A 546452
OHN
SURVENC
DLEYS
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La. State University.
No
Class
LIBRARY.
14p9 12
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Division
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Shelf
Class
500
Book
M36
Library of the
Louisiana State University,
A. and M. College.
v, 2
A
Accession No.
3
1
A NEW and COMPREHENSIVE
SYSTEM
OF
MATHEMATICAL INSTITUTIONS,
Agreeable to the
PRESENT STATE
OF THE
NEWTONIAN MATHESIS.
MO Le
ΟΙ JIN'
Containing the Inftitutes, or Principles of
I. The Phyfico-Mechanical MA-IV. OxODROMICS, or NAVI-
THESIS.
II. Univerfal OPTICS.
III. Univerfal PERSPECTIVE.
GATION.
V. BALLISTICS, or GUNNERY.
VI. HOROLOGY,
or CLOCK-
WORK.
By BENJAMIN MARTIN.
LONDON:
Printed and fold by W. OWEN, near Temple-Bar, and
by the AUTHOR, at his Houſe in Fleet-ftreet.
MDCCLXIV.
ARTES
1817
VERITAS
LIBRARY
ՀԱՍԱ
SCIENTIA
OF THE
UNIVERSITY OF MICHIGAN
TUEBOR
SI QUERIS PENINSULAM AMⱭNAM,
CIRCUMSPICE
RECEIVED IN EXCHANGE
FROM
Louisiana
State University
!
510
A
M 360
TO THE
KIN
JUTON
RAV
3.5-
M38
G
This NEW and COMPREHENSIVE
WAR
Y ST E M
SY
OF
MATHEMATICAL INSTITUTIONS,
Publiſhed,
By his ROYAL PERMISSION,
Under His Moft Gracious and Aufpicious
PATRONAGE,
Is now, with all Humility,
Infcribed,
By his MAJESTY'S
Moft Loyal,
Dutiful, and Obedient
Subject and Servant,
BENJ. MARTIN.
Lowserve Eich.
Clutchin
t
Meterisind
INSTITUTIONS
OF THE
PHYSICO-MECHANICAL MATHESIS.
938.
CHA P. I
The MENSURATION of SUPERFICIES.
EFORE we can give any rational Account
of the philofophical and mathematical Sciences,
and their practical Application, it is neceffary
to premife thofe mechanical Principles which are
founded in the Nature of Things, and make the Bafis of all
that Sort of Knowledge; without thefe no Man can underſtand
Philoſophy, or make any Progrefs in the true Mathefis. We
fhall propofe them if a natural Order, and in the moft perfpi-
cuous Method we can, and begin firft with the practical Men-
furation of Superficies and Solids, which is the firft Doctrine
of practical Science.
}
$
939. The Dimenſions or Content of all Superficies are eſti-
mated in that of a Square; as a Square Inch; a fquare Foot; a
Square Yard, Rod, Mile, &c. and what Number of each leffer
Denomination is contained the greater, is fhewn in the follow-
ing Table:
Square Inçlies.
144
1 Foot fquare.
1296
9
1 Yard ſquare.
T
39204
22
1568160
16890
1210
6272840
43560
4840
40
160
1 Rood fquare.
4= 1 Acre fquare.
303- 1 Pole fquare:
4014489600278784003097600
102400 2560 6401 Mile fquate.
= =
940. To measure a SQUARE ACD B.
Suppoſe the Length of each Side be
Inchies, Feet, Yards, &c. then A B=4; and
4
ABXACA B² = 4 × 416, fquare
B2
X
Inches, Feet, Yardi, &c. as is evident, by In-
fpection, in the Figure.
A
C
VOL. II.
B
D
git Is
INSTITUTIONS
941. To measure a PARALLELOGRAM ABDC.
=
Admit the Length AC = 6, and the
Breadth AB = 4; then AC × A B
24 fquare Inches, Feet, &'c. according to
the Meaſure in which the Sides were taken.
A
B
D
942. To measure a TRIANGLE BDE.
A E
C
It is evident by Inſpection, the Tri-
angle A D B is equal to half the Paral-
lelogram ABD C; but becauſe AC
BD, the Triangles ADB and BED
are equal (635). Therefore the Tri-
angle BED (BAD) ABXAC EFX BD.
Therefore if the perpendicular Altitude EF (= A B) = 4:
and the Bafe BD6; then EFX BD 2×6=12=
=
the Area of the Triangle BED, as required.
2
B F
=
943. To measure the RHOMBUS ABDC.
The Rhombus ABDC is equal to the
Parallelogram A E F C, becauſe A C||
BF, and AC-FE (655.) Therefore
the Rhombus is AE XAC-AE
=
✰ BD. Hence let the Bafe B D 4, B
A
ED
D
F
and the perpendicular Height A E = 3,5; then is the Area of
the Rhombus ABDC= 4 × 3.5
14.
944. To measure the RHOMBOIDES ABDC.
Here again the Rhomboides is =
to the Parallelogram AEFC (655)
Therefore let the Bafe B D = 6,
the Altitude AE (CF) = 4;
then is Area of the Rhomboides
A B D C = 4 × 6 = 2:44.
A
BE
D
F
945.
of the Phyfico-Mechanical Mathefis.
945. To measure a TRAPEZIUM A B C D.
Draw the Diagonal AC; on which,
from the Angles B, D, let fall the Per-
pendiculars B g, De; then is the Tri- A
angle ACB Parallelogram Aae C,
.I
2
a E
and the Triangle A CD = Parallelo- 6 D
d
gram Abd C. Confequently the whole Trapezium A B C D
Ι
half the Parallelogram abcd; ab x b d = ÷ Bg + D ¢
= 1 1
Thus, fuppofe the Diagonal A C = 6, and half
X A C.
the Sum of the Perpendiculars =
B g + De
2
=12, the Area of the Trapezium, as required.
2; then 6 × 2
946. To meaſure a CIRCLE EGF H.
A
E
C
We have fhewn the Area of the Circle
whofe Diameter is 1, is 0,785398 (828)
and the Areas of all Circles are as the
Squares of their Diameters (840.) Let A,
D, be the Area, and Diameter of the given
Circle. Then, as 12: 0.7854 DD: A.
Therefore the Area A 0.7854 DD. Thus let D= 4,
= D4,
then DD 16, and 0,7854 DD = 12,5664 = Area
fought.
B HD
947. Let A = Area, D = Diameter, and P = Periphery
of a Circle; then any one of theſe being given, the other
may be found by the following Equations, (fee 824, 830,
840,) viz.
Given
D, then 3.1416 DP; and 0.7854 D² = A.
A
P ÷ 3.1416.≈ D, or 0,3183 P = D.
P² ÷ 12.5664 = A, or 0.07957 P² = A.
A ÷ 0,7854 = D, or /1.2732 A = D.
12.5664 A — P, or ✔ A÷0,07957 – P.
948.
To measure any regular POLYGON.
For Example, the Hexagon ABCDEF. Now fince theſe
Figures confift of as many Ifofceles Triangles ANF as they have
B 2
Sides
4
INSTITUTIONS
Z
C
A
N E
F
Sides; and the equal Sides of thofe Triangles
ANN F, are the Radii of the circum-
fcribing Circles, and the perpendicular Heights B
NG, the Radii of the infcribed Circles; and
becauſe AFX NG Area of the Tri-
angle ANG (942); therefore the Sum of all the Areas (or
Area of the whole Polygon) will be equal to GNX Sum of
all the Bafes or Sides. Now in the Hexagon, the Triangle
ANF is equiangular and equilateral; and therefore putting
ANAF = 1, we have NG ✓ A N
A G2
0,866; let the Sum of the Sides be S6, then GNX
풀
​$ = 0,866 X 3 + 2,598 the Area of the Hexagon re-
quired.
949. But fince regular Polygons are fimilar Figures, they
will be to each other as Squares of their Sides (670.) There-
fore the Area of any Polygon, whofe Side is 1, being mul-
tiplied by the Square of the Side of any other Polygon of the
fame Sort, will give the Area of that other Polygon. Thus,
let the Side of any Hexagon be 15; the Square of which is
225, then 225 × 2,598 = 583,05, the Area of the Hexagon,
whofe Side is 15, as required.
950. And that the Area of any regular Polygon may be
had in the fame Manner directly, the Areas of each, fuppofing
the Side 1, are computed by the above Method, as in the
Table following. (Art. 847.)
1
17
१
Sides.
Names.
Areas.
3
Trigon,
0.433013
4
Tetragon,
1.000000
5
Pentagon,
1.720477
6
Hexagon,
2.598076
Heptagon,
3.633959
Octagon,
4.828427
9
Enneagon,
6.181827
Decagon,
7.694209
Endecagon,
9.365675
12
Dodecagon,
II,197920
CHAP.
of the Phyfico-Mechanical Mathefis.
CHA P. II:
The MENSURATION of SOLIDS.
951. THE Solidity, or folid Content of Bodies is eftima-
ted in that of a Cube; as an Inch Cube, Foot Cube,
c. which are more generally called, a cubic Inch, Foot, &c.
A Table of this Sort of Meaſure from the leaft to the greateſt
Denomination here follows;
Cubic Inches.
17283
46656
7762329:
1963885176=
15711081408
I cubic Foot.
27
4492
то
1136507100
&
9092061
TO
1 cubic Yard.
166_3
42092 8
336743
I cubic Pole,
253
2024
I cubic Rood.
8= I c.Acre.
254358061056000147197952000
5451776000 =32770584=129528=16191=1 c,M.
952. To measure a CUBE A G.
Suppoſe the Length of the Side AB =4
Inches, then A B× ABA B² = 16;
and A BX A B A B³
bic Inches for the folid Content. (See
64 cu-
E
F
A
622.)
B
953. To measure a PARALLELOPIPEDON.
Let the Length AC = 6 Inches,
the Breadth AE=3; and the Depth
AB 4. Then
A
G
E
F
=
Multiply the Length
AC 6
A
By the Breadth
And that Product A C× AE
AF
AE= 3
18
By the Depth
AB
4 B
D
This Product is the fold Con-
tent ACX AEX AB 72 (by 622.)
954
INSTITUTIONS
954. To measure a PRISM AC.
Suppoſe it a triangular Priſm, and let the
Sides of the Bafe be equal, and A B = 4 In-
ches. Then A B² = 16; and 16 × 0,433
=6,928 fquare Inches, the Area of the Bafe
ADB (942.) Then let the Height be alfo
AC4; then 6,928 4
×
27,712 =
27 cubic Inches, the Solidity required.
I
C
A
955. To measure a PYRAMID A E DB.
=
Let it be a triangular Pyramid; and each Side
of the Bafe, viz. A B = 4 Inches, then the
Area of the Bafe A D B 6,928; and fup-
pofing its perpendicular Height ED 4, the
Product 6,928 × 4 = Solidity of a Priſm, whoſe
Bafe and Altitude are the fame as thofe of the
4
A
E
E
B
Pyramid (954), one Third of which, viz. × 6,928 =
Solidity of the Pyramid, (by 833.)
9,237=
3
956. To measure a CYLINDER CF.
C
G
Let the Diameter of its circular Bafe EF-4
Inches, then the Area will be 12,5664,
(by 946.) And fuppofing the Height of the
Cylinder EC 4; then 12,5664. X 4
50,2656 Solidity of the Cylinder required. E
(See 830, 831.)
957. To measure a right CoxE EDF.
A
Let the Diameter of its Bafe be EF-
4, then is the Area of the Bafe = 12,5664.
Suppoſe the perpendicular Height DC =
3,46; then 12,5664 × 3,46 — 43,478;
one Third of which is the Solidity of the
43,478
Cone, viz.
3
835.)
D
H
E
= 14,492. (See 834,
F
H
958
of the Phyfico-Mechanical Mathefis.
958. To measure a SPHERE, or GLOBE.
Let the Diameter of the Sphere be EF
≈ 4; then is the Area of its greateſt Cir-
cle = 12,5664; this multiplied by 4
is = 50,2656 = the circumfcribing Cy-
linder. Then two Thirds of this, viz.
2
3
F
E
H
× 50,2656=33,51, &c. the Solidity of the Sphere, (by 873.)
Superficies of the Sphere,
N. B. 4 × 12,5664 = 50,2656
in Square Meafure, (by 838.)
959. Or thus; becauſe Spheres are as the Cubes of their Di-
ameters (841.) Then fince the Solidity of the Sphere whofe
Diameter is 1, is 0,5236, (837, 847) if the Diameter of any
other Sphere be given, as EF4; fay, as 1:0,5236:: 4ª
=64: 0,5236 × 64 = 335515 &c. the Solidity of the Sphere,
as above.
960. After the fame Manner may be meaſured the Spheroid
of (844.) Alfo the Segment of a Sphere may be eaſily mea-
fured, by the Theorem in (836,) and of the Spheroid by the
Theorem in (844.)
CHA P. III.
The Philofophical PRINCIPLES and Laws of Mo-
TION and GRAVITATION.
961. M
MATTER,
ATTER, or SUBSTANCE, of which Bodies are
compofed, in itſelf confidered, or in refpect of its
Effence is unknown to us; this is acknowledged by Sir ISAAC
NEWTON, and all Mankind befides. All that we know of
Matter is what relates to its various Properties and Qualities
which preſent themfelves to our Senfes.
962. The firft and principal of which is what Sir I. New-
ton calls the vis inertia of Matter, or its natural İnactivity. No
Man ever yet obferved any Part of Matter to have a Principle of
Action in itself; but on the other Hand, it is abfolutely paffive,
and
00
INSTITUTIONS
and fubject to the Influence of every external Agent. And
from hence is deduced the Firft of thoſe, which are called the
General Laws of Nature, viz.
963. LAW I.
Every Body will perſevere in its State of Reft or moving uniformly
in a right Line, unless it be compelled to change its State by Forces
imprefed.
Thus a Bullet would continue at Reft in the Gun for ever,
if it was not expelled by the Force of Powder; but being
thereby put into Motion, it would for ever move on in the
Direction of the Axis of the Barrel, if not retarded by the
Refiftance of the Medium, and carried downwards from that
right Line by the Power of Gravity. We fhew by Experi-
ment, that the lefs the Friction is of the Axis of a moving
Wheel, the longer its Motion continues. And we obferve the
vaft Bodies of the Comets and Planets preferve their Motions
undiminiſhed (as to Senfe) for many 1000 Years in unrefifting
Mediums.
964. Since by Reaſon of the Inactivity of Matter (962,)
there is nothing in any Body that can augment, diminiſh, or
any Ways alter or vary the Action or Effect of any Force im-
preffed, we thence deduce the fecond General Law of Mo-
tion, viz.
5
LAW II.
The Change of Motion is proportional to the moving Force im-
prefs'd, and is made according to the right Line, in which that Force
is imprefs'd.
If any Force generate any Motion, a double, or triple
Force will generate twice or thrice as much. But the Altera-
tion in reſpect of the Direction of the Motlon, is a compli-
cated Affair, which we ſhall farther confider hereafter.
965. When Bodies act upon each other, they do it by
Contact or Gollifion, and fince in this Cafe, the Action itfelf is
but one and the fame between both; the Effects which it pro-
duces muft of Courſe be equally divided between both Bodies,
and thence an equal Mutation of their State previous to the
Stroke
1
of the Phyfico-Mechanical Mathefis.
Stroke muft needs follow. From this Confideration is deduced
the third General Law of Nature, viz.
LAW III.
Re-action is always equal and contrary to Action; or the Actions
of two Bodies mutually upon each other are always equal, and direc-
ted towards contrary Parts.
966. The Truth of this is abundantly confirmed by Experi-
ments. If the Finger preffes a Stone, the Stone re-acts and
preffes the Finger. The Hammer ftriking the Anvil, receives
the fame Stroke from it, and is thereby made to rebound. The
LoadЛtone, if fixed, attacts the moveable Iron; and the fix'd
Iron equally attracts the moveable Magnet. The Horfe draws
the Stone, and the Stone equally draws the Horſe, becauſe the
Action, or Force in the Rope which connects them is one and
the ſame, and muſt act equally at each End, viz, upon the
Horſe and Stone.
967. But before we can reafon well upon the Subject of
Motion, we muſt underſtand by juft Definitions, what Ideas
we are to fix to that Word. For the Word Motion is become am-
biguous, and is uſed in a fimple and complex Senſe. Motion, in its
fimple Acceptation, is only a Change of Place in Bodies. But in the
complex, or phyfical Senfe, it implies all the Change that is made in
the State of a Body in regard both to its Quantity of Matter and
Velocity of Motion. And in this Respect, it is properly called
the Momentum, or Quantity of Motion, and which is always e-
qual to the Force which produces it by Law II. (964.)
968. That the Quantity of Motion is as the Mafs of Mat-
ter, (cæteris paribus) is evident from hence, that the Motion of
the whole Maſs is the Sum of the Motions. of all the Parts,
or Particles; as the Number of Particles moved, therefore, is
greater or lefs, fo will be the Sum of all their Motions, and
confequently fo will be the Aggregate, or whole Motion or
Momentum of the Body.
969. Again, the Momentum, or Quantity of Motion, is
(cæteris paribus) as the Velocity of the fimple Motion. For fince
Velocity has Regard to the Space defcribed in the fame Time, (778)
it is evident, that if a Body deſcribes twice the Space that an-
VOL. II.
other
C
ΤΟ
INSTITUTIONS
other equal Body defcribes in the fame Time, it is plain there
is twice the Change of Place, and therefore twice the fimple
Motion (967) produced in the ſame Time. But the Velocity
is alſo twice as great; therefore the Motion is as the Velocity.
970. Since then the Quantity of Motion is as the Maſs of
Matter fimply, and the Velocity fimply; it will be conjointly
as the Rectangle or Product of both, when no Regard is had
to either fingly. Therefore putting M Maſs of Matter,
V = the Velocity, and Q = the Momentum or Quantity of
Motion; then in Symbols we have Q: M × V; and thence
* V
!
:
O
M
I
I
: ех M; and M: QX V
971. Let S Space deſcribed by a Body in Motion, and
Tthe Time of deſcribing it. Now fince in equal Times
the Spaces deſcribed will be as the Celerities of Motion (969,)
therefore S: V; alfo it is manifeft, that in deſcribing the fame
Spaces, the Times will be greater as the Celerities of Motion
are lefs; that is, the Time is inverſely as the Celerity or T :
I
; and ſo alfo V. Whence fince V is fimply as S, and
V: ㄓ​ˊ
I
S
T'
it will be conjointly V:
or V T : S.
Τ'
S
972. Becauſe V=
M
(970) = — (971,) therefore QT
T
MS
SM; and Q =
=
T
I
MS X
Τ
That is the Quan-
tity of Motion is in the compound Ratio of the Space and Maſs di-
rectly, and inverſely as the Time.
973. The Quantity of Matter (M) is (cæteris paribus) as
the Bulk (B) of a Body; for in twice the Bulk there will be
twice the Matter; therefore it is M: B. Again, the Quan-
tity of Matter (M) will be (cæteris paribus) as the Denſity (D),
that is, as the Number or Sum of the Particles in equal
Bulks:
Q
* The Reader is defired to obferve once for all, that V: M
denotes only, that V is always proportional to
V
is the fame as
M
; but V =
and fhews that the Ratio of 1 to V is
M
I M
the fame as the Ratio of M to Q
of the Phyfico-Mechanical Mathefis.
II
Bulks; therefore M: D. Wherefore conjointly it will be M
: B × D.
QT (972)
974. Becauſe M =
ет
S
I
S B D × —;
(972) = BD; therefore Q
or the Quantity of Motion is compounded of the
direct Ratio of the Space, Bulk and Denfity, and the inverfe Ratio
of the Time.
975. Of Forces that actuate the Particles of Matter, we
obferve the following Variety. (1.) A Force or Power of one
Kind cauſes the Particles to adhere firmly to each other; and
hence it is called the Force or Power of COHESION. (2.)
Another Sort of Power obliges the Particles of Matter, under
fome Circumftances, to recede and fly from each other;
which Power is therefore called the Repellent, or repulfive Force.
(3.) A third Power caufes all large Portions of Matter, or
Bodies, to tend mutually towards each other, in Directions
to their Centers; and hence it is called a Centripetal Force.
976. What thoſe Powers are in themſelves, or how they
differ, I fhall not pretend to enquire; nor alſo what is the par-
ticular Modus agendi, or Manner of actuating the Particles of
Matter, viz. whether it be by Attraction, Implufton, or other-
wife; fince in theſe Reſearches, there is but little Reafon hi-
therto to expect any fuccefsful Discoveries. It will be quite
fufficient, if we can make ourfelves thoroughly acquain-
ted with their Phanomena and Effects, and apply them in a pro-
per Manner to the various Ufes of Life.
977. The PowER OF COHESION refpects the ſmalleſt
Particles of Matter, and extends to but very fmall Diſtances,
as is plain by numberlefs Experiments; it is therefore propor-
tional to the Surfaces in contact between two Corpufcles, or
their Coheſion is fo much the greater, by how much the Sur-
faces are larger in which they touch.
978. Hence, according to the various Figures of Particles,
they touch by different Quantities of Surface, and cohere with
different Degrees of Firmneſs. Thus, for Inftance, if all
were nearly cubical, they would touch by a great Quantity of
Surface, and conftitute a very hard and firm Body. But on
the other Hand, if we fuppofe Particles truly ſpherical, they
C 2
will
12
INSTITUTIONS
will touch but by a very ſmall Portion of their Surfaces, and
cohere very flightly; and of Courſe, will conftitute a Body
whoſe Parts will be very eafily moved among themſelves, and
yield to any impreſſed Force, which Cafe we call the Fluidity
of Bodies. Hence Bodies receive their various Degrees of
Hardneſs and Softneſs, Fixity and Fluidity, Firmness or Looſeneſs
of Texture, and all other Qualities depending thereon.
ड
979. That this Power is exceeding great appears by many
Experiments. Thus two Balls of Lead, having their Surfaces
pared fmooth, will cohere with a Force equal to 150 lb. tho'
they touch upon no more than of an Inch fquare. Two
Braſs poliſhed Planes, 2 Inches Diameter, fmeared over with
Greafe and put together very hot; will, when cold, cohere fo
firmly as to require 950 lb. to ſeparate them. And Wires of
feveral Sorts of Metal of an Inch Diameter required the
Weights to pull them afunder, as in the following Table are
Specified.
I
TO
Lead
Tin
Copper
Brafs
Silver
Iron
Gold
29 lb.
49 lb.
299 lb.
360 lb.
370 lb.
450 lb.
500 lb.
980. Though this Power acts with fuch prodigious Force
near the Surface, it decreaſes in fuch a Manner as to become
nearly infenfible in the leaft fenfible Diſtance from the Surface;
not only fo, but at a certain ſmall Diſtance it is converted into
another Kind of Force, or at leaſt, it acts in a Manner exact-
ly contrary to what it did before; for it now caufes the Parts
of Matter to recede from each other, and to remain at certain
equal Diſtances or Intervals among themſelves. And thus mo-
dified, it is called the Repulfive Power in Matter.
981. That this is Fact we are affured from divers Experi-
ments, and many Phanomena of Bodies. Thus the Magnetic
Needle will be ſtrongly attracted by either Pole of the Magnet
in contact with it, but at a very ſmall Diſtance the fame Power
becomes repulſive in one Pole, and repels the faid Needle from
it.
of the Phyfico-Mechanical Mathefis.
13
it *. So the Particles of Fluids cohere by this Power, while in
a State of Contact, but when feparated by Heat, they repel
each other, and exift in the Form of VAPOUR or STEAM.
Again, 'tis well known by numberless Experiments, that in all
folid Bodies there is a certain Matter which, while in contact
with the other Parts, does firmly adhere, and is ftrongly con-
nected with them by this Force; but when by natural or artifi-
cial Fermentation, it is difengaged or fet at Liberty, it imme-
diately (by a repellent Power among its Particles) expands in-
to a fine etherial Fluid, every Way like common Air. But of this
we ſhall ſpeak more hereafter.
982. The Confequence of ſuch a repulfive Power among
the Particles of Matter, is, that they having attained an Equi-
librium, act mutually upon each other, and become fufceptible
of Compreſſion and Condenſation; and of Expanſion and Rarefac-
tion; as is well known by common Experiments on Air, Va-
pour, &c. Again, as Action and Re-action are equal (966,)
it follows, that when any Force is impreffed upon the Parti-
cles of fuch a Fluid, they all jointly refift the fame; and when
the impreffed Force is removed, by Virtue of this Power, the
Particles all retreat to their primitive equidiftant Stations, with
a Force equal to that impreffed. And this Renitency or refti-
tuent Force, is what we call the SPRING, or ELASTICITY of
fuch Sort of Bodies.
983. We alfo further learn by Experiments of Thermome-
ters, &c. that Heat augments and Cold diminiſhes this elaftic
or expanfive Force in Bodies; and that their natural Dimen-
fions are hereby continually altering, as is evident not only in
Air, but in denfer Fluids, as Spirit of Wine, Water, and
even Mercury itſelf. Yea, folid Bodies difcover the fame Pro-
perties in feveral Degrees; Ivory is found to be very elaſtic;
Metals of all Sorts expand and contract with Heat and Cold,
as we fhew by the PYROMETER, and is otherwiſe known by
common Experience.
984. The third Sort of Power, or Agent, (mentioned 975,)
is
* Though it be denied by fome, that the North Pole of the Mag-
net does attract the North End of the Needle juſt upon the Ends,
it is certain, by Experiments, that this attracting Force appears ex-
tremely near the Ends; and there at a fmall Diſtance it becomes
repulfive.
14
INSTITUTION'S
is a centripetal Force, and this we obferve to take Place among
the largeſt Bodies or Syftems of Matter. The Phenomena
which demonftrate fuch a Power moſt fenfibly to us are thofe of
heavy Bodies falling to the Earth. This Power we call GRA-
VITY, and this Tendency of Bodies to the Earth is called
Gravitation.
985. That this is a centripetal
Force, or that Bodies are thereby
made to tend to the Center of the
Earth is hence evident, that all
Bodies are obſerved to fall in right
Lines to the Surface of the Earth
every where.
But becauſe the
Earth is fpherical, thofe Lines
which are perpendicular to the
Surface do all pass through the
6
E
е
Center of the Earth, as is evident from a View of the Figure
annexed. For let A B D E be the Earth, Cits Center, then
fuppofe a Body (a) falling in the right Line a A to the Earth
in A, that Line if produced muft go through the Center C.
The fame is to be ſaid of any other Bodies at b, d, e.
986. But any Virtue propa-
gated in right Lines to or from a
H
с
I
Center, will have its Energy on
Bodies every where, as the
Square of the Distance from the
Center inverfely. For, fuppofe a
Cone of this gravitating Virtue
be reprefented by IHC, termi-
nating in the Center of the Earth
C. If then we make CA-
AFFH; the Energy of the
Virtue will be at thofe Diftan-
ces inverſely as the circular A- D
F
G
A
B
C
E-
reas A B, F G, and HI; for the more it is expanded and ra-
rified, the lefs will its Effect be upon the fame Body. But
thefe circular Areas are as the Squares of their Diameters
(840,) or of the Semidiameters A a, Fb, He; which are as
the Squares of the Diſtances CA, CF, CH, (by 656.)
Therefore
of the Phyfico-Mechanical Mathefis.
15
Therefore the Energy of this Power decreases as the Square of the
Diſtance from the Center increaſes.
987. But becaufe the Semi-diameter of the Earth is near
4000 Miles, and the greateft Height to which we can elevate
Bodies above the Earth's Surface being but a Mile or two,
'tis evident this Force in fo fmall a Diſtance will not fenfibly
vary; and therefore may be efteemed as acting uniformly through
any Spaces near the Earth's Surface.
988. The Action of this Power is conflant or perpetual;
this appears from hence, that the Velocity of falling Bodies is
conftantly accelerated or increafing, as we know by Experiments.
If a Body be put into Motion by a fingle or inftantaneous Im-
pulfe, the Velocity of that Body would be uniform (by Law I.
963,) and its Motion rectilineal. If the Power which puts a
Body in Motion be temporary, or acts only for a certain Time,
the Velocity during that Time will be accelerated, and af-
terwards become uniform. But if the Power acts inceſſant-
ly, the Body is every Moment impelled, and its Velocity muſt
every
Moment increaſe. Such therefore is the Power of Gra-
vity.
989. Since Gravity acts conftantly and uniformly, the Velo-
city of Bodies will be equably accelerated in their Deſcent to the
Earth; for fince the Impulfe communicated each Moment is
the fame, it will generate equal Velocities in the ſeveral equal
Moments, which conftantly added together, make an uniform-
ly increafing Sum, and therefore an equably accelerated Velocity.
For the Velocity at any Moment is equal to the Sum of all the
momentary Increments of Velocity from the Beginning.
990. Hence the Velocity (V) of the Fall is proportional to
the Time (T). Becauſe fince the Action of Gravity is uniform,
(987), whatever Velocity is generated in one Particle of Time,
a double Velocity will be generated in twice that Fime, a tri-
ple Velocity in thrice that Time; and fo on continually,
therefore it will be always T: V.
991. The Space (S) defcribed by falling Bodies will, in e-
qual Times, be greater as the Velocity is fo; and therefore in
this Cafe, we have S: V. Allo the Velocity remaining the
fame, the Space will be as the Time of deferibing it (971.)
Therefore in this Cafe S: T. Confequently, when neither
the
16
INSTITUTIONS
the Velocity nor Time is given, the Space will be in the com-
pound Ratio of both, viz, S = T V. But it is T : V (990.)
Therefore ST2, or S: V2; that is, the Spaces defcribed by
falling Bodies are as the Squares of the Time or of the Velocity.
And this we prove true by Experiments.
D
992. To illuftrate this Matter o-
therwiſe; let A B repreſent the Time A
of the Fall, and B C the Velocity
acquired at the End of that Time;
and draw A C. Then if we conceive
the whole Time A B to be divided
into an indefinite Number of equal
Moments, the Velocity in each of
thofe Moments will be as the right
Line drawn from A B parallel to BC,
correfponding to the given Moment.
Now for a fingle Moment the Velocity may be efteemed as
uniform, and fo the Space defcribed that Moment will be as
the Velocity (by 971,) therefore the Sum of the Spaces de-
fcribed in all the Moments (or whole Time AB) will be as
the Sum of all the momentary Velocities, or right Lines which
repreſent them, but the Sum of thefe Lines make the Area of
the Triangle ABC; this Area, therefore, is as the whole
Space deſcribed in the Time AB. And becauſe this Area is as
A B², or B C² (670,) therefore alſo the Space (S) is as T², or
V2, as above (991.)
E
C
993. Compleat the Parallelogram A B C D. Then if we
ſuppoſe another Body (A) to commence Motion at the fame
Time with the falling Body (B,) and the Velocity of the Body
(A) to be uniform and equal to the Velocity BC acquired at
the End of the Fall; then it is evident the Space deſcribed by
the Body (A) in the Time A B will be repreſented by the Paral-
lelogram A B C D; which Space is therefore double of that
(ACB) defcribed by the falling Body B.
994. What has been ſaid of defcending Bodies is in the
fame Manner applicable to afcending Bodies. For the Motion
in the latter Cafe will be retarded equably by the contrary
Action of Gravity, as it was in the former Cafe accelerated.
Thus let B C be the Velocity, with which any Body is pro-
jected
of the Phyfico-Mechanical Mathefis.
17
jected upwards from the Point B, then in every Moment of
Time there will be an equal Decrement of this Velocity, fo
that at the End of the Time BA it will be all deftroyed, and
the Space it will defcribe in that Time will be as the Area of
the Triangle ABC. Therefore, &c.
995. Hence it follows, (1.) That the Velocity every where,
at equal Intervals of Time from the Moments B and A, in the
Afcent and Deſcent is the fame. (2.) That the Time of the
Afcent and Defcent is the fame, or half the whole Time of the
Flight of the Projectile. (3.) That the Body by defcending,
acquired a Velocity equal to that (BC) by which it was pro-
jected. All that we have hitherto faid of Motion, is upon a
Suppofition that the Body moves in Vacus, or in a Medium with-
out Refiflance.
996. From what has been faid, it is evident, that if the
Space a Body deſcribes in any given Time in Vacuo be known,
the Space through which it will fall in any propofed Time, will
from thence be known alfo. Thus by Experiments very accu-
rately made*, it has been found, that a Glaſs Globe filled with
Mercury, defcended through the Height of 220 Feet in four Se-
conds; and that a Ball of Lead fell thro' 272 Feet in 41", and
#and
allowing for the Refiftance of the Air, the Motion in both theſe
Cafes was at the Rate of 1934 Inches, or 16,11 Feet in the
firſt Second of Time. For the 4" in Air will make but 3,75"
in Vacuo; then 3,75″*: 220 F.:: 12: 16,11 Feet, for the
Space in the firft Second, (by 991.)
997. Hence 12: 16,11 F.:: T: S; therefore 16,11 T-
S = the Space defcribed in any Time T expreffed in Seconds.
Hence alfo T=
Space S.
√
S
16,11
Time of defcribing any given
998. The Power of Gravity (G) will be as the Velocity
(V) generated in the fame Time, becauſe that Velocity is the
whole Effect of the Power, and Effects are always as their
Caufes; therefore G: V. Again, Gravity (G) will always be
inverſely as the Time (T) in which the fame Velocity is ge-
nerated; for 'tis evident, a double Force (2 G) will generate
VOL. II.
D
See the Principia, Edit. 3. p. 346.
the
18
INSTITUTIONS
the fame Velocity in half the Time (T) that the Force (G)
does; or the greater the Force, the lefs will be the Time for pro-
ducing the fame Effect; therefore G:: Therefore when nei-
V
ther the Velocity nor Time is given, we have G= or GT-
Τ'
I
V, and T = V × 1/1.
G
999. Since when the Time T is given, G is as V, and in
that Cafe V is as S (991,) therefore alſo G will be as S; or the
Power of Gravity will be as the Space deſcribed in any given Time.
1000.
Since GTV=
Q
M
(972,) we have GTM-
Q, and when the Time T is given it is GMQ; that is,
the whole Quantity of Motion in falling Bodies is compounded of
the accelerating Force (G) of Gravity, and the Mafs of Matter
(M) in the Body.
V
G
IOOI. Since T = (998,) =
S M
е
(972,) therefore QV
GSM. Hence whence the Space (S) is given, it is QV
GM. But we have always G: V (998.) Therefore it is
always Q: M; that is, the Quantity of Motion (Q) is always
proportional to the Quantity of Matter (M); and fince Gravity
(G) is always the fame near the Earth's Surface, the Velocity
(V) of falling Bodies will be every where the fame too; be the Quan-
tities of Matter in what Proportion you pleafe.
1002. Since QM, therefore the Momentum, or Force, by
which Bodies tend towards the Earth's Center, is as the Quan-
tity of Matter. But this Force or Tendency of Bodies, is
what we vulgarly call their WEIGHT; hence it appears, that
the WEIGHTS of Bodies are always proportional to their Quantities
of Matter. And thus having premifed the effential Principles of
Philoſophy, we next proceed to the Laws of Motion, obferved in
ftriking Bodies; for nothing ufeful can be known, 'till they are
firſt aſcertained.
CHA P.
of the Phyfico-Mechanical Mathefis.
19
CHAP.
IV.
The Philofophical THEORY of PERCUTIENT BO-
DIES, and of the COMPOSITION and RESOLU-
TION of FORCES.
1003. T
O determine the feveral Particulars relating to the
Motion, Velocities, and Direction of Percutient Bo-
dies A, and B; we reprefent the Quantities of Matter by M-
and m, and their Velocities by V and v; then will Q=MV
(970,) Momentum of A, and q = mv
1004. If the
+
Body B ftrikes X
the Body A in
B
Motion, and
Momentum of B.
+
Z
A
both move the fame Way, or towards the fame Parts, (as from
X to Z) then the Sum of their Motion towards the Part Z, will
be MV + m v, and the Velocity of both the Bodies after
MV + m v
the Stroke towards the fame Part, will be
M+ m
= V.
For the Velocity is always as the Momentum, divided by the Mafs
of Matter (970.)
1005. If one
of the Bodies,
as A, has a con-
trary Direction,
X
B
N
or tend towards
X, in which Cafe the Bodies will meet, then the Momentum of
A will have a negative Sign, viz. — MV; and fo the Sum of
the Motions towards the fame Part Z will be mv- MV; and
the Velocity after Collifion will be
ทย
MV
m+ M
=V.
1006. Becauſe in what we have hitherto faid, we fuppofe
the Bodies A and B deftitute of Elafticity, therefore after the
Stroke, there being nothing in the Bodies to caufe a Refilition
or Separation, they will both go on together with the fame Ve-
locity V.
1007. The Sum of the Motions towards the fame Parts is
the fame before and after the Stroke. For let them both move
the
D 2
20
INSTITUTIONS
the fame Way (1004,) and let B ſtrike A; then by that Im-
pulfe, the Motion of A, viz. Q, will be augmented and be-
come Q+x after the stroke; but becauſe Action and Reaction
is equal (965,) the Body A will re-act upon B, and produce an
equal Effect by the Stroke, that is, it will diminish the Motion
of B by the fame Quantity x, fo that its Motion after the
Stroke will be qx; but the Sum of the Motions of both
Bodies after the Stroke Q+x+4x=Q+q, the Sum of
the Motions before the Stroke, fee (1004.) And the fame is
to be fhewn, if the Bodies meet, as (1005).
1008. The Magnitude of the Stroke will be proportional to
the Quantity (x,) becauſe that is the whole Effect or Mutation
produced in the Motion of each Body. The Greatness of the
Stroke is therefore meaſured by the Lofs (x), which the moſt
powerful, or percutient Body fuftains in its Motion.
1009. In the above Theorems (1004, 1005,) if the Body
A be ſuppoſed at Reſt, then V = 0, and MV vanishes; the
Velocity then after the Stroke is
M V
M+m
=V; and fo mv =
VM+mV. Whence V:v::m: M + m, and y =
X V.
M + m
m
1010. If we fuppofe the Bodies equal, viz. A=B, or
Mm; then if the Bodies tend the fame Way, the Velocity
after the Stroke will be V+vV; or V —v=V, if they
meet.
1011, If the Bodies are equal, and one of them at Reft,
then
m v
M+ m
V
2
=V; or the Velocity after the Stroke is equal to -
half that of the striking Body.
1012. If A at Reft exceed B infinitely in Magnitude; then
becauſe m is infinitely ſmall in reſpect of M+m; therefore fa
is Vin respect of (1009,) confequently will vaniſh, or the
Body B impinging againſt any firm immoveable Object, will
after the Stroke be at Reft.
1013. If equal Bodies moving with equal Velocities, meet;
they will mutually deftroy each other's Motions; for in this
Caſe MV = my; therefore MV — mvo, confequently
M¡V— m v
M+m
of the Phyfico-Mechanical Matheſis.
21
MV
M V
M + m
M+ m
at Reft after the Stroke.
=V=0; or both the Bodies remain
1014. The Momentum of the Body B, after the Stroke is
MV m± m² v
MV m± m² v
mV =
(1004,) therefore m v —
M+m
M+m
MV mm v M
m + M
M m
M+m
xVv=the Lofs of Motion in
the Body B after the Stroke; but
M m
M + m
is a conftant Quan-
tity therefore the Loſs of Motion in B is as Vv, and con
fequently the Magnitude of the Stroke in Bodies tending the
fame Way, is as V — v; and as V+v, if they meet (1008.)
And if A be at Reft, then Vo, the Stroke will be as v,
the Velocity of the percutient Body.
1015. If the impinging Bodies A and B are perfectly elaftic;
then this elaſtic Force is ever equal in its Action to the compreſ
fing Force (982.) And whatever Action is exerted upon A by
the Impulſe of B, the fame is doubled by Virtue of this reni-
tent, or elaſtic Force; and the Re-action of A upon B is dou-
bled likewiſe (965;) and as the Parts of each Body are mutu-
ally compreffed and flatted by the Stroke, fo thoſe Parts are
thrown out again by an equal Force, and by this Means the
Bodies are made to recede from each other, after the Stroke
with the fame Forces, or Momenta, by which they came toge-
ther, or ftruck each other.
1016. But fince the Force of Collifion is as V v, (1014,)
that will alſo be as the Force of Refilition, or that by which
they ſeparate after the Stroke. Let x and y be the Velocities of
the Bodies A and B after the Stroke; and then VvY ;
and y =V‡ v±x, and fo M x = the Motion of A after the
Stroke, and that of B will be my mV mv ±mx. And
fince the Sums of the Motion before and after the Stroke to-
wards the fame Parts are equal (1007,) we have MV ± m v
= Mx + mV Fmv±mx; and thence Mx ± m x = M V
MV±2 m v— mV
fo±
±2m v m V, and fox=
M + m
1017.
22
INSTITUTIONS"
1
¿
1
1017. Alfo the Velocity of B, after the Stroke will be
M V m V ± 2 m v
y = V + v ± x = V + v ±
2 MV + M v ±mv
M + m
M+ m
From whence 'tis evident, if the Bo-
dies tend the fame Way, the Motion of B will be always pofi-
tive; but when the Bodies meet, the Body B will proceed or
recede, according as Mv is greater or leffer than 2 MV-
M U.
{
1018. The Momentum of A after the Stroke, will be
M⋅ m` V ± 2 M m v
M² V
2 M V
m F
M+m.
Mm v + m² v
M + m.
will be MV
: And that of B will be
; Whence the Motion loft in A
M2V-MmV+2 Mmv 2 Mm V2 Mmv
M+m
M+ m
;
and that gained in B will be found the fame; whence it is evi-
dent, that the Effect of the Stroke on each Body is equal, and double
of that in non-elastic Bodies (1014.)
1019. If the Body B be at Reft before the Stroke, then
vo, and the Theorem becomes + x
MV
m V
M + m
Mm
XV. Hence it appears, the Velocity of the Body
M + m
بایر
(A) after the Stroke, will be affirmative or negative, that is,
forwards or backwards, as M is greater or leffer than m, or as
A is greater or leffer than B.
1020.
If B be at Reft and equal to A; then vo, and
M
m, and fox
о
2 M
o; that is, the Body A will in
this Cafe be at Reft after the Stroke; and the Body B will move
on with the Velocity of A, for y= V in this Cafe, (by
1016.) *
I021.
After we have given the Phyfical Principles of the Newtonian Phi
fophy, we fhall illuftrate and confirm every Pofition and Doctrine by
Experiments, and defcribe the Machines by which they are perform-
ed in the belt Manner; and in that Part the Reader will fee how ex-
adly all the Cafes of percutient Bodies here premiſed are verified by
a practical Inftance of each of them.
of the Phyfico-Mechanical Mathefis.
23
1021. Let A, B, C, be three
Bodies; and let A ftrike B at
Reft; the Velocity generated in
B by the Stroke, will be y =
2 M V
M+m.
A
(1017, fince v = 0;) and therefore the Momentum of
B will be
2 MV m
M+m
=my.
With this Momentum B will ſtrike
Cat Reft, and contiguous to it; the Velocity generated in C
will be
2 my
M+ C'
2 my C
and its Momentum will be
;
and if in-
m + C
2 M V
ſtead of y, we reſtore its Value
M+ m
will be
2 m C
m + C
2 M V
X
=
M+ m
; the Momentum of C
4 MV m C
Mm+MC+ Cm + m²
1022. If now we make B a variable Quantity, while A and
C remain the fame, we may determine the Proportion of B to
A and C, that ſhall give the Momentum of C the greateſt pof-
fible, by making the Fluxion thereof equal to nothing, viz.
4 M² C² V m 4 M C m²
MC+Mm+m C + m²
2
772
= 0.
Whence we get MC →
Whence M:m::m:
mm=0; and therefore M C — m m.
C, or A: B: B: C; therefore B is a mean Proportional between
A and C.
1023. Hence if there be any Number (n) of Bodies in a
geometrical Ratio (r) to each other; and the Firſt be A, the
Second will be r A, the Third r2 A, &c. to the laft, which
will be r¹¹A: Alfo the Velocity of the Firft being V, that of
I
the Second will be
that of the Third
2 V
4 V
2 M V
(for
I+r
M+ m
2 V
2
X
I + r
I + r
11
I
2
city of the laft, which will be
I + r
Motions of the Bodies would be,
2 AV r
2 AV
A+rA
I + r
=
2 V
=)
I + r.
&c. to the Velo-
V. The Momenta, or
of the Firſt A V, of the Se-
cond
4
A V
2
of the Third
>
;
and to the laft
IA
I + r
27
24
INSTITUTIONS
2 r
I + r
viz.
AV, as they are exhibited in the following Series,
Bodies, A, r A, µ² A, µ³ A, µª A, &c. gì”—´¹ A.
Itr
4 V 8 V 16 V
2)
3'
itr I+r Itr
4.
2
&c.
V.
I+r
12.
I
}
Velocities, V,
2 V
J
Motions, AV,
&c.
C.
Itr i tr
3
4
I tr I+r
2AVr 4AV, 8 AV r³ 16AVr4
2r
I + r
1024. If the Number of Bodies be 10n, and r =
then will A ¦ r” — ¹ A ¦ ¦ I ¦ p² − 1 :: I: 512.
2
I + r
n
AV.
2
Alſo V:
::1:0.026::38,45: 1. Laftly,
2
V:: 1:
I + r
11 I
2 r
AV:
AV:: 1:
2 r
I tr
: : 1 : 1 37³·
I tr
1025. If n = 100, and r2, then will the firft Body A be to
the laſt "-A, as 1 to 633825300,000000000000000000000
nearly, and its Velocity to that of the laft, as 2710220000000
ooooo to I nearly, and the Momentums will be as I to
2338480000000, very near.
1026. Let
21
I + r
=R, and let the Motion of the firft Bo-
dy be to that of the laſt, as 1 to M, that is, let M
=R”—¹; if L = the Logarithm, then L, M =
Alfo
L, M
L, R
I
+1=n=
L, M x L, R
L, R
f
2 r
I+
— 1×L, R.
1027. By theſe Theorems, whatever relates to the Motion
of Bodies elaſtic, or non-elastic, acted upon by a fingle Impulfe,
may eaſily be determined. I fhall therefore now proceed to
confider the Motion of a Body, as acted upon by two or more
Forces at once, and determine the Direction thereof by the fore-
going Laws; as upon this Doctrine depends the whole Ratio-
nale of the Mechanical Philofophy, (which alone can be true) and
therefore it is of the higheſt Confequence to be thoroughly and
rightly understood.
Thus
of the Phyfico-Mechanical Mathefis.
25
Thus let C be a
Body impelled in
the Direction AD
by a Body A, with
fuch a Force as
fhall cauſe it to
move uniformly o-
ver the Space CD
in a Second of
Time. At the fame
Inftant let it re-
Fore
•G
ceive a Stroke by
F
D
E
another Body B, in the Direction B F, with fuch a Force as
ſhall cauſe it to paſs over the Space C F in the fame Time.
1028. Now it is evident the Body C cannot move in both
theſe different Directions; and therefore will not move in ei-
ther, but in a Direction compounded of them both, which is thus
determined. Draw D E parallel to BF; then, though the Ac-
tion of B prevents the Body from proceeding in the Right Line
CD, yet it can no Ways alter its Velocity of approaching to
the Line D E in the given Time, by Virtue of the Force im-
preffed by A. At the End therefore of a Second of Time,
the Body C will be fomewhere in the Line D E. By the fame
Way of Reaſoning, it will at the End of the fame Time be
found fomewhere in the Line F E, parallel to CD; and there-
fore in the Concourfe of both, in the Point E. Its Courſe
then is the Line CE, which is a Right Line by Law I.
(963).
1029. Hence appears the Method of compounding a direct
Force CE, out of any oblique Forces CD, and DE; and on
the Contrary, of refolving any direct Force CE into two other
oblique Forces CD and DE. Wherefore repreſenting any two
oblique Forces by the two Sides of a Parallelogram, the direct
Force equivalent to them will be the Diagonal thereof. And
the Truth of this Doctrine is abundantly confirmed by Experi-
ments.
1030. Thus for Example; if the Body C be drawn with a
Weight of 3 Ounces in the Direction CD, and by another of
2 Ounces in the Direction C F; then make CD to CF as 3
VOL. II.
E
to
26
INSTITUTIONS
to 2, and compleat the Parallelogram C D E F, and draw the
Diagonal CE, which will meaſure 4 upon the fame Scale,
(when the Angle DCF is of a certain Magnitude) which
fhews the Body C is in the fame Circumftance, as if it
was drawn by a 4 Ounce Weight in the Direction CE; and
this is proved true, by cauſing a Body as G, of 4 Ounces, to
draw the Body C the contrary Way, viz. from C to G, for
then the Body C will remain at Reft, or be in Equilibrio with
all the Forces.
1031. Hence it farther appears, that if a Body C be acted
upon by three different Forces at one and the fame Time, thofe
Forces will be to each other, as the three Sides of a Triangle
CDE, which are feverally parallel to their Directions. This
is fo plain from what has been faid, that nothing more can be.
added.
1032.
Hence
we learn how to
eftimate the Quan-
B
H
tity of any oblique
Stroke. For let the
a
D
Body A ftrike the
Body Cin a Direc-
tion paffing thro'
its Center, as A C.
Then it is certain,
that it acts upon it
F
E
with its whole Force; and the Stroke is faid to be direct. But if
the fame Body A ftrikes the Body C in any Direction AB,
which does not pafs through the Center C, then the Stroke is
faid to be oblique; and its Force to move the Body is thus
At the Point of impact B draw the Tangent ab, paral-
lel to which draw A G to meet B C produced in G.
1033. Let A B repreſent the whole Force of the percutient
Body A; this is refolvable into the two Forces AG and G B
(1029,) of which the Former is parallel to the Tangent a b,
and ſo does not at all affect the Body C; but the other Force
GB pafles through the Center C, and is that alone by which
the Body C is compelled to move. But A B: GB::CB:
BH (657.) That is, The whole Force, or direct Stroke, is to the
refidual
of the Phyfico-Mechanical Mathefis.
27
reſidual Force, or oblique Force, as the Radius to the Sine of the
Angle of Obliquity BCH (710.)
1034. But CB: BH:: CE:EF, or CD; and therefore
if in any Cafe CE denote the whole Force, the diminiſhed
Force or oblique Stroke will be denoted by CD; and will be
greater or leſs, as the Angle of Obliquity FC E = B C H
is fo.
1035. That the extenfive Ufe of this Doctrine of the Com-
pofition and Refolution of Force in Mechanical Philofophy, may ap-
pear, I ſhall ſubjoin the following Examples thereof in the moſt
interefting Parts of the Science.
B
FE
H
D
K
Thus let A G be the perpendicular
Section of any Plane, as the SAIL of
a WINDMILL, &c. expofed directly
to the Stream or Current of any Flu-
id, as Air, &c. reprefented by HIK;
and let BG be the Section of the
fame Plane in an oblique Situation
thereto. Then it is evident (1.) That the Number of fluid
Particles which ftrike upon the Plane in the direct Pofition A G
will be to the Number of thoſe which fall on it in the oblique
Pofition BG, as AG to DG; becauſe all the Particles be-
tween H and I will pafs by the Plane B G, and not touch it.
(2.) The Force with which the Particles ftrike the Plane in the
direct Pofition, will be to that with which they ftrike it in the
oblique Pofition B G, as A G to DG (1033.) Wherefore the whole
Force of the Fluid upon the Plane in the direct Pofition, will be to
the whole Force in the oblique one, as ĀG² to DG², or as the
Square of the Radius to the Square of the Sine of the Angle of In-
clination.
1036. Let A B be the Axis
of the WINDMILL, CD one
of the Sails placed in an ob-
lique Pofition EC to the Di-
rection of the Wind G C,
D
which is parallel to the Axis
AB. If then GC' exprefs
C
AR
B
the abfolute Force of the Wind upon the Sail in a direct Po-
E 2
fition,
28
INSTITUTIONS
fition, GE will exprefs the Quantity of the fame Force in
the oblique Pofition of the Sail, (becaufe GE is the Sine of the
Angle of Incidence GCE to the Radius GC;) but the Force
GE is refolvable into the two Forces EF, and GF, of which
the Latter being parallel to the Axis avails nothing in turning
the Sail about it. But the Force E F being perpendicular
thereto, is wholly fpent in compelling the Sail to turn
round. But the Force GE is to the Force EF as (G C to
CEX GE
CE,) GE² to
GC
2
which therefore will exprefs the
Force which is employed to turn each Sail,
1037. If therefore we put the Radius GC a, and E Ç
aa-xx, and confequently the Force
=, we have GE
2
CEXGE Q a x
x x
GC
a
; which if we make it a Maximum,
its Fluxion aax-3xxx=0, and fo a a≈ 3xx and x =
a a
3
20,00000
which in Logarithms is
0,477121
=9.761439, the
2
Logarithm Sine of 35° 16′ equal to the Angle CG E, and
therefore the Angle ECG is equal to 54°
A
B
44', when the Sail
MUGATURE WH
C D
E
G
receives the greateſt Force from the Wind.
1038. If A B be the Rud-
der of a Ship A H, placed in
the oblique Situation FC,
and the Water ſtrike againſt
it in the Direction GC;
then making CE Radius,
the Sine of the Angle of Incidence will be E F; and fo the
Force of the Water againſt the Rudder in a direct Pofition, is
to the Force against it in the oblique Pofition FC, as CE
to EF; but EF may be refolved into the two Forces ED,
and FD; of which the firft is parallel, and the laſt perpendi-
cular to the Direction of the Ship. Therefore F D is the Force
which compels the Ship to turn. But the Force EF is to the
Force FD (as CE to CF) as EF to
CFXEF²
CE
2
; that is,
(by
of the Phyfico-Mechanical Mathefis.
29
by putting CE a, CF=x). As
a a x-x³
a
P
; whence the
D
C
E
ليم
A
IR
Angle of Incidence ECF-54° : 44′ (as before) when the Force
of the Water againſt the Rudder to turn the Ship is a Maximum.
1039. After a like Manner is determined
the Angle of Pofition BA E ABE of the
GATES AE, BE of a Lock, or SLUICE
upon a River PQRS; fuch that the faid.
Gates fhall refift the Preffure of the Water
with the greateſt poffible Force. For fince
the Reſiſtance of the Gate A E diminiſhes in B
Proportion as the Preffure of Water, and as
the Length of the Gate increaſes; and in
the fame Depth of Water, both theſe are ast
the Line A E; therefore A E will exprefs the whole Refiftance
of the Gate A E. On the Diameter A B defcribe the Semi-
circle A D B, and continue A E to D, and draw BD and E C;
then becauſe A E²: AC²:: A B²: AD' by fimilar Triangles ;
and fince A D² is inverſely as A E², it will expreſs the Force
of the Water upon the Gate, or the Strength of the Gate re-
quifite to fuftain it, which increaſes as the Reſiſtance of the
Gate decreaſes.
2
S
1040. Again, the Force with which the Gates preſs each
other is proportional to the Magnitude of the Angle A E B ;
let B E expreſs the Force with which the Gate BE preffes the
Gate AE obliquely, this is refolvable into the two Forces
DE, which is parallel to A E, and BD which is perpendicu-
lar to it; therefore that Strength of the Gate A E (equal to the
Force of the Water, multiplied by the perpendicular Preffure of
the Gate BE, viz. AD² × BD,) ought to be a Maximum.
Wherefore putting AB-a, and BD, we have DA
= =
aa-xx, and fo A D'x BD aax-x³, whofe Fluxion
a a x − 3 x x x =
o, gives x =
3*x*0,
Jaa
3
; which fhews the An-
gle BA E = 35° : 16', as in the above Examples.
1041.
30
INSTITUTIONS
1041. Let AD be a Beam in the hori-
zontal Pofition, fupported at the End A
by the upright Piece A E, and it is re-
quired to find the Angle of Pofition of a-
nother Piece BC, of a given Length,
fuch that it ſhall ſupport the Beam A D
with the greateſt Force poffible. Let BC
E
་་ལས་པ་་་
F
a, and AC=x, then if BC exprefs
the abfolute Strength of the Piece BC; CF will expreſs fo
much thereof as ſupports the Beam AD, wherefore this perpen-
dicular Force multiplied by the Diftance A C, or Lever*, is to
be determined a Maximum. Now CF AB=√aa
and foCF× AC=x×√ aa—xx,whofe Fluxion ✔✅
x x x
✔aa
xx
and fo x =
=0; which gives a a xx=
a a
2
✯ ✅aa
x x
xx,
XX-
;
Vaa-xx
which fhews the Angle ABC 45 De-
grees, or Half a right one.
A
F
B
I'
E
1042. Let A E be a Beam, or Piece of Wood, fo fixed in
A as to make a given Angle E AB with the Horizon A B; it is
required to find the Poſition of another Piece BC, of a given
Length, fuch that it ſhall ſupport the Beam A E with the grea-
teft poffible Force. From C let fall the Perpendicular CF;
and becauſe the Angle at A is given, the Ratio of C to A F is
alfo given, which let be as n to m; and put BC a, and
BD; then if BC exprefs the abfolute Strength of the
Piece BC, BD will exprefs fo much thereof as fupports the
Beam AE. Now as CF:AF::BD:AD =
m
n
x, and DC
* N. B. What relates to the Distance A C, confidered as a Force
derived from the Lever, will be explained in the next Chapter.
of the Phyfico-Mechanical Mathefis.
3I
a a
xx, whence AC = x ± √ Vaa
aa-xx, which
m
n
Diſtance, or Lever AC multiplied by DB muſt be a Maxi-
m
mum, viz.
n
*√aa-xx
✓
x x
a a
xx±x√ a² —x², whofe Fluxion
सर के
✓
аа
XX
=0, whence
2 m
n
2 m
* * ±
n
A Q -—- X X
o, and by Reduction we fhall have x =
aa±am, which will determine the Angle of Pofition
ABC.
Note, The Equation above requires the acute Angle A BC
to be the Compliment of this obtuſe one A B C to two right
Angles.
1043. If we put this Value of x into the Equation
±x √ aa—xx +
m
n
aam + 2 am m + 2 ann
x x, we fhall have A C × BD =
when the Angle is obtuſe; and
2 n
aam + 2 amm
2 ann
=ACX BD, when the Angle is
2n
acute; whence 'tis evident, the Pofition of the Piece in the firſt
Cafe is much more advantageous than in the laft.
CHA P. V.
The Application of the foregoing PRINCIPLES to
fuch MACHINES as are called MECHANI-
CAL POWERS.
1044. HAVING confidered the Laws of Motion and mo-
ving Forces, under the Circumftances of Collifion
and Percuffion; we now proceed to confider thofe Forces which
are otherwiſe applied; particularly in fuch Cafes where they be-
come ſubſervient to all the Purpoles of moving heavy Bodies, and
overcoming Refiftances. And as this is effected by Machines
properly applied, fo this Part of Philofophy has acquired the
Name of MECHANICS,
1045.
32
INSTITUTIONS
1045. The Force
of Bodies acting
upon each other,
either immediately,
or by Means of a
B
E
Machine, is ftill de-
rived from the fame
Principles of Mat-
ter and Velocity (968,
969.) When therefore any two Bodies A and B act upon each
other, by the Interpofition of the inflexible Rod A B (the moſt.
fimple of all Machines,) 'tis eafy to affign the Quantity of
Motion in each, while they move about any given Center of
Motion F.
1046. For fuppofe the Rod AB to be moved out of its horizon-
tal Poſition into the oblique one a F b, then will the Space deſcribed
by A be the Arch A a, and that deſcribed by B be the Arch Bb.
Theſe Spaces then as they are deſcribed in the fame Time, by
the fimilar Motions of A and B, will be as the Velocities of
thofe Motions (by 971.) But fince the Sectors A Fa, BF b,
are fimilar, it will be A a: Bb::AF: BF (6.57.) Therefore
the Velocities of A and B will be denoted by their Diſtances.
from the Center of Motion, viz. AF and B F.
1047. If the Bodies be homogeneous, or of the fame Kind,
their Quantities of Matter will be as their Bulks A and B (973,)
and fince their Velocities are as AF and B F; therefore the
Momentum, or Quantity of Motion in A will be as the Rect-
angle A × AF; and that of B will be as B x B F (by 970.)
The Expreffions of their Forces as required.
1048. This Theory is general, and holds good for every
Sort of Motion, which Bodies fo circumftanced are capable of.
But that Sort of Motion which refults from the Action of Gra-
vity, ought to have the Velocity expreffed by the perpendicu-
lar Spaces ac and bd, by which one accedes to, and the other
recedes from the Center of the Earth. Becauſe Gravity acts in
thoſe Directions only (985,) and therefore its Force muſt be
eftimated thereby in given Quantities of Matter. But becauſe
of fimilar Triangles a c F, and bd F, it is ac: bd::aF:bF::
AF
of the Phyfico-Mechanical Mathefis.
33
AFBF. Whence it appears that A Xa and B x bd,
are the fame Momenta as thofe above (1047.)
1049. Hence for Bodies in Motion there is a threefold Ex-
preffion of their Forces, as follows:
Q= = A x A a
1. Q:
For A
2. Q
е
A XAF
3. Q = Axa c
B x B b
For BQ
1050. If we fuppofe the Forces in the two
→
BX BF
B x bd.
Bodies equal,
that is, AX AFBX BF, or AXA a B x Bb, or
A x ac ÷ B x bd, then it is, A : B :: BF:AF:: Bb: A a
::bd: ac. Therefore in cafe of an Equilibrium, the Bodies
are interfely as their Distances from the Center of Motion, or as
the circular Arches, or the perpendicular Spaces deſcribed in the
fame Time, and this is the fundamental Principle of every me-
chanical Power, Machine, or Proceſs whatfoever.
1051. To apply this
Doctrine. Suppoſe B a
very heavy Weight laid
upon the End of a long
The LEVER.
A
tained by, and moveable
Pole or Rod A B, fuf-
F
B
upon the Prop, or Fulcrum F; and inſtead of the Weight A,
fuppofe a Perfon's Hand were applied at the End of the LEVER
A, to raiſe or move the Weight B. Then the muſcular Force
of the Arm is now to be compounded with the Velocity of
Motion to conſtitute a Force equivalent to that of the Weight
B. And fince, in fuch a Cafe, the Force of the Arm is to the
Weight of the Body B, as BF to FA, it is evident, that
though the Weight B encreaſes in any Proportion, and the
intenfive Force of the Arm remains the fame, yet by taking
the Diſtance A F to B F, in the fame Proportion in which B is
encreaſed, the Perfon will ſtill have it in his Power to move the
Body B. And this will be the Cafe in what Manner or Form
foever the Lever be applied.
1052. From what has been faid, the Nature of the BA-
LANCE muft fully appear; for this is nothing more than a
Lever, whofe Brachia, or Arms AF, BF are equal; in which
Cafe the Weights A and B appended at each End will have
VOL. II.
F
equal
34
INSTITUTIONS
equal Velocities; and therefore, in Cafe of an Equilibrium, the
Weights will be equal alfo (by 1047.) But the Weights of
Bodies are as the Quantities of Matter (1002,) whence equal
Quantities of any Sorts of Matter are eaſily determined by the
Equilibrium of the Balance, which is its only Ufe.
G
D
E
F.C
A
B
1053. The STEELYARD is evident nothing but a ſuſpended
Lever, where the Weight A is applied at different Diſtances
from F, the Point of Sufpenfion, to make an Equilibrium with
B, the Body to be weighed; and when this happens, then on
the Arm EF is fhewn how many Times the Diſtance C F is
contained in G F, and juſt ſo many Times is the Weight A con-
tained in B, the Thing required to be found *
The PULLEY.
1054. A Tackle of Pullies is another me-
chanical Power, which Action or Force is deriv'd
from the general Principles of (1045,1046,1047.)
For let W be a Weight to be raifed (by Means of
the Tackle of five Pullies AB) by the Power, or
Weight P. Then it is evident, that when the faid
Weight W is raiſed one Inch, each Rope belong-
ing to the lower moveable Box of Pullies will be
fhortened one Inch, and the Rope to which the
Power P is appended will be lengthened juſt ſo
many Inches; confequently, the Spaces paffed
through in the fame Time, and therefore the Ve-
locities of the Bodies W and P will be to each
other as Unity to the Number of Ropes belonging to
A
P
PO
W
the
*Note, The Beam of the Steelyard is fuppofed to be without
Weight in what has been faid; and in Practice, the Weight of the
Steelyard is compenfated by a large Weight D; and further it is to
be
of the Phyfico-Mechanical Mathefis.
35
the lower Sheave of Pullies, which is here as 1 to 5; therefore
by fuch a Structure of Pullies the Force of P is encreaſed five
Times. And in this Manner you compute the Force of every
other Form or Conftruction of Pullies.
The WHEEL and AXLE.
1055. The Axis in Peritrochio,
or WHEEL and AXLE, is another
mechanical Machine, in which the
Power P hanging from the Peri-
phery of the Wheel CD, acts a-
gainſt the Weight W hanging
from the Axis at A; and it evi-
dent when the Wheel moves,
the Spaces deſcribed by the Bo-
dies P and W, and confequently
their Velocities, will be as the Pe-
ripheries of the Wheel and of the
Axle, and theſe are as the Dia-
B
12
E
F
P
meters (823;) therefore in cafe of an Equilibrium, we have
the Power P: the Weight W:: Diameter of the Axis: the Di-
ameter of the Wheel; which in the Fig. is as I to 12; and
thus the Force of P is encreaſed 12 Times. Hence as the Pe-
ripheries RS, TV, are lefs, the Effect of the Machine is
alſo leſs in Proportion: And on the other Hand, the Force of
the Machine is encreaſed by Means of the Spokes I F, in Pro-
portion as the Diſtance IF is greater than CD, as is extreme-
ly obvious from (1047.)
The INCLINED PLANE.
1056. The INCLINED PLANE does not (like other Ma-
chines) become a mechanical Power, by encreafing the Veloci-
ty of the Agent, but by diminiſhing the abfolute Weight of the Body
to be moved; which it does by its Reſiſtance or Re-action, thus;
let the Body A lie on the Inclined Plane BD; and from the
Center C let fall the Perpendicular CG to the Bafe of the
Plane HD, cutting the Plane in F; alfo draw CE perpendi-
cular to the Plane B D. Now fince the Body gravitates to-
wards
F 2
be obſerved, that in Practice we do not regard the Refiftance of the
Air to the Weight B, by which its real Weight will be a fmall Mat-
ter diminiſhed, as we fhewed in our Treatise on AIR, in the YOUNG
GENTLEMAN and LADY'S PHILOSOPHY.
36
INSTITUTIONS
wards the Earth in the Direction CG, let CF reprefent the
Whole, or abfolute Gravity of the Body. This Force is re-
folvable into two others, viz. CE and E F (1027,) of which
CE is that by which it acts on the Plane directly, and is de-
ftroyed by the equal Re-action of the Plane (965,) the other
Part E F is parallel to the Plane, and confequently, that by
which the Body is carried down the Plane, and is called the
refidual Gravity or Weight of the Body.
1057. Now this refi-
dual Weight which is to
B
be overcome, is to the
abfalute Weight, as E Fis
to CF; but becauſe the
Triangles CFE, DFG,
Ac
E
F
D
G
FL
DBH are fimilar, it is EF:CF::FG:FD::BH: BD.
Therefore the Weight of the Body A is diminiſhed by. the
Plane in the Ratio of the Length of the Plane BD to its
Height HP. Hence the more inclined the Plane is, or the
leſs the Angle at D, the more eafily will a Body be moved
thereon. Therefore when BD coincides with 'D H, or when
the Plane becomes horizontal, the whole Gravity of the Body is
deſtroyed; and hence it appears that the heaviest Body laid upon
an horizontal Plane (perfectly smooth) may be moved with the leaft
Force; as having in that Caſe no reſidual Weight to overcome.
1058. The Wedge has been hitherto,
by all Writers, reckoned among the fim-
ple mechanical Powers; but when it is
confidered that any Wedge ACB is only
a double inclined Plane, or compoſed of
the two fingle Ones ACH and BCH,
it will eafily appear that the Wedge is the
fame Thing with the inclined Plane; for to
double any Thing makes no Alteration in
its Nature or Properties, and confe-
quently cannot make it a different Spe-
A
H B
V
F
G
cies; this Sort of Logic does not become Mathematicians whoſe
Characteriſtic it is to uſe the juftest Method of Reasoning. Be-
fides, it is idle to pretend that the Force which overcomes the
Cohefion of Wood is derived from the Wedge, when it is fo evi-
dently
of the Phyfico-Mechanical Mathefis.
37
dently no other than the Momentum of the Bittle, or Sledge
which drives it into the Wood; the Wedge being uſed as a con-
venient Inftrument only, to keep the diffevered Parts of the
Wood, &c. aſunder. Therefore we take the Liberty of dif-
charging the Wedge from the Office and Rank of a mechanical
Power.
1059. We muſt alſo treat
the Screw with the fame Free-
dom, as it is fo evidently no-
thing more or less than an in-
clined Plane of a ſpiral Form
applied to a Cylinder AB,
and 'tis as plain that whatever
moves up and down upon a
Screw moves all that white up-
on an inclined Plane in a circu-
lar inſtead of a rectilineal Di-
T
هلا
B
rection. In this Screw-plane the Length is the Circumference of
the Cylinder, and the Height the Diſtance between two neareſt
Threads or Helices, which is feldom in greater Proportion than
10 to 1, a Trifle to mention for a mechanical Power. The Le-
ver FG added to the Screw-plane, make a compound mechanical
Power of very great Force and Ufe, for Motion, Power, Com-
preffion, &c. as is too well known to be farther infifted upon.
Here the Power is to the Force as the Distance of the Helices to the
Circumference defcribed by the End F of the Lever FG, which may
be encreaſed at Pleaſure. We therefore reduce the Number of
Simple mechanical Powers to four only, making the moſt of them
at the fame Time; for it is certain the Lever, Pulley, and Axis
in Peritrochio, differ only in Form, and not in their Properties
whence their Power is derived, which is the fame in them all,
as we have ſhewn. We therefore refer it to the Metaphyfician
to determine if there be really any more than two abfolutely dif
ferent mechanical Powers, viz. the Lever and the inclined Plane.
1060. It is alfo, on the other Hand, a Wonder that we meet
with nothing in our mechanic Treatiſes, on the Subject of an Arch
confidered as a mechanical Power, as its Nature and the com-
mon Uſe we make of it ſeems to entitle it to that Denomina-
tion, and cfpecially that the Catenarian Curve fhould be paffed
by
38
INSTITUTIONS
by in fuch general Silence, when it is a Subject that merits in
the higheſt Degree the Confideration of every Mechanic, Archi-
tect, or Engineer, as we have heretofore obferved *, and may
more largely fhew in a future Part of this Work.
1061. In the fimple mechanical Powers, I have fuppofed no
Friction, or Impediment thence arifing to the moving Power,
to interrupt the Theory; yet becauſe there are no Bodies in Na-
ture which move one upon another without fome Degree of
Reſiſtance or Friction arifing from the Roughnefs of the Parts,
this muſt be accounted for in Calculation of Forces, and it is
not to be found but by Experiments, by which it appears to be
(at a Mean) about a third Part of the Weight in Machines in
general; in fome it is much more, and in others lefs.
1062. To one or other of thefe Machines, moſt of the
Inftruments uſed in the common Affairs in Life are reducible.
Thus a LADDER to be raiſed up upon one End, is one Sort of
Lever; a WHEEL-BARROW is another; as alfo the CROW,
for forcing up Shrubs and fmall Trees by the Roots. The
HAMMER applied in drawing a Nail is a third Sort; the
SCISSARS, and SHEARS act on the fame Principle. The CAP-
STAN and WINDLASS, is the fame with the Wheel and Axle;
the JACK for raifing up Bodies is the fame in Effect. The
KNIFE and the Ax are both in the Nature of the inclined Plane;
and every Body that acts with any Kind of Force upon another,
will be found when rightly confidered, to do it on the Prin-
ciples above explained.
1063. As to compound Machines, their Force or Power
may eaſily be computed, and the Reaſon of their Effects may
be clearly underſtood, from the Nature of the fimple Machines
of which they are compofed. Or thus, let the Machine be
ever fo complicated, confider the l'elocities of the Motion of the
Power and Weight, and their Ratio will expound the Force of
the Engine, as well as in any of the foregoing fimple Ma-
chines. For the Principle of (1045, c.) holds good of all
Kinds of Motion whatſoever. But left we fhould be thought
deficient in not giving fome further Account of compound En-
gines, the following general Theory is fubjoined.
1064.
See Mifcellaneous Correfpondence for July, 1756, and the Plate of
a Bridge there conftructed on fuch Arches.
}
of the Phyfico-Mechanical Mathefis.
$39
1064. Suppoſe the Power P
by Means of the Wheel AKI
and its Pinion BLO, fuftain
in Equilibrio the Weight W,
hanging from the Axis EM N
of the Wheel GFH; then
putting A Ca, CB = b;
DF = c, D Nd; we
G E
M
N
K
C
FB
L
H
I
W
PXAC
W x DN
Pa
have (by 1045, 1046, 1047,)
or
>
CB
DF
Ъ
= wa
W d
and therefore PacWbd. Or thus, let (w) be
C
,
the Weight equivalent to P on the Wheel AKI, and its Axle
Pa
b
BLO, then P: w::b:a; therefore =w. Again, let
w be now confidered as a Power with Regard to W on the
Wheel GFH, and its Axle DN, then w: W::d:c; and
Wd Pa
as above. Now fince the Ma-
ww
C
b
v =
ย
therefore w=
chine is at Reft, the Point A is urged with the whole Power of
Gravity or Momentum of P; but if the Weight W be dimi-
niſhed to a leffer Weight x, the Machine will move, and the
Point A will have a Motion confpiring with that of P. Let
V Velocity with which the Point A endeavoured to defcend
with the Gravity of P, and Velocity of the Point A ari-
fing from the Diminution of the Weight W; then will V-v
be the relative Velocity with which it would endeavour to de-
fcend by the fame Power P, counterpoifing the Weight x at
Reft. But the Effects of the Power P with the Velocities V,
and V, in a given Time, will be as the Spaces through
which it would defcend in that Time, which are as the Squares
of thofe Velocities, viz. as V to V. And thefe Effects
are as W and x, who counter-act them; therefore fince V2 :
V²
:: W:x, we have V2 x W x V-v', and V V✔
V W X V
V — o = V √W—W x; whence v =
v ↓ Ŵ — V ✓ x
But when the Machine is in Motion,
✓ W
the Velocity of the Power P, or v, is to the Velocity of the
Weight
40
INSTITUTIONS
Weight x, as ACxDF
b d x v
=
b d V
X
ас
Which Velocity
ac, to CB x D Nbd; or ac
W - √ *
✓ W
= Velocity of the
multiplied by the Weight x, will
:bd: :v :
a c
Weight x.
b d V x
give
✓ W
X
W - √x
ac
✓ W
Effect of the Machine.
1065. If now x be a variable Quantitity, and all the reft
conftant, the Value of x may be determined, when the above
Expreffion of the Effect of the Machine is a Maximum, or the
greateſt poffible, by putting its Fluxion equal to nothing.
Now the Fluxion of
b d V
ac✓ w
хх
✓ W
✓
x
2√x
√x=
X
2 √x
2
хх
× × × √ W -√x, is *
O, whence we have ✔ W —
and fo 2 / W x = 3x, and x = W. Hence
+
if any Engine be charged with of ſuch a Charge or Weight as will
just keep it in Equilibrio, it will produce the greatest Effect poffible.
1066. Hence, if inſtead of x, we fubftitute its Equivalent
4W, we have
when the Engine is in its greateſt Perfec-
b d V x
✔
X
ac
W - √x
✔ W
4 b d V W
27 ac
4 V P
27
W b d
(becauſe P =
a c
44
V √ W - V √ x
we
✔ W
4 b d V W
27 ас
by 4 W, we
tion. Alſo putting W for x in v =
have v V. Laftly, if we divide
=
bd V
have
3ac
उ
Velocity of the Weight x, when the Machine is
in its utmoſt Perfection.
1067. If the Power P inftead of a Weight, as here repre-
fented, were a Current of Water, Air, &c. then is V the Ve-
locity thereof V-v, the Difference or relative Velocity
with which the Fluid ftrikes the Floats of the Wheel. Whence
'tis eafy to apply the foregoing Calculus to any mechanical En-
gine whatſoever, as we fhall more particularly fhew under the
Subject of Hydraulics, in the Theory of Mill-work.
1068. The Velocity here mentioned is fuppofed to be that
of an uniform Motion; for though all Machines actuated by a
conftant
of the Phyfico-Mechanical Mathefis.
4 1
conftant Power will have their Motion at firft accelerated for
fome Time; yet will the Increments of Velocity continually
decreaſe by the Friction, or Refiftance of the feveral Parts, 'till
at laſt they become totally deftroyed, and the Motion, by that
Means, rendered equable and uniform.
CHAP.
VI.
The Method of investigating the CENTER OF
GRAVITY in Bodies.
1069. T
HAT Point F, which in the Lever was called the
Fulcrum, (ſee Fig. to Art. 1051,) is otherwiſe
called the common Center of Gravity; for as in every fingle Bo-
dy there is one common Point which tends to the Center of the
Earth, with the united Forces of all the gravitating Particles
which compoſe that Body, and which therefore is called its Cen-
ter of Gravity; fo in any Syftem of two or more Bodies, which
are any how connected or depend on each other, there is one
certain Point in which their whole Forces of Gravity are united,
and which being fufpended, keeps the Bodies in Equilibrio, fuch
as is the Point F with refpect to the Bodies A and B, and it is
therefore their common Center of Gravity.
В
Br
F
E
1070. If to the Bodies A and B, we fuppofe two others
added, as C and D, at fuch Diftances from F, that C: D::
DF: CF; then Cx CFDxDF, (by 1047,) and be-
cauſe it is alfo AX AFBX B F; therefore thefe Equa-
tions added together, make A × AF + D× DF = B × B F
+ CXCF. Thus it appears the Point F is the common
Center of Gravity of all the Bodies.
1071. Hence a general Rule for finding the Diſtance of the
common Center of Gravity from any given Point E in a right
VOL. II.
G
Line!
42
INSTITUTIONS
=
=
Line E B paffing through all the Centers of the Bodies A, B,
C, D. Thus put EA a, ED = b, EF = x, EC = c,
EB d, then AF xa, DF-x-b, CFc-x, BF =
d- -x. Whence the general Equation above (1070) will become
Ax— Aa+Dx — Db — B d- B x + Cc — Cx. And
by tranfpofing the Terms we have A x + B x + C x + D x =
Aa+Bd+Cc + Db
A+B+C+D
Aa+Bď+Cc+D3. Confequently x=
EF the Diſtance required.
-
1072. The general Rule therefore for finding the Diſtance
of the Center of Gravity from the extreme Part of any Body,
is this, divide the Sum of all the Momenta by the Sum of all the
Weights, and the QQuotient will be the Distance required. Hence
in a right Line A B we may confider all
the Particles which compoſe it, as ſo A
many very ſmall Weights, each,
= x.
C
B
which is therefore the Fluxion of the Weights, or Line A B
Therefore the Weight multiplied by its Diſtance
from A, viz. x, is xx, its Momentum; that is, xx is the Flux-
ion of all the Momenta in the Line AB; whofe Fluent xx
is the Sum of all the Momenta, which divided by the Sum of
all the Weights x, gives x = A B, the Diſtance of the Cen-
1×
ter of Gravity C from the Point A.
In a PARALLELOGRAM.
1073. Thus in the PARALLE-
LOGRAM BEGF, whofe Length
AD=x, and Breadth G E — ỹ, if
you draw a b infinitely near E G,
the Areola a bEGyx will be the
Fluxion of all the Weights y, which
A.
F
2
го
D
C
B
& E
make the whole Weight of the Parallelogram, which multi-
plied by the Diſtance A D = x, gives y xx, the Fluxion of
the Momenta; whofe Fluent
ments, which divided by the
2
xxy is the Sum of all the Mo-
Sum of all the Weights (xy),
quotes AD AC, the Diſtance of the common
// x =
2
2
Center of Gravity from A, as required.
In
of the Phyfico-Mechanical Mathefis,
43
In a TRIANGLE.
1074. In the TRIANGLE ABG,
the common Center of Gravity is found
thus; draw A D (a) to biſect the
Baſe B G (= b) in the Point D, and
the Parallel EF in the Point C. Put
E
F
ba
ACx, and we have a b::*:
B
:
D
G
Q
b x x
EF. Which (as a Weight) multiplied by x, gives
Fluxion of the Weights; this again multiplied by xA C
a
b x x x
(the Diſtance from A) gives
a
Fluxion of the Momen-
b x³
divided by the
ta; whoſe Fluent or Sum of the Moments
b x²
Fluent of the Weights › = x =
24
quotes
*
за
AC, for the Dif
tance of the Center of Gravity from A in the Triangle A EF;
and when x = AD, then AD gives the fame for the Tri-
angle A B G.
In a TRAPEZIUM.
1075. To find the Center of
Gravity G of the Trapezium BD.
Let the fame be divided into two
Triangles ABC and A CD, and
find their Centers of Gravity F, E
(448.) Join EF, which divide in
fuch a Manner in the Point G,
that it may be FG GE::
BɅ
G
A
E
ADC: ABC, which is done by this Analogy, ABCD:
ACD:: EFF G, and the Point G is determined as
ed. This is evident from (1050, and 648.)
requir-
G 2
In
44
INSTITUTIONS
In the PARABOLA.
1076. The Center of Gravity
in a PARABOLA BAC. Let AD
=x, and BC= 2yz; then will
px: zz (740;) and putting p =
*
I
1, it is x : zz, and z : ✔✅✅ ✯ ( = x²),
Whence z✰✰✰, the Fluxion of
2
: x x
A
G.
B
D
ť
the Weights, which multiplied by x gives xxx, the Fluxion
5
2 2
x²
x
of the Momenta; whofe Fluent divided by the Fluent of
✯ x², viz. 3׳, gives 4x=ADA G, for the Diſtance
zx
of the Center of Gravity from the Vertex A.
In the ARCH of a CIRCLE.
in
MO
E
N
F
D
1077. To find the Center of Gra-
vity N of the Arch of a Circle
MEF fixed to the Radius C E.
'Tis evident the Particles M, F,
equidiftant from E, have their
common Center of Gravity at D,
in the Radius CE; and fince the fame is true of all the other
Particles, it is manifeft the common Center of Gravity of the
whole Arch MEF is fomewhere in the faid Radius CE. Put
MC a, MD PC = x; then PM (= DC)
=
=
aa-xx (828.)
аа
(a) :: Mo (= *) :
A P
G
DC) =
And PM (aa-xx): MC
Mm, the Fluxion
ай
Jaa
x x
of the Arch ME
multiplied by the
z.
PM, gives
Vaa
a x
Now the Fluxion of the Weights ż
Diſtance of the Center of Gravity CD
Vaa-
xx
X √ a a
xx = ax = Fluxion of
the Momenta; the Fluent of which, viz. ax
divided by the Weights or Arch ME, gives
CN, the Diſtance from C required.
MCxMD,
ax
༢
MCX MD
ME
In
of the Phyfico-Mechanical Mathefis. 45
In a QUADRANT.
1078. Hence when the Arch E M becomes the QUA-
a x
DRANT AE then
M C2
1 A E
-CN, the Diſtance of the
Center of Gravity N of the Semicircle AEG. if CM=1,
then will the Quadrant AME = 1,57079, (fee 826,) and
M C2
AE
=0,6366= C N, the Diſtance of the Center of Gra-
vity from the Center of the Circle C.
In the SECTOR of a CIRCLE.
1079. To find the Center of
Gravity N, of the SECTOR of a
Circle M C F. Defcribe the Arch
mef, and draw the Chord mf.
= a, ME = 2, Ce =
Put CE = a,
M
E
D
m
d.
Z X
x; then a: z::x:
= me; and A
G
a
b x
putting MDb, we have a: b
a:b::x:
m d. Now
a
mdx Ce
bx
is the Diſtance of the Center of Gravity of
me
Z
Z X
the Arch m ef (1077.) And the Arch me —
multiplied by
a
ZXX
a
*, gives = the Fluxion of the Weights, which multiplied
be
by the Diſtance of common Center of Gravity
>
b x x x
gives
Z
a
b x³
Fluxion of the Momenta, whoſe Fluent
divided by the
за
XXX
2 b x
Fluent of the Weights
gives
(or when xa)
2 A
3z
2 ba
Diſtance required.
3%
In
46
INSTITUTIONS
In a SEMICIRCLE.
1080. Hence when the Arch MEF becomes a Semicircle
2 ACXCE
AEG, we fhall have
3 AE
Diſtance of the common
Center of Gravity of the Semicircle A E G, from the Center C.
If Radius A C≈ 1, then that Diſtance will become
=0.424+.
2
4.71237
In a CYLINDER.
1081. To find the Center of Gra-
vity in the CYLINDER ECBF.
Put AD, and the Area of the
Circle on B Fap (830;) then
isap Fluxion of the Weights;
and ½ ap x x = the Fluxion of the
x
A
C
E
B
F
Momenta; whofe Fluent apxx, divided by the Fluent of the
Weights a px, gives
ap
4
X
2
AD
the Diſtance of the
2
faid Center from A.
In a CONE.
1082. To find the Center of Gravity in
a CONE AH B. The Fluxion of the Cone
II
(or Weights) is
pax²x
266
(834,) and the
a
Fluxion of the Momenta
pax³ x
77.
177
; whoſe
266
Fluent
ap x4
8bb
divided by
ap x3
A
B
6bb'
will quote
C
E
x=4HC, the Diſtance of the Center from H.
In the SEGMENT of a SPHERE.
1083. To find the Center of Gravity in any SEGMENT of a
Sphere a Db. The Fluxion of the Segment is p xx —
2
px² x
(ſee
2 a
of the Phyfico-Mechanical Mathefis,
47
therefore be p x² x
p x³ x
fee Fig. to Art. 836,) and the Fluxion of the Momenta will
x3
+
whoſe Fluent p x³ —
A
3
di-
2 a
a
px3
quotes
a
vided by the Fluent (of the Weights) ½ p x² ୪
Diſtance of the Center of Gravity from D.
In an HEMISPHERE.
1084. Hence when xa, or De DC, that is, when
the Segment becomes the Hemifphere AD B, then the Diſtance
of the Center of Gravity will be
or CD from the Center C.
the Diſtance of the Center of
g
of CD from the Point D,
In like Manner you will find
Gravity in the Semi-Spheroid
ABD to be of CB from the Center C.
38
1085. From this Method of finding the Center of Gravity in
Bodies, there reſults a general Rule or Canon for finding the fu-
perficial and folid Content of Bodies, viz. The Periphery defcri-
bed by the Center of Gravity, multiplied into the generating Line or
Plane, is ever equal to the Superficies or Solid generated by the Rota-
tion of the faid Line or Plane about an Axis. For Example; the
Diſtance of the Center of Gravity in a Semicircle is $
(1080.) And it is a :p ::
8 a a 3
3P
=
3
a a
a Periphery defcribed by
the Center of Gravity. Now the generating Plane or Semi-
8
J
circle is =
ap
(830,) then a x pa
=
paa
4
4
of the Sphere, (by 836, 837.)
Solidity
CHA P.
48
INSTITUTIONS
CHAP. VII.
The Method of computing the CENTER of FORCE,
or PERCUSSION, in moving Bodies.
1086. T
HERE is nothing of greater Confequence to be
rightly understood in all mechanic Arts, than the
Doctrine of Momenta and Forces of moving Bodies. The For-
mer of which we have treated of; and the Latter, viz. the
Forces of Bodies in Motion, and what is called the Center of
Force, or Percuffion in a striking Body, comes now to be con-
fidered.
1087. In order to render the Idea, or Notion of this percuf-
five Force as natural and eaſy as poffible, we are to confider,
that what has been hitherto delivered concerning the Momenta
of Bodies, refpects them in a State of Reft, under the Influ-
ence of Gravity; but when we confider the Body in actual
Motion, there will another Force arife from the Velocity of that
Motion; by which Means the Body will be rendered capable
of acting upon another, in the Mode we call Striking, or Percuf
fion.
1088. This percuffive Force, therefore, arifes from three
Sources, viz. (1.) From the Mafs of Matter in the percutient
Body. (2.) From its gravitating Force, in regard to its Dij-
tance from the Center of Motion; and (3.) From the actual Velo-
city of the Motion itſelf. The two firſt of theſe make the Mo-
mentum; and this compounded with the Latter, conftitutes the
percuſſive Force.
1089. Thus, fuppofe any uniform Rod
A B, were to move or vibrate about the
Point or Axis A; let it firft be fufpended
in the horizontal Pofition A B at the Extre-
mity B; then will every Particle D, C, B F
have a Tendency to defcend in Proportion
to the Quantity of Matter in each (968;)
E
ADC B
this Tendency will be farther augmented in proportion to the
Diſtance from the Center of Motion (1047, &c.)
the Momentum of the Particle at D, C, and B. Laftly, if the
and this makes
Rod
of the Phyfico-Mechanical Mathefis.
49
Rod A B be left at Liberty, it will commence Motion, and
the actual Velocity of the Particle D, C, or B will enable it
to ſtrike an Obſtacle at G, F, or E, with a Force propor-
tional to the ſaid Velocity, conjointly with the refpective Ma-
mentum.
1090. Now though each Particle in the Rod has a percuf-
five Force greater on all theſe Accounts, as it is more remote
from the Point A, if we confider it fingly in itſelf, and inde-
pendent of the reft; yet when we confider them all connected
together by the Force of Coheſion, their feveral Forces will
conftitute one compound Force of the whole Line or Rod
A B, which will not be greateft at B, but in fome other Point
C, between the Extremes A and B ; and this Point C will
ftrike a fixed Obſtacle at F in ſuch a Manner, that the whole
Force of the faid Rod will be exerted upon it, and by the
equal Re-action of the Obſtacle, it will be deftroyed (965,)
and fo the Rod in the Pofition AE, will be motionleſs at the
Moment of the Stroke, though difengaged from the Point A.
And this Point C is therefore called the Center of Percuffion in
the Rod A B.
1091. In order to this, it is neceffary this Point C ſhould
have the whole percuffive Force on each Side, in the Parts A C
and C B, equal; for if it were greater in the Part A C, than
in C B, the Part A C would, after the Stroke, move forwards,
not being counteracted by an equivalent Force in the Part
CB; alfo if the Force in the Part C B were fuppofed fupe-
rior to that in AC, then it would continue to move forwards
alſo after the Stroke, and fo the Motion of the whole Rod
would not be ſpent upon the Obſtacle, as it is when the per-
cuffive Force is greatest of all.
1092. Now fince when this percuffive Force is not equal,
the Rod (fuppofed difengaged from the Point A at the Time of
the Stroke) muſt turn upon the Obſtacle as a Center of Mo-
tion, it follows, that the Force of each Particle on each Side
of the faid Center, or Obſtacle, will be as its Momentum multi-
plied into its Velocity, or Diſtance from that Point (1089,) a-
bout which it then vibrates; and that therefore the Momenta of
the Particles must be reciprocally as their Diſtances from the Point
VOL. IL
H
(C)
50
INSTITUTIONS
(C) which ſhall be the Center of Percuffion, or Point of concentrated
Forces.
D
B
N
1093. But to exemplify this Matter, and A
to inveſtigate a general Rule for finding the
Center of Percuffion in all Bodies, let us
firft fuppofe the leaft compound Cafe, viz. of
two Bodies B, C, fixed to an inflexible right
Line A C, and call their Maffes of Matter
Q, 9; let their Tendency to Motion, or the
Diſtance A B, A C, be G, g; and their
actual Velocity in Motion, be V, v. Then
if each of theſe Bodies be confidered by itſelf
in Motion, the whole percuffive Force of B,
and C, upon Obſtacles at D and E would
be denoted by QG V, and q g. And the Sum of thefe For-
ces united, viz. QGV + 9g v, is the whole conjoint Force
exerted upon an Obftacle, ftruck by the Center of Percuffion.
1094. Let G be the Center of Gravity between the two
Bodies, and N the Center of Percuffion to be found; put A G
QG÷98
=m, and A N≈n; then will
9
8
Q+ 9
= m (1045;) alſo
fince B Nn G, and NC=g-n; it will be QG:
98::g―nn~G (1092.) Whence QGn-QG² = 98²
qgn, and fo QGn+qgn=QG²+9g², and fon=
Q G² + 9 82
QG+98
Q G V + q g v
QG + 98
or (becauſe G is as V, and g as v) it is
=AN; that is, the Sum of the Forces of the
two Bodies divided by the Sum of the Momenta, quotes the Dif
tance of the Center of Percuffion from the Point of Sufpenfion A.
Which is the general Cannon required.
1095. Coroll. Put the Maffes of Matter in both Bodies
Q+q=M, and the Sum of the Forces QGV + q g v = F ;
then becauſe Q G + q g = Q + q × m = M m ; we fhall have
QG+98
F
x
M I
Mn, or F=Mm n, whence we have F m
Theorem it will be of Ufe hereafter to remember.
I
X Which
72
•
1096.
of the Phyfico-Mechanical Mathefis.
51
1096. Let the Line A B (in Article 1089,) be denoted
by ; then will the gravitating Force of the Point B be as
*, (1088;) and the Velocity of the ſaid Point, when in
Motion about A, will be as likewife; therefore fuppofing
the Line in its nafcent State at B, it will be then as, and
÷ × × × = i = Fluxion of the Forces; the Fluent of which
+3
3
will be as the whole percuffive Force in the moving Line, or
Rod A B. But the Sum of all the Momenta is as (by 1072.)
2
3
2
Therefore
x = Difiance of the Center of Percuf-
2 3
fion from the Point of Sufpenfion A, by the general Rule at
Art. (1094.)
1097. Thus alfe in the Parallelogram BE GF, (1073)
moving about the Axis B F, the Fluxion of the Momenta was
yxx, and fo the Fluxion of the Forces will be y xxx, whoſe
Fluent y³, divided by the Momenta y x², will give 3 x =
A D, for the Center of Percuſſion from the Axis of Motion.
1098. Again, the Fluxion of the Momenta in the Tri-
angle ABG (1074) was found
3
b x² x
therefore the Flux-
b x*
a
b 23 x
ion of the Forces will be
the Fluent whereof
a
4 a
b x3
divided by the Momenta
will quote = AD, the Dif
за
3 a'
tance of the Center of Percuffion, from the Axis of Motion
at A.
1099.
Thus alfo the Center of Percuffion
is found in a Cylinder E BCG (1081,) vi-
brating about an Axis in the Point A. For
the Fluxion of the Momenta was apxx, and
therefore the Fluxion of the Forces will be
apx³
6
Lap²x, whofe Fluent
divided by
the Moments
ap x²
will quote = AD
4
واست
L
B
for the Center of Force from the Axis A,
the fame as in the right Line (in 1072.)
H 2
G
H
1100.
52
INSTITUTIONS
II00. To find the Center of Percuffion in a CYLINDER
EBCG, vibrating about any diftant Point L.
Here it muſt be confidered, that if two or more Bodies con-
nected together, fo as to move with equal Velocities, then will
the Sum of the Forces of theſe Bodies be equal to the Force of
their common Center of Gravity. And therefore the Sum of
the Forces of all the Particles in the fluxionary Circle (Pa
p x
2
=) CFGH, will be equal to the Force of their common Center
of Gravity D, or the Momentum of that Point multiplied by its
Diſtance from the Center of Motion L D. Wherefore put-
ting AL=b, and A D = x, as before; we have X
a b p x
apx x
b + x =
+
2
2
a p x
2
Fluxion of the Momenta, which
again multiplied by b+x, gives the Fluxion of the Forces =
b b a p x + 2 b p a xx + pa***, whofe Fluent
paхха
xx
>
b b pa x
+
2
2
враха
pax³
+
divided by the Sum of the Momenta,
bap x
2
6
2
pax²
+
will quote
3 b b + 3b x + × ×
*
for the Diſtance re-
4
36 + 3 x
quired from the Point L.
b
2
Or, if we put LD = g = b + x,
then will x = g b, and the Expreffion will become
88 + gb + b b
z X g + b
ΙΙΟΙ.
3
1101. From what has been fhewn, it is evident, that
fince a Walking-cane is generally but little tapering, and
when a Stroke is made therewith, the Center of Motion
is in the Hand, the Center of Percuffion will be near the
Part which is of the Cane from the Hand, but fomewhat
nearer to the Hand, on Account of the Cane's not being a
perfect Cylinder, but really the Fruftum of a Cone. So like-
wife a Stroke made with a Sword, (becauſe the Blade is nearly
of a triangular, or rather of a pyramidal Form) will be in a
Part much nearer to the Hand than in the other Cafe; but to
determine preciſely where that Point is in a Sword, (or any
Bedy not of an uniform Figure) by an analytical Proceſs,
would
of the Phyfico-Mechanical Mathefis.
33
would prove too intricate an Affair for this Place; but it may
be eaſily found by Means of a Pendulum, as fhewn in the next.
Chapter.
CHA P. VIII.
Of the DESCENT of BODIES on INCLINED PLANES,
and the DOCTRINE of PENDULUMS.
HE Doctrine of
PENDULUM s de-
II02.
T
pends on that of the Deſcent of
Bodies on inclined Planes; which
D
C
therefore comes next to be con-
G
玉
​B
fidered. It has been fhewn
(1057) that an heavy Body (as
A) upon an inclined Plane A E,
E
has its Gravity diminiſhed in the Ratio of the Length of the
Plane A E to the Height A B. About the Height of the Plane
A B, as a Diameter, defcribe the Semicircle ADB cutting
the Plane in D, and join DB; then is the Angle ADB a right
one (645,) and fo the Triangles ABD and AEB are fimilar;
and therefore AE:AB::AB: AD: : abfolute Gravity of the
Body: refidual Gravity, by which it defcends on the Plane.
1103. Again, fince Spaces defcribed in the fame Time are
proportional to the accelerating Forces of Gravity (999;) the
Forces which are as A B and AD, will carry the Body A
through the perpendicular Space AB, and the flant Space A Din
the fame Time; that is, in the Time a Body would fall freely
from A to B through the Height of the Plane, another will arrive
from A to D upon the Plane.
1104. By the fame Argument it is fhewn, that a Body will
deſcend on any other Plane A F (of the fame Height) to the
Point G in the fame Time it would defcend freely through the
Height A B; and therefore it follows, any two Chords AD,
AG`of the Semicircle will be defcribed in the fame Time.
1105.
54
INSTITUTIONS
—
1105. Suppoſe AG DB, then becauſe the Angle FAB
ABD, we have the Angle AFBDBF; and fince both
the Planes A G and DB are equal, and both alike inclined to
the Horizon, it is evident a Body will defcribe them both in
the fame Time; and fo the Time of defcribing the Chord D B
will be the fame as is fpent in defcribing AD. Thus the
Time of Deſcent in A G and GB will be the fame. Confe-
quently, the Times of Defcent thro' any Chords D B, GB, will
be equal.
1106. Since the Times of Defcent through AD and A B
are equal (1103,) the Time (t) of Defcent through AD (or
AB) is to the Time (T) of Defcent through A E, as AD
to the VAE (991;) and fo t²: T²:: AD: AE; but (be-
caufe AD:AB::AB:AE) we have AD: AE: AD: AB,
(672) and therefore t²: T²:: A D²: A B²; and fo t: T::
AD:AB::AB: AE. That is, the Time of the perpendicu
lar Deſcent through A B is to the Time through the Plane A E, as
the Height of the Plane to the Length.
AD to ✅✔✅ AE (991;)
1107. The Velocity acquired in falling from A to D is to
the Velocity acquired in defcending from A to B, as AD to
A B, (for fince they are generated in the fame Time, they will
be as the Powers which produce them, (998.) Alſo the Ve-
locity at D is to that at E, as
that is : V:: ✔✅AD: AE, and fo v²: V²:: AD: AE;
but becauſe AD:AB::AB: A E, it will be AD: AE:
A D² : A B³. Therefore v² : V² : : A D² : A B²; and ſo v :
V::AD: AB. Hence, fince the Velocities at B and at
E have the fame Ratio to the Velocity at D, they must be
qual to each other (198.)
1108. In the fame Manner it is fhewn, that the Velocity
acquired in the Point F, in defcending through the Plane AF,
is the fame with that at B. Confequently the Velocities acquired
in Defcents through any inclined Planes AE, AF, of the fame
Height, are equal to each other.
1109.
of the Phyfico-Mechanical Mathefis,
55
D
A F
ہو
1109. Hence a Body deſcend-
ing through ſeveral inclined Planes
AC CD, DE, contiguous to
each other, will acquire the fame
Velocity in the Point E, as it
would have at B in falling through
the fame perpendicular Height
A B. For at C the Velocity is the fame as it would be in de-
fcending through FC (1107;) and at D, it is the fame as it
would be in deſcending through AD; conſequently it is the
fame at E as it would be in defcending through A E, that is,
the fame as it would be in defcending from A to B, the fame
perpendicular Height:
EX
B
B
III. If now we fuppofe the Number of thofe contiguous
Planes infinite, and their Lengths infinitely ſmall, they will
then conftitute a Curve Line; whence it follows, that a Body
defcending through the Arch of any Curve Line A B, will acquire
the fame Velocity at the lowest Point B, as it would have at B, by
defcending through the ſame perpendicular Height A B▾
IIII. On the Diameter A B deſcribe the
Semicircle A GB, and draw any two Chords
D B and G B ; join AD, and from D and G
let fall the Perpendiculars DE, GF to the
Diameter A B. Then will the Velocities ac- G
quired in defcending through the Chords be
as their Lengths refpectively. For the Velo-
cities acquired through D B and GB will be
D
A
E
F
B
the fame as would be acquired in the perpendicular Defcents
through EB and FB (1105,) that is, as
EB to FB
(991.) But fince AB: DB::DB: EB (659,) we
DB2
EB
have A B =
; in the fame Manner it is fhewn, that A B
G B2
1
; therefore
FB
DB2
EB
GB2
FB'
and fo B D G B'::
EB: FB; confequently DB: GB::VEB: VFB:: Ve-
locity acquired through D B : Velocity acquired through G B.
2. E. D.
1112.
56
INSTITUTIONS
III2.
F
A
D
B
If two Planes A E and CD fimi-
larly fituated, or alike inclined to the Hori-
zon BE, then the Time of Deſcent on CD
will be to that on A E, as CD to VAE.
For the Time on CD is to that of the Deſcent
thro' C B, as CD to CB, and the Time on
AE is to that through A B, as A E to AB
(1111.) But the Times through CB and AB are as ✔CB
to AB. But we have (by fimilar Triangles) ✔BC: VAB
::✔DC:✔AE. Confequently the Time on DC is to the
Time on A E, as VDC to the ✔ A E.
1113. APENDULUM (or pendulous
Body) is in any Body A hanging at the
End of a String A C, and moveable a-
bout the fixed Point C as a Center.
Hence 'tis manifeft, if the ſaid Pendulum
be in the Pofition AC, and there left to
move freely by the Force of Gravity, it
will in its Deſcent defcribe the Arch of a
Circle Aa E; and when it arrives at the
loweſt Point E, it will there acquire a Velocity equal to what it
would have acquired, by falling freely through the fame per-
pendicular Height GE (1108, 1109.)
A
a
B
D
C
G
E
1114. The Body having deſcended to E, will, with the Ve-
locity there acquired, continue its Motion forwards (963,)
and deſcribe an Arch E F in its Afcent equal to A a E, (ſuppo-
fing the Motion were in an unrefifting Medium, and without
Friction at C;) for there muſt be the fame Time ſpent in ge-
nerating and deſtroying any given Quantity of Motion by the
fame Agent acting uniformly, and confequently the fame Space
or Arch E F will be deſcribed in the Aſcent, as was before de-
ſcribed in the Deſcent.
1115. Draw the Chord of the Arch A E, then will that be
an inclin❜d Plane, whofe Height is ABGE. Now 'tis evi-
dent, that when the Arch A a E is very fmall, the Chord AE
will nearly coincide with it, and the Motion of the Body de-
fcending on the inclined Plane or Chord, and that of the Pen-
dulum
of the Phyfico-Mechanical Matheſis.
57
dulum in the Arch will nearly agree in their Properties, òr at
leaft be infenfibly different.
1116. Hence, if there be two Pen-
dulums of different Lengths CA, CD,
the Times in which they will defcribe
the Arches of Defcent A B and DE,
will be as the Square Roots of the
Lengths of thefe Arches, or their re-
fpective coincident Chords AB, DE
(by 1112.) But becauſe of fimilar
Triangles ABC and DEC, it will be
✓AB:✓DE::✓CA:✔✅CD
::t: T. Therefore CA: CD :: t² :
T2. That is, the Lengths of Pendulums
are as the Squares of the Times of Vibration.
D
C
B
1117. The Time in which the Pendulum would defcend a-
long the Chord A E, (fee Fig. of Art. 1113,) is eafily deter-
mined, being that in which it would defcend through DE
(1103, 1105,) = 2 A C; and the Time in which it would pafs
from A to F on the two Chords A E and E F, will be equal to
that in which a Body would defcend perpendicularly through
a Space =
8 AC (991, &c.) But this is not
the Time in which the Pendulum will vibrate through the
whole Arch AEF. And therefore to determine what relates
to the Time of a Pendulum's Vibration, (which is a primary
Confideration) we muſt take to our Affiftance the Properties
of the Curve called the CYCLOID, which muft therefore be
next demonftrated.
4 DE
H
}
D
G
E
Z
IL
I.
K
1
it
B
1118. If a Circle A B C infifting on a Right Line A L begin
to revolve from A towards L, the Point A will by its twofold
VOL. II.
I
Motion
58
INSTITUTIONS
Motion defcribe the aforefaid Curve or Cycloid ACDIL.
Whence we obferve, (1.) The Baſe A L is equal to the Peri-
phery of the generating Circle A BC. (2.) The Axis of the
Cycloid FD is equal to the Diameter of the Circle. (3.) The
Part of the Baſe KL is equal to the Arch of the Circle I K.
(4.) Therefore KF (= ME) is equal to the remaining Arch
IHGD. (5.) The Angle HIK being always a right
one, the Chord KI will be perpendicular to the Cycloid, and
the Chord I Ha Tangent thereto in the Point I. (6.) Laftly,
Drawing IE parallel to the Baſe AL, the Tangent IH will
be parallel and equal to the Chord G D; and IK to G F.
1119. Parallel to E I draw e i indefinitely near, and In per-
pendicular thereto; then by fimilar Triangles DGE, DGF,
Ini, we have DE: DG::DG:DF:: In: Ii; that is, (put-
a, D E = x, DI= x) x : √ ax :: Vax:a::
ting DF
* : * = 1; =
I i
ent is 2 ✔
cloid DIL
a x
a x
√ ax
Fluxion of the Arch DI, whoſe Flu-
(803,) = 2 DG=DI. Whence the Semicy-
2 D F, the Diameter of the generating Circle.
A
L
E
I
D
K
R
f
S
B
b
1120. Let A ID be a Semicycloid inverted; and fuppofe a
flexible String faftened at one End in A, and ftretched along
the faid Curve AID, fo that the other End of the String may
coincide with the Point D, and thus the Length of the String
will be equal to that of the Curve. If now a Ball or heavy
Body were tied to the End D, and left freely to deſcend, it
would defcribe in one Vibration the Cycloid D C L, provided
there
of the Phyfico-Mechanical Matheſis.
59%
there be placed on the other Side another Semicycloid A EL.
Of this I might here give a Demonftration, but it is needlefs.
1121. The Velocity acquired by the Pendulum deſcending
through any Arch RC of the Cycloid, is the fame a Body
would have in deſcending thro' the fame perpendicular Height
OC(1109,) and therefore is as✔✔OC; but (becauſe O C:
SC::SC: KC 1,) OC is as the Chord SCRC
(1119.) Therefore the Velocity in the lowest Point C, is ever pro-
portional to the Space paffed through, or to the Arch of the Cycloid
defcribed in the Defcent.
I I
II22. But in all Kinds of Motions, the Space (S) is as
the Rectangle of the Time (T) and Velocity (V,) that is, S:
TV (by 971, &c.) therefore if in any Cafe (as that above 1121)
it be S: V, it will be T: 1, that is, the Time of the Motion
will be a given Quantity, or always the fame. Hence all the
Vibrations through any Arches of a Cycloid, great or ſmall, are per-
formed in equal Time.
a a
1123. If we put CK = a, KO=x, then 2 SCRC
= 2√
ax. If the Defcent be from L to R, the Ve-
locity at R will be as ✔✅KO= √(1111.) Now (by fi-
milar Triangles) it is CO (ax): CS ( √ a a — a x) (: :
CS: CKa): Rq= (*): Rr=
a x
Jaa- ax
== the
Fluxion of the Arch L R, which divided by the Velocity ✔x,
a x
gives
✓
аах
Q X X
ай
I
X
†
= the
✔ax-
x x
1/2 a x
✔ a
✔ax-
--- X X
Fluxion of the Time. But
is the Fluxion of the
circular Arch K S, (875.) (therefore 2 KS is the Fluent of twice
a x
that Fluxion, viz.
Confequently the Fluent of
√ ax-xx
2 K S
the Time of Defcent through LR is
Hence when S co-
Va
incides with C, LR will become L C; and fo the Time of
Deſcent thro' the Semicycloid LC is
I 2
2 KSC
va
1124.
60
INSTITUTIONS
1124. Therefore the Time of a Vibration through the
whole Cycloid LCD is
4 KSC
But the Time of Defcent thro'
а
the Perpendicular KC-a, is as
4 KS C
: 2 Va:: 2 KSC: a.
a
bration in the Cycloid, is to the Time
2 Va*; therefore we have
That is, The Time of Vi-
of Defcent through Half the
Length of the Pendulum, as the Circumference of a Circle to the
Diameter, or as 3,14159 to 1.
1125.
Let the Time of Vibration of the Pendulum A C
be 1 Second, then its Length is thus found. It is known
by Experiments, that a Body defcends freely through the per-
pendicular Height of 193 Inches in a Second of Time. And
fince the Spaces defcended are as the Squares of the Times
(991,) therefore 3.14159: 12: 1931: 19,6 = ; AC;
whence A C = 39,2 Inches. And fince the Lengths alfo are
as the Squares of the Times of Vibration (1116,) therefore
4:139,2 9,8 Inches -
39,2:9,8
Length of a Pendulum vibrating
in a Second.
2
:
ऊ
1126. Hence fince the Length of a Second Pendulum is ſo
confiderable, the Bob will defcribe (without the cycloidal
Cheeks AID, A EL,) an Arch of a Circle ƒC b, which
will nearly coincide with the Cycloid for a ſmall Space h gi
Hence all the Properties of the common Pendulum vibrating
through very ſmall Arches bg, will have the fame Properties
as though it moved in the Arch or Curve of the Cycloid.
1127.
To make this plain, we have S = V T (971.) Therefore S
-VT. Therefore T
S
ent of
VS
is 2 VS
s
V
21
S
(becauſe V:VS) but the Flu-
2 ✔a, when S = a. (See 803, 804,)
of the Phyfico-Mechanical Mathefis.
bi
A
в
1127. Let A B be any Sort of Body go
fufpended from an Axis a b, and move-
able freely about it; then will it be-
come a Pendulum, and vibrate in the
fame Manner with the common Pen-
dulum C F. For if it be taken out of
the perpendicular Pofition AE into
any other A B, and there let go, it
will by its Gravity defcend, and pafs
the faid Perpendicular À E to a Dif-
tance on the other Side equal to E B,
and ſince every Particle in the Rod en-
deavours thus to defcend, and all thofe
Particles are connected together, their
Forces will be all united in one parti-
cular Part as G, and that Force will fo
act upon the Body A B, as if all the
Matter thereof were collected in that Point; and hence the
Point G is called the CENTER OF OSCILLATION in fuch a
Sort of Pendulum.
D
E
G
B
1128. Hence it follows, that any Body A B being hung upon
an Axis to vibrate, the Time of a Vibration will always be e-
qual to the Time of Vibration in a fimple Pendulum FC,
whoſe Length is equal to the Distance A G of the Center of
Oſcillation from the Axis of Motion in the Pendulum AB.
And therefore a Method is hence obvious of finding the faid
Point G, or Diftance A G in any Body whatſoever, viz. by
taking a ſingle Pendulum F C that ſhall vibrate in the fame Time.
1129. From the above Definition of the Center of Ofcillation,
it is eaſy to underſtand that it is the very fame with the Center
of Percuffion in the Body or Rod A B, for fince the Point G is
that in which the Forces of all the Particles are united to gene-
raté Motion in the Body, and the Center of Percuſſion is that in
which alone the Motion of the Body can be all deftroyed (1090;)
it neceffarily follows, they are both one and the fame Point;
and therefore if A B be of an uniform Figure, it will be ifo-
chronal (or vibrate in the fame Time) with the common Pen-
dulum FC AG AB, (1099.)
1130.
621
INSTITUTIONS
1130. From what has been premiſed concern-
ing the Center of Ofcillation, 'twill be eafy to
folve the following Problem, viz. Any Body B being
fixed to the End of an inflexible Line CB void of
Gravity, 'tis required to find the Diſtance of the Cen-
ter of Ofcillation, when another Body A is fixed to the
fame Line, fuch that the Pendulum compounded of
thofe two given Bodies, fhall perform its Vibrations in
the leaft Time poffible. Let CB = a, CA — x,
and CD=n, the Diſtance of the common Cen-
ter of Oſcillation, which must be a Minimum by
the Condition of the Problem ; now 12
a a В + x x A
(1094,) whofe Fluxion muft there-
2 x x a A B + x² x A²
C
D
B
A
a B+ x A
a a x A B
fore be
2,
a B+x A
which will give x x +
2a B
A
a a B
A
; whence by compleating
the Square, and extracting the Root, we fhall find x =
✔ A B+ B B -
a B
2
A
1131. Having thus found x CA, if we fubftitute its Va-
a
A
=
a a В + x x A
lue in the Equation ʼn —
the Distance CD of
B
aВ+xA'
the Center of Ofcillation of the compound Pendulum becomes
known, and thus any fingle Pendulum of the Length CD-n,
will vibrate in the fame Time with the compound one.
1132. Since the Lengths of Pendulums will alter with Heat
and Cold (as we fhall hereafter fhew by the PYROMETER) the
Times of their Vibrations will vary alſo on that Account (1116,)
and therefore when applied to Clock-work and other Ufes, the
Compound Pendulum will be preferable to the fingle one, in as
much as the Body A may be confidered as a Corrector of the
Motion of the Pendulum CB, fince its Center of Ofcillation
D may be always kept on the fame Point by moving the Ball A
up or down by Means of a Screw on the Rod CD. But in
this Cafe great Skill and Caution will be required for a proper
Adjuftment.
1133.
of the Phyfico-Mechanical Mathefis.
63
1133. Hence it appears, the Nature of the Pendulum (of
any Sort) conftitutes it one of the beft Kind of CHRONOME-
TERS, or Inftrument for measuring Time; and alfo by this
Means it is applicable to fome Cafes of ALTIMETRY, LONGI-
METRY, &c. and thofe too, where Trigonometry is either
wholly deficient, or cannot be ſo eaſily applied. It moreover
ferves for the Meaſure of Forces of Percuffion, Refiftance, Velocity,
&c. and in fuch Cafes where the common Methods of Art will
fail us, and which yet make the moſt effential and fundamen-
tal Part of the genuine Theory of GUNNERY. The PENDULUM
is alfo the only Original and Philofophical STANDARD of Mea-
fure of LENGTH; and if the Length of the Pendulum vibra-
ting Seconds, had at firft been made the STANDARD YARD,
to all Nations, we fhould have had no Doubt about their other
Meaſures, or Dimenfions, which now lie involved in the grea-
teſt Uncertainty and Obſcurity; all thefe Particulars will fully
appear in the Sequel of this Work.
CHA P. IX.
The Phyfico-Mechanical PRINCIPLES of BALIS-
TICS, or the Doctrine of PROJECTILES
applied to the Solution of all Cafes in GUN-
NERY.
1134.
IN
N uniform Motion, or that whofe Velocity is always
the fame, if the Time (T) be given, the Space (S)
paffed over will be as the Velocity (V); and if the Velocity be
given, the Space will be as the Time; but if neither the Time
nor Velocity be given, the Space defcribed will be as the Product or
Rectangles under both, viz. S: :: TV: tv. This we have
largely fhewn (971, &c.)
1135. If the Motion be not uniform and equable, but ac-
celerated by a continual Action of the Force which generates
the Motion; and this continual Action of the Force be equa-
ble and uniform, or in every Moment the fame, then will the
Velocity
64
INSTITUTIONS
Velocity be equably and uniformly accelerated, or increaſe e-
qually with the Time, and will therefore be proportional to
the Time; and this is the Cafe of Bodies falling by their
Weight or Gravity, which near the Earth's Surface is every
where the fame. Therefore in this Sort of Motion T:t::
V: v.
1136. Now though in reſpect of any large Interval of
Time, the Motion of falling Bodies is accelerated, yet in the
fluxionary Moments of Time the Acceleration is fo fmall or
inconſidérable, that with respect to any proximate Moments
the Motion may be eſteemed equable; and confequently the
Space deſcribed in thoſe Moments will be as the Rectangle under
the Moments or Fluxion of the Time and Velocity, that is, ŝ s
=tv(1134.) Alfo fince T:t:: V: v; we have t
T
fo i=
V
T
have s
T
V
v; and
, which if fubftituted for i in siv, we fhall
vvv; and therefore S =
t
T
2 V
vv. Whence the Space
defcribed by a Body falling freely by its Gravity, is always propor-
tional to the Square of the Velocity, fince T and V are given Quan-
tities. This agrees with what was ſhewn in (991, &c.)
T
V
v, 'tis plain the
D
I
1137. Since t
Motion is rectilinear, and the Space de-
fçended fuch as may be reprefented by
a Triangle, one of whofe Sides repre-
ſents the Time, the other the Velocity.
Thus, if the Triangle A BC repreſent
the Space defcended through in any gi-
ven Time A B, and BC be the Velocity acquired at the End
of that Time; then will the Triangle ADE be the Space de-
ſcribed in the Time A D (992,) and DE
be the Velocity acquired in that Time.
น
E
C
(parallel to BC) will
And fo the Triangle
A B C (S): A DE (s) : : A B² (T²): AD² (t²) : : B C² (V²) :
2
Dɲ (v²). And therefore A B (T): AD (t) : : BC (V) :
DE (v); (671) which gives the Equation, expreffing the Nature
of a Triangle, as above.
1138.
of the Phyfico-Mechanical Mathefis.
65
1138. Hence we may eaſily compare the Spaces paſs'd thro'
by a Body moving with an accelerated Velocity in the Time
A B, and afterwards with an uniform Velocity, equal to the
laft acquired Velocity B C, in the fame Time. For the Flux-
ion of the latter Space is st VD dg G; and the Fluxion
of the Former, or that defcribed by the accelerate Velocity, is
s =
T
V
T
2 V
vi =D de E, whofe Fluents t V and v v are the
Spaces themſelves; but fince by the Suppofition T=t, and
Vv; therefore t V:
=
T
2 V
v v :: V:
2 T V V : T V V : : 2 : 1.
That is, the Space deſcribed by the equable Motion is to that by the
accelerate, in the fame Time, as 2 to 1; or as the Rectangle ABCF
to the Triangle ABC. (See 993.)
1139. If the Times are not in the fimple Ratio of the Ve-
locities, but as any Power t" to any Power v"; that is, if t":
m
m
v”, and ſo t:v″; then fince this is a Ratio expreffing the Na-
ture of a Paraboliform Figure, 'tis evident the Motion will now
be Curvilinear; and the Tract which the Body deſcribes will be
the Curve of fome Kind of Parabola. If n = 1, and m = 2,
then t:v²; or AD: DE', then will the Curve A EC which
the Body will defcribe, be that of the common Parabola. If
n = 1, and m = 3, then will t: v³; and the Curve will be
that of the cubical Parabola. If n = 2, and m = 3; then t² :
3
v³, or t: v2, and the Curve will be that of the Semicubical Pa-
>
rabola; and fo of others in infinitum.
1140. Here alfo, the Space defcri-
bed is repreſented by the Parabolic A-
rea A B C, which is to the Space
ABCF defcribed by an equable Mo-
tion in the fame Time A B, and with
the laſt Velocity BC, as 2: 3, or as
nin+m, as we have before fhewn
(827.)
1141. This accelerated Motion in the
Curve of a Parabola is the fame with the
TL
A
HG F
I
E
K
D
B
с
Motion of a PROJECTILE. For if the Body be projected (as
VOL. II.
K
from
66
INSTITUTIONS
from a Cannon) from the Point A, in the horizontal Direction
AF, it will proceed with an equable Motion; and therefore
the Spaces AH, AG, AF will be as the Times of defcribing
them; but while the Body is going from A towards F, it is con-
ſtantly acted upon by Gravity, and drawn downwards towards
the Earth's Center, and therefore in a Direction A B, perpen-
dicular to AF; by which Means the Motion towards F is not
at all impeded.
1142. Therefore fuppofe in the Time it would by the equa-
ble Motion arrive at F, it defcends by its Gravity through the
Space AK, then drawing KI parallel to A F, and HI to A B,
'tis evident by this compound Motion the Body will be found
at the End of the Time AH in the Point I. But AH, the
Time, is as✔AK, the Space defcribed by the accelerated
Motion, or A K is as A H², or KI, confequently the Point
I is in the Curve of a common Parabola (740, &c.) and the
fame may be proved of all the other Points E, C. Therefore
the Path of a Projectile is the Apolonian Parabola.
1143. If the Projection be
not in the Horizontal Direc-
tion, but in oblique Direction
AN, then alſo will the Curve
or Path of the Projectile A EM
be that of a Parabola, as is
demonftrable in the
very
fame
Manner, as before in the other
Cafe.
1144. As the equable Mo-
tion, or that in the Direction
N
I
F
GⓇ
H
E
A
M
B
of the Ordinate KI, is in the Point A and every where the
fame; but the accelerate Motion, or that in the Direction of the
Abſciſs A K, is nothing in the Point A, but there begins and
conftantly encreaſes; it must happen at fome Point in the
Curve that the Velocity of the accelerate Motion is equal to the Velo-
city of the equable Motion, viz. there where the Fluxion of the
Ordinate and Abfcifs are equal. Which Point is thus eafily de-
termined from the Equation of the Curve px=yy; which, in
Fluxions, is p≈ 2yỳ, and making ✰ =y, we have p≈ 2 y,
j, =
or
of the Phyfico-Mechanical Mathefis.
67
ory; therefore y ypppx; hence x = ÷ p =
AK; whence K is the Focus of the Parabola.
1145. Hence, if in the Axis produced, we take AL=*=
of the Parameter of the Point A, then will the Point L be
the Altitude, from which if a Body fall, it will in the Point
A acquire fuch a Velocity, that if it be from thence reflected
in the Direction AF, it will defcribe the given Parabola
AEC.
1146. Now fince it requires the fame Force to project a
Body from the Point A to the Altitude L, as the Body will
acquire in the Point A, by falling from the faid Altitude L, it
follows, that this Altitude is given from the Time of the Af-
cent and Deſcent of the Ball projected perpendicularly up-
wards from the Muzzle of the Cannon; and conſequently the
Trajectory or Curve which the Ball will defcribe, when projec-
ted with the fame Charge of Powder in any Direction A F
whatſoever, will be given alfo; and the Solution of the common
Problems of Gunnery very eaſy.
L
i
E
F
H
E
IK
Κ
C
F
H
E
I
A
D
B B
M. M
1147. Thus, for Example, if about A L as a Diameter
you
defcribe the Semicircle A K L; then as AL is the Mea-
ſure of the Effect of the direct Force of the given Charge; fo
the Chord of the Circle A H will meaſure or repreſent the
Effect of the ſame Charge in the oblique Direction AH. For
the direct is to the oblique Force, as Radius AL to the Sine AH of
the Angle of Incidence ALH = HAD; as is evident from the
Refolution of compound Forces, (1027, &c.)
1148. Now fince A H is as the Force of the Engine in
that oblique Direction, fo this may be refolved by two o-
thers,
K 2
68
INSTITUTIONS
thers, viz. HD perpendicular to the Horizon, and AD pa-
rallel to the fame. Therefore will HD BE, be the grea-
teft Height to which the Ball will rife in that Projection;
and 4 A DA M, the horizontal Diſtance, Random, or
Amplitude thereof. For fince AL: AH:: AH: HD; the
Height to which the Ball will rife with the Celerity HD is
to the Height it will rife with the Celerity AH, as HD²
AH: HD: AL; but AL is the Height with the Cele-
rity A H; therefore HD is the Height with the Celerity HD.
Again, the horizontal Velocity AD, being uniform, will carry
the Body through twice the horizontal Distance AD in the
Time it will rife to the Altitude H or E; and therefore thro'
four times AD, while it afcends and defcends thro' the Alti-
tude D H or B E.
1149. But thefe Matters are beft inveſtigated by Fluxions,
in the Method de Maximis & Minimis, by finding the Rela-
tion of the Abfcifs AD to the Ordinate D I. In order to
which let Radius, Sine, and Cofine, of the Angle of Elevation
HAD be called 1, s, &c. and put A D = x.
AD
تان
C: I :::
C
=AH. And c :s::x:
SX
= DH.
C
Then will
1150. Alſo let mSpace which the Body would deſcribe
by an equable Velocity in the Timet, and n = Space it
would deſcend by its own Gravity in that Time. Then m:t
X
C
7722 c²
:
i x
mc
: 7:
= Time of defcribing A H= And as ²:
nx
x²
m² c²
C
HI, the Space through which the Body
defcends in the fame Time. (1136.)
1151. Hence DH-HI=
5x
nx2
C m² c²
c m² s x
x² 12
m² cz
=DI, which is to be determined to a Maximum, by making
its Fluxion
csm² x 2 12 x x
m² c²
=2nx; and therefore x =
a Maximum, and BE
jection.
o, (S18.) which gives cs m²
c s m²
A B, becauſe now DI is
2n
s² m²
the Height of the Pro-
4 n
1152.
of the Phyfico-Mechanical Mathefis.
·69
1152. If we make DI —
c m² s x
n x²
=0; then c5 m²
m² c²
csm²
nx; and therefore : -
= AM, the Amplitude of
72
stm
will become
= the Time
n
the Projection. And then
tx
m C
of the whole Projection.
1153. If we make the Angle H AD a Right one, then will
AH coincide with A L, and AL will be the Height, or Im-
petus of the perpendicular Projection, and will be equal to
m²
4 n
>
s² m²
becauſe in this Cafe sI in the Expreffion of the Height
4 n
7722
4 1
1154. If now we put AL= = a; then will the Height
of the Projection in the Direction AH be B Eas², and the
Amplitude A M=4acs. And fince I:s::a: sa=AH;
therefore I :c::sa:acs=AD
sa: ac s = A D = ÷ AM, as was ſhewn be-
fore, (1148.)
1155. Hence 'tis plain the greateſt Random or Amplitude
will be AM, made upon the Elevation AK, of an Angle
KAD=45°, because then AD becomes a Maximum, or
equal to CK, and the Height of the Projection will be BE —
ACALAM; and therefore the horizontal Ran-
dom AM is in this Cafe the Parameter, and B the Focus of
the Parabola A E M.
1156. In this Cafe only the Perpendicular HD touches the
Circle in the Point K; in every other Cafe it will interfect it
in two Points H, H, which therefore give two Elevations
AH, A H, for ftriking the fame Object M on the Horizon,
either of which may be taken as the Exigence of the Cafe fhall
require.
1157. There is, befides the above, another Method very con-
cife and fimple, and yet general for all Cafes in the Science of
BALISTICS, by Means of the Tangent instead of the Sine
and Cofine of the Angle of Elevation as before.
AL = a, the Velocity of the Projection will
Thus, put
be as
a
(1145.)
7༠
INSTITUTIONS
Xr
*
(1145) Alfo let A H=t, and HI. Then for the Di-
rection A H we have t: 2 z::a: (992.) Whence 2
=40%,
I
√ ≈ z
1158. Again in the Triangle AD H, we have Radius:
Tangent :: 1 ::: AD: DH, and putting A D = x, and
DI=y, we have DHnx; and HIDH—DI=nx
y= 2.
1159. Alfo A H²
HD2+A D²; that is, t²x² + n² x².
Hence ² + n² x² = 4 a z, (1156;) and therefore &
nx y =
x² + n²x²
4a
·
(1157;) from whence we get 4 a n x
— 4 ay = x² + n² x² = x² × 1 + n n. From hence the prin-
cipal Cafes of Gunnery are eaſily folved.
1160. CASE I.
With a given Impetus (or Charge of Powder) a, to strike a given
Point O, at a given horizontal Diſtance A B.
Put A Bb,
and OB = c;
then will xb,
and yc, and
the Equation a-
bove (1159,)
will
become
IL
H
n n + i x b b
=
4 nab.
4
ac, from whence
we get the Tan-
E
gent of the re-
quired Angle of
Elevation n
D
B
M
F
2 a
I
b
+ √ 4 a²
4 ac
b².
b
Whence it appears there are
two fuch Angles or Pofitions of the Cannon that may be taken,
agreeable to (1156.)
1161.
of the Phyfico-Mechanical Mathefis.
71
1161. If the Point O be in the Horizon, then OВ=c=e,
and n
n =
22±√4a²-6²; and if O be below the Ho-
b
+
I
b
rizon, it will be negative, or c; and then n =
2 a
b
+
✓
4a² + 4ac-bb. In every Cafe 4 a a muſt be equal to,
or greater than 4 ac + bb, elſe the radical Quantity will be ne-
gative and render the Solution impoffible.
1162. CASE II.
To find the Elevation that ſhall produce the greatest horizontal Ran-
dom poffible, with a given Charge of Powder.
Here we have A M≈✯
4 n
nn + I
Xa, whofe Fluxion
or n = ✔ I=1, and fhews
madeo, will give n² = 1,
that the Angle fought is 45°, (as in 1155.)
1163. CASE III.
To find the Charge of Powder requifite to strike the given Point ○ on
a given Elevation of the Piece.
nn + I
In this Cafe we have
4nb-40
bba, which gives the Ve-
locity of the Ball at the Muzzle of the Gun, (1146.)
1164. If we make the Fluxion of this Value of (a) = 0,
it will give n✔bb+cc; and if this be ſubſtituted
for n, in the foregoing Equation, we ſhall have a = 1/2 c + 1/3/
bb + c c, the leaft Charge of Powder that will throw the Ball
on the given Object 0.
1165. Join A O= √ b b + c c; then fince the leaft Im-
petus a AL = ÷ BO ÷ ÷AO; if inOB continued out we
—
take O C = AO; then B C is the Tangent of the Angle BAC,
and AC the Elevation of the Mortar, for ftriking the given
Object O̟ with the leaft Charge. For Radius: Tangent :: 1:
7::
72
INSTITUTIONS
n : : b ( = A D) : c + √ b b + c (=BO + O C) whence n=
c + √ b b + c c
b
as before.
1166. In this Cafe, fince AO-OC, the Angle OAC =
OCA, and fince B C and A L are parallel, the Angle AC B
CAL; and therefore the Angle CALCA B, whence
it is evident, the Line of Direction AC bifects the Angle
LAO.
1167. Therefore if E F be a reflecting plain Surface, placed
on the Cannon, perpendicular to its Axis, and the Cannon be
moved up and down, 'till an Eye placed over the Glafs, fees
the Object O in the Pependicular AL (confidered as a Plumb-
Line) then that will be the required Elevation of the Piece for
ſtriking the Object O with the leaſt Charge. And this mecha-
nical Method was firft invented, or obferved from the Theory
by the late Dr. HALLEY.
1168. We might here largely infift on ftating the Ratios,
and finding the Values of the Time, Direction, Height, Ampli-
tude, Impetus, &c. of Projections; but as they are eaſily de-
ducible from the foregoing Theorems, and we are not now on
the practical Part of Gunnery, it will be befides our Purpoſe to
dwell any longer on this Head.
1169. And eſpecially, if it be confidered, that this Theory
of Projectiles in Vacuo is only of uſe in two Cafes, viz. of
Swift Motions in Vacuo, fuch as thofe of the Planets; and Slow
Motion in a Refifting Medium, as that of Spouting Fluids, &c.
but for Swift Motion in a Refifting Medium, fuch as is that of a
Bullet or Bomb in the Air, it can be of little or no Service; as
we purpoſe to ſhew further on, when we ſhall treat profeffedly
on the ELEMENTS of MILITARY PHILOSOPHY, and from
thence deduce a NEW and GENUINE THEORY of GUNNERY,
which though the moſt neceſſary, has hitherto been the leaſt
improved Part of the MECHANICAL PHILOSOPHY,
CHA P.
of the Phyfico-Mechanical Mathefis, 73
CHA P. X.
Of the PROJECTILE and CENTRAL FORCES by
which Bodies defcribe CIRCLES, or any Species of
Circular MOTION; a Comparison and Computa-
tion of the FORCES, VELOCITIES and periodi-
cal TIMES.
1170. LET C be any large
attracting Body, A
any leffer Body attracted by it
at the Diſtance A C; again,
fuppofe a projectile Force im-
preſſed on the Body A in the
Direction A H, perpendicular
to A C, and in fuch Proportion
to the attractive Force of C, as
that the Body being at Liber-
ty, and urged by both, may
I
M
K
G/O
A
H
E
F
P
N
E
D
defcribe the Circle A F P B A about the central Body C.
1171. Let A F➡, be the fluxionary Arch deſcribed in the
firſt Moment; and from F let fall the Perpendicular F E, FG,
on the Right Lines AC, and A H; then will AG be to A E
(=GF,) as the Projectile Force to the Attractive Force, at the
Diſtance AC. Which latter Force is called the Centripetal
Force, as being thereby carried towards the Center of Attrac-
tion C. Draw the Radius CF and the Chords A F and B F,
and put A C=x, and A B = d. Then by the Nature of the
AF2
AB
Circle, AB AF::AF: AE=
:
: Now becauſe the
Sine FE, the Arch AF, the Chord AF, and the Tangent.
A G are all equal to each other in their firft, or nafcent State;
therefore
AF2
AB
××
will in that Caſe be
=*; whence ✰
d
:::d:: the Centrifugal Force: the Projectile Force; and be-
VOL. II.
L
caufe
74
INSTITUTIONS
cauſe ¿ is infinitely leſs than d, therefore & the Projectile Force
does infinitely exceed, the Central Force in circular Motion.
1172. In any other Circle ILD, the fame Things are re-
preſented by Roman Symbols
ż ż
Z Z
i i
d
= x; then we have ✰ : x
22:27; that is, the Central Forces in two different Circles, are
d
d
as the Squares of the Projectile Forces, or circular Velocities applied
to the Diameters of the Circles.
[F, f, reprefent the Central Forces x, x.
the circular Velocities ż, ż
the periodical Times or Revo-
lutions.
the Diameters of the Circles, or
Radii.
V, v,
T, t,
1173. Let
D, d,
P, t
f V V
Then will the Equation above.
become
d
d
Ꭰ
Fvv
I
; and fo F: f:: VV x
:v v x
that is, the For-.
d
D
d
-the Peripheries of the Circles.
X Z ż x ż ż
ces are as the Squares of the Velocities directly, and as the Diameters
reciprocally, as before.
1174. If the Bodies I and A fo move as to deſcribe equal
Areas in equal Times, (viz. ILC = ACP,) that is, if d'=
dź, then ż:ż: :d:d; and fo 2 : 2² :: d² : d²; confequent-
༧
2
ly :x::
*
(
2
d²
d2
::)
d
d
d
d
d³: d³. That is, the Cen-
trifugal Forces are reciprocally as the Cubes of the Diameters, (ox
Radii) of the Circles.
=v;
1175. If D=d, then F: f:: V²: v²; or the Forces are di-
rectly as the Squares of the Velocities in the fame Circle. If V v;
then F: f::
I I
D' d
::d: D; or the Forces are inversely as the
Diſtances or Diameters, when the Velocities are equal.
1176. Since the Motion in a Circle is equable, the Spâces
will be as the Times; and therefore as V: P:: 1:T; hence
P
√d*
TV = P, and fo V ===✔DF (becauſe ✰ = √ d*
T
1171;)
of the Phyfico-Mechanical Mathefis.
75
I
1171;) therefore P² = T²DF = 3,1416 DD; whence TF =
3,1416 D; and fince 3,1416 is a conftant Quantity, we
2
fhall have F always as
D
T2
; or F: f::
D d
Tis that is the
central Forces in different Circles will be as the Diameters directly,
and reciprocally as the Squares of the periodical Times.
