. P. D.
25° 7'), the stars are unequal and situated precisely at the two extremities
of the major axis. In R. A. 13° 47m 33°, N. P. D. 129°9', an oval
nebula of 2' in diameter has very near its centre a close double star, the
individuals of which, slightly unequal, and about the 9-10 magnitude, are
not more than 2" asunder. The nucleus of Messier's 64th nebula is
“strongly suspected” to be a close double star—and several other instances
might be cited.
(881.) Among the nebulae which, though deviating more from sym-
metry of form, are yet not wanting in a certain regularity of figure, and
which seem clearly entitled to be regarded as systems of a definite nature,
however mysterious their structure and destination, by far the most re-
markable are the 27th and 51st of Messier's Catalogue." This consists
of two round or somewhat oval nebulous masses united by a short neck
of nearly the same density. Both this and the masses graduate off how-
ever into a fainter nebulous envelope which completes the figure into an
elliptic form, of which the interior masses with their connexion occupy the
lesser axis. Seen in a reflector of 18 inches in aperture, the form has
considerable regularity; and though a few stars are here and there scat-
tered over it, it is unresolved. Lord Rosse, viewing it with a reflector of
double that aperture, describes and figures it as resolved into numerous
stars with much intermixed nebula; while the symmetry of form by ren-
dering visible features too faint to be seen with inferior power, is rendered
considerably less striking, though by no means obliterated.
(882.) The 51st nebula of Messier, viewed through an 18-inch re-
flector, presents the appearance of a large and bright globular nebula,
nebulae, stars with bars, milky chevelures, short rays, remarkable shapes, &c.;” V.
* Very large nebulae;” VI. “Very compressed rich clusters;” VII. “Pretty much
compressed do.;” VIII. “Coarsely scattered clusters.”
* Place for 1830: R. A. 19h 52" 12", N. P. D. 67° 44', and R. A. 13h 22n 39°, N. P.,
T}. 41° 56'.
NEBULAE OF PECULIAR FORMS. 5}
surrounded by a ring at a considerable distance from the globe, very une-
qual in brightness in its different parts, and subdivided through about two-
fifths of its circumference as if into two laminae, one of which appears
as if turned up towards the eye out of the plane of the rest. Near it
(at about a radius of the ring distant) is a small bright round nebula.
Viewed through the 6-feet reflector of Lord Rosse the aspect is much
altered. The interior, or what appeared the upturned portion of the ring,
assumes the aspect of a nebulous coil or convolution tending in a spiral
form towards the centre, and a general tendency to a spiroid arrangement
of the streaks of nebula connecting the ring and central mass which this
power brings into view, becomes apparent, and forms a very striking
feature. The outlying nebula is also perceived to be connected by a
narrow, curved band of nebulous light with the ring, and the whole, if
not clearly resolved into stars, has a “resolvable” character which evi-
dently indicates its composition."
(883.) We come now to a class of nebulae of totally different character.
They are of a very great extent, utterly devoid of all symmetry of form,
— on the contrary, irregular and capricious in their shapes and convolu-
tions to a most extraordinary degree, and no less so in the distribution of
their light. No two of them can be said to present any similarity of
figure or aspect, but they have one important character in common.
They are all situated in, or very near, the borders of the Milky Way.
The most remote from it is that in the sword handle of Orion, which
being 20° from the galactic circle, and 15° from the visible border of the
Via Lactea, might seem to form an exception, though not a striking one.
But this very situation may be adduced as a corroboration of the general
view which this principle of localization suggests. For the place in ques.
tion is situated in the prolongation of that faint offset of the Milky Way
which we traced (Art. 787.) from a and s Persei towards Aldebaran and
the Hyades, and in the zone of Great Stars noticed in Art. 785. as an
appendage of, and probably bearing relation to that stratum.
(884.) From this it would appear to follow, almost as a matter of
course, that they must be regarded as outlying, very distant, and as it
were detached fragments of the great stratum of the Galaxy, and this
view of the subject is strengthened when we find on mapping down their
places that they may all be grouped in four great masses or nebulous
regions,— that of Orion, of Argo, of Sagittarius, and of Cygnus. And
thus, inductively, we may gather some information respecting the struc-
“This description is from the recollection of a sketch exhibited by his Lordship at
the British Association. Every astronomer must long for the publication of his own
account of the wonders disclosed by this magnificent instrument.
512 OuTLINES OF ASTRONOMY.
ture and form of the Galaxy itself, which, could we view it as a whole,
from a distance such as that which separates us from these objects, would
very probably present itself under an aspect quite as complicated and
irregular. -
(885.) The great nebula surrounding the stars marked 6 1 in the sword
handle of Orion was discovered by Huyghens in 1656, and has been re-
peatedly figured and described by astronomers since that time. Its
appearance varies greatly (as that of all nebulous objects does) with the
instrumental power applied, so that it is difficult to recognize in represent-
ations made with inferior telescopes, even principal features, to say
nothing of subordinate details. Until this became well understood, it
was supposed to have changed very materially, both in form and extent,
during the interval elapsed since its first discovery. No doubt, however,
now remains that these supposed changes have originated partly from the
cause above-mentioned, partly from the difficulty of correctly drawing,
and, still more, engraving such objects, and partly from a want of suffi-
cient care in the earlier delineators themselves in faithfully copying that
which they really did see. Our figure (Plate IV., fig. 1,) is reduced
from a larger one made under very favourable circumstances, from draw-
ings taken with an 18-inch reflector at the Cape of Good Hope, where its
meridian altitude greatly exceeds what it has at European stations. The
area occupied by this figure is about one 25th part of a square degree,
extending in R. A. (or horizontally) 2" of time, equivalent almost ex-
actly to 30' in arc, the object being very near the equator, and 24' verti-
cally, or in polar distance. The figure shows it reversed in both direc-
tions, the northern side being lowermost, and the preceding towards the
left hand. In form, the brightest portion offers a resemblance to the head
and yawning jaws of some monstrous animal, with a sort of proboscis run-
ning out from the snout. Many stars are scattered over it, which for the
most part appear to have no connexion with it, and the remarkable sex-
tuple star 6 1 Orionis, of which mention has already been made (Art.
837), occupies a most conspicuous situation close to the brightest portion,
at almost the edge of the opening of the jaws. It is remarkable, how-
ever, that within the area of the trapezium no nebula exists. The general
aspect of the less luminous and cirrous portion is simply nebulous and
irresolvable, but the brighter portion immediately adjacent to the trape-
zium, forming the square front of the head, is shown with the 18-inch
reflector broken up into masses (very imperfectly represented in the figure),
whose mottled and curdling light evidently indicates by a sort of granular
texture its consisting of stars, and when examined under the great light
of Lord Rosse's reflector, or the exquisite defining power of the great
NEBULA OF ARG.0, 513
achromatic at Cambridge, U. S., is evidently perceived to consist of clus-
tering stars. There can therefore be little doubt as to the whole consist-
ing of stars, too minute to be discerned individually even with these
powerful aids, but which become visible as points of light when closely
adjacent in the more crowded parts in the mode already more than once
suggested. -
(886.) The nebula is not confined to the limits of our figure. North-
ward of 6 about 38', and nearly on the same meridian are two stars
marked C 1 and C 2 Orionis, involved in a bright and branching nebula
of very singular form, and south of it is the start Orionis, which is also
involved in strong nebula. Careful examination with powerful telescopes
has traced out a continuity of nebulous light between the great nebula
and both these objects, and there can be little doubt that the nebulous
region extends northwards, as far as s in the belt of Orion, which is in-
volved in strong nebulosity, as well as several smaller stars in the immedi-
ate neighbourhood. Professor Bond has given a beautiful figure of the
great nebula in Trans. American Acad. of Arts and Sciences, new series,
vol. iii. - -
(887.) The remarkable variation in lustre of the bright star ºn in Argo,
has been already mentioned. This star is situated in the most condensed
region of a very extensive nebula or congeries of nebular masses, streaks
and branches, a portion of which is represented in fig. 2, Plate IV. The
whole nebula is spread over an area of fully a square degree in extent,
of which that included in the figure occupies about one-fourth, that is to
say, 28' in polar distance, and 32' of arc in R. A., the portion not in-
cluded being, though fainter, even more capriciously contorted than that
here depicted, in which it should be observed that the preceding side is
towards the right hand, and the southern uppermost. Viewed with an
18-inch reflector, no part of this strange object shows any sign of resolu-
tion into stars, nor in the brightest and most condensed portion adjacent
to the singular oval vacancy in the middle of the figure is there any of
that curdled appearance, or that tendency to break up into bright knots
with intervening darker portions which characterize the nebula of Orion,
and indicate its resolvability. The whole is situated in a very rich and
brilliant part of the Milky Way, so thickly strewed with stars (omitted
in the figure), that in the area occupied by the nebula, not less than 1200
have been actually counted, and their places in R. A. and P. D. deter.
mined. Yet it is obvious that these have no connexion whatever with
the nebula, being, in fact, only a simple continuation over it of the general
ground of the galaxy, which on an average of two hours in Right Ascen.
sion in this period of its course, contains no less than 3138 stars to the
33
514 OUTLINES OF ASTRONOMY.
square degree, all, however, distinct, and (except where the object in
question is situated) seen projected on a perfectly dark heaven, without
any appearance of intermixed nebulosity. The conclusion can hardly be
avoided, that in looking at it we see through, and beyond the Milky Way,
far out into space, through a starless region, disconnecting it altogether
from our system. “It is not easy for language to convey a full impres-
sion of the beauty and sublimity of the spectacle which this nebula
offers, as it enters the field of view of a telescope fixed in Right Ascen-
sion, by the diurnal motion, ushered in as it is by so glorious and innu-
merable a procession of stars, to which it forms a sort of climax,” and in
a part of the heavens otherwise full of interest. One other bright and
very remarkably formed nebula of considerable magnitude precedes it
nearly on the same parallel, but without any traceable connexion between
them.
(888.) The nebulous group of Sagittarius consists of several conspicuous
nebulae' of very extraordinary forms, by no means easy to give an idea of
by mere description. One of them (h, 1991*) is singularly trifid, con-
sisting of three bright and irregularly formed nebulous masses, graduating
away insensibly externally, but coming up to a great intensity of light at
their interior edges, where they enclose and surround a sort of three-forked
rift, or vacant area, abruptly and uncouthly crooked, and quite void of
nebulous light. A beautiful triple star is situated precisely on the edge
of one of these nebulous masses just where the interior vacancy forks out
two channels. A fourth nebulous mass spreads like a fan or downy plume
from a star at a little distance from the triple nebula. -
(889.) Nearly adjacent to the last described nebula, and no doubt con-
nected with it, though the connexion has not yet been traced, is situated
the 8th nebula of Messier's Catalogue. It is a collection of nebulous
folds and masses, surrounding and including a number of oval dark vacan-
cies, and in one place coming up to so great a degree of brightness, as to
offer the appearance of an elongated nucleus. Superposed upon this
nebula, and extending in one direction beyond its area, is a fine and rich
cluster of scattered stars, which seem to have no connexion with it, as the
* About R.A. 17h 52m, N.P.D. 113° 1', four nebulae, No. 41 of Sir Wm. Herschel's
4th class, and Nos. 1, 2, 3, of his 5th, all connected into one great complex nebula.-
In R.A. 17h 53m 27", N.P.D. 114° 21', the 8th, and in 18* 11", 106° 15', the 17th of
Messier's Catalogue.
* This number refers to the catalogue of nebulae in Phil. Trans., 1833. The reader
will find figures of the several nebulae of this group in that volume, pl. iv., fig. 35, in the
Author’s “Results of Observations, &c., at the Cape of Good Hope,” Plates i. fig. 1,
and ii.figs. 1 and 2, and in Mason's Memoir in the collection of the American Phil. Soc.
vol. vii. art. xiii.
THE MAGELLANIC CLOUDS. 515
nebula does not, as in the region of Orion, show any tendency to congre-
gate about the stars.
(890.) The 19th nebula of Messier's Catalogue, though some degrees
remote from the others, evidently belongs to this group. Its form is very
remarkable, consisting of two loops like capital Greek Omegas, the one
bright, the other exceedingly faint, connected at their bases by a broad
and very bright band of nebula, insulated within which by a narrow
comparatively obscure border, stands a bright, resolvable knot, or what is
probably a cluster of exceedingly minute stars. A very faint round nebula
stands in connexion with the upper or convex portion of the brighter loop.
(891.) The nebulous group of Cygnus consists of several large and
irregular nebulae, one of which passes through the double star k Cygni,
as a long, crooked narrow streak, forking out in two or three places. The
others,' observed in the first instance by Sir W. Herschel and by the
author of this work as separate nebulae, have been traced into connexion
by Mr. Mason, and shown to form part of a curious and intricate nebulous
system, consisting, 1st, of a long, narrow, curved, and forked streak, and
2dly, of a cellular effusion of great extent, in which the nebula occurs
intermixed with, and adhering to stars around the borders of the cells,
while their interior is free from nebula, and almost so from stars.
(892.) The Magellanic clouds, or the nubeculae (major and minor,) as
they are called in the celestial maps and charts, are, as their name imports,
two nebulous or cloudy masses of light, conspicuously visible to the naked
eye, in the southern hemisphere, in the appearance and brightness of their
light not unlike portions of the Milky Way of the same apparent size.
They are, generally speaking, round, or somewhat oval, and the larger, which
deviates most from the circular orm, exhibits the appearance of an axis
of light, very ill defined, and by ,o means strongly distinguished from the
general mass, which seems to open out at its extremities into somewhat oval
sweeps, constituting the preceding and following portions of its circumference.
A small patch, visibly brighter than the general light around, in its follow-
ing part, indicates to the naked eye the situation of a very remarkable
nebula (No. 30 Doradãs of Bode's catalogue,) of which more hereafter.
The greater nubecula is situated between the meridians of 4” 40" and 6*
0m and the parallels of 156° and 162° of N.P.D., and occupies an area
of about 42 square degrees. The lesser, between the meridians” 0* 28”
and 1' 15" and the parallels of 162° and 165° N.P.D. covers about ten
square degrees. Their degree of brightness may be judged of from the
effect of strong moonlight, which totally obliterates the lesser, but not
quite the greater.
* R.A. 20° 49m 20°, N.P.D. 58° 27'.
* It is laid down nearly an hour wrong in all the celestial charts and globes.
516 OUTLINES OF ASTRONOMY.
(893.) When examined through powerful telescopes, the constitution
of the nubeculae, and especially of the nubecula major, is found to be of
astonishing complexity. The general ground of both consists of large
tracts and patches of nebulosity in every stage of resolution, from light,
irresolvable with 18 inches of reflecting aperture, up to perfectly separated
stars like the Milky Way, and clustering groups sufficiently insulated and
condensed to come under the designation of irregular, and in some cases
pretty rich clusters. But besides those, there are also nebulae in abun-
dance, both regular and irregular; globular clusters in every state of
condensation; and objects of a nebulous character quite peculiar, and
which have no analogue in any other region of the heavens. Such is the
concentration of these objects, that in the area occupied by the nubecula
major, not fewer than 278 nebulae and clusters have been enumerated,
besides 50 or 60 outliers, which (considering the general barrenness in
such objects of the immediate neighbourhood) ought certainly to be
reckoned as its appendages, being about 63 per square degree, which very
far exceeds the average of any other, even the most crowded parts of the
nebulous heavens. In the nubecula minor, the concentration of such
objects is less, though still very striking, 37 having been observed within
its area, and 6 adjacent, but outlying. The nubeculae, then, combine,
each within its own area, characters which in the rest of the heavens are
no less strikingly separated,—viz., those of the galactic and the nebular
system. Globular clusters (except in one region of small extent) and
nebulae of regular elliptic forms are comparatively rare in the Milky Way,
and are found congregated in the greatest abundance in a part of the
heavens, the most remote possible from that circle; whereas, in the nube-
culae, they are indiscriminately mixed with the general starry ground, and
with irregular though small nebulae.
