The method of fluxions applied to a select number of useful problems. 1756
by Saunderson, Nicholas.

Publication date 1756
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METH OD of F LUXIONS

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Applied to a ſelect Number of

USEFUL PROBLEMS, &e. .

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METHOD af FLUXIONS

Applied to a ſelet Number of
USEFUL PROBLEMS:

ToGtTHER WITH |
The Demonſtration of Mr. CorEs’s FoRMs

of FILE NT S in the Second Part of his
LoGoMETRIA;
The AN ALYSIS of the Presse N his
SCHOLIUM GENERALE;
AND

An ExPLANATION of the principal PRoPosITIONS of
Sir Is AAc NEwTON’s PHILOSOPHY.

1 + 4 | h
NICHOLAS SAUNDERSON, LL. D.

Late Profeſſor of Mathematics in the e
Cane |

—
—

Illuſtrated 1 TI’WELVE Crop PLATES.

—
— 9— -tans,.

8 M *

LONDON,
Printed for A. Mir ran, in the Strand; J. WHISTON |
and B. WIr E, in Fleetſtreet; L. Davis and

C. REYMERS, in Fleetſtreet, and againſt
Gow Im. Holborn.

‘MDCCLYI

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. — — — . —˙¹ ]. — We ä é PPC — —

7 —— — ; _ 2 a pr _— I

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ADVERTISEMENT.
HE following Pieces of Dr. Saunderſon,
having been long handed about in Munuſcript,

and fome Things, copied. from them, having ap-
; peared in Print, under other Names; it was
thought proper to collect them into one Volume:
As ⁊cell in Juſtice to their excellent Author, as to
prevent the Errors that are unavoidable, and ever
growing, when Warks of thts Kind are often, tran-
ſcribed.
‘ The Reader wi 1, every where diſcover in them
the Hand of a great Maſter ; that Diſtin&neſs
and Perſpicuity for which Dr. Saunderſon was
fo juſtly celebrated, a judicious choice of Examples,
a fimple Analyſis, and an elegant Conſtruction.
And if their Defe&s are ſupplied viva voce by a
ſhilful Tutor, as they ought to be, this Treatiſe of
Fluxions, zf not the completeft, may nevertheleſs
be reckoned the beſt for Students in the Univerſities,
of any yet publiſhed.

What the Doctor has given us upon Mr.
Cotes’s Logometria is particularly valuable;
as, by his intimate Acquaintance with that ex-
traordinary Perſon, he may be preſumed to have
E his Wri tings better than any one at

—
n . 3 n 7 r

• N

ADVERTISEMENT.
that Time living; Dr. Smith only excepted, to
whoſe ſuperior Genius and faithful Care the World
tis fo much indebted for the Improvement, as well
as the Preſervation, of Mr. Cotes’s Works.

This Collection furniſhes likewiſe an Inflance
remarkably curious, and ſuch as may not again
happen; how far the Faculties. of the Mind,
Imagination, and Memory, properly en: ä
may ſupply the Want of a Senſe * which, but for
this Inſtance, might ſeem abſolutely neceſſary to.

the Acquifition of Mathematical Learning.

See Dr. Saunderfor’s Life prefixed to his Algebra.

C O N-

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‘

CONTENTS.

AN Introduction. ix.
Of the Compoſition and Reſolution of 8 –
ibid.
Of the Deſcent of Heavy Bodies, + os
Of Powers and their Indexes, „
O the Algorithm of Fluxions, 8 “©
Of the Method of drawing Tangents, _ 8.
Of the Dottrine de Maximis .. | II.
Of infinite Series, ” “AS
Of the Inverſe Method of F lags, 46
Quadrature of Curves, | „
Of the Computation of Briggs’s Lurie, 34.
Of the Menſuration of Solids, 29—
Of the Surfaces of curvilinear Solids, _ 42.
Of the Method of Reverſing Series, 47.
Of the Center of Gravity of Bodies, 58.
Two Problems concerning Gravity, 60.
Of the Motion of Pendulum, ) 64.
Of the Denſity of the Air, | ” 70
Of the Reſiſtance of Bodies in Fluids, 5

Of the Afion of a Prolate Spheroid upon a Particle
of Matter placed in its Axis produced, 102.
DF Second, Third, and Fourth Fluxions and Fluents,

12. 
    ‘ The Method of extraBing the Roots of *
    Equations, by —_— the Analogous Terms of
    Series, 116.

ö
55

Wo

CONTENTS:

7. be Method of computing the Forms in the ſecond

Part of the Logometria, 1 IBYp

Of the Preparation of Tabular Fluents, 158.
Analyſis of the Problems in the Scholium Generale
e Mr. Cortes’s Logometria, 162._
Two Problems: Of the Oſcillations of a Pendulum;

aud the Rettification of the Ellipſe, 206,
1 8 of the latter Part of Logometria, Prop. 4.

2 of the Scholium to the ſame Propeſition, 216,
——— of the ſecond Part of the ſame Scholium, con-

cerning the Reſiſtance in Aſcents, 218,
2 of the latter Part of the Scholium to Prop. 5;
: 220.

7 wo Theorems for comparing Curvilinear Areas, 223,
A Problem: To meaſure the Surface generated by an
Arc of the Reftangular Hyperbols revolving

about an Aſymptote, 220,

l Theorem: When an Aſymptotic Sagas is finite,

when infinite, | 229.
Nos upon the chief Propofitions of Sir Iſaac

| NEwrox’s Principia, 8 9

AN

4
33
:
|

2 A N

INTRODUCTION:

= Of the CoMPosITION and RESOLUTION of

FORCES.

II. Of the DzscenT of Hravy Bopigs.
III. Of Powers and their INDEXES, |

Plate I; Fig. Q.

N. B. JY the Determination of a Force in the

| following Theorem is either meant its
Direction, or any other Line parallel to it. Thus
if a Force acts from A towards B, and any Line
as E F be drawn parallel to A B, and F be ſitu-
ated the ſame Way with reſpe& to E that B is
with reſpect to A; then may the Line E F be ſaid
to ſignify the Determination of the Force acting
from A, and the Line F E a contrary. Determi-
nation.

ri E

1f there be a Triangle E FG whereof two Sides
E F and F G repreſent both the Quantities and
Determinations of two Forces acting from a given
Point as A; I fay then that the third Side E G
will repreſent * the EY and Determina-
| | tion

v we

XxX – TheComposiTionand

tion of a third Force, which acting from the ſame
Point A, will be equivalent to the other two.
For drawing AB equal and parallel to E F,
and AC equal and parallel to F G, and compleat-
ing the Parallelogram AB DC, the Lines AB
and A C will repreſent both the Quantities and
Directions of the two Forces acting from A,
which two Forces will both together be equivalent
to the ſingle Force: A D, as is evident from the
firſt Corollary of Newton’s Laws of Motion. Now
ſince EF and FG are equal and parallel to AB
and B D reſpectively, the Angle E F G will be
equal to the Angle A BD ; therefore by the fourth
Prop. of the firſt of the Elements, the Side E G
will be equal to the Side AD, and the Angle
FEG to the Angle BAD ; therefore ſince the
Lines E F and F G have the ſame Determination
with the Lines AB and B D, the Line EG will
be alſo parallel to AD; therefore the Line E G
will repreſent both the Quantity and Determina-
tion of the Force AD. Q E. D

Cosa! 1.

In like manner if there be ever ſo many Forces
actiag from the Point A, and if the Quantities
and Determinations of theſe Forces be repreſented
by ſo many contiguous Lines E F, FG, GH, HI,
the Line E I, drawn from the firſt Term E to the
laſt I, will repreſent bath the Quantity and Deter-
mination of a ſimple Force, which acting from A
will be equivalent to all the former put together.

4 5 For

RrSsOLVT IOX T Forces. xi

For the Forces E F and F G are equivalent to

the Force E G by the Propoſition; and for the
ſame Reaſon the Forces E G and GH are equi-

valent to the Force E H; and the Forces E H
and H I to the Force E I; therefore the Forces
EF, FG, GH, HI are equivalent to the Force E.

COR OI. II.

If the laſt Point coincides with the firſt Point E,
it is an Argument that the Forces at A will have
no Effect, but will only HEE one another in

Aeguilibrio. |

Condi: ME.

On the other hand, if EFG HI bea 8
whereof any one Side as E I repreſents both the
Quantity and Determination of a ſimple Force
acting from a given Point as A, the other Sides
EF, FG, GH, HI will repreſent the Quantities
and Determinations of ſo many other Forces, which
acting from the ſame Point A will be all together
equivalent to the former ſimple one.

. *

xii Of the DES ENT

Of the DzscenT of Heavy Bop1es.
PRO P. II.

The Velocity generated in any Quantity of
Matter q by the conſtant Action of an uniform
Force as r Oy for the Time 7, will be as

the Quantity —. For it will be, caeteris paribus,

as 1 directly, ut as | directly, and as 2 inverſely.
— E D.

Conor. I. |
If the moving _ 7 be as the Quantity of

Matter g, the Quantity — will be conſtant, in which

Caſe the Velocity 3 will be as the Time 7,
and in the ſame Time will always be the ſame.

Canes

• . if any Number of Bodies, how dif-
  ferent ſoever in their Kinds and Quantities of |
  Matter, be acted upon by Forces proportionable

to their Matter, they will be equally accelerated,

and conſequently will paſs through equal Spaces in
equal Times.

CoRoL. Bl.

And vice verſa, if any Number of Bodies, how
heterogeneous ſoever, be acted upon by different
Forces, and be found to be equally accelerated,
that 1 is, to pals through equal Spaces in equal

Times –

i

of Heavy BoDIiEs. xiii

Times ; it is an Argument that the Forces whereby
theſe Bodies are moved are proportionable to the

Quantities of Matter they contain.

SCHOLIUM.
The Quantity r is 9 called the Vis norrix,

and the Quantity 7 the Vis acceleratrix; becauſe

the former is always proportionable to the Motion
generated in any given Time, and the latter to the
Velocity; and as the ſpecific Gravity of a Body
is not its Gravity abſolutely, but its Gravity with
reſpect to its Magnitude, ſo the Vis acceleratrix of
a Body is not the Vis motrix abſolutely, but the
Vis motrix with reſpect to its Matter; and in this
Senſe of the Word our Propoſition will ſtand thus,

viz. The Velocity generated by the conſtant Action

of an uniform Force continued for any Time, will

\be as the Product of the Time and the accelerating
F porce multiplied together: And therefore in Caſes

where the accelerating Force is always the ſame,

the Velocity generated will be as the Time.

Pros. III.

That the Wei ghts of all Bodies are proportion-
able to their Quantities of Matter.

This appears from the equal Accelerations of
Bodies of all Kinds in Vacuo, and is further con-

firmed by Experiments made upon the Oſcillations
of Fendulums, whereof more hereafter,

S Prop,

— D

N

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1

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— e Desen

Pxor. IV. |
That the Gravity betwixt the Earth and its
Parts is mutual and equal.
To demonſtrate this, let the Globe of the Earth
be diſtinguiſhed by the Mind into two Segments
equal or unequal. Now if either of theſe Segments
ſhould gravitate more upon the other, than the
other upon it, there would be a greater Preſſure
one Way than the other, and the Earth giving
Way to the greater Preſſure, would be forced out
of its Place ad inſinitum; not to ſay that upon this

Suppoſition a Body at Reſt would be able to put

itſelf into Motion without the Action of any ex-

ternal Force whatever. Q. E. A. See Cotes’s
Preface to the Principia, and Newton? s general

Scholium to the Laws of Nature,

Prop. V.

That the Forces wherewith Bodies attract $a
Earth are proportionable to their Quantities of
Matter. _

For the Forces ns Bodies attract the
Earth are equal to the Forces wherewith the Earth
attracts them by the fourth Prop. But theſe latter
Forces are proportionable to the Quantities of
Matter in the Bodies attracted by Prop. 3, theie-
fore the former are ſo too. Q. E. D.

As the Force wherewith every Body either at-
tracts the Earth, or is attracted by it, is owing,
15 caeterts

of He avy BoDIES, xv

caeteris paribus, to its Matter as ſuch, and not to

any particular Conſtitution of the Body, ſo neither
is the Force where with the Earth attracts Bodies,
or is attracted by them, owing to any particular
Conſtitution of the Earth, or to any magnetic Body

in its Center, but to the Maſs of Matter whereof
it conſiſts, |

Conot:: 11

Since the Attraction of the with Farth ariſes
from the Attraction of all its Parts; and ſince all

Bodies, as well thoſe that lie looſe upon its Surface 5

as thoſe that adhere to it, may be looked upon as
Parts of the whole Earth, it follows that all the
Bodies whereof our Earth is compoſed, mutually
and equally attract one another, though this At-

traction be too ſmall to have any ſenſible Effect.

PRO VE

If a heavy Body as 4 preſſed by its own Weight
r deſcends through the Space s in the Time t, and
at laſt acquires the Velocity v; I ſay that the Space

5 will be but Half the Deſcent it would have made,

had it moved all the Time : with the ſame Velo-
City v.

For ſince the Velocity of a falling Body increaſes –
uniformly with the Time of the Fall, and ſince
between the firſt Velocity which was o, and the
laſt which was v, an arithmetic Mean is 2 v, it
follows that a Body moving uniformly with Half
the Velocity v will paſs through th Space s in

A 4 Ns the

mat

• rr +. © pegs atos oibebor (hc

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4 *

xvi Of the DESCENT
the Time 7; therefore a Body moving uniformly

with the whole Velocity will paſs through twice

that Space in the ſame Time. Q. E. D:

Conn. 3

Since a Body moving uniformly with the Velo-
city v, deſcribes the Space 2 5 in the Time t, it

follows, that 2 3, and conſequently s will be as v ?;

that is, the Space through which a heavy Body
deſcends will be as the Time of the Deſcent, and

the laſt Velocity multiplied together.

ee
Since by the Scholium to the ſecond Prop. v
is as 7 t, we ſhall have v and conſequently. S as

Cool. III.

For che ſame Reaſon, viz. that v is as t, we
ſhall have vv as 1111, and conſequently r.;
VV

whence s will be as —.

7
0 oroL IV.
From the three foregoing Corollaries, and alſo

from hence that v is as 7 +, may be deduced the
following Canons.

US] D

: 4

1. r is as — or as 7 or as .
   . o
   | VU

3 is as 1 f or as vi, or as —,

3.1

. A it Cs Aero db Bi et A A . — _ *

of HEAby Bovpies. xvii

8 * V.

85 #18 88 © Ss “I or as –

4. Vis as rt, or as Vr, or as 7.
   Cool. V.
   If the Vis acceleratrix r be given, it may be
   ſtruck out of the e and then they will ſtand

thus:

2. Wer, or as 7 v, or as v v.
   5 5
3. F will be as >, or as Vs, or as v.
   1 | ;
4. v is as 1, or as Vs, or as

14

SCHOLIUM,

At the Surface of our Earth, a heavy Body
deſcends through I5 r French Feet or 162. *
Engliſh Feet in a Second of Time ; therefore by

this Prop. in a Second of Time a heavy Body ac-
quires a Velocity which would carry it uniformly
through 323 Engliſb Feet in the ſame Time. Let
then this Quantity repreſent this Velocity,. and
every other Velocity will be repreſented by the
Space through which it would carry a Body uni-
formly in a Second of Time. By the Help of
theſe Data, and the Canons delivered in the fifth
Corollary, if in the Caſe of any other particular
Deſcent, at or near the Surface of the Earth,
either the Time or Space, or the Velocity be given,

• 10757 METER 6;
  ©

Xviii Of the DES EN
the reſt may eaſily be found. As for Inſtance, let
it be required to determine how long, and how far
a heavy Body muſt fall to acquire the Velocity of
a2 Sound, that is, to acquire a Velocity which will
carry a Body uniformly through 1142 Engliſh Feet
in a Second of Time. *
Firſt, To determine the Time, I find in the
fifth Corollary, tis as v; therefore I ſay, as 32 2
is to 1142, 10 is one Second of Time wherein the
former Velocity was acquired to the Time ſought
wherein the latter Velocity will be acquired.
Secondly, As to the Space, I find 5 to be as v;

and therefore ſay, as 32 47 1s to 1142 ſo is 162
Feet through which a heavy Body muſt fall to ac-
quire the former Velocity, to the Space ſought,
through which it muſt fall to acquire the latter
Velocity.

If in any particular Caſe the Vis acceleratrix that
cauſes the Deſcent be different from that of com-
mon Gravity, then of the four Quantities 7, 5, 4, v,

two muſt be given to find the reſt, and the Canons

muſt be taken out of the fourth Corollary.

PRO r. VII. Plate I. Fig. R.

The Force whereby a Body endeavours to de-
ſcend freely upon an inclined Plane is as the Ele-
vation of the Plane directly, and as its Length
.

For in the right. angled Triangle ABC, let the
Hypothenuſe A C repreſent an inclined Plane

whoſc higheſt Point 1s A, and loweſt C, .

ns AP.. ov”

of Heavy Bopiss. * xix
BC be parallel, and AB perpendicular to the

Horizon; then will AB be what is called the
Elevation of the Plane AC. Let the ſame Line

A alſo repreſent the abſolute Weight of a Body

lying upon the Plane AC, whereby, if permitted,
it would deſcend perpendicularly : Then drawing

BD perpendicular to A C, the Force A B will be
equivalent to two Forces A D and DB, whereof
A D will be wholly ſpent in accelerating the Mo-
tion of the Body along the Plane AC, but the
other B D will have no Effect, as being oppoſed

by the perpendicular Reaction of the Plane; there-

fore the abſolute Weight of the Body will be to
its Endeavour to deſcend along the Plane A C,

as AB is to > AD, or by ſimilar Triangles as AC

to AB, or as 1 ro : Therefore if the abſolute

1

Weight of any Body be called 1, its Endeavour to

deſcend obliquely along the Plane AC will be
AB

repreſented by the Quantity — AT 0 N. D.

Tens!

The accelerating Force in every Part of the
ſame Plane AC will be the fame ; for i it will al-

ways be repreſented by the .ſame Quantity ww

and therefore whatever has been demonſtrated in
the ſixth Prop. and its Corollaries concerning: the
perpendicular Deſcent of heavy Bodies, will be
equally applicable to the Caſe of oblique Deſcent,

making / = AC. and r = AC; Ca.

xXx Of tbe DESC ENI

Coat, 5
The Velocity acquired in falling from A to C
will be in a ſubduplicate Ratio of A B the Eleva-
tion of the Plane. For by the fourth Corollary of
the ſixth Prop. v is as rs; but in our Caſe

A- ands=AC; therefore r5=A B, and
Nr5s—/ AB; therefore v is as AB.

Coror, I.

Therefore ſo long as the Elevation of a Plane
is the ſame, the Velocity acquired in falling through
the Length of the Plane will always be the ſame,

be that Length what it will.

Denn

Therefore in falling from the ſame Place A to
the ſame horizontal Line B C, the ſame Velocity
will always be acquired, whether the Deſcent be
made obliquely upon an inclined Plane AC, X
perpendicularly } in the Line AB.

ao FF.

The Time of oblique Deſcent from A to C
will be as the Length of the Plane A C directly,
and in a ſubduplicate Ratio of the Elevation A B
X inverſely. For by the fourth Corollary of the ſixth

AC
Pr – t is as — — „that! is in our Caſe gs – VEE:

Coro, |

= p

of HE avy Bonrpies, xxt

Coro. VI. „
Therefore if there be ever ſo many Planes hav-
ing all the ſame Elevation, the Times of oblique

Deſcent through thoſe Planes will be as their
Lengths.

Cook or. Vit

If 5 be the Space through which a heavy Body
falls in a Second of Time, and if m be a mean
Proportional between the Space s, and the Eleva-

tion AB of any Plane; then will the Velocity

acquired in falling obliquely from A to C be ſuch |
as will carry a Body uniformly through the Space
zm in a Second of Time. For by the fifth Corol-
tary of the ſixth Prop. the Velocity acquired in
falling through s will be to the Velocity acquired
in falling through A B, and conſequently through

AC, as ws is to MAB, or as 5 to m: But the
former Velocity will deſcribe the Space 2s in a
Second of Time; therefore the latter will deſcribe
the Space 2m in the ſame Time.

Conrot.’ Vil.
Suppoſing all Things as in the foregoing Coro.
I fay that as mis to A C, ſo is 1 Second of Time
to the Time of oblique Deſcent from A to C. For
conſidering the Line s as a Plane whoſe Elevation
is equal to its Length, it follows from the fifth
Corollary that the Time of perpendicular Deſcent
through 5, that is, 1 Second of Time will be to
| | | the

xxii Of PowER 8 and their IN DEX ES.
the Time of oblique Deſcent through 113

AC AC 25 ;
is to NB or as t = Sn. or as VSX VKB

to A C, or as m to AC.

Of PowEeRs and their IN DEXE 8.

x6 ſignifies x x & * * *

X5 = = -XXXXx
4 =- -XXXxXx

— =
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£ I
2 — —
& *
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22 = =
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0 1
R 4 – —
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3 x.
2 —

• ** NK
  XIX

Os ER.

Of PowERs and their In DEX E s. xxili

Gnas

1. Every ſucceeding Power is ——— —
   a the Power immediately preceding by the
   . Root x, and every ſucceeding Index is generated
   by. ſubtracting Unity from the foregoing.
2. Whatever Number is the Index of any
   Power, its Negative will be the Index of the re-
   ciprocal of that Power, or of Unity divided by
   that Power. 5
3. The Addition of Indexes. ,
   Multiplication of Powers; that is, if the Multi-
   plicator, and Multiplicand be any Powers of the
   ſame Quantity, the Index of the Multiplicator
   added to the Index of the Multiplicand will give
   the Index of the Product.
4. The Subtraction of Indexes 3 to che
   Diviſion of their Powers; that is, If the Index of
   the Diviſor be ſubtracted from the Index of the
   Dividend, the Remainder will be the Index of the
   Quotient. | |
5. The Duplication, Triplication, &c. of In-
   dexes anſwers to the Squaring, Cubing, Sc. of
   their Powers. |
6. The biſecting, triſecting, Sc. of Indexes
   anſwers to the extrating the Square or Cube Root
   of their Powers.

. Any Root of any Power may | be expreſſed
by a fractional Index whoſe Numerator ſignifies
the Power, and its Denominator the Root.

8. Surds

xxiv Of Pow ERS and their INDEXES.

8. Surds may often be reduced to more ſimple
   Terms by a Reduction of their Indexes.
9. Surds may be reduced to the ſame Denomi-
   nation by a Reduction of their Indexes.
10. The third and fourth Obſervations will be
    true in fractional as in integral Powers.
11. In the Common Tables of Logarithms,
    Unity being aſſumed for the Logarithm of 10,
    if 10” is equal to any Number N, then is the
    Logarithm of N— and the Reaſon of the Opera-
    tions by Logarithms is manifeſt.

\

OF

OF THE

ALGORITHMof FLUXIONS,

DEFINITION.

ET AB repreſent any Moment of Time,
whether finite or infinitely ſmall it matters

not, terminated by the two Inſtants A and B: Let

x be the Value of any flowing or growing Quan-
tity at the Inſtant A, whoſe Velocity at that In-

ſtant is ſuch, that if it was to flow during the
whole Moment A B with this Velocity, it would

gain a certain Increment repreſented by &; then is
this Quantity x called the Fluxion of x at the In-

ſtant A, when the Valee of the flowing Quaatity
Was x. 15 A

S HOL IVM.
From this Definition it appears, that if x flows
uniformly, its Increment gained in the Time A B

will be the ſame as its Fluxion above defined: But

if x does not flow uniformly, i. e. if its Velocity
at the Inſtant B be not the ſame as its Velocity at
the Inſtant A, then its Increment gained in the
Time AB will net be the ſame with its Fluxion
above defined, but will differ more or leſs from

3B it,

—— EN —
a 9 3 —— pe \ N I

: 2 = 5 . : — — — a :

2 _ – 8 * „ a _ rg TI TTY 22 er 2 8 . * — _ — NE EC © 2s Bs .
— * 7 2 N 2 n 2 n 5 1 . + —
— — ae Side 7§—˖rFÜ⅛᷑ãtcxé ac ee —— — . > ye ar ec. x _— Z
a — – — a * want – va —— — * *

• ; — — — — –
• * – — — —— – — 44 — — . –

** ” —

— —— ro

• Of the Al GOoRITAHN

it, according as the Time A B is greater or leſs:
But if the Time A B be infinitely ſmall, then
though the Velocity of x at the Inſtant B be not
the ſame, mathematically ſpeaking, with the Ve-
locity at the Inſtant A; yet the Difference being
infinitely ſmall in Reſpect of the whole Velocity,

it may ſafely be neglected, where the finite Ratios

of Fluxions are only conſidered ; and ſo this In-

crement and the Fluxion above defined may be
taken for one another, i. e. the Quantity x, for ſo

ſmall a Time, may be looked upon as flowing

| uniformly. /

Con ol 1. I.
The Fluxion of a conſtant Quantity is no-
thing. FO

Conor. II.

The Fluxion of the whole equals the F lurion
of all its Parts. Thus if’r be a conſtant Quan-
tity, and x, y, 2 flowing ones, the Fluxion of
r+x+y9—2 will be xj — 2.

N. B. When different Fluxions are compared,

they are all fuppoſed to be generated uniformly 1 in

the ſame Time.

Note alſo, That a conſtant Equality of Fluents
always implies an Equality of Fluxions, but nor
vice verſa, unleſs the Fluents be generated in equal

Times.

• LEMNA.

ce

8 WW

Re %

LEMMA.

If two | oat v and x, arithmetically expreſſed, be
ſuppoſed to flow uniformly, the Produtt of their
Multiplication will not flow uniformly, but with
a Felacity equally accelerated; that is, its continual

Increments gained in ſucceſſroe equal Times will not
be equal, but in arithmetical Progreſſion.

| DEMONSTRATION.

Let the Time AD be divided into the equal

Parts A R BC; C D, and ht and} = hecho

Values of the flowing Quantities at the Inſtant A;
then becauſe theſe Quantities are ſuppoſed to flow
uniformly, their Values will be at the Inſtant B,

1 and K ＋ & at the Inſtant C, v + 2v, and

x» + 2×3 at the Inſtant D, v＋ 35, and x 3x:
Therefore their Product will be, at the Inſtant A,

vx; at the Inſtant B, vx TTT; at

the Inſtant C, vx ＋ 2 œœòFw2 Xx 4e; at the
Inſtant D, vx ＋ 3vx +3 xv++9gv x. Subtract

now the Product at the Inſtant A from the Pro-
duct at the Inſtant B, and there will remain
vx x0v+0v x for the Increment gained by the
Product in the Time AB. Subtra& again the

Product at the Inſtant B, from the Product at the

Inſtant C, and there will remain v x x v + 3 &

for the Increment gained in the Time BC: Laſtly,
ſubtract the Product at the Inſtant C from the
Product at the Inſtant D, and there will remain

v X VN 5x for the Increment gained in the
B 2 Time

4 | Of the ALGORITHM

Time CD: Therefore the continual Increments,
gained by the Products in the ſucceſſive equal

Times AB, BC, CD, will net , but in
arithmetical Progreſſion.

N. B. If the Times AB, BC, CD, be ink.
nitely ſmall, the Quantities vx, 3Va, 5 U& will
be infinitely leſs than the other Parts of the In-
crements; as will eaſily appear by comparing them.

P= 091.2. *

T „ find the Fluxion of the Produtt ariſing from the

continual Multiplication of any Number of flowing
\ Nuantities whatſoever. |

Guan s

Let v and x be the Values of two flowing
Quantities at any given Inſtant of Time, and let
theſe Quantities be ſuppoſed to flow uniformly as
in the laſt Lemma; then it is plain that a Mo-
ment before the given Inſtant; their Values were
v — D, and x — x, and their Product vx — vx
KU &; therefore the Increment gained by
the Product in that Moment was vπ + * — Ox:
But the Increment gained by the Product in an
equal Moment immediately following the given In-
ſtant was found to be v x+- xv ox; and this
latter Increment; by the above-quoted Lemma,
exceeds as much the true Fluxion of the Product
as the other wants of it; therefore the true Fluxion
df the Product vx at the Inſtant of Time that the
Factors are v and x is vx x0. Q. E. I.
: „ 1 Ober

OY FLUXIONS. 1 5
Otherwiſe thus.

The Increment gained by the Product in the
Time AB (ſee the Lemma) is vx + xv ＋ ©
Let this Time A B be infinitely ſmall ; then it is

evident that the Quantity v x, being infinitely leſs

than the reſt, may be neglected for its Inſignifi> .
cancy ; ſo that both the Increment and the Fluxion
of the Product will be v x + . Increaſe now
again the Time AB from an infinitely ſmall to a
finite Moment, and v x and xv will increaſe with

the Time, as is evident from the Definition already

given of a Fluxion : Therefore v x + x v will ſtill
be the F luxion * the Product v x.

e 421 Ik
Let there now be three flowing Quantities, v,
andy, and the Fluxion of the Product v x y, or of
the Product vx xy will be the Fluxion of vx

multiplied into . together with the Fluxion of y

multiplied into vx, by the laſt Caſe : But the

Fluxion of v x was there found to be CER ,
and this multiplied into y gives vy x + x y v3 and
the Fluxion of y being mulciplied into vx gives
vy; therefore the whole Fluxion of the Product

Uxyis „ np0e

CASE III.

In like Manner, if there be four flowing Quan:
tities, the whole Fluxion of the Product vx y z,
or the Product vxyX2 will be found to be

1 een, Q. E. I.

B 3 COROLL,

Of the AL GORITRN

| ‘ ‘CaroLL. N
If a be a ſtanding Quantity, and v a variable
one, the Fluxion of the Product à v will be a b.

PROBLEM II.

To find the Fluxion of any Power of any ee
Quantity, whether the Index of that Power be
N or fractional, affirmative or negative.

Cank Li

Loet v, x; y and 2, in the laſt Problem, be ſup-
nod all equal to one another; then will vx &,
and its Fluxion v a#+ x b will be 2X x ; therefore
the Fluxion of x* will be 2xX x, or 2X Xx*—7.
Again, vxy = x3, and its Fluxion vx / v

• xyv will be 3xXx; therefore, the Fluxion of x3 will be 3x Xx,0r 3* X * i. Again vc x.;
  therefore the Fluxion of x+ will be 4x x x4—7.
  And zniver/ally, the Fluxion of x” will be m xX x”,
  provided the Index m be integral and affirmative.

CAS E II.
Let it now be required to find the Fluxion of |

I

• or : In order to which, make — IX #3

then will ZX ij; and taking the Fluxions on
bath Sides, we have SX * m 2 * o,
(for the Fluxion of 1 is o); therefore 2X * =

| „„ | — .
n AXxx ; = MK XA 2

• 
  there ·

=” —

re;

of FL UXxX1ONS. 7
— 1

therefore 2 S – N — MX N „k. Sg

that from theſe two laſt Caſes it is eaſy to perceive,
that if the Index n be a whole Number, whether

affirmative or OTE; the F luxion of x” will be

m x ,

5 CASE III. |
Let it now be required o find the Fluxion

of xn, * |
To do this, let * =2; then will 2 x*, “HEM

“the fifth Obſervation of Powers and their Indexes ;

therefore x 2X2″ —* =mxXXx”—*, from the two
foregoing Caſes ; therefore (dividing Equals by
Equals) we have a1 Xx 2 = m XR , and

N S e Xx—”. Multiply both “_ by 5

and we 25 2 (or the Fluxion ſought) = * X* 2

• — x5 9 So that at laſt it appears that

whether he Index be affirmative or negative, in-
tegral or fractional, the Fluxion of x” will be

mx XN i.

Conealh L

C

„V.

The Fluxion of this Quantity or

2?
*”X)”X Z , by the two foregoing Problems,
will be wx Xx XY NE”. the Be on. NA
XZ? ep EXETPEIINA XY,

: B44 – Cool..

_

. oy TW
o _ – * te
4 i 4 on 22 W
— 2 —
PO — — 8ũ— as T4, + 90, 9 2 22 —
— — r 2 . ev II
— — 2

1 ,
[1
N
oy
bs
[
ol *
i
.
i “2;
;
i
U
4 U
1 b
9 U
‘T ?
: F
«s Ft
‘F:
b $ :
x 1
; $14
1 4 2
. o
—— q .- A A
1 *
OE 4 q
—_ =:
124 4
11
N
179
| 1 1
i 138

19. N ‘
    1

• 5
  4 g
  F p
  [
  11
  *t
  » 8 \
  7
  1 0
  T7 7
  141
  The 1
  a :
  F434
  L 1
  12
  1
  19
  E
• 
  2

9 J

W, IE

—— — — mY

4 | ꝛaes Direct Mathod of

F | 1 Cox ol I. *.

The Fluxion of any Fraction 7 will be

N 5 AS will appear, richer from the

foregoing Corollary, by ii; ＋ as K, ;

or rather thus: Mak = 1

and y x 9 = SD x; therefore y 3 = * 2992 = 4 —
=3 therefore a 2 (or the F luxion ſought)

5

\

Co the Method of drawing TAN G ENTS.
25 Pros. III. Fig. 1.

| * Is ER to draw 4 Tangent to any given

Point M as of any given Curve, as A M, 1 |
er is A P.

SOLUTION…

Draw two parallel Ordinates M P, mp; draw
alſo the Chord M m, and produce it till it meets
the Axis (produced alſo if need be) in T, and

1 complete the Parallelogram Po m R; then will the

Triangle RMm be ſimilar to the TriangleP MT,

and we thall have RM to R (or P as MP
„ e P
te P T; whence P 1 : M P * — — Let

RM

2 . now

FLux1oNs exemplified. 9

now the Ordinate p be conceived to move always

parallel to itſelf, till at laſt it arrives at M P, or at
leaſt to come. infinitely near it; and during this

whole Motion, let the Line M T be ſuppoſed to

turn ſo upon the Point M, as always to paſs
through m; then it is plain that the Line MT

will at laſt become a Tangent to the Curve in the

Point M, and the Line PT a Subtangent; and
if we call AP, x, MP, y; we ſhall have at laſt

Pp=x, R M=y, and the Subtangent T PS To

5
Compute therefore, from the Nature of the Curve,

the Quantity 75 and you will have the Subtangent

P T: Asi in the following Examples. 1

ExanPLE *

Suppof ing all Things as before, let AM 35 the ET

common Parabola, whoſe Parameter let be p; then

we ſhall have p x =” . and p 29, and x =

2

PT, ee, AP; as we and |

2 and y = = 1 5 , or the ee :

in the Conic Sections.

EXAMPLE I.

2 the Nature of the Curve be chprifſed by this
Equation p =y”; Then we ſhall have pm N
ge n and m * R ES = nyX3 wa,

| or

10 TheDiret Method of |

m

or — =— — : Hence m# =” and np.
5

egen. =—===X AP.

Tfpx=y; then will PT = AT and & M will
be a right Line, becauſe x: ::

Call the Power of the Hyperbola 5 and (by

kay Conic Set. Art. 101.) 2 ==); but

2 ? =?X7 N= therefore . 51.

Then wil PT (Fg. 3.) =—AP, the Sign —
ſignifying that P T the Subtangent does not lie on
the ſame Side the Ordinate PM with the Abſciſſe
AP. This is agreeable to Conic Sectious Art. 107,
| where C anſwers to our A, H to our P, D to our T.

If =. then ꝓ & = , and PT=— |
” Sa&P.”

ExAM TI III. Fig. 2.
Let AM B a be an Ellipſe: Call AC, f; CB, c;
C P, x; MP, y: Then we ſhall have, from the

the Nature of the Ellipſe, 6 CT = ,

and eo, dern _ — 29% therefore

— —

, andy x XI , and
tz
. ine | * ; the Sign — ſignifying

only that the Stet P T is not now to be on
the

FLVUXIONS exemplified. 11
the ſame Side of the Ordinate with P C or x, but

: | on the contrary Side. Further, ſince y* = > —
2 443 0 72

905 Y = SE XP, if inſtead of

1 *in ” W l Value of P I, we ſubſtitute its

Equal 7 1. x, and we ſhall now have

If CO ————— ft XxX 7
. P TSA NR XE —* = 3 and if to

this Value * P T we add C P or *, we ſhall have

0 12 = whence we ſhall have the following

Proportion for finding the Point T, viz. C P:
CA:: 1.

Of the Deftrine de Maxis er MIN IBIS.
5 Rei

To determine when a flowing Quantity 2 a
Maximum or a Minimum.

SOLUTION.

As the Fluxion of every flowing Quantity is
affirmative or negative, according as that Quantity
is upon the Increaſe or Decreaſe, it follows, that
when a Quantity becomes a Maximum or a Mini-
mum, its Fluxion will be equal to nothing. Take,
therefore the Fluxion of ſuch a Quantity in general,
and then ſuppoſe it equal to nothing, and the
Equation will determine the Caſe wherein it is a
on Maximum or a Minimum.

he

1

12 The Direct Method o

9

ExAU L I.
Let it be required to divide à given Numer, as a,
into two fuch Parts that the Produt® of the Multi-

• of thoſe Parts may be the greateſt Poſſible,
  Here putting x and @ — x for the two Parts,
  their Product will be a * — x x, whoſe Fluxion
  ax — 2x x (I ſuppoſe) = %, and find xr =3 a;
  which ſhews that the Product of the two Parts will
  be greateſt when they are equal to each rs

EXAMPLE 1

Let it be required to divide à given Number, as a,
into two ſuch Parts, that one Part being multiplied
into the Square of the other may make the greateſt
Product poſſible. –

Here I put x for one Part, and x x for its Square,

and conſequently a— x for the other Part ; then
will the Product be ax* — x3 whoſe Fluxion
24x K $4? x (I ſuppoſe) = &; and then find
x= N or 4 4, which ſhews that if the Part to be
| ſquared be 2 of the whole, and conſequently the
other Part g, the Product will be the greateſt pol-
ſible :: And ſince in the foregoing Equation, x was
alſo found = X, that ſhews, that if x be leſs than
26, the Product _ be leaſt when x (or the Part
to be ee | –

IRE MER III.