1177. When Tt, then Ff:: D: d; that is, when
the periodical Times are equal, the central Forces are as the Diſtances
from the Center directly.
1178. When D=d, F: f::t: T; or the Forces are re-
ciprocally as the Squares of the periodical Times in the fame Circle.
1179. If F = f, then
D
T₂
d
and fo Tt::D:
V2
✔ d; that is, the periodical Times are in the fubduplicate Ratio of
the Distances, when the Forces are equal. Alfo then we have D
and ſo V²: v³:: D: d; or the Squares of the Veloci-
ties are then as the Diſtances directly from the Center. Hence alfo
T:t(:: √D:√) :: V : v; that is, when the cent. Forces
are equal in two different Circles, the periodical Times will be as the
Celerities.
1180. Again; when Vv, we have T: t:: (P:p :) D
d; that is, when the Velocities are equal, the periodical Times will
be as the Diameters directly. And when Dd, then T:t::v
: V; that is, in the fame Circle the Times will be inverſely as the
Velocities.
1181. Since when Ff, we had T:t::✅✔✅D:✔✅✔✅d; and
when Vv, we had T:t:: D: d; therefore when neither
F, nor V are given, it will be T:t::DD:dd, or
T² : ť³ : : D³ : d³; that is, the Squares of the periodical Times
are univerfally as the Cubes of the Distances of Bodies circulating
about the fame Center C.
d
3
1182. Or thus more generally; let F: f:: D: dm ::
I
D
T3
; whence dm Dt=D" dT²; ord"-'t² = D-1 T³, and fo
L 2
T:
76
INSTITUTIONS
#--- I
771
T
171
I
112
T: t(:: d dz 2 :D 2
::) D
Dz : d 2. Now if mo;
then T:t :: D: d: :D:✓; as before when F = f.
When m=1, then T:t:: D° : d° : : 1 : I. viz. when the
Forces are as the Diſtances, the Times of the Periods will be equal,
in any Circles whatfoever. Laftly, if m2; then T:t::
3
2
³ : d², or T² : t² : : D³ : d³ ; but in this Cafe F: f::D-2
:d-²:: d²: D²; that is, when the centripetal Forces are in-
verfely as the Squares of the Diſtances, then will the Squares of the
periodical Times be as the Cubes of the Diſtances. And this is that
univerfal Law of Nature which is found to obtain in the Mo-
tions of all the heavenly Bodies.
1183. In the above Theorems and Analogies, we have ex-
hibited the Ratios of the Forces, Velocities, and Times of
Revolution in different Orbits, but to express them in their
proper Meaſures for any given Orbit whofe Radius is r, we
muft proceed in the following Manner: The Force is meaſured
by the Velocity that may be uniformly generated in a given Time,
1. which let us expound by the Power r" (the n Power of the
Radius (r.) Then the Diſtance through which a Body will free-
ly defcend in the fame Time will be exprefs'd by (993.)
Therefore if AF be an Arch deſcribed in the Time (t,) the
Diſtance AE defcended in that Time, will be found by this
T² × // r"
Analogy; as 12: T² : : ; r" :
2
12
2
2
= AE (992.)
1184. Now from the Nature of the Circle, we have A F2 =
T² t
ABXAE= 2 ACXAE=2AEXAC=- xr=
I 12
n + I
T² get + I
12
therefore A F - ✔ T² p” +
Txr 2
I
=
Space defcribed in the Circle in the Time t.
11
1185. But in equable Motions, the Spaces defcribed with a
given Velocity (") will be as the Times (971;) therefore T:
Tr
n+1
2
" + I
I :r 2
the true Meaſure of the Celerity in the
Circle. This might have been deduced from (1173) where
v = √ + f = √ rx p = y +¹
"+ 1
1186.
of the Phyfico-Mechanical Mathefis.
77
1186. Then by putting 9 = 3,14159, &c. we fhall have
# + I
4 2 :279 1:2qr
cal Time.
• ❤
I
2
the true Meaſure of the periodi-
St
1187. To find the Space S through which a Body muſt de-
fcend, to acquire the Velocity
n+ I
>
we have r²n: jn + s
r x gr2 Th
=
2 n
71
p2 n
2n+1
//r": S =
2
= 1r (992.) The Velocity in
the Circle therefore is acquired by a Defcent through half the Radius.
1188. To accommodate thefe Expreffions for Ufe; fuppofe
the Force we ſpeak of be that of GRAVITY; then its Meaſure
go2s=321 Feet, becauſe it is known by Experience, that
1/2 r" = 8 = 16 Feet for the Deſcent in the firft Second of
77 to I
ΤΖ
Therefore
+ = 2 sr, and r -
Time.
✓ 2sr=
the Velocity per Second, in any given Circle whofe Radius is
(r.)
219
2 r
2rs: 2rq:: 1:
= 9
S
√2rs
1189. Again,
T, the periodical Time, as is evident from (1186); becauſe
2r=D, the Diameter, and 2 r q = C = the Circumference
of a Circle, therefore ✔Ds = Velocity, and
9 C
S
= peri-
odical Time; which Expreffions are now in the moft fimple
Forms.
1190. Suppoſe a Circle equal to 8000 Miles, or 42000000 Feet
in Diameter, which is nearly equal to that of our Earth's Circum-
ference, and the centripetal Force be equal to that of Gravity ; then
will D = 42000000, and s = 16,083; and foVDs = 26000
Feet the Velocity; and
q C
S
= 5075″ = 1¹: 24′ : 35″
nearly the periodical Time, at the Diſtance of the Earth's Sur-
face.
1191. In any Circle, whofe Diameter is d, and t the Time
of Revolution in Seconds, then the central Force in fuch a
Circle may be compared with Gravity. For fince
q C
S
T,
78
INSTITUTIONS
Ꭰ
d
T, we have by (1176,) T
T2
:: F: f, that is, (fuppofing
Ds ď
I :
tt
q C d
Dstt
q² d
f, the
Gravity, or F = 1) C
central Force required.
1192. For Example, fuppoſe A
a Ball of one Ounce, whirled about
the Center C, fo as to defcribe the
Circle A B DE, and each Revolu-
tion be made in Half a Second;
and let the Length of the Cord
AC be two Feet.
Then t =
stt
B
D
C
and d 4, and
92
= 0,6136;
$
therefore
q² ď
stt
E
9,818 the central Force, or that by which
D
String A C is ſtretched, viz. 10 Ounces nearly.
1193. If the String and Ball be fuf-
pended from a Point D, and defcribes in
its Motion a conical Surface ADB;
then putting DC = h, A C = r, and
AD=g; and putting F = 1, the
Force of Gravity, as before; then will
the Body A be affected with three Forces,
viz. Gravity acting in the Direction
DC, a centrifugal Force, in the Direction CA, and the Tenfion
of the String, or Force by which it is ftretched in the Direction
DA; hence thefe, three Powers will be as the three Sides of
the Triangle CAD refpectively, (1031) and therefore as CD
( = h) : A D (= 8) : : 1 :
g)
g
b
B
C
A
Tenfion of the String com-
pared with the Weight of the Body.
1194. And CD (= b): CA (= r) :: 1 :
q² d
I (1191,) that
stt
is, h:r:: 1 :
292 r
whence
2q²r b
stt
=r, therefore 2 h × q²
1
stt
2 b
stt; and fot = q
have t = 1,108 √ to the periodical Time.
or becauſe
292
1,2272, we
S
1195.
of the Phyfico-Mechanical Mathefis. 79
1195. We farther obferve, that stt=S= the Space any
Body defcends in the Time t, fince 12: t²:: s: S (992, )
therefore fince 2 h x q² = s t t x 12, we have 13: q² (=
3,141597): : 2h: (stt) S; that is, as the Square of the
Diameter is to the Square of the Periphery of any Circle, fo is twice
the Height of the Cone to the Space through which a heavy Body will
defcend in the Time of one Revolution.
1196. Since the mean Diſtance of the Moon is nearly 60
Semidiameters of the Earth, or 1257696000 Feet; if C be the
Earth and A the Moon, (Fig. to 1170) then willAB=251539200;
and the Circumference of her Orbit A F B A = 7897834380,
which is deſcribed in 278 : 7h : 43′ = 2360580″. Hence the
Feet paffed over in one Second will be 3346 AFV.
11128976
Therefore A F² = 11128976; whence
= 0.00443
=
=
2515392000
AE, the Diſtance through which the Moon would defcend
in the Time of one Second, if the circular Motion were to ceaſe.
Now on the Earth's Surface Bodies defcend in the fame Time
16,14 Feet; but 0,0c443: 16,14::: 3643; which is very
near as I to 3600, the Squares of the Diſtances from the Earth's
Center reciprocally; and therefore confirms the Law of central
Forces above laid down (1182.)
1197. Again, fince T2: t²:: D3 d³, we have as D³-
60³ d³ 13: T² = 2360580"/2:
: =
2360580
60 32
= t = 83':
53″, the periodical Time at the Earth's Surface, nearly the
fame as in (1190.)
t²
1198. And fince when Dd, we had T2 t2: f: F;
therefore as 24h 1440, the Square of the Time of the
Earth's diurnal Rotation is to 84, the Square of the periodi-
cal Time when the Force is equal to Gravity, fo is 1 to
I or fo is 293,8 to I
293,8'
of the Earth arifing from the Rotation. Hence the centrifugal
Force, viz. that by which Bodies endeavour to fly off from
the Earth's Surface directly from its Center, is to Gravity,
nearly as 1 to 293,8 under the Equator.
the central Force on the Surface
1199.
80
INSTITUTIONS
1199. But at all Diſtances from the Equator to the Pole,
this centrifugal Force continually diminiſhes in the Ratio of
the Velocities of the Places (1177); but theſe Velocities
are as the Diſtances from the Earth's Axis, or as the Co-fines
of the Latitude; therefore the centrifugal Force isevery where to Gra-
vity as the Co-fine of the Latitude to 293,8; the Radius being = 1.
1200. Let AFB, and ILD be the Orbits the Moon deſcribes
about the Earth and Sun (Fig. to 1170); then will the Semidiame
ters of thoſe Orbits be A C = 240000, and IC= 80000000;
and if we put the Earth's annual Revolution, or periodical Time
=1, then will that of the Moon be
27,3
365
0.0748; the
Squares of which Times will be 1, and 0,005594. And there-
fore fince F: f::
D d
T₂ t²
; we have
82000000 240000
:
I
0,005594
80000000 : 43800000 :: 800: 438 :: F: f. That is, the
Gravity of the Moon to the Sun is to its Gravitation to the Earth,
as 800 to 438, or as 1,82 to I nearly.
1201. Laftly, we may compare in this Manner the centrifugal
Forces arifing from the diurnal and annual Motions of the Earth,
for the Times are, as 1 to 365, whofe Squares are 1 and 133125;
and the Diſtances in Semidameters of the Earth are, as I to
D d 20500
20500; therefore F: f::
T₂ tz
133125
:1:: 15: 100.
And fuch are the Forces arifing from the annual and diurnal
Motions of the Earth, by which Bodies endeavour to fly off
from its Surface.
}
1202. What we have faid may fuffice for the Doctrine of
circular Motion and central Forces; in which, I prefume, we
have been as full and as plain as the Nature of the Subject will
admit; for this is a Species of Knowledge of great Delicacy as
well as Uſe; and is to be reckoned among the firſt Principles
of Aftronomy, and fome other Sciences. We next proceed to
confider the Curves, or Trajectories, that will be deſcribed by
a Body urged with different projectile Forces, compounded
with a given centripetal Force.
1
СНАР. .
}
of the Phyfico-Mechanical Mathefis.
81
CHA P. XI.
The general DOCTRINE of TRAJECTORIES, or
ORBITS defcribed about a CENTER, by a Given
FORCE tending thereto, and compounded with a
variable PROJECTILE FORCE: Alſo of the
Proportionality of AREAS and the TIMES in
which they are defcribed.
1203. HAVING difpatched the Doctrine of Circular
Motion, where both the Projectile and Centripetal
Forces were given, or conftant Quantities, in every Part of
the Curve; we will now confider one of them, (viz. the Pro-
jectile Force) as variable, while the other remains the fame,
and obſerve from thence the different Kinds of Trajectories, or
Curves, that a Body impelled by fuch a Compofition of Forces
will deſcribe about a given Center. And fince at any Diſtance
from the Center the Projectile Force, by which it may deſcribe
a Circle, is given (1187.) therefore that Force may be made
the Standard for Compariſon, by which the Quantity of other
Forces whereby other Curves are defcribed may be difcovered;
as we ſhall ſhew in the following eaſy and perfpicuous Me-
thod.
1204. Let a
Body be projec-
BCDFG
X
Hó
ted from
the
Point A, with
S
KLM
N
ſuch a Velocity
AC=1, as com-
pounded
with
the centripetal
Force A E, may
cauſe it to de-
P
9
foribe a Circle
AKP about the
Q
T
R
Sun, or Center of Force S. Then let it be projected from
VOL. II.
M
the
82
INSTITUTIONS
the fame Point, and in the fame Direction A X, with any o-
ther Degree of Velocity AD≈n, to determine the Conic Sec-
tion, or Curve it will deſcribe.
By Suppofition A X is perpendicular to A S, let Eab be
drawn parallel to A X, cutting the Circle and Curve in the
Points a and b. Put A Sd; the Semi-tranfverfe Diameter
a, the Semi-conjugate b; and A Ex. Then will
24 x
=
b
y= Ea in the Circle; and -✔ 2 a x + x x
xxy=
yEb, in the Conic Section.
Ordinates E a and E b, z.
a F x x x
√20 x
2 a x I xx
a
Now the Fluxions of the
d — x x x
√2d
and X
a
x x
will be as the Velocities in every Point of
Curves in the Direction A X; but thefe Fluxions are as
d
X
b
and - X
a = x
(dividing each by
2 d
a
√2a + x
✓
X
and therefore when A Exo, then the Ratio of theſe
Fluxions will become
d
√2 d
b
a
b
X
or as
"
a
✓ 2
2 a
in the Point A. Confequently✔d:
b
I:n; and ſo n
a
b
b b
=
and therefore n n d=
Whence n n ad=
a
a
b b
b
bb. But fince in the Equation above,✓2 axxx=y=
a
a
(whenx=d,) (766) therefore 2 addd=bb=nnad; whence a
I d
=
21 n
; and b
I nd
2
Having therefore the Di-
72 72
2
ameters 20, and 2b, we have the Conic Section given in
Specie.
b b
1205. Since in the Ellipfe 2 a-d=
==
; and in the Hyper-
b b
bola 2a + d =
; and in the Parabola 2 à —
d
b b
d
= Infini-
ty:
Therefore it is evident, that putting n n = AF² = 2,
the
of the Phyfico-Mechanical Mathefis.
83.
1
the Equation above for (a), will be a =
I d
о
; whence a is infi-
nite; and therefore the Curve defcribed with a Velocity in
the Vertical Point, which is as V2, will be a PARABOLA
A MR.
1206. If nn be lefs than 2, then it is plain that a will be
affirmative, and b negative, or + a, and-b; and therefore the
Curve defcribed will be an ELLIPSIS. If n be greater than 1,
that is, if Eb A D, be greater than E a A C, then will
the Ellipfis ALQ be without the Circle, and the Sun, or
Center of Force in the upper Focus. But if n be less than 1,
or Eb AC
AB less than E a = A C = 1,
:
n
then will the Ellip--
fis AIO be deſcribed within the Circle, and have the Sun or
Center of Force S in its lower Focus. Laftly, when » = 0,
the Body A will defcend in a right Line A S to the Sun S.
1207. If nn be greater than 2, then will d be affirmative,
or +d; and confequently the Curve A NT deſcribed will be
an HYPERBOLA.
1208. Hence it appears, that the Curves defcribed with all
Degrees of Velocity n, between 0 and 2, will be ELLIP-
SES, (for the Circle defcribed with the Velocity n = 1, is really
one Species of Ellipfes) and all the Velocities from 2 to In-
finity will produce Trajectories of the hyperbolic Kind, by a
Centrifugal Force. The PARABOLA with the Velocity=
✔2 being the common Limit between them; and where the
Centripetal becomes a Centrifugal Force, or where Gravity may
be faid to be changed into Levity.
1209. In any determinate Orbit, defcribed as abovemen-
tioned, the Areas generated by a right Line connecting the
Center and revolving Body, will always be proportional to the
Times in which they are deſcribed; the Velocity in any Part
of the Orbit; the Times of defcribing equal Parts, or Arches;
and the Expreffion of centripetal Force every where, may all
be determined and demonftrated in a general Manner, as fol-
lows.
1210. Let S be an immoveable Center of Force or Attrac-
tion, and A any Body, urged by a Projectile and Centripe-
tal Force conjointly. In the Point A let the Projectile Force
M 2
be
84
INSTITUTION S
be fuch, as would carry the Body through the Space A B in
an indefinite fmall Particle of Time; then in the fame Time
it would deſcribe BFA B, were it not drawn from the
Tangent AF to fome other Point C, by the Centripetal
Force F C, acting in a Direction parallel to BS. In like
Manner, in the Third equal Particle of Time, it would de-
ſcribe CH CB, but is drawn to the Point D by the Cen-
tripetal Force HD in a Direction parallel to CS; and fo on.
1211. Now all thefe tri-
angular Areas AS B, BSC,
CSD, DSE, &c. defcribed
about the Center S in equal
Times, are equal to each o-
ther. For joining SF, the
Triangle A S B is equal to the
Triangle B SF, as being upon
equal Baſes and of the fame per-
pendicular Altitude. (635) Alſo
the Triangles BSF BSC;
as being both on the fame Baſe
BS, and between the fame
Parallels BS and CF; there-
fore the Triangles ASB =
BSC. In the fame Manner
the Triangle CSD is proved
I
E
F
H
C
B
4
S
equal to BSC, and therefore to ASB; and fo of all the
Reft; whence the Propofition is evident.
1212. Hence, when the Particles of Time, or the Spaces.
A B, B C, &c. are taken infinitely ſmall, the Polygon
A B C DE, &c. will become a Curve; and the Areas defcri-
bed by any Radius AS will be always proportional to the Times of
their Deſcription.
x bb,
1213. Alſo, becauſe in equal Triangles HB = b b,
whofe Bafes B, b, are unequal, thofe Bafes are reciprocally
as their Heights H, h, (viz. B: b::b: H;) it follows, that
the Velocities in every Part of the Orbit of the revolving Body are
reciprocally as the Perpendiculars let fall from S to the Tangenis of
the Orbit in thofe Points.
#214.
of the Phyfico-Mechanical Mathefis.
85
1214.
2
And becauſe Triangles upon equal Bafes are as
their Altitudes, (viz. HB:bB::H:h;) it follows, that
the Times in which equal Parts, or Arches of the Orbit are defcri
bed, are directly as thoſe Perpendiculars to the Tangents.
1215. Draw A G, GC, parallel to BC, BF; then will
BG=FC, or be as the Centripetal Force in the Point B,
and becauſe the Diagonal A C biſects B G in Q, and this is
every where the Cafe; therefore in every Part of the Orbit the
Centripetal Force will be as the Sagitta BQ of the indefinitely
Small Arch A B C.
1216. From what has been faid, it follows, that every
Body which moves in a Curve about an immoveable Point S,
fo that by a Radius drawn to that Point, it defcribes Areas
proportional to the Times, I fay, fuch a Body is urged by a
Centripetal Force tending to that Point. For fuch a Body in
the Point B is drawn from the Tangent by a Force which
acts in the Direction of a Line parallel to CF, that is, in
the Direction BS; and in the Point C it acts in fome Line
parallel to HD, viz. in the Line CS; and therefore in every
Point, it acts in Lines tending to the Center S.
1217. But if the Areas are not proportional to the Times,
'tis evident that CF, HD, &c. are not parallel to BS, CS,
&c. and therefore the Direction of the Centripetal Force is
not to the fixed Point S; but to fome other uncertain and
variable Point, on this Side or that, as the Areas defcribed in
equal Times increaſe or decreaſe, as is evident from the Con-
ſtruction of the Figure.
CHA P.
86
INSTITUTIONS
CHA P. XII.
A Determination of the LAW of the Centripetal
Force, tending to a given Point any where placed
in the Axis of a CONIC SECTION, by
which the Curve pertaining thereto may be de-
fcribed.
1218.
T
O determine what the Law of the Centripetal
Force muſt be, that if the Point to which it tends be
placed any where in the Axis of a Conic Section, the Body
fhall be made to defcribe the Curve proper thereto, is a Pro-
blem of the greateſt Importance, and may be folved either by
Lineal Geometry, or by Fluxions. In the firſt Way it has been
often done, but it is moft general and expeditious in the Lat-
ter, which here follows.
}
1219. Let VAB be a Part of the Curve, TV C its Axis,
C the Center, S a Point in the Axis to which the revolving
Body tends, A the Place of the Body, A B the Arch deſcribed
in an indefinite ſmall Part of Time; AD=y, an Ordinate,
VD = x, the Abfcifs belonging thereto, A T a Tangent, and
AS the Diſtance of the Body; DH a Line drawn parallel
to AS. Let VSd, AS=z; the Semi-tranfverfe Di-
ameter (a,) the Semi-conjugate (b;) and let the Abfcifs V D
(x) flow uniformly, fo that its Fluxion A a, may bẹ
conftant; while the Fluxion of the Ordinate AD, viz. Ba
j, is variable.
I220.
Then the Equation of the Conic Section being
a a
2 a x x x = yy (767,) we have 2 a x 2 x x =
a a
bb
bb
a² y y
2yj; and fox -
Alſo Ae: eg::AD:TD;
a bz
+
b² x
x y
that is, jx : : y
نو
TD=(*
干
​X
3/
=TD, the Sub-tangent. Therefore
a² ÿ
a”
I
TD = (x = - b b x ) =======
I
a F x
abz
X 2 a x I x x =
X
a b² b² x
2 a x + x x
a F ť
a x
1221.
of the Phyfico-Mechanical Mathefis.
87
B
H
A
a
T C
V
D
d
S
Q X
A + x
1221, Again, DT-VD=VT=
ax
; TS
therefore T S =
a F x
2 a x + × ×
a = x
by Reaſon of the fimilar Triangles TSA, and TD H, we
=)
+d=
a F x
a x + ad I d x
Idx
;
and
a x + ad I d x
2 ax + xx
have T S =
:ASZ:: DT=
a = x
a Ix
Z × 2 a x = x²
:DH=
a x + a d = d x
ac
a c
y x
dx
and fo Bej +
1222. And lastly, DS d-x: DAy:: A a = :
yx
dx
now the Triangle Bbc is fimilar to Scd, and therefore to
SAD, whence SA:SD::Bc: Bb; and fo
SA x B b j d ~ j x + y x
2
2
SD x B c
2
Area of the Triangle AS B,
which is the Fluxion of the Area S V A, or the Space that is
uniformly deſcribed in the fame Time with x.
x
j d − j x + y *
d-
b
1223. And becauſe
y =
√2axxx, we have j
a
b x x a = x
a b x x
; and j =
; therefore theſe Va-
a√ 2a x
✔2ax +
3
x x
j
it to this Expreffion
lues of y and fubftituted in the above found Area, will bring
z a √ 2 a x + x x
for the Time of
b x × a x + ad ‡ d x
defcribing
88
INSTITUTIONS
defcribing the conftant Space ; or, if a = x = m, and
2 a x − x x = n, the faid Area or Time will be abridged to
b x x a x + d m
this Form
2 an
1224. Then fince the Body arrives at the Point A with the
Velocity in the Direction of the Axis V S, and with the Velo-
city y in the Direction of the Ordinate DA; let DA be continued
on, and take AC = j in the Point A, and compleat the Pa-
rallelogram e a. And fince the Abfcifs flows uniformly, the
Body will arrive at the Point with a Velocity which will ſtill be
as x, or the Space A a uniformly deſcribed in that Time in the
Direction VS: But the Velocity with which it arrives at B in
the Direction dB or AD is different from j, or Ae; and is ÿ
9 Ba, and fo j-Baq=Bg, which differential
Quantity being the Variation of the Fluxion j, will be the fe-
cond Fluxion of y, or ÿBg=
a b x x
2 ax = x x =
2
a b x²
n v
n
and
therefore will be as the Space paffed over in the Direction AD
or dB, in the fame Time with by a uniform Velocity, gene-
rated in that Time in the Fluxion j.
1225. Since then we have the Space
the indefinite Particle of Time
a b x 2
12 V 12
paffed over in
>
we can find
b x x a x + dm
2 an
the Space paffed through in the conftant Particle of Time 1,
in the Direction DA, by the following Analogy of the Squares
of the Times to the Spaces, viz. As
a b xz
4 a3
d m
2
2
6² +² xax+ d m
4 a² 112
2
: 1
=Space defcribed in the
n ✓ n b √ n x a x +
given Particle of Time 1, by the Velocity that is uniformly
generated in that Time in the Body A, in the Direction A D.
1226. Laftly, fince the Spaces run over in equal Times are
as the Velocities, and the Velocities as the Forces by which
they are generated; 'tis evident, the Force by which the
Body is urged in the Direction AD muſt be every where,
or in any given Particle of Time, proportional to the Space
483
of the Phyfico-Mechanical Mathefis.
89
4 a³
2
which Diſtance or Force let be repre-
b v n x a x + d m
ſented by A k, and through k draw i parallel to the Tangent
T A.
Then will the Force A k be refolved into the two For-
ces ki, and Ai; of which the Latter is in the Direction of
A S, and therefore the fame with the Centripetal Force by
which the Body A tends to S, in every equal or given Mo-
ment of Time. And becauſe the Triangle A ki is fimilar to
Dkl, and therefore to A DH, we have the following Ana-
logy for the Expreffion of the Force Ai, viz. As AD=
b
n
Z n
:DH =
a
a x + d m
4 a4 Z
b² × a x + d m
1227. Becauſe
3
4 a³
b √ n x a x + d m
= Ai, the Centripetal Force required.
4 a4
b b
is a conftant Quantity, the ſaid Centri-
petal Force will be every where as
AS
DC3 x TS
Z
3
Z
a x + dm³ ax + ad I d x
; becauſe DC=ax.
1228. If now we fuppofe da, or V S = V C, then will
the Center of Force be in the Center of the Curve, and in the
Ellipfis we have
Z
a x + da — d x
-
3
(becauſe we may put a = 1.)
a is negative, or a, we have
a a² + ax =
Z
Z
ax+aa-
3 во
6
a
a x
Alfo in the Hyperbola, becauſe
A X - da + dx = ax —
a²; and fo the Force will in this Cafe alfo be as
Z. In the Parabola, a will be infinite, and a x and ± dx
will vaniſh out of the Equation, and leave only
Z Z
: z, as
a
ď
before. And therefore in every Section the Force will be di-
rectly as the Diſtance S A.
1229. If the Center of Force be in the Vertex of the
Curve V, then do; and the Force will be every where as
2 When the Ellipfis becomes a Circle, then x = (660,)
VOL. II.
N
2.
and
90
INSTITUTIONS
and the Force is every where as
8a3
or inverſely as the fifth
Power of the Diſtance from the Vertex.
But if the Center S
be removed to an infinite Diftance, then will the Force be as
Z
I
Z
Z
X
but
3
3
a d I d x
d
=1, as being each in-
a+x
finite and equal; whence, in this Cafe, the Force will be every
where as
I
DC3
1230. If the Center of Force be placed in the Focus of the
Section; then (per Conics) we ſhall have A S✔ AD²+DS³
a x + a d = dx
:
I
Z Z
Q
z; and therefore
Z
a x + ad I d x
3
Force A i; That is, the Centripetal Force is every where
inverſely as the Square of the Diſtance, when the Center of Force
is in the Focus of the Section. And this is the Cafe with reſpect
to the Sun and Planets, and eſpecially the Comets, whoſe Orbits
are very eccentric Ellipfes.
1231. From the above Demonftrations, we may obſerve the
following Particulars, viz. that when the Center of Force S
was fuppofed in the Center of the Curve, then the Force was
every where as the Diſtance z, which was demonſtrated alfo of
the Circle; when, in different Circles, the periodical Times
are equal; whence the perisdical Times alſo in fimilar Ellipfes are all
equal, if made about the fame Center C.
1232. Or thus, more univerfally of all Ellipfes. Let A =
Area of an Ellipſe, T = periodical Time. V Velocity at
=
the Vertex; therefore fince A (S) TV, we have
=
A
ī
A
=T, and fo in any other Ellipfe
7
=T; whence T:T::Ax
I
I
I
I
V
:Ax
: 1× 7:ab xab×; and when the Ellipfes have
the fame common Tranfuerfe, that is, when a = a,
it will bẹ
I
I
b b
T: 7 :: bx
b x
V
: 6 × 1 : : b V : 6 V ; but V² =
and
ad
Ꮴ.
of the Phyfico-Mechanical Mathefis.
91
b b
a d
(1204); and therefore fince a dad, it is V:V::
b : b ; and ſo V b = b; confequently T = T. So that univerſally
Bodies revolving in ſimilar elliptic Orbits about the fame common Cen-
ter C, will all perform their Periods in equal Time.
I
I
≤½³ ad
1233. Since T:T::abab; and a b = (a √ n² a d
3
I
=nd¹a³ = ) V d² a³, and a bn da²; therefore T: T
3
2
3
T²::
až;
a d a²; and fo T²: T² : : da³ : da³. Now let r and
be the Radii of two Circles, and t, t, the periodical Times.
of thoſe Circles; then it will be T2: t::da³: dr³:: a3 : r³,
T²:
(when dr); and in like Manner T::: (d a³ : dr³ : :) a³
: r³, (when d = r.) But fince t² : t² : : r³ : r³; therefore T²:
::a: a³; That is, The Squares of the periodical Times are
as the Cubes of the tranfverfe Axes, in Bodies revolving in elliptic (as
well as circular) Orbits about a Center of Force, pofited in the
Focus.
1234. Hence the periodical Times in Ellipfes are the fame as in
Circles, whofe Diameters are equal to the greater Axes of the Ellip-
fes; and therefore when the conjugate Diameter (or projectile
Force) is 0, the Curve will become a ftrait Line, which
the Body will deſcribe in the Time it would deſcribe a Circle,
whofe Diameter is equal to that Line.
2
1235. Since the Velocity of a Body is every where as the
Perpendicular (= p) let fall from S to the Tangent to the
Curve in the Point A (1213); and when the Point A is the
Extremity of the conjugate Axis, then will the Velocity be V:
I I
b
I
I
Val Va
(by putting the Latus Rectum l= 1). Let
Semidiameter of a Circle; then fince the periodical Times
in fuch a Circle, and the Ellipfis are equal; and in the Circle
when T² is as a³, we have V :
I
a
every where ; it follows,
that the Velocity of a Body revolving in an Ellipfis, is at its mean
Distance from the Focus, or Center of Force, equal to the Velocity of
a Body revolving in a Circle, whofe Semidiameter is equal to that mean
Diſtance.
N 2
1236
92
INSTITUTIONS
1236. The greatest, leaft, and mean Velocities of a Body re-
volving in an Ellipfis ApQ, and s p, will be as SQ, SA, and
(ſee Fig. to Article 1204.) becauſe thofe Lines are the Per-
pendiculars to the Tangents in the Points Q, A, and p. In
the Parabola, the Velocity will always be as the Square Root of the
Distance from the Center of Force; becauſe the Perpendicular to
the Tangent is always as the Square Root of the Diſtance in
that Curve (1205.)
1237. As in the Ellipfis, the Force is Centripetal, fo in the
Hyperbola it will be Centrifugal; but in the Parabola it will be
neither one or the other; for fince the Center of Force is there
at an infinite Diſtance, the Body cannot be properly ſaid to
move to or from fuch a Center. And in this Cafe the Direc-
tions in which the Power acts are all parallel; and therefore,
è converfo, when the Directions in which a Power acts upon a
Body are parallel, that Body will defcribe in its Motion the
Curve of a Parabola. Whence it follows, that fince the Cen-
ter of the Earth is not at an infinite Diſtance, the Directions
in which Bodies near its Surface are attracted towards it are
not quite parallel, and therefore the Curves which Projectiles
defcribe are not truly (tho' very nearly) Parabolas, but really
the Arches of very eccentric Ellipfes.
CHA P. XIII.
The ELEMENTS of a PLANET's MOTION, dedu-
ced from the foregoing PRINCIPLES.
:
1238. W
E have hitherto been confidering fuch Phyfico-Ma-
thematical Principles of Motion, particularly, with
regard to revolving Bodies, as will enable us to account very
naturally and eafily for the Motion of a Planet, or Comet, in its
Orbit about the Sun; and therefore the preceding Inftitutions
are to be regarded as the firft, or elementary Principles in Af-
tronomy, without which, the Rationale of that celeftial Science
can by no Means appear.
1239.
of the Phyfico-Mechanical Mathefis.
93
1239. The Motion of a Planet is known to reſult from a Pro-
jectile Force, in the Direction of a Tangent to its Orbit, and a
Centripetal Force directed to the Center of the Sun, both which
are ſo compounded together, that the Curve which the Planet
deſcribes, in Confequence thereof, approaches very nearly to
the Form of a Circle; becauſe in the Cafe of the Planet, the
Projectile Force is almoſt infinitely greater than the Force of
Gravity, by which the Planet tends to the Sun; in which Caſe,
an Orbit nearly circular must be deſcribed, as was fhewn
(1171.)
1240. But in regard to the Comets, this Difproportion of the
Projectile and Central Forces is much lefs, if we confider it, as
put in Motion at the Aphelion Point, or greateſt Diſtance
from the Sun, as is evident from (1206.) But if we confider
the Comet, as put in Motion at its Perihelion, then will the
Projectile Force be greater, than that by which it would be
carried about the Sun in a Circle at that Diſtance.
1241. Hence we ſee the general Reaſons, why a Comet revolves
in an Ellipfis about the Sun; for when it is in the Aphelion,
the Force in its Orbit is not great enough to carry it in a Cir-
cle about the Sun at that Diftance. It will therefore defcend
from that Point towards the Sun with a variable Velocity in its
Orbit always increaſing (by 1213,) till at Length it arrives to
the Perihelion Point, where its Velocity is greateſt of all.
-
1242. But, as in this Situation its Projectile Force, or Velocity
in its Orbit, is greater than that by which it can defcribe a Circle
about the Sun, it will neceffarily fly off, and recede from the
Sun to greater and greater Diſtances, but with a Velocity al-
ways decreaſing, all which is evident, from the variable Ratio
of theſe two Forces above and below what is neceffary to pro-
duce a circular Motion.
1243. But farther, it has been fhewn, that a Body, actuated by
a Centripetal Force, which is every where inversely, as the Square
of the Distance, muſt deſcribe a conic Section about that Body,
or Point, fuppofed to be placed in the Focus of the Section
(1230,) and in all Cafes where the projectile Force is to that
which would carry it in a Circle, at the fame Diſtance in a Ra-
tio leſs than that of 2 to 1, the Section will be an Ellipfis
by (1206.)
1244.
94
INSTITUTIONS
G
V
N
12
P
A
F
772
H
I
M
E
I
K
S
B
1244. When the projectile Force is nearly equal to that of the
Circle at the fame Diſtance, the Ellipfis will differ but little from
a Circle; and this we find is the Cafe of all the planetary Or-
bits. And, on the other Hand, when the Projectile Force com-
pared with that of a Circle (=1), is nearly equal to ✔2, then
will the Orbit be extremely elliptical, and its extreme Parts, or
Apfides will nearly coincide with a Parabola, and fuch are the
Orbits we find pertaining to all the Comets, and this is the Rea-
fon why Aftronomers generally make uſe of the Curve of a
Parabola for calculating the Motion, Places, and other Pha-
nomena of Comets, in the perihelion Parts of their Orbits,
which admits of greater Eafe, Expedition, and hardly any
fenfible Error.
1245. But to be more particular, with regard to the Mo-
tion of the Comet in every Part of its Orbit, it muſt be remem-
bered, that the Forces, circular Velocities and Distances, were all
defined by the following Equation F D vv=fdV V (1172.)
And fince the Comet is actuated by the Force of Gravity,
which
of the Phyfico-Mechanical Mathefis. 95
which every where increaſes in Proportion, as the Square of
the Diſtance decreaſes; that is, F:ƒ:: d²: D²; this will give
V:v:: √d: ✔D, or the circular Velocity will be every
where inverſely as the Square Root of the Diſtances; but
the Velocity of the Comet in its Orbit is every where inverſely
as the Perpendicular to the Tangent.
1246. From hence it will follow, that the Comet defcends
from the Aphelion A, towards the Sun S, becauſe its Velocity is
there leſs than the circular Velocity; as it defcends to lef-
ſer Diſtances P S, its Velocity in its Orbit increaſes in a
higher Proportion than the circular Velocity at the Diſtance
PS; and this will be the Cafe every where, till the Comet
arrives at the loweſt Point, or Perihelion B, where the Propor-
tion of its Velocity to that which it had at A, will be as AS
to BS, by (1213,) but the Proportion of the circular Veloci-
ties at B and A will be, as AS to BS. Suppoſe S B = 1,
and SA = 4, then will the Velocity of the Comet at B be
four Times greater than at A, but the Velocity in the Circle
at B will only be twice as great as that in a Circle at A.
1247. Whence it eaſily appears, that the Velocity in the Orbit,
getting the better of that in the Circle, will carry the Comet off
again from the Sun, when it has attained the loweſt Point B,
fince there it is greater than the Velocity of the Circle at the
fame Diſtance; and, as it recedes from the Sun, the Velocity
in the Orbit will decreaſe much fafter than the circular Velo-
city; the Latter will prevail by Degrees, and caufe the Co-
met to deſcribe in its Afcent a Semi-ellipfe B D A, equal to that
in its Deſcent A E B, till at laſt, having attained the higheſt
Point A, its Motion is then in the Direction of the Circle AG,
but for want of a fufficient Projectile Force, to continue in
that Circle, it will defcend again from thence towards the Sun
as before. Therefore the Velocity in the Circle prevailing in
the higher Apfis A, and the Velocity in the Orbit in the
lower Apfis B alternately, will occafion the Comet to de-
ſcribe the fame Ellipfis perpetually about the Sun. We here
ſuppoſe the Planet, or Comet, is not affected by any other Force
than that of Gravitation to the Sun.
1248.
96
INSTITUTIONS
1248. To make this Matter ftill more evident, we may confider
the Proportion of the Centripetal and Centrifugal Forces. We have
fhewn (1230,) that the Centripetal Force every where encrea-
fes, as the Squares of the Distances decreafe; alfo it has been
fhewn (1174,) that the Centrifugal Force arifing from the cir-
cular Motion about the Sun S, does increaſe in Proportion, as
the Cubes of the Diſtances decreaſe; ſo that, when the Comet
arrives to the Perihelion B, its Gravity is but 16 Times greater
than at A; but the Centrifugal Force, or that by which it en-
deavours to fly off from the Sun, is 64 Times greater than in
the Aphelion A, fuppofing, as before, that AS be 4 Times
greater than BS.
1249. Hence it will follow, that tho' Gravity prevails in the
higher Part of the Orbit, the Centrifugal Force (as it increaſes
much fafter) will prevail over it in the lower Part, and ſo prevent
the Comet from approaching any nearer to the Sun. But at B,
as the Comet recedes from the Sun, by Virtue of a fuperior Cen-
trifugal Force; fo in its Afcent, this Force will be conftantly
checked by Gravity, which, as it decreaſes in a much lower
Proportion than the Centrifugal Force, will, at length, prevail
over it at the higheſt Apfis A, and there put a Stop to any far-
ther Recefs from the Sun. Here the Comet again begins to
defcend, by Virtue of a fuperior Gravity, and ſo alternately
defcends and aſcends, according as the Action of theſe two
Powers prevails.
1250. Such are the mechanical Laws and Principles of a pla-
netary or cometary Motion; and it may be worth while to obferve,
that were the Centripetal Force to be in any other Ratio than
that of the Squares of the Diſtances reciprocally, fuch a re-
gular and beautiful Order could not have obtained in the
Syftem. Thus, for Inftance, fuppofing that the Centripetal
Force as the Cubes of the Diſtance inverfely, (or that F:f::
d³ : D³) then from the foregoing Equation (1245,) we fhall
have D: d::v: V, or the circular Velocities will be in the
inverſe Proportion of the Diſtances. But that is the very fame
Proportion that the Velocities in the Orbit have at A and B. And
therefore, fince the Velocities in the Circles and in the Orbit
at A and B, vary in the fame Proportion, it is evident, that
the fame which prevails at one Diſtance, muft prevail at the
other;
of the Phyfico-Mechanical Mathefis,
97
others; and therefore, if the Velocity in the Orbit at A be leſs
than the circular Velocity, there the Comet will begin to de-
fcend, and it muft always continue to defcend, for the fame
Reaſon that it first of all began to do fo, and confequently will,
after an infinite Number of Revolutions, fall into the Sun. But
if, on the other Hand, the Comet be ſuppoſed at B, there the
circular Velocity, being greater than that in the Orbit, will car-
ry it off from the Sun and becauſe it continues always in
the fame Proportion greater, the Comet muſt ever keep rifing
in fpiral Revolutions from the Sun. Therefore, in this Law
of Gravity, the prefent Frame of Nature could not in the leaft
exiſt.
1251. The fame Thing would alſo appear from what we
have faid of the Centrifugal Force; for as that Force every
where increaſes in the reciprocal Ratio of the Cube of the Dif-
tance, which is the very fame Ratio as that in which Gravity is
ſuppoſed to increaſe, it muſt follow, that, if Gravity once pre-
vail, as in the higher Apfis A, it muſt ever prevail over the
Centrifugal Force, and caufe the revolving Body conftantly to
deſcend in a ſpiral Orbit toward the Sun. But if, on the con
trary, the Centrifugal Force prevail in any Point, as at B, then
that Force will ever prevail over Gravity, and not only make
the Body begin, but cauſe it continually to recede from the
Sun.
1252. If the Gravity increaſes in a higher Proportion than as
the Cube of the Diſtance decreafes; then will the circular Ve-
locity increaſe in a higher Proportion than the Diſtances decreaſe,
and conſequently, in a higher Proportion than the Velocity in the
Orbit increaſes from A to B; fo that, as the circular Velo-
city exceeds the Velocity in the Orbit at A, it will much more
exceed it at B, and confequently, the Body will every where
continue to defcend to the Sun with an accelerated Velocity; and
the higher the Power of the Distance is, to which the Gravity
is reciprocally proportional, ſo much the quicker, or in a leſs
Number of Revolutions, will the Body defcend to the Center
of Force. On the other Hand, if once the Body recede from
the Center, it muſt continue to do fo for ever.
1253. Again, if Gravity increaſe in the reciprocal Propor-
tion of fome Power of the Distance between the Square and
VOL. II.
Cube,
O
98
INSTITUTIONS
Cube, the Body will take more than half a Revolution to de-
fcend from the higher to the lower Apfis; for it takes half a
Revolution, when the Gravity is reciprocally as the Square of
the Diſtance, and it has no lower Apfis, when it is reciprocally
as the Cube of the Diſtance, whence the above Propofition is
evident.
1254. If the Gravity increaſe in Proportion as fome Power of
the Diſtance less than the Square decreaſes, the circular Veloci-
ties will increaſe in a lower Ratio than that, in which the Ve-
locity in the Orbit increaſes, and confequently, the latter will
more eaſily prevail; alfo the Centrifugal Force will fooner ex-
ceed the Gravity, and therefore the Body will defcend to the
lower Apfis in lefs than half a Revolution, and return to the
higher Apfis, in leſs than a complete Revolution.
1255. From all that has been faid, it appears, that were the
Planets, or Comets, affected in their Motions by one attracting
Body only, whofe Power is reciprocally as the Squares of the
Diſtances, then they would defcribe what one might call a fixed
Ellipfis, whofe Apfides have no Motion at all. But if it hap-
pens, that any foreign, attractive Force be added to that of the
Sun, fo as to make the Sum, or Difference of thoſe Gravities,
vary in a higher or lower Proportion than that of the Squares of
the Diſtances inverſely, it will occafion the Apfides to move
forward or backward, and the elliptic Orbit to become, as it
were, moveable. The Excentricity of fuch an Orbit will alfo
be changed, and the periodical Times confiderably varied; and
this is really the Cafe with regard to the primary and ſecondary
Planets, and Comets; but particularly in the Caſe of our own
Moon, the gravitating Force of the Earth, added to, or ſub-
ftracted from that of the Sun, makes her Phænomena very va-
riable in all the abovementioned Circumftances. Alſo the For-
ces of Jupiter and Saturn will fenfibly diſturb the Motions of the
Comets, in regard to their Velocity, Aphelion Diſtance, peri-
odical Times, &c.
CHA P.
of the Phyfico-Mechanical Mathefis. 99
CHA P. XIV.
The Method of eftimating and comparing the Circu-
lar, Elliptical, Paracentric, and Angular VELO-
CITIES; alfo of CENTRIPETAL and CETRIFU-
GAL FORCES, by which a PLANET is affected in
the different Parts of its ORBIT.
S a farther Illuftration of the aſtronomical Elements
1250 of a Planet's Motion, we fhall next fhew how to
A
eſtimate the real Values of the Velocity of a Planet in its Or-
bit, and of the Velocity in a Circle at the fame Diſtance; alfo,
of the Planet's Paracentric Velocity, and of the Centripetal and
Centrifugal Forces for any given Diſtances of the Planets from
the Sun. It has been already fhewn, in a general Manner, what
is to be underſtood by circular Velocity, and by the Velocity of
a Planet in its Orbit, and how the Ratio may at all Times be
ſtated, but we have as yet faid Nothing of the Paracentric Velo-
city of the Planet's Motion, by which we are to underſtand its
Access to, or Recefs from the Sun, eftimated in a right Line,
joining the Planet and the Sun. Thus, let a Planet be in the
three Points of its Orbit, P, N, in its Defcent towards the
Sun S, and join SP, SN, and SM; on the Center S, with the
Diſtance SN, defcribe the Arch N p, and with the Diſtance
S M, deſcribe the Arch M m, then it is evident, when the Pla-
net comes to N, it will be nearer the Sun than it was at P, by
the Diſtance P p; and at M it is nearer the Sun than at N,
by the Diſtance N m; now theſe two Diſtances, or Lines, P p
and Nm, are called the Paracentric Velocities of the Planet's
Motion in thoſe Parts of its Orbit.
M,
1257. On the Center S, defcribe the Arch P V, and from P
let fall the Perpendicular Pv; alfo, from N and M draw the
Perpendiculars Ng, Mo. Laftly, draw Mn parallel to the
Tangent LP, and L M parallel to S N, then V v, or p q de-
notes the Planet's Centrifugal Force at the Points P or N; alſo,
mo denotes the fame Thing at M: And L M is the Expreffion
of the Centripetal Force, as is evident from what we have faid
in the XIIth Chapter.
0 2
1238.
100
INSTITUTIONS
1258. From
the Points Pand
M, draw the
Lines P v, Mo,
perpendicular
SN; then
نام
P
&
e-
NSM are
qual,(the Times
being fuppofed
angles P S N,
becauſe the Tri-
V
N
L
77.
772
M
2
A
E
equal) therefore
C
(becauſe the
Bafe S N is
common to
F
!
#
D
both) the Alti
I
K
tudes Pv, Mo
S
are equal; take
B
N n = LM,
then the Tri-
angles P N v,
M
no will be equal and fimilar, and PN
again, in the right Line SN (produced)
Mn and Nvon;
fince S V = SP, and
SN, and Nm-SN----
(N v) n o + V v, and Nm =
Nn + no om; therefore N VN m
NmV v + mo
Vo+mo
Sm SM, we have NVSP
Sm=SM,
SM; and confequently NV
A
Nn.
1259. If now we put SPy, then Ppj, and NV (or
Pp)-Nmj; then becauſe Vv=pq=mo, therefore j
2mo
=
−
Nn, or Fluxion of the Paracentric Velocity. Now it
is evident, that while the Paracentric Velocity increaſes, its
Fluxion ÿ will be Negative orÿ Nn 2mo (932) till
at Length it becomes Nothing, or 2 mo-Nno, in
which Cafe 2 moN n. After this, the Paracentric Velocity į
Nn.
decreaſes, and its Fluxion is affirmative, or +ÿ 2 mo—Nn;
till the Planet arrives at B, where it entirely vanifhes. From
hence we learn
1260. First, that in the Defcent of the Planet from its Aphe-
lion Diſtance A, toward the Sun, its Paracentric Velocity, be-
gins and increaſes till it arrives at a certain Point; after which it
j
de-
of the Phyfico-Mechanical Mathefis.
decreaſes continually, till it vaniſhes in the Aphelion Point B, or
its Velocity of Acceſs to the Sun is accelerated in the firſt Part
of its Orbit, and retarded in the laſt; and vice verfâ in regard
to its Recefs in the other Part of its Orbit.
1261. Secondly, the Fluxion of the Paracentric Velocity
-ÿ = N n
— 2 mo fhews, that fo long as the ſaid Velocity con-
tinues to increaſe, or be accelerated, Twice the Centrifugal Force
(2 mo) will be less than the Centripetal Force N n.
1262. Thirdly, when the Paracentric Velocity is a Maximum,
or greateſt of all, the Centripetal Force (Nn) is equal to twice the
Centrifugal Force (m o.)
1263. Fourthly, from the Time the Paracentric Velocity
continues to decreaſe, till it vanifheth at B, the Centripetal Force,
or Gravity (Nn,) will be less than twice the Centrifugal Force (mo.)
But that Gravity Nn can never decreaſe ſo far, as to become
equal to the Centripetal Force mo, even in the Perihelion Point
B, is what we ſhall ſhew by and by.
1264. Let the Triangles P S N and M S N be equal, or de-
fcribed in equal Times; then will SNxNp SMxMm,
then it will be Np: Mm::SM: SN. Let IK be the Latus
Rectum, or Parameter of the elliptic Orbit, and L × a be a
conftant Rectangle, equal to SM x M m. Put S MD,
and I KL.
La
L² a²
and mo=
D2
M m²
2 SM
Then the Arch M m -
a² L²
(1171) =
2 D3
and M m²
D'
=
Centrif. Force.
1265. Again, the Centripetal Force, or Gravity L M, is in-
verſely as D², or directly, as M m², (1264,) or as
that is, (dividing by the conſtant Quantity † L,)
2 La²
D²
a² L²
D¹
D .
is as
the Force of Gravity. Wherefore Gravity is to the Centrifu-
gal Force, as
2 La²
D2
to
L² a²
2 D3
or as D to L. That is, The
Force of Gravity is to the Centrifugal Force every where, as the Diſtance
of a Planet to a fourth Part of the Latus Rectum of the Ellipfis.
1266. Let IO=ISL, then in the Point I, Gra-
vity is to the Centrifugal Force, as IS is to IO, or as 2 to 1,
and conſequently, the Paracentric Velocity in the Point I, will
be the greateſt of all, or a Maximum (1262.)
1267.
102
INSTITUTIONS
1267. That the Centrifugal Force is always lefs than Gravi-
ty, will be evident, when we confider, that the Perihelion.
Diſtance S B, does, in an Ellipfis, always exceed SO, or of
the Parameter, becaufe in a Parabola, which lies without the
Ellipfis, the faid Line S B is then but juft equal to of that
larger Parameter (742.)
4
1268. Hence it will again appear, that a Planet in the A-
phelion Point A, will defcribe a Circle A G, when the Force
of Gravity is there equal to twice the Centrifugal Force. But
if it be greater, the Planet will defcend in an Ellipfis towards
the Sun, and in the loweſt Point B, Gravity, being leſs than
double the Centrifugal Force, can carry the Planet no nearer to
the Sun from that Point; therefore, it muft of Courſe begin to
afcend with an increaſing Paracentic Velocity of Recefs, till it
arrives to the Point K, where Gravity becomes equal again to
twice the Centrifugal Force. After this, the Centrifugal For-
ces leffening much fafter than Gravity, the Latter will prevail,
and the Paracentric Velocity of receding from the Sun will
confequently decreafe, till the Planet arrives to its Aphelion A,
where it will entirely vaniſh.
1269. And thus we fee, more particularly now, by the Dif-
ference of theſe two Forces, how the Planet is made conftantly to
revolve to and from the Sun, and in fo conftant and regular an
Order, as to give us the cleareft Ideas of a Uniformly Vari-
-able, and Perpetual Motion.
1270. Befides the Velocity of a Planet's Motion hitherto
mentioned, there is one other, which is called the Angular
Velocity of a Planet in its Orbit. In order to eſtimate this,
it muſt be confidered, that any Angle is greater in Propor-
tion, as the Arch defcribed with a given Radius is fo.
And alſo, when the Arch is given, the Angle will be lefs
in Proportion, as the Radius is greater; and therefore every
Angle will be in a Ratio compounded of the direct Ratio of
the Arch, and reciprocal Ratio of the Radius, and far-
ther we have juſt now fhewn (1264,) that in the Cafe of de-
ſcribing equal Areas, the Arches are inverſely as the Radii ;
therefore, in this Cafe of a Planet's Motion, the Angles deſcribed
in equal Times will be inverfely as the Squares of the Radii.
1271. But it has been fhewn, that the Force of Gravity is
every
of the Phyfico-Mechanical Mathefis.
103
every where inverſely as the Radius, and confequently, directly
as the angular Motion of a Planet in its Orbit; and therefore
the Latter will be an adequate Meaſure of the Former.
1272. Therefore the Impetus, or Sum Total of all the Im-
preffions of Gravity, which the Planet acquires in moving
from A to P, is to the Impetus acquired at M, as the Angle
ASP is to the Angle AS M. Hence likewife it appears,
that the Impetus acquired in defcending from A to I, is juſt
Half that which is acquired in defcending from A to B; and
therefore the Impreffions of Gravity upon the Planet, as it
paffes from I to B, are equal to all it receives before, in its Paf-
ſage from A to I.
1273. Having thus ftated the Ratios of the Velocities and
Forces concerned in a Planet's Motion, we fhall next pro-
ceed to illuftrate the fame by a familiar Inftance, where all
thofe Quantities will be expreffed in proper Numbers, and
therefore be more eafily comprehended and understood. In
order to this, let AS reprefent the Diſtance from the Center
of the Earth to the Circumference H A G, a Circle on the
Surface of the Earth; it was fhewn, that a Body revolving in
this Circle (1190) with a Centrifugal Force, equal to that of
Gravity, muſt be projected from A with a Velocity of 26000
Feet Per Second.
1274. Now, fuppofe it was required to find the Velocity,
with which a Body ſhould be projected from the ſaid Point A,
to deſcribe the Ellipfes AE BD; fo that SB may be equal
to 1000 Miles. Then as AS is equal to 4000 Miles, the
whole tranfverfe Diameter A B will be equal to 5000, and
ED will be equal to 4000; and I K will be 3200; fuch are
the Dimenfions of the Ellipfis. Then by the Theory in
(1204,) n =
b b
ad
2000 X 2000
4000 X 2500
0,63246.
Therefore I V : : n : v, or 1: 26000 :: 0,63246: 16444
: :
Feet per Second, the Velocity required.
1275. Thus, the Velocity in the Circle and the Ellipfis, at
the Point A, is known, and the Velocity at B in the Circle
is to the Velocity at A in the Circle as AS to ✅✔✅BS, or
as 2 to 1, that is, the Circular Velocity at B will be at the
Rate of 52000 Miles per Hour (1245.)
1276. But the Velocity in the Orbit at B will be to that by
104
INSTITUTIONS
which it was projected at A, in Proportion as AS to S B, or as
4 to I. Therefore the Velocity of the Planet in B will be
16444×4 65776 Feet per Second, which, as it greatly exceeds
the Velocity in the Circle, will prevent the Planet from revolv-
ing in a Circle about the Sun, and carry it off in its own proper
Orbit, in the Manner before-mentioned (1247.)
1277. Then, as to the Forces, that in the Circle at A is e-
qual to Gravity, which ſuppoſe to be 100 Pound Weight, and
we have ſhewn, that Gravity is every where to the Centripetal
Force of revolving Bodies, as the Diſtance to of the Parameter,
that is, in the Point A, it will be as AS to SO, or as 40 to 8,
or 5 to 1. Therefore the Centrifugal Force of the Body in A
is but 20 Pound.
1278. In the Point B, Gravity is 16 Times greater than at
A, or equal to 1600 Pound Weight, and fince there the Gra-
vity is to the Centrifugal Force as SB to SO, or as 10 to 8,
the Centrifugal Force at B will be equal to 1280 Pounds, which
tho' it be confiderably lefs than Gravity, will yet prevent any
nearer Approach of the Planet to the Sun.
1279. Laſtly, it appears, that the angular Velocity of the
Planet at A, is 16 Times lefs than that at B, or to an Eye placed
at S, the Space deſcribed in one Second at A will appear 16 Times
leſs than that which is deſcribed in the fame Time at B, agree-
able to (1270.) Since A S= 4S B, (1274.)
=4SB,
1280. Thus we have applied the Phyfico-Mathematical Prin-
ciples of Motion to the Theory of Aftronomy, as far as it can
be done without the Affiftance of OPTICS; but, as the greateſt
Part of Aftronomy, both theoretical and practical, depends
entirely upon optical Principles, nor can by any Means be un-
derſtood without them, it will be neceffary here to defift, and
proceed to the Elements of the Science of Vifion; nor will the
Principles of common Optics be fufficient to anſwer our Pur-
poſe, with regard to finiſhing a complete Treatife of Aſtrono-
my. The Doctrine of PERSPECTIVE must be well underſtood,
as alfo the general Principles of the Projection of the SPHERE in
Plano, and that too in a different Manner from that in which
they have uſually been treated, theſe will all be found neceſſary in
the various Branches of that Science. We fhall therefore, in the
next Place, proceed to lay down a Series of Inftitutions, con-
taining the Principles of univerfal Optics.
INSTI
B
OPTICS
D
G
1
1
Fig.5.
F
E
Fig.4.
V
B
and
D
PERSPECTIVE,
Fig.3.
D
A
D
C
D
B
B
E
B
D
E
A
C
Fig. 12.
Fig. 11.
Я
B
1
P.-
B
D
E
D
F
with
F
Fig.1
Plate I
I.
a
A
B
B
Fig.2.
c
A
E
D
B
C
A
C
C
B
A
E/F
H
Fig.9.
E
Fig.10
Fig. 8
Fig.7
D
Fig.18.
F
C
Fig.14
୯
A
E
C
M
A
KE H
F
C
Fig.6.
V Fig.13.
B
Fig.16.
с
I
a
M
B
B
Fig.15.
D
IN
M K
D
Fig. 17.
b.
F
DI
E
Fig.22.
C
D
Fig.21
C
C
B
A
A
с
a
B
D
Fig.23.
A
a
B
I
H G
F
G
F
C
I
N
B
M
b
Fig.28.
+
k
m
K
Fig.20.
Fig. 19.
D
O
BY
N
B
с
C.
A
a
C
A
G
G
。 Fig. 25.
Fig.24.
I
K
A
I
A
B
E
D
Y
W
I
X
Fig.31.
Fig. 29.
R
B
LA
B
H
İM
H
a
2
Fig 27
H
K
H
G
A
B
I
Ꮐ
F
D
E
B
B E
Fig.26.
G
C
數
​Fig.30.
.A
D
( 105 )
INSTITUTIONS
O F
Univerfal OPTICS:
CONTAINING
The ELEMENTS of CATOPTRICS, DIOPTRICS,
and PERSPECTIVE.
CHA P. I.
The Phyfico-Mathematical ELEMENTS of CATOP-
TRICS, or VISION, by reflected LIGHT, from
all Sorts of polished SPECULUMS, or MIRRORS.
1281.
S
INCE the Particles of Light are found to
be real Matter, they will obferve the Laws
of Motion common to all other Bodies a-
rifing from Attraction and Repulfion; and
therefore if a Particle of Light proceeds from the Point A to
the Plane BE, and ſtrikes it in the Point C, it will there meet.
with a repulfive Force, by which it will be reflected from the
faid Plane in the Direction CD, making therewith the Angle
ECD. Now it is required, to find the Point C in the Plane
BE, fuch, that the Ray of Light impinging thereon ſhall be
reflected to the Point D, fo that its Paffage from the given
Point A to another given Point D ſhall be the leaſt poffible.
1282. Let the Perpendicular
AB-a, and the Perpendicular
DE = b; BE➡c, and BC
Then CE=G
=X.
x,
and A C = √ aa+xx; allo
CD = √bb + cc-26x+xx.
A
D
B
C
E
P
Now
106
INSTITUTIONS
Now, fince
AC + CD, that is,
✓ b b xc c
x x
fore its Fluxion
+
Vaa
a a + x x +
2 cx + x x muſt be a Minimum, and there-
x x
✔aa + xx ✓ bb + c c
c x
2 c * + x*
26x+
= o, confequentlyx+bb + c c −2cx + xx+x-c+
aa+xxc. Therefore x x b b + c c = 2 cx + x²
-----
✔
= c − x √ aa + xx, or B C + CD CE x AC, and fo
BC:AC::CE: CD. Hence the Triangles ACB and
DCE are equiangular (657;) and fo the Angle of Incidence
ACB = DCE, the Angle of Reflection; and that this is
really the Cafe in regard to the Reflection of Light, we are
well affured from Experiments.
1283. The Nature
of the Curve AMD,
the Diſtance of the lu-
minous Point B, and
the Pofition of the in-
cident Ray BM being
given, it is required to
find in the reflected
Ray M F, the Point,
or Focus F, where all
the Rays iffuing from.
the Point B will be
united. In order to
this, let CM be the
Radius of Curvature to
D
G
O m
.....
F S
MKR
A
E
BI
་་་་་་
C
the Point of Incidence M, and take the Arch Mm infinitely
fmall, and draw the right Lines B m, m F; on the Cen-
ters B and F defcribe the little Arches MR, MO; and draw
the Perpendiculars CE, Ce, GG, Cg, to the Rays of In-
cidence and Reflection; and fuppofe the Diſtance BM = ₫
and ME or MG = a; then 'tis evident, that the Triangles
MR m, MO m, are equal and fimilar, and confequently M R
is MO; and becauſe the Angles of Incidence and Reflec-
tion are equal (1282,) therefore CECG, and C e — Cg,
and confequently CE-Ce, or E Qis to CGC g, br
EQ
SG: And becauſe the Triangles BMR, BEQ, FM O,
FGS
Of CATOPTRICS.
107
FGS are fimilar, it is, BM+ BE (2 da): BM (d) ::
MR + EQ, or MO+GS: MR, or MO:: MG (a):
a ḍ
MF=
2d-a
1284. If the radiant Point B fall on the other Side of the
Point E in reſpect of M, or (which is the fame Thing) if the
Curve AMD be contex towards the Radiant B, then d will
be negative, ord; and therefore M F =
ad
- ad
2 d a
Hence in this Cafe, the Focus F will always be pofi-
2 d + a
tive, or on the fame Side the Curve with the Point C; and
the Rays after Reflection will diverge.
1285. When the Radiant B is on the concave Side of the
Curve, the Value of MF (=
ल
ad
2 d. a
will be pofitive when
d exceeds a ; but negative when dis lefs than a; and infi-
nite, when da.
1286. If the Radius of Curvature M C be infinite, then alſo
ME a will be infinite, and M F =
ad
2 d = a
will become
d. In this Cafe the Curve, or ſmall Arch M m becomes a
ftrait Line. Therefore in both Cafes, when it is
the Rays, after Reflection, will diverge.
d, or + d₂
d,
1287. If any two of
M
the three Points B, C
and M be given, the
Third may be found.
Thus, if the Curve
AMD be an ELLIP-
A
B
F D
SIS, and the Radiant
Point B be in one Focus,
'tis evident all the re-
E
flected Rays MF will
C
be united in the other Focus F (772.) Whence MF-
a d
2 d
=f, and a =
a
2df
d+ f
hence a X
d+ f = dƒ; and
d + f
therefore dar:
2
f; that is, BM: ME :: AD:
2
P 2
MF.
108
INSTITUTIONS
MF. Hence likewife, when the Points B and M are given,
the Point E or C may be found, for then a =
2df
or AD
d+f'
MF: BM: ME; and a Perpendicular on the Point E
gives the Point C in a right Line MC, bifecting the An-
gle BMF. Note, The fame Demonſtration ferves for the
Hyperbola, where a =
2 d f
d-f
1288. In the PARABOLA
AMD, the Focus F is re-
moved to an infinite Diſtance;
and the Radiant being in the
D
E
M
Focus B as before, 'tis evident
the reflected Rays MFA
B
G
will be parallel to the Axis
C
E
a d
AG; and therefore fince MF
=f=zd—a
2
is infinite, we have 2 da, or 2 B M =
ME. Therefore a Perpendicular E C erected on the Point E
will affign the Point C for the Center of Curvature to the Point
M, in the Perpendicular M C.
1289. If AMD be a CIRCLE, fince
ad
f=
2 d.
a
we have 2 d — a: a :: d
f. If therefore in the Line B M con-
tinued out, we make MO 2 d — a;
then it will be MO: ME:: MB: MF,
whence the Point F, or Focus is found.
B
E
A
M
D
1290. Since in the Circle the Radius of
Curvature is a conftant Quantity, a Per-
pendicular to the Point E will ever paſs through C the Cen-
ter of the Circle; therefore when the Point M is infinitely
near to the Vertex D, the Line E Ma, will be equal to
CDr Radius, and the general Theorem for finding the
dr
a d
Focus F, viz. MF =
will become
>
2 d
a
2 d. r
:f,
in this particular Cafe;
the Curve,
and the Focus F will be in the Axis of
1291.
Of CATOPTRICS.
109
1291. If the Radiant B be at
an infinite Diſtance, then d be-
ing infinite, gives
2
dr.
2 d r
r=f, or the Focus is then e-
qual to Half the Radius. If 2 d.
be greater than r, the Focus f is
pofitive, or on the fame Side with
the Radiant. If 2 d be leſs than
r, then the Focus will be negative,
A
D
F
C
M
E
or on the contrary Side, in regard to the Radiant. If d be
negative ord, that is, if the Radiant be placed on the con-
dr
vex Side of the Curve, the Theorem becomes
dr
2 d + r
f, always poſitive, as before (1284.)
·2d-r
1292. Since dr = 2 d f + rf, and confequently df +rf
= dr- df, we have d+r:d::r-f:f; that is, B C:
BD::CF: D F. And therefore the Axis of the Curve, or
Line B C is harmonically divided in the Points C, F, D, B, as will
appear, when we treat of Harmonical Proportions.
1293. Let A MB D
be a Semicircle de-
B
fcribed on the Dia-
M
F
I
meter A D, and ex-
N
pofed to parallel Rays;
then thoſe Rays which
H
fall by the Axis CB
A
will be reflected to ƒ,
E G
C
D
the middle Point of
CB(1291,) and thoſe
which fall at A, as they touch the Curve only, will not be re-
flected at all; and any intermediate Ray EM will be reflected
to a Point F, fomewhere between A and ƒ; and alſo, ſince
every different incident Ray will have a different focal Point,
therefore thofe various focal Points will conftitute a Curve-
line A Fƒ in one Quadrant, and fI D in the other, which
Curve is called the Cauftic by Reflection.
1294
110
INSTITUTIONS
1294. Since d is infinite, we have
d a
2 d a
=MF = a*
ME every where; therefore if we bifect the Radius C M
in H, and drawn HF perpendicular to M F, the Point F will
be in the Cauftic Curve; for the Triangles MFH and MEC
are always fimilar, and give MH: MC:: MF: ME.
E
1295. Alſo, ſince the Angle M F H is a right one by Con-
ftruction, the Point or Focus F will ever be in the Circumfe-
rence of a Circle defcribed on the Diameter M H M C.
Therefore the Cauſtic A Ff is a Semi-epicycloid, deſcribed by
the Revolution of a Circle MF H, on the Periphery of a Cir-
cle G H ƒ about the Center C, whofe Radius CHMH,
the Diameter of the generating Circle.
1296. If the Angle A CM be Half a right one, than be-
cauſe the Angle EMC = CMF = MCE, the reflected Ray
MF will be parallel to A C, and will therefore touch the Cau-
ftic A Ffin the higheſt Point F..
1297. This Theory, with refpect to the Caustic by Reflec-
tion, is moſt evidently confirmed by Experiment; for if a cy-
linder Bowl, or Glafs, be expofed to the Sun-beams, or Can-
dle-light, this Curve A FƒID will appear very ſtrongly deli-
neated on any white Surface placed horizontally within the
fame.
CHA P. II.
The popular Doctrine of CATOPTRICS, deduced
from the foregoing Theory.
1298.
A
(PLATE I.)
CCORDING to the different Modification of
the Rays of Light, they receive a threefold Di-
Atinction. (1.) They are faid to be parallel when they proceed
in Directions equidiftant, or parallel to each other; as at A in
Fig. 1. (2.) Converging Rays are fuch as tend to one Point
F, as thoſe at B. (3.) Diverging Rays are fuch as proceed from a
Point F, in different Directions, as reprefented at C.
1299
Of CATOPTRICS.
IIf
1299. With regard to reflecting Surfaces, uſually called SPE-
CULUMS, or MIRRORS, there are likewife three different Forms,
viz. (1.) A Plane Speculum, or Looking-glafs, as A B in
Fig. 2. (2.) Concave Speculum, which is the Segment of a
hollow Sphere, foliated on the Outſide A V B, or poliſhed on
the Infide, as Fig. 3. (3) Convex Speculum, the fame Segment
of a Sphere, but foliated on the Infide A CB, or polifhed on
the Outfide, as in Fig. 4.
1300. If we confider a fingle Ray of Light DC, falling on
theſe three Surfaces, as it refpects but one fingle Point C in
each of them, fo the Law of Reflection will be the fame in all,
viz. That the Angle of Incidence DCE is equal to the Angle of
Reflection ECF (1282,) EC being fuppofed perpendicular to
the ſeveral Mirrors in the Point C.
1301. Let A VB (Fig. 5.) be a Concave Mirror, E the
Center, V its Vertex, and VG the Axis. Then let G be a
radiant Point taken any where in the Axis, from whence a Ray
of Light G C proceeds to any Point C very near to the Vertex
V, and draw E C, the Perpendicular to the Point C, and make
the Angle E Cf GCE, and Cƒ will be the reflected Ray,
meeting the Axis in f, which is called the proper Focus, or that
which refpects the Diſtance G V only. (See 1283.)
1302. All parallel Rays, DC falling on the Part CV, ex-
tremely near the Vertex V, will be reflected to a Point F, fo
that DCE the Angle of Incidence be equal to ECF the
Angle of Reflection; which Point, or Focus F is the middle.
Point between E and V, or VF Radius V E (1291.)
1303. And fince this is the Cafe, with regard to the Sun-
beams, which, by fuch a Mirror are all reflected or converged
to the Point F, that Point is called the Solar Focus, or Focus of
parallel Rays; and is relative to fuch Objects only as are at a
very great or infinite Diftance.
1304. Again, as the Point F is that where all the Rays
of the Sun, falling on the Speculum, are collected into a
very fmall Space, they will be there greatly condenſed, and
their Action on Bodies, with refpect to Light and Heat, fo
very much encreaſed, as to produce accenfion, or burning of
any combustible Body placed in that Point F, whence it is
called
112
INSTITUTIONS
called the Focus, or Burning Place, and all fuch Speculums are
called BURNING GLASSES.
1305. What we have faid of a fingle Ray holds good for
any Number, or Quantity of Rays iffuing from a given Point;
and hence it will follow, that all the Rays which flow from any
particular Point of an Object on a reflecting Speculum, will all be
converged to one Point nearly, or made to diverge from one Point;
and that Point, therefore, will be a Repreſentation of the faid
Point in the Object, and confequently, fince every Point in
the Object may eafily be conceived to be thus form'd in the
Focus of the Speculum, the whole Object will be there repre-
fented, formed, or depicted in Imagery; or there will be an
IMAGE formed in the Focus of every diftant OBJECT to which
it is expofed.
1306. To explain this Matter more particularly, let OB be
any Object placed before any Speculum C V D, (Fig. 13.) at
the Diſtance AV in the Axis; let E be the Center of the
Mirror, through which, from each extreme Part of the Ob-
jest O and B, draw the Lines, or Rays OED and BEC to
the Mirror, and as they paſs through the Center E, they will
be perpendicular to the Surface in the Points D and C. Alſo
from each Point O and B draw the Rays O V, BV, to the
Vertex of the Mirror V. Laftly, join O C and B D.
OC
1307. Now it is evident, that fince the incident Ray O V
on the Vertex on one Side the Axis A V, makes the fame An-
gle OVA, as the reflected Ray V M does on the other,
therefore the reſpective Focus M will be on the contrary Side
of the Axis from the radiant Point O. And the fame is to be
obſerved with regard to the other extreme Point B, and its Fo-
cus I. Therefore the Pofition of the Image I M before the Concave
Mirror is inverted with respect to that of the Object O B.
1308. The Points O, A, B, in the Object being repreſen-
ted by M, a, I, in the Image, it will be found by Computation,
that the Form of the Image is curvelineal, when the Speculum
CD is large, tho' very little fo, when it is ſmall in Diameter.
The Rule for finding the focal Diſtances MD, a V, IC, for
the refpective Diſtances of the Radiant OD, AV, BC is
this. Multiply the Distance of the Radiant by the Radius of the
Speculum, and divide that Product by the Difference between twice
the
Of CATOPTRICS.
113
the faid Diſtance and the Radius; the Quotient will will be the
Focal Distance required.
1309. The Object and Image fubtend the fame Angle at the Ver-
tex V and Centre E of the Speculum. For at the Vertex they are
both feen under the fame Angle OV B; and at the Centre of the
Angle which the Object fubtends OEBIEM the Angle
fubtended by the Image.
1310. The Lineal Dimenfions, or Magnitude of the Object and
Image, are as their Diftances from the Speculum. For OB:
IM:: AV: a V.
1311. Therefore when the Diſtance of the Object is equal
to the Radius, (viz. when it is placed in the Centre E) then the
Image there meets it, and is equal to it.
1312. When the Distance of the Object exceeds the Radius
EV, then will that of the Image be lefs; and the Image in all
fuch Cafes will be leſs than the Qbject.
1313. On the contrary, when the Diſtance of the Object is
leſs than the Radius, that of the Image will be greater; and
the Image will be in Proportion larger than the Object.
1314. Thus ſuppoſe IM a fmall Object placed at (n)
between the Center and Solar Focus (1291) then will OB
be its enlarged or magnified Image; and this is the Cafe and
Structure of what is properly called a REFLECTING MICRO-
SCOPE, by a ſmall Speculum CV D.
1315. If CVD (Fig. 14.) reprefent the fame Concavę..
Speculum and E its Center, then if I M be any Object placed
nearer to it than the Solar Focus (or half V E,) then by the
fame Reaſoning we ſhall have OB to repreſent its enlarged and
magnified Image on the other Side of the Speculum (1291); which
Image is in this Cafe erect, or in the fame Poſition with the Ob-
ject. And thus it is, that all large Concaves become Magnify-
ING MIRRORS.
1316. Thus it appears that a Concave Speculum has a poſitive
and a negative Focus, and will magnify or enlarge the Appearance of
an Object in either. And alſo, that it will diminiſh Objects in
the pofitive Focus only..
1317. If CVD be a Convex Speculum (Fig. 14) then any
Object O B placed before it will have a Virtual Focus only, or
Vol. I.
the
·
114
INSTITUTIONS.
the Rays will be fo reflected from it as if they came diverging
from a Point behind it (1291) thus the Ray OV will be fo reflected
from V to B as if it came from the Point M, and the Ray BV
will be reflected diverging from the Point I, and the fame may be
faid of all other Rays from the Points O and B; therefore MI
will be the Image of the Object O B.
1318. With respect to this Image we obferve (1.) That it
is always on the contrary Side of the Glafs from the Ob-
ject. (2.) That it is always erect: (3.) That it is ever leſs
than the Object, the Proportion being that of their Diftances
IV to VO from the Vertex, as before. Hence a Convex Mirror,
when large, will exhibit a delightful Landscape of diftant Objects,
which is its principal Uſe.
1319. As to a Plain Mirror or Common LOOKING-GLASS, it
appears from the Theory (1286). (1.) That the Focus is a
ways Negative, or behind the Glafs. (2.) That the Diſtance of
the Image behind is equal to the Diſtance of the Object before
the Glafs. (3.) That it is erect, and fimilarly fituated with the
Object. (4.) That it is of equal Magnitude with the Object.
(5.) Therefore at the Diſtance of the Object, its Image on
the Surface of the Glafs will appear but of half that Length,
and confequently a Perſon of fix Feet Height will require a Glafs
3 Feet long to view himſelf compleatly.
CHA P. III.
The Phyfico-Mathematical ELEMENTS of Drop-
TRICS, or VISION by LIGHT refracted thro' dif-
ferent Mediums, particularly adapted to LENSES
for OPTICAL USES.
1320. WE are taught to underſtand, by Sir J. NEWTON,
that there is a reflecting and refracting Power which
acts near the Surface of every Medium, or Body, in fuch Manner
as to reflect the Rays of Light at ore Inftant, and at another to
tranfmit and refract them thro' the Subftance of the Medium.
And
Of DIOPTRICS.
115
And tho' the Modus Agendi, or particular Action of the Power
be not fo accurately afcertained, yet that Author has made it
moft certain by Experiments, that ſuch a Mode of Action there
is, and that its Effects on the Particles of Light are the Inflec-
tion, Reflection, and Tranfmiffion thereof in and thro' different
Media. And the particular Modifications of Light from thence
arifing, he calls Fits of eafy Reflection, and Fits of eafy Tranf-
miſſion.
D
1321. Then admit DC were a
Ray of Light incident in one Me-
dium X, upon another Y of a
different Denfity and refractive
Power, in the Point C; and fup- H
poſe it there in a Fit of eafy Tranf-
miffion, then if Y be the Denfer Me-
dium, or has the greatest Refrac-Y
tive Power, the Ray, by the Action
of this Power, will be bent or refrac-
ལ་ས་ཡབ
LA
X
L
ted from its firft Direction DCE into another CF, fo as to
make the refracted Angle F CE of a lefs Quantity than the An-
gle of Incidence DCA. The Line AB being fuppofed perpen-
dicular to the Surface HO of the Medium Y in the Point C.
1322. Theſe Things premifed, the next, and indeed the fun-
damental Principle in Dioptrics, is to fhew that the Sine DL of
the Angle of Incidence ACD has a conftant Ratio to the Sine
IG of the Angle of Refraction BCG; and this will appear,
if we confider that when the Particle of Light arrives at or
near C it is affected with a new and additional Force from the
Medium, which Force acting in a Direction perpendicular to the
Surface HO will cauſe the Ray DC to deflect from its Courſe
to E into fome other Direction CG towards the Perpendicu-
lar CB; which may be thus determined. Let CE be the Space
defcribed in a given Time by the Uniform Velocity of Light in
the Medium X; then will the faid Line C E be as the Force
which produces that Velocity, (998). From the Point E
draw E F parallel to C B, and let E F reprefent the new ac-
quired Force of the Medium upon the Particle in the Point C,
and join CF, then will that be the new Direction of the Ray
and
116
INSTITUTIONS
and the Space deſcribed in the fame Time thro' the Medium Y,
refulting from the two Forces CE and EF (1028). But CF
is to CG (or CE) as BF (or KE) is to IG. That is, the
Sine of Incidence KE or DL is to the Sine of Refraction IG as the
Velocity CF, in the Medium Y, to the Velocity of the Ray C E or DC,
in the Medium X. But thele Velocities of Light in different
Media are as conftant as the Powers of Nature which produce
them; therefore fo is the Ratio of the Sines DL to IG in every
Inclination of the Ray D C.
1323. In the fame Manner it is fhewn, that if a Ray of Light
FC be incident from a Denfe Medium Y upon a Rarer Me-
dium X in the Point C, then by the fuperior Force of the
Denfe Medium it will be deflected towards its Surface HC, and
confequently be refracted from the Perpendicular AC into the
Direction CD, making DL to IG as FC to CD, as before.
1324. Let B M be a Ray of Light incident on a Refracting
Medium Y bounded with a curved Surface AM D, and let MF
be the Refracted Ray, and F the Focus to which all the Rays
falling on or near the Point M will be refracted. It is re-
quired to find the Point F by having given the Nature of the
Curve A MD, the Radiant Point B, the Ray BM, and the
Refraction of the Mediums X and Y.
X
R
B
A
e
D
E
Y
F
H
1325. In order to this, let M C be the Radius of Curvature
to the Point M, and draw Bm infinitely near B M, and join
m F and m C; from the point Clet fall the Perperdiculars CE,
Ce, on the Incident Rays continued; and CG, Cg, on the
Refracted Rays; and on the Centers B and F defcribe the fmall
Arches MR, MO. And put B Md, ME a, MG=b;
the
Of DIOPTRICS.
117
the little Arch MR, and MO=y, and the Sine of Incidence
CE to the Sine of Refraction CG as m to n (the Radius being
MCr) then the right angled Triangles M E C, MR m;
MGC, MOm; BMR, BQE, are Similar; becaufe, if
from the right Angles RME, CM m, we fubtract the Angle
EMm, there will remain the Angle R Mm EMC; and
if from the right Angles GMO, CM m we fubduct the An-
gle GMm, there will remain OM m GMC. Therefore
we have the following Analogies, ME: MC:: MC: Mm;
Mm. Again, MG: MC:: MO:
That is, 'a:r:::
x x
a
Mm; that is, b:r::j:
a:b
: * j = MO=
7290
rx
a
; therefore abx; and
b*. Then BM:BQ(=BE)::MR:
a
Qe; that is, d:d+a:::
d x + a x
d
=Qe. But (1322) Ce:
Qe : Sg ::
Cg::CE:CG::m:n::Ce-CE:Cg―CG::
a x + d x n a x + ndx
d
md
=
Sg. Laftly, the Similar Triangles
F MO and FSg, give MO: Sg:: MF: SF or GF; there-
fore MO-Sg: MO:: MF-SG, or MG: MF; that
a a nx andź b b x
amd
b m d x
is
bbm d
b:
a
bmd- —aan—and
MF, the Focal Diftance required.
X D
M
B
A
C
F
f
1326. If the Curve AMD be a Circle, then CM=ris
conſtant, and the Points E and G will be in a Semicircle de-
fcribed on the Diameter MC; and when the Point M is very
near A the Vertex, then ME and MG are both equal to each
other, and to MC; that is, a br; and the Theorem will
become
}
118
INSTITUTIONS
become
rm d
mid-
nd
r n
the Axis of the Sphere.
Mf, in which Cafe the Focus f is in
1327. If the Diſtance of the Radiant B (d) be infinite, or
the Rays parallel, then the Theorem will be
Focus of the parallel Rays.
in r
m 22
=Mf=
1328. If the Rays fall diverging on the Convex Surface A MD;
thend being Negative, ord, the Theorem will become
bbm d
bm d
bbm d
aan + and bmd- and + aan
-MF (Fig.
to 1324) and when A MD is a Circle, and M infinitely near
to A, then will the Theorem be
і
mr d
m d + d n + rn
=Mf.
1329. If the Curve AMD were ſpherically Concave to-
wards the Radiant B, then will the Radius be Negative orr,
and the Theorem is
-rm d
md-
n d + r n
Mf the focal Dif
tance; which, becauſe m is greater then n, will be Negative, or on
the fame Side with the Raidant B, or the Rays after Refraction
will diverge.
1330. In the Cafe of Parallel Rays falling on the Spherical
Goncave, we have
rm
772- n
Mf, Negative alfo, becauſe dis infinite,
1331. But Converging Rays incident on the faid Concave
will have
r m d
nd + nr-
m d
=Mf, which will be Pofitive or Ne-
gative, as n x d + r is greater or leſs then m d.
1332. If the
Surface AMD
be a right
Line, or the
Radius M C
=r infinite, T
then the Theo-
>
rem (1325)
D'
B
X
A
E
G
M
包
​will be
m b b d
ña a
MF; for in this Cafe a and b are alfo infinite;
and
Of DIOPTRICS..
119
and conſequently ſince a a = bb, we have
Y
m a
A
the focal
Diſtance; and therefore md-af, and fo
:m: :d : — f;
whence it appears the Ray will be refracted towards the Per-
pendicular MC, having a negative Focus ƒ on the fame Side
with the Radiant B (1321).
4
1333. Thefe are the feveral Cafes of a fimple Refraction,
and they are the fame when we confider the Ray coming out
of a denſer Medium Y into a rarer X, only the Ratio of n to m
is in that Cafe to be uſed inftead of m to n in this; or in the
foregoing Theorems, putting n far m, and m for n, inter-
changeably, and other Things altered as required.
Y
M
D
X
C
D
1334. Thus for Example; Suppofe it required to find the For
cus of Rays paffing out of a denfer Medium Y into a rarer X in a
converging State, and refracted at a concave Surface of the Medium
X. Then it is plain the Theorem in (1326) will equally
ferve here with the following Alterations, viz. (1.) Becauſe
the Refraction is into a rarer Medium, we muft write n for m,
and m for n. (2.) Becauſe the refracting Surface is Concave,
the Radius will be Negative, and the Sign where (r) is found
muſt be changed. (3.) Becauſe the Rays are converging, the
Diſtance is negative, and the Sign where (d) is found muſt be
changed. (4.) Therefore the Sign where d and r are both
found, will continue the fame. The Theorem therefore, with
r nd
md—nd + r' m
all theſe Alterations, will become
the focal Diftance required.
=fDf
f = D f,
1335. Now it is evident, that if the incident Ray MN in
this Caſe be confidered as Part of the refracted Ray Mƒ in the
former (1326), then will the Focus f in that Cafe be the
radiant Point in this; and therefore if we fubftituteƒ for 4, and
Put
120
INSTITUTIONS
put r for the Radius of Concavity CD, the laft Theorem will
be
nr f
mf→nf+mx
-Dff.
M
N
CA
D C
1336. If now we fuppofe the two refracting Surfaces A M
and ND to be very near together, fo that the Diſtance AD be
inconfiderable, then may the Focus f be determined for the in-
cident Ray BM after both Refractions at M and N; for the
mfr
laft Theorem gives
1326) Therefore
nx + nf — m f = f = Afor Dƒ. (Fig. to
m fr
nr + nf-
m f
m dr
m d — n d — nr
; and con-
fequently we have
nd r r
f
Df, as required.
m
m r d — n r d + mdr - n d r — n r x
1337. This Theorem may be abbreviated by putting
n
n =a, (or if n = 1, and m—1—a) it will become
drr
ard+ard—y r
f; and is thus accommodated to all Opti-
cal Purpoſes, and for a Lens AMN D of any Sort, on which
Rays diverging, parallel, or converging can fall.
1338. Thus if parallel Rays fall on the Lens, then A B=d,
being infinite, we have
ry
artar
= f.
1339. If the Rays B M are converging, then d being nega-
tivę, we have
drr
-ard-ard-rr
drr
= f.
ard + ard +rr
1340, If the Lens be equally Convex on both Sides, or r=r;
ar
then the Theorem for diverging Rays is ad = f; for
2
paralle
Of DIOPTRICS.
121
1'
parallel Rays where d is infinite,f. And for converging
24
dr
dr
Rays, where d is negative,
f, al-
—2 ad — r
2ad + r
ways pofitive.
1341. If one Side ND be plane, or the Radius r infinite,
then the Theorem for fuch a Plano-Convex Lens, for diverging
dr
adr
"
f; for parallel Rays f; and for conver-
Rays, is
ging Rays, adr
=
dr
ad + r
= f.
-
dr
is
1342. If one Radius r be Negative, or the Side N D be Con-
vex towards AM, then the Lens is a Convexo-Concave, or
Menifcus; and the Theorem for diverging Rays, is
ΥΓ
f; for parallel Rays,
drr
adr-ads + re
f; and for converging Rays
arar
}
drr
it is
= f.
adr-adr + rr
1343. If rr, then for diverging Rays,
parallel Rays,
-rr
0
df; for
f; and for converging Rays, d= f.
1344. If both the Radii r and r be Negative the Lens be-
comes a double Concave, and the Theorem for diverging Rays, is
drr
—adr-adr—rr
fr
ar - ar
=f, always negative. For parallel Rays
f, ever negative. And for converging Rays
drr
f.
adr + adr—re
1345. If one Radius →r be infinite, then the Lens is a
Plano-Concave; and the Theorems become
r
verging Rays; == f, for parallel; and
converging Rays.
a
dr
=f, for di-
ad tr
dr
= f for
ad-r
1346. If the negative Radii are both equal, it makes an
'equally Concave Lens; where the Focus of, diverging Rays is
L
VOL. II.
R
fouod
-122
INSTITUTIONS
found by this Theorem.
24
have
- dr
2 ad + r
=f, and of parallel Rays
=f, in each Cafe always negative. Converging Rays
dr
2 ad + r
=f.
1347. Theſe are all the Caſes that can happen in the Theory
of common Dioptrics, and may all of them be very eaſily applied
in Practice by fubftituting ſuch Numbers for the Ratio of m to
n as we find by Experiments agreeing to Mediums we uſe.
Water,
Thus in Glafs,
m:n:: 4 : 3, whence a = 0,3333-
4:3,
m:n:: 31: 20
31:20
Diamond,m: n::52
a = 0,573
a 1,5.
But for a larger View of this Subject fee my New PRINC
PLES of OPTICs, lately publiſhed.
CHA P. IV.
The THEORY of DIOPTRICS continued; the Na-
ture of the Diacauſtic CURVE explained; and the
Method of finding a GEOMETRICAL FOCUS for
Rays ifſuing from a given Point in the Axis of a
LENS of a Mechanical Figure.
I
1348. T has been ſhewn (1325) how the Focus H, F, N,
&c. of any Rays B A, B M, B N, &c. falling on
the Curve AMD may be found, after Refraction, from pro-
Data; and the Curve N F H, which is the Locus of all thoſe
Points, is called the DIACAUSTIC, or Cauflic by Refraction.
per
1349. If
Of DIOPTRICS.
123
4349. If the Cauftic HFN
be involved (fee 914.) be-
ginning at the Point A, the
faid Point A will defcribe the
Curve ALK, the Involute
of the faid cauftic Curve;
whence the Tangent LF+
the Part of the Cauftic H F,
will always be equal to the
right Line AH; that is,
LF+HF (orKN+HN)
= AH.
1350. Suppofe the other
Lines of the Figure drawn
as directed (1325,) and
moreover the Arch AP de-
fcribed on the Point B. Then
becauſe the right-angled Tri-
angles MRm and MO,
are fimilar to the Triangles
MEC and MG C refpec-
tively, they give Rm: Om::
CE:CG:: min.
1351. Now becauſe R m
=d, the Fluxion of B M
=d, and Om = Fluxion
of LM, and becauſe BP
= BA is a conſtant Part of
BM, the contemporary or
B
A
S
II
द
Ju
E
N
K
proper Fluents of the foreging Fluxions are P M and LM;
and fince in this Cafe, the Fluxions and Fluents in the fame
Ratio (788) therefore P M : L M :: m : n :: B M — BA÷AH
- MF-FH; and conſequently nx BM-BA = mx
AH— M F — FH; from which Equation, we get FH-
AK-MF+BA- BM, which is the Rectification
22
in
of the diacauflic Curve.
n
ก
1352. When the Radiant B is at an infinite Diſtance, then
BA BM; and AP is a right Line perpendicular to thoſe
R 2
incident
124
INSTITUTIONS
incident Rays, in which Cafe we have the Cauftic FHAH
-MF, or NH AH-NK; &c.
1353. It is evident from the foregoing Theory, that no
fpherical Surface (or any other we have treated of) can refract
all the Rays incident upon it from a given Point B, to another
given Point F, in the Axis; and therefore it becomes neceflary
to fhew the Conftruction of Curves that will do this. Let the
Curve required for this Purpoſe be A MD, and the incident.
Rays B M, B m be infinitely near each other, and M F, m F,
the refracted Rays; and draw the Tangents Dm and MN
perpendicular to the fame in the Point M. Laftly, draw m C,
m R perpendicular to the incident Ray BM, continued out, and
refracted Ray MF. Then the Angle M m C Angle of Inci
dence CMN, becauſe each being added to the Angle m MC
D
R
P
B
A E
N
makes a right Angle; and, for the fame Reaſon, Mm R = the
Angle of Refraction F MN: Therefore if we make Mm Ra-
dius, MC will be the Sine of Incidence, and MR the Sine of
Refraction; and as thefe are in a given Ratio m to n (1322)
we have MC:MR::m: n. But MC is the Fluxion of the
incident Ray, and MR the negative Fluxion of the refracted
Ray; the contemporary Fluents therefore of theſe Fluxions
will be in the fame Ratio of m to n. On the Point B deſcribe
the Arches P A and M G, and on the Point F deſcribe the Arch
ME, then PM and A E are the Fluents mentioned; and con-
ſequently we have PM or AG: AE::m:n. Which is the
Property of the Curve A MD.
1354. Therefore when the refpective Foci B and F, and A,
the Vertex of the Curve required, are given, the Curve may
be thus conſtructed. Take A G at Pleaſure, and fay min::
AG
Of DIOPTRICS.
125
n
AG:-AG=AE; then on the Center B with the Radius
37
A G defcribe the Arch M G; and on F, with the Radius FE
deſcribe the Arch ME to interfect the Arch GM in M, and
the Point M will be in the Curve A M required; and thus
all other Points of the faid Curve may be found, and the
Curve AMD drawn thro' them,
1355. By cauſing the Point B or F to go off fometimes to an in-
finite Diſtance, or fometimes to lie both on the fame Side the Point
A, we fhall obtain all thofe Oval Figures which Cartefius has
exhibited in his Geometry and Dioptrics, relating to Refractions,
P
P
M
B
A EG
F
1356. Thus if the Rays B M, B A, are parallel, then the
Arch G M becomes a right Line, perpendicular to the Axis
BF, and the Curve A MD will be an Ellipfis, whofe tranf-
verfe Axis AD is to the Distance between the Foci as m to n.
For put A Fa, AP=MG=y, PMAG = x; and
ſuppoſe m:n: 3:2 (1347) Then GF-a-x, and MF
Vaa- 2 ax + xx + yy; and by the Nature of the Curve
(1354) it is PM + MF AF, which gives this Equation
3 x + √ a² −2 a x + x²+y² = a. And by Tranſpoſition
and Involution (212) we have a² —2 a x + x² + y² = a²
fax + x; and confequently y² = ax-xx; and fo yy
=ax-x². Put at AD, then tx-xx= {yy;
6
*
3
therefore the Curve is an Ellipfis (764.) And, confequently
::
P
:p::9:5. Let d Diſtance between the Foci, then :tp
95, and 2-tpd (768) therefore 2- =ď², or
512
9
9 1² - 51² = 9 d², therefore 2:9 :: d² : 9 − 5 = 4,
that is
t: d
126
INSTITUTIONS
# :d :: 3 : 2 :: m :n, or, the tranſverſe Axis is to the Diſtance be-
tween the Foci as the Sine of the Angle of Incidence to the Sine of the
Angle of Refraction, as Cartefius has fhewn in his Dioptrics.
1357. On the other
hand, if the Point F
be fuppofed at an in-
finite Diſtance, or the
Incident Rays FM,
FA, parallel, then
ME becomes a right
Line; then A Ba,
AE=x, and EM-y;
3x, and E Ba+ x;
כם
P/M
F
F
D
AEG
then alfo A G=x, BM= a +
whence (as before) we get a²+3ax
+ 2 x x = a² + 2 a x + x² + y²; and thence a x + { x² = y²;
and putting at, we get tx + xxyy, which fhews the
Curve A M in this Cafe is an Hyperbola (765) whence
t
P
4
and p; therefore t+= d; therefore 4t² +5t²
= 4 d² = 9 ť²; fo that t²: d²:: 49, which gives t:d::2:3
::n: m. Or, the tranfverfe Axis of the Hyperbola is to the Dif
tance between the Foci of the oppofite Sections, as the Sine of Inci-
dence to the Sine of Refraction.
1358. Thus we have fhewn what Curves will refra& all Rays
iffuing from any one Point to any other given in a different
Medium, or by a fingle Refraction. It remains now to fhew
how the fame Thing may be effected by two Refractions; in order
to which it will be neceflary to premiſe the following LEMMA.
1359. To describe the
Curve G M, fuch, that
from any Point M,two
right Lines BM, KM
M
may be drawn to two
given Points B and K,
B
G
RK C
which shall be to each other in the given Ratio of m to n, or that
BM: KM::m: n.
Draw M R perpendicular to BK, and put B K≈a, BR =
*, and M R = y; then becauſe of the right-angled Triangles
BRM, KRM, we have BM√xx+y, and K M=
√ a².
Of DIOPTRICS.
127
a
2
2 a x + x² + y²; and therefore it muſt be✅✔✅x² + y²:
a
2 a x + x² + y²::m:n; which gives yy — =
aam m
xx
m m
nn
2.amm
mm-nn
; and fince the Co-efficient of the Square
of the Ordinate (yy) is Unity, (764) it is evident the Curve
GM is a Circle.
aamm
1360. When y yo, then
2 am n⋅ x
mm — N N
*
IN M
n 13
a m
a m
X
=
カル
​n
m + n
BG × BQ(330.) Therefore GQ
(= BQ-BG) is the Diameter of the Circle which is the
Locus of the Point M.
1361. The Curve A M, the radiant Point B, with the Ratio of m to
n being given, it is required to defcribe the Curve N D, that ſhall re-
fract the first refracted Rays MN, and make them converge to any
given Point C.
M
K
F
C
H
B
A
D
Suppofe FH to be the Cauftic by Refraction to A M, the
radiant Point being in B; it is evident that the fame Curve F H
will alſo be the Cauſtic by Refraction to the Curve N D (1352)
and therefore FH-DH-NF-
n
n
72
DC+ NC =
773
m
AH-MF + BA-BM, which Equation will give
n
n
m
n
711
n
1 AB- BM+ DC+AD = MN↓ NC.
712
m
m
m
1362. Therefore to conſtruct the Curve N D, in any re-
fracted Ray A H, take at Pleaſure the Point D, for one of
the Points in the Curve required. And in any other Ray as
MF sake MK-BA-BM+
77
ZIL
DC+AD; and
find
128
INSTITUTIONS
find the Point N (by 1360) fo that NK: NC::n:m, or NK
n NC, and the Point N will be in the Curve required; and
m
thus a fufficient Number of Points may be found thro' which
to draw the Curve DN.
1363. From the foreging Theory it appears, that it is pof-
fible to determine the Figure for any Mechanical Lenfes, Con-
vex or Concave, by which either fingly or conjointly, Rays of
Light proceeding from any one given Point B in the Axis of
the Lens may be accurately refracted to any other Point or
Focus f, which in this Cafe may be called the Geometrical Focus.
But certain it is, that no Figure for Glaffes can be found for
fuch a Focus of Rays proceeding from a Point out of the
Axis; and therefore it is naturally impoffible that the Defects
of Dioptric Vifion arifing from the Figure of Glaffes, fhould ever
be rectified by Art. We have here given the Subſtance of all
that has been publiſhed by KEPLER, DESCARTES, HUGENIUS,
Dr. BARROW, Sir J. NEWTON, and Dr. HALLEY, the greateſt
Maſters in this Science.
CHA P. V.
The popular Doctrine of DIOPTRICS deduced from
the THEORY; with the Rules for finding the
FOCAL DISTANCES of all Sorts of LENSES, alfo
the Proportion, Magnitude, Pofition &c. of
IMAGES formed thereby.
(Plate I. of OPTICS and PERSPECTIVE.)
1364.Heed to make fuch practical Deductions from it as
AVING thus premiſed the Theory, we now pro-
will be fufficient to acquaint our Readers (not verſed in Ma-
thematics) with all the uſeful Part of Dioptrics, and its Appli-
cation to Optical Inftruments in every Branch of the Viſual Science.
1365. Therefore let A B (Fig. 6.) be the Surface of any re-
fracting Medium BK denfer than Air, as Water, Glass, &c.
and let DC be a Ray of Light incident thereon in the Point C ;
thro'
Of DIOPTRICS.
129
thro' the Point draw E H perpendicular to the Surface A B,
and the Angle ECD is the Angle of Incidence. When the
Ray arrives at C it is refracted or bent out of its firſt right-
lined Direction D CK into another C F towards the perpen-
dicular C H, and the Angle F CH is the Angle of Refraction,
lefs than the Angle of Incidence K CH by the fmall Angle
F C K, which is called the refracted Angle.
1366. On the other hand, if a Ray of Light CF in a
denſe Medium be incident at C upon any rarer Medium, it
will be refracted out of its firft Direction F C into another C D
which will be farther from the perpendicular C E. And the
greater the different refractive Powers of the Mediums, the
greater will be the Difference of the Angles ECD and FCH.
All which is evident from the Theory (1322.)
1367. If the refracting Surface ACB be not plain, but
fpherical; then let E be the Center of the Sphere, and V E F,
the Axis thereof. Let CD be a Ray of Light falling on the
convex Surface (Fig. 7) in the Point C and parallel to the Axis
V.F; from the Center E draw the perpendicular E C H and it
is evident the Ray D C will be refracted in the denſer Medium
towards the perpendicular E C (1365), and therefore can be no
longer parallel to the Axis, but muft interfect it at fome Point
F which will be the Focus of all the Rays parallel to the Axis,
and very near it, as was fhewn (1327.)
1368. In Caſe of a concave Surface A C B (Fig. 8) a Ray of
Light DC parallel to the Axis E V, will, at its Entrance into
the denſer Medium, be refracted alfo towards the perpendicular
CH, in fuch a Manner to F, that were F C to be produced, it
would cut the Axis produced in the rarer Medium (fuppofe
Air) beyond F. (1329) The Diſtance of the Piont F from
the Vertex V in Water, is four Times the Radius of the Sphere,
viz. VF4V E; and in Glafs it is V F = 3VE (1347)
VF
or the Focus is diftant three Semi-diameter of the Sphere.
1369. If a Ray of Light be twice refracted, firſt into a
denſe Medium and then into a rare one, its Courfe after the fe-
cond Refraction will be variable, according to the Figure of the
Surfaces which bound on each Side the denfer Medium, and
may eaſily be determined by the Theory (1336.) This Caſe
in practical Dioptries brings us to the Confideration of Lenſes,
VOL. II.
which
S
130
INSTITUTIONS
which are the effential Parts of Teleſcopes, Microſcopes, and all
other Inftruments of the Dioptric Kind.
1370. Let A VB (Fig. 9) be a Plano-convex Lens, then
Rays of Light DC, DC, which fall upon it parallel to
the Axis VE will, after Refraction thro' it, be converged to a
Focus in the Axis at the Point F, which is nearly equal to the
Diſtance of the Diameter of the Sphere V F (from the Vertex V)
of which the Lens is a Segment (1341.)
1371. On the contrary, it follows that if Rays of Light
diverge from a Point F, at the Diſtance of twice the Radius
VE of Convexity in any Plano-convex Lens AV B, they
will, after Refraction thro' it, proceed parallel to the Axis V E.
1372. In like Manner, if AV B (Fig. 10) be a double
and equally convex Lens. Then parallel Rays D C falling
upon it will be refracted to a Focus F very near the Point E or Cen-
ter of Convexity, as is demonftrated (1340.)
1373. Therefore on the contrary, when Rays of Light F A,
F B, diverge from a Point in any Object in the Focus F of an
equally convex Lens A V B, they will, after Refraction thro' it,
proceed parallel to the Axis of the Lens V E.
1374. Let parallel Rays DC, D C, fall upon a Plano-con-
cave Lens A C B (Fig. 11) then will they be fo refracted thro'
it to f, f, as if they came diverging from a Point or Focus F at
the Diſtance of the Diameter of the Sphere of Concavity of the Lens
(1345) and the Point F is in this Cafe called the virtual
Focus.
1375. If parallel Rays DC fall upon a double and equally
concave Lens (Fig. 12.) then they will be refracted to f and f diver-
ging from a Point E which is the Center of Goncavity (1346) and
the virtual Focus of the Lens.
1376. In each of theſe two laſt Caſes, if Rays f C, f C, con-
verging to a Focus F or E, are intercepted by a fingle or
double concave Lens A B placed at the Diſtance of the Diame-
ter or Radius of Concavity from the faid focal Point, then the
Rays, after Refraction, will proceed parallel among themſelves, and to
the Axis of the Lens.
1377. If the Lens be not equally convex on both Sides, then
the Focus of parallel Rays will be found by the following
RULE.
Of DIOPTRICS.
131
RULE.
Divide the Product of the Radii, by half their Sum, and the
Quotient will be the focal Diſtance required. (1338) *
Example. Suppofe one Radius 15 Inches, the other 9; their
Product is 135, which divided by half the Sum 12, gives
11 Inches for the focal Diſtance of fuch a Lens.
II
TO
1378. If the incident Rays are not parallel, but come diver-
ging from a diſtant radiant Point in the Axis of a Lens of equal
Convexity, then the focal Diſtance is found by this
RULE.
Multiply the Diſtance of the Radiant by the Radius of the Lens 3
and divide that Product by the Difference between the faid Diſtance
and Radius ; and the Quotient will be the focal Distance required.
($340.)
Example. Let the Radius of the Convexity be 15 Inches, and
the Diſtance of the Radiant 60; then their Product is 900,
and their Difference 45; therefore 45)900(20 the focal Diſ-
tance in Inches, for that Diſtance of the Radiant.
=
1379. When the Radii of Convexity are unequal, the Focus
of diverging Rays is found by the following
RULE.
Multiply twice the Product of the Radii by the Diſtance of the
Radiant; and then divide by the Difference between the Sum of the
Radii multiplied by the Distance, and twice the Product of the Radii
and the Quotient will be the focal Diſtance required. (1337)
Example. Let one Radius be 15 Inches, the other 9; and
the Diſtance of the Radiant 60; then twice the Product of the
Radii is 270, which multiplied by 60 makes 16200; the Sum of
the Radii 24 multiplied by 60, is 1440, from which take 270,
there will remain 1170; then 1170)16200(13
focal Diſtance required.
$5
100
Inches, the
1380. If the Rays fall converging on the Lens, then if we
divide by the Sum inſtead of the Difference, the Rule will in
all other Refpects be the fame, in each of the two laft Cafes,
for finding the focal Diſtance for any given Diftance to which
the Rays tend (1339.)
S 2
1381. In
*This Rule is in general exact enough for Ufe, fince in common
Glafs the Value of (a) in (1338) is very little more than (1).
132
INSTITUTIONS
1381. In all that has been hitherto faid, it is ſuppoſed that
the Lenſes are of Glafs, and that the Sine of Incidence is to
that of Refraction in Glafs as 3 to 2; agreeable to (1347.)
But as there is a confiderable Difference in the refractive
Powers of different Kinds of Glafs, if we have Regard to
that (as in fome Cafes will be neccffary) then the Rules above
will be fomewhat more complicated, and we muft operate ac-
cording to the Theorems referred to, and take in the Value of
(a) as it is found by Experiment, in each particular Sort of
Glafs. †
"5
placed at a Diſtance from
Then it is evident that a
1382. Let OB be any Object
the convex Lens CD (Fig. 15.)
Pencil of Rays CAD which flow from the Point A in the
Axis will all be converged to another Point (a) in the Axis (if
the Diameter of the Lens CD be but fmall (by 1336) whoſe
Distance V a may be found (by 1377 or 1378) and this Point
(a) will be the Repreſentation or Image of the Point A in the Object
(1305.)
1383. Let O be a Point in the extreme Part of the Object
which fends a Pencil of Rays DOC to the Lens CD; a-
mong theſe one Ray O V will be refracted thro' the Center of
the Lens, and therefore the Pofition of the refracted Part V M
will be fimilar or parallel to the incident Ray OV (as will ap-
pear from 1332) and when the Thickneſs of the Lens-
CD is inconfiderable (as in moft Optical Cafes it is) then O V
and V M may without fenfible Error be efteemed one right Line;
and therefore the Axis of that Pencil of Rays, in which at
the Point M they will all be united after Refraction, wherefore
M will be the Image of the Point O in the Object.
1384. And in like Manner I will be the Image of the Point.
B in the Object, and fo the whole Object O B will be repreſented
in its Image I M. And by calculating by the above Rules it
will be found that the Image I a M will be curvelineal, more or
lefs, as the Aperture C D of the Lens is greater or ſmaller.
1385.. The Pofition of the Image, with refpect to the fimilar.
Parts of the Obje&, is inverted. The Reafon of this is evi-
dent by Inspection of the Figure, fince the Axis of the Pencils.
from each extreme Part of the Object crofs each other in the
Center of the Lens.
1386. The
+ See my NEW PRINCIPLES OF OPTICS, lately publiſhed.
Of DIOPTRICS.
133
1386. The Object and Image fubtend equal Angles at the Center
of the Lens. For fince OM and BI may be confidered as
ftrait Lines, the Angle B V O, under which the Object ap-
pears, will be equal to the Angle IV M, under which the
Image is ſeen from the Center or Vertex V of the Lens.
1387. The lineal Dimensions of the Object and Image are as
their Diſtances from the Lens respectively; for fince the Tri-
angles O V B and IV M are fimilar, we have OB, the
Length of the Object, to IM the Length of the Image, in the
fame Proportion as OV or AV the Diſtance of the Object,
to IV or a V the Diſtance of the Image. Hence their Surper-
ficies will be as the Squares, and their Solidities as the Cabes of
the Diſtances from the Lens.
1388. The Object and Image are reciprocal; for if I M be
confidered as an Object, then will O B be the Image thereof
and A, a, are called the conjugate or respective Focuſes, for thoſe
Diſtances of the Object and Image.
1389. Hence it appears, that the nearer the Object is to the
Lens, the farther off the Image will be formed; and when the
Object comes to the Focus of the Lens, the Image will ther
be at an infinite Diftance, fince the Rays from every Point.
will after Refraction be parallel among themſelves (1373.)
Laſtly, if the Object be nearer the Lens than the Focus, the
Rays after Refraction will diverge, and in this Cafe no real
Image can be formed of that Object at all by the ſingle Lens.
1390. There is another Form of a Lens called a Meniſcus,
which is concave on one Side, and convex on the other (Fig.16.)
But tho' it was in great Ufe and Efteem fome Years ago, it is
now entirely uſelefs, as the Plano-convex Lens is known now much
to exceed it in thofe Properties for which it was then ſo much
valued.
SCHOLIU M.
1391. The Lenfes hitherto confidered are of a ſpherical.
Form, and will not admit of a Geometrical Focus, not even of
Rays flowing from a Point in the Axis (1348, &c.) and therefore
it is impoffible that any fuch Lens fhould form a perfect Image
of any Object, not fo much as in a fingle Point thereof. We
have fhewn indeed from the Theory, (1361) that a Lens
may
134
INSTITUTIONS
may be formed that ſhall have a Geometrical Focus for any
radiant Point of its Axis, but not for any other; and therefore it
is plain, no Image of an Object can be formed in any Degree per-
fect by any Lens whatſoever; and confequently, that refracting
Teleſcopes will ever be, in their own Nature, imperfect, if conſtruc-
ted with a ſingle convex object Lens. But this Imperfection from
the Figure will admit of a little Correction from the Addition
of another convex Lens, but will be increaſed by joining there-
with a concave one, as I have largely fhewn in my NEW PRINCI-
PLES of OPTICS. *
CH A P. VI.
Of the NATURE and STRUCTURE of the EYE;
and the ELEMENTS of VISION explained from
the foregoing THEORY.
ROM the Elements delivered in the preceding
1392. Chapter we are enabled to explain the true THEORY
OF VISION, as it is performed by the most exquifite of all
Dioptric Inftruments, the EYE. For which Purpoſe it will be
neceffary
*Before we leave this Subject it may be neceffary to obferve, that
the Hypothefis of the Paffage of a Ray of Light by Reflection, from
A to C and D (ſee 1282) being a Minimum, or the leaft Poffible, is
not an arbitrary Pofition; for if we confider the Reflection of Light
as an Operation of Nature, and that perfect Wiſdom has eſtabliſhed
the Oeconomy of Nature's Laws, it is neceffary to conclude, that
every Thing is done in the moſt direct and fimple Manner, and there-
fore that the Paffage of the Ray A C+CD is the leaſt Poffible from
A to D, by Reflection from the given Plane BE; for this cannot
be denied, without afferting, that the Author of Nature has not taken
the most direct and ready Way of doing Things; which is inconfiſtent
with our natural Notion of Deity, and confequently is irreligious, as
well as abfurd. Therefore it ought to be eſteemed an Axiom, or a
primary Poftulate in Optics, on which the whole Science depends.
And notwithſtanding it is a felf-evident Principle, it is at the fame
Time the Reſult of the ſtricteſt Mathematical Theory; it is alfo de-
monftrable on the Principles of Mechanics; and all Experiments teftify
that the Angles of Incidence and Reflection are equal, and therefore
that the Paſſage of the Rays of Light from one given Point to ano-
ther, either by Reflection or Refraction, must be the least poffible.
Of DIOPTRICS.
135
neceſſary to deſcribe the ſeveral Parts that are immediately con-
cerned in producing this wonderful Effect.
1393. And here it muſt be obſerved, that Vifion is effected
by a Refraction of Light thro' the Humours of the Eye to
the Bottom or Fundus, where the Images of external Objects
are formed on a fine Expanſion of the Optic Nerve called the
Retina, and therefore the anterior Part of the Eye muſt ne-
ceffarily be of a convex Figure, and of fuch a precife Degree of
Convexity as the particular refractive Power of the feveral Hu-
mours require for forming the Image of an Object at a given
focal Diſtance, viz. the Diameter of the Eye.
1394. Hence we find, First; The external Part of the Eye-
ball CD (Fig. 17.) is a Pellucid, properly convex, and ſtrong
Subftance, which, when dried, has fome Refemblance to a
Piece of tranſparent Horn, and is therefore called the Cornea,
or horney Coat of the Eye.
1395. Secondly; Immediately behind this Coat there is a
fine clear Humour which from its Likeness to Water (in a general
View) is called the Aqueous or watery Humour, and is contained
in the Space between CD and G F E.
1396. Thirdly; In this Space there is a Membrane or Dia-
phragm, called the Uvea, with a Perforation or Hole in the
Middle as at F, of a muſcular Contexture for altering the Di-
menfions of that Hole (or Pupil) for the adjuſting a due
Quantity of Light.
1397. Fourthly; Juft behind this Diaphragm is placed a
lenticular Subſtance G E, called from its Transparency, the
Gryftaline Humour, tho' it be not a fluid Body, but of a confi-
derable Confiftence. It is contained in a fine Tunic called the
Arachnoides, and is ſuſpended in the Middle of the Eye by an
Annulus of muſcular Fibres called the Ligamentum Ciliare, as at
Gand E. By this Means it is capable of being moved a little
nearer to, cr farther from the Bottom of the Eye.
1398. Fifthly; All the remaining interior Part of the Eye is
made up of a large Quantity of a jelly-like Subftance called the
Vitreous or glaffy Humour, tho' there is not the leaft Likeneſs to
Glaſs in it, except its Tranſparency; it being moſt like the
White of an Egg of any Thing.
1399. Sixthly; On one Side of the hinder Part of the Eye
as at K, the Optic Nerve enters it from the Brain, and is ex-
panded
138
INSTITUTIONS
panded over all the interior Part of the Eye to G and E alf
around. This delicate Part is by Nature appointed the imme-
diate Organ of SIGHT. On this wonderful Membrane the
Image I M of every external Object OB is formed according
to the Optic Laws of Nature, in the following Manner.
1400. Let OB be any Object, placed at a great Diſtance
A L, from the Eye. Then a Pencil of Rays proceeding from
any Point L will fall on the Cornea D C, and be refracted by the
Aqueous Humour under it to a Point in the Axis of that Pencil
continued out. Then fuppofing the Radius of Convexity of the Cornea
to be 3,3 Tenths of an Inch; and the Sine of Incidence in Air
to that of Refraction in the Aqueous Humour to be as 4 to 3
(as it is nearly) then if the Object be infinitely diftant, or the
Rays parallel, we fhall find (per Theorem 1327) that the focal
Distance after the firft Refraction will be 13,3 Tenths of an
Inch from the Cornea.
1401. The Rays thus refracted by the Cornea, fall conver-
ging on the Cryſtaline Humour, and tend to a Point 12,28 Tenths
of an Inch behind it; alfo the Radii of Convexity in the faid
Humour are 3,3 and 2,5 Tenths refpectively; and the Sine
of Incidence is to that Refraction of the Aqueous into the Cry-
taline Humour as 13 to 12. Therefore (per Theorem 1328) the
focal Diſtance after Refraction in the Crystaline will be 10,6
Tenths of an Inch from the fore Part thereof.‡
1402. The Rays now paſs from the Crystaline to the Vitreous
Humour still in a converging State, and the Sines of Incidence.
and Refraction being here as 12 to 13 (as found by Experi-
ment); and fince the Surface of the vitreous Humour is Concave
which receives the Rays, and is the fame with the Convexity
of the pofterior Surface of the Cryftaline, the Radius will be
the fame, viz. 2,5 Tenths of an Inch. Then the focal Dif-
tance after this third Refraction will be found (by 1331)
to be 6,1 Tenths from the hinder Surface of the Cornea.
1403. Now the Diſtance from the hinder Part of the
Cryſtaline to the Bottom of the Eye, or Retina, is nearly equal
to that focal Distance; and therefore all Objects at a great Dif
Lance have their Images formed on the Retina in the Fund of the Eye,
In 1328 for diverging, read converging.
and
Of DIOPTRICS.
137
and thereby diſtinct Vifion is produced by this Organ of Optic
Senfation.
1404. When the Diſtance of Objects is not very great, the
focal Diſtance, after the laft Refraction in the Vitreous Humour,
will be a little increaſed, and to do this we can move the Cryf-
talline a little nearer the Cornea by Means of the Ligamentum
Ciliare (1397) and thus on all Occafions it may be adjuſted for
a due focal Diſtance for every Diſtance of Objects, excepting
that which is lefs than fix or feven Inches, in good Eyes.
Many, I know, are of Opinion, that this is effected by a Power
in the Eye to alter the Convexity of the Cryftalline Humour as
Occafion requires, but this does not very eaſily appear.
1405. By what has been faid, it appears that Rays of Light
flowing from every Part of an Object O B, placed at a proper
Diſtance from the Eye, will have an Image I M formed there-
by on the Retina in the Bottom of the Eye; and fince the Rays
OM, BI, which come from the extreme Parts of the Ob-
ject, croſs each other in the Middle of the Pupil, the Pofition
of the Image I M will be contrary to that of the Obje&, or in-
verted, as in the Cafe of a Lens (1385.)
1406. The apparent Place of any Part of an Object is in the
Axis and conjugate Focus of that Pencil of Rays by which that
Part or Point is formed the Image. Thus O M is the Axis, and
O the Focus proper to the Rays by which the Point M in the
Image is made; therefore the Senfation of the Place of that
Part will be conceived in the Mind to be at O; in like Manner
the Idea of Place belonging to the Point I, will be referred, in
the Axis I B, to the proper Focus B, therefore the apparent
Place of the whole Image I M will be conceived in the Mind to
occupy all the Space between O, B, and at the Distance A L
from the Eye.
1407. Hence likewife appears the Reaſon why we fee an Ob-
ject upright by Means of an inverted Image; for fince the apparent
Place of every Point M will be in the Axis M O at O; and this
Axis croffing the Axis of the Eye HL in the Pupil, it follows,
that the fenfible Place O of that Point will lie, without the Eye,
on the contrary Side of the Axis of the Eye to that of the Point in
the Eye; and fince this is true of all other Parts or Points in
the Image, 'tis evident the Pofition of every Part of the Object
VOL. II,
T
will
138
INSTITUTIONS
will be on the contrary Side of the Axis to every correfponding
Part in the Image, and therefore the whole Object OB will
have a contrary Pofition to that of the Image I M, or appear up-
right.
1
1408. The Dimenfions, or Magnitude, of an Object OB,
we judge of by the Quantity of the Angle OA B which it
fubtends at the Eye. For if the fame Object be placed at two
different Diftances L and N, the Angles O AB and o Ab,
which in theſe two Places it fubtends at the Eye, will be of
different Magnitude; and the lineal Dimenſions (viz. Length and
Breadth) will be at N and at L as the Angle o A b is to the An-
gle OAB. And the Surfaces and Solidities of the Objects will
be as the Squares and Cubes of thofe Angles (670, 675, 1387).
1409. It is found by Experience, that two Points O, L, in
any Object will not be diftinctly feen by the Eye till they are
near enough to fubtend an Angle OAL of one Minute. Hence
when Objects, however large in themſelves, are ſo remote as
not to be ſeen under an Angle of one Minute, they cannot pro-
perly be faid to have any apparent Dimenfions or Magnitude
at all; fuch as is the Cafe of the large Bodies of the Planets,
Comets, and fixed Stars. But the Optic Science has fupplied
Means of enlarging this natural fmall Angle under which moſt
diftant Objects appear, and thereby encreafing their apparent
Magnitudes to a very furpriſing and delightful Degree in that
noble Inftrument we call a TELESCOPE, as we fhall elfewhere.
explain.*
*
1410. On the other hand, we find in the Creation an Infinity
of Objects, whofe Bulks are fo fimall, that they will not fub-
tend the requifite Angle (1409) if brought to the neareſt Limits
of diftinct Vision, viz. 6, 7, or 8 Inches from the Eye, as found by
Experience; and therefore in order to render them vifible at a
very near Diſtance, we have a Variety of Glaffes, and Inftruments
of different Conftructions, which we ufually call MICROSCOPES,
by which thoſe minute Objects appear many Thouſands, yea
Millions of Times larger than to the naked Eye; and thereby
enrich the Mind with Diſcoveries of the fublimeft Nature, in re-
gard to creating Power, Wisdom, and Oeconomy.
1411. If
* The practical Part of Optics containing the Deſcription and Ufe
of TELESCOPES, MICROSCOPES, and other Optical Inftruments, the
Reader will hereafter find in the Young Gentleman and Lady's Philofophy.
aƒ DIOPTRICS.
139
1411. If the Convexity of the Cornea CD happens not
exactly to correfpond to the Diameter of the Eye, confidered
as the natural focal Diſtance, then the Image will not be form-
ed on the Retina, and confequently no diftinct Vifion can be ef-
felled in fuch an Eye.
1412. If the Cornea be too convex, the focal Diſtance in the
Eye will be leſs than its Diameter, and the Image will be form-
ed fhort of the Retina. Hence the Reaſon why People having
fuch Eyes are obliged to hold Things very near to them, to
lengthen the focal Diſtances (1340) and alfo why they ufe con-
cave Glaffes to counter-act or remedy the Excefs of Convexity,
in order to view diſtant Objects diſtinctly.
1413. If the Eye has less than a juft Degree of Convexity, or is
too flat, as is generally the Caſe with old Eyes, by a natural Do-
ficiency of the Aqueous Humour, then the Rays tend to a Point
or Focus beyond the Retina or Bottom of the Eye; and to
fupply this Want of Convexity in the Cornea, we uſe convex
Lenfes in thoſe Frames we call Spectacles, or VISUAL GLASSES.†
1414. Since the Rays of Light OA, BA, which conftitute
the viſual Angle O AB, will, when they are intercepted by a
Lens, be refracted fooner to the Axis, (1380) the faid Angle
will thereby be enlarged, and the Object of Courſe become
magnified; which is the Reaſon why thofe Lenfes are called
Magnifiers, or READING GLASSES.
T 2
CHAP,
+ Theſe Viſual Glaſſes are very different from Spectacles in two
Particulars; for (1.) They are made with proper Apertures to ad-
mit of no more Light than what is requifite. (2.) They are bent to
an Angle, that the Rays may fall directly and not obliquely on the
Eye; both which Precautions are neceffary for eaſy and diſtinct Vi-
fion.
140
INSTITUTIONS
CHA P. VII.
Of the different Refrangibility of Light; and the
DOCTRINE of COLOURS from thence explained
by the PRISM, and applied to refracting TELE-
SCOPES.
1415.
5.T
HE SCIENCE of COLOURS is one of the moſt
delightful Parts of Phyfics; is but of modern In-
vention; depends entirely on Optical Principles; and is the Life
of all painting and pictureſque Arts, both Natural and Artificial.
Therefore it will be neceffary here to lay down the Elements
which explain it.
1
1416. We have already fhewn the general Nature of the Re-
fraction of Light in different Mediums, (1312) from which it
appears that when the two Surfaces of a refracting Medium are
parallel, the Ray after Emergence will proceed in a Direction
parallel to that which it had at its Incidence; ‡ and farther,
we have taken for granted, that all Rays of Light are equally
refrangible, or uniformly refracted by any Medium. But when
we confider that the Rays are really very differently refrangible
in the fame Medium, and that the Direction of the Incident
and emergent Rays will be different alfo, when the Surfaces of
the refracting Medium are not parallel, but inclined to each
other, I fay, when thefe Things are confidered, we ſhall find
Matter for new Speculation, and foon unfold the Doctrine of
Colours.
1417. The Form of a PRISM, it is prefumed, is well known,
confifting of three plain Sides inclined to each other in certain
Angles; and therefore if a Beam of Light, as D C (Fig. 18)
fall on one Side A B of the Priſm in the Point C, it will leave
its firſt Direction, and be refracted to the other Side at E (1312)
At its Emergence into the Air at E it will be refracted from the
Perpendicular PE to the Side of the Prifm, and therefore make
ſtill a greater Angle with the Direction of the incident Ray D C.
(1323.)
1418. And
† This is evident from the Theorem in (1337) by making r, and r,
infinite, in the ufual Method.
Of DIOPTRICS.
141
1418. And becauſe it is found by Experience, that the Rays
of common folar Light are differently refrangible, that is, ſome
Rays are more and others lefs refrangible by the ſame Medium, it
will follow, that at the firſt Refraction at C, and at the fecond
at E, the Beam of Light will be dilated, and rendered of a
different Form from that of the incident Beam D C. Thus
for Inftance, at E the Part of the Beam which is leaft refrangi-
ble, will be refracted to (r) making the leaſt Angle rEP with
the Perpendicular EP; and thofe Rays in the Beam which
are moſt refrangible will be refracted to (v) making the Angle
v EP the greateſt of all; fo that the Beam by Refraction at
E, is dilated or diffipated into the Form v Fr, very different
from that of the incident Beam.
1419. And moreover, we obſerve the different Rays of the
Beam at r, o, y, &c. appear of a different Colour ; thus the
Rays at r are Red; thofe at o are Orange; thoſe at y, Yellow;
at g, Green; at b, Blue; at i, Indico; and at v, Violet-colour'd.
Now its evident, that the different Colours of the Rays must be
owing to fome peculiar Mode of Action in them on the Optic
Nerve,
1420. That the Senfation of Colour is the Effect of Light
alone, is manifeft from hence, that no Sort of Object on which
the refracted Beain falls, nor any Difference in the Mediums by,
and in which it is refracted does ever change the Colours peculiar
to the feveral Parts of the refracted Beam. They are more or
leſs intenſe, according to the greater or leffer refractive Powers
of the Mediums, but ftill the Colours of the fame Parts of the
refracted Beam are always the fame.
1421. Hence then it follows, that the Senfation or Idea of
Colours is excited in the Mind by the Action of Light, as the
efficient Cauſe; and that the different Phænomena of Colours are
the Effects of different Rays of Light, varying in fome Pro-
perty or Quality, which perhaps we do not certainly, if at all,
comprehend. Sir Ifaac Newton fuppofes, with great Reaſon,
that this colorific Quality of the Rays depends on the different Sizes
or Magnitudes of the Particles of Light which compoſe them; but to
this Hypothefis there are fome Objections; and we know of
none without any.
1422. It
142
INSTITUTIONS
1422. It muſt fuffice therefore to know, that Light is the
Cauſe of Colour; and that where there is no Light there can be no
Colour; and as Darkness, or total Shadow, is nothing more than
the Abfence, or Privation of Light, fo BLACKNESS is no other
Thing in itſelf than a Want of the natural Operation of Light;
and with reſpect to us, it is the Want of all Colour in Bodies.
Hence Black is, properly fpeaking, no Colour at all.
1423. Since all the various colour-making Rays r E, 。 E, y E,
&c. before they are ſeparated by the Priſm A B, compound one
common Beam of Light, whofe Colour is White; we may
eafily thence infer that WHITENESS is not a fimple Colour,
but only the Reſult or Compound of all the fimple Colours be-
fore mentioned (1419) blended together.
1424. Therefore fuch Bodies which imbibe all the Light in-
cident upon them, or reflect none, will appear abfolutely black,
or colourless. And thoſe which reflect all the Light which falls on
them, will appear White; and the fame in regard to Refraction.
1425. But fuch Bodies as reflect or refract one fimple Sort of
Light only, will appear of the Colour peculiar to that homoge-
neous Ray; thus if any Object reflects or refracts only the
Rays Er, its Colour will be Red; if the Rays E g, it will be
Green; and the Ray Ev will, when reflected or refracted alone,
thew the Object of a Violet-colour (1419) and fo of the reft.
1426. Again, if Bodies reflect one fimple Colour, and re-
fract another, they will appear of one Colour by Reflection,
and another by Refraction; as is the Cafe of Leaf-gold, De-
coction of Lignum Nephriticum, &c.
1427. Rays of Light are alfo differently reflexible, and thofe
which are moſt or leaft refrangible, are alſo moſt or leaft reflexible;
therefore Bodies will appear of different Colours in the fame Part,
if made to receive the Beam of Light under fuch Angles of
Incidence as are proper to each reſpective Sort of Rays, for a
given Pofition of the Object and the Eye.
1428. Thoſe Objects which reflect or refract two or more
of the homogeneal Rays will appear of a Colour compounded
of them; and it is obfervable, that of three different Rays
(next to each other) the two extreme ones produce nearly the
Colour of the middle One; thus Red and Yellow made an Orange;
Yellow and Blue make a Green; Blue and Purple make an Indigo-
colour;
Of DIOPTRICS.
143
colour; and from hence all the Phænomena of Colours in natural
Bodies ariſe, and are eaſily confirmed by Experiments.
1429. The Image of any Object, formed by reflected or re-
fracted Light, muſt be of the fame Colour in every Part with
the Object; for the Rays of Light which proceed from the fe-
veral Parts of the Object are not altered or changed in their
Nature by Reflection or Refraction; and therefore whatever
Colour they excite in the Object, the fame muft they fhew
in the correſponding Part of the Image; the whole Image will
therefore be variegated and painted with the fame Colours in
every Reſpect as we view in the Object itſelf.
1430. I think, then, it deferves to be confidered, that every
PICTURE formed by an Optic Glass, ought to be looked upon
as the Portrait of Nature itſelf, and confequently deferves a much
greater Regard than we ufually pay to it. The Per-
formance of Titian's Pencil being as much inferior to the PAINT-
INGS of NATURE, as a created Being is below the CREATOR.
1431. For the fame Reaſon that a Priſm ſeparates the Beam
of Light into its original or fimple Rays, fo likewiſe does a
Lens, viz. becauſe its Sides are inclined to each other (1369) and
therefore the Image in the Focus of a fingle Lens muſt be as
compounded as Light itfelf, and confequently in fome Mea-
fure confuſed; for each particular Species of Rays does in
Reality form a diftinct Image in its own peculiar Focus. And
of Courſe, when this compound Image is viewed with a deep
Magnifier, it will appear both coloured and confuſed, as we find
by Experience in all our Microſcopes, Teleſcopes, &c. of the
refracting Sort.
1432. But Images formed by reflected Light are not fubject
to either of thofe Imperfections, becauſe there is no different
Reflection of Light while the Angles of Incidence are the fame;
and therefore only one fimple Image is formed in the Focus of a
Speculum; and fo perfect, that it will bear to be magnified a fe-
cond Time with fufficient Diftinctnefs; and confequently a double
Power of magnifying in a REFLECTING TELESCOPE will have
the fame Effect in a fmall Length as we have in a very great
Length by Refraction; and this is the Reaſon of that noble In-
vention.
1433. There
144
INSTITUTIONS
}
1433. There have been Methods invented to remedy, or
rather to palliate, the Imperfections of a refracting Teleſcope,
which I have confidered at large in my New Elements of Optics,
to which I refer the Reader, as being a Subject too prolix for a
Syflem of elementary Principles only. In that Treatife I prefume,
it is demonſtrated, that as there are two Defects in refracting
Teleſcopes, viz. one from the different Refrangibility of Rays,
and the other from the Figure of Glaffes, fo the very Means of
correcting the former will inevitably augment the latter; || and
in thoſe very Teleſcopes where this Correction has been applied,
by joining a Concave with a convex Object Glass, the Rays are
afterwards made to pafs thro' a fingle convex Lens before the
Image is formed, and therefore if the Colours were taken
away by the two Glaffes, they must be again produced by the
firft of the five next the Eye. And therefore we have not yet
any fuch Thing as an achromatic Refractor, or one that will ſhew
Objects entirely free from Colours.
1434. But as in the abovementioned Treatife I had omitted
one or two Particulars, relative to this new Refractor, and alfo for
the Sake of the Inquifitive, I fhall here give the following Diffec-
tion of the compound Object Glaſs as I found it in one of thoſe
Teleſcopes I purchaſed for 3½ Guineas, and was three Feet long.
1435. The convex Lens was of Crown-glafs, doubly and equal-
ly convex on both Sides; and its folar focal Diſtance was
preciſely 9 Inches. The concave Lens was of White-flint, or Cryſ-
tal; it was a Plano-concave, and its focal Diftance by Reflec-
tion was four Inches from its Surface. The focal Diſtance of
will
I
2
both theſe combined together, was juſt 29 Inches.
1436. The Radius of the convex Lens was 10,1 Inches, as
appear from (1340,) for in Crown-glaſs, a = 0,532, and
2 a 1,064; then 1:2a::f:r, or 1: 1,064::9,5:10,1.
But with regard to the Plano-concave of White-flint, fince the
Radius is double the folar Focus by Reflection, (1291) it is in
that 8 Inches. The two Radii, therefore, in theſe two Ob-
ject Glaffes, are as 10,1 to 8, or as 160 to 127 nearly: That
is
This is to be underſtood of a convex and a concave Lens of the
Jame Sort of Glafs; and how little the Cafe will be altered, by hav-
ing one of Crystal and the other of Crown-glafs, will appear bye
and bye.
Of DIOPTRICS.
145
·
is (if R be put for the Radius of the Plano-concave) r: R::
160: 127.
1437. Now it is known by Experience, and is by all confefs'd,
that two Lenſes, one a Plano-convex of Crown-glaſs, and the
other a Plano-concave of Flint, muſt have their Radii of Spheri-
city very nearly as 2 to 3, in order to prevent the Error of Re-
fraction arifing from the different Refrangibility of the fame
Beam of Rays, and that in fuch a Cafe, we have R:r: : 3:2::8:
5,35; therefore in a double and equally convex Lens of Crown-
glafs to produce the fame Effect, the Radius muſt be r = 10,7;
but that in the Teleſcope is only 10,1 and therefore too convex
to prevent a coloured Image, when compounded with a Plano-con-
cave of Flint whofe Radius is 8 Inches.
1438. Then if the Sine of Incidence be to the Sine of Re-
fraction (of the fame Ray) out of Crown-glafs into Air as n to
m, and out of White-flint into Air as n to M; it is demonftrated
by Sir Isaac Newton* the Error of Refraction arifing from the
Figure of the Lens is
in White-
m² 2,3
4 n² r²
2
in Crown-glaſs, or
M² y³
4722 R2
flint, in a Lens of a Plano-convex Form. In thefe Expreffions,
(y) is the Semi-aperture of the Lens.
1439. Becauſe there is the fame Refraction, and Error from
thence arifing, in an equally Plano-concave Lens, and being
made the contrary Way, therefore when the Errors are equal in
a Plano-convex of Crown-glafs, and a Plano-concave of White-
flint, they will deftroy each other; or when two fuch Lenſes are
combined together they will correct each other, and prevent any
Aberration of Rays from the Figure.
1440. Therefore let
m²
M²
2 R
2
m² y3
M² y³
2
2
4 n² r
pz
4 π² R2
1
from whence we have
or this Analogy M:R::m:r; but by Experi-
ments it appears, that M: m :: 160: 153 in Crown and Cryſtal.