(894.) This combination of characters, rightly considered, is in a high
degree instructive, affording an insight into the probable comparative dis-
tance of stars and nebulae, and the real brightness of individual stars as
compared one with another. Taking the apparent semidiameter of the
nubecula major at 8°, and regarding its solid form as, roughly speaking,
spherical, its nearest and most remote parts differ in their distance from
us by a little more than a tenth part of our distance from its centre. The
brightness of objects situated in its nearer portions, therefore, cannot be
much exaggerated, nor that of its remoter much enfeebled, by their differ-
ence of distance; yet within this globular space, we have collected upwards
of 600 stars of the 7th, 8th, 9th, and 10th magnitudes, nearly 300
nebulae, and globular and other clusters, of all degrees of resolubility, and
smaller scattered stars innumerable of every inferior magnitude, from the
THE MAGELLANIC CLOUIDS. 517
10th to such as by their multitude and minuteness constitute irresolvable
nebulosity, extending over tracts of many square degrees. Were there
but one such object, it might be maintained without utter improbability
that its apparent sphericity is only an effect of foreshortening, and that in
reality a much greater proportional difference of distance between its
nearer and more remote parts exists. But such an adjustment, improba-
ble enough in one case, must be rejected as too much so for fair argument
in two. It must, therefore, be taken as a demonstrated fact, that stars of
the 7th or 8th magnitude and irresolvable nebula may co-exist within
limits of distance not differing in proportion more than as 9 to 10, a con.
clusion which must inspire some degree of caution in admitting, as certain,
many of the consequences which have been rather strongly dwelt upon in
the foregoing pages.
(895.) Immediately preceding the centre of the nubecula minor, and
undoubtedly belonging to the same group, occurs the superb globular
cluster, No. 47, Toucani of Bode, very visible to the naked eye, and one
of the finest objects of this kind in the heavens. It consists of a very
condensed, spherical mass of stars, of a pale rose-colour, concentrically
enclosed in a much less condensed globe of white ones, 15' or 20' in
diameter. This is the first in order of the list of such clusters in
Art. 867.
(896.) Within the nubecula major, as already mentioned, and faintly
visible to the naked eye, is the singular nebula (marked as the star 30
Doradūs in Bode's Catalogue) noticed by Lacaille as resembling the nu-
cleus of a small comet. It occupies about one-500th part of the whole
area of the nubecula, and is so satisfactorily represented in plate V., fig. 1,
as to render further description superfluous.
(897.) We shall conclude this chapter by the mention of two phaeno-
mena, which seem to indicate the existence of some slight degree of nebu-
losity about the sun itself, and even to place it in the list of nebulous
stars. The first is that called the zodiacal light, which may be seen any
very clear evening, soon after sunset, about the months of March, April,
and May, or at the opposite seasons before sunrise, as a cone or lenticu-
larly-shaped light, extending from the horizon obliquely upwards, and
following generally the course of the ecliptic, or rather that of the sun's
equator. The apparent angular distance of its vertex from the sun varies,
according to circumstances, from 40° to 90°, and the breadth of its base
perpendicular to its axis from 8° to 30°. It is extremely faint and ill
defined, at least in this climate, though better seen in tropical regions, but
cannot be mistaken for any atmospheric meteor or aurora borealis. It is
manifestly in the nature of a lenticularly-formed envelope, surrounding
518 - OUTI,INES OF ASTRONOMY.
the sun, and extending beyond the orbits of Mercury and Venus, and
nearly, perhaps quite, attaining that of the earth, since its vertex has been
seen fully 90° from the sun's place in a great circle. It may be conjec-
tured to be no other than the denser part of that medium, which, we
have some reason to believe, resists the motion of comets; loaded, per-
haps, with the actual materials of the tails of millions of those bodies, of
which they have been stripped in their successive perihelion passages
(Art. 566). An atmosphere of the sun, in any proper sense of the word,
it cannot be, since the existence of a gaseous envelope propagating pres-
sure from part to part; subject to mutual friction in its strata, and there.
fore rotating in the same or nearly the same time with the central body,
and of such dimensions and ellipticity, is utterly incompatible with dyna.
mical laws. If its particles have inertia, they must necessarily stand with
respect to the sun in the relation of separate and independent minute
planets, each having its own orbit, plane of motion, and periodic time.
The total mass being almost nothing compared to that of the sun, mutual
perturbation is out of the question, though collisions among such as may
cross each other's paths may operate in the course of indefinite ages to
effect a subsidence of at least some portion of it into the body of the sun
or those of the planets. -
(898.) Nothing prevents that these particles, or some among them,
may have some tangible size, and be at very great distances from each
other. Compared with planets visible in our most powerful telescopes,
rocks and stony masses of great size and weight would be but as the im-
palpable dust which a sunbeam renders visible as a sheet of light, when
streaming through a narrow chink into a dark chamber. It is a fact,
established by the most indisputable evidence, that stony masses and
lumps of iron do occasionally, and indeed by no means unfrequently, fall
upon the earth from the higher regions of our atmosphere (where it is
obviously impossible they can have been generated), and that they have
done so from the earliest times of history. Four instances are recorded
of persons being killed by their fall. A block of stone fell at Ægos
Potamos, B. c. 465, as large as two mill-stones; another at Narni, in 921,
projected, like a rock, four feet above the surface of the river, into which
it was seen to fall. The emperor Jehangire had a sword forged from a
mass of meteoric iron which fell, in 1620, at Jahlinder, in the Punjab."
Sixteen instances of the fall of stones in the British Isles are well authen-
ticated to have occurred since 1620, one of them in London. In 1803,
on the 26th of April, thousands of stones were scattered by the explosion
* See the emperor's own very remarkable account of the occurrence, translated in
Phil. Trans. 1793, p. 202.
*:
METEOROLITES AND SHOOTING STARS. 519
into fragments of a large fiery globe over a region of twenty or thirty
square miles around the town of L’Aigle, in Normandy. The fact occurred
at mid-day, and the circumstances were officially verified by a commission
of the French government.' These, and innumerable other instances,”
fully establish the general fact; and after vain attempts to account for it
by volcanic projection, either from the earth or the moon, the planetary
nature of these bodies seems at length to be almost generally admitted.
The heat which they possess when fallen, the igneous phaenomena which
accompany them, their explosion on arriving within the denser regions of
our atmosphere, &c., are all sufficiently accounted for on physical princi-
ples, by the condensation of the air before them in consequence of their
enormous velocity, and by the relations of air in a highly attenuated state
to heat.”
(899.) Besides stony and metallic masses, however, it is probable that
bodies of very different natures, or at least states of aggregation, are thus
circulating round the sun. Shooting stars, often followed by long trains
of light, and those great fiery globes, of more rare, but not very uncommon
occurrence, which are seen traversing the upper regions of our atmosphere.
sometimes leaving trains behind them, remaining for many minutes, some-
times bursting with a loud explosion, sometimes becoming quietly extinct,
may not unreasonably be presumed to be bodies extraneous to our planet,
which only become visible when in the act of grazing the confines of our
atmosphere. Among the last mentioned meteors are some which can
hardly be supposed solid masses. The remarkable meteor of Aug. 18,
1783, traversed the whole of Europe, from Shetland to Rome, with a
velocity of about 30 miles per second, at a height of 50 miles from the
surface of the earth, with a light greatly surpassing that of the full moon,
and a real diameter of fully half a mile. Yet with these vast dimensions,
it changed its form visibly, and at length quietly separated into several
distinct bodies, accompanying each other in parallel courses, and each fol-
lowed by a tail or train.
(900.) There are circumstances in the history of shooting stars, which
very strongly corroborate the idea of their extraneous or cosmical origin,
and their circulation round the sun in definite orbits. On several occa-
sions they have been observed to appear in unusual, and, indeed, astonish
* See M. Biot's report in Mém. de l’Institut. 1806.
* See a list of upwards of 200, published by Chladni, Annales du Bureau des Lon
gitudes de France, 1825.
* Edinburgh Review, Jan. 1848, p. 195. It is very remarkable that no new chemical
element has been detected in any of the numerous meteorolites which have been sub-
jected to analysis.
520 OUTLINES OF ASTRONOMY.
ing numbers, so as to convey the idea of a shower of rockets, or of snow-
flakes falling, and brilliantly illuminating the whole heavens for hours
together, and that not in one locality, but over whole continents and
oceans, and even (in one instance) in both hemispheres. Now it is ex-
tremely remarkable that, whenever this great display has been exhibited
(at least in modern times), it has uniformly happened on the might be-
tween the 12th and 13th, or on that between the 13th and 14th of No-
vember. Such cases occurred in 1799, 1823, 1832, 1833, and 1834.
On tracing back the records of similar phaenomena, it has been ascertained,
moreover, that more often those identical nights, but sometimes those
immediately adjacent, have been, time out of mind, habitually signalized
by such exhibitions. Another annually recurring epoch, in which, though
far less brilliant, the display of meteors is more certain (for that of No-
vember is often interrupted for a great many years), is that of the 10th
of August, on which night, and on the 9th and 11th, numerous, large,
and bright shooting stars, with trains, are almost sure to be seen. Other
epochs of periodic recurrence, less marked than the above, have also been
to a certain extent established.
(901.) It is impossible to attribute such a recurrence of identical dates
of very remarkable phaenomena to accident. Annual periodicity, irre-
spective of geographical position, refers us at once to the place ocupied by
the earth in its annual orbit, and leads direct to the conclusion that at that
place the earth incurs a liability to frequent encounters or concurrences
with a stream of meteors in their progress of circulation round the sun.
Let us test this idea by pursuing it into some of its consequences. In
the first places then, supposing the earth to plunge, in its yearly circuit,
into a uniform ring of innumerable small meteor-planets, of such breadth
as would be traversed by it in one or two days; since during this small
time the motions, whether of the earth or of each individual meteor, may
be taken as uniform and rectilinear, and those of all the latter (at the
place and time) parallel, or very nearly so, it will follow that the relative
motion of the meteors referred to the earth as at rest, will be also uniform,
rectilinear, and parallel. Viewed, therefore, from the centre of the earth
(or from any point in its circumference, if we neglect the diurnal velocity
as very small compared with the annual) they will all appear to diverge
from a common point, fixed in relation to the celestial sphere, as if ema-
nating from a sidereal apex (Art. 115).
(902.) Now this is precisely what actually happens. The meteors of
the 12th–14th of November, or at least the vast majority of them, de-
scribe apparently arcs of great circles, passing through or near y Leonis.
No matter what the situation of that star with respect to the horizon or
PERIODICAL APPEARANCE OF METEORS. 521
to its east and west points may be at the time of observation, the paths
of the meteors all appear to diverge from that star. On the 9th–11th
of August the geometrical fact is the same, the apex only differing; B
Camelopardali being for that epoch the point of divergence. As we need
not suppose the meteoric ring coincident in its plane with the ecliptic,
and as for a ring of meteors we may substitute an elliptic annulus of any
reasonable excentricity, so that both the velocity and direction of each
meteor may differ to any extent from the earth's, there is nothing in the
great and obvious difference in latitude of these apices at all militating
against the conclusion.
(903.) If the meteors be uniformly distributed in such a ring or ellip-
tic annulus, the earth's encounter with them in every revolution will be
certain, if it occur once. But if the ring be broken, if it be a succession
of groups revolving in an ellipse in a period not identical with that of the
earth, years may pass without a rencontre; and when such happen, they
may differ to any extent in their intensity of character, according as richer
or poorer groups have been encountered.
(904.) No other plausible explanation of these highly characteristic
features (the annual periodicity, and divergence from a common apex,
always alike for each respective epoch) have been even attempted, and ac-
cordingly the opinion is generally gaining ground among astronomers that
shooting stars belong to their department of science, and great interest is
excited in their observation and the further development of their laws.
The most connected and systematic series of observations of them, having
for their object to trace out their relative paths with respect to the earth,
are those of Benzenberg and Brandes, who, by noting the instants and
apparent places of appearance and extinction, as well as the precise appa-
rent paths among the stars, of individual meteors, from the extremities
of a measured base line nearly 50,000 feet in length, were led to con-
clude that their heights at the instant of their appearance and disappear-
ance vary from 16 miles to 140, and their relative velocities from 18 to
36 miles per second, velocities so great as clearly to indicate an indepen-
dent planetary circulation round the sun. [A very remarkable meteor
or bolide, which appeared on the 19th August, 1847, was observed at
l)ieppe and at Paris, with sufficient precision to admit of calculation of the
elements of its orbit in absolute space. This calculation has been per-
formed by M. Petit, director of the observatory of Toulouse, and the fol-
lowing hyperbolic elements of its orbit round the sun are stated by him
(Astr. Nachr. 701) as its result; viz., Semimajor axis = –0.8240083;
excentricity = 395.130; perihelion distance = 0-95626; inclination to
plane of the earth's equator, 18° 20' 18"; ascending node on the same
522 OUTLINES OFSAST
plane, 10° 34' 48"; motion direct. According to this calcul Hl
body would have occupied no less than 37340 years in travelling from the
distance of the nearest fixed star supposed to have a parallax of 1".]
(905.) It is by no means inconceivable that the earth approaching to
such as differ but little from it in direction and velocity, may have at-
tached many of them to it as permanent Satellites, and of these there may
be some so large, and of such texture and solidity, as to shine by reflected
light, and become visible (such, at least, as are very near the earth) for a
brief moment, suffering extinction by plunging into the earth's shadow;
in other words undergoing total eclipse. Sir John Lubbock is of opinion
that such is the case, and has given geometrical formulae for calculating
their distances from observations of this nature." The observations of M.
Petit would lead us to believe in the existence of at least one such body,
revolving round the earth, as a satellite, in about 3 hours 20 minutes, and
therefore at a distance equal to 2:513 radii of the earth from its centre,
or 5000 miles above its surface.”
* Phil. Mag. Lond. Ed. Dub. 1848, p. 80.
* Comptes Rendus, Oct. 12, 1846, and Aug. 9, 1847.
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C2-1 v. cº-º-tº-: 26 /2





NATURAL UNITS OF TIME. 523
PART IV.
of THE Accoun T OF TIME.
i.
CEIAPTER XVIII.
NATURAL UNITS OF TIME.—RELATION OF THE SIDEREAL TO THE SOLAR
DAY AFFECTED BY PRECESSION.—INCOMMEN SURABILITY OF THE DA
AND YEAR.—ITS INCONVENIENCE.-HOW OBVIATED.—THE º
CAHLENIDAR. — IRREGULARITIES AT ITS FIRST INTRODUCTION. – RE-
FORMED BY AUGUSTU.S.–GREGORIAN REFORMATION.—SOLAR AND
LUNAR. CYCLES. — INDICTION.—JULIAN PERIOD. — TABLE OF CHRO-
NOLOGICAL ERA.S. - RULES FOR, CALCULATING THE DAYS ELAPSED
BETWEEN GIVEN DATES. — EQUINOCTIAL TIME.
(906.) TIME, like distance, may be measured by comparison with stan-
(lards of any length, and all that is requisite for ascertaining correctly the
length of any interval, is to be able to apply the standard to the interval
throughout its whole extent, without overlapping on the one hand, or
leaving unmeasured vacancies on the other; to determine, without the
possible error of a unit, the number of integer standards which the inter-
val admits of being interposed between its beginning and end; and to
estimate precisely the fraction, over and above an integer, which remains
when all the possible integers are subtracted. g
(907.) But though all standard units of time are equally possible, the-
oretically speaking, yet all are not, practically, equally convenient. The
solar day is a natural interval which the wants and occupations of man in
every state of society force upon him, and compel him to adopt as his
fundamental unit of time. Its length as estimated from the departure
of the sun from a given meridian, and its next return to the same, is sub-
ject, it is true, to an annual fluctuation in excess and defect of its mean
value, amounting at its maximum to full half a minute. But except for
astronomical purposes, this is too small a change to interfere in the slight-
est degree with its use, or to attract any attention, and the tacit substitu-
524 OUTLINES OF ASTRONOMY.
tion of its mean for its true (or variable) value may be considered as
having been made from the earliest ages, by the ignorance of mankind
that any such fluctuation existed.
(908.) The time occupied by one complete rotation of the earth on its
axis, or the mean' sidereal day, may be shown, on dynamical principles, to
be subject to no variation from any external cause, and although its dura-
tion would be shortened by contraction in the dimensions of the globe
itself, such as might arise from the gradual escape of its internal heat,
and consequent refrigeration and shrinking of the whole mass, yet theory,
on the one hand, has rendered it almost certain that this cause cannot have
effected any perceptible amount of change during the history of the human
race; and, on the other, the comparison of ancient and modern observa-
tions affords every corroboration to this conclusion. From such compari-
Sons, Laplace has concluded that the sidereal day has not changed by so
much as one hundredth of a second since the time of Hipparchus. The
mean sidereal day therefore possesses in perfection the essential quality of
a standard unit, that of complete invariability. The same is true of the
mean sidereal year, if estimated upon an average sufficiently large to com-
pensate the minute fluctuations arising from the periodical variations of the
major axis of the earth’s orbit due to planetary perturbation (Art. 668.)