. ; required to determine when the following

Quantity becomes a Maximum or a mn vi.

• r | 5 5 29

8 | SE. Here

Ball deſcribes the curve Lane AG D, it would, by 55

Fluxlons exemplified. 13
Here we have 3 * * — 36 * * ＋ 96 x .
1 conſequently x * — 12 x + 32 = X, whence
x —40r 8; therefore the Quantity propoſed is a
Maximum in one Caſe, and a Minimum in the other.
But to determine diſtinctly towhich Caſe the Maxi.
mum belongs, it muſt be obſerved, that the Fluxion
of the Quantity propoſed is either affirmative or

negative, according as the Quantity x x — 12 x
＋ 32 is ſo: But this Quantity when K = & is
affirmative z therefore the Quantity propoſed is

upon the Increaſe till x = 4, when it becomes a
Maximum; after this it decreaſes till x= 8, in
which Caſe it becomes a Minimum, then 1 it increaſes

ad n

| EXAMPLE IV.
7, 0 find. the greateſt Random of a Ball ſbot with 4
given Velocity. Fig. 4.

Let ABC be the Direction of the Ball, and let

the Force given it at A be ſuch as alone would have

carried it from A to B, whilſt the Ball by its
Gravity alone would have deſcended from A
through a known Space as 5; then will A B and
be conſtant Quantitics, becauſe the Velocity of
the Ball at A, and the Force of Gravity are ſup-
poſed the ſame, whatever be the Direction AB C.
Let the horizontal Line A D be the Random of
the Ball, and draw DC and B E Perpendiculars

r A&A IK} Lally compleat the Parallelogram

ACDF. This done, it is evident that whilſt the

its

14 The Direct Method of

its projectile Force alone have been carried from
A to C, and by its Gravity alone from A to F :
For the Force of Gravity acting in a Line parallel
to CD will neither help nor hinder the Motion of
the Ball towards that Line CD ; and the ſame may
alſo be ſaid with reſpect to the projectile Force,
and the Line D F ; whence it follows, that whilſt
the Ball by its projectile Force alone would have
paſſed through the Space A C, by its Gravity alone
it would have deſcended through the Space A F or
CD; and ſince the Spaces through which heavy
Bodies fall from Reſt are as the Squares of the
Times, it follows that as the Square of the Time
wherein the Ball by its projectile Force alone would
have deſcribed the Space A B, is to the Square of
the Time wherein the ſame would have deſcribed

the Space A C (i. e. as AB is to Ac or as KE“

to A ſo is to C D; therefore CD .

AE
But as A E is to A D o is B E to CD; therefore

CD is alſo equal to BEI ——- 2 TE therefore BE *
2 AF.
E= N FE: ; therefore AD=AEXEB.
4
therefore the Random AD will be the greateſt
AExE B

when — 5 becomes a An i. e. when
AE x EB becomes a Maximum. | |
Call AB, ; BE, x; AE, ; at in the
preſent Caſe we ſhall have x X +) xXx=*.

• | But

Fu Ux1oNs exemplified. 15

2xx 29% = X; therefore y = — 55 z there-
fore x X 2 S dekees v ** = XJ

XXX 5% -x TTX — XXX _
LEY 7
therefore rr — 2 * &; therefore x * ᷑ .
and yy r; therefore x =y: Therefore to
give the Ball the greateſt Random poſſible, the
Piece muſt be elevated to an Angle whoſe Sine and
Co-: ſine are equal, z. e. to an N of torty-iye |
Degrees. Q E. I.

RBI . —
.

8

If the Piece be elevated to an Angle of above
45 Degrees, the Random of the Ball will be the
ſame as if it was elevated to an Angle as much leſs

AExXEB
than 45 Degrees ; becauſe the Quantity —

will be the ſame in both Caſes.

EXAMPLE V. Fig. 5.

To find the moſt cemmodious Situation Fa Plane to be
turned about an Axis by a Wind toboſe . is in
the Direction of that Axis.

Let both the Plane of this Page, and the Axis
of Motion, ſuppoſed at ſome Diſtance under it, be

imagined in an horizontal Poſition. Let the Line
A B be parallel to the Axis, and let C BD be the
Plane in Queſtion, in a Poſition perpendicular to

| |

But xx «| yy =7 7 a conſtant Quantity; therefore

; . —
rate * ae y * : : ——— \ ems 8 8 , —
— r ———— ͤ ants SI P N — — ——— o — OO .
. _ —_ \ N 8 1 —
wo — ode l — Are vos hh * 2 r
— wa — 2 — Pens * . ene dhe 6 * *
x — — — 4 ——— –
— oo — ” a .
K 11 —— 4b Re et eA yas —— ; — — * – n _ ai

— — ——————— —— —

• – 1 — =?

26 Tue Direct Method of

the Horizon, inſiſting with its lower Edge upon
the Axis of its Motion, and repreſented by its
Interſection with the Plane of the Paper in the

Line CD. Let A E be perpendicular to CD, ſo

that the Angle A BE may be the Inclination of

the Plane to its Axis. Let E F be perpendicular,

and C G be parallel to the Line AFB; and let

DG be perpendicular to GC. Let XB; or the

Cube of A B repreſent the abſolute Force of the
Wind upon the Plane CD, when directly expoſed
to its Current; then will the Force ef the ſame
Wind upon the oblique Plane C D be diminiſhed
in the Proportion of AB to A E or of AB? to
AExAB’ I ſay of the ſame Wind: But the
Number of Particles now acting upon the oblique
Plane C D will by its Obliquity be diminiſhed in
the Proportion of DC to D G, or of A B to AE,
or of A Ex AB to KE x AB; wherefore upon
the whole Matter, the Force of the Wind upon
the oblique Plane C D muſt be repreſented by

AE’x AB. By this Force the Plane C D, if left

to itſelf, would move not directly before the Wind,
as when directly expoſed to its Current, but in the
Direction of the Line A E: This, I ſay, will be
the Caſe if the Plane was abſolutely at Liberty;
but, in the preſent Caſe, the Plane is reſtrained

from being moved in any other Direction but that
of F E perpendicular to A B, by which Motion it

carries round its Axis; and the Conatus of the Plane

to move in the Direction A E will be to its Conatus

AH

Fux TON exemplified, 17

to move in the Direction FE, as A E to E F, or
as AB to B E, or as KEA B to AFEXEB:

Therefore the Conatus of the Plane to move round
its Axis muſt be repreſented by AE* X E B, and
Will be a Maximum when, ſuppoſing A B conſtant,
the Quantity AE xEBis “Hi

Call AB, r; BE, x3 and we ſhall have
AF* X EB = “x — x3, whoſe Fluxion FIX

E ** N (in * Caſe)= * ; whencerr — 33x N

and x = “0 Therefote the Effect of the Wind ;

will be the greateſt when the Plane is inclined to

its Axis in an Angle whoſe Co-ſine is to the Ra-
dius as 1 to V3; J. e. in an Angle of 54 Deg. 44

Min. This is the Caſe when the Plane is in a
Situation perpendicular to the Horizon ; and it will

be the ſame in all others, provided that the Angle

which the Plane makes with its Axis be the lame.

QE. I.

.

a: ExaMPLE V. Fig. 6. |
To conſtru# the Fruſtum of a Cone, of a given Baſe
and Altitude, which moving according to the Direc-
tion of its Axis, with its ſmaller End againſt the
Parts of an uniform Fluid ſhall ſuffer the leaſt Re ſt-

‘ ance poſſible from it.

Let the iſoſceles Triangle ABC repreſent a Cone
generated by its Revolution abour its Axis ADE.

Let FD G be parallel to the Baſe; let FH and

G H be Perpendiculars to the Sides ABand AC
i © | | reſpoc-

18 Tie Direct Method 7
reſpectively ; and let F I be parallel to A E cutting
the Baſe B & in I. This ſuppoſed, let us firſt in-
quire what Effect a Set of Particles moving in the

Direction AE would have upon the Cone when

at reſt: In order to which, let the Point F be

| Tuppoſed to be ſtruck by one of thoſe Particles,
and let A H repreſent the abſolute Force with
which the Particle moves, and with which it would
ſtrike the Baſe of the Cone if directly oppoſed to
its Motion; then it is plain that the Point F re-

ceiving the Stroke obliquely will only ſuſtain the
Force of a Stroke equal to FH. Let this Force
FH be reſolved into the two Forces F D and DH,
and the former of theſe FD being ſuſtained by
another contrary Stroke G D coming from G, will
have no Effect upon the Cone; but the other
Force D H meeting with no Oppoſition will have
its full Effect upon the Cone to move it in the
Direction D H, if not otherwiſe ſuſtained ; there-
fore the Effect of a Particle at the Point F in the
Conic Surface, will be to the Effect of the ſame
Particle at the Point I in the Baſe, as D H is to

AH, or as AHxHDto AH,, or (becauſe of
the ſimilar Triangles HF D, HAF, where HD
is to F as H F to H A) as HF* to K H’, or as
BE* to AB*. But what hath been ſaid of the
Points F and I will be equally true of any other
two correſpondent Points whatever; therefore the

Effect of any Number of Particles upon the Baſe
of the Cone, will be to the Effect of the ſame

Particles upon its Surface, as AB*toBE.

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Fr ux10ns exemplified. 19

And ſince Action and Reaction are equal, if we
now ſuppoſe the Parts of the Fluid at reſt, and the
Cone to move uniformly in it according to the

Direction of its Axis, the Reſiſtance the Baſe of
the Cone would meet with, when moving in the
Direction AE, will be to the Reſiſtance its Surface

would meet with, when moving with the fame

Velocity in the Direction E A, as AB? to BE »
Let then B repreſent the Reſiſtance of the Baſe, .

and conſequently B E* that of the Surface, and let
the Cone FAG be cut off; then will the Ręſiſt-
ance of the Baſe BC in the Direction E A, be
to the Reſiſtance of the Baſe FG, as the

Areæ of thoſe Baſes, i. e. as the Squares of of their

Diameters, 1. e. as BC. to FG’, or as AB to
AF: Therefore ſince AB repreſents the Reſiſt-
ance of the Baſe BC, KF will repreſent the Re-

= Aſiſtance of the Baſe FG; therefore BF will re-
| preſent the Reſiſtance of the Surface F A G. From

B E=, the Reſiſtance of the Surface B A C, ſub.
ſtrat FD” the Reſiſtance of the Surface F A G,

and add A F* the Reſiſtance of the Baſe F G, and
we ſhall have the Reſiſtance of the Fruſtum BF GC

(when moving in the the Direction EA) = BE* +

AF – FDB =BE + AD. But XB.
KE ED! =AFE—2AExED+EPD:
Therefore the Reſiſtance of the Fruſtum will be
E E* Rn 2B.

| .

20 The Direct Method 7

— ak ExED-þ+ED*. This is upon a Suppo-
ſition that the Reſiſtance of the Baſe B C is called

B’; but if this Reſiſtance, as being by the Suppo-
‘ ſition a conſtant Quantity, be called 1, the Reſiſtance

of the Fruſtum will be * AB’—2AExEDB+ED.

DE’—2AE E D *
or 1 14 A = : And therefore when

this Quantity (ſuppoſing DE and E B conſtant as

in the Problem) is a Maximum or a Minimum, the

Reſiſtance of the Fruſtum will be ſo. – _ |
Call DE, a; B E, ; AE, x; and we have

E DE —2AEXED 44 — 24%
1 hw = “Eh — 1+ Pr ? whoſe
24 X * X — 244x X — 24bbx
Fluxion — „ (in our Caſe)
=X; whence x #—ax — bb = N. and x === a +

— * 53. But » the Height of the whole Cone

105 be greater than 4, which is but the Height of

y— z —— —-,—

2 4
HT
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Part of that Cone ; therefore * W 33

whence we have the following Conſtruction.

Since B C the Diameter of the greater Baſe, and
DE the Height of the Fruſtum, are both given;
biſect DE in K, draw K C, and produce E D to A,
ſo that K A may be equal to K C; then will the
Triangle ABC repreſent a Cone, whereof the
Fruſtum BFG C ſhall meet with leſs Reſiſtance,

in

Frux1@Ns exemplified. 21
in an uniform Fluid, than any other of the ſame
Baſe and Altitude. QE. I.

That this Reſiſtance is a Minimum, and not a
Maximum, is evident from hence ; that the Sign of
the Fluxion of this Reſiſtance depends upon the
Sign of the Quantity xx—ax—b 3, which when

• Da is negative; and therefore the Reſiſtance of

2 the Fruſtum BF G C will be leſs and leſs, till the

as Cone AB C, whereof it is a F ruſtum, has ob-

• rained the Height png ages,

: 4 Conal L. J.

25 Therefore the Fruſtum B F G C meets with leſs

mn. Reſiſtance than a Cone of the ſame Baſe and Al.

• | Hoe, 5

ſe) Cone

• The Reſiſtance of the Fruſtum BF G C will be

8 | to the Reſiſtance of any other F ruſtum B L MC

ne of the ſame Baſe and Altitude, as NB B is to

NB ARSE XN. 5 ſuppoſing N to be the ver⸗
tex of Ro Cone, whereaf BLMC is a Fruſtum,

. o — + — ; 8 =
i Y ‘
ry 2 7 | 7 0 12
— —— x .. — 5 * L

• 7 > – * 1
  0 pl 8 TEM “

| The Direct Method off

Of infinite SERIES.
LEMMA.

Let AB 2 ＋ Cx z, Sc. a- b 24 CY 2, c.
whatever Value is put upon 2; then will A = a,
B =b, CS c, Sc. For ſince in all Caſes AC Bz

＋C ZZ S ZT ZZ, let z = X, and we
ſhall have A=a: Therefore BZ TC z xz =bz

PT EZZ; and B4-C z=d-+ cz univerſally. Let
Z again = , and we have B ; therefore
CSK, and C = c, Sc. Q. E. D.

TrzOREM.
M—2 c |
55 5 55

4— — Da ms, A Se. 3 A denotes
M4 5 OR 7
the firſt Term 2”, B the ſecond, C the third, &c,

DEMONSTRATION.

_ Suppoſing 2 to be any variable Quantity, let

pF4z”=A+Bz+Czz+D233+EX &c.
without Reſtriction : Then ſuppoſing z = , we
have p”=A ; therefore, ſuppoſing z =1, we have

PY =p”þB+C+D-E…. Again, fince _
„ .=þ qa”,
we have B A- Cz2+ P +4 EE |
= Mm NPT. 1e

De AE ( TN =mq
| =.

FLUXIONS exemplified. 23

Pz“ _ An B EAC DZ5 Sc.
7 77 72 P ＋72
therefore mg A gm qB z-|-mqC 22+ mgD 23,
=Bp + 3p Boxz+TDpFaCoxz+
4E p+ 3D qx 23 &c, Therefore, by the Lemma,
mgA=Bp; e P- FBA; mgC=3D p+2C g;
mqgD =4E p+ 3D e. Ou Equations in

their Order give — * = I B, — 2 — * — * =.
m—2 Cg 223 Dg |
5” y * = =; Sc. Therefore

Othereiſe thus :

Let pon = AB z+C 22–D A
as before; nei 23 OI Ah.

| *
4+C q 222+ D q 232 Ge. therefore 5 =

2
met AgET: B +3 Cqzi+iDgzttoc.
as will _ AP Ds the 581 3

m 2 1
| AB bes 24D 21 LE 5 =
P ＋ 42 * = —
Ee, 1E ps: :

m -1
C 4 . There-

. — Method of
Therefore Ke 42 Agx i – ana w

mFr

De rt a,

++ Bgz+3CquÞ TIA G Therefore,
3 \Bo+Ag A,

by the Lemma,

i _
Co+By in. e
Ane m1 en “mr –
=xX Dec Therefore, p- 43 2 1=B;
— — 2 C; — = 5 — 2 E
7 F

Se. as above.

Cox or I. I.

Conor. II.

Log a B
6 . 7 „&.
where A=rp Sc. = | ”

. Con Il.

“NF = KX, the whole Series will be compre-

. hended in the firſt Term ; therefore the leſs q is in

lerer to 2, the faſter will the Series converge.
Note,

12
N

FLUX1ONS exemplified. 25.

Note. ‘The latter Demonſtration of this Propo-
fition cannot well be underſtood. till after the fol-

lowing Problem.

of the Inverſe Method of F..vxIONs.

PROBLEM V.

2 0 find the Fluent of this Fluxion (viz) # * x *.
\

| n
N the Fluent of this Fluxion to be * 5
then willpg x X x17 & x: Therefore pg x x75
=1XX”,; therefore p l, and q—1= nz; there-

fore 9 =1 Co, and y =: 2 5 and the Fluent

ſought will be- Pe- 4

Sonor,

• ** ; will have be its Fluent the Quantity
  4 |

© «

PLEAD of Curves.
Pros: VI. Fig. 7.

Let AM be any Curve, whereof A P is the Axis,

and MP is a perpeudicular Ordinate: I is re-

quired to ſquare, or find the Area of the curvilinear
. APM.

SoLiv-

26 The Inverſe Method –

“SOLUTION. ö
Suppoſe. another Ordinate, as mp parallel te

and infinitely near MP, and compleat the Paral-
lelograms PQ, p 9, RS: Then will the inſcribed

Parallelogram PR be to the circumſcribed one
PS as M P to m ps and conſequently in the pre-
ſent Caſe in a Ratio of Equality: Therefore the
Parallelogram P R will be to the curvilinear Space
P M mp more, if poſſible, in a Ratio of Equality;
and therefore in the Caſe where P and p coincide,
that Parallelogram and the curvilinear Space 1
be uſed for one another.

Call now MP, y; AP, x; and we ſhall 3
P p &, and the curvilinear Space PM mp, or
the Parallelogram P R, will be yx. But this cur-
vilinear Space is the infinitely ſmall Increment or

Fluxion of the Area APM : Therefore if the

| Fluent of this Fluxion be found, which muſt be
had from the known Nature of the Curve, ſuch a
Fluent will exhibit the Area ſought , provided i it

begins to flow with the Area: But if it does not, |

the Exceſs at the Beginning muſt be cut off, or the

Defe& ſupplied, by the Subtraction or Addition

of a known Quantity; and then the F luent win
be equal to the Area ſought. _

Examels I.

Let AM be the common Parabola, whoſe Pa-

rameter let be p; and we ſhall have, from the
Nature of the * P * , and y = p Xx Vr,

4 and

applied to Qu A PRATURES. 27

andy x V, whoſe Fluent is à Vp x xt,

or ZX VX V xx, or 4 xy : Therefore this Quan-
tity 2 xy flows with the ſame Velocity with the
Area itſelf. But this Fluent + * is nothing when
x is nothing; ; and the ſame muſt be pronounced
of the Area APM: Therefore the Fluent and the
Area begin to flow together, and conſequently the
Area will be m_—_ to the Fluent; i. e. the Area
APM will be 3 APxPM, or two Thirds of
the drevinitiibed Parallelogram P Q.

Note. It may not be improper to obſerve here,
once for all, that if inſtead of making p ,
we had ſuppoſed 1 X# , and inſtead of making
AP the Axis, we had ſuppoſed it any other Dia-
meter of the Parabola A M, the Proportion be-
tween the Area APM, and the Parallelogram
P Q would ſtill have been the ſame. For how-
ever the ſeveral Areolæ PMR would have been
changed by ſuch a proportionable Alteration, and
the equal Inclination of the ſeveral Ordinates M P;
the ſame proportionable Change (and that for the
| ſame Reaſon) would have happened to the Paral-
lelogram PQ |

EXAMPLE II.

Let it now be required to meaſure the external |
Space A QM. Since as before X AP = PN“,

or PX QMS AG Q*: If we call AQ, x; and QM. .
ye ſhall have p xx, and == „ and * 3

whoſe

— — Ent —
CCC ˙—ü ⁵˙— rs nt MA Ss

‘

28 The Inverſe Method

whoſe Fluent 2 or 7 will be the Area ſought; ſo

that the external Area A Q is a third Part of
the Parallelogrom P Q; which confirms the for-
mer Example.

Hin 4 ER Ul.

Let the Area of the perabolic Segment C M =
be required.

Since A B=BT?, and NAP PR ET 3

we ſhall have px BP, or NMT, =BC*—BT =;

which is equal to the Rectangle CT E. Call

now B C, 3; CT, «; TM, y; and we ſhall have

5 X—
2y = 2b#—=x x, ad ==, 14

By | b
— 2 3 whoſe Fluent — . will be the
Area of the Segment CMT ſought.
Note. When T coincides with B, we ſhall have
x—= BC, and the parabolic Space. C A B will be

2 3
found equal to B C3 . 3 ABN

Bo 7
2 —2AB xBC; which confirms the is

mer Examples.

Ex AM ILE IV.

Let the Nature of the Parabola be expreſſed by
this general Equation 1 Xo =. |

Here

applied 9 QuapratuREs 49

Here we have you 4, and y X x, whoſe

Fluent —— ate n r * will be the Area

ſought APM: Therefore the Area APM will

be to the Parallelogram PQass to mu.

EX AMP IL V. Fig, 8.

Let the Curve XY be referred to the Line AZ
by Means of the Ordinate M P parallel to AV,
both being perpendicular to A Z; and let the
Nature of the Curve be expreſſed by the Equation
A PPM =1 : Then it is eaſy to ſee that the
Lines AZ and A V will be Aſymptotes to the

Curve; and calling AP, x; and PM, y; we ſhall
| : x :
have y = * and v = TI whoſe Fluent 2 /x

1s nothing when x is nothing, and infinite when x

is infinite: which ſhews that all the Area between

the infinitely produced Aſymptote A V and the
Ordinate MP, though infinite in Extent, is never-

2
theleſs finite in Quantity, and equal to 2% or I

or 2 x), or twice the Parallelogram PQ. But

| becauſe the Fluent is infinite when x is infinite, it

ſhews that all the aſy mptotic Space on the other
Side the ordinate M P is infinite as wery in 2
tity as in Extent.

EXAMPLE

30 The Inverſe Method

po ExAMPLE VI.
To confirm the foregoing Example, let the ſame
Curve X V be referred to the other Aſymptote AV
by Means of the Ordinate M Q: Then calling
AQ x; and MQ,y; we ſhall have y .
and y x ==—— KY whoſe Fluent is = Now this
Fluent when x is nothing is = or an infinite ne-

gative Quantity; when x = A Q» the Fluent is

1 6 2 , | | “FOR , FO
Io and when x is infinite, the Fluent is nothing.

This ſhews that all the Area between AZ and M

is infinite, but that all the Area on the other Side

M Q is finite, and 1s equal to 15 or to the Paral-

lelogram P Q, and conſequently that all the Area
between AV and MP is equal to twice the Pa-
rallelogram P Q as before.

ExAMPLE VII.

Op the Nature of the Curve – ip 4 when referred |

to its Aſymptote A be ſuch that y = 2 or y =
n then will yx =p x x N, whoſe Fluent

• er- will determine the aſymptotic Spaces

as follows:

1. If m be leſs than 1, x will be an 1 affirma-
   tive e Power of x, and therefore in this Caſe our

| Fluent
FL

EY

pup a AA © „ wt, +

U > 0.

ED 15
F uent –

x P will be nothing when x is
nothing; and infinite when x is infinite: there-

. fore, in this Caſe, all the Area between A V and
M P will be finite, and equal to the Parallelogram

P Q multiplied into the Quantity == and there-
fore this Area, caeteris paribus, wil I greater or

—is ſo.

leſs according to the Quantity —

20. If m be greater than 1, — will be a nega-

tive Power of * ; therefore in this Caſe the Fluent

1
1— 7

is infinite: Therefore in this Caſe the Space be-

tween AV and MP will be infinite, and all the

Space on the other Side M P will be finite, and
equal to the cen: P Q multiphed into the

Quantity 5

be 3 or a according as that Quantity
g Rage

1

30. From the two foregoin g C aſes, it phones;

5 and cherefore, caeteris paribus; will

is ſo.

| a3 if M=1 as in the Caſe of the common Hyper-

bola, the aſymptotic Spaces on both Sides MP will

I ©:
be infinite; for _ the Quantities 2 —

will both equal = and . will both be

5 infinite. |
Dae

applied to QuADnatynes 34

— * will be an infinite Negative when x

— —— 5 — –
— — — Ar 0 . . —*˙L⁰?̃ můͤi2! -. af w_

— — — —— —

32 The. Inverſe Method

Examyrs VIII. Fig. 9.

Since in the common Hyperbola both the aſymp-
totic Spaces are infinite, all that can be expected
here is to be able to meaſure any particular Part
of theſe Spaces. As for Inſtance: Let MN be a
common equilateral Hyperbola, whoſe Aſymptotes
are AZand AV, and let it be required to mea-
ſure the Space MP QN comprehended between
the two perpendicular Ordinates MP, N Q. This
may be done various ways, but the moſt commo-
dious is that which follows:

Biſect PO in B, and draw the Ordinate B .

Call AB, r; BP or BQ, x; and let the pig of | A

the Hyperbola be 1; then will MP = –

NQ = — Let the Ordinates M P, N Q re- ;

move into the Places p, u infinitely near the
former, ſo that the Space M P N Q. may open

into the Space pn: Then it is plain that the
Increment or Fluxion of this Space will be 2

x & 2 1 X

72

MP NN Qqn= Fs 1 Tx 6 Now 3
as this Fluxion is not of the Gmpleſt Form, neither |

can its Fluent be obtained by the ſin _ Method;

to gain this Fluent the Quantity —

21&

throw into an infinite Series . 3 == + 5
. |

2Xx? 2K 1 2xx* , 2xx

plain

„and

muſt be i

72 T 7 5 + _ ＋ Ee. Here it is 7

S e SRD

.

V

ain

Des

applied to Qu ADñRA TURES. 33
plain that the Fluent of every Term of this Series

may be found by the — Method; ſo that the

3
F _ of the whole will. be — —++=4+22 2X7

373 573 777

Se.” which Fluent will 4 _y to the Area
ſought MPQN. Or thus :

ks Cw.
Make v = A, n — =C =,

=L Sc. and the Area M p N will be “=o |

to the following infinite Series (viz.) 275 5 ＋ 5
L DTE
| He.
N

1 Conn h |

Whatever be the Quantities r and x, ſo long as
they have the ſame Proportion to one another, the
Area M P QN will be the ſame.

Corotk i

Or if the Ordinates MP, NQ be fo taken that
A Q ſhall always have the ſame Proportion to A P,
the Area M P QN will always be the ſame.

For let 7 and 5 be two given Quantities, and let

AQ and AP be two indeterminate Aſciſſae; but
let AQ always be to AP as r to 5; then will
AQ+APbeto AQ —_APasr+5 tors.
But AQ+AP=2A B, and A QA P=2BQ,

and 2A B is to 2BQ as AB to BQ; therefore
A is to BQ as r-Þs tor,; therefore the Ratio
of AB to BQ will Han. be the ſame; therefore,

234 ze Inverſe Method

by the laſt Corollary, the ive MP an will al-
ways be the fame.

Cox 61 1. III.

If the common equilateral Hyperbela, whoſe
Power is Unity, be referred to its Ahmptotes, and
if the D ference between any two Abſciſſae be to
their Sum as 1 to 23 and laſtly, if we make

== Aos A —B; . 4 DE: Then

2 22 * 22
will the Area eee. bediene of the above-

mentioned Abſciſſae be A — ＋ SPE 75 |

For in the foregoing 8 the Difference 5

between F—X and Bhs is 2x, and en Sum is ar,

• 1 2X
  38 and — r A 2
  r *

; of the Computation of Brigg ‘s Log ARI T HMS.

TuEOR Eu.

It che common yperbola be referred to either

• its Ahmplotes by Ordinates parallel to the other;

I ſay that all the aſymptotic Spaces will be Loga-

ritkms of the. Abſeiſſae where they terminate; pro-
vided that theſe Abſciſſae be computed from the
Center; and all the aſymptotic Spaces from one and

the ſame common Ordinate whoſe. Aue 1s U-
iy.

“154 4 ö

• D
  EY * Ka
  — 9
• ES. 7
  — 00
  Y *

yu

—

applied to LoOGARITEMS. 35
Let a and 3 be the Values of two Abſciſſae

reckoned both from the Center; and let s be the

aſymptotic Area reaching from the Abſciſſe 1 to

S the Abſciſſe a, and : be a like Area reaching from

1 to h, or from a to ab (ſince as 1 is to Þ fo is a
to 2b*): Then ſince g reaches from Ito a, and
t from a to ab, the Area 5 will reach from the
Abſciſſe 1 to the – Abſcifle whoſe Value- is 2 5

therefore the Area 5 and ? are ſuch whoſe Addition
will always anſwer to the Multiplication. of the:
Abſciſſae where they terminate; therefore _ are
the Lene of thoſe: Abſciſſae. Q. E. D. =

R EXAMPL E.
To cms “the by perbalic Logarithm of 10, that is,
to find in the equilateral Hyperbola whoſe Power is

Unity all the aſymptotic Area from the “”_ I
to _ Abſciſſe io.
| 1

1 : be made equal to A, 5 5.5 0 =C, — 5 =D,
Se. the FRI a the Abſciſſe 1 to the |

• 2, Of the hyperbolic Logarithm of 2 will
  DiE

be A- ye —+— DIES Sc. But the Logarithm

of 8 is 3 Times the Ld of 2 therefore if

inftead of making A = 2 we make A = 2, and

thence derive the neher Terms B, C, = E, 1 as

before, we ſhall have A = 2.4.3 4-2 . „Ee.
the hyperbolic Logarithm of 8. Again, if we

• See Corll. II. next preceding. |
  D 2 make

We Inverſe Method

may 8 * from 8 to 10 will be

K+ =+=+5 7 . But K or 2 = B in the
foregoing Series; therefore L or 125 or : 15

81
In like Manner M=F, NH, Sc. and therefore
RG 7 a.”

be added BES =+= += &c. we ſhall have the
hyperbolic a of — 302585092994.

A fuller Computation of the H yperbolic
Ne, LoGARITHM of 10.

2,00000000000000 |
522222222222222
2469135802469
274348422497
30483158055
“3287017502
376335285
41815032
4646115
516235
57359 INS

6373. |
      5
      1829.4 9 & Þ a

I

F<

:

4

• 27
  1

8

DO
E

.
1

21 5114

vi: ; ©

AN –
N

wþplied to L,0GARITHM
| 2,00000000000000.

os, 8. 4

So IZ SIE ME SIN I- IN ol lian

—

7407407407407

| 493827160494

39192637785

3387017562
307910687

28948868

2787669 |

273301

% N

3

BIBT (thi)

• 
  277
• 2 28

2,07944154167982 = Log. 5. |

3

—

p
‘
FT:
I
j
Ft
I
.
N
7
N
154
N
}

N.

8

38 The Tnverſe Method applied

Log 8. – 2,07944154107982
e

• * 5

91449174166

— .

677493512
5973576
57353

579

4
% 4

*%.
ws „

6

4 8 /
}

0
5

pg

Log. 2,302585092 99402

To ener the” firſt Term of the foregoing Series,
ſo that Briggs’s Logarithms may be produced by
it; that is, à Scale of Logaiithms wherein the
Logarithm of 10 is 1; fay, As the hyperbolic
Logarithm 36k, 40 is to Briggs Logar thm of |
10, that is, as 2 302 SBSOgZINE 7 is to 1, fo is

2 the Coeficient of the firſt Term = = of the Series

that produges the hyperbolic ihm. to
0, 868588963806 the Coefficient of the firſt Term

of the Series that produces Briggs s Logarithms;

. 685889638 A
bega te make 259 96380Q met; =

Rs, 2s
» do —
n
TS Im

2 :

to ihe Menſuration of SoL * * 29

=C, Ce. ihe Sum of the Series A A == x98,

27

wilt be Briggs s Logarithm of —= * * fo, Sum
of this Series will be the 3 — from the
Abſciſſe z—1 to the Able ſſe z Tr, or which is
the ſame Thing, from “the Abſciſſe 1 to “the Ab-

Ain whoſe value is .

5
9 “=
—

Coder

The hyperbolic Logorithm of any 3b
is to Briggs’s Logarichm of the “ES as I to

| 434294451903:

“of the Menſuration x 80 L1 Ds.

PROBLEM VII. Fig. 7.

To find the Content of a Solid generated by the Revo-
lution of any — Curve AM about its Axis
AT: –

Let the Diameter of a Circle be to the Circum-
ference as 1 to e. Call AP, x; PM, y; and let
the Ordinate M P make an infinitely ſmall Remove
into the Place mp; then it is plain, that as the
Parallelogram P R repreſented the Fluxion of the
Area APM, a Cylinder generated by its Revo-
lution about AP will repreſent the Fluxion of the
Solid generated by the whole Area APM: But
the Baſis of ſuch a Cylinder is ey, and its Height
or ene 5 Compute therefore the Fluent
D 4 n

———

—— —

i

[}

|
|
[;

l f
U

}

40 We Inverſe Method applied

of ey y, and that Fluent will determine the Solid,
juſt in the ſame Manner as wo Fluent -w Js de-
termined che Ars.

— 4 * wv >
$ 4
MW. > £1

Ex AMF i I.

Let the Nature of the Curve by whoſe Area the
Solid is generated be expreſſed by this Equation

p, ſuppoling m and x integral and affirma-

— 2m

tive: Then will 5 ff x, and yy e p K,
and AN 3 the Fluent whereof is

• eff, mn KEI. But 0
  is a Cylinder having the ſame Baſe and Altitude
  with the Solid propoſed: Therefore the Content

of the Solid propoſed is equal to a Cylinder of the

; e * pied into 2 F rac-

tion – 271 1

Ex AM L E II. Fig. 8.
Let it be required to find the Content of the

Solid, or rather Part of the Solid, generated by

the Revolution of the Curve X Y about one of its
Aſymptotes A Z. Let. the Nature of the fat

when referred. to. its Aſymptote A Z, be 22

Sbx re; rden will e e and 2

— — ”

epa, whoſe Eluen: — per- ſhews,

That if 2m bc leſs than Y, all the Solid generated by

• ma

generated by the Area C B Aa ee

Teo the Menſuration of So1:1Ds. TT
the Area between AV and MP is finite, and
equal to the Cylinder generated by the Parallelo-
gram P Q multiplied into the, Fraction W 4

the other Part of-the Solid being infinite. On the
other hand, if zm be greater than 1, all the Solid

between A V and MP will be infinite; and the
infinitely extended Solid on tne other Side PM
will be finite, and equal to the ig

ed Cylinder multiplied into the Fraction –
Of this latter Kind, among many others, is _—

Solid generated by the Hyperbola ; which will al-
ways be equal to the Cylinder generated 9 me

• 71 P 8 *

i Ex AMPLE. UI. Fig. 10. F
7 0 find the Content of a Spheroid generated by the
Revolution of the Semi- _ A MBa n ils
“ws AF
Call CA, t CB, c; CP; wy: MP, „; ati
ſhall have, from the Nature of the Ellipſe, y be
2 and e 55 ese whoſe Fluent

• ece x
  COON

is the Content of the Solid generated

by the Area CB MP; therefore if we ſuppoſe C P

or x CA or 1, we ſhall have the Hemiſpheroid
4 .

8 .

ec xt: Therefore the folid erstes of the whole
N Spheroid

— In TO ens
_ CE ee é⅛[ —— — —

42 Wye Invetſe Method applied

Spheroid is equal to 2 ecx 27: But ec X 21 is the

Content of a circumſcribing Cylinder.

Of the Surfaces of Curvilinear’S o LIDS.

PR OB. VIII. Fig. – ts
As the curvilinear Area AMP revolving about

i Axis A P generates a Solid, ſo the Curve A M

by fuch a Revolution generates the curve Surface
of that Solid. Let it therefore be required Ta find

| the Surface gentrated by any known Curve A M re-
vol ving about its Axis AP.

Loet the Diameter of A Circle be. to “he Circum⸗

Tervrice 4s x tos, and Ce 2 x; N P, 55 then =

— *

M m is the Fluxion of the Curve A M, © A M by
its Revolution about A P generates a Surface,

M will generate the Fluxion of that Surface,

which will be an Annulus whoſe Breadth is Mm,
and whoſe Circumference is equal to the Circum-
ference of a Circle deſcribed hy the Semidiameter
PM: Therefore 2 e VN +55 will be the Fluxion
of the Surface; and its Fluent, when found from
the Nature of the Curve, will determine the Sur-

TA CE aIOT 101 13.317 :;

II! IExAMY EE I. [Fig. 11.