And therefore fuppofing, the Radius R of a Plano-concave to
be 8 Inches, (as in the abovementioned Teleſcope) then 160:
1538:7=r, the Radius of a Plano-convex Crown-lens
that combined with the other, fhall prevent any Error from the
fpherical Figure.
VOL. II.
U
* See alfo Philofophia Britannica, 2d Edit.
1441. Now
146
INSTITUTIONS
1441. Now fince the Effect or Refraction is the fame in a double
and equally convex Lens, of a double Radius, viz. r = 15½
Inches; therefore fuch a Lens only, combined with a Plano-
concave of White-flint, whofe Radius is 8 Inches, can caufe
the Aberration of Rays from the Figure of the Glaffes to
vaniſh. But that in the Teleſcope has a Radius of 10 In-
ches only; whofe Sphericity is therefore much too great to an-
fwer this Purpoſe.
1442. Therefore fince in a Plano-convex of Crown-glafs,
and a Plano-concave of White-flint, the Radii for preventing
the Error from the different Refrangiblity of Rays muſt have
the Ratio following, viz. R:r:: 3:2:: 160: 107; and for
annihilating the Error from the Figure, the Ratio muſt be R:
r::160:153; it appears to be impoffible, that the fame
two Glaffes, viz. Crown and Cryftal, which correct one
Error, fhould at the fame Time correct the other; and farther
it appears, that the Ratio of the Radii of the Object Glaffes in
the Refractor under Confideration, being in a Ratio different.
from either of thefe (1436), can correct neither of the Errors.
1443. But we are told, “That the Surfaces of ſpherical Glaf-
fes admit of great Variations tho' their focal Distances be limited.
With regard to parallel Rays on a Plano-convex, or Plano-con-
cave, we have ſhewn (1341, 1345.) the focal Diſtance is ƒ =
7
a
and for fa; but (ƒ) is limited or given by Suppofition, and
(a) is the Refraction of the particular Species of Glaſs (1347)
and therefore is given of Courfe; how then does it appear, that
the Radius (r) or fpherical Surface of the Glafs can admit of
fuch great (or indeed any) Variation at all? 'Tis true, the fame
Degree of Sphericity may be divided and variouſly proportioned
between the two Surfaces, but while the focal Diſtance is limited,
the Refraction, and the Error occafioned thereby, will be ſtill
the fame.
1444. Therefore the "Poffibility of making the Aberrations of
any two Glaſſes equal, is a Thing that does not appear;" no more
than how, a "Perfect Theory for making Object Glaffes can be
obtained, from any Principles of Optics hitherto publiſhed." A
Demonſtration of theſe great Pofitions is a Satisfaction we have
yet
Of DIOPTRICS.
147
yet to come, and which the Public has long, and with great
Impatience, waited for.
1445. In the mean Time it may not be unacceptable to ma-
ny Perſons to be informed and directed how to make this
compound Object Lens for their own Ufe; and fo procure at
an eaſy Rate a Teleſcope, which they have been told is of in-
finite Service to Mankind. For this Purpoſe, take a double Convex
and a Plano-convex of Crown-glafs, and a double Concave of
White-flint, all ground upon the fame Tool, and let the Con-
cave be put between the two Convexes, and place them in the
End of the Teleſcope with the Plano-convex outwards, and they
make the triple compound Lens for taking away Colours in Refractors
of a ſmall Length.
1446. But if the Teleſcope is to exceed the Length of 18
or 24 Inches, then two Glaffes will do, viz. one Convex of
Crown-glafs, and the other Concave of Flint, whofe folar focal
Diſtances are to each other as 3 to 4† very nicely; and they
will form an Image without Colours, for the Teleſcope propoſed.
If theſe two Lenfes are held together in the Sun-beams, they will
converge them to a Focus, and thereby fhew the focal Dif-
tance of the compound Object Glaſs, and conſequently of the
Teleſcope itſelf.
1447. But as to the Error from the Figure of the Glaffes, I
confeſs it is not in my Power to give any Directions for preventing
or extenuating the fame in any great Degree. If any Perfon
can find among the different Sorts of Glafs, or any tranfparent
Mediums, any two, which being formed into Priſms, ſhall bave
their refracting Angles, which take away Colours, reciprocally pro-
portioned to their Sines of Refraction into Air, reſpectively; then
he may be affured of a perfect Theory of making Object Glaſſes.
And he that fhall do this, erit mihi plufquam Magnus Apollo.
INSTI
U 2
R
+ For let 2
ƒin a Plano-convex of Crown-glaſs, and
= F,
a
in the Plano-concave of White-flint. Then f: F:
R6
N
0,53
:
a
3
0,6
:: 0,376: 0,5:34, as in the Prefcript above.
(148)
INSTITUTIONS
O F
PERSPECTIVE:
CONTAINING
The Mathematical THEORY thereof deduced from
OPTICAL PRINCIPLES, and applied to a General
PRAXIS.
CHA P. I.
The Optical ELEMENTS of Lineal PERSPECTIVE.
1448.
W
E are now prepared to treat of the Elements, or
first Principles of PERSPECTIVE, the moft de-
lightful and neceffary of all the Mathematical
Sciences. Theſe Elements are immediately.
derived from Optics; for the Art of deline-
ating Objects as they appear on a given tranfparent Plane, or Super-
ficies, to an Eye at a given Height and Distance, is the true DE-
FINITION of PERSPECTIVE.
1449. The Want of theſe preliminary Principles has rendered
the Treatifes on this Subject defective in the moft effential Part,
and the Art of Perfpective itſelf very difficult to be underſtood:
In ſhort, it would be abfurd to fuppofe any Man can underſtand
Perſpective, without being acquainted at leaſt with as much of
the Optic Theory as we have premiſed, and fhall here fuperadd
in this Chapter.
1450. It appears by the above Definition, that in order to
delineate the true Appearance of an Object on a given Plane, it
will be firſt neceſſary to know the Law according to which the
apparent linear Dimenfions of Object increafe or decreaſe;
and here we muſt obſerve (1.) That the viſual Angle, or the
apparent Magnitude of a Line will be lefs at a greater Diftance;
and vice versa. (1408) (2.) That it will be lefs as the faid Line is
viewed
INSTITUTIONS, &c. 149
viewed more obliquely. (3.) Therefore the Law of Diminution will
be nearly in Proportion to the Distance and Obliquity of the View
conjointly.
1451. It is true, the Diminution of any Line viewed direct-
ly increaſes with the Diſtance, but not exactly in Proportion
thereto, unleſs the Diſtances be very great. Let OB (Fig.19)
be an Object viewed directly by an Eye at C. On the Center
C, with the Radius C B defcribe the Semi-circle A D B, and
join CO; then will O C B be the Angle under which the Object
O B appears at C. Again, fuppofe the Eye at A, and draw AO;
then the Angle OA B is the viſual Angle under which it ap-
pears at A. Now A B is double the Diſtance B C, but the An-
gle OA B is more than half the Angle OCB; for the Angle
OAB or E AB is juſt half the Angle O CB or ECB (642.)
But at great Diſtances, the differential Angle E AO will be-
come inſenſible; and therefore in fuch Caſes, the Diminution will
be directly as the Diſtance of the Object.
1452. When any Object as BD (Fig. 23) is viewed oblique-
ly (that is, when the Angles A D B and A B D are not equal)
then a Diminution of its apparent Magnitude will enfue; for
in all fuch Cafes, the Length B D is reduced to B O, which
fubtends the vifual Angle DAB at the Diſtance A B. There-
fore the Propofition is evident. (1408)
1453. It is further evident, that if the fame Object BD were
removed to twice the Diſtance of A B, then the Angle of ap-
parent Magnitude O AB would be but one half fo large (642);
and therefore the apparent Magnitude of any Object on Ac-
count of its Obliquity, is alfo diminiſhed in Proportion to its
Diſtance; and therefore the whole Diminution of Magnitude
on Account of its Diſtance and oblique Pofition jointly, is pro-
portional to the Square of the Distance
1454. But we have a more direct Demonſtration of this
fundamental Propofition in Perſpective, in the Diagram of (823)
which we fhall here infert.
P
If TP be any Object viewed oblique- T
ly by an Eye at C, at the Height of CD
above it, then the vifual Angle is PCT,
and its Meaſure the Arch Bb to the
Radius C B, or P Q to the Radius C P.
D
And
150
INSTITUTIONS
And becauſe TP:QP::CP:CD; and PQ:Bb:: CP:
CBCD; therefore (c Equo 652.) it is T P: Bb:: CP2:
CD2. But when T P is a given Quantity, then (fince Radius
I
CD = 1) we have Bb: p2 or the vifual Angle B b of apparent
CP2
Magnitude is ever in the inverse Proportion of the Square of the Dif-
tance C P.
1455. Hence it appears, that in any Line AF (Fig. 20.)
equal Parts A B, BC, CD, DE, E F, are feen by an Eye
at C under fuch Angles, whofe Meaſures gh, hi, ik, kl, lm,
are reciprocally as the Squares of the Diſtances A C, BC, C C,
CD, CE. Therefore the Circle I K may be called the Line of
Meaſures in Optical Perspective, or fuch Delincations as are made
on Spherical and cylindrical Surfaces.
1456. If the right Line G H be parallel to the Line A F,
then will the viſual Rays A C, BC, &c. divide the Line G H
in a fimilar Manner to that in which they divide the Line A F;
that is, if the Divifions in A F are equal, they will alſo be equal
in GH; but if they are unequal in A F, they will have the
very fame Ratio of Inequality in G H. For the Triangles
FCE and fCe, alfo E C D and e C d, are fimilar; which give
theſe Analogies ef: EF:: eC:: EC; and ed: ED::e C:
EC; confequently ef:ed:: EF: ED; and fo of the
Reft.
are ſeen by the
will they be as the
For in the fore-
1457. If any Objects A B, CD, (Fig. 21.)
Eye at C under equal Angles a b, and cd, then
Squares of their Diflances from the Eye directly.
going Figure to (1454) we had TP: Bb:: CP2: CD2; and
therefore fince in this Cafe Bb and C D are conftant Quantities,
TP will be directly as CP2; that is (in Fig. 21.) A B:CD::.
B C D C².
1458. The Object B D (Fig. 22.) of a given Length, placed
at a given Height AB will be feen by the Eye at C in a right
Line perpendicular to A B, under an Angle BCD, which will
be of a variable Magnitude as the Eye approaches to, or recedes
from the Point A ; and there is one Diſtance A C where that
Angle of apparent Magnitude will be a. Maximum, or greateſt
of all others; and is determined in the following Manner.
1459. Every
Of PERSPECTIVE,
151
1459. Every Angle PCQ (fee Fig. to 1454) is as the
Arch P Q directly, and the Radius PC inverfely or as
alſo we
· PQ
PC
TP XCD
But TC: CD::TP:PQ=
TC
D²
TP x CD
PCQ is as
That is, if (in Fig. 22)
TCX/CD² + DP²
have PC/CD2+ DP. Therefore the vifual Angle
we put A b = a, B D = b, A D = a+b=c, and AC=
x; then will the Angle BCD be as
x
b x
√ √2
a² + x² × √ c² + ü
then by taking the Fluxion thereof (800) and making it equal
to Nothing (818) we ſhall get ²=ac, whence a : x : : x : c, or
the Diſtance AC is a geometrical Mean between A B and A C,
when the viſual Angle B C D is the greateſt poffible.
then
1460. If any Line (Fig. 25) O B appears to an Eye at A,
under the fame Angle with another Line CD, then at any o-
ther Point E, the two Lines will have the fame apparent Mag-
nitude (however the vifual Angle CAD may vary) if the Dif
tances of the Eye from each Line preferve the fame Ratio. For
leto b be equal and parallel to O B, and draw CE and DE;
by fimilar Triangles, we have AH:AK::CH: OK; and
EH: ELCH:0 LOK; therefore A H:AK::EH:
EL. Alfo a right Line paffing from the Eye to thofe Lines di-
vides them in the fame Ratio; for it is GH:HD::OK:
KB:: L:Lb.
1461. The Appearance of any diftant Line or Object O B
upon another Line or Plane CE given in Pofition, will in-
creaſe or decreaſe with the Diſtance from the faid Line or Plane,
tho' not in the fame Ratio; for at the greater Diſtance A it will
occupy the Length CE, but in a lefs or nearer Diſtance at D,
it occupies only the Space ce, much lefs than before, as is evident.
by Inſpection.
1462. Having thus premiſed fuch Optical Principles of lineul
Perſpective, or the Appearance of LINES among themſelves
with regard to their Pofitions, and Diſtance of the Eye; we
ſhall now proceed to a generally THEORY of univerſai PER-
SPECTIVE, and demonftrate the fame from its genuine Princi-
ples. And tho' the common Methods of making Perfpective
Draughts
152
INSTITUTIONS
1
Draughts are very eaſy, and in every ones Hands, yet as the judi
cious Architect, Defigner, Painter, &c. will find it no eafy Matter
to come at a concife and plain Theory, or eafy Rationale of this ne-
ceffary or fundamental Part of his Art, in any of the Books hither-
to publiſhed; it is prefumed, the following new Method of de-
monftrating the Reafon of fuch uſeful and common Rules will
not be unacceptable to the ingenious Artifts of all fuch Profef-
fions, as require the Affiftance of this excellent Science.
CHA P. II.
The THEORY of PERSPECTIVE demonftrated from
the PRINCIPLES of OPTICS and GEOMETRY.
1463. F
[PLATE II. of PERSPECTIVE. ]
ROM the Definition of Perſpective, (1448) it ap-
pears, that in order to give a Rationale of the com-
mon Practice, we muft premiſe a Theory of the Vifion of Objects
on a tranſparent Plane at a given Diſtance and Height of the
Eye. Thus fuppofe the Eye of the Spectator at I, and E F G H
the Plane on which it obferves the Appearance of Objects,
(Fig. 1.) This is called the Perſpective Plane.
1464. As the Appearance of Objects will be variable ac-
cording to the Situation of the Planes in which they are pofited
in regard to the Perspective Plane; theſe particular Situations of
Object-planes are principally to be confidered. And, firſt,-it
is evident, that any Plane paffing thro' the Eye cannot be ſeen
as a Plane, but as a Line only on the perspective Plane; for
the Eye having no Elevation above fuch a Plane, car fee no
Part of its Surface; the Edge or bounding Line of ſuch a Plane
being all the Appearance it can have to the Eye.
1465. Of theſe Ocular Planes, there are two of principal
Note, viz. the horizontal Plane O K L M parallel to the Hori-
zon; and the vertical Plane N R QP, which is perpendicular
thereto. The first interfects the perſpective Plane in the Line
IK,
G
Y
Fig.3
H
"
Fig.5
G
Fig.1
L
Υ
· B
K
I
E
H
m
P
K
D
T
I
V
L
E
Z
1
\M
R
Alz
B
IN
1
ए
Fig. 4
B
Fig.6.
I AR
A
k
2
P
ME
P
M
A
H
G
B 1 2 3 4 5
/
Fig. 8
A
a
I
K
BR
E
L C
B R
Z
G
F
k
Q
The THEORY of
PERSPECTIVE
Plate II
Fig. 2
H.
X
Z
M
5
4
B 3
B
R
H
L
Fig. 9.
B
3
D
B
4 3
2
C.
Fig. 7.
D
め
​F
K
P
M
Fig. 10
D
K
Of PERSPECTIVE.
153
LÊ, which is therefore called the horizontal Line; and the other,
in the Line QP, which is called the vertical Line, on the per-
fpective Plane.
1466. Secondly; thofe Planes which do not paſs thro' the
Eye, will have a direct or an oblique Situation with reſpect thereto.
If it be a direct Situation, it will be parallel to the perspective
Plane which is ſuppoſed to be placed directly before the Eye.
Thus the Plane A B C D is a direct one, and parallel to the per-
ſpective Plane H F. And among all Planes fituated obliquely to
the Eye, that which is in the Plane of the Horizon, as A E HD,
is the moſt confiderable; and is called the Ground Plane. This
is perpendicular to the perfpective Plane HF.
1467. From what has been faid, it appears, that Objects in
the Surfaces of the horizontal and vertical Planes cannot be ſeen
at all by the Eye at I; and therefore they are not to be regard-
ed in Perſpective. We must therefore confider of Objects in
and upon the direct and oblique Planes, and the Appearance
they make on the perſpective Plane.
1468. Thus let O B be an Object in the direct Plane; and
from the extreme Points O and B draw the vifual Rays OI, BI,
to the Eye at I. They will pafs thro' the perſpective Plane in
the Points o and b; and by joining thoſe Points with the
tight Line o b, that Line will be the Picture of the Line
or Object O B, upon the perſpective Plane. Thus alſo the
Perſpective of the Line OA is oa; and the Proportion of
Objects and their perſpective Appearances is the fame in this
direct View, viz. that of their Diftances from the Eye; for
OB:ob::OA: 0a :: OI: 0I, by fimilar Triangles.
1469. Thus (br) and (an) will be the Perſpectives of the Lines
BR and A N in the object Plane; and rban will be the Per-
fpect of RBA N of half the Plane A C, and a b c d the Per-
ſpective of the whole Plane A BCD. And all Lines parallel to
A B or C B in the object Plane, will have their perſpective Lincs
parallel to a b and c d in the Picture on the perfpective Plane.
And in what Manner foever the object Plane AC is divided by
Lines drawn upon it, their Repreſentatives will divide the Pic-
ture (ac) in a fimilar Manner. (1456.)
1470. Any Point B in a direct Plane has the fame Ratio of
Diſtance from the horizontal and vertical Planes, as its perfpec-
VOL. II.
X
tive
154
INSTITUTIONS
tive (b) has from the horizontal and vertical Lines, viz. that of
the Diſtances of the Planes from the Eye. For BR:br:: BO:
bo::10:10:: Iz: Ii, from the Nature of fimilar Triangles;
hence it is eaſy to delineate the Appearance of any Objc&s, or
form their Pictures on the perspective Plane when they are pre-
ſented in a direct View; as in the Front of Buildings, &c. as
will be exemplified hereafter.
1471. The laft Sort of Plane, in and upon which we are fup-
poſed to view Objects, is that of the Horizon itfelf, as ADHE,
above which the Eye has an Elevation, more or leſs, as i P =
IY. This is therefore called the ground Plane, and its Inter-
fection H E with the perfpective Plane, is called, the ground
Line. And this is the most confiderable of all others, as being
the common Table or Plan of all perspective Views, Land-
ſcapes, and picturefque Draughts of every Kind.
1472. With regard to this horizontal Plane, it has been ſhewn
(1469) that the two remote Angles thereof A and D are reprefent-
cd by a and d, in the perfpective Plane; and the other two Angles
E and H, are in the faid Plane alfo, as being common to both;
therefore by drawing the Lines a E and d H, there will be form-
ed the Figure E and H on the perſpective Plane which will be
the true perspective Appearance of the ground Plane ADHE.
1473. Hence it follows, that a E is the Perſpective of A E,
n P of NP, and H of DH. Whence it appears that Lines
which are parallel in the ground Plane, and perpendicular to the
perfpective Plane, are not fo in their perſpective Picture, but
they all converge to a Point i, which is called by the Point of
Sight in the perſpective Planc as being exactly oppoſite to the
Eye, or that Point in which a Perpendicular from the Eye falls
on the Plane.
1474. In the ground Plane draw V W parallel to A D; its
perfpective vw will be parallel to ad, in the Picture; and
adwo will be the Perfpective of the Part ADW V in the
original Plane. But for a Demonftration of what relates to forming
the Picture or perſpective Appearance of the ground Plane and
Objects upon it, and confequently, of the common Practice of
Perſpective, we muſt have Recourfe to the following Method of
Reprefentation, which is the most natural, concife, and per-
fpicuous, of any I have been able to think of.
1475. Let
Of PERSPECTIVE.
155
1475. Let ABCD be a right-lined Figure in the ground
Plane VGKT, contiguous to and at right Angles with the
perſpective Plane YZSR; FH the Diſtance of the Plane;
and H I the Height of the Eye at I. HE is parallel to G B or
CK and bifc&s A D and B C in the Points F and E. On the
Point E raiſe the perpendicular EM HI, and draw the Lines
BM, CM, GI, KI.
1476. Draw the vifual Lines IA, I B, and I M, which is
called the principal Ray, and is perpendicular to the perfpective
Plane in the Point i. Then it is evident the Plane I G B M in-
terſects the perſpective Plane in the Line Ai, and the Ray BI
being in the faid Plane G M, muſt inerfect the Line A i in ſome
Point (b) which is therefore the Perfpective of the Point B ;
and of Courſe Ab is the Perfpective of the Line A B.
1477. In like Manner it is fhewn, that as the Plane IK CM
interfects the perfpective Plane R Z in the Line DI; and the
Ray IC being in that Plane and interfecting the Line ID in the
Point (c), that Point (c) will be the Perſpective of the Point
C; and D c the Perſpective of the Line DC. And joining the
Points b, c, the Line be will be the Perſpective of the Line B C
in the ground Plane.
1478. Let A B = DC, then B C will be parallel to AD,
and fince in this Cafe AbD, therefore be will be parallel
to A Dalfo. And hence it appears, that all right Lines, as B C,
in the ground Plane which are parallel to the ground Line A D, will
alſo be parallel to the fame in their Picture on the perspective Plane.
1479. Hence alio it is evident, that the Perspectives Ab, Fe,
Dc, of all Lines AB, FE, DC, which are perpendicular to the
ground Line A D, do converge or tend to the Point of Sight (i) in
the perspective Plane.
1480. If the Line A B be continued out in Infinitum, towards
V, then fuppofing the Point B to move along that Line con-
tinually, the vifual Ray BI, will keep rifing on the Plane IGBM
towards IM making the. Angle BIM ftill lefs and lefs, till
the Point B arrives to an infinite Diftance, and then the Ray
I B will coincide with IM; and confequently the Line A i will
be the Perſpective of A B continued to an infinite Length.
Thus alfo D i will be the Perspective of the Line D C infinitely
continued towards T. And therefore the Triangle AiD will
X 2
#56
INSTITUTIONS
on the perſpective Plane be the Picture, or true Perspective, of
the Plane ABCD continued out upon the Plane of the Horizon to an
infinite Length.
1481. Hence the Line Y i Z, is the Perspective of the Horizon,
or Boundary of the Sight at an infinite Diftance; and therefore all
Objects on the Plane of the Horizon, will, in their perſpective
Appearance, or Landſcape, keep rifing from the ground Line,
or Bafe AD, towards the Point of Sight (i); and leffen in
their Dimenſions as they are more remote, till at laſt they vaniſh
in the horizontal Line Y Z.
1482. We have feen how Lines parallel or perpendicular to
the ground Line AD are to be delineated or drawn in Perfpective;
and we are next to fhew how thoſe right Lines appear, or are to
be drawn upon the perfpective Plane, which lie oblique to, or
make any Angle with the ground Line A D, or any other
parallel to it. In order to this, make AL = AG = 1 i, and
draw Ap to make any Angle p AR or pAD with the Baſe
A D, acute or obtufe. Then in the horizontal Line Y Z,
take iXLp; and drawp X and IX; and the Plane IXpA
will interfect the perfpective Plane in the Line A X. Draw the
viſual Ray I p which as it is in the Plane I Xp A must go thro'
the perſpective Plane fomewhere in the Line A X, which fuppofe
at (r), then is the Point (r) the Perſpective of (p); and fince
while the Point p is fuppofed to paſs from A top in deſcribing
the Line A p, its Perſpective (r) will move in the Plane A Z
from A to r, and deſcribe the Line Ar; which therefore will
be the Perſpective of the Line Ap.
1483. If the Line Ap were continued out to an infinite
Length, and the Point (p) ſuppoſed to move conſtantly therein,
its perſpective (r) will appear to move towards X, till at length
the Point (p) being at an infinite Diftance, the Point (r) ar-
rives at, and coincides with X, in the horizontal Line; the Line
A X is therefore the Perfpective of the Line Ap infinitely continued;
and X is called the accidental Point, to which the Perſpectives of all
Lines parallel to A p tend.
1484. Let LP be taken equal to AL; and iZ equal to
i I, and then joining A, P, and I, Z; we have the Triangles
APL, and iZI, equal to each other; then will the Plain
IAP Z interfect the prefpective Plane in the Linę A Z, which
will
Of PERSPECTIVE.
157
will be the Perfpective of the Line A P continued out to an in-
finite Diſtance.
1485. But fince ALLP, and LP is parallel to A D ;
therefore A P is the Diagonal of a Square, and contains an An-
gle DAP of 45 Degrees with the ground Line AD; there- .
fore the Point of Diſtance Z is that to which all Rays parallel to AP
tend in the perspective Plane.
1486. Let A B AD, then is ABCD a geometrical
Square, and its Diagonal A C, of which the Perſpective is Ac;
and the Point (c) is therefore that in which the perſpective
diagonal A Z interfects the Ray or radial Line i D. Make
¡Y =iZ, = i I; and join D Y, then will that Line DY be
the perſpective Diagonal of D B, (the other Diagonal of the
Square A C) infinitely continued, and Db the Perſpective of
the Diagonal DB determined by the Interfection of the Lines.
DY and Ai, as before.
1487. Thus it is demonftrated that A b c D on the perſpective
Plane A SZY is the true Picture or perſpective Delineation of
the original Square A B CD, on the ground Plane, as required.
And from hence is deduced the Rationale of the common
Method of drawing the Perfpective of a given Square or any
Figure infcribed therein; as alfo of a ſtill more univerſal Me-
thod of affigning the Perfpective of any Point, Line, Superficies,
or Solid, on the Plane or perfpective Table, which we fhall
next proceed to explain.
CHA P. · III.
The RATIONALE of the common METHODS of
drawing the PERSPECTIVES of OBJECTS, ex-
plained from the preceding Theory.
[PLATE II. of PERSPECTIVE. }
1400. TIVE are much better known than the Reaſon of
TH
HE common Methods of Drawing in PERSPEC-
them; we ſhall here give both together, and then our Endea-
your will at least have the Face of Novelty. Let T G (Fig. 3.)
be
158
INSTITUTIONS
be the Bafe or ground Line, in any perspective Plane or Table
TGYZ. The Method of drawing the perſpective Appearance
of any Object on this Table is as follows.
1489. Let it be required to draw the Perspective of the given
SQUARE A BCD and all its Parts, in a front View. The Square
is fuppofed to be contiguous to the Perſpective Table, and there-
fore its Side A D will be in the ground Line. As the View is
direct, or in Front, the Eye muſt be fuppofed directly againſt
the Middle of the Side AD; and therefore in the Perpendicular
EF continued out, take F I equal to the given Height of the
Eye, and (I) will be the Point of Sight (1476) fet off the given
Diſtance of the Eye from the Table each Way from I, to Y
and Z; and theſe will be the Points of Distance (1486) in the
horizontal Line Y Z.
1490. From the Points A and D draw the Radials I A, ID;
and the Diagonals A Z, and DY, interfecting the Radials in the
Points b, and c; then draw be; and the Figure Abe D is the
Perſpective of the given Square ABCD as required. For Ac
is the Perſpective of the Diagonal A C, and D b that of Db (by
1486); therefore Ab and Dc are the Perfpectives of the Sides
A B and DC (1479) and be of the remote Side BC (1477.)
Alfo the Point n is the Perfpective of N the Interfection of the
Diagonals in the Square; and oq, Fe are the Perſpectives of
OQ and FE which bifect the Square orthogonally, (1478,
1479.)
1491. It is not neceffary to make ufe of more than one Point
of Diſtance, Z; becauſe one Diagonal A Z, gives the Point.
(c) in the Radial ID which determines the Perſpective of
the Diagonal A C; and then by drawing thro' the Point (c)
the Line bc parallel to the ground Line A D, the Perſpective
of the Square ABCD is compleated as before (1477.)
1492. From hence we may derive a Demonſtration of an
univerfal Rule for putting all Objects into Perſpective from
given Point therein. The Rule is this.
!
From the given Point in the ground Plane, let fall a Perpendicular
on the ground Line, from whence draw a Radial to the Point of
Sight; then from the Radial fet off that perpendicular Diſtance in
the ground Line, and from thence draw a Diagonal to the Point of
Diſtance, the Interfection of the Radial and Diagonal will give the Seat
gr
Of PERSPECTIVE
159
or Place of the Perspective of the given Point. And thus the Per-
Spectives of any Number of Points may be found, which being con-
nected will form the Perfpective of the given Figure which they ter-
minate.
1493. To exemplify this Rule; let C be any given Point in
the ground Plane, from which let fall the Perpendicular CD to
the ground Line TG; and from the Point D draw the Radial DI
to the given Point of Sight I. Then in the ground Line make
DA=DC, and from the Point A draw the Diagonal A Z to
the given Point of Diftance Z, which will croſs the Radial in
the Point (c) the Perſpective of the Point C as required. This
is evident from the common Method beforegoing (1476), be-
cauſe AC is the Diagonal of a Square.
1494. Suppoſe B any other given Point; let fall the Per-
dicular BA, and draw the Radial A I; then make AG AB,
and drawn the Diagonal G Z, it will alfo interfect the Radial
in the Point (b) the Perſpective of B as required. To prove
which, let us ſuppoſe B C = AD=DC=AB. Then join-
ing (bc) that must be the Perfpective of B C, the fame as be-
fore determined by the commen Method; and fo it will prove.
For in the fimilar Triangles AID, b Ic, we have AD: bc::IA:
Ib (656,) according to the common Method; alfo from the
fimilar Triangles Z GA, Zbc, we have Z G: Zb::
AG: be, according to the Rule; but from the fimilar Triangles
b GA and bZI, we have I A:Ib:: ZG: Zb; and therefore
it is AD: bc:: AG: be; but AD AG, by fuppofition;
therefore the Line (bc) is the fame in both Cafes.
1495. Hence we have eafy Methods of drawing the Perſpec-
tives of any fuperficial Figures whatſoever; for if they are cir-
cumſcribed with a Square, and Lines drawn thro' their termina-
ting Points and Angles to the Sides of the Squares, then by
taking the Perſpective of the Square you will at the fame Time
have the Seat of every Point or Line therein which bounds
the Figure propoſed, and theſe being connect.d, the Perſpective
of the Figure is formed as required.
1496. Thus fuppofe any Line BK were to be put into Per-
fpective, then one Extreme being placed on the ground Line at
K, let the Angle B of the Square be placed on the other; then
the Points K and (b) being connected in the Perſpective, will
give
160
INSTITUTION 9
give Kb, for the Perſpective of the Line K B required. Or if
from the Point B you let fall the Perpendicular B A, and make
AGA B; then drawing the Radial A I, and Diagonal G Z,
they will interfect each other in the Point (b), which therefore
is the Seat of the Point B, and fo K b is of Courſe the Seat of
the Line K B, according to the Rule (1492.)
1497. Since any right Line is fo eafily put into Perſpective,
there can be no Difficulty in finding the perſpective Appearance
of any given right-lined Figure. Thus the Perfpective of the
Parallelogram CEFD is ceFD; the Perfpective of the
Square D FN Qis DFng, and the Perfpective of the remote
Square CEN Qis ceng, all determined by infcribing them in
the geometrical Square A B C D.
1498. If the Figure be not right-lined, but bounded by a
Circle or ſome other Curve, yet if they are properly circumscribed
by a Square, the Interſections of the Diagonals and other Lines
with one another in the feveral Parts of the Perimeter of the
Curve, will give a fufficient Number of Points in the Perfpec-
tive Table, thro' which to draw the Perfpective of the Circle or
Curve propofed. Thus for Example (Fig. 4), if in the Square
ABCD you draw the Circles OFQE, and TVWS; then
by drawing the Diagonals of the Square and other proper Lines
GH, IK, LM, RP, OQ, EF, &c. their Interfections
in the Periphery of the Circle, being all found in the Perſpec
tive Table, and a curve Line drawn thro' them; fuch curve
Lines are the Perſpectives of the Circles, as required.
1499. The Square OFQE with an Angle F in Front, has
its Perſpective Foeq at the fame Time determined. Alſo the
Triangle X V Z, is projected by the vifual Rays into its Per-
ſpective xv z on the Table; and the Perſpectives of its ſeveral
Parts (which are fix Triangles and a Trapezium) are formed by
the Lines which divide it; all which is evident by Inſpection of
of the Figure.
1500. If it be required to delineate the Perſpective of a
Square A B CD viewed obliquely (as in Fig. 5.) then the given
Obliquity of the View will determine the Pofition of the
Point of Sight I in the horizontal Line, and the given Diſtance
being ſet off from I to Y; then if the Radials I A, ID, are
drawn, and one Diagonal DY to interfect the Radial I A in ( 6 )
1
that
Of PERSPECTIVE.
161
that will be the Perfpective of the Point B; be drawn parallel
to AD, will be the Perfpective of BC; and A be D will be
the Perſpective of the Square ABCD as required.
Theory (1475) being in this Cafe the fame as in a direct View,
the Practice must be conducted by the fame Rules of Courfe.
The
1501. We muft not quit this Subject without a Remark on
the Abfurdity of feveral Writers on Perſpective, viz. that
tho' they give the practical Rule of drawing the Perfpective of any
Superficies right, yet they place the Perspective itſelf wrong. Or
they place that Part of it nearest the ground Line, which is in
Reality moft remote from it; for Example (in Fig. 4.) the
Perſpective vzx of the Triangle V Z X, they place contrary
to its Pofition here, viz. with its Bafe z towards the ground
Line AD, and its Vertex v towards the Point of Sight I. In
fhort, they give the fame Pofition to all the Parts of the Per-
fpective, which the Originals themfelves have in the ground
Plane, contrary to what really appears to the Eye viewing thofe
Objects thro' a tranfparent Plane, as we have fhewn. This er-
roneous Repreſentation of Gardens, Fortifications, &c. gives
fuch an unnatural Idea thereof as must be very difagreeable to a
nice Genius. Many Inftances of this you will find in the
JESUITS PERSPECTIVE. And even Pozzo himfelf has puzzled
his Readers with this Piece of Nonfence.
CHA P. IV.
The THEORY of the PERSPECTIVE of CIRCLES,
and circular AREAS, demonftrated; with the
Solution of fome very useful PROBLEMS relative.
thereto.
(PLATE I. of OPTICS and PERSPECTIVE.)
1502. A
S there is fomething very curious in the Speculation
of the Perspective of a Circle, and very little known,
as being rarely found in Treatifes on this Subject; I prefume it
will be agreeable to the ingenious Reader, if I employ a Page
or two, in order to fet that Matter in a clear Light.
VOL. II.
Y
1503. I
*
162
INSTITUTIONS
1503. In the first Place, then, it is to be confidered, that
the Rays which proceed from a Circle to the Eye (in any Ele-
vation above the Plane in which it lies) do form a Cone; whofe
Bafe is the Circle, and whofe Vertex is in the Eye. (625.)
1504. If this Circle be viewed thro' a tranfparent Plane
pual to the ground Plane of the Circle, then its Perfpective
will be a CIRCLE alfo; becauſe in this Cafe, the viſual Cone
is cut by a Plane parallel to its Baſe, and therefore the Figure
of the Section will be the fame as that of the Bafe, in any Po-
Tuon of the Fye above the pcrfpective Plane.
1505. If the perfpective Plane pafs thro' any Part of the Cir-
cle or Bafe of the Cone, and the Pofition be parallel to either
Side of the Cone, then that Part of the Circle's Periphery which
is feen through, or under the Plane, is projected on the faid
Plane in the Figure of a PARABOLA, (by what has been faid
of the Genefis of that Curve in (740); for the Eye at F will
view the Pointe at R on the Plane, and the circular Periphery
Be B will by the viſual Rays of the Cone be depicted in the Curve
of the Parabola BR B, which will therefore be the true Per-
fpective of that Part of the Circle in this Cafe. *
1506. If the perfpe&uve Plane paffes thro' the circular Baſe
of the vifual Cone, in a Pofition not parallel to either Side, and
yeɩ lo as not to pass thro' both in the fame Cone if they were
continued out beyond the Circle or Bafe, then fo much of the
Circle as could be feen thro' the Plane would be projected there-
on in the Form of an HYPERBOLA (by 765); for the Eye at
C will project the Point c on the Plane at V, and the circular
Periphery Be B into the hyperbolic Curve B V B on the Plane,
which therefore will be its perfpective Appearance.
1507. In all other Pofitions of the perfpective Plane with re-
gard to the Sides of the Cone of Rays, the Perſpective of the
Cicle will be an Ellipfis therein (by 763.) For fuppoſe the
Plane to cut the Cone in the Line TV, then to an Eye at the
Vitex L, the Point F will appear at T, and the Point P at
Von the faid Plane; and of Courfe the whole circular Baſe will
be projected i. the Ellipfis T B V B, which is therefore its per-
fpective Appearance.
1508 It a Circle ED be placed on an horizontal Plane A E,
and viewed thro' a perpendicular transparent Plane DG (Fig. 28.)
* See the Figures of the conic Sections here referred to.
by
Of PERSPECTIVE.
163
น
by an Eye at F at the Diſtance D B, and Height BF; then its
Perſpective D C will be an Ellipfis alfo. But becauſe the viſual
Cone EFD is in this Cafe in an oblique Pofition (and what
they call a ſcalenous Cone), and as the Obliquity of the Cone,
or its Inclination to the Horizon, will vary with the Distance
and Height of the Eye, fo it will be cut in a various Manner
by the Plane D G, producing as many different Ellipfes by thofe
Sections in fome of which the Diameter ED of the Circle will
be the longer Diameter of the Ellipfe, in others it will be the
ſhorteſt; confequently, between both there will be one Cafe
were the Ellipfe will become a Circle, or have all its Diameters
equal on the perſpective Plane.
1509. Now this will happen when the Cone E F D is cut in
a fimilar Manner by both the Planes, viz. the ground Plane
A E, and the perfpective Plane D G, or when the Angle.
GDF is equal to the Angle DEF; for if we fuppofe the
Eye to move thro' the Arch of a Circle GFA from the ver-
tical to the horizontal-Plane, it is evident, when the Eye is at
G the Angle DEC will be of fome Quantity lefs than a right
Angle; and the Angle FDC will be Nothing. But as the
Eye departs from G, the faid Angle FDC begins, and en-
creaſes to a right Angle when the Eye arrives at A; on the
other hand, during all that Time the Angle DEC decreafes,
and at laft vanifhes; and therefore at fome Moment of that In-
terval the two Angles must be equal, which let us fuppofe to
happen when the Eye is in the Pofition F.
1510. Then are the Triangles E F D and DCF fimilar
for the Angle CDF DEC, and the Angle CFD is
common to both; therefore the Angle E D F D C F,
and
confequently the vifual Rays of the Cone make fubcontrarily
the fame Angles with each Plane A E and DG; therefore the
Section of the Cone made by each Plane muſt be of the fame
Figure, and confequently fince ED is a Circle its Perſpective
DC will be a Circle alfo.
1511. From hence fome curious Problems wil arife of the
laft Importance in Perfpective. The first is, To determine the Po-
fition of the Point F, or Point of Sight, from the gin ņ
of the original Circle DE, and Defiance from the
In order to this, let D E = D, DB = s; alſo pr
Y 2
1
양
​>
164
INSTITUTIONS
of the Eye B F = b; the Diameter of the perfpective Circle
DC=d; and let E Fe, and DFf. Then D:e::d:f
(1510) let D + s = r; then e = p² + b² and ƒ =
✓
✔ 3²+s² (636) alfo D: d::r: h, therefore d=
Dh
Whence D:
*
r
we get b²
b
f
ᎠᏏ
I: : : √ r² + b² : √h²+s²; whence
=rs =D + s X s; and therefore h
= BF, the Height of the Eye at F required.
*
D + sx s
1512. The above Equation gives this Analogy, D+s:b::
hs, whence it appears (645, 660) that the Locus of the Point
F is the Circle EF A. Hence A B BDs; AF=DF;
=
the Diameter A E≈ D + 2s; and the Radius A N
D+25
2
DE + 2 DB
2
the Problem is evident,
from whence the Conftruction of
1513. The fecond Problem is, the Diameter of the Circle DE.
and Height of the Eye B F being given, to find the Distance DB
from the perſpective Table. The Solution is thus; h²- Ds + ssg
(1511); let the Square be compleated (338) then b² + D=
ss+Ds + D, and therefore s+ D = √ b² + ÷ D; con-
fequently s = b² + D D = DB the Diſtance re-
quired.
1514. A third Problem, is to determine the Distance D B, at
which the Perspective DC of a given Circle D E ſhall be a Circle,
and its Area be to the Area of the given Circle in any given Propor-
tion of z toy. To folve this Problem, it muſt be confidered that
fince D BBA, (1512) we have D F F A; and fince the
Areas of Circles are as the Squares of their Diameters (840)
we have DE: DC:: EF2: DFF A² (1512) but E F2;
FA²:: EB: A B, (660) therefore DE: D C²:: (y: z::)
EB: AB( :: D+s:s) whence we have ysz D+z5;
zsz D, therefore y- z:z:: D:s=
and y s
DB, the Diſtance required,
2
z D
Y-Z
1515. The Center of the original Circle will not be projec-
ted into the Center of the perſpective Circle; neither will the
Diameters
Of PERSPECTIVE.
165
Diameters of the latter be the Perfpectives of the Diameters of
the former. Nor will any right-lined Figure (as a Square, Tri-
angle, Polygon, &c.) infcribed or circumfcribed about the original
Circle have its Perſpective in, or about the perſpective Circle of
the fame Kind or fimilar to it; for that of a Square will be a
Trapezium; that of a Triangle a diffimilar Triangle, that of a
Pentagon will be an irregular Polygon; and fo of others. All
which the Reader will be eaſily affured of by drawing this Cafe
in the ufual Manner and Form as directed (1492.)
CHA P. V.
The Theory of CATOPTRIC PERSPECTIVE; or the
perspective Appearance of Objects on a REFLEC-
TING PLANE.
[PLATE I. of OPTICS and PERSPECTIVE.]
1516. W
E have confidered the Dioptric THEORY of PER-
SPECTIVE ſo far as it relates to Lines and Super-
ficies; and before we go farther it will be proper to confider the
perſpective Appearance of the fame Objects by Light reflected
from a polifhed Plane; in as much as we ſhall find the Perſpective
of any Line or Superficies is identically the fame on the Catoptric
or reflecting Plane as it is on the Dioptric or refracting one.
1517. Preparatory to a Demonftration of this Pofition, let
HD (Fig. 26) be a Line or Section of the Speculum, perpen-
dicular to the ground Line A F, and B the Place of the Eye at
the given Diſtance A D, and Height A B; from any Point E
in the Line A D, let a Ray of Light proceed to the Speculum
in the Point C, and be reflected from it to the Eye at B; then
continuing the reflected Ray out beyond the Speculum, it will
interfect the Line A F in the Point F, making the Line DF =
DE; and confequently, the Point E by reflected Light will ap-
pear upon the Speculum in the fame Point C with another Point F
viewed
166
INSTITUTIONS
viewed by the Eye thro' a tranſparent Plane HD, ana placed at the
fame Distance behind it.
1518. The Demonftration is eafy; let C G be perpendicular to
the Plane H in the Point C, it therefore will alſo be parallel to the
Line A F; then becauſe the Angle GCB GCE (1282) and
GCBCF D, and GCE=CED (631) therefore we
have the Angle CED-CFD; and therefore the Line CE=
CF, and confequently the Line DE DF. Wherefore the
apparent Place of the Point F, by Reflection, is the Point F at an equal
Distance beyond the Speculum.
1519. Proceeding in the fame Steps of Demonftration here as
in Dioptric Perſpective (1475), we ſhall find the fame Appearance
alfo of Lines and Superficies on a reflecting Plane as we did there on
the refracting one. Let A YZD (Fig. 27.) be the Speculum,
ABCD be a Square placed before, and contiguous to it; ex-
tend the Sides A B, and D C indefinitely on each Side, and
draw EH between and equidiftant from both. Let I be the
Place of the Eye at the Diſtance F H, and Height HI; take
FEFE; and thro' E and H, draw BC and GK; alſo on
the Point E erect the Perpendicular E MH I, and draw IM
paffing thro' the Speculum in the Point i; and join IG, IK;
M B, M C.
1520. Then becauſe the Point B of the Square is in the
Plane IG B M which inter..&ts the reflecting Plane A Z in the
Line Ai; and because the incident and reflected Rays are in the
fame Plane (1282) therefore the Ray of Light which proceeds
from the Point B and is by the Speculum reflected to the Eye
at I, muft imping on the Speculum in fome Point (b) in the
Line Ai; and the reflected Ray will be bI. If this Ray be
continued out, it will meet the Side of the Square A B produced
in B, making ABA B. (1518)
1521. Now becauſe the fame Thing is demonftrable of every
other Point in the Side of the Rectangle A B, therefore the
Perspective of the Line A B by Reflection is the Line A b, or that
very Part of the Radial A i which is the Perspective of the equal
Line A B by Refraction thro' a tranſparent Plane (1476).
1
1522. In like Manner it is demonftrated, that the Perſpective
of the Side DC by Reflection is Dc, the fame Part of the
Radial as is that of the equal Line D C by tranfmitted Rays.
By
Of PERSPECTIVE
167
By joining bc, the Line (b c) is the Perfpective equally of BC
by reflected, and B C by refracted Light. And confequently,
Abc Dis the perspective Appearance of the Rectangle ABCD by
Reflection, and every Way the very fame as it was fhewn to be of an
equal Rectangle A B C D beyond the pellucid Plane (1487).
1523. Therefore whatever was demontrated relative to the
Perfpective of Lines and Superficies any how pofited or formed
with respect to a common tranfparent Plane in the Sequel of the
Dioptric Theory, will hold equally true here in Catoptric Per-
ſpective. And tho' this Part of the Science has been the leaft of
all confidered by Writers on this Subject, yet when compared
with common Perfpective it will be found not only more delicate
in its Nature, but alſo more adapted to Ufe. This we fhall
illuftrate more particularly when we come to treat of Pictu-
refque Perſpective, or that Part of the Science which treats of the
PRINCIPLES of PICTURES or LANDSCAPES formed by
OPTICAL GLASSES both of the reflecting, and refracting Sort.
CHA P. VI.
Of INVERSE PERSPECTIVE; where the Nature ana
Principles of the ANAMORPHOSIS or Deforma-
tion of Figures, and their Rectification, are ex-
plained.
1524.
ΤΗ
HE inverfe Method of Perſpective is not leſs pleaſant
than uſeful, nor is it in the leaſt Degree difficult,
as it is only going backwards in the fame Steps we took in the
direct Method; and it must be obſerved, that with regard to
any Science, unleſs the Student can go backwards as well as
forwards, he makes but an imperfect Proficiency.
1525. Inverfe Perspective fhews how from a given Piece of
Perſpective to determine the Original or Prototype thereof under
the Circumstances of Size, Situation, Distance, &c. And
here, as in the direct Method, we must begin with finding the
Original of a Point, then of a Line, and lastly of a Superficies;
and
168
INSTITUTIONS
and being able to determine theſe, there will remain no Dif-
ficulty with reſpect to Solids, becauſe the Perſpective of a Solid
is only the complex Perfpective of its fuperficial Parts; and for
this Reaſon, we think this the moſt proper Place to treat of
the Inverfion of a perfpective Piece, and the Nature of Anamor–
phofis or the Deformation of Pictures, which naturally reſults
from thence.
1526. For by the direct Method it appears, that the Per-
ſpective is very diffimilar to its Original, and therefore if the
latter be formofe and regular, the former will be diform and
irregular, or in the ufual Phrafe, deformed and diftorted. So
on the other Hand, if any natural and well-formed Picture be
confidered as a Piece of Perfpective, formed on a perfpective Plane,
then when it is refolved into its Prototype, that muſt be a very
unnatural, miſhapen, and ill-proportioned Figure; all which
will be evident from the following Proceſs.
1527. Let TV W X be a Perſpective Table (Fig. 29.)
ſtanding on the ground Line S V. Alfo let I be the Point of
Sight, and Y the Diagonal Point of Diſtance; and then fup-
poſe it required to find in the ground Plane the Seat of the
Original of a given Point P in the Perſpective Table. You
proceed thus; from the Point of Sight I, draw thro' P the
Radial I K, and on K in the ground Line erect the Perpen-
dicular KG; then from Y draw thro' the given Point P, the
Diagonal Y L to meet the ground Line in L.
Laftly, on the
Point K, with the Diſtance K L, defcribe the Arch IH to in-
terſect KG in H; and the Point H is the Original of the Point
P, as required (1476, 1493.)
1528. In the fame Manner any other Point Qin the Table
will have its Prototype determined in C; and the Distances
of any of thoſe original Points H, and C, from the ground
Line SV will always be equal to the Diſtances K L and LS con-
tained in the ground Line between the Interfections of the re-
ſpective Radials and Diagonals.
1529. To find the Original of a given Line P Q in the Per-
fpective Table; nothing is required but to determine the
Original Seats H and C of its two extreme Points P and Q
(1494) then by connecting the Points H, C, the Line H C is
formed on the ground Plane and is the Original of the Perfpec-
tive
Of PERSPECTIVE.
169
tive PQ, as required. The Diſtance of this Original at
each Extreme from the ground Line, as alfó it's Magnitude
or Length, is meaſured on the faid Line SV, in Inches, Feet,
&c. according to the Meaſure by which that Line is di-
vided.
1530. In the fame Manner any other Perfpective Line OP is
found to have it Prototype BC, on the ground Plane; and thus
any Perspective Angle OPQ on the Table V X, will have its
original Angle BCH determined on the ground Plane; whofe
Quantity is there alfo meaſured in the ufual Manner.
1531. Therefore, to find the Original or Prototype of any given.
Perſpective Figure or Superficies, OP QR on the Table, nothing
more is neceffary than to find the original Seats E, B, C, H,
of it's angular Points O, P, Q, R, (1493) for thefe being
connected, from the Figure EBCH, which is the true Origi-
nal of the Perſpective given; whoſe Diſtance and Dimenfions
will then be eafily known by common Geometry.
1532. As in the direct Method, the Perſpective or every regu-
lar Figure is irregular and diffimilar, (1487) fo in the Inverſe
Method, the Original of every regular Piece of Perſpective, muft
itſelf be irregular and of a different Form; and therefore, if the
Perfpective Figure OPQR be a Square, the Original EBCH
is a Trapezium, (1531) and if that Square be divided into equal
Parts, by Lines drawn thro' the equidiftant Points f, e, d, the
Original will be divided very unequally by correfpondent origi-
nal Lines drawn thro' the original Points a, b, c, at unequal
Diſtances in the Line HC, the Original of P O.
1533. Therefore it will follow, that if any Portrait, or other
Picture be drawn on the Perfpective Square, it will, when pro-
jected by the viſual Rays in the original Trapezium, be there
deformed, or appear of a monftrous Shape. For the natural
Form and juſt Proportion of Parts in the Portrait, which de-
pend on equal and fimilar Spaces which they fill in the Square,
will be all deſtroyed, when projected on the Trapezium into
unequal and very diffimilar Spaces. And this Distortion of the
Figure is called, an Anamorphofis, Deformation, or monftrous
Projection.
1534. The Anamorphofts is eaſily effected in any Degree by
the Rules of Art. For if the Perfpective Square P R be divided
VOL. II.
Z
inte
170
INSTITUTIONS
into a Number of finall Squares, or other equal or fimilar Spa-
ces; then fince by the Rules above the Original of the Square
and all its Parts are to be found, and drawn on the ground
Plane, it remains only to draw the fame Parts of the Picture in
each irregular Space of the Trapezium, as you fee in the cor-
refponding Part of the Perfpective Square; and the Deformation
will be compleated, as required.
1535. The Diftortion or Deformation will be in Proportion
to the Height and Diſtance of the Eye from a given Perſpective
Figure OPQR. For if the Diſtance of the Eye IY continue
the fame, the Breadth of the Projection at each End, viz. EH
and BC will leffen as the Height of the Eye WV increaſes,
and confequently, the Deformity of the Picture will be the
greater, as its Length continues the fame nearly.
1536. Alfo, if the Height of the Eye be the fame, but the
Diſtance greater, then will the Deformation or Trapezium vaſt-
ly encreaſe, both in Length and Breadth, in all the remote
Parts towards BC, in Compariſon of thoſe towards HE; and
therefore the Deformation becomes greatly augmented in this
refpcct alfo; and, indeed, much more than by altering the
Height of the Eye at the fame Diſtance.
1537. Hence it appears, that there is a certain Pofition of
the Eye, in which it will view any given artificial Deformation
or monftrous Picture E BCH, fo that it fhall appear perfectly
natural, or in a juſt Proportion and Symetry of all the Parts of
the Object it is intended to reprefent. For it is evident, from
the foregoing Theory, if the Deformation EBCD be placed
on the Horizon, at the Diſtance ME, from the Perſpective
Table V X, then an Eye placed at I, at the Diſtance IY, and
Height TX, will view that monstrous Projection on the ground
Plane, as a regular and perfett Square on the Table; and the de-
formed Image contained in it; as a well-proportioned and natural
Portrait or Picture.
1538. Now, becauſe the Image in the Eye is every Way
fimilar to the Object it views on a Plane placed before it, and
parallel to the Fundus or Bottom of the Eye; therefore, fince
this Image on the Retina in the Eye is the fame as would be
formed of a Picture well drawn or defigned on the Plane
or Table VX, and fince it does no ways depend on the
faid
Of PERSPECTIVE
171
faid Table, but on the peculiar Pencil of Rays paffing through
it only, which is ftill the fame when the Table V X is removed;
it follows, that the Eye in the Pofition affigned at !, will view,
by itſelf alone, the Deformation contained in BEHC, as a juft
and well drawn Picture.
1539. This Invention has given Rife to thoſe common Ana-
morphofes of King Charles's Head, St. George and the Dragon, &c.
on long Slips of Paper (like Fig. 30.) fold at the Shops, which
being held in a Pofition parallel to the Horizon, and viewed at
the proper Diſtance ED, and ſmall Height of the Eye at I, do
very agreeable furprize the Spectator (inconfcious of the Defign)
with a regular and beautiful View of thofe Objects, of which in
the Deformation they faw fcarce any Appearance.
1540. From what we delivered in the Theory of Catoptric Per-
fpective, it is evident, that if T V W X be confidered as a re-
flecting Speculum, and EBCH a Deformation placed before it,
then its Appearance OPQR, in the Speculum, will be a regu-
lar Picture; for that will be the true Perſpective of the Figure
on the ground Plane, in the fame Manner by a reflected, as by a
tranfmitted Pencil of Rays (1522)
1541. Since the Law of Reflection is the fame in Surfaces of
every Figure (1300) it will follow, that if we conceive the Spe-
culum TV W X to be pliable and formed into a Cylindrical Sur-
face, as ABCD (Fig. 31.) There will ftill be a regular Per-
ſpective Picture formed by Reflection from a proper Deformation
EFGH; the greater Divergency of Rays in this Cafe, caufing
only a greater Diſtortion in the Anamorphofis, and a Diminution
of the Picture in the Speculum.
1542. Hence arife all thofe Experiments of polished Cylinders,
Cones, and other figured Speculums, which rectify the Appear-
ance of thoſe ſeemingly unmeaning and ludicrous Deformations
painted round about them, on the Planes on which they are
placed; any of which are eafily drawn by any Perfon fkilled in
the Theory of Catoptrics; but thofe who are not, may very fuc-
ceſsfully uſe the following practical Method which is univerſal
for all Speculums.
1543. Let a fquare Hole be cut in a Piece of Paper, Paft-
board, Vellum, &c. and fine Threads pafted on the Paper
over the Hole, horizontally and perpendicularly, fo as to divide
Z 2
the
172
INSTITUTIONS
the fquare Hole into a proper Number of leffer Squares; then
paſt this Paper with the Lattice-Square on the Surface of your
poliſh Cone, Cylinder, Looking-Glaſs, &c. and place a Can-
dle at a proper Diſtance and Height (ſuch as you intend the Eye
fhall have to view the Picture) then will that polliſhed Lattice
be reflected on the Table in the Form of the required Anamor-
phofis, with the Shadow of the Threads, dividing it in a pro-
per Manner for drawing the Deformation of any propofed Ob-
ject, whoſe Picture is drawn in a Square, of the fame Size with
that on the Speculum.
CHAP. VII.
Of SCENOGRAPHIC PERSPECTIVE; or the ME-
THOD of drawing FIGURES of THREE DIMEN-
SIONS in PERSPECTIVE,
1544.
THE
HE Superficies which terminate or bound a Solid,
conftitute its Form, or external Appearance; and
therefore the PERSPECTIVE of a SOLID, is nothing more than
the complex Perſpective of all its fuperficial Parts; ſo that, pro-
perly ſpeaking, there is nothing beyond fuperficial Perspective in
Nature, for that rightly applied, gives the Perſpective of every
Figure or Form of Solids, or rather Figures of three Dimen
fions*; as will appear in the following Articles,
1546. We ſhall begin with the moft regular and fimple Form
of a Solid, viz. that of a CUBE, whofe Sides are fix in Num-
ber, and all equal; but if a Cube be placed direct before the
Eye, only one or two of the fix Sides can then appear. For if
Fig. 1. (Plate II.) be confidered as a Cube, then, becauſe its
Height
We have thought proper here to make a Diſtinction between a
Solid, and a Figure of three Dimenfions; for though every Solid has
always three Dimenfions, yet a Figure of three Dimenfions may not
be a Solid, but a hollow or concave Body; thus there may be a fuper
ficial Cubic Inch, as well as a Solid one, but the Perfpective of both is
the fame; for this Art is not concerned with the Solidity or internal
Parts of Bodies that do not appear to the Eye.
Of PERSPECTIVE.
173
Height QP exceeds the Height of the Eye IY, it will be im-
poffible for the Spectator to ſee more than the Side in Front,
EG. For Rays of Light proceeding from any Point in the two
fide Planes EB, HC, and that on the Top GB, can never
come to the Eye at I; and the Bottom ED, and fartheft Side
DB, are precluded from the Sight by the Solidity of the Cube,
ſo that one Side only can in this Caſe appear.
1546. But if the Height of the Eye IE (Fig. 4.) exceeds the
Height of the Cube A B, then not only the Front-fide ABCD,
but likewife the Top or uppermoft Side will appear to the Eye.
And only theſe two can appear for the Reafons before-mention-
ed (1545.) The Perfpective of the Side A B C D of the Cube
in Front, will be fimilar to the Side of the original Cube, and
therefore a Square, as is evident from (1468, 1469, 1470.)
1547. But fince the Top, or upper Side of the Cube is below
the Eye, and parallel to the Horizon, it may be conceived as
placed on a ground Plane at the Height of the Cube above it ;
and its Perſpective on the Table, will therefore be the fame as
of any Square ABCD for the Height IF, and Diſtance IY
of the Eye, (1487) that is, Abc D will be the Perfpective of
the upper Side of the Cube; and confequently, the Perſpective
of the whole Cube, viewed in Front, will confift only of thoſe
two Sides, as in the Figure.
1548. A Cube of a lefs Height than that of the Eye, placed
with one Side parallel to the perspective Table, but removed
from the direct Ray IF towards either Side, to more than half
it's Width AF or FD, will difcover a third Side to the View;
and in a folid Cube, three Sides are the moſt that ever can be
feen; this is called a View in Profile.
1549. Therefore to give the moft general and eaſy Idea of
the Perſpective of a Cube (or any folid Body) it will be beſt to
diveſt it (as it were) of its Solidity, and reprefent it as hollow,
or confifting only in Form, or the external Sides or Super-
ficies.
1550. Wherefore (Fig. 7.) let ABCD be the Side of a
hollow Cube, placed contiguous the perſpective Table, and
viewed in Profile. Then will the Perfpective of the Bafe be
BbcC (by 1487.) This is called ICHNOGRAPHY or perspective
Plan of the Cube. The Perfpective A adD of the Top of the
Cube
174
INSTITUTIONS
Cube is determined in the fame Manner, by fuppofing AD the
ground Line in regard thereto. The Perfpective of the Front
ABCD, as it coincides with the Table, is the very Side itſelf.
And the Perfpective of the Oppofite or remote Side, as it is
parallel to that Front and to the Table, will be of the fame Fi-
gure, viz. a Square abcd (1468, 1469.)
1551. Therefore the fame Part A a of the Radial AI which
bounds the Perſpective of the Top of the Cube, terminates alſo
the upper Part of that Side of the Cube which is in View, as be-
ing the common Interfection of both thofe Sides or Planes. For
the fame Reaſon Bb terminates the lower Part of the viſible Side
of the Cube on the Table; therefore the Perfpective of that
Side of the Cube is Aab B. And fince the Cube is hollow, and
the Sides open, the other Side will appear internally; and its
perſpective Profile will be D de C.
1552. All we have faid is evident from the Confideration of the
mutual Interfection of Planes, as indeed from the Figure itſelf;
for if we turn the Figure Side-ways, and look upon CD conti-
nued out to M as a ground Line, then will the two Sides,
which are now perpendicular to the Horizon, be parallel to it,
or they will be the Bottom and Top of the Cube, and the Height
of the Eye above thofe Sides or Planes will be I M, as it is now
CM. And CedD, Aab B, will be their Perfpectives, drawn
by the common Rules (1489, 1492.) as Bbc C, AadD,
are for the lower and upper Sides in the prefent Pofition of the
Cube.
1553. If the Bafe of the Cube be divided by Lines parallel to
the ground Line, like the Square in Fig. 5. then by fetting off
thofe equal Divifions in the ground Line from B to 4, 5, 6; and
from them drawing Diagonals to the Point of Diſtance Z, they
will cut the Side Bb in the Points I, O, G, thro' which the
Lines IK, OQ, GH, are to be drawn for their Perſpectives
in the ichnographic Plan BbcC (1478, 1479.) and they
will be all parallel to the ground Line BC, as in the original
Baſe:
1554. In the fame Manner, if CD be confidered as a ground
Line, and the equal Divifions in the Side CD, be fet off front
D towards M, then Diagonals drawn from them to the proper
Point of Distance in this Cafe, will interfect the Side D d in the
Points
Of PERSPECTIVE.
175
Points F, q, k, through which the Lines FK, qQ, Hk, be-
ing drawn parallel to DC (1552) will be true Perſpectives of
the original Lines in the Side of the Cube.
1555. Hence we ſee that the perſpective Parallels IK, OQ,
GH, on one Side, contain right Angles with the perſpective
Parallels KF, Q, HK, on the other adjacent Side, and are
therefore perpendicular to each other, as in the original Cube.
And confequently, the perſpective Sections of the Cube through
thofe Lines, viz. EIKF, OQ9, IGHk, will be all of
them Squares, and parallel to the Side ABCD, as they are in
the Cube itſelf.
1556. If each Side CB, CD, be divided by perpendicular
Lines in the Points 1, 2, 3; then if from thofe Points, Radials
are drawn to the Point of Sight I, they will in the Plains Bbc C,
CedD, be the Perfpectives of the original Lines in the Sides of
the Cube reſpectively (1552). And thus it appears, that to put
a Cube into a Perfpective, is nothing more than to find the Per-
ſpective of a Square for its feveral Sides feparately, and that
thoſe fix perſpective Plans together compleat the Perſpective of
the Cube, which is ufually called the SCENOGRAPHY of it.
1557. The Cube having an Angle placed in Front, or in
fuch Manner that a Diagonal of its Baſe may be perpendicular
to the ground Line, is put into Perfpective by the fame Rules,
and with the fame Eafe, as the other; in this Cafe we ſee the
upper Side of the Cube, and have an equal Profile of two of its
Sides. Thus Fig. 8. is the Scenography of a Cube, whofe Side
is equal to the Square EOFQ in Fig. 4. Then abbg is the
Perſpective of the upper Surface (the fame as Foeq in Fig. 4.)
and EOFQ is the Ichnography of its Bafe, found as directed
(1485) for the Height EI, and Diſtance IZ, of the Eye:
Having thefe two Plans, the Perfpective of the Cube is com-
pleated by drawing the Lines a E, b0, hF, gQ
1558. Becauſe the Sides of the Cube, or parallelopiped in this
Cafe, make an Angle of 45 Degrees with the ground Line,
therefore their Perſpectives ag, eq, will converge to the Point of
Distance Z, (as in the other Cafe they converged, to the Point
of Sight I.) As will all other Lines drawn parallel to them on
the Sides of the original Solid. (1485)
1559.
176
INSTITUTIONS
1559.
If on
any one Part, the Solidity of the Cube be ſuppo-
ſed to be uniformly continued for any given Length Ee, then
the Cube becomes a Parallelopiped, the Perfpective of which is
ſtill determined by the fame Rules, as is evident from the
Figure.
1560. And if any Side of the Parallelopiped be divided by
parallel Lines perpendicular to the Baſe; then by ſetting off the
Diſtance of thofe Lines from e to C in the ground Line to the
Points 3, 2, 1, reſpectively; if a Ruler be laid from each of
thoſe Points, to the Point of Sight I, it will give the Points in
the Line eq of the perfpective Bafe, through which, if Lines
are drawn parallel to ae, the Side of the Perfpective of the
Cube or Parallelopiped, will be divided in the fame Manner as
the Prototype. All which is evident from the Figure, and what
we have before taught (1552). For the Radials divide the Baſe
Lines in this Cafe, as Diagonals did in Fig. 7. where the Cube
was viewed by the Side. But this Divifion is to be made equally
by Radials or Diagonals at Pleafure, as the Reader will eafily
underſtand from the Theory.
CHA P. VIII.
The RATIONALE of the practical METHODS of SCE-
NOGRAPHIC PERSPECTIVE, with the THEORY
of Perspective MEASURES for that Purpoſe.
1561. A
S in fuperficial Perfpective there are two practical
Methods of putting any plane Figure in Perſpective,
viz. one by infcribing them in a Rectangle, (1488) the other
by given Points (1492). So in like Manner we here proceed to
draw the Perſpective of any Upright, Solid, or fuperficial Figure
of three Dimenſions, two different Ways, viz. The first by
inſcribing the given Figure, if regular, în a Cube, or proper Pa-
rallelopiped; the Second, is by finding the Perſpective of the ſe-
veral Lines which are formed by the Interfection of Planes, com-
pofing

Fig.3
m
Fig 7
n
h
K
Fig.1
2
BAB
9
a
b
h
d
Fig.10
Fig.4
100
m
d
b
a
Fig. 8
3
Fig. 2
Plate III.
Fig.5
Fig.6
a
D
F
E
B
A
Fig.n
Fig.9
b
Of PERSPECTIVE.
177
pofing the Superficies of the given Figure; and theſe Lines con-
nected, from the Scenography thereof. This fecond Method is
general for all figured Bodies, regular, or irregular.
1562. As a Specimen of the firft Method, let it be required
to put a PRISM into Perſpective; to do this, we may conceive
it circumfcribed by a hollow Parallelopiped, whofe Bafe and
Height are equal to thofe of the given Prifm; then draw the
Scenography or Perſpective of this Parallelopiped, which let be
AB bade CD, the Bafe of which Bb c C is alſo the Perſpective
of the Baſe of the Priſm; and fince the Edge of the Prism tou-
ches the Surface of the Parallelopiped, let the Diſtance thereof
be meaſured, and ſet off on the ground Line from B to 4; then
a Ruler laid from Z to the Point 4, will cut the Side B b in I:
Draw IE parallel to A B, and it will be the Perſpective of the
Perpendicular let fall from the Edge or Angle of the Priſm on
the Rafe. Therefore drawing EF parallel to AD, it ſhall be
the Perſpective of the Edge of the Prifm. Then drawing the
Lines BE, bE, and CF, F, they compleat the Perspective
of the Priſm in the Figure BE bc F C, which is fhaded for the
Sake of DiftinAneſs.
C
1563. After the fame Manner, a Pyramid, Cone, Globe, &c.
may be drawn in Perſpective, which cannot be difficult to thoſe
who underſtand the foregoing Principles. What I here fpeak
of relates to Matters merely Perfpective; but to give a natural Re-
lievo to conical and ſpherical Figures, whether Convex or Con-
cave, requires more a ſkilful Management and Difpofition of Lights
and Shadows, as we ſhall hereafter ſhew.
1564. The fecond Method for determining the Scenography of
Solids, is by finding the Perſpective of Lines which are formed
by their interfecting Planes. To underſtand the Reafon of this
Method, the following Theorem muſt be premiſed. Let ID be
a given Line, placed on the ground Line CD (Fig. 10.) then from
any two Points A, B, in the horizontal Line, draw the Radials
AD, AI; BD, BI; and parallel to CD draw any Line LK,
cutting the Radials AD, BD, in G and E; from thofe Points
in H and F. This
done, I fay, the Line GH is equal to the Line EF. The Demon-
ftration is from fimilar Triangles (656.)
draw G H, EF, cutting the Radials AI, BI,
VOL. II.
A a
For
178
INSTITUTIONS
For
AG: AD:: GH: DI.
And
BE:BD:: EF: DI.
Alfo
AD: BD:: AG: BE.
Therefore
ADXBE BDXAG.
Therefore alfo DIxEF=DI × GH. (652.)
Confequently
EF=GH. 2 E. D.
1565. If FD (in Fig. 9.) be made equal and parallel to ID
(in Fig. 10.) and from the Point of Sight I, the Radials IF, ID,
be drawn; then if any Line LM be drawn parallel to the common.
ground Line DCD, interfecting the Radials ID, AD, in the
Points L and M; and LG and MO be drawn parallel to F D
or ID, they will be equal to each other alfo, as is demonftrable
in the fame Manner as the above Theorem,
1566. Therefore let BL KI be the Perfpective of the Baſe
of the Solid (Fig. 9.) drawn for an oblique View; and neither
Side or Angle in Front. Then when the oppofite Sides of the
Solid are equal and parallel, and one Angle placed in the
ground Line at B; if AB be the Height of the Solid, then
through L draw the Radial ILD, meeting the ground Line in
D; on the Point D draw D F equal and parallel to A B, and
join IF; on the Point L erect the Line LG parallel to DF in-
terfecting the Radial IF in G; and join A G; then is A BLG
the Perſpective of one Side of the Solid. In like Manner you
find the perfpective Lines IE, KH; and drawing A E, EH,
and GH, the Perfpective of the whole Solid is compleated.
1567. Or thus more univerfally, let ID be the Meaſure of
the Solid's Height, and placed on the ground Line at D (Fig.
10.) and make any Triangle AID at Pleafure, whofe Vertex
A is in the horizontal Line; then from the Angles of the per-
fpective Baſe I, L, K, (Fig. 9.) draw Lines IC, LM, KP,
interfecting the Radial A D in C, M, and P, on which Points
erect the Lines CN, MO, PO, and they will be the perfpec-
tive Heights of the Solid at the Angles I, L, K; that is, CN
=IE, MO= LG, and PQ=KH; (by 1564) which
Heights are all determined by the parallel Lines EN, GO,
and HQ, (in Fig. 9, 10.)
1568. Hence it appears, that if any Line be perpendicular to
the ground Plane, and its Diftance from the ground Line, and
Height,
Of PERSPECTIVE.
179
Height, be given; then its Perfpective may be eafily drawn.
For the perſpective Seat of any given Point on the ground Plane
is found by (1492) and the Perfpective of any given Height
or Line is found for that Seat or Point, by (1494.)
1569. If the given Line be perpendicular to the ground Plane,
but placed above it, at a given Height; then that Height is to
be added to the given Length of the Line, and both confidered
as one Line; whofe Perſpective is to be found for the Whole
firft, and then for the lower Part or Height above the Plane;
then this latter fubducted from the Whole, leaves the Perſpective
of the given Line.
1570. A Line of a given Length, Pofition, and Inclination
to the ground Plane, if placed upon it, is thus put into Per-
fpective. First let the Perfpective of its Seat on the ground
Plane be found (1492) then from the elevated End let fall a
Perpendicular to the ground Plane, and find the Perfpective of
that Perpendicular (1494). Laftly, draw a Line from the
perfpective Seat to the Top of the perfpective Perpendicular, and
that will be the Perfpective of the given Line.
1571. For Example; fuppofe (c) the Seat of the Line in the
perfpective Plane (Fig. 7.) and K the Seat of the Perpendicular,
then through K draw the Radial IC, cutting the ground Line
in C; upon the Point C erect CD equal to the Perpendicular,
and draw ID; and from the Point K draw KF parallel to CD;
and it will be the Perfpective of the faid Perpendicular; then
join c F, and it will be the Perſpective of the Oblique Line, as re-
quired.
1572. If the Line be wholly elevated above the ground Plane,,
then its Perspective may be drawn two Ways; for, firſt, you
may continue the given Line to the ground Plane; and then
finding the Perspective of the Whole, and its Parts, that of the
given Line will be known of Courſe. Or; fecondly; from each
End of the elevated Line let fall a Perpendicular to the ground
Plane, and then find the Perfpective of thofe Perpendiculars, the
Summits of which being joined, give the Perfpective of the
Line, as required.
1573. Having thus fhewn how all Lines may be drawn in
Perfpective of any given Length, Pofition, or Elevation on, or
above the ground Plane, it follows, that all Bodies of what Form
A a 2
Of
180
INSTITUTIONS
or Figure foever they may be, are drawn in Perſpective at the
fame Time, and by the fame Rules. Since the Lines formed
by their various interfecting Planes will be given in Length, Po-
fition, and Inclination from the Nature of the Solid; their Per-
fpectives therefore are eaſily drawn by the Rules above (1492 to
1495) and the Extremities of thofe perfpective Lines being
joined, give the Scenography or Perſpective of the Body re-
quired.
•
1574. It farther appears how, by the menfural Triangle AID
(Fig. 10.) the perſpective Heights of Men, Houſes, Trees,
Beafts, Birds on the Wing, &c. may all be duely adjuſted and
meaſured; and their proper Diminutions proportioned to their
Distances from the ground Line in Landſcapes, Prints, and alk
perspective Views. Thus fuppofe ID were a Tree on the firſt
Ground of a Landſcape, then the Radials AI and AD deter-
mine the Diminution of Trees of the fame Height in all remote
Diſtances from the ground Line. So that at C the Height is
CN; at P, it is PQ; at G, it is GH, and fo on, till it va-
niſhes in the Horizon at A.
1575. But by fimilar Triangles we have CN: ID:: AC:
AD; and PQ:ID::AP: AD; and GH:ID::AG:
AD, &c. Therefore CN : GH:: AC:A G ; or the Linear
Dimenſions of Objects, in a Piece of Perſpective or Landſcape,
are directly as the Distances from the Point of Sight.
1576. Again, the Diminutions of the Heights of Objects in
a Landscape, will always be as the Diſtances from the fore
Ground, or ground Line; for AD AC: ID-NC::
AD-AG:ID-HG (649.) That is, ID- NC: ID
-GH:: CD: DG.
1577. Hence it appears, that while Objects are at the ſame
perpendicular Diſtance from the perſpective Plane, their per-
ſpective Appearance or Magnitude will always be the fame;
and therefore if a Bird, or any other Body afcend or defcend in
a perpendicular Direction through any Height, it will ftill have
the fame Dimenfions in every Part of the Afcent or Defcent on
the perfpecuve Plane, and muſt be ſo repreſented in Drawing,
as we obferved before. And thus we have premifed every
Thing necefiary in the THEORY and PRACTICE of common
PERSPECTIVE: We now proceed to illuftrate the fame by a
Variety
Of PERSPECTIVE
181
Variety of uſeful Examples; with Obfervations thereon, as
Occafion requires.
CHA P. IX.
The RULES of PERSPECTIVE, illuftrated by a Va-
riety of ufeful EXAMPLES, with proper Obferva-
tions thereon relative to a general PRAXIS.
1578.
[PLATE III. Of PERSPECTIVE.]
A$
As we
S we have given the fundamental Rules for drawing
all Kinds of Superficies and Solids in Perfpective; it
remains now, that we illuftrate the fame by proper Examples,
with fuch farther Obfervations and Inftructions as may tend to
facilitate the Practice of thofe Rules to the young Deſigner.
1579. In Fig. 1. we have Inftances of a long Parade, gravel
Walks, grafs Plats, a Canal, &c. all put into Perſpective ac-
cording to the Rules laid down (Chap. III.) by drawing the pro-
per Radials and Diagonals from the Point of Sight (i) and Di-
ftance (z); as alfo the Scenography of a whole Row or Street of
Buildings on one Side, and of Trees on the other, by the Rules
contained in Chap. VII. Here we may obſerve, that in order
to preſerve Uniformity, and Identity of Meaſures, in any Per-
ſpective or Landſcape, there ought to be but one Point of Sight,
and Distance; for if the Breadth or Diſtances of the Houfes were
determined from the Point of Diftance (z) and the Diſtances of
the Trees from another (y) there could be no Meaſure found,
nor any Judgment formed, of any Diſtances at all in fuch a
Piece.
1580. To the fame Point of Sight (i) the fame Rules of Per-
ſpective are applicable for Objects on all Planes in any Situation
about the Eye; for the Elevation of the Plane above the Eye, and
the Elevation of the Eye above a Plane, are the fame Thing in Per-
ſpective; and confequently the Diminution of Objects in the
Air follow that fame Rules as thofe on the ground Plane. Thus
Birds
182
INSTITUTIONS
Birds (Fig. 1.) diminiſh as they recede from the fore Ground of
the Sky (if we may uſe that Expreffion) to remoter Diſtances,
and at Length vaniſh at the Point of Sight in the horizontal
Line.
1581. Thus alfo (in Fig. 2.) which is a direct perfpective.
View of the Infide of a Room, the CIELING above, the Floor
below, and each Side, are to be confidered as ſo many Planes
put into Perſpective from the given perpendicular Diſtance of the
Eye from each (called the Height of the Eye) and the given Di-
ftance (iy) from the Front of the Room. For fince the Inter-
fections of the fide Planes with the Cieling above, and the Floor
below, are fuppofed perpendicular to the ground Line, or per-
fpective Plane, they will converge to the Point of Sight (i) ac-
cording to (1479) and thereby terminate the Perſpective of the
whole infide View; which, indeed, cannot be difficult, as it
contains the plaineft Cafes of direct, fuperficial, and Scenographic
Perspective.
1582. In Fig. 3. you have a fide View of the PEDESTAL Of
a Pillar, whofe Front being formed by the Meaſures proper to
the Order, you proceed to form the Perſpective of its Bafe and
Capital for the given Height and Diſtance of the Eye, as in the
Figure. Then the Scenography of the Whole is finiſhed by the
Rules in Chap. VII. and VIII.
1583. This Figure is added to fhew how the Pillar is to be
placed on its Pedeſtal in Perſpective; for it is eafy to obferve,
that the Axis or central Line of the Pillar muſt be perpendicu-
larly over the perfpective Center (c) of the Bafe, and comfe-
quently the Perpendicular (cd) is that Axis or Line which de-
termines the perſpective Pofition of the Pillar on its Pe-
deftal.
1584. Fig. 4. contains the Perſpective of two PILLARS
or COLUMNS in Front; one in a Direct, the other in a Side
View; the geometrical circular Bafe (a b d) being given, the per-
Spective elliptic Baſe of each is determined by (1498) and then the
Pillars themſelves are drawn or defined by the two parallel Lines
which are perpendicular to the ground Line, and touch the El-
lipfis on either Side.
1585. Hence it appears, that a Colonade of Pillars, viewed in
Front, will not have all an equal perfpective Magnitude, or they
muft
Of PERSPECTIVE.
183
muft not be drawn all of an equal Size; but that which is near-
eft to, or directly before the Eye, has the leaft perfpective Mag-
nitude; and others appear larger on the perſpective Plane, as
they are farther removed from the Eye or Line of direct View.
However paradoxical this may feem, it is thus eafily demonftra-
ted The Ellipfis (gchk) of the fame Circle on the fame
ground Line is greater or lefs in Proportion to the Obliquity or
Diſtance of the View, as is evident from the Figure here, but
more ſo in Fig. 4, 5. of Plate II. And fince the Sides of the
Pillars are Tangents to thofe Ellipfes, and perpendicular to the
ground Line, they muſt be at a greater Diſtance in an oblique
than in a direct View, and therefore give a greater perfpective
Magnitude of the Pillar.
1586. It may be proper alfo to give an optical Demonftration
of the Truth of this Pofition; let (ad) be the Diameter of the
geometrical Bafe; this in the direct View fubtends the greateſt
Angle (aid) under which the Baſe of the Pillar (gchk) can
appear, and the perfpective Diameter is gh; but ac and g h are
the fame in both Views, and a Pillar on that Diameter would
be the fame in both Cafes. But the apparent Magnitude of a
Pillar is in Proportion to the Angle which is fubtended by that
Diameter that is perpendicular to the vifual Ray (i c.) Suppofe
(ef) to be that Diameter, and draw the Rays (ie) and (if)
which produce to the ground Line at (m) and let (i e) crofs it in
(1) then it is evident the Diameter (ef) will eclipſe from the
Eye a Part of the ground Line equal to (lm) in the oblique
View; and therefore the perfpective Magnitude of the Pillar
will be as much greater here than it is in the direct View, as
(m) is greater than (a d.)
1587. Hence it appears, that though it is always the Prac-
tice of Architects to reprefent all Pillars in a front View of an
equal Size, yet the Abſurdity of fo doing is as great as it would
be to give the fame Objects the fame Heights at different Di-
ftances from the ground Line. And thoſe who vindicate fuch a
Proceedure, muſt be looked upon as either very ignorant, or elſe
prevaricating with the Rules of Art in a moft licentious De-
gree.
1588. The Axis of a Pillar in a direct View is equidiftant
for either Side; but it is not fo in the oblique View; for as it
rifes
184
INSTITUTIONS
rifes from the Center of the elliptic Bafe, it will, in that Cafe,
be nearer to that Side next the Eye than to the other, as is evi-
dent by Inſpection of the Figures. The fame Thing is to be
underſtood of any other regular Body in an oblique View.
1589. In Fig. 5. we have two PARALLELOPIPEDS, or ſquare
Columns, put into Perſpective, on their proper Bafes (abcd,
eƒgh) which are ſuppoſed to be fquare; one in a direct View,
in which only one Side can be feen; the other in an oblique View,
fhews the fame equal Side in Front, and a Profile of the other
Side (defh) next the Eye at (i) the Point of Diftance being
(z). Theſe are Cafes fimilar to thofe before obſerved of the
Cube (1546 to 1549) in the fame Points of View.
1590. The Manner of placing a SPIRE OF STEEPLE on the
Top of the Tower of a Church, &c. in Perſpective, is ſhewn in
Fig. 6. The Spire is fuppofed to be a Square Pyramid, the Per-
ſpective of whofe Bafe is (a b de) (being a fide View, like that in
Fig. 1.) Here alſo you can fee only two triangular Sides (viz.'
bgd and a gb) fo far as they rife above the Top of the Tower.
If from the Center (c) of the perfpective Bafe a Perpendicular
be raiſed, and its perfpective Height (cg) proper to the perpen-
dicular Height of the Pyramid, than (g) will be the Vertex of
the Spire, which will be much nearer to that Side of the Tower
next the Eye, than to the other, for the Reafon before-men-
tioned.
3
1591. To place a TABLE, CHAIR, &c. on a Floor, as
(abcd,) in true Perſpective, is no difficult Thing by the Rules
prefcribed. For Example, let ad be 10 Feet, and fuppofe the
Table 4 Feet long, and to ftand 3 Feet from the Side of the
Floor (a b) and 1 Foot from the ground Line (a d) as in Fig.
7. is evident; let the Height of the Table be 21 Feet, and its
Width 16 Inches, then on (a) crect the Perpendicular (a m)
and from (a) to (e) fet off 2, the Height of the Table, and
draw (ei) to conſtitute the menfural Angle (cia). Then draw
Parallels to the ground Line at the Distance 12, and 3 Feet;
they will cut the Radial (i a) in (g and h) then the Perpendicu-
lars gf, kl, will be the perſpective Heights of each Side of the
Table; all which, with every other Circumftance, is directed
by the Rules in Chap. VIII.
3
1592;
Of PERSPECTIVE.
185
1592. As to the CHAIR, if the Perfpective of its Bafe be
found, then the Height of the Seat and Back are in the fame
Manner determined by an Angle of Meaſures, or by the com-
mon Rules. The Bottom or Seat of the Chair not being a
Square but a Trapezium, will have its Perfpective determined by
the Point of Sight (i)-and accidental Point ≈ (1483) with re-
gard to the two Sides which are parallel; but the other two which
are inclined converge to a Point before they arrive at the hori-
zontal Line, which is fuppofed at an infinite Diſtance from the
Eye, and where only parallel Lines can meet, nor they neither
but in Perſpective.
1593. If a Box is to be opened in Perfpective, then the Pofi-
tion with regard to the ground Line is to be confidered. If the
End be parallel thereto, then the Corner of the Lid, in opening,
will defcribe a Circle in Perſpective, as (a df) Fig. 8. (1469,
1504) and therefore if a Circle be deſcribed on each End of
the Box, with a Radius equal to the perfpective Width of the
Box, and the given Angle to which the Lid is opened, be fet
off from (a) to (b); then you will have the four angular Points
of the Lid by which it may be drawn as required.
1594. But if the Side of the Box be placed parallel to the
ground Line, then the angular Point of the Lid, in opening, will
deſcribe a Semi-ellipfis in Appearance on the perſpective Plane,
as (abdf), fuppofing the End of the Box contiguous to the Side
of a Room. Here the elliptic Angle (acb) is determined from
the given circular Angle in which the Lid of the Box ftands.
open, as directed (1498), or if the Angle (b) has its perpendi-
cular Height meaſured, and its Perſpective found (1570) the
Perſpective of the Lid or Cover is thereby determined.
1595. Again, if Doors, Windows, &c. are to be put into
Perſpective as they ftand open in any given Angle; the Rules
for doing it are ſtill the fame. Thus fuppofe the Door be 2
Feet wide AC (Fig. 9.) then on the Center C defcribe the
Semicircle ADF, upon which the angular Point of the Door
moves; ſuppoſe the given Angle to which it is opened be ACB;
then if on the perſpective Floor (hefd) you find the Perſpective
of the Semicircle ADF; and therein make the elliptic Angle
(a cb) correspondent. to the given Angle ACB in the Proto-
type; you will have the Pofition of the Point (b) on which you
VOL. II.
B b
raife
1186
INSTITUTIONS
raiſe the perpendicular Perſpective (bo) of the given Height of
the Door (1568) and then, laftly, by forming the Perſpective
of the Door-place or Aperture (am ne) in the Side of the Room,
you will have the four angular Points of the Door determined,
by which it may be readily drawn in the Pofition required.
1596. In the fame Manner the Windows on the upper
Part
of the Room may be drawn. But it is to be obſerved, that fince
the Angle is given in which the Door or Window is opened,
and alſo the Diſtance on, and Height, above the ground Plane,
the Angles of the Door, or Cafement, may be moſt eafily de-
termined by the Method of Perpendiculars as directed in
(1567.)
1597. If a Perfon in a Room, on one Side of a Street, views
through the Windows the Buildings on the other Side, the
Window then becomes the perſpective Plane or Table for the
orthographic Delineation of whatever appears in Front on the
other Side of the Way, (1649,) as is here reprefented in Fig.
IO.
1598. The whole Scenography or perſpective View of the In-
fide of a Shop is prefented in Figure 11. in order to compare it
with the Picture made thereof by a Speculum or Lens which we
fhall hereafter have Occafion to take Notice of.
CHA P. X.
The PRINCIPLES of Catoptric and Dioptric PER-
SPECTIVE confidered with regard to VIEWS, PIC-
TURES, LANDSCAPES, &c. formed by MIRRORS
and LENSES; with the RULES for exhibiting them
in any required Proportion to the OBJECTS,
1599.
E have already confidered fo much of Catoptric Per-
Spective as relates to reflecting Planes; but it was
neceffary to treat of Scenographic Perspective before we could pro-
perly handle the Doctrine of Optical Perspective univerfally as con-
cern'd in the Formation of Images, Pictures, Landscapes, and all
Kind
Of PERSPECTIVE.
187
Kind of Views by reflected and refracted Light from and through
Convex and Concave Glaffes of every Kind.
1600. The Subject we are now entering upon, though the
moft effential and exquifite Part of the Science, has not (that
we know of) been touched upon by any Author on Perfpective,
and therefore we fhall be the more particular and explicit on this
Head. In order therefore to determine how far the Picturesque
IMAGES of OPTICAL GLASSES agree and coincide with the
Drawings made by the Rules of commm Perfpe&ive, it must be re-
membered, that the Linear Dimenfions of an Obje& and its Per-
ſpective are proportioned to their Diſtance from the Eye (1468)
that is (in Fig. 1. Plate II.) RN: rn::IZ: Ii:: AB: ab
::AD: ad.
1601. Now fuppofe A G were an open Cube or other rectan-
gular Body whofe oppofite Sides are equal, and let the Side E G
in Front be called the Profcene; the hinder Side AC the Post-
fcene; the Sides E B, HC, the Lateral Scenes; the ground Plane
AH, the Primary SCENE, and CB the Erial Scene, if placed a-
bove the Eye; or Secondary Scene, if below it. Then conceive a
perſpective Plane placed contiguous to the Profcene, on which
the whole Scenography of the Body will be viewed in Perſpective
by the Eye at I.
1602. Put the linear Dimenfion of the Profcene E H or EF
P, and the Perfpective of the Poftfcene ad or abp; alio
let their Distances from the Eye be Ii D, and IZ = d.
Then we have Pp::d: D.
d: D. And this Analogy will hold for
any Situation of the perſpective Plane, between the Eye and the
given Cube.
1603. If inſtead of the Eye, a Convex or Concave Speculum
were placed at I, then an Image will be formed of the faid
Cube or Body, and the linear Dimenſions of every Part will be
proportioned to the Diſtance from the Glafs, as we have
fhewn (1310) Then, if we call AB or AD an Ob-
ject, and put it equal to O, and its Image = i; we ſhall
have O:i:: df = focal Diftance of the Image; and therefore
fin all Kinds of Speculums.
di
O
Bb 2
1604.
188
INSTITUTIONS
1604. In a CONVEX MIRROR, we had
dr
therefore
dr
2d + r
=f(1291)
id
=
; from whence we get Or
2 id +
2 d + r
ir, which gives this Analogy, O:i:: 2d+r: r. Or, the
Proportion of the Length or Breadth of the Poftfcene is to that of the
Image thereof, as twice the Distance added to the Radius, is to the
Radius of the Speculum.
16c5. The Profcene being equal to the Poftfcene (1601) its
Length or Breadth EF or EH will ftill be denoted by O, but
its Diſtance being lefs, let that be = D; and becauſe the Image
will be larger in Proportion, put it I, and proceed as be-
fore; we have Or 2 ID + Ir, and the fame Analogy O:
I :: 2 D + r:r.
=
1606. But becauſe in both Cafes Or is the fame, therefore
we have 2 id + ir = 2 ID + Ir, and therefore I :i:: 2 d
From hence we have the Ratio of the linear
+ r: 2 D + 1.
Dimenſions of the
Profcene to that of the Poftfcene; and con-
fequently the Optic Scenography of the Solid is thereby determined,
and its Difference from that in common Perſpective.
1607. In the CONCAVE Speculum,
=
becauſe
dr
2d-
=f,
(1291) we have Orir 2 di Ir-2 DI; whence
1:i::r 2 d: r-2 D. From whence it appears, that when
the Radius of the Speculum exceeds twice the Diſtance of the
Objects, then the Images will be pofitive or behind the Glafs;
but otherwiſe negative or before it, agreeable to (1315, 1316.)
1608. In Cafe of a LENS, it is every where O :i:: d:f, and
id
о
= f(1340). Alfo for a double and equally Convex Lens we
dr
id
have ƒ==(1387) and therefore Or = id—ir
=ID—Ir; whence we have I:i::d-r: D—r; or in
the Scenography of the Image of fuch a Lens, the linear Dimenfions
of the Profcene and Poftfcene are as the Differences between the Di-
ftances and Radius of the Lens respectively.
1609. After the fame Manner the Scenography of Images for all
different Forms of Lenfes may be found, and compared with
thoſe
Of PERSPECTIVE.
189
*
thofe of common Perfpective. And fince, in all, the Terms of
Compariſon confift of the Sums or Differences of the Radius
and Equimultiples of the Diſtances; therefore while the Radius
of the Glaſs bears any confiderable or fenfible Proportion to the Diſtan-
ces of Object, the optical SCENOGRAPHY of their IMAGES will dif-
fer more or leſs from the common Perſpective thereof.
1610. But it is evident, when the Diſtances are fo great that
the Radius of the Glafs bears no fenfible Proportion thereto, it
will then vaniſh out of the Terms of the Compariſon, and then
the Analogies
Convex Mirror, I:i:: 2d: 2D::d : D.
become in the Concave Ditto, I:i:: -2d: - 2 D:: d: D.
Convex Lens, I:i:: d: D.
1611. Confequently, in all fuch Cafes the optical Scenogra-
phy of the Image is the very fame with that of common Perfpec-
tive on the tranſparent Plane; for by all the Glaffes, we have
Iid: D, and on the perfpective Table, we have P:::
d:D (1468) therefore I :i:: P: p. Whence it appears,
that all the Parts in the optical and perſpective Scenography are per-
fectly fimilar, and that, therefore, the IMAGES of OPTICAL
GLASSES in fuch a Cafe, are PICTURES, PROSPECTS, or
LANDSCAPES in true Perspective.
1612. Hence the excellent Ufe of a Convex Speculum, in exhi-
biting to the Eye a genuine perſpective View of all diftant Objects,
as a perfect Copy for the Artift to draw by. It prefents
him with an inſtantaneous Conftruction of the perfpective Sceno-
graphy of the interior Parts of a Room, and all its Furniture,
Tables, Chairs, Scrutores, Book-cafes, Pier-glaffes, Pictures,
&c. juft as they ought to be drawn. An Inftance of its Ufe
in this reſpect, is Fig. 11. Plate III. which is a perſpective View
of a Mathematical SHOP, with its Counters, Spheres, Globes, Air-
pumps, Teleſcopes, Glafs-cafes, &c. actually delineated from the
perspective PICTURE of a CONVEX MIRROR (1598.)
*
1613. Another great Uſe of ſuch a Speculum, is to exhibit the
true Perfpective of an Object diminiſhed in any given Degree.
For, becaufe Or = 2id + ir (1607) it is Or-ir=
2 id,
190
INSTITUTIONS
2 id, and therefore d =
For Example, fuppofe.
Orir
2 1
i
•
4 I; and let the Radius of the
it required to draw a Microſcope, Air-pump, &c. 4 Times lefs
than the Life; then O: i
Speculum ber 12 Inches; then d
=
48- 12
2
= 18 In-
ches, the Diſtance from the Mirror to make the Image 4 Times leſs
than the Object.
1614. Since the Image is larger in Proportion as the Radius
of Convexity is fo, it appears how excellently well adapted fuch
Mirrors are, when very large, for exhibiting the moſt perfect
LANDSCAPES of diftant SCENES, whether Gardens, Fields, Lawns,
Woods, Mountains, Vales, Rivers, Sea, &c. with all the natural
COLOURS, LIGHTS, and SHADES, MOTION, and every other
Incident which can tend by this perſpective Miniature, to improve
and out-vie even NATURE itſelf.
1615. The fame Things may be faid of a Concave Mirror,
with regard to the Form and Proportion of the feveral Parts of
the Image, and the juft Perfpective of the Whole, but then the
inverted Pofition of the Picture before the Glafs, and the Incon-
venience of obſerving it, renders it not fo uſeful in the Arts of
drawing and defigning, as that of the Convex Form.
1616. But then it has this moft entertaining Property of re-
folving the perſpective Picture, or Landſcape formed by the Con-
vex Mirror into its Original or Prototype, and gives each Part
Diſtances, Size, and Situation. In this Cafe it is ſuppoſed that
the Radius in both Speculums is the fame. Thus, for Illuftra-
tion, fuppofe a View of 7 Miles round St. PAUL'S were drawn
from a Convex Mirror of 10 Feet Radius, that Drawing held
at the fame Diſtance before an equal Concave Mirror as it appear-
ed to be behind the Convex One, the faid Concave would revert
the Landſcape, and prefent the Eye with a delightful Fiew of the
large City of London, and County about it, as large as the Life,
and at the fame Diflance in every Part, as appears to the naked
Eye. The Reafon of all which, is very evident from the Theo-
ry we have above premifed. Hence the great Ufc of Concave
Mirrors in viewing perſpective Prints, in the portable Camera
OBSCURA.
1617.
Of PERSPECTIVE
191
1617. From the fame Theory (1608) it appears, that a
Convex Lens does alſo preſent us with the fame perspective Figure
or Image of diſtant Objects in the Focus, which as the Radius
is longer, will be larger in Proportion; and therefore, in a
real Camera Obſcura, when fuch a Lens is applied in a ſcioptric
Ball and Socket, you view upon a Screen, at a proper Diſtance,
the Scenography of Buildings, and a LANDSCAPE of every rural
Scene, fo heightened by Colours, and animated by Motion, as
justly excite our Admiration, and we readily pronounce the
Pencil of Nature perfect, and all her Paintings inimitable.
1618. Upon the Whole we may conclude, that as all artifi-
cial Paintings are but Copies of Nature, the more they ap-
proach to, and are regulated by the Art of Perſpective, the more
natural and valuable they will be, and Beauty and Harmony will
fo much the more evidently appear; for we have ſhewn in the
Theory (1292) that in all Nature's Painting the ftricteft barmo-
nical Proportion is obferved; and confequently NATURE is all
PERSPECTIVE and HARMONY.
CHA P. XI.
Of the PERSPECTIVE APPEARANCE of OBJECTS
on INCLINED PLANES.
[PLATE IV. Of PERSPECTIVE.]
1619. Ware fuppofed to be taken on Planes perpendicular
WE have hitherto treated of ſuch perſpective Views as
to the ground Plane; but as it will, on many Occafions, be ne-
ceffary to confider them on Planes or perſpective Tables inclined
in any Angle thereto, I ſhall here deliver the Theory and Rules
for that Purpoſe.
1620. Let A DNR be a perfpective Table inclined to the
ground Plane BCGK in a given Angle (Fig. 1.) A B C D is
a Rectangle viewed upon the Table by an Eye at I, at the Height
ĮH, and Diſtance from the ground Line HF. Biſect BC in
E,
192
INSTITUTIONS
E, and draw EH, and I i parallel thereto, then will (i) be the
Point of Sight on the Table, and RY, drawn through it paral-
lel to AD, will be the horizontal Line, in the fame Manner as
in the upright Plane (1465).
=
1621. Produce CK and E F indefinitely, and make the An-
gle IMH Angle of the Plane's Inclination; then will I M
be parallel to i F. Draw the Radials i D, ¿A; make KL-
HM, and join IL, which is then parallel to iD. From the
Points B and C, draw the vifual Rays IB, IC; they will in-
terfect the Radials i A, iD, in the Points S and P, then join-
ing SP, the Area ASPD will be the Perfpective of the Rect-
angle ABCD; the Demonftration is the fame as was uſed for
the upright Table (1475, &c.)
1622. But for a practical Method, the following Theorem in
this Cafe may be preferable. In the ground Line A D produ-
ced, take DO DC; and in the horizontal Line RY, take
¿Y = iI, the Diſtance of the Eye from the Table; draw YQ
parallel to iD, and draw YO, which will interſect the Radial
iD in p, the fame as the Point P found by the other Theory.
For in the fimilar Triangles L CI and DCP, we have IC:
DC::LI: DP; and in the fimilar Triangles YOQ and
pOQ, we have QO: DO: YQ: Dp. But the three firſt
Terms of each Analogy are feverally equal to each other; for
LCLD + DC QD + DOQO; and DC-
DO, by Conftruction; alfo LI= iD=YQ. Therefore,
alſo DP = Dp, and conſequently the Points P and p coincide
or are the fame. 2. E. D.
1623. Whence the practical Rule for finding the Perfpective
of a given Point C on the ground Plane, upon a given inclined
Table is this. From the given Point draw a perpendicular CD to
the ground Line, and draw the Radial iD; in the ground Line A D
produced, take DO=DC, and iY = iI, and draw YO,
which will cut the Radial iD in the Point P, the Seat or Perspective
of the given Point C, as required.
·
1624. The Diſtance of the Eye from the Plane is equal to its
perpendicular Diftance from the ground Line, together with
the Cotangent of the Plane's Inclination; for i I=FM =
FH + HM, but H M is the Tangent of the Angle HIM,
the

Y
Z
T
W
I
V
Plate IV
D
Fig.1
10
B
N
E
A
R
S
LA
F
D
O
I
Fig.2
H
M
R
770
B
K
L
M
N
Fig.4
G
F
d
Fig.3
H
C
10
D 20
30
40
20
Fig.7
30
40
B
50%
бо
60%
A
Fig. 8
a
K
D
E
H
C
P
19
Fig.5
G
E
50
60
E
H
Fig. 6.
TU
0
H
B
Of PERSPECTIVE.
193
the Complement of the Inclination HMI to the Radius IH the
Height of the Eye.
1625. Hence it will be eafy to draw the Perfpective of any
given Object upon an inclined Table BCTV (Fig. 2.) Let
the Object be a hollow Cube whoſe Side is M N O P; ſuppoſe it
to touch the Plane on its upper Part; and that the Inclination is
equal to the Angle W CV. I is the Point of the Sight on the
Plane, and X, Z, the Points of Diftance, as in the upright Plane.
Then, becauſe the Diſtance of the Cube from the Table at Bottom
is CP, if we draw the Radials I C, I B, and the Diagonal Y P, the
Point of Interiection F will be the Seat of the Point P; and
drawing the Line YO, it gives the Point H for the Seat of the
Point O. Therefore drawing EF and GH parallel to the
ground Line B C, the Area EFGH will be the Perſpective of
the Square Baſe of the Cube, by the Theory (1487.)
1626. The inclined Table being juft the Width of the Cube,
we have A D = MN; continue AD to S, which is now to
be confidered as a ground Line, for the Top of the Cube; there-
fore taking AR MN, and drawing RZ it will cut the Ra-
dial IA in K; and then drawing KL parallel to AD, it will
give AKLD for the Perfpective of the Top of the Cube.
Laftly, by joining A E, FD, GK, and HL, the whole Scens-
graphy of the Cube will be compleated.
1627. From this Procedure for the Cube, I prefume the Me-
thod of finding the Perſpective of any Points, Lines, Superficies
and Solids, will not be difficult to the young Drafts-man; for
this one Example includes them all, either contiguous to, or at
a Diſtance from the Table; and parallel and perpendicular to
the ground Plane, ground Line, &c.
1628. I need not obferve, that the Sides of the Cube in Front,
are not Squares in the Perſpective, as they are in upright Planes,
but Trapeziums; for EF is less than AD; and G H leſs than
KL. And this will be the Cafe of all Lines and Superficies not
parallel to the perspective Table.
1629. The Inclination of the Plane may be fuch that the Per-
fpective of a Circle on the ground Plane fhall be a Circle alío. Let
DE (Fig. 3.) be the Diameter of the given Circle; HD the
Diſtance, and HI the Height of the Eye; draw the Rays ID,
IE; and let the Plane HF cut them fubcontrarily in e and d;
VOL. II.
(1510)
Сс
194
INSTITUTIONS
(1510) then fince all the Angles in the Triangles DHI, and
EHI, are known; and by Suppofition the Angle HD e=
HdE, and the Angle H Ed He D; therefore fince He D
= IHe+HIe; we have HeD-HIe 1 He, the Incli-
nation of the Plane HF required for that Purpoſe.
=
1630. Hence alſo it is eaſy to affign the Inclination of a Plane
wherein the Seat or Perſpective of a given Point fhall be the
fame as in the perpendicular Poſition. Thus (Fig. 2.) ſuppoſe
CV the upright Plane, the Eye at Y views the Point O thereon
at (a); on the Center (C) with the Distance C a, crofs the Ray
YO in (b,) and through that Point draw CQ which is the Po-
fition of the Plane required, fince Cb Ca.
1631. Let CW be perpendicular to the Ray YO, then is
the Angle VC WaOC; and confequently the Angle a Cb
=2aOC.
1632. Since the right Line CO occupies the fame Space Cb,
Ca, on the inclined and upright Planes, it will follow that any
Line CP at a lefs Diſtance will take up a lefs Space, and any
Line CX at a greater Diſtance will take up a greater Space on
the inclined Plane, than on the fame in the erect Pofition. Much
more might be faid on this Head, which we fhall leave to the
Reader's Invention, having premiſed the Principles, which is
here all that is propofed.
CHA P. XII.
Of Theatrical PERSPECTIVE; or, the RULES of PER-
SPECTIVE applied to the SCENERY of a THEATRE.
[PLATE IV. Of PERSPECTIVE. }
1633. THE Wed in PERSPECTIVE. For as we are to be
•THE
HE whole Artifice or Deſign of a THEATRE, is
founded
entertained with an Imitation of ſome memorable Actions paſſed,
fo in order to render the Performance more natural and pleaſing,
the Scenes on which they were tranſacted ought to be reprefen-
ted as they appear to the Eye of a diftant Spectator, that is, in
Perspective. For whether there be a tranfparent Plane, or not,
be-
Of PERSPECTIVE.
195
between us and the Scene of Action, the Image in the Eye is
equally in Perſpective, and excites the fame Idea in the Mind,
as if the whole Affair were viewed on a perſpective Plane.
1634. Hence it is, that the whole HOUSE, internally, is only
one large Scenography, or perſpective View of an hollow Paral-
lelopiped, or large Room, about three or four Times as long as it
is Wide or Deep. Hence it is, that you fee the Body of the Houfe
fo contrived, as to reprefent the Perfpective of the firft Part of
fuch a large Room, the Ceiling, Sides, and Floor, all Converge,
by the Rules of Perſpective, to a Point of Sight at the fartheft
End, which is the remoteſt Part of the Stage.
1635. For the STAGE is only the Floor of the other Part of
the Scenography of the large Room, ſuppoſed to have been the
Scene of Action; and therefore alfo the Sides and Ceiling, Pro-
fcenes, and Poftfcenes, are all in regular Perſpective. But in or-
der to give a more exact Idea of this Matter, we muſt have Re-
courfe to perſpective Reprefentation alfo.
1636. Therefore let ABCD (Fig. 4.) reprefent the ground
Plane or Floor of a long and large Room, on one Part of which
toward A B, fuppofe fome great Action or Converſation to have
happened, and the fame to have been obſerved by the Eye Y at
the other End of the Room. Then it is evident, if in any in-
termediate Part a tranfparent Plane E F GK had been erected,
the Obferver would have viewed the Whole tranſacted perfpec-
tively on the Plane EG, in that Part ac EK which includes
the Scene of Action AEK B. (1487.)
1637. Now though we have fhewn how fuch an Event may
be truly repreſented in a perſpective View, yet this is ftill but a
Picture of it; there, it is true, we fee the Place, the Perfonages, the
Manner, the Time, and fome other concurring Circumſtances,
that altogether, give fome Idea of the Thing; but real Life,
Action, Voice, and Variety of Scenes, Attitudes, Paffions, &t.
are wanting to animate and realize the perfpective View, of
Picture.
1638. Now all this is effected by the ARTIFICE of a STAGE
or THEATRE. For let Y X be the Height of the Eye placed
over the middle Line of the Room, and drawing Y I parallel to
HX, the Point of Sight will be I in the horizontal Line (/m) of
he perſpective Plane. And now, inftead of the upright Plane
Cc 2
EG,
196
INSTITUTIONS
EG, we fuppofe another Plane fEK g placed on the fame
Line EK, and fo far inclined, that its Elevation iL be juft
equal to the Eye's Height Y X, it is evident, the perfpective
Horizon (m) will coincide with the End fg of the inclined
Plane; the perſpective Scene a c E K will be projected into the
large One e E Kk, and all above it on any other perpendicular
Plane at (ek) or (no) behind.
1639. This new inclined perfpective Plane fE Kg is the
Stage or Theatre we ſpeak of, and is a Scene of real Action in a
perspective View at the fame Time. The Area e E Kk is here
the perſpective Scene of Action (or Scena Dramatis) and inſtead
of Men and Women in the Picture acEK, we have fufficient
Room for real Perfons (Perfonae Dramatis) to act the feveral
Parts of the intended Repreſentation (or Drama); here now are
the Profcene, the lateral Scenes, the Poftfcene, &c. all in Per-
fpective, and moveable to the moſt convenient Situations. Not
only the whole Scenography of the Stage is by this Means formed
and adjuſted by the Rules of Perſpective, but the very Actors
themſelves, as they advance or retreat, have their Appearance.
encreaſed or diminiſhed to the View of the Spectators (especially
thoſe remote in the Pit and Gallaries) according to the Laws of
optical Perspective.
1640. On this inclined Plane, or perspective Stage, the
dramatic Scene may be encreaſed at Pleafure, and thereby very
diftinct Views and Profpects agreeably rife to the Sight, by the
artful Paintings and Defigns on the Lateral and Poftfcenes;
thus e E Kk may be enlarged to oEKn; and fekg which was
before the Poſtſcene, may now be confidered as the Profcene, to
a diftant Dramatic Scene (oe kn) whofe Poftfcene is (nopq.)
And thus the Stage, or Scene of the Drama, may be extended
to comprehend the moft diftant View or Scenes of Action, even
to the Horizon itſelf, for the remote End of the Stage, viz. the
Line fg, is the horizontal Line in this theatrical perſpective
Plane.
1641. As the Dramatic Scene e E Kk is variable according to
the Nature and Circumftances of the particular Parts or AЯions
of the Play, fo the Lateral and Poſtſcenes must be conformable
thereto, and confequently variable, and of different Sorts; thus
if the Stage repreſent the Infide of any particular Building, as a
Ca-
+
Of PERSPECTIVE
197
Caftle, Prifon, Church, &c. the Side and Back-fcenes muſt
correfpond to the fame in all the Characters appropriate to fuch
Edifices. If the Stage be a Street, the lateral Scenes repreſent
Houſes, &c. on each Side, and the Poftfcene gives a perfpec-
tive View of all at the far End, or beyond it. If you fuppofe
the Scene a Vista, then Rows of Trees, &c. defcend in Perfpec-
tive on either Side. If Fields or Lawns be the primary Scene,
and Swains, Shepherdeffes, &c. the Perfons of the Drama,
then Trees, Woods, Cottages, &c. make the Perspective of
the Sides; and diftant Views of Countries, funny Hills, and
horizontal Clouds are portray'd on the Poftfcene.
1642. With regard to the Scenes, they are variable and dif-
ferent in the different Parts of the Scenography; when the Stage
is not open, the Profcene is only a plain Curtain, to be drawn
up and let down as Occafion requires. The Poftfcene is either a
Curtain with Defigns or Drawings in Perfpective; or elſe it con-
fifts of two fliding Parts, which being put together, from each
Side the Stage, make one uniform perſpective Piece ſuitable to
the Nature of the Part of the Play then acting.
1643. The lateral Scenes, and thofe above the Stage, confift
of many different Parts; thofe on the Sides are moveable in
Grooves made in the Floor of the Stage; and may by this Con-
trivance be variouſly changed, and being placed oppofitely one
a little before the other, they make together but one united
View or perſpective Scene, and by this Means produce an agree-
able Deception, and at the fame Time give an Opportunity for
the Actors to come on, and go off, on any Part of the Stage, as
the Circumſtances of the Action may require. In the fame
Manner the aerial Scene may be compoſed of ſeveral Parts of fpar-
ticular Curtains, or moveable Pieces, defcending one below the
other, and uniting in one perfpective Defign.
1644. But to ſee more clearly how thefe Things are contrived
and difpofed, let Fig. 5. reprefent a front View of the Stage in
Perſpective. One Half of which is plain and naked, and the
other Half diſguiſed and decorated with perfpective Paintings,
&c. EFGK is the Front; Eek K, the Stage; GgkK the
plain Side, confifting of ſeveral lateral Planes projecting beyond
each other in their proper Grooves; (ef g k) is the Pofticene, half
plain, and half painted; E Ffe is the other Side, covered, as
it
198
INSTITUTIONS
1
it were, with perſpective Drawings and Defigns; and making
but one piece in Appearance. Ffg G is the aerial Scene, open
on one Part, and properly formed on the other; (i) is the Point
of Sight by which the Whole is regulated. This Piece of Per-
ſpective is the proper Scenography of that Part of the Stage in
Fig. 4. which is denoted by EFGK kgfe.
1645. Hence it is evident, that the Heights of the ſeveral
Planes, or Sliders, on the Sides of the Stage, are to be adjuſted
by the Inclination of the Line Fi, or Gi, to the Line Ei, or
Ki, on the Stage; fo that the Height EF or GK, of the firſt
Sliders, and (ef) or (gk) of the laft, in the perſpective Stage
Fig. 5. are the fame as the Lengths of the Lines denoted by
the fame Letters in Fig. 4. And the fame is to be obferved, for
the Sliders in any other Scene (ek no) in a remote Part of the
Stage.
1646. Another Figure is yet neceffary to fhew how the
Grooves are to be made on the Sides of the Stage, and their Diſtan-
ces aſcertained and proportioned. In order to this let Kk (Fig.
4.) be the Perſpective of the ground Line for the Length of the
Stage divided into equal Parts (as directed in 1554,1555.) Then
let Fig. 6. be the whole Length of the Stage; A B the Width
of the Houſe, and EiK the Area of the Stage, as in Fig. 4.
and draw the Lines Ai, and Bi. Then from the Points of equal
Divifion in the perſpective Line Kk (Fig. 5.) raiſe Perpendicu-
lars, and they will terminate the fore Parts of the lateral Sliders-
reſpectively, in that Perſpective of the Stage.
1647. Let the middle Line Hi of Fig. 6. be in a right Line
with Hi in Fig. 5. Then from the feveral Points of equal Divi-
fion in the Line Kk of Fig. 5. let fall Perpendiculars to the
Line K k in Fig. 6. and they will interſect it in the Points K, r, s,
t, u, w, x; through which, if Lines are drawn parallel to the
ground Line AB, they will affign the Places and Diſtances of
the Grooves (r 1, ƒ2, t 3, &c.) on each Side the Stage K k and
Ee. And the Figures 7 and 8 denote the Grooves of the Poft-
ſcene. But there is in different Stages, a different Difpofition of
the Parts and Machinery; what we have faid is fufficient for a
general Rationale of theatrical Perspective, which is all that is here
intended.
CHAP.
T
Of PERSPECTIVE.
199
CHAP. XIII.
The Doctrine of PERSPECTIVE DECEPTION EX-
1648. WE
plained, and exemplified.
[PLATE IV. Of PERSPECTIVE.]
E have already confidered the various Illuſions and
Deceptions which the viſual Senfe is ſubject to from
Catoptrics and Dioptrics; that is, how different the Magnitude,
Pofition, Place, Diftance, &c. is of any Object, as it is con-
ceived in the Mind by the vifual Faculty, from Rays reflected or
refracted to the Eye in regard to the Idea of the fame Thing ex-
cited by direct Rays.
1649. Of theſe Deceptions fome are very ufeful, witneſs the
Looking glafs, Reading-glafs, &c. Others afford us a very rati-
onal Amuſement; as thofe Lenfes and Speculums which give
an agreeable Relievo to perſpective Views, Prints, &c. ufed in
optical Machines. And fome on the other Hand gives us a great
deal of Trouble, as in Cafe of Refractions through the Atmof-
phere in Matters of Aftronomy and Navigation, by which the
Celeſtial Phænomena are fhewn in very different Circumftances.
from thoſe which are real.
1650. But the Deceptions we here intend to ſpeak of are
purely Perſpective, and are defigned to impoſe on the vifual Fa-
culty in the moſt agreeable and advantageous Manner. In-
deed all Perſpective is a Sort of Illufion, as it repreſents Things
not as they really are, but as they appear on an intervening
Plane by refracted or reflected Light. And therefore, if in any
particular Cafe Things are not juſt as we could wiſh to have
them, we can, at leaſt, by this Art, make them appear to be
fuch, and to an inadvertent Obferver, the Difference from the
Truth will not be difcovered, and that which is only Perspective
fhall be taken for the real Thing it reprefents.
1651. We fhall illuftrate this Matter by an Example or two.
Let A B C D be the ground Plane of the perfpective Vista Cb
a D, (Fig. 7.) at the far End of which, ſuppoſe a high Wall
(bcde) at right Angles thereto; and let the Length of the Walk
BC
200
INSTITUTIONS
BC be 40 Yards. Now if it be defired to make this Viſta or
Walk to appear longer than it really is, it may be thus effected.
Suppoſe it were required to lengthen it by one Half, or to make
it appear to be 60 Yards long. Then fince on the ground Line
CE, the Diſtance C 40 determines the Length of the Viſta Cb;
if from the Point 40 you fet off 20 Yards more, it will give the
Point 60, from whence drawing a Line to the Point of Diſtance
Z, it will give the Point (f) in the perfpective Side Cb of the
Viſta continued out; by which Means the perſpective Cfg D
may be compleated for a Walk CF on the ground Plane 60
Yards in Length.
1652. Therefore fince the additional Part of the perſpective
Vista (abfg) may be painted on the Wall contiguous to the
End (ab) and as it will be a perfect Continuation of the faid
Viſta, it will exhibit to the Eye of a Spectator, at a proper Di-
ftance, the View of a Walk A FGD juft 60 Yards long; nor
will he be ſenſible of the Deception, or that the Part (abfg) is
drawn or painted on the lower Part of the Wall. The Truth
and Pleaſure of fuch perfpective Illufions any one may be con-
vinced of by the notable Inftances of fuch Pieces of Art in Vaux-
hall Gardens, and in many other Places.
1653. After a like Manner you may rectify the Appearance
of an irregular Room. Thus for Inftance, fuppofe the Floor
of the Room was in Form of the Trapezium AEFKB (Fig. 8.)
deficient from a Quadrangle by the Part EIF; then in order to
make the Room appear of the ufual Form to a Perſon viewing
it at ſome Diſtance, it will be neceffary to compleat the perfpec-
tive Trapezium, or Floor A E F K Binto the Rectangle AIK B,
which is done by drawing the Line B Z, interfecting the Side
AE continued out in I, and joining IF. Therefore on
the Side of the Room E F G H, you draw the Perſpective of the
feveral Parts deficient, viz. (1.) The perfpective Triangle EIF
for the Floor. (2.) The correfponding Triangle G HM for the
Cieling. (3) The Part E HMI to compleat the Perfpective
of the Side. (4.) IMGF to compleat the farther End or
Side of the Room. And (5.) on theſe ſeveral perſpective Sup-
plements, are to be drawn Windows, Doors, Glaffes, Pic-
tures, &c. to make the Perfpective reprefent the Room in the
Man-
Of PERSPECTIVE.
201
Manner you propoſe to compleat the View of it; and then it
will have the agreeable Effect defired.
1654. We have already taken Notice of the great Uſe, yea,
abfolute Neceffity of perfpective Diſguiſe in all theatrical Pur-
poſes; the very Effence of a Theatre confifting in that Sort of
Illufion. Here all the Scenery is a Contrivance to repreſent
Things which are not as tho' they really were; and, in fhort,
all the Incidents to a Draina, are one united Syftem of Decep-
tion; and by the Price that is paid for it, one would think there
was no Pleaſure fo great as that of being deceived; and which,
therefore ought to be confidered as a high Recommendation of
Perſpective, which above all other Arts does moſt agreeably
and innocently impoſe upon our Senſes.
CHA P. XIV.
The RULES of PERSPECTIVE applied to ARCHITEC-
TURE, in raiſing the Perſpective ELEVATION of
BUILDINGS.
1655. T
[PLATE V. Of PERSPECTIVE.]
HERE is certainly no Art in which the Science of
PERSPECTIVE is fo immediately concerned as Ar-
chitecture, fince whatever Edifice, or Fabric, is propofed, the
Ground Plan thereof muſt firſt be made; then the Ichnography or
Perſpective of the Plan muſt be drawn; and, laftly, the Scenogra-
phy or perfpective Elevation of all the Parts muſt be compleated,
before a proper Idea can be given of the Defign; and therefore
the Architect, above all Men, is under a Neceffity of under-
ſtanding the Rules of Perspective, and the Reafon of them like-
wife, if poffible.
1656. Wé fhall illuftrate this Matter in regard to the perſpec-
tive Elevation of an Edifice by the following Example. Let
ABCD be a geometrical Square, on the Ground Plane, in
which another Square EFGH is infcribed; and in that a
CROSS, KL MN, which is the Form of the Plan on which
VOL. II.
Dd
28
202
INSTITUTIONS
1
an Edifice is to be erected. Suppoſe it to be viewed obliquely,
with an Angle in Front; then, by the Rules delivered (Chap.
III.) you find the perspective Plan (klmn) for the given Point
of Sight I, and Diſtance I Z.
1657. Let us now fuppofe this perſpective Plan, or Croſs,
drawn more at large, and for a lefs Height of the Eye in Pro-
portion to the Diſtance, as (k l m n) in Fig. 2. Then it is
evident, that if from the feveral angular Points you raife Per-
pendiculars, and give to each of them their proper Heights (ac-
cording to their respective Diſtances from the Ground Line
A D) by the Line of MEASURES, (as directed in Chap.
VIII.) You will then, by joining their Apices in a pro-
per Manner by right Lines, form the Scenography of the
Building in its linear Extremities; after which its front Sides,
gable Ends, &c. are to be filled up, and pannelled with Win-
dows, Doors, and architectural Ornaments, with a juft Dif
pofition of Light and Shadow upon the Whole; in all
which there can be no Sort of Difficulty, if the foregoing Pre-
cepts be underftood, as appears by this Example, wherein the
feveral Parts of the Edifice receive their Form and Proportion
immediately by the Radial and Diagonal Lines, drawn to the
Points of Sights and Diftance, I, Z; it would therefore be
fuperfluous to ſay any Thing more on this Head.
1658. And becauſe the above Rules of Perſpective are in the
higheſt Degree neceſſary in truly defigning and repreſenting the
Ruins of Building and Monuments of Antiquity, we thought it
would be proper, likewife, to give a Specimen of fuch a per-
ſpective View in all the Variety of Architecture.
1659. In Figure 3. there is a Side View of a Colonade
of Pillars, all diminiſhing in juſt Perſpective to the Point of
Sight at I. There are alfo three Pillars in a Front View, but
entirely out of Perſpective, being all of an equal Size, (fee 1587)
this is one Reaſon why I made Choice of this Piece (for all the
Figures of this Plate except the firft, are borrowed,) being
willing the Reader fhould fee in what an imperfect State the
Practice of Perſpective is in at this Day, and how ridiculouſly
the Laws of this Science are tranfgreffed, even by the Profeffors
themſelves.
1660. Having
Of PERSPECTIVE.
203
"I
1660. Having faid all I think can be neceſſary for perſpective
Draughts and Deſigns, either of Superficies or Solids; I have
added, for further Illuftration, two other Figures (viz. Fig. 4,
and 5.) to fhew how vaulted Arches, Roofs, the interior Parts
of Churches, winding Stairs, diftant Views thro' Buildings,
and many other Particulars, relative to Architecture, appear
in perfpective Defigus. In thefe the Reader will find not
a fingle Article but what is ftrictly conformable to, and ex-
ecuted by the Methods and Rules of Perſpective above laid
down.
CHA P. XV.
Of Sciagraphic PERSPECTIVE, or the ART of
SHADOWS.
1661. it is a mere negative, and therefore can not, pro-
AⓇ
S SHADOW is nothing but the Abfence of LIGHT,
perly be the Subject of Art. However, there is what we ufually
call the Art of Shadows, and as it is a neceſſary Article in Paint-
ing and Deſign, and is, for the moſt Part, conducted by the
Rules of Perſpective, we shall here treat of it in a few Words.
1662. Since whatever intercepts the Rays of Light muft pro-
duce Shadow, it is evident, that all Bodies, or Objects which
are opake, muſt be attended with Shadow, if placed in the
Light which can fall obliquely on them.
1663. Again, it is neceffary that the Pofition of the Shadow
be contrary to that of the luminous Object, whether a Candle,
the Sun, a Window, &c. becauſe the Rays of all Light are
rectilineal.
1664. The Figure of the Shadow depends partly on the Fi-
gure of the Object, and partly on the Form of the Rays. For
it is evident, that in any Light the Shadow of a Triangle will
be different from that of a Circle, or a Square; and the Sha-
dow of a Cone will not be the fame as that of a Globe or
Cube. And, on the other Hand, it is as certain, that the fame
Dd 2
Body
204
INSTITUTIONS
Body will have its Shadow varied according to the Form of the
Rays. For a Cone will caft a Shadow of a greater Length by
intercepting diverging Rays, as thoſe of a CANDLE, than it
will do from parallel Rays, as thofe from the SUN or a Window.
1665. But it will be fufficient here to obferve, that the Laws
of Shadow, proceeding from a Privation of diverging Rays,
are ſtrictly conformable to, as they refult immediately from, the
Rules of common Perspective; by ſuppoſing a Candle inſtead of the
Eye, and confidering the Rays of this Light in lieu of the viſual
Rays coming from the Object to the Eye.
1666. To demonftrate this we need only have regard to the
foregoing perſpective Figures. Thus, in Plate II. Fig. 1. if a
Candle be fuppofed to be placed at I, and (antv) an opake
Object at any Height, Pt, above the Horizon, or Ground
Plane, then, fince all the Rays which fall thereon are intercept-
ed, it is evident, if the Rays which pafs by the extreme Parts
thereof be continued to the Ground-plane, as I a A, InN, It T,
Iv V, they will include a Space A NTV, which will be wholly
deprived of Light; and that dark Space is therefore called the Sha-
dow of that Object, and gives, of Courſe, its true Limits, Fi-
gure, and Dimenſions.
1667. Again, (in Fig. 2.) let a Candle, or radiant Point be
fuppofed at I, and let Abc D be any opake Object on the
Ground or horizontal Plane, then the Rays IcC, IbB will
project its Shadow into the Space ABCD, as is evident by
Inſpection.
1668. The Figure and Dimenfions of the Shadow is deter-
mined in the fame Manner, in any Pofition of the Plane whatſoever.
Thus fuppofe (abcd) (Plate II. Fig. 1.) were an opake Surface;
then from a Radiant at I, the Rays, Ia A, IbB, IcC, IdD,
will project its Shadow into the Space ABCD on any Plane
ftanding on the Ground Plane; and if the Plane which receives
the Shadow be parallel to that in which the Object is, then the
Shadow will be fimilar to the Object.. In short, every Thing,
with regard to the Object and its Shadow, will be inverſely the
very fame as has been demonftrated of the various Relations
between the Object and its perfpective Repreſentation, and
therefore, if thefe be undertoo the Doctrine of Shadows can
admit
Of PERSPECTIVE
205
admit of no Difficulty, unleſs in that Cafe, where the Shadow
is not projected wholly on one Plane, but partly on feveral.
1669. For Example fuppofe (brtva) were an irregular, opake
Object, and ſo fituated, with regard to the Radiant at I, that
its Shadow falls partly on the Ground Plane, and partly on fome
other Plane elevated above it. Then even in this Cafe, you
have nothing more to do, than, by the Rules of Inverſe Perfpec-
tive, to determine the Prototype of the given Perſpective (brtva)
and how much of it is on one Plane, and how much on the
other. See Chap. VI.
1670. Or thus; ſuppoſe Rays drawn thro' all the angular
Points a, b, r, t, v, to the feveral Planes, they will there de-
termine the Extent and Figure of the Shadow; thus It T and
Iv V will determine its Limit TV on the Ground Plane; and
the Rays Ir R, IbB will mark out the fame on the other Plane,
as RB; and therefore, all the Space between R B on the ele-
vated Plane, and TV on the Ground Plane, will be occupied
by the Shadow.
1671. Or, laſtly, by Calculation, thus; let the Diſtance of
the Object Y Pa, the Height of the Radiant or Candle YI
H; the Height of the Point t, or Pth, and PT = x,
the Diſtance ſought of the Point T, where the Shadow com-
mences. Then, by fimilar Triangles, IYT and PT, we
have IYP: TY: TP; that is, H:b:: a + x:x;
ab
= x. In Words thus, multiply
H— b
whence this Theorem
મ
the Distance of the Perpendicular Y P by the Height tP of the Sha-
dowing Point; and divide that Product by the Difference between the
perpendicular Height of the Radiant and given Point t, and the Que-
tient will be the Distance PT of the Shaddow T, from the Foot of
the Perpendicular P, as required.
1672. In the fame Manner the Point N is determined for the
Shadow of the Point (") in the Object; and from thence the
Length of the Shadow TN for the Part of the Object (tn).
Alfo, if the Shadow falls upon a Wall, you find its Height on
the faid Wall by the fame Rule, knowing the Diſtance of the
Wall from the Radiant, and the Point to which the Ray tends
beyond it, found as above.
1673. It
206
INSTITUTIONS
1673. It is evident, that the Shadows of all perpendicular
Lines converge to a Point perpendicularly under the Radiant ;
and confequently, if we fuppofe the Radiant removed to a very
grea: Ditance thofe Shadows will become parallel, which,
therefore, is the Cafe of all fuch Shadows produced by inter-
cepting the Rays of the Sun.
1674. Hence the Shadow of a Parallelogram will diverge
from the Point under the Radiant, if a Candie; but if the Sun,
the Sides of the Shadow will be parallel. The fame may be
obferved with Regard to the Shadow of a Cylinder.
1675. It is alſo evident, that the Shadow of a round Table
on the Floor will be circular, both from the Candle ftanding upon
it on any Part, and alfo from the Sun-beams (1504.) only in
the firſt Cafe, the Shadow will exceed the Dimenfions of the
Table; in the latter, it will be juft equal to it.
1676. The Shadow of a Globe lying on a Table muſt ne-
ceffarily be elliptical; for the Cone of Rays are intercepted by
a Plane perpendicular to its Axis, and paffing thro' the Globe
in ſuch Fofition as to make an Angle with the Horizon, or Ta-
ble, juft equal to the Co-Altitude of the Radiant whether Can-
dle or Sun. In this Caſe the Cone is never Scalenous, and can ad-
mit of no Subcontrary Sections from the two Planes, therefore the
Elliptic Shadow can never become Circular. (See Chap. IV.)
1677. As we have fhewn a general Method how the Sha-
dow may be determined for any given Line on given Planes, and
as the Shadow of any Superficies is determined by that of each
bounding Line, and the Shadow of a Solid refults from thoſe of
its connected Superficies, it is evident that the fame general Per-
fpective Rule for Shadows (1671.) of Lines extends to the De-
termination of the Shadow of any Body whatſoever; the parti-
cular Application of which is left to, and is no fmall Part of the
Praxis of the young DESIGNER.
1678. The Art of Shadows, in one particular Branch, con-
ftitutes a Science of itſelf alone, viz. GNOMONICS, or the ART of
DIALLING; to which therefore we proceed to apply it; but it
will be firſt neceſſary to teach the Projection of the Sphere from the
Principles of Perſpective above laid down; and then the Ratio-
nale of making a DIAL will appear in a new Light.
CHAP.
Of PERSPECTIVE.
207
CHA P. XVI.
The RULES of PERSPECTIVE applied to the Geogra-
phical PROJECTION of the SPHERE, for the Con-
Atruction of MAPS, CHARTS, &c.
1679. Charts by a PROJECTION of the Sphere in Plang is
T HE Method of making Geographical and Aftronomical
as ancient as any
Mathematical Science; but as it is in that Way
limited and intricate, we fhall here fhew how general and eaſy it
is by the Principles and Rules of Common PERSPECTIVE, which
is not only new, but the natural Source of fuch a Doctrine.
1680. For if (Fig. 1. Plate II.) the Plane of Projection HF,
the Vertical Plane QN, and the Horizontal Plane L O, were all
of a circular Form, and of an equal Radius, they would repre-
ſent the Planes of three great Circles of the Sphere whoſe common
Interſections would all paſs thro' the Center (¿).
1681. Let the Planes of three fuch great Circles of the Sphere
be EQTR, QKPR, and EPLT (Plate VI. Fig. 1.) of
which the firſt is fuppofed perpendicular to the Viſual Ray pro-
ceeding from the Center C to the Eye at Y, and is therefore the
Plane of Projection. The Second is the Vertical Piane, and the
Third, the Horizontal Plane, as all the three are ſuppoſed to in-
terfect each other at right Angles. Confequently QR is the
Vertical Line, and ET the Horizontal Line of the Projection.
1682. Let O be the Place of any Object on the Surface of the
Globe or Sphere, then its Diſtance from the Plane of Projection is
the Sine OB of the Arch O G, which meaſures its Latitude or
Declination from that Plane. Its Distance from the Vertical
Plane is the Sine OD of the Arch O K of a lefler Circle KOL,
whoſe Plane is parallel to the Plane of Projection, and its Dif-
tance from the horizontal Plane is the Co-Sine ON of the fame
Arch OK, or the Sine of the Arch O I. All which Sines or Di-
tances bear a known Proportion to the Radius of the Globe or
Sphere, and therefore are to be eftimated in that meafure, as
found in the Trigonometrical Canon. (7c8 to 716.)
1683. The Diſtance of the Eye CY is alfo eftimated in the
fame Meaſure, viz. in Parts of the Radius: That is, if Radius
208
INSTITUTIONS
=r, then CY = 11, 1, 21, 31, &c. or nr, which may re-
preſent any Diſtance of the Eye indefinitely, and thereby render
the Theory of Projection, this way confidered, univerfal. There-
fore let us fuppofe,
1. The EQUATOR, the PLANE of PROJECTION.
1684. Since the Circle F QT R, in this Cafe, is the Equator,
the Vertical Circle QP R, and the Horizontal One E PT will
be Meridians; and P the Pole in the Hemifphere remote from the
Eye; and if O be the Place of an Object in that Hemiſphere,
then we have CG:GH::CB (MO): BI(= OD.)
BI{=OD.)
that is, as Radius is to the Sine of the Difference of Right Afcenfion
or Longitude fo is the Co-Sine of Declination or Latitude to the Dif
tance of the Point O from the Verticle Plane.
1685. Again, we have CG: GF :: CB (= MO): BS
(=ON.) That is, as Radius is to the Co-Sine of the Difference of
Longitude or right Afcenfion, fo is the Co-Sine of Latitude or Declina-
tion, to the Diſtance of the Point O from the Horizontal Plane.
1686. Or thus in Symbols; let H G = p, G F = 4, O B÷
x, OM = y; Then by (1684.) DO Diſtance of the
Ру
=
7
-
Point O from the vertical Plane; and by (1685.) ON =
-
97
r
Diſtance from the horizontal Plane; and x = Diſtance from
the Plane of Projection. Having theſe Diſtances from the
three Planes, we determine the Diſtances from the vertical and
horizontal Lines, in the Plane of Projection, by the common
Rules of Perspective.
1687. Thus, fuppofe the Equator between the given Point
py npy
O and the Eye Y, we have nr + x : nr :: :
ĵ
nr + x
a Diftance of the Point O from the vertical Line of the
Projection; (1470.) whence we have npy anr + ax;
and therefore nr + x: ny::p:o; the Analogy for find-
ing ..
1688. Again, we have nr + x: nr::
gy 12 qy
:
7 nr + x
b;
(1470.) which gives nrxny: qb, the Analogy for
finding b, the Diſtance from the Horizontal Line.
1689. If
Fig.1
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Fig.2
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B
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BUIBIRUAI
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Fig.4
ભારતમાં વર
J
1
Fig.3
5000001
Fig.5
Plate V.
of
PERSPECTIVE.
ןןןןןןם
D
Of PERSPECTIVE.
209
1689. If the Eye be placed in the Pole oppofite to P, then
n = 1; and the two Analogies become (1.) r + x : y :: p:a,
扎​=I
the Diſtance from the vertical Line; and (2.) r + x : y :: q: b₂
the Diſtance from the horizontal Line; in this Cafe, the Projec-
tions is that of Ptolemy; and is reprefented in Fig. 2.
=
1690. If the Eye be at an infinite Diftance, then n is infinite,
and the two Analogies become (1.) ry::p: a Distance
from the Vertical; and (2.) ry:: q:b Diftance from the
horizontal Line. In both theſe Cafes the Meridians are projec-
ted into right Lines (1464) and the Parallels into concentric Circles
(1504).
1691. If the Eye were placed in the other Part of the Axis at
z, then the Point O is between the Eye and Plane of Projection;
and in ſuch a Caſe the Analogies will be (1.) nr x: ny :: p
:a; and (2.) nr — x: ny :: q: b; for the Diſtances from the
vertical and horizontal Lines, as is evident from (1470). The
firft of theſe Cafes is called the STEREOGRAPHIC, and the
other the ORTHOGRAPHIC PROJECTION of the SPHERE.