(909.) The mean solar day is an immediate derivative of the sidereal
day and year, being connected with them by the same relation which de-
termines the synodic from the sidereal revolutions of any two planets or
other revolving bodies (Art. 418.) The exact determination of the ratio
of the sidereal to the solar day, which is a point of the utmost importance
in astronomy, is however, in some degree, complicated by the effect of
precession, which renders it necessary to distinguish between the absolute
time of the earth's rotation on its axis, (the real natural and invariable
standard of comparison,) and the mean interval between two successive
returns of a given star to the same meridian, or rather of a given
meridian to the same star, which not only differs by a minute quantity
from the sidereal day, but is actually not the same for all stars. As
this is a point to which a little difficulty of conception is apt to attach,
it will be necessary to explain it in some detail. Suppose then rº
the pole of the ecliptic, and P that of the equinoctial, A B C D the sol-
stitial and equinoctial colures at any given epoch, and P p q r the small
circle described by P about n in one revolution of the equinoxes, i. e. in
25870 years, or 9448300 solar days, all projected on the plane of the
* The true sidereal day is variable by the effect of nutation; but the variation (an
excessively minute fraction of the whole) compensates itself in a revolution of the
moon's nodes.
NATURAL UNITS OF TIME,
5
2
5
Fig. 112.
ecliptic A B C,D. Let S be a star anywhere situated on the ecliptic, or
between it and the small circle P q r. Then if the pole P were at rest, a
meridian of the earth setting out from P S C, and revolving in the direc-
tion CD, will come again to the star after the exact lapse of one sidereal
day, or one rotation of the earth on its axis. But P is not at rest. After
the lapse of one such day it will have come into the situation (suppose)
p, the vernal equinox B having retreated to b, and the colure PC having
taken up the new position p c. Now a conical movement impressed on
the axis of rotation of a globe already rotating is equivalent to a rotation
impressed on the whole globe round the axis of the cone, in addition to
that which the globe has and retains round its own independent axis of
revolution. Such a new rotation, in transferring P to p, being performed
round an axis passing through ºt, will not alter the situation of that point
of the globe which has n in its zenith. Hence it follows that p v c pass-
ing through it will be the position taken up by the meridian Prº C after
the lapse of an exact sidereal day. But this does not pass through S, but
falls short of it by the hour-angle r. p S, which is yet to be described
before the meridian comes up to the star. The meridian, then, has lost
so much on, or lagged so much behind, the star in the lapse of that in-
terval. The same is true whatever be the arc Pip. After the lapse of
any number of days, the pole being transferred to p, the spherical angle
rt p S will measure the total hour-angle which the meridian has lost on the
star. Now where S lies any where between C and r, this angle con-
tinually increases (though not uniformly), attaining 180° when p comes
to r, and still (as will appear by following out the movement beyond r)
increasing thence till it attains 360° when p has completed its circuit

526 OUTLINES OF ASTRONOMY.
Thus in a whole revolution of the equinoxes, the meridian will have lost
one exact revolution upon the star, or in 9448300 sidereal days, will have
re-attained the star only 9448299 times: in other words, the length of the
day measured by the mean of the successive arrivals of any star outside
of the circle P p q r on one and the same meridian is to the absolute time
of rotation of the earth on its axis as 9448300: 9448299, or as 1.00000011
to l.
(910.) It is otherwise of a star situated within this circle, as at 6. For
such a star the angle ºr p o, expressing the lagging of the meridian, in-
creases to a maximum for some situation of p between q and r, and
decreases again to o at r ; after which it takes an opposite direction, and
the meridian begins to get in advance of the star, and continues to get
more and more so, till p has attained some point between s and P, where
the advance is a maximum, and thence decreases again to o when p has
completed its circuit. For any star so situated, then, the mean of all the
days so estiniated through a whole period of the equinoxes is an absolute
sidereal day, as if precession had no existence.
(911.) If we compare the sun with a star situated in the ecliptic, the
sidereal year is the mean of all the intervals of its arrival at that star
throughout indefinite ages, or (without fear of sensible error) throughout
recorded history. Now, if we would calculate the synodic sidereal revo-
lution of the sun and of a meridian of the earth by reference to a star so
situated, according to the principles of Art. 418, we must proceed as fol-
lows: Let D be the length of the mean solar day (or synodic day in
question) d the mean sidereal revolution of the meridian with reference
to the same star, and y the sidereal year. Then the arcs described by the
sun and the meridian in the interval D will be respectively 360° #and
$/
360° # And since the latter of these exceeds the former by precisely
360°, we have
T)
!-360° "1360°,
$/
360 7 =
whence it follows that
*=1 + "=10027.3780,
d 3/
taking the value of the sidereal year y as given in Art. 383, viz. 365’ 6”
9- 9.6°. But, as we have seen, d is not the absolute sidereal day, but
exceeds it in the ratio 1:00000011 : 1. Hence to get the value of the
mean solar as expressed in absolute sidereal days, the number above set
down must be increased in the same ratio, which brings it to 10027.3791,
NATURAL J.XITS OF TIME. 527
which is the ratio of the solar to the sidereal day actually in tse among
astronomers.
(912.) It would be well for chronology if mankind would, or could
have contented themselves with this one invariable, natural, and conve-
nient standard in their reckoning of time. The ancient Egyptians did
so, and by their adoption of an historical and official year of 365 days
have afforded the only example of a practical chronology, free from all
obscurity or complication. But the return of the seasons, on which de-
pend all the more important arrangements and business of cultivated life,
is not conformabe to such a multiple of the diurnal unit. Their return
is regulated by the tropical year, or the interval between two successive
arrivals of the sun at the vernal equinox, which, as we have seen (Art.
383), differs from the sidereal year by reason of the motion of the equi-
noctial points. Now this motion is not absolutely uniform, because the
ecliptic, upon which it is estimated, is gradually, though very slowly,
changing its situation in space under the disturbing influence of the planets
(Art. 640.) And thus arises a variation in the tropical year, which is de-
pendent on the place of the equinox (Art. 383.) The tropical year is
actually about 4.21" shorter than it was in the time of Hipparchus. This
absence of the most essential requisite for a standard, viz. invariability,
renders it necessary, since we cannot help employing the tropical year in
our reckoning of time, to adopt an arbitrary or artificial value for it, so
near the truth, as not to admit of the accumulation of its error for several
centuries producing any practical mischief, and thus satisfying the ordi-
nary wants of civil life; while, for scientific purposes, the tropical year,
so adopted, is considered only as the representative of a certain number
of integer days and a fraction—the day being, in effect, the only standard
employed. The case is nearly analogous to the reckoning of value by
guineas and shillings, an artificial relation of the two coins being fixed by
law, near to, but scarcely ever exactly coincident with, the natural one,
determined by the relative market price of gold and silver, of which
either the one or the other — whichever is really the most invariable, or
the most in use with other nations, – may be assumed as the true theo-
retical standard of value.
(913.) The other inconvenience of the tropical year as a greater unit
is its incommensurability with the lesser. In our measure of space all
our subdivisions are into aliquot parts: a yard is three feet, a mile eight
furlongs, &c. But a year is no eacact number of days, nor an integer
number with any exact fraction, as one third or one fourth, over and above;
but the surplus is an incommensurable fraction, composed of hours,
minutes, seconds, &c., which produces the same kind of inconvenience in
528 OUTLINES OF ASTRONOMY.
in the reckoning of time that it would do in that of money, if we had
gold coins of the value of twenty-one shillings, with odd pence and far-
things, and a fraction of a farthing over. For this, however, there is no
remedy but to keep a strict register of the surplus fractions; and, when
they amount to a whole day, cast them over into the integer account.
(914.) To do this in the simplest and most convenient manner is the
object of a well-adjusted calendar. In the Gregorian calendar, which we
follow, it is accomplished with as much simplicity and neatness as the
case admits, by carrying a little farther than is done above, the principle
of an assumed or artificial year, and adopting two such years, both con-
sisting of an exact integer number of days, viz. one of 365 and the other
of 366, and laying down a simple and easily remembered rule for the
order in which these years shall succeed each other in the civil reckoning
of time, so that during the lapse of at least some thousands of years the
sum of the integer artificial, or Gregorian, years elapsed shall not differ
from the same number of real tropical years by a whole day. By this
contrivance, the equinoxes and solstices will always fall on days similarly
situated, and bearing the same name in each Gregorian year; and the
seasons will for ever correspond to the same months, instead of running
the round of the whole year, as they must do upon any other system of
reckoning, and used, in fact, to do before this was adopted as a matter of
ignorant haphazard in the Greek and Roman chronology, and of strictly
defined and superstitiously rigorous observance in the Egyptian.
(915.) The Gregorian rule is as follows:–The years are denominated
as years current (not as years elapsed) from the midnight between the
31st of December and the 1st of January immediately subsequent to the
'oirth of Christ, according to the chronological determination of that event
by Dionysius Exiguus. Every year whose number is not divisible by 4
without remainder, consists of 365 days; every year which is so divisible,
but is not divisible by 100, of 366; every year divisible by 100, but not
by 400, again of 365; and every year divisible by 400, again of 366.
For example, the year 1833 not being divisible by 4, consists of 365
days; 1836 of 366; 1800 and 1900 of 365 each; but 2000 of 366. In
order to see how near this rule will bring us to the truth, let us see what'
number of days 10000 Gregorian years will contain, beginning with the
year A. D. 1. Now, in 10000, the numbers not divisible by 4 will be # of
10000 or 7500; those divisible by 100, but not by 400, will in like manner
be # of 100, or 75; so that, in the 10000 years in question, 7575 consist
of 366, and the remaining 2425 of 365, producing in all 3652425 days,
which would give for an average of each year, one with another, 365" 2425.
The actual value of the tropical year, (art. 383) reduced into a decimal
NATURAL UNITs of TIME. 529
fraction, is 365-24224, so the error in the Gregorian rule on 10000 of
the present tropical years, is 2-6, or 2° 14" 24"; that is to say, less than
a day in 3000 years; which is more than sufficient for all human purposes,
those of the astronomer excepted, who is in no danger of being led into
error from this cause. Even this error is avoided by extending the
wording of the Gregorian rule one step farther than its contrivers probably
thought it worth while to go, and declaring that years divisible by 4000
should consist of 365 days. This would take off two integer days from
the above calculated number, and 2.5 from a larger average; making the
sum of days in 100000 Gregorian years, 36524225, which differs only by
a single day from 100000 real tropical years, such as they exist at present.
(916.) In the historical dating of events there is no year A. D. 0. The
year immediately previous to A. D. 1, is always called B. C. 1. This must
always be borne in mind in reckoning chronological and astronomical
intervals. The sum of the nominal years B. C. and A. D. must be dimin-
ished by 1. Thus, from Jan. 1, B. C. 4713, to Jan. 1, 1582, the years
elapsed are not 6295, but 6294. -
(917.) As any distance along a high road might, though in a rather
inconvenient and roundabout way, be expressed without introducing error
by setting up a series of milestones, at intervals of unequal lengths, so
that every fourth mile, for instance, should be a yard longer than the rest,
or according to any other fixed rule; taking care only to mark the stones
so as to leave room for no mistake, and to advertise all travellers of the
difference of lengths and their order of succession; so may any interval
of time be expressed correctly by stating in what Gregorian years it begins
and ends, and whereabouts in each. For this statement coupled with the
declaratory rule, enables us to say how many integer years are to be
reckoned at 365, and how many at 366 days. The latter years are called
bissextiles, or leap-years, and the surplus days thus thrown into the
reckoning are called intercalary or leap days.
(918.) If the Gregorian rule, as above stated, had always and in all
countries been known and followed, nothing would be easier than to reckon
the number of days elapsed between the present time, and any historical
recorded event. But this is not the case; and the history of the calendar,
with reference to chronology, or to the calculation of ancient observations,
may be compared to that of a clock, going regularly when left to itself,
but sometimes forgotten to be wound up; and when wound, sometimes
set forward, sometimes backward, either to serve particular purposes and
private interests, or to rectify blunders in setting. Such, at least, appears
to have been the case with the Roman calendar, in which our own origi-
nates, from the time of Numa to that of Julius Caesar, when the lunar
34
530 OUTLINES OF ASTRONOMY.
year of 13 months, or 355 days, was augmented at pleasure to correspond
to the solar, by which the seasons are determined, by the arbitrary inter-
calations of the priests, and the usurpations of the decemvirs and other
magistrates, till the confusion became inextricable. To Julius Caesar,
assisted by Sosigenes, an eminent Alexandrian astronomer and mathema-
tician, we owe the neat contrivance of the two years of 365 and 366 days,
and the insertion of one bissextile after three common years. This im-
portant change took place in the 45th year before Christ, which he ordered
to commence on the 1st of January, being the day of the new moon im-
mediately following the winter solstice of the year before. We may judge
of the state into which the reckoning of time had fallen, by the fact, that
to introduce the new system it was necessary to enact that the previous
year, 46 B. C., should consist of 445 days, a circumstance which obtained
for it the epithet of “the year of confusion.”
(919.) Had Caesar lived to carry out into practical effect, as Chief
Pontiff, his own reformation, an inconvenience would have been avoided,
which at the very outset threw the whole matter into confusion. The
words of his edict establishing the Julian system have not been handed
down to us, but it is probable that they contained some expression equi-
valent to “every fourth year,” which the priests misinterpreting after his
death to mean (according to the sacerdotal system of numeration) as
counting the leap year newly elapsed, as No. 1 of the four, intercalated
every third instead of every fourth year. This erroneous practice con-
tinued during 36 years, in which therefore 12 instead of 9 days were
intercalated, and an error of three days produced; to rectify which, Au-
gustus ordered the suspension of all intercalation during three complete
quadriennia, – thus restoring, as may be presumed his intention to have
been, the Julian dates for the future, and re-establishing the Julian
system, which was never afterwards vitiated by any error, till the epoch
when its own inherent defects gave occasion to the Gregorian reformation.
According to the Augustan reform, the years A. U. C. 761, 765, 769, &c.,
which we now call A. D. 8, 12, 16, &c., are leap years. And starting
from this as a certain fact, (for the statements of the transaction by clas-
sical authors are not so precise as to leave absolutely no doubt as to the
previous intermediate years,) astronomers and chronologists have agreed
to reckon backwards in unbroken succession on this principle, and thus to
carry the Julian chronology into past time, as if it had never suffered
such interruption, and as if it were certain (which it is not, though we con-
ceive the balance of probabilities to incline that way') that Caesar, by way
* With Scaliger, Ideler, and all the best authorities. Yet it has been argued that
Casar would naturally begin his first quadriennium with three ordinary years, defer-
LUNAR CYCLE. 531
of securing the intercalation as a matter of precedent, made his initial
year 45 B. c. a leap year. Whenever, therefore, in the relation of any
event, either in ancient history or in modern, previous to the change of
style, the time is specified in our modern nomenclature, it is always to be
understood as having been identified with the assigned date by threading
the mazes (often very tangled and obscure ones) of special and national
chronology, and referring the day of its occurrence to its place in the
Julian system so interpreted. -
(920.) Different nations in different ages of the world have of course
reckoned their time in different ways, and from different epochs, and it is
therefore a matter of great convenience that astronomers and chronologists
(as they have agreed on the uniform adoption of the Julian system of
years and months) should also agree on an epoch antecedent to them all,
to which, as to a fixed point in time, the whole list of chronological eras
can be differentially referred. Such an epoch is the noon of the 1st of
January, B.C. 4713, which is called the epoch of the Julian period, a cycle
of 7980 Julian years, to understand the origin of which we must explain
that of three subordinate cycles, from whose combination it takes its rise,
by the multiplication together of the numbers of years severally contained
in them, viz.:-the Solar and Lunar cycles, and that of the indictions.
(921.) The Solar cycle consists of 28 Julian years, after the lapse of
which the same days of the week on the Julian system would always
return to the same days of each month throughout the year. For four
such years consisting of 1461 days, which is not a multiple of 7, it is
evident that the least number of years which will fulfil this condition
must be seven times that interval, or 28 years. The place in this cycle
for any year A. D., as 1849, is found by adding 9 to the year, and divi-
ding by 28. The remainder is the number sought, 0 being counted as 28.