115 AM be a Semicirele upon the Center c Z
pd let it be required to find the Surface of a Sphere
generated by the Revolution of that Y bci about its
WN N44. a

«-

0 al

fo the Menſuration of Curve SURF ACES. 43
Call AC, 7; AP,x;MP,y; Pa, 2r—x; and

we ſhall have H araxx, and eee

a N

and — 7

— —
| . , ; — &. 2 4828
0 * »- g 1 *
53 iS mm

19x? — 22 —

= — -K: Therefore * ==

WW

2 . IF)

| therchregy x2 L = and 0 Heere,

and 2c X Vi. (the Fluxion of the Surface
ſought) =eX 27x, whoſe Fluent ex 27x will be the

Surface of a Segment of the Sphere, whoſe Axis

is AP (x). But from the ſimilar Triangles A PM,
AMa, we have AP AM as AM \M to A, 5

| conſequently APxAa, or ar x=AM: 3 therefore

e N 2r38=eXA M:*: But ex AM is the Area of a
Circle whoſe Radius is AM; therefore the Surface
of any Segment of a Sphere is equal to a Circle
*whoſe Sen neter is a Line drawn from the Pole
of the Segment to the Circumference of its Baſe;

therefore the Surface of the whole Sphere is equal

to a Circle whoſe Radius is the Diameter of the
Sphere, or, which i is all one, is equal to four great

Circles of the Sphere.

Exam ris: II. Fig. v2. ds

; Lt it be required to eftimate the Surface of the parabolic

Solid generated by the common Parabela A M about
its Axis AP.

Loet F be the Focus of the Parabola, and wrt

wal m; AP, x3 MP, ; and we ſhall have

2 x, and 2yy Amn x, andyy = 2m x, and
*P

4
14
_ :

I

+}

• Þ
  [2
  |
  f
  o
  h

44 + The Inverſe Method

M Am, andy X x , , 1
=4mxX* AS,, or amx + A XK therefore

• Tan n, and 26 K
  =20Xx nN Lam. Now, to reduce this Fluxion
  2c X 4m K- Fam into a more convenient Form,

for the better finding of its Fluent, make „Am = 4
then we ſhall have N =, and 4m x m m Am 2,

and 28 x * VImx 4am ex Vn, whoſe
Fluent is 2c * X 2XV4amz. But at the Beginning
of che Surface of the Parabolic Solid. that is, when
K , this Fluent is 20 X Vn n; which Part
being ſubtracted ſrom the e Fluent, to
render it equal to the Surface ſought, we ſhall have
chat Surface Sue 102eX 4 in 2—20 * + n MX
VI. !

Nox to conſtruet 5 Fluent, that | is, to find

„

pls, ſet off from the Point F. on n the Aziz con-
trary to FA the Diſtance FQ=AP,; and through

F and Qdraw the Chords LI, Nu, perpendicular l

to the Axis; and ſince FQ AP, we ſhall have
AQ=A P–A F=x-|-m=z, and N Qn A
and 32x% 4nz=2%A Qx N’Q whichis equal to the

patabolic Space A MN; therefore 2X K Vn
is equal to the whole Area AN; and for the

ſame Reaſon 2 x 2 Am m is equal to che Area
AL l, and 2 * 2 2 21 * Vn n is equal

to the Area L oy ul, and 2c X 2 ;x/4 Mm E— 20% 3-the

Vn n

| “er We Y —_—— 1 1

s ace

ofplied t ov eοπ¹ο) . 45

a 4mm is equal to the foreſaid Area multiplied in-
to e: Therefore, as the Diameter of a Circle is to
the Circumference, ſo Is che Area 5 NET, to the

Surface e

Pros. IX. pig. L3-

It is required baving given A B the Rodi us os any
Circle, together with B C the Tangent of any Ark
as B D, to find that Ark ;, and conſequently from the
given Tangent of ſome known Ark to find the whole
Circumference.

Draw the Secant ADC, and! imagine “Inother
as Adc infinitely near it, and let C E be perpen-
dicular to Ac: Then from the ſimilar Triangles
CcE and Ac B, as alſo ACE and A D d, we
have the following Proportions; Ce to CE, as

Ac or AC to AB; or as AC to AB AC;
and CE to D d, as A C to AP, or as A C to AB,
or as AB x AC to AB“; therefore ex aequo Cc
is to D 4% as AC to AB“. | |

i Now to render the Calculation more ſimple,
call A B, 1; BC, f; then will Cc= 7; AC III,

and D N 2 _ W Sc.
whence B 1 744.9 — Sc.

Now to find the 1 ee ae let BD be
fome known Ark whoſe Tangent is given. As for
Example, let BD be an Ark of 45 Degrees; then
will 1 its “Im B C=B A or 15 and we ſhall have
B D

V

46 _ . The Inverſe Method

B D 21-44 — „ — &c. and conſequen: ly
8B D, or the whole Circumference will be 8 Times
the foregoing Series. But as this Series, how
ſimple ſoever, converges too ſlow for Praftice, it
will be proper to take a leſs Ark than that of 45
Degrees, whoſe Tangent can be computed. As |
for Inſtance, let BD be an Ark of 30 Degrees,
and produce its Tangent BC oF; ult B F=BC;

then drawing A F it is evident that A C F will be
an equilateral Triangle, and conſequently that
AC=2C Bor 27; whence AC (45) C B (7)

ks Ba (1), and conſequently ft ; whence

I I I I
| “Na | —”A3X3 f 3×9. a3 * 47

| * 51 Sc. Therefore in this Caſe BD=

(hon OR I “1A 3 1

3. ate e e, 32

— &c. 5 i- Tf T IT)

— Oc. Therefore 6BD = – multiplied into the
3 Series. But 6B D is the Semi- circum-
ference of a Circle whoſe Diameter is 2, or the
whole Circurſerence of a Circle whoſe Diameter

65
is * and g == for if we make x = = 77 —7 we

ſhall h. ave e and x Via, Thete- F

410 re

applied to Cy CLOMETRY. 47
fore the Circumference of a Circ ircle whoſe Diameter
4 9

hang Tae

—ͤꝛ

is 1is equal t to 1

„ or if we make 1 Sa, 4 * —

== Sc. we ſhall have the Circumference of a

Circle whoſe Diameter is 1 equal ro 4.— A6
Ng e, Sc. |
7 the Work in Ns Tables, Page 55.
The Diameter being 1, the Circumference is

3.14 1592635359. |
Of the Method of Reverling SERIES,

This will beſt be ſeen by the two NY
Examples. 8

1 I. Fig. 9.
het MCN be the common Hyperbola referred

to its A/ymptate AB. Let B C be a fixt Ordinate,
and NQ a moveable one: It is required, having

given the Baſe BQ, to find the Area BC NQ;

and vice verſd, having given the N B CN Qto
N _ ah Bake: B _ | |

— .

SOLUTION.

Ln I 1 the Power of the Hyperbola. Call

alſo A B, Tz BQ * N Qz, d 15 Area
B CN O K*; then we ſhall have y =

yn oe =» and

x
& or 2 = 2 jo” OR” XX —. x. — c.
| Therefore

18 Reverſion SERIES.

Therefore 2 – * 4 3 — KKL, c.
And thus the firſt Part of the Problem is diſpatched.
Let it now be required, having given z, or the
Area BCN Q to find x, or the Baſe BQ; which
is às much as to ſay, that as we have already ex-
prteſſed the Value of z by an infinite Series wherein
the ſeveral Powers of x are concerned; it is now
required to expreſs the Value of x by an infinite
Series wherein the ſcveral Powers of 2 are con-
cerned. In order to which it is neceſſary that the
ſeveral Powers of z ſhould firſt be found, to as
many Terms as the Series is intended to be con-
tinued. Thus:
Equation f 1. Z=xX— EI PETE. 5 1 2.95
„Eq; 2. 2 = Xbox? — l * — + X35 |
Eq. 3. = % XK +#x— +3 xt+ A K
Eq. 4. 21 = & N * F- „ — 2×5
Eg. 5. 1 . % N N N + x5. Having got theſe Equations, the whole Buſineſs of this Problem is from – theſe Equations to find others, out of which all the Powers of x; except the firſt ſhall be expunged ſucceſiively, fo that at laſt x may be found equal to àn infinite Series compoſed of the ſeveral Powers of x. As for In- ſtance. Equation the firſt is & — 4 x Cc. =2.
Now the Term — 4 * is to be expunged, v”y Means
‘of Equation the fecond, thus: & — -F AA x= x5
*=22 ; thetefore 4 = xt £2 . b 225
which laſt Equation being added to n the firſt:
wil give
| vn fm 6 A= 5 — 2 ZZ ..
Here

eſs

Se. or thus:

. Reverſion of 8 E R IE S. 49
Here it is eaſy to ſee, that the next Term to be
expunged is — 2×3, which is done by Equation 3,

thus: x3 — 2 #4- 2 * a3; therefore’+ & +4 x4,

A x5 = ＋ 2 Ih laſt Equation added to
Equ. 6 gives |
Equ. 7. xx X- – x5 = ET TIN
The next Term — 22 & is cxpupged by £4. 4.
thus: x- 2K ; therefore 2 75, x5= 24.

This added to Equation 7 gives
Equ. 8. «& X—7pix5=2+ = E- -A“.
The laſt Term to be expunged is — r x3, which
is effected by * 5, thus: x=25; therefore 74 X58 54, 25, which added t© Equ. 8 gives Equ. 99% X X=2+4 224.7 234-25 Ar
1 24 1
ii 1* 2 + 1 8 1X2X 3X4X5″

Sir IS AAC NE w ron. Theorem for rever/ing

Series’s of the foregoing Kind, viz. N the
Quantity is concerned in every Power.

Sit Z2=4X-b x He 8 A Texs Sc, et viciſſh n erit

Z
x ==

a

42
2 62 —4 c
23
45
5a b c— Gb3—azd4
A * 2+
1 14 ET 6a*b Ad 12 2 Ge

49
E P rom

50 Reverſion of SERIE 8.
From the Solution of the firſt Part of this

Example may be drawn the following uſeful
e (viz.) That the. Fluent of this

Fluxion = Zis the hyperbolic Los richm of x. For
if, inſtead | of BQ we call A wh x; the F luxion of
the Area B C N * be = — : Therefore e converſs

the Fluent of = — 1s the * BCNQ, which

(ſuppoſing A 3 is the hyperbolic n
of the Abſciſſe A Q as . |

ExamMPLz 2. Fig. 14

It is required, having given A B the Radius of any
Circle, together with C D the Sine of any Ark as
BC, 10 find that Ark, and vice verſa.

Let cd be an Ordinate infinitely near and par-
allel to C D, and let c E be parallel to AB. Call
AB. 15 S BC, 2; CE, ); Cc, 2; and
Ab, vi—y5: Then from the ſimilar Triangles
ADC, CEC we ſhall have this Proportion.

AD(Vi—yyistoAC(1),asCE )) is to Ce (2

=—L==j +2534 499+ 55/+ G6
Therefore 2=14- 2 27 “32 52 {= 57+ Sc. which
is the firſt Part of this Problem.

The other conſiſts in the Computation of the
following Equations.

| Equ. 1.

Reverſion of SER1 1 8. 51

3. 25 NK N NX 55 ＋ 5 97
   4+ „ *
4. 2 — 4 BIN 2 -N
5. xX — 2K CTT ZA YK NN NZ
6. Z rz -r 27+ Cc. =).
   Or thus: 1
   FE * x. 27
   7 8 1 bh IX2X3 TN 88 IX2 x ZX4X5XOX7

• &c.

Sir Is AAC NEWTOR’s Theorem for rever/ing

Series s of this Kind, (viz.) where the Quantity is

concerned but in every other Power, * with
the firſs Power.

Sit X y-+b y3-c y*4-dy7 be y9, Sc. et viciſſim
erit

&

2b —a C
a7
8a b c—-22d— 1233

• .
  3 5554.5 3a bac CIO d- ga. – aze

413

• &c, 5

25

E 2 5 PROBLEM

52 Reverſion of SERIES.

PROBLEM X.

I is required to find by an infinite Series the Cofine of
of any propeſed Ark in a Circle whoſe Radius is 1.
Let the Ark propoſed be z, and its Sine y;

then will its Coſine be I). But by the laſt

Problem J=Z—7 23] = 25— 555 -c. There-

fore yy=2%— 4 244 2; 26 rtr 52+ Sc. There-

fore 1—y y=l——a 2 I Bhs oy 264- Arc.

Therefore /1—y (or the Coſine ſought) =1—z

22
1 * 2

• 22 tm FET 26 + r- 25— Sc. 21

hog 8 26 25

IX2X3 x 5 — RF ITE ;
— cc.

School lu u.

Fre rom the Solution of the laſt Problem but one
it appears, that if y be the right Sine of an Ark of
a Circle whoſe Radius is 7, the Fluxion of that

Ark will be ———_ and vice verſa, the Fluent
dan:
of 1.” WP is an Ark whoſe right Sine is y of a
Vr r- .
Circle whoſe Radius is r. And from hence again
it follows, that if x be the verſed Sine of an Ark
of a Circle whoſe Radius is r, the Fluxion of that
r | |
Ark will be EE , and, vice ve,. For in
this Caſe: 21x X N=}; cherefore 27 X=—=2XX=2y 75
a there-

of the Center of GRAVITY. 53
—

therefore 7 _ Subſtitute Vr * X in-

ſtead of y on one Side, and Vr ry inſtead of

rx
Vr xx *
is the Fluxion

7 on the other, and we ſhall have

©
Irr- y 4 therefore: Var xxx
of an Ark whoſe verſed Sine is x of a Circle whoſe

Radius is r. Or more briefly thus: In the fore-
going Figure, CD (Var x—x x) is to AC (r), as

“Ei is to Co (EE 2. ).

Var X—x x

Of the Center of Gzaviry of Bodies.
LEMMA.
Plate III. Fig. 1 git

Let a, b, c, d, be any Number of Bodies whoſe Weights
and Places are all given, and let their Centers be
Placed in the ſame right Line: it is required to find
e their common Center of Gravity.

Let the Line ad | in which the Bodies are all
ſuppoſed to be placed be conſidered as inflexible,
or as a mathematical Lever ſupported at e, but
moveable upon that Center: Then it is plain that
“as e is the common Center of Gravity of the whole
Syſtem, and the whole Syſtem will be atreſtupon that
Point ; therefore the Sum of the Momenta where-
with the Bodies @ and 5 endeavour to turn the
Lever one Way will be equal to the Sum of the
E 3 Momenta

54 Of the Center of GRavirty.
Momenta wherewith the Bodies c and d endeavour
to turn it the other Way. Let axe a repreſent the
Momentum wherewith the Body @ endeavours to
turn the Lever on the Side of a, and the Momenta
of the reſt will be h x eb, cxec, dx e d, and we ſhall
have this Equation @ x ea-Þbxeb=cxecÞdxed.
Aſſume now ſome Point as õ any where in the Line
a d produced, that the Bodies may lie all on the
ſame Side of s; then we ſhall have ea=s es a,
e Be- b, e c c- e, ed=sd—se. Subſti-
tute theſe new Values inſtead of the old ones in
the former Equation, and it will be transformed
into this a x õ – XA , XS e- XS c XC
Dex eds d—dxs ez which Equation being re-
| 2 Xa NSU Tex 4.
ſolved gives e * ES « ,
In Words thus. Multiply the Weight of every
Body into its Diſtance from the Points 5, divide

the Sum of the Products by the Sum of the

Weights, and the Quotient will be the Diſtance
of the common Center of Gravity from the ſame

Point 5. Q. E. I.

S

Hence if any Number of Bodies a, b, c, d, be
ranged upon a Lever, they will have the ſame
Effect upon it as if all their Weights were accu-
mulated into e their common Center of Gravity.
For if we ſuppoſe the Lever sd now to turn up-
on , the Sum of their Momenta in the former Caſe
(I mean when placed at a Diſtance one from an-
| other)

gw

Of the Center of GRAvity. 335
other) will be ax 5 a + $X 56 + cX5c+ dx ds
and their Momenta when placed at e will be

42 PIYTe + dxsez and theſe two Sums were

found equal in the Demonſtration of the foregoing
Lemma.

SCHOLIUM.’
If the Point 5 be taken any where among the
© Bodies, and not in the Line ad produced; then
muſt the Sum of the Products on one Side or the
other be conſidered as affirmative, and conſequent-
ly the Sum of the Products on the other Side will
be negative. Add theſe together according to the
Method uſed for adding affirmative and negative
Quantities together, and the Center of Gravity
will be found to lie on the affirmative or negative
Side, according as the whole Sum was affirmative
or negative. If the whole Sum be nothing, the
Point s will be the common Center of Gravity.

InsTANCE.

There is a certain cylindrical Rod 36 Inches

long, and weighing one Pound; at whoſe two
Extremities are ſuſpended two Weights, the one
of five Pounds, the other of ſeven: I demand the
common Center of S of the Rod and theſe
two Weights.

The Momentum of the Rod is equal to the
Momentum of 1 Pound at the Center; (by the
Coroll.) whence ſuppoſing q at the Place of the
Pound Weight, and working as the Lemma directs,

E 4 we

56 Of the Center of GRaviry.
we ſhall have the Diſtance of the Center of Gra-
vity from that Weight equal to 152 Inches.

Pros. X. Fig. 16.

To find the Center of Gravity of any curvilinear Space
ABC that hath an Axis, and whoſe Parts may
be ſuppoſed to be endued with Gravity. To find

. © alſo the Center of Gravity of a Solid generated by
the Revolution of ſuch a Space about its Axis.

Imagine the whole Space ABC to be reſolv-
ed into an infinite Number of infinitely narrow
Trapezia, ſuch as MN n m, whoſe proper Centers
of Gravity are all in the Axis AD, and let all the
Chords M PN, N Q # be at firſt ſuppoſed perpen- 8
dicular to the Axis. Call AP, x; MP, y; P Ox;
and ſuppoſing the Weights of the ſeveral Trapezia
MNun, as repreſented by their Areæ, to be
multiplied into their reſpe&ive Diſtances from the
Point A, call the Sum of the Products S; and
call the whole Space A B C, repreſenting the Sum
of the Weights of all its Parts, ; and we ſhall
have S=2y x x, and. y; therefore the Fluents
of theſe Fluxions being found will determine the

| 8
Quantities S and 5; and conſequently 5, or the

Diſtance of the Center of Gravity 1 oo *. Axis
from the Vertex KA.

Let now the ſeveral Chords Mn, Nu be ſup-
poſed to turn upon their reſpective Points P. Q, .

ſo as to become inclined | to the Axis, but ſtill
parallel

Of the Center of G RAVI TY. 57
parallel to one another; and the Center of Gravity
of the whole Space A BC will ſtill continue in
the ſame Place: For though by this Means, the
Weights of all the Parts be diminiſhed, they are
diminiſhed in the ſame Proportion.

Laſtly, Let all the Chords Mn, Nu, c. re-
turn again into their perpendicular Poſition, and
let the Space A BC generate a Solid about its
Axis A D, and if we ſuppoſe the Diameter of a
Circle to be to the Circumference as 1 to e, and
uſe the Quantities S and s as before, we ſhall have
in this Caſe g= eyyx x, and;=eyyx; whence S
and 5 _ be determined as before.

ExamMPLE I.

Let the Nature of the Curve be expreſſed by
this Equation px x”= =7″s then we ſhall have

I 1 .
„ Fx, and 20 * K or $ = 25 1 XX3z
whence S _ K ==”, or me — XX
Fi 8

Moreover or 2y * 2 – Xx» XX3 3

21 mn
— * 15 te es r* ＋ — ; therefore
the Diſtance of the Center 5 Gravity of the whole

Space AB C from the Vertex A is A D x IS

S

Ex AML E III.
For the Center of Gravity of the Solid we ſhall
have according to the foregoing Notation, eyyxx

an. * of the Center of Gn av 1 v, |

RA. x therefc S =
E xXX = _ * X; Ore —
9 = .

• or . x EPR, Again wes er
• — _ * 82 s
  =ep XX= X x 3 therefore E there-
  | fore >= 2m -1 7

2 3 z and the Diſtance of the Center
of —— of the whole Solid will be the Axis A D
multiplied into the Fraction 2

SCHOLIU M.

As the two former Examples will give a Solu-
tion to this Problem in an infinite Number of
curvilinear Figures, ſo will it alſo in Triangles
and Parallelograms, and the Solids generated by
their Revolutions about their Axis, ( viz. ) Cones
and Cylinders. For in a A Px h, and in
a Oran P. , or px%=y |

TRHEORE M. Fig. 18.

If in 2 Triangle as ABC, and from any Angle
as A be drawn a Line as A E biſecting the op-
| Polite Side; and if upon that Line be ſet off

| AF=2AE, the Point F will be the Center of

Gravity of the Friangle.

For as the Line A E biſects 8 Line B * it

muſt biſect every other Line parallel to it; and

conſequently the Center of Gravity muſt be ſome-

where in that Line AE. For the ſame Reaſon if

BD be drawn biſecting AC, the Center of Gravity
muſt

Of the Center of GRAv1iTY. 59
muſt alſo be ſomewhere in the Line BD ; there-
fore it muſt be in the common InterſeQion F.
Join D E, which as it biſects both A C and BC
will be parallel to AB; and we ſhall have, from
ſimilar Triangles, A F to FE as AB to D E, or
as BC to BE, or as 2 to 1; and (componendo)
A F to A E as 2 to 3. Q. E. D.

E x 4 M P Ls III. Fig, 17.
To find the Center of Gravity of an Hemiſphere ge-
nerated by the Revolution of a Semicircle B A C
about its Axis AD.

Here it will be proper, for a more 1 |
culation, to call DP, x; and MP, y; as before:
and if we call the Radius 7; we ſhall have
yy r rx x, and eyyxx, or Ser rXX—Ex3X 3
therefore in the Caſe of the Solid generated by the

Area MB 1 we have S THE, and in

4
the Caſe of the whole Hemiſphere .

3 4 4
Again 7 or eyy e e -e’, therefore 5=erx —7ex; which in the Caſe of the whole Hemi-

ſphere is equal to er3—4eri=23 er3; therefore
—=+7: That is, in an Hemiſphere, or Hemi-
ſpheroid, the Center of Gravity is + of the whole
Axis from the Baſe. For if all the elementary
Laminæ be contracted in a given nn they
will form an Hemiſpheroid.

Eri

60 Two PROBLEMS

Ex Ar IE IV.
Let us in the next Place enquire into the Center of
Gravity of the generating Semicircle,

And here ſince yy=r r—x x, we ſhall have
2x &= —2y y, and 2yx x Or S= —2y yy, Whoſe
Fluent — when x=0 will be , and when
t=7 will be o; therefore 8 : Buts is the

e err r

Area of the Semicircle A B 82 —; 88

S —47 3

5 3e

Times the Circumference of any Circle is to 4

Times the Diameter (or as 33 to 14 very nearly)
ſo is the whole Axis of any Semicircle, or Semi-
ellipſe, to the Part between the Baſe and che Center
of Gravity.

; which is as much as to 4 that as 3

Two PROBLEMS concerning Gravity,

PROBLEM >. & ih

To find the Weight of an infinitely tall Priſm or
Cylinder ſtanding upon the Surface of the Earth,
upon a Suppoſition that the Force of Gravity in all

Places is reciprocally as the Squares of their Di 72
tances from the Center of the Earth. |

Let the Cylinder in queſtion be diſtinguiſhed
into an infinite Number of infinitely thin 3
and let x be the Thickneſs of any one of theſe
Laminæ at any indeterminate Altitude x from the

8 | Center

concerning G RA
N of the Earth; Then will its Weight be as

— —(7 i. e. )= will be the Fluxion of the Weight of
| 5 Cylinder; whoſe Fluent is _ Call the Se-

| midiameter of the Earth r; and — Value of this
Fluent at the Earth’s Surface will be , and at
an infinite Height it will be ad ; OP |
the Weight of this infinite Cylinder will be = =”

Let p be the Weight of a ſmall Part of this cy.
linder, at the Surface of the Earth, whoſe Altitude
is a; and its Weight, according the 2 Pro-

portion, ought to be repreſented by – —; therefore
the Weight p will be to the Weight of the in-
finitely call Cylinder, as = to 5 i. e. as à to r.
PROBLEM XII.

Plate IV. Fig. 21.

Let S be the Center of the Earth, B any Point
in its Surface, and let the Force of Gravity in all
Places be reciprocally as the Squares of their Diſ-
tances from the Center of the Earth: If is required
zo determine the Velocity of a beavy Body at the Sur-

face of the Earth, which it — in Falling Joes
any given Altitude A B.

Let x be any indeterminate Diſtance from the
Center of the Earth, and let v be the Velocity of

the

62 Two PROBLEMS

the falling Body at that Diſtance. Let == 5 == re
| * the Force of Gravity at B, and as
ly — — ; its Force at the Diſtance x. Firſt then it is

plain that after the falling Body is arrived at the
Diſtance x, and then deſcends further through any
infinitely ſmall Space as x, the Time of that in-
finitely ſmall Deſcent will be as ©, that is, it will
be as the Space directly, and as the Velocity in-

verſely; and the infinitely fmall Acquiſition made
by the Velocity in that Deſcent will be as that

Time < and the accelerating Force — Jointly ;
that is, © will be as — ; therefore v d will be as
2 therefore the Fluents of theſe Fluxions that
are generated in equal Times will be 8

1
S- * be as SB — FF or as
SBYSA =”. or as ELIE. Therefore, becauſe of

the conſtant Quantities 2 and S B, v will be as

AB
AS
This being diſcovered, let DB be the Height
from whence a Body will fall to the Surface of the
Earth in 1 Second of Time ;. and ſince during ſo
ſmall a Deſcent, the Force of Gravity may be
looked upon as uniform, it is evident that a Body
; I | falling

concerning GRAviTY. by
falling from D to B will acquire a Velocity which
will carry it uniformly through the Space 258 BD

in a Second of Time. Let 2BD repreſent this
Velocity ; then muſt every other Velocity be re-
preſented by the Space through which it will carry
a Body in a Second of Time. Now to find the

Velocity acquired in falling from A to B, I ſay, as
DB

AB
Hg is to 8 fois 4D 1 B* (the Square of the Ve-

locity acquired in falling from D to B) to 4D B

xDSxX5= (the Square of the Velocity acquired

in falling = A to B) = 4m mx > ſuppoſing

| u to be a mean Proportional between DB and DS:
Therefore a Body falling from B to A acquires
a Velocity that would carry it through the _

q Vg in a Second of Time. Q. E. I.
SCHOLIUM.

D is found by Experiment to be 1 FB Paris
5 Feet, and DS 1961581575 of the ſame Meaſure;
1 | therefore zm is about ſeven Exgliſb Miles.

Conor. 1.

t — A B be infinite, the Quanity 2B KS > becomes
equal to 1, and therefore goes out of the Queſtion; ;

and therefore a Body falling from an infinite Height
will n at the Surface of the Earth but a finite
100 Velocity,

=

=

64 Of the Motion of PE NDUL v MS.
Velocity, (viz.) ſuch a Velocity as will carry a
moving Body uniformly through. 4 Miles 6

Second of Time.

Cents: .16

Therefore, e converſo, if a Body be thrown up-

wards with ſuch a Velocity, it will never return,

but aſcend ad infinitum.

HAIR Ill. | |
Aſter the ſame Manner may be determined the

Velocity of a falling Body, whatever be the Law

of G ravity. .

hap ag the Moron of PENDULUMS.
Luna.

Plate III. Fig. 19.

Tf two Bodies fall from equal Altitudes, their Velocites

at all -other equal Altitudes will be equal, though
one wn co obliquely, and the other perpendicularly.

De F449-4 2304:

Let ABC be an horizontal Chord of any Curve
AD C whoſe loweſt Point is D, and whoſe Axis

B D is perpendicular to the Horizon. Let E be

any Point in the Curve AD, and e another in-
finitely near it, and above it, and draw horizontal

Chords E E and ee cutting the Axis B D in F
and F reſpectively: Then imagine a Body as p to

“_ from reſt at A Hong the Curve AD, and an- |
other

\

Of the Motion of PENDULUMS. 65 |
other as q to fall perpendicularly from a Point H
in the Axis BD; and let the Point H be ſo taken,
that the Velocity of the Body q in the Point F
acquired in falling from H to F may be preciſely

equal to the Velocity of the Body p in the Point e

acquired in falling obliquely from A toe: Then
it is plain that the Time wherein the infinitely ſmall
Space Ee is deſcribed will be to the Time the
Space F is deſcribed in as E e to F; andifEe
be conſidered as an inclined Plane, it is further
evident, from the Nature of ſuch an inclined
Plane, that the Force wherewith the Motion of
the Body p is accelerated in falling from e to E is
to the Force whereby the Body q is accelerated in
falling directly from F to F as F/, is to Ee.

Therefore the Time Ee is deſcribed in, is as much

greater than the Time FF is deſcribed in as the

Vis acceleratrix in f is greater than the Vis acce-
leratrix in e: Therefore the Increments gained to
the Velocities of the Bodies p and q in falling from
eto E, and from f to F (which are as the Times

and the – reſpective Forces conſidered together )
will be equal: Therefore the. Velocities of the

Bodies p and q in the Points E and F will ſtill be

equal; which is as much as to ſay, that if the
Velocities of the two Bodies p and ꝗ falling from
A and H be in any one particular Caſe equal at
equal Altitudes, they will always be fo : But at

the Point A the Velocity of the Body p was o;

therefore at the Point B the Velocity of the Body
q was o; therefore the Point H coincides with the
| $- =: Point

n—_

{|
|
|
|
|
,

3 Of the Mor ion

Point B: Therefore if two Bodies fall from equal

Altitudes, their Velocities at all other equal
Altitudes will be equal, though one deſcends
obliquely, and the other „ pra Q,

E. P.

Se.

| The Velocity of the Body p in the Point E will
be in a ſubduplicate Ratio of B F the Diſtance
between the two horizontal Lines AB and E F,

becauſe the Velocity of the Body q in the Point F
is ſo.

Conor, I.
If the Ark ADC be Part of a Circle whoſe

Diameter is DI, the Velocity of a Body in the

loweſt Point D, acquired in falling through the
Ark AD, will be to the Velocity of a Body in

the ſame Point by falling through the Ark E D, as
the Chord of the Ark AD is to the Chord of the

Ark ED; for the wg Velocity is to the latter

as 5B is to HD F, or as /VIXDB is to
DIXD F. e eee

Ark AD, and /DTXDF is the Chord of the

Ark ED.

e PA o-

(NY | wes

4] w_

ww TT =” OO 991

„J PENDULUMs. “op

PROBLEM XIV.

To find preciſely the Time of a leaſt Oſcillation of a

given Pendulum ſwinging in an Ark of a Circle;
and to find, without any ſenſible Error, * tbe Tine
x any other, |

Retaining the Conſtruction of the laſt Lemma; |

and ſuppoſing moreover that the Curve ADC is

an Ark of a Circle whoſe Diameter is DI, and
wherein the Pendulum performs its feveral Oſcil-
lations, being now ſuppoſed to be at the Point E

in its Aſcent from D to C. Let VB F expreſs the
Too acquired by a heavy Body in falling from

to F, and conſequently the Velocity of the

5 at the Point E; then will 2

expreſs the Velocity acquired in falling through a
Space equal to 4 1D: And fince a Body by this
Velocity would be carried uniformly through a

Space equal to +I D in the ſame Time as it would
fall through 3 ID; divide the Space £ ID by the

Velocity æ M5, and let the Quotient VI B ex-
preſs the Time wherein a heavy Body would fall
through 41D, i. e. through half the Length of

the Pendulum ; then will 5 expreſs the Time

| E the Ark E e is deſcribed by the Pendulum.

t the Ark Ee is the Fluxien or Increment of
the Ark D E, and (by the Scholium to the tenth
F 2 Pro; em)

68 Of the MoTton

IDXF ID
Problem) is _ to = ah XVID

Ef | E e * | ;
Me =a/FD*. Therefore TEE TFN V5 |

RX 2 Ee Biſe& BD in K, and KD in L,

und if the Ark ADC be not very large, the
Quantity I F cannot differ ſenſibly * its middle

. . therefore
IF rom . ce

2 IL Ro TR}
=_- equal very nearly to IKID**J7FFaFD:
Upon the Diameter B D deſcribe a Circle R GD GB

cutting the * E E and ee in G and ꝗ re-

KDE 8
ſpectively, and = En will be equal to Gg the

=Ff
Fluxion of the Ark — ; therefore — EBSED

Gg
BP therefore TF; or the Time where-

| Quantity I K, nor

12 the r aſcends from E to e will be

77 L x * ID x 28 55 Therefore 12 whole Time of its

— . D to C will be = * VT DX ALS

and the Time of an entire ie through the

11. FX B GDGB.
    whole Ark ADC will be TK XVIDX 5

Let the Ark ADC be infinitely ſmall and the

the Quanity [4 will become equal to Unity, and
the

%F

he

of PENDULUMS. 69

the Time of the leaſt Oſcillation will be /ID

• 2 Chats 5 That is, As the Diameter of any

Circle is to the Circumference, ſo is (V I’D) the
Time of a heavy Body’s falling from Reſt through
half the Length of a Pendulum, to the Time of
a leaſt Oſcillation of that Pendulum. Call this

Time thus found 7, and if the Ark ADC be not

very large, the Time of an Oſcillation through

that Ark will be f 5+ K os or it will be ben. |

• I.

SCHOELIU M.
If the Ark ADC were a Semicircle, the Time

of a leaſt Vibration would be to the Time of a
Vibration through that Semicircle according to the

Proportion above laid down as 6 to 7, or as 29

to 33+; and by other Means we know that the

Conus of this laft Proportion ought to be leſs

than 342;, and greater than 3445. If therefore
the Proportion above found approaches ſo near the
Truth in ſo large an Ark as a Semicircle, and is
abſolutely true in an Ark infinitely ſmall, it will
be eaſy to conjecture how inconſiderable the Error

will be in ſmall Arks ſuch as are uſually deſcribed

by Pendulums.

Conor. I.
If two pendulous Boxes of the ſame Figure and

5 Magnitude be, together with their Contents, of the

F 3 lame

70 Of the Mor io

ſame Weight, and at equal Diſtances from their
Points of Suſpenſion perform their Oſcillations in
the ſame Time; it is an Argument that the Quan-
tities of Matter of both theſe Pendulums are the
ſame, let the Parts be ever ſo heterogeneous; and
conſequenfly that the Matter in thoſe Bodies will
be proportionable to their Weights : For if theſe
two Pendulums were to be let fall from Reſt, they
would fall through half their Lengths in equal
Times, and conſequently would be equally ac-
celerated-

C OROLL, II.

The Space chrough which a heavy Body falls
from Reſt in any given Time is to half the Length
of a Pendulum that oſcillates in that Time, as the
Square of the Circumference of any Circle is to

the Square of the Diameter: For as the two firſt
in the Proportion are Spaces, the two laſt repre-
ſent the Squares of the Times wherein thoſe Spaces
are deſcribed.

bk.

cones mt

Small Vibrations of the ſame Pendulum are all
performed i in the ſame Time nearly : For the ſmall

2 of Time : * 4. which is to be added |

to t the Time of a wi a will (in this
Caſe) cauſe no ſenſible Variation i in the whole.

COROLL.

oF PENDULUMS, 128

Co Rol T. IV.

The 1 of any Pendulum is as the Square
of the Time of a leaſt Oſcillation of that Pendulum
multiplied into the abſolute Force of Gravity:

For if a heavy Bedy falls through any Space as

| in any Time, that Space / will be as the Square
of the Time multiplied into the Force of Gravity,
But the Time of Deſcent through I is as the Time

of Deſcent through +1, and this again is as the Time

of a leaſt Oſcillation of a Pendulum whoſe Length
is /; for the ſame Reaſon that the Circumference
of a Circle is as the Diameter; Therefore the

Length of any Pendulum is as the Square of the

Time of a leaſt Oſcillation of that Pendulum mul-
tiplied into the abſolute Force of Gravity.

N. B. If this Corollary be applied to the Oſcll-
lation of a Pendulum upon an inclined Plane, in-
ſtead of the abſolute Force of Gravity, underſtand

that Force wherewith the Body endeavours to de-

cend in a right Line along that Plane,

Conor, V.
Therefore the Time of a leaſt Oſcillation of any

Pendulum is in a ſubduplicate Ratio of the Length

of the Pendulum directly, and in a ſubduplicate
Ratio of the Force of Gravity inverſely. For if v
be the Force of Gravity, / the Length of any
Pendulum, and t the Time of a leaſt Oſcillation ;
v will be as /, by the laſt Corollary ; therefore

| 5
e will be as V>- F 4 Cast

72 Of the MoT1on

CoroLrL. VI.
T herefore, if the Time of – leaſt Oſcillation be

given, 2 and conſequently = = will be a given

Quantity, in which v will be as /; that is, the
Force of Gravity will be every where as the *
of an iſochronous Pendulum. |

a . VII.

If the Force of Gravity be given, the Time of
a leaſt Oſcillation will be in the ſubduplicate Ratio
of the Length of the Pendulum.

Cox ol 1. VIII.

Univerſally, If v be the Force of Gravity, I the

Length of any Pendulum, » any Number of Vi-
brations of that Pendulum, and ? the Time where-

in that Number of Vibrations is performed; the
„ 4 will always be as the Quantity /* :

For if Ve — repreſents the Time of a ſingle Oſcil-
lation; a Vx u will repreſent the Time of

any Number 1 ow thoſe Vibrations, that is, the

—, and v1 as ln

| “7 |
Time # will be as V-x n; therefore ?* will be as

CoroL:L.