II. Of the PROJECTION of the SPHERE on the Plane of the
MERIDIAN.
1692. If the Meridian be chofe for the Plane of the Projection,
as EP TR, (Fig. 3.) then in the Cafe of common Maps of the
World, the Equator EQT is the horizontal Plane; and the
Six o'Clock Hour-circle PQR is the vertical Plane. P and R
are the two Poles; and the Eye Y is placed in the common In-
erfection QC of the vertical and horizontal Planes continued
out.
1693. Let O be the given Point, as before, through which
draw the Meridian POG; then will the Sines and Co-Sines
GH, GF, O B, OM, be the fame as before, and denoted by the
fame Letters. Alfo ODP the Diſtance from the Vertical
7'
Plane, ON= = 22 = Diſtance from the Plane of Projection
?
x
as in the laſt, and OB = the Diſtance from the horizontal
Plane. Then by the Analogies of Perspective (1687, 1688.)
91
we have nr + ar::*:
VOL. II.
nr 7x
Diſtance from the
n r r + q y
E e
hori-
210
INSTITUTIONS
y
horizontal Line. And as ur +22
nr
Diſtance from the vertical Line.
: nr::
ру
:
nr py
r n r r + q y
IT
1694. If the Diſtance of the Eye nrr, as in the common
Stereographic Projection, then n = 1; and the two Expreffions
become
rr x
rrt gy
and
2
r py
rr + qy
And when rx
; which are to each other, as
rx to py.
py; then r:p : : y : x : : r : {
(712); that is when of the Tangent Latitude O G is equal to
the Sine of Difference of Longitude, then the Point O in the
Projection will be equally diflant from the vertical and horizontal
Lines.
1695. When the Point O is in the Equator, xo and y = r;
and the Distance from the vertical Plane is then
rp
r + q
But the
Ratio of rq to p, is the fame as that of p to r
q; as will
appear if we place the Arch QG in another View; let QTR
(Fig. 4.) be the Plane of Projection, or primitive Circle, as it is
ufually called. Then by the Eye at R the Point G is projected
into the Point O in the horizontal Line CT.
GH = p, G F = HC = 4, and QH = r
Line RG. Then by the Property of the Circle
RH: HGHG : HQ; that is, r+q:pp: r-q, and
therefore prr-q².
p²
Put CR = 1,
9; and draw the
(658) we have
1696. Now becauſe QH = and CO =
r + q
rp
r + q
>
there-
and confe-
fore QHCO:: p²: rp :: p÷r :: GH: QC;
quently QO is the Arch of a Circle. Therefore all great Cir-
cles of the Sphere are projected into Arches of Circles of different Di-
menfions.
1697. Let the Radius of the Circle DOE be R, the Cen-
ter B, and Diameter AO. Alfo put CQ= P, and BC = Q,
then becaufe Cơ -
да
rp
P₁
(1696) we
R + Q R + Q
r + q
p
and fo R + Q: r::r + q :p :: QC
1 +9
have R + Q
: CO; therefore AC: CQ:: RH: GH; and confequently
the Angle QAC = QRG (657) and the Angle QB C =
GCH
Of PERSPECTIVE.
211
GCH (642) therefore making the Angle LCTQCG,
the Line LC will be parallel to B Q, and give the Center B.
1698. Or thus; CO is the Tangent of half the Angle
QCG (705); whence it appears that the Tangent of half the
Angle or Arch GQ fet from the Center C, gives the Point O;
then through the three Points QOR the Meridian is drawn
with Eafe (695).
1699. The Projection, as hitherto confidered, is the vulgar
Stereographic One, which on every Account, is the worst that
can be for Maps and Charts; becauſe of the great Diſproportion
and Diſtortion it occafions in all the Parts of the Earth's Surface
repreſented upon it, as appears from the very unequal Parts of the
horizontal Line or Equator intercepted between Meridians
equally diftant on the Globe; therefore this Projection muſt
give a very falfe Idea of the geographical Relations of Places,
and is confequently the most unfit for fuch Purpoſes. (See
Fig. 5.)
1700. To remedy this Imperfection in a confiderable Degree,
we may ſuppoſe the projected Quadrant of the Equator CT, or
CE divided into equal Parts, of any Numbers, and let CO
be any Number of them denoted by t; that is, let CO be the
S
1
† q
npr
t
Part of r, then will
or
S nr + q
snp = ntr +1 9, and we get n =
s nr q
np
; whence
= Diſtance of
sp - tr
the Eye to project that Meridian; for Example, let the Meri-
dian of 10 Degrees from the vertical Plane be projected by the
Eye; then s 90, and r = 10; n =
Parts of which r = 100.
90p
109
= 175
10 r
1701. In Practice, by dividing the Diameter or horizontal
Line into equal Parts; you have the Points given thro' which to
draw the Meridians (695). And this is uſually called the
GLOBULAR PROJECTION; forafmuch as the Meridians and
Parallels are circular, and at equal Diftances in the Equator and
Meridian or vertical Line; and confequently fuch a Projection is
greatly preferable to the former for general Maps of the World;
E e 2
212
INSTITUTIONS
a Specimen of which fee in that Map we have prefixed to our
PHILOSOPHICAL GEOGRAPHY.*
1702. There is another very uſeful geographical Projection,
viz. on the Plane of the Horizon, of which we have given a Spe-
cimen in Fig. 6. and though any particular Point may be deter-
mined on the faid Plane by the Rules of Perſpective, yet the
Whole of it is with more Eafe and Readinefs laid down from
Calculations founded on the Principles of ſpherical Geometry
hereafter to be explained.
CHA P. XVII.
The RULES of PERSPECTIVE applied to Aftronomi.
cal PROJECTIONS for the Conftruction of Celeſtial
PLANISPHERES, the ANALEMMA, &c.
INCE the Surface of the Celestial Globe may be pro-
1703. SINCE
jeed in Plano, in the fame Manner as that of the
Terreſtrial One, it follows, that the fame Theorems which ferve
for the Conftruction of MAPS, may alfo be adapted to CELE-
STIAL CHARTS or PLANISPHERES.
1704. For this Purpoſe, inſtead of the Latitude and Longitude
of Places on the Earth, we muft, in regard to the Sun and Stars,
ufe the Words Declination, and Right Afcenfion; and then the
fame Symbols will ferve; that is,
Sine of Right Afcenfion.
q= Co-Sine of that Right Afcenfion.
x= Sine of Declination.
y
Co-Sine of that Declination.
1705. Alfo the fame Diagrains (in Fig. 1, and 3.) may here
be ufed; if O be the Place of a Star, then its Diftance from
ру
the vertical Plane is P
9.1
}
Ĵ
= DO; from the horizontal Plane
NO; and from the Plane of Projection = OB; the
fame as before (1686).
1706.
* Alfo in a New Map of the World entitled, The PRINCIPLES of
GEOGRAPHY and ASTRONOMY explained, &c.
Of PERSPECTIVE.
213
1706. If the Equator be the Plane of the Projection; then
the Pole of the World is the Center, and the Circles of Declina-
tion are projected into right Lines, alfo the Parallels of Declina-
tion are concentric Circles; all as before (1685.) But in this
Planifphere the Pole of the Ecliptic is alſo projected, and all the
Circles of Latitude, together with the Ecliptic itſelf, are pro-
jected into circular Arches, by the ufual Methods.
1707. In this Projection the folftitial Colure is the vertical
Plane, and the equinoctial Colure the horizontal Plane; the Di-
ftance of any Star from the former will be
the latter,
ngy
nr + x
пру
nr + x
; and from
But if the Eye be placed in the Pole on
the Globe's Surface, the Diſtances of the Star becomes
gy
and
1 + x
py
r + x
; and thus all the Stars may be readily laid down
from a Table of their Declinations and right Afcenfions. Such
celeſtial Planiſpheres we have by the late Mr. Senex, of 24 Inches
Diameter.
1708. If the ECLIPTIC be the Plane of Projection, then is
its Pole the Center, and all the Circles of Latitude are right.
Lines; alfo O B = x, and O M = y, are the Sine and Co-
Sine of the Star's Latitude; GH = p, and GF 4, are the
Sine and Co-Sine of its Longitude (Fig. 1.) and the vertical and
horizontal Planes are thofe of two Circles of Latitude paffing
through the folftitial and equinoctial Points, as before. And
the Stars may
therefore by the fame Formula
py
r + x
and
qy
r+x
be laid down in this Projection from Tables of their Latitude and
Longitude. Such Planifpheres we have of all the Stars in the Bri-
tish Catalogue by the fame Hand.
1709. There is alfo a very uſeful Projection of the Heavens
on the Plane of the Horizon. In this Cafe the Meridian is the
vertical Plane, and the prime Vertical the horizontal Plane.
Here the Star's Diftance from the vertical Line is
rpy as be-
rr + qy
fore (1695). Then if a, b, t be the Sine, Co-Sine, and Co-
Tan-
214
INSTITUTIONS
Tangent of the Star's Azimuth, we have a:b:: rit::
tpy
rpy
r r t qu
Star's Diſtance from the horizontal Line of the
rr + qy
Projection. But when the right Afcenfion and Declination of a
Star is known, together with the Latitude of the Place, or Pole's
Height, then the Azimuth of a Star is known alfo; whence its
Place on this Projection is eaſily affign'd.
1710. It is not common to project the ſtarry Heavens on the
Plane of the Meridian, or Circle of Latitude paffing through the
Poles of the World; but if it be required, it may be done by the
Formulæ in (1694.) by fubftituting as directed (1704). How-
ever, this Projection is not confined to Geography, but is the
Ground-work of that celebrated aftronomical Inftrument called
the ANALEMMA, or ORTHOGRAPHIC PROJECTION of the
SPHERE.
1711. For if the Eye be ſuppoſed at an infinite Diſtance, then
n r p y
n r r + q y
the Stereographic Forms (1694)
ру
nr rx
nr r + q y
and
,
become x and for the Diſtances of any Point from the hori-
zontal and vertical Lines of the Projection. And in this Cafe,
the Meridians, Parallels, and other Circles, Oblique to the
Plane of Projection, are projected into ELLIPSES. A Specimen
of this Projection you have in Fig. 7.
1712. That every Oblique Circle PGS is projected into an
Ellipfis Pa F S by parallel Rays is thus demonftrated (Fig. 8.)
Let CP CG a, CFb Semi-Conjugate of the Curve
PFS. Alfo put CM, and OM
and N My.
Then by the Property of the Circle (658) we have SM ×PM
=
=O M², that is, a + x × a—x = a²
ха
a
1-2
y,
x²= y². Hence
y MO. Draw OB parallel to CM, and
Bb parallel to GF; then OM: NM::
FC; that is, y: y::a: b; therefore y =
which is the Property of an Ellipfis (765)
by
py
↑
a
(BC; ¿C : : ) G C :
b
b
y
= =
a
a
but y = NM ==
; forra, pb, (766) therefore the Curve PFS
is an Ellipfis.
1713.
Of PERSPECTIVE.
215
1713. When the Point O coincides with G, or is in the E-
quator, then y = r, and the Diſtance CF or Semi-Conjugate
of the Ellipfis,
r
== Sine of right Afcenfion from the
Point Q
1744. Hence, as r:p::p:
r
Half the Latus Rectum of
the Ellipfis (766); but in this Cafe; therefore y = p
**
=OM, when M is the Focus of the Ellipfis; whoſe Diſtance
therefore C M from the Center, is equal to the Co-Sine right
Afcenfion, which is then equal to the Sine (x = OB) of Decli-
nation.
1715. Hence if CF, the Semi-Conjugate of the Ellipfis, be
made Radius; then, fince in this Cafer = p, we have the Di-
ру
ftance from the vertical Line every where =y; that is, all
T
the Semi-Ordinates N M of the Ellipfis will be equal to the Co-Sines of
Declination, to the Radius C F.
1716. And whatever has been demonftrated of the MERIDI-
AN POG, and its orthographical Projection, may be applied.
in the fame Manner to any PARALLEL AOL when pofited
obliquely to the Plane of Projection. For in this Cafe, PCS
may be confidered as the horizontal Line, and AML as the
vertical One; and then, if ML Radius, O D and
= OD
ON will be the Sine and Co-Sine of right Afcenfion from K;
and confequently the Distances from the horizontal and vertical
Planes. Alſo O D p will be the Diſtance from the hori-
zontal Line, and its Diſtance from the vertical Line will be
(1711).
↑
ド
​1717. Hence when O coincides with R, then q=r; and
*, the Sine of Declination (or Obliquity) from the vertical Plane
is then the Semi-Conjugate of the Ellipfis of the projected Paral-
lel. And therefore, becaufer:x:x: -Latus Rectum
**
g
2
(766) = 2, it is q = x, at the Focus of the Ellipfis; where
the
216
INSTITUTIONS
the Diſtance of the Focus, from the Center M, is equal to the
Sine of right Afcenfion, or Co-Sine of Declination.
1718. From all which it appears, that fince the Diameters,
and Foci, of the Ellipfis of any projected oblique Circle are given
by this Method, therefore fuch an Ellipfis may be drawn, by
Points, or mechanically (785) or from a Table of Sines, or
laftly, by the Sector, which is the moft ready Method of all.
1719. The ANALEMMA (Fig. 7.) is that particular Species
of the orthographic Projection, where the Eye is placed in the
common Interfection of the Ecliptic and Equinoctial, at an infi-
nite Diſtance. Thefe Planes will therefore be projected into right
Lines; and the Equinoctial will be the horizontal Line; and
the equinoctial Colure the vertical Line; as the other Colure is
the Plane of Projection.
1720. Here alfo the Parallels of Declination are all right Lines,
as they ſtand at right Angles on the Plane of Projection. The
Meridians here drawn, are thoſe of 15 Degrees interval; and
confequently are the Hour Circles of the Sphere. As to the Ufes of
the Analemma in Aftronomy, fuch as are confiderable will appear
in the Sequel of this Work, it fuffices, at prefent, that we have
fhewn its Nature and Conftruction from the Principles of Per-
fpective. In the mean Time we refer the Curious to Sutton's
Analemma of 18 Inches Diameter, in which the Meridians and
Parallels are drawn thro' every Degree, and many of the princi-
pal Stars laid down; but this Inftrument is yet capable of great
Improvements, as we may hereafter fhew.
CHA P. XVIII.
The PRINCIPLES of PERSPECTIVE applied to the
Computation of SOLAR and LUNAR ECLIPSES.
1721. T
H IE PRINCIPLES of PERSPECTIVE are applica-
ble to Projections for Solar and Lunar ECLIPSES;
and the fame Rules ferve here as in all the foregoing Cafes. For
let S be the Center of the Sun A C B (Fig. 9.) E the Center of
the
Of PERSPECTIVE.
217...
"
*
the Earth QW; and LEP a Part of its Orbit. Alfo let
XIMF be a Circle defcribed about the Earth's Center in the
Plane of its Motion, or of the Ecliptic, at the Ditance of the
Moon. And let HK be the Orbit of the Moon, making the
given Angle HK G, with the Ecliptic, at the Time of a folar
Eclipfe; and K the defcending Node.
1722. Now in Projections of this Kind, the moft-commodi-
ous Pofition of the Eye is in the Center of the Sun at S, in
which Cafe the horizontal Plane will be that of the Ecliptic; the
vertical Plane will be that Circle of Latitude which cuts the E-
cliptic in the Points of Conjunction X, and Oppofition M;
`and the Plane of Projection is that which paffes thro' the Earth's
Centre E, and touches its Orbit, when it is in the Line of Eclip-
tic Conjunction or Oppofition S M.
1723. It is evident from the Diagram that the Angle AQS is
that under which the Sun's Semidiameter appears at the Earth.
The Angle ESQ is that under which the Semidiameter of the
Earth appears at the Sun, and is call'd, the Sun's horizontal
Parallax. The Angle E,XQ or EM Q is that under which
the Earth's Semidiameter appears at the Diftance of the Moon,
and is call'd the Moon's horizontal Parallax. And the Angle
XQS is the Difference of the horizontal Parallaxes of the Sun and
Moon. For XSQ+XQ S =Q XE, (632) therefore XQS
=Q XE XS Q
1724. By the Word Parallax, is only meant the different
Place in which the Sun, or Moon appears, when viewed from
the Center of the Earth at E, and from any Point of it's Surface
as Q And, therefore, at the Diſtance of the Phænomenon,it
must be equal to the Angle under which the Semidiameter of the
Earth appears. It muft, therefore, be greateſt of all when in
the Horizon of the Place Q; and vanifh for Objects at an
infinite Diftance; fo that the Stars have no Parallax at all; and
that of the Sun ESQ is almoft immeasurably fmall.
1
1725 Thefe Things premifed, we may proceed: Let K be
the Place of the Node, and O that of the Moon in her Orbit
HK, then it is plain, her Distance from the horizontal Plane is
the Sine of her Latitude OG; her Distance from the vertical
and her Distance from the Plane of Projec
py.
Plane is OD
VOL. II.
Ff
&
tion
}
218
INSTITUTIONS
tion is ON = 22; for here we retain ftill the fame Symbols as
before (1686, 1704.) only OG is here the Moon's Latitude
from the Ecliptic, and G K the right Afcenfion.
1726. Then becauſe the Place of the Moon O is between the
Eye at S, and the Plane of Projection at E; the perſpective A-
nalogies (1693.) will be for an ECLIPSE of the Sun.
1727. As nr
ау
ру
n r p y
: nr::
:
=
r
nr r
qy
ftance of the Moon from the vertical Line.
nr rx
7
to the Di-
And as nr
: nr::x:
rrr
- her Diftance from the horizontal Line.
97
But in Eclipfes, the Latitude and Diftance of the Moon, from
the vertical Plane, are fo very ſmall, that we have yr, very
nearly; therefore in fuch Cafes the Analogies become nr
nr ::p:
pnr
9
nr
111 x
; and nr
q : nr : : x:
72 1*
1728. Now becauſe nr
-
9:
ES the Diſtance of the Sun, and
9 = r = EX nearly, the Diſtance of the Moon; and ſonr
q=SX; therefore fince SX: SE (SQ) :: Angle SQX:
Angle SXQ or EXQ (718) the first Analogy is, in Words;
as the Difference between the horizontal Parallaxes of the Sun and
Moon, is to the Parallax of the Moon; fo is the Moon's Longitude
(at the given Inftant) from the vertical Plane, to her Distance from
And, fo is her Latitude, to her Diſtance
from the horizontal Line. Note, the Longitude and Latitude are
here put for their Sines, there being no fenfible Difference in
fuch very fmall Arcs.
the vertical Line.
1729. If therefore from the aſtronomical Tables you take
the Moon's horizontal Parallax, and with it, as a Radius, you
deſcribe a Circle on E, as QYWZ; then will QW be the
horizontal Line, and Y Z, the vertical Line, on the Plane of
Projection, which if it be about 10 or 12 Inches in Diameter
will be fufficient for the Delineation of all the Phafes of a folar
Eclipfe.
1730. For fince the Diſtance of the Sun SE is vaftly great in
reſpect of the Earth's Semidiameter EQ, therefore the Hour
Circles and Parallels will all be orthographically projected, and
may be ſo drawn on the ſaid Plane (by 1711 1718.) and fince
by
Of PERSPECTIVE.
219
by the above Analogies, the Path of the Moon over the Diſk of
the Earth (or Plane of Projection) is determined by finding her
Place thereon at two Inftants of Time, one juft before the Be-
ginning, the other after the End of the Eclipfe; it will be eaſily
feen how the Moon, and any particular Place, as LONDON, are
fituated in reſpect of each other, and what are their Diſtances a-
part at any given Moment, during the Paflage of the Sha-
dow.
1731. For if from aftronomical Tables you take the Sun's
Semidiameter in your Compaffes, and with one Foot placed on the
given Moment of Time in the Parallel of London, vou deſcribe a
Circle; and then from the faid Tables you take the Moon's appa-
rent Semidiameter, and with one Foot of the Compaffes, placed in
the fame Moment in the Moon's Path, you defcribe another Cir-
cle; theſe two Circles will fhew how the Diſks of the Sun and
Moon are related to each other in the Heaven, in regard to a
Spectator at London.
1732. Thus if the Circles do not touch, the Eclipfe is not
begun, or over; if the Circles juft touch, the Eclipfe is then
juſt beginning or ending. But if the Circle of the Moon lies
over that of the Sun, there is an Eclipfe. If the Diameter of
the folar Diſk or Circle be divided into 12 equal Parts, or Di-
gits, then the Number of thefe covered by the lunar Circle,
will fhew the Phaſe and Quantity of the Eclipfe. But on theſe
Particulars we muft not enlarge any further at prefent.
Of a LUNAR ECLIPSE.
1733. When the Moon is fo near the Node at the Oppofition
M, that her Latitude is lefs than the Semidiameter V M of the
Earth's Shadow QT W, it will be more or leſs immerſed into,
and eclipsed by it. And then by having the Moon's Latitude,
and Longitude (or Diſtance reduced to the Ecliptic) from the
Plane of the Circle of Latitude paffing through M, the Point of
Oppofition, you will by the fame Analogies find her Place for
any Inftants of Time on the Plane of Projection, and thereby be
able to delineate her Path. Only here, becauſe the Plane IF is
between the Eye and the Object, the Sum of the Parallaxes is
to be taken; for now it is SM:SQ:: SQM (or R QM) :
F f 2
SMQ:
220
' ';
INSTITUTIONS
و
}
SMQ; but RQMQMS + Q'S M (632)
the Parallaxes of the Sun and Moon.
·
Sum of ',
1734. But to reprefent the lunar Eclipfe, the Diameter of the
Earth's Shadow at the Point M must be known. In order to
that, it is to be obferved, that the Angle M QR confifts of two
others, viz. RQV and V QM; of which R QV = AQ M
apparent Semidiameter of the Sun; and the other V QM=
apparent Semidiameter of the Shadow at M. But R Q M
QMS+QSM; therefore QSM + QMS-RQV =
VQM. That is, from the Sum of the horizontal Parallaxes
of the Sun and Moon, fubtract the apparent Semidiameter of
the Sun, the Remainder is the Semidiameter of the Earth's Sha-
dow at M, and is therefore given from the aftronomical Ta-
bles.
1735. With this Semidiameter, as a Radius, defcribe a Cir-
cle on the Plane of Projection about the Point E, and fhade it
to reprefent the Section of the Earth's Shadow. From this Point.
E, let fall a Perpendicular to the Path of the Moon, and that
will fhew the neareſt Diſtance of the Centers of the Shadow and
Moon. And by defcribing feveral lunar Circles at given In-
ftants of Time in the Moon's Path, the Phafe and Quantity of
the Eclipfe will appear at each particular Inſtant (1732.) And
as the Shadow of the Earth, at the Moon, is in Diameter near
three Times as large as the Moon, therefore a lunar Eclipfe
may be total to all the World for a confiderable Time, whereas
an Eclipfe of the Sun can be little more than momentarily fo, as
the Difk of the Moon but very little exceeds that of the Sun, and
that too but really happens in Eclipfes.
1736. We have now explained and applied the perspective
Principles to the Doctrine of Eclipfes, and though there may
be readier Methods of conftructing an Eclipfe, there is none fo
punctual and exact as this, when applied by the Method of Inter-
polation; but that will be the Subject of a future Part of this
Work.
+
1737. The fame perfpective Analogies are applicable to the
TRANSIT of VENUS and MERCURY over the Sun's Difk; and
as there is no Difference in the Method of delineating the Path
of the Planet, and that of the Moon in a folar Eclipfe, but
what
L
Of PERSPECTIVE.
221
what is merely verbal; and that the Difk of the Sun is to be put
for the Diſk of the Earth, it is prefumed what has been ſaid is
fufficient for the Conftruction of any Tranfit of thofe Planets.
CHA P. XIX.
GNOMONICS; or the PRINCIPLES of PERSPECTIVE
applied to the ART of DIALLING.
1738. N the Application of PERSPECTIVE to DIALLING,
IN the
EN
the Pofition of the Eye and Plane of Projection are in-
terchanged; or the Eye is in this Cafe placed in the Center of
the Sphere, and the Planc in Contact with its Surface. For in de-
lineating the Hour-lines on a given Plane, it must be confidered,
that the vifual Ray which connects the Sun, and the Eye de-
fcribes the Plane of each Hour-circle, and of Courfe, in its Mo-
tion through any Plane on the Surface of the Sphere, it will trace
out the Hour-line on that Plane.
1739. And it is quite the fame Thing whether we confider
the Plane of any Hour-circle projected on a given Plane at the
Surface of the Sphere by Rays from the Sun, or by viſual.
Rays from the Eye in the Center; for fince both the Sun and
the Eye are in the projected Hour-plane, the Projection itſelf
muſt be a Right-line on the given Plane (1464).
1740. And, in fhort, fince all great Circles of the Sphere
pafs through the Eye in the Center, therefore they will all be
projected into Right-lines on a Plane in any Pofition whatfo-
ever. But with regard to leffer Circles, they will be of one or
other of the Conic-fections, as they are pofited relative to the
Plane.
1741. Let AP QS (Fig. 1.) be a great Circle of a Sphere,
then will it be projected into the Right-line A B, infinitely con-
tinued on a Plane touching the Sphere in the Zenith Z. And
any Part or Arch ZP is projected into Z D the Tangent there-
of.
1742.
222
INSTITUTIONS
1742. Let IL be the Diameter of any leffer Circle parallel to
the Plane A-B, then will its Center K be in the Perpendicular
N Z, and is projected into the Point Z; its Semidiameters I K
and KL into the Tangents ZE and ZF, which as they are.
equal to each other, fhews the Projection of a leffer Circle paralle!
to the Plane, is a CIRCLE upon the Plane.
1743. Let LG be a Diameter of a leffer Circle in fuch an
oblique Pofition to the given Plane A B, that it may be wholly
projected upon it within a finite Diftance Z B; then it is evident
the Center M will be projected into the Point D on the Plane,
the Semidiameter I M into the Right-line FD, and the Semi-
diameter M G into the Right-line DB; and fince it is evident
that D B muſt exceed DF, the Projection cannot be a Circle.
But confidering LNG as a right Cone, its Section (if continued
out) by the Plane A B, muſt be an ELLIPSIS (763).
1744. Again, it is as evident, that if the Diameter GO of
any leffer Circle paffes through O, the extreme Part of the Dia-
meter HO parallel to the given Plane A B; then, becauſe the
Cone GNO (infinitely continued) is cut by a Plane parallel to
one of its Sides NO, the Projection of any Circle GO on that
Plane will be a PARABOLA (740).
1745. A fmall Circle pofited in any other Manner than what
is above ſpecified, will be projected into an HYPERBOLA, as is
manifeft from the fame Principles of the Section of a Cone
(765).
any
1746. We have fhewn that the Interfection of the Plane of
Hour-circle, with a given Plane on the Surface of the Sphere,
is the true Hour-line on that Plane (1738) and the Cafe is the
fame if the given Plane paffes through the Sphere in any Part,
provided it be in a parallel Pofition; for the Planes of the fame
Hour-circles will interfect two or more parallel Planes in the
fame Manner; and therefore the Hour-lines will be the fame in
all; this is evident from (615 and 631.)
1747. Hence it is common to repreſent the given Dial-plane
as paffing thro' the Center of the Sphere, as in Fig. 2. where it
is denoted by DW BE. The Sphere with its Hour-circles, is
there orthographically projected (1711) EQ is the Equator;
ÆP QS is the Meridian, or Hour-circle of XII; and a, b, 6₂
d, e, f, N, are the Elliptic-projections of the Hour-circles of
12,
Of PERSPECTIVE.
223
12, 11, 10, 9, 8, 7, 6, in the Morning, which are all at the
equal Diſtance of 15 Degrees on the Sphere; for 24) 360° (=
15° per Hour in the Sphere's Revolution.
1748. The Planes therefore, of thefe Circles, will interfect.
the given Plane DW BE in the Lines N 12, N 11, N 10, N9,
&c. which will therefore be the Hour-lines on that Plane; and
thus they are to be confidered in all the other Quadrants of the
Plane.
t
1749. But when the Hour-lines are affigned or delineated on
a Plane, it is neceffary there fhould be fome Contrivance to in-
dicate or point out the Moment of Time when the Sun is fuc-
ceffively in thoſe horary Planes; now if the Plane of the Dial
paſs through the Center of the Sphere, the Expedient we
require is found in the Axis of the Sphere PS. For fince all the
Planes of the Hour-circles interfect each other in the faid Axis,
the Sun when it comes on any one horary Plane will project
that Axis into the Hour-line on the Dial-plane proper to that
Hour-circle.
1750. Moreover, as the Axis PS of the Sphere is that Line
about which its Motion is performed, it muft in it's felf be
confidered as abfolutely at Reſt. For any Line parallel to the
Axis, in the Plane of any Hour-circle, has lefs Motion in Pro-
portion as it is nearer to the Axis; and confequently when it
coincides with, or becomes the Axis, it can have no Motion at
all.
1751. The Axis therefore of the Sphere retaining always the
fame Poſition, if it be fuppofed to confift of a fine inflexible Wire,
it will intercept the Sun's Rays in the Planes of the Hour-cir-
cles, and therefore its Shadow muft conftantly fall on the Hour-
lines of the given Dial-plane, correfponding to the reſpective
Planes of the faid Hour-circles; fo that the Axis of the Sphere, by
Means of its Shadow, will conflantly indicate the Moment of Time
when the Sun is in any Meridian.
1752. Now the Plane DW BE is called the horizontal
Plane with reſpect to the Place Z, becauſe it is parallel to the
horizontal Line AB, touching the Sphere in that Point; and
when all the Hour-lines are drawn upon the Plane for all the
Meridians or Hour-circles the Sun can be in above the Plane,
then
224
INSTITUTIONS
then this Plane with its Hour-lines, and Semiaxis N P com-
pleats the HORIZONTAL DIAL in the Sphere.
1753. The Semiaxis N P is called the GNOMON or STILE of
the DIAL; and its Elevation or Height BNP above the Plane
of the Dial, is always equal to Z the Latitude of the Place.
For EP ZB Quadrant; from each of which take away
the common Arch ZP, and there will remain AZ PB.
Therefore alfo the Stile's Height is equal to the Elevation of the
Pole in the Latitude Z.
t
1754. If the Plane be in a vertical Pofition, or paffes thro'
the Zenith and Nadir Points Z and D, and is the Plane of the
prime Vertical (as Fig. 3.) then fince the Sun cannot be on or
above fuch a Plane, from VI to VI; no other Hour-lines
need be drawn upon it. And in this Cafe, the Oppofite or South
Pole S is elevated above it in the Angle DNS = ZNP
Complement of the other Elevation PNB to a Quadrant. Here
alſo the other Semiaxis NS is the Stile; and this is called an
erect and dire& SOUTH DIAL. And the other Side of the Plane
is a North Dial, for the Hours before VI, and the Stile is
NP.
»
the
1755. It is eafy to obferve, that this vertical Dial is an hori-
zontal One in the South Latitude D, which is the Co-Latitude
of the Point Z, fince Z D 90 Degrees. The Meridian
Line N D or Hour-line of XII is directly under the Gnomon or
Stile NS, and is therefore called the Subftilar-line or Sub-
Stile.
1756. If we fuppofe the Plane to continue vertical, but to de-
cline or move towards the Eaft or Weft; then it becomes an
Eaft or Weft declining DIAL; and it is evident in fuch Cafes,
the Angle which the Stile NS makes with the Plane is leffened,
and the Subftile departs from the Hour line of XII, or rather,
this departs from that, towards the Eaft or Weft.
1757. When the vertical Dial has a Pofition directly Eaft or
Weft, then the Axis PS of the Sphere is in the Plane; and con-
fequently the Stile of an Eaft or Weft Dial, muſt be parallel to
the Plane thereof, and the Height of it may be taken at Pleaſure,
and will be always equal to the Diſtance of the Hour-lines of III
and IX, from the Middle-line of VI. And as the Plane of the Dial
is
E
Z
F
A
D
B
I
'P
V
Æ
H
--
ལ་
བ་མའ
If
****
****
-
འ མ
A
τη
Fig
Fig 2.
D
IV
T
N
R
A
1
Æ
H
H
F
P
G
Τ
VI
V
Prime
Meridian
XI
12
VII
Vertical
B
Plate VII
PERSPECTIVE
DIALLING.
D
B
୯
Z
– '
A
B
H
R
a
N
Fig
6.
V
VI
ச
IV
એ
M
IX
X
d
Z
XI XII
4.
G
Fig
I
II
e.
d
Fig. 5.
W
P
Fig. 3.
B
20
00
B
H
G
D
ΠΛ
ДА, М
T
DO
B
A
ZH MÁ ITA
E
E
1
Fig 7
୯
Of PERSPECTIVE.
225
is now in the Plane of a Meridian, it is an horizontal Dial under
the EQUATOR every where, and is therefore called an EQUA-
TORIAL DIAL.
:
1758. With refpect to the horizontal Plane D W BE (Fig.
2.) it is evident, that as the Point Z approaches the Pole P,
the Plane will approach to that of the Equator Q in which
Cafe the Axis PN becomes perpendicular to the Plane, and
marks out the Hour by dividing the Plane into 24 equal Parts;
and fuch an One is called a POLAR DIAL, as being an horizon-
tal Dial under the Pole P.
1759. This POLAR DIAL (commonly called an equinoctial
Dial) with a perpendicular Stile, being drawn on a Plane, and
that Plane elevated to the Latitude of a Place, the Hour of the
Day will be fhewn upon it, by the Shadow of the Perpendicular,
with the fame Exactnefs as on a common horizontal Dial for
that Latitude; and therefore this Dial is, in its own Nature,
an UNIVERSAL DIAL, for all Latitudes or Parts of the
World.
1760. The Eye being in the Center N of the Sphere projects
the Parallels of the Sun's diurnal Motion on a given horizontal
Plane into one of the Conic Sections; for let EC be the Sun's
Declination, then CR is the Diameter of the Parallel of its
Motion on that Day, and CNR the Section of the Cone de-
ſcribed in the Sphere by the vifual Ray drawn from the Eye to the
Sun. Now fince CR is parallel to the Equator EQ, the Axis
PS will be perpendicular thereto; and therefore upon a hori-
zontal Plane at the Pole P, which is parallel to CR, the Pro-
jection of the Parallel, which is a Circle, will be a CIRCLE
alfo (1504).
1761. Produce the Side of the Cone R N to V; then ÆC
=ÆV; and on V R erect the perpendicular L N on the Cen-
ter N. Then upon an horizontal Plane in the Latitude L,
which will be parallel to the Side of the Cone N R or VR, the
Projection of the Cone's circular Bafe, or parallel of the diurnal
Arch, will be a PARABOLA (1505).
1762. Hence upon every horizontal Plane between the Pole
P and the Latitude L, the circular Bafe, or Parallel muſt be pro-
jected into an ELLIPSIS, more eccentric in Proportion as the
VOL. II.
G g
Plane
226
INSTITUTIONS
Plane is nearer the Point L (1507) for every fuch Plane will
cut both the Sides of the Cone CNR produced.
1763. And on the other Hand, upon every horizontal Plane,
between the Equator and the Latitude L, the Baſe of the Cone
will be projected by the Eye into a HYPERBOLA (1506) be-
caufe fuch a Plane muft cut the Side CN of the Cone CNR,
and the Side VN of the oppofite Cone V NT, when pro-
duced.
1764. Becauſe EC LP, therefore L = CP =
Complement of the Sun's Declination; therefore in all Lati-
tudes less than the Co-Declination, the Projection is an Hyper-
bola; in all Latitudes greater, it is an Ellipfis; and in the Lati-
tude equal thereto, it is a Parabola. In the Latitude 90°, the
Ellipfis becomes a Circle; and in the Latitude oo, or in the E-
quator, the Hyperbola degenerates into a Right line.
1765. Hence we have an eafy Tranſition to the Shadows caft
by a Gnomon on the Plane of any horizontal Dial. For the faine
Ray which projected the Sun's diurnal Path or Parallel on a
Plane, projects alfo the Shadow from the Point of the Gnomon
on the fame Plane. Thus fuppofe a Plane (abc d) placed pa-
rallel to A B, below the Center N of the Sphere, cutting the
Axis in E, fo that the Part of the Axis EN becomes the Gno-
mon on that Plane, then it is evident, the fame Ray NC pro-
jects the Parallel of CR on the Plane at A B on one Hand, and
the Shadow of the extreme Point N of the Gnomon on the Plane
(abc d) on the other; and that when the Sun is in the Meridian
at C, the Shadow of the Gnomon falls in the Subftile Ef on
the Point I, at the Diſtance F I from the perpendicular F N.
•
1766. Now becauſe of the Parallelifmn of the Planes, the
Curves defcribed by the Shadow will be exactly the fame as thoſe
of the projected Parallel of the Sun's diurnal Path; that is to
fay, (1 Under the Pole P, the Shadow of the Point N will
defcribe a CIRCLE. (2.) At the Latitude L, equal to the Co-
Declination of the Sun, it will defcribe a PARABOLA. (3.) On
all other Planes between P and L it will defcribe ELLIPSES.
(4.) In all Latitudes from the Equator to the Point L, the Path
of the Shadow will be a HYPERBOLA. And (5.) When they
Sun is in the Equator at E, the Shadow will fall on K, at
Noon,
Of PERSPECTIVE.
227
Noon, and its Path will that Day be the Right-line g Kh.
(1760 to 1765.)
1767. Hence it appears, that when the Sun enters Cancer,
and ÆC = 23° 30', then L is in Latitude 66° 30°, and confe-
quently, the Shadow can defcribe Ellipfes only in the frigid Zone
LP, when the Sun is altogether above that Horizon; and its
Altitude on the South and North Part of the Meridian un-
equal.
1768. If the Height of the Gnomon N F be made. the Radius,
then the Angle NIF CNH
is the Sun's Meridian Altitude,
and the Angle IN F its Co-Altitude; then fay, as Radius to
NF, fo is Co-Tantent of the Meridian Altitude, to the Di-
ftance of the Shadow FI; when the Sun is in C. If CI be the
Ecliptic, then the Points I, K, and W, will be found by this
Analogy for the folftitial and equinoctial Days.
1769. Hence if the Dial-plane be of a fufficient Length, all
the Parallels of the Sun's Declination may be deſcribed thereon
at his Entrance into each of the 12 Signs. Thofe of the firft Six,
or Summer half Year, will fall between K and I; and thofe for
the Winter, between K and W. In our Latitude of 51° 30', we
have Radius Tangent of 75° :: FN FW:: 13,7
ncarly.
:
1770. Suppoſe ÆZQS a graduated Brafs Meridian, ſuſ-
pended from a Ring at Z, which is moveable and fet to the La-
titude of the Place; then if CR be the Tropic of Cancer, and
VT the Tropic of Capricorn; and (nm) a Plate of Braſs in the
Axis of the Sphere, with a moveable Piece containing a ſmall
Hole which may be placed any where between n and m; the
Ray of Light paffing through that Hole, when properly ad-
jufted, will, from the Meridian Sun, always fall on the North
Point of the Equator, for when the Sun is in Cancer at C,
and the Hole at (n) the folar Ray will be # Qparallel to CN;
and when the Sun is in Capricorn at V, and Hole at (m) the
Ray will be in Q parallel to V N, and therefore every Day, at
Noon, the Ray will pass through the Hole to the Point Q, or
Hour of XII in the Equator.
11
1771. Alſo when the Sun is on any other Hour-circle before
or after Noon, it will ftill pafs through the Hole, (adjuſted to the
Day,) to the Equator, but just as far from Q, as the Hour-cir-
Gg 2
cle
228
INSTITUTIONS
cle is from the Meridian; becauſe the diurnal Motion of the
Ray N C, and the Motion of the Ray nQ parallel to it, will
neceffarily be the fame; therefore the Ray through the Hole in
the Axis will ever point out the Hour on the Periphery of the
Equator. And this is the Rationale of the Conſtruction of the
univerfal RING-DIAL, the Form of which is exhibited in
Fig. 5.
1772. We fhall finiſh this Theory of perspective Dialling,
with obferving that the Hour-lines of any Dial are laid off by
the Tangents of the Angles which the projected Meridians
make on the Horizon of the Place. For let W NES (Fig. 6.)
be the Horizon of LONDON, Latitude 51° 30', on which pro-
ject the Sphere, and the horary Meridians will interfect the Ho-
rizon in the Points 1, 2, 3, 4, &c. The Eye being in the
Center C; let the Plane of Projection touch the Sphere in N;
then is (ab) the horizontal Line, and NC the vertical Line;
and then to fhew from the Rules of Perfpective, that the Hour-
line of II paffes thro' (2) the Point of Interſection of the 2 o'Clock
Meridian with the Horizon; we have CN=r, c2 = Ng=
verfed Sine, or Diſtance from the horizontal Line; and
Nc = g 2 = s = Sine of the Arch N 2, or, Diftance of the
Point (2) from the vertical Plane. Then becauſe that Point is
between the Eye and Plane of Projection, the Analogy is (1691)
as nr - piri:s:
rs
* S
(becauſe here n = 1)
nr
V
V
== (putting = Co-Sine Cg) which therefore is the Di-
C
c
ftance Nd of the Hour-line of II from the vertical Plane CN.
rs
But
But
=t=Nd
Tangent of the Arch N2; (for s:c:: r
CN:Cd(711.)
it, or Cg : C 2 :: CN: Cd (711.) Therefore the Hour-lines of a
horizontal Dial are drawn from the Center C thro' the Points of Inter-
fection of the Hour-circles with the Horizon of the Place.*
INSTI
* N. B. Fig. 7. is here inferted from Bion, repreſenting a general
borizontal Dial; a particular Defcription of which will hereafter be
given.
(229)
INSTITUTIONS
O F
1
SPHERICAL TRIGONOMETRY.
CHA P. I.
Of the Nature of SPHERICAL TRIANGLES, with
a SOLUTION of all their CASES.
1773. WE have heretofore confidered the Dimenfions of a
SPHERE with regard to its fuperficial, and folid
Contents in the Inftitutions of plain Geometry; that which we call
Spherical Geometry being converfant chiefly about the Relation,
Magnitude, and Inclination of the GREAT CIRCLES of the
SPHERE; and more particularly the various TRIANGLES that
are form'd by their Interfections on its Surface; whofe Properties
and Affections we now proceed to treat of.
And firft of thoſe
which are Rectangular.
1774. A Spherical Triangle, is that whoſe three Sides are feve-
rally the Arches of three great Circles of the Sphere, interfecting each
other in the three Angular Points. And the Angles of every
Triangle are meaſured by the Arches of Circles; therefore the
Arches of Circles are the Meaſures of every Part of a Spherical
Triangle.
1775. And as the Quanity of the Arches themfelves is deter-
minable only from the Relations of thofe right Lines which
are called their Sines, Tangents and Secants, as they ftand con-
ſtructed with the Radius in a plain Triangle, (706 to 708.)
Therefore the Solution of a ſpherical Triangle, differs in no-
thing
230
INSTITUTIONS
thing effential to the Operation, from that of a common plain
Triangle,
1776. Let ABC (Fig. 1.) be a right-angled ſpherical Tri-
angle on the Surface of the Sphere, and ſuppoſe the Sides con-
tinued out indefinitely in the Baſe AB to E; the perpendicular
AC to H; and the Hypothenufe BC to J. On the Angle B
draw the great Circle GFE thro' D the Pole of the Baſe AB;
and on the Angle C, draw the great Circle G H I thro' G, the
Pole of Hypothenufe B C. Then will there be form'd the four
Triangles E B F, EDA, ICH and IGF, each having two
right Angles, and two quadrantal Sides, as is evident by Infpec-
tion.
1777. There are alfo form'd two other right Angled ſperical
Triangles. viz. CDF right Angled at F, and G D H, right
Angled at H, which may be called complemental Triangles with
reſpect to the Original, or given Triangle A B C.
1778. For in the first CED the Bafe CF is the Comple-
ment of the Hypothenufe BC, and the Angle at the Perpendi-
cular, DA E, is the Complement of the Baſe A B. And
the Hypothenufe CD, is the Complement of the Perpendicular
A C. And, laftly, the Perpendicular D F is the Complement
of EF B, the Angle at Bafe; the Angle C at the Perpendi-
cular, being the fame in both.
1779. And in the other Triangle DG H, the Bafe G H is the
Complement of HIC in the given Triangle ABC. The Per-
pendicular DHAC, becaufe A D and CH are Quadrants.
The Hypothenufe D G EF = B the Angle at Bafe.
=
The
Angle at Bafe GIF BC the Hypothenufe of the given
Triangle. And the Angle at the Perpendicular, D AE, the
Complement of the Bafe A B.
1780. Now by help of thefe Quadrantal and Complemental
Triangles, the Canons or Analogies for the Solution of all the
Cafes of a right Angle, fpherical Triangle ABC, may be eaſily
derived from the two following Theorems.
THEOREM I.
In a right Angled Spherical Triangle, the Radius (or Sine of 90 De-
grees) to the Sine of the Bafe, as the Tangent of the Angle at Bafe
is to the Tangent of the Perpendicular. For let BP (Fig. 2.)
be a fourth Part of the Orthographic Projection of the Sphere on the
Plane
Of SPHERICAL TRIGONOMETRY.
231
Plane of the Meridian (1710.) And let A B C be a right-an-
gled ſpherical Triangle form'd on the Globe, by the three great
Circles B, BF, and AP. Then is B C the Sine of the Hy-
pothenufe, and A B the Sine of the Bafe (1690.) CH is the
Sine of the Perpendicular A C, and F E the Sine of the Angle at
Bafe B; alfo A D, and Æ G, are the Tangents of the fame Parts.
And becauſe of fimilar Triangles A BD and ÆBG, (657)
we have as E B Radius is to A B the Sine of the Baje, ſo is Æ G the
Tangent of the Angle at Bafe, to AD the Tangent of the Perpendi- ·
cular A C. Q. E. D.
1781. THEOREM II.
The Radius (or Sine of 90 Degrees) is to the Sine of the Hypothenuſe,
as the Sine of the Angle at Bafe is to the Sine of the Perpendicular,
in every right-angled ſpherical Triangle. For by Reaſon of the
fimilar Triangles CBH, and F BE, we have FB: CB::
FE: CH. 2. E. D.
1782. By theſe two Theorems, all the Cafes of a right An-
gled ſpherical Triangle are folved. That is, of the three Sides
and three Angles, if any two are given, befides the right An-
gle, the other three Parts may be found; and here it must be ob-
ſerved, that either of the two Sides A B or A C is reckon'd the
Baſe, as the Angle B or C is given. And therefore there can be
but a fixfold Variety, or different Cafes in regard to any right-
angled Triangle, which here follows.
1783. CASE I.
Given the Bafe AB, and Angle B;
Perpendicular AC. In the quadrantal
(Fig. 1.) to find the
Triangle EBF; it is
SBE: SBA::tFE:tAC; that is R:BA:: 1B: tAC
(1780).
1784. To find the Hypothenuſe BC; in the quadrantal Trian-
gle ADE, we haves DE: s DF::t AE : ¡CF; that is, R :
cs B: ct A B: ct BC.
1785. To find the Angle C. In the quadrantal Triangle F G I,
we have, as s FG: DG; SD:sGH; that is, R: B :
cs AB :cs C (1781).
s
S
1786. CASE
232
INSTITUTIONS
1786. CASE II.
Given the perpendicular AC, and Angle at Baſe B; to find the
Baſe AB. In the quadrantal Triangle FBE we have EF:
tAC::SBE: SBA; .tBtAC:: R: sBA.
1787. To find the Hypothenufe BC; we have sEF: SAC::
SBF: SBC; .. SB: SAC:: R: SBC.
1788. To find the Angle C. In the quadrantal Triangle HCI,
it is, s DCs DH::s DF:SHI; . cs AC: R::cs Ba
Bi
SC.
1789. CASE III.
Given the Hypothenufe BC, and the Angle at Bafe B; to find the
Bafe A B. In the quadrantal Triangle ADE, we have s DF
:S DE CF tAE; .. csB: R:: ct BC ct AB,
or Rcs B::tBC: tA B.
:s :: :
AB.*
1790. To find the Perpendicular AC.
angle EBF, there is sBF: SBC
SBCSB: SAC.
In the quadrantal Tri-
EF: SAC; .. R:
រ
1791. To find the Angle C. In the quadrantal Triangle HCI,
it is CFCI:: DF:tHI; that is, csBC: R:: ct B
s SCI::tDF:
:t C.
1792. CASE IV.
Given the Bafe A B, and Perpendicular AC; to find the Hypo-
thenufe BC. In the quadrantal ADE, there is sAD: DC
:: SAE: SCF; that is, R: csCA::cs AB: cs BC.
1793. To find the Angle B; we have s A B R::tAC:t B
(1786).
1794. To find the Angle C; s AC: R::AB: C; by the
fame Reaſon as in the laſft.
1795. CASE V.
Given the Hypothenuſe BC, and the Perpendicular AC. T
find the Bafe AB. Ascs AC:R::cs BC: csAB (1792).
1796. To find the Angle B; s BC: R: :SAC: SB (1787).
1797. To find the Angle C; in the quadrantal IGF, we have
tIF : ƒ DH ::sGI:sGH; that is, t BC: R::tAC:
CSC.
1798. CASE
* Becauſe the Co-Tangents of any two Arches are inverſely as their
Tangents, as is fhewn hereafter at (1831).
Plate I. Of Spherical TRIGONOMETRY.
H
Fig. 1.
B
X
H
B
Fig.3.
E
I
D
F
A
2
E
A
a
I
C
M
R
a
F
D
À
Fig.5.
B
D
N
A
tal
P
Fig.
2.
AH
B
B
Fig.4.
H
F
G A
P Q
P
Fig.9
B
F
E
Fig. 6.
A
E
れ
​D
A
୯
Fig. 8.
B
ក្ម
H
・Fig.7
D
M
R
H
m
I
1
K
P
Of SPHERICAL TRIGONOMETRY. 233
1798. CASE VI.
Give the Angles B and C; to find the Bafe AB. As sB:
CSC:: R:cs AB (1785).
1799. To find the Perpendicular AC. As sC: csB::R:
ESAC (1785).
1800. To find the Hypothcnufe BC; in the quadrantal IC H,
we have tHI:tDF:: SCI:CF; which gives t C: ctB
::R:cs BC.
1801. The two Angles B and C of every right-angled ſphe-
rical Triangle, are together greater than the right Angle A;
for the Angie B EF GD; and the Angle CIH;
is greater than IH + HG 90 Degrees,
but I H + DG,
or a right Angle, whence the Theorem is evident.
CHA P. II.
The Method of determining the FLUXIONS of the
SIDES and ANGLES of a right-angled SPHERICAL
TRIANGLE.
1802.
[PLATE I. Of SPHERICAL TRIGONOMETRY.]
F the Semi-diameter of the Earth bore no fenfible Pro-
portion to the Diſtance of the heavenly Bodies; if we
could view them not through a refi acting Medium; if the Pofi-
tion of the Earth's Axis were immutable, and laftly, if there were
no Obliquity of the Ecliptic, then the Subject of this Chapter
would be unneceſſary.
1803. But asthe Cafe now ftands, we find a fenfible Difference
between the true and apparent Place of the Sun, Moon, and Planets,
on Account of their Parallax (1724.) Refraction (1321.) Motion
of the Earth's Axis, &c. by Means of all which the Latitude,
Longitude, Right-Afcenfion, Declination, Altitude, Azimuth, Am-
plitude, Hour-Angle, &c. will all be affected in the fare Man-
ner as if the great Circles by which they are aſcertained were
continually moveable through a very finall Space, one Way and
the other, about their Axis in the Sphere.
Hh
VOL. II.
1804.
234
INSTITUTIONS
1804. Therefore it becomes neceffary, where Exactneſs is
required, (as in Aftronomy, Navigation, &c.) to make a proper
Correction of the Computations made by ſpherical Triangles,
by fuppofing fuch a fmall Motion in any one of the Circles of
which that Triangle confifts, and from thence to find the Fluxi-
qnary Increaſe or Decrease of its other Sides and Angles refpective-
ly, occafioned thereby; which will exprefs the Quantities fought
from the Parallax, Refraction, &c. given.
1805. Therefore (in Fig. 1.) if we fuppofe the Circle HD A
to be moveable upon the Pole D; and to change its Situation
from HDCA to bDca; then it is evident, that in the three
variable Triangles ABC, DCF, and DH G, there will (be-
fides the Right-angle) be one Part conftant in each, and the
other Parts all variable; thus the Angle B EF, is conftant
in ABC; the Side DF in DCF; and the Hypothenufe GD
in DG H.
=
1806. For Diftinction Sake, let X, Y, Z, denote the Hy-
pothenufe, Bafe, and Perpendicular of the Triangle ABC, and
let Cd be drawn at Right-angles to the Circle bDq. Then is
AaY, the Fluxion of A B; but Ỷ is the Fluxion of
and dc =
and deŻ. Alio the
X,
its Complement AE; Cc = X,
Fluxion of the Angle CIH, is He
onary Parts are thus determined.
C. Which Fluxi-
1807. In the fluxionary quadrantal Triangle A D a, we have
(1780.) $ DA: SDC:: Aa: Cd, or RsDC:: Aa: Cd
s DC x Ỷ cs AC X Y
R
R
1808. The ſmall fluxionary Triangle Cd may be eſteemed
Rectilineal, which gives this Analogy. As tC (= c) : R :: Cd
CSAC XY SAC
Ý
: cd; that is, tC: R::
R
× Y=cd=
tC
Ż (1807).
1809. Alfo, in the fame Triangle, as sC: R:: Cd: Cc::
es AC
CSAC
X
x Ÿ=Cc = X.
7
R
SC
1810.
Of SPHERICAL TRIGONOMETRY. 235
1810. Laftly, s DA: sDH:: Aa: He (1781); but DH
}
AC; therefore R: SAC:: Ý:
SAC
R
He e
X × Ỷ = H = C,
or the Fluxion of the Arch IH which meaſures the Angle C.
1811. In the Triangle CDF, having one Side DF conftant,
the Fluxions of the other Parts are derived from the above Equa-
tions by Subſtitution of equal Parts; for becauſe cs AC =
¿DC, and + X=-X in Quantity; therefore
Ý
Ỷ =
Bafe CF.
SDC
SC
X (1809), the Fluxion of the decreaſing Side or
1812. For the fame Reafon
s DC
tC
× Ÿ
-Ż = cd, the
Fluxion of the decreafing Hypothenufe D C (1808).
CS
1813. Laftly, we have
SCD
R
× Ỷ = Ċ=He, the Flux-
ions of the increafing Angle C, or Arch IH. The Fluxion of
the Angle D, or Arch A E is negative, or Aa- Ÿ
R
cs C D
× Ċ.
1814. In the Triangle DGH, the Hypothenufe DG being
conftant; the Fluxions of the variable Parts are found by Subfti-
tution of Equals as before. Thus
C S DH
ctGH
× ÝŻ, or Fluxion
of the Side D H, which is Pofitive (1808).
C S DH
csGH
1815. Alfo, it is
x Ý X, the Fluxion of the An-
w
gle G, which is alſo poſitive (1808).
1816. And then for the Fluxion of the Side GH, (which
will be, in this Cafe, negative) we have
s DH
R
× Y = ¿.
1817. In the Demonftrations hitherto, the Letters X, Y, Z,
are uſed in Reference to three Triangles, and fometimes they
ſeparately denote an Angle, and fometimes a Side. But if we
have regard to one Triangle only, we muft ufe fuch Symbols as
will denote the feveral Parts abfolutely. Therefore let H be
the Hypothenufe; A and a, the two Sides; and D, b, the two
Hh 2
An-
236
INSTITUTIONS
Angles adjacent to them refpectively. Then will the Equations
in each Cafe or Triangle be transformed, in the following
Manner.
1818. CASE I.
X
When one Angle B is invariable, we have c 1 A C × Ỷ =
† C × Ż (1808.) become c sa × Å = tb × a, and fo of the
Reft, whence we derive the following Equations and Analo-
gies.
1. csaÀ = tba; . . Å ‡ à : : t b : c sa (1808).
2. c s a Å — s b È ; ... A: H:: sb: csa (1809).
3. sa Å = .'. Ab::
Rb; .. A : b :: R : sa (1810).
4. tba sb H; .. sb: tb :: csb: R:: a: H.
=
1819. CASE II.
When one Side A is invariable.
1. s H B
SHB –
sb a; .. В :
a :: sb: sH (1811).
s
2. SHB =
tb Ĥ; .. В
B:
H::tb: sH (1812).
B
3. cs HВ = R b; .. B: b:: R:cs H (1813).
1820. CASE III.
When the Hypothenufe H is conftant.
•
1. c s AВ = c t a Å‚ ‚ À : В ¦ ¦ e s A : cta (1814).
2. cs AB
=
csa b;.. Bb:: esa: es A (1815).
3. sA B = — R à; ... B: — à :: R: SA (1816).
-
a
:
4. cta Acsa b;..csa:cta:: sa: R:: A b.
1821. From thefe Equations, it is evident, there is a three-
fold Value or Expreffion for the Fluxion of every variable Part;
and many Equations to exprefs the Ratio of any two Fluxions,
befides thofe here fpecified. The great Ufe of this Doctrine of
Fluxionary Trigonometry we fhall hereafter particularly illuftrate
and exemplify in all the above-mentioned Cafes.
CHAP.
Of SPHERICAL TRIGONOMETRY.
237
CHA P. III.
THEOREMS for the SOLUTION of the CASES of Ob-
lique Spherical TRIANGLES.
1822. THEN each of thefe three Angles of a ſpherical
Triangle is greater or lefs than a Right-angle, it is
called an Oblique Triangle, as BCD (Fig. 3.) In fuch an one
all the Angles are uncertain, and therefore three Parts out of the
Six muſt here be given, to find the Reft; and confequently there
will alſo be Six Cafes in the Solution of an Oblique Triangle. In
order to which, the Ten following Theorems must be pre-
mifed.
1823. THEOREM I.
In every oblique ſpherical Triangle, the Sines of the Sides are pro-
portional to Sines of the oppofite Angles. To demonftrate this; let
fall the Perpendicular CA on the Side 'BD (continued out Fig.
3.) and it will be as Rs BC:: SB: SAC; and again, R:
SCD::SD: SAC (1781). Therefore RX SACSBC
X SB SCD X D; whence 3 C: DC: D:s B.
Q; E. D.
S
$
1824. THEOREM II.
In an oblique ſpherical Triangle BDC, drawing the Perpendicu-
lar CA, the Tangents of the Sides BC and CD are reciprocally
proportional to the Co-Sines of the vertical Angles. For in the
right-angled Triangle ACE, it is, R:csACB::BC:
tAC (1789.) and in the Triangle ACD, it is R: cs DCA
:: tDCA C. Therefore csACBXtBC-RX1AC
= c s DCA x tDC; confequently t BC: tDC :: cs A CD
:csACB. 2; E. D.
1825. THEOREM III.
In the oblique Triangle BDC, with the Perpendicular AC, the
Co-Sines of the Sides B C and CD are directly propertiened to the
Co-Sines of the Parts of the Bafe A B and AD. For in the right-
angled Triangle ACB, it is, ResAC::csAB:es BC.
(1792)
238
INSTITUTIONS
1
(1792) and for the fame Reafon R: cs AC::es AD:cs DC..
Therefore (as above) cs A B
AB:
Q; E. D.
csBC
1826. THEOREM IV.
cs AD: cs DC.
In the fame oblique Triangle B D C, the Co-Sines of the Angles at
Baſe are directly as the Sines of the vertical Angles. For as R:
SACB:: SAC:cs B (1785) and as Rs ACD::csAÇ
cs D. Therefore cs B: csD:: SACB: SACD. 2. E. D.
1827. THEOREM V.
In the fame oblique Triangle B CD; the Sines of the Parts of the
Baſe AB and A D, are inversely as the Tangents of the Angles at the
Baſe. For Rs AB
For RsABtB: tAC (1783.) and R: SAD::
tD: tA C. Therefore s A B XtB (RXtAC) SAD
XtD, whence s AB: SAD::tD: t B. 2; E. D.
1828. THEOREM VI.
As the Sum of the Sines of two unequal Arches is to their Diffe-
rence; fo is the Tangent of half their Sum to the Tanget of half their
Difference. For let AC M be a Quarter of a Circle (Fig. 4.)
in which take the two unequal Arches AB, AC; their Diffe-
rence is BC, which is bifected by the Radius O D. Then
AD is half their Sum, and D C DB half their Difference.
The Sines of the Arches are BG and CH; and the Co-Sines
OG and OH. Through the Points B and C draw the right
Line IP, and parallel to it draw KQ touching the Circle in
the Point D; alfo draw NE parallel to A O, and E F perpen-
dicular thereto. And lastly, through C draw OL, then is
DQ the Tangent of half the Sum of the Arches AD, and DL
the Tangent of half their Difference CD; and DK is the Tan-
gent of D M the Complement of A D. Then becauſe CD
BD, or CE EB, we have alfo HF FG, and therefore
=
CH + BG=2EF, and CH-BG2 CS (221). But
2 EF: 2CS: EF: CS (656.) :: EP: EC:: DQ:DL
(657.) therefore CH + BG: CH-BG :: DQ (Tangent
of A D) : DL (Tangent of DC). Q; E. D.
•
1829. THE-
Of SPHERICAL TRIGONOMETRY.
239
1829. THEOREM VII.
The Sum of the Co-Sines of two unequal Arches is to their Diffe-
rence, as the Co-Tangent of half the Sum of the Arches to the Tangent
of half their Difference. For OG + OH = 20 F, and OG
- OH = 2 HF (221.) Therefore 2OF: 2 HF:: OF:
HF ( EN: SE): EI:EC:: DK: DL; therefore the
(::EN:
Sum of the Co-Sines OG + OH: OGOH (their Diffe-
rence) :: DK (the Co-Tangent of A D): D L, the Tangent
of DC. 2, E. D.-
1830. THEOREM VIII.
In any oblique Triangle B DC, (having drawn the Perpendicular
AC, and bifected the Bafe in E,) it will be, as the Co-Tangent of
balf the Sum of the two Sides is to the Tangent of half their Difference,
fo is the Co-Tangent of half the Bafe, to the Tangent of the Diſtance
(AE) of the Perpendicular from the middle Point (E) of the Baſe.
For we have csAB: csAD :: csBC csDC (1824).
Whence es BC + cs DC csBCcs DC :: csAB
:
+csAD:cs AB-csAD, (648, 649.) But cs BC +
BC + DC BC-DC
cs DC: csBC-cs DC :: ct
(1829.)
2.
: t
2
Alſo es A B + cs AD: cs A B — cs AD:
AB+ AD AB-AD
ct
: t
::ctBE:tAE; and hence by
2
2
BC + DC
BC-DC
Equality (652) c t
: t
:: ct BE:
2
2
tAE. 2. E. D.
1831. THEOREM IX.
In any oblique ſpherical Triangle, the Tangent of half the Bafe is
to the Tangent of half the Sum of the Sides, as the Tangent of half
the Difference of the Sides is to the Tangent of the Diſtance of the Per-
pendicular from the middle Point of the Bafe. For fince KOQ
(Fig. 4) is a Right-angle, and OD perpendicular to the
Bafe KQ; it is DQ:DO:: DO: DK (660.) Therefore
DO2
DQ'
DK=
or becauſe DO2 is conftant, it is DK:
I
DQ,
There--
or the Co-Tangent of Arches are as the Tangents reciprocally.
240
INSTITUTIONS
BC + D C
BC+ DC
Therefore ct
:ctBE::tBE: t
2
2
BC-DC
t
2
1
: tAE (1829.) Q; E. D.
1832. THEORem X.
As the Co-Tangent of half the Sum of the Angles at Bafe, is to the
Tangent of half their Difference, fo is the Tangent of half the verti-
cal Angle to the Tangent of the Angle which the Perpendicular makes
the Line bijecting the vertical Angle. Let the Circle C F bifect the
vertical Angie BCD (Fig. 3.) Then it is, R: SACB:: cs AC
:csB (1798) and R: SACD::cs AC:cs D. Therefore by
Equality (652) and Permutation, we have cs B: csD::SACB
SACD; and therefore it is csB+cs D:cs Bcs D: :
B. D
s AC B÷s A CD: SACB — SACD
SACD (:: ct
B-D
::
2
2
(1829):: t BCF: 1 ACF (1828.) 2; E. D.
Q;
CHA P. IV.
The SOLUTION of the SIX CASES of Oblique Sphe-
G
rical TRIANGLES.
1833. CASE I.
}
IVEN the two Angles B and D; and the Side B C
oppofite to one of them; to find the other Angle and Sides
(Fig. 3.) The Analogy for the Side CD is, as s D: s BC:: SB
:s CD, (1823.)
1834. To find the Angle BCD; we have cs BC: R::ct B
:tACB (by 1791.) Then, it is, csB: cs D: SACB
:s ACD (1826); whence the whole Angle BCD is known.
1835. To find the Side or Bafe BD; we have R:cs B::
tBC:t AB (1784, 1831.) Then, as tD:t B:: SAB:
SAD (1827.). Whence AB+ AD = BD required.
1836. CASE
Of SPHERICAL TRIGONOMETRY.
241
1836. CASE II.
Given two Angles B and B C D, and a Side B C between them; to
find the other Angle D. First, R: cs BC:t B: ct BCA (1791,
1831); whence ACD is alſo known. Then s ACB: SACD
:cs B: csD (1826).
Then
1837. To find the Side DC. Say as, R:cs BC::tB;
et ACB (1791, 1831) whence A CD is known.
es DCA:cs BCA::tBC: tDC (1824).
1838. To find the Side BD. The Procefs is here the fame,
as for the Side CD, if from the End B of the given Side B C,
you let fall the Perpendicular B A oppofite to the given Angle
BCD.
1839. CASE III.
=
Given two Sides BC and CD, and an Angle B oppofite to one of
them; to find the other Side B D. Say, as Rcs BtBC:
tAB (1789, 1831). Then cs BC: cs DC::csAB: csAD
(1825). Whence AB+ AD BD ſought.
1840. To find the included Angle BCD.
es BC: B: ctACB (1791,
::cs ACB:cs ACD (1824).
BCD.
Firſt ſay, as R':
1831). Then t DC: tBC
Whence ACB + ACD=
1841. To find the oppofite Angle D. Say, as s DC: SB::
$ B C : s D (1823).
N. B. It muſt here be obferved, that when the Side DC is
lefs than BC, this Cafe will be ambiguous in all its Parts;
fince it will be uncertain from the Data, whether the Angle D iş
Obtufe or Acute; fince the Sine of an Angle, and of its Comple-
ment to 180 Degrees, is the fame.
1842. CASE IV.
Given two Sides BD and BC, and the included Angle B ; to find
the other Side CD. Say, as Rcs B :: t BC: t A B (1789,
1831.) whence AD is also known; then es A B csAD::
c s BC : c s CD (1825).
1843. To find the Angle D.
(1791, 1831.) whence A D
tBtD (1827).
VOL. II.
: c5
Say, as Rcs B: BC:t AB
is known; then s AD: SBD::
I i
1844.
242
INSTITUTIONS
1844. To find the Angle C. Let fall the Perpendicular Da
on the given Side BC, and oppoſite to the given Angle B; then
R:cs BtBD: 1 Ba, and then a C is known; confequent-
ly s a C : s a B : : t B: tC as above (1843).
2
1845. CASE V.
Given all the three Sides BD, BC, and CD; to find an Angle,
fuppofe B. Say t BD:t BC + CD::BC-CD
1 ½
: tAE, the Diſtance of the Perpendicular from the middle.
Point E of the Bafe BD (1831.) whence A B is known.
Then t BC:AB:: R:cs B (1797).
=
1846. CASE VI.
Z
Given all the three Angles, B, C, and BCD; to find a Side, fup-
pofe BC. Then fay, as ct B+ D
B + D : 1 ÷ B — D
B-D::BCF
ACF Angle included between the Perpendicular A Ç,
and the Line CF, which bifects the vertical Angle C; from
whence the Angle BCA is known. Then fay, as 7 B: ct BCA
:: R:cs BC (1800, 1831).
N. B. There are other Methods of ſolving the Cafes of oblique ſphe-
rical Triangles, but they are more complex and difficult both in Theory
and Operation, and therefore are not to be expected in an Elementary
Inftitution. Nor fhall we here infift on thofe Solutions of Triangles
which depend on the given Sums and Differences of Sides and Angles,
or of their Sines, Tangents, Verfed-fines, &c. as they are very in-
tricate, and rarely neceflary in Practice.
CHAP.
Of SPHERICAL TRIGONOMETRY. 243
CHA P. V.
The Method of determining the FLUXIONS of the
SIDES and ANGLES of Oblique Spherical TRIAN-
GLES.
1847. CASE İ.
L'
ET ABC be an oblique ſpherical Triangle, (Fig. 5.) and let
the Side AC move on the Point or Angle A, and by its
Motion defcribe a fmall Portion of a Parallel Cr; at the fame
Time let Cd be Part of a Parallel defcribed by BC on the Pole
B; and draw the great Circle Be interfecting Cd in d. Then
let D, E, and F, denote the three Sides AB, AC, and BC
refpectively, of which AB and AC are confiant, and all the
other Parts variable.
1848. Then we fhall have R: SE :: SCAc: sCc (1781)
for the fluxionary Triangle CcA is right-angled at (c); and be-
cauſe the Sine of a very fmall Arch is nearly equal to the
Arch itfelf, therefore R sE: CA (A): C c =
X À.
1849. Alfo, R: SF :: C B d ( = B) : C d =
=
SE
R
s F
R
× B. Again
becauſe A Cr=dCB Right-angle (1847); therefore ACB
=dC, and becauſe of the Right-angle at d, and the fluxion?
ry Triangle d Cc Rectilineal, we have R: sd Cc (ACB):
SE
R
Cc: cd :: ×À: F =
SEX SC
R*
1850. But sC:s D :: s B : §E (1823); therefore sE × sC
=SD x sB; confequently it is alſo ƒ =
× â (1848).
SD x SB
× Å.
R2
SE
Cd:: X A:
R
¡EX SC
R
SEX csC
XA=
F
R
× B (1849). Whence we have B
X A.
Rx F
1851. Again; R : cs.d Ce ( = C) :: C c
Ii 2
1852.
244
INSTITUTIONS
1852. And lastly, ct.c Cḍ (= C): R :: Cd: cd ::
X B: F F =
SF
ct C
X B.
s
R
1853. In the fame Manner it is fhewn (by making the like
Conftruction at B) that C
s D X cs B
× Å; and
Rx s F
s F
c t B
X ċ
Co-Secant of an Arch E B
1854. Let f Secant, and cf
(Fig. to Inft. 705) then it is, CH (= ED): CE :: CG
(=CE): CF; that is, sEB: R: R/GE, or cfBE;
therefore R² - sBE × c/B E.
cЛBE.
1855. Hence becauſe R2 =
s D x s B
F
XA (1850) = sD
× cfD (1854.) Therefore we have, from the above Equa-
tions, the following Analogies for the Fluxions of the Sides and
Angles of this Triangle, viz.
1. A: FcfD:s B.
I.
sF
2. A BR× SF:SEX csC (1851).
3. A:ċ :: Rx sF: SD x csB (1853):
4. BF: ctCsF (1852).
5. ċ : F :: c t B : s F (1853).
:
6. BċctCctB (1851, 1853).
:: tB: tC (1831).
CASE II.
1856. If one Side D, and an adjacent Angle B be invariable,
then this Cafe will be reduced to that of a right-angled Triangle,
by letting fall the Perpendicular AG (from the End of the given
Side, and oppofite to the given Angle B) on the Side BC con-
tinued out. For then there will be found the right-angled Tri-
angle Á CG, in which the Side AG (and the Segment GB) is
conftant; and ſo this Cafe becomes the fame with Cafe II. (1811,
1819).
1857. CASE III.
If one Side D, and an oppofite Angle C, be invariable (Fig. 6.);
then are the other Parts variable by the Motion of the given Side
Dor AB. Let the Side AB move into the Situation (ab) and
be-
Of SPHERICAL TRIGONOMETRY.
245
becauſe it is ever ab A B, it is impoffible that (ab) ſhould
be parallel to AB (631); therefore ab will meet A B, conti-
nued out, in fome Point Q. On the Point Q, as a Center,
defcribe the ſmall Arches bn, am; then becauſe AB = ab
nm, if from the firſt and laſt, you take away the common Part
An, there will remain Bn =Ạm.
1858. Then fince the fluxionary Triangles Am a and Bnb
may be confidered in their Nafcent State as Rectilineal, and right-
angled at m and n; therefore we ſhall have Am: Aa::csA:
R; and Bn: Bb:: csB: R; whence Am x R = A a × X
csA BnX R= Bb xcs B; therefore A a: Bb::cs B:
CSA: E: F.
s
1859. Again, we have s C: SD::s B: s E (1823) :: B:
; E. For becauſe the flowing Quantities or Sines of B and E are in
the conſtant Ratio of s Ctos D(1856). The Fluxions of the Sines,
viz. ; B and ; E will be in the fame conſtant Ratio (788). But
the Fluxion of the Sine is to the Fluxion of its Arch as Co-Sine
CS B
to Radius (874). Therefore B=
R
CS E
R
δ
X B, and E-
s
XE; Whence s C: SD::cs BX B: csEXE, which
gives BESCxcsE: SDXcs B.
1860. In the fame Manner it is proved that AFCX
CSF: SDXcs A.
1861. By comparing the Analogies in (1858, and 1859,) we
get BF:CxcsE:sDXcs A.
1862. And by comparing this with the Analogy in (1860) we
have AB csF:cs E.
1863. The Analogies in (1859, 1860) may be more fimply
exprefs'd by Tangents; thus, it is B:::
sD i
SC:SD: B: SE; therefore
SC
s D
cs B
; but
CSE
s B
E
::
s E
cs B CSE
::B
:t E, becauſe the Sine has the fame Ratio to the Co-Sine as the
Tangent has to Radius = 1. (ſee Fig. to 705.) Thus alfo,
it is At At F.
1864.
246
INSTITUTIONS
1864 CASE IV.
When two Angles A, and D, in the Triangle A E D (Fig. 7.)
are invariable. But this is reduced to the firft CASE (1847) by
confidering, that the Angles of the Triangle A E D, are equal
to the Sides of another Triangle F G H, formed by Circles con-
necting the Poles G, F, H, of the Sides of the Triangle AED.
For let the Sides DE, DA be produced to Quadrants in O and
C, and on the Point D deſcribe a Circle BKO G. Let G be the
Pole of the Circle C D B, F the Pole of the Circle OD I, and
H the Pole of the Circle A E Q. Then is GC FO= 90°,
from which take the common Part F C, and there remains F G
= CO = Angle D. In the like Manner F N = HM = 90°;
Subduct HN, and there remains F H NM = Angle A.
Laftly, GL=HP = 90°, take away H L, and we have GH
=LP Angle PEL, the Complement of the Angle A E D.
1865. Hence alſo it appears, that the Sixth Cafe of oblique
Triangles (1846) is reducible to the Fifth, by changing the
given Angles A, E, D into Sides of another Triangle FGH.
1866. I have now premiſed all that I judge does properly be-
long to the Elementary Part of the Doctrine of fpherical Triangles.
As to the Five circular Parts of Lord Neper, I think it an Artifice
of more Ingenuity in the Invention, than of Ufe in Practice, and
have here omitted it. Alfo thofe Methods of folving oblique
fpherical Triangles by given Sums, Differences, Products, &c.
of their feveral Parts, are not to be confidered as fift Principles,
but rather the Inventions of Art refulting from thoſe Principles,
and may be explicated by them when ever they occur in Practice.
However we cannot think theſe Elements compleat without ſuch
as determine the Arca of a fpherical Triangle, which therefore
we fhall add in the next Chapter.
CHAP.
Of SPHERICAL TRIGONOMETRY.
247
CHAP. VI.
The Method of determining the AREA of a SPHERI-
CAL TRIANGLE.
1867, TN Fig. 8. Let A CF be a Quadrant of the Equator,
the Pole D; SBT a Parallel to the Equator; ABE
an oblique Circle croffing the Equator in A; AK V an oblique
Circle below the Equator; DBK a Meridian, and D L O an-
other drawn indefinitely near to it. By this Means there will be
form'd an oblique fpherical Triangle A BK which is divided
into two right angled Ones ABC, and A CK, by the Equa-
tor A F.
1868. Let p
Periphery of a great Circle, and xs BC,
or Sine of BC, the Latitude of the Zone A HEF, whofe Sur-
face is x x AFX p; becauſe xp Surface of the whole
Zone continued round the Globe, (838.) therefore alfo the Sur-
face of the indefinitely fmall Part LOCB is × OC =
$ BC X OC. But this is also the Fluxion of the Triangle
ABC; for that of the Zone, and of the Triangle at the Point
B will be the fame (792) as is evident from the Reaſoning there
ufed.
*
SBC
1869. The Fluxion of the Angle ABC is =
XOC
R
I
(1810.) But s B C × OC :
× OC:
I :
::R: 1.
R
SBC
R
That is, the Fluxion of the Triangle A B C is to the Fluxion
of the Angle B, is in the conftant Ratio of Radius to Unity. And
therefore the contemporaneous Fluents will be in the fame
Ratio.
1870. In the Nafcent State of the Triangle, when the Sides
may be confidered as Rectilineal, the Angle at B is equal to the
Complement of the Angle A to a Right-angle (633). But as
the Triangle flows, or encreafes, this Angle B alfo flows and
encreaſes to a larger Quantity, till at laft the Triangle ABC
becomes the quadrantal Triangle A E F, and the Angle B be-
comes a Right-angle at E, the whole Increafe therefore of the
Angle
248
INSTITUTIONS
Angle B is equal to the Angle A. And it is this increaſing Part
of the Angle B, which is the Fluent of the Fluxion
$BC
R
*
OC. Therefore the Area of the Triangle ABC is to the Increase
of the Angle B, or the Excess of A + B above a Right-angle, in
the conftant Ratio of R to 1.
1871. The fame Thing is fhewn with regard to the Triangle
ACK; therefore it is evident, that the Area of any oblique ſpheri-
cal Triangle A B K is conftantly proportional to the Encrease of the
three Angles above two Right-angles.
-
1872. Hence in regard to the Triangles A B C and A EF,
it will be as A: AEF:: A + B 90: ABC; and in the
fame Manner in the Triangle CA K and F A V, it is A: FAV
:: A+ C— 90: ACK. Therefore with regard to the whole
oblique Triangle A BK, we have the Angle B A K to the whole
quadrantal Area EA V, as A + B + K 180° to the Area
of the Triangle ABK.
=
1873. Let S Surface of the Globe, T Area of the
Triangle A B K, Q Area of the quadrantal Area EA V, N
Angle BAK≈ EV, and M = Sum of the three Angles
of the Triangle N+B+ K, then it is felf-evident, that the
quadrantal Space EAV is fuch a Part of a Hemisphere as the Arch
EV is of a great Circle; that is, Q: S :: N: p; but N: Q
:: M-2p:T (1870); therefore p:S:: Mip: T,
or 2p: M-2p:: S: T; that is, as 720: N+B+K-
180: the Surface of the Globe S: Surface of the Triangle A BK.
2
1874. But the Surface of a Globe or Sphere is equal to four
Times the Area of its great Circle, or S = 4 A (839); there-
forep: S: 180: A :: M 180: T.
But Apr,
or half the Radius (r) multiplied into the Periphery (830); there-
fore (180 180r::) Ir:: M 180: T.
—
1875. The Radius (r) expreffed in Degrees is = 57°, 2957795
(884); therefore if the three Angles of a Triangle leſſened by
180°, be multiplied by 57,2957795, the Product will be the Area
of that Triangle in Square Degrees. And becauſe in one Square De-
gree, there are 3600 Square geographical Miles, or 4830,25 Eng-
lifh Miles, therefore the Area of the Triangle may be expreffed
in Miles of either Sort.
CHAP.
Fig. 1.
Plate II. Of Spherical Trigonometry.
Z
2
H
H
72
F
E
Z
Fig. 4.
B
Z
H
B
B
Fig. 6.
Fig. 8.
F
하
​H
I
I
A
A
A
P
Fig. 2.
B
H
F
D
D
L
P
Fig. 5.
H
C.
B
Z
F
D
е
W
T
1
M
Fig. 7.
B
1
F
B
F
Z
Fig. 3.
G
D
H
N
N
D
72
G
E
F
Fig. 9.
A
Of SPHERICAL TRIGONOMETRY. 249
CHAP. VII.
The foregoing PRINCIPLES applied to the SOLUTION
of PROBLEMS in ASTRONOMY, GEOGRAPHY,
NAVIGATION, DIALLING, &c.
HE Conftruction of a Diagram for prefenting at
1876. TH
one View the greateſt Numbers of ſpherical Trian-
gles, in order to the Solution of aftronomical Problems, is that of
Fig. 9. which is a ſtereographical Projection of the Sphere on the
Plane of that Meridian ENQS, which is called the folftitial
Colure (1707).*
1877. In this Scheme, ÆQ is the EQUINOCTIAL, and its
two Poles N, S. EL is the ECLIPTIC, and its Poles P, T.
HO is the HORIZON, its Poles Z, the Zenith, and D the Na-
dir. Z D, the prime VERTICAL, and its Poles H, O. E
is the Tropic of Cancer; L, the Tropic of Capricorn. O is
the Sun's Place in the Ecliptic; NOS a Hour-circle, and
ZOD a vertical Circle paffing through the Sun →. PRT a
Circle of Longitude, and NRS a Circle of Declination paffing
through a Star at R.
1878. Now thefe various Circles of the Sphere by their Inter-
fections, form many ſpherical Triangles, both Rectangular and
Oblique. In a right-angled Triangle, if any two Parts are given,
it is a Problem to find the reft. And in oblique Triangles, the
Problem requires three different Parts to be given, as we have
fhewn (717-721). But one Angle in moft Triangles, Right
or Oblique, is of no Confequence in aftronomical Problems, and
therefore does not enter the Data.
1879. In a right-angled Triangle, there are, therefore, four
fignificant Parts, viz. three Sides, which call a, b, c; and one
Angle, which let us denote by (n). Then fince there are fix
Combinations of two Quantities in four, viz, an, bn, en, ab,
be, a c, there will be at leaft fix Problems refulting from every
right-angle Triangle.
VOL. II.
Kk
1880.
*It is here fuppofed, the Reader is acquainted with the Names
and Ufes of the CIRCLES of the SPHERE, as they have been at large
explained in the GENTLEMAN and LADIES PHILOSOPHY.
250
INSTITUTIONS
1880. Again, in an oblique Triangle there will be five figni-
ficant Parts, viz. three Sides, a, b, c; and two Angles, m, n;
and there will be ten Combinations of three Quantities in five, viz;
abm, bcm, a cm, abn, ben, acn, amn, bmn, cmn, abc;
therefore ten aftronomical Problems will arife from every oblique
Triangle.
1881. We fhall now ſpecify thoſe right-angled Triangles
that are of moft Ufe in Aftronomy, which are as follow, first,
the Triangle BCO, right-angled at B; in which BC is the
Sun's right Afcenfion, or its Complement to the Point C. Co
is the Sun's Longitude, or Complement to C. BO is the Sun's
Declination; and the Angle BCO the Obliquity of the E-
cliptic.
1882. Secondly; the Triangle AO C, right-angled at A ;
in which the Side CO is the Sun's Longitude in the Ecliptic
from C; the Side A O, its Altitude above the Horizon. AC
is the Azimuth from the Point C. And the Angle ACO is
the Sum of the Co-Latitude and Obliquity of the Ecliptic, viz. ÆH
+ E.
1883. Thirdly; Let the Sun be in the Tropic of Cancer, and
in the Point G in the prime Vertical, when due Eaſt or Weft.
Then in the Triangle V CG, right-angled at V, the Side V C
is the Hour from Six. The Side VG EE, the Sun's De-
clination. The Side CG the Sun's Altitude when East or Weft.
And the Angle VCG ÆZ, the Latitude of the Place.
1884. Fourthly; On the fame Day the Sun is at I, at the
Hour of Six precifely; then in the Triangle ICM, right-an-
gled at M; the Hypothenufe CI is the Sun's Declination; IM,
the Sun's Altitude at Six; and CM the Azimuth from C. And
the Angle ICM = NO=ÆZ the Latitude of the Place.
1885. Fifthly; On the fame Day the Sun is in the Horizon at
K, and there is formed the Triangle W CK right-angled at
W; in which WK is the Sun's Declination. CW is the afcenfio-
nal Difference, or the Hour from Six of its Rifing and Setting;
CK, the Amplitude of Rifing and Setting from the Eaſt or
Weft. And the Angle WCK OQ, is the Co-Latitude of
the Place.
=
1886. Sixthly; In the Triangle NKO, right-angled at O,
the Side NO is the Latitude; the Side N K the Co-Declination;
and
Of SPHERICAL TRIGONOMETRY.
251
and the Side KO the Co-Amplitude or Azimuth from O. And
the Angle KNOW Q the Hour from Midnight.
1887. What we have faid with Refpect to the Sun in Cancer,
is the fame for its Place in any other Part of the Ecliptic; and
the fame Triangles are to be drawn for the fame Purpoſes in the
Winter Signs, as we have here done for the Summer. And fur-
ther, it is to be obſerved, that fince any Star may be fuppofed
at O, it is evident all the fame Triangles ferve for a Star as for
the Sun. Therefore in the fix Triangles already enumerated, no
Hefs than feventy-two Problems may be ftated in Reference to the
Sun and Stars; beſides many other Triangles that might be ad-
ded to thefe, to encreaſe the Number of aftronomical Pro-
blems.
1888. In the oblique Triangle OZN, the Side O Z is the
Co-Altitude of the Sun; ON, the Co-Declination; ZN, the
Co-Latitude of the Place. The Angle ZNO is the Hour from
Noon, and the Angle O ZN or OZE is the Azimuth from
the North or South Part of the Horizon.
1889. This Triangle, therefore, affords ten Problems more
in reſpect of the SUN (1878) and if we confider the oblique
Triangle Z RN, we fhall find the Sides and Angles the fame in
regard to a STAR at R, and confequently other ten Problems will
thence be produced.
1890. Again; in the oblique Triangle RNP, the Side RN
is the Star's Co-Declination; the Side RP, its Co-Latitude; and
the Side NP = Æ E the Obliquity of the Ecliptic. Also the An-
gle NPR is the Star's Longitude from the Point E; and the
Angle RNP its Azimuth. From hence we have ten Problems
more; and therefore this and the last Triangle afford twenty prin-
cipal Problems about the Stars only.
1891. In the laft Place, let o be the Moon or a Planet, then
if we draw a great Circle through O and R, the Place of a Star;
we ſhall have an oblique Triangle o ZR, in which the Side
OZ is the Moon's Co-Altitude; ZR, that of the Star, and
OR the Distance of the Moon from the Star. Alfo the Angle
OZR the Difference of the Azimuths of the Moon and Star.
This Triangle affords four very ufeful Problems relative to the
Moon and Stars. Upon the Whole, in theſe few Triangles in
this one Projection only, more than one hundred aftronomical Pro-
blems are contained.
Kk 2
1892.
252
INSTITUTIONS
1892. In regard to GEOGRAPHY, if we ſuppoſe Z and R to
repreſent two Places on the Surface of the Terreftrial Globe;
then in the Triangle Z RN, the Side ZN is the Co-Latitude
of the Place Z; the Side RN is the Co- Latitude of the Place
R; and the Side Z R is the Diſtance of the two Places on the Sur-
face of the Globe. The Angle ZNR EV, is the Diffe-
rence of Longitude of the two Places, and the Angle RZN is
the Bearing of the Place R from Z. This one Triangle there-
fore furnishes ten principal geographical Problems, which depend
on the Doctrine of the sphere.
1893. In NAVIGATION, particularly that Part of it called
ORTHODROMICS, which treats of the Art of Sailing on a great
Circle of the Sphere, a fpherical Triangle is concerned. Thus
let Z be the Port from whence a Ship is to fail to another Port
X; then in the Triangle ZNX; the Side ZN is the Co-Lati-
tude of the Port Z; the Side N X is the Co-Latitude of the Port
X; the Side Z X is the nearest Diflance between them or the
Ship's Way. Alfo the Angle N Z X is the Courfe to be ſteered
at Z; and the Angle Z N X is the Difference of Longitude be-
tween the Ports.
1894. But in failing from Z to X, the Courfe is perpetually
altering, or the Angle made by the Ship's Way on the great Cir-
cle Z X with every Meridian is different, or the Latitudes always
changing, therefore it is neceflary to calculate this for every 5 or 6
Degrees of Longitude, that the Ship may be kept upon, or near
the circular Arch ZX, every where; and this is the whole Art
of great Circle-failing, which is entirely converfant in the Solu-
tion of spherical Triangles. A great Variety, therefore, of nautical
Problems here offer themfelves from this one View of a ſpherical
Triangle.
1895. As to LoXODROMICS, or failing upon a Spiral, or
Rhumb-line, as the Theory of that Curve has not yet been con-
fidered, nor the Rationale of that Sort of Sailing been fully ex-
plained by Writers on Navigation, we have determined to treat
of that Subject more particularly in another Chapter.
1896. In DIALLING, the Triangle NOK, right-angled
at O, is uſed in finding all the Angles N KO, which the feveral
Hour-circles NKS make with the Horizon. For NO being
the Latitude of the Place, and the Angle ONK being given,
} 鼈
​the
Of SPHERICAL TRIGONOMETRY.
253
the Arch of the Horizon OK is found by Cafe I. of right-an-
gled Triangles (1783). But this is only touched upon here, to
fhew how very uſeful and extenſive the Doctrine of spherical
Trigonometry is in moſt Arts and Sciences. What relates to the
Conftruction of the Scale of Latitudes and Hours in practical Dial-
ling, as alſo of a new Dialling Sector, and a new univerſal Ho-
RIRONTAL DIAL, fhewing the Hour by the fame Guomon in
all Latitudes, will be treated of more fully hereafter.
CHA P. VIII.
The Application of FLUXIONARY SPHERICS in
aftronomical COMPUTATIONS, relative to
PARALLAXES, REFRACTIONS, equal AL-
TITUDES, PRECESSION of the EQUINOXES,
&c.
1897. W
E have fhewn how the Places of the heavenly Bo-
dies, the Times of their Rifing, Setting, c. and
their various Phænomena in regard to their Right-afcenfion, Decli-
nation, Amplitude, Azimuth, Longitude, Latitude, &c. are to be
calculated by the common Proceffes of spherical Trigonometry.
But the minute Variations and Alterations which happen in
thoſe Quantities by Means of a Parallax, Refraction, Receſſion of
the Equinoxes, ſpiral Motion of the Sun, &c. are of too much Mo-
ment in Aftronomy not to be moft fcrupulouſly attended to;
and as they are best of all computed by Fluxionary Spherics, the
Principles of which have been explained, we therefore now
illuftrate that Method by fome Examples.
1898. As the moft confiderable of thefe Variations are of
that Sort which ariſe from a Parallax, and greatly affect the moſt
interefting Subjects of this Science, viz. the MOON and the
PLANETS, it will be neceffary to premife the following Theorem,
viz. The Parallax of a Planet is always proportioned to the Sine of its
apparent Distance from the Zenith, which is thus fhewn. Let C
be
254
INSTITUTIONS
be the Center of the Earth, P the Place of a Spectator on its
Surface; Z the Zenith, and let N and n be two Places of a
Planet in an Azimuth-circle ZO; draw PN and CN, alfo
Pn and Cn, then are the Angles PNC, Pn C, the Parallaxes
of the Planet at the Altitudes NO, nO, (1724). To com-
pare which, we have PC: CN :: SPNC: SNPCs NPZ.
Alfo PC: Cn (= CN): : s Pn C:s CP n s n P Z; there-
fore it is s PNC: Pn C::sZPN:s ZP n.
But theſe pa-
rallatic Angles at N and n, being very fmall, are as their Sines;
whence the Propofition is evident.
1899. Hence it follows, that any Parallax at N is to the hori-
zontal Parallax at O (viz. that of the Planet feen from P in the
Horizon at O) as the Sign of its Zenith Diftance (nearly) to the
Radius. Alſo it appears, that the parallatic Angle vanishes at
the Zenith Z; and alfo when the Diſtance P N or CN becomes
immeaſurably great. So that the fixed Stars can have no Parallax
of this Kind. Laftly, we have, the Distance of a Planet N C
from the Sun, to the Semidiameter of the Earth CP, as the Sine of
the Parallax CNP, to the Sine of the Zenith Distance Z P N.
1900. As the Angle ZPN is greater than ZCN by the
Quantity of the Parallax P N C, it is evident the apparent Zenith
Diſtance of a Planet exceeds the true, by juft that Quantity. There-
fore let A be the true Place of a Planet; Z A its Zenith Diſtance,
P the Pole of the Ecliptic; and PA a Circle of Latitude paffing
thro' the Planet. Alfo let (a) be the apparent Place, and draw
the Circle of Latitude P a b thro' the Planet A. Let E A L be
drawn parallel to the Ecliptic; and let HLO be the Horizon.
(Fig. 2.)
a
1901. Then in the oblique Triangle A Z P, the Side Z P, and
the Angle Z adjacent, are conſtant (1855). And the fluxionary
Parts are (1) Aa = à (1817, 1818) the Parallax in Altitude.
(2.) The Parallax in Longitude APd B. And (3.) The
Parallax in Latitude ad = ǹ. Then becauſe the Parallax in
Altitude is known from the horizontal Parallax (1898) in aftro-
nomical Tables; therefore we have sH: sb: :
(1819). That is, in Words, as the Co-Sine of the Planet's La-
titude, is to the Sine of the Angle ZAP, fo is the Parallax in Alti-
tude to the Parallax in Longitude.
a
AP
: B = A P d
1902.
Of SPHERICAL TRIGONOMETRY.
255
1902. Again, to find the Parallax in Latitude; it appears
from the fluxionary Triangle A a d, right-angled at d, that
Aaad: R: sa AdsEAZ, that is, R: csZAP::
Aa: ad: :: H; which is in Words, Radius is to the Co-Sine
of the Angle ZAP, as the Parallax in Altitude, is to the Parallax
in Latitude.
a
1903. If EL be the Ecliptic itſelf, and the Planet be in, or
very near it, then s HS APR; and the sbsZAP =
csZAE; therefore sH: sb:: R:csZAE (1899): tZA:
t EA (1797): å: B; that is, as the Tangent of the Planet's Ze-
nith Distance is to the Tangent of its Distance from the Nonagefima
Degree E, fo is the Parallax in Altitude, to the Parallax in Longitude.
a
1904. Alfo, in this Cafe, we have Rcsb::RsZAE
:: sZA:s ZE (1795) :: å: Ĥ; that is, the Sine of the Zenith
Difance, is to the Co-Sine of the Altitude of the Nonagefima Degree
a
as the Parallax in Altitude to the Parallax in Latitude.
1905. If we put M = horizontal Parallax; then, becauſe
SZA
Ꮓ Ꭺ
R
(1898) xM=i=
sZAXtAE
1
tZA
AE
X B, (1903) we have i
s Z A x s ZAP
XM =
RXtZA
R²
× M = (1823)
SZPXsZPA
Ꮲ Ꭺ
× M.
Therefore RsZP x sZPA::
R²
M: B. That is, the Square of Radius is to the Rectangle of the
Sines of Altitude of the Nonagefima Degree, and the Planet's Lon-
gitude from thence, as the horizontal Parallax is to the Parallax of
Longitude.
1906. Again, × M =
SZA
R
SZA
SZE
Xi (1904) whence
SZAXSZE
s Z E
A =
x M =
R
X M. Which gives R:
3
RX SZA
sZĘ:: M: H; that is, Radius is to the Co-Sine of the Altitude of
Ꮓ Ꭼ
the Nonagefima Degree, as the horizontal Parallax to the Parallax
in Latitude. Hence becaufe R and M are conftant Quantities,
the Parallax in Latitude will ever be as the Co-Sine of the Alti-
tude of the Nonagefima Degree.
1907 As to the Parallax in Right-afcenfion and Declination, the
Analogies are the fame as before; for H ZO. being the Meri-
dian, EQ, the Equinoctial, its Pole N, HO the Horizon, A
the
256
INSTITUTIONS
the Place of the Planet, ZD a vertical Circle, and NF a
Hour-circle paffing through it (Fig. 3.) it is evident the Trian-
gle Z AN, and its fluxionary Parts A a, a d, Ad, are all the
fame as in the foregoing Figure, and therefore the fame Analo-
gy as was uſed (1903) for finding the Parallax of Longitude, finds
here the Parallax of Right-afcenfion, viz. sH (SAN): sb
(= s Z AN) : : å ( = A a) : Ẻ ( = BN b). Alfo for the Pa-
rallax of Declination (ad) it is (1904) as R : csb::a: H=
a d.
a
1908. The Effects of Refraction of Light thro' the Atmosphere,
are next to be confidered; the general Nature of refracted Light
has been ſhewn at large (1321, &c.) and that Effect is to ele-
vate Objects in Appearance, or to make them appear higher than
their true Places, which is juft contrary to the Effect of the Pa-
rallax which depreffes them (1899).
1909. It appears alfo from the Theory, that the more oblique
the Rays are, the more they will be refracted; or the Rays
will be more refracted as they are nearer to the Horizon;
therefore the horizontal Refractions are greateſt of all, and in
the Zenith there is no Refraction at all, which is the Caſe alſo
in regard to the Parallax.
1910. Therefore if (a) be the true Place of a Planet (Fig. 2.)
then by Refraction it will be elevated to A in the vertical Z D,
which Refraction a A in Altitude being found by Obfervation,
you will from thence find the Diminution d PA in Longitude,
and the Alteration (a d) of Latitude, correfponding to the fame
by the Analogies for the Parallax (1903, 1904). And alfo .
thofe of Right-aſcenſion and Declination (Fig. 3.)
1911. The horizontal Refraction makes a Difference in the
Time of Rifing and Setting of the heavenly Bodies, and alſo of
their Amplitude from the Eaft or Weft Points of the Horizon.
Thus let (cc) be a Parallel of the Sun or Star's Declination,
then without the Refraction P would be the Point in which it
would aſcend the Horizon, QP the true Amplitude; and PÑO
the Hour from Midnight. But if the horizontal Refraction be e-
qual to R M, then will the Sun or Star be thereby elevated to
the Horizon in the Point M, and its apparent Amplitude will be
QM, and the apparent Time of Rifing will be M N O.
1912.
Of SPHERICAL TRIGONOMETRY.
257
=
D, the Side NP-E,
B, the Angle PNZ,
C. And we have
1912. In the Triangle ZN P, the two Sides ZN and NP
are conſtant (1847). The Side ZN
the Side ZP F; the Angle P ZN
or PNO A, and the Angle ZPN
sB: cfD::: A (1855) that is, the Co-Sine of the true Ampli-
tude QP is to the Secant of the Latitude EZ, as the horizontal Re-
fraction R M = is to the Angle PN M, which is the Difference
in Time of the Rifing or Setting of the Sun er Star thereby occafioned.
Or otherwife thus; becauſe s B: sE :: sA: sF = R (1828)
therefore s B =
R²
D
SEX SA
R
R²
Again, cfD =
SD (1854) there-
R² X F
SEX SA
¿DX A
fores B: ::FA, hence s B =
R
3
which Equation gives this Analogy sD x sEx SA: R³ ::
F: A, as before.
1913. Again, for the fluxionary Variation of the Amplitude
PM, we have sF: ct CF: B, (1855) but in this Cafe,
SFR, and ctCtNPO, therefore R:NPO::sB
(= SPO) NO (1780). Therefore s B: t NO :: F: B ;
that is, the Co-Sine of the true Amplitude is to the Tangent of the
Latitude, as the horizontal Refraction is to the Difference PM be-
tween the true and apparent Amplitude.
1914. This apparent Amplitude is neceffary to be known, ef-
pecially by Navigators, becauſe the Variation of the Needle de-
pends upon it, fince that important Article is nothing more
than the Difference between the obferved Amplitude on the real
Horizon, and that of the magnetical Card.
1915. It appears from what we have formerly fhewn, that
the Orbit of a Planet about the Sun is elliptical; and fince
that is the Caſe of the Earth, its Motion will be variable, fome-
times quicker, and fometimes flower, fuch as is expreffed in Se-
conds of a Degree in the following Table per Hour, for the ſeveral
Months of the Year.
January
December
153"
February - November
152
March
October
150
April
September
148
May
Auguft
145
June
July
143
VOL. II.
LI
1915.
}
258
INSTITUTIONS
1
1916. In Fig. 4. let Æ Q be a Quadrant of the Equinoctial,
EC of the Ecliptic; BAN an Hour-circle paffing through the
Sun at A, and in one Hour after let its Place be at (a). In the
right angled Triangle ABQ, the Angle at Q is conſtant; the
Declination is AB; whofe Fluxion is found from the Equation
tba sbi (1818) which gives tb: sb:: R: csb:: H(Aa)
AB ab. In Words, Radius is to the Co-Sine of the An-
gle QA B, as the horary Motion in Longitude A a to the horary Va-
riation in Declination.
a
=
•
1917. Since à is always as cs.b, it is evident that the hourly
Increaſe or Decreaſe of Declination will be leaft of all when the
Sun enters Cancer and Capricorn, and greateſt of all at the E-
quinoxes Q, where it amounts to a whole Minute of a Degree.
1918. Therefore fuppofe HPOQ (Fig. 5.) be the Hori-
zon, Z the Zenith, N the Pole of the Equinoctial HCO; alſo
let A B E be an Almicanter or Circle of Altitude interfecting the
Equinoctial in the Points A and a ; and draw Z A D and Z a F;
alſo the Hour-circles N A G and Na I at equal Diſtances from
the Meridian P N Q. Then if the Sun or Star had no Motion
in Longitude, its Altitude above the Horizon at equal Times
before and after Noon, would be the fame, that is A DaF;
alfo the Declination would continue the fame, or NA N a.
But, as we have fhewn, the Declination alters hourly; and
when the Circle of Declination NA is thereby leſſened, as in all
the afcending Signs, it is evident the Sun which croffed the Almi-
canter at A in the Morning, will not crofs it at (a) in the Af-
ternoon, but at a Point more wefterly, becaufe Nd is by Suppo-
fition less than Na; therefore the Time or Hour-angle CNd
will exceed that of the Forenoon CNA. On the other Hand,
when the Declination N a from the North increaſes to Nb (as in
all the defcending Signs) then the Sun comes fooner to the Almican-
ter at (b) and ſince the Decrease or Increaſe of Declination is
known for any interval of Time in defcribing the Arch AC a
by (1916) therefore the fluxionary Angle a N d, or a N b is
known from the Triangle Za N, where the two Sides Za and
ZN are conftant.
1919. For by Cafe I. (1855) we have this Analogy for find-
ing the Fluxion of the Angle N (there called B) viz. s F: ct C
: : Ŕ ( = ad) ; В (≈ aNd) that is, as the Co-Sine of the Sun's
Decli
:
Of SPHERICAL TRIGONOMETRY.
259
Declination (saN F) is to the Co-Tangent of the Angle ZaN
(C) fo is the horary Difference of Declination (ad) in the Time
between the two equal Altitudes, to the Seconds contained in the fiuxiona-
ry Angle aNd. On the equinoctial Days N A≈ 90°, or s F
becomes Radius.
1920. If the Moments of equal Altitudes at A and d before
and after Noon, be obferved by a Clock or Watch, then the
fmall Equation of Time aNd, juft found, is to be fubducted
from that interval of Time, and the Remainder will be the
Time of moving through Aa; and the Half thereof will give
the Moment of the Sun or Star's Appulfe to the Meridian at C,
or the equated Time at Noon. And this muſt be done in the fix
Signs from Capricorn to Cancer, and in the other Six, the Equa-
tion will be a Nb to be added to the obferved Time of equal
Altitudes at A and b.
1921. To find the Variation of the Azimuth a Zd or a Z b,
we have the firſt Analogy (1855) viz. s B (= a NZ): cfD
(= ZN) :: F (= aNdN): AbZF. In Words, as
the Sine of the Angle A N C, or Time before Noon, is to the Secant
of the Latitude, fo is the Variation of Declination in the Interval of
equal Altitudes to the l'ariation of the Azimuth Fh, or F k.
1922. The Poles of the World, or Equinoctial, are found
to have a Motion about the Poles of the Ecliptic, contrary to the
Order of the Signs, and therefore the Interfections of thefe two
Circles or equinoctial Points muſt have a real retrograde Mo-
tion; and conſequently the fixed Stars will have their Diſtances
from the equinoctial Coloure continually increafing. Now this
Motion is at the Rate of 50" per Annum, at a Mean; and ſo
much therefore will the Longitude of the Stars be annully aug-
mented. The phyfical Caufe of this Motion will be hereafter
explained.
fo
1923. Let HZO be the Meridian N, the Pole of the equi-
noctial Æ Q, and P the Pole of the Ecliptic E Q. Alſo let A
be the Place of a Star, PA Ca Circle of Latitude, and N A B a
Circle of Declination paffing thro' it, make CPc 50″, and
let e q be parallel to the Ecliptic EQ, cutting cP in (a). Then is
A a the Space through which the Star advances in one Year in its
Parallel, and Ce its Difference of Longitude in the Eclip-
tic. And the Difference of Declination, viz. A Babis
found
LI 2
260
INSTITUTIONS
found by the first Analogy (1855) viz. cƒD (≈ NP) : s B
(≈ ANP) :: Å (= Cc) ; F = Ą B — a b, the Difference of
Declination required.
1924. And to find the Difference of Right-afcenfion, we make
uſe of the fourth Analogy of (1855) VIZ. SF (AN): ctC
(A): FB:: Difference of Declination: Difference or Increaſe
of Right-afcenfion.
N. B. Moſt of thoſe fluxionary Quantities, which we have
given Rules for calculating here, are exemplified in Numbers,
and applied to Ufe, in my NEW PRINCIPLES OF GEOGRAPHY
and NAVIGATION.
CHA P. IX.
Theorems for the Stereographic PROJECTION of the
SPHERE in Plano.
1925. T
HE Doctrine of Spherical Projections, whether Stere-
ographic, Globular, Orthographic, or Gnomonical, iş
wholly derived from the Principles of Perfpective as we have
largely fhewn, and from thence demonſtrated the Properties of
each particular Species of Projection in a general Manner; but
before we can proceed to an Application of this Art, to the
Projection of spherical Triangles, Dialling, Aftronomy, &c. we
muft premiſe a few more Principles, efpecially with regard to
the ftereographic Projection, in this Place.
1926. If we look back to Fig. 4. Plate VI. of perſpective
Projections, we ſhall obſerve, that if any great Circle of the
Sphere be oblique to the Plane of Projection, or of the primitive
Circle QTR; then if the Arch G T meaſures its Obliquity or
Elevation above the Plane of the Primitive, that Circle will be
fo projected by an Eye at R, that its higheſt Point at G will bẹ
in the Diameter at O, diftant from the Center C, by the Tan-
gent CO of half the Complement QG of its Elevation
(1698).
/
1927:
Of SPHERICAL TRIGONOMETRY.
261
1927. Again, becauſe the Arch QL = GT (1697) and
QB parallel to CL, the Angle BQC = QCL = Elevation
of the Circle; therefore the Diſtance of the Center B, from the
Center C of the Primitive, is CB = Tangent TD of the Circle's
Elevation TCD.
1928. Let the Arch QOR (Fig. 7.) be compleated into a
Circle A O EP, and draw the Diameter A BE; then this may
be confidered as another primitive Circle, on which any oblique.
Circle may be projected by fetting off the Tangent of its Eleva-
tion from the Center B either Way in the Diameter A E (1927).
Thus becauſe the Tangent EK (= BO) of an Arch ES of
45° is equal to Radius B E or AB, therefore the Point A is the
Center of the oblique Circle O HP elevated above the Primitive.
in an Angle of 45° = ES = EOH. If the Elevation be of a
lefs Number of Degrees than 45, the Center will fall in the Di-
ameter within the Circle; if greater, without. Thus let BL be
the Tangent of 30°, then L is the Center of the Circle O M P,
whofe Elevation is the Angle A O M = 30°. If BV be=
Tangent of 75 Degrees, then V is the Center of the Circle
OY P, making the Angle YOE = 75° its Elevation above
the Primitive; and fo for any other oblique Circle.
1929. Hence appears the Method of drawing two oblique
Circles through any given Point O in the Primitive, to contain
a given Angle with each other, by Means of the Tangents of
their Elevations. But the Centers of thofe Circles are alſo found
by the Secants of their Elevation, fet off from the Point O, the
extreme Point of the projected Diameter OP; for fince the
Angle QBO = GCQ (1697) the Angle CQB = Q BA
= DCT; and therefore the Triangles CTD and CBQ are
equiangular and equilateral; for CTCQ = Radius; and
TD=CB = Tangent of GT (1927); therefore CD=
BQ=OB = Secant of GT the Elevation; whence the
Center B is thereby given.
1930. Hence the Angle OQT made by the Interſection of
two Circles QT and OQ is equal to the Angle made by their
Radii; for CQ= Radius of QT, and B Q= Radius of QO,
and the Angle CQB=TCD=0QT the Elevation.
1931. Since, therefore, the Angle contained by the Radii of
the two Circles in the Projection is equal to the Inclination of
the
262
INSTITUTIONS
the Planes of thofe Circles, it is evident, the Planes themselves
in the Projection muſt contain the fame Angle as they do in the
Sphere; that is, any Angle OQT is of the fame Quantity in
the Projection as it is on the Globe itſelf.
1932. But as this is a principal Point in the Doctrine of Pro-
jections, we fhall give another Demonftration of it, thus. Let
the Eye at A (Fig. 8.) project the Angle S BR upon the Plane
ES placed right before it; and ſuppoſe BD a Tangent to the
Circle S B, and BC to the Circle R B, in the Point of the Inter-
fection B. Therefore the Plane in which the Tangents BD and
BC are contained, as alſo the Plane of the Circle ES are both
perpendicular to the Plane of the Circle BE AS; and fo their
common Interfection CD will be perpendicular to the Line ES
produced. Now the Eye at A projects the Tangent BD into
FD, and the Tangent BC into FC, becauſe the Point B is
projected into F by the viſual Ray A B. Through the Center
G draw the Diameters AH and BL, and BI parallel to ES,
and join AL, BH. Then is the Angle DBA = BLA
(665) = A HB (645) — A BI (659) = BFD (631); that
is, DBA BF D, and therefore B D = DF. Then in the
Triangles CDF, and CD B, fince the Angles DBC and
DFC are ſubtended at the fame Diſtance by the fame Line
DC, they muſt be equal. But the Angle D B C = SBR on
the Globe; which therefore is equal to its Projection D F C on
the Plane ES continued.
1933. Any Circle BC (Fig. 9.) placed oblique to the Plane of
the Circle E G is projected into a Circle upon that Plane by an Eye
placed at A in the Pole of the Primitive E G.
parallel to EG, then becauſe the Arch D A =
For draw DC
CA, the Angle
ABC ACD (643) and the Angle A is common; there-
fore the Triangles A B C, and A Cb or Ace, are fimilar (621).
Therefore the viſual Cone A CB is cut by the Planes CD and
EG fubcontrarily, and confequently the Projection of its Baſe
BC will be a Circle at b C, or ec (1510).
N. B. BC is a small Circle, but had it paffed through the
Center F it would have been a great Circle, and the Demonftra-
tion the fame.
*
1934. Hence as e F is the Tangent of half the Arch BH,
and Fc Tangent of half HC; the Points e and c are given ;
there-
Of SPHERICAL TRIGONOMETRY.
263
therefore the Line ec bifected, gives (n) the Center of the Circle
ec in the Projection.
1935. A fmall Circle BI, perpendicular to the Plane of Pro-
jection EG, will be projected by the Eye at A into the Circle
Ke, whoſe Center N is in the Line EG, produced, Diſtant from
the Center F of the Primitive, by the Secant of the Arch BE,
(or the Diſtance of the Circle BI from its Pole E) and the Ra-
dius thereof will be equal to the Tangent of the fame Arch. For
draw A B cutting the Line GE in e; and through I draw A K
interfecting the Line GE (produced) in K; then the Line Ke
bifected, gives the Center N, on which let the Circle K BI be
defcribed. Draw NB, KBH, and FB; the two Triangles
ABH and KH F being right-angled at B and F, and having
one Angle at H common, have alſo the Angle B AH=FKH.
But BAH BFH (642) and FKH FNB; there-
== //
44
fore BFHFNB; and becauſe FNB + BFN
=
BNF
+ BFN to a right Angle; therefore N B F is a right An-
gle; and confequently N B is perpendicular to F B, and is the
Tangent of the Arch EB; and NF is the Secant of the fame.
2. E. D.
1936. From theſe Theorems, it is evident, we have certain
Rules for defcribing any Circles great or ſmall, on a given Plane;
and therefore the Sphere may be projected on the Plane of any
one of its Circles at Pleaſure. We have already given a Speci-
men of three of theſe Projections in Plate VI. of our Inftitutions
of Perspective, in Fig. 2, 5, and 6. with their Rationale, as de-
rived from the Principles of that Science. But the Praxis or Me-
thod of drawing the Circles in each, are more immediately de-
duced from the Theorems we have juft now premifed.
1937. Thus the PROJECTION on the Plane of the EQUATOR
(in Fig. 2. of that Plate) confifts wholly of right Lines, and con-
centric Circles. The firft of which are the great Circles, or Me-
ridians, at right Angles to the Plane of Projection, and whofe
Planes all paſs through the Eye; they are therefore all projected
into right Lines. The Circles are the Parallels of Latitude,
whoſe common Center is the Pole C, or Center of the Primitive,
and their Diſtance from the faid Center is equal to the Tangent
of half their Diſtance from the Pole, and are therefore eaſily
drawn by (1934).
1938.
264
INSTITUTIONS
1938. The PROJECTION of the SPHERE on the Plane of a
MERIDIAN, is that of Fig. 5. where all the Meridians are great
Circles oblique to the Plane of Projection, and drawn by finding
their Centers as directed in (1927) in the fame Manner as the
oblique Circles OHP, OYP, and O M P, were drawn (1928);
and all the Parallels of Latitude, being fmall Circles, perpendi-
cular to the Plane of Projection, are drawn by finding their Cen-
ters as directed (in 1934 or 1935).
1939. A PROJECTION on the Plane of the HORIZON, is
that of Fig. 6. of Plate VI. and the great Circles or Meridians
are drawn by the fame Rules in this as in the former Projection;
for let TQFR (Fig. 7.) reprefent the Plane of the Horizon
where the Elevation of the Pole O is TGTQO, then it
is evident, the Center of the Six o'Clock Hour-circle QOR is
at B, the Center of the Primitive A OEP, on which, if the
Meridians or Hour-circles are drawn, as above directed (1938)
they will, if continued beyond the Pole O, be the proper Meri-
dians or Hour-circles for the horizontal Projection TQFR,
Thus for Inftance, the Hour-circle of XII is OP in one, and
TF in the other; that of Six o'Clock is AOEP in one, and
QOR in the other; the Hour-circle of II. is OZP in one,
and ZON in the other; and fo of the Reft; therefore they are
the fame in both Projections, and are drawn for both at the ſame
Time, all which is evident by Inſpection.
1940. The Parallels of Latitude are all oblique to the Horizon
or Primitive, but are all Circles in the Projection (1933) and
are drawn by ſetting off the Tangent of half their leaft and great-
eſt Diſtances from the Pole of the Primitive, (1934, or 1935)
or Zenith of the Sphere; and bifecting that Interval, you will
have the Center of every Parallel in the Line F T continued out
beyond T. The Equator is drawn by the fame Rule; and the
prime Vertical is a right Line QCR, as its Plane paffes through
the Eye in the Nadir or lower Pole of the Horizon.
1941. Hence it appears how great the Affinity is between
Projections of the Sphere of different Denominations, or rather
that they are all but the fame Thing in different Views, and all per-
formed by the fame Rules. It is hoped the Reader will, from
the Method we have here taken, have a clear Idea of the Ratio-
nale and Praxis of this moſt uſeful Branch of Science.
INSTI
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40
The THEORY of NAVIGATION.
( 265 )
INSTITUTIONS
O F
LOXODROMICS;
O R,
The THEORY of NAVIGATION on a RHUMB
LINE demonftrated. And from thence
each particular SPECIES of SAILING is
deduced.
CHAP. I.
The THEORY of NAVIGATION on a RHUMB LINE
demonftrated, from its Genuine PRINCIPLES;
with the Nature and Conftruction of MERCATOR'S
Chart.
1942. Tof a GLOBE; and therefore the shortest Way from
HE WAY or COURSE of a Ship, is on the Surface
one Place to another, is in the Arch of a great Circle; and when
a Ship is conducted on fuch an Aich, it is called ORTHODRO-
MICS, or GREAT CIRCLE SAILING.
1943. This Method of Sailing, were it eafy and convenicut,
would certainly be the beſt of all others; but fince a great Circle
makes different Angles with the Meridians it paffes over, there
fore the Line of the Ship's Courfe will be perpetually varying;
and there will be no conſtant Rule for, her Conduct. But for eve-
VOL. II
M m
ry
266
INSTITUTIONS
ry fmall Diſtance fhe fails, a new Calculation for the Angle of
the Courſe muſt be made, and this would render Navigation ex-
tremely difficult and perplexed.
1944. The eafieft Method, therefore, of Sailing, is to fteer
her Courſe in fuch a Manner, that it thall every where make the
fame Angle with the Meridians fhe paffes over; and then there
will be a conftant Guide to direct her Progrefs, viz. the Coм-
PASS Box and NEEDLE.
1945. For fince by the Property of Magnetism, the Needle
does in every Place make a given Angle with the Meridian, if a
Ship be fteered upon a given Point of the Compaſs, the Courſe
will always make a given Angle with the Meridian alſo; that
is, it will croſs every Meridian under equal Angles, while the
Needle continues in the fame Pofition with regard to the Meri-
dian.
1946. But if in different Places, the Needle makes different
Angles with the Meridian, that is eaſily diſcovered by the Man-
ner, and the Quantity of its l'ariation obferved, and then the
Navigator can readily keep the Ship's Courfe ftill upon the
ſame Point of the Compafs. And this is the moſt effential Part
of the Art of Sailing.
1947. Therefore it will be neceſſary to enquire into the Na-
ture of that Line which a Ship deſcribes in her Courſe, or which
interfects all Meridians under equal Angles, and is uſually call-
ed the RHUME, or Rhumb-Line. It cannot be a Right-line, as
being on the Surface of a Globe; it cannot be a Circle, as we
have fhewn (2.) it must therefore be a particular Curve which
conftantly approaches the Pole, and by cutting each Meridian
under the fame Angle, it must make perpetual Gyrations about
the Pole, but can never terminate in it. The RHUMB, there-
fore, or Ship's Courſe, is of that Species of Curves called SPI-
RALS.
1948. Of SPIRALS, there are many different Sorts, one of
which is called the Equiangular or Logarithmic SPIRAL; and
this is the Nautical SPIRAL, or RHUMB-LINE, as we fhall
here demonftrate. Let EQ be the Quadrant of a Circle, P
the Center, and PE, PA, PB, PC, feveral Radii drawn
very near to each other, and equidiftant; that is, let E A =
ABBC. Then let the equiangular Spiral E W X be drawn,
making
Of NAVIGATION.
267
making equal Angles with the Radii every where; viz. ADE
= B F D = CGF, &c. Then fince the Triangles DPE,
FPD, GPF are equiangular, and therefore fimilar; we ſhall
have EPDP:: DP: FP::FP: GP, &c. therefore EP, DP,
FP, GP, are in geometrical Proportion, while their correfpond-
ing Arches EA, E B, EC, are in arithmetical Proportion;
wherefore the latter are Logarithms of the former, (140) that is,
A E is the Logarithm of the Ratio of DP to EP; and BE is
the Logarithm of the Ratio of FP to EP; and CE, the Lo-
garithm of the Ratio of G P to EP, and fo on. Therefore the
Curve E W X is the Logarithmic SPIRAL.
1949. Let a Tangent LV be drawn to the Curve at L, and
draw the Radii PLH, PMI, PNK very near to each other,
and from the Points M, N, let fall the Perpendiculars M O,
NR; and upon PH erect the Perpendicular PV to interfect the
Tangent in V. Then becauſe the final Triangles LMO,
MNR, are fimilar, we have LO:LM:: MR: MN::LO
+ MR: LM + MN:: LP: LWX:: LP: LV. There-
fore the whole Length of the infinite Spiral L W X is equal to
the finite Line LV.
1950. From P take any Distance PW, and defcribe the
Arch of a Circle W SZ, and at S erect the Perpendicular ST,
cutting the Tangent in T, then in the fame Manner it is fhewn
that the intercepted Part of the Spiral L W is equal to the right
Line LT, and of Courſe the remaining Part W X is equal to
TV.
1951. Now let the fame Figure repreſent the common fterco-
graphic Projection of the Surface of the terraqueous Globe on the
Plane of the Equator, of which E Q is a Quadrant; P the Pole
of the World; HP, IP, KP, feyeral Meridians; and L W X,
the Rhumb-Line, or Ship's Courſe, making equal Angles with
them all. Then let H L be a given Latitude, and L P will be
equal to the Tangent of half the Complement of that Latitude (1698,
1926); and a Ship paffing from L to M makes her Difference of
Longitude IH, Departure MO; and Difference of Latitude LO.
1952. Then it is evident, that the Differences of Longitude A E,
BE, CE, are Logarithms of DP, F P, GP, the Tangents of
half the Complements of the Latitudes AD, BF, CG; and HE,
IE, KE, are Logarithms of LP, MP, NP, the Tangents of
balf
M m 2
268
INSTITUTIONS
half the Co- Latitudes of the Places L, M, and N, in respect to the
Radius PE.
1953. Then, fuppofe a Ship fails from the Latitude L on the
Rhumb ELMW to the Latitude M in the Parallel MO (Fig.
2.) the Logarithm of the Ratio of MP to EP is the Arch IE;
but if ſhe fails on any other Rhumb L w, fhe will come to the
fame Parallel in the Point (m,) and then the Logarithm of the
fame Ratio m P (— M P) to E P, will be the Arch i E. Confe-
quently as there may be an Infinity of Rhumbs fuppoſed to be
drawn through the Point L, there will be an infinite Variety of
Scales of Logarithms to be found in the Arches of the Equator for
expreffing the fame Ratios.
1954. In failing on the fame Rhumb L W, (Fig. 1.) it is,
as Radius to the Secant of the Angle of the Course OLM, fois LO
: LM:: MR: MN: LO MRL M+MN:: LP:
LO+MR:LM+MN
LWX; fo is the Arch of the Meridian equal to the Go-Latitude,
to the Length of the whole Spiral or Rhumb from the Latitude or Point
L. For the Triangles L M O, MNR, are equiangular and
fimilar.
1955. Hence if LO MR, then L M MN; therefore
the Length of the Rhumb between any two equidiftant Parallels is the
fame; or the Length of the Rhumb LM is always proportioned
to the Difference of Latitude LO. For LO: LS:: LM:
LW. (1950.)
1956. Again, as Radius to the Tangent of the Courſe
OLM, ſo LO:MO:: MR: NR:: LO + MR: MO
+ NR:: LP or Co- Latitude, to the Sum of all the indefinitely
ſmall Arches of the Parallels, or whole Departure made in fail-
ing obliquely from one Meridian to another.
1957. Hence in very fhort Courſes, where the Diſtance fail-
ed LM, and the Difference of Latitude L O may be eſteemed
rectilineal; we have Radius to the Tangent of the Courfe, fois Dif-
ference of Latitude LO to the Departure M O.
1958. In failing on different Rhumbs L M and Lm to the
fame Parallel m O, (Fig. 2.) it will be as the Tangent of the Courfe
MLO to the Tangent of the Courfe mLO, fo is the Departure
MO to the Departure m O, and fo is the Difference of Lon-
gitude HI to the Difference of Longitude Hi.
1959.
Of NAVIGATION.
269
1959. And hence it appears, that the Differences of Longitude
made in Sailing upon two different Rhumbs LW, and Lw, are as
the Tangents of the Angles, which they make with the Meridians;
and therefore the Differences of Logarithms (I H, iH), of the
fame Ratio (viz. of MP or m P to EP) in the two different
Scales appropriate to the two Rhumbs, will be likewife as
thoſe Tangents reſpectively.
1960. In very fhort Voyages, the fmall Triangle L M O
may be eſteemed Right-lined, and ufed as fuch without fenfible
Error, becauſe a ſmall Portion of the Surface of a Globe, near
8000 Miles in Diameter, will differ infenfibly from a Plane.
Hence all Cafes of Sailing (except that for the Longitude) are ſolv-
ed by a fimple plain Triangle, this is therefore called PLAIN SAIL-
ING, whofe Principles we have now premiſed.
1961. But when the Length of Voyages makes it neceflary
to confider the faid Triangle really as it is, viz. a curvilineal One
throughout; and that every Part confifts of a Curve of differing
Species, viz. LO the Arch of a great Circle; MO, the Arch
of a fmall Circle; and LM the Arch of a Spiral, it will cafily
appear, that under ſuch Circumſtances it will be no eaſy Mat-
ter to refolve Cafes of Sailing on the Surface of the GLOBE it-
felf, or by the Globular Chart.
1962. Therefore to facilitate the Praxis of fo excellent and
neceffary an Art, Methods have been contrived to delineate the
Surface of the Globe on a Plane, in fuch Manner, that all the
Calculations relative to NAVIGATION, ſhould ſtill be ſubject to
plain Trigonometry, and yet productive of Accuracy and Truth
beyond all neceffary Degrees.
}
1963. To illuſtrate this Affair, let C (Fig. 3.) be the Center
of the Globe; P, the Pole; A P, BP, two Meridians ftand-
ing on the Arch of the Equator QR. Let ED, ST, be Ar-
ches of Parallels whofe Radii are DO, TV. At the Points.
A, B, are erected the two Perpendiculars or Tangents A M,
BN; and through the Points E and D, are drawn the Sccants
GC, FC. Then is the Arch of the Parallel E D thereby pro-
jected into the Right-line GF on the Plane between the two
Tangents. Therefore GFA B, in the fame Manner the
Arch ST is projected into MNA B, all which is evident
from the Doctrine of Gnomonical Projection (1738, St.)
1964.
270
INSTITUTIONS -
1964. But theſe Arches ED, ST, are when compared with
the correfponding Arch AB of the Equator, as their respective
Radii DO, TV to the Radius A C of the Globe. Or A B is
to ED, as Radius to Co-Sine of the Latitude; for DO is the
Sine of DP the Complement of the Latitude BD or A E.
1965. Again, the Arches ED, ST, are to their Projections
GF or MN, as EC or SC, to OC and MC; or they are en-
larged in Proportion of the Secant of the Latitude to the Radius.
1966. Suppoſe a Ship fails on the Rhumb A DL from A to
D; then is AE the Difference of the Latitude, AB the Diffe-
rence of Longitude; the Angle EAD the Courſe; AD the
Distance failed; and D E the Arch of the Parallel at D. Now
it is required to delineate this mixtilineal Triangle A ED on a
Plane in fuch Sort, that all the Parts may have the fame Ratio of
Magnitude and Pofition as they now have on the Surface of the
Globe.
1967. In order to this, it muſt be confidered, that in what-
ever Proportion any one Part is altered by projecting the Trian-
gle EAD on the Plane AMNB, the other Parts must be al-
tered in the fame Ratio, to produce Similarity, and thereby to
keep the given Pofitions of all the Parts to each other. But by
the Projection, the Part or Perpendicular ED is enlarged to
GF, in the Ratio of EC to GC (1965.) and therefore the Side
or Bafe A E must be enlarged in the fame Proportion on the
Line A M.
1968. Now it is evident, the Arch A E defcribed with the
Radius AC is to the Arch G K defcribed with the Radius GC,
as Radius to the Secant of the Arch A E, or Angle A CE; if
therefore in the Right-line AM, we take AH equal to the
Arch GK, and draw HI parallel to AB for Chord of the
Arch A B) and draw the diagonal AI; we fhall have the right-
lined Triangle on the Plane, every Way fimilar and proportional
to the Triangle EAD on the Globe: For it is HI (= GF)
:ED:: GC:EC:: AH(≈ GK): AE :: AI: AD.
(=
1969. Hence the Angle of the Courſe H AI is the very fame
as the Angle EAD on the Globe; and therefore the Pofition
of the Point I with regard to the Point A in the Plane, is the
very fame as that of the Port D in refpect of the fame Point A on
the Globe; ſo that the Eafting or Wefting on a Rhumb A D mak-
L
ing
Of NAVIGATION.
271
ing any Angle EAD with the Meridians, is truly preſerved
and repreſented in the Triangle HAI in Plano; and confequent-
ly the whole Affair of Navigation is by ſuch a Projection reduced
to the Solution of a plain right-angled Triangle, and thereby be-
comes a very eafy Tafk as far as Calculations are concerned.
And this Projection of the Surface of the Globe on a Plane, is
called MERCATOR'S PROJECTION, or CHART.
i
CHA P. II.
The THEORY of NAVIGATION by a TABLE of
Logarithmic TANGENTS demonftrated.
1970. L
ET Radius of the Equator PE = r (Fig 1.); the
Latitude of the Place H L z; the Radius of the
Parallel at L = x; and the Longitude or Arch of the Equator
EHy; then will HI; and LO= 2; and we have
PH: PL::r:x::: -Lm, the Fluxion of the Parallet
at L.
t
xj
And putting = Tangent of the Angle LMO or
x y
Whence =
LM m, we have Lm: m M ::
: £: 1.
x;
; and fo
t
1 x
X
rtż
j; and
rj.
X
1971. To conftruct this Equation, let the quadrantal Arch-
of the Meridian be extended into a Right-line A B (Fig.4.); at A
erect a Perpendicular AC = t; and make A D=z, the Lati-
tude of the Place; upon Derect the Perpendicular DE, which is
to AC
t, as r to x, fo that it may be every where DE =
tr
x
; and thus by conceiving Perpendiculars to be raiſed on every
Point of the Meridian A B, we fhall have the Curve CEH
produced, and of fuch a Nature that the curvilineal Space
ADEC, divided by Radius, will be equal to the Arch of the
Equator expreffing the Longitude E H made in failing from the
Equator E to the Latitude Z on the given Rhumb ELW
(Fig. 1.)
Y
1972.
+
272
INSTITUTIONS
1972. For let de be drawn infinitely near, and parallel to
DE; then AD = 2, and D d = 2, and DE x D d =
trz
tr
30.
=ry (1970.) But the Fluent of is the Space ADEC,
and that of ry is ry; therefore
rỳ
upon the Courſe EL.
x
ADEC
y the Longitude
r
AFGC
1973. After the fame Manner we have
r
= the Lon-
gitude anſwering to the Latitude AF upon the fame Courſe.
the Difference of Longitude corref
Confequently
DFGE
r
ponding to the Difference of Latitude DF upon the fame
Rhumb.
1974. It is eafy to conceive the Space A DE C, and its
Fluxion Deed may be transformed into equal rectangular Spa-
ces or Parallelograms, as AIK C, and IK ki, and becauſe in
this Cafe IK = AC=t, therefore Ii =
* នំ
7*
X
which is to Dd
=, as to x, that is, as Radius to the Go-Sine of the Latitude,
or as the Secant to the Radius. But this is the Proportion of the
protracted Arch to the Length of the natural Arch of the Meri-
dian (1967, 1968.) And fince A D is the Length of the latter,
AI will reprefent the former, viz. the enlarged meridional Arch,
ufed in Mercator's Chart. (1969.)
1975. Hence fince the Rectangle AIKCAIXA Ċ =
ry, or, (putting AIZ) Ztry, therefore rt:: Z: y,
or Radius is to the Tangent of the Courfe, as the protracted Arch or
Degrees of Latitude is to the Arch of the Equator expreffing the Longi-
tude.
1976. Hence when the Rhumb or Spiral ELW interfects
the Meridians at an Angle of 45 Degrees, we have rt; and,
then zy; the Degrees of Latitude, therefore, in the protracted
Meridian of Mercator's Chart, are a SCALE of LONGITUDES in
failing upon a Courſe or Rhumb of 45°.
1977. Becaufer:x:: Sr (putting S for Secant of the Lati-
tude) therefore
J
S
trz
and fo
४
t S ż
=rj(1970); and
r
when
1
Of NAVIGATION.
273
tr, as in Cafe of the Rhumb of 45°, then Szry; and
if r 1, as in the Canon of natural Sines, Tangents, and Se-
cants; then Sj; whence it appears, that the Fluxion of the
Longitude (ỳ) is equal to the Fluxion of a Space or Aren conſiſting of Se-
cants fucceffively erected on the fluxionary Points of the Meri-
dian.
=
1978. Becauſe z is conftant in the Equation Sj, we
have Sy, or the Fluxion of the Longitude () will be as the
variable Secant S; and fo the Sum of all the Fluxions or the
Longitude itſelf, will be as the Sum of all the Secants to any given
Latitude. And hence the Reaſon why Mr. Wright took this Me-
thod to conftruct his Line of ineridional Parts, viz. by the con-
ftant Addition of the Secants, as they are found in the Canon. See
his vulgar Errors in Navigation corrected; where the nautical
meridian Line firft made its Appearance.
1979. It has been fhewn, that the Fluxion of the Logarithm of
any Number is equal to the Fluxion of the Number divided by the
Number itfelf (849). But the Longitudes are Logarithms of
the Tangents LP, MP, &c. to the Radius EP (1952) ſuch a
Tangent therefore is a Number, which put = n; then, when
r = 1, we have t S & == (1977) which gives ny =n,
:
and therefore in :: j;
(Fig. 1.)
72
n
or PH (PE): PL::IH: Lin
1980. And alfo, whent I, as upon a Rhumb of 45°, we
= ", whence it appears, that the Fluent of S&, which
have S,
n-
is an Arch of the nautical Meridian, (1978) is equal to the Fluent
of, or the Logarithm of the Tangent of half the Co- Latitude of that
n
Arch. Therefore the whole nautical meridian Line is a Scale of
Logarithmic Tangents of the half Complements of the Latitudes, upon
that Rhumb which makes an Angle of 45° with the Meridian. There-
fore this we ſhall call the nautical Scale of Logarithmic Tangents.
ADEC
1981. But it is evident from the original Equation
да
Longitude EH, or Logarithm of the Ratio of PL to
PE (1972.) that the Logarithm of the fame Ratio will be vari-
VOL. II.
N n
1
able
274
INSTITUTIONS
1
able according to the different Values of r, or the Radius of the
Globe PE. For HI is the Logarithm of the Ratio of the Tan-
gent PL to Radius PH or PE, when r = PE = 1, and the
Angle MLO 45°. But if r or PE be expounded by
=
any Number leſs or greater than Unity, it follows, that the
Length of H1 y will be encreaſed or diminiſhed in the fame
Proportion; for the Area ADEC being confidered as con-
ftant, the Logarithm (y) will be always as
Radius.
trz
I
r
or inversely as the
1982. From the fluxionary Equation =rj (1970), it
x
appears that near the Beginning of the Rhumb Line in the Equa-
tor, where x =r, that Equation becomes tż ry, and then r:t
:::. And therefore upon the Rhumb of 45°, where t=r,
we have ży. Therefore the Fluxion (3) of the Logarithm of
the Ratio of the first mean proportional Tangent to the Radius is equal
to the infinitely fmall Difference (≈) between that proportional and
Radius.
1983. Hence if Radius PE = 1 (Fig. 1.)
portional be PD = 1-2, the fecond PF
3
and the firſt Pro-
1−Ż 22, the third
PG=1—2³, &c. then will the Logarithms be AE,
=
BE=2*, CE 3, &c. therefore I
tional I-z, and its Logarithm is nż.
#
I-, it is I Z = I 2 = I
=ż.
I
n
is any Propor-
But fince I-
•
Z
12
Z whence 1-1 — Z
I
. But by the Newtonian Theorem for extracting the Roots
of Binomials (306) we get 1-
N
I
I
Z3
371
4n
I
I
I
I
2" = 1
Z
11
2n
z¹ + }z³, &c.
24, &c. or -x z + ½ ≈² + 1 ≈³ +
A
2
I
2
3
I z"; whence z + 2 + 3 ≈³ + = 24 +
2³, &c.nz Logarithm of the Number I
z.