(922.) The Lunar cycle consists of 19 years or 235 lunations, which
differ from 19 Julian years of 365% days only by about an hour and a
half, so that, supposing the new moon to happen on the first of January,
in the first year of the cycle, it will happen on that day (or within a very
short time of its beginning or ending) again after a lapse of 19 years, and
almost certainly on that day, and within an hour and a half of the same
hour of the day, after the lapse of four such cycles, or 76 years; and all
the new moons in the interval will run on the same days of the month as
in the preceding cycle. This period of 19 years is sometimes called the
Metonic cycle, from its discoverer Meton, an Athenian mathematician, a
ring the rectification of their accumulated error to the fourth, by inserting there the
intercalary day. For the correction of Roman dates during the fifty-two years between
the Julian and Augustan reformations, see Ideler, “Handbuch der Mathematischen
und Technischen Chronologie,” which we take for our guide throughout this chapte:
532 OUTLINES OF ASTRONOMY.
discovery duly appreciated by his countrymen, as ensuring the correspon-
dence between the lunar and solar years, the former of which was followed
by the Greeks. Public honours were decreed to him for this discovery,
a circumstance very expressive of the annoyance which a lunar year of
necessity inflicts on a civilized people, to whom a regular and simple
calendar is one of the first necessities of life. The cycle of 76 years, a
great improvement on the Metonic cycle, was first proposed by Callippus,
and is therefore called the Callippic cycle. To find the place of a given
year in the lunar cycle, (or as it is called the Golden Number,) add 1 to
the number of the year A. D., and divide by 19, the remainder (or 19 if
exactly divisible) is the Golden Number. *
(923.) The cycle of the indictions is a period of 15 years used in the
courts of law and in the fiscal organization of the Roman empire, under
Constantine and his successors, and thence introduced into legal dates, as
the Golden Number, serving to determine Easter, was in to ecclesiastical
ones. To find the place of a year in the indiction cycle, add 3 and divide
by 15. The remainder (or 15 if 0 remain) is the number of the indic-
tional year.
(924.) If we multiply together the numbers 28, 19, and 15, we get
7980, and therefore, a period or cycle of 7980 years will bring round the
years of the three cycles again in the same order, so that each year shall
hold the same place in all the three cycles as the corresponding year in
the foregoing period. As none of the three numbers in question have any
common factor, it is evident that no two years in the same compound
period can agree in all the three particulars: so that to specify the numbers
of a year in each of these cycles is, in fact, to specify the year, if within
that long period; which embraces the entire of authentic chronology.
The period thus arising of 7980 Julian years, is called the Julian period,
and it has been found so useful, that the most competent authorities have
not hesitated to declare that, through its employment, light and order
were first introduced into chronology.' We owe its invention or revival
to Joseph Scaliger, who is said to have received it from the Greeks of Con-
stantinople. The first year of the current Julian period, or that of which
the number in each of the three subordinate cycles is 1, was the year 4713
B. C., and the noon of the 1st of January of that year, for the meridian
of Alexandria, is the chronological epoch, to which all historical eras are
most readily and intelligibly referred, by computing the number of integer
days intervening between that epoch and the noon (for Alexandria) of the
day, which is reckoned to be the first of the particular era in question.
The meridian of Alexandria is chosen as that to which Ptolemy refers the
commencement of the era of Nabonassar, the basis of all his calculations. '
* Ideler, Handbuch, &c., vol. 1, p. 77.
CHRONOLOGICAL ERAS. 533
(925.) Given the year of the Julian period, those of the subordinate
cycles are easily determined as above. Conversely, given the years of the
solar and lunar cycles, and of the indiction, to determine the year of the
Julian period proceed as follows: — Multiply the number of the year in
the solar cycle by 4845, in the lunar by 4200, and in the Cycle of the
Indictions by 6916, divide the sum of the products by 7980, and the
remainder is the year of the Julian period sought.
(926.) The following table contains these intervals for some of the more
important historical eras: —
Intervals in Days between the Commencement of the Julian Period, and
that of some other remarkable chronological and astronomical Eras.


First day. | Chronolog’l Current year Interval
Names by which the Era is usually cited. current of designation of the Ju- davs
the era. of the year. lian Period. yS.
Julian Epochs. Julian Dates.
Julian period ................................ Jan. 1. B. c. 4713 I. ()
Creation of the world (Usher) ......... (Jan. 1.) 4004 710 258,963
Era of the Deluge (Aboulhassan Kus-
chiar)....................................... Feb. 18. 3102 1612 588,466
Ditto Vulgar Computation............... (Jan. 1.) 2348 2366 863,817
Era of Abraham (Sir H. Nicholas)....| Oct. 1. 2015 2699 985,718
Destruction of Troy, (ditto)............. July 12. 1184 3530 1,289,160
Dedication of Solomon’s Temple....... (May 1.) 1015 3699 1,350,815
Olympiads (mean epoch in general
use) ......................................... July 1. 776 3938 1,438,171
Building of Rome (Varronian epoch,
U. C. ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . April 22. 753 3961 1,446,502
Era of Nabonassar ........................ Feb. 26. 747 3967 1,448,638
Metonic cycle (Astronomical epoch)... July 15. 432 4.282 1,563,831
Callippic cycle, do. (Biot)......... June 28. 330 4384. 1,599,608
Philippic era, or era of Philip Aridaeus Nov. 12. 324. 4390 1,603,398
Era of the Seleucidae...................... Oct. 1. 312 4402 1,607,739
Caesarean era of Antioch.................. Sept. 1. 49 4665 1,703,770
Julian reformation of the Calendar...] Jan. 1. 45 4669 1,704,987
Spanish era................................... Jan. 1. 38 4676 1,707,544
Actian era in Rome................. © tº e g tº º º Jan. 1. 30 4684 1,710,466
Actian era of Alexandria. ................. Aug. 29. 30 4684 1,710,706
Vulgar or Dionysian era.................. Jan. 1. A. D. 1 4714 1,721,424
Era of Diocletian........................... Aug. 29. 284. 4997 | 1,825,030
Hejira (astronomical epoch, new
moon)....................................... July 15. 622 5335 1,948,439
Era of Yezdegird........................... June 16. 632 5345 1,952,063
Gelalaean era (Sir H. Nicholas)......... March 14. 1079 5792 2,115,285
Last day of Old Style (Catholic na-
tions)....................................... Oct. 4. 1582 6295 2,299,160
Last day of Old Style in England)....] Sept. 2. 1752 6465 2,361,221
Gregorian
Gregorian Epochs. Dates.
New Style in Catholic nations ......... Oct. 15. 1582 6295 2,299,161
Ditto in England.................... Sept. 14. 1752 6465 2,361,222
Commencement of the 19th century,
epoch of Bode's catalogue of stars...| Jam. 1. 1801 6514 2,378,862
Epoch of the catalogue of stars of the
R. Astronomical Society............... Jan. 1. 1830 6543 |2,389,454.
Epoch of the catalogue of the British !
Association ............................... Jan. 1. 1850 6563 |2,396,759]
534 OTTLINES OF ASTRONOMY.
N. B. The civil epochs of the Metonic cycle, and the Hejira, are each one day later
than the astronomical, the latter being the epochs of the absolute new moons, the former
those of the earliest possible visibility of the lunar crescent in a tropical sky. M. Biot
has shown that the solstice and new moon not only coincided on the day here set down
as the commencement of the Callippic cycle, but that, by a happy coincidence, a bare
possibility existed of seeing the crescent moon at Athens within that day, reckoned from
midnight to midnight.
(927.) The determination of the exact interval between any two given
dates is a matter of such importance, and, unless methodically performed,
is so very liable to error, that the following rules will not be found out of
place. In the first place it must be remarked, generally, that a date,
whether of a day or year, always expresses the day or year current and not
elapsed, and that the designation of a year by A. D. or B. c. is to be re-
garded as the name of that year, and not as a mere number uninterrupt-
edly designating the place of the year in the scale of time. Thus, in
the date, Jan. 5, B. C. 1, Jan. 5 does not mean that 5 days of January in
the year in question have elapsed, but that 4 have elapsed, and the 5th is
current. And the B. C. 1, indicates that the first day of the year so
named, (the first year current before Christ,) preceded the first day of the
vulgar era by one year. The scale of A. D. and B. C. is not continuous,
the year 0 in both being wanting; so that (supposing the vulgar reckon-
ing correct) our Saviour was born in the year B. C. 1.
(928.) To find the year current of the Julian period, (J.P.) corre-
sponding to any given year current B. C. or A. D. If B. C., subtract the
number of the year from 4714: if A. D., add its number to 4713. For
example, see the foregoing table.
(929.) To find the day current of the Julian period corresponding to
any given date, Old Style. Convert the year B. C. or A. D. into the cor-
responding year J. P. as above. Subtract 1 and divide the number so
diminished by 4, and call Q the integer quotient, and R the remainder.
Then will Q be the number of entire quadriennia of 1461 days each, and
R the residual years, the first of which is always a leap-year. Convert
Q into days by the help of the first of the annexed tables, and R by the
second, and the sum will be the interval between the Julian epoch, and
the commencement, Jan. 1, of the year. Then find the days intervening
between the beginning of Jan. 1, and that of the date-day by the third
table, using the column for a leap-year, where R = 0, and that for a com-
mon year when R is 1, 2, or 3. Add the days so found to those in
Q -- R, and the sum will be the days elapsed of the Julian period, the
number of which increased by 1 gives the day current.
CHRONOLOGICAL INTERVALS. 535

TABLE I. Multiples of 1461, the days in a TABLE 2. Days in
tº s & Residual years.
Julian Quadriennium. 0 0
l 1461 4 5844. 7 10227 1 366
2 2922 5 7305 || 8 11688 2 73L
3 4383 6 8766 9 13149 3 1096 |
TABLE 3. Days elapsed from Jan. I to the 1st of each Month.
In a com- || In a leap In a com- In a leap
mon year. year. | mon year. year.
Jan. 1............... () 0 || July 1......….. 181 182
Feb. 1............... 3 31 Aug. 1.............. 212 - 213
March 1............ 59 60 Sept. 1.............. 243 244
April 1 ............. 90 91 Oct. 1............... 273 274
May 1....... ........] 120 121 Nov. 1.............. 304 305
! June 1.............. 151 152 Dec. 1...............! 334 335

EXAMPLE.-Włat is the current day of the Julian period correspond-
ing to the last day of Old Style in England, on Sept. 2, A. D. 1752?
1752 1000 1,461,000
4713 600 876,600
6465 year current. . 10 14,610
1 6 8,766
4)6464 years elapsed. R = 0 0
Q =1616 Jan. 1 to Sept. 1, 244
R = 0 Sept. 1 to Sept. 2, 1
2,361,221 days elapsed.
Current day the 2,361.2224.
(930.) To find the same for any given date, New Style. Proceed as
above, considering the date as a Julian date, and disregarding the change
of style. Then, from the resulting days, subtract as follows:—
For any date of New Style, antecedent to March 1, A. D. 1700........ 10 days.
After Feb. 28, 1700, and before March 1, A. D. 1800. . . . . . . . . . . . . . . . . . 11 days.
§ { 1800 £ & 4 & 1900. . . . . . . . . . . . . . . . . . 12 days.
& & 1900 4 & é & 2100. . . . . . . . . . . . . . . . . . 13 days, &c.
(931.) To find the interval between any two dates, whether of Old or
New Style, or one of one, and one of the other. Find the day current
of the Julian period corresponding to each date, and their difference is
the interval required. If the dates contain hours, minutes, and seconds,
they must be annexed to their respective days current, and the subtraction
performed as usual.
(932.) The Julian rule made every fourth year, without exception, a
bissextile. This is, in fact, an over-correction; it supposes the length of
the tropical year to be 3653", which is too great, and thereby induces au
536 OUTLINES OF ASTRONOMY.
error of 7 days in 908 years, as will easily appear on trial. Accordingly,
so early as the year 1414, it began to be perceived that the equinoxes
were gradually creeping away from the 21st of March and September,
where they ought to have always fallen had the Julian year been exact,
and happening (as it appeared) too early. The necessity of a fresh and
effectual reform in the calendar was from that time continually urged, and
at length admitted. The change (which took place under the popedom
of Gregory XIII.) consisted in the omission of ten' nominal days after
the 4th of October, 1582, (so that the next day was called the 15th, and
not the 5th,) and the promulgation of the rule already explained for future
regulation. The change was adopted immediately in all Catholic coun-
tries; but more slowly in Protestant. In England, “the change of style,”
as it was called, took place after the 2d of September, 1752, eleven no-
minal days being then struck out; so that, the last day of Old Style being
the 2d, the first of New Style (the next day) was called the 14th, instead
of the 3d. The same legislative enactment which established the Gre-
gorian year in England in 1752, shortened the preceding year, 1751, by
a full quarter. Previous to that time, the year was held to begin with
the 25th March, and the year A. D. 1751 did so accordingly; but that
year was not suffered to run out, but was supplanted on the 1st of January
by the year 1752, which it was enacted should commence on that day, as
well as every subsequent year. Russia is now the only country in
Europe in which the Old Style is still adhered to, and (another secular
year having elapsed) the difference between the European and Russian
dates amounts, at present, to 12 days.
(983.) It is fortunate for astronomy that the confusion of dates, and
the irreconcilable contradctions which historical statements too often ex-
hibit, when confronted with the best knowledge we possess of the ancient
reckonings of time, affect recorded observations but little. An astrono-
mical observation, of any striking and well-marked phaenomenon, carries
with it, in most cases, abundant means of recovering its exact date, when
any tolerable approximation is afforded to it by chronological records;
and, so far from being abjectly dependent on the obscure and often con-
tradictory dates, which the comparison of ancient authorities indicates, is
often itself the surest and most convincing evidence on which a chrono-
logical epoch can be brought to rest. Remarkable eclipses, for instance,
now that the lunar theory is thoroughly understood, can be calculated
back for several thousands of years, without the possibility of mistaking
the day of their occurrence. And, whenever any such eclipse is so inter-
* See note at the end of this chapter, p. 540.
LUNAR YEAR. 537
A
woven with the account given by an ancient author of some historical
event, as to indicate precisely the interval of time between the eclipse and
the event, and at the same time completely to identify the eclipse, tha
date is recovered and fixed for ever." -
(934.) The days thus parcelled out into years, the next step to a per-
fect knowledge of time is to secure the identification of each day, by im-
posing on it a name universally known and employed. Since, however,
the days of a whole year are too numerous to admit of loading the
memory with distinct names for each, all nations have felt the necessity
of breaking them down into parcels of a more moderate extent; giving
names to each of these parcels, and particularizing the days in each by
numbers, or by some special indication. The lunar month has been re-
sorted to in many instances; and some nations have, in fact, preferred a
lunar to a solar chronology altogether, as the Turks and Jews continue to
do to this day, making the year consist of 12 lunar months, or 354 days.
Our own division into twelve unequal months is entirely arbitrary, and
often productive of confusion, owing to the equivoque between the lunar
and calendar month.” The intercalary day naturally attaches itself to
February as the shortest.
(935.) Astronomical time reckons from the noon of the current day,
civil from the preceding midnight, so that the two dates coincide only
during the earlier half of the astronomical, and the later of the civil day.
This is an inconvenience which might be remedied by shifting the astro-
nomical epoch to coincidence with the civil. There is, however, another
inconvenience, and a very serious one, to which both are liable, inherent
in the nature of the day itself, which is a local phaenomenon, and com-
mences at different instants of absolute time, under different meridians,
whether we reckon from noon, midnight, sunrise, or sunset. In conse-
quence all astronomical observations require, in addition to their date, to
render them comparable with each other, the longitude of the place of
observation from some meridian, commonly respected by all astronomers.
For geographical longitudes, the Isle of Ferroe has been chosen by some
as a common meridian, indifferent (and on that very account offensive) to
all nations. Were astronomers to follow such an example, they would
probably fix upon Alexandria, as that to which Ptolemy's observations
* See the remarkable calculations of Mr. Baily relative to the celebrated solar eclipse
which put an end to the battle between the kings of Media and Lydia, B. c. 610,
Sept. 30. Phil. Trans. ci. 220.
* “A month in law is a lunar month or twenty-eight days, (!!) unless otherwise ex-
pressed.”—Blackstone, ii, chap. 9. “A lease for twelve months is only for forty-eight
weeks.”—Ibid.
838 OUTLINES OF ASTRONOMY.
and computations were reduced, and as claiming on that account the re-
spect of all while offending the national egotism of none. But even this
will not meet the whole difficulty. It will still remain doubtful, on a
meridian 180° remote from that of Alexandria, what day is intended by
any given date. Do what we will, when it is the Monday, the 1st of
January, 1849, in one part of the world, it will be Sunday, the 31st of
December, 1848, in another, so long as time is reckoned by local hours.