. oP *

of PEN DULU MSV. 3
CoroLL. IX. |

The Times of ſimilar Oſcillations (by which
is meant ſuch as are made in ſimilar Arks)
will always be in a ſubduplicate Ratio of the
Length of the Pendulums, whatever be the
Quantities of thoſe Arks, provided the Force of
Gravity continues the ſame. For if the Arks
AD, ED, eD, and conſequently E e be as the
Diameter DI, the Lines BF and Bf will he fo

too; p and = or 15 Time wherein the Ark Ee

is deſcribed will be 157 4 8 as /D1, and

(componendo) the whole Time of an Oſcillation
through the whole Ark AD C will be as DI, or

as VE , that is, in. a ſubduplicate Ratio of the

Length of the Pendulum.

| COoROLT. X.
The Force wherewith any Pendulum whoſe

1 Length is / is accelerated in any Point E of its

E F
Ark will be as wo 08; For if the Weight of the
Pendulum, or Fo abſolute Force of Gravity, be

called I, the accelerating Force at E will be — 4

DIxXEF Ff Ee
But * w * JJFxXED == X X EF Therefore
Ke. 1 * EF.

FF EF. and F. = FF

CoRoOLL.

74 Of the MoTron

Cox ol I. XI.

Therefore in ſmall Oſcillations, the Fe orce
wherewith the ſame Pendulum is accelerated in its
Motion will always be as its Diſtance from the

loweſt Point of its Ark; for in this Caſe the Sine
E F differs nothing mo the Ark ED.

| Coro. XII.
If ghe Ark ADC be ſuppoſed infinitely wall

te Velocity of the ſame Pendulum at any Point

as E will be as DX. —5 E’; for this Velocity
will always be as BF, that is, as VBB FP, that
is, in the ſame Circle as /BDxDI—F DDI.

But B DX DI is the Square of the Chord of the
Ark A D, and therefore in our Caſe B DDI is
the Square of the Ark AD; and for the ſame Rea-
fon F DxD1 is the Square of the Ark ED: There-
fore the Velocity of the Pendulum in the Point

E is as VD A* —D E*:

N. B. The following Theorem with its Corol-
laries are added for the better underſtanding of
what is delivered in the Principia concerning the
Motion of Fluids, and of Sounds; and that with-
out the help of the cloid.

“TT

Fig. 20.
If: any Body as P be made to oſcillate in a \ ſtrair
Line A D C, whoſe middle Point 1 is D, being ac-
4 | cCelexated

• 
  i

of PENDULUMS. 75
celerated in its Motion from A to D, and equally
retarded in its Motion from D to C by Forces
every where proportionable to the Diſtance of the
Body from the middle Point D; I fay then that
the Body P will perform its Ofcillations in the
ſame Time, whatever be the Length of the Line
ADC.

To demonſtrate this, imagine two ſuch Bodies
P and p beginning their Oſcillations at the ſame

Time, and both oſcillating in the ſame Line (viz.)

P in a large Space ADC, and P in a more con-
tracted Space aDc: Then it is plain, that the
Accelerations and Velocities of theſe two Bodies,
and conſequently the Spaces paſſed through in any
given Time will always be in the ſame Proportion
(vix.) in the Proportion of DA to D: There-
fore the Bodies P and p will both come to the
Point D at the ſame Time, and conſequently both
perſorm their Oſcillations in the ſame Time.

For the better conceiving this, imagine the
Force whereby the two Bodies P and p are agitated
to act not continually but by ſucceſſive finite Im-
pulſes at equal Intervals of Time, and always pro-
portionable to the Diſtances of the Bodies from
the Point D; and then imagine thoſe Impulſes
and Intervals to be diminiſhed ad inſinitum.

Conotkh L
Suppoſing the Body P to oſcillate in the Line

ADC as above, and to be every where accelerated

or retarded by a Force which is to the abſolute
Force

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76 of de Mer 10
Force of Gravity as the Diſtance DP is to a given

Line L., or as 5 2 to 1; I fay then, that the

Time of an Oſcillation through that Space AD C

will be equal to the Time of a leaſt Oſcillation of
a Pendulum whoſe Length is L.
For ſuppoſing the Line A D C to be infinitely |
ſmall, and ſuppoſing a Pendulum whoſe Length
is L to ſwing in an Ark equal to the Line ADC,
the Vis acceleratrix of a Body P at any Diſtance
DP will be equal to the Vis acceleratrix of the
Pendulum at the ſame Diſtance from the loweſt
Point of its Ark, by the tenth and eleventh Corol-
laries of the foregoing Propoſition : Therefore
their Accelerations and Velocities in correſpondent
Points will be equal, and conſequently alſo the
Times of their Oſcillations. But the Time of an
Oſcillation through the Line A D C will always be
the ſame, whatever be the Length of that Line:
Therefore univerſally the Time of an Oſcillation
through the Space ADC will be equal to the
Time of a leaſt Oſcillation of a Pendulum whoſe

7 Lengrh 1 is L.

Conor. II.
Since the Force whereby the Body P is accele-

3

• or . is every where as > Wh, and con-

| ſequently as L if the Diſtance DP be a given

Quantity; it follows, that if the accelerating or
retarding Forces in every Point of the Line A D C

be

Py wm _.

C’s

APD s | 77

5 be proportionably intended or remitted, che Length
IV will be reciprocally as the Force at any given

Diſtance. But the Time of a leaſt Oſcillation of
a Pendulum whofe Length is L, and conſequently
the Time of an Oſcillation through the Space
ADC, is as VL, by the ſeventh Corollary of the
foregoing Propoſition: Therefore the Time of an

5 Oſcillation through the Space AD C is reciprocally

in a ſubduplicate Ratio of the accelerating or re-

tarding Force at any given Diſtance from *

Ein III.

By the Demonſtration of the foregoing Theorem;
it appears, that if a Body as P oſcillates in a larger
Space ADC, and another as p oſcillates in the

fame Line, but in a more contracted Space a Dc,

and if the Diſtances D P, D p, be taken in the ſame

Proportion as is D A to D a, that then the Velo-

city in P will be to the Velocity in p as DA to Da:
Therefore if the Space ADC be ſometimes in-
larged, ſometimes contracted (the accelerating
Forces at the ſame Diſtances continuing always

the ſame) and if the Diſtance DP be always taken

in the ſame Proportion to D A, the Ve! e in

the Point P will alſo be as D A.

cebit. IV.

Theſe Things ſuppoſed, and alſo that the ac-

celerating and retarding Forces at the ſame Diſt-

ances from D continue always the ſame: If upon
the Line ADC as a Diameter be deſcribed a

Semi-

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758 Of tbe Mor To
Semicircle AM C, and a Line as MP be drawn

any where perpendicular to AC in P; I ſay, that
the Velocity of the oſcillating Body in the Point P

will always be as that Perpendicular PM: For if

we ſuppoſe the Line A PDC to be diminiſhed in
infinitum, the Velocity in any Point P will then be

D A*—P PB, by what was demonſtrated in
the firſt Coroll. compared with the twelfth Coroll.
of the laſt Propoſition but one. Take now DP
toD A in the ſame Proportion as before thoſe
Lines were diminiſhed ; and then preſerving this
Proportion, reſtore again DA and DP to their

former Magnitudes, and by the foregoing Corol.

lary, the Velocity in P will be increaſed in the ſame

Proportion as t the Line D A is increaſed : But the

Quantity VBK P- will alſo be increaſed in
the ſame Proportion ; for ſince DP is as D A,
D Þ* will be as BA, and BAF as BA. and
VDA—DP as DA: Therefore this Quantity
VBA P’, or its Equal P M, will ſtill expreſs

the Velocity of the oſcillating Body in the Point PF;
whatever be the Length of the Diameter AD C.

Conoii V.

The Time wherein the oſcillating Body paſſes

over any Part A P of the whole Line ADC will

be to the Time of an entire Oſcillation through

that Line as the Ark AM is to the Semicircle
MC.

of PENDULUMS, 79

For drawing another Perpendicular N Q in-
finitely near to M P, the Time wherein the in-
finitely ſmall Space QP is deſcribed will be as
N that is, as the Space P Q directly, and as
the Velocity PM inverſely. Join D M, and draw

| NR equal and parallel to P Q cutting M P in R,

and the ſimilar Triangles DMP, and MNR

WW gb MP, as MN is to MD;

PQ_MN,
therefore PN N Therefore the Time

| wherein the Space P Q is paſſed over is as the Ark

MN: Therefore the Time wherein the Space
AP is paſſed over is as the Ark A M; and there-
fore this Time will be to the Time of an entire
Oſcillation as the Ark AM is to the Semicircle

Ane

| of th DznetTyY of the Ai

HyPoTHESIS.
« That the Denſity of the Air is proportionable

eto the compreſſive Force with 2 it is con-

« denſed.“ 3

Though in extreme Caſes this Hypotheſis be
attended with ſome Difficulties; (for it ſuppoſes the

Air capable of being both condenſed and rarified

in inſinitum, with ſome other Abſurdities) yet as
far as any Experiments of both kinds hitherto

made can reach, it is not found liable to any ex-

ception p
LEMMas

• 
  ” r
  4

a — —_— —
— PPP n *

80 Of the DENSITY of the AI R.

LEM MA. Fig. 17.

Let S be the Center of the Earth, S B its Semi-
diameter, BC the Height of an homogeneous
Atmoſphere of the ſame Weight with our own,

and every where of the ſame ſpecific Gravity with
that of common Air at the Surface of the Earth,
and let AB be an infinitely ſmall Part of BC:
I fay then, that the Denſity of our Air at the

Point B will be to its Denſity at the Point A, as
B C to AC. |
For ſuppoſing our Atmoſphere to be annihilated,

let the homogeneous Atmoſphere above deſcribed

be placed inſtead of it, and let the Line B C be
conſidered as the Axis of a Column of this Air;
then it is plain, that as this Column is a homo-

geneous Body, the Weight incumbent upon B

will be to the Weight incumbent upon A as B C
to AC: But the Weight incumbent upon B from
the former Atmoſphere was the ſame as the Weight
incumbent upon B from the latter, becauſe both

• Atmoſpheres were ſuppoſed to have the ſame

Weight. Moreover, the Weight of the Air in–
cluded between A and B was the ſame in the for-
mer Cale, as it is in the latter; becauſe both the
Space AB, and the ſpecific Gravity of the Air
included within it was ſuppoſed in both Caſes to be
the ſame. Subtract the Weight of the Air between
A and B from the Weight at B, and then will
remain the Weight at A in both Caſes the ſame.
Since then the Weight of Air incumbent upon B

was

Of the DEn$SITY of the AIR. 81

was to the Weight incumbent upon A as BC to
AC, it follows, from the Hypotheſis already aſ-
ſumed, that the Denſity at B is to the Denſity at
A as BC to AC. Q. E. D.

COR OLI.

Whence (dividendo) the Denſity at B will be
to the Exceſs of the Denſity at B above the Denſity
at A as BC is to B A; that is, calling B C, 5
S A, æ; and the Denſity at A, v; we have V to B,

| 8 * x
as I to &; and therefore at the Point B, 52 7

PROBLEM XIV. |
To compare the Denſity of the Air at any given A.
titude with its Denſity at the Surface of the Zarth,

upon a Suppoſe tion that the Force of Gravity 7s
every where the ſame.

Let S be the Center of the Earth as before,
SB its Semidiameter, BA any given Altitude
above the Surface B. Call SA, x; the Denſity
of the Air at A, v; and call the Height of an
hom@eneous Atmoſphere of the ſame Weight
with ours, and every where of the ſame ſpecific
Gravity- with that of common Air at the Surface
of the Earth, /, Let the Altitude B A alſo re-
preſent Part of the Axis of a Column of the At-
moſphere incumbent upon B; and ſuppoſing this
Column to be reſolved into an infinite Number
of infinitely thin Lamine, call the Thickneſs of
that Lamina which is contiguous to the Point A, x;

n G and

82 Of the DENSIT YF the AIR.

and its Weight will be as v x, that is, as its Den-
fity and Magnitude conſidered together. But the
Weight of every Lamina is the Fluxion of the
Preſſure in the Place where it exiſts; and as the
Denſity is every where proportionable to the Preſ-
ſure, the Fluxion of the Denſity will be propor-
tionable to the Fluxion of the Preſſure, that is,

o 0 5 Y 5 V .
© will be every where as vx; therefore – will be
. v

”-— 2 and conſequently as 7. But at the Point

B, – 5 = * by the Corollary to the OS Lemma;

therefor Aſo where elſe = — => But the Fluent

SB
of 7 s 5, whoſe Value at the Point B is > and

at te Point A, = and the Difference of theſe

two Values is 5 Again, the Fluent of – 2 is
the hyperbolic Logarithm of v (by what has been
formerly demonſtrated) the Value whereof at the
Point B is the Logarithm of the Denſity at B;

and at the Point A is the Logarithm of the Den-
ſity at A; and the Difference of theſe two Values
is the Logarithm of a Fraction whoſe Numerator
is to the Denominator as the Denſity at B to the

Denſity at A; therefore oy is the hyperbolic Lo-
garithm of ſuch a Fraction. Make 0,4342 94481903
BA $
= m, and mx ? will be Briggs’s Logarithm of
te

” BY

1C

‘ F.0d

Of the DEns1Ty of the AIR. 83
the ſame Fraction. Therefore whenever the Al-
titude B A is given, find the Fraction whoſe Lo-

38 B A |
garithm is mx e, and the Denſity of the Air at

B will be to the Denſity at A as the Numerator of

that Fraction is to the Denominator. Q. E. I.
N. B. Every whole Number muſt here be con-

ſidered as a Fraction whoſe Denominator is U-

nity.
Cone.

If we call the Denſity of the Air at the Surface
of the Earth, 1; its Denſity at any other Altitude
B A will be expreſſed by a Fraction whoſe Lo-

garithm is * For if mx == be the
Logarithm of the Fraction —, the Denſity at B will
be to the Denſity at A as 1 tos, or as 1 to 55

| BA
whoſe Logarithm is — m x _

: Conatt: I

| Whence ſince » and / are conſtant Quantities,
the Logarithm of rhe Denſity of the Air at A will
be as the Altitude B A.

Conor rt. III.
Whence it follows, from the Nature of Loga-

_ rithms, that if any Number of different Altitudes

be taken in an arithmetical Progreſſion the Den-
| G 2 ſity

8 Of tbe DENSIT YF the AIR.
ſity of the Air at thoſe Altitudes will be in a geo-

metrical Progreſſion; that is, it will be diminiſhed
in a continual geometrical Proportion.

COR OIL I. IV.

Tf the Denſity at A be given, the Altitude B A
may be found thus : Let the Denſity at B be to
the Denſity at A asr to 5, and let , be the Loga-
rithm of the F ration © 7 then will = *I

and conſequently B A will be equal to / * =.

SCHOLIUM,

The ſpecific Gravity of well FTE) Mercury
is to that of Water as about 13+ to 1 ; and the
ſpecific Gravity of Water is to that of Air when
the Mercury in the Barometer ſtands at 3o Inches
as 870 to 1. Put theſe Ratios together, and the
ſpecific Gravity of Mercury will be to that of Air,
when the Mercury in the Barometer ſtands ar 30
Inches, as 11890 to 1. Therefore the Height of
. an homogeneous Atmoſphere of the ſame Weight
with ours, and every where of the fame ſpecific
Gravity with that of common Air at the Surface
of the Earth; which in the foregoing Computation
we called I, will be 356700 Inches, or 29725 Feet,
or 5,63, or 5,629735 Engliſh Miles very nearly.
Wherefore if the Altitudes I and B A be taken in

Miles and Parts of a Mile, we ſhall have 72
3 BA 3
©0,07714, and 7X —=BAX0,07714

PA o-

the Point A 18 1 and at the Point B

. agg 95

PROBLEM XV.

To compare the Den/ity of the Air at a given Altitude

with its Denſity at the Surface of the Earth, upon

a Suppofition that the Force of Gravity in all Places

is every where reciprocally as the Squares of their
Diftances from the Center of the Earth,

SOLUTION.

Uſing the Notation of the foregoing Solution, :
the Weight of an infinitely thin Lamina of Air

ö ; UV Xx |
contiguous to the Point A will now be as _ be-

I
cauſe the Force of —_ is as — ; therefore d

or as —
v xx [ Ny”

4 5
vill now be as —,- and = as 4 „

5 —
But, at the Point B, hs 51 BS ; becauſe] in this

Caſe =. =1: NS wad wa — MG

But the F luent of © _ is _ whoſe Value at the

SA : IB

and the Dilfrence of theſe two Values is vu

S BY. AB SB AB..
Therefore — T XN or SAX To is the hy-

perbolic Logarithm of a Fraction whoſe Nume-
rator is to the Denominator, as the Denſity at B

to the Dentiry at A: Therefore 8 AN N will

G 3 –

86 Of the DENSITY of the AI x.

be Briggs’s Logarithm of the ſame Fraction. Di-
miniſh the Altitude A B into the Altitude DB,
by making as S A is to S B ſo A B to DB, and

We ſhall have Ax ABD; meats none?
DB SA SA 1
= X-: Therefore if the Denſity at D be

[

found by the Method of the foregoing Problem,
upon a Suppoſition of uniform Gravity, the ſame
will now be the Denſity at B upon a true —
of Gravity.

S I.

The Altitude BD may be found a little more
expeditiouſly, by taking AS, A B, AD continual
Proportionals, and then 8 AD from
AB.

ee .

If SA, SB, SE be taken cuntinngl Propor-
tionals, B E will be equal to BD; for ſince SA
is to SB as SB is to S E, we ſhall have (divi-
dendo) S A to AB as SB to B E, and alternative-
ly S A to SB as AB to BE: But as SA is to
SB, ſo is AB to BD, by the Conſtruction:
Therefore BE equals BD.

Coro. III.

Tf ſeveral Altitudes S A be taken in muſical
Progreſſion, the Denſity of the Air at thoſe Al-
titudes will be in geometrical Progreſſion. For
Quantities are ſaid to be in a muſical Progreſſion

1 | whoſe

Of the DE NSITY of the AI R. 87
whoſe Reciprocals are inan arithmetical Progreſſion;
therefore if the ſeveral Altitudes S A be taken in a
muſical Progreſſion, their Reciprocals S E will be

in an arithmetical Progreſſion; therefore the ſeve-
ral Altitudes B D will be in a contrary arithmeti-

cal Progreſſion ; therefore the Denſities of the Air

at their reſpective Altitudes S A will be in a geo-
metrical Progreſſion.

SCHOLIUM.

To apply the Numbers of the foregoing Problem
to this, the ſeveral Altitudes SB, SA, AB, BD
muſt be taken in Miles and Parts of a Mile: In
order to which obſerve that S B the Sem diameter
of the Earth is about 3968 Miles.

• Here it may not be amiſs to take Notice, that
  if our aſſumed Hypothe/is, that the Denſity of the
  Air was proportionable to the compreſſive Force,
  was ſtrictly and univerſally true, our Atmoſphere
  muſt be inveloped in no leſs than an infinite Quan-
  tity of circumambient Air to prevent its flying
  away; or which is the ſame Thing to preſerve a
  finite Denſity of the Air at the Surface of the

Earth. For ſince S A is to SB as SB to SE,

if S A be infinite, that is, if SB be infinitely leſs
than SA, SE will be infinitely leſs than S B, in

which Caſe E will coincide with S, and BE or BD
will be equal to BS: Therefore BD will be finite

even when AB is infinite, and conſequently the
Denſity of the Air at the infinite Altitude A B will
be finite, and may be determined by the Solution
of this Problem.

A, .
G 4 Of

‘ Of theRnsI5TANCE

Of. tbe REz15TANCE of Bopixs in Furs.

F.zmama LL

Let s hs the Space through which a heavy Body
. deſcends from Reſt in a Second of Time, acquiring
a Velocity that would carry it uniformly through
25 in the ſame Time; and let 2c be the Space de-
ſcribed in a Second of Time by a Globe whoſe
Diameter is d, being made to move uniformly in
an uniform unelaſtic Fluid: Laſtly, let e be an
accelerating Force, which acting uniformly upon
any Body will generate a Motion equal to that of
the Globe, in the ſame Time as that wherein the
Globe deſcribes + of its own Diameter: I ſay then
that this Force e will be to the Force of the Re-
ſiſtance the Globe ſuffers from the fluid Medium,
as the Denſity of the Globe is to the N of s
Fluid.

This is the 38th Propoſition of Book II. of
Newton’s Principia, and is demonſtrated by the
Help of the two foregoing.

LEMMA II. APROBöL IE M.

To find the Reſ france of a given Globe being ae
to move with a given Velocity in a 1 Medium
of a given Denſity.
Uſing the Notation of the foregoing Lemma,
fince the Globe, by its uniform Motion, deſcribes
the Space 2c in a Second of Tiews, it will deſcnibe

| 4
2 “di in the Time a c Let

5 gas LS | ä — 1

1 W

of Bodizs in Flur ps. 39
Let @ be the Weight of the Globe in Vacuo;

then the Velocity generated in the Globe by the
Weight à in a Second of Time, will be to the

Velocity generated by the Force e in the ſame

Globe, and in the ſame Time, as à is to e.

Again, the Velocity generated in the Globe by the
Force e in a Second of Time, will be to the Ve-
locity generated in the ſame Globe by the ſame

Force e in the Time = as one Second is to = or

as 3c is to 4d. Put theſe two Ratios together
(viz. ) the Ratio of à to e, and the Ratio of 3c to
44, and we ſhall have the Velocity generated in
the Globe by the Force @ in a Second of Time to
the Velocity generated in the ſame Globe by the

Force e in the Time 75 as 3a c is to 4 de. But

the former Velocity is that whereby the Space 25
may be deſcribed uniformly in a Second of Time,
and the latter Velocity is, by Suppoſition, that of

| the Globe whereby 2c is deſcribed in the ſame
Time; therefore the former Velocity is to the

latter as 2s to ac, or as g to c therefore as is to c

ſo is 32 c to Ade; therefore e af ; that is, the

Force e is equal to the Weight of the Globe mul-

tiplied by the Fraction bs 2 But, by the forego-

ing Lemma, this Force is to the Reſiſtance the
. Globe meets with from the Fluid as the Denſity of:

a

=] 2 — Lad — — —— * — ar * *
; r MOEN TT PA: PR

90 Of theResr5sTANCE
the Globe is to the Denſity of the Fluid ; or as the

Weight of the Globe is to the Weight of ſo much

of the Fluid as is equal in Bulk to the Globe;
therefore the Reſiſtance the Globe meets with is

equal to the Weight of ſo much of the Fluid as

anſwers in Bulk to the Globe multiplied into the
_ Fraction Xt that is, if 5 be the relative Weight

of the Globe in m6 Fluid, the Robiſtance ſought
wall be DN as 5 QE. L

CoRoLL.

The Reſiſtance the Globe meets with from the
Fluid will be as the Square of its Diameter, as
the Square of its Velocity, and as the Denſity of

the Fluid all together. For the * “Md

| conſtant, the Reſiſtance will be as a—b X 7. Let

m be the Denſity of the fluid Medium; then will
a—b, or the Weight of ſo much of the Fluid as
is equal in Bulk to the Globe, be as m di; that is,
as its Denſity and Magnitude 1 There-

fore the Reſiſtance will be as m ds XS 7˙ that is, as

md *.

LEMMA,

\,

ere in FVI DS. 91

LEMMA III. A PROBLEM.

To find the greateſt Velocity a given Globe can ac-
quire in deſcending by its relative Wei ght in a
Fluid of a given Denſity.

This is nothing elſe but to find what Velocity a

Globe ought to move with to meet with a Reſiſt-
ance equal to its own relative Weight in the Fluid:

For it is certain, that wherever this happens, the

Globe muſt be incapable of all further Accelera-
tion. Therefore ſuppoſing all Things as in the

two foregoing Lemmata, let 2c be the Space de-
ſcribed in a Second of Time by the IM ſought,

and the Reſiſtance will be 2 ** 25 „as above.
Make this Reſiſtance equal to i the relative Weight

b
| of the Body, and we ſhall ave dee gx di.
Let m be a mean Proportional between 4 d, and

3

the Space s, and we ſhall have 4c c=4mmX wok

| Eb PRET
and 2c D m X :

er Therefore the greateſt Ve-
locity the Globe can acquire will be ſuch as will
carry it uniformly through the Space 2 mxV
in a Second of Time, Q. E. I.

Conor.

92 Q/- 40A $616 TANET,

CoROLL.
In the ſame Fluid, the Square of the grenelt

Velocity will be as 5 LY J: For in this Caſe — will
be as di; and therefore the Quantity — – 3X 0 4

5
will be as 7. Ta

LEMMA IV. A PROBLEM.

To find how long, and bow far, a given Globe ought
zo deſcend by its comparative Weight in a Medium
of a given Denſity but without Reſiſtance, to ac-

_ quire the greateſt Velocity it is capable of in de-

• ſeending with the ſame Weight, and in the “oy
  Medium, with Reſiſtance.

Let f be the Space and g the Time required; |

then will the Square of the Velocity acquired in

falling through the Space s by the Weight @ be

to the Square of the Velocity acquired in falling
| through the Space F by the Weight b as 4 fis to B,.
(by the Theory of the Deſcent of Bodies.)
But the Square of the former Velocity is to the

| Ek 5
Square of the latter as 45s is to ——7X *d s by

| the third Lemma: Therefore as is to 5 hf. as 455

is to = we *

the ee F is to 4 of the Diner of the Globe

as the Denſity of the Globe is to the ö of che
Fluid. QE. L

Again,

K M; whence fn x td; that i 1

A ww — 2 ·

7 A =

FY IN — Cn 44 ted. f— „

of Bovrns m FLutDs of ,
Again, ſince the Space through which a Body

is uniformly accelerated is as the accelerating Force

multiplied into the Square of the Time of its Con-
tinuation, we ſhall have the Space 5 to the Space

F as a X 1 (that is, as @ multiplied into the Square

of the Time of one Second) to bXgg; whence g
will be found equal to 7 * and g equal to Vi of

3
that is, the Time ſought will be to the Time of one

Second, in a ſubduplicate Ratio of af to bs. QE. I.

CoOnxOtk

If a given Ball was to move in a ſtrait Line
through a fluid Medium of a given Denſity, and
if its firſt Reſiſtance was to continue all along the
ſame till it had quite deſtroyed the Motion of the
Ball, the Space through which the Ball would paſs
to ſuffer this Effect would be always the ſame:
For it would always be as the Square of the firſt
Velocity directly, and as the Reſiſtance, that is,
as the Square of the ſame Velocity inverſely.
Therefore if this Space be known in any one par-
ticular Caſe, it will be known in all others. But in
the Caſe where the Square of the firſt Velocity is

: Sz ds, this Space is known to be the Space f

2 a Continuation of the firſt Reſiſtance in

this Caſe will have the ſame Effect as a like Con-

tinuation of the relative Weight 5; therefore in all
other Caſes the Space abovementioned will be /.

Col e

“= Of the RESISTANCE

Cororr. II.

The Time wherein the abovementioned Effect
will be produced will be that wherein the Space 2 f
will be uniformly deſcribed by the firſt Velocity.

Lemma V.
The Fluent of a Fraction whoſe Numerator is

the Fluxion of the Denominator, is the hyperbolic
Logarithm of the Denominator : w or it has been

proved already chat the Fluent of = – is the hyper-

bolie Logarithm of z.
CororLrs, I.
Therefore the F luent of this Fluxion —
m m.

is the negative hyperbolic Logarithm of mm—vv,
that is, it is the hyperbolic Logarithm of

mm —

CoROLL. II.

2m v
—

_— Logarithm of pun For — =

+—: and therefore the Fluent will be
—

the ah Logarithm of uu, minus the
hyperbolic Logarithm of m—v ; that is, it will be

the hyperbolic Logarithm of the F raction

And the Fluent of this Fluxion >

is the

Pay

PR 0-

—

of BO DIES in Friul. 95

OW DUE; = XVI.

Let a given Ball paſſing in a ſtrait Line through

2 fluid Medium of a given Denſity, be reſiſted
every where in Proportion to the Square. of its
Velocity: It is required, having given the Velocity
of the Ball at its firſt ſetting out, to find not only the

Motion it will loſe, but alſo the Space it will paſs z
through in a given Time afterwards. |

Loet d be the Diameter of the Globe, Ne let the

Space f be taken in Proportion to 44, as the Den-
ſity of the Globe is to the Denſity of the Fluid ;

let M be the firſt Velocity, and R the firſt Re-

ſiſtance of the Ball, and let T be the Time wherein
the Space 2 f may be uniformly deſcribed by the
firſt Velocity M; then will T be alſo the Time

wherein the firſt Reſiſtance R, if uniformly con-
tinued, would entirely deſtroy the firſt Motion of

the Ball; by the Corollaries to the fourth Lemma.

Let s be the Space deſcribed by the Ball in any

indeterminate Time t, from its firſt ſetting out,
and let v be the Velocity and r the Reſiſtance at the
End of the Time ?: Then ſince the Velocity gene-

rated or deſtroyed by any uniform accelerating or
retarding Force is as the Force multiplied into the
Time of its Action, we ſhall have the Velocity M
to the Velocity V as Rx T the Product of the
Time and Force whereby the former Velocity may
be deſtroyed, to X: the Product of the Time and
Force whereby the latter Velocity is deſtroyed.

But R is to r as MM is to vv by the Hypotheſis ;

there

96 Of the RESISTANCE
therefore R Tistor;as MMI is to vv;; there-
fore M is to V as MMT is to v7; therefore

MT x But the Fluent of tis t, and the
Fluent fMTxL iS TO, whoſe Value at
the Begining of he rern bi rx. and at
the End of the Time 7 is M Tx; and the Dif-

Mev TM To

ference is M Tx Mo = – ; therefore
Boer of oy at ”

IS ———; and conſequently v, or the Velo-
locity at the End of the Time will be LEP and

therefore the Velocity deſtroyed by the 8
in the Time 2 will be ſuch a __ ng the whole M

as is expreſſed by the Fraktion; L — Q. E. I.

Now to find the Space we have the following
Proportion (viz.) As the Space 2 f is the Space ;,
fois MT the Product of the Time and Velocity
whereby the former Space may be uniformly de-

ſeribed to v? the Product of the Time and Velocity

whereby the latter Space is deſcribed ; therefore
= N W“ U 7 Inſtead of { ſubſtitute its Value
MTxZ, and me-thall have Saxe Bur the

the F Oo of; is 5 and the F luent of = — Is the

< 1 *

.

E

| rithm op

of BOoDI RS. In FLULDS 970
ol the Time is the hyperbolic Logarithm of M,
and ar the End of the Time ? is “the hyperbolic”

Logarithm of v; . the Nifference is the, hyper-
bolic Logarithm 4 * cherefora s 8 the hyper-

bolic Logarithm of 8 multiplied into 2 f. In-

ſtead of <Y ſubſtirure its vad 2

LE as 20 fon id,

and we ſhall Rave equal to the hyperbolic Logi- R

E x 27. Now as the Ball in the

Time T by its firſt velocity M would deſcribe”
uniformly the Space. 2 Hut would in the A

by the ſame Velocity M deſcribe the Space 2f x: 7. |

therefore the Space s actually deſcribed by the Ball
in the -Fime g, with Reſiſtance, will be to the Space. |
it would have deſcribed i in the ſame Time, had it
moved all along with its firſt Velocity, without any *

Reſiſtance, as r hyperbolic Logarithm of 2 8
is to the Fraktion 5: Make m , 43429448 190 ll

and we Mall We the hyperbolic Logarithm of

The

• equal. to. Briggs Logarithm of the ſame
  lA into — chat is, into 2,302585093 3

therefore the Space deſtribed in the Time ? with”
Reſiſtance is to the Space that would have been de-
icribed in the ſame Time without Reſiſtance, as the

H | common

0 amanda
— — , ̃ – ‚ , 2 — — emo os _
— — —

os. Of theRfsisTancey

5 ME Ae: n
common Logarithm of 5 5 multiplied into

2, 302 38 093 is to the Fraction = E. I.

See Newtow’s Prizcigis, Edit. 4 eg 3 55.
abc.

In the foregoing Solution the Space 5 was found
T 2 3

equal to the hyperbolic Logarithm of
multiphed 1 into 2 f, which is equal to Briggs’s Lo-

garithm of the ſame Fraction multiplied into 2

Let t, and let the Denſity of the Fluid the
Ball moves in be the ſame with that of the Ball,

then will che Logarithm of = be the Logarithm :
| 2,30103 3 ; and the Space a will be equal to 7 4,
an 2 z=6 $4022 45 therefore the Logarithm

oe 1 | EL; or the Space s will be 1,8484 4,

3 is ſomewhat leſs than 2 d. Again, The

t M
Motion loſt which was II vil in this Caſe be

M; therefore a Ball projected in a Fluid of the
ſame Denſity with itſelf will loſe half its Motion,
before it has paſſed through a — * to two
of its own Diameters.

PR o-

Pen. in FLuI DoS 99

PROBLEM XVIL

Let a given Ball be ſuppoſed to fall from Reſt
in a Fluid of a given Denſity, being every. where
accelerated by the Exceſs of its comparative
Weight above the Reſiſtance of the Fluid: Ir is
required to aſſign the Space through which it will fall,
and the Velocity it will acquire in any given Time.

Let 5 be the Deſcent made, and v the Velocity

dos in any indeterminate Time p, and let

Unity, or 1, repreſent both the relative Weight
of the Ball, and the greateſt Velocity it can ac»
quire in deſcending through this Fluid: There»
fore 1 will alſo be the Reſiſtance of the Fluid an-
ſwering to the Velocity 1, and vv the Reſiſtance
anſwering to the Velocity v, and 1—v v will be the
unequal Force whereby the Motion of the Ball will
be conſtantly accelerated. Let f be the Space, and
g the Time of the Deſcent neceſſary to be made
for the ſame Body to acquire by its relative Weight
the ſame Velocity 1 in the ſame Medium without

Reſiſtance : Then, ſince the Velocity 1 is gene-

rated in the Time g by the Force 1, and ſince the
Velocity v is generated in the Time 5 by the Force
1— v, we ſhall have 1 to vas 1X# is to mY

1 25
whence = . i But the Fluent of = fu * |
and cheFluent of — — the byperbolie Logarichm |

of the-Fraction = j (by the ſecond Corollary to

H 2 che

fob Of the ns 15 TANCE
the fifth Lemma ) which, at the Beginning of the

Fall, is nothing; cherefore =, is the hyperbolic
Eogarithm of — — and if (as before) m be put

equal to Afi2gd43 9% hw 57 will be che com-
mon Logaritim of A. Let u be che natural

„Werne a 20 n
Number anſwering to the Logarithm + = and
we ſhall have j _ 15 whence v will be equal to

KEE chat! is, the Velocity 1 acquired in the

Time 2 wilt be ſuch a Part of the Lana Veloci-

• 1s expreſſed by the Fraftion T 22 3 Or (which

is the ſame thing) the Velocity v will be to the
reateſt Velocity : as 1—1 is to *I. Q. E. I.
233 ſince the Space 2 f- can be deſcribed by
Velocity 1 in the Time g, and ſince the Space ;

is Greg by the Velocity v in the Time 5 55 we –
ſhatl have 2f to; Y; 2K is to v; whence r

| 90 27. Ariſtead of * put its Value above found

(viz) uy cf and we ſhall have; = Fx. —

us the Fluent, of vis 5, ind the Fluent of 200

12 v

= by the firſt
Corollary to the fiſth Lemma; therefor the Space 5
is the bypettitic Lohn ithm “I by multiplied 15
yy into

15 the hyperbolic Logarithm e of –

1 b | I—vVy 58

c

of = will be

EI

/ BO DIES in FLuId Ss 101
into fF; or it is Briggs’s Logarithm of the ſame
Fraction 1 in 2 dard has * been
is equa} to
e, cher fore non we won Jay that: we
wry is the Logarithm of © _ “FE muliplied

into 2 But this Fiaction = ==) 18 the Pro-

D-
duct 1 two others; whereof one is —— a ee

proved equal to == =; rherefore – 1 LA,

the ber. is = ieee of I >

Ls 1 2
then will Ty be the Logarithm of * I Let 7

be the Logarithm of 45 then ſince the Loguithd
of 1 was before found een 0 the Logarihm

2pm

— Put theſe two Logarithms

&:
rogerher, and we ſhall have the Logarithm of the

Eraftzon e kat, therefore

os Space FSI PE kn! Ix- fb uf
5 1 —— —, 386294361 dees oeh

3

See Newton’s Principia, L i. Fro 40.

H 3 Sh.

102 2 ottrackive Fox cr

i be n

By ſeveral Experiments made upon the Deſcent
of Bodies both in Water and Air, this Theory of
Reſiſtances is not only ſufficiently confirmed, but
it is alſo further demonſtrated that, in theſe Fluids
at leaſt, the Reſiſtance that might ariſe from other
Cauſes is either none at all, or very inconſiderable
in Compariſon of that here conſidered which ariſes
from the Vis inertiae of the ſeveral Particles whereof
all Bodies, as well Fluids as others, are compoſed.
See the Scholium to the fortieth Prop. of the ſecond
Book of a Principia. 8 |

Of the Acvion of a prolate SpHEROID ufon a
Particle of Matter placed any where in its
Axis produced; neceſſary for underſtand-
“Ing the Figure of the Earth,

W 2 A Turo u. Fig 22.

Loet EF ike) be à Line given in Poſition,
and let FG () be ſuppoſed to move always
perpendicular to E F, and let a—2 bc K c
the Quantities a, 5, and c being conſtant: I ſay
then that the Point G will deſcribe the Circum-
ference of a Circle whoſe Center is ſomewhere in
the Line E F, or.in the Line E F produced.