Z.
I
1984. This Series is the fame with that derived from the Hy-
perbola (829), and when computed in Numbers gives 2302585
for the Logarithm of the Ratio of 10 to 1, or of 1 to 0,1. Con-
fequently
Of NAVIGATION.
275
fequently 2302585 = n; or fo many mean Proportionals
there are in the Ratio of 10 to 1 in this Scale of Logarithms
which are thoſe firft invented by L. Neper.
1985. But in Practice it was found much more convenient to
put the Logarithm of the Ratio of 10 to 1 = 1000000 = nż,
as in the common logarithmic Canon. Hence we have n: n:
2302585 1000000 :: :: 0,43429448.
I
Thefe Numbers,
therefore expreſs the Ratio of the Logarithms in each Syftem
refpectively.
1986. From what foregoes, it appears, that the infinite Series
juft now found, anfwers to the Space ADEC = y (1972)
when r = 1; and nży the Length of the nautical Meri-
y=the
dian for the Latitude z. Then putting R for the Logarithm of
the Tangent of 45° or Radius, and T for the tabular Tangent
of any other Arch, as that which denotes the half Co-Latitude
of HL; we have, as EP: LP:: 1:1
I I z = t = natural
Tangent of LP. Therefore the Ratio of
but the Logarithm of
I
t
lefs the Logarithm of t.
I
t
R-T.
Z
I
EP
LP I- Z
I
is equal to the Logarithm of Radius 1
=
That is, the tabular Logarithm of
1987. But the tabular Logarithm of any Ratio is to Neper's
Logarithm of the fame, as I to 2,302585 (1985); therefore
R—T X 2,302585 the Length of the protracted meridi-
onal Arch HL = z, in the Scale of Neper's Logarithms. Put
N=2,302585, and T = Logarithm Tangent of any other
Arch, as that of NP. Then becaufe NR-NT is the
Length of the Arch HL, and NR-NT is the Length of
the Arch KN, we have NT-NT=T-Tx N, the
Difference of the Latitude L and N expreſſed in Parts, of which
the Radius EP is 1.
1
2
1988. If the Radius be expreffed in Minutes of a Degree,
then the protracted meridian Arch will be had in Minutes likewife.
Now the Circumference of a Circle is to Radius as 6,28318 to
But in that Circumference there are 360° or 21600′; there-
fore fay, as 6,28318 : 1 :: 21600′: 3437′,747, which is the
Nn 2
Num-
I.
276
INSTITUTIONS
Number of Minutes in the Radius. Put this Number 3437′,747
=m; then mN × T-7 will expreſs the Length of the Dif
ference of Latitude in Minutes of the protracted Meridian; and
thus a Table of meridional Parts or Minutes, may be computed
from the common Table of logarithmic Tangents.
1989. For Example, let it be required to find the meridional
Parts in the Arch of the Meridian contained between the Tropic of
Cancer and the arctic Circle. Then the Latitude of the Tropic
is 23° 30', its Complement is 66° 30', and the Half thereof is
33° 15', whofe logarithmic Tangent is 9,816658 T; the
Latitude of the arctic Circle is 66° 30', its Complement 23° :
30, and the Half of that 11° : 45', whofe logarithmic Tan-
gent is 9,318064; then T-T 0,498594; and mN =
7915,704; therefore mNx T-T= 0,498594 × 7915,704
3946,722, the meridional Parts or Minutes in that protracted
Difference of Latitude.
=
1990. As in all Circles, the Radii P A, PE, PI, (Fig. 6.) are
proportioned to fimilar Arches in their Peripheries A B, EF, IK;
and as thoſe Arches are Logarithms of the Ratios of DP to AP,
GP to EP, LP to IP, in failing on the Rhumbs A N, EQ,
İQ, making the fame Angle with the Meridians in all, there-
fore it follows that the Radii are the Modules or conſtant Expref-
fions of the Ratio of the Logarithms pertaining to their feveral
Peripheries or Syftems refpectively. Thus if IP be Radius or
the Module for the Logarithms of Neper's Sort, and be expref-
fed by Unity, then the Module or Radius for the Logarithins of
the common Tables will be 0,43429448 = EP (1985) and
then if the Arch IK be Neper's Logarithm of that Ratio of 10
to 1, EF will be the Logarithm of the fame Ratio in the tabular
Syftem. So that if I K = 2,302585, we have EF = 1.
1991. Again, if AN be the Rhumb which makes an Angle
of 45° with the Meridians, and AB an Arch of one Minute
then making A C A B, we have A BDC a Square, and DC
= A C
AC — AB; and fince the Radius AP is in this Cafe to be
confidered as divided into Minutes and equal to 3437′,747,(1988)
which is but a Part of PI; the faid Radius PI, if it be divided
throughout into the fame equal Parts or Minutes, will contain
juſt 10000. For IP: AP :: I: 0,3437747, which therefore
is
Of NAVIGATION.
277
is the Module for the Syftem of Logarithms pertaining to the
Rhumb of 45°, or the nautical meridian Line.
1992. The Modules, therefore, or Ratios of the three Sy-
fems of Logarithms are theſe, viz. for Neper's Logarithms
PI= 1,000000; for the Tabular or Brigg's Logarithms PE
0,434294; and for the nautical Logarithms we have PA
0,343775. Therefore if E O and IQ be the fame Rhumb
of 45°, then by making EH EF; and IM=IK; the
Triangles A CD, EHG, and I M L will be fimilar; and we
have AC DC:: EH: GH::IM: LM.
=
1
—
1993. Hence becauſe A C A B = 1 Minute, we have as
0,343775: I (:: PA: PI::AB: IK) :: 1: 2,90888IK.
And this is the Meaſure of the natural Sine and Tangent of 1 Mi-
nute in the Tables. Again, AP: EP:: 0,343775:0,434294
:: (AB: EF::) 1 : 1,2633114 = EF, the Logarithm of the
fame Ratio in Brigg's Syftem.
1:
1994. Since then the Logarithms of the fame Ratio pertaining
to the fame Rhumb, are expreffed by the different Numbers 1,
1,263, 2,909, in the three different Syſtems above-mentioned.
It is eaſy to conceive, that upon different Rhumbs in the ſame
Syſtem, the fame Numbers will exprefs or be the Logarithms of
the fame Ratio. For let PI be Radius = 1; and take Ic =
AC, and Ig = EH, and draw the Parallels ce and gi; and
the Squares Icab = ACBD, and IGbf= EHGF; then
it is Ic ca: Ig: gb:: IM: ML ce; whence Ic: IM
::ca: ce I: 2,909: Tangent of a Ic = 45°: Tangent of
elc 71° 1' 42". And therefore if a Ship fails on the Rhumb
Iel making that Angle with the Meridians, Neper's Logarithms
will be a Scale of logarithmic Tangents of the half Co-Latitudes every
where.
=
•
1995. In the fame Manner we have Ic: Ig :: ca:gh = cd,
that is 1: 1,2633:: Tangent of alc = 45°: Tangent of dIg
=51° 38′ 9″. Confequently, if a Ship fails on a Rhumb Idk
making that Angle with the Meridians, then (becauſe c d — EF)
the logarithmic Tangents, in the common Canon, are thoſe which
every where express the Ratio of the Tangents of half the Co-Latitudes
to Radius; and therefore the Difference of the logarithmic Tan-
gents of the half Co-Latitudes will be as the Difference of Longitude
CO3*
278
INSTITUTIONS
correfponding to the given Difference of Latitudes on the Rhumb of
51° 38′ 9″.
1996. But this Difference of logarithmic Tangents or Lon-
gitudes, is an Arch of the Equator, whofe Radius, or natural
Tangent in the Tables, is 1. Therefore, putting T-T for
this Difference of logarithmic Tangents (1987) we muft fay, as
0,434294 : 1 : 4342′,94 : 10000′ — Minutes of a Degree in
I (1990). Then ſay, as the Arch 1 :
10000 :: T—T: T-T × 10000 = Minutes of a Degree
in an Arch of the Equator expreffed by T—Ï.
the Radius or Arch
1997. Hence the principal Cafes of Mercator's Sailing are fol-
vable by the common Tables of logarithmic Tangents; for if
the Difference of Latitude be given, the Quantity T-T x
10000, or Difference of Longitude correfponding thereto on the
given Rhumb of 51° 38′ 9″ will be known alio; and then, if
the Difference of Longitude be given on any other Rhumb or
Courſe, the Tangent thereof will be known likewife. For, as
the Difference of Longitude T-Tx 10000 is to the conftant Tan-
gent of 51° 38′ 9″", fo is the given Difference of Longitudes to the
Tangent of the Courſe on which the Ship jails (1958, 1959).
rtż
X
1998. From the fluxionary Equation =rj (1970) it
appears, that upon the Rhumb of 45°, when tr, the Equa-
tion
rz
x
j, will then exprefs the Fluxion of the nautical meridian
Line, which in its nafcent State, or very beginning (where x = r}
will become, or the Fluxion of the natural Meridian it-
felf.
1999. Now the nautical meridian Line
begins from the Radius with the natural
Meridian; but the artificial Line of Tan-
gents begins from the Tangent of 45°.
But the Fluxion is equal to, or half the
Fluxion of the Tangent of 45°, as is thus
fhewn. On the Center C defcribe the Cir-
C
cle DB E, and draw the Tangent DP,
T
E
P
D
the
N. B. Fig. 7. was forgot to be put in the Plate, and is here added.
Of NAVIGATION.
279
the Secant CP,
and infinitely near it CT; and deſcribe the
Then put Radius CD
fmall Arch PQ.
r, DP = t, DR
= 2, TP = ¿, Bb ż; and by fimilar Triangles we have
TP:QP::CP:CD; and PQ: Bb: CP: CB = CD.
Therefore TP : Bb :: CP² : CD. That is, rr + tt: rr
:;:ż; and when t=r, then 2 : 1 ::¡: ŵ, or ¿ — 2 ¿.
2000. Becauſe z is the Fluxion of the nautical Meridian
(1998) it is the Fluxion of the logarithm Tangents proper to the
Rhumb of 45° (1980). And fince the Fluxion of the Loga-
rithm of any Tangent is
i
therefore when fuch a Tangent be-
comes equal to Radius, or tr, then the Fluxion of its Loga-
rithm is barely; but we have fhewn that in fuch a Cafe i = 2.
Therefore the Fluxion of our artificial Line of logarithm Tangents is
double the Fluxion of the nautical meridian Line of logarithm Tan-
gents. And, fo the Fluents or Degrees of logarithm Tangents in one
will be double to thoſe of the other every where.
2001. Hence it appears, that any Line of artificial Tangents
is the fame with the nautical meridian Line, if we take every half
Degree for a whole one, and number them accordingly. Hence any
fuch Line of logarithm Tangents may be uſed or fubſtituted for a
Line or Table of meridional Parts, and all the Problems of Merca-
tor's Sailing are folved thereby. The Identity or Coincidence of
theſe two Lines is repreſented in the double Scale (Fig. 8.) where
AB is the common Line of artificial Tangents, and CD the
nautical Meridian; one to 25 Degrees, the other to 50.
CHA P. III.
The different SPECIES of Loxodromic SAILING,
deduced from the preceeding THEORY, and illu-
Arated by proper DIAGRAMS.
2002.
FR
ROM a particular Confideration of the Nature of what
is called Departure will refult another Species of Navi-
gation called MIDDLE LATITUDE SAILING. It is evident, by
In-
280
INSTITUTIONS
Inſpection (Fig. 1.) that which is called Departure is only the
Arch of a Parallel contained between two Meridians, as P H,
PI, paffing through the two Latitudes L and M; and is al-
'ways to be reckoned in that Parallel to which the Ship arrives.
Thus if fhe fails from L to M, the Departure is the Arch MO,
paffing thro' the Latitude M. But if the Ship fails from M to
L, then the Departure is the Arch L m, fo that there is in Re-
ality, two different Departures belonging to the fame Courſe,
Diſtance, or Difference of Latitude.
2003. Therefore to avoid all Ambiguity and Falacy in Sail-
ing, it is neceſſary to make conſtant Computations of the De-
parture upon Parallels fo near together, that the Arches MO
and L m may differ but by an infinitely ſmall Quantity, or that
the Triangle L O M may be eſteemed rectilineal; and then we
ſhall have the ſeveral Departures M O, NR, &c. computed
as they riſe in paffing over the ſeveral Meridians, as from L to
M, from M to N, &c. till at laft the Ship arrives at ſome Diſtant
Port W in Latitude QW, and the Sum of all theſe Departures
will be the whole Departure made in the Voyage from L to W.
2004. But fince the particular Departures M O, NR, &c.
proceed continual decreafing from L to W, 'tis evident their Sum
will be greater than W S, which is the Sum of the fame Num-
ber of Departures each of them equal to the leaſt of thoſe made
in the Voyage. And on the other Hand if the Ship fails from
Latitude W to Latitude L, the Sum of all the particular De-
partures will be the fame as before, but lefs than the Arch Y L,
which is the Sum of the fame Number of Departures, each equal
to mL, the largeſt of thoſe made in the Voyage.
2005. Confequently, the Departure made in the Voyage from
L to W, or from W to L, being the fame; and confifting eve-
ry
where of Quantities equidifferent, the greateſt and leaſt of
which are as LY to WS; the Sum of all the Departures will
LY+WS
be a Mean between the two, or as
2
= de,
de, the cor-
refponding Parallel, or mean Arch of the middle Latitude He
(136), which therefore will nearly reprefent the whole De-
parture of the Voyage either Way between the two Ports L
and W. (318.)
2006. Hence
Of NAVIGATION.
281
I
2006. Hence we deduce this fundamental Theorem, The Co-
Sine of Middle Latitude P e is to Radius PH as the Departure de is
to the Difference of Longitude HQ. Note, If the Sum of the
Co-Sines of the Latitudes L and S be taken, it will be nearer the
Truth than the Co-fine of Middle Latitude.
2007. Put Radius = R; Middle Lat.
D; Diff. of Lat. L; Diff. of Long.
=
Courſe C; and Diſtance run S.
=
2
M; Departure =
G; the Angle of the
Then by the laſt, we
have cs M: R :: D: G; and by (1957) it is R: tC :: L: D.
Therefore R x D cs Mx GtCx L, which gives
this Theorem, the Co-Sine of Middle Latitude is to the Tangent of
the Courfe as the Difference of Latitude is to the Difference of Longitude.
2008. Then becauſe the Diſtance failed LM is to the De-
parture MO, as Radius to the Sine of the Courſe; that is, as
S: D::R: SC; we have R x DSC XS = cs M x G;
which gives this Theorem, as Co-Sine of Middle Latitude is to the
Sine of the Courſe, ſo is the Diſtance run to the Difference of Lon-
gitude.
2009. Having thus premiſed the Principles of each Species of
Sailing, we fhall illuftrate the fame by Figures adapted to them
feparately. And firft, as to PLAIN SAILING, we have ſhewn
(1957, &c.) that it confifts only in the fix Cafes of a plain right-
lined Triangle; which therefore let be denoted by ABC (Fig. 9.)
in which AC is the Difference of Latitude; AB, the Departure;
BC, the Diſtance failed; and AC B, the Angle of the Course;
any two of which being known, the reft are found by plain Tri-
gonometry (711 to 716.)
2010. MERCATOR'S SAILING has all its Cafes expreffed in
two fimilar Triangles in one Figure, viz. ACB and DCE
(Fig. 10.) in which AC is the Length of the Arch of the Meri-
dian in proper or natural Degrees of the Latitude, or Difference
of Latitude between two Ports. CD is the fame enlarged ac-
cording to (1968) and is called the meridional Difference of Lati-
tude. AB is the Departure. DE the Difference of Longi-
tude. CB, the natural, and CE the enlarged Diſtance run, and
C the Angle of the Courfe. Here are allo fix Varieties or different
Cafes of Sailing from two Data, as before.
2011. In MIDDLE-LATITUDE SAILING, two plain right-
angled Triangles are alfo ufcd, but not fimilar ones, viz. ACB
VOL. II.
O o
and
282
INSTITUTIONS
1
and DFE (Fig. 11.) in which ACFD is the Difference of
Latitude. AB the Departure, DE the Difference of Longitude,
CB is the Distance failed; and the Angle of the Courſe. Two
of theſe being given, the others are found by the Theorems
above (2006, f.)
2012. But that the Reaſon of this complex Figure of Triangles
may appear, it is to be obſerved, that if GH be drawn parallel
to A B or DE, it will be the Tangent of the Courfe C, and GF
the Co-Sine of the Middle Latitude to the Radius CG. For on
the Center G defcribe the Semicircle CLN; and from L fet-
ting off the Middle Latitude to M, we have MK the Sine, and
KG = FG the Co-Sine thereof; and therefore by fimilar Tri-
angles GFH, DFE, we have GF GH::FD: DE;
that is, the Co-Sine of Middle-Latitude is to the Tangent of the
Courfe, as the Difference of Latitude to the Difference of Longitude;
as before in (2007).
J
2013. Again; we have CG: GH;: CA: AB; and GF
: GH:: DF (= AC): DE: confequently it is GF CG
::AB: DE, or Co-Sine of Middle Latitude to Radius, as the
Departure to the Difference of Longitude, agreeable to (2006).
And fo of the other Theorems.
2014. Another Species of Navigation, is called, PARALLEL
SAILING; which confifts in conducting the Ship due EAST or
WEST upon a PARALLEL to the Equator. The Diagram by
which this is reprefented, is that of Fig. 12. Where P is the Pole
of the World; PC, PD, two Meridians; GH an Arch of
the Equator, and EF a fimilar Arch of a Parallel of Latitude.
Then we have PC:CD:: PA: AB, or Radius to the Diffe-
rence of Longitude as Co-Sine of the Latitude to the Distance failed in
the Parallel (1964).
2015. This Sort of Sailing therefore admits of three Cafes
only, viz. (1.) When AB and AP are given to find CD.
(2.) When AB and CD are given to find PA. And (3.)
When AP and C D are given to find AB. All which is evi-
dent by Inspection.
CHAP.
Of NAVIGATION.
2,83
CHA P. IV.
The THEORY of NAVIGATION by Logarithmic
TANGENTS exemplified in a Solution of the feveral
Cafes of SAILING by that Method:
S all the above Methods of computing the Way and
2010. Courfe of a Ship confift in the Solution of a plain
As
right-angled Triangle, they require no Example; but the Man-
ner of folving the Propofitions of Sailing by the Tables of Loga-
rithmic Tangents is not quite fo common, nor fo clear, in what we
have had hitherto publiſhed on that Subject. It may be proper
therefore to illuſtrate the Theory we have given by a Solution of
fuch Cafes in which the Longitude is concerned, eſpecially as it
will from thence appear that this Invention fuperfedes the Ne-
ceffity of a Table of meridional Parts, and is when joined with
plain Sailing, a compleat Compendium of practical Navigation.
2017. If we retain the Notation in 1987, &c. and, more-
over, put L for the Difference of Longitude, C for the Ship's
Courſe, and G for the conftant natural Tangent of 51°: 38:
9"; then we have this Analogy T-T × 10000 : G :: L:
¿C (by 1997) which admits of three Cafes, befides thoſe of plain
Sailing, viz.
2018. CASE I.
Given the Difference of Latitude, and Difference of Longitude,
GX L
to find the Ship's Courfe; the Theorem is
t C.
2019. CASE II.
T--TX 10000
Given the Difference of Latitude and Courfe, to find the Diffe-
rence of Longitude; the Theorem is
= L.
2020. CASE III.
T-TX 10000 × ✰C
G
Given the Courſe, and Difference of Longitude, to find the Diffe
A
rence of Latitude; the Theorem is
о
© 2
GX L
tC x 10000
=T-T
20218
284
INSTITUTIONS
2021. For an Example of the firft Cafe, fuppofe a Ship fails
from the Lizard in Latitude 49° 55′ North to Barbadoes in Lat.
13° 10' North; the Difference of Longitude being 53° 00', re-
quired the Rhumb on which the Ship was ſteered?
The Solution of this Cafe is as follows:
Lat.
Comp.
Halves
Barbadoes 13 10
76 50
38 25
Lizard 49 55
40 05
20 21 —
Log.Tang.
9,8993082 = T.
9,5620477 = T.
The Difference is
T—T = 0,3372605
Multiply by
10000
The Product is T-TX 10000
=3372,605
Then the Tang. of 51° 38′ 09" = G =?
= 10,1015104
12633210
Mult. Diff. of Long. L= 53° 00′ 3180-3,5024271
The Product is G x L
Which divide by T-T x 10000 =
3372,605
The Quotient is the
13,6039375
弓
​} 3,5279654
Tangent of the tC = 49° 59′ 10″ 10,0759721
Courſe required
2022. Then to find the Diſtance failed, ufe this Proportion
of plain Sailing, viz. As Co-Sine of the Course is to Radius, (or as
Radius to the Secant of the Course) fo is the Difference of Latitude
AC to the Diſtance run CB 3429,38 nautical Miles or Mi-
nutes of a Degree. (See Fig. 9.)
2023. Example to CASE II.
Suppoſe a Ship fails from Latitude 49° 55′ to Latitude 13° 10′,
on a Rhumb or Courſe of 49° 59′ 10″; to find the Difference of Lon-
gitude. The Solution will ſtand thus. (See Theorem 2019).
Multiply T-T× 1000 = 3372,605
3,5279654
By Tang, of the Courfet C 49° 59′ 10″- 10,0759721
=
13,6039375
Divide by conftant Tang. G 51° 38′ 9″
=
10,1015104
The Quotientis Diff, of Long. E53°00 -
=
3,5024271
Thus,
Of NAVIGATION.
285
"
Thus, by two good Obfervations of the Latitude and the
Courſe fteered, the Reckoning of a Ship's Way is beſt aſcer-
tained, eſpecially if you fail near the North or South.
2024. Example to CASE III.
Admit a Ship fails from Latitude 49° 55′ on a Courſe of 49°
59′ 10″ foutherly, till her Difference of Longitude be 53° 00′, or
3180 Miles; to find the Latitude fhe is then in. The Solution of
this Cafe is thus (by Theorem 2020).
To the Log. of the conftant Tang. G
Add the Log. of the Diff. of Long, L =
3180
The Sum is the Log. of G x L
Then to the Tang. of Courſe = 49° 59′ 10,0759721
IO"
10,1015104
L= }}
3,5024271
13,6039375
}
4,0000000
14,0759721
G x L
— T—T
=}-
9,5279654
Add the Logarithm of 10000 =
The Logarithm of tC × 10000
Then
✰ C x 10000
0,3372605
Now one Latitude being given 49° 55', the Logarithm
Tang, of half its Complement is
To which add
9,5620477 = T.
0,3372605=T-T
The Sum is the Log. Tang. 38° 25′- 9,8993082 — T.
The Double of which is 76° 50', which is the Complement of
13° 10′, the Latitude of the Ship required. (See 2021).
2025. From hence it is evident, that theſe three Cafes, add-
ed to thoſe of plain Sailing, are fufficient for all practical Compu-
tations of a Ship's Ways, or for keeping a Reckoning at Sea,
which are all performed by a Table of Logarithmic Tangents, or by
the common Gunter properly made, as we fhall hereafter fhew,
when we come to give the Theory and Conftruction of mathemati-
cal Inftruments.
2026.
286
INSTITUTIONS
2026. What has been hitherto delivered is upon the com-
mon Hypothefis, that the terraqueous Globe is of a fpherical Figures
but that it is not fuch is well known; and not only fo, but the
Difference therefrom is fo confiderable, as not to be without
its fenfible Effects in Aftronomy, Geography, and Navigation, as
I have at large fhewn in my NEW PRINCIPLES of GEOGRA-
PHY and NAVIGATION, and have there conftructed large
Charts to 70 Degrees Latitude, and new Table of meridional
Parts adapted to the true fpheroidical Figure of the Earth; by
which the Solutions of all Cafes of Sailing are rendered juſt as
eafy as by the falſe Charts and erroneous Tables in preſent
Uſe, which are no ſmall Diminution of the Glory of naval Sci-
ence in ENGLAND. What farther relates to this Subject we
propoſe to deliver in the Theory and Conſtruction of Inftruments
uſed in Navigation (in which there is great Room for Improve-
ment) but this we muſt defer to a future Part of this Work,
and proceed with our Inftitutions relative to the Theory of the
Sciences, which ought to precede the mechanical and inftrumen-
tal Part.
PHY
(287)
PHYSICO-BALLISTICS:
O R,
INSTITUTIONS of GUNNERY.
CONTAINING
The philofophical and mathematical ELE-
MENTS of that Art, connected with,
and illuſtrated by the Experiments pub-
liſhed by the late Mr. ROBINS.
CHAP. I.
The THEORY of RESISTANCE to BODIES moving in
a refifting Medium.*
AVING in the
preceding Part delivered the true
2027 Principles of the Art of navigating a Ship, which is
HA
to be confidered as a Part of Military Science, we think it may be
very properly connected with its Sifter Science, BALLISTICS,
or
It appears, (fays Mr. ROBINS) that the modern Writers on the
Art of Gunnery have been very much deceived, in fuppofing the Re-
fiſtance of the Air to be inconfiderable, and thence afferting, that the
Track of Shot and Shells of all Kinds is nearly in the Curve of a Pa-
rabola. That by this Means it has happened, that all their Determina-
tions about the Flight of Shot diſcharged with confiderable Degrees of
Celerity are extreamly erroneous, and confequently that the preſent
Theory of Gunnery in this its most important Branch is uſeleſs and
fallacious. Preface to principles of Gunnery.
288
INSTITUTIONS
1
or the Doctrine of Projectiles (not in Vacuo, but) in a refifting Me-
dium, eſpecially as nothing confiderable has been wrote directly
on this Subject in our Language to explain the true PRINCIPLES
or THEORY of GUNNERY, nor any Thing of a practical Na-
ture, but a Treatife of Gunnery confirmed by Experiments by
Mr. ROBINS, an excellent Performance it is true, but unleſs the
phyfical PRINCIPLES or THEORY on which the greateſt Part de-
pends, be explained and connected with the Experiments, the
Art of GUNNERY will, after all, be deficient in the moſt effen-
tial Part, as we prefume will be fufficiently evident from the en-
fuing Diſcourſes.
2028. Since the Path of Projection is through a refifting Me-
dium, the firft Thing neceffary is to eſtabliſh the THEORY of
RESISTANCE to Bodies moving in a fluid Medium, where the
Particles, both in the Body and the Medium, are fuppoſed to be
abfolutely void of any elaſtic Force, and to act upon each other
only by Virtue of the Vis infita. We ſhall here alſo ſuppoſe the
Fluid to be quite perfect, or free from all Tenacity of Parts, and
that the Friction is of Courſe nothing at all.
2029. Therefore let AM B be any
Curve moving in a Fluid in the Di-
rection of the Axis CA; then to any
Point M let be drawn a Tangent, in
which take Mm ¿ Fluxion of the
Curve; and let LMP be drawn per-
pendicular to the Bafe B C, and draw
the Perpendiculars Lm and mr; and
put CB4, CP = x, PM = y;
then Mrj, and r m = x; and be- B
cauſe of fimilar Triangles CMP and
L
M
in A
D
P
C
Mrm, we have CM (≈ a) : PM (= y) : : M m (= ±) : r m
(*); whence =2, and 2
ax
y
a² x²
**
==
yy
a direct Stroke of a Particle
2030. Let the abfolute Force of
of the Fluid be repreſented by L M, and fince the Stroke at M is
oblique to the Curve, it muſt be refolved into two, viz. M m, and
Lm; the first of which does not affect the Curve, but the latter
acts perpendicular to it. And therefore the whole Force is to
the oblique Force as L M to Lm, or as Radius to the Sine of
In-
Of GUNNERY.
289
Incidence (1033). Wherefore the whole Force being = 1,
that by which the Point M is ftruck will be as
x
Z
; for becauſe of
fimilar Trian. Lm M, Mmr, we have Mm (2) : m r ( = x)
; : LM ( = 1) : Lm =
1; .
Z
mr (=)
2031. But this Force Lm must be reduced to another acting
againſt the Point M in the Direction in which it moves, that is,
it muſt be reſolved into the two Forces Lr and mr; of which
the latter being perpendicular to PL does no Ways hinder its
progreffive Motion; but the former Lr acts wholly againſt it.
But Mm (= *) : mr (=*): Lm (= 2) Lr=
Z
which
therefore expreffes the Refiftance to the Particle M moving in
the Direction M L.
2032. But as this Force acts perpendicularly (not to the
Curve, but) to the ordinate M D, it muſt be multiplied by the
Fluxion thereof, and the Product, viz.
will be the
x x =
3
Expreffion for the Flux. of the Refiftance of the Arch A M. Then
px
ä
fince a p::x: = Circumference of a Circle whofe Radius
a
is; if we multiply this Circumference by the Fluxion of Refi-
ftance, the Product
px x³
will be the Fluxion of Refiftance to
the Surface of the Solid generated by the Revolution of the Curve
AM about its Axis A.C.
2033. In order to find the Fluent or Reſiſtance for the Sur-
face of a Globe, we have
x ż
=
a² x²
وو
(2029) therefore 23 x
pyy, but yyaa-xx (828) whence Py² xx
3
aaxx-xx, whofe Fluent is
Q3
63
X
x = a²x 4*, for the
Reſiſtance; and fo when xa, we ſhall have
4
+
ap for the Ex-
preffion of the Refiftance to the Surface of the Hemifphere, or
Globe moving in the Fluid,
VOL. II.
PP
2034.
4
}
!
1
1
290.
INSTITUTIONS
2034. If a Cylinder were to move in the faid Fluid in the
Direction of the Axis, fince the Stroke of every Particle would
be direct, and the whole proportional to the Area of the Bafe;
it is evident (fuppofing the Diameter of the Cylinder and Globe
equal) that the whole Reſiſtance to the Cylinder will be (as the
Area of its Baſe)
ap
(fee 830) whence it appears, the Refiftance
to the Globe is to that of a Cylinder (of equal Diameter) as 1 to 2.
2
2035. A Cylinder falling upon fuch a Fluid, will in the
Time of paffing through its whole Length, communicate a
Motion to the Particles, as will be to the whole Motion of the
Cylinder, as the Denfity of the Medium to the Denſity of the
Cylinder. For let Q, M, V, B, D, be the Expreffions for
the Quantity of Motion, Matter, Velocity, Bulk, and Denſity of the
Cylinder; and 4, m, v, b, d, dénote the fame Things in the
Fluid. Then the Motion of the Cylinder will be Q= MV,
and that communicated to the Fluid will be q = vm (970);
but it is MBD, and m bd (973). Whence Q= VBD,
and qvbd; and fo Q::: VBD: vbd; but V B vb,
therefore 9 Q::d: D.
q:
I
2
=
2036. If the Axis and Diameter of the Cylinder be equal to
the fame in a Globe; it is evident, fince the Refiftance to the
Globe is but that of the Cylinder, the Globe, that it may
communicate the fame Motion to the Fluid, muft move through
twice the Length of its Diameter. If the Velocity and Denfity
of the Cylinder and Globe be equal, (and Q, M, V, B, D,
ſtand for the fame Things in the Globe) then, fince V = V,
it is 2: Q:: (M : M :: B : B) :: 2:3 (837). Alfo the Mo-
tion of the Cylinder is to that communicated by the Globe in
- paffing through two Diameters, as the Denfity of the Cylinder
(or Globe) to that of the Medium, viz. Q: 4 :: D: d (2035)•
Laftly, let q be the Motion communicated to the Medium by
the Globe in paffing through of it's Diameter; then q:q: 2
:::32. And by compounding thefe Ratios, we fhall have
2:q: Dd; that is, the whole Motion in the Globe is to that
communicated to the Medium (equal to the Refflance of the Medium
to the Globe) in paffing through of its Diameter, as the Density of
the Globe to the Denfity of the Medium.
t
2037.
Of GUNNERY.
291
▸
2037. If the Particles of the Cylinder, Globe, and Medium
be ſuppoſed perfectly elaſtic, then the Velocity of the Parts, af-
ter Percuffion, will be double to what it is in the Medium be-
fore mentioned (1015) and therefore the fame Effect will be
produced in half the Time, or in paffing through half the Space.
Confequently, in a perfectly elaftic Fluid, a Globe, in the Time
of paffing through Parts of its Diameter, will communicate a
Motion to the Particles that will be to the whole Motion of the
Globe, as the Dènfity of the Fluid to the Denſity of the
Globe.
2038. There is no fuch Thing in Nature as a perfect Fluid,
or fuch whofe Parts are abfolutely free, and unaffected by any
external Force; for all Fluids are heavy Bodies, that is, their
Parts gravitate towards the Center of the Earth, and confe-
quently on one another. Therefore the lower Parts will be
compreffed by the Weight of thofe above them; and therefore,
when a Body is confidered as moving through fuch a compreffed
Fluid, the Motion or Velocity of the Body will be lefs than that
with which the Particles will rufh into the Space relinquished
by the Body, by Virtue of the Compreffion, I fay, in this Cafe
we are to conceive the Force of Re-action on the fore Part of
the Body, in fome Degree abated by the Particles circulating
round to fill up the relinquished Space, in order to reſtore the
Equilibrium (that would otherwiſe be deſtroyed) by the conftant
Influx of the Fluid behind the Body.
2039. This complexed Cafe of real Fluids makes the Com-
putation of their Refiftance to Bodies very difficult and tedious,
but Sir Isaac Newton has fully confidered the Thing, and found
that the Refiftance to a Cylinder moving in fuch a compreffed
Fluid, is but a fourth Part of what we have fhewed it would be
in a perfectly non-elaftic Fluid (2035.) fuppofing the Veloci-
ty of the Body and Denfity of the Fluid the fame in both Cafes ;
that is, the Motion imparted to fuch a Fluid in the Time it paſſes
through four Times its Length, is to the whole Motion of the Cylinder,
as the Denfity of the Fluid to that of the Cylinder.
2040. In fuch a compreffed Fluid, Bodies of every Form (if
their tranfverfe Sections are equal) will be equally refifted; for
„úf a Cylinder, Globe, Cone, circular Plane, &c. of equal Diame-
Pp 2
* See his Principia, Lib. II. Prop. 37, &c.
ters
1
292
INSTITUTIONS
ters were placed in a Canal of fuch a current Fluid, it is evident
the Fluid would be equally obftructed by them all, or the in-
creaſed Velocity of the Fluid paffing by each will be the fame;
they therefore equally refift the Fluid, and, were they to move
in the Fluid, would be equally refifted by it. Confequently,
fince the Motion of a Globe is but of that of the Cylinder cir-
cumfcribing it, the Force that will generate or deſtroy the
whole Motion of the Cylinder while it moves through 4 Times
its Length, will generate or deſtroy the whole Motion of the
Globe while it moves uniformly through of 4 Diameters, or
Parts of its Diameter, that is, through 2 3 Diameters.
2
2041. It will be neceffary here to obferve, that the Theory of
Reſiſtance in a rare elaftic Medium is, by the Newtonian Mathefis,
arrived to its Perfection, fince thereby it is determined what the
Figure or Form of a Body muſt be, which, moving in ſuch a
Medium, fhall meet with the leaft Refiftance poffible. And as
this Part of mathematical Philoſophy is of primary Confideration
in Naval Architecture, and Military Science, we fhall here ex-
plain the Principles of the Calculus, as follows.
2042. On the
Right-line BC,
ſuppoſe the Pa-
rallelograms B
D
n N
n
Gg b, MN nm,
of the leaft
Breadth, to be C
erected, whofe
m. M
6 B
R
Heights BG, MN, their Diſtance M b, and half the Sum of
their Bafes M m + ½ Bb =
2
rence of the Baſes 1 M m
Points in the Curve GND;
and n, (fo that gg=nn
be in the fame Curve.
a, are given: Let half the Diffe-
Bb be called x: Let G and N be
and producing bg, and mn to g
b,) the Points g and ʼn may alſo
2043. Now if the Figure CDNG B, revolving about the
Axis BC, generates a Solid, and that Solid moves forwards in
a rare and elaſtic Medium from C towards B, (the Poſition of
the Right-line BC remaining the fame;) then will the Sum of
the Refiftances againſt the Surfaces generated by the Lineola
Gg,
Of GUNNERY.
293
Gg, Nn, be the leaft poffible, when Gg is to Nn as BG
x Bb to MN x Mm.
2044. For the Force of a Particle on G g and N n, to move
I
I.
them in the Direction B C, is as and (2031); and
Gg
N n
the Number of Particles that ftrike in the fame Time on the Sur-
faces generated by Gg and Nn, are as (the Annuli defcribed by
gg and nn, that is, as BG x gg and MN Xn, or as) BG
and MN; therefore the Refiftances againft thofe Surfaces
BG
are as
G g
BG
as
y
to
MN
MN
2
Nz²
to ———.
-
Z
Nn
>
2
that is (putting y for G g, and % for N n²,)
2045. But the Sum of thefe Refiftances
ર
BG
+ MN) is
is a
MN X
ZZ
لولو
-BG x
L: But
: But y =
yy
Therefore - BG X
1
Minimum.
or MN x
ZZ
gg
(Gg² = Bb
=) aa— ·2ax + xx + bb; and z = (Nn
nn² =) aa + 2 axxx
bb; therefore
MN
and ¿ = 2a x + 2 xx: Confequently
ZZ
2
Bb +
= Mm² +
2xx — 2 ax,
X 2 * × a + x
BG
× 2 x X a x; or
(MN
× a + x =
)
MN
yy
ZZ
xMm=
(BG
ха
) BG
yy
× Bb. (2042.) There-
fore (yy) Gg* : (zz) Nn* :: BG × Bb : MN × Mm.
2046. Confequently, that the Sum of the Refiftances againſt
the Surfaces generated by the Lineola Gg and Nn, may be the
leaft poffible, Gg muſt be to Nn as BG x Bb to N M
× Mm.
=
2047. Wherefore, if gg be made equal to gG, fo that the.
Angle gGg may be 45°, and the Angle BGg 135°; alſo
Gg² = 2gg', and Gg 4gg; then 4 gg: Nn :: GB
хвъ
294
INSTITUTIONS
× Bb: NM x Mm; and fince GR is parallel to Nn, and
BG, BR parallel to nn, Ñn; alſo n n = gg = gG; it
follows, that (nn gG) Bb: (Nn) Mm :: BG: BR;
=
BG x M m
BR
t
; alfo (nn) gG: Nn:: BG:
therefore Bb
4
GR. Confequently,
488
N 22+
==
4 B G +
BG x Bb
4
GR
MNx Mm
=)
BG2
MN X BR
MN.
Therefore 4 BG X BR is to GR3 as GR to
2048. How this Curve DNG is to be conftructed by Means
of the Logarithmic Curve, and from thence a practical Method
of forming the SOLID of leaft RESISTANCE, we may hereafter.
fhew, in its proper Place, when we ſhall have premiſed and ex-
plained the general Doctrine of Curves.
CHA P. II.
The DOCTRINE of abfolute, fpecific, and relative
WEIGHT in BODIES, explained on the ftatical
PRINCIPLES of refifting FLUIDS, as WATER,
AIR, &C.
2049⋅ B
EFORE we proceed any farther, it will be neceffa-
ry (in the THEORY of BALLISTICS) to premiſe a few
Things relating to the various Diſtinctions of GRAVITY, or
Weight in Bodies, viz. abfolute, relative, and ſpecific Gravity.
Abfolute Weight is the whole Weight of a Body in Vacuo. Relative
Weight is that which a Body has when weighed in a reſiſting
Medium. Specific Gravity is the comparative Weight of Bodies of
equal Bulk. Thus if the Weight of a Cubic Inch of Water be to
that of a Cubic Inch of Copper, as I to 9, we fay the Specific Gra-
vity of Copper is 9, or its Weight is 9 Times greater than that of
Water.
2050.
Of GUNNERY……
295
any
2050. But to illuftrate this
Affair, let A B be a Ballance,
and from the Scale D let
Body F be ſuſpended by a fine
Horfe Hair E, and equiponde-
rated in the Air with Weights
А
Ay
B
put into the other Scale C. C
G
After this, immerſe it in the
Fluid I H, contained in the
Jar GH, and the Equilibri-
um will be deſtroyed. For the Body F will now appear lighter,
and afcend, while the Scale C defcends.
2051. That the Weight of the Body F fhould be diminiſhed
in the Fluid, is hence evident, that it cannot defcend therein.
without raiſing at the fame Time an equal Bulk of the Fluid,
which Quantity of the Fluid, thus raifed, will refift the Body F
with all its Force (equal to its Weight) and therefore will de-
ftroy juſt as much Motion (or Weight) in the Body F (965).
And therefore, by putting Weights in the Scale D, till the E-
quilibrium is again reftored, we fhall have the Compariſon of
Weight between the Body F, and the Fluid, in equal Bulks: For the
Weight of the Solid is in the Scale C, and that of an equal Bulk
of the Fluid in the Scale D; and hence the ſpecific Gravity of
the Solid and Fluid becomes known; for which Reaſon this In-
ftrument is called the HYDROSTATIC BALANCE.
2052. If from the Weight of the Solid, in Air, we deduct the
Weight of an equal Bulk of the Fluid, the Remainder is the
relative Gravity of the Body; and is all that Force by which the
Body finks or defcends in the Fluid. But if the Weight of the
Solid be less than that of an equal Bulk of the Fluid, it is evident
fuch a Solid cannot be totally immerfed by its Weight, but will
fwim with a Part extant above the Fluid. If the fpecific Gravity
of the Solid and Fluid be equal, the Solid will retain any Poſition
in the Fluid, and neither fink nor fwim. Hence the Rationale of
SINKING and SWIMMING is evident.
2053. Not only the fpecific Gravity of Solids, but alſo of
Fluids become known by the fame Balance; for if the Solid F
be weighed in two different Fluids, the Weights put into the
Scale D to restore the Equilibrium, each Time will exprefs the
Gravities of equal Bulks of thofe Fluids; and hence a Table
of
296
INSTITUTIONS
of ſpecific Gravities for Solids and Fluids may be made, a Speci-
men of which here follows, where the Compariſon is made,
with the Gravity of Rain-water, put = 1,000.
2054. Rain-water
1,000 Ebony
1,177
Sea-water
Aqua-fortis
1,030 Cork
0,240
1,300 Good Wheat
0,757
Oil of Vitriol
Spirit of Wine rectified
Burgundy Wine
1,700
White Peafe
0,807
0,840 Bone
1,656
Spirit of Nitre rectified
1,610 Ivory
1,826
0,953 Horn
1,840
Canary
1,033 Adamant
3,400
Red Wine
0,993 Glafs
2,666
Diftilled Vinegar
1,030 Flint
2,542
Cow's Milk
1,030 Marble
2,700
Urine
1,030 Mundick
4,430
Mercury Crude
13,593 Chalk
2,379
Amber
1,040 Magnet
1,840
Sulphur
1,800
Newcastle Coal
1,272
Borax
1,720 Oil-ftone
2,380
Red Coral
2,689 Slate
2,740
Cinnabar natural
7,300 Alabafter
1,875
Tartar common
Camphire
1,849 Copperas Stone
4,300
0,995 Copper Ore
3,775
Vitriol of Dantzick
1,715 Lead Ore
6,800
Sal Gemma
2,143 Bismuth
9,700
Allum
1,714 Speltar
7,065
Nitre
1,900 Tin
7,550
Gum Arabic
1,375 Iron
7,645
Verdigreaſe
1,714 Brafs, wrought
8,000
Bees Wax
0,960 Copper
9,000
Pitch
1,190 Lead
10,131
Rofin
1,100 Silver, Sterling
1
Honey
1,450
Pure
10,000
11,091
Dry Box Wood
Dry Fir
0,950 Gold, Sterling
0,546
Pure
17,150
19,640
Lignum Vita
1,327
2055. Becauſe it is found by Experiment, a Cubic Foot of Rain-
water weighs very exactly 1000 Ounces Averdupois, therefore
the
Of GUNNERY.
297
the Numbers in the Table will exprefs the Ounces contained in
a Cubic Foot of any of the other Subſtances. An Ounce Aver-
dupois
437 480 51 56, nearly.
437, and an Ounce Troy
:
:: :
Troy Pound, as 437 x 16
that is, as 17 to 14 nearly.
480 Grains. But
The Averdupois Pound is to the
7000 to 480 X 125760;
Therefore a Cubic Foot of Water
weighs 62 lb. Averdupois, and 76 lb. Troy nearly.
2
2056. What relates to the abfolute and ſpecific Gravities, the
Magnitudes, Density, &c. of Bodies, will beft be underſtood
by fymbolical Computation; in order to which let A and B be
two Bodies of equal Bulk, but different Quantities of Matter;
and let B and C be two other Bodies with equal Quantities of
Matter, but of different Bulks:
D= Denſity
Bulk
And let B
M
Alfo
Quantity of Matter
D =
B = Bulk
Denſity
M Matter
d = Denfity
And
b
- Bulk
?
>in the Body A.
>in the Body B.
S
in the Body C.
m = Matter
2057. Then, becaufe the Denfity of any Body is proportional
to the Quantity of Matter under equal Bulks, we fhall have
D :D :: M: M; and, becauſe when the Quantities of Mat-
ter are equal, the Bulks must be reciprocally as the Denfities,
D M
therefore we have D:d :: b: B. Whence D =
M
B
M=mj
; confequently DMB = db M. But B = B, and M = m;
therefore D Bm = db M.
Whence we have D: d::bM:
m B; and B: b::dM: Dm; and M: m ::DB: db,
2058. The Specific Gravity of Bodies being as the Weights,
that is, as the Quantities of Matter, in equal Bulks, will be as
the Denfity: Therefore D: d:: S: s; and by Subftitution of
Ratios, we have the general Theorem above become S B ” —
sb M. And fince the abfolute Weights (A, a,) of any two Bo-
VOL. II.
dies
Qq
772
298
INSTITUTIONS
dies are as the Quantities of Matter, we have SB a As b.
Wherefore Ss:: Ab: aB; that is, the fpecific Gravities
will be as the abfolute Weights directly, and the Bulks inverſely,
or as the abfolute Weights divided by the Bulks.
2059. Alfo A:a:: SB: sb; that is, the abfolute Weights
of Bodies are in the compound Ratio of their ſpecific Gravities
and Bulks. Or the abfolute Weight of any Body is had by mut-
tiplying its Bulk and ſpecific Gravity together.
2060. Again; becauſe B: b:: As a S, it appears that the
Bulk or Magnitudes of Bodies will be as the abfolute Weights
directly, and ſpecific Gravities inverſely. Or the Magnitude
of any Body is had by dividing its abfolute Weight by its ſpecific
Gravity.
2061. From what we have faid, it is evident we have not the
abfolute Weight of Bodies in the Air, but only the Relative. Let
Aabfolute Weight, and B relative Weight; then A B
B the Weight of an equal Bulk of Air, (2051) and the fame
may be faid for any other Fluid. Wherefore let the Motion of a
Globe moving in a Fluid be ſuch as will be generated by the
Force of its relative Weight falling in Vacuo through a Space (S)
that fhall be to of its Diameters (D) as the Denſity of the
Globe (D) to the Denfity of the Fluid (d), then will the Velocity
it will acquire by the Fall be the greatest it can poſſibly acquire by de-
fcending in the Fluid; and the relative Weight of the Globe will be
equal to the Refiftance, arifing from the Medium to the Globe, as
mentioned in (2039, 2040).
2062. For let R Refiftance, F Force that will generate
or deſtroy the Motion of the Globe while it defcribes & Parts of
RD
d
its Diameter; then F: R:: D: d (2039) and ſo F = -
And fince S Space defcribed in the Fall, the Globe by an
uniform Motion with the Velocity acquired by the Fall, will de-
ſcribe a Space = 2S in the fame Time (993) and ſo, becauſe
S: 4 D :: (2 S: D) D: d. But the Times T, t, in which
3
8
the Spaces 2 S and
3
D are uniformly deſcribed are as thoſe Spa-
8
ces, viz. Tt: 2S: D, hence Tt:: D: d.
3
2063. Again, fince in the Times T and t equal Quantities
of Motion are produced (for F produces the whole Motion of the
Globe in the Timet, and the relative Weight (B) of the Globe
pro-
Of GUNNER Y.
299
produces the fame in the Time T, (2061). And fince the lefs
the Time is, the greater muſt be the Force to produce a given
Effect; therefore F: B:: T:t:: D: d. Confequently F =
BD
d
RD
d
ز
whence RB, and the Refiftance being equal
to the relative Weight, the Body can be no longer accelerated, but
muſt deſcend with an uniform Motion in the Medium, with a
Velocity equal to that acquired by the Fall through S in
Vacuo.
2064. Hence if the Denfity of the Globe and of the refifting
Medium be given, as alſo the Velocity of the Globe in the Be-
ginning of its Motion; then the Refiftance it meets with may
be computed as follows. Let A abfolute Weight of the
Globe in Vacuo, and B Weight of an equal Bulk of the Me-
dium; then A - B B the relative Weight of the Globe.
BB =
in the Medium, (2061).
2065. And fince the Spaces defcribed in the fame Time in Va-
cuo are as their accelerating Forces (999), therefore as the Space
deſcribed in Vacuo by the Weight A is 16,2 Feet in one Second,
the Space defcribed in Vacuo by the Weight B in one Second
will be known, for as A: B:: 16,2:
Space.
16,2 B
A
to the ſaid
2066. Moreover, the Time of defcribing the Space S in Vacuo
by the Weight B is thus found; as
16,2 B
A
AS
:S:: 1/2 :
16,2 B
the Square of the Time required; or
AS
16,2 B
= T the
Time. But S-
4 D D
(2061) whence T =
4 ADD
48,6 B ď
3d
2067. The Velocity of the Fall will carry the
Globe with a
uniform Motion over a Space = 2 S in the Time of the Fall T
and the Velocity of uniform Motion is always as the Space divid-
ed by the Time (991); therefore
2 S
T
=V = the Velocity ac-
quired by the Fall.
2068. The Globe moving in the refifting Medium meets with
a Reſiſtance = B (2063); if any other Velocity (1) be given
Qq 2
the
}
300
INSTITUTIONS
the Reſiſtance agreeing to that Velocity will be thus found. As
V² : V² : : B :
1/2 B
V2
Reſiſtance to the Velocity V. For that
the Reſiſtance is as the Square of the Velocity in general, is
hence evident; becauſe the Refiftance will be greater in Propor-
tion to the Number of Particles which ftrike againſt the Globe
in a given Time, and alfo to the Intenſity of the Stroke of each
Particle; and each of theſe will be as the Velocity, and therefore
both together as the Square of the Velocity.
2069. But, as I faid, this Ratio of the Refiftance to the Velo-
city is only general, and agreeing to flow Motions; becauſe in
very ſwift Motions the Circumftances will be altered, on which
the Reſiſtance depends, both in regard to the Medium, and to
the Figure of the Body moving in it. For with refpect to the
Medium, though compreffed, yet if the Velocity of the Body
be fo much greater than that of the Particles rufhing in behind
the Body, that a Vacuum is left, the Cafe of this compreffed
Fluid will be nearly the fame with one that is free (2028); and
confequently the Refiftance to a Cylinder moving fo very fwift.
as to leave a vacuous Space behind, will be near 4 Times as
great as when it moves flowly through the fame compreffed
Fluid (2039).
a
2070. Again, the Refiftance will vary in fwift Motions ac-
cording to the Figure of the Body. In fuch a Cafe the vacuous
Space will be more or lefs, and the compreſſed Fluid approaches to
the Nature of one that is free, wherein the Reſiſtance to à Globe
has been fhewn to be but half what it is to a Cylinder (2034);
therefore the Refiftance to a Globe moving very ſwiftly, even in
a compreffed Fluid, (becauſe of the Vacuum behind) may meet
with little more than half the Refiftance of the Cylinder moving
with the fame Velocity.
2071. But we cannot fuppofe the Compreffion of the Medium
not at all to affect the Body, for even to a Globe, the Particles
which ſtrike it obliquely, will be in fome Degree confined by
the compreffing Force, and therefore give more Refiftance than
when they are free to move. Alſo if the Medium be Elaſtic,
this Reſiſtance will be thereby farther increaſed (2037), fo that
upon the Whole we may conclude that the Reſiſtance to a Globę
moving ſwiftly in a compreffed claftic Medium (fuch as our Air)
will
Of GUNNERY.
301
will be between that of a Globe and of a Cylinder in a free and per-
fect Fluid, and that arifing to either in a flow Motion from a
comprcffed Medium; that is, it is more than twice and less than
4 Times the Refiftance the fame Globe would meet with moving
flowly in a compreffed Medium.
2072. Hence then we may take it for granted, that in very
fwift Motions, a Globe is refifted about three Times more, in
Proportion to its Velocity, than when its Motion is floweft; and
that the Refiſtance will decreaſe as the Velocity decreaſes, and
as the Circumftances of the Medium return to their natural State..
Let this fuffice at prefent, for the Theory of the Refiftance of
Fluids, which is a fundamental Doctrine in the Science of BAL-
LISTICS.*
C H.A P. III.
Of the NATURE and PROPERTIES of AIR; its
ELASTICITY, DENSITY, abfolute and ſpecific
ĠRAVITY explained by COMPUTATION and Ex-
PERIMENTS.
2073. Thas been already, in general, fhewn (975–980).
THE
HE Nature of ELASTICITY, and whence it arifes
We now propoſe to confider the Phyfico-mathematical Theory
thereof; and apply it to the Air, and other elaſtic Fluids, par-
ticularly that which is generated by firing GUNPOWDER, that
the ENGINEER may be fully acquainted with the Philoſophy of
this important Subject.
2074. In order to this, let there be
placed any Number of Particles in the
right-lined Diſtance AB at an equal
A
C.
C......
. B
..D
Distance
*It is neceffary here to correct an Error in the Table of ſpecific
Gravities, Pape 296, where that of a Magnet is faid to be 1,840,
whereas it fhould be 4,840, as I have found by Experiment. This
Error alfo paffed unobſerved in the larger Table of the PHILOSOPHIA
BRITANNICA, 2d Edition.
302
INSTITUTIONS
Distance from each other; and in any other equal Diftance
CD let there be placed twice as many Particles, at equal In-
tervals alfo, then it is plain the Intervals between the Particles
in CD will be but half fo great as thofe between the Particles in
the Line A B. Hence the Number (N) of Particles in the Lines
A B, CD, will be inverſely as the Interval (L) between each,
I
that is N will be always as I'
2075. In a Superficies, fuppofing the Intervals of all the Par-
ticles equal, we ſhall have Nas (670); alſo in a Solid it
3
I
I
L2
will be N³ as (675). But N, and N3 is as the Denſity
3
(D) of the Surface, and of the Solid (973); therefore D is as
in a Solid.
L3
I
I
L
in a Surface, and as 13
2076. Let us now fuppofe each Particle repels the Particles
next to it (and thoſe only) with a Force as (F) which is as any
I
12
LA
Power (n) of the Interval inverſely; that is, let F be as
Now the Sum or whole Force in the Superficies will be as the
Denfity D, and as the centrifugal Force F; or as DXF =
I
I
X
Ln
of the Fluid.
I
L+7. Which therefore will expreſs the elaſtic Force
2077. But in the Body of the Fluid the Denſity being as
I
I
3
(2075),
(2075), we haveD L³ as 1, L³
3
as and L as
D'
임금
​which
D
I
1 x + 2
gives
fubftituted in the Expreffion of the elaftic Force
7 of 2
D 3 ; therefore the elaftic Force of the Medium is as the Cube
Root of that Power of the Denfity whofe Index is n + 2.
2078. Hence if the elaftic Force (E) in any Fluid be as the
Denſity (D) then the Expreffion D
fo
n + 2
3
x + 2
3
71 + 2
=E =E; and
3
=1, whence n + 2 = 3, and therefore n = i;
where-
Of GUNNERY.
303
I
wherefore in fuch a Fluid F will be as
L
(2076); that is, the
Particles repel each other with Forces that are inverfely as the Diſtances
of their Centers.
2079. And this is the Property of our Air A
whoſe Denſity is always as the compreffing
Force, as is proved by the following Experi-
ment. Let Mercury be poured into an inflec-
ted Tube A BCD, open at both Ends, to a
fmall Height BC; then ſtopping the Orifice
D very cloſe, meaſure the Length of the con-
fined Air DC very nicely; this done, pour
Mercury into the other Leg A B till its Height
above the Surface of that in CD be equal to
the Height at which it ſtands in the Barometer.
Then it is plain the Air in the ſhorter Leg will
be compreffed with a Force twice as great as
at firft; for then it was preffed only with the
Weight of the Atmofphere, but now it is.
preffed with that Weight, and an additional
equal Weight of Mercury. With this double Force the Air in
DC is now compreffed into the Space DE
by meaſuring it.
B
O H.
E
C
DC, as appears
2080. Hence it appears, that the Spaces S = DC, and s =
DE, which the fame Quantity of Air poffeffes under different
Preffures p and P, are as thofe Preffures inverſely, viz. S:s::
P: p. And becauſe the Denfities d and D (where the Quantity
of Matter is given) are inverſely as the Bulks (973); therefore
d: D::s: S ::p: P; that is, the Denfity of the Air is directly
as its compreffing Force, or as the Elaflicity, which is equal thereto
(982).
2081,
*
394
INSTITUTIONS
1
2081. The ſpecific Gravity of Air, and thence its abfolute
Weight may be diſcovered by the following Experi-
ment. Let A B be a Glafs Tube, open at both Ends,
and the End B immerfed in the Water ED of the
Phial CD, and let the Mouth of the Phial С be
fealed cloſe. Then let a little Air be blown through
the Tube A B, and it will condenſe the Air CE in
the Phial, and thereby encreaſe its Elafticity, which
will raiſe a Column of Water B F in the Tube, whofe
Weight together, with that of the Air, preffing on
the Surface F, will be equal to the encreaſed Spring
of the Air CE. If now this Phial be carried to the C
Height of 72 Feet, or 864 Inches above the Earth's
Surface, the Water will rife from F to G, through
a Space F G 1 Inch; whence it appears, that I
Inch of Water is equivalent in Weight to 864 Inches
1
A
G
F
E
D
of Air, of the Denfity it has at the Earth's Surface;
and fo I Inch of Air is equivolent, or of the fame Weight with
Part of an Inch of Water. Whence the Weight of Air
is to the Weight of Water (in equal Bulks) as to I, or as
1 to 864. Sir Iſaac Newton has ſtated it as 1 to 860.
804
I
864
2082. Now becaufe it is found by Experiment, a Cubic Inch
of Water weighs 253,3 Grains; therefore fay, as 864: 1 ::
253,30,293 = 1 of a Grain nearly, for the Weight of a
cubic Inch of Air; and becauſe a Cube is to its infcribed Sphere
as I: 0,524 (847); therefore fay, as I: 0,524 :: 0,293 :
0,1535 = 10% of Grain, the Weight of a Globe of Air 1 Inch
in Diameter.*
I
2083. The fame Inftrument (viz. the Phial with the Tube.
of Water (2081) being held near the Fire, the Water will afcend
to the Height of feveral Inches in the Tube, which fhews the
Expansion of the Air, or its increafed Elafticity by HEAT. On
the other Hand, if the Phial be immerfed in cold Water, the
Water in the Tube will defcend fome Inches below F, which
will demonftrate the Contraction of the Air, or the Diminution of
its Elafticity, by COLD.
1
I
2084.
*The Weight of a Pint of Air, Wine Meafure, or 231 cubic In-
ches, I have conftantly found by the Ballance to be 8 Grains, which
is at the Rate of of a Grain, per cubic Inch, as above.
TO
Of GUNNERY.
305
CHA P. IV.
Of the NATURE and PRODUCTION of artificial ÁIR ;
and a COMPUTATION of the explofive FORCE of
GUNPOWDER thence deduced.
HE Properties of common Air hitherto conlidered
2084 are not fo much the Subject of our prefent Specu-
T
lation, as the fame Properties in another Subject which the mo-
dern Philoſophy has diſcovered, and which from its perfect Si-
milarity to common Air, is ufually called artificial or factitious
AIR. This Subſtance, in its natural State, is no other than
the common Subſtance of all Sorts of Bodies, and which being
by chymical Operations, or otherwife, fet at Liberty, acquires
the Form of Air in every Refpe&t, becoming a fine, tranſparent,
inviſible, elaftic, ponderous Fluid.
A
C
2085. One eafy Method of obtaining this elaftic Fluid is
thus; let A B be a tall cylindric Glafs, with a very
fmall Hole E at the Bottom; in the Glaſs put any
alkalinous Subftance D as a Piece of Chalk, Tartar,
&c. then fill the Glafs to the Top with common
Water (ftopping the Hole E with the Finger) and
to the Water put a little Aqua Fortis, and then flop
the Orifice A clofe with the Cork C; this done,
you will fee the Acid act upon the Alkali, and pro-
duce a Fermentation which confifts in the violent
Eruption and Ebullition of this new generated Fluid,
which will keep conftantly afcending to the Top of
the Veffel, in Form of fmall Bubbles of Air; and
by its Elafticity, it will, by Degrees, force all the
Water out of the Tube through the Hole E.
E
B
2086. By this Experiment the Elafticity of this new genera.
ted Air is fhewn to be greater than that of the Atmoſphere;
and by many other Experiments, it appears to be extremely
great, and to produce Effects thereby fuperic. to thofe of any
other Force we know of in Nature; but in order to make Efti-
mations of this Kind, we muft firft know what Quantities and
VOL. II.
Weights
Rr
6
306
INSTITUTIONS
Proport. Part
Weights of this Fluid are produced from Bodies of feveral Sorts;
and as theſe have been determined by Experiments, by feveral
eminent Philofophers, I fhall here fubjoin a Specimen thereof
from the late Dr. Hale's Vegetable Statics.
Several Sorts of
Matter.
유
​2087. Dear's Horn
Øyſter Shell
Heart of Oak
बाल बाल बाली
117
24-I
33
7
162
266
46.
I
☎
108
135
30
Indian Wheat
270
388
77
Peaſe
I
396
318
113
3
Muftard Seed
270
437
77
Amber
14
135
135
38
I O
Dry Tobacco
35
I
153
142
44
3
Honey with Calx of Bones I
144
359
4I
Yellow Wax
I
54
243
15
T6
Coarfe Sugar
I
126
373
36
I
TO
Newcaſtle Coal
I
180
༣
158
I
51
3
Nitre with Bone Calx
I
2
90
211
26
Rhenifh Tarter
I
504
443
144
Calculus Humanus
31
+
515
230
147
2088. By this Table it appears how greatly this Fluid is con
denſed in its natural State in Bodies; thus 3 of an Inch of Cal-
Culus Humanus, produced 650 Times its own Bulk of Air,
which made nearly one Half of its fixed Subftance. When this
Air, therefore, comes to be fet at Liberty, no wonder we fee
fuch violent Expanfions, Exploſions, Incalefcences, &c. as
happen in firing the Pulvis fulminans, GUNPOWDER, and in the
Mixtures of various Sorts of Fluids. For fince it has been found
by many Experiments, that the Elafticity and Weight of this
Air is the fame with that of the common Air; therefore its
Elafticity being as its Denſity (2078, 2079) will be as much greater
in its fixed State in Bodies than that of the common Air, as its
Bulk
Of GUNNERY.
307
Bulk when fet at Liberty (or in a State of Expanfion) exceeds.
the Bulk of the Body which produced it (973).
2089. But to apply this to GUNPOWDER (with which we are
here more immediately concerned) we muſt firſt confider its
.conftituent Parts which are Nitre, or Salt-petre, Sulphur, and
Charcoal Powder. As to Charcoal, it appears, by Experiment,
to afford none of this Auid elaftic Air; and as to Sulphur, it
is fo far from yielding any of this Fluid, that on the contrary,
it abſorbs or attracts, and fixes the common Air in which it is
fired, and thereby diminiſhes its Quantity and Elafticity, as is
well known by Experiment. The elaftic Fluid, therefore, that
is produced, by firing Gunpowder, muſt be derived chiefly from
the Salt-petre; and this is known to be a Subſtance imbibed
from the Air by the Earth, becauſe thoſe who make it, extract
it from the fame Parcel of Earth many Times one after another,
after it has been duly prepared, and expoſed to the Air for a
proper Time.
2090. But becauſe Nitre produces only 180 Times its Bulk
of Air (2087) and Mr. Haukfbee, by Experiment, found Gun-
powder produced 232 Times its Bulk; and by moft accurate
Experiments, Mr. Robins has found it to produce 244 Times
its Bulk; therefore it ſeems, that fome additional Quantity of
Air is produced by compounding theſe different Subſtances to-
gether in this Sort of Powder. This Quantity of Air, from
Gunpowder, was determined in the following Manner: Mr.
Robins fired of an Ounce, Averdupois, in a Receiver of about
520 cubic Inches Capacity; this funk the Mercury in the Gage
2 Inches; and as the Height of the Mercury was then near 30
Inches, therefore 18 of an Ounce would have produced fo much
Air as would have preffed all the Mercury out of the Gage; that
is, 1 of an Ounce would produce 520 cubic Inches of Air of
the fame Elafticity of common Air. But 1: 520 :: 1: 575,
the cubic Inches that one Ounce of Powder would produce.
I
ठ
5
2091. But as Heat augments the Elafticity of Air, and this
Powder was fired upon a red-hot Iron; Part of the Expanfion
was owing to that Heat. But this Heat was lefs than that of
boiling Water which encreaſes the Expanfion of Air to more
than a Part of the Whole; if, therefore, the Number 575
be leffened by Part, it may be nearly the fame as would be
4
I
Rr 2
pro-
308
INSTITUTIONS
P
3
produced in the Receiver not heated, viz. 460 cubic Inches;
but 17 Drams of Powder will fill 2 cubic Inches, therefore 16:
17: 460: 488 cubic Inches of Air from 2 cabic Inches
of Powder; therefore I cubic Inch of Powder will produce 244
of Air.
Ι
2092. Mr. Robins, to try how much this Elafticity or Expan-
fion of Air would be augmented by the Flame of the fired Pow-
der, fuppofes the Heat of this Flame nearly the fame with that
of the extreme (or white) Heat of red-hot Iron; and this he
found to augment the Elafticity of common Air in Proportion of
194 to 796. If then we fay, 194 796: 244 999;
this laft Number will fhew how much the Elafticity of the Air
produced from Powder, when inflamed, is greater than that of
the Air in its natural State.
मु
3
2093. Since I cubic Inch of Mercury weighs very nicely
348,1 Ounce Averdupois Weight, a Pillar of Mercury, whofe
Bafe is one Square Inch, and Height 29, will weigh 14 Pound
15 Ounces; therefore as the Air fuftains by its Preffure or na-
tural Elafticity, a Weight of 15 Pound (at a Medium) upon a
Square Inch; the Elafticity of the inflamed elaftic Air of Gun-
powder, which is 1000 Times as great, will act upon every
Square Inch (at its firft Accenfion) with a Force which is
15000 lb. or fomewhat above 6 Ton Weight.
CHA P. V.
From the given DIMENSIONS of any PIECE of AR-
TILLERY, the DENSITY of its BALL, and the
QUANTITY of its CHARGE, are determined the
VELOCITY which the Ball will acquire from the
Explofion; and alfo the Length of the Charge pro-
ducing the greateſt Velocity poffible.
2094 N the preceeding Article we have fhewn what the Force
2094 - NO
of the Powder is at the Inftant of its Accenfion or
taking Fire, but fince this Action of the Powder is not inſtanta-
neous
Of GUNNERY.
309
neous, but continues to urge the Bullet all the Time it is in the
Barrel; and fince this Force is not uniformly the fame, but de-
creaſes from firſt to laft, and is every where inverſely as the
Space which the elaftic Fluid fills, that is, as its Bulk, (2080)
it will be neceffary in order to make a Computation of the Mo-
tion or Velocity of the Bullet, to premife the following Lem-
mata.
2095. Let AC be the Space
thro' which a Body is propel-
led by any Force whoſe Action
is continued and determined,
G
F
B
A
and which may be reprefented
E
by the Ordinate GE moving C
D
&
on the Abfcifs A C, and let B G be the Curve defcribed by the
Point G; draw D F indefinitely near to EG, and put AE=
x, EG = y; the Velocity of the Body in Ev, the Time
in which A E is deſcribed = t; then will (ED) & and ¿
(the Fluxions of the Space, Time, and Velocity) be as the naf-
cent or evanefcent Increments of thoſe Quantities x, v, and ↑
(788).
2096. Now though the Velocity upon the Whole is not uni-
form yet for a Moment, or while the Lineola ED is deſcribed,
it may be eſteemed ſo (992) and therefore we ſhall have v
x
i
(by 991). Alſo we have the moving Force y = (998);
wherefore it is i =
வ்
V
وتو
whence y xvi = EG × ED
= the Fluxion of the Space ABGE; therefore v² = ABGE
(804), and ſo we have v = √✓2 ABGE; that is (fince 2 is a
conſtant Quantity) the Velocity will be every where in the ſubduplicate
Ratio of the Area ABGE.
2097. Since the Quantity of Motion Qis always proportional
to the moving Force GE, or y; therefore Q== vm (970)
whence t
=
I
ข
; alfo, becauſe when Q=1, a given Quantity;
then, alfo v =
m
I
I
2
and tas v; therefore
= 1 x v = v¹;
732
M
con-
310
INSTITUTIONS
confequently v
>
m
that is, the Velocity of any Body urged
through a given Space by a given Power acting with a determinate
Force in every Part of that Space will be reciprocally in the fubdupli-
cate Ratio of the Quantity of Matter.
J
S
K
HI
L
R
P
о
IB
M
2098. Theſe Things premifed, let A B reprefent the Axis of
any
Piece of Artillery, A the Breech, and B the Muzzle; DC
the Diameter of the Bore, and DE GC a Part of its Cavity filled
with Powder; O the Ball that is to be impelled thereby, lying
with its hinder Surface at the Line GE. Then, becauſe the
Force (F) of the Powder againſt an Inch Square is known (2092,
2093) the Force (F) exerted on the Area of a Circle, one Inch in
Diameter, will be known alfo; for it will be F: F:: 1:0,7854,
whence F = 0,7854 F (269). Now let D 1 Inch, and
d = Diameter of any other Circle as that of the Ball O, that
is, let d EG, and (f) the Force of the Powder exerted on
this laſt Circle; then F:ƒ:: D² : d² (840) therefore ƒ = F d³
=
J
0,7854 F d² the Force of the Powder acting at the first
Inftant of its Accenfion on the Ball O in the Direction of the
Axis A B.
}
2099:
Of GUNNERY.
311
2099. Let this Force then be repreſented by any Line F H,
perpendicular to A B in the Point F. Then, becauſe the Den-
fity of the elaftic Fluid confined in the Space EDCG is to the
Denfity thereof when expanded into any other Space a DC b, as
thoſe cylindric Spaces inverfely (2072) or as the Lengths A M,
A F; and fince the elaftic Force is in the fame Ratio with the
Denſity; therefore by making MN: FH::AF: AM; and
BQ:FH::AF: AB; then fhall M N and BQ be as the
Force of the Powder upon the Ball at M and at B, and the
Curve which connects the Points H, N, Q, will be an Hyper-
bola (by 778).
3000. To the Point A draw the Right-line A I parallel to FH,
and SH parallel to AB; then will AI, A B be the Afymptotes of
the Hyperbola, and the Rectangles AFX FHAM × MN
= AB × BQ = 1, the Power of the Hyperbola (779); and
from what has been faid, it is eaſy to underſtand that the whole
Force of the Powder exerted on the Ball while it moves from F
to B, is as the hyperbolical Space HFBQ, and therefore as the
A B
AF
hyperbolical Logarithm of the Ratio
(829, 850).
3001. Since the Force impelling the Ball at F is known (2098)
and the Weight of the Ball is fuppofed given, the Ratio between
the faid Force and the Weight of the Ball is known, which
let be as F H to FL; then if the Ball were to be impelled to
the Diſtance FB by a Force equal to its Gravity, becauſe
the Force of Gravity through fo fmall a Space is uniform, or
acts in every Point with the fame Tenour, therefore in any Point.
M or B the Force will be as M R FL, and BPFL, and
confequently the Line which joins the Points L, R, P, is a
Right-line, and parallel to the Axis A B, and the whole Ac-
tion or Force of Gravity impelling the Ball through A B will be
as the Rectangle FLPB FB x FL.
3002. But the Velocities acquired by the Ball when propelled
by the Force of Gravity, and by the Force of the Powder, thro'
the Space FB, are as FLBP, and FHQB (2096).
And fince F B is a given Length, the Velocity acquired in falling
through that Space is known. Thus fuppofe the Length of the
Bore AB = 45 Inches, and the Length of the Charge AF =
2 //
312
INSTITUTIONS
2
= 3
ठ
2 & Inches; then AB-AF FB 42 Inches (as in
Mr. Robins's Example). Then 16,2 Feet: 42 (=3,53 Feet)::
32,4: V² (by 996). Whence 64,8 x 3,53 = V = 15,07
Feet per Second, the Velocity acquired by the Body falling thro'
the Space F B.
ΤΖ
3
3003. Again, to find the Ratio of F L to FH, fuppofe the
Ball O be of Lead, whofe Diameter d of an Inch; then
will its Weight be of a Pound Averdupois. Therefore the
Gravity of the Ball is to the Force of the Powder at F, as to
0,7854 F d² (by 2098), that is, as 1 to 79521,6; and ſuch is the
Ratio of F L to FH, fuppofing the Preſſure of Air on a ſquare
Inch juſt equal to 15 lb.
1
I
3004. But the Number 79521,6 is fomewhat too large, fince
a Column of Mercury, whoſe Baſe is 1 fquare Inch, and Alti-
tude 29, weighs but 14 lb. 15 Ounces; and I find Mr. Robins
has taken for a mean Altitude 28,2 Inches of Mercury, or 33
Feet of Water, whofe Weight is but about 14lb. 4 Ounces,
and he alſo makes the ſpecific Gravity of Lead to that of Water,
as 11,345 to 1, conſequently a Column of Lead, 34,9 Inches Al-
titude, will have the fame Weight or Preffure; and multiplying
this by 1000 (2093) the Product 34900 Inches, will be the
Height of a Column of Lead, whofe Preflure is equal to that
exerted on the Bullet at the Moment of Accenfion.
3005. Now the Bullet being of an Inch in Diameter, is
equal to a Cylinder on the fame Baſe, and 1 Inch in Height; for
the Solidity of the Sphere is
3
4
pd²
6,
I
ठ
2
and that of a Cylinder is
pdb
4
(by 836, 831). Wherefore if pd2 = pdh, then 4 d = 6b;
whence hd of = 4, the Height of the Cylinder,
equal to the Ball in Weight and Magnitude. Therefore
34900 × 2 = 69800; whence the Ratio of the Force of Pow-
der is to the Weight of the Bullet, as 69800 to 1. And ſo F L :
FH:: 1:69800.
I
3006. But we have FB: FA : 42:23: 339: 21.
Hence the Rectangle FL BP is to the Rectangle AFSH, as
IX 339: 21 × 69800: 1: 4324. And fince the Rectangle
AFSH is equal to the Power of the Hyperbola (853) it is to
any hyperbolic Logarithm FHQB, as 0,43429, &c. to the
tabu-
!
Fig 2.
E
F
R
H
U
I
N
M
B
W
K
A
C
Fig 1
2.
D
Of GUNNERY.
313
tabular Logarithm of the fame Ratio
AB 360
AF 21
>
which is
1,2340579 (by 854). Therefore the Rectangle FLBP
1
: Space FHQB:: I x 0,43429
::1:12263. Confequently v: V ::
=
4324 X 1,2340579
FLBP:/FHQB
(2096): ✔✅I : √12263 :: 1 : 110,7 :: 15,07 Feet: 1668
Feet V Velocity of the Ball required; that is, the Velo-
city communicated to the Ball by the whole Force or Action of
the Powder will be that of 1668 Feet per Second, of uniformi
Motion.
CHAP.
VI.
The THEORY for determining the VELOCITY of the
BULLET at the MUZZEL of the GUN arising from
the QUANTITY of the CHARGE, the LENGTH
of the PIECE, and DIAMETER of its Bore; alſo
the CHARGE producing the GREATEST VELOCI
TY in a Piece of given Dimenſions.
3007:
As
S the THEORY of military PHILOSOPHY confifts
chiefly in two great Points, viz. (1.) To fhew
how far the Velocity of the Bullet or Cannon-ball is affected,
whilſt in the Piece, by the Force of the Powder, the Diameter of
the Bore, and the Quantity of the Charge, under any given Varia-
tions; or (2.) after it is out of the Piece, by the Refiflance of the
Air; and having premiſed the neceffary geometrical Princi-
ples, we ſhall now proceed directly to an Illuftration of them
Both.
3008. Let a = A B, the LENGTH of the PIECE under Con-
fideration, and b Length of any other Piece of Artillery.
Then if the Length of the Charge A F and the Bore be the fame
in each, we fhall have the Area FH QB, or L
VOL. II.
s f
a
Vi
AF
b.
L
AF
314
INSTITUTIONS
L
b
AF
: V² (3006). But the three firſt Quantities are known,
and therefore V = Velocity of the Bullet through the Length of
the Bore denoted by (b) will alſo be known. Hence it appears,
that the longer the Barrel is, the greater will be the Velocity of the
Bullet.
3009. If the BORE of the GUN be varied, all other Things
remaining the fame; then, putting D Diameter of the Bore
in the preſent Inftance, and D = Diameter of any other Bore;
it is evident the Quantity, and, conſequently, the Force of the
Powder in each will be in the Ratio of D2 to D, and the Force
of the Powder in the Bore, whofe Diameter is D, will be refift-
ed in the Ratio of the Weight of the Ball or of its Magnitude,
that is as D³ (by 841). Wherefore the Area F HQB will be
D2
I
varied in the Ratio of D3 or that is, D :D :: V² : V²;
Di
Ꭰ
: =
and foD:D :: VV the Velocity of the Bullet
fought. Hence the fmaller the Bore is, the greater will be the Velo-
city of the Bullet.
3010. If A F, or the QUANTITY of the CHARGE be varied,
other Things remaining the fame; then the Rectangle A FSH,
and, conſequently, the Area FHQ B will alſo vary, but not in
the fame Proportion; for there is a certain Quantity of Powder
or Length of A F, which with Refpect to the Length of the
Piece A B, will give a greater Velocity to the Bullet than any
other.
Then will
3011. Therefore let A B = a, and AF = x.
the Velocity be as the Force of the Powder at the firſt Inſtant of
Accenfion, viz. as AH, or AF = x; and alſo as the whole
AF-x
Action of the Powder upon the Ball while in the Barrel, or as
a
the Area F HQB = L; confequently, the Velocity will be as
a
* L²/
x
X
>
or is x LaxLx. Letz any other Length of the,
a
Charge, then we ſhall have x L:zL :: V² : V².
a
2
3012. Hence the Velocity of the Bullet at the Muzzle of
Guns of a different Length, Bore, and Charge, will be as in
the following Analogies, viz.
Va
Of GUNNERY.
375
V²: V2 :: L²: L², for Lengths as a to b.
2
x
x
V² : V² :: D: D, for Diameters of Bore as D to D.
V:
a
a
V : V :: xL²:zL2, for Lengths of Charge as x to z.
x
Confequently, V : V ::
5
a
5
DxL2
S/DEL
། ཁྱ
Q
L
X
X
3013. But fince it is evident from the Expreffion of the Velo-
city x L
a
— xLa—xLx, there will be one determinate Va-
lue of x which will make it a Maximum, therefore by putting its
Fluxion=0, we have La — x L x
x- o (becauſe the
Fluxion of L ≈ = -,
Lx ~~ (849); wherefore La-1- Lx. Now
x
X
I
let a = 10; then, becauſe, the hyperbolical Logarithm of 10
is 2,302585; therefore Lx = 1,302585; but as I: 0,4342448
:: 1,302585 : 0,565710 = tabular Logarithm of x (853).
Therefore 3,679 A F, when the Velocity is a Maxi-
x =
mum.
3014. And becauſe in this Caſe L
a
a
X
1
=I (for La-Lx=
L2) the Velocity x L when a Maximum, will be as
I = L
=
x
X
* or 3,679. Alfo when or AF
a
1
of AB, then becauſe
* = 1, we have the Velocity * L = La=2,302585, which
x
X
is not quite of the Maximum Velocity. And thus the Velocity
acquired from any other Charge of Powder may be compared with the
greatest Velocity.
3015. It is obfervable, that when the Velocity is greateſt,
then AF:AB:: (3,679 : 10 ::) 1: 2,71828; which Ratio
is called the modular Ratio, becauſe its Meaſure (L) is 1 =
the Module of the Syſtem, or Power of the Hyperbola (779).
I
I
3016. This Maximum Charge is to that in common Uſe,
(in Art. 3003) as to, that is near 7 Times as
large. The Reaſons why we do not, or rather care not to uſe
2.7
Sf 2
this
816
INSTITUTIONS
1
this Charge for the greateſt Velocity, are many; for first, it
would make too great a Confumption of Powder, viz. 6 or 7
Times more than what we now ufe. Secondly, the Force of the
Powder in the Gun, would in that Cafe, be fo great, that the
Barrels must be 5 or 6 Times as thick as they now are to avoid
bursting; by which Means they would be rendered too expen-
five and unweildy. Thirdly, the Velocity of the Bullet, from
common Charges, is fufficient in long Guns; and the greatest
Velocity would be much too great. Lastly, in fhort Guns, as
Piftols, &c. we may ufe this Maximum Charge very well on
many Occafions, for I 2,7 27 Inches, the Length of
§
:
the Barrel, whofe Charge AF2 Inches.
3017. If instead of one Bullet, there be 2, 3, or 4, laid be-
fore the Charge, then will the Velocities of the Bullets, in each
reſpective Cafe, be as
fince
I
√ √
39
(574). Therefore,
, the Velocity of a Bullet when there is 4, will
be but that of a ſingle one.
Hence alſo the Velocity of a
leaden Bullet is to that of an Iron one, as
or √3 to 2.
11,345 to ✔✅✔✅7,645,
3018. In this Theory, Mr. Robins has ſuppoſed two Things,
viz. (1.) That the Action of the Powder upon the Bullet ceaſes
as foon as it is got out of the Piece; and (2.) That all the Pow-
der of the Charge is fired before the Bullet is fenfibly moved out
of its Place. The firft of thefe Poftulates is felf evident; and
the Second is very nearly true, that the few Grains uſually
thrown out unfired, can but little affect the Theory; and he
found it fo by many Experiments upon Guns of different Lengths,
and diſcharging a different Number of Bullets from the fame
Gun. But this was more particularly confirmed afterwards by
many Experiments made by Appointment of the Royal Society.
The general Refult of which proved, that not above Part of
the whole Charge was collected in Grains unfired; and this, in
Reality, could be reckoned no more than a Part. For theſe
Experiments were made with common Powder, which produ-
ced a larger Quantity unfired, than the fineft grained Powder
will do (fuch as Mr. Robins uſed) in the Proportion of 5 to 3, as
was found by Experience. And again, the Saltpetre extracted
from the Powder, thrown out unfired, compared with that of
1
the
Of GUNNERY.
317
IZ
I
ठ
the Charge, was found to be nearly the Ratio of 7 to 9; and,
therefore, though 2 Penny-weight of unfired Grains were col-
lected from a Charge of 12 Penny-weight, which was ½ Part,
yet this reduced in the Proportion of 5 to 3, will be but; and
this again reduced in the Ratio of 9 to 7, is nearly of the
Whole. In fome Trials, the Deficiency was but the
I Part
of the whole Charge, from whence it will appear, how inftan-
taneuofly the whole Body of the Charge is fired, and how little
the Velocity of the Bullet muſt be affected by the ſmall Part un-
fired, as will be evident from the enfuing Experiments in
Chap. VIII.
18
CHAP. VII.
The THEORY of the MACHINE contrived by Mr.
ROBINS, for determining the VELOCITY which
any BALL moves with at any Distance from the
Piece it is diſcharged from.
3019. HAVING afcertained by the THEORY what the Ve-
locity of a Bullet ſhould be under any given Quan-
tities of Charge, Length of Barrel, and Diameter of Bore; it re-
mains in the next Place to confirm this excellent Theory by
Experiments. For this Purpoſe, Mr. Robins invented a Me-
thod, which by Means of a compound PENDULUM, and its Ap-
paratus, gives the Velocity of any Bullet iffuing either from the
Muzzle of the Gun, or at any Diſtance from it. The Ratio
nale of which Machine, and the Manner of applying it for fuch
Ules, are now to explained.
3020. To this End the Machine must be first of all defcribed,
of which the Body confiits of three ftrong Poles or Legs B, C, D,
fpreading at Bottom, and joining at Top in the Head-piece A.
On two of thefe Legs, towards the Top, are fcrewed on two
Sockets R, S, in which a Pendulum EFHI is hung, by Means
of the Cross-piece EF, which is the Axis of Sufpenfion on which
the
3 18
INSTITUTIONS
the Pendulum vibrates very freely. The lower Part of the
Pendulum is a thick fquare Piece of Wood G HIK, faftened
to the Back-part (which is of Iron) by Screws. A little below
the Bottom of the Pendulum, there is a Brace OP fixed to the
Legs B, C; and to the Brace is fixed a Part M N U made with
two Edges of Steel, bearing on each other in the Line U N,
fomewhat in the Manner of a Drawing-Pen; the Strength with
which they prefs on each other being diminiſhed or encreafed at
Pleaſure by a Screw Z going through the upper Piece. At the
Bottom of the Pendulum is faftened a narrow Ribbon L N W,
which paffes between the Steel-Edges, and paffing thro' a Hole
in the lower Piece of Steel hangs looſely down as at W.
3021. The Artifice of this Contrivance, tho' fimple, is ad-
mirable; the Bullet diſcharged from the Gun againſt this Pen-
dulum puts it into Motion; and both the Bullet and Pendulum
are to be confidered as Non-Elaflic Bodies; the Bullet being of
Lead, is evidently fo; and the Wood tho' to flow Motions it
may be confidered as fomewhat elaftic, yet with refpect to very
fwift Motions it cannot be conceived in the leaft Degree fo;
fince there is no Time for the Parts to act by their natural
Refort.
3022. The Bullet and Pendulum, therefore, will in their
Motions and Action on each other, obferve the Laws of Percuf-
fion laid down for Non-elaftic Bodies (from Art. 1003, 1012.) But
before we can come to apply them in the preſent Cafe, we muſt
firft ſhow how the Momentum or Quantity of Motion is to be ef-
timated in the Pendulum when put into Motion by the Bullet,
and alfo with what Velocity it moves after the Stroke.
3023. In Order to this we muſt know that the Weight of the
Pendulum (Mr. Robins made Uſe of ) was 56lb. 3oz. The Center
of Gravity was diſtant from the Axis of Suſpenſion 52 Inches. And
200 of its ſmall Vibrations were performed in the Time of 253
Seconds; wherefore the Time of one Vibration is 23 Parts
of a Second; and therefore fince the Length of a Pendulum which
vibrates in one Second is 39,2 Inches (1125) and the Lengths of
Pendulums are as the Squares of their Time of Vibration (1116.)
Therefore fay, as 12:
253″2
200
200
39.2 Inches: 62,73 Inches,
the Diſtance of the Center of Oſcillation from the Axis of Vibra-
tion.
Of GUNNERY.
319
tion. Laftly the Center of the Piece of Wood GHIK (on
which the Bullet is fuppofed to impinge) is diftant from the faid
Axis 66 Inches.
3024. Now if all the Matter of the Pendulum were concen-
tered in the Center of that Piece or Wood, it would refift the Bul-
let with all its Force, viz. of 561b. 3oz. becauſe that would then
be the Center of both Gravity, and Ofcillation; but fince the Cen-
ter of Gravity is fhort of this Point, the Refiftance will be dimi-
nifhed in Proportion, viz. in the Ratio of 66 to 52. And again,
another Diminution of the Force will arife on Account of the
Velocity of the Motion, becauſe alfo the Center of Oſcillation
(by which that is eftimated) is fhort of the Center of the Wood
where the Stroke is made in the Ratio of 66 to 62 . Therefore
the Force or Refiftance will be diminiſhed in the Ratio of 66 X
66 to 62 × 52; that is, it will be 662: 62 X 52::564.
30z.: 42 lb. Loz.
T
3025. What has been hitherto faid, is rather by Illuftration
deduced from the Confideration of a compound Pendulum of the
moft fimple Form, but fuch was not the Pendulum which Mr.
Robins uſed in his Machine, the Theory of which we fhall there-
fore give with its Demonftration, as follows:
3026. Let FC be the Shaft, and
ABED the Weight of the Pendulum,
both Parallelopipeds, and HI the Axis
of Motion; then is FG (FC),
the Diſtance of the Center of Gravity of
the Shaft, and FN (FC), the
Diſtance of the Center of Ofcillation, and
put the whole Length FC a. And
let the End C be the Center of the
fquare Parallelopiped ABDE, whoſe
Weight let us call W, and the Weight
of the Shaft w.
3027. The two Centers of Gravity,
G and C, will have a common Center
at g; and it will be w: W:: Cg:
Gg. Put Cg = d; then is Gg =
다
​H
I
A
闆
​G
N
B
n
C
D
W d
ՂԱ
320
INSTITUTIONS
+ wd
a w, whence d =
W d
W
d; therefore W d = ½ a w
a w
二
​2 x W + w
ลบ -
wd; and W d
Confequently,
a w
Fg=a-d=a —
2 a W + aw
2 W + 2 w
8 =
the
2 x W + w
Diſtance of the compound Center of Gravity from the Axis of Me-
tion.
1
3
3028. By the Addition of the Weight AD, the Center of
Ofcillation will be drawn down from N to fome other Point n;
to determine which, we proceed thus. Since FNa (1097)
and the Momenta of the Weights in the Shaft is aw; there-
forea × wa = a aw = Momenta, or Sum of all the
Forces in the Shaft. Alfo the whole Force of the Body A D
(confidered alone, vibrating at the Diſtance F C a) is a a W,
and the Momenta of its Weight will be a W. If therefore the
Sum of the Forces of the Shaft and Body AD, be divided
by the Sum of the Moments, we fhall have a aw + a² W
Zaw + a W
त्रु
Law + a W
½ w + W = n = Fn, the Diſtance of the Center of Of-
2
cillation from the Axis HI (1094).
3029. From what we have fhewn, it appears that
a:g:: W+w: W + w (3023).
2
a: n :: W + w: W + 3w (3024).
2
+ =
Wherefore by Compofition of Ratios, we have a a: gn :: W+w
W + ;w; and therefore gn × W + w = aa×
W + w.
3030. But gn x Ww F, the Force of the Pendu-
lum expreffed in a general Way (1095) and therefore with Re-
But were
gard to the Point C, the Force is a a x W + w.
3
the Body AD a fimple Pendulum of the fame Weight, its Force
at the faid Point would be a ax W+w, which is to the Force
in its preſent State, as W+w to W+w; and which,
therefore, is the Ratio of Diminution.
3031,
Of GUNNERY.
321
3
3031. To apply theſe Theorems to the Machine here uſed,
we have a 66, g = 52, n =
=
52, n = ½
62,73, (or 62 to uſe Mr.
Robins's Number) and the Weight of the whole W + w
56lb. 30%. = 899 Ounces. Then we have 66² (≈ 4356) :
623 × 52 (= 32583): 56 lb. 3oz. (= 899 oz.): 671
— =
Ounces, or 42 lb. nearly (3025). As 899 671 228 oz.
=w; therefore w 342 oz. and W = 557 oz.
ΤΖ
lb. as 504 to 1.
I
3032. This compound Pendulum, then, is reduced to the
Cafe of a fimple Pendulum whofe Length is 66 Inches, and
Weight = 42 lb. oz. which is to the Weight of the Bullet =
Let this equivalent.
fimple Pendulum then be repreſented
by AC 66; and on the Center C
and Diameter DE defcribe the Semi-
circle D A E. Let Aa E be the Arch
deſcribed by the Pendulum impelled by
the Bullet; and A E the Chord of that
Arch. Then we have fhewn (1113)
that the Velocity of the Pendulum by
which it fhall defcribe the Arch E A is
the fame as would be acquired by an
heavy Body falling through the fame
perpendicular Height GE; which Ve-
locity is the next Thing to be deter-
mined. In order to this, it is eafy to
وح
a
E
underſtand, that when the Pendulum is in its perpendicular Po-
fition in a State of Reft, and the Ribbon drawn ftrait, with a
Pin put through that Part which is contiguous to the Edges UN,
that when the Bullet impinges on the Pendulum, the Rib-
bon will be drawn out in fuch Manner as to meafure the Chord
of the Arch deſcribed by the Point L; for it will be equal to the
Interval between the Pin, and the Edges U N.
T2
3033. Now by an Experiment made with a Gun 45 Inches
in Length, 2 Inches Charge of Powder (weighing 12 Penny-
weight) the Bullet Inch-diameter, and lb. Weight; and
the Muzzel of the Gun at the Diſtance of about 16 or 18 Feet;
it was found that the Ribbon was drawn out 17 1 Inches. Now
the Diſtance of the Point L is 71 Inches from the Axis, and
therefore if we fay, as 71: 174 :: 66: 16 Inches nearly; it
VOL. II.
I
Tt
7
will
322
INSTITUTIONS
will appear, that the Chord of a fimilar Arch defcribed by the
Center of the Piece GHIK was 16 Inches nearly.
4
3034. Wherefore (in the Fig. 3026) A E≈ 16, and DE
= (2 AC =) 132. And drawing AD, we have DE: AE
:: AE: GE, that is 132: 16 :: 16: 1,939 = GE. If then
we fay, as 193† : 32 :: ✔ 1,939: 3 Number of Feet
✓
per Second, that a Body would deſcribe by an uniform Motion
with the Velocity acquired in falling through GE. The Velo-
city therefore with which the Pendulum moved after the firſt
Moment of Impact was that of 3 Feet per Second.
4
3035. The Cafe of the Bullet and Pendulum is therefore re-
duced to that of the two Bodies B and A (in Ic03); and fince
m: M:: 1½ lb.: 42 16. 1/2 oz.:: 0,083: 42,031 : 1 : 504,6;
and becauſe the Pendulum was at Reft, we have (1009) the Ve-
12
I
locity of the Bullet = v =
M + m
m
(= 505,6) × V = 505,6
× 31643 Feet per Second. That is, by Experiment,
the Bullet moved at the Rate of 1643 Feet per Second, and by
the Theory it was determined 1668 Feet (in 3006); and as this
was at the Muzzel of the Gun, and the other at the Diſtance of
16 or 18 Feet from it, the fmall Difference of 15 Feet is no more
than the Allowance which ought to be made for the Refiſtance
of the Air to the ſwift Motion of the Bullet paffing through that
Space. Whence appears the wonderful Degree of Exactnefs be-
tween the Theory and Experiments made quite independent of it.
3036. Having thus determined the Velocity for the Length of
one Chord; the Velocity of the Pendulum for any other Length
of a Chord (or Ribbon drawn out) will be known; becauſe the
Velocities have the fame Ratio with thoſe Chords, for fince DE
× GE = A E², and DE is a ſtanding Quantity; it will al-
ways be GE as A E², and confequently AE as GE, that
is, as the Velocity acquired in falling through GE (by 991).
And hence, becauſe the Velocity of the Bullet v=VX 505,6;
and this Number 505,6 (for the fame Bullet and Pendulum) is
conftant, we have the Velocity (v) of the Bullet always propor-
tional to (V) the Velocity of the Pendulum, and conſequently to
the Chord of the Arch in any Vibration.
CHAP.
Of GUNNER Y.
323
С НА Р. VIII.
An Account of the EXPERIMENTS made for com-
paring the actual VELOCITIES with which BUL-
LETS of different Kinds are diſcharged from their
respective PIECES, with their VELOCITIES COM-
puted from the THEORY.
3037. THE THEORY for determining the Velocities of Bul-
lets difcharged from given Pieces of Artillery having
been fully deſcribed; and in the laft Chapter it has been fhewn
how the actual Velocities of Bullets are meaſured and afcertained
by Means of a Machine; it now remains to compare the Reſult
of the Theory with Experience, and thereby to evince how ac-
curately this Theory agrees with the real Motions of Bullets
though founded on Principles no ways connected with thofe Ex-
periments. Some of theſe we fhall felect for illuftrating the
Theory by a particular Application; and then give a general
View of the Experiments as we find them tabulated by Mr. Ro-
bins himſelf.
3038. By Experiments made on a Barrel only 12,375 Inches,
the Theory was farther confirmed. For (cæteris paribus) we
have by the Theory (3008) the L
45
2,625
2
: 17,25 : L
12,375
2,625
: 162,37 Square of the Chord whofe Length is fought in
the prefent Cafe, which therefore would have been 12,74 In-
ches, had the Weight of the Pendulum been the fame, but as it
was fome finall Matter lighter, than that in (3020), the Chord.
was by the Theory a little larger, viz. 12,8 Inches; and by
Experiment it was 12,7, 12,6, 12,4; on three different Trials.
Here the very ſmall Differences are little more than what muſt
refult from the Bullets lodging in the Pendulum and making it
thereby heavier each Time.
3039. By Experiments made with a Barrel 24,312 Inches in
Length, all other Things the fame; the Length of Chord, by
the Theory, agreed exactly with the Length of the Ribbon drawn
Tt 2
out
324
INSTITUTIONS
I I
out being both 14,4 Inches. Whence the Velocity of the Bullet
from the prefent Gun was to that from the Gun of 45 Inches, as
14,4 to 17,1; that is, as 17,1 14,4: 1643":1544" nearly.
And for the Velocity of the Bullet from the Gun of 12,375, we
have by Theory 17,1 : 12,8 :: 1643″: 1432″. Whence it
appears, the Velocity in a fhort Gun is much greater (cæteris
aribus) than in a long one.
3040. Again, by varying the Quantity of Powder in the
fame Space or Cavity DEG C the Theory is ftill farther con-
firmed; for if inſtead of 12 dwt. you put only half that Quan-
tity, viz. 6 dwt. Then by the Theory (2996) the Velocity by
6 dwt. is to that of 12 dwt. in the fubduplicate Ratio of 6 to 12
(becauſe thofe Numbers are as the Forces or Elafticities impel-
ling the Ball) that is, as ✅/12: ✅/6 :: 17,1:12; the Chord
or Length of Ribbon therefore ſhould be nearly 12 Inches when
the Bullet was difcharged with 6 dwt. of Powder, and by two
Experiments the Length of the Ribbon drawn out was 11,2;
12,2; and the Deficiency was owing to this, that the Heat in
firing fmall Quantities of Powder is not proportionally ſo great
as in firing larger Quantities; and therefore the Impulſe is not
to be expected quite fo great as by the Theory. Thus I dut.
of Powder by the Theory gives a Velocity of 482 Feet per Se-
cond, but by repeated Trials with that Quantity, the Velocity
was not quite 400 Feet per 1".
}
3041. The lefs Space the Fire or inflamed Powder has to
act in the more Denſe it is, and confequently the ſtronger its
Action; therefore when the 6 dwt. of Powder was rammed into
but half the Space DEG C, it was found in two Experiments
to produce Lengths of Ribbon which meaſured 13,2 and 13,9
Inches, whereas by the Theory it ſhould be 13,6 Inches.
3042. Let us now apply the Theory to determine the Velo-
city of a Ball of 24 lb. fhot from a Cannot of 10 Feet
3
120 In-
ches Length, with of its Weight (viz. 16 lb.) of Powder.
The Diameter of an Iron Globe of that Weight (and confe-
quently of the Bore of the Piece) will be found to be 5,325 In-
ches, allowing the (pecial Gravity of Iron to be as in (2054).
Alfo 16 lb. of Powder will fill a Cylinder of that Diameter to
the Height of 21,63 Inches, allowing 8 Drams to a Cubic
Inch.
Of GUNNERY,
325
Inch. Hence (in the Fig. of Art. 2098.) we have A B 120;
AF = 21,63; and F B 98,37. Whence (computing as in
3002, 3003,) we fhall find F L: FH::: 14732,4.
3043. Therefore FP: FS:: IX 98,37: 21,63 × 14732,4
•
=
AB
::: 4978,1. Alſo, we have ES: FQ:: 0,43429: L AF
= 0,744125. Confequently it is FP: FQ:: 1 × 0,43429:
4078 x 0,744125 I 6987,4. The fubduplicate of this
I:
Ratio is that of 1 to 83,58. And the Velocity acquired in fall-
ing through the Height F B 98,37 Inches is 23,05 Feet per
1". Whence 83,59 × 23,05 = 1926,6 Feet per 1" which
the faid Bullet acquires at its Exit from the Muzzel of the Can-
And ſince the ſpecific Gravity of Iron and Lead are (as
the Quantities of Matter under equal Bulks) as 2 to 3 nearly ;*
therefore fay (by 2097) as √3:√///2
√3:2::1926,6": 1573",
the Feet per 1″ fuch a Bullet would move through if made with
Lead.
non.
3044. In theſe great Guns it will be neceffary to underſtand
what Space the Piece will recoil through in firing it, or rather
what Space the Gun will move through Backwards, while the
Ball is carried forwards to the Muzzel thereof. In order to
this, we must confider, that it is one and the fame Power (viz.
the Elafticity of the Powder) that acts both on the Gun and
Ball; and therefore in the Theorem QV GSM (1001) G
is a given Quantity; and fo in the prefent Cafe, we have QV
=SM, but Q V M in every Cafe (970) therefore S V²;
that is, the Spaces defcribed in a given Time, by given Forces, will
be as the Squares of the Velocity.
3045. Let S (FB) Space defcribed by the Ball, and V its
Velocity; and let s Space through which the Gun recoils.
with the Velocity v; M Quantity of Matter in the Gun,
and m the fame in the Ball. Then we have S : s :: V² : // ² : :
=
I I (by 2097). Whence S:s:: M: m. Now the Weight
771 M
lb.
of the Gun is about 47 Cwt. or 5264 lb. and that of the Ball
24 lb.alſo S=98,37 (3042). Therefore we have 5264 lb. : 241
:: 98,37
*The Reader is defired to correct an Error in the Table of ſpeci
fic Gravities (2054) where that of Lead is 10,131 inftead of 11.131.
326
INSTITUTIONS
2
:: 98,37 : 0,448 = Space of the Recoil, which is not quite an
Inch, when the Ball is juſt out of the Gun.
3046. Hence we ſee the uſual Methods of conftructing Plat-
forms for Cannon with fo much Strength and Firmness, and
conſequently with ſo great an Expence, are not at all neceffary;
fince if it be but fufficiently ſteady at the Beginning of the Re-
coil, the remaining Part may be much flighter (as Mr. Robins
obferves) fince its Unfteadinefs or Shaking beyond the first Inch
can have no Influence on the Ball, (which is then out of the
Gun) nor any how alter the Direction of the Shot.
3047. Having thus finiſhed a particular Application of theſe
Experiments to elucidate the THEORY of GUNNERY according
to the new Principles of Mr. Robins; we fhall now give a gene-
ral View of the many valuable Experiments he made, and di-
gefted into proper Tables, by which it will appear how furpriz-
ingly the Theory agrees with and is corroborated and confirmed
by thoſe Experiments. In thefe Trials it became neceffary
fometimes to change the Board of the Pendulum (when too
much battered and over-charged with Bullets) but when its
Weight varied any Thing confiderable, Mr. Robins has taken
Care to acquaint us with it. The following Account we ſhall
give in his own Words.
3048. The firft Table contains three Experiments only, made
with a Barrel 45 Inches in Length, and the Board on the Pen-
dulum was 4 lb. lighter than that deſcribed (3023).
Quantity of
Powder.
Chord of afcending
Arch meaſured on
the Ribbon.
The fame by
the Theory.
Error of
Theory.
No.
Dw.
I
12
2
12
18,7
19,6
3
6
13,6
19,0
+3
19,0
-,6
13,4
-,2
3049. The next Experiments were made with the fame Bar-
rel, but the Board on the Pendulum was now of little more
Weight than that in the Example of (3023).
1
N°.
Of GUNNER Y.
327
Length
of the
Cavity
A F.
Quan-
tity of
Powder.
Chord of afcend-
ing Arch mea-
The fame by
Theory.
Error of
Theory.
fured on the
Ribbon.
Dw.
Inch.
Inch.
Inch.
6
11,9
12,1
+,2
6
12,2
12,1
—, I
6
13,2
13,6
+24
6
13,9
13,6
3
12
16,7
17,2
+›5
12
17,5
17,2
3
12
16,9
16,8
,I
12
17,0
16,8
,2
6
11,7
11,5
-,2
6
II. I
,
11.5
+4
12
16,7
16,3
4
No.
Inches.
4
ठ
in/00 in 100 = 1+ Huponpompom pom pom poms/as
221
I
I
1 2 2 2 2 2 22
56 78
9
10
II
12
13
14
The laſt five Numbers refulting from the Theory are correct-
ed from the Quantity of Bullets lodged in the Board, which, as
many other Experiments of a different Kind were tried in the
Interval, amounted at laſt to above two Pounds; whence the
Weight of the Pendulum being increaſed, its Vibration with the
fame Blow muſt be proportionably diminiſhed.
3050. The next Experiments were made with a Barrel of the
fame Bore with the laft, but only 12,375 Inches in Length :
To diftinguish them, we fhall for the future denominate the
firſt Barrel by the Letter A, and this ſhort one by C. The
Board on the Pendulum was at firft rather lighter than in (3023).
Extent
of the
Cavity
contain-
ing the
Quan
tity of
Chord of af-
cending Arch
The fame
by Theory.
Error of
Theory.
Powder.
meaſured on
the Ribbon.
Powder.
Nº.
Barrel
Inch.
Dw.
Inch.
Inch.
15
C
16
C
2 5
17
C
18
A
19
A
2 2
20 A
21 A
22 A
2
in 100 in 100 in 100 in too in po info in 100 in 100
2 2 2 2 2 2 2 N
12
8
12,7
12,8
12
12,6
12,8
+, Z
12
12,4
12,8
+24
12
17,0
17,3
+,3
12
17,2
17,2
I 2
17,I
17,2
+, I
12
17,2
17,2
12,4
12,2
,2
3051.
328
INSTITUTIONS
3051. In fome of the following Experiments a third Barrel
was uſed of the fame Bore with the other two, but 24,312 In-
ches in Length: This Barrel 1 denominate B; the Board fixed
on the Pendulum was at firft but little heavier than that in
(3023); and when in the Courſe of the Experiments it is fen-
fibly increaſed in the Weight, I diminiſh the Numbers arifing
from the Theory by a correfponding Part.
Extent
of the
Cavity
contain-
Quantity
of Pow-
der.
Chord of af-
cending Arch
meaſured on
The fame
by Theory.
Error of
Theory.
the Ribbon.
ing the
Powder.
No.
Barrel
Inch.
23
A
24
A
2222
25
A
26 C
27
28
29
C
BABA
30
31
В
32
В
33
Α
34
A
4
2 2 2 2 2 2 2 2 2 2 -
inpompoenp000 in poi po un po info in 100 in 100 m/+
Dw.
12
Inch.
Inch.
17,1
I
17,2
9
15,2
15,0
11 +
+ , t
›2
9
15,4
15,0
,4
12
11,5
12,8
+1,3
12
11,5
12,8
+1,3
6
8,7
9,
+3
12
12,3
12,5
+,2
12
14,4
14,4
0,0
I 2
14,4
14,4
0,Q
6
10,3
10,5
+,2
8
14,7
14,5
,2
12
15,7
15,3
›4
The Error in the 26th and 27th Experiments being much
greater than what has occurred to me in any other Trials, I fuf
pect, that fome Miſtake was made in the Weight of the Powder,
or that the Barrel (which had indeed lain by in a moift Place) was
very damp; which Circumftance, I know by Experience, will
confiderably diminiſh the Action of the Powder.
3052. The following Experiments were made with a Pen-
dulum much heavier, it weighing in the whole 97 lb. its Center
of Gravity was 55,625 Inches diftant from its Axis of Sufpen-
fion, and 200 of its fmall Swings were performed in the Space of
255", whence its Center of Ofcillation is 63,9 Inches diftant
from the Axis of Sufpenfion. Alfo fometimes another Barrel
was uſed 7,06 Inches in Length, and ,83 in Diameter, its Ball
was exactly fitted to the Bore without any Windage, ſo that
it went in with Difficulty, the Weight of this Ball was 33%
dw. This Barrel we fhall denominate D.
N°.
Of GUNNERY.
329
Extent
of the
Cavity
contain-
ing the
Powder.
Quantity
of Pow-
der.
Chord of af-
The fame
cending Arch
meaſured on
the Ribbon.
by Theory,
Error of
Theory.
ردرد
N°.
35
36
37
38
39
40
4. I
42
43
44
45
46
47
>>>>WWOO>>>>>>]
Barrel
Inch.
Dw.
Inch.
Inch.
Inch.
5
48
49
50
2
2
555
56
57
58
59
51
52
53
54
D
55
D
D
60
AAAA
70
22572
56 7∞
1 2 2 2 2~ ~~~~ 2 2~~~~~~~
12
9,2
9,2
I 2
9-5
9,2
4
24
36
I
+6 20 2
23
11,7
11,3
13,2
12,6
,6
12
9,3
9,1
2
8
7,6
8,1
+,5
12
6,1
6.6
+,5
12
6,5
6.6
+, I
12
8,0
8,2
+,?
I 2
8,3
8,2
; I
12
9,5
9, I
34
2
12
9,1
9, I
2.
6
7,2
6
6,7
8
5
I 2
6,8
2 70
6,5
6,5
6,7
7
2
, I
12
7,5
6,7
8
6
4,7
4,8
+, I
6
5,0
4,8
,2
I 2
7,0
7,2
+,2
I 2
7, I
6,8
3
6
427
4,8
+, I
6
4,8
4,8
,0
I
2.
216
215
6
6,4
6,5
18
I
6
6,4
625
+,I
+, I
6
6,6
ठ
6,5
I
216
6
6,7
6,5
,2
61
A
273
12
9,0
I, I
+, I
The Error in the 50th Experiment, the greateſt in this Set,
was doubtlefs owing to the Wind; for the 49, which was made
immediately before it in the fame Manner, and with the fame.
Quantity of Powder, differs but little from the Theory. The
Excefs of the 38th Experiment above the Theory was in Part
occafioned by the Impulfe of the Flame on the Pendulum, which
in this large Quantity of Powder was plainly to be difcerned.
VOL. II.
U u
CHAP.
339
INSTITUTIONS
*
CHA P. IX.
>
The foregoing PRINCIPLES applied to investigate the
VELOCITY which the FLAME of GUNPOWDER
acquires by expanding itself, fuppofing it be fired in
a given PIECE of ARTILLERY without either
Bullet or any
other Body before it,
IN
3053. N order to experiment the Velocity with which the Par-
ticles of Gunpowder expand themſelves (in the Ex-
plofion) at the Muzzle of the Gun, Mr. Robins charged the
Barrel of 45 Inches (Art. 3033) with 12 pwt. of Powder, and a
finall Wad of Tow only; and then placing the Muzzle 19 In-
ches from the Center of the Pendulum (mentioned Art. 3023)
it was fired, and the Impulfe of the Flame on the Pendulum
made it afcend through an Arch, whofe Chord was 13,7 Inches.
Now fince a Chord of 17 Inches agrees to the Velocity of 34
Feet per 1, (fee 3034) therefore fay, as 17 Feet: 3 Feet
13,7 Feet: 2,6 Feet; that is, the Velocity of the Pendulum
4-44
was at the Rate of 2,6 Feet per 1". ·
1
1
I
3054. Now the Weight of the Powder and Wad was about
13 pwt. and that of the Pendulum 42 lb. oz. (when reduced
to the Center 3024). Whence m: M:
1
2
I 3
320
42,031
:: 1: 1034,6; therefore the Velocity of the Powder v
M + m
77
4 x × V = 1035,6 × 2,6 = 2692,56; (fee 1009 and
3034). Hence the Velocity of the Particles of Powder, (ſup-
pofing the Whole of it (together with the Wad) impinged upon
the Pendulum,) was at the Rate of 2692 1 Feet per 1" of uni-
form Motion.
ΤΟ
3055. And this is the leaft Velocity the Particles of Powder can
be ſuppoſed to acquire in the Expanfion. For firft we may ob-
ferve, that not more than of the Whole is converted into
this claftic Fluid. Since I oz. 437 grs. produced 460 Cubic
Inches (2086) and each Cubic Inch weighs 0,29313% of a
Grain (2082). Whence 460 x 0,393 = 134,78 Grains of
[
1
×
elastic
1
Of GUNNERY.
331
laftic Air.
But
13
437
7
ΤΟ
3
TO
of the whole Powder, nearly.
Therefore the other muft, in mixing with the elaftic Part,
greatly impede the Action, and retard the Motion or Velocity
thereof in Exploſion; eſpecially if it be confidered that this inert
Part is in fome Meaſure of an unctuous Nature, and will not
be thrown out, but ftick or adhere to the Infide of the Barrel an
Impediment to the Reft.
3056. Again, fome Part of the Flame must be loft in Expans
fion Sideways through 19 Inches of Air; for an elaſtic Fluid ex-
pands itſelf equally every Way, and confequently only a Part
thereof can impinge upon the Pendulum. And even that Part
will meet with a great Reſiſtance from the Air, and fo will have
'its Velocity diminiſhed on that Account. The Quantity there-
fore of the Powder, and its Velocity, eftimated this Way, is
fhort of what it is in the Barrel, and at the Exit from the Muzzle
thereof. Now to difcover what that is very accurately, Mr.
Robins contrived the Experiment in a different Manner, which
I fhall here explain, with a Variation of fome Circumftance to
render it eaſier to be underſtood and practifed.*
3057. The Method of proceeding is this; let the Barrel be
fixed to the Center of the Pendulum, and let the Weight of the
Pendulum, Barrel and all, be 56 lb. (or 42 lb. reduced to the
Center, as per Art: 596); in this Situation let it be charged with
12 pwt. of Powder only, put cloſe together with the Rammer,
and then upon diſcharging the Piece, the Pendulum will afcend
through an Arch, whofe Chord, at a Medium, will be 14 In-
Now fince in this Cafe the Powder (or rather its elaftic
Air) acts equally on its felf and on the Pendulum, therefore
the Quantity of Motion or Momentum of the Powder (viz. q
vm) will be equal to that of the Pendulum (QVM) that is,
VM vm (970). Alſo, fince to 14 Inch Chord there cor-
=
refponds the Velocity of 2
therefore fay, as m=
32: M = 42 :: V2 3082, nearly; fo that the
3
4.
Feet per 1";
: V
Velocity is by this Means determined for the whole Mafs of
Powder to be at the Rate of 3082 Feet pér Second.
U u 2
3058.
* This Pendulum with the Barrel fixed upon it in the Manner re-
prefented by Fig. 2. of the preceding Plate, we apprehend is the
moft facile and fimple of any
>
1
t
!
332
INSTITUTIONS
3058. If now fo light a Body as 1 pwt. of Tow be placed be-
fore the Powder contiguous to it, it will preſently acquire the
Velocity with which the elaſtic Part of the Powder will expand
itſelf when uncompreffed, and by this Means we may be able
to meaſure that Degree of Velocity pretty nearly. Thus if the
Barrel be charged with 12 pwt. of Powder, and I of Wadding,
and thoſe fired, the Pendulum will afcend through an Arch,
whofe Chord is 17,3; from which, if we fubduct 14,5 for the
Powder, the Remainder 2,8 will be owing to the Wad; and
therefore fince 2,8 Inches of Chord gives the Velocity 0,53 of a
Foot per 1"; therefore fay, as ¸½: 42:: 0,53: 7128 Feet,
the Velocity of the Wad, or that which the fwifteft Part of the
Flame moved with per 1".
IT
I
3059. In this Way alfo the Velocity of the Bullet may be de-
termined to a greater Exactneſs than by the former Method
where the Barrel was fired, and at a Diſtance from the Pendu-
lum. Thus let the Barrel be charged with 12 pwt. of Powder,
and Bullet lb. (as at 3033); then upon diſcharging it, the
Pendulum will afcend through an Arch, whofe Chord will be
32,3 Inches. Now 14,5.of this Chord is owing to the Impulfe
of the Powder (as above 3057); therefore 17,8 is occafioned
by the Motion of the Bullet, and is fomewhat greater than that
determined (in 3033). Now to a Chord of 17,8 Inches, there
anſwers the Velocity 3,45 Feet (as per 3034); whence the Ve-
M + 72
locity of the Bullet o=
772
XV=505,6×3,45
1734
Feet, nearly. Whence we find the Velocity of the Bullet, at
the Muzzle of the Gun, is at the Rate of (at leaft) 1700 Feet.
per I".
3060. From this Experiment it appears, that the Action of
the Powder on the Gun is the fame, whether it impels a Bullet
before it, or whether it be fired alone; and it is therefore a con-
vincing Proof, that the whole Quantity of Powder is fired in the
latter Cafe, as well as in the former. In all the Experiments.
hitherto mentioned, the Bullet has been fuppofed contiguous to
-the Powder; and Mr. Robins found that it was laid at a finall
Diſtance from it (as an Inch, or two, at moft) the Theory will
agree very nearly with the Experiments. But when the Bullet
is
Of GUNNER Y.
333
is laid at a confiderable Diſtance from the Powder, as 12, 18,
or 24 Inches, the Cafe will be very much altered, and the Bul-
let will be impelled or acted upon in a Manner very different from
what it was before.
3061. For now we are to confider the Powder, by the Time
it reaches the Bullet, as acting with two Forces, viz. one by
Pulfion or Percuffion, which it receives from the great Velocity
with which it's parts expand, and ftrike against the Bullet, as
fhewn above (615), and the other Force is by Preffure; for
when the inflamed Powder has expanded itself into all the Ca-
vity behind the Bullet, it will, after the firft Impact, continue to
prefs on the Bullet with all the Force of its Elafticity, till the
Bullet be out of the Barrel; and hence it will be eaſy to under-
ſtand that the Velocity of the Bullet will be in this Cafe greater
than if it were impelled by its elaftic Preffure of the Powder on-
ly; and it is demonftrably fo by Experiment.
+
be
3062. For Mr. ROBINS charged the Barrel of 45 Inches with
12 put. of Powder as ufual, and then placed the Bullet at the
Diftance of 11 from the Breech (or 8 from the Powder) and
upon diſcharging it againſt the Pendulum, he found that it had
acquired a Velocity of about 14co Feet per ". Whereas if it
had been acted upon by the Preffure of the Flame only, it would
not have acquired a Velocity of 1200 Feet per 1", as may
eafily made appear by the Theory, thus; let M be the Place of
the Bullet (fee Fig. to 2097) then will M N reprefent the Force
of Preffure on the Ball the firft Inftant; but it is AM = 11,25:
AF = 2 { :: HF 69800: MN = 16131, (2099). Then
we find (by the Methods in 3006 and 3002) the Rectangle
MRPB Space MNQB:: MR x MB x 0,43429: A M
X MN X L
A B
АМ
=
AM; which Quantities are all known; the Sub-
duplicate of this Ratio is that of 1 to 86,338; now the Velocity
acquired in falling through the Space MB, is 13,51 Feet per
1". Therefore fay, as 1 : 86,338 :: 13,51: 1166 Feet, the
Velocity acquired by the Force of Preffure only (by 2096).
=
3063. And to this the Experiment agrees; for the fame Gen-
tleman (with his ufual Sagacity) in order to feparate the two dif
ferent Actions of Powder upon the Bullet, and to retain that
only which arofe from the continued Preffure of the Flame, con-
trived
334
INSTITUTIONS
II
dife
trived the following Method. He no longer placed the Powder
at the Breech from whence it would have full Scope for its Ex-
panfion, but fcattered it as uniformly as might be through the
whole Cavity of 11 Inches left behind the Buliet, conceiving
that by this Means the progreffive Motion or Velocity of the
Flame in each Part would be prevented by the Expanſion of the
neighbouring Parts. And he found upon difcharging the Bar-
rel, the Velocity of the Ball was (inftead of 1400 Feet per 1")
no more than I 100 Feet per 1". Which was 66 Feet ſhort of
what it ſhould be by the Theory.
3064. This Deficiency he ſuppoſes was owing to ſome inte-
ftine Motion of the Flame; for the Powder being kindled in a
Space much larger that it could fill, muft have produced many
Reverberations, and Pulfations of the Flame; and from thefe
internal Agitations of the Fluid, its Preffure on the containing
Surface, he fuppofes, muſt be confiderably diminished; and
from hence he judges it neceflary, to avoid any fuch Irregulari-
ty, to take particular Care to have the Powder confined cloſely
in as ſmall a Space as poffible, even when the Bullet lies at fome
little Diſtance from it.
3065. From what has been ſaid, it will be apparent, that
when the Ball lies at a great Diſtance from the Charge, the
Action of the Powder will be fo greatly augmented in the Space
behind, and will be fo accumulated and condenfed by the Velo-
city each Part has acquired by the Time it comes to the Ball,
that if the Barrel be not of an extraordinary Firmneſs in that
Part, it must by this reinforced Elafticity of the Power infallibly
burit. And the Truth of this Reafoning he experienced in an
exceeding good Tower Mufquet, forged of very tough Iron; for
charging it with 12 put. of Powder, and placing the Ball at 16
Inches from the Breech, on firing it, the Part of the Barrel juſt
behind the Bullet was fwoln out to double its Diameter, like a
blown Bladder, and two large Pieces of two Inches long, were
burst out of it.
3056. From what has been faid, we fee the Reaſon of all the
extraordinary and enormous Effects of warlike Engines in which
Gunpowder is ufed; as in Grenades, Bombs, Petards, Mines,
&c. For as we have fhewn the Force of this Powder upon every
fquare Inch is 15000 lb. (2093), and that it expands itſelf with
2
Of GUNNER Y.
335
a Velocity more than 3000 Feet per 1" (3057) it is no Wonder
it fhould burft the Shells of Bombs and Grenades with fuch
Force and Violence, and rend the Planks and Parts of Gates,
Bridges, Walls, &c. in Petards, with fuch fudden infupera-
ble Power and Impetuofity. For the Powder in the Petard
weighing 5 lb. will have an Effort equal to that of a Cannon-
Ball of 10. moving with half the Velocity, viz. at the Rate
of 1500 Feet per ". For the Momentum is the fame in both
Cafes (970).
CHA P. X.
The QUANTITY of the AIR'S RESISTANCE to PRO-
JECTILES, and BULLETS in particular, deter-
mined by EXPERIMENTS on the Balliftic PENDU-
LUM.
IVE
3067. E have demonftrated the Principles on which the
Computation of the Velocity and Refiftance of
Bullets depends; and have confined the Theory by Experiments
in reſpect of the former: We now proceed to do the fame
Thing for the latter, that is, we ſhall fhew how the Quantity of
the Air's Refiftance to Bullets is to be computed from the The-
ory joined with the Experiments made by Mr. ROBINS for that
Purpoſe.
3068. We have already fhewn, that when two Bodies im-
pinge on, or ſtrike each other, the Magnitude of the Stroke is
proportional to the Lofs of Motion in the percutient Body (1008).
Now in the Cafe of a Shot made with a Gun, Bow, &c. the
two impinging or percutient Bodies are the Bullet and the Air;
which, indeed, are very different in their Natures; the leaden
Ball being a continuous and unelaftic Subftance; but the Air a
difcontinued elaftic Body. The former in Motion, the latter
at Reft. Yet this, notwithstanding their Actions on each other,
may be eaſily eftimated both by Theory and Experiment as fol-
lows.
3069.
336
INSTITUTIONS
!
3069. Mr. ROBINS charged the Mufket-barrel of 45 Inches
with the Bullet and Powder as uſual, and fired it againſt the
Pendulum at the Diſtance of 25, 75, and 125 Feet from the
Muzzle of the Piece at three feveral Times refpectively; and he
found that it impinged against the Pendulum in the firft Cafe,
with a Velocity of 1670 Feet per 1"; in the fecond Cafe, with
a Velocity of 1550 Feet per 1"; and in the third Cafe, with a
Velocity of 1425 Feet per 1". Therefore in ftriking againſt
50 Feet of Air, it loft a Velocity of about 120 Feet per 1".
Now fince the whole Motion of the Bullet in the firft Cafe was
1670 X 1/2 (by 970, 3003) and in the fecond Cafe it was 1550
therefore the Difference 120 × 1/2
I
120
ΤΣ
X 12;
= 10 lb. will
be the Lofs of Motion in the Bullet which it fuftained in paffing
through the 50 Feet of Air; but this Lofs of Motion was the
Effect of the equal Reaction or Refiftance of the Air; confe-
quently the Refiftance of the Air to a Bullet moving with the
mean Velocity of 1610 ( = 1670 +1550) Feet per 1" is a-
bout 120 Times its Weight.
2
3070. To find the Time which was ſpent in paffing through
this 50 Feet of Air; fay, as the mean Velocity 1610 Feet: 1
60" 50 Feet: 1,87"". And therefore to find the Refiftance
::
of the Air to the Bullet paffing through the fame Space with any
given Velocity, and Time will be eafy as follows. In a fecond
Experiment made with all poffible Care, the Mean of three Dif-
charges againſt the Pendulum placed at 25 Feet Diſtance was
the Velocity of 1690 Feet per 1", and of 5 Shot against the
Pendulum at the Diſtance of 175 Feet, the Mean was a Velo-
city of 1300 Feet per 1". The Velocity loft in this Cafe in
paffing through 150 Feet, was that of 390 Feet per 1", or 130
Feet per 1" for 50 Feet of Air.
3071. Now the Mean Velocity
1690 + 1300
24
)
in this Cafe was
1490 Feet per 1". And fince the Refi-
ſtance is at leaſt as the Square of the Velocity (2068): There-
fore fay, as 1490² : 1610² :: 130 Feet: 152 Feet, nearly; fo
that the Lofs of Motion would be 152 Feet per 1", with the
Velocity 1610 Feet per 1". But the Time ſpent in paffing this
50 Feet of Air is about 2""; and fo this Lofs of 152. Feet
in
Of GUNNERY.
337
in 2″ will be but 142 in 1,87″, the Time of paffing through 50
Feet with the Velocity of 1610 per 1". Confequently the Lofs
of Motion or (its Cauſe) the Refiftance of the Air in this Cafe
was 142 X
ilb.
lb. = 11,83 lb. or almoſt 12 lb. viz. 142 Times
the Weight of the Bullet.
ΤΣ
3072. Let us next ſee what Reſiſtance a lefs Degree of Velo-
city met with from the following Experiment made by the fame
Gentleman with great Accuracy; he charged the fame Gun
with the fame Bullets, but with a lefs Quantity of Powder;
and the Mean of 5 Shot made againſt the Pendulum, at the Di-
ſtance of 25 Feet, was a Velocity of 1180 Feet per 1"; and
then of 5 others againſt the Pendulum removed to the Diſtance
of 250 Feet, the mean Velocity was that of 950 Feet in 1".
Whence the Ball in paffing through 225 Feet of Air loft a Ve-
locity of 230 Feet per 1", or 51 Feet in ſtriking againſt 50 Feet
of Air. Now the mean Velocity is that of 1065 Feet per 1";
whence the Time of defcribing that Space 225 Feet was about
14″; and that of defcribing 50 Feet, about 3,1"; conſe-
quently in 1,87" the Lofs of Motion will be about 31 Feet per
i"; wherefore 31 × = 26. 10 oz. nearly, which is the
Refiftance of the Air to this Velocity.
I
3073. Having thus determined by Experiment what Refi-
ſtance the Bullet meets from the Air with two different Degrees
of great Velocity; let us next fee what will refult from a Com-
putation made from the Theory eſtabliſhed by Sir I. Newton
for flow Motions, the Principles of which we have already
explained in Chap. V. and fhall now apply them. Since
the Weight of a Globe of Air, equal to the Bullet, is incon-
fiderable in Comparifon of the Weight of the Bullet itfelf,
we have (2061) A = B. And conſequently
S
J
16,2
= T,
(2066). Alſo becauſe it is D, we have 4 D = 1; and
ſo S =
D
d
; but Lead is 11,345 Times heavier than Water,
and Water is 860 Times heavier than Air; whence D:d::
9756,7: 1. Confequently S = 9756,7 Inches = 813 Feet;
VOL. II.
Xx
whence
338
INSTITUTIONS
S
whence
= 7" =T, the Time of the Fall through S.
16,2
Then the Velocity V = 25 = 229,5 Feet per 1".
I 2
T
=
3974. Now this is the greateft Velocity the Bullet can ac-
quire by falling in the Air (2061) and the Refiftance it then meets
with is equal to B A. Weight of the Bullet;
if we would have the Refiftance, therefore, to any other Ve-
locity, as that of 1600 Feet per I', we muft fay, as 229,52
: 1½ :
16002 lb. 4 lb. nearly. But we have before found by
undeniable Experiment, that a Velocity of about 1500 Feet per
1" (3072) meets with a Refiftance of 12 lb. nearly; therefore the
Refiſtance to ſwift Motions is greater than that to flow Motions
(which is as the Squares of the Velocity) in the Ratio of 12 to
4, or 3 to I; according to what was delivered in the The-
ory (2072).
3075. If we enquire by the Theory what Refiftance a Velo-
city of 950 Feet per 1" will meet with, we fhall find it to be
about 1,427 lb. but by Experiment it was found to be in Reality
2 lb. 10 oz. which is nearly as 1 to 2. We fhall fee that this
Refiftance is very fenfibly encreaſed, even in fo finall a Velocity
as that of 400 Feet per 1". For by Experiment, a Bullet dif-
charged with that Velocity ranged but 319 Yards, or 957 Feet,
on the Surface of Water. Whereas by the Theory for flow Mo-
tions, it ſhould have ranged to the Diſtance of 1109 Feet, as I
fhall now demonftrate.
·
3076. The Reſiſtance to a Velocity of 400 Feet by the Theo-
ry is 0,253 lb. (found as above); then fay, as 10,083 lb.
: 0,253 lb. 16,2 Feet: 49,4 Feet, the Space a Body would
defcend through in one Second if urged by a conftant uniform
Force equal to the Refiftance 0,253 lb. (999). Whence 49,4
X 2 = 98,8 Feet is the uniform Velocity acquired in 1". And
fince the Times are as the Velocities (1000) when the Force is
given; therefore fay, as 98,8 Feet: 400 Feet: 1": 4″,049
the Time in which the Velocity of 400 Feet per 1" will be
generated by the Force 0,253 lb.
3077.
Of GUNNERY.
339
3077. Now let ICF be an Hyper- 1)
bola, whofe Afymptotes are A E, AD.
Let A B be the Time juft now found
=4",049; and BC the given Velo-
city of 400 Feet per 1", generated in
that Time by the Force 0,253 lb. Alfo
let BE 4", draw CG parallel to A
A E, and EG to BC, then will the
I
H
C
G
B
Rectangle BCGE be as the Space defcribed by the conftant
uniform Velocity BC in the Time BE; and the hyperbolic
Space BCFE will be as the Space defcribed with the fame Ve-
locity BC decreaſing by the Refiſtance (2096).
3078. Now AB × BC = 1 = Power of the Hyperbola,
(779) whence BC = ; therefore the Rectangle BCGE
I
A B
= (BCX BE =) BE X = ; and the Area BCFE
I
BE
AB
AB
is the hyperbolic Logarithm of the Ratio
AE
AB'
which is equal to
the tabular Logarithm of the fame Ratio multiplied by 2,302585
BE
(854).
But
AB
of
A E
AB (3)
А В
4
4,049
=1, nearly; and the Logarithm
=0,301030; therefore 0,301030 x 2,302585
×
X
= 0,6931, &c. Therefore fay, as I : 0,6931 :: (BC × BE
=) 1600 Feet: 1108,96 1109 Feet nearly, as before affer-
ted (3075). The Refiftance therefore by this Theory is
much leſs than what it really is, in as much as it gives a Velo-
city of 152 Feet in 1109 more than the Truth, in this ſo flow a
Motion.
:
3079. Where this Theory can take Place, we have FE-
Velocity at the End of the Time BE; and is eafily had thus, as
AE (2) AB(1): BC (= 400 Feet): FE( 200
Feet). Alfo, by drawing A F, it will cut BC in H; and BH
will be as the Refiftance at the End of the Time BE, in the
Hypothefis of the Refiſtances being fimply as the Squares of the
Velocity, which is very erroneous as we have fhewn in all the
X x 2
above
340
INSTITUTIONS
above Inftances, and therefore cannot be applied to any actua
Cafes of Gunnery.
3080. As in all the foregoing Cafes of fmall Shot,
the greateſt Velocity has not exceeded that of 1700
Feet per ; and as it has been fhewn that the Re-
fiftance to fuch a Velocity is three Times more
in Proportion than what it is in flow Motions (3074)
it will be ſtill neceflary to fhew what Proportion of this
increaſed Refiftance belongs to all other leffer Degrees
of Velocity in order to compleat the Theory, and
render it of general Ufe. For this Purpoſe let A B be
a Right-line divided into 100 equal Parts, and let
CDAD, be divided into 1700 equal Parts;
then ſince DB:AB:1:3:: Refiſtance to a Velo-
city of 1700 Feet per 1": Refiftance given to the
floweſt Motion; it is evident, that to a Velocity of
any given Number of Feet in the Line C D, there
will correſpond a Number in the Line A B, which in
refpect of the whole Line, will fhew what Ratio of
the encreaſed Refiftance belongs to that given Velo-
city; thus for Inſtance, to the Velocity of 1500 Feet
per 1" correſponds the Ratio of 4%, nearly; to
the Velocity of 1000 Feet per 1", the Refiftance is
nearly in the Ratio of or; and that of 500
Feet per 1" has the Ratio, nearly. And
fo for any other Velocity propoſed.
60
100
3081. Hence then if the Refiftance for any propo-
fed Degree of Velocity be calculated by the Theory,
the Error of the Theory in fwift Motions may be
hereby corrected, and brought pretty near the
Truth. Thus, fuppoſe I find by the Theory that
to a Velocity of 1000 Feet per 1", there is a Refi-
ftance of 1,5 lb. to a Ball of lb. wt. if this be en-
I
12
B
70
310
700
D
وا
018
06
い​た
​2007
creaſed in the Ratio of 3, by faying, as 35: 1,5 lb.: 2,5 lb.
this 4th Number 2,5 = 2 lb. 8 oz. will be nearly the fame as
was found by Experiment (3072) to be the true Quantity of Re-
fiftance. So that by this Contrivance the Theory for flow Mo-
tion may fill be applied to very good Purpoſe.
3082.
Of GUNNERY.
341
2
3c82. We have fhewn that when a Cannon Ball of 24 lb. is
impelled by a full Charge of Powder, it acquires a Velocity of
1900 Feet per 1″; but if it were impelled with only a common
Charge (viz. 12 lb. of Powder = the Weight of the Ball) its
Velocity will then be about 1700 Feet per 1". With this Ve-
locity we have ſeen a Bullet of 3 of Inch Diameter will meet
with a Reſiſtance of 10 lb. (3096). And fince the Reſiſtance is
(cæteris paribus) in the duplicate Ratio of the Diameter, or as
the Surfaces; and the Square of 5,325 Diameter of the Can-
+
non Ball, is equal to about 50 Times the Square of 3, the Dia-
meter of the Bullet; therefore the faid Cannon Ball will meet
with about 50 Times the Refiftance of the Bullet, or 500 lb. wt.
or about 20 Times its own Weight.
I
3083. That the Refiftance is in the duplicate Ratio of the
Diameter in Globes or Balls of different Size, is deducible from
(2033) where it was fhewn to be asap, which is half the Area
of a great Circle (830) and of the Superficies of the Sphere.
(839); therefore the Refiftance is in the Ratio of the Superfi-
cies of Globes, or in the duplicate Ratio of their Diameters
(842).
CHA P. XI.
The THEORY of RESISTANCE, VELOCITIES,
TIMES and SPACES defcribed by PROJEC-
TILES in their perpendicular Ascent and De-
SCENT in refifting Mediums.
3084. BEFORE
EFORE we can proceed further, it will be neceffary
to raiſe Theorems for afcertaining the Spaces, Times,
and Velocities in the perpendicular Aſcent and Deſcent of Projectiles,
in a Medium refifting in any multiplied Ratio, of the Velocity.
In order to this, let s Space, t
Time, v = Velocity,
r = Refiſtance. Then (by 971) we have vts; and fince
this holds in every Cafe of t and s, therefore v i = i,
3085.