This equivoque, and the necessity of specifying the geographical locality
as an element of the date, can only be got over by a reckoning of time
which refers itself to some event, real or imaginary, common to all the
globe. Such an event is the passage of the sun through the vernal
equinox, or rather the passage of an imaginary sun, supposed to move
with perfect equality, through a vernal equinox supposed free from the
inequalities of nutation, and receding upon the ecliptic with perfect uni-
formity. The actual equinox is variable, not only by the effect of nuta-
tion, but by that of the inequality of precession, resulting from the change
in the plane of the ecliptic due to planetary perturbation. Both varia-
tions are, however, periodical; the one in the short period of 19 years,
the other in a period of enormous length, hitherto uncalculated, and
whose maximum of fluctuation is also unknown. This would appear, at
first sight, to render impracticable the attempt to obtain from the sun's
motion any rigorously uniform measure of time. A little consideration,
however, will satisfy us that such is not the case. The solar tables, by
which the apparent place of the sun in the heavens is represented with
almost absolute precision from the earliest ages to the present time, are
constructed upon the supposition that a certain angle, which is called “the
sun's mean longitude,” (and which is, in effect, the sum of the mean side-
real motion of the sun, plus the mean sidereal motion of the equinox in
the opposite direction, as near as it can be obtained from the accumu-
lated observations of twenty-five centuries,) increases with rigorous uni-
formity as time advances. The conversion of this mean longitude into
time at the rate of 360° to the mean tropical year, (such as the tables
assume it,) will therefore give us both the unit of time, and the uniform
measure of its lapse which we seek. It will also furnish us with an epoch,
not indeed marked by any real event, but not on that account the less
positively fixed, being connected, through the medium of the tables, with
every single observation of the sun on which they have been constructed
and with which compared. -
(936.) Such is the simplest abstract conception of equinoctial time. It
is the mean longitude of the sun of some one approved set of solar tables,
converted into time at the rate of 360° to the tropical year. Its unit is
EQUINOCTIAL TIME. 539
the mean tropical year which those tables assume and no other, and its
epoch is the mean vernal equinox of these tables for the current year, or
the instant when the mean longitude of the tables is rigorously 0, accord-
ing to the assumed mean motion of the sun and equinox, the assumed
epoch of mean longitude, and the assumed equinoctial point on which the
tables have been computed, and no other. To give complete effect to this
idea, it only remains to specify the particular tables fixed upon for the
purpose, which ought to be of great and admitted excellence, since, once
decided on, the very essence of the conception is that no subsequent altera-
tion in any respect should be made, even when the continual progress of
astronomical science shall have shown any one or all of the elements con-
cerned to be in some minute degree erroneous (as necessarily they must,) .
and shall have even ascertained the corrections they require (to be them-
selves again corrected, when another step in refinement shall have been
made.)
(937.) Delambre's solar tables (in 1828) when this mode of reckoning
time was first introduced, appeared entitled to this distinction. According
to these tables, the sun's mean longitude was 0°, or the mean vernal equi-
nox occurred, in the year 1828, on the 22d of March at 1%. 2n 594-05
mean time at Greenwich, and therefore at 1" 12" 20° 55 mean time at
Paris, or 1* 56* 34°55 mean time at Berlin, at which instant, therefore,
the equinoctial time was 0° 0' 0" 05:00, being the commencement of the
1828th year current of equinoctial time, if we choose to date from the
mean tabular equinox, nearest to the vulgar era, or of the 6541st year of
the Julian period, if we prefer that of the first year of that period.
(938.) Equinoctial time then dates from the mean vernal equinox of
Delambre's solar tables, and its unit is the mean tropical year of these
tables (365-242264.) Hence, having the fractional part of a day ex-
pressing the difference between the mean local time at any place (suppose
Greenwich) on any one day between two consecutive mean vernal equi-
noxes, that difference will be the same for every other day in the same
interval. Thus, between the mean equinoxes of 1828 and 1829, the
difference between equinoctial and Greenwich time is 0°-956261 or 04
22* 57* 0°-95, which expresses the equinoctial day, hour, minute, and
second, corresponding to mean noon at Greenwich on March 23, 1828,
and for the noons of the 24th, 25th, &c., we have only to substitute 1d,
2d, &c., for 0", retaining the same decimals of a day, or the same hours,
minutes, &c., up to and including March 22, 1829. Between Greenwich
noon of the 22d and 23d of March, 1829, the 1828th equinoctial year
terminates, and the 1829th commences. This happens at 04:286003, or
at 6* 51* 50" 66 Greenwich mean time, after which hour, and until the
540 OUTLINES OF ASTRONOMY.
next noon, the Greenwich hour added to equinoctial time 3644-956261
will amount to more than 365-242264, a complete year, which has there-
fore to be subtracted to get the equinoctial date in the next year, cor-
responding to the Greenwich time. For example, at 12* 0° 0' Greenwich
mean time, or 0+500000, the equinoctial time will be 364-956261--
0-500000–365.456261, which being greater than 365-242264, shows
that the equinoctial year current has changed, and the latter number
being subtracted, we get 0"-213977 for the equinoctial time of the
1829th year current corresponding to March 22, 12° Greenwich mean
time.
(939.) Having, therefore, the fractional part of a day for any one year
expressing the equinoctial hour, &c., at the mean noon of any given place,
that for succeeding years will be had by subtracting 0°-242264, and its
multiples, from such fractional part (increased if necessary by unity,) and
for preceding years by adding them. Thus, having found 0.198525 for
the fractional part for 1827, we find for the fractional parts for succeeding
years up to 1853 as follows':—
1828 •956.261 1835 | 260413 1842 564565 1848 || “110981
1829 || 713997 1836 || 018149 1843 || 322301 1849 || 868717
1830 | *471733 1837 || 775885 1844 || 080037 1850 | *626453
1831 || -229.469 1838 • 533621 I845 •837773 1851 || 384.189
I 832 || -98.7205 1839 •291.357 1846 •595.509 1852 | "141925
1833 | "744941 1840 || 049093 1847 || 353.245 1853 | 899661
1834 || “502677 I841 || -80.6829
* These numbers differ from those in the Nautical Almanack, and would require to
be substituted for them, to carry out the idea of equinoctial time as above laid down.
In the years 1828–1833, the late eminent editor of that work used an equinox slightly
differing from that of Delambre, which accounts for the difference in those years. In
1834, it would appear that a deviation both from the principle of the text and from the
previous practice of that ephemeris took place, in deriving the fraction for 1834 from
that for 1833, which has been ever since perpetuated. It consisted in rejecting the
mean longitude of Delambre's tables, and adopting Bessel's correction of that element.
The effect of this alteration was to insert 3" 3*68 of purely imaginary time, between
the end of the equinoctial year 1833 and the beginning of 1834, or, in other words, to
make the interval between the noons of March 22 and 23, 1834, 24h 3° 35'68, when
reckoned by equinoctial time. In 1835, and in all subsequent years, a further depar-
ture from the principle of the text took place by substituting Bessel's tropical year of
365-2422175, for Delambre's. Thus the whole subject has fallen into confusion.
[Note on Art. 932.
The reformation of Gregory was, after all, incomplete. Instead of 10 days he ought
to have omitted 12. The interval from Jan. 1, A. D. 1, to Jan. 1, A. D. 1582, reckoned
as Julian years, is 577460 days, and as tropical, 577448, with an error not exceeding
0"-01, the difference being 12 days, whose omission would have completely restored the
Julian epoch. But Gregory assumed for his fixed point of departure, not that epoch,
but one later by 324 years, viz. Jan. 1, A. D. 325, the year of the Council of Nice;
assuming which, the difference of the two reckonings is 9°505, or, to the nearest whole
number, 10 days.] *
A PPENDIX.
I. LISTS OF NoFTHERN AND SOUTHERN STARS, WITH THEIR APPROxIMATE MAGNITUDES, ON THE
WULGAR AND PHOTOMETRIC SCALE.
1. NoFTHERN STARs.
MAG. MAG. MAG. MAG
STAR. STAR. STAR. STAR.
Vulg. Phot. Wulg. Phot. Vulg. Phot Wulg. Phot
Arcturus........ ..] 0-77 | 1.18 y Cassiopeiae.......| 2:52 2-93 m Draconis.........| 3:02 || 3:43 e Aurigae (War.). 3:37 || 3:78
Capella............ 1-0 : | 1.4: a Andromedae ..... 2'54 || 2-95 (3 Draconis......... 3-06 || 3-47 y Lyncis............ 3.39 || 3.80
Lyra....... tº º º e s tº e a 1-0 : | 1.4: a Cassiopeiae....... 2-57 2-98 8 Arietis............ 3-09 || 3•50 & Draconis......... 3-40 || 3-81
Procyon.......... . 1-0 : | 1.4: y Geminorum...... 2.59 || 3:00 y Pegasi............ 3-11 || 3•52 tr Herculis.......... 3.41 || 3'82
a Orionis........... ..] 1-0 : | 1.43 Algol (War.)...... 2.62 || 3:03 e Virginis?.........| 3:14 || 3:55 8 Canis Min. 2.....| 3:41 || 3-82
Aldebaran......... |-1 : | 1.5 : * Pegasi............. 2-62 || 3:03 0.Aurigae .......... 3-17 | 3•58 § Tauri...... § tº $ tº e º e g 3.42 || 3-83
a Aquilae............| 1:28 | 1.69 y Draconis ......... 2-62 || 3:03 || 3 Herculis.......... 3-18 3•59 d Draconis......... 3.42 || 3-83
Pollux............. 1-6 : || 2:0: § Leonis............. 2-63 || 3-04 a Canum Ven..... 3-22 || 3:63 p Geminorum......| 3:42 || 3-83
Regulus........... 1-6 : 2-0 : a Ophiuchi.......... 2-63 || 3-04 9 Ophiuchi......... 3°23 || 3-64 y Boötis ............ 3-43 || 3-84
a Cygni........ & e º e s a I-90 2-31 6 Cassiopeiae....... 2.63 3-04 6 Cygni............. 3°24 || 3°65 & Geminorum .....] 3'43 || 3-84
Castor ............. I '94 || 2:35 y Cygni.............. 2-63 3-04 e Persei ............ 3.26 3-67 & Herculis ......... 3-44 || 3-85
e Ursae (Var.)...... 1-95 || 2:36 a Pegasi ............ 2-65 || 3:06 m Tauri’............ 3.26 || 3-67 & Geminorum .....! 3:44 3-85
a Ursae (Var.)......| 1.96 || 2:36 9 Pegasi ............ 2.65 || 3:06 § Persei............ 3-27 | 3.68 q Orionis....... .....| 3:45 3'86
a Persei..............| 2:07 || 2:48 a Corona ........... 2.69 || 3-10 § Herculis......... 3-28 || 3-69 8 Cephei............ 3.45 || 3-86
m Ursae (Var.)......| 2:18 || 2:59 y Ursae ............... 2-71 || 3-12 * Aurigae ........... 3.29 || 3-70 0 Ursae ............. 3-45 3-86
y Orionis............ 2- 18 || 2:59 8 Ursae ............... 2-77 || 3-18 y Ursae Minor ....] 3-30 || 3-71 t Ursae ............. 3-46 || 3.87
9 Tauri............... 2-28 || 2:69 * Boötis.............. 2-80 || 3-21 m Pegasi............ 3-31 || 3-72 m Aurigae........... 3-46 || 3.87
Polaris ............ 2-28 || 2.69 * Cygni.............. 2-88 || 3-29 Ś Aquilae........... 3.32 || 3-73 y Lyrae.............. 3-47 3-88
y Leonis ............ 2-34 || 2.75 a Cephei ............ 2.90 3-31 8 Cygni ............ 3-33 || 3-74 m Geminorum..... 3-48 || 3-89
a Arietis ............ 2'-10 || 2:81 a Serpentis......... 2-92 || 3-33 Y Persei............. 3-34 || 3-75 y Cephei............ 3-48 || 3-89
& Ursae ............... 2°43 || 2:8.4 & Leonis............. 2-94 || 3:35 u Ursae.............. 3.35 | 3.76 & Ursae ............. 3.49 || 3-90
(3 Andromedae...... 2.45 2'86 Y Aquilae............ 2.98 || 3-39 {3 Triang. Bor.....| 3:35 | 376 e Cassiopeiae..... 3.49 || 3'90
8 Aurigae............ 2°48 2.89 ô Cassiopeiae........ 2.99 || 3:40 & Persei............ 3.36 || 3-77 0 Aquilae............| 3:50 | 891
y Andromedae......| 2'50 | 2'91 7. Boötis.............. 3-01 || 3:42 W Ursae ....... © tº e º tº º 3.36 || 3-77
§


#
i
MAG. MAG. MAG
STAR. ,--~~-y STAR. A----——, STAR. ,-\–y STAR.
Vulg. Phot. Wulg. Phot Vulg. Phot
Sirius...... .........| 0-08 || 0:49 a Hydrae .......... ...] 2-30 || 2.71 k Scorpii ........... 291 || 3:32 d Corvi..............
m Argås (War.)..... «Pº tº-e & Canis............... 2-32 2-73 m Centauri......... 2.91 3-32 8 Eridani...........
Canopus .......... 0-29 0-70 a Pavonis.......... ..] 2-33 2-74. * Argūs............. 2-94 || 3:35 6 Argūs.............
a Centauri........... 0.59 || 1:00 9 Gruis............. ... 2-36 2-77 9 Corvi ............. 2-95 || 3:36 8 Hydri.............
Rigel............... ()-82 || 1:23 a Sagittarii........ ..| 2°41 2°82 6 Scorpii............ 2.96 || 3:37 e Corvi .............
a Eridani........ ....] 1 Q9 1:50 8 Argūs............ ..| 2°42 || 2-83 § Centauri......... 2-96 || 3:37 8 Arae...............
6 Centauri...... .....] 1 °17 | 1:58 8 Ceti............... ...] 2.46 || 2:87 § Ophiuchi......... 2.97 || 3:38 a Toucani..........
a Crucis.............. 1-2 | 1.6 ^ Argºs.............. 2-46 2.87 a Aquarii........... 2.97 || 3:38 || 3 Capricorni ......
Antares............] 1 2 | 1.6 6 Centauri........... 2.54 2.95 T Argūs............. 2.98 || 3:39 || 0 Argås.............
Spica........ {º} tº tº º is ſº tº 1:38 | 1.79 8 Canis .............. 2.58 2-99 6 Centauri......... 2.99 || 3:40 | r Scorpii............
Fomalhaut........ 1°54 || 1-95 k Orionis............ 2.59 || 3:00 a Leporis .......... 3-00 || 3:41 | 8 Leporis...........
§ Crucis......... ....| 1:57 | 1.98 & Orionis............ 2-61 || 3:02 & Ophiuchi......... 3-00 || 3:41 y Lupi....... e e o e e s s a
a Gruis ........ & g º 'º tº a 1.66 2-07 y Centauri........... 2-68 || 3:09 § Sagittarii......... 3.01 || 3:42 v Scorpii ...........
y Crucis ............. 1.73 ſ 2-14 & Scorpii............. 2.71 || 3-12 T Ophiuchi......... 3-05 || 3:46 t Orionis............
e Orionis............ I-84 || 2:25 Ś Argºs ............. 2.72 || 3-13 9 Librae............. 3-07 || 3:48 a Arae...............
e Canis .............. 1-S6 2-27 a Phoenicis......... 2.78 || 3:19 y Virginis.......... 3-08 || 3:49 tr Sagittarii.........
A Scorpii ........ ....| 1-87 || 2:28 t Argūs ............. 2-80 || 3:21 u Argūs ............ 3-08 || 3:49 a Musca?............
& Orionis............ 2-01 || 2:42 a Lupi............... 2.82 || 3:23 6 Sagittarii......... 3-11 || 3:52 a Hydri 2"...........
8 Argūs .............. 2-03 || 2-44 e Centauri........... 2.82 || 3:23 a Libra?............. 3. I2 || 3:53 7 Scorpii ...........
y Argūs ............. 2-08 || 2:49 m Camis .............. 2.85 || 3:26 A Sagittarii ........ 3-13 3'54 § Hydra ............
e Argūs ........ ..... 2°18 2°59 8 Aquarii............ 2-85 || 3:26 6 Lupi ....... tº e º 'º e º 'º 3-14 3•55 y Hydrae ...........
a Trianguli A...... 2-23 2-64 6 Scorpii.............] 2.86 || 3:27 a Columbae......... 3-15 || 3:56 8 Trianguli.........
e Sagittarii..........| 2:26 2’67 m Ophiuchi ....... ...] 2.89 || 3:30 t Centauri.......... 3'20 || 3:61 || a Scorpii............
6 Scorpii............ 2.29 2-70 y Corvi............... 2.90 || 3:31 § Capricorni....... 3:20 || 3:61 || 7 Argås........... •:
P
h
O
t.