9

„77VFV op

| PP Wk

Make — 2 *

_ of a SPHEROID oy

DzMoNSTRAT LON.
Produce the Line 1 E out from E to D, ſo that

ED may be equal to >, and call D F,: z; then will

2bb

E F or X=Z— > and —23 x=—2b24— „and
— * * b PA… and a—2Þ * c x, or
E 6b bb

| EY J==&——2 ba — ct nnn

3 |
Mhor therefore = EE as.

rr, and we ſhall have rz; 3
which is 2 Nature of a Circle whoſe Center is D

and Semidiameter 7. Q. E. D.

Note. If the Coefficient of x is + 2 b, ht Point
D; is tO be taken on the other Side of E.

PRO P. II. A Taxxorem. Fig. 23.

Let ABB be an Ellipſe, A a the greater Axis,
B 6 the leſſer, C the Center, F and F the Foci, Pa

Point any where in the leſſer Axis B5 produced

from B, PD a Line meeting the Ellipſe any where
in the Point D, ED G a Line drawn through D

perpendicular to the Axis B & in E; and let the

Length E G be ſo taken that the Line PD may
always have the ſame Proportion to E.G, that FC

hath to CB: I ſay then, That the Locus of the
Point G will be in the Circumference of a Circle whoſe

Center is ſomewhere in the Line PU, or in that Line
bm from 2

its 1 1 ET,

Bo Of the atriiHve Pon 2
rA 72.5 87 41 + ff
ub & d o. DaxonsT rar Aq abo:
Call C A. ü; CB, c; CE, n; CP, * CE. >
band Wie mal fave PE. or-, add BE
i r. 2IdxÞ-xx,.. Moreover i it is evident. —

Itxx

che · Nature af the Elli ‘»ſe, “that: EDT, i eee

“OE

IL |

Uberefore- EI) or FE. ED: . or
„ xx EExxX—I xx

_ PPS od. og -CC —_ |

3 PD= or EN — d

£4 4 8 * Wo ng

« 19 55 |
— + or —24 45 — —.—. 4 Therefore by the fore.

going Prepolition, heli doi of the Point G is
the Cirtumference of a Circle whoſe Center is in
the Line PS, or inthat! e QE. D.

28 3811 COR. 3

AS o Fonſtrüct “the Wins 6f che Point G;
BK, CI, an d * de Pethendlculars to the ow

2014 8 Cie 1 215

-24. LetgK bedakenequal eo P x, gand ö
<quat’is PIN; hen ir it ris evident that the Points

Nand will be in the! Cirolanfiredce of the Circle
Tegoirtcd.: And | ſince BK ĩs to h has PB is to PS,
dit is alſo further evident, that the Line P K. that
paſſes chrongb the Points P and K will alſo paſs
thrbiph k. Let the Bine P & cut the Perpendi-

Falar Clin], *. * 1H perpendicular co PI
cutting

5

Conſtruction. ;

ef a SPWEROTD:” Tot

cutting the Line P (produced if. need be) in H:
then will a Circle upon the Center H, and wake
Diftance H * ur kik be the pans Io:

e

8 II.

The Trthrighes P’BK,P3k,*P CI, PIH, BOB
are all fimilar to one another; for they have all
otie right Angle, and the Sides about it propor-
tionable. PB is to B Kas FC to 3 |

; : 1 * *
— 1 7 1 I + © 2 8 : 8
. | þ1 ? * * *
0 WA . 4 > i

8 L 5 Coro L. III. 1
“as PF. and Hk. and the Trang le HIK

will be fimülar to the Triangle PC; For Hf I is
to IP, as B C is to CF; and P is to K as PC
is to BC; wherefore, ex aequs! | re gi * n

be to I K 4s PC to OF.

„ „For- III. A Pan. re
Let the Ellipſe A Bas by a Revolution about
its leſſer Axis B þ generate a prolate Spheroid whoſe

equal Particles attract at all Diſtances with Forces
reciprocally as the Squares of thoſe Diftances :

i is required to determine the Law of the Force
_ whenewith. à Particle of Matter, placed any where in

Se Axis Bb rag: will be won by the whole
Spheroid.

| ter | |
Reta; ning the Notation and Conſtrustion of the

foregoing Propoſition 3 From the Point e in the

Line

106 Of the; attraftive Force
Line C E e let a new Ordinate e dg be drawn infi-
nitely near the former EDG, and let the Spheroid
be imagined to bexrefolved into an infinite Number
of infinitely thin circular Laminae, whoſe Planes
are all perpendicular to the Axis Bb, and let the
Space E D de by its Revolution about E e generate
one of theſe Laminae: Then it is evident from the
firſt Corollary of the ninetieth Propoſition of the firſt
Book of the Principia, that the Attraction of this

| Lomina will be as Ee——pT—: But PE=PH

—f E; | therefore the Attraction of the Lamina

wil be as KR D Therefore

the Fluent of this Fluxion, 1 is to repreſent
the Attraction of the whole Spheroid, will conſiſt
of three Parts, which muſt be found ſeparately
mw:
Firſt, The Sum of all the F004 E e, at leaſt
ſo much of 8 Sum as makes for our 1 9285 is

BS or 2c.

HEE?
h PD:
drawgR 30 and parallel to Ee, and cutting
E & in R, and join G H; Then will the Triangles
G Rg, and HE G be ſimilar, and we ſhall have
Rg, or E e, to G R as E G is to HE: whence

H EXE e E GIG R:=P DxG R ð ; therefore
FE: |
5 RN; and the Fg of all the

Secondly, In order to * the Fluent A

Fluxions

of aSPHEROID; 107

— — — g will be equal to the Sum of all

the Fluxions G Rx — — generated | in the ſame Time,
But as much of this 1 latter Sum as makes for our
Purpoſe is =D N X<:
Therefore the ſecond Part of our Fluent is
BY x—. Add this to B 5 the firſt a already
found, v3 they will make Bby 1 5 X

CC | ‘ & 2006 WL of
BI =. Call cr, which is

Fluxions

always proportionable to the Magnitude of -the
Spheroid, q 3 and then the two firſt Parts of the

Fluent added together will make – .

Thirdly, We come now to the third Part of the
Fluent, which is to be ſubtracted from the former,
PHxEe

and which is the Fluent of PFF and this

alſo will be obtained by the Help of the former
ſimilar Triangles G Rg and HE G; where Rg,
or Ee, is to Gg, as E G is to GH; therefore
G HxE eZE GxG cr D G N therefore

Ee Gg c P HxEe c 62 |
FD =GH * an- DP Hx xf

and the Fluent of the former is equal to the Fluent

K G
of the latter, which is P H * * GH:

Join

108 Of the altractive Fo RC E

Join PF and H , and upon the Center P, and
with the Radius PC, ſweep the Ark L CI, cutting
PF and EF. in L and I reſpeCtively ; and becauſe
of the Angle LP equal to the Angle K H *, (by
the third Coroll. to the laſt Prop.) the Arks K gk
and LC will be ſimilar, and we ſhall, have
KGk Doe c KGE PH
H= PN nd PHXAXGH PEN.
xLCz. But P H is to PI, as f to m; therefore
F

PHis t t to nm, and 8

dankee Set C N C i= E L.

In the Tangent C F, let a Line as C O be taken
equal to the Ark CL, and the third Member of
the Fluent to be ſubtracted from the two former

will be 74 XC O. Subtract therefore this Fluent
ace O from the Sum of the two former, which

m3
vas or 21,0) F. and late will remain „ XFO

for the Ai of the whole Soherold,: There-
fore the Attraction of the Spheroid will be as its
Magnitude, and the Line FO direCtly, ad as s the
Cube of the Line CF inverſely, $1

This is upon a Suppoſition that the 3 of
the Spheroid is: uniform, and that its Denſity con-
tinues always the ſame: But if we ſuppoſe the
Denſity to vary, then the Attraction of the Sphe-
roo will be as its Magnitude. and * and the

5 Line

of aSPHEROID, © 109

Line F O directly, and as the Cube of the Line
CF inverſely; that is, the Attraction will be aa
the Quantity of Matter in the F and the

Quantity Cr © —_ together. * 1.

eee, 1

Since C O is equal to an Ark ai Tangent
is CF; by a Series formerly computed 1 it follows,

1

that CO will | be equal to cr “FEB
PE FF 7 ©

— &c. Therefore 799 E pr Se.

1 3 Be
— — — Ge. Let CF
0 CF ge res con g va-

niſh, unc meh the Fetm p A all that fullow

it, repreſented by c. wil vaniſh with it, an we

h FO I
mall have == 8 8 561 SE.

nee l.
The Attraction of che Spheroid will be to the

Attraction of the ſame Matter condenſed into an

infinitely ſmall Particle in the Center, as 8 F is

This follows from the foregoing Ca-

to

Pz

rollary ; ; for if the Fi igure of the condenſed Particle

was

116 Of thattraffiveForer

was a Spheroid, C F would be evaneſcent in reſpect
ef PC, which is ſuppoſed to continue the ſame :
But the Attraction of the Particle will be the ſame
in this Caſe, whatever Figure it puts on.

3 III.

The Attraction of every Sphere whoſe Matter
is uniform is as the Quantity of Matter it contains
directly, and as the Square of the Diſtance of the
attracted Particle from che Center of the I

• :

Cool. IV.

Every Sphere attracts with the — Force as if
all its Matter was accumulated into its Center:
For, by the foregoing Corollary, the Attraction of
a Sphere depends only upon the Diſtance of the
attracted Particle from the Center of the Sphere,

and upon the Quantity of Matter in the Sphere,
and not at all upon its Extent. |

Conor *

Since the Quantity of Matter in the Spheroid i is
to the Quantity of Matter of a homogenous Sphere
upon the ſame Axis Bb as BCyAC*i is to BC

or as A AC* is to BC; it follows that the Attrac-
tion of the Spheroid will be to the Attraction of

a homogenbus | Sphere upon the * Axis as

FO .

3 ET SCHOLIUM»

of aSpuBnoID Tit

SCHOLIUM 1.

If the two Axes of the generating Ellipse be
not very unequal, and the Proportion of the laſt
Corollary is to be computed, there is no Neceſſity
of having Recourſe to trigonometrical Tables:

For the Computation may be performed much
more expeditiouſly by the Series of the firſt Coroll.

2— CF
thus: Make A C*=#p, 2 . R =,
£425 Sc. Then will the Attraction of the

Spheroid be to the Attaction of a homogeneous

Sphere u upon the ſame Axis, as 2 tat Sc.

T0 #. 3. © ®

is to . Ex. Gr. Let B C be 100, and AC 101,

and Sa F 201, and let the Particle P

be placed at the Vertex B; then we ſhall have
?=10201, qz=205, and 7=4, and B C*— I0000,

| and the Attraction of the Spheroid will be to the

Attraction of the Sphere as — Zis to
10000 8 +

that is, as 10201 1232 is to 10000,

that is, as 10080 is to 10000, or 126 to 125,

ScnoLliuvm II. Fig. 24.

Let the attracted Particle P be placed within the
Spheroid, ſuppoſe between B and C, and let BK

and 5k be taken as before both on the ſame Side

the

112 Of the atiraflive FOR Ex
the Axis, and the Lines PK and P will now be

different Lines. Moreover i in the Segment whoſe
PEXEs :-

bann BA, the Flax e gil,

— —_PH5 — l

bar i in ” other gen whoſe Axis 1 B, the |

— PH E.

1 luxion i is 3

whoſe Axis is P & will be PBC 1 TN EH
X=X IF Hk and the Attraction of the other Seg-

ment whoſe Axis is EB will, be E BEB NPT |

TK .
N P Hx XH K. Subtract the Attraction of
tis latter Segment from that of the former, that: i 18

ſubtract PB. from Pb, and there will remain 8

Again, ſubtract B K— PTx> from 7. PTX- “—_

and there will remain b k—B 3 —aÞ xt.

Laſtly, ſubtract PHx 0 — * Th

and there will remain —P Hg RAD

Now chat thĩs laſt Part i is as the Diftance PC as

well as the two former, I thus demonſtrate : Pro-

duce che Chord K E, “nd the Axis Bo till they
; meet

“PK–* Therefore
drawing PT Ts oe to Bb as far as the

Circumference K I &, the Attraction of the Se .

• 
  — —

—— FH

— —

%%. SS DS 5 S ©

295

of a SPHEROID, 113

meet in Q, and H will be to I Q, as CI to CQ

and I Q will be to I K, as CQto CB; therefore,

er degue, HI will be to K, as C 1 to C B; that

DON

is, in a conſtant Ratio; for Clor — —½ or

2B 5 xX= is a conſtant Quantity; nab the

Triangle HI, and conſequently alſo the Tri-
angle H K & will be given in Specie : Therefore the

Angle K H I will be the ſame, wherever the Par-

ticle P is placed between B and 5 Therefore

aA will be a conſtant Quantity ; ; “Therefore

tt TE
PH x 0 or 2PC rer will be

as PC.

Of Sec onD, 11 and FoURTH
FLuUx1o0NsS and FLUENTS.

Let a & repreſent any finite Quantity of Time
terminated by the two Inſtants @ and 5, and let x
be the Value of any variable Quantity at the In-
ſtant a, whoſe Velocity at that Inſtant is ſuppoſed
to be ſuch, that if the Quantity was to flow uni-

formly with that Velocity, during the whole Time

ab, it would gain in that Time &: Then will x
be the Fluxion of the Quantity x, according to the
Idea of Fluxions formerly given; and every other
Quantity depending upon x will have a different
Velocity at the Inſtant 3. Thus if y be made equal
| * | to

114 Of the higher ORDERS
to x”, the Velocity of y at the Inſtant à will be
ſuch as in the Time a4 would gain uniformly
m xXx H; and therefore m x X A is called the
rj Fluxion of y.

Again, the Velocity of this Quantity mx Xx mo
at the Inſtant 2 is ſuch, as in the Time ab would
gain uniformly n iK A; and therefore
this is called the ſecond Fluxion of y.

Thus again, the Velocity of this Quantity
m m1 AAN XxX at the Inſtant @ is ſuch,
as would gain uniformly in the Time 45 the Quan-
tity mmi Xm—2Xx3*X”=3; and therefore
this Quantity is called the third Fluxion of y.
And for the ſame Reaſon, the fourth Fluxion of y
will be mx m—I1xX m—2Xm—3 XxX * I and
ſo on ad infinitum ; except when m is a whole
Number and Affimative, in which Caſe the Series
of Fluxions will at laſt break off.

On the other hand, let us in the next Place
inquire, what Quantity will have ſuch a Velocity
at the Inſtant a, as that, if it was to flow uni-
formly with that Velocity during the whole Time

42 b, it would gain , or x”. Now that Quantity

we ſhall he to be? N Ni K E; according to

what * formerly been ſhewn concerning Fluxions;

1

and therefore Ix 7 n is called the firſt

Flucnt of y.
Againl

.

IJ .
N *
”
OY
I. o

• 
  —
  • 
    4
    2 — Pr .
    1
    : 1
    |
• 4 K
• 
  : 2
  ”y
  N
  D
  /
  /
  — — „ ² A A
  ” . \
  © 5 7a q _
• : | p
  7
  /
  /
  ; , % oo
  / by
• N f {1
  / f 3
  3 4 4 iS 4 4 |

7 r * ” *

; 4 * *

: * —— —
; —— — — + ab | a+ 1 as
p ‘ * „ E
j *
1
AMY F
4

—
—
r

of FLUXIONS and FLVEN TS. 115
Again, if it be inquired what Quantiry at the
Inſtant à has ſuch a Velocity as in che Time a 5

1
would gain emily = ha I will be

found to be © Xx t; and there-

N m1 NE
fore this Quantity is called the ſecond Fluent of y.
In like Manner, the third Fluent of y will be
I

• X Xx m2 x m3.
  infinitum; unleſs mis whole Number and Negative;
  in which Caſe the Series of Fluents will alſo d
  off. |

= X x*+3, and ſo on ad

ScHOL IVM.
The radical Quantity of all theſe Pe which
in the preſent Caſe is x, is generally ſuppoſed to
flow uniformly, and conſequently to have no
Fluxions beyond the firſt : And in this Cale it is
uſual to put Unity for. the Fluxion of x, that its
ſeveral Powers may not come into the Computa-
on. If this be done, the frft, ſecond, third, &c.
Fluxions of y will be mx e i, m% .I XK ,
n IX m—2 X e , 6c. reſpectively: And
the firſt, ſecond, third, 2 Fluents of y will

be „ x=+1, —

mi 5 x m + 2

E
= 27 x x*+3, Cc. reſpectively.
| 8 Fs T his

116 The Roo rs of fluxional. Equations

This Doctrine thoroughly underſtood will be :
very uſeful in extracting the Root of a fluxional

pp as will hereafter be ſhewn.

The Method of vail ng the Roots of fluxional
Equations, by comparing the analogous Terms

of Series, illuſtrated by Examples.

„

In an equilateral Hyperbola whoſe Power is Unity,
bs 2 be the aſymptotic Area comprehended be-
tween two Ordinates whoſe Abſciſſae are 1 and
1x, and we ſhall have this fluxioual Equation

0
(viz. ) 3 == Js” Let it now be required to ex-

tract the Root z of that fluxional Equation ; that
is, let it be required to expreſs the Value of z by
2 Series conliſting of the ſeveral Powers of x.

Here becauſe x is the variable Quantity that is
ſuppoſed to be known, let its Fluxion be expreſſed

by 1, and you will have 2 2 = 55 Or 2x S—1=0.

This ſuppoſed, let z be expreſſed by the following
Series a x”-|-b x – e di, Fc. where
a, b, c, d, Sc. are Coefficients, and mz and x Indexes
of Powers, all hereafter to bedetermined : Then it
will always be beſt to determine the firſt Term of
the Series from the Nature of the Problem the
fluxional Equation was drawn from.
| Thus

pa — Gatos 12

expreſſed in 8 E RI E * * np
Thus let x=0; then it is plain from the Nature
of this Area that z=x : But when x, 2 will

be accurately equal to a x”, all the other Terms of

the Series vaniſhing in reſpect of this : Therefore,
in this Caſe, ax*=x; and conſequently in all others:
Therefore m=1, and the Coefficient @ ( if there
were any Occaſion to determine it t here) is alſo
equal tO I.

Next to mine # the Difference of the In-

dexes; let us reſume the fluxional Equation before

found viz. x Z—1=0, and ſince z=ax”, Ec.
we have $=am xn, and x San x: Therefore

ſince we have here two different Powers of x, viz.

am n 1, and a mx”, and ſince 1 is the Difference
of the Indexes of theſe Powers, I make n=1 in
order to preſerve the Form of the Series; as will
abundantly be ſhewn by this, and the following
Operations. So that at laſt the Series is determined

fo be of the following Form, viz. X a x-þbx*

b+cx3Þ-@ e K Fx, &c. and the Coefficients
will be determined by the fluxional Equation, as
follows:

S 2 cx 1 ;
ent Cf; = %
AS YH ax T2535 * ＋ 3c Sc. =0
f ＋AdN t 5 exs.
—1I=—1i+0+o-þo+o-o..

Where we have the following Equations a—1=0,

2b4-a=z0, 3 (5+2 b=0, 4d 3 =0, 5 6-4 9

13

118 The RooTs of fluxional Equations
6f4-5e=0, Sc. Therefore a= +1, b=— 4,
c=+3, d= -. = ＋ , fF=—7z3 and
r r- 3% x x3 -E — 7 x5 Sc.
N. B. The Root > in this Example might alſo

be extracted by the Means of ſecond F luxions, or
firſt Fluents.

Ex AML II.
In the foregoing fluxional Equation let now z
be known, and let it be propoſed to extract the

Root x, or to expreſs x by an infinite Series con-
ſiſting of the ſeveral Powers of z.

Since then in the foregoing Equation —— =2,

if we make Z=1, we ſhall have 1-x—x=0,
Let x=a2+b2*Þ 234 ANA ez3+ f ze, Ge.
for this Series muſt be of the ſame Form with the
foregoing, otherwiſe they would not be convertible
into one another. This ſuppoſed, reſuming Our
fluxional Equation IFxX— X==0,

a=r+$0o$+0+0+6

• x = * T TU Ic -d

| ez. e,

• 4 – 2E ατεεν
  | — 5e2* — 6f 2p.

Whence we have the following Equations ;
I—& = O, 4—2 b=o, b—3c = = 0, —-4 d Ho,

seo, 6 That is, 1= a, 5 =

6 —

| expreſſed in SERIES. 119
ſi. d = 4 e = K * – 45 So that at laſt
3 + 38

we have the following Series x = 2 Mx 2 4 EO

D TAN dZzS TZ ez, Oc. if the Letters
a, b, c, d, Sc. repreſent only Coefficients; but if
they ſtand for whole Terms, as they come up,

then we ſhall have x 2 e IF

r Sc.

ExAMPLE III.

“it #8 required to find any Ark as 2 from its Sine x,

which is ſuppoſed to be known, in a Grele whyſe

Radius is 1.

Here the fluxional Fra expreſſing che
Relation between the Ark 2 and its Sine & is

• I

= 2, or

Es 3 82 Ma <=1 :
— 1 ke & == 1, and

we ſhall have

To render this laſt Equation yet more ſimple,
throw it into ſecond Fluxions, and we fhall have

I—XXX2 22—2X2=0; or dividing both Sides

by 2 Z, we have I—=XxXz—XZ=0:; or thus
„A* L—X o. Having thus brought our
Equation into the ſimpleſt Fo orm, let æ a ——
unt, Fc. and when x=0, 2 will be equal to x,
becauſe every infinitely ſmall Ark is equal to its
Sine: Therefore a x”=x, and conſequently a= 1,
and m =1.

=, or IM 2 = = I.

14 Nov

120 The Roors of fluxional Equations
Now to determine ; ſince z—xxz—x3Z=0,
and ſince z = ax nearly, we ſhall have z am

X X . , and x x Z = 4 mMXM—1IX Xx”, and —xZ = —a MXx”. There are therefore two different Powers of x concerned in our fluxional Equation, viz. x#—?, and x; therefore I make
1 22, and the Series will ſtand in the following

Form, viz. z=a xþ+b xc * dx ge 9 F,
Sc. Therefore

z=2 eee

S Xen + 10x 11fx9

—X = %—2X 3Z0x3—4X 5 25

| —6X7dx/ 8X9 ex9. 0
2 =- X —-3bx 53 c —7d x7
E29. | |
Whence we have the following Equations for
determining the Coefficients; 2X 36 —a=o,
4X5c—3X 3 b=0, bX7d—5 X 5 c , 8X9 e—
** o, IOX II- NQ e == o, c. that is,
2X3 „ 7 85

: 9X9 ;
f= 7 N e. So that if x be — Sine of ; any Ark,

the Ark itſelf will be x + E E 2233 «5 4-
5X5, 0-17 gd 9X9 e . oo ho
fo To Tix. uppoſing
a, B, c, d, Sc. to be Coefficients : But if they be

the whole Terms, we ſhall have z= x 2 7 5 a x2
3X3, , 5X5 , 2 24 9X9
b,
r To T T 10XL1 2 85

BE .

12

—

expreſſed in SERIES 121
N. B. If x be the Coſine of any Ark, the Fluxion |

—X
of that Ark will be „ and its Square
my 1 .

So that, the ſame dae Equation that

finds the Ark from its Sine will Rs find it from

its Coſine; but here the Series will have a different
Beginning from ‘the Series in the former Caſe :

For if q be a quadrantal Ark, when x 1s infinitely
ſmall, the Ark whereof it is the Sine will be x;
and therefore the Ark whereof it is the Coſine will
be -&: Therefore -* wilt make the two firſt

Terms of this Series; after which the Series will

aſcend regularly by Powers of x, whoſe Indexes
will have 2 for their common Difference.

But the Invention of the Ark from the Coſine
will beſt be effected thus: If z be the Ark whoſe

Sine is x; then z will be the Ark whoſe Co-

ſine is x. Za – ts already determined to be

3X3 =, 2
qu: hi TI * n

poſing an .

Ex AML E IV.
Let now the Reverſe of the former Example be

propoſed ; z that is, from the Ark 2 being given to

find its Sine or Coſine x, ſince we find the lame

fluxional Equation ſerves for both.

x2

The fluxional Equation is

22, or

—_

making 2=1, we have l, or I- * &.
| or”

Throw

——ꝛ— ̃ lX—ů 3

— . ec — > Pre l — CDT be _ —
Aon OA . A RWO L. 4 64
y — ce ACRE — n
E * 1% ITY WA ” n e

1 “RO
mm 1

—
—ä— ee”
mm

1
Ht
Il

f
i

t

{

.
|
N
|
N
N

o
| ——— —L ER 4 — —

122 The Roors of fluxional Equations
Throw this laſt Equation into ſecond Fluxions,
and we have —2x X=2z x, or — x , or x; =o,
Let x=a 2″Þb , &c. and firſt let x be the
Sine ſought : Then when Z=0, x will be equal
to x: Therefore a2z””=2z: Therefore m=1, and
al, as in the former Example. Again, in the
fluxional Equation x ＋ x =o, ſince z=2″ ſetting
aſide the Coefficient, and x=2—2?, we have n=2.

Let therefore x =az+b23+ 25+ 427, Ec.
and the Equations. will be as follows :

| x =a2+ b23+c25þ 25.
3 $2: = 0.

x =2X362+4X5c2BÞ6X7425
+8 xX9ge27/+10X11f2%. _
Whence, and from what has been proved before,
— 4 — 6 0
we have a=, = >= I Sox?
d

72 ; 22 7
8X9? 2 — rere, &c, Let à, b, c, d, Sc.

1

repreſent the ſeveral. Terms of the Series ; then we

have x2 : 4 22 5 —
_ 7. 0 , 2’Se9
d22— ——ez?, Sc. | :
” JOWEH

If x be the Coſine fought, then, when z go, the
Coſine x will be 1, v2. the Radius: Therefore the

firſt Term, viz @2″=1; therefore a=1, and m O:
And ſince #=2, as before, let x==a|b 2?-+c 24

d e + fzie, Sc. and the Equations will
be as follow : :- |

x4

Sine or Coline is x is

of the Ark whoſe Sine or Coſine is 2 18

expreſſed in SERIES, 123

Nt —=a+b*4c #442 Tel.

1X2b-+-3X 425X642 S0

Qt +1x8e=+9xwfz. \

Whence, and from that a=1, we have the follow-

5 b
ing Equations; 4 1, b= —— 8 cx —_——
c ” e |
4 — Ih = r e Sc. Make
2, b, c, 5 Sc. repreſent the Terms of N Series;
then we ſhall have x==1— — 22— — 322—
I I 3 0
— 2 —— 42 — e, &c.
5X6 7X8 NN

EXAMPLE .
In a Circle whoſe Radius is 1, let * and 2 be

the Sines or Coſines of two Arks, whereof the

former Ark is to the latter as 1 to p a conſtant

Quantity : It is required having given x to find z

Now becauſe the Proportion of theſe two Arks
is that of 1 to p, let the true Quantities be what

they will, it follows, that if theſe Quantities be

made to vary, their Fluxions alſo mult be to one

another as 1 to p, and their Squares as 1 to p*.

But the Square of the Fluxion of the Ark whole

x? I
: if x be

-z. or
1X X I—Xx

made equal to 1; and the Square of the Fluxion
22

1— · 2

Whence

;
j
;

e att co SE po ir

1
|
4 +8xgex’+10X11f #9.
|
|

124 The Roors of fluxional Equations

TINS n 8 I ET in
Whence we have this Proportion — to
I—XX 1—2 2

as 1 to p, and this Equation p- x23.

Throw this Equatioe into ſecond F 80776 and
you will have — 2 S = 2X T2 * 2 2—=2.X 27,

Divide all by 2 2, and you will have — N
—X X Z—X , or -& X X- S TFpHh = .

Let z=a K ex, Cc. and firſt let x and ⁊
be Sines: Then it is plain that when x is infinitely
ſmall, the Ark whereof it is the Sine will alſo be
equal to x; and therefore the Ark whereof z is
the Sine will be px. And ſince px as well as x
muſt be infinitely ſmall, we ſhall have the Sine

‘ z=þx, and conſequently ax ; whence a=

and m 1. Again, in the fluxional Equation
M & p- , 2 Will be as “2 ;, x2e,
and xx, and paz will all be as x” ; wherefore u=2 ;
Therefore z = x+b X- x- Ed x -A- ,- far,

Sc. Whence we have

= Tax c Axe x- CEN d

— NK —2 X 3b x- 4 X 5c K
—bX74d x! – dee &“. “No
e 26 88285

—)ex?ꝰ. |
[+ PL = + fax + PDR + pcs
N 2

Whence, and from what has before been prov-
ed that ap, we have the following Equations,
| : | 8=),

” expreſſedin 8 x KES. = 2
Wi MX 9.

— — D, $ < i,
a=, = Tn 6X7
i= * * 22 e, &c. Whence if

. 8×9 _

a, b, c, d, Sc. inſtead of Coefficients ſtand for
the Terms of the Series, we ſhall have z=px+
1X — . “M 3X3—b bx? 3X5—P Xx?

• r = * OX7 *

1. . 4.222 2857 De-, Sc. Where it may
   be ye” that if p be an odd Number, the Series
   will break off, and ſo be finite; otherwiſe it will
   run on ad infinitum. *
   | Before we can apply our Manner of Reaſoning

to Cofines, we are e obliged to premiſe the following
Lemma.

LE MMA. Fig. 25.

In a Circle whoſe Radius is 1, let q be a quad-
rantal Ark, let x be any other Ark infinitely
ſmall, and let p be any affirmative whole Num-
ber: I ſay then, that the Coſine of the Ark pg—p x,
or of the Ark q—axp will be +1, or &: That
is, If p be an even Number, the Coſine will be + 1,
according as the Number p is what they call pari-
ter par or otherwiſe : But if p be an odd Number,
then the Coſine will be + px, according as the
Number p—-1, is pariter par or otherwiſe.

This will beſt appear by dividing the Circle |
abc # whoſe Radius is 1 into four equal Quadrants

a b,

126 The RooTs of fluxional Equations

ab, bc, cd, da, and drawing the croſs Diameters
a c and 5d: For then if we call all the Coſines
that fall within the firſt and laſt Quadrants 45
and ad affirmative, we ſhall have the Coſine of
24—2 X= —1, that of 4 q—4x=+1, that of
64—6x= —1, that of 8 q—8x= +1; and ſo on
alternately : We ſhall have alfo the Coſine of
q—x=—+x, that of 3 q—3g x= —3x, that of
57-5 K* = +5 x, that of 5 q—7 x= 3 and ſo
on alternately ad infinitum.

This premiſed, let now x and z be Coſines; and
let x be infinitely fmall : Then it is plain that the
Ark.whereof x is the Cofine will be gx; therefore
the Ark whereof z is the Coſine will be p-.
But the Coſine of the Ark pg Tx, or ,
by the Lemma; therefore 2 or az”=+p x, or +1,
according to the Conditions already laid down in
the Lemma. Now if z ＋ x, the ſame Series that
was before computed for the Sines will alſo ſerve
for the Coſines, making a= +, as Occaſion re-
quires. But if z, or az”==+1; then we ſhall have
a= +1, and mo, and a new Series muſt be
drawn out of the fluxional Quantities by making

x -b 22-þc x- d x ge -, Fx, Cc. as fol-

lows :

\F 1

1 ,

d
| ther 7X8 e x9–gx10 f x8.

• ane

„ eee,

33). | IHE. 4

expreſſed in SERIES. 127

• =1X2b+3X4cx*+;x64dx)
  E Sy”
  — 5X6 d x6—7X8 e > As pans

—

—8 e x*.

• N 62

| n

From theſe Equations deducing; the Coefficients
a, b, c, d, Sc. and then making them ſtand for
whole Terms, the new Series will ſtand as follows:

2=+ 1 , 2, «1

• IX2 3X4 5X0
  1 N 8X8 —p?
  cx >*8 d xa 9700 e *, Cc. There-

fore the Series relating to the Coſines always
break off.

N. B. o muſt be reckoned amongſt the Num-
bers that are called Pariter par.

Ex AMP LE VI.

To throw the following Quantity Pg V into
a Series finite or infinite, as a * = xt, Ge.
Make D* D, and making x=1, we ſhall
g — 4%.
phos p

Therefore p 29x en z=0, Moreover lince

pes”

have Sm p- EY N M N —

128 The Roo rs of fluxional Equations, &c.

AY =2, when x= o we have z =D : There-

fore = : Therefore nnd =.

Again, In the fluxional Equation p 2 92

=0 we have p ⁊ as x.; and & Z, and m gx will

both be as x7. Therefore 5s = 1, and z=a+bx

Ted xs Te 18 80 fx, Sc. hence we have

as follows: |
. =

+5Pf xt.

J&qxz2 = % bqbx+29c* =

Js zA LA e.
— — ec

—mqd x3—m ge.

Hence making a=” as above, and determining |
the other Coefficients, as uſual, and then making
them ſand for whole Terms, we ſhall have

3 Xx m1 bax mM—2 |
-x rn 3 | Ws
„ dex mes een .
1

= — = m—_ — –

{
k
i

1
i |
« |

• 
  |
  4
  t
  |

=
t +
5

The

cd V4 Us

A

Wor =

into one. But, by the firſt

The CoTESIAN Ferne Ge. 129

The Method of computing the Fo R Ms in the
ſeeond Part of tbe Lo GO MET RIA.

PROBLEM I.

LEMMA.

5 BY 1D 236:
7 find the Fluent of this Fluxion TN where d

and R are conſtant, and T variable. 3
Make T +R &, and we have 1 D K, and

ar – whoſe Fluent i is the Meaſure of the

Ratio of x to R, or of x to any other conſtant – –
Quantity, ad Modulum d“: Therefore the Fluent |

of T R is the Meaſure of the Ratio of T+ R to :

| Rad Modulum d. Q. E. I.

Fenn ..
LEMMA.

| : Ra 4
To find the Fluent of this Fluxion . where d

and R are conſtant, T variable, and 8 ST 2—Rz,

d Rar dRzT 3 :z4RT
Nor e
as will be evident by ws two Fractions
roblem, the Fluent

• By Prop. i. Coroll K Logometriae. =

.
2% 7 * 2
äꝛZ?])P—— 2X — uo

rr é . ⁵— Xu . ]⅛²Üuy. 0 ˙⁵—’n W
. — aos. ‘

Pr “wag A
—— 2 2

rr r ̃ m ꝗ% ⁵.: ³ꝗ OO,

—

• —ͤ— . on CNT — — Fn ni WF Ee 7 he 2
  TT nA SER

= == my == of — _ = 5 — „
» — 8 — ,] «% ůRlnl e

}
16.”
1
!

— — — Ay ** — 3 0
0 N 4 ——— — —— . — —
Ld

3
32322

— — —————— ̃ . e IN OE 182 — —— edt eras tet…
— * 2 – . — — – – bar

p – rr
wore St Fs S

130 The Cor ESTA Fokus
of this latter Fluxion is che Meaſure of the Ratio

of TR to R, ad Modulum — 2 d R; and the

Fluent of the former Fluxion is the Meaſure of

the Ratio of T—-R to R ad Modulum ⁊ d R, or of
R to T- R, ad Modulum — d R. Now as theſe
Meaſures belong to the * Modulus, and conſe-

quently to the ſame. Syſtem, if put together they
will conſtitute the Meaſure of the two Ratio’s put

together, or of enn 1 to T -R.

Wherefore the Fluent of SE or 5 * “5 5

S—_R 2
2 dRT;
TTR is the Meaſure of the Ratio of TR to

T—R ad Modalum — 2 d R, or it is the Meaſure

of half that Ratio ad: Modulus — 4 R. But half

the Ratio of TIR to T—R is the Ratio of
T-+Rto’S, ſince by Hypotheſis S S=T TR R

TR xT—R: Wherefore the F luent of 2 —

is found at laſt to be the Meaſure of the Ratio of
T+R to S ad Modulum — d R. * I.

PROBLEM III.

L 2 MM A Fig. 26.

To find the Fluent of this Fluxion 2 * , where d

and R are conſtane, ‘E variable, and-SS=P’T
—RR. | |
Make hs: Line AP=R the Radius, Boat
the Tangent, and ADC=S the Secant of the Aro
I | | | BD;

4 1725
reſpectively: Therefore the Fluent of

of FrTUr Vys computed, 141

BD; and produce BC to c, ſo that Ce may be

equal to T, and draw A c cutting the Arc BD –
produced in d, as alſo CE perpendicular to
AE c: Then the ſimilar Triangles A D d, ACE,
as alſo CcE, Ac B or ACB will give the follow-

ing Proportions, viz. d to C E as A Dor AB to

AC; and C E to Cc as AB to AC, and conſe-

quently Pd to Cc as A AB* to AC, Therefore

AB Nr. =
; — — — B a 1
Dd=Cc X == SS ut D d is the Fluxion

of the Arc BD, which’is the Meaſure of the Angle.

BAC ad Radium vel Medulum A B. Therefore

the Fluent of dh S is the Meaſure of an Angle,

whoſe Radius, Tangent, and Secant are R, T and S

Si the
Meaſure of the ſame Angle 4d 3 d R,
that is, it is the Meaſure of an Angle, whoſe

Radius, Tangent and Secant are as R, T, and S

reſpectively, but whoſe. true Radius is d R.
R e

K 2 PRO

8 als ne = a- = r_ \

• — — N J
  — —— UU IT A a

— —
1 * — I” I
EEE ²˙ hee ee at ere Dc Bl ee A = a SINE :
—— * —— alov. wie. ates nll. fxg Cf x e’s RT 7 YE — — .
e .
— n — – K –
D — — 7 W 1 K 2 * 1 .