342
INSTITUTIONS
3085. The Refiftance being the Reaction of the Medium,
will be proportional to the Decrement of Motion it caufes in the
moving Body; that is, r =—=— — i m (970); and when
mis given, it is r = , or the Refiftance is as the Decre-
ment of Velocity. The Refiftance (r) will alſo be inverſely as
the Time (t) in which it produces a given Effect (—), there-
I
forer ==; wherefore in general r =
Hence i =
J
ข
orri-
(3084) therefore r ; —— vi.
=
ல்.
3086. If the Body afcends, it will be retarded by two Forces,
the Refiftance (r) of the Medium, and the centripetal Force,
or Gravity (g); and this Retardation will produce a Decrement
of Velocity (—) which will be as the Moment of Time (i),
and the Sum of the retarding Forces r+g, (as is evident from
the Nature of the Thing); therefore in general ri÷gi =— i.
And for a deſcending Body, if the Reſiſtance be leſs than Gra-
vity, it will be gi-riv. But if Gravity be leſs than the
Refiftance, then rigi
v.
3087. We have fhewn that vis, whence v = there-
fore for the afcending Body, we have rigi x
,
t
xv=rs + gi. But for a defcending Body, if Gravity be
greater than the Refiftance, the Theorem will begi-r;=
v v ;r ; —g ;=vi, when the Reſtſtance exceeds Gravity.
3088. If the Body afcends in an unrefifting Medium (or in
Vacuo) then r = 0;
o; and gi
and gi= vė; but
if the Body defcends, it will be gi = v, and g¿ = vv. When
the Refiftance becomes equal to Gravity, then rg, and gi—
ri=&=0; that is, the Velocity will then become conftant or
uniform. And fince till then the Velocity is continually in-
creafing (for there will be gr, in any given Time,) it
is evident, the Velocity is in that Cafe a Maximum, as we have
before fhewn (2063).
v
3089. Suppoſe the Refiftance (r) be as any multiplied Ratio
V"
of the Velocity
11
>
- I
that is, let r =
a
a"
I
where (a)
is
Of GUNNERY.
343
is a given Quantity.
a"
72
cauſe i =
น
v'
71
за
$
Then for a defcending Body it will be gi
= v(644) and fo;=
we have; -
an
I
Q"
Alfo, be-
gan
1 ல்
And for the afcend-
2013
ing Body, ;=
an
and i =
t
& a"
+ &n
& Gr
3090. If we fuppofe the Reſiſtance to be as the Square of the
7
Velocity, then » = 2, and r = (3089) let V = greateſt
a
Velocity acquired in the Defcent, and becauſe in that Cafe r = g
V2
(3087) and v becomes V, we have g=—, and ſo ag = V³.
a
Letz Space which a Body falls through in Vacuo, to ac-
quire the greateſt Velocity V; then g = VV (3088), and
(taking the Fluents) gz V²; therefore 2 g z= V² = ag⋅
Whence a = 22.
=
2
i
3091. Therefore in the defcending Body, it will be s
a v i
บบ
2 Z v i
V V — v v ; let V² — v² = x², and taking the
ยข
a g
Fluxions, we ſhall have v = xx; and fos =
2 z x
x
· 2 z * *
Xx
And taking the Fluents, s=Q — 2 z L x.
(Here Qis fome conftant Quantity, and L. x is the Fluent of
x
* by (849). But Q — 2 L. × × ≈ = Q — z L. x² = Q —
x
z L. V² — v = ‹
v². Now when so, and then the Cele-
rity with which the Body begins to defcend, we ſhall have Qin
that Cafe z L. V² — c². And confequently sz L.
V²
2
Va
3092. LetL.d= 1; and it will be s L. d = z L
and
V2
223
V₁
- L.d = Ld == L
2; and therefore dz =
V2
Z
V²
V²
whence
344
INSTITUTIONS
S
whence we get v² =
V² dx + c²
V2
And for the Time
dz
Z
7
+
R=
L
V -
V
V + c
L
V
C
(t) of the Deſcent, we have ; =
V
i
V
V + v V - V
Fractions to a common Denominator.) Therefore taking the
Z
2
Fluents, we have t =Q+ — L.V + v − Z L. V —v =
ет
Z V + v
; let to, then ve; and we fhall have
a i
22
(3089)=
ад
V²
22
(as will appear by reducing the two laft
Z
Confequently, we have t =
L.
V
V + v x V
V + c x V — v
3093. If the Body falls from a State of Reft, then the incep-
tive Velocity c = 0;
and the above Equations will become for
; for the Velocity at the End of the
V2
the Spacesz L
V²
Ꮩ
V₁ dz
V²
Fall v² =
2
and for the Time of the Fall t =
dz
V + v
I
L
If we put dễ = m, then v = V
I
772
V V
ข
3094. In the fame Manner, we find the Theorems for an
afcending Body are s≈ L
and i=
a i
V² +
V² + c²
༣
V² + c² - V dž
dz
2zi
V V + v v
(3089, 3090). Now this
a g + v v
Auxionary Equation is analogous to that which is found for the
Fluxion of an Arch of a Circle, by Means of the Radius and
Tangent of that Arch; and therefore the Fluent of the Time
may
Of GUNNERY.
345
may be found in the Meaſure of the Arch of a Circle, as fol-
lows.
3095. On the Center D, with
the Radius ADV, defcribe the
Quadrant ATE; let the initial
Velocity c be expounded by the gi-
ven Tangent AP, the refidual Ve-
locity v by a Part of that Tangent
AM; and draw Dm infinitely near
to DM; and then
M m
m;
A
M m
P
N
•
ma
T
E
draw D P ; and from the Point M let fall the Perpendicular M R
upon Dm; then are the Triangles DN n, and DMR fimi-
lar; and ſo DM: DN (or D A) :: MR: Nn.. Alſo the
Triangles m RM and M AD are fimilar; and fo DM: DA
mM: MR. The reſpective Terms of theſe two Analogies
being multiplied together, give D M²: DA2 :: Mm: Nn;
V V ¿
vv:
MA
i:
that is, VV + v v : V V : : & : N n
Fluxion of the Arch AN.
2
the
V V + v v
3096. If this laft Equation be multiplied in each Part by
2 Z
2Z X N n
it will become
V V
V V
2 z ż
V V + VV
; therefore
2 z X N n
V V
2 x X AN
V V
(3094); and taking the Fluents, it is tQ-
Letto, then will A MAP, and AN=
AT, and in that Cafe, therefore, Q
2.Z XAT
There-
V V
AN
22 × TN
TN
=
V V
. V V
2 Z XAT
fore it will be t =
becauſe 2 g.
V", by (3090).
3097. To apply thefe Theorems to the Motion of a Bullet
fhot from a Gun in a perpendicular Direction upwards, on Sup-
pofition that the Refiftance is always in the duplicate Ratio of the
Velocity. Let us take the Example of (3059) where the Bullet
is projected with a Velocity c AP 1700 Feet per 1". To
determine the Height or Space of that Projection, we have s =
VOL. II.
Y y
≈ L
1
346
INSTITUTIONS
z L
V² + c²
V2
=
(by 3094) becauſe in this Cafe v = o. Now
fincec 1700, V = 229,5 (3073); and z = 813 (ib.) we
fhall have s 1420 Feet, for the perpendicular Altitude of
the Projection in a refifting Medium. Now this is confiderably
greater than the real Altitude, becauſe we have fhewn (2072)
the Reſiſtance to the initial Velocities is three Times more than
what we have here fuppofed.
3098. But let us fee what Height the Bullet will afcend to in
Valuo, projected with the fame Velocity of 1700 Feet per i".
This is found by the Theorem
1700, and n
n
m²
4 n
16,11, and t=1".
=a (1154) where m =
Whence m22890000,
and 4 × = 64,44; and ſo a = 45675 Feet, which is about 32
Times higher than before; and very likely 40 Times higher
than it does really afcend in Air with the given Velocity.
3099. The Time in which this perpendicular Altitude in Va-
cus is deſcribed, is found by Theorem (1152)
st m
n
T, for
in this Cafe sr1, and t=1"; therefore the Time of
m
Afcent and Deſcent is
2n
1700
32,22
= 53″ nearly.
3100. Now the Time in which the Bullet afcends perpendi-
cularly to the Height of 1420 Feet in the Air is equal to
ΤΑ
g
(by 3096), becauſe in this Caſe TN becomes TA (ſee Fig.
to 3095). To find the Arch AT we have the Radius A D =
V = 229,5, and AP c = 1700; wherefore fay,
As AD
Is to A P
So is Radius
2.360783
3.230447
229,5
1700
90°00′
- 10.000000
To the Tangent of the Angle PDA — 82° 18′ —— 10.869664
3101. The Diameter is 2 AD = 459; therefore fay, As
1: 3,14159:: 459: 1442, the Circumference of the Circle.
Then fay, As the Circumference 360° is to the Arch 82° 18′
fo is 1442 to the Length of the Arch A T
:
329,8.
Now
Gra-
Of GUNNERY.
347
Gravity produces an uniform Velocity g = 32,22 Feet per 1″.
Whence
329,8
32,2.2
に
​AT
=10″ nearly, the Time of the
g
Afcent in the Medium of Air, refifting according to the dupli-
cate Ratio of the Velocity. The Time therefore in this Cafe is
not more than of that which is fpent in the Afcent in Vacuo
(3099).
5
3102. The Time of the Defcent through the Air from the
Z
faid Altitude of 1420
Feet, is L
V
V + v
V. - V
= (by 3093).
E
But in order to determine this, we muft firft find the Velocity
772
(v) acquired in the Defcent from the Theorem v = V I
(in Art. 3093) where V = 229,5; d= 10, (becauſe its Lo-
garithm=1
= 1 (by 3092), s=1420,5; 2=813;
=55,86. Whence we find the Velocity v
per 1". Therefore Vv = 56,94; and V
and L
V + v
V
V
Z
and m = d≈
227,44 Feet
v = 2,06;
=2,345992; alſo = 3,543; therefore r =
8,"31. Whence it appears that the Times of the Afcent and
Defcent are in the Ratio of 10 to 8,3; or nearly as 5 to 4, and
both together make but 18″,3 which is 35″ leſs than the Time
in Vacuo (3099).
3103. Hence it appears they are widely miſtaken who affert,
that if an heavy Body be projected upwards with a Velocity
greater than that which can be acquired in falling, the Time
of the Defcent will be greater than that of the Afcent; fince it
is demonſtrated, that a longer Time is required to deſtroy a
great Velocity in afcending, than in generating a much ſmaller
one in defcending through the fame Space.
3104. From this Theory alfo it is manifeft, that if a Ball or
Shot be projected downward with a Velocity equal to the great-
eft that can be acquired in falling, the Motion will be uniform;
but if it be projected with a Velocity greater than that, the
Motion will be retarded; if with a lefs Velocity, it will be ac-
celerated, but never will become equal to that greatest Veloci-
ty, all which is evident from the Theorem in Art. 3092.
Y y 2
3105.
348
INSTITUTIONS
3105. Hence alſo we fee the Reafon why the Force acquired
by a Bullet in falling is ſo much less than that with which it is
projected; for fince the Mafs of Matter continues the fame, the
Forces will be as the Velocities, that is, in the foregoing Ex-
ample (Art. 3097, 3102). The Force of the Bullet, when pro-
jected, from the Muzzle of the Gun, is to the Force it acquires
in falling, as 1700 to 227,5; or nearly as 8 to 1.
3106. Indeed, with refpect to very light Bodies, the Cafe
may be a little different; and the Times of Defcent of an Arrow,
a Ball of Wood, &c. may be greater than that of the Afcent,
fince they are refifted by the Air in a much greater Proportion
to their Quantity of Matter, on Account of their larger Quan-
tity of Surface; what this may be, as nothing of Conſequence
depends upon it, I have not here calculated; but it may be
cafily done by any one who underftands the foregoing Theory,
and has Curiofity and Leifure for fuch Amufements.
3107. In all that has been hitherto faid of the perpendicular
Projection in a refifting Medium, we may obferve what an egre-
gious Difference there is between ſuch a Shot and one made in
Vacuo in regard of the Height, the Time, and the Velocity and
Force of the PROJECTILE; and how widely the latter is different
from the Truth.
CHA P. XII.
The common practical RULES of GUNNERY, de-
rived from the parabolical HYPOTHESIS, Com-
pared with EXPERIMENTS, and thereby fhewn
to be extremely fallacious, and of no Ufe in PRAC-
TICE.
3108.
Uſe
E next proceed to fhew how very fallacious all the
Rules and Cafes of practical GUNNERY are, as they
are derived from the Principles of the parabolical Hypotheſis, which
we have explained (in 1141, &c.) and fhall here apply, in or-
der to confute them by Experiments. After this we ſhall confi-
der the Nature of the Motion of a Body projected obliquely in a
re-
Of GUNNER Y.
349
1
T
refifting Medium, and the Form of the Path it defcribes, and
fhew it to be no PARABOLA (as is always fuppofed) nor any
other regular or geometrical Curve.
3109. It was fhewn (1155) that the greateſt Amplitude or
Random of a Shot (in Vacuo) was that which was made upon
an Elevati-
on of 45°.
And it ap-
pears, that
AD is equal
to half A L;
----,
H
E
E
K
1
*
and there-
F
fore fince
H
E
AD = I
4
E
B
A M, we
fhall have AM
=
M Ат
2AĻ the greatest Random poffible.
But we have fhewn (in our common Example (3098) that
AL 45675 Feet; and therefore 2 AL 91350 Feet=
AM, the Random on 45°. Now this is about 17,3 Miles;
but all experienced Gunners, and practical Writers affure us
that this Range is actually not quite a Mile; and fo Merfennus
found the horizontal Range of an Arquebuſe of 4 Feet to be leſs
than 800 Yards, which is not quite a Mile. The Range
therefore by this falfe Theory is about 35 Times greater than the
Truth.
I
2
3110. The Cafe is the fame with respect to large Shot as in
fmall ones; for St. Remy (as quoted by Mr. Robins) tells of fome
Experiments made by Mr. Du Metz, in which the Range at
45°, of a Piece 10 Feet in Length, carrying a Ball of 24/6, and
charged with 16lb. of Powder, was 2250 French Fathom, which
is not quite 3 Miles. But fuch a Ball was projected with a Ve-
locity of about 1900 Feet: per 1", (Art. 3043.) and therefore
772
2
per Theorem (1154.) =AL=56021 Feet; and fo 2 AL
4 n
=AM
= 112042 Feet — 21 Miles, which is more than
Times the real Random, in the refifting Medium of Air.
3111. The fame Difference between this Hypothefis, and
the Truth, will be found in Proportion to obtain in fmaller De-
grees of Velocity, and on any other Elevations, as is evident fron
Mr.
1
211
་
350
INSTITUTIONS
Mr. Robins's Experiments compared with this erroneous Theory.
For a leaden Bullet, of an Inch Diameter, diſcharged with a
Velocity of 400 Feet per 1", and in an Angle of 19°: 05′, of
Elevation ranged on the horizontal Plane no more than 448
Yards, whereas by the Theory it ought to have ranged 1023
Yards.
3112. For in this Cafe AL (
(m²
4 n
=2483 Feet; and 2 A L
=AM=4966 Feet 1655 Yards
the greateſt horizon-
tal Random. But all Randoms, in Vacuo, are as the Sines of
double the Angles of Elevation; as is thus fhewn. Draw CH,*
then is the Angle of Elevation HAM ALH (665) = ½
x
2
HCA (642) therefore F H is the Sine of double the Angle of Ele-
vation; but FH ADAM, the Random. Confe-
quently the Randoms are as the Sines of double the Angles of
Elevation. Therefore fay,
As the Sine of double 45°
90°
T
To the Sine of double 19° : 5′ = 38° 10′
So is the greateſt Random 1655 Yards
=
10.
9.790954
3.218798
3.009752
31
To the Random on 19° : 5′ 1023 Yards
So that the Range made in the refifting Air is but about of
what it would be in the parabolic Hypothefis."
3113. Again, a Ball was diſcharged with the fame Velocity
as in the laſt Experiment, but on an Elevation of 9° : 45′ ; and
its Range on the Horizon was at a Medium 990 Feet, or 330
Yards. Now if this were to be deduced directly from the The-
orem (in 1152)
c s mn²
2
12
=AM, we ſhould find A M = 1655
Feet, or 552 Yards nearly, as in the following Operation.
Sine of 9° 45′
Add the Logarithms of Cofine, 80° 15'
9.228784
9 993681
And of m² = 160000
5.204120
From that Sum
4.426585
Subduct the Value of n = 16,11
1.207095
The Remainder is the Range A M = 1665,6
3.219490
3114.
The Reader is defired to draw this Line in the preceding Figure,
as it is there (by Accident) omitted.
Of GUNNERY.
351
3114. But if this Random were to be computed from the
Random on an Elevation of 19° 30′ found in (Art. 3112.) it
would come out a very different Number from either of the
foregoing; for fay
As the Sine of double 19° : 5′ = :
38° : 10′ — 9.790954
}
Is to the Sine of double 9° : 45′ = 19:30
So is the Random on 19°: 5′
To the Random required
****
448
212
1 1
9.523495
2.651278
2.383819
proves
Now this is as much too little as the other was to big, and
the Theory falfe on every Account.
3115. Again, for greater Variety, a Ball was fired at an Ele-
vation of 8°, and with a Velocity of 700 Feet in 1", and the
horizontal Range at a Medium was 690 Yards; but if this
Range be computed from the common Theory, we ſhall have
AM 15208 Feet, and then by (Art. 1154.)
2 AL
772
น
4 77
=
we ſhall find the Random at an Elevation of 8° to be 4192
Feet, or 1400 Yards nearly, which is more than double the
real Diſtance.
3116. Laftly, a Ball being fired at an Elevation of 4° with
the fame Velocity as in the laſt Experiment ranged 600 Yards
on the Horizon. Now this Range if deduced from the laſt Ex-
periment, and the Theorem (in Art. 1154.) fhould not have
been more than 350 Yards. From all, which Inſtances of the
Inconſiſtency of this vulgar Theory, with Facts and Experi-
ments, we obſerve the Falfity of the Hypothefis on which it is
founded, and may justly wonder to fee every Day Books pub-
lifhed relating to GUNNERY, and the Doctrine of Projectiles,
with as ftrong a Prefumption of theſe falſe Principles, as though
Philofophy were not in the leaft understood, and the Nature of
thefe Things had never been enquired into by any one.
CHAP.
→
352
INSTITUTIONS
i
CHA P. XIII.
Sir ISAAC NEWTON's Method of investigating the
PATH of a PROJECTILE in a refifting Medium
illuftrated, applied to practical GUNNERY, and
exemplified by EXPERIMENTS.
3117. B
UT now that the ENGINEER may be affured from a
genuine Phyfico-mathematical THEORY, that the
Path which a Projectile defcribes, when thrown from the Gun
in any oblique Direction, even when the Refiſtance of the Me-
dium is no more than proportional to the Square of the Velocity,
cannot poffibly be a PARABOLA, nor any regular geometrical
Curve, he is to confider in the firft Place, that fince the Body
is prevented going on in the Right-lined Direction A B by the
perpendicular Action of
Gravity by which it is
brought down from any
Point C to the Point D in
the Curve of the Parabola
ADF in Vacuo, ſo if we
fuppofe the Projection to
be made in a refifting Me-
dium, the Refiftance will
•
D
IH
E
F
leffen the Velocity of the
Ball, and prolong the A
Time of its arriving to the
Diſtance of D, and therefore as Gravity acts all the Time uni-
formly, it will producé in the Projectile a greater perpendicular
Deſcent from the Point C, and carry it from C to fome Point
E below D; and fince this is the Cafe every where, it follows.
that the Curve A E G defcribed by the Projectile, cannot be the
Parabola ADF, but fome other Curve contained within it.
3118. Again, fince the Velocity of the Projectile in the de-
ſcending Part of the Curve HG will be always much less than
that in the afcending Part A H, as we have above demonftrated
(3102).
Of GUNNERY.
353
养
​(3102) therefore Gravity will in this Part carry the Ball through
much greater perpendicular Spaces (in the fame Time) from
the parabolic Curve; and fo the Curve in the Part HG will
have a lefs Degree of Curvature, and therefore the Point G,
where it falls on the Horizon, will be much nearer to the Line
HI of the greateſt perpendicular Altitude, than the Point A
from whence it was projected, different from what happens in
the Parabola.
V
а
X
H
I
B
T
MA
E
D
k K
KN
C
h
*
3119. And Sir ISAAC NEWTON has fhewn, that the Curve
AHG approaches nearer to the Form of an Hyperbola than that
of a Parabola; for an Hyperbola will be truly defcribed in a refift-
ing Medium, whofe Denfity is every where inverfely as the
Tangent to the faid Curve, as he has proved in Prop. X. Lib. II.
of the Principia. And therefore if AGK be the hyperbolic
Trajectory of the Ball, projected from the Point A in the,
Direction AH, (in a Medium of a variable Denfity) whofe
Afymptotes are MX, XN, cutting the horizontal Line M N
in the Points M and N; and of which the latter XN is
perpendicular thereto; then fince in this Cafe the Denfity
in any Points A and G is inverſely as the Tangents A H and
VOL. II.
ZZ
GT,
354
INSTITUTIONS
1
GT, it is manifeft if the Medium be now fuppofed of an
uniform Denfity, fuch as is equal to the mean Denfity of the
Medium in Queſtion, then in ſuch an one, on Account of
the greater Denſity at A, the Velocity of the Projectile will
be more diminiſhed than in the other Cafe, and confequent-
ly the Trajectory will be continued within the Hyperbola
AGK (by 3117) in the aſcending Part AG. But in the de-
fcending Part G K it will approach nearer to the Afymptote X N
than the Hyperbola, (which can never meet it, by 776); con-
fequently in the uniform Medium, the horizontal Motion of the
Projectile (fetting afide Gravity) will be infinite; and therefore
after a determinate Time, the Projectile will arrive at the ſaid
Line XN (3097).
3120. But notwithſtanding the real Trajectory is not truly
an Hyperbola, yet it approaches fo nearly thereto in the fmall
Part of it above the Horizon, that in practical Affairs the Conic-
Hyperbola may be uſed without any fenfible Error, for eſtimating
the Motion of a Projectile in an uniform refifting Medium. Now
fuppofing AGK fuch an Hyperbola, and drawing A I and
HC parallel to NX and MX, then Sir Ifaac Newton has proved
(in the above cited Propofition) that the Velocity of the Projec-
tile will be as
I
ΑΙ
I
or V² = that is, the Velocity at
ΑΡ
A is to the Velocity at any other Point G, as
√
ΑΙ
to
I Alſo, that the Refiftance of the Medium is to the
GV
3
KN; and AI
Force of Gravity (in any Point A) as A H to A I. More-
over, from, the Nature of the Curve, AM
= AC; and by Construction IC
(2 AI =) HX.
3121. Thefe Things premifed, it is evident that if the Lines
A1 and AH be given in Magnitude and Pofition, the Hyper-
bola A G K may be defcribed, becauſe in this Cafe we have gi-
ven H X --- 2 A I, and therefore the Center of the Hyperbo-
la X. Hence alfo the Afymptote XN, and fince the Point I is
given, we have also the other Afymptote X M cutting the hori-
zontal Line in M. Hence, laftly, we have given the Point K, by
taking
Of GUNNERY.
355
taking KNAM (776 to 780). It remains now to fhew
how the Lines AI and AH may be determined in the prefent
Caſe of the Problem by Experiments, and thence an expeditious
Method of defcribing the Hyperbola A G K, or Trajectory of
the Projectile; and thence a Solution of all the Cafes of the
Problem.
3122. In order to this, let two equal Balls be diſcharged
with the fame Velocity on two different Angles of Elevation
HAK, and h Ak; and let the Points K and k where they
fall on the Horizon be obferved, and the Proportion of their
Diſtances A K and Ak be found by Meaſuration. And let A K
: Ak::d: e. Then having erected the Perpendicular AI of
any affumed Length, take A H or Ah alfo of any Length what-
foever, and then by Scale and Compaffes find the Lengths A K,
and A k, as directed in the laſt Article; and if they are found to
be in the fame Ratio as d toe (by Experiment) then was the
Length AH rightly affumed, and the Hyperbola AGK every
Way fimilar to that deſcribed by the Ball in the Air.
3123. But if not, take in the indefinite Right-line S M, the
Length SM equal to the affumed Length AH; and erect the
perpendicular M N equal to the Difference of the Ratios of A K
to Ak, and of d to e; that is, let MN=
AK
A k
d
e
And in
like Manner, affuming feveral Lengths of A H, find ſeveral o-
ther Points N, thro' all which draw a regular Curve N N X N,
cutting the Line S M M M in X. Laſtly affume AHSX,
and thence again find
the Lengths A K and
Ak; thefe fhall have
the fame Ratio with
d and e, or the fame
Lengths found by
Experiment; and AI,
and this laft found
S
N
N
X M
M M
N
AH, fhall be fimilar
to, or have the fame Proportion with thoſe which belong to the
Hyperbola deſcribed by the Ball in the Air. For fince
AK
Ak
d
Zz2
356
INSTITUTIONS
d
MN, where SM SX there MN = 0, and confe-
quently there
AK
Ak
d
and fo AK: Ak:: d: e.
e
3124. To illuftrate this by an Example; let it be required to
find AI and AH for the two Shots made upon the two Eleva-
tions of 19° 5′ and 9° 45′ (in Art. 3112, 3113) where the ho-
rizontal Ranges were 448 Yards d, and 330e. There-
d
=
fore = 448 1,36. Then affuming AI
45
e
10, and A H
70, you find A K = 45, and Ak= 37,8, and fo
=
37,8 1,19 (by 3122). And then
AK
A k
AK d
A k
1,36 — 1,19
e
=0,17 MN. In like Manner, if you affume other Lengths
AH 60, 50, 45, 40, refpectively you will find other Values
of MN = 1,23. 1,28. 1,39. 1,50. Then drawing a Curve
through the extreme Points of all the Ordinates MN, and it
will interfect the Line S M in X, fo as to give the Abfciffa S X
=46 AH fought; and in this Cafe A K will be to Ak as
2/20
448 to 330; or meaſured upon the Scale by which AI and A H
were laid down, they will be A K = 25,7 and A k = 18,8.
I
3125. Having thus obtained the Proportion or Values of the
Lines AI and AH in this one Cafe, they remain unalterably
the fame for all other Angles of Elevation, for Difcharges of the
fame Velocity. And indeed in all Caſes whatſoever, the Value
of A H will remain as being reciprocally as the Denfity of the
Medium, which is ſuppoſed to be uniform; but if the Velocity
of the Projection be varied, then will alſo the Value of AI. For
fince it is V =
will be always in the inverfe duplicate Ratio of the Velocity
greater or lefs. And therefore in any Cafe of a given Velocity
you have the Lines AI and AH, and ſo the Trajectory may
deſcribed for any Angle of Elevation HAN, according to this
Theory.
I
ΑΙ
I
(3120) it will be AI = √₂, or AI
be
3126.
Of GUNNER Y.
357
4
3126. Hence likewife the Ratio of the Refiftance (R) in the
Point A to Gravity (G) is given. For R: G:: AH: AI
(3120): +65 = 1,333 (3124). Therefore if the Weight
4,65: + 3
I
IZ
of the Bullet were. then : 4,65 :: 1½ :
1395
48
0,288 lb.
the fame nearly as found by the common Method to the fame
Velocity (in 3076).
=
b, and
3127. Having now found the Values of A I and A H for any
propoſed Velocity of Projection, we may proceed to find the
Random A K generally for any given Velocity and Angle of
Elevation HAN; in order to this let A Ha, A I = b,
then H X = 2 AI 2 b. Alfo let AK, AN z, and
NHy; and we fhall have z x = KNAMA E,
and AC AI b. Alfo EN AKx. Then by the
fimilar Triangles EAC, EN H, we have this Analogy A E
(≈ — x): EN (x) :: AC (b): HN (y). And thence com-
=
pounding, zxb+yy; whence x =
b x
x + xy
y
x x
; and z z = × × ×
ZY
b + y
and z =
b + y
y y
But becauſe of the right
2
b + y
And fo
yy
a² y²
2
Angle ANH, it is aayyzz=xxx
a² yz
b + y
2
14
xx; and therefore x AK =
b + y
2
3128. Hence alfo if the Amplitude A K be given, then may
the Sine HN of the Elevation neceffary for ftriking any Object
K at the given Diſtance AK be found by the fame Theorem.
-
2+2 X 32
For we ſhall thereby get b²x² a² X 12 — 14 — 2 b x² y;
from which Equation by a few eafy Trials, the Value of y =
NH may be found. Or in a Method eaſy for Practice thus.
On the Center A with the Radius AH defcribe a Circle, as
(ab); and on the given Point K erect the Perpendicular K F;
then apply a Ruler to the Point C in ſuch Manner that the Part
FH intercepted between KF, and the Circle ab may be equal
to CE. Then through the Point H, draw the Right-line A H,
and it fhall be the Direction or Elevation required. For fup-
pofing the Angle given, we have always CE FH, becaufe
(by
358
INSTITUTIONS
(by Construction 3120) the Angles ACE and HEN are fini-
Jar, and therefore CE: AE:: EH:EN:: FH: KN;
hence fince AE (= MA) KN, we have CEFH.
And the Point H muit be in the Chicle & b, becauſe of the given
Length A H.
3129. By putting the above Equation (3127) into Fluxions,
and making the Fluxin of x AK equal to nothing, we fhall
get an Equation for the Sine NH of the Angle or Elevation
which of all others will produce the greateſt honzontal Random
AK; for then we fhall get 2 a² ÿ ÿ- 4 y³ j = 2 b x² j +
y; and dividing by 2y, we have ay
2 x¹yj;
x² 9 = x² × b + y.
y
3
3
2 y³ = bx² +
a² x²
2 2
34
But xx=
; wherefore a²y——
b + y
az y z
34
a² y²
34
Whence by Re-
2 y³ = b + y ×
b + y
b + y
I
2
2
duction, and by dividing by y, we get a ab 2 byy + y³.
3130. Here it is plain the Quantity 2 byy is lefs than a ab,
and therefore y y less than aa, or HN lefs than A H²; and
confequently the Angle HAN is leſs than half a Right-angle,
which it would be in an unrefifting Medium, (fee 1155) and in
that Cafe, we have yyaa, and ya, which Valuc
of y, if it be fubftituted in the above Equation, in order to re-
duce it, it becomes a ab 2 byy yya ; whence yy =
+yya;
a a b
26 + a √
a a b
26 + a √ 1
and fo y
3131. Let us now fee what the Quantity of this Angle is for
the Values of AI and A H as determined above (3124) where,
if we put A I= b = 1, then A Ha= 4,65, and the Equa-
tionisaa=2yy+y³ = 21,6225. Nowy
21,6225
5,255
=
=
21,6225
2 + 4,65 √
2,03. But it is plain this Value of y is too
fmall, for it makes 2 y y + y³ little more than 16, whereas it
must be 21,6255; this was occafioned by making ya
=3,25 according to the cuftomary Method, which makes one
of the Roots too large, and confequently the whole Quantity
2 yx + y³ too ſmall. The beſt Way, then, to approximate to
the Value of y is (Tentando) by Trial, for by this Means we
fhall eafily find that y 2,26 nearly. Therefore fay, as AH
= 4,65: HN = 2,26:: Radius: Sine of 29° nearly; where-
as in Vacuo we have fhewn this Angle is precifely 45 Degrees
(1155).
=
3132.
Of GUNNER Y.
359
3132. If it be required to find what this Maximum of Am-
plitude is for the above found Values of AI and AH, you have
it determined by the Theorem in (3127) (and alſo for any other
Degree of Velocity, as fhewn in 3125). So according to the
prefent Cafe (3124) we fhall find the greateft Value of AK =
2
a² y²
b
y
2
Jut
2,83 nearly; and therefore fay, as A K
2,57 (by Experiment) : A K = 2,83 (a Maximum) :: 448
Yards: 490 Yards nearly, the greateft Amplitude.
3133. If the greateſt Ordinate or Height of the Projection be
required, let the Line DN be taken in the horizontal Line A N,
a Mean proportional between AM and AN, and through the
Point D, draw the Ordinate G D, and it will be the Maximum
required. This Ordinate GD is equal to the Difference be-
tween the vertical Line N X, and a 4th Proportional to the
Lines DN, AN, and 2 AI = IC; or GD = NX-
2 AI × AN.
DN
The Demonſtration hereof may be eaſily dedu-
ced from the abovementioned Propofition of Sir Ifaac's Principia
(3119). But it is too prolix and intricate to be here inferted.
3134. To exemplify this in the laft Cafe of the greateſt Am-
plitude AK; we have AK = x 2,83, and y=HN=
b x + x y
= 4,1
=
2,26 (3131). Alfo we have z AN=
=
y
nearly (3127); then ANAK KN = 1,27 = A M¸
Therefore DN = √✅AN × AM = 2,28. Now NX=
NH + HX = 2,26 + 2 = 4,26; and
2 X 4,I
2,28
2 AI × AN
X
DN
= 3,6. Therefore GD = 4,26 — 3,6 = 0,66;
=
which gives about 110 Yards for the greateſt Height of the Pro-
jection. It is obfervable alfo, that DK (DN
KN =)
1,01 and AD (AK—DK) 1,82; and fo the Point D
is nearly of the whole Random from the Point A ; whereas in
Vacuo it is always the Middle Point between A and K.
3135. From a bare Inſpection of the Trajectory ABGK,
it is evident, the Path of the Projectile will nearly coincide with
the Tangent AH for a confiderable Diftance A B, which in
a Shot of 24 lb. will be about 500 Yards (as the Angle there
will not exceed a Degree;) and hence we fee the Reafon
why Mr. Anderſon, the moft eminent of all practical Au-
thors on Gunnery, found himfelf obliged to fuppofe that
the Track of Shells and Cannon Balls was much lefs incur
vated,
360
INSTITUTIONS
vated, than what it ought to be on the parabolical Hypothefis
and in order to reconcile this Circumftance with the faid Theo-
ry, he imagined that every Shot was impelled to a certain Di-
ftance A B in a ftrait Line, or that in ſuch a Diſtance it was not
affected by the Power of Gravity. But fo ftrange and abfurd a
Suppofition, as the Sufpenfion of the Power of Gravity for a
Moment, plainly evinced how ignorant he was of the great Di-
munition of the Velocity of the Shot in its Flight from the Refi-
ftance of the Air. The Notion, therefore, of a Point-blank
Shot, as they call the Diſtance A B, is entirely groundleſs, and
owing to the vulgar Error of the Air's Refiftance being inconfi-
derable.
3136. We have now delivered all we can think neceſſary relat-
ing the Theory and Practice of Gunnery; and it is with fome
Regret we are obliged to conclude, that from a true and genuine
Theory, the practical Part of this moſt neceffary Science ap-
pears very perplext and difficult; it is for this Reafon we defift
from purſuing it any further. The celebrated Mr. J. Bernoulli
has given a moſt exquifite Theory of the Path of a Projectile in a
refifting Medium, but as the Practice reſulting thence depends
upon the Quadrature of mechanical Curves, it can by no Means
be rendered uſeful.
3137. The late learned Mr. Simpson has alfo given us an ad-
mirable Theory, but the practical Rules that might be from
thence deduced would prove io difficult, that few of our ENGI-
NEERS, I fear, will care to be at the Trouble either to under-
ftand, or reduce them to uſe. Mr. Muller has fupplied us with
converging Series for the various Cafes of Projectiles in a refifting
Medium; but then he has given no Demonſtration of them, and
they are upon Suppofition that the Refiftance is as the Square of
the Velocity only, which is far from the Truth, as we have fhewn;
befides infinite Series feem but little adapted to Caſes of practicable
Gunnery.
3138. In the laſt Place, Mr. ROBINS himſelf, in a poſthu-
mous Tract, has left us fome of the beſt Methods of reducing
the Genuine (though complicated) Theory of military Projectiles
to practice, but even theſe will be attended with Trouble and
Difficulty enough; and beſides, being publiſhed without any In-
veſtigation, or Rationale, which he propofed doing if he had
lived, they will be found as little fatisfactory as ready in Practice.
And I think we may conclude, Sir Ifaac Newton would not have
given us the above Method for deducing the practical Rules of
Gunnery if he had known of any other that was better.
INSTI-
( 361 )
INSTITUTIONES HOROLOGICÆ,
OR, A
PHYSICO-MATHEMATICAL THEORY
O F
CLOCK-WORK.
CHAP.
I.
Of the Nature and Defign of CLOCK-WORK in
general; and the PRINCIPLES on which it de-
pends.
3139. A THEORY OF CLOCK-WORK, I am inclined to think
will be looked upon as a Novelty by the English
Mechanic; as I have feen nothing of that Kind publiſhed in our
own Language. Indeed I have met with but one Treatife on
the Subject of Clock-work in English, viz. The artificial Clock-
maker; but this is altogether practical and refers in every Parti-
cular, almoft, for the Rationale, to a Treatife in Latin, entituled
Horologium Ofcillatorium, by Mr. HUGENS.
3140. But this Treatife of Mr. HUGENS is upon the Theory
of one Sort of Clock only, viz. that which moves with a Weight,
and is regulated by a ſingle Pendulum. This great Author being
the firſt that applied a PENDULUM to a Clock for this Purpoſe;
though Pendulums were long before in Ufe as CHRONOMETERS,
to meaſure Time by their equable Vibrations in aftronomical
Obfervations, and on many other Occafions. But more of this
VOL. II.
A a a
here-
362
INSTITUTIONS
hereafter, when we come to treat directly of Nature, Ufe, and
various Forms of the Pendulum.
3141. The Defign of Clock-work is twofold, viz. (1.) To
meaſure Time exactly, or to divide a given Portion of Time in-
to very ſmall equal Parts; for Inftance, to divide the Time of a
Day into Hours, Minutes, Seconds, &c. (2.) To produce Mo-
tions, or periodical Revolutions fimilar to any given ones, as
thofe of the heavenly Bodies, together with their Phaſes, Aſpects,
Pofitions, &c. at a determinate Time. But Clock-work of
this Sort is uſually called a Planetarium, Orrery, &c.
3142. A CLOCK is the principal Machine, or Capital of all
mechanical Contrivances, for measuring TIME. Its Principle.
of Motion is derived from a two-fold Power or Force, viz. that
of a Weight, or a Spring. For either of theſe Forces are fufficient
to actuate, or put into Motion, the Syftem of Wheels and Pinions
which compoſe the intermediate Parts or Body of the Machine;
the Indexes fixed on the Axles of the Wheel point out the pro-
per Divifions of the integral Portion of Time on appropriated
Circles upon the Face of the Clock, and a Pendulum or Balance
is added, to regulate the Motion communicated to the Ma-
chine, or render it uniform.
3143. The two phyfical Principles of all Automata, viz. the
WEIGHT and the SPRING are now to be confidered. With
Reſpect to the firft, as its Force is derived from the Power of
Gravity only, and this Power being always the fame in a given
Quantity of Matter (968) it follows that the Force of a given
Weight is a conftant Quantity, or always remains the fame in the
fame Medium, and therefore in fuch Cafe, this becomes abfo-
lutely a uniform Power or Principle of Motion; fuch as is ne-
ceffary in perfect Clock-work.
3144. The ELASTICITY of a well-tempered Steel Spring is a
fit Power or Principle of Motion in a Clock, for when it is bent
or coiled to any given Degree, the Intenfity of its Force conti-
nues the fame; but as it is coiled more or lefs about its Axis,
the Force becomes increaſed or diminiſhed; and fince the Action
of a Spring in Clock-work is by unbending itſelf by Means of
its renitent Force, the Force it exerts on the Wheels would
gradually decreaſe; and therefore if it was at firft fufficient to
keep the Clock in Motion, it could not long continue fo, but
the
Of CLOCK-WORK,
363
A
the Pendulum would at Length ceafe to move, and the Clock
ſtand ſtill.
3145. When a SPRING, therefore, is applied to Clock-work,
it becomes neceffary to contrive a Method by which its variable
Force may be rendered equable or uniformly the fame on the
Wheels of the Clock from its firſt or greateſt to its laſt or leaſt
Degree of intenfive Force. And this is done by giving a requi-
fite Form to the Barrel on the Axis of the firft Wheel, which it is
connected with by a proper Chain, or String.
3146. The Force, therefore, of the Weight or Spring is the
Primum Mobile, or firft Mover, in Clock-work; and it
may be
proper here to obſerve that the Force they communicate to the
Machine is continually diminiſhed by the Wheel-work, till at
Length upon the ferrated Teeth of the Crown-wheel it is but
juft fufficient to keep the Pendulum in Motion; that is, the
Force is there required to be equal to the Reſiſtance the Pendu-
Jum meets with from the Air and the Axis of its Motion, for
then its Motion will be continued in the Arch of Vibration pro-
pofed.
3147. But if the Force of the faid Crown-wheel on the Pal-
lets of the Verge of the Pendulum be fuperior to the Refiftance,
it will only cauſe the Pendulum to vibrate farther, or in a fome-
what larger Arch than the given one. But if the Force be lefs
than the Reſiſtance of the Pendulum, then the Arches of Vibra-
tion will decreaſe gradually to nothing, or the Pendulum and
Clock will ceaſe to move.
3148. The more perfectly the component Parts of the mov-
ing Syſtem are wrought and finiſhed, the leſs Force of a Weight
or Spring will be required to keep it going. But when the Work
is coarſe, and clogged with Duft, infpiffated Oil, &c. the
Force of the Weight or Spring will be fo much diminiſhed thro'
the Train of the Work, that unleſs it be in Proportion encreaſed
it will not keep the Clock in Motion. But whatever be the
Condition of the Clock, while it does move, it will meaſure
Time equally; for on Suppofition the Pendulum continues of
the fame Length, it makes all its Vibrations, in larger or ſmall-
er Arches, in the fame Time, as we have formerly fhewn
1122, 1126); and, therefore, if once a Clock be ſet right, or
Aaa 2
has
364
INSTITUTIONS
has its Pendulum duly adjufted, it muft (as Hugens fays,) always
meaſure Time truly, or not meaſure it at all.
3149. Upon the Whole we may conclude, that fince a Weight
is, in its own Nature, a conftant Principle and muft neceffarily
act with an uniform Tenour; and on the other Hand, the Ac-
tion or Force of a Spring is in itſelf alway variable, and can only
be rendered of an equable Tenour by an Artifice, it muft fol-
low that the former is more eligible in Clock-Movements
than the latter; and that a Spring is preferable to a Weight
only on a fingle Account, viz. of its bringing the Clock into a
more compendious or portable Form than can be admitted where
a Weight is made ufe of for giving it Motion.
*
CHA P. II.
The NATURE, FORM, and ACTION of the FUSEE,
explained from Mechanical and Mathematical
PRINCIPLES,
3150. WE
E are now to explain the Theory of that Invention
by which the SPRING is made to act with an equa-
ble Force on the Syftem of Wheel-work by the Mediation of a
Part called the FUSEE, which for that Purpofe is required to
have a peculiar Form; every one knows how a Weight acts
upon the Cylinder, and thereby communicates an equal Force
and Movement to the Machine. But the Manner in which
the Spring and Fuſee do the fame Thing conjointly, is not ſo ob-
vious, but yet will be eaſy to conceive by attending to the fol-
lowing Particulars.
3151. The Chain being fixed at one End to the Fufee, and
at the other to the Barrel, when the Machine is winding up,
the Fufee is turned round, and of Courſe the Barrel; on the
Inſide of the Barrel is fixed one End of the Spring, the other
End being fixed to an immoveable Axis in the Center. As the
Barrel moves round, it coils the Spring feveral Times about the
Axis, thereby increaſing its elaftic Force to a proper Degree;
all
Of CLOCK-WORK.
365
all this while the Chain is drawn off the Barrel upon the Fufee
and then when the Inftrument is wound up, the Spring by its
elaftic Force, endeavouring conftantly to unbend itfelf, acts up-
on the Barrel, by carrying it round; by which Means the Chain
is drawn off from the Fufee, and thus turns it about, and con-
ſequently the whole Machinery is put in Motion.
3152. Now as the Spring unbends by Degrees, its elaſtic
Force, by which it affects the Fufee, will gradually decreaſe;
and therefore unleſs there were fome mechanical Contrivance in
the Figure of the Superficies of the Fuſee to cauſe, that as the
Spring is weaker, the Chain fhall be removed farther from the
Centre of the Fufee, fo that what is loft in the Spring's Elafti-
city is gained in the Length of the Lever; Ifay, unleſs it were
for this Contrivance, the Spring's Force would always be une-
qual upon the Machine, and fo would produce an unequable
Motion of the Parts thereof.
3153. The Figure of the Curve, which fhall form the Super-
ficies of the Fufee by a Revolution about its Axis, may be in-
veftigated as follows. (Fig. 1.) Let BCD be the Curve, AL
the Axis of the Fufee produced; let D be the Point where the
End of the Chain is fixed on the Fuſee when the Watch is down,
or the Spring uncoiled, and B the Point where it touches it
when the Spring or Machine is wound up. From the Points B
and D let fall the Perpendiculars to the Axis B A and DH; in
which produced, let there be taken A E and HI proportional to
the Force or Strength of the Spring, when the Chain is at B and
D. Through E, I, draw the Right-line EIK interfecting the
Axis fomewhere in K; and from any Point C in the Curve,
draw CF perpendicular to the Axis in G ; then will F G be as
the Strength of the Spring when the Chain is at G.
3154. Now fince the Force acting on the firſt Wheel ought
always to be uniformly the fame; and this Force being always as
the Strength of the Spring expreffed by FG, and the Diſtance
at which the Chain acts from the Axis of the Fulee conjointly:
Therefore the Force at any Point C will be as the Rectangle F G
× GC, and fince this is a given Quantity it may be made F G
a b
× GC ab, and fo we have F G =
GC
=
3¹55.
366
INSTITUTIONS
3155. Therefore to determine the Equation of the Curve
BCD, let KH='a, HI = b, HG = x, and GC = y.
Then becauſe of the fimilar Triangle HKI and G K F, we
ab
have HK HI:: GK: FG-; that is, a: ba+x:
a b
y
y
; whence we have aaay + xy, which is the Equation
of the Curve, and ſhews it to be that of the Hyperbola with Re-
ſpect to the Space between the Curve and its Afymptotes, as is
evident from (779).
3156. Hence when xo, then ay, or HK = HD;
alſo when the Point G arrives at A, then y = AB. And be-
cauſe EA XABIH×HD, we have EA:IH::HD:
ABay fo is the greateſt Force of the Spring to its leaft
Force, on the Fuſce (3154).
3157. Becauſe the Ordinates HD, AB, &c. are at Right-
angles to AL, (3153) the Curve BCD is that called an equi-
lateral Hyperbola (fee 780). By the Revolution of which about
its Axis or common Afymptote A K the true Form of the Solid
or Fuſee is generated, as in Fig. 2.
√2
3158. In that Figure A DE, FG H, and I KL, MNO,
are oppofite equilateral Hyperbolas deſcribed about the Afymp-
totes PQ, R S, interfecting at Right-angles in the common Cen-
ter C. Put CT (TD) = 1, then CD= ✓2 = Radius
of the Circle D KGN touching the four equal Hyperbolas in
their reſpective Vertices, And CF (=CY = √✓✓CQ² + QY¹)
-2, is the Diſtance of the Focus of each Hyperbola from the
Center. Laftly, the Parameter ab KN, the Diameter of
the Circle. All which is evident from what we have heretofore
demonftrated of the Properties of the Hyperbola in general
(765, &c.)
3159. It is therefore demonftrated that the Section of any gi-
ven Fufee BDK X through its Axis Z T is determined by two
equal Arches B D and K X of two equal and adjacent Hyperbo-
las beginning from their Vertices D and K.
3160. It is alfo evident, that fince TD and ZB do repre-
fent the Force of the Spring, when it is wound up, and when
quite down, therefore in every Fufee truly made the Diameters
DK
Of CLOCK-WORK.
367
DK and BX of the greatest and leaft Ends thereof must be cxactly
proportionate to the greatest and leaft Force of the Spring.
3161. Alfo it follows, that when the Proportion of the great-
eft and leaft Force of the Spring is known, or the Ratio of TD
to Z B is given, then alſo the Length of the Fufee T Z is a gi-
ven Quantity; or there can be but one determinate Length for the
Fufee, to answer to the two given Forces of the Spring.
3162. Laſtly, it is evident, that when the Length of the Fu-
fee, and one of the Forces, TD or Z B are given, then the
other Force is given or determined, and not to be affumed at
Pleaſure.
3163. Having thus determined the geometrical Form of the
Fuſee, we next proceed to illuftrate the Theory of the ſeveral
Cafes by Examples. Therefore put TD a, ZBy, and
TZ = x; and then the Equation aaay.+ yx (3155) will
appear in its uſual Form. Whence (1.) If a and y are given, to
find x; we have
aa
y
a = x. (2.) When x is given, or x =
I, we have given the Ratio of a to y; for then a a = ay † ją
confequently ay::a +1:a. (3.) When a and x are given,
then
a²
a + y
= y.
(4.) Laftly, when x and y are given; we
have a² ayy, and (compleating the Square) a =
√xy + + y² + y..
*
3164. The proper Numbers for expreffing the Forces (a) and
(y) of the Spring will be in Ounces and Drams Averdupois Wt.
which Ounces may be made Tenths of an Inch, in the Meaſures
of the Fufee TD, TZ, and Z B. Thefe Forces are thus de-
termined. In Fig. 3. let A B C D be the Barrel containing the
Spring, and let FBX be the Pofition of the Chain or Cord upon
the Barrel and Fufee when the Spring is wound up; then ſup-
poſe the Chain difengaged from the Fufee, and carried under the
Barrel in the Direction EH to the Pulley at H, over which it is to
be hung with ſuch a Weight W appended, as will juſt counter-
act or balance the Force of the Spring coiled up. After the
fame Manner, if CDK be the Chain when the Clock is down,
then if this be taken from the Fuſee, and paffed under the Bar-
rel to the Pulley K, and a Weight L hung on to the End, fuch
as
368
INSTITUTIONS
as fhall juſt keep the Barrel in the fame Poſition; then this
Weight L will be equal to the Force of the Spring fo far uncoiled.-
Therefore TD=a:ZB=y : : W : L.
3165. CASE I.
Suppoſe the Weight W be 63 Ounces, and the Weight I be
21; then a= 63 and y = 21;
=21; or becauſe 63: 21:: 3 : 1,
we have a = 3, and y = 1; and then we find x =
aa
y
a
6 = 2 a, that is when the Forces are as 3 to 1, the Height or
Length of the Fufee TZ is equal to the Diameter of its Baſe or
End DK. If W:L::2:1::TD: ZB; then x = a, or TZ
=TD. When W: L:: 3:2; then x= a; and univer-
fally if W: L:: m : n :: a ¦ y ¦ ¦ a + x: a, then it will be
a = x.
=
3166. CASE II.
m
72
n
Given the Length of the Fufee TZ = 6, to determine the
Ratio of the Forces of the Spring, or the Weights W, L, which
will give the Diameters DK, and X B of the Ends of the Fuſee.
Since a² ay + 6y (3163) we have a: y:: a +6: a; then
by affuming the Value of a you have that of y. Thus ſuppoſe a =
3, then ay::3 +6:3:3: 1. And in this Cafe the Diame-
tér DK = 3X B. And becauſe a:y :: m:n, (3165) there-
XB.
fore for any affumed Ratio, we have
n
m - n
xa: thus if m
:n::2:1, then xa, or if m:n::
a, or if m:n:: 3:2, then 2x = a.
Confequently if x 6, we have a 12, and y = 8.
3167. CASE III.
Given the Length of the Fufee, and the greateſt Force of the
Spring to find what the leaft Force muft be. Suppoſe x=TZ=
6, and a=TD=3; then y =
a²
૦૨
a + x
21.
63 Ounces, we have L21.
or DK = 2 X B.
== 1; fo that if W
If x = a, then y = 1ª,
3168.
Of CLOCK-WORK.
36g
3168. CASE IV.
Given the Length of the Fufee, and the leaft Force of the
Spring, to find the greatest. Let x = 6, and y = 1; then
√ xy + + y² + 1 y = √6,25 + 0,5 = 3 = a (3163) ſo
3=a
that if y = 21 Ounces, the greateſt Force of the Spring will be
63. Therefore in every Cafe the Form and Dimenſions
of the Fuſee are geometrically determined.
3169. It only remains now to fhew the Method by which
the Hyperbola ADE (Fig. 2.) is to be deſcribed in Plano in or
der to be made a Gauge for giving the true Form of the Fufee re-
quired in any particular Cafe. Thus fuppofe it be found by
Experiment (3164) that the greateſt and leaſt Force of the Spring
to be uſed is 63 and 21 Ounces. Then having drawn two Lines
PQ and RS at Right-angles in C for Afymptotes, let the An-
gles PCS and R CQ be bifected by a Right-line G D continu-
ed each Way indefinitely.
3170. Then having determined the Diameter of the Baſe
DK of the Fufee, take that Extent in your Compaſſes and ſet
it off each Way from the Center C in the Line DG to F and C
(in Fig. 4.) then will thoſe two Points be the Focuffes of two op-
pofite Hyperbolas ADE and F G H (774).
i
3171. Having provided a Ruler A B C of the Form (in Fig.
4.) you fix one End of a String A B F on the End A, and the
other End in the focal Point F of the intended Hyperbola. This
Chord ABF muſt be juſt fo much leſs in Length than the Ru-
ler ABC, as is equal to the Diameter G D of the Circle
that is, ABC-GD=ABE. Then if with a ſteady Hand you
move the Ruler about a Pin fixed in the oppofite Focus C, and at
the fame Time keep the Chord nicely to the Edge of the Ruler,
as at B, with a proper Pencil or drawing Point, that Point B
will defcribe the required Hyperbola ADE (Fig. 2.)
3172. For the Ellipfis in the Figure to (768) becomes a Circle
in the prefent Cafe, and the tranfverfe Axis T V there becomes
the Diameter DG here; but in every Cafe the Difference of
two Lines CB, FB, drawn from any Point B in the Curve of
the Hyperbola will be equal to the tranfverfe Axis, as is fhewn
(769) and therefore univerfally CB-BF GD; or the
VOL. II.
Bbb
Point
370
INSTITUTIONS
Point B is conſtantly in, and therefore deſcribes the Hyperbolic
Curve required.
;
3173. When the Curve is thus drawn on Paper or Paſt-
board, it will be eaſy to transfer it to a Plate of Brafs, or Steel
and thereby form a Gauge, for giving a true Figure to the Fu-
fee propoſed. In fuch Fufees of the larger Sort, where the Di-
ameter of the Chord or Chain is large enough to be confidered,
it muſt be added to the Diameters DK and XB of the greateft
and leaft Helix of the Fufee.
3174. In order to draw the Fufee by Scale and Compaſſes, it
has been fhewn that there is a ftated Proportion between the Di-
ameters, and Length of the Fufee; and therefore in whatever
Numbers one is expreffed, the other may be expreſſed in the
fame. Thus for Inftance; if TD be to Z B, as 63 to 21, then,
becauſe, in this Cafe, 2a, or TZ 2 TD (3165);
* =
therefore TZ = 126, of the fame equal Parts. So that if
TZ= 1,26 Inches, or 12,6 Tenths of an Inch, then TD-
0,63, or 6,3 Tenths; and ZB = 0,21, or 2,1 Tenths of an
Inch, which are laid down from any decimal Scale, or other Scale
of equal Parts.
3175. Theſe Things are, I think, all that belong effentially
to the Theory of the Fufee in Clock-work; and could the Artiſt in
Practice execute this Part to the Perfection of Theory, it would
then communicate a Motion to the Machine as equable as that
produced by a Weight itself. And by this geometrical Conftruc-
tion of the Fuſee, thoſe which are uſually made by Trial with
the Lever, may be compared and corrected in regard to their
Figure and Dimenfions.
CHAP.
Of CLOCK-WORK.
371
CHA P. III.
The Rationale of CALCULATION in CLOCK-
WORK; the fame applied and illuſtrated in a
DESCRIPTION of the original AUTOMATON
invented by HUGENIUS.
317 being, as the Origin or Principles of Motion in this
3176. THE Nature of the Weight, and the Spring with its Fufee,
Kind of Machinery, explained; it remains to fhew the Conftruc-
tion and Difpofition of the Syftem of Wheel-work to anſwer the
general Purpoſe of a Clock. But as little of a mathematical Theo-
ry is here required, or employed, it will the fooner be diſpatched.
3177. The Communication of Motion being by Wheels and
Pinions, it is in the firſt Place neceffary to take Care, that the
Diameters of the Wheel and Pinion it drives, have the exact
Proportion of their Numbers of Teeth refpectively, that the
Teeth of one may exactly correfpond to the Cavities or Interfti-
ces of the other; thus if a Wheel of 80 Teeth be propoſed to
drive a Pinion of 8 Teeth, then the Diameter of the Wheel
muſt be to that of the Pinion exactly as 80 to 8, or as 10 to I.
If this be done throughout the Syſtem, the Movements will be
every where natural and exact.
3178. In the next Place, from the Nature of the Lever and
Axis in Peritrochio, it appears that the Power or Force on
the Pinion is to that on the Circumference of the Wheel on
the fame Axis, as the Diameter of the Wheel is to that of the
Pinion; confequently by this Means, and by the Friction of the
Parts, the Force at firft impreffed by the Weight or Spring is
conſtantly diminiſhing through the whole Compages, till at laſt
it is but very ſmall on the Pendulum, viz. juft enough to conti-
nue it in Motion.
3179. The Revolutions of two immediate Axis in a given
Time are inverſely as the Number of Teeth in the Wheel of one
to that Number of Teeth in the Pinion of the other, which it
drives, thus if the Number of Teeth in the Wheel and Pinion
B b b 2
be
372
INSTITUTIONS
be 80 and 8, then the Axis of the Pinion will be turned round
Io Times to I Turn of the Axis of the Wheel; therefore the
Quotient of the Wheel, divided by the Pinion it drives, is the Ratio
of Turns to Unity. Thus 8) 80 (10, as before.
3180. Hence if there be any Number of Wheels A, B, C, D,
&e. acting upon fo many contiguous Pinions a, b, c, d, &c.
and let the Quotients of each Wheel, by the Pinion it drives, be
m, e, f, g, &c. then
A
a
m; ora: A:; 1 : m.
Again, B
is the Wheel on the fame Axis with the Pinion a, and drives the
Pinion b on the next Axle; then
B
b
e, is the Number of Re-
volutions in this third Axis to one on the Axis of the Pinion b;
and therefore me
m B
в
n the Number of Turns in this
third Axle to one of the first Axle, or that of the Wheel A, ſɑ
that b : B :: m :n, Let the Wheel on the Axis of the Pinion b
C
drive another Pinion c, then =f with Refpect to the Turns
C
of the preceding Axis which are expreffed by n; therefore
n C
C
=nfo; and fo we have c: C::n:; and for the next
Axle, we have d: D::o: p, and fo on for as many Axes as are
requined in the Train of the Work.
ABCD
a b c d
3181. Then placing thefe Analogies one un-a: A :: 1 : m
der another, and multiplying all the Antece-b: B::m: n
dents and Confequents together, we have abc: C:: no
cd: ABCD::mno:mnop:: 1 :p. Where- d: D::0: R
fore
=p; whence we have this Rule. Divide the
Product of the Number of Teeth in the Wheels by the Product of the
Teeth in the Pinions, the Quotient will be the Number of Turns of the
Axle of the laft Pinion d in one Turn of the first Axle of the Wheel A.
A B C
3182. Since the Expreffions
a b' c
of the Numbers of the Teeth in the Wheels and Pinions, it is
evident, any Numbers having the fame Ratio will answer the
fame Purpoſe, or give the fame Number of Revolutions to an
&c. are but Ratios
!
Axis
な
​..
Of CLOCK-WORK.
373
}
Axis at a given Diſtance from the Firft.
60
Thus
A
~^ a¨
ย
may be go
or, or 50, &c. (as they all give the fame Quouent, viz 10)
and therefore ſuch Numbers are to be chofen by the ſkilliul Ar-
tift as will beft fuit the general Defign and Circumftances of the
Machine.
A B
1
3183. If the Number n = X ㄜ​ˊ be given, it is not ne-
a
fhould be whole Numbers; but either.
A
B
ceffary that
and
a
b
A
one or both may be Fractions; thus
may be 33
5
= 6,6, and
a
B
A
B
b
40 = 5; and fo
a
6
=
n = 6,6 × 5 = 33. Or
A
if = = 45 = 7,5, and
B
B
a
b
I 8
5
J
3,6; then X
b
A
a
7,5 × 3,6 = 27, the Turns in the third Axle, or that on which
is the Pinion b, to one Turn of the firſt Wheel.
3184. Becauſe we may put
A R
S
X
n, it is evident
a
b
S
one Wheel SAB will, by driving a Pinion sab, pro-
duce the fame Number (n) of Turns in the next Axle to the
Wheel A. Thus let A B = 45 x 18 =
45 × 18 = 810, and ab - 6
× 5 = 30 then $10
=27=n, as before.
But fuch large
Numbers can be admitted only in very large Works; therefore
the equal Ratios 1, 162, 243, &c. may be taken as Occafion
requires (3182).
30
3185. We have now confidered how Wheels and Pinions are
to be conſtructed as far as the Pendulum or Balance which regu-
lates the Motion; the Conftruction for Application of the Pendu-
lum is now fomewhat different from what it was in the original
Invention of the Pendulum Clock by Mr. Chriftian HUGENIUS of
Zulichem in Holland, which he first defcribed and publiſhed in a
Diagram cut in Wood, in the Year 1657. And as this
And as this may be
justly eſteemed one of the greateſt Curiofities of Art, and was
never (that we know of) exhibited to the
Reader, we fhall here preſent him with it,
from the Original.
View of an English
cut in Wood exactly
3186.
.1
!
Y
374
INSTITUTIONS
Y
A
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L
P
+
о
T
L
M
•
P
J
4
K
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15°
}
T
24
H
G
•
S
(R
18
48
દ
દ
72
30 m
•
E
C
V
6.39
Y
(OLEREINDSINTAR...
80
C
B
V
Of CLOCK-WORK.
375
3186. The View is a Section through the Axles of the feve-
ral Wheels, which, in this firft Automaton, were all placed in a
perpendicular Line through the Middle of the Clock, as here
repreſented. The parallel Plates of the Frame are A A and B B,
and the first or great Wheel CC is placed on the Arbor of the Bar-
rel DD, to which is the Chord applied with the Weight append-
ed. This Wheel has 80 Teeth, and drives a Pinion E with 8
Teeth, (or Leaves as they are ufually called) placed on the 2d
Axle, which therefore moves ten Times round in one Turn of
the firſt Wheel.
3187. The Second Wheel F has 48 Teeth, and drives a Pinion
G of 8; therefore turns round fix Times in one Turn of the
Wheel F; and 6 x 10 = 60 Turns in one of the first
Wheel.
3188. The third Wheel H (on the Axle of the Pinion G)
has alfo 48 Teeth; but the Form of this Wheel is different from
the other, being like a Hoop, and the Teeth cut on one Edge.
gives it fome Refemblance to a Crown, and is therefore ufually
called the Crown Wheel. This Wheel drives a Pinion I of 24
Leaves placed horizontally on an Axis, of Courſe, perpendicu-
lar to the Horizon. And as it turns but twice in one Turn of
the Wheel G, it will have 120 Turns to one of the firſt
Wheel C.
3189. The fourth Wheel K on this Axis has an horizontal Po-
fition, and is, like the laft, in Form of a Hoop on the upper
Edge of which 5 Teeth are cut like thofe of a Saw, except that
the upper or oblique Part is a Curve of a peculiar Form, neceffa-
ry on Account of giving the moft natural Motion to the Pendu-
lum by Means of two Parts (called Pallets) L, L, which alter-
nately play in the Teeth of the Wheel, at the oppofite Sides K,
L. This is commonly called the Swing-wheel.
3190. For theſe Pallets L, L, being fixed to an Axis move-
able on its extreme Parts in a firm Piece of Braſs N, P, will by
the gentle and alternate Impulſes they receive from the Wheel
K, keep the faid Axle L M in a conftant vibratory Motion,
which being communicated to a flender Rod MS, fixed to it at
M, and having a Fork V to receive the Rod VV of the Pen-
dulum TV, it is plain, that vibrating Motion will be at laſt
impreſſed on the Pendulum itſelf.
3191.
376
INSTITUTIONS
'
1)
3191. But this Motion thus communicated to the Pendulum,
muſt be fuch as nicely quadrates or coincides with the proper
Motion of the Pendulum itſelf, which depends on its Length.
Now as the Wheel K has 15 Teeth, and each Tooth ftrikes
each Pallet 15 Times in one Turn, both the Pallets together
muft receive 30 Impulfes, and confequently the Pendulum muft
have 30 Swings or Vibrations in one Turn of the Wheel K;
and therefore it must vibrate 30 × 120 3600 Times in one
Turn of the great Wheel C.
3192. Now 3600 is the Number of Seconds in one Hour;
therefore a Pendulum vibrating Seconds, whofe Length is 39,2
Inches (1125) was applied to the Clock; and then the Axis of
the firft or great Wheel C turned round once in an Hour.
3193. As the Axis of this Wheel paffed through the Platé
A A, then an Index put on the End was carried round
once in an Hour, and over a large Circle (on the Face of the
Clock) divided into 60 equal Parts, thereby indicating the Mi-
nutes at its extreme Part.
22
This Pini-
3194. On the fame Axis (near the Plate A A) was fixed a
Wheel BB of 30 Teeth which drove another Wheel of the
fame Number of Teeth, and a Pinion of 6 Leaves.
on drove the Wheel of 72 Teeth; and therefore it moved but
once round in 12 Turns of the Pinion or Axle of the Wheel C.
Confequently an Index placed on the Socket 0 of the Wheel
(moveable about the common Axis) will fhew the HOURS as it
paffes over a Circle divided into 12 equal Parts, on the Face of
the Clock Y Y.
it has 60 Turns in
The Axis of this
A A had a Plate
3195. Laftly; the Axis of the Crown-wheel H has 60 Turns
to one of the great Wheel C, (3187) that is,
an Hour, or it turns once round in a Minute.
Wheel, therefore, paffing through the Plate
λ a fixed upon its End with a Circle divided into. 60 equal Parts
denoted by their proper Numbers, which fhewed the SECONDS
of Time as they paffed by a Hole z made in the exterior Plate or
Face of the Clock Y Y.
י
11.
3196. This original Pendulum clock, conftructed with fuch
Simplicity as to have no more than four Wheels and Pinions in the
Body of the Clock, fhewed HOURS MINUTES, and SECONDS
of Time, with all the Exactnels and Equability that a mecha-
1
や
​nical
Of CLOCK-WORK.
377
nichal Exegefis can admit of. And I think we may conclude
that it is not only the first but the moſt perfect Pattern of a
CHRONOMETER that has been or can be propoſed; for though
it be poffible with three Wheels and Pinions only to produce this
threefold Divifion of Time; yet the great Difproportion, and
inconvenient Difpofition of the Parts, would render fuch a Con-
ſtruction inelegant and immechanical, and therefore not to be
admitted as a Work of Art. They who would ſee a great deal
wrote on this Subject to very little Purpoſe, may confult the
voluminous Tracts of Cafp. Scottus, and others of the like Ge-
nius.
CHA P. IV.
Hugenius's INVENTION for applying a WEIGHT to
a CLOCK, that ſhall act upon it inceſſantly, and
render its MOTION conftant, explained and illuf-
trated; with its Improvement in ROYAL PENDU-
LUMS.
3197. HE celebrated Author of the foregoing Invention,
ſo uſeful in public and private Life, did afterwards
improve it by another, which no lefs demonftrated his fingular
Genius and Sagacity for automatical Machinery. This fecond
Invention was to render the Motion of the Clock, or rather of
the Pendulum, more equal than before; but to underſtand the
Reaſon of it, the Manner in which he ſuſpended the Weight for
keeping the Clock in an unintermitting Motion, is firſt to be ex-
plained.
3198. It is eaſy to underſtand that a Weight fufpended upon a
Clock in the Manner it is upon a Jack, will keep the Clock going
till it wants winding up; but during the Time of winding it up,
the Action of the Weight is taken off from the great Wheel of
the Clock, as it is all that while difengaged from the Barrel on
which the Weight hangs; and therefore the Clock not being
impelled by the Weight, will, during that Time, ftand ftill,
and confequently fo much Time will be loft.
Ccc
VOL. II,
3199.
378
INSTITUTIONS
3199. Our Author therefore tells
us, that he invented the Method
which follows, of a perpetual or
endleſs Line for that Purpoſe, which
being of a proper Length, the two
Ends were nicely fpliced and con-
nected together; it was then ap-
plied in the Clock as reprefented in
the Figure adjoined, by the Letters
ABCDĄ. Where A repreſents the
fulcated Wheel DD (in the Figure
of the Clock) with feveral Spicule or
ſmall pointed Pins fixed in the Sur-
face of the evacuated Part, as is there
fhewn.
3200. The Pulley C is fixed to
a Rachet-wheel G, but both moveable
on a fixed Axis, from the Left-hand
towards the Right only; for all Mo-
tion the contrary Way is prevented
M
K
A H G
กา
15
FO
من التاني
D
L
by the Trigger or Catch at H falling conſtantly into the ferrated
Teeth of the Wheel G. The Surface alfo of this Pulley C is
fpiculated, like that of the Wheel DD on the Arbor of the firſt
Wheel of the Clock,
3201. The endleſs Line or Cord, being put over the Barrel
A, and Pulleys B and C, will, by Reafon of the Spiculæ or poin-
ted Pins, hang firmly on A and C, fufpending the large Weight
E affixed to the Pulley B. Now it is evident from the Nature
of the Pulley (1050) that the Weight E equally stretches the
Parts of the Chord I and K; and therefore acts with an equal
Force, viz. half its Weight, on the Wheel A, and Pulley C.
3202. But fince this Force upon the Pulley C tends to move
it from Right to Left, and all Motion that Way is ſtopped
(3200) it follows, that the Line I is to be confidered as fixed
all the Time the Clock is going; but the Parts K and M, by
Vertue of the Weight E, will be conftantly moving over the
Surface of the Wheel A, and thereby communicating Motion to
it, and, of Courſe, to all the Machinery of the Clock.
·
3203,
Of CLOCK-WORK.
379
2203. As to the Pulley D, and the fmall Weight F, it is
plain they are defigned only to give free Motion to the Parts of
the Chord L, M; and keep them in a perpendicular Pofi-
tion.
3204. Thus it appears, that as long as the Weight E can
deſcend, fo long the Clock will keep going; and therefore as
many Times as the Circumference of the Wheel or Barrel A is
contained in the Diſtance through which the Weight can de-
ſcend, ſo many Hours will the Clock go without drawing up,
becauſe the ſaid Wheel A goes once round in an Hour (3192).
In this original Clock, the Wheel DD is one Inch Diameter,
or bout 3 Inches in Circumference, and went 30 Hours without
drawing up; therefore 90 Inches, or 7 Feet was the perpen-
dicular Defcent of the Weight, or Height of the Clock.
2
3205. When the Weight was down, it was eaſily drawn up
again by applying the Hand to the Part of the Line at L, and
drawing downwards it will move the Pulley C, till the Weight
E afcends to the Top, during which Aſcent it conftantly acts or
gravitates on the Wheel A, by Means of the Rope K, with the
fame Force as when at Reft, and therefore the Clock goes con-
ftantly by this Contrivance without ever loofing a Moment of
Time, if it be not neglected.
3206. Hugenius informs us that in thoſe of his Clocks
which were eſteemed the beſt, the Weight E was fix Pounds;
the Power therefore by which the Clock was animated, was
that of 3 Pounds. The Weight of the Pendulum was alfo 3
Pounds; and of the Length, to fwing Seconds.
3207. By an Addition of Pulleys, it would be eaſy to make
the Weight defcend more flowly, and, of Courſe, to make the
Clock go longer before it wants to be drawn up. In the Mo-
dern Conſtruction of Clocks, a different Method of applying the
Weight is uſed, and it is wound up upon a Barrel with a Handle
or Winch; by this Means, the ufual Time the Clock will go,
is 8 Days in thofe called Royal Pendulums, in which the Swing-
wheel K has a Pofition (not parallel, as here, but) perpendicular
to the Horizon, or the fame with all the other Wheels. Every
Thing of this Sort is evident by Inſpection of any 8 Day Clock,
and needs no other Inſtruction.
Ccc 2
3208.
380
INSTITUTIONS
3208. I fhall only here obferve, that the Form of the Teeth
in this Wheel ſhould be exactly circular on that Side or Part by
which acts on the Pads or two Arms of the Clock of the Pendu-
lum ; in the extreme Parts of thoſe Arms there fhould be a ſmall
cylindric Pin fixed, always touching and moving in the circu-
lar Surface of the Teeth of the Wheel; the Radius of the
Circle for the Teeth is the neareſt Diſtance of the Pin in the
End of the Pad to the Axis or Arbor of the Pendulum on which
it is fixed.
3209. Moreover it is farther neceffary, that the Teeth on
this Wheel be of fuch a Number and Length, that the Pins up-
on the Ends of the Pads, may very nicely take and eſcape them
alternately, or without the leaft Intermiffion of Action. Such a
Conftruction is both natural and mathematical, but it is not in
Ufe, that I have feen; I know of no Method that can be ſubſti-
tuted that is eaſier, or that will produce a more fteady or equable
Motion of the Second-hand fixed on the End of this Arbor, or
(in the Clock-maker's Language) fo accurately cauſe the Pen-
dulum to beat dead Seconds.
CHA P. V.
The Rationale of HUGENIUS's Invention for con ·
tinuing and regulating the MOTION of the PEN-
DULUM by a fingle Wheel only; thereby render-
ing it most EQUABLE for USE at SEA.
3210. HUGENIUS having confidered (and probably found
by Experience) that the Action of a Weight being
propagated through a Multiplicity of Wheels and Pinions to the
Pendulum, could not be fo conftant, and agitate the Pendulum
with a Force fo equable, as it would do if the Number of Wheels
were lefs; and of Courfe concluded, that if the Weight could
be applied to the Arbor of that Wheel only which actuated the
Pen-
Of CLOCK-WORK.
381
Pendulum, it would then communicate a Motion to it that
would be the moſt ſimple and conftant that the Nature of Things
admits of in this Kind of Machinery.
-
3211. At Length he invented the Method of fuch an Appli-
cation, which was as follows. An endless Line with its proper
Weights and Pulleys, (every Way fimilar to that defcribed in
Fig. to Art. 3199, only as much leſs in Proportion as the
Force of the Weight was lefs,) was applied to the Arbor
of the Swing-wheel in a fmail. pinnulated Grove of a Bar-
rel, (which we now fuppofe to be A) on one Part, and to a
Ratchet-wheel C on the other, fupporting the fmall Wheel E,
with the other much ſmaller at F.
3212. It is manifeft that the Weight E depending on the
Line IBA, confidered as fixed on the Pulley C, does now fole-
ly act on the Arbor of the Swing-wheel by the Part AB, and
therefore the Motion communicated to the Pendulum, must be
the moſt Uniform poffible, as we cannot conceive any Thing
incident to the Weight E that can make any Alteration in its
gravitating Force, and as there can be no Caufe for any Irregu-
larity in the Motion of the Pendulum.
3213. As the Body of the Machinery in the Clock is in this
Conftruction difingaged from the Swing-wheel and Pendulum,
fo none of its Anomalies can affect either; the proper Office of
all this Part being now only to raiſe up the Weight E as often
as it is down, which the Author tells us, was every Half-minute,
or 30 Seconds. For fince in every Revolution of the Wheel A,
the Weight E muft defcend through a Space equal to the Cir-
cumference of that Wheel; therefore it muſt want very frequent-
ly to be drawn up, which could not be done but by Means of the
Machinery of the Clock, which by its Nature becomes a con--
ſtant Agent for that Purpoſe.
3214. Where it not for the Frequency of drawing the Weight
up, the Pendulum, with one Wheel to continue its Motion,
would be the most perfect Chronometer that could be made; but
as it is thus connected with the Clock-part, it becomes juft
as uſeful as if it wanted no Attendance to draw it up at
all.
3215. By this Invention, therefore the Clock with all its
Wheels, is, in Effect, reduced to one fingle Wheel; and the
Mo-
382
INSTITUTIONS
Motion juſt as equable and perfect as if it confifted of no more;
and the Author affures us, that the Equability of Motion in
Clocks of this Sort was apparently greater than in Clocks of the
common Conſtruction.
;
3216. As a Proof of this, he mentions a very extraordinary
Circumſtance, which he diſcovered by Accident; it was this
as two of thoſe Clocks ftood together on a Shelf, he obferved
the contrary Vibrations of each of them did fo exactly agree and
coincide, that they never in the leaft receded from it. And if
this Concordance of their Motions was on Purpoſe difturbed,
yet in a very little Time it would of it's one Accord return, and
continue, in fuch perfect Coincidence, that only one Sound could
be heard of both their Vibrations.
3217. Having wondered fome Time at fo unuſual a Phæno-
menon, he refolves to enquire into its Caufe with the utmoſt
Diligence, and at Length found that it proceeded from a Mò-
tion of the Shelf or Plank on which they ftood, though in itſelf
altogether infenfible. For the Reciprocations of the Pendulums
communicated fome Degree of Motion to the Clocks, firmly
placed as they were; and this Motion being impreſſed on the
Board, did neceffarily cauſe that, if the Pendulums vibrated
otherwiſe than by contrary Strokes very exactly, they must at
Length do fo, and then the Motion of the Shelf would entirely
ceaſe.
3218. For whatever Motion be communicated to the Shelf
by the contrary Motion or Vibrations of the Pendulums as thofe
Motions are impreffed in contrary Directions, they will, if equal,
deſtroy each other ( ) and if they are not impreffed equally,
and at the fame Inftant on the Board, yet as the Difference equal-
ly affects the Board and the Clock, it will, by this Alternation,
conſtantly tend to an Equibrium both in the Clocks and the Shelf,
till at Length that being affected, the Pendulums muſt move bý
contrary Strokes very nicely.
3219. But our Author concludes, that notwithſtanding the
Fact is evident, yet a Cauſe of fuch a tender and delicate Na→
ture could never have Efficacy enough to produce fuch an Effect,
unleſs the Motions of the Clocks had been previouſly, and by
other Means rendered moft equable and confentient between
thern-
Of CLOCK-WORK.
383
themſelves; and which therefore is by this Experiment fully af-
certained and demonftrated.
3220. It is above 100 Years fince this Difcovery was made
and publiſhed to the World, and was intended, by the Inventor,
to render his Clock more immediately uſeful at Sea; for this
Purpoſe he contrived it in a particular Form with a triangular
Pendulum, and a Conſtruction which gave it always an upright
and fteady Pofition on Ship-board; and thus many of them
were made and ufed at Sea for the better difcovering of the Lon-
gitude; and with confiderable Succeſs, as we find by an Account
thereof (in feveral Voyages) by the Author in his Horoing Ocil-
lator. Pag. 17, &c. where alfo you have Iconiſms of the Pen- `
dulum, the Clock, and its Frame in which it was fufpended in
the Ship.
3221. But after all, we find the Uſe of this moſt promifing
Invention diſcontinued at Sea, which would never have been,
were it poffible for any Thing of this Nature to fucceed there;
and if this be the Fate of a CLOCK actuated and regulated by
the conftant Power of a WEIGHT, it may tend to moderate our
Expectations of the Longitude by Means of Watches, wherein
the fame Artifice has been attempted by Means of Springs. But
how much Inferior all Automata by Springs are to thoſe which
move by Weights, common Experience can fufficient teſtify,
and will be farther evident by confidering their Nature and Con-
ftruction, explained in the next Chapters.
3222. I fhall conclude this with a Query to the ingenious
Artificer, viz. whether it is not practicable to move a Pen-
dulum by a ſingle Wheel, and a Weight that ſhall not need wind-
ing up at all, by the Addition of another equal Pendulum to the
Clock? I fee nothing to contradict it in the Theory; if
this were done, the utmoft Equability of Motion from Clock-
work would thence be derived to the Pendulum, and fuch a
double Pendulum Clock would become the moſt perfect Automaton
in Nature.
CHAP.
384
INSTITUTIONS
1
CH A P. VI.
СНАР.
Concerning the Invention of portable AUTOMATA,
or WATCHES; of the BALANCE, and REGU-
LATOR; and the THEORY of Ifochronal VIBRA-
TIONS in SPRINGS.
3223.
IN
N the Invention and Application of Pendulums to Ho-
rological Automata, Mr. Hugens ftands unrivaled; but
with regard to the Invention of Clocks in Miniature, or of a por-
table and pocket Form, which we generally call WATCHES,
there is (at least, there has been) great Difpute. Of theſe
Watches there is not the leaft Mention or Intimation in the
Horologium Ofcillatorium by Hugenius, though printed in 1673,
from whence it ſeems probable, that at that Time he knew of no
fuch Thing.
3224. It is alſo certain, that many Years before, our cele-
brated Countryman, Dr. HOOKE, exhibited feveral of thoſe
Watches as his own Contrivance and Invention, and a Patent
was offered him for the fame in 1663, but not liking the Con-
ditions, he refuſed it. Dr. Derham (in his Artif. Clock-maker)
tells us, he ſaw a Watch prefented to King Charles II. on which
was this Inſcription, viz. Robert Hook, inven. 1658. T. Tompion,
fecit, 1675. Mr. Ward, in the Life of Dr. Hooke, fays, he
faw the fame Infcription on a round Braſs-plate, which former-
ly had been a Cover to the Balance of one of Mr. Hooke's Wat-
ches; and he farther adds, that Mr. G. Graham, informed him,
he heard Mr. Tompion fay, he was employed three Months that
Year by Mr. Hooke, in making fome Parts of thoſe Watches
before he let him know for what Ule they were defigned; and
that Mr. Tompion ufc to fay, he thought Mr. Hooke was the In-
ventor of them.
3225. From hence it appears, that thefe Watches were in-
vented by Dr. Hooke, within one Year after the Clock itſelf,
which (as before obferved, 3185) was not made public till the
Year 1657. Notwithſtanding this, in the Year 1674, Mr.
Hu-
The THEORY of SPRINGS & PENDULUMS.
A
l Fig. 7.
α
Fig. 2.
E
α
d
Fig. 6. F
C
D
B
Tig. 1.
C
B
ທ
rrrrr!
i
C
F
rrrrr
B
a
A
A
G
E
C
Fig. 11.
Fig. 10.
N
G
D
Fig. 3.
K
K
K
K
K
A
I
い
​I
M
л
M
www
Fig. 4.
d
B
f
F
C
C
C
C
Fig.5.
I
Fig. 9.
R
F
Philosophical FOOT
The Philofophical YARD or STANDARD.
Paris FOOT
English FOOT
Fig. 8.
E
M
75
N
T
GV
P
K
Ꮐ
C
H
e g
B
:
385
Of CLOCK-WORK.
Hugens publiſhed a Pocket Watch of his own Invention, as he aſ-
ferts, but it was in feveral Reſpects different from Dr. Hooke's.
And as it is inconfiftent with the great Character of Hugenius,
for Learning and Probity, to fuppofe him capable of Plagiariſm,
therefore it muſt be allowed, that the Reduction of Clock-work,
to the Size of a Pocket-watch, was feparately the Invention of
each of theſe ingenious Competitors.
3226. This Invention confifts in applying a Balance-wheel,
to beat the Time inftead of a Pendulum; for though they are
called Pendulum-watches, it is only becauſe of the alternative
Motion of the Balance being fomewhat like that of a Pendu-
lum.
3227. This Balance has its Motion regulated by a Spiral-fpring
properly applied, an Idea of the Balance and Spring is eaſy to be
formed from the View of any Watch-work. As alſo the Man-
ner in which they are actuated by the Machinery of the Watch;
for this being the fame as in a Spring-clock, only in ſmall, there
is nothing new till we come to the Regulation by a Wheel and
Spring instead of a Pendulum.
3228. From barely confidering the Size of a Watch, it is
plain no Pendulum can be admitted into its Conftruction; for
in the firſt Place, a Pendulum cannot be made of fo fhort a
Length as is requifite, with any Degree of Exactneſs; and ſe-
condly, the Pendulum always muft have a perpendicular Pofi-
tion, which cannot be allowed in the Ufe of a Watch, which is
incident to all Kinds of Pofitions.
3229. As the Balance-wheel has in itſelf no Principle of, or
Difpofition to Motion, but is actuated folely by the Machinery
of the Watch, it is to be confidered alone as no other Thing
than a Check upon the Motion of that Syftem, or a Reſervoir
in which it is ultimate received and abforbed, like the Fly of a
Jack in a great Meafure, only the Fly has a circular Motion,
whereas that of the Balance is an Ofcillatory one.
3230. The Motion of the Balance and the Fly is therefore e-
qually fubject to the Inequalities of the Motion of the Wheel-
work of the Watch; but this can be regulated in the Vibrations
of the Balance by Means of a Spring, whereas a circular Mo-
tion of a Wheel admits of no fuch Regulation or Correc-
VOL. II.
Ddd
tion,
386
INSTITUTIONS
tion, and therefore can have no Place in this Sort of Mecha
niſm.
3231. Hence then it is evident, that the whole Artifice of a
Watch confiſts in giving Motion by the firſt Spring and Fufee to
the Balance, or ofcillating Wheel, as nearly equable as poffi-
ble; and then to correct the Irregularities of thofe Ofcillations,
and render them ifochonal as far as may be, by a fine Spring
properly applied to the faid Balance, though not by its vibrating
Property, as many have fuppofed.
3232. But as this Doctrine of the isochronal Vibrations of a
Spring is a curious Point, I fhall be a little more particular and
explicit in deriving it from its firſt Principles, and then illuſtrat-
ing it by a Figure. Therefore let G be any Force (whether
Gravity, Elafticity, &c.) which conftantly acts on a Body and
produces an accelerated Velocity; and let S, T, V, be the
Space, Time, and Velocity of fuch Motion; then we have
fhewn (991) that it is S: TV, univerfally; and alſo, that it
is GTV (ib.) therefore we have S x V: GT2 V, or S:
G x T²; and conſequently we have S: G:: T2: 1; therefore
when the Ratio of S to G is given, the Time T will be a given
Quantity or always the fame. So that if G be the Force of E-
lafticity, and S the Space through which it vibrates by that Force,
then, however thoſe Quantities vary in Magnitude if they keep
the fame Proportion, the Vibrations of the Spring will all be
ifochronal, or performed in equal Times.
2
3233. For a farther Illuſtration of the Similarity of Motion
in a Pendulum and Spring; let Aa (Fig. 1.) be a Spring, or a
ftrait elaſtic Wire fixed at the End A, and at the other End fup-
pofe it drawn out of its natural or perpendicular Situation, by a
Line paffing over a Pulley B, with a Scale Cat the End, and
Weights put into it for trying the Experiments. Then admit
we put a Dram into the Scale C which draws the Wire or Spring
from A a to the Site Ab; and then we add another Dram, and
it draws it into the Pofition A c; and a third Dram being put
into the Scale, draws the Wire into the Situation A d; and fo
on, as long as the Wire or Spring can retain its rectilineal
Form; then will the Spaces a A b, a A c, a A d, be as 1, 2, 3;
that is, as the Weights applied which retain the Spring in thofe
Pofitions refpectively; therefore the elaftic Force of the Spring
at
1
Of CLOCK-WORK.
387
at a, b, c, being alſo as thoſe Weights, and confequently as
the Spaces defcribed by the Wire, viz. the fmall right-lined
Triangles a A b, a Ac, a Ad, it follows, that the Wire will
defcribe each of them in the fame Time; and therefore all its
Vibrations, as long as it continues rectilineal, will be performed
in equal Time.
1
If 3
3234. But if upon adding 3 Drams more, we obferve the
Wire drawn into the Pofition Ae, and there appears incurvated,
it is evident then, becaufe the Space a A e is not equal to twice
the Space a A d, the Time in which it moves from e to a can-
not be equal to the Time in which it moves from d to a.
more Drams be added, fuppofe all the Nine draw the Spring to
the Pofition Af, which is there more curved than before; and
therefore a A ƒ is ftill more deficient from 3 Times the Space
a A d, and conſequently the Time of defcribing it will differ ſtill
more from the Time of defcribing the Space dA a. Upon the
Whole, then, it appears, that after the ftrait Spring begins to be
bent into a Curve, the Times of Ofcillation are no longer ifachro-
nal.
3235. In like Manner, with regard to a Pendulum Aa (Fig.
2.) we have formerly demonftrated (1105) that the Times of
Defcent through any Number of Chords (ad, ae, aƒ,) of a
Circle are all equal. Therefore if we take very finall Arches
ab, ac, ad, they will, as to Senfe, coincide with their Chords,
and ſo of Courſe the Time of defcribing each of them will be e-
qual; but if the Arch be fo large that it can no longer be efteem-
ed rectilineal, as a e, or af, then the Times of defcribing them
ceafe to be ifochronal. So that very ſmall Diftances only on each
Side the Perpendicular a A, both in the Spring and Pendulum,
will admit of ifochronal Vibrations.
3236. But what has been hitherto faid of the Spring and its
Vibrations has been with a View rather to fhew its Nature and
Congruity with the Pendulum, than its Application to Uſe in
automatical Machinery on that Account; for though the Pen-
dulum by Means of its ofcillatory Property, becomes fuch an ex-
cellent Regulator in all Kinds of Time-pieces, yet the Spring
poffeffed of the fame Property, will anſwer no fuch Purpoſe on
that Account; becauſe the Time of a Vibration is fo exceeding
fmall. Now the Vibrations of a Spring (fuch as we have confider-
D dd 2
edł
388
INSTITUTIONS
I
ed) ought to coincide with thofe of the Balance, if its Regula-
tion were to be produced by them; but the Vibrations of the Ba-
lance, or Beats of a Watch, are not more than about 16000 in
an Hour, that is, about 4 per Second; whereas a ftrait Spring,
of the Length adapted to a Watch, will vibrate many hundred
Times in a Second; and, indeed, a Spring of any Size, or Kind,
can have no Ufe in that Refpect, all its Effect, as a Regulator,
being derived from its elaſtic re-active Force, by Way of conftant
Preſſure (and not Ofcillation) which therefore must be more parti-
cularly explained, and is the Subftance of the following Chapter,
CHA P. VII.
The THEORY of SPRINGS confidered as REGULA
TORS of the BALANCE of CLOCKS and WAT¬
CHES.
3237. A
3
Sa ftrait Spring, in Length, cannot be more than
about of the Diameter of the Watch-plate, it
will be found too ſhort to admit of a proper Tenour of Action
as a Regulator of Motion in fuch fmall Balances; for the Ac-
tion of a Spring ought to be very free and eaſy, but yet at the
fame Time ftrong enough to govern the Motion of the Balance,
and controul the Irregularities of its Vibrations derived from the
Syftem of Machinery. Therefore a Spring of a Spiral, undula-
ted, or fome other Form, wherein there is a confiderable or
fufficient Length in a concife Space for anfwering its Purpoſe in
a WATCH, muſt be choſen.
3238. As the Action of the Spring under all thoſe different
Forms will be the fame, Ifhall chufe that of (Fig. 3.) to explain
it by. Thus let CL be the Spring in its natural Situation, or
fuch as it has when left to its itſelf; at the End L ſuppoſe it to
reft againft, or be fixed to, an immoveable Support K, then by
its elaſtic Force it will make Refiftance to any Power or Weight
by which it is preffed inwards, or drawn outwards. Thus let
P be
A
389
Of CLOCK-WORK.
M
be a Force applied which bends it through any Space or Length
"CI; then from the Nature of the Spring (3233) 2 p will
bend it through a Space equal to 2 s, and 3p will bend it thro'
3s; and ſo on. So that
So that will be every where proportional
to p.
$
3239. Then if P denote the Power by which the Spring is
wholly compreffed, and S Space of total Compreffion; then
it will be SP: p; and the fame Analogy will refult from a
partial and total Extenfion. This Principle is not only deduced
from a phyfical Theory as before-mentioned, but is confirmed
moft accurately by Experiments on Springs on every Form.
*
3240. And here I may obſerve, that uſeful Inftrument called
the Spring (or Cylindric) BALANCE for weighing Bodies, is no-
thing more than the Practice of this Principle; for the Number
of pound Weights in the Body appended is indicated by the
Number of equal Divifions on the Bar, which is the Space thro'
which the Spring is compreffed by the Weight of the body, as
each fingle Divifion correfponds to that of a pound Weight.
1
3241. This is the Effect of a Spring in regard to Preffure;
we fhall next inftitute a Compariſon between this claftic Force and
a percuſſive Force, or that of a ſtriking Body moving with'a certain
Degree of Velocity. For if a Body whofe Weight or Mafs of
Matter is M, and moving with an uniform Velocity V, ftrike
upon the End C of the Spring, it will bende fame through a
Part or the whole Space S; and the Quanuty of Motion by
which it affects the Spring at first will be MV (970); but by
the Reaction of the Spring, the Velocity V will be gradually
diminifhed; fo that when the Spring is bent through any Space s,
the original Velocity V will be reduced to v, and the Quantity
of Motion or Momentum of the Body will there be only Mv, fo
that the Lofs of Motion in bending the Spring through the Space
s, will be MV — M v.
3242. Let A = Space through which a Body defcends in Va-
cuo in one Second, by the Power of Gravity; and C = Celerity
acquired in that Defcent. Then we have C² : A ¦ ¦ V² : a
Space of the Body M muft defcend thro' by Gravity to acquire
the Velocity V. (991) Alſo ✅Ã: 1 :: √a: T = Time
of the Defcent through a
}
1
+
J
3243.
390
INSTITUTIONS
3243. Suppoſe, then, by the Stroke of the Body M moving
with the Velocity V, the Spring be bent through a Space s in
the Time t. In this Cafe we have S: s:: P: p =
s P
S
elaftic Force of the Spring at 1, or at the End of the Space s.
to the
3244. Now let the momentary Decrement of Velocity in the
percutient Body be, then will the inftantaneous or fluxionary
Momentum produced by the elaftic Force of the Spring be-
M in the Moment of Time ;; and we have ſhewn (1000)
that in Caſe of Forces (M and acting uniformly, the Quan-
tities of Motion (MV, M,) generated are proportional to
the generating Forces, and the Times (T, i,) conjointly;
•
s P
SP
M & M x T: x ; ; which gives
and therefore MV : — M÷
ல்=
VPsi
MST
S
3245. Again, in the fame Cafe of Forces acting uniformly,
the Spaces are as the Velocities and Times conjointly (971)
ᎢᏙ ;
therefore 2 a: VT :: ; : vi; whence i-
3246. Therefore
S
2 av
VPS TV;
ல்
X
MST
; which gives
200
V2 Psi
2 v i =
2 ບ ຕໍ່ -
;
MS a
V2 Ps2
the Fluents of which are v² and
; but when the former of thefe was V2, the latter was
2 M Sa
V2 P s²
= 0, becauſe of so;. therefore v² V²
2 MSa
Conftruction, let
3247. To reduce this Equation to more fimple Terms by ą
2 a MS
Va sa
R², then will v² V²
P
R²'
2
R2
or v² = V² X
Now if R Radius CG of a Cir-
R2
cle, (Fig. 4.) and sCBg F, the right Sine of the Arch
GF; then by the Property of the Circle, it is R2-♪² — BF²;
therefore v² = V² ¨×
BF2
R²
ز
BF
and ſo v = VXR,
=
that is, the
Ve-
Of CLOCK-WORK.
39*
Velocity v is to the original Velocity V, as the Cofine BF to the Ra-
Aus CG.
TV;
3248. Becauſe =
>
and V ×
v
X
24บ
/R²
R
TV;
R
T
R;
therefore i
X
X
2 a
R
2
2 a V × √ R²
But drawing bf indefinitely near to B F, and fd perpendiculat
to the fame, we have from the fimilar Triangles BCF and
R
dfF, as BF: df:: CF:ƒF
=
CF x df
=
BF
√
R2-
T
Therefore i
t =
2a.
T
2 a
2 a
xfF; and the Fluents of this Equation are
x GF; which gives this Analogy : T :: GF:
3249. Thus the Times, Velocities, and Spaces, reſpecting the
Motion of Springs, are eaſily affignable by a trigonometrical Cal-
culus in the Parts of a Circle. But in most practical Cafes, par-
ticularly Watch-work, the Space through which the Spring is
bent by the Stroke is but a Part of the whole Length, and
therefore the Quantity of Motion MV, or Force of the Stroke
being confumed in bending the Spring, through the Space s in
the Time t, the Velocity will there ceaſe, or vo; in which
Cafe the Time may be expreffed independent of any particular
Arch of a Circle, as follows.
1
3250. It is evident from the Theorem in (2247) that R2-
s², when vo; therefore s being now equal to Radius CH
CB (Fig. 4.) and alfo the Sine of the Arch GF, that Arch
now becomes the Quadrant of a Circle HB. Then if m Peri-
phery of a Circle whofe Diameter is I, we have I : m :: 2 s: 2 sm,
Quadrant HB. Wherefore in this Cafe, t-
2 sm
T
therefore
4
T
X
IxGF = 1
T
2 5777
T
But
24
20
24
; allo
Ꭲ .
1"
; whence t
ณ
A
A:
4
X:T:: √Ā: ✔a; therefore
√ax 2√√a
I
*
392
INSTITUTIONS
I
4 X 2 Va
2 s m
X
VA
S
we have
m
a
But it beings R=
2 MS
P. Confequently t =
Р
in Seconds of Time.
SMa
/25 M4 (3247)
812
X
P
MS
2 PA
Ms
||
2
2 p A
BF2
3251. Becauſe v² V2 x
R2
(3247) we have V² ²
บ
BFZ
R² - BF²
- V² - V² x
V₁
- V² x
R2
R²
CB2
R2
V² s²
2
2 MS a
; then becauſe R² =
R²
(2247) we have V22
P
P
= √² s² X
2 M S a
Vz
3.
C2
C² Psz
3252. But
(2242) therefore V — ¿² —
a
2 MSA
And therefore M V2
M &² =
C² P s2
2 SA
Ce ps
This The-
2 A
orem may be of Ufe in that Part of Mechanics concerned in the
Doctrine of the Vires Viva & Mortuæ.
C
3253. Hence when vo, we have the initial Velocity V
J
J
P
ps
m
= C
And becauſe t =
2 MSA
2 MA
2
MS
2 PA
SIN
; therefore the Product of the Velocity and Time will
m Cs
be V t =
MSP
m Cs
X
4 A
MSP
4 A
3254. And becauſe
m C
4 A
is a conftant Quantity, we have
Vts; or the Space through which the Spring is bent always as the
Velocity and Time conjointly, the fame as in the Cafe of Bodies
defcending by Gravity (971).
3255. Wherefore in the fame, or different Springs, the Spaces
through which they are bent in a given Time will be as the Velocities;
and with a given Velocity, they will be as the Times employed in
bending them.
3256.
Of CLOCK-WORK.
393
m
3256. Becauſe in the Expreffion of the Time, t = 2
MS
2 PA
the Quantities m and 2 A are conftant; therefore it
will always be t
S
MS
P
; or t² P: MS. But we had alſo t :
Confequently MSV: ts P.
MS S
✩ i
(2254) therefore { :
tP V
Hence we obtain a Compariſon between theſe fix principal
Quantities in a Spring, and a Body bending it with a percuſſwe
Force.
3257. Thus if the Length S and Force P of a Spring be given
we have MV:ts; but in this Caſe ť² : M (2256); therefore
tV:s, or the Space is as the Rectangle of the Time and Velocity, as
in Falling-bodies (991).
3258. Likewife in the fame Spring, ſince 2: M, therefore
when M, or the Body is given, the Time (1) of bending the Spring
will be the fame, whatever be the Degree of Velocity V, or the
Spaces through which it is bent.
3259. But in a given Spring S, P, and Body M, fince t will
be conftant, we have Vs; that is, the Space through which the
Spring is bent, will always be proportional to the Velocity of the Body 3
which is another Cafe fimilar to defcending Bodies urged by dif-
ferent accelerating Forces.
3260. It was proper on fome Accounts to give the foregoing
Theorems expreffing the Velocity, Time, &c. in the Form they
there have; but in the particular Cafe of the Space deſcribed,
or Deſcent in one Second, we have 2 AC; thofe Theorems,
therefore, will be expreffèd more fimply for Calculation, thus
772
(putting = 3,1416) t == X
MS
M S
= 1,57
2
2 AP
CP
=1,57
✓
MS
; and V CX
xd
Ср
ps
2 AM
ps
=0
Сх
CM
m Cs
Cps; and MV² = Cps; and lafſtly, Vi=
M
VOL. II.
E e c
4 A
===== 1,575 3
394
INSTITUTIONS
}
S
= 1,57 s; or t = 1,57, in Seconds. And in all theſe Cafes,
C = 32 Feet, or 386 Inches.
CHA P. VIII.
The foregoing THEORY of SPRINGS farther confi-
dered, and exemplified by CALCULATIONS in their
various APPLICATION to WATCH-WORK.
a
3261. T has been already fhewn, that an Automaton put into
I I
Motion by a Weight, and having that Motion regula-
ted by a Pendulum, is the moſt perfect that can be in Nature;
and that in the next Degree of Perfection is the Spring-clock ;
but when we confider the Structure of a WATCH, left wholly
to the Power and Action of Springs, we fhall find it naturally
deftitute of any abfolute Principle of equable Motion.
3262. The Action of a Weight is conſtant, and the Oſcillations
of a Pendulum are equable, from an innate Principle; but we
cannot conſider the Action of a Spring either as a Fir-mover
or Regulator of Motion, as conftant and equable in its artificial
Application in Watches; this we have alfo fhewn in regard to
the Firſt, in the Theory of the Spring and its Fufee; and in reſpect to
the Second, the Office of the Spring as a Regulator, is not de-
rived from the Natural, but what we may properly call, the ar-
tificial Vibrations thereof. The natural Vibrations of all Springs
are in themſelves as perfectly ifochronal as thoſe of Pendulums
(3232); but the Cafe is quite otherwiſe with reſpect to the arti-
ficial Vibrations, whofe Equability of Motion, being the Reſult
of the compound Action of two Strings together, cannot be fup-
pofed fo perfectly conftant, and regular, as no mechanical Combi-
nation of Caufes can act with the Simplicity and Uniformity of
Nature itself.
3263.
Of CLOCK-WORK.
395
3263. If the Watch be required to beat Quarter-feconds, or
to oſcillate 14400 Times in an Hour, theſe Ofcillations by a
Pendulum are rendered equable without any Trouble; but to
effect this by a Spring, or Number of Springs, will require
much more mechanical Skill and Contrivance, and at laſt be
attended with fome Degree of Inconftancy or Irregularity.
3264. Indeed, the fine Spring connected with the Balance is
to be confidered as fomewhat more than a Regulator; for it does,
as it were, form or modulate thoſe Ofcillations, as well as regu-
late them; without fuch a Spring the Balance would ofcillate,
it is true, but fince the Motion produced by the Action of the
Crown-wheel, on one Pallet, must be in a Moment ſtopped,
deitroyed, and generated anew in a contrary Direction by its
Action on the other Pallet, and this in fo quick and conftant an
Alternation, the Effect would be too violent and fhocking to be
fufferable, and too irregular to be of any Uſe.
3265. But by the Application and Co-operation of the Spring,
the faid contrary Motions are gradually generated and deſtroyed s
and the Times of the Oſcillations of the Balance-wheel render-
ed as equable and uniform as the niceft Mechaniſm can admit
of. To effect this, many different Methods have been invented,
principally by Dr. Hooke, Mr. Hugens, Mr. Leibnitz, &c. Some
of theſe were by ſingle Balances, others by double Ones; fome
had only one regulating Spring; others had two, or more.
The
Attempt to regulate a Watch by a Loadſtone inſtead of a Spring,
is not worth mentioning.
3266. There were two Ways in which the double Balance
was applied; in one of them each fingle Balance had a Verge
with one Pallet only, placed on each Side of the Crown-
wheel diametrically oppofite to each other, for that Wheel had
the fame Poſition with the contrate IVheel, or as it now has in the
prefent horizontal Watches. On the Verge of each Balance was
fixed a ſmall Wheel; thefe Wheels were proportioned to the
Diameter of the Crown-wheel; having the fame Number of
fine Teeth, they played in each other, and fo gave an equal
Motion to the larger Balance-wheels juſt above them.
3267. The other Way, with two Balances, had the two
fmall Wheels, by which they moved each other, and more-
E ẹ ẹ 2
over
396
INSTITUTIONS
over to each Balance-verge, there was added a ſpiral Spring as
a Regulator. In this Method, one Balance only had a Verge on
which both the Pallets were, and it was moved by the Crown-
wheel, placed in the fame perpendicular Poſition it now has in
common Watches. With refpect to theſe two Inventions, Dr.
Derham fuppofes the firft was never profécuted fo far as it
deferves; and the Second has this Excellency, that no Jerk or
the moſt confufed Shake can in the leaft alter its Vibrations,
And that the Reafon why this Method of conftructing Watches
came into Difufe, he judges was the great Trouble and vaft
Nicenefs required in it. If this was the Cafe, it reflects no great
Honour on the Reputation of Artifts in this Way.
3268. The Spring, as a Regulator, is applied in Watches,
with one End fixed to the Verge of the Balance, and being coil-
ed feveral Times round, to give it a fufficient Length, it has its
other End faftened to a Part towards the Extremity of the
Watch-plate, which has lateral Teeth, and is moved by a ſmall
Pinion on an Axis in the Center of a fmall filvered Plate on one
Side of the Cock or Balance, divided into 30 fmall equal Parts.
This Plate is alfo fixed on the faid Axis, and by the Key is moved
againſt an Index to the Right or Left, thereby increafing or re-
mitting the Force of the Spring when the Watch goes too flow
or too faſt; and this Apparatus for correcting the Regulator,
and thereby the Time of the Watch, we find by common Ex-
perience, is but too frequently neceffary in the moſt excellent
Pieces of this Sort of Machinery.
3269. Mr. De la HIRE, in a Memoir of the Academy of Scien-
ces, objects very much to the common fpiral Form of the Regu-
lator, and eſpecially to the fixing one End thereof to, or near
the Verge or Axis of the Balance. He thinks, if the End of the
common fpiral Regulator were faftened to a Part of the Radius of
the Balance inftead of the Axis, it would have more Force to
govern the unequal Movements thereof, and at the fame Time.
be lefs fubject to its Irregularities; and farther adds, that the
Spring bent into an undulatory or wave-like Form (as in Fig. 5.)
is much better than the Spiral, in that a greater Length may be
had in a lefs Space, and the Spring thereby fuftain itſelf with
greater Eafe, and act without that Compound and diſtorted
Mo
Of CLOCK-WORK.
397
Motion which the Spires of the common Regulator are ſubject
to, in their lengthening and fhortening. And what he has afferted,
he tells us, he has fully evinced by Experiments with Watches
of this new Conftruction.
3270. But I have conftantly obferved, that theſe Inventions,
Alterations, and Innovations concerning the Balance and its
Regulator, have all proceeded (rudi Minerva) from the natural
Force of mechanical Genius, with very little, if any, Rationale
from Principles of a philofophical or mathematical THEORY of the
Nature and Action of SPRINGS; which certainly muſt reflect very
great Light on a Subject that has always been looked upon as
overwhelmed in Difficulty and Obfcurity. For, (if I judge
right in theſe Matters) there feems to be a much better and more
natural Method of governing and regulating Watch-work poin-
ted out by the foregoing Theory, than that of the Balance-wheel
and its ufual Regulator.
3271. For by this Theory, it is evident, that with Reſpect
to the Power of the Spring, the Momentum of a Body which ſtrikes
it, the Space through which it is bent, and the Time of bending
it, if any of theſe Quantities are known or given, the reſt may be
found; becauſe their Relations are all determined by the Theo-
remst 1,57
Ms
Cps
and V =
T P
(2250) as alfo the
M
real Quantities of each reſpectively..
3272. Therefore fuppofe the Train of a Watch be 14400,
or 4 Beats per Second; then let A be a Body fufpended by a
Thread AC from the Center C, (Fig. 6.) and being raiſed to
the Point D let it defcend through the Arch DA and ftrike the
Spring BF with the Velocity acquired in that Deſcent; and
then A a is the verfed Sine, or perpendicular Defcent of the Bo-
dy to acquire the fame Velocity. Laftly, the Momentum of
the Body A will by its Impulfe on the Spring bend it through a
certain Space B H. Now for the given Length and Strength of
the Spring, and the given Space and Time of bending it, the
other Quantities may be found by Calculation, as follows.
3273. Confidering this Spring BF as a Regulator, it must be
very fine and tender; and therefore we will fuppofe it ſuch that
a fingle Scruple only, or 20 Grains, fhall be juft equal to its
whole
398
INSTITUTIONS
whole Force, or be able to keep it bent through its whole Length,
which may be one Inch, and let the Time of bending it through
half its Length be of a Second; then there is given BF=S
= 1, P = 20, s = 0,5 AH, p 10, and to",25;
to find M, the Weight of the Body A; V, the uniform Velo-
city of its Motion; and a = Aa, the Height it muſt deſcend
perpendicularly to acquire that Velocity.
=
3274. To find the Weight of the Body A, we have from the
Theorem (3260) t = 1,57 MS
386p
, this Equation 386 p t² =
2,4649 Ms; and therefore this Analogy, p: M::s: 156,6 ť²;
156,6 t² p
2
which will give M =
196 Grains, in the prefent
S
Cafe; fo that the ſtriking Body A is near 20 Times the Weight
of p.
3275. The uniform Velocity of its Motion is V =
ps
M
3,138 Inches per Second, this is equal to the Circumference
of a Circle A E F whofe Diameter is just one Inch.
A V2
V2
3276. Laftly, we find A a = a =
C² (3242) =
4 A
V2
=0,0127, or about
1 3
ΤΟ
of an Inch; and therefore the
772
Arch AD fo exceeding ſmall as to be altogether inconſider-
able.
3277. If therefore the Pallets of the Verge be fo proportioned
to the Teeth of the Crown-wheel as to move the Body A thro'
of an Inch each Stroke without the Spring; it will, when the
Spring is added, be moved through only half an Inch, or bend
the Spring from B to H in of a Second as required.
3278. If the Mafs of Matter in the Body or Globe A were
difpofed into the Form of a Circle A EF, or fo as to make the
Perimeter of a fine Wheel, it would then become the Balance-
wheel of a common Watch; and being connected with the
End of the Spring BF, that Spring would alſo become the Re-
gulator; and the Watch thus conſtructed, would beat Quarter-
feconds.
3279.
Of CLOCK-WORK.
399
3279. Hence in Watches of a large Size, and efpecially
Table-clocks, where more Springs than one may conveniently
be applied, this Regulation of Time might be moſt commodi
oufly performed by a fingle Balance-lever with two Springs,
or a double One with four; the Reaſon of which will appear
from (Fig. 7.) where AI and PQ are two Levers, whoſe
Weights A, I, P, Q, are equal to each other, and to the
Weight A in Fig. 6th. Alfo the Springs B F, IE, PH, GQ,
are all feverally equal to BH in that Figure. Theſe Levers
croſs each other in the common Center of Gravity C, where they
are fixed at Right-angles-to the Verge.
3280. As the Weights and Springs are increaſed in the Ba-
lance, it is neceffary the Force of the Spring at the Fufee ſhould
be encreaſed in Proportion. And it is further to be obſerved,
that in the quiefcent State of the Levers, the four Springs are
in a State of Compreffion; the two Springs fixed at F and H
being compreffed on each Side from (a) and the other two from
the Point (b). If a B be one Half of the Arch through which
the Levers ofcillate, then it is plain, that in each Ofcillation,
while two Springs are compreffed by one Lever, the other two
Antagoniſt Springs are. relaxing; and therefore as one Pair re-
tards, the other equally accelerates the Motion of the Levers ;
and ſo no Inequality of Motion can arife from the joint Action
of Springs; but on the contrary, as perfect a Correction of the
Irregularities of the Watch-work is obtained, as can be produced
by the Agency of Springs.
CHAP.
400
INSTITUTIONS
CHA P. IX.
The METHOD of finding the CENTER of OSCILLA
TION in all Kinds of PENDULUMS, deduced from
a New THEORY. Alſo the NATURE of a uni-
verfal MEASURE of LENGTH, or philofophical
FOOT, explained and exemplified.
3281. W thing valuable with regard to the Equation and
now quit the Subject of Watch-work, where no-
Regulation of Time can be hoped for, or expected; and return
again to the farther Confideration of thoſe Automata which are
regulated more by Nature than Art, viz. by the Power of Gra-
vity governing the Ofcillations of Pendulums. But here the
Artiſt muſt follow pretty clofely the Dictates of the omniscient.
Mechanic, and work, if he propoſes to merit Applauſe, by the
Rules of divine Geometry.
3282. Now as the PENDULUM is the Principle of Truth and
Perfection in Clock-work, all Circumftances relative to it fhould
be confidered with the greateft Attention, and principally that
which concerns the Center of Ofcillation; and here will arife the
following Queſtions, viz. what this Center of Ofcillation is in the
BALL OF BOB of a common Pendulum? What is the Diſtance
thereof from the Point of Sufpenfion? And how that Diſtance is
to be preſerved unaltered?
3283. I believe very few Mechanics in this Way know fo
little of Art or Nature, as to fuppofe that the Center of the Bob,
is the Center of Ofcillation. But fewer ftill know where it is,
or how to find it in the Pendulum. Yet the Knowledge there-
of is of the laft Confequence in very large Balls, and fome Mr.
GRAHAM had which exceeded 60 Pounds; alfo in fhort Pen-
dulums in Table Clocks, this Center of Ofcillation fhould be
nicely aſcertained. Mr. Hugens lays the greateſt Strefs on this
Point; and all his Folio-treatife is wrote profeffedly on this
important Subject.
1
3284.
Of CLOCK-WORK.
401
3284. But the Method he pursued to diſcover it, will give
us too much Trouble and Fatigue; a nearer and eaſier Way
has, fince his Time, been found out; and which we ſhall now
elucidate in that Cafe firft, where the Weight of the Ball or Bob
is fo great, that the Weight of the Rod by which it is fufpended is
inconfiderable in Compariſon of it. In order to this, a Retroſpect
to a few Things already demonftrated in theſe Inſtitutions will be
neceffary, and tend greatly to ſhorten the Operation.
3285. Let A B be a Line ofcillating about a Point or
Center A; the Diftance of the Center of Gravity A G
=g, and of the Center of Oſcillation ANn; x = a
fmall Particle or Weight; x = its Diſtance from A ;
and let the Sum of all the Particles or Weight of the
whole Line be S. Then x is the Moment, and xxx
the Force of the Weight; and the Sum of all the x2 is
*3
and when
3
S3
S, we have for the Force F of the
S, we have
✰
3
whole Line. All which is evident from (1086 to 1098).
S3
A
939
3286. But
S × 15 × 3 S = Sgn=F, agreeable to
3
what was fhewn in (1095). Alfo when
x3
S3
x =
S3
S, then the
Sum of all the xx
will be
; and twice that
3
3 × 8
24
=
I 2
Sum will be 2 S3 S3; which therefore will be as the Sum
24
of the Products of all the Particles (*) each multiplied by the Square
of its Diſtance from the Center of Gravity G.
3287. Therefore put S39; and we fhall have Sgn=
q+z; and Sgn — q =
I 2
=
S2
z = 1 S³ SxSg². There
Sg
4
fore Sgn — Sg² = 9 = $g X-g. And confequently
n
9
Sg
n gGN; which gives this general Rule for finding
the Diſtance of the Center of Ofcillation N from the Center of
Gravity G, viz. divide the Sum of the Products of the Particles or
Weights, feverally multiplied by the Squares of their Diſtances from
the Center of Gravity, by the Solidity or Maßs multiplied by the Dif-
tance of the Center of Gravity from the Axis of Ofcillation, and the
VOL. II,
Fff
Qua-
لية الان
402
INSTITUTIONS
唇
​Quotient will be the Diſtance of the Center of Ofcillation from the Cen-
ter of Gravity.
9
d
3288. If we put GN= d; then becauſe q = Sg d it will be
F
72
Sg=
(2286) therefore qn Fd, and we have F: q
::n:d::AN: GN. The reaſoning is the fame in regard to
Planes and Solids, and confequently the Rule is the fame in all;
but I have chofen the eafieft and moft fimple Method by which
it may be demonftrated. As a Proof of its Univerſality I ſhall ap-
ply it to find the Center of Ofcillation in a Globe, and fhew it
to be the fame as Hugenius found it with fo much Labour and
Prolixity.
y, GP
3289. Let DE d be the Section of a Globe through its Axis;
Dd the Diameter perpendicular to the Axis K L of Oſcillation;
GE the Radius at Right-angles to Dd; G the Center of Gra-
vity, and V the Center of Ofcillation in the Line of Suſpenſion
OG; and let PFp be any concentric Circle. Let the Ordi-
nate PM
x, and n: I: Circumference of a
Circle to Radius; and draw N M parallel G D. Then fup-
pofe a cylindric Surface to ftand on the Circumference PF p
perpendicular to the Plane D E d, and terminated by the Sur-
face of the Sphere, then becauſe b: 1 :: p:x, we have nx = P
Circumference PNpP, and therefore 2ynx to the faid
cylindric Surface (832).
3290. Now GP = x, is the Diſtance of the Particles in
each Section of this Surface (perpendicular to the Axis of Ofcil-
lation) from the Center of Gravity of the Section; therefore
2 nyx × x² = 2 nyx x, is the Fluxion of the Surface or Weight,
which multiplied by (the Square of the Distance from the Center
of Gravity) GP² — x², gives the Fluxion 2 ny x3 x, whoſe
Fluent is nyx+.
2
3291. The Sine and Có-Sine of an Arch are inverſely as their
Fluxions; for (in Fig. 4.) the Triangles fd F and F Cg are
fimilar, therefore fd: dF :: Cg: gF; that is,
whence we have x =
X
::: y : x ;
; and putting the Radius of the Globe GD
2
=a, we have xay; therefore 2 nyx³ = 2 ny ×
Va
yi
شرو
x-
x
× ×³ = 2 nj²j × a² — y² = 2n a² y² j—2nyªj,
·
whoſe
Of CLOCK-WORK.
403
3
whofe Fluent is na² y³ — 2 nys; and when ya, this Fluent
4
3
5
is nas, analogous to the Quantity (q) in the general Expref-
fion (2287).
6
3292. Again, the Solidity of the Sphere is pd² (836) but
p = na, and d = 2 a, therefore S
= 1/3 × na × 4 a²
Zna³,
and putting OG = z, we have na³× z analogous to Sg in
उ
2
(2287); therefore na³ z) nas (= }}
3
4
I5
Z
GV, the Dif-
tance of the Center of Ofcillation from the Center of the Sphere,
as required. And is the very fame with that found by Hugenius,
Bernouilli, Varignion, Mac Laurin, Simpfon, Emerfon, &c. by
tedious and obfcure Methods.
3293. If Odd, then O Gd+az; and then alfo
2a2
5%
-
2a2
5d+5a
; and when do, or the Sphere ofcillates
2 a²
about a Point d in its Surface, we have dV = = 2a, or
5 a
7% of the Diameter dD, and not, as it has been determined by
Mr. Carrè, Hayes, Stone, and others.
3294. Put
2a2
5%
(GV =) v, then 2 a² = 5 zv, and aª
vz; therefore z: a::a: 5 v. Now the Length of a
Pendulum vibrating Seconds is 392 Tenths
ſuppoſe the Diameter of the Globe or Ball
a = 20, and z + v=OV =
20
OV = 392.
of an Inch. Then
4 Inches, we have
And then 392 + v:
20 I, very nearly; fov = 1, or 5 v = 2, and v = 3
of the Tenth of an Inch, or of an Inch.
I
25
3295. In a whole Day, or 24 Hours, there are (24 x 60 =)
1440′; and if we put OV = 1440, it will be eaſy to find the
Length of a Pendulum that ſhall gain juſt one Minute per Day.
For fince the Times of a Vibration in the two Pendulums will
be inverfely as the Number of Vibrations performed in a given
Time (or one Day,) thofe Times will be as the Numbers 1440
and 1439. Again, the Lengths of Pendulums are as the
Squares of the Times of their Vibrations, or as 1440² to 14392;
but 1440²: 1439²:: 1440: 1438, very nearly; then 1440:
1438: 392 391,46, then 392 391,460,54, or a lit-
tle more than of an Inch; therefore 25: 20:
20 25::
20
-
Fff 2
60"
404
INSTITUTIONS
60" 48". Whence it appears, that the Diſtance GV, ſmall
as it is, in a Second Pendulum, is enough to make the Clock er
48 Seconds per Day, if not regarded.
I
3296. A Cubic Inch of Braſs weighs 4,4 Oz. Troy, and the
Cube being to its infcribed Sphere nearly as 2 to 1 (846). A
Sphere or folid Braſs Globe of one Inch Diameter, will weigh
2,2 Oz. Troy; therefore as 13:4³:: 1: 64 :: 2,2: 141 Oz.
or 11 lb. Troy, (841) the Weight of the Ball of the above-
mentioned Pendulum. But if it were 5 Times heavier, the
Quantity G V would ſtill be greater in the Ratio of the Squares
of the Diameters of the Globes; and hence it appears, of how
great Conſequence it is, in the Science of meaſuring Time, to be
able to aſcertain this Point or Genter of Ofcillation in fimple Pen-
dulums.
3297. Before we quit this Theory, it will be very material to
obferve, that the Doctrine of a perpetual and univerfal MEASURE
is founded in it; and that what is called a phyfical or philofophical
Y YARD, is nothing more than the Length of a fimple Pendulum
vibrating in a Second of Time. That is, the univerſal YARD is
equal to 392 English Lines, or Tenths of an Inch. And the uni-
verfal Foor is a third Part of that Number, or 130 of ſuch
Lines. But this Notion of a HORARY YARD, and FooT,
and the Manner of aſcertaining it by the Pendulum of a Clock
well adjuſted to equal Time by the Revolution of the Stars, as
defcribed by HUGENIUS, is to tedious here to infiſt on.
3
3298. I fhall offer the following Method as the moſt conciſe
and eaſy for this Purpoſe. Let any Meaſure propoſed be confi-
dered as a uniform folid Body, whoſe Length is the general Stan-
dard. If fuch a Meaſure or Solid were fufpended on an Axis to
vibrate freely on one End (paffing through the central Line of
the Axis) it has been fhewn, that the Center of Oſcillation in
fuch a Body will be juſt of its Length from the Axis of Sufpen-
fion (1097). And therefore any fuch Body equal in Length to
392 + 196 = 588 Tenths of an Inch, will vibrate exactly in
one Second of Time; and is of Courfe, the univerfal or philofophical
YARD.
मु
3299. This YARD muſt be divided into 10000 equal Parts,
in which the Length of other Standard Meaſures are to be ex-
preffed
Of CLOCK-WORK.
405
preffed by comparing the Times of their Vibrations with that of
the univerfal Yard. For this Purpofe, let its Length be L=
10000, and N 3600, the Number of Vibrations in an Hour;
and let / and n denote the fame Quantities in any other Meaſure
propoſed; then we have T : t :: √✓L : ✅/l ::n: N, the Times
of a fingle Vibration T, t, being inverſely as the Numbers N and
» performed in a given Time. Whence this Theorem VLX N
n
=
n
3300. Then as the English Standard Foot is 120, fay, as
588: 10000 :: 120: 2040; ſo that the English Foot is 204
of the fame Parts of which the univerfal YARD contains 1000.
Now this is diſcoverable from its Number of Vibrations in an
Hour, which ſuppoſe were found, by a well adjuſted Clock to be
✓10000 × 3600
7969; for by the Theorem
1=2040, as before.
7969
=✓ whence
3301. After the fame Manner the Paris Royal Foot would be
found 2179,6 Parts of the univerſal YARD; and therefore as
2040: 2179,6 :: 1000: 1668 :: 12 Inches: 12,8 Inches; or
more accurately the Paris Foot is 12 16 Inches. The Philofo-
phical, Engliſh, and Paris Feet are therefore as 333,3, 204,
218; and after this Manner, by counting the Vibrations made
in a Minute or an Hour by any other Standard Meaſure, may its
Length be ascertained, and Ratio expreffed, in Parts of the uni-
verfal YARD.
3302. And as a farther Illuſtration of this Matter, I have re-
prefented thefe Meaſures feparately in the Copper plate (Fig. 9.)
where A B is the philofophical YARD; CD the philofophical Foot;
EF the English Foot; and GH the Parifian Foot; all in their
due Proportion of Length. From what has been faid, it is evi-
dent of how much Importance the Theory of a fimple Pendulum
is, not only as the moſt perfect CHRONOMETER, but as a STAN-
DARD for Meaſures of LENGTH of every Kind, and to all Ages.
CHAP.
406
INSTITUTIONS
CHA P. X.
The THEORY of a Compound PENDULUM explain-
ed, and its CENTER of OSCILLATION invefti-
gated.
•
3303. T next in Courfe, nor is the Theory thereof lefs cu-
HE Confideration of a Compound Pendulum comes
rious or neceſſary than that of a fimple One, fince in the com-
mon Conftruction of Clocks, the Weight of the Rod or Wire is
too confiderable to be neglected; and then the Rod and Ball to-
gether are to be eſteemed as a Pendulum confifting of two diffe-
rent Weights, whoſe compound Center of Gravity, and Center
of Ofcillation are to be determined.
3304. Again, after all the Efforts of human Art or Skill, the
Pendulum can never be perfectly equable in its Motion, but will,
from the very Condition of Nature itſelf, be liable to have the
Times of its Vibration altered, by an infenfible Acceleration or
Retardation from the variable State of natural Caufes. There-
fore, for common Ufe, HUGENIUS recommended a moveable
Ball, or Weight for correcting the Irregularities of Vibration,
as by its different Pofition on the Rod, it will cauſe a ſmall Alte-
ration of the Diſtance of the Center of Ofcillation, by which Means
the Clock may be adjuſted to true Time.
3305. In a Pendulum of this Sort there are three different
Weights to be confidered, viz. that of the large Ball or Bob,
that of the Rod, and that of the Corrector, moveable upon it.
The Center of Ofcillation of all which muft be found, which
will not be difficult after we have found that for two Weights.
Therefore let AC (Fig. 10.) be a Pendulum, Bifect its Length
in G; and put AG g, and A Ca. Alfo call the Weight
of the Bob (c) and that of the Rod or Wire (b).
3306,
Of CLOCK-WORK.
407
I
3306. Now the Momentum and Force of each Part are ſe-
parately to be eſtimated. The Momentum of the Rod is uni-
verfally ab (1072); the Moment of the Ball C is (a c), being
as the Weight multiplied by its Diftance (1089). The Force
of the Rod is b × 1 a × 3 a (1095) = ½ ba²; and the Force
of the Ball C is ca² (1089); therefore the Sum of the Forces di-
vided by the Sum of the Moments, gives the Diſtance (2) of the
} ba² + ca²
2
Center of Oscillation, that is,
as in (1094). And from
AC a
༢་
b n + n c
3/3 b + c
उ
3
3
½ ba + ca
-n=
3
ba+ca
= b + c
hence the Length of the Pendulum
which is therefore determined for a
Second-pendulum, by putting n 39,2 Inches; or for Half-fe-
conds, if n = 9,8 Inches. Note, in all thefe Cafes, the Length
A C is the Diftance between the central Line in the Axis of Suf-
penfion, and the Center C of the Bob.
3307. Now let AC (Fig. 11.) be the Pendulum with the
Addition of the Corrector D, or ſmall moveable Weight = d.
And let its Diſtance be AD = f. Then will its Momentum be
fd, and its Force dff; and now the Sum of the Forces of each
Part divided by the Sum of the Moments will be a²bca²+df²
Lab + ac + df
the Distance of the Center of Ofcillation or Length of an ifochronal
Pendulum, which for the Future we will call p.
3308. Then if it be required to find (f) or the Pofition of the
Corrector D, we have
ན
this Equation² = ƒp +
½ ab + ac + df
a² b + ca² + dƒ³
=p, which gives
2
1 abp + cap —
— a² b — a²c
then
ď
2
by compleating the Square, and extracting the Root, we have
f
ƒ = { p ± √ √ = p² + ÷ abp + cap —
2
— 3
d
a² b
a² c
3309. Hence it is obvious, there will always be two real
Roots or Values of (f) while
abp + cap is less than a ab
+ aac, or while the Length p is leſs than
3
{{ a² b + a² c
which
÷ ab + ca
is
408
INSTITUTIONS
7
is the Length of the ifochronal Pendulum (~) confifting of the
Rod AC, and the Weight C only (3306).
3310. When, therefore, we would accelerate the Motion of
the Pendulum by the Application of the Weight D, we have
the Choice of two Places for it between A and C, viz. D or E,
which Places are equally diſtant from N, the Point which Bifects
the ifochronal Pendulum p. For for the radical
Quantity as above (3308).
3311. Therefore, when fpAN, it will accelerate
the Motion of the Pendulum the moſt of all; in which Cafe we
1 p² + ½ abp + cap — — a² b. — a² c
have
1
- 3
d
@, which
½ abp + cap
— a² b + a²c
उ
d
d
will give the Equation 1 p² +
from which, by Reduction, compleating the Square, &c. we
hall find p =
ab
z d
2 ac
a
zd
2 d
4 bd + 4 cd + b b + 4 b c + 4 cc-
; and from hence the Value of fip, is known
when its Effect is a Maximum.
3312. For accommodating this Theorem to practice it will
be moſt convenient to affume b dI; then we fhall have
the Equation for AD = ƒ=ip±√ 1 p² + 1 ap + acp -
f
}; a² — a² c; and when ƒ =p; then p =
a x I- 2c.
ax
2
4
a
2
√2+ 8c + 4 6 6
3313. To give an Example of the Ufe of thefe Theorems;
let the Pendulum of the Clock be required to beat Seconds pre-
ciſely, and ſuppoſe the Weight C =
50lb. and b
= d Ilb.
Then we have n≈ 1440 (2295), c=50, and the Theo-
rem in (2306) will become
½ n + nc
3/31/3 + c
=a= 1444,8. And by
ſubſtituting theſe Values of a, b, and d, in the Equation for (ƒ),
we get ƒ = p±√ p² + 72962 p — 105061210.
f
2
=
3314. Therefore when the Accelleration is greateſt of all,
we have fp; and p²+72962 p 105061210, or p² +
291848 p 420244840; whence we find p≈ 1436, nearly;
but
Of CLOCK-WORK.
409
but 144014364', or the greatest Acceleration is 4 Mi-
nutes per Day, by placing the Weight D at N, or makingƒ:
AN = 718 = 1, when a Minimum.
3315. Since by fhortening the ifochronal Pendulum (p) only
4 Parts of the 1440, the Length of ƒ AD, or the Space
through which the Weight D moved upwards is 718; it is evi-
dent that a ſmall Alteration in the Length of A C, will allow of
a very large Scale of Correction on the Rod of the Pendulum both
in regard to retarding as well as accelerating its Motion.
3316. For, to accelerate the Motion of the Pendulum one
Minute per Day, the Center of Ofcillation moves through but
2 Parts of the 1440, or the Length of the Pendulum is then p
1438 as we have fhewed (3295). Therefore putting n
1438, we ſhall have by Theorem (3306)
1 b n + n c
2
3 b + + c
2
酆
​5/5
- a
1441,8 which is juſt 3 leſs than 1444,8, the Length for fwing-
ing in a Second precifely (3313). Hence the Motion of the
Center of the Weight C is to that of the Center of Ofcillation
as 3 to 2. And it is univerfally an:: b + c : } b + c ;
or, in the preſent Cafe, an:: 50,5 50,3: 1444,8: 1438.
392 391,46, as we have before obferved (3295).
3317. From all which it appears, that the Center of Oſcilla-
tion is but 5 Parts of 1440 diſtant from C the Center of the Ball,
and alſo, that when the Corrector D has its Pofition, ſuch as to
produce no Acceleration, it muſt be either upon the Center of
Ofcillation, or at the Axis of Sufpenfion A; or ƒ pa
=AC, very nearly. But in the common Conftruction of
the Pendulum this Difpofition of the Corrector D can have no
Place, becauſe it cannot be brought nearer to the Center of the
Weight C than the Length of its Semidiameter, which, in a
Ball or Globe of 12lb. only, is 2 Inches (3296); but in a Globe
of 50lb. it is more than 3 Inches. Now this is more than the
Diſtance to which the Weight D muſt be removed to accelerate
VOL. II.
Ggg
7
the
N. B. The Reader is defired to correct the Expreffion of the Radical Symbols
in Institutions 3308 and 3311, thus ;
1
2
+ p² + ½ a b ; + ca p
I
Jazb
#
INSTITUTIONS
410
|
=
the Motion I Minute per Day; for by the Theorem (3313) if
we put p 1438, we fhall get f= 1331,5; if then we fay,
as 1440 1331,5: 392: 362,5, we have the Diſtance of the
Weight D from the Center C, but 29,5, or not quite 3 Inches.
3318. Therefore in order to have a Scale of a fufficient Vari-
ation for the Pofition of the Weight D, we must give to the
Rod AC a greater Length in Compariſon of the Length (p) of
a Pendulum vibrating Seconds, and confider the Weight C or
Bob of the Pendulum as moveable upon it in the fame Manner as
the Weight D is. Thus let AB = a, be the Length of the
Rod or Wire; and A Ce, the new Diſtance of the Weight C
upon it; then will the Theorem (3306) become
87
p; and therefore a² b + e² c = 1 abp +
I
8² c + ecp = ½ pab-a2b; and e²ep-
ž
-
Q
Z
I 2
ङ्कु
a²b+ e² 6
½ ab + ec
ecp, whence
½ pab — — a² b
2.
3
pa —², by putting b = 1, as before. Then com-
C
pleating the Square and extracting the Root, we get e =
= p² + ½ pa — — a²
3
C
And the Theorem in (3312) for A D
will now become ƒ = ? p± √ = p² + { pa + pec = {a²
— — e²c.
Theſe Theorems for AD and A C afford a new Conſtruction
of the Pendulum whofe Nature and extenſive Ufe will be fully
declared in the enfuing Chapter. (See the Plate of the Univer-
fal Pendulum, Fig. 1, 2.)
1
CHAP.
!
Of CLOCK-WORK,
411
i
'
CHAP. XI.
The THEORY and CONSTRUCTION of an Univerfal
PENDULUM for measuring the Time of Solar,
Lunar, and Planetary DAYS very correctly.
IN order to fhew the Univerfality of the new Structure
3319. Norte Pendulum premifed in the foregoing Chapter,
it will be neceffary to confider that the Defign of a Clock
- is to meaſure conftantly, that equal Portion of Time which is
called a Mean Day, and is divided into 24 equal Parts or Hours.
As Time in itſelf flows equally, the Clock, if it could be kept.
to an equable Motion, would be an adequate Meaſure of it;
and this would be the greateſt Excellency and Uſe of ſuch a
Machine.
3320. But this, however, has, in my Opinion, been very
indifferently provided for in the governing Principle of a Clock's
Motion, viz. a Pendulum of ifochronal Vibrations; and it muſt be
obferved, that after all the Precautions and Inventions for ren-
dering this Ifochroniſm of Vibration permanent, yet fuch a Dif-
covery ſtill remains too obviouſly the great Defideratum of Clock-
work, and therefore Artifts are obliged to have recourfe to the
beft Methods of remedying fuch Irregularities, and of checking
them even in their Nafcent State.
3321. How grofs and indirect the Methods of rectifying the
Pendulum in general are, if compared with that of the fecondary
or moveable Weight D, invented by Hugonias, will be very evi-
dent on mature Confideration. And as it is here improved, I
flatter myſelf it will be allowed the moit eafy and extenſive that
can be defired, when it is confidered, that the Scale of Variation
for the Corrector D is by this Means made fo large as to render
any Clock capable of meafuring not only the Mean Time or
Day, but alſo that meaſured by the Motion of any of the cele-
Ggg 2
ftial
412
INSTITUTIONS
ftial Bodies, whether Moon, Planet, or Star; as we now pro-
ceed to fhew.
!