i
§
II. SYNOPTIC TABLE OF THE ELEMENTS OF THE PLANETARY SYSTEM.
Mean Distance from the
Mean Sidereal Period in Mean
Bccentricity in parts of
Inclination of Orbi the Longitude of the Ascendin
Name of Body. || “...is. Solar Dāys. the Semi-axis. #. it to g Node. g
Sun O / // O f //
Mercury ..... 0.3870.981 87-969.2580 0-20551.49 7 0 9-1 45 57 30-9
Venus......... ()-7233316 224-7007869 0-0068607 3 23 28°5 74 54 12-9
Earth ......... 1-0000000 365-2563612 0.0167836 -
Mars.......... 1-5236923 686-9796458 0.0933.07.0 I 51 6-2 48 0 3-5
Flora.......... 2-2016870 1193,249 0-1565570 5 53 4-8 110 18 12-0
Victoria...... 2-3348770 1303-148 0-2179220 8 22 15-3 235 28 25°3
Vesta ........ * 2-36108.10 1325-147 0'0895694 7 8 29-7 103 23 31-6
Iris............ 2-3850809 1345°403 0.2322236 5 28 14-0 259 42 47-6
Metis.......... 2.3868970 1346'940 0-1228221 5 35 54-7 68 28 57.7
Hebe.......... 2°426II 20 1380°184. 0-2005582 14. 47 1-8 I38 30 32°5
Parthenope . 2°4465,170 1397.719 0-0963995 4 36 44°l 125 1 32°1
Astraea........ 2-5770470 1511-095 0-1880586 5 19 22.7 141 25 14-6
Egeria........ 2:57.83650 I512-217 0-III5620 I5 55 44-9 43 37 7-1
Irene.......... 2.5849050 1517.975 0-1678530 9 5 47-0 86 51 28-7
Juno .......... 2-6708370 I 594-296 0-2548847 13 3 22-1 170 54 45-6
Ceres.......... 2.7680510 1682-125 ().0766523 10 37 4-4 80 48 46.6
Pallas 2-7728580 1686-510 0.2398150 34 37 33-1 172 43 59-7
Hygeia ....... 3-12089.00 2020-761 0.0892461 3 47 14-2 287 52 25-2
Jupiter....... 5-2027760 43.32°5848.212 0-0481621 1 18 51.3 98 26 18-9
Saturn........ 9-5387861. 10759-2198.174. 0-0561505 2 29 35-7 111 56 37-4
Uranus....... 19-1823900 30686-820.8296 0°0466794 0 46 28-4 72 59 35-3
$ Neptune ..... 30'0368000 60126-7.100000 0-0087195 1 46 59°0 130 5 11-0
i

ELEMENTS OF THE PLANETARY SYSTEM.
Longitude of the Peri- Mean } Longitude (L) at Epoch of the Elements in M.T. Mass (denominator Diameter º Time of Rota-
Name of Body. e Anomalv (A Greenwich (G). of fraction, the II]. Density. . . g
y helion. º (A) | at | Berlin (B). (G) } Sun tº: 1.) Miles. 7 | tion on Axis.
O / A/ O p // h. Iſl.
Sun .................. . .................. . .................. 1 882000 0-25 607 48
Mercury ..... 74 21 46.9 I66 0 48-6 1801. Jan. I. Oh G. 4865751 3140 1-12 24 5
Venus......... 128 43 53.1 11 33 3-0 T}o. 401839 7800 0-92 23 21
Earth ......... 99 30 5-0 100 39 10-2 Do. 3895.51 7926 1.00 24 0
Mars.......... 332 23 56-6 64 22 55°5 Do. 2680337 4100 0-95 24, 37
Flora.......... 33 0 40-8 35 48 7-0 1848. Jam. 1. Oh B.
Victoria...... 301 56 31-9 40 22 I5-1 1850. Oct. 0. Oh Par.
Vesta ......... 250 46 32°2 225 44, 18.8 I850. Jan. 9. 0h B. tº Q & © tº º º tº © 2502
Tris............ 41 24, 16-1 190 15 34°8 1850. Mar. 31. 0h B.
Metis.......... 71 33 10-9 I83 39 45-7 1851. June 4. 0” B.
Hebe.......... 14. 49 23.9 275 13 30-0 1847. July 10. Oh B.
Partkenope 317 0 42-7 28]. 34. 23°4 1850. May 11.0% G.
Astraea........ 35 20 47-0 318 45 3-3 1846. Jan. 1. Oh B.
Egeria........ I15 28 40-3 291. 14 2-4 1850. Nov. 2. Oh B.
Irene.......... 179 10 19.3 55 4 49-7 1851. July 1. 10h B.
Juno .......... 54 24 12-8 124, 31 10-8 1850. Apr. 8. Oh B. º e º e º 'º e º e 79? tº ſº tº tº º 27 0?
Ceres.......... 147 46 12.4 219 6 29-5 1850. Sept. 25.0". B. ......... 163?
Pallas.. 121 21 48-5 217 31 10-6 1850. Aug. 23.0" B.
Hygeia....... 221 22 49-9 344. 57 38-5 1849. June 6. 0h G.
Jupiter....... 11 8 34-6 L = 112 15 23:0 1801. Jan. 1. Oh G. 1047-871 87000 0-24 9 56
Saturn........ 89 9 29-8 135 20 6'5 Do. 3501-600 79160 0-14 10 29
Uranus....... 167 31 16-1 177 48 23-0 Do. 24.905 34,500 0-24. 9 30?
Neptune ..... 47 12 56-7 330 44 41°8 1848. Jan. 1. 0h G. 18780 41500 0-14
APPENDIX. 545
Note.—The elements of the orbits of Mercury, Venus, the Earth, Mars, Jupiter,
Saturn, and Uranus, are those given by the late F. Baily, Esq., in his “Astronomical
Tables and Formulae,” and are the same with those which form the basis of Delam-
bre's tables, embodying the formulae of Laplace. The elements of Uranus and Nep-
tune can only be regarded as provisional; those of the former requiring considerable
corrections, necessitated by the discovery of Neptune, but which, not being yet finally
ascertained, by reason of the uncertainty still attending on the mass and elements of
the latter planet, it was thought better to leave the old elements untouched than to give
an imperfect rectification of them. The masses of the planets are those most recently
adopted by Encke (Ast. Nachr. No. 443), on mature consideration of all the autho-
rities, that of Neptune excepted, which is Prof. Pierce’s determination from Bond's
and Lassell’s observation of the satellite discovered by the latter. The densities are
Hansen's (A. N. 443.)
The elements of Vesta, Juno, Ceres, and Pallas, are the osculating elements for
1850, computed by Encke (A. N. 636.) [Those of Flora are from the computations
of Brunnow (A. N. 645); of Victoria, Villarceaux (A. N. 741); of Iris, Schubers
(A. N. 730); of Metis, Wolfers (A. N. 764); of Hebe, Luther (A. N. 721); of Par-
thenope, Galen (A. N. 757); of Astraea, D'Arrest (A. N. 626); of Egeria, D'Arrest
(A. N. 749); of Irene, Vogel and Rümker (A. N. 765); and of Hygeia, Santini (A.
N. 702.)
Of these last-mentioned small planets, Hygeia, Parthenope, and Egeria were dis-
covered by Dr. Gasparis, at Naples, on April 12, 1849, May 11 and Nov. 2, 1850,
respectively; Iris, Flora, Victoria, and Irene, by Mr. Hind, on Aug. 13 and Oct. 18,
1847, Sept. 13, 1850, and May 19, 1851, respectively. The elements of the recently-
discovered small planets may undergo material corrections from further observation.
Irene has a blue colour and a faint nebulous envelope. The orbits of Astraea and
Hygeia approach at one point (their common node) within 0-006 of the radius of the
earth's orbit. It will not be long before the planets themselves come within that prox-
imity to each other (A. N. 752.) Victoria and Astraea are subject to variations of
brightness, which indicate rotations on their axes, and dark spots (A. N. 760.) D’Arrest
(A. N. 752) remarks that a relation subsists between the excentricities of the orbits of
the small planets, and the inclinations of the planes in which they lie to the sun’s
equator, the more excentric orbits being the more inclined. While these sheets pass
through the press, another, yet unnamed, is announced by M. de Gasparis.]
III.
SyNOPTIC TABLE OF THE ELEMENTS OF THE ORBITS OF THE SATEL-
LITES, SO FAR AS THEY ARE KNOWN.
1. THE Moon.
Mean distance from earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59r-96435000
Mean sidereal revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27°321661418
Mean synodical ditto. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29*530588715
Excentricity of orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0054844200
Mean revolution of nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . .6793°39'1080
Mean revolution of apogee . . . . . . . . . . . . . . . . . . . . . . . . . . . 3232d-575343
Mean longitude of node at epoch. . . . . . . . . . . . . . . . . . . . . . . . 13°53' 17”7
Mean longitude of perigee at do. . . . . . . . . . . . . . . . . . . . . . . .266 10 7 '5
Mean inclination of orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...5 8 47 9
Mean longitude of moon at epoch . . . . . . . . . . . . . . . . . . . . . . 118 17 8 °3
Mass, that of earth being 1,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0011399
Diameter in miles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2153
Density, that of the earth being 1, . . . . . . . . . . . . . . . . . . . . . . . .0.5657
* The distances are expressed in equatorial radii of the primaries. The epoch is Jan. 1,
1801, *; otherwise expressed. The periods, &c. are expressed in mean solar days.
§
2. SATELLITEs of JUPITER.
Inclination of Orbit to | Inclination of the fixed | Retrograde Revolution s
Sat. Sidereal Revolution. Mean distance. a fixed Plane proper Plane to Jupiter's of §. on the fixed *: * of Jupiter
to each. Equator. Plane. eing 1,000,000,000.
d h m 8 O f // O / // Years
1 1 18 27 33'506 6-0.4853 0 0 0 0 0 6 ...................... e e º O & 17328
2 3 13 14 36-393 9-62347 0 27 50 0 1 5 29-9142 23235
3 7 3 42 33-362 15°35024 0 12 20 0 5 2 141°7390 88497
4 16 16 31 49-702 26-99.835 0 14 58 0 24 4 531°0000 42659
The excentricities of the 1st and 2d Satellites are insensible, those of the 3d and 4th small, but variable, in consequence of their mutual
perturbation.
3. SATELLITEs of SATURN.
5-
H3
- - * º Ho
Nangº of Sidereal Revolution. Mean distance. Frº. ple Mean º ** | Excentricity. Perisaturnium. É
d h m s d O / // O / // E
1. Mimas................ 0 22 37 22-9 3-3607 1790.0 256 58 48 :4
2. Enceladus............ I 8 53 6-7 4-3125 1836-0 67 41 36 -
3. Tethys........... © e º e º e 1 21 18 257 5-3396 Ditto 313 43 48 0-04 2 547
4. Diome.................. 2 17 41 8-9 6'8398 Ditto 327 40 48 0-022 42 2
5. Rhea .................. 4 I2 25 10-8 9°55’28 Ditto 353 44 0 0-022 952
6. Titan .................. 15 22 41 25°2 22-1450 1830-0 137 21 24 0-029314 256 38 11
7. Hyperion ............ 22 12 7 28–H ! ...... tº e º e º e e s º
8. Iapetus................] 79 7 53 40-4 64-3590 1790-0 269 37 48


The longitudes are reckoned in the plane of the ring from its descending node with the ecliptic. The first seven satellites move in, or very
nearly in, its plane; that of the 8th is inclined to it at an angle about half-way intermediate between the planes of the ring and of the planet's
orbit. The apsides of Titan have a direct motion of 30'28" per annum in longitude (on the ecliptic).
The discovery of Hyperion is quite recent, having been made on the same night (Sept. 19, 1848), by Mr. Lassell, of Liverpool, and Prof.
Bond, of Cambridge, U. S. Its distance and period are as yet hardly more than conjecture. , Messrs. Kater, Encke, and Lassell agree in repre-
senting the ring of Saturn as subdivided by several narrow dark lines, besides the broad black divisions which ordinary telescopes show.
i
§

4. SATELLITEs or URANUs.
Sat. sidereal Revolution, Mean distance. *º: * Nodes and Inclinations.
d h m §
1 ? The orbits are inclined at an angle of about
2 8 16 56 31°3 17.0 1787. Feb. 16th, Oh 10m 78° 58' to the ecliptic, in a plane whose
3 10 23 ° 19-82 l ascending node is in long. 165° 30' (Equi-
4 13 11 7 12.6 22-8 1787. Jan. 7th. Oh 28m. nox of 1798). Their motion is retrograde.
5. 38 2 7. 45-52 The orbits are nearly circular. - -
6. 107 12 2. 91:0 - -
5. SATELLITEs of NEPTUNE.
One only has been certainly observed,—its approximate period being 5°20′ 50° 45'-distance about 12 radii of the planet.
i

IV.
ELEMENTS OF PERIODICAL COMETS AT THEIR LAST APPEARANCE.
Halley’s. Encke's. Biela’s. Faye's. De Wico’s. Brorsen’s.
1835. Nov. 15.
1845. Aug. 9.
1846. Feb. 11.
1843. Oct. 17.
1844. Sept. 2.
1846. Feb. 25.
7- 22h 41m 22° 15h 11m 115 0h 2m 50s 3, 42; 16. 11h 36m 53° 9h 13m 35s
(*) 3040 31' 3.2// 157 o 44' 21” 109 o 5' 47// 19° 34 1977 342° 31' 15"/ 1160 28' 34”
Q 55 9 59 334 19 33 245 56 58 209 29 19 63 49 31 102 39 36
t 17 45 5 13 7 34 12 34 14 11 22 31 2 54 45 30 55 7
0. 17-98.796 2°21640 3-50182 3-81179 3.099.46 3•15021
& 0.967.391 0-847436 0-7554.71 0°555962 0.617256 0.793629
P 27865d.74 1205d.23 2393d-52 2718q-26 1993k•09 2042-24
Retrograde. Direct. Direct. Direct. Direct. Direct.
r is the time of perihelion passage; w the longitude of the perihelion; and Q that of the ascending node for the epoch of the perihelion;
t, the inclination to the ecliptic; a, the semi-axis; e, the excentricity; P, the period in days.
N. B. The reader will find a complete list of elements of all known comets up to June, 1847, by all their several computors, in Prof. Encke's
edition of Olbers’s “Abhandlung über die leichteste und bequenste Methode die Bahn eines Cometen zu berechnen.” The list is compiled by
Dr. Galle. It contains orbits of 178 distinct comets. From an examination of these orbits we collect the following, as a more correct statement
of cometary statistics than that in art. 601, viz.:- Retrograde comets under 10° inclination, 3 out of 15; under 20°, 9 out of 29. Retrograde
comets, moving in orbits sensibly elliptic, under 17° inclination, 0 out of 9. In such orbits, of all inclinations from 0 to 90°, 11 out of 37.
Thus we see that the induction of that article is materially strengthened by the enlarged field of comparison.
INDEX.
N. B. The references are to the articles, not to the pages. -
... attached to a reference number indicates that the reference extends to the article cited, and
several Subsequent in succession.
A.
Aberration of light explained, 329. Its
uranographical effects, 833. Of an
object in motion, 335. How distin-
guished from parallax, 805. System-
atic, 862.
Aboul Wefa, 705.
Acceleration, secular, of moon’s mean
motion, 740.
Adams, 506. 767.
Adjustment, errors of, in instruments,
136. Of particular instruments. (See
those instruments.)
AEtna, portion of earth visible from, 32.
Height of, 32. note.
Air, rarefaction of, 33. Law of density,
37. Refractive power affected by
moisture, 41.
Airy, G. B. Esq., his results respecting
figure of the earth, 220. Researches
on perturbations of the earth by We-
nus, 726. Rectification of the mass
of Jupiter, 757.
Algol, 821.
Altitude and azimuth instrument, 187.
—s. Equal, method of, 188.
Andromeda, nebula in, 874.
Angle of position, 204. Of situation, 311.
Angles, measurement of, 163. 167. Hour,
107.
Angular velocity, law of, variation of,
350.
Anomalistic year, 384.
Anomaly of a planet, 499.
Annular nebulae, 875.
Apex of aberration, 343. Of parallax,
343. Of refraction, 343. Solar, 854.
Of shooting stars, 902. 904.
Aphelion, 368.
Apogee of moon, 406. Period of its re-
volution, 687.
Apsides, 406. Motion of investigated,
675. Application to lunar, 676 ...
Motion of, illustrated by experiment,
692. Of planetary orbits, 694. Li-
bration of, 694. Motion in orbits
very near to circles, 696. In excen-
tric orbits, 697...