— ——ä— OS. AY — OR

{
| |
|
!
|
‘
[
|

;
|
|
ü
[
i
‘Þ

ol
_

70 OR. Os et: rn

133 The CoTesIian Forms

PROBLEM IV.
To find the Fluent of . this Fluxion in the firſt Foxm

75 > where, as well as in all the Problems that
follow, d, e, f, and > are conflant, and 2 vari-
able.
Make A 3 and we ſhall have » Fuze,
„F522 4 d &
and = hr 5 and = by fx pour whoſe

Fluent is the Meaſure of the Ratio of x to e, or

and is uſually ex-

nf

preſſed thus © | QE. I.

7

PROBLEM V.

To find the Fluent of this Fluxion in tht Sound Form
d æ T

T
Sorurron I.
k —— == — 5 = 5
| Make IRR. orf RR, 2 T2, and
ef 2

FEB, ä or T.— R. ==, and we mall have

. and 4 4Z235—”=7. Multiply one Side
—— — =p * and he other into its Equal —
Lye 2I—1 DART

• 
  and we ſhall have 7 r = Again,

multiply

. * ; whoſe Fluent ©
IRT

e+f2
F

of FLueNTSs cou 133

nu both Sides into 2 75 and we ſhall have
4d 1 Jh3

ne ye

is the Fluent ſought by Prob. II. Q. E. I.

| SoLuTION II.
Make Tom = R, and == T=; a = F, or
, or R. – T = 82, and we fall then have

dæ a 2 dR*7
fe r = I whoſe Fluent, by the third

Lemma, is . Meaſure of an Angle, whoſe Radius,
Tangent, and Secant are as R, T, 2 S, reſpective-

| ly, but whoſe true Radius is 7 JR QEL

Whence it appears,

Firſt, That this Flyent, and canſequently all
others that depend upon it, has two different Va-
lues, one logometrical, and the other trigonometrical.
Secondly, That one of theſe Values will always

be poſſible, but that they can never both be poſ-

ſible at once; and conſequently that whenever one

fails, Recourſe muſt be had to the other: For if
the Equation – — 7 <=R* in the logometrical Expreſ-

ſion be poſſible, then the other Equation 7k.

in the trigonometrical one will be impoſſible z and
vice verſa, | ;
K 3 Thirdly,

|

4

?

1
.

D
$

L

4

‘

1

5 1

1

ns 4

1h

134 The Cor Es IAN FoRMS
Thirdly, That though this Fluent, as well as all

| others that depend upon it, is only expreſſed in

the logometrical Style, yet when this fails, the
trigonometrical Fluent will be ealily had by changing
the Sign of Ra, and ſo making R, as well as the
Modulus that depends upon it, poſſible ; and then
making R, T, and 8, inſtead of expreſſing the

Terms of a Ratio, to expreſs the Proportion be-

tween the Radius, Tangent, and Secant of an Angle,
whoſe Meaſure, according to this new corrected
Modulus, will be the Fluent ſought.

PROBLEM VI.

To continue the Series’s of Fluents belonging to the
firſt and ſecond Fox us, from the Fluents found in
the fourth and fifth Problems, upwards ad infi-
nitum.

The general Characteriſtic of the two firſt Forms
42 20 1
18 77 Tar where 6 is either a whole Number as
in the firſt Form, or a Fraction whoſe Denominator
is 2, as inthe Second, and anſwers to the Author’s

“I
Let = — 5, and the Fluxion next above

: dx ⁊ -r |
it, viz. 7 : or 2 ;=5; and we ſhall have,

from the frft Equation, dA I f2’a

=ea=+f by the Second; and if we take the Fluents
on en Sides, and upon them equal, we ſhall

have

of FiugnTs compared. 135

4 4 ea
have ; 7 Se af b: 53 whence F N 5 3 7

by = a of which Equation, if the Fluent a

LET

ds

da d R

be known, the next Fluent above it may be found;
and then making this equal to a, the next above
this will alſo be found; and ſo on ad Infinitum.
2 F.

| EXAMPLE I.

Let 0 RY and conſequently. — AE.
| | 4 f 61 77 | _
and we ſhall have = according to the
19 „„ 1
fourth Problem, and © g- 2 L 2
ea i F þ nf fl _ <

2 bas or 55, or the Fluent of the Fluxion
dz 2m—= 4 de e+fz *

Y = * 5 5 And if now we

make this laſt Fluent equal to a, and j—2, we
ſhall find after the fame Manner the F “Hig of

; and ſo on ad infinitum.

FJ

Eau = II.

—— i =
8 7 = *

: 4 48 11
and i > or B, or the Fluent of « eee,

MITE.

K 4 P R0O-

e

r

— wy 2 N= mu EE —

136 The Cor ERSIAN Forms

PROBLEM vn.

20 continue the Series’s of Fluents belonging to the
firſt, and ſecond Forms, from the Fluents found

in the fourth and fifth Problems, downwards ad
infinitum.

This may be done by the Reverſe of the ſixth
Problem, that is, by ſuppoſing 2 known, and 4

d
ſought; in which Caſe à will be equal to _ ——
: But becauſe there are ſome Caſes oY

2 1 of this Nature from Fluent to Fluent

—5

cannot without ſome Difficulty be made, I ſhall
rather propoſe the following Method, which is
much eaſier and more expeditious.

42

This Fluxion of the firſt Form A=

the Quantities e and F for one another, becomes
d æ x2 e-

Te = = (by multiplying the Terms of the

Fraction into 2 which does not affect its Value)
d 2 K.

323 which is a Fluxion of the ſame fr
Form, whoſe Fluent has not yet been computed,
and where [17 TY In like Manner this Fluxion

43 2— A n—1

of the ſecond Form — 9 wherein the

Author’s (=a; this TOR, I ſay, will by a

Y 2 where
, by changing the Sign of 3, and changing

like

of FLUENTS computed. 137

ike Ch rorned e — —

another Fluxion of the ſame Form, whoſe Fluent
has not yet been computed, and where {= —a.
But if this be the Caſe of the Fluxions, it muſt
alſo be true of the Fluents themſelves; that is,
Every Fluent of the firf Form wherein 8=a may,
by changing the Sign of , and the Quantities
e and f for one anather become a Fluent of the
ſame Form, wherein -; and every F luent
of the ſecond Form, wherein 8==a is by this Means
convertible into another of the ſame Form wherein
= —a. And thus may all the Fluents of the two
firſt Forms, which have not yet been found, be
with Eaſe computed.

EXAMPLE I.
7 he Fluent of this F * of thi firſt Form

ds —
| = — wherein f==1 13 = , — Therefore

Af

5 bs Fluent of this Fluxion of the ſame Form

dS:
z TFE

where O or 1—1, is _ n

ExamPeLe II.
ERV, T=2z 3″, S =; then

the F luent of this Fluxion of the ſecond Form,
d& ⁊

138 The Corzsian Forms
d aa

N

P : Therefore if r= — ‘# N= or ——_

; where J=1 wil be. 7 SHR

—

2 :

af} f- 0 * a =
and « 5= FF”, or — then the Fluent

43212

of this Fluxion of the ſame Form

where (=—1, will be 2-2 4.
ye

1e A
After the ſame Manner may the Fluent of
dz 2 2

EN of the frft Form be computed from the
48 22—1
Fluent of N Fa of the ſame Form; and the

d 22—22—

Fluent = T Ip = of the ſecond How from the
d 2 fn

Fluent of =] of the . Form, and fo on
ad inſinitum.

Scholl lu u. 2
From what has been delivered it plainly ap-

pears, that the Fluent of this Fluxion of the ſecond _

dr Zz Au- 1 iN
Form F where 6=o0o, is convertible into

itſelf; that is, ſuppoſing all Things as in the

ſecond —_— the Fluent of this Fluxion will

be either 774 *I, or yg r _ |
nf $

PRO

e

18. A

2

41

Ly

$44

| -—— , and we ſhall wwe BI

| of FLUENTS computed. 139

PAOELEZNM VII.
70 find the Fluent of this Fluxion of the e. fifth Foxy

1
.
z Ve v
Let Ve FY P, and we have e -F N= Pa,
_ Pad
x” 0 and 1 fre. and
3 4 dr ad.

— 2s, en sb. Divide
Vs i,

the firſt Side by 27, and * Our by its 7 rag
Pt |

F NU N ot
= — R a Fluxion of the ſecond Form, where

(=o, and whoſe Fluent, mutatis mutandis, will be

the F luent ſought ; ; that is, if for d in the ſecond

2 4
Form, we write 8 for 2, P; for », 2; for e, — 3

and for F, 4-1 ; and moreover, if we make R==«/e,
=P or Je N, and S IN ord fz’1

the F luent ſought will be —_— 1 R P .
E. I,

Pro-

140 The CoTEslAN Forms

ProplLEM IX.

To continue the Series of Fluents belonging to the
‘ fifth Fox, from the Fluent found in the eighth
Problem, upwards and downwards ad infinitum.

42 20 —1 ; 5 42206 —＋4—1

D en

let alſo Ve f2x=P, and 2%*=N, and from this

laſt Equation we have g N, and
ondxa2—2

d N
, or 0 2 =- Multiply the firſt Side
8 29495 , rf A py 1 = Pay

by e /, and the 3 by its Equal P“, and we
ſhall have 6 ye a0 »f Za, or by eaÞ+0 „FAN P.
which I ſhall call ihe Fluxion of the firſt Part of the
general Equation. Again, ſince ef 2″=P?, we
have » fS2>—?=2 PP, and i: f =P Pp, and
| 3 nf 22m 2d z —1
Je +fz 1 Ve FN |
which I call the Fluxion of the ſecond Part of the
general Equation, Join this with the firſt Part,
and we have 8 ye ab n+: nx f6=dNP+dNF,
and if we make the . . on both Sides qual,

we have hne IE 42 dN . P; and

conſequently $£a+1FExfÞ= dP by the

Means of which Equation a * 5 may either
of them be found when the other is known,

QE. r.

ExXAMPLE

or +1 dN,

le

ans oo hwidqw$o$oA —_ wt,

LE

= that RY —

of FLUENTS computed. 141
E x A mm 1 1
Let the Fluent of —> E be Sven, and

2 ＋＋ — = be ſought.

Here j= —41, and the general Equation, when

• to this particular Caſe gives – 4 24 f 5

=—dP=——0 P, whence ea; FAP.

RT 8
But 4 R KI by the eighth Problem ;

R+T | |
wherefore a EY == — l and

RET
teen 9

3

WM he ious of 77 De-
ſuppoſed equal to a, and that of

: _ Tus be

5

7 = yon
ſought ; then will the Quantity ve @ vaniſſi out of
the general Equation, when applied to this par-
ticular Caſe; and the other Terms will be

2 — 14 P, and b or the Fluent of 5 e
—4F.
gf”

Pzo-

x2 The CoTESIAN Forts

PROBLEM X.

To compute any Number of Fluents at pleaſure of the

ſixth Form,

The Fluents of this Form are derived from
thoſe of the Fifth; for any Fluent of the fifth
Form, where bn, by changing the Sign of „,
and the Quantities e and f for one another, is
converted into a Fluent of the ſixth Form, where

42 SN —1

(AN: For I by this Means becomes

42 22 1

• „ that is, (by multiphing the Nume-

Ve

rator into 2 25, and the Denominator into vB)
d & XA 4 n=!

—

PROBLEM XI.

To compute any Number of Fluents at Pleaſure f

the third Form.

Call-any Fluent of the third Form where fax, c,

and call the two Fluents of the fifth Form, where
, and a-1,. a and reſpectively; and we

ſhall always have ea+fb=c. For ſince by

42
Hypothefis rr * a, if we multiply both

42 2 —-1

Sides by e H x, we ſhall have === JET Ne

F or S whence
fee 8 D.

Po-

11

of FLUENTS computed. 143
PROBLEM XII.

To fnd any Number of Fluents at pleaſure of the

fourth Form.

This i is done by obſerving that every Fluent of
the third Form, where = N is, by changing the
Sign of » and the Quantities e and F for one an-
other, converted into a Fluent of the fourth Form,
where = —x—1, and every Fluent of the third-
Form, where j= —a is converted into a Fluent
of the fourth Form, where {= T: For
d M Ve- by this Means will be changed
into d& z K off * e 2, that is, (by dividing
one of the Factors, viz. d 3 2 . 2 and
multiplying the other, viz. 4/ [ETFS into Y
which cannot affect the Product) d ===
X TJ Ts, or 4 ; T yowT * N |
Q.E. IA

1 XIII.

To find the Fluents of this Fluxion of the ninth
d 1

4 F D Ye CFL
Let eU “=P, and we ſhall have FN P-,

= and Nb , We

=P IE an
148 ; OT 3
y YZ 2

ee er,

and

Fe oRM =

mall have moreover 3 2.1

75, and

| 1 —

144 The CoTEsIAN Forns
d 2 2 — ; | 26. p

d ——— —— —
Rr
FPlusien of the ſecond Form, where So, and

. tk N ad

= 22: : So chat the Fluent dught is

— 4 a rr = I

2

PROBLEM XIV.

To compute any Number of Fluents at Pleaſure of

the ninth Fogm.

Call any Fluent of the fifth Form, where b, , |

and call the two Fluents of the ninth Form, where

(=a and XI, @ and 5 reſpectively, and we
ſhall always have ga -D c (See Prob. XI.)

Whence if any one Fluent of the ninth Form be
_— all the reſt may be eaſily computed,

Penn .

To compute any Number of Fluents at Pleaſure of the
tenth Form.

This is done by obſerving that every Fluent of

the ninth Form where b, by changing the Sign
of u, the Quantities e and F for one another, as

alſo the Quantities g and 5 for- one another, is

converted into a Fluent « the tenth Form, where
4 | 5 : | E 1 A 7

_ tenth Form, where = 1 IN: For

of FLUEkN TS computed. 145
4 21 and every Fluent of the ninth Form,

where = —a, is converted into a Fluent of the
d DD =

Tb V.

-=”< os Dx
by this *, 1M becomes —o— fn

Fe

| Multiply b + g 2—* into z”, and alſo /fÞez—

into Ye, and laſtly multiply the Numerator into

their Equivalent NT, and the Fluxion will now

d T- 2001
become —

g+b a

PROBLEM XVI.

To compute any Number of Fluents at pleaſure of the
ſeventh Form.

Call any Fluent of the ſeventh Form, where
Se c; and call the two Fluents of the ninth
Form, where = and a1, a and þ reſpectively 3
and we ſhall have r

an XVII.

To 0 Compute any Number of Fluents at pleaſure * the

eighth FoRM.

| This is done by obſerving, that every F Juent of
the ſeventh Form, where 8a, by changing the
Sign of y, and the Quantities e and F for each
other, as alſo the Quantities g and 5 for each
other, is converted into a Fluent of the eighth

Fern, where (=—\ as is eaſy to ſee.

• PR-

146 The Cor ESIAN Forms
PROBLEM XVIII.

6 – find the Fluent of this Fluxion of the 3

e

g+b2″
Make I 52 2 =Q, and we fhall have z’=

VE, and e+f2 (= P)= D 2288 85

over we ſhall have 22 QQ and conſe-

quently MINED? = — = e 2 Q

a Fluxion of the fourth Form, where a o, where

| | 5
4, . A235 , A R=\//,

ForM dS Y R

| wo READ 8 4
1 22 Sb 8 V Tbe

So that the Fluent fought is — dP Wh 2
. :
E. QOE-L
PROB I EM XIX.

To inveſiigate a GENERAL THEOREM for continoin
upwards and downwards all the Series’s of Fluents

comprebended under the eleven firſt Forms; and

an infinite Number of others under the ſame Che-

rater; but which is rig intended — the eleventh
Fou.

The general Character of che F Wied of the ele-

ven firſt Forms’ is q & ,] -i bz rr,
Ss: obſerving

Multiply both Sides into e + 7 2, and we ſhall
L 2

of FLU ENTS computed. | 147

obſcrving that in the ſix firſt Forms – f , or

4=r. Letthends x. H ö,

da 2) MIX FDM ＋ þ z2»4—1> j = bs aa
dz t – FFT —1 Xe+ bai: 5 La
moreover æ N, P, and g=

and from the Equation z. N, we ſhall have

W N, and 64e g-,

or ba = JN NFF Multiply

both Sides i into FY APbz, and we ſhall have
d.
tegiddxeb fox b +6 f bc == P Q, which I

call he Fluxion of the firſt Part of the general Equa-
tion. Again, ſince TFT P (by conſider-
ing ef 2” as one ſingle Quantity) we ſhall have
Z Z & Tf, and f Ken
NN NXT, and

4
un ee 472 -N -*2_ or of ==

NPT. Multiply both _ by g–b 25

and we ſhall have „fg ef u p Q a

is the Fluxion of the ſecond Part ww the general
Equation. Laſtly, ſince TED Q. we have
db D & PZH, and hd 22%tH—=IX

DT 4—1, or 1b3=”Nx0FB- .

have

143 The Cor ES TAN Forms

d
have le bg T = NP which is he

Flurion of the laſt Part of the gencral Equation.
Put now theſe three fluxional Parts together,
and make the Fluents equal, and we ſhall have

: eg TLC N- Pe UNT be
3 ———— |
— r

| PrROKEE MN Ak. –
To apply the foregoing Theorem to the Computation of
Fluents of the eleventh ForM.

In order to do this it muſt be obſerved, that in
this Form on, 4=2, and that every Fluent of
this Form where 8=a, by changing the Sign of ,
the Quantities e and f, as alſo g and & for each

other, is converted into a Fluent of the ſame
Form, where f=—a. This premifed, let qSz——X

2 — DSN 42 5 2 and d ⁊ i

e -F
=> = we ſhall have = —1, and the
WAI

general Equation applied here gives — eg a+

• 254777 c _ x e+f 2? e =

1 45 xd P Q, making PV NÆr, as the Author

2 |

has here defined it, and not equal to ef z”\;, as
above. But of theſe three Fluents, a, &, and c,
. PR – c is

© wv

40

of FLuzn Ts computed. 149
z 1s given by Problem X VIII. whence @ will eaſily

| be found; becauſe, from what has already been

obſerved, c is convertible. into a; and theſe two
a and c being obtained, the middle Fluent þ will
be had by the Equation above; after which all the
reſt will be eaſily computed. E. F.

PROBLEM XXI.
7 o compute the Fluents of the twelfth Form.

Make YYY = Q, and by the 2
Problem, we have d M M. * 3d ax 0

S ww
5 bf — K AS ind MT Ex
2 22, 8 K Q
W I D 1 | .

ebx, X 4 2 20 — rn

=
+4 7 m which Fluent being now known,

the reſt will be eaſily deduced from the eleventh

Form, by the Method of the 14th Prob. Q. E. F.

L- 9 P Ro-

750 The CoTzs1an Forms

PROBLEM XXII.

To find the R. of the two following Fluxions 1 F
| d > 0 8

the thirteenth F ORM VIZ. 0 2
| da : ; TX Sr

In order to deſtroy the middle Term of the

Trinomial ef -g 22″, and ſo to turn it into 3
Binomial, that the Floxibn may be reduced to
ſome of the foregoing Problems: Let gz T,

and we ſhall have 2 ff+fgabggzr=T
TT ;
and r , and ebf2Þg 21m

*$ — Put 781 2 and we

ſhall have ef g z20= 7 => > putting

S: for T:— RE. Further, ſince 3 f g ,
d ⁊ 2

we have ng2 n=” ma” i FT ET FI

. 3 T 4 RT
os as _—— Dr putting wy for Ra; whoſe

F “Is by the ſecond Problem, is ne “IR *

Now to find 5, make f 27-|-g Z 155
we ſhall have; 32 8 —1 —— 2 1 ary and |

42 Y- -2 gd A : _
2 . 8 3 or Fa 18

E
| 5

Make

therefore Fa 2 x7 87 =

already

of FLUENTS computed. 151

Make Nah J tn oh N and we ſhall have

\ fa4-212b=dN ; by which Equation, ſince « if
r 5 will eaſily be known.

2 PROBLEM XXIII. :
To compute the Fluents of the thirteenth FORM.
F © 2 Ads
T ©» of fofrmn—

| 42 2 295-2 —1

NN t and we ſhall have ea Ae —

© 2%”, by Means of which Theorem, the Series

of Fluents from thoſe found in the laſt Problem
may be continued upwards ad :nfinitum ; and the

reſt may be found by conſidering that every

Fluent of the thirteenth Form, where = is by
changing the Sign of », and the Quantities e and g
for one another, convertible into a Fluent of the
fame Form, where {=—2—a.

N. B. Since in this Form, the Fluent where

” k=1 is lx = „ and, ſince this Fluent is

by the uſual tad convertible into itſelf, we

RLT
ſhall * 2 „ ar =, and

nq
i

“4

6 IRS

L 4 Po-

73: Ts en FO RMS

PROBLEM XXIV.
. To find the Fluents of the fourteenth Fox.
Let k+z=y, and we ſhall have FA 22.
Gs

and

• 
  d x X 3 *
  Ne yxell—f RA EHF 2 2g 2. ä
  1

a F en of the thirteenth Form; where do,

. _ fl—2gk

ne” = —_

Reg, Ter, S=vaxetfz +22″
ell—- Rl EAI FI 29kX

en Gli 2 £ AN

it 89

• . From the Meaſure of this
  DP

Ratio ſubtract the Meaſure of che given Ratio of
{Ito g, which cannot affect the Fluent as ſuch;
Au ;

K-24 |
and this. Fluent being once found, the reſt are
eaſily derived from the thirteenth Ferm, according
to the Manner of the fourteenth Problem.

. Every Fluent of this fourteenth Form,
where 6=a is by changing the Sign of y, the
Quantities e and g for one another, and & and / for
one another, convertible into a Fluent of the ſame
Form, where (=3 A.

bbc 1=1, e

and then we may make ==

Scho-

\ So. No JA.

of Flux NTS computed, 153

ScnoLlium.

At the End of his eighteen Forms, ths Author
gives us other Values of R, T, and 8, at leaſt of

I and S for every Form, which may be uſed at

pleaſure inſtead of thoſe put down in the reſpective
Forms: And before we can proceed any further,
it will be neceſſary to ſhew whence, and how theſe
Values are derived. Let then in any of the Forms

R=a, Tb. S c, and we ſhall have ITO
: 4 S
3 KL: ar |
=*q— — 2 Tee 6 Make now Ra,
ET #< ac 8

• 
  1g: Se. 9, and ve hall have EFT in ths
  Caſe equal to 4, in the 3 and R 2 5 .

in this Caſe equal to RI in that.

But though the logometrical Fluents thus ex-
preſſed agree entirely with thoſe of the Forms, yet
the trigonometrical Fluents deduced from them will
be very different; for that a logometrical Expreſ-
fion may, when Occaſion requires, be converted
into a proper trigonometrical one, it is neceſſary
that T. -R ſhould be equal to 88 8, whereas
according as R, T, and S are here defined,
T*—R> is equal to S:; therefore to anſwer both

dhe Ends of the Forms, we mult correct the Value of

S thus,

rr

N and the Fluent K — 3 S= ——_

154 The CorksiAx FoRMs

S thus, Inſtead of makingS=** 5 „ and 5.
let us make SS = —— =—_— „and S= — 85

This ae. the Fluent R FE „ when S =

• will

both be Fluents of = ſame Fluxion, as 3
from one another only by a conſtant Quantity,
viz. by the Meaſure of the Ratio of 1 to /—1;
and therefore if the logometrical Part was true be-
fore this Correction was made, it will be ſo ſtill ;
and as T T-RR is now equal to ＋- 88, it is
plain that the /ogometrical Expreſſion will now be
convertible into a proper trigonometrical one.

PROBLEM XXV.

To find the Fluent of this Fluxion of the 4 xteenth

d 2 21

D N
Make 54 JW, and we ſhall have + TH
ee and La BTg g, and

FoRM

ef 2g —— ＋ Sg. We have more-

over, from the grit Equation, 132 n= „ and
there –

• * * k A. 4

en — 155

herefore La K -;

there e Te |

a Fluxion of the ſixth Form, —_— So, d=,
e— 4 ff N

T ), | y==2, ==

„. RSV, T=g x

. Teta

1 F
the Values of R, T, and S as they immediately
flow from our Notation ; but if we take the Values
of R, T, and S according to the laſt Schalium

we ſhall have R= Ng, 18 HAZ

— — —

ve TYAN

92 * — : And theſe Values (for their
ef wolg 20

greater Simplicity) the Author chuſes rather

than the other, and the F luent of this Fluxion

.
A 2 ER.

PROBLEM XXVI.

To continue the Series of Fluents of the ſixteenth
FoRM.

3 „
Lr Jefferies
dz 1 3
= > N, and * Gries P,
T c | 4

ns Wy CorESIAN Fo RMS

and we ſhall have dds 2 en and 6 FE 2 22

• 225 or ddz -M. |

4 Fg
or de FA -N P; which will be the
Fluxion of the firſt Part of the general .
Again, ſince e|fz’+gz2»=PP, we have 2 F 2

＋ ug rr, and 8 .

Ve AF Tg 221 2 =
da —— 4

== Nfz;

and Ve FAT 22 5 or fil-g; R

which is the Fluxion of the 3 Part of the 1

Equation. Join theſe two Parts together, and

make the Fluents equal, and there will ariſe this

general wy 5 deat ZX JU LIE: Nec=
“N Pa= 1 P; which Equation, by the Help of

| = Rs fifth Problem, will find any Number of

Fluents of this Form propoſed.

N. B. Every Fluent of this Form. where {=a

is by changing the Sign of y, and the Quantities
e and g for one another, convertible into a Fluent
of the ſame Form, where j=1—a.

PROBLEM XXVII.
To find the Fluents of the Jfteenth Form.
The Fluents of this Form are derived from thoſe

of the ſixteenth in the ſame Manner as the Fluents

of the third and ſeventh Forms are derived from
thoſe of the fifth and ninth.

he

432—1

17 > — 8

of FLUENTS computed. oy

PROBLEM XXVIIL
To find the Fluents of the eighteenth Form.
Let KIU z y, and we ſhall have ef g 22″
e er 2

12
| 0

De

e, l fe en

aFluxion of the ſixteenth Form, where b 2, d

T Ay, y=1, e e- k Ak, or p, a |

= 7 $7. e TLF TA x3/+gz
and g g; r Vp. [== Je- F

• XV ſeg ; and the Fluent itſelf

equals — „ But if, as the Author has

1p

done, we aſcribe ie aforeſaid H to R, T, l
Ss the ent will then be 2 77 13 R

R- T |

• and

this Fluent being obtained all the reſt are deduci-
ble from the a Form; after the ſame Man-
ner as the ſeventh was deduced from the ninth.

N. B. Every Fluent of this Form where 9 N,

by changing the Sign of „, and the Quantities e
and g for one another, and & and / for one an-

; reden becomes a Fluent of the ſame Form, where

Pro-

138 Of the PREPARATION

PROBLEM XXIX.
To find the Fluents of the ſeventeenth Form.
This Form is derived from the eighteenth, juſt
in the ſame Manner as the third is derived from

the fifth, as is eaſy to ſee; and every Fluent of
this Form, where d= is convertible into a Fluent

bf the ſame Form, where b —a.

Of the PREPARATION of TABULAR
FLUENTS.

(Tranſl ted from a Latin Manuſcript of the Author.)

The Fluents of fluxionary Expreſſions, as they
ſtand in the Tables, do not always come out in
their ſimpleſt Form; but require, before they can
be of Uſe, to be ſome how prepared or tranſ-
formed; either by adding or ſubtracting ſome
given Quantity; by increaſing or diminiſhing the
Modulus; by exterminating an impoſſible Quantity;
or, perhaps, by reducing the Tabular Ratio to more
fimple Terms. I have thought proper to give an
Inſtance or two, from Mr. Cotes’s Tables, wherein
a Difficulty of this kind occurs.

E x AMP L E I.

Let the Fluent propoſed be that of the Fluxion
2 44 *
a a- & x?
the former variable, and the other invariable; and
4 is likewiſe ſuppoſed to be greater than K-
not 2 | Now,

in which x and a repreſent right Lines,

2
I

285

nd

or 4

I TABZVULAIA FruznTs, 159
Now, as the Denominator of this Fluxion is
rational, it muſt belong to the firf, or to the

ſecond Form, according as the Number 9 is a whole

Number, or a Fraction. But, in our Example,
ſeeing x=2, and 0 1=1, Or 2 — Iro, we have
D, and Form II. is that which we are to uſe.
In which, d=2 aa, z x, n=2, do, eman,

f= —1, R= 2, T=x, S S = = and
tle pg os 8
R 2
UR * a Maa—xx
—1

Andi now, to exterminate the impoſſible Quan-
| ape __

tity in the Denominator S; to the Ratio Vea aa –
1 e.

] add the Ratio Dany (whoſe Meaſure, whatever

it be, 18 at leaft determinate and invariable) and

the Sum is the Ratio DE or Tau- UN 2

Naa—xx Jaa x

whence the Fluent fought i is 24 vs . a N x
aN. a A -N

For the Meaſure of a Ratio to 5
a a- xXx &

Modulus is equivalent to the Meaſure of the Du-
plicate of the ſame Ratio to half that Modulus.

But the Ratio — on hte dividing both Terms

by a+, becomes 2 : And the Fluent thus
prepared

— — — 2 — — a = _

IT

GX

A Ame

160 Of the PREPARATION

prepared is a , or the Meaſure of the Ratio

ax
of ax to a—x to the Medulus a.

1 Kean 8
Let it now be Propoſed to find the Fluent of

by Form VI. where we have d ga, z=x,

1=2, bo, e=aa, f=—1, R= 1, T=
Ma% xz a |

55
Notation, the Quantity R, and thence the Modulus

—4R, R, comes out impoſſible, we infer that the

Fluent ſought is not the Meaſure of a Ratio, but
of an Angle : *: And that the Modulus of the trigo-

nometrical M ea ure, is the Modulus of the —_—

cal Meaſure made poſſible.

I change therefore the Sign of —1 in the

Value of R, OY N, or 1; and thence

a * —
NT — 1 9. —
RE == 2 , or —3 1

and from this Notation conclude, that the Fluent
fought is the Meaſure of an Angle to the Modulus

—a, the Radius, Tangent, and Secant of that Angle
being as x, – and a; or where the Radius

is to the Secant as & to a: for of the three Terms,

OM See Logometria, Page 45.

Radius,

„S = And becauſe, according to this

—_ — Wand . w@u.

of TABULAR FLUENTS. 161

Radius, Tangent, and Secant, any two determine

the Magnitude of an Angle. Our Fluent there-

fore is the affirmative Meaſure of an Angle whoſe
Radius is to its Secant as & to a, the abſolute
Magnitude of the Radius being — 4, or it is the

negative Meaſure of the ſame Angle to the Radius
+a: for an affirmative Meaſure (whether. /ogome-
trical or trigonometrical) to a negative Modulus, is
the ſame I hing as a negative Meaſure to an affirm- |

ative Modulus;

Now to conſtruct this Fend in its ſimpleſt

Form, in Fig. 27. Plate V. from the Center C,
at the Diſtance CA a deſcribe a Quadrant AMB;

and in the Radius C A having taken C P x, and
erected the Perpendicular P M. the Arc AM

will be the Fluent ſought. For joining C M, the
Radius of the Angle A C M will be to the Secant
as x to 4: To this Fluent add the Quadrantal
Arc AMB, and the Arc -A M will be changed
into BM. And therefore the Fluent of the Fluxion
propoſed 1s a circular Arc whoſe Radius is 4 and
its right Sine x. |

NM ANALVLIS

162 ANALYs1s of

ANALYSTIs of the PROBLEMS in tle
SCHOLIUM GENERALE of Mr.
Cotes’s LoGoMETRIA *

PROBLEM I. Fig. 28.
To find the Length of the parabolic Curve A P.

AN ALLVSIS.

Cal the Parameter, a; the Abſciſſe A Q. K; and
the Ordinate P Q, y; and we ſhall have a x=y,

and a . and & 27 and V x 55. or
the Fluxion of the Curve, Au. * 4 22400, a

Fluxion of the fourth Form; where 6=o, and

• 4P, or the rational Part of the Fluent, =7x)X

, and © A RIN EIL

“f S
“ny equal to the Meaſure of the _ ad Mo-

dulum: a, between 42 — and ©

Now to conſtruct this Fluent, let PT bs a Tangent
to the Parabola in the Point P, cutting QA pro-

duced in T; and ſince 8 x, or >, :
Tangent P T will be IX * 4.4 7. whence the
rational Part of our Fluent 7 + „ 2 2 will be

„or the 7rration!

the

Harmonia 8 Pag. 22.

equal

ld >

ff} we

— i —— —

£ f\

the
Ar,

and

N
or
1
and
JX
onal

Mo-

zent
pro-

the
the
be

ual

Mr. Cor E Ss PROBLEMS. 163

equal to PT or A L*. Now as to the irrational
Part, which is # Meaſure of the N ad Modulum

between 2+ Na fe L’s 3 = , or between “=

| +3 7 — 2 and =; it is evident that ”

Modulus + – F, F a the Focus; that = 4
or x=AQ, that 2 5 DAL. *

laſtly that – San LQ. 50 da if to the Line A 3
be added the Line LM equal to the Meaſure of the
Ratio between A QA L and L Q in a Syſtem _
whoſe Modulus is AF, the whole Line A M will be

the Length of the Curve AP required. Q. E. I,

PROBLEM II. Fig. 29.

Jo meaſure any Part QP of theArchimedean SPIRAL.

ANALYSIS.

Draw the indefinite Line QZ e the
Poſition of the Radius primus, and the Nature of

this Spiral is, that the Length of any other Ra-

dius QP muſt always be as the Angle PQ.
This premiſed, draw the Radius Qy greater than,
and infinitely near QP; and ſettting off upon
p, QU equal to Q, and joining P U; ſince
the Angle PQZ is always as QF, the Angle
PQp will always be as Up; and conſequently
P U which will ever be as the Angle P Qp, and
the Radius PQ together, muſt always be as
*The Conſtruction being AL drawn parallel to TP, or QL=LP.

1 QPxUP;

—

yy — 5 — — o –
mo
— —— 1 – —_

| U
QPxU p; and conſequently QP Kb will be a

: Again, ſince + 4 —QPxU2 or V . or

164 ANALYSIS of

coaltant Quantity. Call this Quantity + a, and
QP, , and UP, x; and we ſhall _ ==, we
and a/x xy y, or the Fluxion of the Curve alre
| + the
| A Va as in Prob. I. and therefore the bet
Fluent will be 5 2 together with the the
| – FF h Vai HAN Cu
Meaſure of the Ratio between EX |
and 2), the Modulus being 3 a. | –
Jo conſtruct which, let PT be a Tangent to a
the Spiral in the Point P, cutting a Line as QT | ‘Y

perpendicular to Q in T, and the ſimilar Tri-
angles U y» P, QP T give us Up» () to Pp F
. : 0

( a* 4-4 y*) as Q (0) is to Pr Y
and therefore + yX * EDT ara L K

J

. a, or the Modulus, will be equal to

A
= „or a third Proportional to the Lines AQ

and QL; and therefore will be equal, by the Con-
ſtruction, to AF. Laſtly, for the Terms of the

• AL parallel to PT biſects Q in L, and AF is a third
  Proportional to QA, QL, by Conſtruction. 5
  | | : Ratio,

the
third

tio,

Mr. Cor Es’s PROBLEMS. 165

Ratio, ſince 4 a = . G and conſequently .

I
we ſhall have r , or N, and the reſt are
a

a

already defined. So that if to the Line AL you add

the Line LM equal to the Meaſure of the Ratio
between ALTA Q and LQ ad Modulum A F,
the whole Line A M will be the Length of the
Curve Q required. Q. E. I.

N. B. This Affinity between the Spiral in this
problem, and the Parabola in the laſt, ariſes from
hence, that the latter is nothing elſe but the In-
volute of the former.

PROBLEM III. Fig. 30.

To find the Length of any Part as Ee of the Reci-
procal SPIRAL,

ANALYSIS,

Draw the Radius Ae greater than A E, but
infinitely near it; and upon the Center A, and
with the Radius A E deſcribe the Arc EG H
cutting As in G, and AB in H; and that Arc
EH will be to the Arc G H as the Angle E A H
is to the Angle GAH; that is, (from the Nature
of the Spiral) as As to AE; and by Diviſion
EH is to EG as Ae or AE is to Ge; and by
Alternation E. H ;is to AE as EG is to Ge, or as
AF is to AE; and therefore the Subtangent AF
will always be equal to the Arc E H: But that

M 3 Arc

166 ANALYS1S of

Arc being as the Radius AE, and the Angle
EAB together, will be a conſtant Quantity,
ſince the Angle E AB is reciprocally as the ſame
Radius AE; and therefore the Subtangent A F
malt be fo tos.

This premiſed, call AF, a; AE, ; and

E G, K 3 and we ſhall have x= 27 and * :

or the Fluxion of the Curve . R a

Fluxion of the third Form, — o, and

R+T
who Fluent 4P—24R 22 0 a) 8

a : yu : whore t * a a ayy 1 1S the Tangent

EF, and 1 VaaTy; * 2 beser of oh

Ratio ad Modulum A F, between E F-4-F A and
E A, or between E A and E F-F A (by Con-

ſtruction) LM: For ſince EA=FF—F A*=

EF -F EFF AXEF—FA, it follows, that E F+F A
is to E A, as EA is to E FF A: So that the
Fluent at E is EF – LM; and for the ſame Rea-

ſon, the Fluent at e is em, and if this latter
Fluent is ſubtracted from the former, the Dif-
ference EF – ef n- LM will be the Length
of the Curve Ee required. Q. E. I.