3322. The Time that intervenes the departing of any cele-
ftial Body from the Meridian and its Appulfe to it again, is call-
ed the Day peculiar to that Luminary; but fince none of the
Planets defcribe, circular Orbits, but Ellipfes, their Motions.
will be variable, and the Times of their meridional Revolutions,
or Days, will be unequal; therefore the Mean Revolution, in
all, muſt be taken for the Mean Planetary Day; and theſe col-
from the aftronomical Tables are for the feveral heavenly
Bodies, as in the Table here fubjoined.
le
H.
3323. A fixed STAR
23 56 3 28
1436
SATURN
23 56 11 00
О
8 О
1436 /
JUPITER
23:56 23 00
O 20 О
1436/-/3
MARS
23 58 08 58
2
5 30
1438
SUN
24 00 00 00
3 56 32
1440
VENUS
24 02 28 00
6 24 32
14421
MERCURY
24 12 25 28
16 22
O
14523
Leaſt
(Leaſt 24 43 08 28
47 5 0
14833 3/30
52 40 0
1488,7%
Greateſt 24 57 07 28
61 4
4 0
1497/1
}
MOON Meán 24 48 43 28
3324. The firft Column in this Table fhews the Hours, Mi-
nutes, Seconds, and Thirds in each planetary Day, the Mean So-
lar Day of 24 Hours being the Standard. In the ſecond Column
are contained the Minutes, Seconds, and Thirds, by which each
following Day exceeds the first or Sidereal Day; thus the Mer-
curial Day is longer by 16 Minutes, and 22 Seconds; and the
greateſt Lunar Day by 61 Minutes, and 4". In the third Co-
Jumn, the Quantity of each Day is expreſs in the Minutes of a
Mean Solar Day.
3325. If therefore the Length of the ifochronal Pendulum (p)·
in the foregoing Equations be found for the Numbers in the 3d
Column refpectively, then will the Distances A D, and A C,
or the Pofitions of the Weights D and C be found by the The-
orems in (2318), fo that the Clock fhall keep Time with the
2
Pla-
>
Of CLOCK-WORK.
413
#
Planet, and Hour-hand point to the Hours when the Planet
fhall be on the correfpondent Hour-circles. So that by this
the Time of
any Planetary Day, or the Pofition of the Planet in any Part of
its Revolution.
manada *
Means any Clock may readily be adjuſted to thew
3326. But for this Purpoſe the Length of the Rod A B must
be fufficient to allow of a Scale of Pofition to the Corrector D,
while the ifochronal Pendulum (p) encreaſes from its leaft Length
for the Sidereal Day of 1436′ to its greateſt Length for the
longeft Lunar Day of 1505', in which Cafe the Weight or Bob
C will poffefs the Extremity of the Rod, or A Ce will
become AB = a. And becauſe as the Days encreaſe, the
Hours, Minutes, and Seconds increaſe in the fame Proportion (as
there is a conſtant Diviſion of each Day into the fame Number,
viz. 24 Parts), therefore, alfo, the Time of the Pendulum's
Vibration will encreaſe with the Length of the Day. And con-
fequently the Numbers in the 3d Column of the Table will be
as the Times of Vibration in the Pendulums appropriated to the
Clocks for fhewing the Time of the Days refpectively.
=
3327. Therefore it will be neceffary in the firft Place to de-
termine the Length of P the ifochronal Pendulum whofe Time
of Vibration is as 1500, by faying, as 14402: 1500²:: 1440:
1562 P. And becauſe the larger the Weight C is, the lefs
will be the Scale of Acceleration, therefore we must take the faid
Weight, fuch as will allow this Scale to be fufficiently large to
comprehend all the Variation of the Moon's Motion; and by
Trial, it will be found that 10lb. will be the greateſt Weight
that can be allowed to the Bob; therefore c = 10; b and d be-
ing, as before.
d: 3328. Hence we determine A B, the Length of the Rod, by
the Theorem
P + ¿P
3 + c
a=1587,8 as in (3306). Alfo
by the Theorem in (2311) we find the Length of an ifochronal
Pendulum, when a Minimum, to be p 1526,2; therefore
1562,5 — 1526,2
1526,236,3; and half this Number, viz. 18
fubducted from 1500 leaves 1482 for the Expreffion of the Time
of a Vibration of the Pendulum () compared to the Time 1500
of the Pendulum P.
3329.
می
}
1
*}
414
INSTITUTIONS
3329. That the Reafon of this Affertion may appear, we
fhall demonftrate this Lemma, viz. when Numbers are very large,
and their Differences very small, then any three or four of them that
are in geometrical Proportion are alſo in arithmetical Proportion.
Thus let a, and ad, be two large Numbers, for Inftance 1440,
and 1439, then d
Part of a; and let it be a: ad :
adiz; then
སང
***
E
i
1440
a²
2 á d +
d d
a 2 d, becauſe
a
7
1
1
4
!
(dd) vanishes in Compariſon of the reft; therefore a, ad,
a2d, are in geometrical Proportion; they are allo evident-
ly in arithmetical Proportion; and hence 1440, 1439, 1438
are Numbers that have the fame Property; for fince 14402;
14392: 1440 : 1438, therefore
:
14392
1440
1438, and confe-
quently 1440 1439: 1439: 1438, and fo they. are geome-
trical and arithmetical Proportionals at the fame Time.
3330. Since the Number 1483 anſwers to the fhorteft Lunar
Day in the Table (3323) and the Difference between that and
the longeſt Day is but about 14 Minutes; it is evident, this
Scale of 18 will ferve for all Lunar Days. And the Values of
P and f may be found for every Number betwixt 1483 and 1497
by the Theorems in (3307, 3308). And the Scale on the Röd
of the Pendulum may be graduated for Ufe and the adjuſting
Weight D put to its proper Place for the due Rectification of
the Clock to the Length of the refpective Lunar Day propofed,
which is always known from an Ephemeris.
3331. From the Table (3323) it appears, that the ſhorteſt
Lunar Day 1483 exceeds the longeft Planetary Day 1452; and
confequently the Lunar Scale on the Pendulum will be of no Ufe
for the Planets. A planetary Scale therefore muft be conftruc-
ted that fhall have an Extent fufficient for the Movement of the
Corrector D from the Sidereal Day of 1436', to that of the Pla-
net Mercury of 1452; whofe Difference is 16 Minutes. And fince
the Mercurial Day of 1452 exceeds the Solar Day 1440 by 12′,
if we allow 4 more for the Diameter of the Weight C, we fhall
have 1456′ for the longeſt Day in the Planetary Scale, for which
(having the Length of the Rod A Ba 1587,8) we can
t
find P and e, by the Theorems already premifed.
1
1
3332.
Of CLOCK-WORK.
415
y
3332. Thus per Theorem (3329) we have 1440 1456::
1456 : 1472 = P, the Length of the fimple Pendulum proper
for that Length of Day.
3333. Again, having given AB = 1587,8 and P — 1472,
p a I
3
we have P+P + 2,14 — a²
e➡ P✈
C
1494—AE,
by the Theorem (3318). Therefore, ABAE — EB —
94, which is a little more than an Inch, through which the
Weight C is to be raiſed on the Rod of the Pendulum AB,
from C to E.
I
:
2
½
3334. We are next to enquire the Length of an ifochronal
Pendulum p correfponding to the greateft Acceleration, when
fp. In this Cafe we fhall have p² + pa + ecp =
a² + e² c, by the Theorem ſo often quoted in (3318). If we
put = a² + e² cs, and a + e c =
1
t, we fhall have p = 2
√s + tt2t1433, and of Courſe AN
1p716,7k to ma
1
=t,
AD=ƒ=
3335. But the Time of Vibration of this Pendulum being a
Mean proportional between 1440 and 1433,5 by the Lem-
ma in (3329), that is, 1440 *::*:1433,5, whence
1440 X 1433,51436,75, which fubducted from. 1456
the floweſt Vibration (3331) will leave 19′,3 which will be juſt
large enough to take in the Planetary Days as far as they are fen-
fibly different from each other: For the Sidereal, Saturnian,
and Fovian Day differ fo little, as not to be difcernable in the
Scale, as is evident from the Numbers in the third Column of
the Table (3323).'~~
}
3336. In this Scale the Length of the ifochronal Pendulum
for the Mean Day of the SUN being 1440, we fhall find
d
+
ƒ = AD = ¦ ¿±√ p°
I
7
tps 929, wherefore the
Place of the Corrector D is given for regulating the Clock for
Ule, or to fhew Mean Time:
common
2
3337. In like Manner, by having the Numbers un
1438 for Mars, you find p1436.
1442 for Venus
1452 for Mercury
→p = 1445. }
1465:
1
*
嘴
​*
7
+
1
And
416
INSTITUTIONS
And from thence, by the above Theorem, the Pofition of D, or
AD=f, will be found for each Planet reſpectively. And thus
the Lunar and Planetary Scales of Acceleration are compleated;
and the Clock fitted to fhew Time univerfally.
N.B. For the better Illuftration of what we have delivered
concerning this new constructed Pendulum, we have (in Fig. 2.
of the following Plate) added fo much of the lower Part of the
Half-fecond Pendulum, as contains both the Lunar and Planetary
Scales; but thefe Scales will be four Times as long in a Pendu-
lum that beats Seconds.
CHA P. XII.
Concerning the BOB or WEIGHT of the PENDULUM;
and the THEORY of fuch a FORM or SHAPE there-
of, as ſhall meet with very small RESISTANCE
from the AIR.
I
3338. N a Treatife on the Rationale of CLOCK-WORK, 1
judge it will be expected that fomething ſhould be faid
concerning the FORM of the BOB, or WEIGHT of the Pendu-
lum, with regard to the Refiftance of the Air in which it moves,
and the confequent Irregularity in its Motion, which will be
thereby unavoidably produced; and the rather, becauſe fo great
a Judge in theſe Matters as HUGENIUS, has afferted it to be a
very interefting Point, nam plurimum refert, are his Words.
3339. But as we have fhewn, the only Defign of the Con-
trivance and Mechaniſm of a Clock is to annihilate the Refi-
ftance which the Pendulum meets with from the Air, and every
other Caufe, it is plain, if it were poffible to conftruct a Clock
with ſo much Accuracy as to do that, the Motion of the Pen-
dulum would be as equable as if it moved without any Refi-
ſtance at all; for the THEORY of Clock-work would be a very
imperfect Thing, if it could not provide against the Effects of
Re-
The Universal PENDULUM.
T
Fig. 3.
D
Ꮐ
H
C
Sidereal
Gay
Fig. 2.
S
I
M
I
B
Fig.
+
I
G
K
G
-I
D
E
D
E
Fig. 5.
I
B
N
C
Fig. 6.
D
I
I
E
:
1
}
I
T
B
Solar
Day
Shortest
Gay
PLANETARY
SCALE
גי
S
Mercurial
Day
SCALE
Hean
Day
LUNAR
a
N
F
E
Longest
Day
H
I
D
A
Of CLOCK-WORK.
417
Refiftance of every Kind, and caufe the Clock to go with an
equable Motion, whether the Pendulum moved in Vacuo, in Air,
or in Water, and even in Quickfilver itself.
3340. But it may nevertheleſs be uſeful in fome Cafes of a
Pendulum, to know what Form of the Ball will be moſt conve-
nient and leaft refifted in its Motion through the Air, which
though infenfible in a few Vibrations, or in a fhort Time, may
yet, by a conftant Accumulation, produce Effects too fenfible
and molefting in long Run, maugre all the Ingenuity and
Diligence of the Artift. This Subject therefore we fhall now
proceed to confider more particularly.
3341. It is well known that the Refiftance of a Body in Mo-
tion regards the Quantity and Figure of its Surface, and not the
Quantity of Matter moved; for a CUBIC INCH of Matter may
be difpofed into the Form of a SPHERE, whofe Diameter will
then be 1,247; and its Superficies = 4,836 Cubic Inthes (836)
whereas the fame Matter in the Form of a CUBE had a Quantity of
Surface 6 Cubic Inches. And the SPHERE has the leaft
Quantity of Surface that it can poffibly be contained under ; and
therefore has leſs Reſiſtance in Proportion to its Weight, than any
other Body, and confequently is the beft Form, for the Ball of a
Pendulum in general. And becauſe the Surfaces of Spheres are
proportioned to the Squares of their Diameters (842), larger
Spheres will have much lefs Surface than fmall Ones, in Propor-
tion to their Weight, and will therefore meet with lefs Refiftance.
3342. The SOLID of leaft RESISTANCE (whofe Theory you
will find in (2042, &c.) would undoubtedly merit the firſt Conti-
deration, were it not, that its Conſtruction will be much too diffi-
cult for Practice; and this Solid is in itfelf but little preferable to
the Fruftum of a Cone of leaſt Reſiſtance, which is not only made
with the utmoſt Eafe, but is at the fame Time belt fitted for the
Purpoſes of the planetary Pendulum and Clock before deſcribed.
The Theory of which we fhall, therefore, here deliver.
3343. Let ABC (Fig. 3.) be the ifofceles Triangle genera-
ting by its Revolution a Cone, whofe Fruftum CGFB has the
leaſt Reſiſtance for a given Bafe CB, and Altitude DE. Let
the Fruftum be fuppofed to imove in the Medium in a Direction
la
parallel to its Axis; let SV be drawn parallel to A E, to repre-
ſent the Direction in which the Particles of the Fluid ftrike the
Hhh
VOL. II,
Cone.
418
INSTITUTIONS
Cone. Then V I being perpendicular to the Bafe CB will re-
prefent the Particles ftriking the Bafe with their whole Force,
fuppofing the Fruftum moved from D toward E; but in moving
from E towards D, the Particles S F will come obliquely on the
Side A B, and therefore cannot ftrike any Particle F with their
full Force.
3344. Let any Line LF reprefent the whole Force of the
Stroke againſt the Bafe BC, and through the Point F draw
MH perpendicular to the Side AB; let MF FH; and
compleat the Parallellogram ALFM; then the whole Force.
LF will be refolved into two Forces LM and MF, of
which one, viz. LM being parallel to the Side AB of the
Cone, cannot affect its Motion; the other Part M F is, there-
fore, all the Force of the Fluid on that Point, that can cauſe any
Refiftance.
3345. We are next to enquire what the Effect of this Force
MF or FH is, in the Direction DE; to this End let FH be
refolved into two Forces FD and DH, of which the former
FD is equal to, and its Effect wholly deſtroyed by the antagoniſt
Force G D, arifing from the Action of the Fluid TG on the
other Side; the Force DH, being all exerted in the Direction.
DE, is therefore all the Force of the oblique Stroke that directly
oppoſes the Motion of the Body, and conſequently that can cauſe
Refiftance to the Point F in the Side A B.
any
3346. Therefore the Effect or Reſiſtance of a Particle of the
Medium at the Point F in the Surface, is to that at the Point Iin
the Bafe, by a direct Stroke, as DH to L For A H, and therefore
as A H x D H to A H². But by fimilar Triangles HFD,
HAF, it is HD: HF:: HF: HA, whence AH x DH=
FH2, therefore A H is to F H, as the Refiftance at I to that
at F: But AH: FH::AB: BF; therefore the Refiftance
at I is to that of F, as A B²: BE³. And fince what has been
faid of the Points F and I hold equally true of all other Points in
the Surface and Baſe of the Cone, therefore the Refftance to the
Baſe will be to that upon the Surface of the Cone as AB² to B E².
3347. Hence then the Refiftance to the Surface FAG of the
Cone cut off, is to that against its Bafe F G, as FD2 to A F²;
therefore B E - FD2 is as the Reſiſtance to the Surface of the
Fruftum; and by adding A F², (which is as the Reſiſtance upon
the
Of CLOCK-WORK.
419
the Bafe or End F G) we have B E² + A F² — F D² for the
Expreffion of the Refiftance to the whole Fruftum, when moving
in the Direction E A. But A FFD A D'A E'
-FDA D²
E²
E²
- 2 AEX ED + ED² (637). Therefore the Refiftance of
the Fruftum will be B E + AE-2 AE x ED + ED²
AB² 2 AE x ED+ED².
3348. But fince A B expreffes the Refiftance to the Baſe
BC (3346) and is a conftant Quantity, it may be repreſented by
Unity, or A B² = 1; and as any Quantity may be confidered as
divided by Unity, the Refiftance of the Fruftum may be alfo
2 AE x ED + ED²,
or thus, I+
thus expreffed
A B²
DE² - 2 AEX DE
A B÷
A B²
3349. Therefore putting DE = a, BE = b, and AE = x,
we have the laſt Expreffion for the Refiſtance in Symbols thus
I+
02
2
- 2 ax
b² + x²
whoſe Fluxion (when a Maximum or Mini-
2 ax² x
mum) is
202 x x
xx + bb²
2 ab² x
= 0; whence we have
x² — ax — bb = 0; and by compleating the Square and ex-
tracting the Root, we have x = ÷ a± √
4
+ b². But x
being the Height of the whole Cone muft be greater than a,
which is but a Part of that Height, therefore x = a +
4
+
+ bb.
3350. Hence we have the following eafy Conftruction; biſect
DE in K, draw KC; and produce ED to A, ſo that KA
may be equal to K C. Then will ABC be a Cone, whofe
Fruftum BFGC fhall meet with leſs Reſiſtance, in an uniform
Fluid, than any other of the fame Bafe and Altitude. For becauſe
AK = KC = -4, and KE = 4, and EC=b, we
have KC² = a² + b b = x² — ax + 1 a², which gives x²
—ax — bb = 0, as required in the Property of fuch a Fruſ-
tum (3349).
4a²
*
Hhh 2
335I.
420
INSTITUTIONS
3351. In order to be certain that this Refiftance is a Minimum,
and not a Maximum, we are to confider this Principle, that while
a Quantity is increaſing, its Fluxion will be pofitive; but negative while
it is decreafing. Now the Quantity xxax bbo, is
analogous to the Fluxion from whence it proceeds, and there-
fore when xa, this Quantity becomes-bb, which is plain-
ly negative.
3352. Since then the Refiftance decreaſes from the Va-
+bb, where it is a Mini-
lue of x = a, to x = ½ a +
2
4
mum, it is plain that the Fruftum BCGF is lefs refifted than a
Cone of equal Bafe and Height, or than any other Fruftum of a
larger Cone, whofe Baſe and Height are the fame with theſe,
viz. CB and D E.
3353. The above Demonftrations will hold for any Propor-
tion of CE to DE; and the lefs CE is in regard to DE, the lefs
will A D be in refpect of A E; fo that at Length the Fruftum
of a Cone, of leaft Refiftance, will come near to the Form of
the Solid of leaft Refiftance, and ferve almoft equally as well for
the Purpoſe of the Bob of a Pendulum, and is excellently well
fitted for that of the planetary Pendulum defcribed in the preceed-
ing Chapter, which confifts of two fuch Fruftums, whofe Cen-
ter is that of their common Bafe. And further, as ſuch a double
Fruftum of a Cone is lefs refifted than a Globe of the fame
Weight, fo the middle Segment or Proportion of fuch a Fruſtum
cut longitudinally, will be better and more artificial, than a Bob
of the common lenticular Form.
It is fomewhat remark-
able, that the French and Spanish ACADEMICIANS in the De-
partment to PERU, for meaſuring the Length of a Degree of
the Meridian at the Equator, had in their Clock a Pendulum of
two truncated Cones, but placed juſt in a contrary Direction to that
which the Theory above requires, and is here fhewn in Fig. 2.
CHAP.
of CLOCK-WORK.
421
R
**
CHAP.
XIII.
أزو
?
7
The THEORY for finding the TIME of a leaft OSCIL-
LATION of a given PENDULUM fwinging in the
ARCH of a CIRCLE and alfo the Time of any other:
with proper CANONS for CALCULATION.
3354.
13
W
E have formerly fhewn (1119) that the Time of the
leaft Vibration of a given Pendulum is equal to the
Time fuch a Pendulum would take to vibrate in the Arch of a
Cycloid. Alfo, that when the Arch of a Circle, in which fuch
a Pendulum vibrates, is fo fmall, that it differs not fenfibly from
the Cycloid, the Time of a Vibration will not be affected there-
by; and this Confideration makes cycloidal Cheeks unneceflary in
long Pendulums, or fuch as fwing Seconds, where the Arch of Vi-
bration is not more than 3 or 4 Degrees of a Circle.
3355. But in Table-clocks, whofe Pendulum is not more than
about 6 or 8 Inches long, the Cafe is different; for here the
Space can be but fmall, in which there can be any fenfible Coin-
cidence of the Circle and Cycloid, and yet the Arch of Vibration
in theſe fhort Pendulums is generally much longer in Proportion
to their Lengths than in the long Ones; whereas they ought to
be proportionably fhorter. Thefe Clocks muft therefore be fub-
ject to an Error that cannot be avoided but by cycloidal Plates,
fuch as Hugenius invented and prefcribes for this Purpose; the
Nature of which fee already largely explained (1120), (or by any
of the new Species of Pendulums hereafter to be defcribed) for the
common Method of moving the Bob up and down upon the Rod,
by a Screw at the Bottom, is very fallacious and inartificial, and
can never procure an equable Motion to the Cock, unless the
Arch of Vibration be very fmall indeed.
3356. When the circular Arch defcribed by the Pendulum
begins to deviate from the Cycloid, the Times of Vibration will
begin to encreaſe or exceed the Time of a leaft Vibration, and
therefore it will be neceflary to thew how this Error arifes, and
in what Proportion it encreaſes, that the ingenious Tyro, in Clock-
work,
y
422
INSTITUTIONS
work, may not be left in the Dark in that Part of the Theory
which is moſt effential to his Art, or conducive to the Truth of
his Work. And this we fhall be able to do in the plaineft Man-
ner, by following the Footfteps of the late learned Profeffor
SAUNDERSON.
3357. The general Problem is, to find (independent of the Cy-
cloid) the Time precifely of a leaſt Ofcillation of a Pendulum of a given
Length fwinging in an Arch of a Circle; and alſo to find, without
any fenfible Error, the Time of any other. Let the Curve A DC
be the Arch of a Circle, whofe Diameter is DI, and in which
the Pendulum N D is fuppofed to vibrate. Then in deſcending
from C to any Point E, it will acquire a Velocity which will be
as EM= √BF (having drawn the Chords A C, EE in-
terfecting the Diameter ID in B and F). Now✔✅FD is as
the Velocity acquired in defcending through ID, and by that
Velocity it would defcribe uniformly a Space equal to ID
in the Time of the faid Defcent through ID (993). Then
I
the Space ID divided by the Velocity
Time
2
ID of the Defcent through
the Length of the Pendulum (971.)
I
2
ID will give the
ID, i.e. through half
3358. Draw ee very near to EE; then may Ee be confider-
E e
ed as the Fluxion of the Arch DE; and
will be the Time
✔BF
wherein the ſmall Arch Ee is deſcribed by the Pendulum, or the
Fluxion of the Time of a Vibration. But Ee=
2
ID × Ff
IFXFD
VID
VIF
× VID × 1/
Fƒ
VFD
; therefore we have the
E e
Fluxion of the Time
√BF
VID
VIF
× VID
ID X 1/1/0
Ff
✓ BF XF D
3359. Bifect BD in K, and KD in L; and when the Arch
ADC is finall, the Quantity IF cannot differ fenfibly from
IK,
Of CLOCK-WORK.
423
IK, nor
VID IL
from
VIF
IK
I L
IF
(becauſe, ID IL :: IL:
2
IK or IF, therefore ID: IF: IL²: IK² (by 672). There-
E e
fore
✔BF
Ff
✔BF × FD
IL
is very nearly equal to X VID X
ΙΚ
3360. Upon the Diameter BD defcribe a Circle cutting the
Chords EE and ee in G and g reſpectively, then will the Fluxion
of the Arch DG be G =
2
BD x Fƒ
(875) confequently
√F B × F D
and therefore the Fluxion of the
Gg
BD
2
Ff
;
✔F B × F D
Ee
IL
Time of Vibration through DE will be
X.
✔BF
IK
VIDX
G g
BD'
IL
3361. But the Fluent of this laft Fluxion is
× VIDX
IK
DGB
Ᏼ Ꭰ
2
tion of the Pendulum from D to C. The Time of a whole
which is therefore the Time of half a Vibration or Mo-
Vibration therefore through ADC will be
IL
IK
× VID
X
BGDG B
BD
I L
IK
become T-VID ×
3362. Let the Arch ADC be indefinitely ſmall, then the
Quantity
= 1, and the Time (T) of a leaſt Vibration will
BGDG B
Ᏼ Ꭰ
And therefore BD:
BGDGB::VID: T; that is, as the Diameter is to the Cir-
cumference of a Circle, fo is the Time (VID) of the Defcent through
half the Length of the Pendulum to the Time of a leaſt Oſcillation of
that Pendulum which is the very fame Analogy as we found from
the
424
INSTITUTIONS
the Cycloid (1124). Wherefore the Time of Ofcillation in a
Cycloid, and in an indefinitely finall Arch of a Circle is the fame;
viz. T=1 Second, when the Length of the Pendulum is
Inches.
39,2
3363. Therefore the Time of an Ofcillation or Vibration in
a circular Arch in general, is TX
IL
IK'
,
or becauſe ILIK
+ KL, therefore the general Expreffion of the Time of Vibra-
tion through any Arch A D C will be T + Tx
KL
ΙΚ
KL
IK
3364. But the Exceſs Tx of the Time of Vibration
in a circular Arch above the Time in the Arch of a Cycloid, or
the leaft of all, is next to be transformed and fitted for Computa-
tion. In order to this put the verſed Sine of the Arch of Vibra-
tion BD; then will KL (3359) 2 L = IK
(LND, being the Length of the Pendulum). And TX
KL
HM Tx
==
IK 2 L
2
Tx
*
2
*
8 L-2
the Exceſs or Error above
T, the Time of Ofcillation in the Cycloid.
3365, Let the Time be expreffed in Seconds; and let S
3600" the Seconds in an Hour, or 86400" the Seconds in a Day
TS x
8
L — 2 x
=r, the Seconds loft in an
or 24 Hours. Then
Hour of a Day, by the Pendulum vibrating in a circular Arch of
Tx
a given verfed Sine x.
For it is plain, that as
is the
8 L
Q X
TS x
Increaſe of T the Time of Vibration in a Cycloid, fo
8 L
2x
will repreſent the Dimunition of the Number of Seconds S in a
Day or an Hour; or S
TSX
8 L
2x
in the fame Time in the Arch of a Circle.
Number of Vibrations
3366. If the Quantity (») be given, we have the verfed Sine
of the Arch of Vibration x =
8 L r
TS-27
But in all thefe The-
orems, the Value of is found in Parts of L; and fo muft be
reduced to the tabular verfed Sine by this Analogy as L:x::
R:
}
>
The THE ORY of the Circulating PENDULUM, Oscillating ROTULA &c
K
B
D
Fig: 2.
Fig: 1.
E
Fig: 7
B
B
P
E
H
R
Fig: 4.
T
B
E
1.
M
D
Fig: 6.
Ω
N
B
I
Fig: 3.
T
M
C
E
B
d
B
t
N
LA
72
H
OF
Fig
D 10.
K
от
Z
ADB
Fig: 9.
▲ Fig: 5.
T
D
M
F
E
772
H
P
Fig: 8.
E
B
A
HO
I
G
F
Of CLOCK-WORK.
325
Rx
TS-r
Rx
R:
L
R; therefore putting B = 1, we have
45
tabular verfed Sine of the Angle required.
L
3367. That the young Automatift may fee the Use of this Doc-
trine, I ſhall illuftrate this Theorem by an Example There-
fore let it be required to find how many Vibrations are made in a
circular Arch of 120 Degrees, by a Pendulum whoſe Length L=
39,2, and which vibrates 86400 Times in one Day, in the
Arch of a Cycloid. Here L = 39,2, T = 1″, S = 86400",
19,6, being the verfed Sine of 60 Degrees, then
and x
= 2
TS x
8 L 2 x
L
=
6278", which deducted from 86400, leaves
8c122, the Number of Swings required. Whence fuch a Clock
will loofe daily 6278" 1 Hour: 44: 24".
=
3368. Again let it be required to find in what Arch of a Cir-
cle a fecond Pendulum muft vibrate to loole one Minue, or 60"
per Day. Then r = 60″, T = 1″, and ½ TS≈ 43200. And
therefore
4 "
1 TS-7 γ
2
=
2
0,0028367, the tabular verfed Sine of
4° : 19′; the Arch of Vibration therefore is 8° 38′ required.
In like Manner it is found, that the verfed Sine of half the Arch
in which ſuch a Pendulum will loofe but one Second per Day, or
in 24 Hours, is 0,0000896, correfponding to 46′ of a Degree,
the Arch therefore is 1° : 32′ ; which is equal to one Inch in
Length.
3369. Hence it appears, that in Second-pendulum Clocks, if
the Arch of Vibration does not exceed two or three inches, the
Error will be very fmall, and may be eafily corrected, either in
the fimple or compound Form, by the common Methods. But
it is otherwiſe with Table-clocks, whofe Pendulums are fhort,
and the Arch of Vibration long. Thus for Example, let the
Pendulum of fuch a Clock be 9,8 Inches, to fwing Half-
feconds. And let z =
2
4 r
TS
then
/ T S ≈
4 + z
=r, the Er-
ror of the Clock in Seconds per Day. Therefore T = 1,
S =
172800, and z = 0,060307, the tabular verfed Sine of 20 De-
grees; and
VOL. II.
1TS Z
2
44
4+2
=1283 Half-feconds, or 10: 40″ is the
Iii
Error
326
INSTITUTIONS
Error of fuch a Clock per Day; which is ftill much greater when
the Pendulum is but 6 or 7 Inches long, as in moſt of thoſe
Clocks, and therefore no great Exactnefs in them can be ex-
pected.
N. B. The Arch of Vibration in Table-clocks is confiderably
more than 20 Degrees, and of Courfe the Error greater on that
Account.
CHA P. XIV.
The phyfical THEORY of the FIGURE of the EARTH;
the fame demonftrated to be a SPHEROID; and from
thence a SOURCE of ERROR in the MOTION of
PENDULUMS.
3370. B
Efides the Ofcillation of Pendulums in circular Arches
inſtead of cycloidal Ones, there are other Sources of
Error, which are unavoidable, as they are founded in the very
Conftitution of the Earth itfelf, and its Appendages. That
which arifes from the Earth itſelf is the different Force of Gravity
on different Parts of the Earth's Surface. For fince the Velocity
in a given Time is always as the accelerating Force (998);
therefore, in fuch Latitudes, where the Power of Gravity is
ftrongeft, there the Velocity of the Pendulum will be quickeſt,
and vice verfa; confequently the fame Length of a Pendulum
will not perform its Vibrations in the fame Time in different
Regions of the Earth.
3371. This variable Power of Gravity arifes from two Cauſes:
(1.) The fpheroidical Form of the Earth, by which Means a
Body on different Parts of the Surface is not equidiftant from the
Center. (2.) The centrifugal Force arifing from the Rotation of
the Earth upon it Axis, by which the Force of Gravity is very
unequally diminiſhed from the Equator to the Poles. A Pendu-
lum, therefore, upon different Parts of the Earth will be vari-
oufly
Of CLOCK-WORK.
327
oufly influenced by this Power, and confequently can never vi-
brate equally but in the fame Place.
3372. We fhall lay open theſe Sources of Error in the Motion
of Pendulums by tracing them from their first Principles, and
thereby convince the young Automatiſt that they neceſſarily ariſe
from the natural Conftitution and Difpofition of the Earth, tak-
ing it for granted, that the Earth in its firft Formation, and at
the Commencement of its Motion, was in a fluid State, or at
leaft fo far, that its Parts could yield to the Force impreffed
upon them by that Motion, according to the Laws of Nature
(964, &c.)
3373. Therefore let NS (Fig. 5.) be the Earth's Axis, and
EQ the Diameter of the Equator, in the Earth conſidered as an
Ellipfoid. And let B C be a Column of fluid or yielding Particles
gravitating to the Center C, which (becauſe of an Equilibrium
between all the Parts) muft be of equal Weight with any other
Column of Particles CN or CE. Put CE a, CN b,
= CN=b,
CB, and ABs Sine of the Angle BCN; and laftly,
let the Power of Gravity (g) be every where as the (2) Power
of the Diſtance from the Center C.
3374. Therefore the Gravity at E will be to that at B, as
CE" to CB"; and therefore
Gravity at B.
& X C B n
C En
& x
a'
the Power of
3375. Again, the centrifugal Force (f) at E is to that at B,
as CE to AB (1177). But AB: CB::s: 1
S: I = Radius,
whence AB = s × C B = s x; therefore CE:AB::a:sx
:: f: fsx
::f:
a
the centrifugal Force at B.
3376. But fince this Force acts in the Direction AD, let its
abfolute Quantity at B be denoted by BD = fsx; this is re-
a
folvable into two other Forces FD and F B, of which the latter
is all that Part which oppofes Gravity in the Direction BC;
whence, fince BD: BF:: 1 :::*: fsx the centri-
fsx
a
a
=
fugal Force at B, by which Gravity is there diminiſhed.
Iii 2
3377.
328
INSTITUTIONS
(*) at B will be
a
Q"
a
3377. Hence the Power of Gravity on a Particle of Matter
fx, impelling it towards the Center
C; therefore its Momentum or Weight will be
8.
x
f s² x x
a
(1072); but this is the Fluxion of the Weight of the whole Co-
lumn of Particles BC, and therefore the Fluent
2 2
fs²x²
f.
24
8x" + =
n + I × an
will be the Weight of the faid Column of Particles BC.
3378. But this is likewife the Weight of every other Column
of Particles (3373), and therefore of the Column CE; and be-
cauſe in this Cafe the Angle NCE is a right One, we have
x = a, and s = 1; wherefore the Weight of the Column CE
will be
nf. f.
fa
23
8x" + 1
ха
Xa=
12 + 1 X an
2
2 × 12 + I
n + I X an
f s² x²
2 a
; this Equation by proper Reduction, and putting s
= 0, will give 2 g — nƒ — ƒ × a" + ¹ = 2 g *" +¹; and thence
T
we get a:x:: 2g": 28-nff::CE: CN ::
is the DIAMETER of the EQUATOR to the Earth's Axis.
3379. If Gravity be fuppofed uniform, then no, and then
28:28f:: CE CN.
ƒ::CE: But gƒ:: 294:1, (1198);
therefore CE: CN:: 588:587, in fuch a Cafe.
3380. If Gravity were proportional to the Distance from the Cen-
Ι
2
ter C, then n = 1, and we ſhould have g: g-ƒ³:: CE:
CN.
3381. But in the prefent Conftitution of Nature, it is well
:
: 28 + f
28+ ƒ=1
known that n = 2 (1230); therefore 2 g
:: 28+ f2g: 589: 588: CE:CN.
g
3382. What has been hitherto faid, fhews the Figure of the
Earth is not spherical, but must be that of a SPHEROID, flatted
at the Poles. And when it is faid, Gravity is inverſely as the
Square of the Diſtance from the Center (or as x-2) it is to be
underſtood that (x) is the Diſtance from the Center of Curvature,
which is the true Center of Attraction to a Particle on the Surface
of
Of CLOCK-WORK.
329
of the Spheriod, and not the Center of the Spheriod itself, as is
the Cafe of a Globe or Sphere.
3383. Now the Radius of Curvature at the Equator E is
2 CN
CE2
and at the Pole N is
2 CE
CN2
(933,935) but the inverfe
Ratio of the Squares of theſe Quantities is CN to CE;
and therefore Gravity at E and N is not inverſely as the Square,
as the fixth Power of CE and CN.
3384. Let D Length of a Degree at the Pole, and d = a
Degree at the Equator; now it is plain thofe Degrees will be as
the Radii (R, r,) of Curvature in thofe Places, viz. D: d::
R:r::
2 CE
CN=
2. CN
CE²
:: CE CN³.
Confequently VD:
/d::CE:CN.
3385. Therefore if a Degree could be accurately meaſured
at the Equator E, and at the Pole N, the Ratio of the Diameter
of the Earth at the Equator to that of its Axis would be known.
Indeed, if two Degrees be meaſured in any two diſtant Parts,
that Ratio is equally known from them. But a Degree has
been meaſured at the Equator, and found to contain 56767
Toifes or French Fathoms.* Alfo, a Degree has been meaſured
under the arctic Circle, and was found to be 57438; laftly, a
Degree in Latitude 45 was meaſured, and found 57050 Toifes.
From any two of theſe, the Ratio of CE to CN, may be found;
and from a Mean of them all, it appears that CE: CN :: 266
: 2640.
7
3386. From a View of this Theory, it is plain, that there is
a Diminution of Gravity arifing from two Caufes, viz. the cen-
trifugal Force, and the Spheroidical Figure of the Earth. Let CB
be produced to G, fo that BG may reprefent the centrifugal
Force at the Equator, and B D that in the Latitude B. And
draw G D parallel to C N, then it is GB: BD::CE: A B.
And BD BF :: (CB: AB ::) r: s. Whence BD
GBX AB
CE
ABX
BF xr
S
; whence GB: BF:
да
× CE:
fo is the whole Diminution of Gravity under the Equa-
tor at E to the Diminution at the Latitude B.
* See my new PRINCIPLES of GEOGRAPHY and NAVIGATION.
3387.
330
INSTITUTIONS
3387. As the Diminution of Gravity decreaſes from the Équa-
tor to the Pole, fo there must be an Increment of Gravity conftant-
ly attending the fame, which will be reprefented by BG-
BFGF, becaufe G B is the whole Diminution of Gravity
at the Equator E. But becauſe BG: GD :: GD: GF
(660) we have BG GF BG²: GD (672) :: BC²:
A C²; that is, the Decrement of Gravity at the Equator is to the
Increment thereof at the Latitude B, as the Square of Radius to the
Square of the Sine of the Latitude.
:
3388. And therefore, becauſe BG and BC, are conftant
Quantities, the Increment of Gravity will ever be proportioned
to the Square of the Sine of the Latitude in the Sphere; and in the
Spheroid of the Earth, it will be very nearly the fame, as AC
will differ infenfibly from the Sine of the Latitude. Therefore
fince by Menfuration, it appears that CE is to C N, as 266 to
264,7 (3385) which is very near the Ratio of 230 to 229, as
determined by Sir ISAAC NEWTON; it will follow, that the
Ratio of the centrifugal Force at the Equator is to Gravity as I
to 230, or that the Decrement of Gravity at the Equator is 230
Part of the Whole.
3389. Therefore if we put this Diminution of Gravity =
10000; then will the Gravities at the Equator E, London B, and
the Pole N, be as 2290000, 2296124, and 2300000. And
confequently the Lengths of Pendulums, vibrating in equal
Times at thofe Places, will be reſpectively proportional to thoſe
Numbers. For the Times of Vibration in Pendulums are in a
given Ratio to the Times of Defcent through half their Length
(1124) and, of Courſe, to the Times of Defcent through their
whole Length. But if a Body defcends through different Spaces
in the fame Time, the Forces. impelling it will be as thoſe Spa-
ces (999); conſequently, the Lengths of Pendulums will increaſe
with the accelerating Forces of Gravity from the Equator to the
Pole, for the Times of Ofcillation to be equal.
3390. For Example; it will be 2296124: 2290000 :: 39,2
:: 39,1 = the Length of a Pendulum vibrating Seconds at the
Equator; again, 2290000: 2300000 :: 39,1 : 39,27 = the
Length of a fecond Pendulum at the Poles. And this Propor-
tion in the increafing Lengths of Pendulums has been verified by
Experiments made in all Parts from the Equator to the polar Circle.
3391.
Of CLOCK-WORK.
331
3391. Mr. MAUPERTUIS (in his TREATISE of the FI-
GURE of the EARTH) has given us a Table of the Length of
an ifochronal Pendulum for every 5th Degree of Latitude, as
deduced from their Meaſures of a Degree in different Latitudes,
and from a great Number of Experiments on Pendulums in all
Parts from the Equator to the polar Circle; theſe I have reduced
to English Meaſure for the Benefit of fuch who do not underſtand
thofe of the French. If the Pendulum, which, at the Equator,
is exactly adjuſted by the Motion of a fixed Star, be carried to
any other Latitude, it will be accelerated, or gain upon the
mean Time (3389). And the Quantity of this Acceleration
upon one Revolution of the Stars, is here alfo fhewn to every
5th Degree of Latitude.
3392. TABLE
Of the ACCELERATIONS of a CLOCK, and of
the Lengthenings of a PENDULUM from the
Equator to the Pole.
ACCELERATION
LATITUDE of
in one Revolution
the Place.
of the fixed Stars.
Length of the
PENDULUM.
O"
39,0754
5
1,6
39,0768
IO
6,4
39,0811
15
14,3
39,0873
20
24,9
39,0980
25
38.1
39,1097
30
53.3
39,1236
35
70,2
39,1389
40
98,1
39,1551
45
106,6
39,1718
50
125,1
39,1903
51:30
39,2
55
143, I
39,2049
60
159,9
39,2202
65
175,1
39.2338
70
188,3
39,2459
75
198,9
39,2554
80
206,8
39,2626
85
211,6
39.2669
90
213,2
39,2684
332
INSTITUTIONS
-1
1
וי
}
CHA P. XV.
The EFFECTS of HEAT and COLD in varying the
DIMENSIONS of BODIES; and thereby proving
another natural SOURCE of Error in PENDU„
LUMS.
3393. THE other natural Source or Caufe of Error in the
Time of a Pendulum's Vibration, is the different
Temperature of the Air or Climate; for it is found by Experience
that all Bodies are fubject to be expanded by Heat, and contrac-
ted by Cold, in all their Dimenfions. And therefore if the Pen-
dulum by theſe natural Caufes be conftantly varying its Length,
the Time of Vibration' will be always inconftant; and fo, on
this Account, can never (without a proper Rectification) be an
equable Meaſure of Time.
I
700
3394. For fince the Lengths of Pendulums are inverſely as
the Squares of the Numbers of Vibrations in a given Time, (a
Day, for Inftance), if we exprefs the Length of a Second-pen-
dulum in 100ths of an Inch, it will be 3920; and then fuppofe
it contracted by Cold of an Inch, its Length will be 3919;
and fince in one Day there are 86400"; we fhall have, as
3919 3920 :: 8640c": 86411". Whence it appears,
that a Clock will gain or lofe every Day 11 Seconds for every
100th of an Inch it is contracted or lenthened by Cold and Heat.
(See Table 3392).
3919:
3395. This is by far the most general Cauſe of Error in Time-
pieces; for the different Power of Gravity affects the Pendulum
in this Reſpect only in different Latitudes or Climates, but this
Effect of Heat and Cold equally affects it in every Region, and in
any particular Place. And as fuch minute Variations in the
Length of Pendulums are productive of fuch fenfible Errors in
the Account of Time, they ought to be at all Times accurately
known that the Clock might be properly rectified to every varia-
ble Degree of Heat and Cold.
វ
''
3396.
Of CLOCK-WORK.
433
3396. Hence it appears, that a proper Regulation of Time-
Pieces with Pendulums is clofely connected with the Nature of
THERMOMETERS, whofe fole Ufe is to indicate, at all Times,
the comparative Degrees of Heat and Cold by a proper Scale of
Variation. And if by Experiments we can find how much a
given Rod of Iron or Braſs extends or contracts with given Dif-
ferences of Heat and Cold, denoted by the Thermometer, then
ſuch an Inftrument being fixed by the Clock, will always fhew
what Degree of Correction is neceffary at all Times of the
Year for keeping the Clock to true Time.
ΤΖ
I
an
3397. Now by many Experiments made on purpoſe to de-
termine this Affair,* it appears, that a flat Rod of Braſs
Inch wide, and of an Inch thick, and in Length 384 Inches,
will expand or extend in Length juſt of an Inch with ſuch a
Difference of Heat as will caufe the Mercury in Farenheit's
Thermometer to move through 224 Divifions; therefore — of
an Inch will correſpond to 114 Divifions; but the Extenfion
of of an Inch will occafion the Clock to lofe 11" per Day.
(3394.) Confequently each Divifion of Farenheit's Termome-
ter correſponds to an Extenfion of Part of an Inch, in the
Rod of the Pendulum, and to one Second of Time which the Clock,
by that Means, lofes per Day.
I
1000
For Inftance;
3398. Therefore, if the Clock be adjuſted to equal Time,
when the Mercury ftands at 55, (the Temperate Degree) in
the Thermometer ; it will be ſeen in the faid Inftru-
ment for any other Time, how much the Clock has loft
or gained upon the equal Time, by the Height of the Mer-
cury at that Time.
fuppofe the Mercury ſtands
in a warm Summer's Day at 70, that is 15 Diviſions above
Temperate; therefore the Pendulum is lengthened of
an Inch, and confequently the Clock will lofe 15" per Day,
if not properly rectified. On the other Hand, if the Mer-
cury ſtands at 40, (as far below Temperate) then it will fhew
the Pendulum is contracted as much, and the Clock now gains.
15" per Day.
I S
TCOO
3399. Hence it appears that as the greatest Summer Heat
in England never raifes the Mercury higher than abour 80 De-
VOL. II.
Kkk
grees,
*See a Treatife entituled, OBSERVATIONS Aftronomiques & Phy-
fiques par G. JUAN & ANT, de ULLOA. Page 86.
434
INSTITUTIONS
grees, our Second Pendulum Clocks, of brass Rods, can loofe at
`moſt but 25″ per Day by Extenfion from Heat. But they may gain
30″ or 40″ in a Day by extreme Cold, contracting the fame.
3400. If the Rod of a Second-pendulum be made of Iron, it
will vary lefs in Length by Heat, and Cold, for by the fame Ex-
periments it was found, that (cæteris paribus) the Extenfion of
Brass was to that of Iron as 20 to 131, which is extremely near
the Proportion of 3 to 2.
And the Extenfion of Iron to that of
Steel was found to be as 13 to 12. It was alfo found that the
Expanſion of Brass to that of Copper was as 20 to 194.
3401. From what has been faid it appears, there is no fuch
Thing in Nature as an equable Time-piece; that a Pendulum-
Clock is the only Machine which, by Art, can be made to ap-
proach near it; and laftly, that fuch a Clock will ſtand in conftant
Need of a proper Adjuftment or Rectification by the fixed Stars,
or other aftronomical Means. To contrive to correct thefe Er-
rors, as faft as they arife, by fome artificial Conftruction of the
Pendulum has been the Endeavour of many Artiſts, and with
fome Degree of Succefs; but none have come to my Knowlege
that appears to anſwer this Purpoſe fo well as the following Me-
thod, which I am informed is put in Practice by an ingenious
Artiſt in the North of England; and is as follows.
3402. A Bar of the fame Metal with the Rod of the Pendu-
lum, and of the fame Thickness and Length, is placed againſt
the Back-part of the Clock-cafe; from the Top of this a Part
projects to which the upper Part of the Pendulum is connected
by two fine pliable Chains, or filken Strings, (as in the Cut at
Fage 374) which just below pafs between two Plates of Brafs,
whofe lower Edges will always terminate the Length of the
Pendulum at the upper End. Thefe Plates are fupported on a
Foot fixed to the Back of the Cafe. This Bar refts upon an
immoveable Bafe on the lower Part of the Cafe, and is braced
into a proper Groove, which admits of no Motion any Ways but
that of Entenfion and Contraction in Length by Heat and Cold.
3403. Now 'tis evident, that the Extenfion or Contraction of
this Bar and the Rod of the Pendulum, will be equal, and in
contrary Directions; and therefore, fuppofe by Heat the Pen-
dulum is increaſed of an Inch in Length, below the Edge
of the brafs Plates or Cheeks, then becauſe the Bar is lengthened
I
700
just
of CLOCK-WORK.
435
juft fo much upwards it will raiſe or draw up the Pendulum juſt
1 of an Inch, and thereby make its Length below the Plates
I
시이이
​ſtill the fame as before. The cafe is the fame in regard to Con-
traction by Cold; for as the Pendulum is thereby fhortened gra-
dually, it is as gradually lengthened by being let down between
the Plates, by the equal Contraction of the Bar behind. Whence
it ſhould ſeem that this is a conftant and adequate Rectification
of the Pendulum by which it will always keep true Time.*
3404. If the Clock be of the common Conftruction, viz. with
a Pendulum confifting only of a Rod and Bob; then we muſt
be content with a Method of correcting it as foon as its Quan-
tity is diſcovered. Thus, if the Bob be made to reft or depend
on a Nutt and Screw upon the loweft Part of the Rod; then
if there be 25 Threads to an Inch, and the Nutt be of a cir-
cular Form, and its Perimeter divided into 45 equal Parts, it is
evident from (3397,) that each of thofe Parts will correfpond
to a fecond of Time in the Clock, and to a fingle Divifion of
the Scale of Farenheit's Thermometer; and therefore by moving
the Nutt one Way or the other, and fo many Diviſions, as the
Thermometer directs, we fhall be able to correct the Error
fhewn thereby to a great Degree of Exactnefs.
3405. If the Pendulum be of the compound Sort, viz. with
a Ball and a Corrector, and conftructed as directed (in 2334,
2335, 2336,) then it will be eafy for the Artift to divide the
Rod of the Pendulum into Minutes and Seconds by the Theorems
there given, and confequently, if by the Revolution of a Star,
he finds how much his Clock loofes or gains upon the Mean
Time in a Day or ſeveral Days; or if by equal Altitudes of the
Sun, he knows at any Time how much the Clock is too faſt or
too flow, he can very readily adjuſt it to the Mean or true Time.
It is not worth while here to take Notice of the Ufe of the BA-
ROMETER in eftimating the Errors of the Pendulum arifing
from the different Weight or Denfity of the Air, becaufe the Dif-
ference of Refiftance to the Bob is on that Account fo exceed-
ing ſmall as ſcarcely to come under the niceft Obfervation, and
which we have before provided againſt, (3342, &c.)
Kkk 2
CHAP.
*The Reader may meet with a larger Account of this Invention
in a French Treatise on Clock-work by Mr. THIOUT, with a Print
thereof; but as I have had no Opportunity of perufing that Book,
I do not know who was the Author of this Contrivance.
436
INSTITUTIONS
CHA P. XVI.
The THEORY of the CIRCULAR PENDULUM demon-
ftrated, with its peculiar ADVANTAGES in CLOCK-
WORK.
3406.
WE
E have now delivered all we think neceſſary con-
cerning the Structure of Clocks and Watches of
the ordinary Sort, whofe Motions are governed and regulated
by Springs and Pendulums in the ufual Way. But here
we are debarred from the Ufe of a Second-pendulum Clock of a
portable Form; though from what we have faid, it appears, no
other Sort of Automata can be regarded as equable Time-keepers,
in any tolerable Degree. But Nature has fupplied a Method,
and Art has difcovered it, by which a Pendulum may be made
very eafily to circulate Seconds in the fame Length that a com-
mon Pendulum vibrates Half-feconds; which is as follows.
3407. We have formerly fhewn, that a Weight at the End
of a String might be made to move in fuch a Manner, that the
String fhould defcribe the Surface of a Cone, while the Weight
moved through the Circumference of its Bafe, the Vertex of the
Cone being the Point to which the String is fixed [See Fig. to
1193.] Or thus, let D [Fig. 6.] be a Weight at the End of
a String DE fixed to the Point E; then if the faid Weight be
moved in the Circle DbFa D, the String will defcribe the
Surface of the Cone DEF; and this Motion being continued
would conftitute a Circular, or rather Conical Pendulum.
3408. Now a Pendulum of this Form that fhall perform its
Circulations in a Second of Time is required to be but of the
Length of a common Pendulum to vibrate in a Second, as we
fhall bye-and-bye demonftrate. The Invention of this Form
of a Pendulum the celebrated HUGENIUS claims alfo as his
own, and tells us he found it nearly at the fame Time he in-
vented the common long Pendulum. His Words are, " Unde
aliud quoque horologii commentum deduximus, eodem fere tempore quo
prius illud.
3409. I
Of CLOCK-WORK.
437
3409. I have been the more particular in mentioning this
Gentleman's Claim to this Invention, becaufe I find it is alfo
claimed by Dr. Hook, who we have before obſerved was his
Competitor for the Invention of Watches. (3324.) Dr. Hooke has
not, I think, a very clear Title to the Invention, nor has faid
any Thing relative to the Theory thereof. Mr. Hugens has only
given the Theorems on which it depends, but without any
Demonftration. So that the Rationale of a Clock with a Circu-
lar Pendulum is yet a Novelty in our Tongue; nor has any fo-
reign Author wrote expressly on this Subject, that I know of.
I fhall therefore proceed to explain the Rationale of fuch a Clock
in as plain and conciſe a Manner as I can.
3410. Let NAM [Fig. 6.] be a Parabola inverted, its Ver-
tex A, and Axis AO. Then ſuppoſe a Veffel or Bowl was
excavated to the Figure of fuch a Parabola, ſo that its concave
Surface might be that of a Paraboloid; if fuch a Bowl were pro-
perly agitated, a heavy globular Body within it might be made
to circulate round it in any Part of the Surface as at D or G,
and there to defcribe the Circles Da Fb, or GcHd, whofe
Diameters are DF and GH.
3411. The Ball D is fupported or ſuſtained in the Circle by
an Equilibrium of three different Forces, viz. (1.) The Force
of Gravity, or that of its own Weight. (2.) A Centrifugal
Force which is always a neceflary Confequence of circular Mo-
tion: And, (3.) The Re-action of the Side of the Veffel in any
Point D, which is always equal to the Force of its Preffure
againſt it, in a perpendicular Direction, arifing from the other
two Forces.
3412. The Circle which the Ball deſcribes being ſuppoſed
parallel to the Horizon, OC will be perpendicular to its Plane
in the Center C; and let DE be drawn perpendicular to
the Side of the Veffel in D. Then becauſe Gravity acts per-
pendicularly to the Hooizon, let it be reprefented by the right
Line E C. Again, fince the Centrifugal Force is in a horizon-
tal Direction always from the Center C, it will be properly re-
preſented by the right Line CD; then, laftly, becauſe the Bo-
dy D is fuftained in Equilibrio at D, its Preffure or Re-action di-
rectly against the Side at D will be denoted by the right Line
DE.
438
INSTITUTIONS
1
DE. All this is very evident from the Mechanical Doctrine of
Compound Forces. (1027, &c.)
3413. In like Manner, if the Body circulates in any other
Part of the paraboloid Veffel, as at G, the three Powers will
there alſo be repreſented by the three Sides of the Triangle IGB;
that is, Gravity by the Side IB; the Centrifugal Force by the
Side BG; and the Refiftance of the Veffel by the Line GI,
perpendicular to the Tangent at G.
3414. As by theſe three Forces the Body is kept in a conſtant
Equilibrium in every Point of the Circumference it defcribes, it
will be ſuſceptible of any Degree of projectile Force impreffed
upon it by an horizontal Agitation of the Veffel, and thereby
be put into Circular Motion, every Way fimilar to that by a cen-
tral Attraction (1170,) or to that of a Body annexed to the End
of a String, (1192,) and will therefore be fubject to all the
fame Laws of Motion.
3415. Confequently, fince the Body in deſcribing the Cir-
cles DaF b, and GcHd, has Centrifugal Forces proportional
to the Radii CD, and B G, it will deſcribe thoſe Circles in equat
Times, (1177.) Therefore, all circular Revolutions in every
Part of fuch a Veffel are ifochronal, or made in equal Times.
3416. If a Plane be freely moveable about a Center, and a
heavy Body lies upon it, connected by a String to the Center,
then as the Weight of the Body is fuftained by the Plane it
may be confidered as without Weight, and in Equilibrio.
Therefore if the Plane be moved horizontally, it will imprefs a
projectile Force on the Body, and it will begin and continue to
move round the Center, with a circular Motion. Now this
projectile Force being in the Direction of a Tangent, the Body
will endeavour to proceed in fuch a Line, but it is conſtantly
checked and drawn therefrom into the Periphery of a Circle bý
the String, which String will therefore be ſtretched with a cer-
tain Force by which the Body endeavours to receed directly
from the Center; and which is therefore the Centrifugal Force
generated by the Motion of the Plane.
3417. As the Velocity of the Plane or circular Motion is
greater, the Centrifugal Force will encreaſe and that in Propor-
tion to the Square of the Velocity, (1175.) So that the Centrifu-
gal Force will begin from Nothing and encreaſe to any Degree;
and
Of CLOCK-WORK.
439
and confequently will, in one Degree of Velocity, be equal to
the Weight of the Body, or the String will, in that Cafe, be
ſtretched juſt as much as it would be by the Weight of the
Body hanging freely to it at reft.
3418. If, therefore, in this particular Cafe, a Power of At-
traction to the Center, equal to the Weight of the Body, was
fubſtituted inſtead of the String, it would make no Alteration in
the Motion of the Body; and hence it appears, that when a
Body is moved in a circular Orbit by a projectile and gravitating
Force, there will be a Centrifugal Force produced juft equal to the
Power of Gravity.
3419. But, in fuch a Cafe, the Velocity of Motion is ſuch
as the Body would acquire by defcending through half the
Radius of the Circle (1187.) Therefore in the Time of that
Defcent, the Body (with the faid Velocity uniformly continued)
would defcribe twice that Space, or a Space equal to the Ra-
dius, (993.) Therefore the Time (t) of Defcent through of
the Radius (R) is to the periodical Time (T) in the Circle, as
Radius (R) is to the Periphery (P) of the Circle, or as the Dia-
meter (D) is to twice the Periphery (2 P). That is, t:T::D
: 2 P.
3420. Now in the paraboloid Veffel, fince CE, or BI, is
a conſtant Quantity, being ever equal to half the Latus Rectum
of the Parabola (747,) there will be one Part where the Cen-
trifugal Force or Radius of the Circle BG will be equal to
Gravity BI. And it is evident this muſt be when the Diame-
ter of the Circle GH paffes through the Focus B of the Pa-
rabola, becauſe then BG BIGH, the Latus Rectum.
3421. By the Nature of the Parabola we have A B equal
BIBG (742.) Therefore, the Time of defcribing
the Circle GcHd (or any other) will be to the Time of De-
ſcent through BI≈ A B, as 2 P is to D. (3419.)
N4
3422. Bifect BI in Q, then AB = BQ; and let A B be
the Diameter of the generating Circle of the Cycloid KAI.
Now we have fhewn that the Time (T) in which a Pendulum
QA vibrates through the Cycloid K L is to the Time (t)
of Defcent through half its Length A B as the Periphery of a
Circle (P) to its Diameter (D,) (1124.) Whence we have t
7D
P
=
TD
2 P
which gives 2P TTP, or T: T:: 2P: P::
2:1;
440
INSTITUTIONS
I
2:1; confequently, the Time of revolving in any Circle in the
Paraboloid is double the Time of Vibration in the Cycloid, in a Pen-
dulum whofe Length AQ is the Latus Rectum of the Parabola.
*
2
3423. It remains now to fhew how a Pendulum may be
conftructed fo that it may always defcribe a Conical Surface,
and its Ball perform its Gyrations in a parabolical Superficies. To
this End let K H✶ be a Verge or Axis perpendicular to the Ho-
rizon with a Pinion at K moved by the laſt Wheel in the Train
of the Clock; at H it has a hardened fteel Point in a Pivot of
Agate, to render the Motion as free as poffible.
3424. Let it be propofed that the Pendulum fhall perform
each Revolution in a Second of Time; then it is plain, the Pa-
raboloid Superficies it moves in, must be fuch whofe Latus Rectum
is double the Length of a Pendulum vibrating Half-Seconds in a
Cycloid, (3422.) Let O be the Focus of the Parabola MEC,
and MC the Latus Rectum; and make AE=MO=
2
MC
the Length of a common Half-fecond Pendulum.
3425. At the Point A of the Verge, let a thin Plate A B be
fixed at one End, and at the other End B let it be faftened to a
Bar or Arm DB, ftanding out from the Verge at right Angles
and to which it is fixed at D. This Lamina or Plate A B is the
Semi-cubical Parabola or Evolute of the given Parabola MEC,
fuch as deſcribed (921 to 926.)
3426. The
Equation of this Cubical Parabola AB was
Let P, then Pxxy3 and in the
23
27
22 P
that Cafe, 2 x x = y² = 4 P²; therefore
27 px x = y³.
Focus, P2y; in
27
}} {√ the Diftance of
16
8
By affuming the Value of x
you will find all the corref
*²={P², and x = P√//} =
the Focus from the Vertex A.
(or putting of an Inch)
pondent y's or Ordinates of the Curve A B, by which it may very
eafily be drawn.
x
ΤΟ
1
2
3427. If the Pendulum is to make its Gyrations in Se-
conds, then the Parameter is MC = 4,9 Inches, or 49
•
27
Tenths. P 82,7 Tenths, and by affuming x-
IE P = =
to the Number in the firft Column of the following Table,
we fhall have y to thoſe in the ſecond refpectively.
Abfciffe
Fig. 1. in the PLATE of the THEORY of Circulating PENDULUMS,
c.
Of CLOCK-WORK.
441
Abfeffa
Ordinates,
Abfciffe
Ordinates.
0,05
0,274
1,4
2,531
O, I
0,435
I,7
2,880
0,2
0,692
2,0
3,213
0,3
0,906
2,3
3,526
0,4
1,098
2,6
3,823
0,5
1,274
2,9
4,113
0,6
1,438
3,2
4,392
6,8
1,743
3,6
4,748
1,0
2,023
4,0
5,096
1,2
2,284
3428. The String of the Pendulum muſt be of fuch a Length
that when one End is fixed at B it may lie over the Plate AB,
and then from A hang perpendicular with the Center of its Bob
in the Point E (or Vertex of the Parabola MEC) when at reft.
Then the Verge KH being put into Motion, the Ball of the
Pendulum will begin to gyrate, and thereby conceive a Centri-
fugal Force which will carry it out from the Axis to fome Point F,
where it will circulate Seconds or Half-Seconds, according as the
Line A E is 9,8 Inches, or 2 only; and A B anſwerable to it.
3429. HUGENIUS tells us, many Clocks of this Conftruc-
tion were made, and with Succefs; but that they prevailed not
fo much as the common Sort, on Account of their not being
ſo eaſily and expeditiously made
ciple of EQUABLE MOTION,
Circle has not perfectly
lum.
That they depend on a Prin-
which the long Pendulum in a
That the Index fhewing Se-
conds moves with a moft regular and uniform Motion, and not
(Subfultim) by Jerks and Stops as in common Clocks That
this Pendulum is entirely filent, or without that conftant and me-
lancholy Click Clack, which neceffarily attends the long Pendu-
Moreover, it may be obferved that the Pendulum
to circulate Seconds is but a fourth Part of the Length to vibrate
Seconds, or but juft the Length of a common Half-Second Pen-
dulum. -And, laftly, a Contrivance may be added to ftop
the Pendulum in any Part of the Circumference, and thereby
render it capable of fhewing Thirds.
3430. This Conſtruction of a conical Pendulum is undoubtedly
the beſt Form for a Temporary CHRONOMETER, for meaſuring
VOL. II.
LII
The
1
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ร
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}
442
INSTITUTIONS
the ſmalleſt Parts of Time occafionally.
A
But the Part A B
ſhould be divided into two from the Point A, and proceed diva-
ricating to the End B, where the Ends of two fine Chains (ufed.
in Watches) or filken Strings are to be fixed, which Strings
ſhould alſo unite after they pass the Point A in the Verge or a
little before they come to the Bob F. By this Means its Mo-
tion will be rendered more fteady and certain than by a ſingle
Chain or String. The Arm DB fhould alſo be nicely balanced
by a Counterpoife on the other Side. And an endleſs Screw at
K will do better than a Pinion with Teeth; but this is fubmit-
ted to the Experience and Ingenuity of the Artificer.
CHA P. XVII.
The THEORY of Mr. SULLY'S INVENTION of a Ho-
rological ROTULA, instead of a PENDULUM, for
regulating CLOCK-WORK; with an IMPROVES
MENT thereof.
3431. A
S this INVENTION of Mr. SULLY is upon an en-
tire new Principle, found out by Trial and Expe-
riments directed by a natural Sagacity, without any Affiftance
from a Phyfico-mathematical THEORY, I have Reafon to think it
may prove a Novelty equally acceptable, as fingular, to the Eng-
lifh Automatift; not only fo, but the Omiffion of fa curious an
Improvement in Clock-work, muft certainly be thought a Defect
in a Treatile wrote profefledly on the THEORY thereof.
3432. Mr. HENRY SULLY was an English Watch maker, and
lived many Years at PARIS; he wrote, in the French Language,
à Treatife on Clock-work, which he entitled Regle artificielle de
Temps, printed at Paris, A. D. 1717. I never. heard that this
Artificial Rule of Time, was ever tranflated into English, nor have
I feen the Original; it fhould feem by the Title as if it was a
Deſcription of his new Clock; but we are told by Profeſſor
EULER (in the Petropol. Comment, for the Year 1727,) that
Mr. Sully had publifhed an Account of his new Clock but a little
moré
1
Of CLOCK-WORK.
443
more than a Year before, viz. in the Year 1725. But leaving
the Date of its firft Publication, this learned Profeffor thought it
an Invention worthy of an Explication, which he has accord-
ingly given us under the Title of A Differtation on a certain
new Kind of Tautochronal Curves; and which I ſhall here tranf-
late and abridge for the Uſe of the English Artiſts.
3433. SULLY'S Invention confifted in a Kind of Rotula or ſmall
Wheel* A B made to ofcillate about its Center C (Fig. 2.) by
Means of a String with a ſmall Weight CP, playing between
two curved Plates CE, CF, fixed to the Surface of the Rotula.
But what Degree of Curvature was neceſſary for thoſe Plates, that
the Oſcillations might be all isochronal among themſelves, and
to the Vibrations of a Pendulum of a given Length, was what
Sully wanted to know, and could only find by frequent Trials,
and that very imperfectly.
3434. But to fhew what this particular Species of Curvature
is from Phyfico Mathematical Principles, the Procefs is as follows.
Let CM (Fig. 3.) denote one of the Plates in a Situation not na-
tural, or when the Rotula is turned on one Side; and let C B be a
vertical Line, parallel to the Direction of the Thread MP touch-
ing the Curve in M; then from the Point of Contact M draw
MT perpendicular to CB, which will alſo be perpendicular
to the Curve in M. Draw the Right Line C O containing the
Angle BCO by which the Rotula is carried out of its natural
Pofition. On the Center C with any Radius CB deſcribe the
Arch BO, meaſuring the Angle BCO.
3435. Then it is evident the Weight or Power P at the End
of the String will endeavour to reſtore the Rotula to its natural
Situation by a Force which will be as P × TM (1049.) But
as the Application of this Force is continually altering with
reſpect to the Fulcrum C, we must find one equal to it to be
applied to the Radius CO, and acting normally at the Point.
O. Produce PM to N in a Right Line drawn horizontally
through the Center C. Then becaufe CN TM, the Ef-
fect of the Power P to turn the Rotula will be the fame as if it
was applied to the Radius CN in N, and acting normally
thereto.
LII 2
3436. But
* See Plate entituled "The THEORY of the Circulating PENDU-
■UM, Ofcillating ROTULA, &c.",
44.4
INSTITUTIONS
3436. But from the Nature of the Lever (1051) we have
CO: CN:: P:p the Weight to be applied at O, whofe
Force fhall be equal to that of P applied at N. But
CNTM, therefore CO: TM:: P:p, and fo p =
PxTM
CO
; therefore becauſe P and CO are conftant, we have
p every where proportional to T M.
3437. We are next to confider that the Power (p) acting
normally at O, will make CO a Pendulum; and farther, that
if the faid Power (p) (which is the accelerating Force,) be pro-
portional to the Space paffed over, the Ofcillations of the Pen-
dulum will be isochronal (999.) But the Space to be run over
is the Arch OB, or the Angle OCB; therefore in order to
produce this fochronism in the Pendulum (and confequently in
the Rotula) the Affair is reduced to this Problem, viz. To find a
Curve CME of fuch a Property, that having a Right Line CO
given in Pofition, and in it, from a given Point C, another Right
Line CT be drawn perpendicular to the Radius of Curvature in any
Point M, the Part TM fhall be always proportional to the Angle
TCO.
are proportional, their
Therefore let M and m
indefinitely near to each
3438. Now when any Quantities
Fluxions have the fame Ratio (789.)
(Fig. 4.) be two Points in the Curve
other; and draw CM, Cm, and the two Normals MT and mt,
occuring in R the Center of Curvature at the Part M. Alfo let fall
the Perpendiculars CT, Ct, on the faid Normals. Then is
1
TCp the Fluxion of the Angle TCO; and Tp, the Fluxion
of the Normal TM. Now it is plain, that in Regard to the
Auxionary Angle TCp, the Side T C is Radius, and Tp, the
Tangent. Therefore the Fluxion of the Arch O B, which is
Bb will conftantly be as Tp, the Fluxionary Tangent of the
fame.
3439. Let CB be equal to CT Radius, and as the Ra-
dius of a Circle is conftant, the Fluxion thereof is nothing,
that is pto; therefore CT is perpendicular to T M in the
Point R, or Center of the Circle of Ofculation in the Point M.
3440. In this Cafe, the Angle OCT has its Fluction
B not only proportional but equal to the Fluxionary
Tangent Tp. For the Fluxion (2) of any Arch z and the
Fluxion of the Tangent (i) thereof, are expreffed by this
Equa-
Of CLOCK-WORK.
445
Equation =
a² i
a² + t²
(823.) and therefore when tơ, or
the Tangent is in its nafcent or fluxionary State, that Equation
becomes =
a² i
a²
-i.
3441. But the Fluxions of contemporaneous Fluents being equal,
hew thoſe Fluents are alfo equal, or that TM the Radius of
Curvature is ever equal to the Arch of the Circle TO de-
ſcribed in the fame Time.
3442. Hence the Nature of the Curve CM is manifeft, it
being no other than the EVOLUTE of a Circular Arch OT
(Fig. 5.), whoſe Beginning is at the Point of the Circle O, and
is deſcribed by the Evolution of that Circle whofe common Ra-
dii are CTCO.~ In this Proceſs, we have uſed a dif-
ferent Method of Demonftration from that of Profeffor EULER,
which we think is more natural, clear, and concife.
3443. Hence then it appears that the Lamina or Plates OF,
OE, [Fig. 5.] are formed by the Evolution of the Circle
ADO equal to the Periphery of the propoſed Rotula, and to
begin from the Point O in the Circumference, and not from the
Center C, as in Sully's Conftruction. In the fame Point O
likewife the String OP is to be fixed, which by its Weight
P will move between the Plates O E, OF, (like a Pendulum
between the Cycloidal Cheeks) and thereby continue the Motion
of the Rotula, once begun; and render the Ofcillations ifochronal,
by a Force ever proportional to the angular Space deſcribed in each
Ofcillation.
3444. The next thing to be determined, is, the abfolute Time
of an Oscillation in a given Rotula; this is done by finding the
Length of a Pendulum vibrating in a Cycloid, ifochronal to the Ro-
tula. The Method for doing this our celebrated Profeffor has rather
pointed out, than demonftrated; we ſhall therefore here ſupply the
Principles which he affumes as known, and on which this Part
of the Theory depends.
3445. To this End we muſt confider that as in reſpect to
the Endeavour or innate Force of Bodies to defcend, it is all
one whether we confider the Quantity of Matter as diffuſed
through any Space or all collected into a Point, this Effort to
deſcend being conſtantly proportioned to the Mafs of Matter into
the
446
INSTITUTIONS
the Force of Gravity (1000.) So with Regard to an Oscillatory
or Angular Motion of Bodies about a Center or Axis, it matters
not how the Maſs of Matter is diſpoſed, whether we conſider
it in the Form of a Circular Area, or in that of a Right Line
but this latter is moſt natural, as it is that by which the ſaid An-
gular Motion is deſcribed.
3446. If therefore the Radius of the Rotula be confidered as
charged with its whole Weight (Q) we ſhall find its Force of
Refiſtance to fuch Angular Motion will be expreffed by Qgn,
(2286,) where g== Diſtance of the Center of Gravity,
and n Diſtance of the Center of Ofcillation, the Length
= 3
of Radius being = 1. The whole innate Force, therefore,
of the Radius or Rotula to reſiſt to a Circular Motion is as Q.
उ
3447. But if we confider the Rotula in Motion its active
Force then (called the Vis viva) will be as the innate Force mul-
tiplied by the Velocity of the Center of Force. But this Center
of Force, in the Sector of a Circle is diftant from the Center of the
Circle, of the Radius (1098,) it is therefore the fame in the Cir-
cular Area, or Rotula; therefore if the Velocity of a Point in the
Circumference be fuch as is acquired by the Deſcent of a
heavy Body through the Space S, which will be equal to 2 S
(993;) then the Velocity of the Center of Force will be S
(for 1 2S::: S), confequently the whole Force of the
moving Rotula will be Q× SQS, and therefore the
Fluxion of its Motion will be us.
Z
3
4
3448. But this Fluxion of Motion in the Rotula is the Ef-
fect of the Action of the Weight P by Means of the String
PO upon the Plate O E. Let v = Velocity, (or Space de-
ſcribed) in an infinitely ſmall Particle of Time by the Weight
P; then its momentary Force is Pv; and as this produces the
momentary Motion Qs in the Rotula, it must be equal to it,
that is, PXvQs, and fo v =
14
Q s
2P'
3449. This is the Cafe of the Rotula and its Weight P, at
the Commencement of an Ofcillation, when the Pofition of the
Whole is as fhewn in Fig. 6. Where BDT is the Rotula,
T A one of the Plates, when the Radius of Curvature BA is
in a Horizontal Pofition, A P is the perpendicular Direction
of the Weight P; and let fall CB perpendicular to A B. Then
let
Of CLOCK-WORK.
447
let the Rotula by the Action of the Power P move in the firſt
Moment with the angular Motion T Ct, and then will ta be
the Part of the Plate on which the Power acts, and Ba the
horizontal Radius of Curvature; the Line of Direction is now
ap, and the Power P has moved from P top, through the Per-
pendicular Space P q
9 = v.
3450. On the Center C with the Radius Ca deſcribe the
fmall Arch a d, and draw Ca, Cd; then is Ad Pq; and
the right-angled Triangles Aad and B a C are fimilar; there-
fore Ad: A a:: Ba: BC; hence Adv
But A a Tt, whence
Bax Tt
BC
= 2 =
therefore 2 Px BaxTt=BCxQx s =
е
s
Bax Aa
B C
Q$ (3448 ;)
2 P
(becauſe of Ba
Ba=
ŝ 2 PX BT
BT) 2 PXBT x Tt. Therefore =
Tt Q× BC •
3451. But Tt is the Space deſcribed by the Point T in the
firft Particle of Time, and is therefore as the inceptive Velo-
city v; confequently
s
Ś
is as
Tt
but =T, the Fluxion
of the Time of an Ofcillation, fince in all Cafes of defcending
Bodies, we have
S
V
= T (991.)
3452. Having thus determined the Fluxion of the Time of
Ofcillation in the Rotula, we ſhall readily diſcover the Fluent,
or Time itſelf, by comparing fuch an Ofcillation with a Syn-
chronal Vibration in the Cycloid. Thus (Fig. 7.) let AN be the
Semi-cycloid juft equal to the Semi- arch of Ofcillation BT,
and therein take Nn Tt and through N, n, draw PN,
pn, perpendicular to the Axis of the Cycloid, and let AO be
the Length of the Pendulum defcribing the fame by Evolution.
from the Cycloidal Cheeks, as taught (1120.)
3453. Then from (n) draw nt perpendicular to PN, and the
Fluxion of the Time of a Vibration will be as
in
N(3451,) which
n
therefore muſt be equal to the Fluxion of the Time of the Ofcil-
t n
lation, viz.
N n
S
2 P x BT.
Tt
QxBC'
But from the Nature
of
448
INSTITUTIONS
¡
of the Cycloid (1117 to 1125,) we have
fore becauſe AN=BT, it is
I
AO
nt
ΑΝ
there
N n
AO
2 P
Qx BC (3450)
which gives Q× BC = 2P × AO, and therefore this Analo-
gy Q: 2P :: AO: BC; that is, The Weight of the Rotula is to
twice the Weight P, as the Length of a Pendulum vibrating in a
given Time, is to the Diameter of the Rotula, which shall ofcillate in
the fame time.
3454. For Example, fuppofe it be required to find the Radius
BC of the Rotula that fhall ofcillate precifely in a Second of
Time. Then A O = 392 Tenths of an Inch; and ſuppoſe
2 P x AO
Q: P:: 100: 1, then BC
= 7,84
Q
Tenths of an Inch. Or the Diameter 2 B C = 1,568 or a lit-
tle more than 1 Inch.
784
100
3455. It was neceffary to fuppofe the Weight P very ſmall
in Compariſon of the Weight Q of the Wheel or Rotula, that
its own Vis Inertia might be inconfiderable; for where that to
be taken into the Account, it muſt be deducted from the Force
which we have all along appropriated to move the Rotula; and
this would embarraſs the Theory, or leave the practical Ex-
ecution of it imperfect.
3456. Another Caution neceffary in a Regulator of this Na-
ture, is, that the Direction of the String below the Cheeks be al-
ways perpendicular, or that the Weight P has not the leaſt
ofcillatory Motion in itſelf, as it would make a Difference in its
Force upon the Rotula, and induce an Irregularity in its Mo-
tion, and Time of its Ofcillations. A principal Thing to pre-
vent this is to have the String of a confiderable Length below
the Plates that the Radius BC of the Wheel may bear but a
ſmall Proportion to it.
3457. But as a long String will be inconfiftent with the
Defign of a Clock in a portable Form, there is another Remedy
for both the forementioned Evils, and which will at the fame
Time be a great Improvement upon SULLY's Invention. This
confiſts in adding a very large Wheel or Rotula ADG [Fig. 8.]
to Sully's fmall Rotula BHI; and then as a much greater Force
will be required to move the compound Retula, fo there will be
a greater
Of CLOCK-WORK.
449
a greater Diſparity between. its Weight and that of P, and a
much leſs Motion of the String BP becauſe of the very flow Mo-
tion of the Rotula thus altered.
3458. That the Reafon of this may better appear, let TCT
(Fig. 5.) be the fluxionary Sector of the Rotula, and Tỉ = ż
the Fluxion of the Periphery, then the Area of the faid Sector
will be CT × 2, the Diſtance of the Center of Gravity
CT, and that of Oſcillation CT; then (putting CT=r)
we have its Force to refift eircular Motion (3287,) thus ex-
preffed, Qgn=rz × 3r × 3r (1098) 3. Hence the
x x = 4
Force of the whole Ratula will be 1r3z. But it is 1:C::r
4
Z.
::: Radius: Circumference; therefore z
the Force of any Wheel or Rotula, will be
4th Power of the Radius.
3459. Therefore if R
Cr, and fo
C, or as the
Radius of the large additional Ro-
tula, the Force of the Whole will be increafed in Proportion to
R, that is, the Length of the Synchronal Pendulum will be as
Qx BC x R
10=
2 P
AO × R+. (3453) -
I
2, R
3460. Since the Time of a Vibration of the Pendulum A O
is I, we fhall have the Time T of a Vibration in the Pendu-
lum AO x R4, thus 12 T:: AO: AO x R4 :: 1: R4,
therefore T_R✨, and T = R. So that if T = 2,
=√2, if T = 3, R√3; if T4, R = 2; and fo
on. Whence it appears how very flow the Rotula will move
when thus enlarged, and how little the Spring BP will on that
Account be liable to be put out of a vertical Pofition.
3461. Such is the THEORY and CONSTRUCTION of the
Automaton Sullianum with its improved Rotula; and befides, this
Kind of Regulation, which is entirely new and peculiar to this
Piece of Clock- work, there is one Property of it which no other
Time-Piece has, and that is, the flow Motion of the Rotula,
by which the Time of the Ofcillations become very long, and
confequently by that Means more equable; which induced Mr.
EULER* to think a Time- Piece conftructed in this Manner
VOL. II.
might
M m m
*For referring to this Improvement of an additional large Rotula,
he fays, Et tandem alium evolvam Cafum, qui non contemnendum in
RE NAUTICA ufùm mihi præftare videtur.
450
INSTITUTIONS
might be of Service at Sea in finding the Difference of Time or
Longitude, beyond any other Sort of Clock-work yet attempted.
CHA P. XVIII.
The CONSTRUCTION of an ANGULAR PENDULUM,
of a general NATURE, and adapted to a Planetary
TIME-PIECE, which, in the fame LENGTH, ſhall
vibrate in any given TIME.
3452.
Τ'
HERE has been, as yet, no other Sort of Pen-
dulum in Uſe, but fuch whofe Time of Vibration
depends upon its Length only; for this is the Cafe of the Circular, as
well as the Cycloidal Pendulum. However the Phyfical Theory
leaves us not deftitute of a Method of regulating Clock-work,
by a Pendulum, which fhall in any given Length vibrate in any
Time required.
X
3
3463. To demonftrate this we need only confider the Nature.
of a common Pendulum of the moft fimple Form, viz. that of
an uniform Rod or Parallelogram whofe Length fuppoſe = a,
and Weight = w; then the Force of the Rod is
=w; a²w.w
× 1a × 3a. (3286.) Now if the Pendulum be transformed, or
divided into two equal Parts, and both theſe Parts move or oſcil-
late about the fame Center as before, then if thoſe Parts be ſe-
parated from each other to an equal Diſtance from the Vertical
Line, the Center of Gravity (which we will now called x) will
approach nearer to the Point of Sufpenfion, and the Center of
Ofcillation (n) will recede farther from it in the fame Propor-
풀요​?
tion, for it will always be 3
a², a given Quantity.
as (x).
I a² w
x w
X
fox
= n; and fo x n =
Therefore (n) will be inverſely
3464. Therefore alfo the Time of a Vibration will be greater,
in the fame Pendulum of two Parts, as thofe Parts contain a
greater Angle with each other; or ftrictly speaking, the Time
of
of Ciock-WORK.
451
of a Vibration will be in the inverse duplicate Ratio of the Co-
Sine of half that Angle.
3465. To illuftrate this by an Example; let ACB (Fig. 9)
be a common Joint-Rule, whofe two Parts ACD and BCD
are cloſe together; through the Center C of the Joint, let a
fmall Hole be drilled, and the Rule fufpended thereby on a po-
liſhed Pin or Wire, to vibrate freely Let AC = a, the Dif
tance of the Center of Gravity CG
。
a; and that of the
Center of Ofcillation CN a. Alfo let the Weight of the
whole Ruler bew. Then (as was faid) the Length of an
ifochronal fimple Pendulum will be 3
// a²
a
n =
a. (1095)
3466. But now let the Ruler be opened, or the two Legs
A C and B C removed from the Perpendicular C D to an equal
Diſtance on each Side, viz. to HC and IC; and let CK =
CL (= AC) = CG; and on the Center C deſcribe the
Arch KGL; and draw KL cutting the vertical CD in O.
Then is the Point O the new Center of Gravity in the Rule
thus opened; and CO =
{
ter of Ofcillation will be
13
x;
and the new Diſtance of the Cen-
1
a² w
3
= 11.
x
X 20
3467. Examples will make this Doctrine plain; fuppofe it
required to open the Legs of the Ruler to fuch an Angle that
the Time of a Vibration fhall be juſt equal to that of a fimple
Pendulum whofe Length is equal to that of the Rule CD; in
this Cafe we have na, therefore x = a = CO; which
will give this Analogy, as CO: CK (:: ¦α : ¦ a) :: 10:15
:: Radius: Secant of 48° 12′ = HCD, half the Angle HCI,
as required.
3
a
3468. For a fecond Example, fuppofe CN 9,8 Inches
or CD = 14,7; then will the Ruler clofed vibrate Half-Seconds
preciſely; let it now be required to find what Angle it muſt be
opened to, that each Vibration may be performed in One Second?
In this Cafe it is evident = 39,2 Inches. Confequently
117,6x, confequently 117,6
Then CO: CK:: 1,837
a² = 39,2 x ; therefore a
:
:: = x =
14,7 14,71,837 CO.
:
=
::
7,35 Radius Sccant of 75° 32′ HCD, half the Angle
required.
Mmm 2
3469. When
452
INSTITUTIONS
0,
3469. When xa, the Diſtance of the Point N becomes
infinite, and the Time of Vibration alſo; that is, the two Legs
CH and CI are then in a Right Line, and at Reft. But this
Form of a Parallelogram angular Pendulum will meet with too
much Refiftance from the Air; and befides, is a third Part
longer than neceflary; therefore that fuch a Pendulum may
have all Advantages it must be in the common Form of a Rod
and Ball at the End.
3470. The Rod and Bob of this new Pendulum muſt there-
fore be double. The Bob may coufift of two Hemiſpheres or
two Plano convex Lenfes, put together by their plain Sides, ſo
as to make a compleat Sphere, or elfe a double and equally
Convex Lens. Let A B, A C, (Fig. 10.) be the two Parts of
fuch an angular Pendulum fufpended on the Point A, and open-
ed to the Angle BA C, exactly bifected by the vertical Line
AD. But when the two Parts are cloſed, they make the fim-
ple Pendulum A E.
And here let the Weight of the Rod A E
(= Weight of AB + AC) = b, and the Weight of the
Bob = c.
3471. Then when the Pendulum is opened to the Angle
BAC, the Center of Gravity is raiſed from a Point near E to
O, in the vertical A D, as appears by drawing the Line BC;
then the Center of Ofcillation (n) is removed to a greater Dif-
tance in Proportion; for putting A E (AB) = a, and AO
= x; we have a² b x a²c
xa a = x n.
кай
X
} {
x xb+c
= n (1094,) whence
=
3 b + c
b+c
3472. Put BO=y, then a (A B² =) x² + y², and fo
3 b + c
b + c
x² + y² as xn; confequently y=xnx which is an Equa-
tion to a Circle (819,) and fhews the Locus of the Bob B is the
Periphery of a Circle whofe Diameter is n = AD the Length of
a Pendulum ifochronal to the angular Pendulum BAC.
× x² + y² = x n; therefore it will be every where
3473. Hence alfo it appears that the whole Circle ABDC,
(ſuppoſed heavy) or any Fart BAC (the Points B, C, being
equidiftant from A), will vibrate in the fame Time with a Pen-
dulum equal to the Diameter AD. All which Hugenius has
demon-
Of CLOCK-WORK.
453
demonftrated from far different Principles, which he himſelf un-
derſtood very well.
3474. Now it is manifeft the Time of Vibration in this an-
gular Pendulum may be altered at Pleaſure, and be made greater
or lefs as occafion requires; and confequently, that it is capable
of being adapted to meaſure Time univerfally in a Clock from
the Sidereal (or ſhorteſt) Day, to the longeſt Lunar One, and
thereby to answer the fame Purpoſes as the univerſal Pendulum
before deſcribed in Chap. X. and XI.
3475. In order to this, we are to confider the Value of x,
when the Pendulum is cloſed in A E, is
2ac+ab
2 Xc+b
=$, (3027)
Xaa (3471,) therefore
I b + c
but it is always equal to
n x b + - c
उ
3 b + c
2actab
X
x a² =
>
nx b + c
2 xc+6
which will give this Analogy
a: n :: c + ½ b : c+b.
Alfo by the fift Equation it is
a: 8 :: c + b:c+b; the fame Analogies as before in
(3029.)
2
3476. Let t Time of the Pendulum's Vibration when
accommodated to the Sidereal Day of 1436'; and T Time
of its Vibration when adapted to the Solar Day of 1440′ (3313)
T₂
t
Then it is : T² :: t : = N,
chronal Pendulum for the Solar Day.
c + ÷ b
3477. Let p =
c+ b
T2
=N
===
the Length of the ifo-
pa²
; then pa² = Nx; therfore Pa
Z
; therefore t paTx; but we have g:x:: Radius
: Co-Sine of the Angle BAO: rc; hence
& C
::
t pa²
p; therefore gCT² = rtpa²; confequently
1 pa²
= C,
& T
the Radius (r) being Unity.
Hence fince t, þ, a, g, are
given Quantities, we have C as
I
T
,
or the Co-Sine of the An-
gle BAO as the Square of the Time of Vibration inverfely.
3478. If
454
INSTITUTIONS
3478. If we make b: c :: 1 : 10 then we ſhall have c + b
: c+bta :: 1436: 1463,4a (3313) = Length of
the Rod A E, when the Pendulum is cloſed; and the Length as
well as Time of Vibration of an ifochronal Pendulum are both
reprefented by (t) for the Cafe of the Sidereal Day; and T is
= 1440.
3479. Again, fince in that Cafe, c + b c +
1 b :: a : 8
≤ p a²
= C
(2475): 1463,4 1397 g. Whence we have
: =
8 T²
83° 58′ ABO, whofe Complement BAO = 6° 02′;
therefore BAC = 12° 04′, the Angle to which the Sidereal
Pendulum muſt be opened to vibrate Solar Time, for Uſe in com-
mon Clocks.
3480. Let the Time of Vibration of the Pendulum for the
longeft Lunar Day be T 1500; then the Angle to which the
Pendulum muſt be opened will be C
T2 C
-
= 66° 25′,
T2
whofe Complement is 23° 35′; the double of which is 47° 10′
BAC, the Angle to which the Pendulum muſt be opened
for the longest Lunar Day.
3481. By the fame Theorem, putting T to any Number
of Minutes in the Day propofed for a given Planet (3323,) you
find the Angle to which the Pendulum muſt be opened to vibrate
true Time for that Planetary Day. But to fave the Reader
Trouble, I have here ſubjoined a Table by which the Planetary
and Lunar Scales may be added to a Pendulum of this new Form.
•
1436
143611
O O
1455
13 5
I 31
1460
14 40
1436/1/2
2 8
1465
16 5
1436 3/
2 37
1470
17 24
1437
3 3
1475
18 36
1438
4 17
1480
19 42
1439
5 14
1485
20 45
1440
6 2
1490
21 45
1445
9
2
1495
22 41
1450
11 15
1500
23 35
3482. Ac-
Of CLOCK-WORK.
455
be
3482. According to this Table the Arch of a Circle may
divided into Degrees on one Part, to 23° 35′ on each Side the
Vertical Line; and then into 64′ of Time on another Part,
contiguous to the Degrees; by which a proper Scale will be
conftructed for the Pendulum which may thereby be adjuſted
to the Time or Length of any given Planetary Day.
3483. It muſt be left to the Ingenuity of the Clock-maker,
to contrive its Application in the beft Manner, and to deter-
mine the Length of the Pendulum, which is done by taking the
Length in Tenths of an Inch proportional to the Numbers before
made uſe of for the Lengths of fimple and compound Pendu-
lums. Thus the Length of Second Pendulums being 98,
you fay, as 1444: 1463,4:: 98: 99,32 = a AB, in Tenths
of an Inch; the Length A B is therefore 9,23 Inches.
I
2
3484. Hence it appears how well adapted this Angular Pen-
dulum is for meaſuring Planetary Days; and perhaps it may be
found in fome Refpects more uſeful for that Purpoſe, than
that we have formerly deſcribed (Chap. XI.) However, this has
a Property which that had not, viz. of having the Time of a Vi-
bration protracted or augmented to what Degree you pleaſe. For
Example; ſuppoſe the above Pendulum A B were required to vi-
brate Seconds exactly. Then T:T:: 1:2; and T² : T² ::
T2 C
I: 4; therefore = C (3480,) the Co-fine of the Angle
72
BAO 76° 54' in this Cafe; in that of the uniform Rod or
Ruler it was 75° 32′. (3468.)
CHA P. XIX.
The THEORY of feveral new Eliptical and Horizontal"
PENDULUMS of different FORMS, which ſhall vi-
brate in any given TIME.
3485.
BE
ESIDES the Methods laid down for conſtructing a
Univerfal or Planetary Pendulum, there is yet ano-
ther, which as it contains fome other Properties very fingular
and
456
INSTITUTIONS
and curious, I imagine the ſpeculative as well as the practical
Horologist will be pleaſed with the following Account of it.
3486. Suppoſe A B an inflexible Line (without Weight;)*
and in any Point taken therein, as L, let it be propofed to af-
fix thereto at right Angles, a heavy Rod or Bar FLG, bi-
fected in L, which being moved (in latus) fideways, fhall of-
cillate in the fame Time with the fimple Pendulum A B, the
Point A being the Point of Sufpenfion to both. Let m be the
generating Point by which the Rod is deſcribed; and join Am.
Put A L = x, Lmy, Am=
y, Amd, and A B = n.
3487. Then dd = x² + y²; and dj j + y²y the
x² ÿ
Fluxion of the Particles in the Rod FG multiplied by the
Square of its Diſtance; the Fluent whereof is ² y + — y³
which therefore is as the Sum of all the Particles, each multi-
plied by the Square of its Diſtance from the Point of Suſpen-
fion A; the Weight of the Rod is as its Length, or half its
Length, viz. as (y;) therefore
= A B. (1094.)
x²y + 1
3
xy
درو
-n-
3
x² + = y²
3
x
I
3
3488. Hence we have 2 + 2 =nx; and therefore yy
x² J
=x-x². Now this Equation is evidently that of an EL-
LIPSIS (767,) as that of the Angular Pendulum was an Equa-
tion to a Circle (3472.) Whence it appears the two Extremes
F and G of the Rod are in the Perimeter of an Ellipfis.
3489. To determine the Species of this Ellipfis, in the
Equation y²=3nx-3x², let x = AC =
Conjugate Diameter, and then y = CE
2
n, the Semi-
2
a, the Semi-
Parame-
Tranfverfe, which Values fubftituted for x and y in the Equa-
tion give 3n² = a², whence a²: n² :: 3 : 1 :: a : p
ter of the Ellipfis (764) Therefore p = 1; a= 3
1 ; a = 3 = DE
and n = √3 = A B; by which the Ellipfis is truly limited
or ſpecified.
3490. It may be proper here to obferve to the young Reader
that there is a Diſtinction made (by Mathematicians) of Analytic
Problems, viz. into Plane and folid Problems They call that a
Plane Problem when the Quantities in the Equation are but of
two Dimenfions (as a Plane has only Length and Breadth ;) thus
the
*See Fig. 1. in the Plate entituled, New PENDULUMS of different
KINDS.
New PENDULUMS of different KINDS.
P
Fig
*
B Q
2.
R
D
C
D
P
B Q
M
HI
E
A
C
D
H
Fio: 4
•
1
>
1
I
C
E
M
A
F
m
L
H
Fig: 3.
R
K
E
O
P
B
D
H
B
C
K
Fig:5
Fig. 1.
N
Ꮐ
N
D
S
A
E
T
K
I
D
a
FRIISIDIEMATTUMUTHBUMS
C
Ꮮ
F
E
Fig 7
03
M
d
H
***
།
P
Ꮐ
E
I
P
B
Fig:6
H
N
A
A
B
C
N
Fig:
8.
Ꮐ
B
Fig: 9.
F
Of CLOCK-WORK.
457
the Equation 2 xnx², is a Plane one, becauſe y² + ²
y²
= nx= Rectangle of two Dimenfions, n and x. But the
Equation we have juft now found yy = 3nx 3xx (or y²+
3x² = 3nx) (2488,) has three Dimenſions, (viz. 3, n, x;)
and is therefore a folid Problem, as a Solid has Length, Breadth,
and Thickneſs.
3491. The Plane Problem, we obſerved (3473) makes the
whole Circle, or any two Points or Parts of it equidistant from
A (Fig. 11.) ifochronal Pendulums. But the prefent folid Pro-
blem, fhews that any Line terminated by the Ellipfes at Right
Angles to the Conjugate A B, and confequently that the Plane
of the Ellipfis itſelf, or any Part of it between one or two right
Lines, (normal to the conjugate Axis A B) will all perform
their Oſcillations in equal Time.
3492. Now this Form of a Pendulum is very well fuited for
ſhort Vibrations, and may therefore be applied to Clocks of a
portable or Table Form, and in a Size as ſmall as you pleaſe. For
there are three different Sorts of Pendulums that you may make
out of this Elliptic Plane.
3493. The First is a Pendulum AMN with horizontal
Length only, ofcillating on the Point A; and which therefore
has this peculiar Excellence, that its Vibrations are not liable
to be diſturbed by either the different Force of Gravity in different
Regions of the Earth, or by different Degrees of Heat and Cold,
as thofe of all long Pendulums are. Confequently this Pendu-
lum does not ftand in Need of that conftant Correction neceffary
in all that have vertical Length. It is alfo on this Account bet-
ter adapted to Sea-ufe; and is much leſs liable to be agitated or
put out of Order than long Pendulums by the Motion and Tof-
fings of the Ship.
3494. The Second Pendulum which this elliptic Plane affords,
has the Length of the Conjugate A B, and a Bob OPB, which
is a Segment of the Plane on its loweft Part. This Bob has two
Peculiarities, viz. (1) it has properly no Center of Magnitude
or Ofcillation which the Clock-maker has any Need to concern
himſelf about, as in common Bobs (3292 to 3296.) Then (2)
it has a Form extremely well fuited to avoid the Reſiſtance of
the Air, as is evident by Infpection. On both thefe Accounts
VOL. II.
Nnn
this
458
INSTITUTIONS
L
{
!
1
this Pendulum (which is both vertical and horizontal) may claim
Preference of all fimple Pendulums of the fame Length A B.
2
*
}
continues
=
3495. The Third Pendulum derived from the Ellipfis is one of
the univerfal Kind, or that may be adapted to a Clock for fhew-
ing Planetary Time. For this Purpoſe we are to take the longeſt
Line or Tranfverfe Diameter DE of the Ellipfis. For it is evi-
dent from the Equation (2487,) that while AC
the fame, the Length of an itochronal Pendulum (n) will en-
creaſe with an Increaſe of CEy; for Example, if AB 98
Tenths of an Inch, (to fwing Seconds,) then CE = 84,7;
and AC 49. for Solar Time; but if it be required to fhew
the Time for the longeſt Lunar Day, the Time of Vibration
muſt be increaſed in the Ratio of 1440 to 1500 (3323,) and
confequently the Lengths of the ifochronal Pendulums in the
duplicate Ratio thereof, or as 14402 to 15002, as fhewn (1116,)
this will give A B≈ 106 = n, for the faid Lunar Day when
we get y = CQ = 91; then CQ-CE=EQ = 6,3=
the Length of the Part to be added at each End to accommodate
the Pendulum for the Extent of the Lunar Scale. And thus
may the Increaſe of Length be found for Mercury, Venus,
Mars, Sc.
3496. I have already obferved, that a Fourth Sort of Pendulum
refulting from this Elliptic Equation is the Plane AE BD of
the Ellipfis itſelf; nicely fufpended on the Point A; for then
vibrating Sideways from C towards D and E, it will be perfect-
ly ifochronal to the fimple Pendulum, whofe Length is AB.
And thus much, at prefent, may fuffice for the Theory of Ellipti-
cal and Horizontal Pendulums, of which we fhall make fome fur-
ther Uſe hereafter.
}
}
1
CHAP.
N. B. The Reader is defired to correst an Error in Inflitution 3384, by writing
CE2
;
for
CN
2 CN
CE2
and
>
ing an Over-fight.
NC
CE
2 CE
for
N CZ'
Also to crafe Inflit. 3383 entirely, as be-
}
}
Of CLOCK-WORK.
459
CHA P. XX.
The THEORY and CONSTRUCTION of different Sorts
of UNIFORM PENDULUMS, which, in a given
LENGTH, ſhall vibrate in any given TIME; with
feveral curious Particulars relative to CLOCK-
WORK, regulated by fuch Pendulums.
3497. MR.
R. GRAHAM (ever memorable for his fingular
Skill and Execution in Clock and Watch Work) hav-
ing tried many Experiments with different Sorts of Metals
combined in the Rod of a Pendulum to correct their Irregula-
rities by their different Expanfion or Contraction with Heat and
Cold, found them all to little or no Purpoſe; he thought of
another Method of effecting the fame, which was by Means of
a Pendulum confifting of one Part folid and the other Fluid,
viz. a Glaſs Tube filled with Quickfilver.
3498. This Mercurial Pendulum, 'tis easy to underſtand,
would have its Dimenfions encreaſed or contracted each Way
from the Center of Ofcillation by Heat and Cold; for the Tube,
as a Solid, would encreafe a little downwards, and the Mercury,
being a Fluid, would have its Column encreafed upwards from
the fame Cauſe; and therefore the Center of Gravity in one
being carried down, and in the other raifed upwards, 'tis plain,
if theſe Increments different Ways of the folid and fluid Parts
could be nicely proportioned as they ought, the Center of Of-
cillation, would not be in the leaft affected thereby, but would
conſtantly be at the fame Diſtance from the Point of Suſpenſion,
and the Pendulum vibrate, of courſe, in equal Time.
3499. Accordingly he conftructed a Pendulum of this Sort,
and after many Trials, fucceeded fo well, that when he com-
pared one with a Clock which he kept for a Standard or Regulator
(whofe Weight of Pendulum was 60tb) and which he obferved
altered
Nnn 2
460
INSTITUTIONS
altered not more than 12 or 14" in 24 Hours between Winter
and Summer, he found by 3 Years and 4 Months conftant Ob-
fervation, that the Irregularity of the Clock with the Quickfilver
Pendulum, did not, when greateft, exceed a 6th Part of that of
the other Clock, and for the moft Part of the Time it did not
exceed an 8th or 9th Part. In this Clock, the Tube was Braſs
and varniſhed on the Infide.
3500. But as a Pendulum of this Form, to beat Seconds,
must be near 5 Feet long, it will be worth while here to fhew
how any uniform Solid Pendulum may be conftructed, that, in
a given Length, fhall vibrate in a given Time; especially as many
curious Particulars will refult from the Demonftration of the
Theory, and fuch as we think not a little interefting in a Trea-
tife of Clock-work.
3501. Let CDEH (Fig. 2.) be a Rectangle fufpended on
the Point B by a Thread A B, in its middle Point A; produce
BA to F; and drawing OL M parallel to CD; put A Ba,
AL, LM = y, and B Md. Then, ſuppoſing the
Rectangle DH to vibrate (in latus, or) Sideways about the
Axis PQ, the Force of all the Particles in the Line LM will
be a + x² × y + y³ (3487) =
which multiplied by x gives a² y
whofe Fluent ay x + a x²y +
the Rectangle CM, or of the Rectangle CE, when x or A L
becomes A F.
1
3
3
2
I
3
a²y + 2 a xy + x² y + {y³,
+ 2 axxy + xx² y + } y³ x‚
x³y+y³x is the Force of
.3
I ,3
3
3502. Then let G be the Center of Gravity in the Rectangle,
and becauſe the Fluxion of the Weight xy multiplied by the
Diſtance a +x is a xy + yxx, the fluxionary Moment of the
Rectangle, therefore the Moment itfelf will be the Fluent
axy + Lyx². Confequently
2
3
a²y x + ay x² +
I
3
{ x³ y + — y³ x
3
ay x +
2y x²
a² + a x + x² + { y²
I
I
3
a + 1/2 x
Bn, the Diſtance of the Center
of Ofcillation from the Point of Sufpenfion B, when x =
AF.
3503. Let a = 0, or let the Rectangle be fufpended in its
Vertex
Of CLOCK-WORK.
461
Vertex A, then will the Diſtance of the Center of Ofcillation
22
ΑΝ
3
1 x ² + 1 / 1 ²
1 x
3
x² + y²
2
= 3/4-x
=+, and when
x
x
the Breadth of the Rectangle is infinitely ſmall, or yo, it
X
X
3
becomes a Right Line A F, and then =}=} AF, as
before fhewn (1129.)
3504. If the Point of Suſpenſion B be within the Rectangle,
then A B = a, will be negative, but a² will be poſitive; and
the Diſtance of the Center of Ofcillation will be
a²
2
ax + 3x² + 3y²
a
उ
3505. If 2y=x, then the Rectangle becomes a Square; and
the Equation for the Center of Ofcillation becomes
5
a²±ax + 3/3/2²
I
2
1 x ± a
+
and if a 0, or the Square be fufpended in the vertical Point A,
the Theorem is
5+2
I 2
*
2 X
Diſtance of the Center of Of
cillation in the Line A F, from the Point A.
3506. If we fuppofe x lefs than 2y, then is the Rectangle to
be confidered as a Horizontal one; but the fame Equation
(3504) gives the Diſtance of the Center of Ofcillation in the
Line A F, continued out, if required.
3507. If x = 0, this Rectangle then becomes an horizontal
a² + = y ²
Line CD = 2y; and the Equation (3502) becomes a
the fame Expreffion as was found for the Center of Ofcillation
in fuch a Line before. (3488.)
3508. Let us now ſuppoſe the Plane CDEH (Fig. 3.) to
move parallel to itſelf in the Direction of the Line 1 K, and
thereby generate the Solid or Parallelopiped C S in flowing from
A to K, or the Solid CT in moving from A to I. Put A I, or
AK, z; then the Force of the Rectangle or Plane CE,
which is a² yx + ax²y+}x³y + 3x (3501) being multiplied
by ż, will be the Fluxion of the Force of the Solid, viz. a²yxż
+ a²x² y z + 3 x³y ż + ¦ µ³ x ż. And axy + ½ x² y ż is
the Fluxion of the Momentum thereof; therefore the Fluent of
the former divided by that the latter, is
उ
3
2
462
INSTITUTIONS
a² x y z + a x²yz + {x³yz + + y³ xz a² + a x + = x² + y²
a x y z + = x² y z
==
a + 1/2 x
3
the Diſtance of the Center of Ofcillation from the Point B, and
is the very fame as was found for the Rectangle or Plane
CE (3502.)
3509. From all which it is evident that the Center of Ofcilla-
tion is not affected by the Thickness of the Parallellopiped, but
only by its Length AF, and Breadth CD = 2y; alſo,
that the Increaſe of its Diſtance from the Point of Sufpenfion is
always as 32, or 3 of a Third Proportional to its Length and half
x
Breadth (2503.) Therefore in an horizontal Plane, when yx,
we have its Diſtance +
3/4 x 3/3/ = x, or 1x below the
x
Plane in the Line A F continued out.
उ
3510. If z = 0, and y = 0, the Solid degenerates into a
Line or Rod A F, fufpended by a String (without Weight) A B
a; and then the Diſtance of the Center of Ofcillation will
a² + a x +
be
a + ½ x
3
x²
(3502.)
3511. When the Rod AF is fufpended at one End A, then
the Diſtances of this Point A and the Center of Ofcillation N
from the Center of Gravity G, we called g and d (3288.) And
if the faid Rod be fufpended from the Point B by the inflexible
Line A B, put the Diſtance B G = G and the Diſtance Gn =
a² + a x + 3*
D; then will GD = gd, in all Cafes. For
= B n = N.
3
a + /= x
Therefore, a² + a x + ÷ x² = GN; and
호
​a² + a x + — x² — a + 1/1 +
a + = x
=
GN-G
G
=D; whence
12
GN-G² GD, but G N— G² = ‚½*² = 1/2 × × / x =
gd; therefore G D = gd; and confequently G: g:: d: D.
3512. After the fame Manner it is proved for the Parallello-
CE; and all other vibrating Bodies, that the Diſtances of
gram
the Points of Sufpenfion and of the Centers of Ofcillation from the Cen-
ter of Gravity are in an inverfe Ratio, and are therefore, mutually
interchangeable, or convertible into each other. Thus for Example,
if
Of CLOCK-WORK.
463
if the Rectangle CE was to be ſuſpended upon the Point N
(which is the Center of Ofcillation when fufpended at A) then
the Plane will be inverted, and the Center of Ofcillation will
be in the End or loweſt Point A of the middle vertical Line.
3
उ
3513. Or thus; Let O be the Center of Ofcillation in the
Plane or uniform Pendulum CE; then AOA F, when
the Point of Suſpenſion is in the Vertex A. Then let the Pen-
dulum be fufpended on the Point B, fo that AB AF
(Fig. 4) then will the Center of Ofcillation be removed from
O to the End or Point F in N, and the Diſtance BN being
equal to AO, the Time of Vibration on the Centers A and B
will be the fame.
3514. If the Point of Sufpenfion be taken nearer to the Cen-
ter of Gravity G, the Center of Ofcillation will go out of the
Plane to fome diftant Point in the Right Line A F produced.
Thus if (b) be the Point of Sufpenfion, B will be the Center
of Ofcillation. And if Gb Gn (in Fig. 2.) then will GB
here be equal to GB there (according to 3511, 3512.) As
it will ever be AG x GNBG x Gn Ghx GB,
and ſo on.
× −
3515. Hence it is evident, fince in the Fraction
AG × GN
Gb
GB, the Numerator is conftant, we fhall have G B as
I
Gb
and therefore G B will be infinite, when Gbo, or when the
Plane is fufpended on the Center of Gravity G. Confequently,
in any given uniform Pendulum CE a Point of Sufpenfion (h) may
be found or affigned between B and G, fuch that the Time of Vibra-
tion fhall be equal to that of a common Pendulum of any given Length
not less than the Minimum of this Sort.
3516. As the Time of a Vibration in a Pendulum of this
Form, is, when a Maximum, infinite; it may be proper to de-
termine its Quantity when a Minimum or the leaft it can be. To
this End put BG = x, and Gny (Fig. 2.) and x+y=p,
an ifochronal Pendulum; this, when a Minimum gives y
= 0; and therefore y = -;
yj; but fince xya conftant
Quantity, (3514,) we have xy += 0; and hence xÿÿÿ
aji
=0, and fo we find xy or BG =Gn, when the Vibra-
tion is of the leaft Time poffible.
3517. But
404
INSTITUTIONS
I
3517. But it is always xy dg (3511,) and therefore in this
Cafe x = y = ✔gd = √125² (3287,) therefore x + y = Þ
= 2x = 2√2 putting S AF = 1; that is, the
whole Length AF is to the Length of the ſhorteſt Pendulum
(p) as I to 0,577.
=
3518. That this Form of a Pendulum may be adapted to ge-
geral Ufe, or to meafure any Time from the Sidereal to the Lu-
a²—a S + } S² + {W²
nar Day, in a given Length thereof S, Let
2
S a
=p, the Length of an ifochronous Pendulum proper for the
Time of the given Day (3504.)
and W = y = CD, half the
lum.) Then a² — a § + } S²
a² — a S + ap=pSS2
as+ap
2
S
3
3
S²
(Here SxAF the Length,
Breadth of the given Pendu-
+ W² pS - ap; and
W² = s; and putting p
3
1 = 1 /
3/3/
2
S=t, we have a² ta = s; and a² ta + = t² = s + 1 tt;
2
and therefore a √ √ + = 1² - ÷t;
=
√s 1/1/
4
and thus the Diſtance of
the Point of Sufpenfion from A is found for any Value of (p)
from that of a Pendulum of 1436' for a Sidereal, to 1500 Jor
the longeſt Lunar Day (3323.)
3519. Hence then it appears, (1) That an uniform Pendulum
entirely folid, may be made to vibrate in a given Time. (2)
That it may be of any given Length and Width. (3) That it may
be adapted to regulate Clocks for meaſuring different Lengths
of Days. (4) And that the Errors they are fubject to from
Heat and Cold, or Difference of Gravity in different Regions of
the Earth, are very fmall, if not altogether inconfiderable; ast
the Point of Sufpenfion is here at fo fmall a Diſtance from the
Center of Gravity; and the Length of the Pendulum itſelf fo
much leſs than an ifochronal One of the common Sort of 58,8
Inches (3298.)
CHAP.
The Principles of CELESTIAL MECHANIC´S.
S
P
P
E
NKURUELVE ARTSUENA ALQUIMIGRESEN
G
F
Fig. 1.
S
R
1
A
P
S
:
Fig. 2.
C
I
G
意
​F
C
D
味
​E
H
I
M
K
K
S
I
M
i
K
T
Fig. 4.
W
C
I
C
R B
G
B
E
E
H
P
E
D
Fig. 3.
K
I
A
X
T
D
A
Pla
Z
H
M
N
P
е
B
D
Fig. 5.
E
F
Of CLOCK-WORK.
465
CHA P. XXI.
The THEORY of Rotulary PENDULUMS, conftructed
in a new METHOD, which, in very ſhort
LENGTHS, fhall vibrate in any given TIME.
WE
3520. is rendered more compleat by a few Theorems
JE cannot difiifs the Subject of Pendulums, till it
and Obfervations which yet remain to be demonftrated, as up-
on them, fome confiderable Improvements in the Doctrine
of Pendulums will appear to depend. The great Lengths
of a Second Pendulum requiring fo large and cumberſome a
Clock-cafe has long been an Objection, which the Rotulary
Pendulum of Mr. SULLY was defigned to remove; but this may
be done by Methods more facile and natural, as we ſhall now
proceed to fhew.
3521. Though our Methods of ſhortening the Second Pendu-
lum will require the Uſe of the Rotula, yet its Application will
be to the natural, and not to an artificial Pendulum, as Sully's
was. And theſe Methods will admit of as much eafier Con-
ftructions as they are in themſelves more natural and better
adapted to Uſe, eſpecially at SEA, where Sully's Pendulum will
be much more affected by the Motion of the Ship, than any of
thefe, which we have now to propoſe.
3522. The THEORY on which thefe Methods of reducing
the Size of Second-pendulum Clocks are founded, are as follow.
Let it be required to find the Center of Ofcillation of the Periphery
ADHC of a Circle fufpended at B by an inflexible Line BH
and vibrating (in latus) about an Axis PQ Suppoſe C and D
two Particles or Weights in the Periphery, and draw the Lines
CB, D B, and the Diameter CD, which continue out to E,
and let fall the Perpendicular B E upon it; laftly, through the
Center G draw the Vertical BHA.
=
3523. Then (putting CG GDr, GB = a, and
DE e,) we have CBCG² + BG² + 2CG x GE
(639) = p² + a² + 2 ″ × r + e.
Alio we have B D² =
VOL. II.
BG=
466
INSTITUTIONS
BG - G D²
2GDX DE BG² + GD² — 2 GD
× GE = a² + r² — 2 r x r +e; therefore C B² + BD² =
2 a² + 2,²; or the Sum of the Forces of the two Particles C
2
and D is a² + y² x 2.
Therefore putting P = Periphery of
the Circle, the Force of the whole Periphery will be a² + r²
× P.
3524. In a Circle whofe Diameter is 1, the Periphery is
3,14159; then Ip: 2r: 2rp P, which fubftituted
::
for P in the foregoing Expreffion, gives 2r pa² + 2 pr³, for
the Force of the Periphery to refift to angular Motion from
the Diſtance B G. When ar, or the Sufpenfion is at the
Vertex H, the Force is, as r³ × 2p, or as 2rr x P. But
when a =
o, or the Circle is fufpended on its Center G, the
Force thereof is r3 x 2p, or 2 x P.
3525. The Momentum of the Periphery is a P; therefore
a² + r² x P
a P
BNC, the Diſtance of the
a² + y²
=
a
When ar, then 2r
Center of Ofcillation.
HA = C,
as we before fhewed from other Principles. When a = 0,
then
at reſt.
a² + y²
о
2
=C, infinite; or the Circle will in that Cafe be
3526. If we confider the Radius A G as a variable Quan-
tity then fince the Force of the Periphery is 2pra² + 2pr³,
or a² P + Pr²; therefore Pa²; + Pr²;, or 2 pria² + 2p
3 will be the Fluxion of the Force of the circular Area
ACHD (Fig. 6.) The Force then of that Area will be Pra²
+ Pr³, or pr² a² + ½ pr^. Wherefore when ao, or
the Circular Area or Rotula A DHC moves on the Center of
Gravity G, the Force is pr as we formerly found by ano-
ther Method. (3458.)
3
I
2
2
3527. Then becauſe the Area ACED = {r P=rrp,
therefore apr² will be the Momentum thereof; confequently
a² pr² + ½ pr+
-BN-
ΕΝ C, the Diſtance of the
ap 7.2
9.2
=a+
20
Center of Ofcillation from the Point of Sufpenfion B.
3528. What we have hitherto faid is upon Suppofition that
the Rod BG, by which the Circle is fufpended is without
Weight;
Of CLOCK-WORK.
467
Weight; but if its Weight be confiderable, call it (b); then
a² b (3306.) And the Diſtance of the Cen-
its Force will be
3
I
ter of Oſcillation will, in this Cafe, be 3 a² b + 2 a² + y²
= C. And, ifro, then
3
— a² b + 2 a²
26+2a²
ab + 2 a
Zab + 2 a
= C. Which Ex-
preffion is analogous to that for the Pendulum formerly de-
fcribed (3306.)
3529. We may now proceed to the Application of the
Wheel or Rotula to the natural Pendulum for reducing Clocks
to a concife and portable Form, and at the fame Time to have
the Advantage of a flow Vibration, and, of Courſe, a much
greater Equability of Motion, than common fhort Pendulums
can be expected to have. Firſt, Let a WHEEL be made uſe
of for this End (as in Fig. 7;) then if the Thickneſs of its Peri-
meter be inconfiderable, let the Weight thereof be called
P, and the Radius R, and we fhall have its Force as
RP (3524.)
3530. Again, the Force of any heavy uniform Rod S fufpended
on its Middle Point or Center of Gravity will be as
Alfo the Weight of the Rod is as S, and S
3
I 2
S³ (3286.)
R, therefore
the Force of fuch a Rod or double Radius to refift angular
Motion will be RRS; therefore, if the Weight of all the
Radii A, D, E, F, &c. be called W; then their Force will
be RRW.
is
3
3531. The central circular Part of the Wheel (abcd) may
be confidered as a ſmall Rotula whofe Radius is r, and its Force
wr² (3529) putting w = to its Weight: For the Weight
is as the Area, which is rP (3458 ) = pr² = W There-
fore the Sum of the Forces of all the Parts of the Wheel will
be R² P + R² W + ½ w r².
43
Ι
2
3532. And becauſe the Weight of the Pendulum muft, in
this Cafe be confidered in every Part, and particularly expreffed,
we will put the Weight of the circular Area MNO = P =
pr² (3527.) Whence the Center of Ofcillation of the whole
Pendulum will be 3a² b + a² p + ½ pr²
2
Lab ap
2
3533. Therefore by connecting the Wheel and Pendulum
0002
together,
468
INSTITUTIONS
R² R² 1
R2 P+R2 W +
I
together, the Center of Ofcillation refulting from both will be
wr² + — a² b + a² p + 1/2 pr²
Lab + ap
I
2
3
= C.
3534. We may hereafter fee more of the Ufe of the forego-
ing Theorem in other Parts of Mechanics; but in the Bufinefs
of Pendulums the Rotula is much preferable to the Wheel and
the Theorem is more fimple; for putting the Weight of the
Rotula W, and its Radius R, then its whole Force of
Inertia will be WR2 (3530,) and when the Pendulum is con-
44
nected with it in the beſt Manner, its Weight will be inconfide-
rable, therefore b =0; and the Theorem for the Center of
Ofcillation is
2
1 WR² +
a² p
I
2
+ ½ pr²
= C.
a p
14
3535. If we put a² +½r
2
r²=n, we have
WR2 + np
ар
2
= C; and therefore W Rap C-n p; whence this
Analogy, p: WR2: a C-12.
2
3536. If we put WR +pr2 S, we have
4
S+ a² p
a p
=C, therefore S + a² pap C, therefore ap℃ — a² p =
S
p;
$, and a C — Q²
a² = and by compleating the Square, and
extracting the Root, we have a C-
S
P
+ C².
3537. In the fame Manner may the Angular Pendulum (de-
fcribed Chap. XVIII.) be connected with the Rotula, and its
Time of Vibration thereby prolonged, or its Length very much
fhortned for a given Time of Vibration. For
± WR² + ½ a ab + a² c
2
उ
x b + x c
=n(3471,) or putting b+c=
उ
s, and c + b =v; we have
¹WR² + sa²
2
บท
11
T₂
2
t
(3477.)
Whence WR² + sa² xt = T² vx = T² v Cg; there-
14
! WR² + sa² x't
fore
T² v
v z
C=Cofine of the Angle to which
the Pendulum must be opened, that its Vibration may be as
the given Time T (3478, 3479.) This Conftruction of the
Pendulum and Rotula is fhewn Fig. 8.
3538. Laftly,
Of CLOCK-WORK.
469
3538. Laſtly, the uniform Pendulum, defcribed in the laſt
Chapter, may with Eaſe be adapted to the Rotula (as in Fig. 9.)
and the Theorem (3504) may be accomodated thereto; for
2yx is as the Area or Weight of the Pendulum itſelf, whence,
putting 2yx=w, we have y²x²× w², and putting 1 +
I IV R² + a² — ax + sx²
2
3
w²s, we ſhall have the Theorem
=C, the Length of an ifochronal Pendulum.
*
a
x
T
3539. And thus it appears how eaſy it is for the expert Clock-
Maker to chufe what Lengths and Forms of Pendulums he
pleaſes, to vibrate in a given Time; and by this Means to ren~
der his Clocks portable and concife in any Degree. And fur-
ther, that by fuch compendious Conftructions they are in the
beft Manner fitted for Uſe at SEA.
CHA P. XXI.
The PRINCIPLES of Celeſtial MECHANICS explained,
and applied to the Conftruction of a New HE-
LIOSTATA, or Planetary CLOCK for fixing the
RAYS of LIGHT proceeding from the SUN, MOON,
and PLANETS, by which thofe Objects are ren-
dered apparently at REST for Aſtronomical ОB-
SERVATIONS.
3540. HAVING fhewn the Conftruction of a Variety of
Planetary Pendulums, we now proceed to confider
the Nature, Form, and Difpofition of a PLANETARY CLOCK,
that is of a much more general Nature than the Helioftata of
S'Gravefand, which I have at large explained in my Philofophica
Britannica; for that is wholly confined to the Sun, as its Title
implies; and is of a much coarſer, and more cumberſome Con-
ftruction, than that we have here to propoſe.
3541. It muſt be confeffed the Helioftata has been greatly
improved by the learned C. G. KRATZENSTEIN in what he
calls Mechanicæ Cæleftis Specimen Primum, or Firft Specimen of
Celeſtiai
470
INSTITUTIONS
Celestial MECHANICS, in which he has in fome Meaſure pro-
vided for viewing the Moon, and Planets as well as the Sun;
but we can by no Means think the Mechaniſm of his Clock, or
the Manner of applying it, ſo concife and delicate as the Nature
of the Subject both require and will admit of; this we preſume
will be readily granted by any one who will pleaſe to compare
the Account we here give of the Clock of our own Conftruc-
tion with that you find of his in the Petropolitan COMENTA-
RIES, for the Years 1747 and 1748.
3542. The Rationale he has given of this new Kind of Celef-
tial Mechanifm (as he very properly calls it) is exceeding good;
and as it only wants to be a little foftened and dilated, to fuit it
to an English Genius, we fhall here undertake that Office, and,
we hope, to the Satisfaction of the curious Reader.
1
3543. The Deſign of ſuch a Celeſtial MACHINE being to ren-
der the SUN, MOON, and PLANETS, apparently quiefcent, or
at Reft to the View, it cannot but be eſteemed the moſt uſeful
Invention in Mechanics, and best adapted for aftonomical Ob-
fervations; where the conftant Motion of the celeftial Bodies
and the difficult and irkfome Application of long Teleſcopes
create perpetual Moleftation to the ingenious Student; and ge-
rally, not only difcourage, but very often entirely defeat the
Enterprizes of induſtrious Obfervers.
3544. The first Principle of this new Theory of Celeſtial
MECHANICS is that of Catoptrics, where it is fhewn, that the
Angles contained by the incident and reflected Ray with the Perpendi-
cular Plane are always equal to each other (1282.) Hence it will
follow, that if the incident Ray be confidered as fixed, and the
reflecting Plane be moveable, then the reflected Ray will alfo
be moveable and its angular Motion will be juſt double to that
of the Plane.
3545. For fuppofe the Radiant Object at S, and SD a Ray
of Light, incident perpendicularly upon a reflecting Plane A B;
with any Radius CD defcribe the Circle DIKG to touch the
Plane in the Point of Incidence D; and let DK be the vertical
Diameter, and IG the horizontal one. Then it is evident that
SD will reprefent not only the incident Ray, but alfo the
reflected Ray as well as the Perpendicular to the Plane, for in
this Cafe they all coincide.
3546. Then
t
471
Of CLOCK-WORK.
K
1
(2.) The
(3.) That
3546. Then ſuppoſe the reflecting Plane A B moveable about
the fixed Point D by an inflexible ftrait Wire DL, and that it
was thus moved about, till it came into the Pofition EF making
an Angle ADE with its firft Pofition, of 45°; laftly, ſuppoſe
this Motion was produced by an Arm CK moveable on the
Center C, and having a Fork at the End K to receive the
Wire DL; then it is plain, (1.) That while the Plane
is moved from the Situation AB into EF, the Wire DL
is moved into the Pofition DM interfecting the Circle in
the Point I, or End of the horizontal Diameter.
Arm CK has moved into the .Pofition CI.
becauſe in the Triangle DCI, the Sides DC and CI are
always equal, therefore the Angles at D and I will be ever
equal; and conſequently (4.) The Angle KCI will ever be
equal to 2 CD M, or the angular Motion of the Arm C K is
twice that of the Wire DL or of the Plane A B. (5.) That
the incident Ray SD, the Perpendicular PD, and the reflected
Ray DO, will ever be reſpectively parallel to the three Sides
CD, DI, and IC of the Triangle CDI. And that there-
fore (6.) The angular Motion of the reflected Ray O D will
be double that of the Plane or Glafs A B.
}
-
3547. And (vice versa) the fame Things are equally evident
with regard to the incident Ray being confidered as moveable;
and then, if its Motion be juft twice as much as that of the
Plane A B, the Pofition of the reflected Ray will be conftant,
or always the fame. Thus for Example, if E F be the Pofition
of the reflecting Plane, and O the Radiant be ſuppoſed to move
through a Quadrant from O to S, then in that Time the Wire
DM will move into the Pofition DL and the Plane into the
Pofition A B, which angular Motion MD or ED A being but
half ſo great as the angular Motion of the Radiant ODS (or
ICK) it is evident, the reflected Ray DS will be always in the
fame Pofition, and fo may be confidered as fixed or immoveable.*
3548. This is the fundamental Principle on which the The-
ory of the Machina Cæleftis, as alfo that of the Newtonian SEA-
OCTANT (uſually called Hadley's Quadrant) do entirely depend.
i
'
This,
*See Fig. 1 and 2 of the Plate entituled The PRINCIPLES of
Celestial MECHANICS.
472
INSTITUTIONS
This, with Regard to the faid OCTANT, has been already ſhewn
in a ſmall Treatise on that Subject; † and with respect to this
CELESTIAL CLOCK, we fhall now a little expatiate in its Il-
luftration.
3549. If the SUN or PLANET were to move or circulate
about us in the Horizon, it is evident, fince the Wire or Ray
DM is, in this Cafe, always in the Plane of the Horizon; and
perpendicular to the reflecting Plane, therefore the reflecting
Plane will always be Vertical to the Plane of the Horizon;
and its Perpendicular DP will defcribe a Quadrant while
the Sun or Planet defcribes a Semi-circle in the Horizon, at the
fame Time that it appears quiefcent or fixed, when viewed in
the Reflecter EF, in the Direction SD (3547.)
3550. But if we fuppofe the Sun or Planet to move in a Circle
parallel to the Horizon, and at a given Altitude above it, while
the Eye continues in the horizontal Plane; then the Motion of
the Radiant and Reflecting Plane will be in the fame Ratio as be-
fore, but the Poſition of the ſaid reflecting Plane will now be
oblique to the Plane of the Horizon, in order that the reflected
Ray may be parallel to it, and continue fixed and immoveable
as before.
3551. Thus if HO be the Horizon, (Fig. 3.) and the Angle
SRO be that of the Sun's Altitude above it, then if we take any
Diſtance R Q for Radius, and on the Point Q erect the Perpendi-
cular QD, interfecting the Ray SR in D, it will be the Tan-
gent of the Sun's Altitude to the Radius QR. Through the
Point D draw GC parallel to OH; and laſtly, draw PDH
to biſect the Angle of Altitude SDG. Then it is evident, if
a Plane Speculum EF be placed with its Center on the Point
D, and in a Pofition perpendicular to the Line PH, it will be
that which is required for reflecting every incident Ray SD
from every Part of the diurnal Circle of the Sun into one con-
ſtant Direction DG parallel to the Horizon OH.
3552. For continue GD to K, and take DC = DR =
Secant of the Altitude, and then C will be the Center of a Cir-
cle parallel to that of the Sun's Motion; and drawing CH
(CD) it will reprefent the Arm, which by its Fork at the
End will always keep the Wire DH conftantly bifecting the
+ The THEORY of HADLEY'S QUADRANT demonſtrated.
Angle
Of CLOCK-WORK.
473
Angle SDG, and thereby render the reflected Ray DG per-
manently fixed. The above Phænomena belong to the Pa-
rallel Sphere, or to a Perfon under either Pole.
3553. But to the Inhabitants of an oblique Sphere the Equator
itſelf and all its Parallels, make an Angle with the Horizon;
and thereby caufe that every Day, and every Part of every Day,
the Altitude of the Sun above the Horizon will be variable; and
conſequently there muft, in this Cafe, be another Figure for the
Conftruction or Difpofition of the Speculum for fixing the
reflected Ray in the Plane of the Meridian, and parallel to the
Horizon.
3554. Thus let AV (Fig. 4.) be an horizontal Line, and
the Angle E A V be the Elevation of the Plane of the EQUATOR
EA above the fame. Alfo let the Angle E AG be the North
Declination of the SUN or PLANET, and the Angle EAC the
Declination, South. Then, of Courfe, EXA is the Latitude of
the Place. Now A E being Radius, E B is the Tangent and A B
the Secant of the North Declination or Elevation of the Planet
above the Equator EA; therefore by drawing BR parallel to
A V, and therein taking BIA B, it will give the Point I
for the Place of the Center of the Speculum (3551,) by which
the Sun-beam SI will be reflected into IR parallel to the Ho-
rizon by a Wire I A perpendicular thereto, and which continued
to (i) bifects the Angle of the Sun's Altitude SIR = BAV =
2 BAI.
3555. In like Manner, if through the Point C, a Line CT
be drawn parallel to the Horizon AV, and in that we take CK
= AC, then K will be the Place of the Speculum to reflect
the Sun-beam SK parallel to CA, (in South Declination) into
the Direction KT, by the Wire KA, bifecting the Angle
CAV SKT.
3556. From whence it is evident, the Center of the Circle
parallel to the Sun or Planet's diurnal Motion, and in which the
Speculum is placed, will always be in the Line BX parallel to
the Axis of the World; and therefore if the faid Line be continu-
ed each Way to G and Z indefinitely, it may repreſent the Axis
ZH of a Clock LM NO, continued out; which Clock, being
placed in a Pofition parallel to the Equator A E, if it be furniſh-
ed with an Arm HP moving on the Center H, and carrying on
VOL. II.
Ppp
its
1
"
J
474
INSTITUTIONS
its Extremity P an upright Wire or Stem P A with a Fork at A
to receive the Wire IA of the Speculum, then by the Mécha-
nifm of the Clock the Index or Arm HP will be carried with a
Motion fimilar to that of the Sun or Planet whofe Beam, SI,
will thereby be always rendered permanent in the fame horizon-
tal Direction, from the rifing to the fetting of the fame; all
which is evident from the preceding Articles.
3557. It is manifeft alfo from the Conftruction of the Figure
that the Pofition of the Speculum being conftantly variable, the
Diſtance thereof from the Point A in the horizontal Line A V,
and its Altitude above the faid Line, will be variable likewife,
with the different and variable Altitude of the Luminary above
the Horizon of the Place; and therefore it is neceffary to cal-
culate the horizontal Diſtances and Altitudes of the Speculum.
from the greateſt to the leaft Quantities thereof for any parti-
cular Latitude, that it may be readily adapted to any given De-
clination of the Sun or Planet for the Day it is uſed.
3558. With Refpect to the Firft of thefe, viz. the Diſtance
of the Foot of the Speculum I from the Point A in the horizon-
tal Line AV, it is eafily computed, thus; from the Center or
Point B let fall the Perpendicular BD; then in the Triangle
ABD, the Angle B A D is known, being the Sum of the Co-
Latitude (E A D) and the Declination (EAB); and the Side
AB (BI) being the Secant of the given Declination to the.
Radius A E, from whence AD (the Bafe) is found; which
added to BI, gives (BI+DA) the horizontal Diſtance re-
quired, fuppofe for the greatest Declination propoſed E A B.
Thus in the Triangle CAF we find AF; and then AF +
CK, is the horizontal Diſtance for any other Declination EA C
on the other Part, or below the Equator.
시
​3559. The Altitude of the Speculum above the horizontal
Line AV is the Perpendicular BD, for the Declination E A B;
and CF for the Declination EAC; which are Parts of the
given Triangles; and therefore are readily found for all Declina-
And thus the Speculum is adjusted for ready and con-
ftant Ufe at all Times.
tions.
3560. If it be required to conftruct this Celestial CLOCK uni-
verfally, or fo as to adapt it for all Latitudes, it is only making
it to move on a ftrong and well wrought Hing on the Fore-part
*
1
at
Of CLOCK-WORK.
475
at N, and then it may be elevated to any Degree of Latitude on
a graduated Arch on the back Part at O, to which it may be faf-
tened by a Screw.
3561. But then it muſt be confidered, that as the Perpendicu-
lar Height A P, of the Fork A, above the Arm HP, is the
Tangent of the Co- Latitude, or Elevation of the Equator, viz.
ALP LPW to the conftant Radius LP; therefore that
Height (AP) will be conftantly variable with the Latitude in
an inverſe Ratio, or is greater as the Latitude is lefs; and vice
verfa. And this muſt be provided for by Calculation alſo, and
conſtantly adjuſted to the Latitude by a graduated Șcale. All
which is eaſily done by the Triangle ALP, wherein LP is of
a given Length, and PA is found for any given Latitude
LAPAPQ So that for every particular Latitude, the
three Quantities AD, BD, and AP muft be calculated, and
the Clock and Speculum thereby adjuſted.
3562. According to the Luminary you propofe to view or
obferve, the Pendulum of the Clock muft be peculiarly adapt-
ed; the Methods of doing which, for the various Sorts of Pla-
netary Pendulums, we have already defcribed in their feveral Con-
ftructions, and muſt leave the Artiſt to chufe out of them all that
which he thinks beft; for by this Time we prefume he muſt be
very fenfible the SUN, the MOON, a STAR, and each particular
PLANET must have a Pendulum properly qualified to render it
quiefcent to the View.
3563. In this Celestial CLOCK it will morcover be very ex-
pedient to have the Hour-circle upon the Face of the Clock
moveable, and divided into 24 Hours, or twice XII; for by this
Means the Hour and Minute of the Luminary's Culminating,
or being on the Meridian, may be brought into the Vertical
Line or Meridian of the Clock; and then the Hands being placed
to the preſent Hour and Minute in that moveable Circle, will
conftantly fhew the Time and Motion of that Luminary; for
the Hour-hand will always keep Pace with it in the Heavens,
and fhew its Right Afcenfion, or Difference from Solar Time,
which will be ſtill more evident if the moveable Hour-Circle be
added to that which is fixed, and both divided into twice XII
Hours. (See 3325, &c.)
Ppp 2
3564. Be-
476
INSTITUTIONS
3564. Before any Obfervation is begun, the Clock muſt be
very nicely placed in the Meridian Line, or fo as that the Meri-
dian Line of the Clock may coincide with the Meridian of the
Place; to which End this Clock muſt be furniſhed with a very
good Magnetical Compafs and Needle, and the VARIATION thereof
muſt be accurately found by Experiment for the Latitude where
it is uſed. And by that the Clock may be readily adjuſted to
the Meridian. Alfo two Spirit Levels must be placed in a pro-
per Part of the Machine at right Angles to each other, by which
it may be always reduced to a truly horizontal Pofition.
3565. By Means of this Machine duly adjufted to the SUN,
if the Room be darkened and a Hole made in the Window-
Shutter to let in a Beam of the Sun's Rays upon the Speculum,
it will continue to be reflected in one conſtant Direction all the
Time it can fall on the Speculum, which may be fo contrived
as to be long enough for moft Optical Experiments, either with
PRISMS, or the SOLAR MICROSCOPE, &c. Befides it will be
no difficult Matter to follow the incident Beam with the Speculum,
by having Caſtors at the Bottom, or Foot of the Frame, fup-
porting the Clock, and thereby moving and adjuſting it, as
required, with the greateſt Eaſe.