Areas, Kepler's law of, 490.
Argelander, his researches on variable
stars, 820..., on Sun’s proper motion,
854. -
Argo, nebulae in, 887.
in constellation, 830.
Ascension, right, 108. (See Right ascen-
sion.)
Asteroids, their existence suspected pre-
vious to their discovery, 505. Ap-
pearance in telescopes, 525. Gravity
on surface of, 525. Elements, Appen-
dix, Synoptic Table.
Astraea, discovery of, 505.
Astrometer, 783, 784.
Astronomy. Etymology, 11.
notions, 11 g
Atmosphere, constitution of, 38... Possi-
ble limit of, 36. Its waves, 37. Strata,
37. Causes refraction, 38. Twi-
light, 44. Total mass of, 148. Of
Jupiter, 513.
Attraction of a sphere, 445—450. (See
Gravitation.) -
Augmentation of moon’s apparent dia
meter, 404.
Augustus, his reformation of mistakes
in the Julian calendar, (919). Era
of, 926.
Australia, excessive summer tempera-
ture of, 369.
Azis of the earth, 82. Rotation perma-
nent, 56. Major of the earth's orbit,
373. Of sun’s rotation, 392.
Azis of a planetary orbit. Momentary
variation of, caused by the tangential
force only, 658. 660. Its variations
periodical, 661... Invariability of,
and how understood, 668. -
Azimuth, 103.—and altitude instrument,
Irregular star n
General
(549)
550
INDEX.
B.
Baromºter, nature of its indication, 33.
Use in calculating refraction, 43. In
determining heights, 287.
Belts of Jupiter, 512. Of Saturn, 514.
Benzenberg's principle of collimation,
179.
Bessel, his results respecting the figure
of the earth, 220. Discovers parallax
of 61 Cygni, 812.
Biela's comet, 579...
Biot, his aéronautic ascent, 32.
Bode, his (so called) law of planetary
distances, 505. Violated in the case
of Neptune, 507.
Borda, his principle of repetition, 198.
Bouvard, his suspicion of extraneous
influence on Uranus, 760.
C.
Caesar, his reform of the Roman calen-
dar, 917.
Calendar, Julian, 917. Gregorian, 914...
Cause and effect, 439, and note.
Centre of the earth, 80. Of the sun, 462.
Of gravity, 360. Revolution about,
452
Centrifugal force. Elliptic form of earth
produced by, 224. Illustrated, 225.
Compared with gravity, 229. Of a
body revolving on the earth’s surface,
452.
Ceres, discovery of, 505.
Challis, Prof., 506, note.
Charts, celestial, 111. Construction of,
291... Bremiker's, 506, and note.
Chinese records of comets, 574. Of ir-
regular stars, 831. -
Chronometers, how used for determining
differences of longitude, 255. -
Circle, arctic and antarctic, 94. Verti-
cal, 100. Hour, 106. Divided, 163.
Meridian, 174. Reflecting, 197. Re-
peating, 198. Galactic, 793.
Clepsydra, 150.
Clock, 151. Error and rate of, how
found, 253.
Clouds, greatest height of, 34. Magel-
lanic, 892...
Clusters of stars, 864... Globular, 867.
Irregular, 869.
Collimation, line of, 155.
Collimator, 178...
Coloured stars, 851...
Colures, 307.
Zſomets,' 554. Seen in day time, 555.
&
590. Tails of, 556.566. 599. Ex-
treme tenuity of, 558. General de-
scription of, 560. Motions of, and
described, 561... Parabolic, 564. El-
liptic, 567... Hyperbolic, 564. Di-
mensions of, 565. Of Halley, 567...
Of Caesar, 573. Of Encke, 576. Of
Biela, 579. Of Faye, 584. Of Lex-
ell, 585. Of De Vico, 586. Of Bror-
sen, 587. Of Peters, 588. Synopsis
of elements (Appendix). Increase of
visible dimensions in receding from
the sun, 571. 580. Great, of 1843,
589... Its supposed identity with
many others, 594... Interest attached
to subject, 597. Cometary statistics,
and conclusions therefrom, 601.
Commensurability (near) of mean mo-
tions; of Saturn's satellites, 550. Of
Uranus and Neptune, 669, and note.
Of Jupiter and Saturn, 720. Earth
and Venus, 726. Effects of, 719.
Compensation of disturbances, how ef-
fected, 719. 725.
Compression of terrestrial spheroid, 221.
Configurations, inequalities depending
on, 655 -
Conjunctions, superior and inferior, 473.
Perturbations chiefly produced at, 713.
Consciousness of effect when force is ex-
erted, 439.
Constellations, 60. 301. How brought
into view by change of latitude, 52.
Rising and setting of, 58.
Copernican explanation of diurnal mo-
tion, 76. Of apparent motions of sun
and planets, 77.
Correction of astronomical observations,
324... s. Uranographical summary,
view of, 342...
Culminations, 125.
126
Cycle, of conjunctions of disturbing and
disturbed planets, 719. Metonic, 926.
Callippic, ib. Solar, 921. Lunar 922.
Of indictions, 923.
Upper and lower,
D.
Day, solar, lunar, and sidereal, 143.
Ratio of sidereal to solar, 305. 909.
911. Solar unequal, 146. Mean
ditto invariable, 908. Civil and astro-
nomical, 147. Intercalary, 916.
Days elapsed between principal chrono-
logical eras, 926. Rules for reckon-
ing between given dates, 927. -
INDEX.
551
Declination, 105. How obtained, 295.
Definitions, 82...
/Jegree of meridian, how measured, 210...
Error admissible in, 215. Length of
in various latitudes, 216. 221.
Diameters of the earth, 220, 221.
planets, synopsis, Appendix.
also each planet.)
Dilatation of comets in receding from
the sun, 578.
Dione, 548.
Discs of stars, 816.
JDistance of the moon, 403.; the sun, 357.;
fixed stars, 807. 812...; polar, 105.
JDistricts, natural, in heavens, 302.
Disturbing forces, nature of, 609... Ge-
neral estimation of, 611. Numerical
values, 612. Unresolved in direction,
614. Resolution of, in two modes,
615. 618. I. revts of each resolved
portion, 616... On moon, expressions
of, 676. Geomtrical representations
of, 676. 717.
Diurnal motion explained, 58. Paral-
lax, 339. Rotation, 144.
Double refraction, 202. Image micro-
meter, a new, described, 208. Comet,
580. Nebulae, 878.
Louble Stars, 833... Specimens of each
class, 835. Orbitual motion of, 839.
Subject to Newtonian attraction, 843.
Orbits of particular, 843. Dimen-
sions of these orbits, 844. 848. . Co-
* loured, 851... Apparent periods af-
fected by motion of light, 868.
Pove, his law of temperature, 370.
Of
(See
E.
Barth. Its motion admissible, 15. Sphe-
rical form of, 18. 22... Optical effect
of its curvature, 25. Diurnal rotation
of, 52. Uniform, 56. Permanence
of its axis, 57. Figure spheroidal,
219... Dimensions of, 220. Elliptic
figure a result of theory, 229. Tem-
perature of surface, how maintained,
866. Appearance as seen from moon,
* 436. Velocity in its orbit, 474. Dis-
turbance by Venus, 726.
Eclipses, 411... Solar,420. Lunar, 421...
Annular, 425. Periodic return of,
426. Number possible in a year, 426.
Of Jupiter’s satellites, 538. Of Sa-
turn’s, 549.
Feliptic, 805... Its plane slowly varia-
ble, 306. Cause of this variation ex-
plained, 640. Poles of, 307. Limits,
solar, 412. Lunar, 427.
Egyptians, ancient, their chronology, 912.
Elements of a planet’s orbit, 493. Waria-
tions of, 652... Of double star orbits,
843. Synoptic table of planetary,
&c., Appendix
Ellipse, variable, of a planet, 653. Mo-
mentary or osculating, 654.
Elliptic motion a consequence of gravi-
tation, 446. Laws of, 489... Their
theoretical explanation, 491. -
Ellipticity of the earth, 221.
Elongation, 341. Greatest, of Mercury
and Wenus, 467.
Enceladus, 548, note.
Encke, comet of, 576. His hypothesis
of the resistance of the ether, 577.
Epoch, one of the elements of a planet's
orbit, 496. Its variation not inde-
pendent, 780. Variations incident
on, 731. 744.
Equation of light, 335. Of the centre,
375. Of time, 379. Lunar, 452.
Annual, of the moon, 738.
JEquator, 84.
Equatorial, 185. -
Equilibrium, figure of, in a rotating body,
224.
Equinoctial, 97. Time, 935.
Jºquinox, 293. 303.
Equinoxes, precession of, 312. Its ef.
fects, 313. In what consisting, 314...
Its physical cause explained, 642...
Eras, chronological list of, 926.
Errors, classification of, 133. Instru
mental, 135... Their detection, 140.
Destruction of accidental ones by tak-
ing means, 137. Of clock, how ob.
tained, 293.
Establishment of a port, 754.
Ether, resistance of, 577.
Evection of moon, 748.
Excentricities, stability of Lagrange's
theorem respecting, 701.
Eaccentricity of earth’s orbit, 354. How
ascertained, 377. Of the moon’s, 405.
Momentary perturbation of, investi-
gated, 670. Application to lunar
theory, 688. Wariations of, in orbits
nearly circular, 696. In excentrie
orbits, 697. Permanent inequalities
depending on, 719.
F.
Faculae, 338.
Faye, comet of, 584, and Appendix.
552
INDEX.
Flora, discovery of, 505.
Focus, upper. Its momentary change of
place, 670, 671. Path of, in virtue of
both elements of disturbing force, 704.
Traced in the case of the moon’s vari-
ation, 706... And parallactic inequa-
lity, 712. Circulation of, about a
mean situation in planetary perturba-
stions, 727.
Force, metaphysical conception of, 439.
Forced vibration, principle of, 650.
JForces, disturbing. See Disturbing force.
G.
Galactic circle, 793.
£b.
Galaxy composed of stars, 302. Sir W.
Herschel’s conception of its form and
structure, 786. Distribution of stars
generally referable to it, 786. Its
course among the constellations, 787...
Difficulty of conceiving its real form,
792.
some directions unfathomable,
others not, 798.
Galle, Dr., 506. Finds Neptune in place
indicated by theory, 768.
Galloway, his researches on the Sun's
proper motion, 855.
Gasparis, Sig. De, discovers a new pla-
net (Appendix).
Gauging the heavens, 793.
Gay Lussac, his aéronautic ascent, 32.
Geocentric longitude, 503. Place, 371,
497.
, Geodesical measurements, their nature,
247...
Geography, 111, 205:
Globular clusters, 865.
cal stability, 866.
867.
Golden number, 922.
Goodricke, his discovery of variable stars,
821...
Gravitation, how deduced from phaeno-
mena, 444... Elliptic motion a con-
sequence of, 490...
Gravity, centre of, see Centre of gra-
vity.
Gravity diminished by centrifugal force,
231. Measures of, statical, 234. Dy-
namical, 235. Force of, on the moon,
433... On bodies at surface of the
sun, 440. Of other planets, see their
Ilā, Iſle3.
Gregorian reform of calendar, 915...
Polar distance,
in
Their dynami-
Specimen list of,
Telescopic analysis of, 797. In -
H.
Halley. His comet, 567. First notices
proper motions of the stars, 852.
Hansen. His detection of long inequa-
lities in the moon’s motions, 745...
Harding discovers Juno, 505.
Heat, supply of, from sun alike in sum-
mer and winter, 368. How kept up
400. Sun’s expenditure of, estimated
397. Received from the sun by dif.
ferent planets, 508. Endured by co
mets in perihelio, 592.
IIebe, discovery of, 505.
Heights above the sea, how measured
286. Mean, of the continents, 289.
IIeliocentric place, 500.
IIeliometer, 201.
Hemispheres, terrestrial and aqueous, 284.
IIerschel, Sir Wm., discovers Uranus,
505, and two satellites of Saturn, 548.
His method of gauging the heavens,
793. Views of the structure of the
Milky Way, 786. Of nebular subsi-
dence, and sidereal aggregation, 869,
874. His catalogues of double stars,
835. Discovery of their binary con-
nexion, 839. Of the sun’s proper mo-
tion, 854. Classifications of nebulae,
868, 879, note.
Horizon, 22. Dip of, 23, 195. Rational
and sensible, 74. Celestial, 98. Arti-
ficial, 163.
IIorizontal point of a mural circle, how
determined, 175...
IIour circles, 106; angle, 107; glass,
150.
IIyperion, Appendix, Saturn's satellites.
I.
Iapetus, 548.
Inclination of the moon's orbit, 406. Of
planet's orbits disturbed by Orthogo-
nal force, 619. Physical importance
of, as an element, 632. Momentary
variation of, estimated, 633. Crite-
rion of momentary increase or dimi-
mution, 635. Its changes periodical
and self-correcting, 636. Application
to case of the moon, 638.
Inclinations, stability of, Lagrange’s the-
orem, 639. Analogous in their per-
turbations to excentricities, 699.
Inductions, 923.
Inequality. Parallactic of moon, 712.
Great, of Jupiter and Saturn, 720...
INDEX.
553
Inequalities, independent of excentricity,
theory of, 702... Dependent on, 719.
Intercalation, 916.
Iris, discovery of, 505.
Iron, meteoric, 888.
J.
Julian Period, 924.
formation, 918.
Juno, discovery of, 505.
Jupiter, physical appearance and de-
scription of, 511. Ellipticity of, 512.
Belts of, 512. Gravity on surface,
508. Satellites of, 510. Seen without
satellites, 543. Recommended as a
photometric standard, 783. Elements
of, &c. (See Synoptic Table, Appen-
dix.
Jº and Saturn, their mutual pertur-
bations, 700, 720...
Date, 930. Re-
K.
Rater, his mode of measuring small in-
tervals of time, 150. His collimator,
178.
Repler, his laws, 352, 487,489.
physical interpretation, 490...
Their
L.
Lagging of tides, 753.
Dagrange, his theorems respecting the
stability of the planetary system, 669,
639, 701.
Laplace accounts for the secular accele-
ration of the moon, 740.
I assell, his discovery of the satellite of
Neptune, 524. Of an eighth satellite
of Saturn, Appendix. Re-discovers
two of the satellites of Uranus, 551.
Latitude, terrestrial, 88. Parallels of,
89. How ascertained, 119, 129. Rö-
mer’s mode of obtaining, 248. On a
spheroid, 247. Celestial, 308. Helio-
centric, how calculated, 500. Geo-
centric, 503.
Daws of nature how arrived at, 139.
Subordinate, appear first in form of
errors, 139. Kepler's, 352, 487...
Level, spirit, 176. Sea, 285. Strata, 287.
Leverrier, 506, 507, 767.
Lezell, comet of, 585.
Libration of the moon, 435. Of apsides,
694.
Iight, aberration of, 331. Velocity of,
x * ~ j-a- - cº-
#vºrſ.
331. How ascertained, 545. Equa-
tion of, 335. Extinction of, in tra-
versing space, 798. Distance mea-
sured by its motion, 802... Of certain
stars compared with the sun, 817...
Effect of its motion in altering appa-
rent period of a double star, 863.
Zodiacal, 897.
Local time, 252.
London, centre of the terrestrial hemi-
sphere, 284. -
Longitude, terrestrial, 90. How deter-
mined, 121, 251... By chronometers,
255. By signals, 264. By electric
telegraph, 262. By shooting stars,
265. By Jupiter's satellites, &c., 266.
By lunar observations, 267... Celes-
tial, 308. Mean and true, 375. He-
liocentric, 500. Geocentric, 503. Of
Jupiter’s satellites, curious relations
of, 542.
Lunation (synodic revolution of the
moon), its duration, 418.
M.
|Magellanic clouds, 892...
Magnitudes of stars, 780... Common
and photometric scales of, 780... and
Appendix.
Maps, geographical, construction of, 273.
Celestial, 290... Of the moon, 437.
Mars, phases of, 484. Gravity on sur-
face, 508. Continents and seas of,
510. Elements (Appendix).
Masses of planets determined by their
satellites, 532. By their mutual per-
turbations, 757. Of Jupiter’s satel-
lites, 758. Of the moon, 759. -
Menstrual equation, 528.
Mercator's projections, 283.
Mercury, synodic revolution of, 472. We-
locity in orbits, 474. Stationary points
of, 476. Phases, 477. Greatest elon-
gations, 482. Transits of, 483. Heat
received from sun, 508. Physical ap-
pearance and description, 509. Ele-
ments of (Appendix). -
Meridian, terrestrial, 85. Celestial, 101.