PR o-

Mr,CoTEss PROBLEMS, 167

PROBLEM IV. Fig. 3

To find the Length of am Part as Ee 4 the Loga-
rithmic Cunvs.

AnaLYSsSis.

Call the Subtangent AF, which is a conſtant
Quantity, a; call alſo Aa, x; and AE, yz and

we ſhall have & = 2 as before; and conſequently

the Length of the Curve Ee will be E F—ef+1 m
LM, as before. Q. E. J.

Only here it muſt be obſerved, that to find the
Lines LM and Im, there is no Neceſſity of having
Recourſe to the Canon, ſince this Curve, of its
own Nature, affords us a perfect Syſtem of Meaſures:
For if LM be taken equal to E F— F A, LM
will be the Meaſure of the Ratio of A E to AL,
or of AE to EFF A ad Modulum AF, from the
very Nature of the Curve; and the ſame may be
| faid of Im, which is the Meaſure of the Ratio of

ae to 21 or ef—f a ad Modulum af B
N. B. This Curve is the Evolute of the Spiral

in the laſt Froblem,
PROBLEM V.
To find the Fluxion of this Fluent ef 2.
| | $oLivTION. |
Make e–fz’=y, and we ſhall have -F,

and the Fluxion of ef = 5 .
” M 4 But

168 ‘ AnaLys1s of

But if ye v, y will be F ue ; which being
ſubſtituted inſtead of y in the former Value, we

• | ſhall have the Fluxion of e+f2 2% =@ y Fm 2 $=1

r-. QF. I.

PROBLEM VI.
To find the Fluxion of this Fluent d 2 “xe–f D*.

SOLUTION.
Make 2%=x, and ibf2=y, and the Fluxion

required will be dq K* d xx. But, by the

common Methods, x—0y zi; and therefore

XXy=0 nZ2m—1×6 FA dy „eZ – FZH
NE FD. 5 by the laſt Problem,
-. * z -I EF 2 and therefore j X x=
wy wnfs LK ; therefore dx XX Y dy x,
or the Fluxion required, will be 0 e d x 2911

my 45 ef. QE. I

DEFINITION. Fig. 32.

If in the Diameter A B of any Circle is taken
any where a Point as Q, and from thence a Per-
pendicular as QO is erected, cutting the Circle in
the Point O; and if in the Line QO, or in that

Line produced, is taken QP a third Proportional

to QO and QA; and laſtly, if the ſame Con-
ſtruction is made for every Point of the Diameter,

and on each Side of it, the ſeveral Points P thus

found

JU 1

‘ Mr.CoTrss PROBLEMS. 169

found will form a Curve, which is called a Ciſſoid x
and the Circle whoſe Diameter is AB is called
| the generating C rrele.

CoRoLL. I.

If to the End B of the Diameter A B is erected |
a Perpendicular as B C, that Line BC will be an

• Aſymptote to the Crſſord.

| Seni II.
If we call the Diameter A B, a; BQ, x; and
PQ, y; the Nature o oF the Ci id will be expreſſed

HY But if A Qis called

2 3

x

x, the Nature of the Curve will be y== = |
G—x

by this Equation 4x

PROBLEM VII.
Jo find the Length of any Part, as AP, of the

Cis$S01D.

ANALYSIS.

Call AB, az BQ, x; and P Q, y; and ſince,

_—

| according to the Nature of the Ciſſoid, y= ——,
| GE.

or 2 Xa—x”, we ſhall have, by the ſixth Pro-

3 “y–
blem, jy =—= AN 2 and 5 2
r But * 3X 4x* „ therefore

K 2 K
rr e, or 3 7 X a+3 x3;
and

170 AnaLys1s of
and therefore rt , or the Fluxion of the
Curve, 3

prefixed, 3 the Part AP of the Curve now

inquired into flows contrarily to B Qor x, whoſe
Fluxion is + x.

Ve

But the foregoing Fluxion —.— ME * is

a Fluxion of the fourth Form, fo (= —1, and
whoſe Fluent conſiſts 5 two Parts; J the _

| is * or = Pz, or 2

ITT or —X CIA
4

To conſtruct Kg ſuppoſe the Ordinate QP
to cut the generating Circle in O, and biſect both
QP in F, and QO in G, and draw the Lines

AFE, BG: This done —
Y-, and xx to QB ; and therefore V e

will be equal to the Line B G, and we al have

3 ax – Xx AM

—X 7 Jax = == —X B G. But now ſince,

according to the 83 of the Ciſſoid, QP is to
QA as QA to QO, and ſince from the Nature
of the Circle QA is to QO as QO to QB, it
follows, ex aequo, that Q is to QA as QO to QB;
and biſecting the Antecedents, that QF is to QA
as Q to QB; and therefore the Triangle B Q

• =

“x Ja Tzx; the Sign — being |

XX will be equal to

Mr.CoTrss PROBLEMS. 171.

is ſimilar to the Triangle A QF, and conſequently –
alſo to the Triangle AB E; wherefore as B Q (x)

is to BG, ſo is AB (a) to AE=”xBG: So that
the rational Part of our Fluent is at laſt found to

be 2 AE, |
In the irrational Part R is to S as „ is to

V. or ny is to Va, or as Vg ax is to a, or as |
x.
Nax is to T7 But x is a mean Proportional

Ws a and x, or between B A and B 22305

and 7: is the Tangent of an Angle of thirty

Degrees ad Radium a B C; and therefore R is
to S as BD to BC, and conſequently R T is to
Sas B DAD C is to BC: For in all /ogometricat
Fluents whatever, T is the Hypothenuſe of a
right- angled Triangle whoſe other two Sides are
R and 8, as will be evident from the Conſtruction
of the Forms themſelves : And fince the Modulus

of this Part 1s AN xa, or —= „or —3BC,

B DDC.
| ge

the irrational Part will be —3 B 4

and the whole Fluent will be 2AE—3BC | |

BD
WT

«Ip ConfſtruQion. E |
| Now

= AnaLvs1s of
Now to determine exactly by the Help of this

Fluent the Length of the Curve A P, it muſt be

obſerved, That when the Point Q and conſe-

quently P coincides with the Point A, the Line
AE will become AB, and 2 AE, 2 AB: Sub-

tract therefore 2 A B from the rational Part 2 AE,

and the reſt 2 AE —2 AB will be for our Purpoſe.

In like manner it muſt be obſerved, That when
the ſame Point Q: and conſequently D, coincides

with A, C will become — nc

1B ALA C.
[FC

former leaves |-3BC

: and this latter ſubtracted from the

BALAC __|BD4-C
1

alſo for our Purpoſe. But the Meaſure of the

Ratio of BD–D C to B C being ſubtracted from
the Meaſure of the Ratio of B AA C to BC,
in the ſame Syſtem, will leave the Meaſure of the
Ratio of B AA C to B DD C. So that the
Length of the Curve AP is at laſt found equal

BAHAC _
to 2A E—2A FR. BD4-D G. QE. I.

| P R O-

Mr. Cor ESS PROBLEMS. 173

PROBLEM VIII.

To find the Altitude of a Cylinder equal to, and upon
the ſame Baſe with the Solid generated by the Re-
volution of be ciſſoidal Area A PQ about its
Axe A Q.

.
: ANALYSIS.

Call A B, a; AQ, x; and PQ, y; and we

mall have, from the Nature of the Ciſſoid,

„ *

N and yy ==

the Diameter of a Circle to be to the Circumfe-
rence as I to d, we ſhall have d 50 x, or the Fluxion

d x
of the Solid, = — a Fluxion of the firſt Form;

where 924. and whoſe Fluent is „ . —

to 3
A -A £ –

; and if we ſuppoſe

2

4 |
But * irrational Part may be expreſſed thus,

—+d a | ; ſince ſubtracting the Meaſure of a

an Ratio is the ſame as adding the Meaſure
of a poſitive one; and as this Fluent when the
Point Q coincides with A is apparently equal to
nothing, it will be a true and adequate Repreſen-
tative of the Content of the Solid now inquired

into.
Put now 2 for the Altitude of a Cylinder equal
to this Solid, and upon the ſame Baſe with it;
and

7 AnaLiYeisof

• :
  and we ſhall have 455 — —

—4 a*x

Aa —_ z whence z will be found by applying

the ſeveral Members of the “I Series to the

= — —
Quantity 455. or – 5 34% 3
| 1 42 AA
205 and 2782 “Le Ll

245) * 2
N “> 2 |
a = —C at ARR”
RW i343 © X

So that the rational Part of the Altitude is equal

to s. equal to the * Line
— Z. Again, 4 — ——=—=AT—AS=S T3
6 * x3 x2

and therefore the irrational Part of the Altitude
will be S T|— „ (ſince whatever Change is
made in the Modulus 9a a like Change in
the Meaſure ) or ST 80 equal to the Right

Line QX: So that the 8 ſought will be
QX-XZ=QZ+. QE. I.

• By Conftrafion AQ, AB, AR, AS, AT, are =>;

“QX=TS 30 50 and X Z=S * A

• See Logometr. p. 24

4+ | 5

2 . 3 OO EIT WR re 1

Mr. CoTress PROBLEMS. 175

PROBLEM IX. Fig. 32.

To meaſure the conchoidal SoLips generated by the

Revolution of the two Areas A ED C, ae DC
about the Axe A P.

a

Draw the Line PT UK H greater than P E,
but infinitely near it, cutting the conjugate Curves
AE and à e produced in H and þ reſpectively, the

Line CD in K, and the Circle RS (deſcribed from
the Pole P at the Diſtance PSC A=Ca) in T.
Draw alſo SQ cutting PR at right Anglesin Q, and
call PC, az PR,r; PQ, x; SQ, y; the Area
of the whole Circle RS T, 5; the Solid generated

by the Area AEea; or the ſum of the Solids

generated by the two Areas A ED C, aeDCs5
the Difference of thoſe Solids, d; and the Solid
generated by the Sector PRS, 2; and we ſhall
have, firſt, The Circumference of the whole Circle

RS T== : and conſequently the Circumference
; of the Circle whoſe Radius is SQ equal to —= 2,
and ſince S T, or the Fluxion of the Arc RS. is
equal to — » We ſhall have the 2 gene

rated by the Line 8 T equal to — But «,

or the Solid generated by the . Area PST
is equal to a Cone whoſe Baſe is the above- men-
tioned Annulus, and . Altitude is PS;

where –

176 AN ALVYS1Iò of
ee e and u= 1
3 3
„
x x, the Fluent at firſt being 3 Again,

| fince the Areas PST, Pe b, and PEH are the

ſame as to Magnitude as if they were ſimilar, the
Solids generated by them will be as the Cubes of
the Lines PS, Pe, and PE reſpectively ; and „

wherefore «=

or the Solid generated by the Area EH be will be

to 4 as PE’_P& to PS: But ÞE’=PD’+
3PD xDE+3PDxDE+DE”, and Pe = 55 —3 PDxD E + 3PDxDE—D D E?, and PE-—P#=6PD’XDE2DE3; wherefore 51s to zas 6 PDXDE+Þ2DE? to P Ss, and 45 is to
2 AS 35 DEAT＋D DE? ; is to PS?, and by Diviſion

11 is to 4 as 3PD* XDE i IS to PS}, Or AS 35D.
1s an PS”, or as 3 PC“ is to PQ, that is, as 3 4a

is to x Xx; wherefore ; — | 2 Xs — = (ſubſtitu-
| 5

ting inſtead of « z its Value — — LEE 25

-” – AS =, 2 a*b ——
therefore L 5 —u =—<= = Ir;

• r rx

| 2 a*b 2
wherefore ; 2 -. is to u as hug r- is to 23
rx

—, or as 344 is to 7x, or as 2 FC is to
_PRxPQ, or as 3PCxPD is to PN.. Since
therefore 2 is te 8 gPCxPDis to PR),

1 it

: to po — HW — LAS

Mr. Cor Es’s PROBLEMS. 177

it follows, by Compoſition, that + 5 or the Semi- ſum

of the Solids generated by. the Areas A EDC,

and ae DC is to u, or the Solid lid generated by the

Sector PRS as 3 P CP DP R* is to PN
Again, ſince the Solids generated by the Areas
PST, EH K D, and eK are to one another as

as PS, PEP P’, and PB. Pe reſpettively,
we ſhall have z, or the Difference of the Solids
generated by the two Areas EH K D, and eh KD

to as PE ＋P-2 FDB; to PBS, or as 6P DXDE*
is to P S5, or as 6 PD is to PS, or as 6 PC is to

| 64:
“FU chat! is, as 64 is to x; e
F as – 2abx | )J

• 2 aſe and therefore the Altitude of a Cylinder

equal to the Semi-difference of the two conchoidal

| Solids, and whoſe Baſe is b, viz. a Circle upon 2

Diameter A a will be equal to 2 4

“=2PC| pe:

And thus having found both the Semi-ſum Ta

Semi-difference, whe Solids themſelves will eaſily
be found. Lo E. J. 5

— —ñ—4—̃ —— —— —— T — 0.459 * 80 4 _
.

wo. ANALYSIS of

PROBLEM X.

It is required to meaſure the conchoidal Areas A EDC .

| and aeDC, Jointly and ſeparately.

Z ANALYSIS.
| Suppoſe all Things as in the laſt Problem, ex-

cept that now s repreſents the Area AE ea, or the

Sum of the two Areas A E D C and aeDC, that

4d is their Difference, and « the Area of the Sector

PRS; and the Area EHS or ; will be to the
Area PST or 4 s PE to FS, or as

4PSXDE is to PS or as 4 PP is to PS, or as
4 C is to PQ, that is, as 4 à to x; wherefore

| 44 42 2 2 arr Xx
= N — — —

• Ro Irre r

a Fluxion of the fifth Form, where b So, and
whoſe Fluent, when made poſlible, gives 5 =
,PS+SQ

rr r- xX | 95
27 | | – — =AaxPC “20. –
AoxPC|—E Rata xC M*. Q. E. ;

Again, 4 or EHKD—ebKD is to 4 4s

PEI PD> stoPS, or as 2 DE is to
PS”, or as 2 to 1; wherefore = 2 2, and 4 d is

equal to the Area PRS; or if through M a Line

as Gg is imagined to be drawn parallel to A a,
and cutting che Perpendiculars A F and 2 of in G

• By Conftrudtion ſee Harm, Menfar. Þ. 39.
• WY | | and

—

• * +

S =

“© i AE

Mr. Corrs PRoBLEMS. 179

and g reſpectively, 2 d or the Sector PRS will be

equal to each of the Friangles MG F or Mg f;
for MG is to GF as P C is to CN; and there-
fore as CN is the Meaſure of the Angle CPD
ad Modulum P C*, G F will be the Meaſure of the
ſame Angle ad 1 MG or PR; wherefore
the Area AFM Cor ; 5 +24 will be equal to the
Area AE DC, and the Areaaf MC, or. 52d
vill be equal to the AreazeDC. Q. E. 1.

PROBLEM XI. Fig. 34.

To meaſure the interior hyperbolic Area ADB:
The Center of the Hyperbola . C, and the con-
Jugate Axis PQ.

ANALYSIS.

CASE-L

Let AB be perpendicular to CE, and call
CD, t; and C P, e; Ce, x; and EB, y; and
we ſhall have, from the Nature of the Hyper bola,

„* =I, and y= * =I, and 2y 4,
or the Fluxion of the Area propoſed, =——X
VXN Tt, aF juxion of the fourth Form where
o; the rational Part of whoſe Fluent is —

VX X rt, and the irrational Part, when the

Terms of the Ratio are made poſſible, and reduced
| * By the Conſiruction. |

N 2 to

180 Ax ALYSIS of

NV x—A
F.
both which Parts are equal to nothing, when E

the ſame Denomination, is —c 7

• coincides with D: So that this Fluent is a true
  Repreſentative of the Area ſought ADB. But

| | — <>. 7 REM
the rational Part of this Fluent — X/x*—f2, or
XY, is equal to E BXCE, and “4 Modulus c t, or

COXC
aaa btn BE = ER 5 or E BX n
SE

is 22 to E B XC 8, and the Ratio —

1
K | Cc E4CR_ _C ep.

3

4
4 2
ſince CR. c Xt CE ‘—CD), we ſhall
have CD E CR —=CE+CRxCE<CR,

whence CE+CR will be to CD as CD is to
CE-CRorER; whence upon the whole Mat-

ter it follows, that the Area ADB is equal to

; CD
EBxCE—E BxCS 5 =EBxXCELCN=

EBXEN equal to the Area of the Triangle ANB.
KI.

© It being CR: CD: CD: CS:: AB: PQ and C N=C$

. EE ; by Conſtruction.

C481

—

E

Mr. Cor ESs’s PROBLEMS 181

Cn

1 now the Ordinate AB be ſuppoſed to turn
upon the Point E, ſo as to become oblique to the

Line CE, and let all the reſt of the Ordinates
| parallel to it be ſuppoſed to turn with a like an-

oular Motion upon the ſeveral Points of the Axe;
all other Things remaining the ſame as before; and
the Figure will ſtill be an Hyperbola, becauſe the
Expreſſion of the Relation between the ſeveral
Ordinates, and their reſpective Abſciſſae will be
the ſame as before; but the Elements of the Area

A B, or the Intervals of theſe Ordinates, will

now be leſſened in the Proportion of the Radius to
the Sine of Inclination : But as the Elements of

the Triangle AN will alſo be leſſened in the ſame

Proportion, the Area A D B will ſtill be equal to
the Triangle ANB; and therefore is found in all

Caſes. k. 1.

PROBLEM XII.

To find the Center of Gravity of the interior byper-
_ bolic Area ADB.

Ana 18

Cass *
Let A B be perpendicular to C E, and let C E

be conſidered as a Lever moveable about the Axis

P q and uſing the Notation of the- laſt Problem,

the F luxion of che Momentum of the Area ADB
Nt. –

__ r

$5; —
to turn the Lever will be XI XXNVX X,

as is evident from Stalics. But this is a Fluxion
ol the bird Form, where 0=1 ; and whoſe Fluent

EXE! xg X—t f is equal to Eh XE B;

and as this Fluent at the Point D is equal to no-

thing, it will be a juſt Repreſentative of the
Momentum of the Area ADB : But as this Area
is equal to E NE B, the Quantity C ZE NXEB

will alſo repreſent the Momentum aforeſaid, as is

plain from the Nature of the Center of Gravity;

2CR EB=CZxXENXEB, and
3
2CR =CZx3EN, and conſequently C Z is to

EC R as 2 CR is to 3 EN. Q. E. I.

E |
Let now the Ordinate A B, and all thoſe that

wherefore Z

are parallel to it become inclined to tne Line CE,
as in the ſecond Caſe: of the laſt i roblem; and

the ſeveral Elements of the Area AD B will now
be diminiſhed in their Gravity as well as in their
Quantity; but as the Loſs will be every where
proportionable to the Whole, the Center of Gravity

cannot be affected by it, and therefore muſt be

found by the ſame Methods as before. Q. E. I.

• As in the Confirudtion, >

Pk 0-

Mr. CoTEess PROBLE MS. 183

PROBLEM XIII. Fig. 35.
To meaſure the exterior perbolic Area AP QB.

ANALYSIS.

Cann L

Let AB be perpendicular to CE, and call
CD, ; C CE, x; and EB, y; and we ſhall

„ 775 >
have yy xXx x- whence proceeding as in
the eleventh Problem, we have the Area AP QB
equal to the Triangle ANB; CN being taken
contrary to C E, becauſe now the irrational Part
of the Fluent is affirmative. Q. E. I.
Cd E II.
See Caſe II. to Problem. QEE

Lotion XIV.

To find the Center of Gravity of the ſame byperieli
Area AP Q.

ANALYSIS. |

Here when CE , AB will be equal! to PQ,
and CR to SD, and the Fluent 20 xE B

3
(See Problem XII. ö to — D* XC P 2 EDX S
t CINCK.
xXE B= — — X . B; and therefore the Mo-

N 4 mentum

184 AAT of

rw ofthe Arca A PQB eil be. N20 TXCR R

R R 3
wt + xEB=CZxXENXEB:

Wherefore CZ is to CR as2TR to 3EN,
DEE
PROBLEM XV. Fig. 36.
To find the Surface generated by the Revolution of the
Hyperbola AN about its primary Axe AC. Its
Vertex being A; its Center C; the Aſymptote CB;

the Focus F ; the Semi xes AC, AB; X N an
Ordinate.

ANALYSIS.

Since C F is to CA as CA to CE (by Con-
ſtruction) or as C B to C G, and ſince C BSC F,
C G will be equal to C A. Call then CA or CG, ft;
F ere. AB, co Tas; and X N, 73

and we ſhall have y y== Xx2 – , and y 25 xs,

222 *2

Nc. Again, yy x*=:

— —

5

and >

2 XK4ͤ*ͤ kꝗk—ͤ—ñ— , . YT”
Cy * * — 1+ wherefore yy x* +2 5, or

3 x? *

• A Xx2 .. 5 74 I OXONIS PINE IEEE m N.
  em x* A i 1

=—— Wa = a TY putting r for

—c or CE: Therefore yxVx* + * =— dh, ..

gat therefore if we put d for the oeemicinewans
rence

Mr.CoTrss PROBLEMS. 185
rence of a Circle whole Semidiameter is c or A B,

2 4
we ſhall have —=xviF97, or the Fluxion of

2d «x —
the Surface ſought, = * , a Fluxion

of the fourth Form; where 6==0 3 the rational Part
tx ee ua
of whoſe Fluent is XV – A NE Z.

XxX O; and the irration al Part, when the
Terms of the Ratio are made poſſible, and reduced’

to the ſame Denomination, is — dr. 1
EZ4-ZE 8

S 25 But when X coincides

with A, CZ (or CX*) will become CG or CA, and

XO will become A B, and O 7 7 — become

Ore = Subtract then AB from XO, and

En CGAGE
there will remain KL, and ſubtract CE =
CZ+ZE CE
from CE 52 ER and there will remain

CE 86816 5 or L MF, and the Surface fought
will be d X KL —dXLEM, or dx RK M, which
Surface therefore will be to a Circle whoſe Semi-
diameter is AB or cas dX KM is to dXc, that is,
as K M to AB. Q. E. I.

• By Conſtruction.

2 — — ,
88
— 3 — — +

.

• 
  — — — — = — ==
• N / K — — eg ee NES eng
  — as mw” os ns uy = — —
• C —jp— — — ũ ‚—

2 * 4 we
— a PD LE ICE * —— — = * 2 2 2 24 < rw, Jet 1 s 4 — — T = ” A L —— — = = LN — —8 g * A . + 8 5 —— — — — — — ͤ60-m eat,” + ee 0” 9 —— 3 2 n ” — a a ERIE 8 r — — — . — re rr r A ww ou ow, er FRA IRE = [ on = = 3 = * 4 * “Sy af = * =_ — 3, o — > — „ . * 3 441 + rh 0
— cg — 22 2 r : – 3 4

|
1
{
1
j

186 AN ALVYSIS of

PROBLEM XVI. Fig. 37.

It is required to find the Surface generated by the
Revolution of the Hyperbola B N about its ſecondary
Are CC.

ANALYSIS.

ene! cn.

XN, y; and the Semi- circumference of a Circle
whoſe Semidiameter is C 1 d; and then proceeding

upon the Equation y y N +7, as in the laſt

Problem, we ſhall have the rational Part of the

Fluent for determining the Surface equal to

1 C | |
«x $7.5 Ll L. and the

irrational Part will be XL M; and as both theſe.

Parts, when C and X are coincident, are equal to

| nothing, the Surface ſought will be XK L+-L M

=dXxK M*. Q. E. I.

PROBLEM XVII. Fig. 3

To find the Surface generated by the 1 of the
elliptical Arc B N about its ſecondary Axe AC.

A .

Apply the Notation of the laſt Problem to this
Scheme, and the Equation yy = — X * K x
pourſued as in the fifteenth Problem, will give the

Surface fought equal to dx KM as before. Q.E. I.

• By cogr. CE:XE::XC: KLand LM=CE| Ce

PR o-

=?
——

Mr. CorEs’s PROBLEMS. 187

PROBLEM XVIII. Fig. 39.

70 find the Surface generated by the Revolution of the

elliptical Arc B N about i:s primary Axe AC.

ANALYSIS.

Apply the Notation of the ſixteenth Problem to
this Scheme, and then proceeding as in the fif-
teenth Problem, we ſhall have the rational Part

of the Fluent for determining the Surface equal to

ax K L as before, which begins to flow with the
Surface ſought. But the irrational Part will be
the Meaſure of an Angle ad Modulum — ar, whoſe

Secant is to the Radius as 7 to x, which is the

Caſe of the Angle E C X, and which I thus ex-
preſs — dr [ECX, or thus + dr | ECX. But
when X coincides with C, and conſequently CE
with C B, the Angle — E CX becomes – B CX;

which therefore being ſubtracted from — E CX

leaves BC X ECX BCE CEN; and the

irrational Part, at leaſt ſo much of it as ſerves our

Purpoſe will be dr CE XX LM. Q. E. .

1 Fig. 40.

If two 8 be added together, whereof the
Radius, Tangent, and Secant of one are as R, T,

and S; and thoſe of the other as , 7, and 5; thoſe

of the Sum will be as Rr IT, and oe and

| BH reſpectively.

DE mon-

—— — 4/6 —
r —

0 * 1
e el
8

— ̃ ß—

Prey r g

— —ä—ä
2

c Nee

TN K —ů;ͤ——C a 2 2 ——— ty pat SAR} — 2 1 2. —

• r —
• o ws 4 SS.
• ” CNET ee

|
t
F
A
1
|
9
1
k –
:
A

188 ANALYSIS ef

DEMONSTRATION.

Draw any Line as AB to repreſent R the

greater of the two Radii, and out of it take A5

to repreſent 7 the leſs; and perpendicular to A B,
but in contrary Directions draw B C=T, and
b c=t, and joining AC and Ac, let Ac and CB
when produced meet in D, and draw CE per-
pendicular to A D in E: Then it is plain that
the Angles BAC and B AD will be equal to the
Angles firſt propoſed ; that the Sum will be the
Angle CAD; and that the Radius, Tangent, and

Secant of this Sum will be as AE, E C, and AC;

or as AEX A D, E CXA D, and A CX AD re-
ſpectively. But E CAD is equal to A BC D,
ſince each is equal to twice the Area of the Tri-
angle ACD: And moreover, ſince by Prop. XIII.

El. II. CD+2A EXA D=A AC’+AD D=2 AB*
+B C+-B D if from one Side we take away

SP, and from the other B B C? 2B CXBD+—-BD*,
which by Prop. IV. El. II. is equal to it, and
laſtly, if we take half the reſt, we ſhall” have

AEXAD=A A B*—B CxBD. Subſtitute there-

fore the latter inſtead of the former i in the Pro-

portion aforegoing, and alſo A BO D, or A BBC

LABXBD inſtead of E CXAD, and we ſhall
have the Radius, Tangent, and Secant of the Angle

CAD as A AB*—BCXBD, and ABXBC-ABXBD,

and A C XA D, reſpectively. Out now for A B=,
A B X Ab,

4 b

1 3 OE ER “WY —_— 1 7588 9 — 1 rere A — —

hoo LY

JW” OP

. ow. (DD LU Of | ww – and of

1 |

M,. CoTress PRoBLEMs. 189
ABA b, or Ry; for BCXBD, B Cx, or Tz;
for A BNB D, AB XG, or R.; for ABXBC,
AX BC, or r IT; and laſtly for ACX AD,
ACX Ac, or S5; and ſince the Terms are all
leſſened in the ſame Proportion, they will ſtill
continue in the ſame Proportion to one another;
and the Radius, Tangent, and Secant of the Angle
CAD will be as RT, and RH T, and S,
reſpectively, QE. D.

School run.

For the better Application of the urging
7 heorem it will be neceſſary to obſerve, firff, That
if the Radius, Tangent, and Secant of any Angle
have all the ſame Sign before them, the Angle
will be acute. That if the Sign of the Radius be
different from thoſe of the Tangent and Secant, it
will be obtuſe. That if the Sign of the Tangent be
different from thoſe of the Radius and Secant, the
Angle will be acute and negative. That if the Sign
of the Secant be different from thoſe of the Radius

— — Dc

1 dis
2 2 3 p
ISO ene 66523 — — p

MAGE M6 > = ty

A Lan
—— —y—’ — — — 2 =_

—̃

» 12 Tn

1 r _
F

— . ͤÄ r 2 —

A „

:

«a 4 \

3 „„ wa
1

v——

: – * _ _—— —·˙ ——
» <T

— — oo, 7

n

and Tangent, it will be negative and obtuſe: And

laſily, That if the Radius be nothing, or the Tan-
gent and Secant, infinite, the Angle will be a right
one. Theſe are the Marks whereby Angles are
to be diſtinguiſhed one from another, and which
muſt be duly obſerved, not only before Addition,
in order to put proper Signs before the Radius’s,
Tangents, and Secants of the Angles to be added,
but alſo afterwards, in order to judge aright of the

Aggregate.
4 : __ Secondiy,

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• *
  .
• 
  9
  .
  9 *
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  5 0
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• ©
  ny”.

5 (4% þ
4.44,
Bo
1
gi
we 1
e
PL
“3
ES 5

• 7 4 4
  —_—
  **

Py 4 p
„
4:
_.
x: *

• 
  .
• 
  0
  —

190 AnALYSs1s of

Secondly, It muſt be obſerved, that if the Radius,

Tangent, and Secant of any Angle be as R, T,

and S, thoſe of its Complement to a right one

will be as T, R, and S; and thoſe of its Com-
plement to two right ones as R, —T, and —8
reſpectively. Not chat I would here be underſtood
as if the Tangent and Secant of this Complement
were always to be negative, but that they muſt
always be contrary to thoſe of the Angle firſt
propoſed. .
Many other Obſervations might be drawn from
the foregoing Principle; as chat we are furniſhed

with a Method for ſubtracting one Angle from

another“; for inveſtigating a general Theorem
whereby ay Number of Angles may be added
. together ; for multiplying, or dividing an Angle
by any given Number whatſoever, Sc. But theſe,
however uſeful they may be in themſelves, being
not ſo much for our preſent Purpoſe, we ſhall not
inſiſt upon them, but proceed to apply what has
already been taken Notice of.

• By Alike the Sign of + tin the Lemma.

PR o-

ae a Md hot i — tl ̃˙—waʃ ˙ it £1

(6

EE

Sh K 7 . 8 8 . 8

Mr. Cor Rss PROBLEMS. 1917

PROBLEM XIX. Fig. 41.

Ii is required to determine the Law of Attraction
wwherewith a Spheroid attratts any Particle of

Matter placed in the Direction of its Axis produced;
ſuppeſing the whole Spheroid to be of the ſame
Denſity, and to conſiſt of Particles which conſidered
by themſelves attract at all Diſtances with Forces

reciprocally Proportionable to the Squares of their
Diſtances.

CONSTRUCTION.

Let AD B E be the generating Ellipſe, AB the

greater Axe, D E the leſſer, C the Center, and F

the Focus; and if A B be the Axis of the oblong

Spheroid in the Direction whereof the attracted
Particle Q is ſuppoſed to be placed; then in the
Angle FCD inſcribe a Line as FP equal in

Length to the Diſtance of the Particle from the

Center of the Spheroid : But if the Spheroid is a
prolate one whoſe Axis is DE, and the Particle
be ſuppoſed to be placed in the Direction of that
Axis; then let FP be a Line drawn from the

Focus to the Center of the Particle itſelf; and in
the Line C F produced, in the firſt Caſe, take

CG equal to the Meaſure of the Ratio between

PF+FC and PC: But in the latter Caſe, ſet off
upon the Line CF the Part CG equal to the

| Meaſure of the Angle CPF; and let the Diſtance

of the attracted Parick from the Center C be the
2 : 7 Modulus

192 ANALYS1s of
‘ Madulus in both Caſes, namely, PF in the former
Caſe, and PC in the latter; and the Attraction

of the Spheroid will be as the Quantity of Matter

in the Spheroid and the Line F G directly, and the
Cube of the Line C F inverſely.

ANALYSTS
| C Fes 1 ©
Loet Q be the attracted Particle in the Direction
of the Axis B A produced, and aſſuming any
where in that Axis AB, a Point as X, draw the
Ordinate XN, and join N Q. Call CA, t;
CD, c; CF, m; COQ a 8 and N Qs 2

“£2”
and we ſhall have X KN 8
CA

CA?—CY X =

5 ce — 201-2 c*a — c
l 7 , and conſequently 22

Lc ＋2 ca == CPC E 0 ax4mx*

71 N 77 5

. 2a AN But by the
ninetieth and ninety firſt Prop. of the firſt Book
of the Principia, the Force wherewith any
circular elementary Lamina whoſe Semidiameter

Fea” Z=

is X N attracts the Particle Q will be as

x x

r ; and therefore the Fluent of the F Juxion

: | txX

X—,/ —= (I mean ſo much
Can . ca XK &

of

te

of

VE – fεe COCENXN/a%—m*

Mr. Co F ESs’s PROBLEMS, 193

of it as lies between A and B) will repreſent

the whole Force of Attraction required. But &

is a ſimple Fluxion whoſe Fluent is x, and

ci 2 2a s

is a F bes of the

ſixteenth Form, where b 2; the rational Part of

. — 222 „
whoſe Fluent is . and che N Part the

n of a Ratio ad Modulum 5 ef whoſe An-

3
cadkmx __tmz+fa-mx
x tz

| * – ca c 208 —— Pm. FH
whoſe Conſequent 1s * ＋ 7 3
; * 2 d

1 is mm „ and

— and therefore the

1 2 42
irrational Part of this Fluent will be 2 —
acc EH EE. a
c M — me |

2

| .
Now the Quantity x— , when X coincides

N 2

— —

4

with A will be equal o i—Zxa=7=— x77
7 I

hos when X coincides with B will be equal to

e, and the former ſubtracted from the |

2 C2 2 7M

Ws leaves — , OF — for our Purpoſe,

im en

A : : . –

6 A be

• at the Point.

U

ns: ANALYSIS of

tma—PmS+ a+ u a – T

A becomes
ct V –
t ma- HEN LHA a- Nn
— . —— — i ; and at the
Ct af min c Jai.

Point B the aforeſaid Ratio becomes ENI,

c az
and the former ſubtracted from the latter leaves

2

— and therefore ſo much of the irrational

i . as lies between A and B will be tal am
mM? 1}a—n

cet PF-+FC_ 2.0% PF-FC. 242
FFF PE—FC m — PE PC 7 M:

But the other Part of the Fluent was found to be

. 2 C2
—— or — XCF; and therefore the
26027

Attraction of the e will be . of

as its Magnitude and the Line FG directly, and

the Cube of the Line C F inverſely.

Hitherto we have ſuppoſed the Denſity of the
Spheroid to be given; but now if we ſuppoſe that

to vary, and all ether Things to continue as be-

fore, the Attraction will be as the Denſi ity: And

therefore univerſally it will be as the Denſity and
Magnitude of the Spheroid, and the Line FG

directly, and the Cube of the Line CF inverſely.
But the Quantity of Matter contained in any Body
is as its Magnitude and Denſity together: And
therefore now the Attraction of the Spheroid will

2 | 32

1 Or Tn > on Bon

Mr. Co TE S’s PROBLEMS. 195

be as its Quantity of Matter and the Line F G
8 and the Cube of the Line C F inverſely.

P. I
[> 25 E l.
Let now P be the attracted Particle in the Axis
E produced, and from any Point as x in that
Axis draw the Ordinate x , and join » P. Call
CD, t; CA, c; CF; n; CP, a; xP;x; and
P, z; and becauſe now m=:2—7?, and con- 02 tac a +2 -m x
7 \

tional Part of the Fluent for determining the Ads

traction of the Spheroid will now be x +=,

2 cat in
whereof is for our Parpalh. And moreover

tht * of an Anse ad Medulum 5 whoſe

n
Radius, 7. angent, atid Secant will be to one another
as the Quantities f 2, caa-πιπ , and c . En;
which at the Point D will be as 4 -n,
Ca—mad-nt, and c . TEm:; or as f me -n,
Pam, and c 42 Tm; ; or as amt, 44. Ema,
and I H Em. Let there be now two

Angles, whereof the Radius, Tangent, and Secant
of one, which we ſhall call p, are as t, m, and

Ii Pm, and thoſe of the other, which we ſhall
call g, as a, m, and Jans; . and (by the laſt
Lemma) thoſe of the Angle p— will be as ? a+,

O02 and

ſequently 22

o the ra-

RN athens ——— AGO CIRICEY 1

196 „AN ALVYSIS of

and amm, and V n=; and thoſe
of its Complement 1—p–9,” putting 1 for a right

Angle, will be as am—mt, and t a+m, and

IV x Vn. reſpectively So that the ir-

rational Part of the Fluent at the Point D is
cz} al
—[1—pdg; and by a ike Computation, at the

Point E it will be found to be C48

Fog B7

== Of == . EEE

m3.
2

or — a ‘xCG; which joined Soy the Part
ph 7 2 Ot

above found —— > XC F will make — N F G,

2 6214 2 c*t a

for the Aon required : Whence i it lows,
chat in this Cale alſo the Attraction of the Sphe-
rod will be as the Quantity of Matter, and the

Line FG directly, and the Cube of the Line CF

inveriely. Q. E. I.

| PROBLEM XX. 5
Suppoſeng all Things as in the laſt Problem; let it
be required to compare the Attraction of ihe

Spheroid with that of a Sphere of the ſame Denſs uy,

and upon the ſame Axis.