3566. But the nobleft Purpoſe to be anſwered by this Celestial
MACHINE is fixing the heavenly Bodies, or rendering them
Stationary or quiefcept in the Field of the Teleſcope for aftrono-
mical Obfervation; an Advantage hitherto wanted (and unat-
tainable by any other Means,) for advancing the Science of
Aftronomy to its true Summit of Perfection. The Teleſcope
for this Purpoſe ſhould be of the reflecting Sort, and furniſhed
with a Micrometer of different Forms, viz. the Lattice, the
parallel Wires, the divided Object-glass, the fine Screw, &c. for
meaſuring and delineating the Surfaces of the SUN, Moon, and
PLANETS.
3567. I need not add, that the Foot and Stem of the Telef-
cope fhould be to contrived, that it may be always placed truly
horizontal, exactly in the Meridian of the Clock, and juft of
the fame Height of the Center of the Speculum, by this Means
the Obferver will fit at Eafe, and, without the leaft Diſturbance,
furvey the various and wonderful Phænomena of each celeſtial
Body.
Of CLOCK-WORK.
477
Body. And thus we have finiſhed the Theory, or general
Principles, of the fublimeſt Piece of Machinery, that the In-
vention of Man has yet been able to produce.*
3568. It may alſo be obferved, that if the Speculum at I or
K be placed in the Direction perpendicular to the former, or
parallel to that of the Wire AI or AK then will the Ray SI,
SK be reflected the contrary Way, or in the Direction I B, or
K C, in the fame Right Line R B, or TC as before; and in
many Cafes fuch a Reflection or Pofition of the Ray will be more
convenient than the other on the Part towards the Sun.
3569. Laftly; The Speculum C to be uſed in all Obferva-
tions of the heavenly Bodies muſt be a plain One, and as per-
fectly true as Art can make it; and indeed unleſs it be extreme-
ly true and well polifhed, it cannot answer any fuch Purpoſe.
No Concave or Convex Speculum can in fuch a Caſe be uſed, for
Reaſons mentioned in the Theory of Catoptrics; though in
many Experiments which require the Convergency of Rays for
illumination only, a Concave Reflector may be applied to an-
ſwer many uſeful Defigns, which the Ingenuity of Artiſts will
naturally fuggeft.
CHA P. XXII.
The PRINCIPLES of Celeſtial MECHANICS applied to
the Conftruction of a MICROCOSM, confifting of a
PLANETARIUM, TELLURIAN, and LUNARIUM
for exhibiting the Celeftial MOTIONS.
3570. The properly, eftcemed the Second Part of CELES
HERE is another Branch of Clock- Work, which may
be
ȚIAL MECHANICS; for by this we exhibit the MOTIONS of
the Heavenly Bodies, as well as the TIME thereof, by the
Clock
*See a further Account of the practical Uſe of this new Helioftata,
or Planetary Clock in another Part of this Work, viz. The Young
GENTLEMAN and LADY'S PHILOSOPHY, Page 302, &c.
478
INSTITUTIONS
Clock, and proper Mechaniſm for that Purpoſe, to be added
thereto.
3571. The Celestial Bodies whofe Motions are fhewn by
Clock-Work are the PLANETS, the EARTH and the MOON;
the Motions of the Secondary Planets, or Satellites of Jupiter and
Saturn, are fometimes fhewn in fuch a Syftem of Mechanifin
as is ufually called an ORRERY; but the common Methods
of conftructing thefe large Inftruments render them very ex-
penfive, and confequently they are but rarely made. But we
fhall endeavour to remove this Difficulty by propofing other
Conſtructions and Forms of Orreries that will be lefs coftly, and
perhaps more elegant and natural, as well as much more con-
cife.
3572. The Calculations of the Celestial Motions depend on
confidering the Times of the Revolutions of the heavenly Bodies
about their proper Centers, and taking the Ratios of thofe Pe-
riods in fuch Numbers as will beft anfwer for toothing the
Wheels and Pinions of the Work by which thofe artificial Mo-
tions or Revolutions are to be produced; and then how or
in what Manner they are to be connected with the Movement of
the Clock for their Continuation.
3573. We shall firſt begin with the Motions of the Primary
PLANETS about the SUN; and by the beft aftronomical Obfer-
vations, their mean Revolutions or periodical Times are as fol-
low, viz.
SATURN revolves in 10759,3 Days.
JUPITER
MARS
EARTH
VENUS
MERCURY
4332,5
686,9
365,25
224,7
88.
3574. Now the Ratio of the Earth's Period to thoſe of the
Reft of the Planets is hereby given, and therefore may be ex-
preffed in leffer Numbers which may be at the fame Time In-
tegers, and conveniently adapted for the Teeth of the Wheel-
Work for the intended Movement. And upon Trial thofe
"N
which
Of CLOCK-WÓRK.
479
which are found to anſwer beft are as in the following Table,
Viz.
As 365,25:
88 :: 83 :
20 for MERCURY.
365,25
224,7 :: 52:
32
VENUS.
365,25
686,9 :: 40:
75 MARS.
365,25
4332,5:: 7
83
83 - JUPITER.
365,25: 10759,3 :: 5: 148
SATURN.
And according to thefe Numbers, the Wheel appropriated to
the EARTH muft have 50 Teeth.
3575. Then if on one Arbor you fix fix Wheels with the fol-
lowing Number of Teeth, 83, 52, 50, 40, 7, 5, to drive
another fet of Wheels each moveable upon another Arbor,
and with the Number of Teeth in the other Column refpec-
tively, 20, 32, 50, 75, 83, 148; and laftly, if on the feveral
moveable Wheels proper Sockets are fixed (moving within, and
independent of each other,) then will round ivory Balls, at the
End of flender Wires fixed to thofe Sockets, be carried about
the Arbor in the fame Periods of Time, refpectively to each
other, with thofe of the real Planets in the Heavens.
3576. If this Syftem of Movements be properly difpofed in a
Box; and on the Top or Cover be drawn or engraved the Ca-
lendar and Ecliptic, and a brafs or lacquered Ball be placed on
the Top of the Arbor of moveable Wheels, then, by a Handle
or Winch, the Whole may be put into Motion; and it will
then become a Planetarium or manual ORRERY of a moft uſeful
Form, and exhibit all the Phænomena of the SOLAR SYSTEM,
or that according to Copernicus.
3577. If instead of the lacquered Ball, you put a three-inch
terreftrial GLOBE on the faid Arbor, and a fmall brafs Ball upon
the Arm carrying the ivory Ball, reprefenting the Earth among
the Planets, then will the Machine be changed into the Ptolomaic
Syſtem of the World; by which all the Abfurdities of that ſenſe-
lels Hypotheſis may be eaſily exhibited and refuted.
3578. Such a Syftem of Planetary Movements may cafily be
applied to a Piece of Clock Work by which it may be made to go
any required Time, as an Hour or two; in which the Earth
may revolve two or three Tunes about the Sun (or the Sun
about
480
INSTITUTIONS
!
about the Earth,) which is a fufficient Interval for exhibiting
all the confiderable Appearances in either Syftem, relative to
the Primary Planets. But if it be required that the Motions of
this artificial Syftem fhould be the fame with thoſe in the real
Syftem of the World, then we muſt proceed as follows.
3579. Let it be required to conftruct a Movement for a
Mọ-
tion to be performed in the Space of one Year, or 365 Days;
for we muſt not pretend to the Accuracy of an Hour or two
in a Year. In 365 Days there are 730 Half-days of 12 Hours.
This Number is produced by the Factors 181, 8, and 5; for
5 × 8 × 181 = 730. Therefore any Number of Wheels
and Pinions which will produce thefe Quotients, will produce
the Motion defired.
2
For Example,
4) 73 (181
4) 73 (18.
I
8) 146 (18/1/
4) 40 (10
4) 32 (8
8)
64 (8
5) 20 (4
4) 20 (5
12)
60 (5
3580. Thus fuppoſe a Wheel (A) with a Pinion of 5 Teeth
drives a Wheel (B) of 20; and this Wheel carrying a Pinion
of 4 Teeth drives a Wheel (C) with 40; and this with a Pi-
nion of 4 drives the laft and largeft Wheel (D) with 73 Teeth.
Then it is evident (3180.) that in one Turn of B there will
be 4 Turns of A; and 40 Turns of A in one Turn of C;
and 730 Turns of A in one of the laft Wheel D. And there-
fore this Wheel D moves round once in a Year, and confe-
quently if the firft Wheel A be connected with another equal
Wheel fixed on the Arbor of the Great Wheel (or Fufee) of
an 8 Day Clock, then by Circles on its extreme Parts the Day
of the Month, or Sun's Place in the Ecliptic will be very exact-
ly fhewn, through the whole Year.
3581. Now this little Syftem of Wheels may be very eafily
connected with the Movement of a Clock on one Part, and
with that of the Planetarium on the other; fo that the Clock
(with a proper additional Weight,) fhall conftantly keep the Pla-
nets moving about the artificial Sun juft with the fame angular
Velocities as the real Planets themſelves move with in the
Heavens; and if in the Beginning of the Year the Places of
the
15
10
کی
ot
29
25
2
XI
m
II
ΤΑ
V
P
H
<d
DINI
The MICROCOS M.
Of CLOCK-WORK.
481
the Planets in the Orrery be duely adjuſted or rectified by an
Ephemeris, they will nicely correfpond to thoſe they reſpectively
repreſent in the Heavens, and continue fo to do ever after.
3582. It will be readily allowed, that this Method of con-
ſtructing a Planetarium is the moft natural and fimple that can.
be, as not a Wheel, or even a Tooth, need here be uſed more
than what is abfolutely neceffary. By this Planetary CLOCK,
it will every Day be feen, what Part of the Ecliptic the Sun, the
Earth, or any Planet is in the various mutual Aspects of
Planets — their Conjunctions, Quadratures, and Oppofitions
-the apparent Direction of their Motion, direct, stationary, or
retrograde whether they are above or below the Horizon
whether they riſe or ſet before or after the Sun:
you have by this Means the whole Syftem of Celeſtial Phanomena
conftantly in View.
In fhort
3583. By the fame fmall Compages of Wheels, deſcribed
(3575,) placed horizontally, it is evident the Axis of the Ter-
reftrial Globe may be made to move about the Axis of the Eclip-
tic once in a Year; and thereby all the Phænomena of the Earth's
ANNUAL MOTION, with Regard to the Seafons of the Year in
different Latitudes and Climates, may be ready fhewn, and ob-
ſerved. For fince the Pofition of the Wheel D is fuppofed ho-
rizontal, if an Axis be fixed in its Center perpendicularly, it will
repreſent the Axis of the Ecliptic. But if at a proper Height
above the Wheel D the faid Axis be bent out of the Perpendicular
ſo as to make an Angle therewith of 66° 30′ in the Form of an
Arm, and on this Arm another Axle be placed at right Angles,
this will make, with the forementioned Axle, an Angle of
23° 30′, and confequently will reprefent the Axis of the Earth;
and will be carried on round the Perpendicular Axis of the
Ecliptic once in a Year, if properly connected with the Clock.
3584. Then for the Diurnal MOTION of the EARTH at the
ame Time, it may be eaſily effected thus; let the Axis of the
Earth be moveable in a Socket fixed Perpendicular to the fore-
nentioned ſtrong Arm; on the under Part of which (below the
Arm,) let a Wheel (G) be fixed, ſuppoſe of 72 Teeth; then
et the Teeth of this Wheel play in the Teeth of another Wheel
F) which must be but half the other Number, viz. 36; and
his Wheel muſt be placed horizontally about the Stem or per-
VOL. II
pendicular
Qqq
482
INSTITUTIONS
}
pendicular Axis juft below the Arm, and fupported on a Bridge
to render it independent of the Work for the Annual Motion
below. Laftly, this Wheel F muft have another equal Wheel
(E) of 36 Teeth, to connect it with the Wheel A on the Ar-
bor of the great Wheel of the Clock, and this Wheel A muft
also have 36 Teeth.
3585. Therefore, if upon the Terreftrial Axle, the Globe
of the Earth be placed, fo as to be moveable thereon, or fixed
with a Nutt and Screw at the End, as Occafion requires, then
it is plain, it will be carried about once in 24 Hours, becauſe
the Wheels A, E, F, each move round in 12 Hours, and the
Wheel F of 36 Teeth driving the Wheel G of 72 muft move
it round, (and the Earth on the Axis to which it is fixed,)
once in 24 Hours.
3586. With Regard to the Wheel E, as it drives the Wheels
above the Bridge for the diurnal Motion, fo by an Arbor going
through, and carrying a Pinion of 5 Leaves, it dives the
Wheels B, C, and D, below for the annual Motion at the
fame Time. Alfo by the Interpofition of this Wheel E, the
Globe is made to revolve the right Way, viz. from Weßt to Eaſt,
as the Earth itfelf really does.
3587. Then by a Circle of Illumination, and an artificial Sun at
a proper Diſtance in the Axis thereof, with fome other Appara-
tus, all the Phenomena of the annual and diurnal Motions of the
Earth are moft naturally exhibited for every Day throughout
the Year. Thus you fee the North and South Pole alternately
in the illumined and dark Hemiſphere ; ——————— the Sun rifing and
fetting each Day upon different Parts of the Horizon; the
variable Lengths of the diurnal and nocturnal Arches defcribed by
the City of LONDON or any other Place: If a Crepufcular
Circle be added, the Time of Beginning and End of Twilight
will be fhewn for each Day; with many other important Parti-
culars in Geography and Aftronomy.
3588. 'Tis moreover eafy to contrive, that this Mechaniſm
for the Globe may be difengaged from the Clock, or be put on
and taken off at Pleaſure; in which Cafe it may be turned with
Winch fo as to reprefent all the Phænomena of a Year or a
A
Day, in a very fhort Time; and then fuch an Apparatus may
be added, as will exhibit the Solar and Lunar Eclipfes, &c. the
Manner
८
Of CLOCK-WORK.
483
Manner of doing this, together with a Print of the Globe thus
conſtructed, may be feen in the Young GENTLEMAN's and
LADY'S PHILOSOPHY.
SCHOLIUM.
3589. We fhall hereafter give a Conſtruction of a Lunarium,
to fhew the Monthly MOTION and PHASES of the Moon, as
alfo the Motion of her APOGEE and NODES; if then this lunár
Movement be connected with that of the Clock (as it eaſily
may) and the whole be difpofed in a proper Form, it will make
what may be more truly called a MICROCOSM, than any thing
which has hitherto borne that Name; and to affift the young
Artiſt's Imagination I have preſented to his View, in a Copper-
Plate, the Form of a Specimen of fuch a Piece of Celeſtial
ARCHITECTURE in which are the following diftinct Parts, viz.
(1.) A large CLOCK for a Primum Mobile. (2.) Two PILLARS,
(or Pilafters) on one is placed a Barometer, and on the other a
Thermometer. Theſe Pillars fupport an Architrave on which is
placed (3.) An ORRERY or Planetarium, fuch as before de-
fcribed. (4.) A TELLURIAN, or Terrestrial GLOBE with the
annual and diurnal Motions. (5.) A LUNARIUM, fhewing
the Motion, Phaſes, and Age of the MOON. Such a ſuperb
Structure, we prefume, would well become the Study or Mu-
fæum of the Opulent and the Great; but fo little Genius, Tafte,
or Encouragement is now found for fuch Works of Art, that it
is not worth while to infift further on their Ufe or Excellency.
By ARTS and ARTISTS we, at this Time, underſtand only
Engraving, Painting, and Sculpture, and thoſe who practice
them; and LEARNING and LEARNED MEN, mean no more
than Novels, Romances, Plays, and their Authors, in the pré-
fent Trifiram-Shandy Age.
Q99 2
CHAP.
484
INSTITUTIONS
CHA P. XXIII.
The MECHANISM and CONSTRUCTION of a TEL-
LURIAN, with Three Terreſtrial GLOBES; illuf-
tracting the THEORY of the EARTH'S MOTION;
the Rationale of the SEASONS; and the Inequality
of DAYS and NIGHTS.
IN
3590. N the Conſtruction of the ORRERY defcribed in the
laft Chapter, the Earth's annual Motion was fhewn,
but not the diurnal Motion at the fame Time; we fhall now ex-
plain a new Piece of Celestial Mechanifm, which being applied
to the foregoing ORRERY (now ſuppoſed in a Horizontal Pofi-
tion) will not only produce both theſe Motions of the Earth
together, but at the fame Time, by Means of two other Globes,
the Rationale of all the various Phænomena and Affections of the
Earth will at once moſt evidently appear.
;
3591. I prefume it is needlefs to mention to the intelligent
Reader, that if any Body defcribes the Circumference of a Circle,
with the fame Part always directed to the Center, it will neceffarily be
moved or turned once about its own Axis or Center in the fame Time
for by this Means the Body has that very Part directed to every
Point in the Circumference of the Circle, which would be impof-
fible without one entire Revolution about its Axis; this is therefore
an Axiom or felf-evident Truth.
3592. Alſo, it is farther evident, that the Direction of the
Motion of fuch a Body, both about the Center of the Circum-
ference it defcribes, and about its own Center or Axis, is the
fame, or towards the fame Part. But to illuftrate theſe Poſt-
tions, let ABD be the Circle defcribed by any Plane FEG
fixed upon an inflexible Rod or Arm E C, moveable about the
Center C.* Let this Arm (or Radius Vector) revolve from the
Situation EC to any other IC, then will the angular Mo-
*See Fig. 1. in the Plate of the TELLURIAN with three GLOBES.
tion
Of CLOCK-WORK.
485
tion of the Body FG be ACH about the Center C in that
Time.
3593. Upon the Plane in the Situation IH, draw ae parallel
to A E, and it will make the Angle Ice with the Radius IC;
and thereby it appears that the Line AE (or ae) has deviated
from its firft to the prefent Pofition IH by the Quantity of the
Angle ICE; and therefore by fo much has the Plane moved
about its own Center (c) in the Time of defcribing the Part of
the Circumference A H. But fince CE and ae are two paral-
lel Lines cut by a right Line IC, the Angles ACH and
Ice will be equal; that is, the angular Motion of the Plane
about the Centers C, and c, will be conftantly equal; and the
Direction of the Motion from e to I is the fame as that from
A to H.
7
3594. Hence it will follow, that if any luminous Body be
placed in the Center C of a Circle ABD, and a Globe F G be
moved in that Circle fo as to have the fame Part always turned to
the Center, then will the fame Hemifphere or Part of the Globe
be conftantly enlightened, and the other Part will be always
dark. - Thus fuppofe NQS (Fig. 2.) were a Planet
(our Earth, for Inſtance) moving in the Manner abovemention-
ed about the Sun in the Center at an immenſe Diſtance, then
would the fame Hemiſphere SÆN be perpetually illuminated;
and the other SQN conftantly in the Dark. Whence it is
evident, a Planet in fuch Circumftances could never be a pro-
per Seat, or Place for Habitation, for fuch a Species of rational
Creatures as the Human Race.
3595. Let fuch a Globe, then, be fuppofed to move about
its Axis NS placed perpendicular to the Plane of its Motion
about the Sun. Such a Motion we call its Diurnal Motion, as
it produces DAY and NIGHT alternately to all Parts of the
Surface, in the Space of one Revolution which in our Globe is
24 Hours. Then it is plain, that every Part on the Surface of
fuch a Globe would be one half of the Revolution in the illu-
mined Hemiſphere, and the other half, in the dark One; fo
that in fuch a Condition of the Globe, there would be a perpetual
Equality of Day and Night.
3596. Again, fince the Axis NS of the Diurnal Motion is
perpendicular to the Plane of the Annual Motion, which is fup-
pofed
486
INSTITUTIONS
poſed to pass through the Centers both of the Earth and Sun
then it is evident, the Rays of the Sun will be perpendicular to
the middle Part of the Globe at the Equator E Q, and will fall
obliquely on every other Part; and the more fo as the Part is
nearer the Poles N, S, on either Side the Equator. Whence
it will follow, that there will be one and the fame Degree of Light
and Heat to the fame Part of the Surface of the Globe conftantly;
and therefore no Variation of Seafons throughout the Year. Hence
fuch a Pofition or Conftitution of the Globe of the Earth would
by no means fuit with the Condition of the Human Species.
3597. Let us next fee what will be the Cafe of a diurnal
Motion of the Globe about an Axis in a Pofition inclined to the
Plane of its annual Motion, fuch as SN (Fig. 3.) and further
let us fuppofe that the Axis NS is conftantly in the fame Planet
with the Line ECL which connects the Centers of the Sun
and Globe. Then it will follow, that the Parts of the Globe
about the upper or North Pole N, will be conftantly inclined
to the Sun, while the oppofite Parts about the South or lower
Pole S are as conftantly turned from it.
3598. The Confequence of this inclined Pofition of the Globe
will be, an Inequality of Days and Nights; and different, or
contrary Seaſons to Parts at an equal Diſtance from the Equator
on either Side. This is evident by Infpection; for EC is the
Semidiurnal Arch in the Equator; Ef is the fame in the Parallel
Ee, which is longer (in Time) than the other by the Time of
defcribing F f.
But the femidiurnal Arch (1g) in the fame
Parallel /L of South Latitude is fhorter than C by the fame
Quantity g G F f. And the Parts about the North Pole N
to the Diſtance of NB (which meaſures the Inclination of the
Axis, or Angle NC B) are perpetually in the illumined Hemif-
phere and have conſtant Day; and on the Contrary, thoſe
Parts at the fame Diſtance about the South Pole S have an eter→
nal Night.
3599. Then with Refpect to the Seafons, they are by this
Means very different in oppofite Parallels of Latitude; for now
the Parts at E receive the perpendicular Rays, and have the
Summer Seafon; but fince the Parts within the Arctic Circle Bb
have an inoccidual Sun, or enjoy perpetual Day, they will
have the botteſt Seaſon, and the North Pole N will be the hotteſt
Part
E
487
Of CLOCK-WORK.
Part of fuch a Globe. On the other Hand, the Parts within the
Antarctic Circle A a will never feel the enlivening Beams of the
Sun, and muſt be doomed to perpetual Winter and the intenſeſt
Cold. Such a State of the Globe would therefore be very un-
eligible to every Sort of Inhabitants of the prefent Terrestrial
Globe.
3600. Now it will be readily allowed, that an Alternation and
Variety in the Seafons of the Year and Lengths of Days and Nights,
will be the most defirable and delightful State to rational Inha-
bitants of the Globe, and fupply them with every Bleffing
equally and interchangeably through every revolving Year. And
this wonderful and univerſal Effect is produced in our Terra-
queous Globe by one fimple Contrivance (a moft ftriking In-
ftance of divine Mechanifm,) viz. by giving the inclined Axis.
SN of the Earth's Motion an equable Movement about the
Axis of the Ecliptic A B once in the Time of its annual Revo-
lution about the Sun, and in a Direction contrary to the Earth's
Motion; and this will produce the abovementioned neceffary
Effects.
3601. For we have fhewn (3593) that while the Globe is
carried about the central Sun, with the Axis SN, or the fame
Side A EB always turned towards it, the faid Globe and its
Axis, will in that Time be alfo turned once about their proper
Center C and in the fame Direction with the Globe's annual
Motion; therefore, if the fame Motion be impreffed on the
Globe in a contrary Direction, it will have all the Parts of its
Surface turned to the Sun, of Courſe, in the Space of one Re-
volution or Year, and its Axis SN will always keep a Direc-
tion parallel to that which it had at firſt, ſo that all Effects and
Phænomena will be the very fame as if the Earth were really at
Reft on its inclined Axis, and the Sun itſelf were to move
about it (as it appears to do) in the Ecliptic Circle ECL.
3602. For then, when the Sun was in the Equinoctial Point
or Beginning of the Ecliptic at C, it will, at the fame Time,
be in the Equator Æ Q, and confequently illumine the Globe
from Pole to Pole and produce an Equality of Days and Nights
in every Part thereof; and this is called the SPRING-SEASON of
the Year.
3603, Alter.
488
INSTITUTIONS
3603. After the firſt Quarter of the Revolution, or the Sun
has paffed from C to E, and reached the Tropic of Cancer Ee,
then the Globe will be illuminated from A to B, and the North
Pole N, with all northern Latitudes from E to B, will be now
turned towards the Sun and receive the greatest Quantity of Light
and Heat they can ever have, and confequently this will be
the Midft of their Summer Seafon. The Day Ef in every
northern Parallel Ee, is longer than the Night fe; and all
within the Arctic Circle bB have no Night at all on that Mid-
fummer Day.
3604. Juft the Contrary happens in fouthern Latitudes; for
there the Sun-beams reach no farther than to the Antarctic Cir-
cle Aa; and all within it are in total Darkneſs for one whole
diurnal Revolution of the Globe. And the Length of the Day
(1g) in any Parallel /L, is fhorter than the Night g L by the
Quantity gG; confequently in theſe Latitudes, the Light and
Heat of the Sun is now the leaft, or the Cold and Darkneſs the
greateſt it can be, and therefore make their WINTER-SEASON.
3605. When the Sun apparently defcends from the northern
Climates to the Equator again on the other Side the Globe to
the firſt Point of Libra oppofite to C, it will then again make
equal Day and Night throughout the Globe, by enlightening it
from Pole to Pole, as before (3602.) The Light and Heat is
now at a Mean, and make the other temperate SEASON called
AUTUMN.
3606. As the Sun paffes from the Autumnal Equinox thro'
the next three Signs of the Ecliptic, the fouthern Parts of the
Earth will be turned gradually to the Sun; till, at Length, the
Sun being arrived at L, the first Point of Capricorn, it will
then be neareſt of all to the South Pole S and illuminate all
that Hemiſphere A LB, which before was dark when the Sun
appeared at E; and confequently every Thing with reſpect to
Day and Night, and the Seaſons, is juft the reverſe of what
it was then. All Southern Latitudes now enjoy the Midſummer
Seaſon, and in the northern Latitudes it is Winter. What was
before the Length of their Night, viz. g L, is now the Length
of their Day, and vice verfa.
3607. The Sun returning again to C, renews the Spring and
begins the future Year. -Thus by giving the Earth an
annual
Fig. 3.
B
A TELLURIAN with three GLOBES. MECHANISM for a LUNARIUM .
Fig. 3.
N
B
N Fig. 2
B
Fig. 4.
F
& E
T
E
E
A
C
G
A
a
A
R
R
A
a
W
OLS
Fig. 6.
59
10
62
N
D
F
40
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I
G
E
G
F
A
a/H
t
Fig. 1.
M
Fig. 4.
N
D
A
C
B
A
S
dadf
The MOON'S
Node
B
20
bo 70
C
Fig.
Ꮐ
10
H
301
F
The MOON'S
Synodical
Revolution
2
ماه
I
N
K
48
The Moon's
า
D
B
20
E
A
59
Fig. 1.
D
365
Fig. 5.
B
20
A
69
10
E
365 C
Of CLOCK-WORK.
48.9
1
*
}
annual retrograde Motion about the Axis A B of the Ecliptic,
all Parts of its Surface become equally expofed to the Sun on
each Side the Equator, and equally participate the Advantages
which redound from fo wife and beneficent a Contrivance, in
the Courſe of one whole Year..
+
3608. Now this whole Theory (as above explained) of the
Earth's annual and diurnal Motions may be very eafily reprefented
and illuſtrated by a Piece of Mechanifm, which I call a TEL-
LURIAN; confifting of three fmall Terreftrial Globes, placed
in the Manner fhewn in Figs. 2, 3, and 4. The firſt of theſe
(Fig. 2.) is moveable with the Hand about an Axis N S, which
is perpendicular to the Plane of its Motion about the Sun, and
is fixed to the Radius Vector, and therefore will exhibit the
Phænomena of equal Day and Night, and Identity of SEASONS
through the Year, and nothing more, as before obſerved (3596.)
3609. The fecond Globe (in Fig. 3.) has the Axis SN of its
diurnal Motion inclined to the Axis A B of the Ecliptic in an
Angle BCN = 23° 29′. But this Axis SN being fixed to
a Part POR, which is itſelf fixed to the Radius Vector, will
indeed produce a Variety of Seafons, and Difference in the
Lengths of Days and Nights; but then as the Axis SN is
conftantly turned to the Sun, there can be no Viciffitudes of
Scafons nor any Variation of the Lengths of Days and Nights,
which remain conftantly the fame, in the fame Place through-
out the Year, agreeable to (3599.)
3610. But the third Globe (in Fig. 4.) is not fixed to the Radius
Vector, but is connected with the Mechanifm contained therein,
in fuch Sort, as to qualify it for exhibiting the real Phænomena
of our own Globe, and juft as they happen in Nature. To
this End the first Thing to be done is to counteract or nullify
the direct annual Motion of the Globe (in Fig. 3.) about the
Axis AB of the Ecliptic, by fixing its Stem O P upon a Plate
that has an equal Motion in a contrary Direction at the fame
Time.
3611. Thus; Let, A CB (Fig. 5.) be a Part of the outward
Rim of the Plate of the Orrery (3575), divided into 365
Teeth; and upon the lower Plate of the Radius Vector, let a
ſmall Wheel CD be placed with 59 Teeth to move in thoſe of
the Orrery-plate. Upon this Wheel a Pinion E must be fixed
VOL. II.
}
Rrr
with
490
1
INSTITUTIONS
$
t
with ΙΟ Teeth to turn another Wheel FG of 62 Teeth; and:
then if the Shaft or Stem of the Globe (Fig. 4) be fixed in
the Center N of this Wheel or Plate, it will communicate to
the Globe the Retrograde Motion required. For 59)365(= 6,2,
or the Wheel CD is moved round its Center 6,2 Times
while it revolves once round the Orrery-plate; and becaufe
10)62(6,2 alfo; it is evident, that the Wheel FG moves
once round in one Revolution of the Radius Vector, and be-
caufe the Wheel CD moves directly, from Weft to Eaft, 'tis
plain the Wheel F G moves retrograde, or the contrary Way
as the Arrows (4) fhew in each.
Į
3612. Hence by thefe two Wheels only, the annual Motion
of the Globe (Fig. 4) about the Axis A B is annihilated, and
a conftant Parallelifm of the Axis NS of its diurnal Motion is
produced; and therefore not only the Difference of the Seafons
and Days and Nights is effected, but all the Change or Variety
thereof that happens to the natural Globe for every Latitude,
and every Day of the Year.
ΟΙ
3613 The next Thing to be confidered, is, the Syftem of
Wheel-work for giving the fame Globe its proper diurnal Mo-
tion about its Axis NS, and in the hatural Direction from
Weft to Eaft. Here we find 365 of theſe diurnal Revolutions
to one annual One; and therefore fuch a Syftem of Wheels
muft give 59 Turns of the Globe in one Turn of the Wheel
CD, which muft be the Primum Mobile, or firft Mover to that
Syftem.
a
3614. Therefore the Number 59 muft be broke into three
Quotients, becauſe the laſt Wheel muſt have a retrograde Di-
rection, as will appear by and by; fuppoſe theſe Quotients are
10)20(= 2; 8)40(= 5; and 10)59(= 5,9; then it is evi-
dent, if upon a Bridge or Plate fixed over the Wheel-work of
Fig. 5. the following Wheels are properly difpofed, viz.
Wheel. A of 20 Teeth (Fig. 6.) fixed on the Arbor of the
Wheel CD (Fig. 5.) to drive a Pinion B of 10 Teeth on a
Wheel C of 40, which drives a Pinion Dof 8 Teeth fixed upon
a Wheel of 59, and this at laft drives a fmall Wheel F of 10
Teeth; then, I fay, it is evident this laſt Wheel F will be
turned round 59 Times while the firſt Mover CD turns once;
and its Duechon will be retrograde, as fhewn by the Arrows..
1
!
3615. The
Of CLOCK-WORK.
491
3615. The Wheels being fo proportioned, that the Center
of the laſt Wheel F (Fig. 6.) lies exactly over that of the Wheel
FG (Fig. 5.) a Socket is to be fixed into the ſaid Wheel F,
moveable about the Stem or Fulcrum of the Globe which is fixed
firmly in the Center N of the Plate FG. This Socket paffing
through a Plate (faftened over all the Machinery) will appear
above it, as at PT. And if on this upper Part of the Socket
there be fixed a Wheel TV with 10 Teeth playing in another
V W of an equal Size and Number of Teeth fixed to a Part of
the Axle NS below the Piece PR (in which the faid Axle
moves) then will the Globe (Fig. 4.) be turned round 59 Times
in each Revolution of the Wheel CD of Fig. 5. and confe-
quently it will turn upon its Axis once every Day or 24 Hours.
And this Motion will be direct, becauſe that of the Wheel TV
(or F below) is retrograde.
3616. And thus the TELLURAN applied to the ORRERY
will by its three Globes together give a full Demonſtration and
Illuftration of the Genuine THEORY of the annual and diurnal
Motions of the Earth in every Refpe&t; one Globe ſhews (Fig. 2.)
that by a parallex Axis without Inclination, there can be no Dif-
ference in the Seaſons or Quantity of Day and Night ———————— A
Second Globe fhews (Fig. 3.) that an inclined Axis without Paral-
lelifm, can only fhift the Scene of the Seafons, and produce a
different Length of Day, and Night in contrary Latitudes;
but can never make any Alteration in either
But the
third Globe with an Axis, at the fame Time, always inclined and
parallel to itſelf, produces every Variety we find in Nature, and
exhibits an exact Conformity to the Motions and various Phæ-
nomena of the natural Terraqueous Globe.
Rrr 2
CHAP.
492
INSTITUTIONS
CHA P. XXIV.
The CONSTRUCTION and MECHANISM of a
LUNARIUM, for fhewing the MOTION and
PHASES of the Moon, as alfo of her APOGEE, and
LINE of NODES,
3617. THE Conftruction of a LUNARIUM, or Inftrument
to fhew the Phænomena of the Moon in regard
to her own Motion and Phafes, and the Motion and Poſition of
the Apfides and Nodes of her Orbit, will be more complicated
than that of the Tellurian; but as the periodical Revolutions are
known, it will be eaſy to contrive fuch Syftems of Wheel-Work
as fhall reprefent them adequately, and produce Appearances.
every Way fimilar, and correfponding to Nature itſelf.
3618. The LUNARIUM, to exhibit the Phænomena of Lunar
Motions, may be contrived and conftructed in the following
Manner. ACB here again repreſents the dentated Rim of the
Plate of the Orrery of 365 Teeth, in which the Wheel CD
(Fig. 2.*) moves as before; and is now alſo to be confidered as
the Primum Mobile, or that which gives Motion to all the Ma-
chinery of the Lunarium. As this Wheel has 59 Teeth, it is
evident, that if to its Arbor below, there be fixed a Wheel A of
20 Teeth, to drive another equal Wheel B of 20, and this to
move a 3d Wheel C of 10 Teeth; then this laſt Wheel C will
tuin round once in the Time of a Lunation, or 29 Days; for it
will turn twice in each Revolution of A, which is in 59 Days.
And the Direction of its Motion will be the fame, viz. from
Weft to Eaft.
艺
​3619. In the Center of the Wheel C (Fig. 1.) let there be
fixed a ſtrong perpendicular Wire to pass through, and to a con-
venient Height above, the Upper Plate or Top of the Lunarium.
Then let there be provided another Wire EM (Fig. 4.) and to
one End E thereof, let therebe fixed an Ivory Ball ÆN QS, with
proper Circles to reprefent the EARTH; and at the other End,
See the Plate of MECHANISM for a LUNARIUM,
1
ano-
-
Of CLOCK-WORK.
493
another Ball M to reprefent the MOON, one fourth Part of the
Earth's Diameter. For the Diameters of the Earth and Moon are
known to be nearly as 7964, to 2192 Miles.
3620. The Magnitude and Denfity of the Earth and Moon
being duely confidered, the common Center of Gravity A be-
tween them will be found to fall very near the Surface of the
Earth; this then will be the Point on which thofe two Bodies
will be in Equilibrio; and therefore if this Point A be fixed
upon the Top of the Wire or Axle of the Plate C, both the
Moon and Earth will be carried round their Center of Gra-
vity in 29 Days; and in 12 Lunations or 354 Days they will
be carried once about the Sun, which Revolution is called the
Lunar YEAR, which is 11 Days fhorter than the Solar Year of
365 Days; and this Deficiency of 11 Days is what the Chro-
nologifts call the EPACT.
3621. This Conftruction of a Lunarium prefents us at the
fame Time with another Motion of the Earth, viz. its Menftrual
Motion about the common Center of Gravity A; and hence we
learn (1.) That it is not the Center of the Earth C which de-
ſcribes what is commonly called the Orbis Magnus or annual Or-
bit about the Sun, but the Center of Gravity A between the
Earth and Moon. (2.) That the Moon alfo in its menftrual
Motion does not regard the Center C of the Earth, but the Cen-
ter of Gravity A, as the Center of her proper Motion. (3.)
That the Center C of the Earth is fartheft from the Sun at the
New Moon, and neareſt at the Full Moon. (4.) That in the
Quadratures the Menftrual Parallax of the Earth is fo fenfible as
to require a particular Equation in Aftronomical Tables. Thefe
Points, though of ſo great Importance, we have not yet ſeen re-
prefented in Orreries or Lunariums hitherto made.
3622. Again, if a Candle or Lamp be placed in the Center of
the Orrery, then will the artificial Moon, revolving about the
Earth, be enlightened in the fame Manner as the real Moon in
the Heavens is by the Sun, and all her Phaſes in each Lunation
with respect to the Earth will be the very fame as we obſerve
them in the Heavens.
3623. But further, if the Balls for the Earth and Moon were
properly furniſhed with Magnets within Side, and then inſtead
of being fixed to the Ends of the Wire EM, they were nicely
fufpended
494
INSTITUTIONS
•
fufpended on fine Points of Needles placed there, with magne-
tical Cards fhewing the feveral Points of the Compafs, as might
eafily be done, then we might agreeably obferve that the Earth
and Moon would both conftantly fhew the famé Part or Face to
each other as they moved about the Center A; and that in turn-
ing once round, the Ends of the Wire E M would apply fuccef-
fively to every Point of the Card or Compafs, and thereby fhew
that each Ball in one Revolution about the Center A, did alfo
move once about its own Axis.
کو
3624. But though this be the real Cafe of the Moon M, it is
not of the Earth, which (to answer other Purpofes) has ano-
ther Motion communicated to it, viz. the diurnal Motion in
24 Hours, every Way independent upon the menftrual Motion
about A. And therefore that the Lunarium may be quite fimi-
lar to Nature, the Ball ENQS fhould be ftill poffefied of an
internal Magnet, and fo placed upon an Axis fixed in the
End of the Radius Vector, as to be voluble about it at Pleafure.
13625. In this Conftruction, the general Phænomena of the
Earth and Moon will be mutually evident; for hence it will-
appear (1.) That the Moon muft always neceffarily fhew the
fame Face or Side to the Earth. (2.) That the Earth turns every
Part of its Surface to the Moon once in 24 Hours: (3.) In
Confequence of this, the light and dark Parts of the Moon ap-
pear permanent and fixed to us. (4) But a great Variety of
luminous and opake Parts prefent themfeives in a conftant Ro-
tation over the Surface of the Terreftrial Ball to the Inhabitants
of the Moon, if any there be.
Moon appear as in the Heavens.
(5.) All the Phafes of the
(6.) Alfo the fame Lunar
Phafes will be obferved in the Earth by the Lunarians, for we
are mutually a Moon to each other. (7.) Thofe Phales are
contrary in the Earth and Moon, for when the Moon is new or
dark to us, our Earth is wholly enlightened or full to them..
(8.) Our terreſtrial Ball being four Times larger in Diameter,
will prefent the Lunarians with a Full moon 16 Times as large as
ours. (9) Only one Hemifphere of the Moon being turned to
the Earth, the Inhabitants of that alone can ſee our Globe..
(10.) Our Earth appears to them always in one Place, or fixed
in the fame Part of the Heavens.. (11.) The Moon has an ap-
}-
parent
{
495
Of CLOCK-Work.
ד
parent diurnal Motion through the Heavens, by the Earth's
real Motion on its Axis. (12.) The Lunarians in the oppofite
Hemiſphere never ſee our Terreftrial Globe, as we can never
fee them.
3626. With reſpect to the Sun, it is evident by the Lunarium,
(1.) That it muſt always enlighten one half her Globe. (2.).
That every Part of the Lunar Ball is turned to the Sun in the
Space of her Monthly or periodical Revolution. (3) Therefore
the Length of the Day and Night in the Moon is ever the fame,
and equal to 144 of our Days. (4.) When the Sun fets, the
terreftrial Moon rifes to the Lunarians, in the Hemifphere next
the Earth. (5.) To them therefore there can never be any
dark Night at all. (6) While thofe in the other Hemifphere
can have no Light by Night but what the Stars afford them.
3627. Not only the Moon herſelf but even the very Orbit ſhe
deſcribes about the Earth, has a fenfible Motion, and is found to
make one Revolution in a little leſs than Nine Years; for by the
Aftronomical Tables the Place of her Apogee was
S D
"
in the Year $1755 in o: 15: 38: 20
2. 1764 0: 21: 50: 17
Y
1
Motion in Years
-9-12 6: IL: 57
366,2
Therefore ſay, as 366º,2 : 9 :: 360°: 9 × 360
Time of
366,2
one Revolution of the Apogee; then
Days.
ях збо
366,2
X 365 3229,5
1
3628. Now in order to compute the Syftem of Wheel-work
for producing this Motion of the Apogre, we muſt divide
32291 Days by 59, the Teeth in the firſt Wheel CD (Fig. 2.)
and the Quotient is 54,7; this must be broke into two other
Quotients, viz. 7 and 7%, both on Account of its being too
large in itſelf, and eſpecially becauſe it is neceffary to have two
Wheels move, viz. FG and IK, that the latter may have a
direct Motion, as the Apogee really has. Therefore a Pinion E
of 10 Teeth on the Axis of the Wheel C D muft drive a Wheel
FG of 70 Teeth; and another Pinion of 10 on that Wheel
muft
>
}
496
!
INSTITUTIONS
muft drive a Wheel IK of 78 Teeth; and then a brafs Socket
fixed about the Center of this laft Wheel, will carry about an
Ellipfis, reprefenting the Moon's Orbit, in the Time required.
3629. Again, the Orbit of the Moon is not parallel to the
Ecliptic or Earth's Orbit, but makes an Angle therewith of
5° 18′ at a Mean; now from different Forces of the Sun and
Earth in different Situations, the Moon will in each Revolu
tion be various attracted, and her Orbit agitated; being fome-
times drawn down, at other Times elevated; fometimes it is
moved forwards, and at others, backward; and thus there will
be a variable Motion generated in the Line of the Nodes, but
upon the whole the Retrograde Motion is greateft, and confe-
quently the Nodes will move backwards in the Ecliptic, and
make one Revolution in, nearly, 19 Years.
1
3630. For the Place of the afcending Node & in
1
S
the Year $1746 was 11: 27:
{
I : 4
21765 is II: 19: 30: 32.1
Years, 191 1
11: 22: 38: 28352
Degrees.
好不
​19 × 360
Therefore fay, as 352°,5:19:: 360°:
Time of
352,5
19 × 360
× 365
7082,5
352,5
one Revolution in Years, or
Days.
3631. Then 59)7082,5(= 120, very nearly; this 120 muſt
be broke into three Quotients, that the laft Wheel may have a
retrograde Motion. Thefe Quotients may be 3, 5, 8; and then
a Pinion A (Fig. 3.) fixed on the Axle of the Wheel CD
(Fig. 2.) having 6 Teeth may drive a Wheel B of 18; this
with another Pinion C of 6, drives a Wheel D of 30; and
this alfo with a Pinion of 6 moves a Wheel E of 48 Teeth; and
in a retrograde Direction; and therefore a Socket fixed in this
Wheel will carry a Line pointing to the Place of the Nodes
through all the Signs of the Ecliptic in Antecedentia, in the Space
of 70821 Days as required.
11
کے
3632. In the laft Place, upon the upper Part of the Axis of
the firſt Mover CD (Fig. 2.) let a Pinion of 10 Leaves be
fixed to drive a Wheel of 62; then. a Socket (including all
}
the
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!
t
The THEORY of the EQUATION of TIME.
Fig. 3.
α
A
Fig. 7. D N
Р
B
Fig.
F
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R
A
EFNG O
C
B
A H
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I
I
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P
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A
XI
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F
H
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I
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Fig. 5.
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Fig. 8.
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B
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D
IX
I
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G
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M
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F
S
W MKVII
B
C
D
S
P
F
E
A
C
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B
Fig. 2.
;
P
VI
N
K
me
A
Fig. 4.
F
D
B
Fig. 6.
H
E
·B
C
I
F
H
S
V
P
12
K
Of CLOCK-WORK.
497
the reft) being fixed in this Wheel will carry an Ecliptic Circle
upon the Plate of the Lunarium, which may at any Time be
placed in a Pofition fimilar to the Ecliptic on the Plate of the
Orrery, and then it will always continue fo by Virtue of its
conftant Paralleliſm produced by this Mechaniſm. (3611.)
3633. If a Circle be placed about the Line of Nodes (3629,)
and moveable upon it as an Axle, it may be occafionally placed
horizontally, or inclined to the Horizon in an Angle of 5° 18′;
and then if the Moon M (Fig. 4.) be fixed to the End of a fine
fteel Wire, paffing freely though a flender Socket on the End of
Radius Vector E M, fo that it may be over the Middle Part of
the laſt mentioned Circle and conftantly move upon it; it will
be easy to underſtand how the Moon may be made to move
either horizontally or in her proper Orbit; in order more par-
ticularly to fhew whatever relates to the Nature, Caufe, and
various Phænomena of ECLIPSES.
3634. Alfo, being now furnished with a Solar and Terrestrial
Ecliptic it will be very eaſy to underſtand the Rationale of the
general Deceptic Vifus, or Illufions of Sight in regard to the Mo-
tions, Places, Magnitudes and other Affections of the heavenly
Bodies, and to refolve them all confiftently with Truth. Alfo
from hence the Heliocentric and Geocentric Places of the Moon
and Planets will readily, at any Time, appear. And thus all
the Great Points and Pofitions in the Solar and Lunar Aftronomy
become eafily explicable by fuch a Conſtruction of an ORRERY,
TELLURIAN and LUNARIUM, as has been defcribed.
3635. After the fame Manner a Jovian LUNARIUM may be
conſtructed with Eafe, by knowing the Periods and Diſtances of
each of Jupiter's Moons, which are as in the following Ta-
ble.
Satellite. Diſtance.
Days. H.
Ratio to I Day.
I
5%
1: 18: 27724:
42
234
9
14,3
3 13: 13 20: 71
12: 86
25,3
7: 3 3420
16: 16: 32
6: 100
3636. The Numbers in the Ratio of thefe Periods to one Day
will fhew the Diameters and Number of Teeth in a fixed and
a moveable Set of Wheels for thefe Satellites in the fame Manner
VOL. II.
Sss
as
498
INSTITUTIONS
as before in the primary Planets (3574.) That is, upon one Ar-
bor there must be fixed four ſmall Wheels, or Pinions, with the
Numbers of Teeth 24, 20, 12, and 6, to drive four moveable
Wheels of 42, 71, 86, 100 Teeth refpectively. Upon thefe,
Sockets are placed to carry the Secondaries round their Primary
Jove, in Distances meaſured in Semi-diameters of his Globe ex-
preffed by the Numbers in the fecond Column.
3637. But now to adapt thefe to the dentated Edge of the
Orrery, and from thence derive a proper Motion we need only
confider, that if a Pinion of 8 be fitted to the Teeth of the Or-
rery below, and fixed to a Wheel of 64 Teeth above, playing
in another Pinion of 8 fixed on the Arbor of the fixed Set of
Wheels, then will that Arbor, and all its Wheels, be turned.
once round in every natural Day of 24 Hours; and confe-
quently, all the Satellites will move about their Primary in
fuch a Number of Days as they really do in the Heaven, and
at proportionable Diſtances.
3638. If now a wax Candle be placed in the Center of the
Orrery, and a Piece of white Paper on the Radius Vector be-
yond the Satellites perpendicularly; then the Motions of the
Shadows of the Moons will naturally repreſent the apparent
Motions of thefe Moons as feen in the Field of a Teleſcope,
viz. a Rectilineal Motion through the Diameter of the View; as
alſo their direct, retrograde, and flationary Phænomena, and laſtly,
thoſe of their Immerfions and Emerfions in Eclipfes; their Occul-
tations, and Diſapparition on the illumined Difk of their Primary;
all which are of fuch Confequence in finding the Difference of
Meridians, or Longitude of Places; and many other Points of
Aftronomy, Navigation, &c.
3639. A Saturnian LUNARIUM may be conftructed in the
very fame Manner for 5 Moons; whole periodical Times are as
follow, from Dr. Halley's Tables.
Days. H.
The Firft revolves in I : 21 : 18
Semid. of Ring.
2,097
The Second
2: 17
17 41
I
2,686
The Third
4 12 25
39752
The Fourth
15: 22: 41
8,698
The Fifth
79: 7:48
25,348
i
3640. The
Of CLOCK-WORK.
499
3640. The four inmoft Satellites defcribe their Orbits very
nearly in the Plane of the Ring produced; which Plane makes
an Angle of about 31 Degrees with the Plane of the Ecliptic;
and the Nodes are in 19° 45′ of Virgo and Pifces. The Orbit
of the fifth Satellite is a little wide of the Reft. Which Parti-
culars being obferved in the Conftruction of this Lunarium, it
will very naturally exhibit the various Phænomena of this Satur-
nian Syſtem of Moons and Ring, for every Year of Saturn's Pe-
riod. Note, in this Lunarium, the Line of the Nodes muſt be
kept always parallel to itſelf, as was fhewn for the Line of the
Earth's Nodes (3611.) And then, as Saturn paíles from the
Node in Virgo to the oppofite one, the Sun will enlighten the
northern Plane of the Ring; as it will the fouthern Plane in
paffing through the other fix Signs.
CHA P. XXV.
The Aftronomical DOCTRINE of the EQUATION of
TIME explained. A CALCULATION of that Part
arifing from the OBLIQUITY of the ECLIPTIC to
the PLANE of the EQUATOR; with a TABLE of
this EQUATION for every DEGREE of the
ECLIPTIC.
IF
3641. TF a Clock or Watch could be constructed with Me-
chanifm abfolutely perfect, it would always fhew or
keep equal Time, or go truly; but as this is not the Cafe it will
often want rectifying, or being fet right. Now as we have no
direct or immediate Standard or Index of equal Time, by which
the Error of a Time-Piece can be pointed out and inſtantly
corrected, we muſt be content, or rather, we ought to felici-
tate ourſelves that we have it in our Power to do this by any
Means at all. For this Purpoſe we have recourfe to two Me-
Sss 2
thods
500
INSTITUTIONS
thods generally; one is the Sun-Dial and an Equation Table;
the other is by aſtronomical Obſervation and Calculation.
3642. The first Method is very eaſy, and therefore adapted
to common Uſe, which is now to be explained; as we have
before given the Theory of the 2d Method largely from Inft.
1915 to 1920 inclufive. The Equation of Time is a Doctrine
refulting from two Principles, viz. (1.) The Sun's apparent
diurnal Motion not being in the Equator, the only great Circle
of equal Motion with respect to the Meridian; and (2.) The
annual apparent Motion of the Sun not being in a Circle, but an
Ellipfis which cannot admit of Equal Motion; as we have abun-
dantly fhewn (1236.)
3643. We fhall confider each Part of the Equation of Time
feparately, and then both together. Therefore, firſt with
reſpect to the Motion of the Sun being in the Ecliptic and not
in the Equator; it is evident from the View of any Globe re-
volving upon its Axis under the Meridian, that the Motion,
or rather the Velocity of the Motion, by which the feveral
Points of the Ecliptic and Equator pafs the Meridian will be
very different; and that, that of the Equator will be conſtant
and equable, and that of the Ecliptic always variable and un-
equal. But this Matter will be eafily elucidated by a Figure.
3644. Therefore let EI (Fig. 1.) be the firſt Quadrant of
the Equator, and E H the firſt of the Ecliptic; the Angle IEH
being 23° 29′. Alſo, let P be the Pole of the Equator PHI,
a Quadrant of the Solftitial Colure and PFG a Poſition of the
Meridian very near to it; let PA B be the Meridian very nigh
to the Equinoctial Point E, and PE the Meridian paffing thro'
that Point; then it is evident, that upon the first momentary
Motion of the Globe, the very fmall Arches EB and EA will
pafs under the Meridian in the fame Inftant, and therefore
thofe Arches will adequately repreſent the Fluxions or Ratio
of the inceptive Velocities of Motion in the Equator and Eclip-
tic at the Equinox E.
3645. Again, as PHI is the Solftitial Colure and PFG a
Meridian exceeding near it, the Arches GI and FH, will be
as the Fluxions or ultimate Velocities with which the two
Arches EG and EF become equal in the Solstice H.
But
the infinitely ſmall Arches E B and EA may be confidered
{
as
譬
​Of CLOCK-WORK.
501
as right Lines, or the Angle AEB as rectilineal; and there-
fore EB EA :: Co-fine of AEB: Radius :: j. And
GI FH Radius: Co-fine of IH (AEB): : .
v.
Therefore, we have ỷ:: ¿: x, or ÿ¿x², a conftant
Quantity; therefore the Velocity of Motion in the Beginning of the
Ecliptic E exceeds the conflant Velocity of the Equator just as much as
it falls fhort of it at the End of the Quadrant in H.
✯:ÿ
3646. There is, therefore, fome intermediate Point C where
the Velocity in the Ecliptic is equal to that in Right Afcenfion in
the Equator, which is now to be inveſtigated; make the Arch
Ed EC, and D d will be the Difference of the two Arches
EC and ED, which, in that Cafe, muſt be a Maximum ;
for let the Arch EC≈≈, and E D = x; then will D d = z
-x, whofe Fluxion, when a Maximum, iso,
(818,) or =; that is the Fluxions or Velocities of the
Arches EC and ED will be equal, when their Difference D d be-
comes the greatest of all.
ż
3647. Now to determine the Quantity of the Arch EC,
when Dd is a Maximum, we have (by Fluxionary Spherics
1818,) the Fluxion of E C (ż) to the Fluxion of ED (*) as
the Co-fine of CD to the Sine of C. Alſo by common Sphe-
to the Co-fine of E as
Or thus in ſhort, *:*::
rics (1799,) we have, the Sine of C
Radius to the Co-fine of CD.
:
cs CD: SC; and sC cs D: ResCD; therefore
* x c s CD
ż
= sc =
CSEX R
CS CD
; which will give this Analo-
gy, csCD²: cs Ex R:::; and therefore when Cc =
Dd, or x, or when D d is a Maximum (3646,) we have
cs Ċ D² = csEx R; and fo csCD = √esE × R.
Whence E C, ED, and Dd, are all known.
3648. The fame Arch E C may be found without Fluxions
by premiſing the following Lemma, viz. The Sum of the Tan-
gents of any two Angles BAC, BAD, is to their Difference, as
the Sine of the Sum of thofe Angles is to the Sine of their Difference.
Let BC and BD (Fig. 2.) be the two propofed Tangents to
the Radius A B; and take B d = BD, join Ad, and draw
DE and F perpendicular to A C. Then it is manifeft, be-
caufe Bd BD, that AD Ad, and dAB DAB, and
confequently, that CAd is the Difference of the two Angles
1
BAC
'.
502
INSTITUTIONS
BAC and BAD. Then are the Triangles CDE and CdF
fimilar, and fo we have C D (= CB + BD): Cd (= C B ·
BD) :: DE: dF; but DE and dF are Sines of DAE
and d AF to the equal Radii AD and Ad, whence the Truth
of the Lemma is evident.
3649. Then (per Spherics, 1789) Radius: Co-fine E ::
Tangent EC: Tangent ED (Fig. 1.) and by Compofition
and Divifion (648) we have Radius +cs E: Radius-cs E
:: Tangent EC + Tangent ED: Tangent E C-Tangent
ED::sEC + ED : sEC — ED (per Lemma.) But the
Ratio of the firſt two Terms of thofe Analogys is conftant be-
cauſe the Angle E is fo, therefore that of the two laft Terms
will be fo likewife; and confequently, s.ECED will be a
Maximum, when the s. EC + ED is fo, that is, when EC +
ED = 90° for then the Sine thereof will be Radius.
Whence we have Radius + csE: Radius CSE: Radius
: s.EC — ED; whence becauſe the three firſt Terms are
known the fourth s.ECED = s.D d, is known alſo.
3650. But there is yet a more direct and fimple Method of
coming at this Equation, becaufe when a Maximum it is known
to be a Third Proportional to Radius and Tangent of half the Angle
E; as will thus appear. The Radius is the Sine of 90°, and
therefore is the Co-fine of o°. Therefore Radius + Co-fine E
Co-fine o + Co-fine E: Co-fine o
E
: Radius
Co-fine E
E ↓ 0
Co-fine E
Co-tangent
Tangent
2
—
(1829) ::
О
2
Co-tangent E: Tangent E. But it is tr::r:ct (1831),
therefore r²ct X t, therefore r² X t = c t x t x t; whence
ct:t::r² : t², that is Co-tangent
2
: Tangent El. Therefore Rad²:
2
2
ct Xt;
E: Tangent E :: Rad.²
.E:: Radius: Sine of
EC —ED=sDd, and dividing by Radius, we have Rad.
Tang. E Tang. E: Sine of D d, as was to be demon-
ftrated.
3651. The Quantity of this Equation computed is D d =
2° 28′ 34″, which converted into Time, is 9 Minutes 54 Se-
conds; and this is the greateſt Difference of Time that would
ever be found between the Sun-Dial and a Clock if the Ob-
liquity
Of CLOCK-WORK.
503
liquity of the Ecliptic or Sun's Path were the only Cauſe there-
of. Now fince EC + ED 90, and D d = (EC — ED
=) 2° 28′ 34″, therefore E C = 46° 14′, therefore the
Equation begins from nothing in Aries or in the Point E,
and increaſes to its Maximum in 8
decreaſes till the Sun arrives at
again becomes nothing.
16° 14′ and from thence
in the Point H, where it
3652. In this firft Quarter of the Ecliptic, it is evident,
that had the Sun moved in the Equator it would have been in
d at the Moment it is in the Ecliptic at C, and confequently
the Sun at C is in the Meridian P D, which is before the Time
the Equinoctial Sun arrives to it, at (d), as being then Eaſtward of
it by the Difference in Time every corresponding to the Mo-
tion in Right Afcenfion of the fmall Arch Dd. Whence it
appears that the apparent Time of the Ecliptic Sun, or that fhewn
by a Dial, is in this firft Quadrant always before the Mean or
Equal Time in the Equator by the Difference belonging to the
Equation D d, which, therefore, when found, muſt be fub-
ducted from the Solar Time to have the mean Time for the Watch
or Clock.
3653. By making the fame Conftruction in the fecond Qua-
drant of the Ecliptic, it will appear, that the Point (d) is Weſt-
ward of the Point C, and therefore comes first to the Meridian
Arch. Whence the mean Time is now before the apparent or So-
lar Time, and confequently the Equation Dd now becomes
Addititious, or muſt be added to the Solar Time to have the
Mean, which now precedes it. Thus when the Sun is in
43° 46′ (= C H,) if we add 9′
the Dial it will give the Mean Time for the Watch; for ſo
much does the true Noon of that Day exceed the apparent Noon
on Account of the Sun's oblique Motion.
54″ to the Time ſhewn by
3654. It is evident the fame Equation will be produced in the
3d and 4th Quadrants of the Ecliptic, and will accordingly
be Ablatitious and Addititious, as is fhewn in the following Table,
where the Quantity of that Equation is computed for the Sun's
Place in every Degree of the Ecliptic.
Subtract
504
INSTITUTIONS
Subtract from the Apparent Time.
The true Place of the Sun.
8 m
II
I
Signs | r
Deg.
/ //
"/
//
1°1
O
8
23
CO
45 30
I
О
20
8
2
О
40
∞ ∞ ∞
34
8
44
3
I
O
8
53
4
I
19
9
2
5
I
39 9
10
~∞∞ co co
8
3529
8
24/28
8
1327
8
I26
7
48 25
6
I
589
17 7
34124
7
2
18
9
24 7
2023
8 2
37 9
30 7
6/22
9
2
569
35
6
5021
10
3
15
9
40
5
35 20
I I 3
34
9
44 6 18 19
12
3
52
9
486
218
13 4
I I
9
50 5
44117
141 4
29
9
52 5
2716
15
4
46 9 54
16
5
41.9
17
18
19
534
54
9 51
20
6
21
22
23
5 21 9 54 4
5 37 9
5
6
6
10 9 49
251 9 46
40 9
42
9 38
6 55
24 7 9 9 33
25 7 23 9 26
50 14
3113
II 12
52 II
3210
I l
51
98 76
30 7
9
сара
I 48
26 7
369
19
I
26
27 7
48
9
12
I
5
288
I 9
4
O
43
29
8 12 8
55
O
22
20
823
8
45
О
432 O
I
//
//
//
Deg.
X
MAN
Signs
54 4
5+++ 3 3 3 2 2 2 -
သ
15
Add to the apparent Time.
Of CLOCK-WORK.
505
CHA P. XXVI.
ACALCULATION of that Part of the EQUATION of
TIME which arifes from the Elliptic FORM of the
EARTH'S ORBIT; with a proper TABLE thereof
for every DEGREE of the EARTH'S ANOMALY.
3655. T
“HE ſecond Cauſe of the Inequality of Time fhewn
by a DIAL and a CLOCK, was faid to be owing
to the Elliptic Form of the Earth's annual Orbit (3642), and
this is known to be Fact from common Obfervation; for through
every
Year the Diameter of the Sun is found to fubtend a varia-
ble Angle, being fenfibly greater at one Time than at another;
for a few Days after the Winter Solstice it meaſures, by the Mi-
crometer, about 32′ 43″, but the fame Diſtance after the Sum-
mer Solftice it is no more than 31′ 38″.
3656. Now were the Orbit of the Earth truly circular it
would be always at an equal Diſtance from the Sun, which
would therefore always appear of an equal Bignefs; confequent-
ly, as it does not ſo appear, it muſt be at unequal Diſtances
from us to caufe thofe unequal apparent Magnitudes (1451.)
And becauſe the greateſt and leaft folar Diameters are as 32′ 42″
= 1963″, and 31′ 38″≈ 1898; the greateſt and leaſt Diſ-
tances of the Earth from the Sun will be inverſely as the
fame Numbers 1963 and 1898.
3657. Let this Ellipfe Orbit of the Earth be denoted by
ABPD, (Fig. 3.) in whofe Focus S is the Sun; and let the
other Focus be F. Upon the Center S defcribe a Circle GIKL,
whoſe Diameter G K is a Mean Proportional between the two
Axis AP and BD of the Ellipfis; then will the Area of that
Circle be equal to the Area of the Ellipfis (894.) And we
may now compare the Motions in the Ellipfis and Circle as they
would each be deſcribed in the fame Time; and thus inveftigate
an Equation by which the unequal Motion of the Earth in
the former, and the equal Motion of a Point in the latter, may
be always equated, or adjufted to each other.
Ttt
VOL. II.
3658.
506
INSTITUTIONS
3658. Suppofe now the Earth begins its Motion from the
Aphelion A of the Ellipfe, at the fame Time that the imaginary
Point fets out with an equable Motion in the Circle from G; then
becauſe the Diſtance AS exceeds the Diſtance SG, the Velo-
city of the Earth will be less than that of the Point (1213)
and therefore in the Time that the Earth deſcribes the Arch
A a, the Point will have defcribed Gg, making the Area G Sg
equal to the Area A Sa, purſuant to the general Law of
Nature. (1212
•
3659. Becauſe the Area AS a
GSg, fuduct from each
Side the common Part or Area GS d, and there will remain
the Area AG da equal to that of the Triangle dSg; there-
fore the faid Area AGda will be ever proportional to the
Equation of the Orbit, or the Arch (gd). Confequently when
this Area becomes a Maximum, there the Equation will be
greateſt alſo, which is in the Point L where the Circle inter-
fects the Ellipfe, for then the faid Area becomes ALG.
3660. As in the Point L, the Earth in its Orbit, and the ima-
ginary Point in the Circle have both the faine Diſtance from the
Sun, LS; they have there the fame Velocity of Motion; and
after they have paffed that Point L, the Earth approaching con-
ftantly nearer the Sun, will have a Velocity greater and con-
ftantly gaining upon that of the Point in the Circle, but ftill the
Point in N will be before the Earth at H, till at laft the Angle
HSN or Equation of the Orbit will vaniſh in the Line SPK,
where the Earth arrives to its Peribilion P, and the Point in the
Circle to K.
3661. The Area A SH being fill equal to the Area GSN,
we have at length, the Semi-Ellipfis ALP = the Semi-circle
GLK; and fubducting the common Area GL P, there will
remain the Area ALG the Area PLK. And as the Equa-
tion aroſe from nothing in the Point A to its greateſt Magnitude
at L by continual Increments; fo after paffing the Point L, it
muft leffen by decreafing Quantities, which conftitute the
Area PLK; fo that what was gained in the firft Part, is loft
in the latter, and the Equation becomes nothing in the Line
SK.
3662. Or thus; continue the Ray SH to M; then the Elip-
tic Sector ASH GSN, as being defcribed in the fame.
Time;
Of CLOCK-WORK.
507
=
=
Time; from each, take the common Part GLHS, and there
will remain A LG NLO + OSH MHL + MSN;
therefore A LG-MHL KMHP MSN, the Mea-
fure of which Angle is the Equation MN; and is therefore
nothing when M H L becomes equal to A L G or P LK.
3663. In the firſt half of the Ellipfis, or while the Earth de-
ſcends from the higher Apfis or Aphelion A to the lower P, the
Mean Anomaly will ever be greater than the true, or the Place
of the imaginary Point (g) will be to the Eaſt of the Point (d).
And therefore if we fuppofe the Earth at reft in the Center at
S, and the Circle GLKI to be the Primum Mobile, then (d)
will be the Place of the Sun in the Ecliptic, and (g) that of the
Point of Mean Motion; alfo ASP will be the Meridian, to
which when d arrives, it will be NooN by the Sun Dial, but
when g comes to it it will be XII by the Watch. And fince
thefe apparent Motions of the Heavens are from Eaſt to Weft,
it is evident, the Time of Noon by the Dial, will precced that by the
Clock; and confequently the Equation or Arch g d (turned in-
to Time) muſt be fubtracted from the Time by the Dial, to have
the Mean Time by the Clock or Watch, as that is now fiswer
than the Dial.
3664. But all the Time the Earth afcends from the lower
Apfis or Perihelion P, to the Point of Interfection I, it will be
nearer to the Sun at S than the Ecliptic or Circle, and confequent-
ly its Motion will be fwifter than that of the imaginary Point;
therefore its Place p at any Time will be before that of the ſaid
Point at k, and confequently will come after it to the Meridian at
P, in the apparent Motions; therefore the Mean Time will be be-
fore the folar Time, or the Clock will now be fafter than the Dial; and
the Equation ep muſt be added to the Time by the Dial to have
the mean Time by the Watch.
3665. This will alfo continue to be the Cafe till the Earth
arrives at the Aphelian A; for though in paffing from the mean
Diſtance at I to the greateft at A, the Velocity of the Earth at
Qis lefs than that of the Point R in the Ecliptic, and is con-
ſtantly decreaſing, yet it will always be before the ſaid Point;
and therefore, with Refpect to the apparent Motion, it must come
later to the Meridian than the Point of equal Motion R, and fo
Ttt 2
the
508 ·
INSTITUTIONS
the Equation R q is ftill to be added to the folar or apparent Time by
the Dial, to have the mean Time by the Clock.
3666. Here Rq is the Equation; and the Area or Sector KSR
- PSQ, from which take the common Area PSRI and there
will remain PKI=QIq+ RS q, and therefore P KI-
Q!q RS q, is proportional to the Equation R 4, (ſee 3662;)
therefore when QI becomes AIG (=PKI), the Equation
of the Orbit vaniſhes again.
9
3667. Let GN be any given mean Motion (Fig. 4.) and
AH the correſponding true Anomaly; draw Sh, Sn, indefinitely
near to S H and SN; and put Nny, and H h = ż; then
becauſe the fluxionary Triangle HSNS n, we have the
Angle HSb: NSn:: NS2: HS (1270):::j. Therefore
NS² x j
=2, Fluxion of the true Anomaly A H.,
HS²
3668. When the Fluxions of Quantities are equal, their
Difference will be a Maximum (3646;) therefore in L, where
SHSN SL we have ; and confequently the
Equation of the Orbit GNAH, will there be the greateft
poffible, as we before ſhewed, (3659.)
j;
3669. We have found Areas, Angles, and Arches, pro-
portional to the Equation of the Orbit; but to compute the
real Quantity of this Equation, we muſt firſt folve the fo much
famed Problem of KEPLER, viz. To cut off a Sector A Sa by the
Right Line Sa (Fig. 3.) that ſhall be to the whole Area of the Ellipfe, as
the Time of defcribing the Arch A a, to the Time of a whole Revolution
in the Elliptic Orbit. But as the moft geometrical Way of doing
this, is by infinite Series; and there is a much eaſier and
very exact
Method of computing the Equation invented by the late Biſhop
Ward we ſhall here explain that, with Bullia/dus's Correction
thereof.
3670. This Method depends on an Hypotheſis, that a Ray F L
(Fig.5.) drawn from the upper Focus F to the Planet at L, defcribes an
Angle A F L proportional to the Time of the Planet's paſſing through
the Arch AL. Therefore let ABP be the Ellipfe the Planet
defcribes; AP the Line of the Apfides; S the Focus in which
is the Sun, F the other Focus or Center of equal Motion. The
Angle AFL being as the Time, the Place of the Planet will
be
Of CLOCK-Work.
509
be at L, and the Angle ASL is the true or coequate Anomaly.
3671. Produce FL towards E, and make FE AP, and
join ES.
Then is LELS (769,) and the Triangle
ELS being ifofcles, the Angle LES=LSE, and both to-
gether equal to the Angle FLS (632.) Therefore in the
Triangle EFS, having the Sides FE and F S, and the Angle
of mean Anomaly A FE, or EFS, we find the Angle E,
and 2 EFL S, which is the Difference between the mean
Anomaly AFL, and the True ASL, and is therefore
the Value of the Equation fought; for this taken from the mean
Anomaly AFL leaves the True one ASL.
3672. Alfo in the Triangle FLS, having all the Angles
and the Side FS, the Side SL is known, which is the Diſtance
of the Earth from the Sun, in its Orbit at L. And thus the
Equation of the Orbit, and Diſtance from the Sun, is found in
this Hypothefis very exactly for the Earth; but for the other
Planets, eſpecially Mars and Mercury, it requires fome Correc-
tion, which it received from the celebrated Aftronomer Ifmael
Bullialdus, as follows.
3673. Upon the tranfverfe Axis AP defcribe the Circle
ADP, and let AFL be the mean Anomaly, as before. Thro'
L draw the Line QLG perpendicular to the Axis, meeting
the Circle in Q; join FQ, cutting the Ellipfe in M, and M
will be the Place of the Planet in its Orbit for the mean Ano-
maly AF L. Let BC be the Semi-conjugate, which continued
to D, we have CB: CD:: GL: GQ, which is therefore
known; join S M, and then the Angle ASM will be found in
the fame Manner as before the Angle A S L was found, (3671.)
3674. This Correction of Bullialdus accelerates the Motion
of the Planets in the first and third Quarter; and in the Second
and Fourth retards them a very little Matter, in Refpect of Ward's
Hypothefis, which makes their Places by this Theory agree
much better with Obfervations. But the late Mr. SIMPSON
has given us a Conſtruction which finds the Planet's Place ftill
much nearer the Truth than either of the foregoing.*
3675. The general Reaſon of this Hypothefis is this, that
the Velocity in the Orbit being every where inverſely as a Per-
pendicular
N
See his Mathematical Effays, p. 41.
510
INSTITUTIONS
pendicular on the Center of Force S upon a Tangent to the
Orbit in a given Point or Place of the Planet (1213), it will
follow that Aa, Bb, Pp be Spaces paffed through in equal
Times; then drawing the Lines Fa, Fb, Fp, the Angles
A Fa, BF b, PFp, will in the Earth's Orbit, be extremely
near equal. For at A and P we have Aa: Pp:: PS: SA
:: AF: FP; therefore A Fa PFp. Again, Aa: Bb
:: BS:SA:: FB: FP; but in the Earth's Ellipfe F is very
near to C, and we fhall have FB: FP:: AF: FB, very
nearly; confequently the Angle BFb AFb, extremely
near. Therefore the angular Motion about the upper Focus F is
very near equable.*
=
3676. When the Planet is in B, the middle Point of its
Semi-Ellipfe, then F E becomes FN, and FB BN AC
=SB; and FSN is a Right Angle, and fince the Point B
very nearly coincides with the Point 1 (Fig. 3.) where the
Equation of the Orbit is a Maximum, therefore the Angle
FBC will infenfibly differ from the Half of that Equation.
And becaufe (3656) AS is as 1963, and SP as 1898, there-
fore AC FB, will be as 1930,5, and AS-AC=CS
= CF, will be as 32,5; therefore fay As FB
·
=
: FC: Radius: Sine of FBC
1930,5
58' 9", which doubled,
1° 56′ 19″, = F BS, the greateſt Equation of the Earth's Or-
bit; and fuch you find it in the Aftronomical Tables of Dr.
HALLEY. This Equation turned into Time for every Degree
of the Earth's Anomaly, is as in the following Table.
* Thro' Inadvertence, the Lines, Fa, Fb, Fp, were forgot to
be drawn in Fig. 5. but they can be eafily fupplied by the Reader,
and it is therefore hoped will occafion no Obftruction to his under-
ftanding this material Point.
Subtract
Of CLOCK-WORK.
511
Subtract from the Apparent Time.
The Mean Anomaly of the Sun.
2
3
4
5
Signs
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I
Deg.
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4328.
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9
6
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16
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54 7
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19 7
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//
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Deg⋅
I I
10
9
8
Add to the apparent Time.
7
6
Signs
512
INSTITUTIONS
CHA P. XXVII.
The THEORY of the Compound EQUATION of TIME,
F
and the CONSTRUCTION of one General TABLE
thereof depending on the SUN's PLACE in the
ECLIPTIC.
3678. THE two Equations of Time we have now confider-
ed muſt be both applied to obtain a general or ab-
folute Equation, confifting of the Sum or Difference of theſe
particular Ones, according to their affirmative or negative Qua-
lity; but becauſe One requires the Knowledge of the Sun's
Place in the Ecliptic, and the other, the Sign and Degree of
the Earth's or Sun's mean Anomaly; therefore to render this
Matter lefs troubleſome or difficult, Aftronomers have compof-
ed one general Table out of the two, making it depend en-
tirely on the Sun's Place in the Ecliptic.
3679. But this general Table can be only a temporary One,
as ferving with any Exactneſs only a certain Time, becauſe the
Line of the Apfides A P, from which the Numbers in the fe-
cond Table begin, is not fixed with refpect to Signs of the
Ecliptic, but has a direct flow Motion through the fame at the
Rate of one Degree in 59 Years, or 1° 41′ 7″ per Century. Con-
fequently in about 30 or 40 Years this general Table will re-
quire to be renewed where any Computations of Accuracy are
concerned, but for barely fetting common Clocks by a Dial, it
may ſerve much longer.
3680. By Dr. Halley's Tables, the Pofition of this Line
AP is in 8° 42′ 52" of Cancer and Capricorn at the Begin-
ning of the Year 1764. But as the Theory of this compound
Equation is not quite fo fimple and evident as every young Ho-
rologiſt might wish, I ſhall endeavour to render it plain by an
algebraic Calculus, and to illuftrate the fame by a proper Di-
agram; both which have been hitherto wanting in Books
treating of this Subject. But to facilitate thefe Demonftations
the two following Lemmata are to be premiſed.
3681. In
Of CLOCK-WORK.
513
"
3681. In any Triangle FHS (Fig. 6.) we have FH+SH
× FH-SH = FS x 2CV; fuppofing HV perpendicular
¿o FS, and FC CS. For with HF as a Radius, on the
Point Has a Center, defcribe the Circle A BF and continue
the Side S H each Way to the Circle, and the Bafe FS to the
Circle in B. Then is AH = HF, and AS FH + SH,
=
and SE - FH - SH; and becauſe it is ES × SA = FSX.
SB (658,) and SB FV + VSFS + 2 VS; there-
fore SBFS+ VS CV, confequently 2CV =
S B.
2 CV.
Therefore FH + SH x FH SHFS X
3682. The Rectangle of the Sines of two Arches added to the Rect-
angle of their Co-fines, make a Sum equal to the Rectangle under the
Radius and Co-fine of their Difference. (Fig. 7.) Let the two
Arches be A C, and CD (=CB); their Sum AD, and
Difference A B; let CF and O F be the Sine and Co-fine of
the greater Arch A C; and let m D (= m B) and Om be thoſe
of the leffer Arch CD, or C B. Alfo let BE and OE be thoſe
of the Difference AB. Draw mn parallel to CF; and mv pa-
rallel to A O; then it is plain the Triangles OCF, Omn,
and Dmv are fimilar; therefore we have OC: OF:: Om
: On, whence OC x On OF x Om. Again, OC
: CF :: Dm mv, therefore OC × mv = CF × Dm.
Confequently OC × On + mv
×
+ CF × Dm. QE. D.
OCXOE=OFXOm
3683. Theſe Lemmas premiſed, if we look back on Fig. 4-
we fhall there find the fame Triangle FHS (as in Fig. 6.)
by drawing the Line FH, and letting fall the Perpendicular
HV. Put ACCP = a, C B = b, CS = CF = c,
SH=v, and the Co-fine of the Angle HSP = x, to the
Radius SK SN=1. Then I : * :: v :v x = SV, alſo
FH = AP — SH (769) = 2 a
and FS= 20;
wherefore FH + SH × FH - SHFS x 2CV; that
v,
is, 2 a × 2a ---- 2 v = 2c × 2 × C + x v; from whence
we have v
Q2
b2
=
SH, the Diſtance of the
a + c x
a+ cx
Uuu
3684. And
Earth from the Sun.
VOL. II.
514
INSTITUTIONS
3684. And becauſe SN² SK² = ab (3657), we fhall
have
SN 2
SH2
× j =
a x a + c x
623
xy, the Fluxion of the true
Longitude AH, or Distance from the Aphelion A. Having
thus obtained an Expreffion of the Increment of the Longitude
for the Elliptic Anomaly, and having before found it for the Ob-
liquity of the Ecliptic (3647), we can find how far both thefe
Caufes together will affect the Motion of the Earth in Right
Afcenfion, fince in that alone confifts the whole Ground and
Reafon of the Equation of Time; and if we thus unite both their
Effects we ſhall have the whole or abfolute Equation in one The-
orem, for any Value of x, or Longitude of the Earth from
either Apfis A or P.
3685 For the Fluxion of the Longitude or true Anomaly
being to that of the Equator or Motion in Right Afcenfion al-
ways as esCD: csEx R :::*, (3647.) Therefore
2
fay, As csCD: csEx R::
ХХ
C SEX R
es CD²
ахатся
63
2
xj:
ax a + c x
63
the Fluxion of the Motion in the Equa-
tor, arifing from both the Caufes of Irregularity united.
3686. Now this being compared with the equable Motion in
Right Afcenfion of the imaginary Point in the Circle G L PI,
it will eafily appear when their Difference is the greateft or
a Maximum, becauſe in that Caſe their Fluxions will be equal
ax a + c x²
(3646,) viz.
63
Хух
c s E X R
j; and con-
c s C D ²
2
fequently in that Cafe
ax a + c x
b3
cs D C
CSEX R
3687. But as this latter Part of the Equation is not of the
fame Form with the firft, it muſt be reduced to algebraic Terms;
therefore put m = Sine of K S
Sine of KS, the Diflance of the Perihelion
P from the Solftice; its Co-finen; then the Co-fine of
the Angle PSH being, its Sine will be xx; and
by the Lemma (3682) we have nx + m√ I — x x = Co-fine
of HS, or CS Sine of C or Longitude of the
Earth at H reduced to the Ecliptic GLĶI.
3688. Let
-
Of CLOCK-WORK.
515
3688. Let EN M (Fig. 8.) be the Projection of that half of
the Equator which is above the Plane of the Ecliptic, and
EQM of that half which is below it; and CD a Perpendicu-
lar thereto; then is the Right-angled Triangle CED the fame
with that in Fig. 1. Put p= Sine of the conftant Angle E,
Co-fine, and we have (by Spherics,) Radius (1)
: s.EC (nx + m√T — x²) :: p : p n x + pm√x² = Sine
of CD, whofe Square Co-fine is therefore 1-pnx+p m √ I — x x
= cs.CD; and the csEq; thefe Values fubftituted in the
a x a + c x 1 − p n x + pm vi—xx
63
and
q
Equation above, give
2
I
9
This Equation reduced, gives the Value of x, or Quantity of
the Angle HSP.
3689. While the Angle HSP is acute, x will be affirmative,
or + x; but when obtufe it will be negative, or X. Alfo
while His in the Semi-ellipfis ALP, m will be affirmative or
+m; but when H is in the other Half P1A it is negative, or
m;
and the Equation there becomes
a x a + c x²
63
p n x
pm Vi
I
xx
9
3690. From all which it is evident, that this Compound Equa-
tion of Time, as it arifes from the Sums and Differences of the
two fingle Equations must be of a different Value in different
Parts of the Orbit; and that there will be four of thoſe Maxima
in all, viz. two arifing from the Sums, while x is pofitive;
and two from the Differences of the fingle Equations, when x
is negative, or -x.
X.
3691. But to make this Matter yet plainer, let Athe
Equation for the Earth's Elliptic Orbit, and B≈ that for the
Obliquity of the Eclipic; then all the Time the Earth H is defcend-
ing from A to P, the former is negative, or A (3663.)
And in afcending from P to A, it will be affirmative or + A.
Alfo in the 1ft and 3d Quarters of the Ecliptic we have — B;
but in the 2d and 4th it is + B (3653, 3657.)
3692. Now 'tis evident, in the Diagram (Fig. 8.) that the
firſt Quadrant of the Ecliptic wholly coincides with the
Sewi-
Uuu 2
516
INSTITUTIONS
Semi-Ellipfe ALP, and therefore the Maximum Equation will
here arife from A+ B, which, as they are both Negative, thews this
Equation is fo too, or that it must be taken from the Time by
the Dial, to have the mean Time by the Watch or Clock,
which is then too flow. This Equation is 16′ 13″ and is when
the Earth is in 8 : 10°, or the Sun in m: 10°, on November 28.
3693. Again, almoſt the whole of the 2d Quadrant of the
Ecliptic is in the other Semi- Ellipfe PIA; confequently the
two fimple Equations here being both pofitive their Sum A + B
will be ſo too, or muſt be added to the Solar Time to get the
Mean by the Watch, which is here too faft. This Maximum
amounts to 14′ 49″ on February 10th.
PAN
When the Earth is in
N: 21° 35′, or the Sum in ≈ 21° : 35′•
NA
3694. The third Quadrant of the Ecliptic is wholly on the
fame Side with the Semi-Ellipfe AIP, and the Earth will enter
it before the third Maximum Equation; therefore B will be
Negative and that Equation will be A-B4' 5", to be
fubducted from the folar Time-by the Dial, becauſe now B is
greater than A, and the Clocks are too flow, again. This hap-
pens on May 15th, when the Earth is in m: 24° 18′ or the
Sun in 8 24° 18′.
3695. Laftly, the laft Quadrant of the Ecliptic is nearly the
whole of it on the fame Side of the Line of the Apfides A P
with the Semi-ellipfe A LP; here B will be pofitive, or + B,
and A negative, or
greatest, will be
A, and therefore, the Equation, when
BA 5' 55"; and becauſe B is here
=
alfo greater than A, therefore the Equation is to be added to
the apparent Time by the Dial to have the true or equal Time
by the Clock or Watch which is now too fast, or before the
Dial. This Maximum happens on July 26, the Earth being
in 3° of, and the Sun in 3° of 2.
3696. If the Reader attentively confiders the feveral particu-
lar Circumftances and Relations of thefe two fimple Equations
he will fee how this Compound Equation will gradually arife by
Addition and Subtraction, and increaſe to its various Maxima, and
then alternately decreafe to nothing, according as the Numbers
appear in the following Table, which is here adjuſted for ready
Ufe to the Signs and Degrees of the Ecliptic.
Of CLOCK-WORK.
517
A Table of the Compound Equation of Time depending on the
Sun's Place.
Signs ရာ
ŏ
II
Add.
Sub. Sub. Add.
Add. Add.
Deg. "/
//
//
97
7 40 I
93 56 I
6 5 49 2
14
17
21
27
1 2
I
233
52 I
19 5
51 I
58
I
36 3
47 I
33 5
52 I
41
36 43 I
49 3
43
I
47 5
53 I
24
46 24 2
I
3
38 2
0 $
53 I
7
56 5 2
13 3
32 2
13 5
521 0
50
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2313
25 2
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32
75
27 2
34 3
18 2
38 5
491 0
13
8
5 812
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5 5
47
5
94 49 2
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3 3
45
440
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303
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44
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56 3
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462
46 2
301 I
9 3 56 I 65 49 2
14 7
42
The Equations following the Sign + to be added, and the Sign-
to be fubducted from the apparent Time, to get the Mean Time.
45 7
307 21
518
INSTITUTIONS
A Table of the Compound Equation of Time depend-
ing on the Sun's Place.
दा
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↑
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Sub.
Sub.
Sub.
Sub. Add.
Add.
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7 42 15 3213 29 I
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40
The Equations following the Sign + to be added, and the Sign
to be fubducted from the apparent Time, to get the Mean Time.
Of CLOCK-WORK.
519
}
CHA P. XXVIII.
Of the best METHODS of drawing a MERIDIAN LINE
by Concentric CIRCLES, by the HYPERBOLA, &C.
The Theory of a New EQUAL ALTITUDE IN-
STRUMENT for that Purpofe.
3698.
AS
S CLOCKS and DIALS are of a like Nature and
Ufe, fo they are mutually fubfervient to each
other, and ferve to correct one another, when either fhall chance
to be at Fault; but to this End it muſt be ſuppoſed that they
are originally each of them properly conftructed, and the Dial,
particularly, fixed in a true Pofition. And as this is fo important
a Point, we here propofe to give fuch Directions and Precau-
tions as are neceffary to be obſerved in that Affair.
3699. The first Thing to be obſerved in fixing a Horizontal
Dial is, that the Surface of the Pedaftal be truly level or hori-
zontal, and this must be examined and thoroughly rectified by a
Plumb-line, or Spirit Level. The Reafon of this Injunction is
becauſe the Gnomon of the Dial cannot be parallel to the Axis of
the Earth without fuch an horizontal Pofition; as we have fhewn
when we treated of Dialling.
3700. The fecond Thing neceffary is to place the Meridian
of the Dial truly North and South, or exactly in the Meridian of the
Place; for this Purpoſe it will not be ſufficient to ſet it by a Mag-
netical Needle in a Rectangular Box, neither by a Clock, Watch,
or any other Dial; but quite independently of any other Time-
Piece, by a true MERIDIAN LINE. This is upon Suppofition
the Dial is a very good one, and intended to go very correctly.
3701. As a MERIDIAN LINE is of the moſt extenfive Uſe,
we ſhall here give ſome of the principal Methods for drawing
The moſt fimple and practical of theſe is the following.
Upon a Point A (properly chofen) as a Center, defcribe feveral
concentric Circles as NH E, OID, PKC; and in the faid
Center A fix a ſtrong Wire, as truly upright or perpendicular
to the Horizon as poffible, the Height of it fix or eight Inches;
one.
then
520
INSTITUTIONS
then obferving, in the Forenoon, very nicely where the End of
the Shadow touches each Circle, and there make fine Marks, as
fuppofe in the Points C, D, E ; and do the fame for the After-
noon Shadows, as at H, I, K. Then bifect the Arch EH in
F, and the Arch CK in G; and through the Points A, F, G,
draw the Right Line A FG, and it fhall be the Meridian-Line
required.
3702. It has been formerly fhewn (1766,) that the Curve
LR M defcribed by the End of the Shadow is an Hyperbola (in
any Latitude leſs than 66 Degrees,) whofe Vertex is R and
its Axis the Meridian Line PR G. Further, it is eafy to con-.
ceive, that the Sun in the Horizon, projects the Shadow to an
infinite Diſtance, and there the Curve of the Hyberbola coin-
cides with the Afymtote; and that this is at a Point in the Ho-
rizon juft oppoſite to the Sun; confequently the infinite Line
joining theſe two oppofite Points will be an Afymptote to the hy-
perbolic Curve of the Shadow defcribed on the given Horizon-
tal Plane, and will contain with the Meridian of the Place an
Angle equal to the Co-Amplitude for that Day.
{
3703. Therefore from the given LATITUDE of the Place,
and the DECLINATION and AMPLITUDE of the Sun, the Hy-
perbola for the given Day may be defcribed, and the Meridian
Line thereby found by a fingle Obfervation of the Shadow; and
verified by any Number of them you pleaſe. In order to this
Practice, it is to be confidered, that on any Day in the Summer
or Winter when the Sun's Declination is the fame, there will
be the fame or equal Hyperbolas defcribed by the Shadow,
whofe Vertices will be determined by the Shadow of the Me-
ridian Altitude on each Day.
AD
a
3704. Thus, for Example: Let A B be a Gnomon of
given Height, and let A CB be the Meridian Altitude of the
Sun, equal to the Sum of the Co-Latitude and Declination, whence
the Diſtance AC is known. Again, the Angle A D B, on
the Winter Day, is the Meridian Altitude, equal to the Diffe-
rence of the Co- Latitude and Declination; and therefore A D
is known; from which take A C, and the Remainder CD is
the Tranfverfe Axis of the Hyperbola's.
蹁
​3705. Bifect CD in E, and that will be the Center of the
Hyperbola's; thro' which draw two Right Lines GP and L M
making
Of Cioek-WORK.
521
making with the Line A D Angles GEA and MED equal to
the Co-Amplitude and they will be the Afymptotes to the Hyperbolas.
3706. On the Vertical Point C erect the Perpendicular C K
meeting the Afymptote EG in K, and CK will be the Semi-
Conjugate of thefe Hyperbolas; therefore make EF and EQ
equal to EK, and the Points F and Q will be the Foci of the
Hyperbolas, (768.)
3707. Thus having the Diameters and Faci, the Curves HCI
and NDO may be readily defcribed, either by the Inftrument
in the Plate of the Fufee, or by finding a Number of Points S in
the Curve which is very eafy to do, becauſe it will be every
where QSFS CD (769;) and fo for every Point S
you have QS CD + FS, or FS QS
=
CD, by
which any Number of thoſe Points are found, and the Curve of
the Hyperbola drawn through them as required.
3708. Having thus conftructed the Hyperbola HCI for the
given Day, let the Paper on which it is drawn, be fo placed on
the Plane, that the Point A may exactly coincide with the
fmall Hole in which the Wire. or Pin was fixed, and then fa
moved or adjuſted, that the Length of any obſerved Shadow A R
taken in a Pair of fine Compaffes may be applied therein, that
is, when one Point of the Compaffes is in A the other may fall!
precifely in the Curve of the Hyperbola at R; then will the
Axis of the Hyperbolas F E Q be in the true Meridian of the.
Place. And by obferving a Number of thefe Shadows, its Po-
fition may be verified to great Exactneſs.
3709. The Time moſt proper for this Procefs is at or near
the Solftices, and the Meridian determined at or about the Summer
Solftice, will be moft exact, as the Shadows A R are then fhort-
eft and beft defined at the Ends. The Reafon of chufing thefe
Times of the Year, is becauſe the Hyperbolas have now the
greateſt Degree of Curvature, from which they degenerate gra-
dually till, in the Equinoctial Day, they become a Right
Line, (1764.)
3710. But when the Meridian Ling is to be determined of a
confiderable Length, other Methods may be more readily ap-
plied; that by an Equal- Altitude-Teleſcope is certainly a very good
one, but the Expence of fuch an Inftrument, the Skill of the
Perſon,
VOL. II.
Xxx
522
INSTITUTIONS
}
1
!
Perfon, and Difficulty of the Performance, will confine this
Method to very few Hands.
1.
3711. The Method of Tycho Brahe was much efteemed by
himſelf and others in his Time; it confifted in having an In-
ftrument fo placed on a Plane that he could eaſily thereby obferve
the greatest Elongation of any circumpolar Star towards the Eaft
and Weft; confequently the End of this Inftrument would de-
fcribe an Arch, the Half whereof would meafure thefe equal
Elongations from the Meridian, which therefore became known
or determined; viz. by bifecting that Arch, and drawing a Line
through that Bifection and the Center on which the Inftrument
moved. But this Method is more fuited to Aftronomers than
for common Ufe.
{
·
3712. Dr. DERHAM's Meridian Inftrument is well known as
a very ufeful One for drawing a Meridian Line with tolerable
Exactnefs. But I no not think any Method fo good, (that is,
fo exact,) as thofe which have their Points and Centers upon
(and not above) the Plane on which the faid Line is to be drawn,
and which alſo are entirely independent of Shadotus, becauſe in
great Lengths Shadows always become penumbral, or ill defined,
and therefore not fit for an Affair of fuch Precifion.
age
3713. I fhall therefore propofe a new Method of conſtruct-
ing an equal Altitude Inftrument that moves immediately on the
Flane,, or Floor, or Pavement where the Meridian Line is re-
quired to be drawn, and which will give as little Trouble, and
as great Accuracy, as can be expected in fuch an Operation.
This Inftrument confifts of a Ruler ABC made of Mahogony,
or other Wood, not fubject to warp; At Alisa fmall Hole
in a Piece of Brafs in which the Ruler is moveable about-a-
flender fteel Pin. The central Line AC is fuppofed to be
drawn with great Care and very ftrait; the Part B C is a fidu²
cial Edge of Ivory, exactly coinciding with the faid Line AC;
and at D is a fall, black Line, which, upon the Motion of the
Ruler, will defcribe any Arch of a Circle, as DEF 10 am-1
3714. Upon this Piece of Wood for Metal) there is placed
an Inftrument for taking equal Altitudes of the Sun, con-
ftructed in the following Manner. GH is the Bafis of it,
and from the Center N move the two Indices, NP and NL,"
upon two Arches of a Circle, viz. I K fixed to the immoveable
1
A
1
Bafe
'
Of CLOCK-WORK.
523
Baſe GH; and the Arch MON, fixed to the Alidade or Index
LN; and confequently moveable with it. Upon the Index
NP, is a Sight-Vane at R, with Croſs-Hairs in its Perforation,
and a Shadow-Vane NQ is fixed at the Center N, exactly pa-
rallel to the former.
3715. The peculiar_Artifice of this Conftruction is, that in a
Small Form, it performs the Office of a large Quadrant. For the
Arch MO is divided into every 10 Degrees only, and therefore
is not required to be large; the Index PN is placed to that Di-¯
vifion which is next lefs than the Sun's Altitude; and then it is
evident, the Remainder in Degrees and Minutes is meaſured by
the Index L N, and its Vernier Plate at L, upon the Arch IK,
which is divided into 15 or 20 Degrees at moft, and therefore
but of a ſhort Length.
3716. This Inftrument properly fixed upon the large Alidade
AC (as in Fig. 3.) it is plain, that if in the Forenoon it has
that Situation AC when the Indices are fo adjuſted that the Sha-
dows of the Crofs-Hairs, in the Vane R, fall precifely on the
black Lines croffing in the Center of the Vane QN, and a fine
Point or Mark be made at D; and then, in the Afternoon, the
faid Alidade AC be brought to fuch a Situation, that the central
Line AC be that of AF, when the Shadows of the Crols Hairs
exactly coincide again with the Lines on the Vane Q, I ſay
then, it follows, that if the Arch DE be bifected in E, and the
Line A E be drawn, it fhall be the Meridian Line required.
3717. This is all fo plain as to want no farther Explication ;
and I find it fo eafy in Practice to take any Altitudes, and to
meaſure any Angles in general with this compendious Inftru-
ment, that I can venture to recommend it to the ingenious practi-
cal Geometrician, as worthy of his Notice. It may be furniſh-
ed with a Spirit Level to place it horizontally, and if a Lens of
a proper focal Diſtance be placed in the Vane R, it will not
only fhew the Height of the Sun at any Time to a Minute, but
allo that of any other Object by its Picture, on the Vane NQ,
properly darkened. As HADLEY's Quadrants are capable of
the fame Improvement and concife Form, I fhall give a fur-
ther Account of them at another Time; only observing, they
may be applied in the above Method alfo for drawing a Meridian
Line.
X X X 2
CHA P.
524
INSTITUTIONS
$
1
ار
CHA P. XXIX.
The THEORY and CONSTRUCTION of an ELLIPTI-
CAL, CIRCULAR, and DIAMETRAL DIAL, which
by Means of a common HORIZONTAL DIAL, will
find the true MERIDIAN.
3718. WE
¿
६
ม
7E have fhewn the beft Methods of drawing a Meri-
dian Line, by which a Dial may be fet truly upon
a horizontal Plane. We fhall next fhew how Dials may be
conftructed in fuch a Manner as to fet themfelves, or to find their
own Meridian by Means of a common Horizontal Dial only. One
of this Sort is called an ELLIPTIC DIAL, the other a CIR-
CULAR DIAL, becaufe in the firft the Hours are the unequal
Divifions of an Ellipfis, and in the latter they are the equal Di-
vifions of a Circle; in both which Refpects they differ from com-
mon horizontal Dials, where the Hours are the unequal Divifion
of a Circle, as we have fhewn (1772).
3719. The Theory of the Elliptic DIAL we fhall here deliver
with all the Perfpicuity we can from a New, and (we preſume)
moft natural Projection of the Sphere orthographically, in what is
ufually called the Analemma, explained form 1703 to 1720.
Therefore (in Fig. 1.*) let HZOY be the general Meridian ;
EQ, the Equator; PX, the Hour Circle of Six; ZY, the
prime Vertical; and HO, the Horizon. AF, LK, two Pa-
rallels of Declination.
3720. Then in our INSTITUTES of Gnomonical Perspective, it
has been fhewn (1738, &c.) that the Sun having any Declina
tion EG, will, by its Ray GC, defcribe that Day a Cone ACF,
whofe Bafe is the Parallel AF upon the Surface of the Sphere,
whole Axis is that of the Sphere PC, its Vertex C, and Height
CB. This Cone we fball (for Diftinction's Sake) call the Ra-
dial Coned sign
•
1
3721. Then if the Axis ZCY of the prime Vertical be con-
fidered as an opake Line, it is plain, that Point of it at C, will
to intercept the Ray GC, and thereby produce a lineal Shadow
CK in the fame Direction, which Shadow will therefore de-
fcribe
*In the Plate entituled the Theory of Elliptical, Circular, and Dia-
netral Dials.
:
of CLOCK-WORK.
525
1
fcribe an equal and oppofite Cone LCK, whofe Bafe in the
Surface of the Sphere will be the oppofite equal Parallel LK.
This therefore we fhall call the umbral Cone.
JOTA
3722.
Then as the 24 Hour-circles divide the Equa-
for EQ, and all its Parallels AF, LK, into fo many equal
Parts, it is evident the Ray GC in applying to thofe Circles fe-
verally, will affign the Pofitions of 24. Lines dividing the Sur-
face of the Radial Cone into 24 equal Parts. And alfo, of
Neceffity, the Shadow CK will at the fame Time give the
Pofition of 24 correfponding Lines on the Surface of the umbral
Cone; which therefore may be confidered as Hour Lines on the
Surfaces of theſe Cones, interfecting each other in the common
Vertex C.
1
}
3723. As the Sun's Declination EA decreafes, the circular Bafes
AF and LK of theſe Cones will encreaſe, till at Length the Sun
being the Equator at E, both thofe Bafes coincide, and the
Cones degenerate into the Plane of the Equator EQ; where
thofe 24 Hour-Lines alfo feverally unite and form the polar Dial,
defcribed (1759) whofe Gnomon is P C the Axis of the Sphere.
3724. It is farther evident, that, in the Cafe of the Sun's be-
ing in the Equator, the Hours will be indicated by twr Shadows
united into one, projected from the two opake Gnomons PC and
ZC, on the Plane and graduated Perimeter of the Equator.
For whatever Hour-Circle the Sun may be upon, the Shadow of
the Stile CP (as being its Axis) will be in the Plane of that
Circle (1751). For the fame Reaſon, the vertical Circle paffing
through the Sun at the fame Time, will have the Shadow of its
Axis ZC projected in its own Plane; therefore, becauſe the
Planes of both of thofe Circles interfect each other, and alfo the
Plane of the Equator, in one common Line the Shadows of
both thofe Gnomons will, of Courfe, I be projected into that
Line, which will be the Shadow pointing out the preſent Mo-
ment on the Dial. Indeed, propelly speakingjathe Shadow in
the Cafe now ſpecified, is'only that of the Point Gin the Inter-
fection of the two Gnomons. ~ zixA oni li agdς10:
3725. Again, ſuppoſe the Sun has any Degree of Declina-
tion EA, the Paralle AF will be its Path forthat Day; now
ſuppoſe it in any Hour-Circle, as that of Six PCX, in the Point
B; and let Z BY be the Vertical paffing through the Center of
the
s j
}
526
INSTITUTIONS
the Sun: Then it is eafy to underſtand, that the Ray proceeding
from the Sun's Center at B, will defcribe the Line BC on the
Side of the radial Cone ACF, becauſe that is the common In-
terfection of the two Planes on the Side of that Cone.
Here the
faid Ray is obftructed by the opake Particle C in the Interfection.
of the two Gnomons or Wires, ZY and P X, and by that
Means the Shadow of that Point will be a Line on the Umbral
Cone LCK, exactly in the fame Direction C R, and therefore
on the contrary Side.
3726. Hence then, if the Bafe LRK of the Umbral Cone
were a Circle divided into 24 Hours, and the Cone itſelf were
away; then it is plain, the Point R will be the Hour of Six on
the Weſtern Side of that Circle, if B be the Hour of Six in the
Morning, upon that Point will the Shadow of the Interfection
Cprecifely fall. If we ſuppoſe not only the Perimeter, but alſo
the Plane of the Bafe (or Parallel) LK to be there placed, then
would the two Axes P X and ZY pafs through it in the Points.
R and V, of which the former is the Center. And on that
Plane would be projected two Shadows, one of the Part RC,
and the other of the Part CV, of the faid two Axes.
3727. And in this Manner the Plane of any particular Paral-
lel LK may be confidered as a Dial-plane, having two Stiles or
Gnomons RC, V C, whofe two Shadows conftantly interfect
each other on the Hour-circle in its graduated Circumference,
and thereby fhew, the Hour for that particular Day.
3728. But as the Periphery of a Parallel is a variable Quanti-
ty, increafing or decreafing daily, it can by no Means anſwer
the Intention of an Hour-circle, which ought to be a fixed or de-
terminate Thing; and therefore if inftead of the Parallel of De-
clination, we fubftitute a moveable Equator, it will answer the
Purpoſe of an univerſal Dial for finding the Meridian in any
Latitude, and confequently, independent of the Magnetic
Needle, will fet itſelf, "or fhew the true Hour of the Day; as
will thus eafily appear."
3729. Let the given Parallel of Declination be A F, and draw
IG and M N Tangents to the Equator EQ; and then ſuppoſe a
moveable Circle, juft equal to the Equator, and divided into 24
Hours, Minutes, &c. were to flide up and down upon thefe
Tangent Lines of the Sphere, fo as readily to be placed in a
Situ-
Of CLOCK-WÓR K.
527
Situation fimilar to that of the Parallel LK in the Umbral
Cone, correfponding to the Parallel AF of the Radial Cone.
Such a moveable Equator will be reprefented by the dotted Line
or Circle I W N; for the Lines CL, CA, CK, CF, conti-
nued out, will cut the Tangents in I, G, and N, M; there-
fore EI CW = QN, confequently the Plane of the
moveable Equator IN, is parallel to the Plane of the Parallel
LK, and the Peripheries of each are parallel Circles of the
fame Cone ICN; therefore they are divided fimilarly by the
Shadows of the two Gnomons WC, and UC, or it will be
every where RV: VK:: WU: UN. And thus the Hour
will be fhewn truly, and alike in each.
3730. If therefore a Dial be conftructed according to this Ana
lemma, it will be the Plane of this moveable Equator; and fhew
both the Hour, and the Meridian-Line at the fame Time: For the
equational Hour-circle being rectified to the Tangent of the Sun's
Declination (for any given Day) on the Lines GI, MN, the
Hour fhewn by the Gnomon W C (as being Part of the Axis
of the Sphere) can never be true, but when HZO is in the
Plane of the Meridian; but the Hour fhewn by the Gnomon
UC, muft alfo in that Cafe be true; therefore, when the In-
terfection of the Shadows of thefe two Gnomons fall precifely
on the Hour-circle, then both the Hour and Pofition of the Meri-
dian-Line will be given by the Dial.
3731. The fame Thing will be effected by a fixed Equator
EQ, and a moveable Gnomon Z C; for while the Equator is
carried from EQ to IN, the Index or Axis ZI, is moved
through the Plane of the Equator from W to U; and therefore,
if we take CS = WU, and from the Point S draw Sa parallek
to CZ, and alfo drawing Gb M, we have the two Gnomons
Cb, and Sb exactly equal to the former two, WC and UC, and
alike fituated to the Plane of the fixed Equator EQ, as they
were to the moveable One IN. And therefore the Hour will be
fhewn in this Cafe, the fame as in the other.
3732. Lastly, the fame Purpoſe will be anſwered by a Gno-
mon Db, moveable upon a horizontal Line HO, through the
Space CD, fince the Point (b) in the Interfection, gives the
Hour, and is the fame in both Cafes.
3733.
528
INSTITUTIONS
3733. If therefore upon the Plane of the Horizon HO in any
Latitude EZ PO, the Equator EQ be projected into
an Ellipfis, and it be properly divided in every 15th Degree by
a Table of Sines as directed (1718) thefe Divifions will be the
Hours, and then an upright Stile Da, moveable in the Hour
Line of XII, or Meridian HO, will conftantly fhew the true
Hour on this Elliptic Equator, as before it did on the Circular
One. Becaufe in the Projection there is a perfect Similarity of
all the Parts in One, to all the correfponding Parts of the other,
as we have fhewed (1715).
3734. In this Cafe, the Shadows of the two Gnomons Db,
and Cb, require two different Hour-circles, and fo will be no
longer one Dial, but two of a diftin&t Nature. Thus let Fig. 2.
be the orthog. Projection on the Plane of the Horizon FOM;
and let A QB be the Ellipfis into which the Equator is project-
ed, and divided into its proper Hours; alfo let P be the project-
ed North-pole, Z the Zenith, and D the Foot of the perpen-
dicular Stile Sb, for any Declination of the Sun EG in Fig. 1.
Then fuppofe the Shadow of that Stile falls upon the Hour XI,
it will be projected into the Line DXI, and this must be the
Hour-Line of XI on the horizontal Plane, as is evident from
the Nature of the Projection; for the Space ZD here, is equal
to CD there; and D is the Projection of the Foot S of the
faid Stile in the Plane of the Equator.
-
3735. But with Refpect to the Stile Cb (in Fig. 1.) its Sha-
dow here cannot pass through the Point (or Hour of) XI in the
Ellipfis A QB; for the Foot C being in the Center Z of the
horizontal Plane, and the Hour-circle of XI cutting the Hori-
zon in E, making EO= 11° : 51′ (1772) it is plain, ZE
will be the Hour Line of XI on the horizontal Plane, or the
Shadow of the Stile Cb. And drawing the Line Z XI, there
is formed the fpherical right-angled Triangle XI Z Q, wherein
ZQ= the Latitude, and XI Q the Hour, being given, the
Angle at Z will be found = 18° : 55′, which is 7° : 4′ greater
than the Angle EZO. Confequently the two Gnomons will,
in this Projection, require two different Curves for the Hours,
viz. the Ellipfis A QB, and the Circle FO M, which therefore
conſtitute two different Dials.
3736.
Of CLOCK-WORK.
529
}
3736. When EG (EI) is the Tangent of the Sun's
greater Declination EA, then AF and LK are the two Tro-
pics; and CD = ZD (Fig. 2.) is the greateſt Space through
which the perpendicular Index Da is to be moved in the hori-
zontal or meridian Line HO, which is to be graduated into
23° : 30', to correſpond to the Sun's Declination in this Man-
ner. In the Triangle & CD, we have Cb Tangent of 23°
: 30′, and the Angle at b ZCP Co Latitude. And as
Radius: s. C ¿D :: C¿: CD; therefore fince C b is the Tan-
Cb
gent of 23° 30′ to the Radius CP, CD will be the Tangent
of the fame Arch to the Sine of ZP made Radius, or applied in
the Sector from 45 to 45, upon the Lines of Tangents. Thus
ZD and ZH is graduated for the Summer and Winter Half-
Year, by which, to fet the Index or Gnomon for the Declina-
tion of the Sun, proper to the given Day.
+
:
Cb
3737. Therefore it is evident, that when the fame Minute, is
fhewn by the Elliptic and Horizontal Dial, the Hour Line of
XII, viz. HO, will be in the Plane of the Meridian of the
Place, and confequently a Right Line drawn parallel to it will
be a true Meridian-Line. And thus fuch a double Dial will at
all Times fet itself without any Affiſtance of the Magneti-
cal Needle, or any other Meridian Line than its own.
靠
​1
CHAP. XXX.
The THEORY and CONSTRUCTION of CIRCULAR
A and DIAMETRAL DIALS, which, with a
Horizontal DIAL, find a true MERIDIAN LINE.
3738. BEfides the foregoing Elliptic Dial, there is yet another
Sort which by Means of a common Harizonal Dial,
will fet itſelf, or fhew the Poſition of the Meridian Line without
the help of the Needle; and by which, of Courſe, the Variation
VOL. II.
YYY
of
530
INSTITUTIONS
of the Needle is alſo eaſily diſcovered in the given Place. But
this Dial is of a circular Form, and what may ſeem a little ſtrange,
is, that the Hours on this Circle are all equidiftant, as in the
Equator itſelf. The whole Artifice or Conſtruction of this
Dial depends upon a due Confideration of the Properties and
different Sections of a Scalenous Cylinder which will be eafy to
underfland after what we have premiſed of the Scalenous Cone
(1508,) and its Subcontrary Sections (1510, &c.)
3739. First then let it be confidered that if Tangent-lines
were drawn through the feveral Points of the Equator, (or any
Circle) and at right Angles to its Plane, they would conftitute
a right Cylinder, whofe Section by a Plane perpendicular to its
Axis will be a Circle, but in all other Cafes an Ellipfis, as we
have fhewn (1715.)
3740. Secondly; if thefe Tangent Lines are now fuppofed
gradually to incline or lean towards the Plane of the Equator,
then they form what we call a Scalenous Cylinder, being com-
preffed by this Inclination of the Lines into a flattiſh Form;
and whofe Section by a Plane perpendicular to its Axis will be
an Ellipfis, one Diameter of the right Cylinder being contracted
gradually into a lefs Length, while the other continues the
fame; alfo all other Sections will be Ellipfes, except two, viz.
one Parallel to the Equator (or original Circle) and the other
by a Plane in a Subcontrary Pofition.
كم
3741. Thirdly; theſe Tangent Lines conftantly inclining to
the Plane of the Equator will at laft perfectly coincide with it,
or the Scalene Cylinder now degenerates into a Plane coincident
with that of the Equator, which, in this Cafe, is projected by
theſe Lines or Rays into a frait Line, viz. one of its Diameters.
And this Confideration lays the Foundation of another Species
of Dialling, viz. Rectilineal, which we fhall enlarge upon by-
and-by.
3742. But to come directly to the Point, and for Illuſtration
of what has been faid, let ABFD be the Scalenous Cylinder
whofe Axis is IG, perpendicular to which let the Line RS be
drawn; this will be the ſhorteſt Diameter of the Elliptic Section
and ſuppoſe the longeſt Diameter be HO, which alfo is conceiv-
ed to pass through the common Interfection C of the Axis, and
the other Diameter RS (See Fig. 3.) Upon the Point C,
with
Of CLOCK-WORK.
531
with the Radius CH defcribe the Circle H ZOY, cutting the
flatted Sides of the Cylinder in the Points E, H, and O, Q;
and draw the Line ECQ.
3742, Then it is evident, that fince HỌ is equal to the
longeſt Diameter of the Elliptic Section by a Plane through
R S, it is therefore equal to the Diameter of the original Circle or
Baſe of the inclined Cylinder A BF D, and alfo parallel to the
Diameter through whofe Extremities the Lines A B, F D paſs.
Confequently, the Section of the Cylinder by a Plane (through
HO) parallel to the Baſe will be alſo a Circle, equal to the Cir-
cular Bafe.
3743. And becauſe the Angles REC and SQC are equal,
as alſo R HC = SOC, (by reafon of parallel Lines AB, DF
interfecting the equal Lines HO and EQ) therefore the Section
of the Cylinder through E Q will be fubcontrary to that through
HO, (1509,) and confequently alſo a Circle, agreeable to (1510.)
3744. Now. if FPQX be confidered as a Meridian of the
Sphere, which is orthographically projected thereon, and EQ the
Equator; we may then look upon HO as the Horizon of fome
Place Z, whofe Latitude is EZ, and whofe Co-latitude ZP
is biſected by the Axis of the Cylinder IG, becauſe the Section
through RS is ſuppoſed to be perpendicular thereto, and con-
fequently equally divides the Angle ECH in the Point K, or
OCQ in the Point L.
3745. Then (from what we demonftrated in the laſt Chap-
ter) it is plain, that if KL be confidered as a Plane having a
Perpendicular Stile or Gnomon CN, the Projection of the
Equator E Q thereon by Rays A B, FD, parallel to CN,
will be an Ellipfes, whofe ſhorteſt Diameter is R S, and longeft
KL. And that as the Sun advances to the Parallel T V,
the perpendicular Stile being parallelly removed from C to (a)
will fhew the Hour on that Ellipfis; and if TV be confidered
as the Tropic of Cancer, then Ca will be the Length of the
Summer Half of the Zodiac on the Dial-plate.
3
fo
3746. Suppoſe now the Plane KL were taken away, but
the Gnomon NC to remain; then it is evident the Rays A B,
GI, DF, will project the Equator EQ into a Circle on the
horizontal Plane HO (1510.) And the faid Rays will divide
the Horizontal Circle in the fame Manner or Ratio, as they do
Y y y 2
the
532
INSTITUTIONS
the Equator itſelf. For let (eh) be any one of thofe Rays,
which paffing through the Point (g) of the Equator, projects it
into the Point (k) in the horizontal Circle HO; then becauſe
eh is parallel to A B, we have CE Cg :: CH: Ck.
Wherefore the Horizontal Circle will be divided into 24 equal
Parts by the Rays which paſs through every 15th Degree of the
Equator.
:
3747. Therefore 'tis evident, that if upon the Horizon HO,
a Circle be drawn, and divided into 24 equal Parts; thofe Di-
viſions will be the Hours truly marked out by the Shadow of a
Stile or Gnomon CN elevated above the Plane thereof in the
Angle NCO PO+PN the Sum of the Latitude, and
half the Co-Latitude of the Place Z. So that at London, for Ex-
ample, the Angle NCO 51° 30′ + 19° 15′ = 70° 45°.
3748. 'Tis farther evident, that this Stile CN is a moveable
One, as it originally and properly belongs to the Elliptic Dial on
the Plane K L (3745;) where its parallel Motion, Northward,
is Ca; but this is now to be eſtimated in the Plane of the Hori-
zon from C towards O, where it will be expreffed by the Line
Cc whofe Value is thus found. In the Triangle b Ca (right-
angled at a) we have Rad. (CP) Sine of Cba (= OCL)
:: Cb: Ca. Again, in the fimilar right angled Triangles
Ca, OCS, and fCL, we have Ca: Cc:: CL (Rad.)
Ccx Rad.
: Cf; therefore Ca=
O S x C b
Rad.
Cb: C: Rad.: OS x Cf:: Rad. :
; therefore
Cf
OS × Cf
=Tan-
Rad.
gent of LCf. Whence the Value of Ce is known.
3749. But becauſe it will be moft convenient to ſet the Index
CN by the Sun's Declination, therefore the Semi-zodiac C c
muſt be confidered as a Tangent of 23° 30 to fome Radius,
which from the Analogy above is evidently the Line Lf, viz.
the natural Tangent of the Half Co-Latitude of the Place z. There-
fore by applying the Line Lf from 45 to 45 in Lines of Tan-
gents on the Sector for Radius; you may then take the paral-
lel Diſtance from 23° 30′ to 23° 30′, and that will give the
Length of Ce from the Center C each Way in the Meridian
Line for the Summer and Winter half Year. And thus your
Circular Dial will be compleated, as in Fig-4.
*
3750. There
Of CLOCK-WORK.
533
3750. There are many other Particulars obfervable in this
Sort of Dials, but we muſt pafs them by at prefent as not ne-
ceffary to our Purpoſe, which is only to fhew hoto by two Dials of
a different Conſtruction, and placed upon one Right Line, a true
Meridian Line may be drawn, and the Hour heron, without any
other Affiftance. This we have fhewn may be done by a com-
mon horizontal Dial combined with the Elliptic, and Circular Ones;
we ſhall next fhew the Nature and Conftruction of a Rectilineal
or DIAMETRAL DIAL for the fame Purpoſe.
บ
3751. Suppoſe (Fig. 3.) that the Tangent Rays AB, GI,
DF, were (in the firſt Place) all at right Angles to the Plane
of the Equator EQ; an infinite Number of them would, in
this Cafe, conſtitute a perfect Cylinder whoſe Section, perpendi-
cular to the Axis would every where be a Circle, as being paral-
lel to the circular Plane of the Equator.
3752. Let the Rays now gradually deviate from that rectan-
gular Pofitition to one more and more inclined to the faid Plane
of the Equator in a Direction from P towards E. Then it is
evident, all that Time the Radial Cylinder will become more
and more flatted or ſcalenous; and every perpendicular Section
RS will become of a leſs and leſs Diameter; and, of Courfe,
the Hour-Circle in the Equator degenerates into an Ellipfis, and
becomes more and more fo; 'till at length, the Rays all ar-
rive at the Plane of the Equator, and thence the fcalenous Cylin-
der itſelf becomes a Plane, and coincides with that of the Equa-
tor; alſo the horary Ellipfis now Collapſes, and both Sides unite
in a Right Line which is the Diameter of a greater Circle PX
paffing through the Poles.
3753 During this fuppofed Motion of the Tangent Rays
the Dial Plane KL has defcribed the Quadrant EX, and its
Gnomon or Index CN the Quadrant PE, and its Hour- Circle
has paffed thro' every Degree of Elliptic Curvature from a Cir-
cle to that of a Right Line; and the Pofition of its Gnomon,
which at firſt was in the Axis of the World CP, is now CE
in the Plane of the Equator.
3754. This Pofition of the Dial-Plane PX and its Gnomon
CE fits it for An univerſal Right-lined DIAL in every Latitude ;
for from the Nature of the Analemma, it is evident, that the
24 Hour-Points of the Equator are now projected into the El-
liptic
534
INSTITUTIONS
liptic Line (as it may be called) or the Eaſt and Weft Diameter
of the Plane DX. Which Line is equally a Diameter in the
Plane of any great Circle, as that, for Inftance, of the Horizon
HO, becauſe it is the Common Section of them all, and fo is
⚫qually as much in one as in another.
3755. Therefore fince the Rays which paſs through the Be-
ginning of each 15th Degree of the Equator, will project thoſe
Points into the faid Common Section or Hour-line of the Dial,
this Line will be thereby divided in the Manner of a Line of Sines
each Way from the Center, as repreſented in Fig. 5.
3756. The Stile of this we have fhewn (3753) must be pre-
cifely elevated to the Plane of the Equator, viz. the Angle
ECH (Fig. 3.) for any propofed Horizon HO; and as this
Dial is of the Elliptic Kind, its Stile or Gnomon CE is a move-
able One. And therefore if TV be the Parallel of the Sun's
greateſt Declination, it will alſo repreſent the Pofition of the
faid Gnomon for the tropical Day, as being parallel to CE,
and interfecting the Axis CP in the Point (b) making Cb
Tangent of 23° 30′. Therefore, fince in the Trian-
gle CbV, it is Cb: CV:: Radius: Secant of the Angle
PCO, the Diſtance CV is known for any Latitude PO or
EZ.
3757. And fince Cb is the Tangent of 23° 30′ to the Radius
CP; therefore C V will be the Tangent of 23° 30′ to a Radius
equal to the Secant of the Latitude, and hence CV the Semi-Zo-
diac for this Dial, is eafily graduated for the duly adjuſting the
Index to the Sun's Declination, as directed (3736,) and as
fhewn in Fig. 5.
3758. But this Rectilineal Dial is not fo conveniently
combined with the common horizontal Dial for finding the true
Hour or Meridian Line as the Elliptical Dial is, becauſe in this
the Hours at the extreme Parts of the Line run fo near toge-
ther, as not to admit of fufficient Accuracy in obferving and
comparing the Time in each. Nor is the circular Dial (Fig. 4.)
fo fit for this Purpoſe as the Elliptic One, becauſe of the ſmall
Length of the Zodiac for fetting the Stile CN (Fig. 3.) not al-
lowing a fufficient Eftimation of the Sun's Declination.
3759. 'Tis very obfervable that both the circular and right-
lined Dials are univerfal Ones, or will ferve for all Latitudes, be-
cauſe
Of CLOCK-Work.
535
cauſe the horary Divifions of each always remain the fame, and
the Indices or Stiles only require to be properly rectified, viz.
that of the Circular Dial as directed (3749,) and that of the
Rectilineal One to the Co-Latitude HE, be it what it will.
3760. Laſtly, in the Right-lined Dial (Fig. 5.) the Reader
will obferve that the Shadow of the Stile muft go backward and
forward twice a Day in the Summer Half-year, viz. Morning
and Evening, as is evident by Inſpection. What remains on
this Subject of DIALLING will be delivered in the Third Vo-
lume; wherein it is propofed to give the THEORY and CON-
STRUCTION of all the uſeful Mathematical and Philoſophical
INSTRUMENTS, as alſo all the neceffary TABLES uſed in thoſe
Sciences, with the Rationale and particular Uſes of each.
END of the SECOND VOLUME.
INDEX
To the MATHEMATICAL INSTITUTIONS.
A
A.
IR, Nature and Properties
of
>
301
Elafticity, Denfity, and
Gravity
ibid. & feq.
artificial, Nature and Pro-
duction of
tiles,
305
its Reſiſtance to Projec-
ble
when as great as poffi-
ibid. & feq.
Bodies, Defcent down inclined
Planes
53
the Doctrine of abfolute,
fpecific, and relative Weight
C.
294
Illuftrated by Expo Atoptrics, Elements of 105
335
Experi-
ments on the Ballittic Pendu-
CA
Doctrine of ΠΟ
Illuftrated by Plate I. ex-
ibid.
plain'd
112
167
lum
Anamorphofis, Nature and Prin-
ciples of
Appearance, Perfpective, of Ob-
jects on inclin'd Planes
Illuftrated by Figures
Application of fluxionary Spherics
Areas, Proportionality of
~
191
ibid.
253
81
201
Architecture, Rules of Perfpec-
tive applied to it
Automata, portable, or Watches,
the Invention, &c
384
Automaton, invented by Hugenius
371
A Diagram for Illuftration
}
adapted to Lenfes, &c.
Catoptric Perfpective, confider'd
with regard to Views
Central Force confider'd
Center of Gravity in Bodies
of Percuffion
114
186
73
41
48
15
85
of Ofcillation
Centripetal Force, conſtant
Law of it illuſtrated
Centripetal and centrifugal Forces
confider'd in regard to a Planet
99
Centrifugal Force, compaired with
the Centripetal
IOT
exemplified by Figures
ibid.
Clock-work, Nature and Defign
374
B.
of
BA
Aliftics, or Projectiles, Doc-
trines of
301
, Rationale of, Calcula-
8.1
ibid.
applied to Gunnery
Ball, Cannon, Velocity of, from
its Denſity, and Quantity of
its Charge, how determined.
308
tion of it
Illuftrated by a Diagram
Cohefion, Power of
Colours, Doctrine of
-, explain'd by the Priim
ibid.
Co.
371
374
II
140
4 N
INDE
X.
Colours, applied to refracting Te-
leſcopes
ibid.
explained from mechanical
and mathematical Principles
D.
G.
365
DIacauftic Curve, explained
T22
Dial, the Theory and Conftruc-
tion of an elliptical, circular and
diametral, and their Ufe524
Dialling Principles of Perfpec-
tive) applied to it
221
Dioptric Perfpective, confider'd,
with regard to Views, &c. 186
E.
Lements of Catoptrics
Viſion
105
134
Illuftrated and explained by
G
Nomonics, or Perfpective,
I applied to Dialling
Gravitation, Principles of
,
Laws of
-, Power of
Illuftrated
Gunnery
Inftitutions of
221
7
.00
8
17
18
63
287
Sir Ifaac Newton's Me
thod of inveftigating the Path of
a Projectile in a refifting Medi-
um applied to it
352
Gunpowder, the Velocity which
a Flame of it acquires by ex-
panding itſelf
330
Mr. Robins's Machine for
Experiments of, explained 317
a Print of Ditto ibid.
E
Figures
ibid.
Equation of Time
499
Table of
5.04
Plate for illuftrating ibid.
tic Orbit of the Earth
calculated for the ellip-
Table for every Degree
H.
595
HEL
Elioftatic, or Planetary Clock
469
of the Earth's Anomaly
511
Compound Ditto
512
chanics
ibid.
ཀ = -
a general Table thereof,
depending on the Sun's Place
Principles of celeftial Me-
I.
Experiments, Account of, 517 Mages, the Proportion, Mag-
for
compairing the Velocity of Bul-
lets
323
Illuftrated by Tables 327
F.
128
nitude, Pofition of, &c. form-
ed by the Lens
Inflitutions of the mechanical Ma-
thefis
I
Inftitutions of univerfal Optics
F
404
TOOT, Philofophical
explained by a Plate ibid
Force, exploſive of, Gunpowder
305
Forces, centripetal, their Compo-
fition, c
·, · Projective
Central
fugal, compared
105
of Perſpective
145
of Loxodromics
265
of Gunnery
287
Horologica
361
L..
19 IGHT, different Refrangibi-
73
L
ibid.
Centripetal and Centri-
140
Loxodromic Sailing, Spècies of
279
101
364
Lunarium, Conftruction and Me-
chanifm of, for fhewing the
Phafes of the Moon, &r. 492
ME-
Fufee, Nature, Form, and Action
ofic
INDE X.
cofm
469
applied to the Micro-
477
to Matter and Motion,
7
to mechanical Powers
M.
Echanics, Principles of ce-
Mleftial, explained by the
planetary Clock
jectile in a refifting Medium
applied to practical Gunnery
Motion, Principles of
Quantity of
2
352
7
9
of Bodies Elaftic, or
Non-elaftic
24
Principles of, applied to
Machines
31
3x
to the Lever
35
Menfuration of Superficies
I
to the Steelyard
·34
a Square
ibid.
Pulley
ibil
Parallelogram
2..
Wheel and Axle
35
Triangle
ibid:
Inclin'd Plane
ibid.
Rhombus
ibid.
Rhomboides
ibid.
Wedge
Screw
36
37
Trapizium
3
Circle
ibid.
chines
ibid.
of Solids
5
a Cube
ibid.
Parallelopipedon ib.
――― applied to common Ma-
Compound Machines ibid.
N.
Prifm ibid. Nga Rhumb-line, demon-
TAvigation, the Theory of,
Pyramid
Cylinder
Cone
on
ibid.
ftrated
ibid.
Illuftrated by Figures
Sphere or Globe ibid.
Mercator's Chart, Nature and
Conftruction of
265
Method of eftimating elliptical,
paracentric, and angular Velo-
cities
Focus
9
of finding a geometrical
122
265
270
by a Table of Loga-
rithmic Tangents
271
Navigation exemplified by Solu-
tion of Cafes of Sailing by that
Method
O..
283
of drawing the Perfpective Bjects, their Appearance on
of Objects
157
of drawing Figures of
three Dimenfions in Perspective
practical of, Scenographic
Perſpective
Objects his
a
165
Oblique fpherical Triangles, Cafes
of, and Theorems for explain-
237
the fix Cafes folv'd
ing
172
176
Method of deter-
mining the Fluxions and Sides
of determining the Fluxi-
ons of the Sides and Angles of
=
240
243
a fpherical right-angled Trian- Optical Elements of Lineal Per-
gle
233 fpective
1
148
of determining the Fluxi Ofcillation of Pendulums 400,
ous and Sides of oblique fpheri-
cal Triangles
243
of determining the Area
of a spherical Triangle 247
Sir Ijaar Newton's, for in-
veftigating the Path of a Pro-
PE
421
explained by a Plate ibid.
P..
PEndulum, angular, explained
4 N z
450
Pen-
INDE
X.
Pendulum, compound, explained
4c6
circular, explained, and
its peculiar Advantages 436
Pendulums, Dodrine of
53
Hugenius's Invention for
regulating the Motion of 380
Method of finding the
Center of Ofcillation
1
400
Univerfal Theory, and
Conftruction of, for mealuring
Solar, Lunar, and Planetary
Days
411
Of the Bob, or Weight
4:6
Errors arifing from the
Spheroidical Figure of the Earth
326
>
Heat and Cold another
Source of Errors in Pendulums
Rotula, horological, for regulating
Clock Work, by Mr. Sully 442
Rotalary Pendulums, Theory and
Conſtruction of
465
128
Rules for finding the focal Diſtance
of all Sorts of Lenfes
of Perspective
181
for exhibiting Objects in any
Proportion thereto
186
Applied to Views, Landscapes,
ib. & feq.
Pictures, &c.
of Perſpective applied to The-
atrical Scenery
194
of Perſpective applied to Ar-
chitecture
201
of Perfpective, to the Geogra-
phical Projection of the Sphere
207
applied to the Conftruction of
Maps, Charts, &c.
207
332
jections
455
Eliptical and Horizontal
Uniform Pendulums, their
4:9
Conſtruction and Uſe
Percutient Bodies, Theory of 19
Perspective, lineal, Elements of 148
Of Circles and circular
Areas
161
172
Scenographic
Appearance of Objects on
191
Inclined Planes
Theatrical, explained 194
Deception, explained 199
Sciagraphic, or Art of
Shadows
203
Principles of catoptric and diop-
tric Perfpective, in regard to
Views
186
Applied to the Art of
221
of Celeſtial Mechanics
applied to Aſtronomical Pro-
212
of practical Gunnery, derived
from Parabolical Hypothefis,
proved to be falſe
S.
348
Ailing, Loxodromic, Species
SA
of
279
Sphere, Stereographic Projection
of, in Plano. Theorems for
260
Spherical Triangles, Nature of
229
Solution of their Cafes ibid.
& feq.
Spherical Right-angled Triangles,
Method of determining the
Fluxions of the Sides
233
Spherical Trigonometry, applied
to the Solution of Problems in
Astronomy, Geography, Na-
vigation and Dialling
Dialling
469:
Projectile Forces
73
249
Proportionality of Arcas
81
R.
Spherics, Fluxionary, Application
of, to Parallaxes, Refractions,
Altitudes, &c.
253
Springs in Watches, Theory of
R
Ationale of Calculation in
Clock Work
371
388
TABLE
INDE X.
T.
ABLE of the Specific Gra-
vities for Solids and Fluids
TA
296
of the Accelerations of a
331
Clock, &c.
Tellurian, Conftruction of, and
Ufe
Illuftrated by a Plate ilid.
of Rotulary Pendulums.
465
and Conftruction of an
elliptical, circular, and diame-
tral Dial, and their Ufe
Time, Equation of
524
499
504
4S4, Table of
Theorems for the Solution of
Cafes of oblique fpherical Tri-
angles
for the Stereographic
Projection of the Sphere in
Plano
Theory of Catoptrics
cles
of Optics
237
-, Plate for illuftrating the
Theory of
ibid.
calculated for the elliptic
Form of the Earth's Orbit 505.
Triangles, Spherical, Nature of
260
105
128
Cafes
of Perſpective Lineal 152
of the Perſpective of Cir-
161
of Catoptric Perspective
165
of Inverfe Perſpective 167
of Scenographic Perſpec·
tive
172
of perspective Meaſures
of Scenographic Perspective 176
of Theatrical Perspective
194
of Perfpective Deception
199
tive
of Sciagraphic Perfpec-
203
265
of Navigation
Exemplified in a Solution of
Cafes
of Gunnery
>
229
Solution of all their
ibid. & feq.
Oblique, Spherical
Cafes of, and Theorems for ex-
plaining
237
a Table for every De-
gree of the Earth's Anomaly
511
Compound Ditto 512
general Table thereof,
depending on the Sun's Place
>
517
Trigonometry, Spherical, Doc-
trine of
İlluftrated by 2 Plates.
V.
Velocity, circular
233
$6
——, elliptical, paracen-
tric, and angular ibid. & feq.
Method of determin-
283
287
ing that of a Bullet
313
Mr. Robins's Machine
for determining it
Vibrations, Tochronal, of the
Spring of a Watch, defcribed
317
W.
of the Refiftance of Bo-
dies moving in a refifting Me-
dium
26 & feq.
for determining the Ve
locity of a Bullet
313
of Mr. Robins's Machine
for determining the Velocity of
a Ball
386
317 WEIGHT, in Bodies, abfo-
hute, fpecific, and relative
of Refiftance, Velocities,
Times and Spaces defcribed by
Projectiles
341
of the Springs in Clocks
and Watches
204
explained on the Princi-
ples of refilling Fluids, as Wa-
. ibid. & feq.
388
ter, Air,
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tions on the Fair Sex. Tafte and Riches, &c. 12mo. Price 38 6d.
17. A Letter to a Young Lady, concerning the Principles and Con-
duct of the Chriftian Life. By Lawrence Jackson, B. D. Prebendary
of Lincoln, 2d. Edit.
Price 13.
Mathematical, Philofophical, and Optical Inftruments, in-
vented or improved, by B. MARTIN.
A
New manual Orrety
Ditto with Wheel-work
Planetarium thewing the Phænomena of the Ptolemaic and Coperni-
can Syftem, from 51. 55. to
A new Portable Air-pump, exclufive of any Apparatus
Ditto, with a Clamp
Ditto, fmafler
Ditto, with a Clamp
Barometer, Thermometer, and Hydrometer, all in one Frame
Hydrostatic Balance, from 15. to
A new and most accurate Hydrometer for proving of Spirits, in a
Mahogony Box and Glass Jarr
Hadley's Quadrant of a new Conftruction, for making it uſeful by
Land as well as Sea
A new Navigation Rule
A Dialing Sector
A Marine Plain Table
s. d.
2126
8 8 Q
22 I O
660
6 16 0
440
4 14 6.
212 6
2126
1 70
4 14 6
O
4 6
O
1
5 6
11 6
20
5
5
A new Protracter, for fetting off Angles to a Minute, with a Nonius z
Portable Electrical Machines, from 37. 135. 6d. to
A new univerfal compound Microfcope, all in Brafs, Shagreen Cafe 5 50
A Pocket compound Microfcope in Brafs
Solar Microſcope in Brafs and Wood
Megalafcope for magnifying large Objects
A new proportional Camera Obfcura, with a Solar Microſcope
A compleat Apparatus of Optical Infiruments in Brafs, confifting of
a new univerfal compound Microfcope, a Solar Microfcope of the
lateft Improvements, a fmall Reflecting Telefcope. The Stand
of the Compound Microſcope is adapted alſo as a Reſt for the Tele-
fcope. The whole is furnished with every Thing neceflary for the
niceſt Obfervations with the Microſcope.
I
}
A portable optical Apparatrs, consisting of a Scioptric Ball and
Socket, a folar Microfcope, Wilfons Microfcope, a Pocket com-
pound Microfcope, a Pocket Telefcope and folar Teleſcope, in
fpotted Fish-fkin and Brafs
Vifual Glaffes or Spectacles of a new Conftruction, beft Sort in Tem-
ple Frames
Nofe ditto
Ditto, with the best Pebbles, in Temple Frames.
2126
3136
I IO
2126
12 12 0
3136
O
5 6
O 26
0 16 0
At the fame Place may be had all other Inftruments in SURVEYING, DI-
ALLING, NAVIGATION, . at the moft reaſonable Prices.
UNIVERSITY OF MICHIGAN
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شهری
3 9015 06389 5745