Line, 87, 190. Circle, 174. Marc,
190. Arc, how measured, 213. Arcs,
lengths of, in various latitudes, 216.
Messier, his catalogue of nebulae, 865.
Meteors, 898. Periodical, 900... Heights
of, 904.
Metis, discovery of, 505.
Micrometers, 199.

554
INDEX.
Milky way. (See Galaxy, 802.)
Mimas, 550, and note.
Mira Ceti, 820.
Moon, her motion among the stars, 401.
Distance of, 403. Magnitude and ho-
rizontal parallax, 404. Augmenta-
tion, 404. Her orbit, 405. Revolution
of nodes, 407. Apsides, 409. Oc-
cultation of stars by, 414. Phases
of, 416: Brightness of surface, 417,
note. Redness in eclipses, 422. Phy-
sical constitution of, 429... Destitute
of sensible atmosphere, 431. Moun-
tains of, 430. Climate, 431... Inha-
bitants, 434. Influence on weather,
482, and note. Rotation on axis, 435.
Appearance from earth, 436. Maps
and models of, 437. Real form of
orbit round the sun, 452. Gravity on
surface, 508. Motion of her nodes
and change of inclination explained,
638... Motion of apsides, 676... Wa-
riation of excentricity, 688... Paral-
lactic inequality, 712. Annual equa-
tion, 738. Evection, 748. Variation,
705... Tides produced by, 751.
Motion, apparent and real, 15. Diurnal,
52. Parallactic, 68. Relative and
absolute, 78... Angular, how mea-
sured, 149. Proper, of stars, 852...
Of sun, 854.
#ſountains, their proportion to the globe,
29. Of the moon, 430.
Mowna Roa, 32.
Mural circle, 168.
N.
Nabonassar, era of 926.
Madir, 99.
Webulae, classifications of, 868, 879, note.
Law of distribution, 868. Resolvable,
870. Elliptic, 873. Of Andromeda,
874. Annular, 875. Planetary, 876.
Coloured, ib. Double, 878. Of sub-
regular forms, 881, 882. Irregular,
883. Of Orion, 885. Of Argo, 887.
Of Sagittarius, 888. Of Cygnus, 891.
Nebular hypothesis, 872.
Nebulous matter, 871. Stars, 880.
Neptune, discovery of, 506, 768. Pertur-
bations produced on Uranus by, ana-
lysed, 765... Place indicated by the-
ory, 767. Elements of, 771... Per-
turbing forces of, on Uranus, geome-
trically exhibited, 773. Their effects,
774...
Newton, his theory of gravitation, 490...
et passim.
Nodes of the sun’s équator, 390. Of the
moon’s orbit, 407. Passage of pla-
nets through, 460. Of planetary or-
bits, 495. Perturbation of, 620...
Criterion of their advance or recess,
622. Recede on the disturbing orbit,
624... Motion of the moon’s theory
of, 638. Analogy of their variations
to those of perihelia, 699.
Nomenclature of Saturn’s satellites, 548,
note.
Nonagesimal point, how found, 310.
Normal disturbing force and its effects,
618. Action on excentricity and pe-
rihelion, 673. Action on lunar ap-
sides, 676. Of Neptune on Uranus,
its effects, 775.
Nubeculae, major and minor, 892...
Number, golden, 922.
Nutation, in what consisting, 321. Pe-
riod, 322. Common to all celestial
bodies, 323. Explained on physical
principles, 648.
O.
Obliquity of ecliptic, 303. Produces the
variations of season, 362. Slowly
diminishing, and why, 640.
Observation, astronomical, its peculiari-
ties, 138.
Occultation, perpetual, circle of 113.
Of a star by the moon, 413... Of Ju-
piter’s satellites by the body, 541.
Of Saturn’s, 549.
Olbers discovers Pallas and Westa, 505.
His hypothesis of the partial opacity
of space, 798.
Opacity, partial, of space, 798.
Oscillations, forced, principle of, 650.
Orbits of planets, their elements (Ap-
pendix) of double stars, 843. Of
comets. (See Comets.)
Orthogonal disturbing force, and its ef-
fects, 616, 619.
Orthographic projection, 280.
P.
Palitzch discovers the variability of Al-
gol, 821.
Pallas, discovery of, 505.
Parallactic instrument, 185. Inequality
of the moon, 712. Of planets, 713.
Unit of sidereal distances, 804. Mo-
tion, 68.
INDEX.
555
Parallaz, 70. Geocentric or diurnal,
339. Heliocentric, 341. Horizontal,
355. Of the moon, 404. Of the sun,
357, 479, 481. Annual, of stars, 800.
How investigated, 805... Of particu-
lar stars, 812, 818, 815. Systematic,
862.
Peak of Teneriffe, 32.
Pendulum-clock, 89. A measure of gra-
vity, 235.
Penumbra, 420.
Perigee of moon, 406.
Perihelia and excentricities, theory of,
670...
Perihelion, 368.
Passage, 496.
mets in, 592.
Period, Julian, 924. Of planets (App.).
Periodic time of a body revolving at the
earth’s surface, 442. Of planets, how
ascertained, 486. Law of, 48. Of a
disturbed planet permanently altered,
734...
Periodical stars, 820... List of, 825.
Perspective, celestial, 114.
Perturbations, 602...
Peters, his researches on parallax, 815.
Phases of the moon explained, 416. Of
Mercury and Venus, 465, 477. Of
superior planets, 484.
Photometric scale of star magnitudes, 780.
Piazzi discovers Ceres, 505.
Pigott, variable stars discovered by,
824...
Places, mean and true, 374. Geometric
and heliocentric, 371, 497.
JPlanetary nebulae, 876.
Planets, 456. Zodiacal and ultra-zodia-
cal, 457. Apparent motions, 459.
Stations and retrogradations, 459.
Reference to sun as their centre, 462.
Community of nature with the earth,
463. Apparent diameters of, 464.
Phases of, 465. Inferior and superior,
467. Transits of (see Transit) Mo-
tions explained, 468. Distances, how
concluded, 471. Periods, how found,
472. Synodical revolution, 472. Su-
perior, their stations and retrograda-
tions, 485. Magnitude of orbits, how
concluded, 485. Elements of, 495.
(See Appendix for Synoptic Table.)
Densities, 508. Physical peculiarities,
&c., 509... Illustration of their rela-
tive sizes and distances, 526.
Plantamour, his calculations respecting
the double comet of Biela, 588.
Longitude of, 495.
Heat endured by co-
Pleiades, 865. Assigned by Mädler as
the central point of the sidereal sys-
tem, 861.
Plumb-line, direction of, 23.
observation, 175.
Polar distance, 105.
circle, 170, 172.
Poles, 83. Of ecliptic, 307.
Pole-star, 59. Useful for finding the
latitude, 171. Not always the same,
318. What, at epoch of the building
of the pyramids, 319.
Pores of the sun’s surface, 387.
Position, angle of, 204. Micrometer, ib.
Precession of the equinoxes, 312. In what
consisting, 314... Effects, 313. Phy-
sical explanation, 642.
Praesepe, Cancri, 865.
Priming and lagging of tides, 753.
Principle of areas, 490. Of forced vibra-
tions, 650. Of repetition, 198. Of
conservation of vis viva, 663. Of col-
limation, 178.
Problem of three bodies, 608.
Problems in plane astronomy, 127...
309...
Projection of a star on the moon’s limb,
414, note.
Projections of the sphere, 280...
Proper motions of the stars, 852. Of the
sun, 853.
Pyramids, 319.
Use of, in
Point, on a mural
R.
Radial disturbing force, 615...
Radiation, solar, on planets, 508. On
comets, 592.
Rate of clock, how obtained, 293.
Reading off, methods of, 165.
Reflexion, observations by, 173.
Réfraction, 38. Astronomical and its
effects, 39, 40. Measure of, and law
of variation, 43. How detected by
observation, 142. Terrestrial, 44.
How best investigated, 191.
Repetition, principle of, 198.
Resistance of ether, 577.
Retrogradations of planets, 459. Of
nodes. (See Nodes.)
Rhea, 548, note.
Right ascension, 108. How determined,
293.
Rings of Saturn, dimensions of, 514.
Phenomena of their disappearance,
515... Equilibrium of, 518... Mul-
tiple, 521, and Appendix. Appear
556
INDEX.
ance of from Saturn, 522. Attraction
of on a point within, 735, note.
Rittenhouse, his principle of collimation,
8
Rosse, Earl of his great reflector, 870,
882
Rotation, diurnal, 58. Parallactic, 68.
Of planets, 509... Of Jupiter, 512.
Of fixed stars on their axes, 820.
S.
Saros, 426.
Satellites, of Jupiter, 511.
518, 547. Discovery of an eighth
(Appendix). Of Uranus, 523, 552.
Of Neptune, 524, 553. Used to de-
termine masses of their primaries,
532. Obey Kepler’s laws, 533. Eclipses
of Jupiter's, 535... Other phaenomena
of 540. Their dimensions and masses,
540. Discovery, 544. Velocity of
light ascertained from, 545.
Saturn, remarkable deficiency of density,
508. Rings of, 514. Physical descrip-
tion of, 514. Satellites of, 547, and
Appendix. (See also elements in Ap-
pendix.)
Sea, proportion of its depth to radius of
the globe, 31. Its action in modelling
the external form of the earth, 227.
Seasons explained, 362... Temperature
of, 366.
Sector, zenith, 192.
Secular variations, 655.
Selenography, 437.
Sextant, 198...
Shadow, dimensions of the earth’s, 422,
428. Cast by Venus, 467. Of Jupi-
ter’s satellites seen on disc, 540.
Shooting stars used for finding longi-
Of Saturn,
tudes, 265. Periodical, 900. (See
Meteors.)
Sidereal time, 110, 910. Year. (See
Year.) Day. (See Day.)
Signs of zodiac, 380.
Sirius, its parallax and absolute light,
818.
Solar cycle, 921.
Sphere, 95. Projections of, 280. Attrac-
tion of, 735, note.
Spheroidal form of Earth (see Earth) pro-
duces inequalities in the moon’s mo-
tion, 749.
Spots on Sun, 389...
Stars visible by day, 61. Fixed, 777...
Their apparent magnitudes, 778...
Comparison by an astrometer, 783.
Law of distribution over heavens,
785... alike in either hemisphere, 794.
Parallax of certain, 815. Discs of,
816. Real size and absolute light,
817. Periodical, 820. Temporary,
827. Irregular, 830. Missing, 832.
Double, 833... Coloured, 851, and
note. Proper motions of, 852. Irre-
gularities in motions not verified, 859.
Clusters of 864... Nebulous, 879...
Nebulous-double, 880.
Stationary points of planets, 459.
determined, 475.
Venus, 476.
Stereographic projection, 281.
Stones, meteoric, 898. Great shower of,
65.
Struve, his researches on the law of dis-
tribution of stars, 793. Discovery of
parallax of a Lyrae, 813. Catalogue
and observation of double stars, 835.
Struve, Otto, his researches on proper
motions, 854.
Style, old and new, 932.
Sun, oval shape and great size on hori-
zon explained, 47. Apparent motion
not uniform, 34. Orbit elliptic, 349.
Greatest and least distances, 350.
Actual distance, 357. Magnitude,
358. Rotation on axis, 359, 390.
Mass, 360. Physical constitution,
386. Spots, ib... Situation of its
equator, 390... Maculiferous zones
of, 393. Atmosphere, 395. Tempe-
rature, 396. Expenditure of heat,
397. Eclipses, 420. Density of, 447.
Natural centre of planetary system,
462. Distance, how determined, 479.
Its size illustrated, 526. Action in
producing tides, 751. Proper motion
of, 854... Absolute velocity of in
space, 858. Central, Speculations on,
861.
Sunsets, two, witnessed in one day, 26.
Survey, trigonometrical, nature of, 274.
Synodic revolution, 418. Of Sun and
moon, ió.
How
Of Mercury and
T.
Tangential force and its effects, 618.
Momentary action on perihelia, 673.
Wholly influential on velocity, 660,
Produces variations of axis, ib...
Doubles the rate of advance of lunar
apsides, 686. Of Neptune on Uranus,
and its effects, 774.
INDEX.
557
Telescope, 154. Its application to astro-
nomical instruments, 117.
Telescopic sights, invention of, 158, note.
Temperature of earth's surface at differ-
ent seasons, 366. In South Africa
and Australia, 369. Of the sun, 396.
Tethys, 548, note.
Theodolite, 192. Its use in surveying,
276.
Theory of instrumental errors, 141. Of
gravitation, 490... Of nebulous sub-
sidence and sidereal aggregation, 872.
Tides, a system of forced oscillations,
651. Explained, 750... Priming and
lagging of, 753. Periodical inequali-
ties of, 755. Instances of very high,
756.
Time, sidereal, 110, 327, 911. Local,
129, 152. Measures angular motion,
149. How itself measured, 150...
Very small intervals of, 150. Equi-
noctial, 257, 925... Measures, units,
and reckoning of, 906...
Titan, 548, note.
Titius, Prof., his law of planetary dis-
tances, 505, note.
Trade winds, 239...
Transit instrument, 159...
Transits of stars, 152. Of planets across
the sun, 467. Of Venus, 479... Mer-
cury, 483. Of Jupiter’s satellites
across disc, 540. Of their shadows,
549.
Transparency of space, supposed by Ol-
bers imperfect, 798.
Transversal disturbing force, and its
effects, 615...
Trigonometrical survey, 274.
Tropics, 93, 380.
Twilight, 44.
|U.
Umbra in eclipses, 420. Of Jupiter, 538.
Uranography, 111, 300.
Uranographical corrections, 342... Pro-
blems, 127... 309...
Uranus, discovery of, 505. Heat received
from sun by, 508. Physical descrip-
tion of, 523. Satellites of, 551. Per-
turbations of by Neptune, 760... Old
observations of, 760.
W.
Wanishing point of parallel lines, 116.
Line of parallel planes, 117.
Variation of the moon explained, 705...
Variations of elements, 653. Periodical
and Secular, 655. Incident on the
epoch, 731.
Velocity, angular, of sun not uniform,
350. Linear, of sun not uniform, 351.
Of planets, Mercury,Venus, and Earth,
474. Of light, 545. Of shooting
stars, 899, 904.
Venus, synodic revolution of, 472. Sta-
tionary points, 476. Velocity of, 474.
Phases, 477. Points of greatest
brightness, 478. Transits of, 479.
Physical description and appearance,
509. Inequality in earth's motion
produced by, 726. In that of the
moon, 743... e
Vernier, 97.
Vertical, prime, 102. Circles, 100.
Vesta, discovery of, 505.
W.
Weight of bodies in different latitudes,
ºf 822. Of a body on the moon, 508.
On the sun, 450.
Winds, trade, 240...
Y.
Year, sidereal, 305. Tropical, 383.
Anomalistic, 384, and day incommen-
surable, 913. Leap, 914. Of confu-
sion, 917. Beginning of, in England,
changed, 932.
Z.
Zenith, 99. Sector, 192.
Zodiac, 305.
Zodiacal light, 899.
20mes of climate and latitude, 382.
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Two remarkable meteorites have fallé.
in Iowa within the last twenty years. Feit.
ruary 12, 1875, an exceedingly brilliant me
teor, in the form of an elongated hors;
shoe, was seen throughout a region of a,
least 400 miles in length and 250 in breadth, , ,
lying in Missouri and Iowa. It is de-
scribed as “without a tail, but having a
flowing jacket of flame. Detonations were
heard so violent as to shake the earth and 4
to jar the windows like the shock of an *
earthquake,” as it fell about 10:30 p... xm, a |
few miles east of Marengo, Ia. The
ground, for the space of some seven miles
in length by two to four miles in breadth,
was strewn with fragments of this meteor,
varying in weight from a few ounces tº i
Seventy-four pounds. . .
On May 10, 1879, a large, and extraor-
| dinary luminous meteor exploded with ter-
rific noise, followed at slight intervals
with less violent detonations, and struck.
the earth in the edge of the ravine near
Estherville, Emmet Co., Ia., penetrate
ing to the depth of fourteen feet. Vith-
in two miles other fragments were found,
| One of which weighed 170 pounds, and an-
other thirty-two pounds. The principal
mass weighed 431 pounds. All the discov-
ered parts aggregated about 640 pounds.
The one of 170 pounds is now in the cabinet
of the state university of Minnesota. The
composition of this aerolite is peculiar in
many respects; but, as in nearly all aero-
lites, there is a considerable proportion of
iron and nickel.—Iowa. State Register.




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