In the former Caſe of the laſt Problem, the Aide
_ am _ z af

—— 4

C G was taken equal to a
42 — 112 14 —M

= r the fifth Corel. of the firſt Prop. of the
8 | oo

_ wot WRV’..2&

1—9—43 *

and the former ſubtracted from the 3 leaves

| 2 — .

• 27 Fa to

—

——

of

| 7 and —

Mr. Cores’s PROBLEMS. 197

Logometria ) ahh 55 Se And i in the ſecond Caſe
mz
CG will be Ga to m — a Sc. (See the Series

for finding an Arc . the Tangent given.)

Wherefore in both Caſes F G will be equal to
M3 2cct

xF o ae Suppoſing now

hs . a and t to continue, let the Quan-

tity m, and all its Adherents ſignified by the
Note Sc. vaniſh ; and the Spheroid will, by this

Means, be changed into a Sphere upon the ſame

3
Axis, whoſe Attraction will be 2 whence it

a?
follows, |
Firſt, That the Ae of a Sphere at all
Diſtances will be as the Quantity of Matter therein

contained directly, and as the Square of thoſe

Diftances reciprocally : And

Secondly, That the Attraction of the Spheroid
is to that of a Sphere of the ſame Denſity and

2c e 2 73
upon the ſame Axe as —— Xx F G to, or as
| N53 n 3a? :

3 ca Xx FG to xx m3; and conſequently at either

Extremity of the Axis, the Attraction of the
Spheroid will be to that of the Sphere as 3 c X F G
to n. Q. E. I.

O 3 Pa

198 ANALYSIS of

PROBLEM XXIX. Plate VII.

Ws o find all the Variety of Trajefories, which a Body
projefted from a given Place with a given Velocity,
and in a given Direction may deſcribe when afted
upon by a centripetal Force reciprocally propor-
tionable to the Cube of its ang from the Center
/ Altraction.

SOLUTION.

ow… SAS} 3. FR 42- |
Let SP, SQ, PQ, Sc. be conſidered as in-
determinate, and draw the Line Spr infinitely
near the Line SP R, cutting the Line P 2
and the Arc A Re, i inp and r reſpectively; and
taking Sax Sp, join p. Call S C, a; SP, x;
and if the centripetal Force at any given Diſtance
d be called 2 d, its Force at any other Altitude x

| os :
will be ==, and (by the thirty ninch and fortieth.

Prop. ft he firſt Book of the Principia) the Velo-
City of the Body at the ſame Altitude x will be as

— a2 —x*
24.74 &—X X, or as V —— and the Perpendi-
ax | | x

cular SQ, which is always reciprocally as this

Velocity, will be as TEES Let us now aſſume

See Mr. Cotes’s Conſtructions i in Harm. Menſurarum, P- 31.

et ſeqq.
a given

Mr. CoTEes’s PROBLEMS. 199

a given Quantity as & of ſuch a Magnitude that
at the firſt ſetting out the Perpendicular SQ may

• 
  be equal to ., and it will be fo ever after-
  Bmx

wards, and we ſhall have P Q=z 1 K . 2
and QT = KY ee — QT

44— 22 52— 22 K ä |
aa—=xx and SQ +Q T=az, and and /S Q+Q T* or SR =4aa—Fb a given
Quantity, which for the future we ſhall call z.
Now ſince Pr is to x p as PQ is to QS, that is,
as Va- to 5, or asa/nn—xx to 5; and
ſince 7p is to Ry as SP is to SR, or x to , it
follows, joining the Ratio’s together, that P 7 or

—xistoRras x x V-: is to bn; ; and there-

7

fore Ry, or the Fi juxion of the Arc AR will be

——# x x
equal to * a Fluxion of the fifth Form,
where go, and whoſe Fluent is þ .

—c
SJ M]——_—— = – . D 3 making 8 M nn SC2—_SR?,
and taking SD always equal to the Altitude S P,
From which Conſtruction it follows,

Firſt, That the Trajectory AP will touch the
Circle A R in the Point A. |

Secondly, That the Altitude SA or u is this
higheſt to which the Body can aſcend ; for ſhould
it aſcend higher, the Quantities QT, Sc. wherein

Nm. P—x? is concerned, would become impoſſible.
O 4 | Thirdly,

200 ANALYSIS of

Thirdly, That therefore the Point A is the
Apfis ſumma of this Trajectory, and that i it has no
other Apis but the Point A.

Fourthly, That the Ratio 2, 55 , and its

Meaſure A R, and the Angle ASP may be of
any Magnitude frem Nothing to Infinity, and
conſequently, that the Body P i in its oblique De-
ſcent muſt make an infinite Number of Revolutions
before it can come into the Center S. |

FI ifthly, That therefore the Spiral AP muſt
neceſſarily cut the Line S A in an infinite Number
of Points, which may alſo be ſaid of the Spiral on
the other Side of the LineS A; and as theſe two
Curves are ſimilar, they muſt neceſſarily cut the
Line SA in the ſame Points, which therefore muſt
be the Points where they cut one another : So that
the Nodes muſt all be in the Line S A; and thus
much for the Form of the Trajectory. BY

Now as for the Area SAP, ſince r

5 Uu x
„and conſequently the Area SRr=

• . 8 5 5 7255 LEGS
  DX
• Ju — **

Area S A P will be equal to =

, the Area 8 P p, or the Spin of the |
xd x
Vn nx

of the ſame fifth Form, where 6=1, whoſe Fluent

EPR In , or SMXED will be equal to
the Arens AP. 1 ;

71 a F Lexicon

CA

Mr.CorTrss PROBLEMS. 201

CASE II. Fig. 43.
Draw 8 Q perpendicular to the Tangent PM,
and the Velocity in this Caſe will be reciprocally,
and conſequently the Perpendicular SQ dire&ly

as SP; wherefore the Triangle S P Q will always

be ſimilar to itſelf, and the Angle S PM always

the ſame; therefore the Curve in this Caſe will be

the equi-angular Spiral, whoſe peculiar Property

it is, that the intercepted Arc PX ſhall be the
Meaſure of the Ratio between S Z and SP, or

S D and SP, ad Modulum S M; whence it follows
alſo, calling S Z, x, that the Fluxion of the Arc

PX is equal to . : and the Fluxion of the
Sector S PX * to 2 N, and the

r of the Area SP Z equal to Tos XXX,

whoſe F luent es . gives che Area 8 PZ

4 SN, SPP,

• 2

; or which is all one,
E —_—

equal to

the Area SPZ will be to. – as 2 SM is

As IE F
Here applying the Notation of the firſt Caſe

to this Scheme, the Velocity of the Body at the
| e * will be as —— a 5 and Na.. or.

will

202 ANALYSIS of

will be equal to VO T*—SO . Whence pro-
ceeding as in the firſt Caſe, we ſhall have the Arc

BR=SM| 5g

the Line SP in any given Time during the oblique
Aſcent of the Body P will be equal to a Rectangle
under 2 SM, and the Increment of the Line D 5E
generated in. the ſame time. Q. E. 5

CASE IV. Fig. 47. |

Draw Sp infinitely near SP, cutting the Curve
PpZ in p, and the Arc PB in x, and upon the

Center 8, and with the Radius SC ſweep an Arc
of a Circle cutting the Lines SP, Sp, and SB,

produced if need be in K, L, and M reſpectively;
and ſince in this Caſe V- 5, or Va.. ,
and conſequently a==b, we ſhall have 7p to 2
{that is PQ to QS) as æ to 4: But x P is to KL

alſo as x to 4; wherefore putting both Ratio’s

together 7p or —x wiil be toKLasxxistoaaz
wherefore K L, or the Fluxion of the Arc KLM
will be 3 and therefore if we conſider the

Line S B as a Radius primus to which the Body P

uill not come but after an infinite Aſcent, the Arc

K L M will be equal to =, and conſequently the

Arc P B will be equal to à or SC; and the ſame
| witty be ſaid of any other Arc D Z; and therefore

the

— and the hon deſcribed by –

th

te

D

.

M.. Cor Es’s PROBLEM S. 203

the Nature of the Curve PZ is Mt: de-
termined.

Again, the Fluxion of the Area SKM “IS os
K

and therefore the F We of the Area 8 PZ will
be z a *, whoſe Fluent 3 ax gives the Area
SPZ=43SCXSZSP. E. I.

CASE V. Fig. 46.

Call afSC?— = or V3, z; and we
ſhall have now the Fluxion of the Arc AR=
In
We and the Arc itſelf equal to the Mea-
ſure of an Angle ad Modulum b, whoſe Radius,
Tangent, and Secant are to one another as ,
Vi-, and x: =SM|DSE, taking SM or &

equal to Y N and SD=S P;
whence it follows,
Firſt, That the Point A will be the only Apſis
of the Trajectory AP, and that the Body P can-

not poſſibly deſcend lower than the Altitude SA
or u, becauſe if it ſhould, the Quantity –
would become impoſſible.

Secondly, That the Angle A’S P will always be
A ASE SM. |

| Thirdly, And ſince the Angle ASE can never

exceed a right one, that the greateſt Angle ASP,

when the Body P has aſcended ad infinitum will be

to a right one as SM to S A. 5

| | As

204 ANALYSIS, Se.

As for the Area 8 A P, that will be found FO
to + SMX D E, as in the firſt Caſe. Q. E. I.

If in the ſecond Caſe, the Angle SPM is a right
one, the equi-angular Spiral deſeribed i in that Caſe
will become a Circle, and the Body P will now
deſcribe a Circle at the Diſtance S P with the ſame
Velocity with which it before deſcribed the equi-
angular Spiral, that is, with the Velocity which
it would acquire in falling from an infinite Height
to the Altitude SP; and therefore the Velocity
of the Body P in all Caſes at the indeterminate
Height SP, will be to the Velocity of a Body
deſcribing a Circte at the ſame Diſtance, and with
the ſame centripetal Force, as it is to the Velocity
acquired in falling from an infinite Height to the
ſame Altitude SP, that is, in the firſt Caſe as
dd

— 9 — 2— 22
ax Ve x2 is to or as Vaz— xz, is to a,

or as 4/S SC?._SP is to SC; and in the third,
fourth, and fifth Caſes, as /S C*+S P’ is to S C.

\ $oLUTION of fav0 uſeful PROBLEMS. 205

Two PROBLEMS: Of the OsciLLATIONS
of a PENDULUM ; and the RECTIFICATION
of an ELLIPSE.

| LEMMA I. |

Let there be two Series’s propoſed, out of
which a third is to be formed, by multiplying the
Terms of one into the reſpective Terms of the
other; and let the ‘Terms of the frſt Series ariſe
from a continual Multiplication of the Quantities
a, b, Sc. as allo thoſe of the others from a con-
tinual Multiplication of the Quantities c, d, Sc.
I ſay then, that the Terms of the third Series will
ariſe from a continual Multiplication of the Quan-
tities ac, bd, Sc. For the two firſt Terms of
the firſt Series will be a and a6, thoſe of the ſe-
cond c, and cd, and thoſe of the third ac, and
ach d, Oc

LEMMA II. –
Lab be a conſtant Quantity, EY x and y

variable ones; and let V x—x R ; let =; P,

IO * k, Bs, x S=XT; Ec. Then we
ſhall have __ ihþQ—zy9x=R,
Res, 155 Sc.

Ds MO N-

f

—

266 Sor.vTI0N of two uſeful PROBLEMS;

DEMONSTRATION:

If d -i p, and p G

we ſhall have the n Equation de P+

| Todo

iT X. F Q= — Xe T7 D. See Cotes Logometria,

p. 66. T beor. L But in our Caſe wane _
.

axed „r p; therefore according to our
Netation z =, d=1, e=b, f=’—t, , n
a=2, and xÞP=Q: Therefore the above- quoted

Equation in our Language furniſhes us with the
following ones + 5 P=Q =y, 25 .- 2 R=y x,
2 5R—3 Sry xx, 4bS—4 T=y x, Sc. Whence

from P may be derived the Quantities Q, R, S, T

as in the Lemma.

SCHOLIUM.

From the ſame Principle, if a
| PPI -&

• PE x2 Q=R, xX* R=S, x2 S IT, Sc. we ſhall
  ve — 30. x .

5

2 4

RN 3 w_— EEE ——

2

which may ſerve for the Rectification of the |

unt,

PRO-

• : 6
  — j
  .

—

SOLUTION of {00 uſeful PROBLEMS. 207

PROBLEM IJ. ig. 47%

To fad the Time of a ſingle Oſcillation of a Pendulum
about the Center C made in any given circular Arc
HAL, whoſe loweſt Point is A.

Let the horizontal Line H L cut the Line C A
in N, and let the Pendulum be ſuppoſed aſcending

in the Arc ADL at the Point D ready to de-

ſcribe the infinitely ſmall Arc Dd, and draw the
Line DE B perpendicular to AB C, meeting in
E a Semicircle whoſe Diameter is NA. Let

2 C A (or the Diameter of the Circle deſcribed by

the Pendulum) be equal to 4, its Semi- circum-

ference equal to c, N A=b, AB=x, B Ey.
and the Arc AE =v: Then will the Arc

24

Di/= ==, and the Line NB=h—x or = :
Vd xx x ED | x

and the Velocity acquired in falling through the

Space d will be to the Velocity acquired in falling
through the Space N B, or to the Velocity of the

Pendulum in the Point D, as * is to =, or as

dx to y: Therefore if /dx repreſents the Velo-
city acquired in falling through the Space 4, y
will repreſent the Velocity of the Pendulum in the
Point D. Moreover the Time wherein the Space

2d is uniformly deſcribed by the Velocity YA

will be to the Time wherein the Space D d is uni-

| d
tormly deſcribed by the Velocity y, as as as

5

S is to — |

• —
  —

——

4
PI
—— — 922 — en —
= l —B – . =- ti os, + 1
— — 2 — — ——

—

— gy

\ you —U—ʒeüd — bg

208 SOLUTION of 70 uſeful PROBLEMS:

ch 24 2d
that is, as — is to — 1 is to
TT. vie. 95

—— — “id

3X1 3 N **. Therefore

if : be the Time * the Deſcent of a heavy Body
; through the Space d, and conſequently the Time

wherein the Space 2 4 will be uniformly deſcribed

with the Velocity /dx; then the Time of the

Aſcent of the Pendulum through the Arc Dd will

= *

5 — 4
be ee R . La the Quantity I
J

be thrown into an infinite Series, and then we

mall have the . of Aſcent om D tod 8 17

n, + EX 2 2 7 * 25 *

5 .
8955 12 84% 9 b x
x R, ‘xR=S, x$=T, Sc. then will the Time
of Aſcent from A to D be equal to 4 x P +

T 1 1 3: | 5 5
irn eden
35

TXT; &c, But =. and by the

8 Lemma æ 5 P—y=Q, 4 þQ—zyx=R,
£hR—+yx*=S, + S- T, Sc. Therefore

at the 9 L, where 7 8 —> and y vaniſhes,
we have —= P, 2 UP Q. N. SRS,

25S T,

• x 2 V .
  Sc. Let = or —=Þ, xP=Qz

ah EE errant

2 1 18

19 WV

| SOLUTION of two uſeful PROBLENMS. 209

28 T, Cc. Therefore in this Caſe, the Quan-

tities P, 8, T, Ce. 1 from a continual
Multiplication of theſe, viz. 5 , 1h, 46,34 3%

Sc. But the Terms of the other Series into which

the Quantities P, Q, R, 8, T, Sc. are to be re-
ſpectively multiplied, viz. it; * — *

. 35 5 8 « 44
51 16 5 X EIT &c. _ 2 continual
| f
Multiplication 5 theſe, 55; 77. 7 4 8 25 Ge.

If therefore A Series be formed out of both theſe

Fe

by a continual wms, 5 5 — , or 2 ＋,

5 3
In * 7 4 274, 447 Sc. the Sum of the Series

will exhibit the Time of Aſcent from A to L,

1. if the firſt Term, inſtead of +* 7 be made equal

to ed? and the reſt derived as above, the Series

d
will exhibit the Time of an entire on

through the Arc HAL, and the firſt Term 7

will be the Time of the leaſt Oſcillation of that

Pendulum. Q. E. I.

CoROLLARY. 5
The Time of the Deſcent of a heavy Body

through Half the 3 of any Pendulum falling

3 : from

— – 2 — — — —

— ä̃x— rr ,,⏑‚—⏑—rܾ]. —⅛«‚ .

• –
  —U— — a — — Crs — —— 3

210 Sox T ION of 229 uſeful PROBLEMS.
from Reſt is to the Time of a leaſt Oſcillation of

that Pendulum as the Diameter 04 a Circle is to
the Circumference. 7

SCHOLIUM.

If 2 v be an Arc whoſe Sagitta is , and Dia-
meter d, we have from the Doctrine of Fluxions

5 o* 4 v4 4X4 vs
this Equation – 1 7 5

s |
2 – = 7 Sc. whence the Series

in the firſt Problem, viz. 4 SR X 224 2

4

—

b+
36 ds = * 2 16 * * 5 x2, &c. becomes equal

36 64 4+?

: 1 V2 II = 173 6 22931
to this, viz. bs 5 T 192 & I Igo. 5160960
oY Sc. whence may be deduced the following
Theorem.

Make A=,4112325 in 3 Parts of one
e of Time, its Logarithm —1, 614088,
B, oo000 17942 10, its Logarithm —6, 2538748,
C = ,00000000000895 3766, its Logarithm —
12,9520057, D=,0000000000000000504 3650,
its Logarithm —17,7027454, and let » be the
Number of Degrees, and Parts of a Degree con-
tained in any Arc: I fay then that the Time of
any Number of leaſt Vibrations of any Pendulum

will be to the Time of the fame Number of Vibra-
tions made by the ſame Tendulum in an Arc of
: | n Degrees,

„„

SoLUT10N of tawo uſeful PROBLEMS, 211

n Degrees, as one Day is to one Day ++ A nn

EBM CA Ds, Sc. Seconds. As for Ex-
ample. Let the Proportion be required in an
Arc of 120 Degrees. Here n= 120, Ar =

5921,8 Seconds, B n* = 372,0 Seconds, C5 =

26,7 Seconds, Das = —=2% Seconds. The

whole Augmentation amounts to 1 Hour, ‘5921,8
45 Minutes, 23 Seconds: Therefore if 372, o
it was poſſible for the Pendulum of a2 26, 7

Clock to meaſure Time by its leaſt Vi- 2,2
brations, and that Pendulum were after- 6322,7

wards made to ſwing in an Arc of 120 6333, 0
Degrees, it would loſe 1 Hour, 45
nutes, 23 Seconds every Day.

N. B. That the ſecond Term of the Series in

an Arc of 15 Degrees does not amount to a tenth –

Part of a Second, nor the third in an Arc. of 47
Degrees, nor the fourth in an Arc of 81 De-
grees.

| PROBLEM II.
” o meaſure the 88 of an ELLIPSE. Fig. 48.
Let ABab be an Ellipſe ſimilar to that which

is propoſed, wherein let A C Half the tranſverſe

Axe be equal to 1, FC Half the Diſtance of the
Foci equal to n, MP an Ordinate to the Axe

equal to y, and PC its Diſtance from the Center

equal to x : Then will the Square of Half the leſſer
Axe B C be equal to 1 m2 and from the Na-

ture of the Ellipſe we ſhall have P=1—m XI—#23
from whence may eaſily be deduced this fluxional
| 1 Equation,

212 SOLUTION of wo uſeful ProBt Ms.

Equation, viz. V2 y2,’ or the Fluxion of the

=. Throw

elliptical Arc B M, N m
| I — * * N

=P,

ES

Ii: into an infinite, Series, and let 3

• P=Q, * Q=K, Sc. as in the Scholium to the
  the ſecond Lemma, and we ſhall have the elliptical
  Arc BM=P—+mQ —+4mR= e m8 — ip
  m*T, Sc. in which Series the firſt Term P, or the

Fluent of = is a circular Arc, whoſe Radius

is 1, and whoſe right Sine is x, as is evident both
from its own Nature, and from that of the Series.

Let the Point M now coincide with A, and

the circular Arc P will become a quadrantal Arc,
and the Quantity /i—x x now vaniſhing, it is

evident from the Scholium abovementioned, that

the Quantities P, Q. R, 8, T, Sc. will now be
derived from one another by a continual Multi-

plication of theſe, viz. P, 3,4, 4, 2, Sc. But the

Terms of the other Series 1— . 4 4 16
— m, Sc. ariſe from a continual Multipli-
cation of theſe, viz. 1, — 3m, + im, +2 un,

• £m, + 7; m, + Sc. Therefore by the firſt
  Lemma, the elliptical Arc B A will be equal to a
  Series formed by a continual Multiplication of

theſe Quantities, viz. P, — 3m, n, -m,

35 m, + £3 mm, Sc. This is upon a Suppo-
\ ſition that the tranſverſe Axe of the Ellipſe is

equal

ue. * aa. ron Grd

Nota in LoGoMETRIAE, Prop. 4 213

equal to 2; but. if that Axe be of any other
Length and bears the ſame Proportion to the

Diſtances of the Foci as 1 tom; then making P

equal to the Circumference of a Circle whoſe
Diameter is the ſame with the Axe of the Ellipſe,
the whole Perimeter of the Ellipſe expreſſed in

Newton’s Way will be P— 2 A+ ie 1 BA- 34 £9

mC + 2: . D- ger . E, Sc. which Series —

alſo be _ thus, P * Dn AZ __ |
1
„ 75 Em Co Im nn m. E,

Sc.

In Partem eee Propofitionis quartae

LoGoMETRIAE. Fig. 49.

Sit altera Aſymptotos C Þ, et producatur BAB,

producatur etiam Z Q & occurrens reliquo Hy-
perbolae Cruri in æ, et Aſymptoto Cg (quan-

tum opus productae) in Z; et per Articulum 93.

Conicorum Hoſpitalii, erit Rectangulum Px P “
aequale Rectangulo B Ax Ag; ſed per Articulos

83, 118, etQ@s=—=QP, et per Articulum 95,
1 E PZ: Unde et Q, QZ aequantur, et P &,
aequalis eſt ipſi QZ P, et PZ ipk QZ—QFP,
et Rectangulum PE Xx PZ aequatur Rectangulo
QZ FQPXQZ—QP,; aequatur etiam A g ipfi

AB per Art. 109. ergo Rectangulum QZ+QP

XQZ — QP acquatur Quadrato ipſius A B, ergo
Rationes QZQP ad AB, AB ad Q-,
| V3. et

214 Nota in LocoMeTRIAR, Prop. 4.

et ſubduplicata Ratio QZFQP ad NZ —

eaedem ſunt. Porro, eſt AB ad QZ ut C A ad

-CQ, et AD ad Q ut CA ad CQ ergo eſt
AB ad Q ut AD ad QP, et permutando A B

ad A D ut QZ ad QP, et permiſcendo A BAD

ad ABA D ut QZ – QP ad QZ — QP; ergo
Rationes omnes QZ QP ad AB, AB ad

QZ —QP, ſubduplicata Ratio QZ ＋ QP ad

QZ—QP, et ſubduplicata Ratio A B AD ad
AB – AD acquantur inter ſe : Haſce Rationes

pricres nominabimus.

Rurſus, Rectangulum RPA-RX x R P—RX

aequale eſſe Quadrato ex CA ex iiſdem Arti-
culis conſtat, quibus oſtenſum eſt Rectangulum
QUZFQPXQZ—QP acquale eſſe Quadrato
ipſius A B; quare eſt RP RX ad A Cut AC
ad RP RX, quare Rationes omnes R P–R X
ad AC, AC ad RPR X, et ſubduplicata Ratio
RPR X ad RP—RX aequantur inter ſe; haſce
pofteriores nominabimus. Reſtat ut oſtendamus
Rationem AE ad PF aequalem eſſe Rationibus
ſingulis tum prioribus, tum poſterioribus, et prop-
terea Rationes ſingulas tum priores tum poſteriores
acquales eſſe inter ſez id vero fic evincitur:

Triangula AEB, PF 2, ſimilia ſunt propter

Paralleliſmum laterum, adeoque eſt AE ad PF

ut AB ad PZ, vel AB ad QZ—QP, ſed Ratio

AB ad QZ—QP ͤ una eſt ex Rationibus prioribus,

ergo Rationes omnes priores aequales ſunt Rationi

ipſius AE ad PF. Rurſus Triangula AEC,

„

. co HH oo cnc 2M.sLgD

A OO

2 2 A

5

Nota in Loco T RIAN, Prop. 4. 215

PF X etiam ſimilia ſunt propter Paralleliſmum
laterum, ergo eſt AE ad PF ut A Cad PX;

ſed Ratio AC ad PX una eſt ex Rationibus

poſterioribus, ergo Rationes omnes poſteriores
aequales ſunt Rationi AE ad PF, et Rationes

omnes tum priores, tum poſteriores, aequantur

inter ſe. Haec de Rationibus.

Oſtendendum ſecundo Sectorem Hyperbolicum

CAP aequalem eſſe Spatio mixtilineo AE F P,
quod fic demonſtrari poteſt. Triangulum C A E
aequatur Triangulo CPF, per Art. 99, priori
Triangulo CAE, addatur Triangulum A E B, et
fiet Triangulum C A B, ſubducatur Triangulum
CD ; et relinquetur Triangulum C A D, addatur
Trilineum D A P, et conficietur Sector Hyper-
bolicus CAP: Alteri Triangulo CP F ſubdu-
catur Triangulum CDB et manebit Trapezium
DBFP, adjiciatur Trilineum DAP et fiet
Figura mixtilinea A BF P, quae adſciſcens Trian-
gulum AE B, evadet Figura mixtilinea AEF P.
Ergo Sector Hyperbolicus C A P aequatur Spatio
AEF P cum ex aequalibus Elementis tum addi-
titiis tum ſubductitiis componatur utraque Figura.
Poſtremò oſtendendum Sectorem Hy perbolicum
CAP menſuram eſſe Rationum aequalium prae-

‘ finitarum ad Modulum aequalem Triangulo CAB;

id vero conſtat ex priori Parte hujus Prop. Nam

Spatium AE FP eſt menſura Rationis inter A E

et PF ad Modulum aequalem Parallelogrammo,
quod duplum eſt Trianguli CAE. Eſt autem
Spatium A E F P aequale Sectori C AP, et Trian-

| P 4 gulum

= — gs — — —

216 In Scbol. Prop. 4. LocoMeTRIAE..
gulum C A B duplum eſt Trianguli C AE, prop-

terea quod Longitudo C B dupla eſt Longitudinis
CE, per Art. 107. Quare Sector Hyperbolicus

CAP menſura eſt Rationis inter AE et P F, &c.
ad Modulum aequalem Triangulo CAB. Q. E.D.

In Scholium ad Propofitionem Quartam
L0GOMETRIAE Fig. $6. ©
LEMMA.

Si in dato aliquo Syſtemate fit 5s Menſura Ra-

tionis alicujus ad Modulum S, fueritque in alio

quovis Syſtemate 7 ad T ut 5s ad S, erit ? Menſura
Rationis ejuſdem ad Modulum T. Nam in omni
Syſtemate Modulus et Menſura Rationis ejuſdem
ſunt proportionales. Hoc praemiſſo, pergo ad alia

quae hoc in loco conſideranda veniunt, et fic ſe

habent. Solverat Newtonus Problema de Deſcenſu
Gravium, quibus reſiſtitur in duplicata ratione

Velocitatis per Conſtructionem Hyperbolae, cujus

vis aequilaterae A T, Centro D et vertice Princi-
pali A deſcriptae, et capiendo Sectorem D A T

ad dimidium Trianguli DAC ut f ad T: Verum

cum iſtiuſmodi Sector Hyperbolicus deſumi nequit
niſi prius habeatur Punctum P, per quod Semi-

diameter altera D T tranſire debet, ut abſcindat

aream propoſitam, cumque poſt inventum Punc-

tum P, nihil opus eſt, ut Problema ulterius re-
feratur ad Hyperbolam, itaque Cotęſus Conſtruc-

tionem Nerotonianam de Hyperbolis ad ſimplicem

Lineam rectam more ſuo tranſtulit in hunc modum,.
| | Area

77ͤͥͤ ets Shot” tc. 20nd

1

n n ww cd 8 —

od

In Scbol. Prop. 4. LocoMETRIAE. 217

Area Sectoris DA T Menſura eſt rationis ſab-
duplicatae inter A CAP et A C—A P ad Mo-

dulum aequalem Triangulo DAC, vel eſt Menſura
Rationis illius Integrae ad Modulum DAC;

itaque cum fit ex Hyp. f ad T ut Menſa Ra-

tionis inter A CAP et AC—AP ad Menſurae

Modulum +4 DAC, erit per Lemma praemiſſum, .

Menſura Rationis inter A CA P et AC—AP

ad Modulum T: Fiat ergo ut T ad ? ita Modulus
Briggianus ad quartum, ſitque quartus ille .Lo-

garithmus Fractionis – 3 t ent AC AP ad

AC — A Put Rad S, et prodibit APA Cx

R—S
R * Quare fi exponatur recta quaevis A C et

ita abſcindatur longitudo AP, ut fit ad AC ut
R—S ad RS, recta illa A C ita ſecabitur in P,
ut f fir Menſura Rationis inter AC+AP et
AC—AP ad Modulum T. Invento jam hoc
modo Puncto P, reliqua facili Negotio eruuntur,
ex Propoſitionibus octava et nona Libri ſecundi
Principiorum; erit enim v ad U ut AP ad AC,

item 7 ad R ut AP* ad AC*; vel fi capiantur

AC, AP, A K continuae Proportionales, erit 7
ad Rut AK ad AC ; denique Spatium s erit

Menſura Rationis inter A Cet C K, nam ex Pro-

poſitionibus modo citatis, eſt s ad S ut Area altera
Hyperbolica AB N K ad Rectangulum CK KN,
, e, ut Menſura Rationis inter C A et C K, ad
Modulum Syſtematis.

Haec

218 In Scbol. Prop. 4. LoGoMETRIAE.

Haec utique eſt Conſtruftio Cotgſiana ſimplici
Linea recta ut ſupra diximus expoſita ; ſed et illud
tamen obiter notandum, fi harum Demonſtrati-
onem aggreſſus eſſet Cotęſius, id ex notis Geome-
triae Vulgaris principiis minime praeſtari poſſe, niſi
adhibitis Hyperbolis Newtonianis ; unde ſatis liquet
Conſtructionem hanc Cotgſianam nihil aliud eſſe

quam Newronianam illam ad praxin deductam; ſi-

cut et ipſe Newtonus eandem ad Numeros reduxit,

aquam eo Loci perventum eſſet, ut ſolveretur
Problema ex Phaenomenis. Vide Prop. quadra-
1 Libri enn Principiorum.

In dicti Scholii e dp de R ESIS-

TENTIA in ASCENSU.

LEMMA.

Arcus Circuli 57,3 Graduum fere equalis eſt
Radio, nam Circumferentia Circuli eſt ad Semi-

diametrum ſuam ut 355 ad = vel ut 710 ad 1 13,
vel ut 360 ad 57,3 fere.

CoROLLARIUM.

Hinc fi detur Numerus Graduum in Arcu quo-
libet, invenietur Longitudo Arcus in partibus
Radu, dicendo ut 57, 3 ad Numerum Graduum
datum, ita Longitudo Radii ad Longitudinem
Arcus: Et e converſo, fi detur Arcùs Longitudo et

quaerantur Gradus, dic ut Longitudo Radii ad

Longitudinem Arcùs datam, ita 37, 3 ad „
Graduum quaeſitum.
C aetera

„

In Sobol. Prop. 4. LocoMETRIAE,. 219

Caetera quod attinet animadvertendum, Prima
Si detur v Velocitas prima Corporis aſcendentis,
et quaeratur tum reſiſtentia prima 7, tum tempus ,
et Spatium s totius Aſcenſus, accepta utcunque
Longitudine EG, Fig. 51, ut et huic aequali et
perpendiculari G O, Capiatur G F vel G f ad GE
ut v ad V, ſitque G h ipſis G E, G F tertia pro-
portionalis, ſed ad partes contrarias ſita; et erit
ad R ut Gh ad GE. Jungantur OF, Of et
ex datis OG, G F, quaeratur Angulus FOG,
ut et hujus duplus F Of; et invenietur Tempus ,
dicendo ut 57,3 ad Angulum FOf, ita T ad ❓
Invenietur denique Spatium s, fi fiat ut Modulus

Briggianus ad Logarithmum Briggianum Fractionis
E

FC ita S ad g. Secundo, Si detur r et quaerantur

reliqua, fiat G ad GE ut r ad R, et habebitur

Punctum , Capiatur GF vel Gf media propor-

tionalis inter G h et G E, et habebitur Punctum F;
quibus datis reliqua elicientur ut prius. Tertio,
Si detur ſolum Tempus ? et deſiderentur reliqua;
fiat ut T ad t, ita 57,3 ad Angulum FOf, et
hujus dimidium FO G etiam innoteſcet; dato
autem Angulo FO G et Radio OG, inveniatur
Tangens G F, et dabitur Punctum F, et reliqua
omnia ut ſupra.

Poſtremo, Si detur folum Spatium s, fiat ut O
ad s ita Modulus Briggianus ad Logarithmum

Fractionis wn et erit EH ad G E ut R ad S unde

dabitur Punctum , et reliqua omnia.

C@R 0 L-

220 In Scbol. Prop. 5. Lo ‚ RAE.

.

Ex jam Demonſtratis conſequens eſt, ſi Corpus
e quiete demittatur, Deſcenſurum illud in Aeternum
et per Spatium infinitum, prius quam poſſit ac-
quirere Velocitatem, Velocitati finitae V aequalem ;
et ſi. ſurſum pro jiciatur cum Velicitate infinita,

Aſcenſurum illud per Spatium infinitum prius

quam ad Altitudinem Summam pervenire poſſit,

cum tamen Tempus totius Aſcenſus finitum ſit, et

ad Tempus T ur Peripheria Circuli ad Diametrum.

In partem poſt: -rtorem Scholii ad Prop. qu: ntam
LoGOMETRIAE.

LEMMA.

Si duae Quantitates Homogeneae et indeter-

minatae A et B ita inter ſe comparatae fuerint ut
fit A ſemper ut A—B, erit etiam A ſemper ut B;

mutetur enim A in 4, et ſimul B in 54, et cum fit

ex Hyp. Aut AB, 5. e. cum fit A ad A—B,
ſemper in data Ratione, erit A ad AB ut à ad

a—b, et Dividendo A ad B ut aad5; 5. e. erit
A ut B. Q. E. D.

In parte hujus Scholii poſteriore, Solvitar Pro-
Blema de Denſitate Aeris ad quamvis Altitudinem
invenienda, ubi Vis Gravitatis diminuitur in du—
plicata Ratione auctae Altitudinis a Centro Terrae;

et certè Solutio hic allata perquam ſimplex eſt et

elegans; haud tamen inducar ut credam demon-
ſtrationem hujuſce Solutionis aut adeo concin-

• natam

In Schol. Prop. 5. LocoMRETRIA E. 221
natam eſſe, aut Tyronum mentibus accommoda-
tam, ut ulteriori illuſtratione non ſit opus; hanc
itaque pro Modulo meo aliquanto dilucidatiorem
dare conabor, et ſumma _ poſſum brevitate
expediam.

Concipe ergo Columnam Aeris Cylindricam a
Superficie Telluris ad Summam uſque Atmo-
ſphaeram pertingentem, et, ſervata Auctoris con-
ſtructione, Fig. 52. cum fit SF ad SA ut SA ad Sf
erit S A aequali Rectangulo S Fx S/; ſed et idem
SA atque eadem Ratione etiam aequatur Rectan-
gulo SMx Sm: Quare Rectangula SF xSf et
SMxSm, aequantur inter ſe, eſtque SM ad SF
ut SF ad S m; et dividendo, FM ad SF ut fm ad
Sm vel Sf; unde viciſimeſt FM ad fm ut S F ad
Sf, ſed propter SF, SA, Sf, continue propor-

tionales, eſt SF ad SF ut SF* ad SA*, quare FM
eſt ad fm ut 8 F. ad SA, eſtque fm aequalis

• X FN, unde cum detur S A- eee, 4

9 Mxfg9 IF
et Fm, g, live Areola fg n, ut = : 5

FMxfg

|
ut Denſitas ejuſdem et Vis Gravitatis in Loco F j
quae ſimul ſunt ut pondus Aeris inter Feet M;
quare pondus Aeris inter F et M eſt ut Area
Fgum. Rurſus, denſitas et preſſio Aeris in Loco
F exponitur per Ordinatam fg, et in Loco M,
per Ordinatam u, ergo differentia preſſionum in
Locis F et M. exponenda eſt per differentiam
Ordi-

eſt ut moles Aeris inter F et M, atque

222 In Schol. Prop. 4. Loco rf TRIA R.
Ordinatarum nempe per Fg ; ſed haec dif-
ferentia nihil aliud eſt quam pondus Aeris inter
F et M; quare pondus Aeris inter F et M eſt
ut Fg . Sed oftenſum eſt et hoc pondus eſſe
ut Area fg nm, quare Area fg nm eſt ut Fg n.
Haec ita ſunt univerſaliter : Quod ſi detur Dil-
tantia fm, erit Area fgnm ut fg, quare in hoc
Caſu erit fg ut fg—mn, et proinde etiam Fg ad
mu, ſemper in data Ratione; ſicut in praecedenii
Lemmate. Exponat jam data illa diſtantia f
datam hanc Rationem inter fg et u, et Summa
omnium Fm exponet ſimilem Summam Rationum

omnium 5 ea eſt itaque indole haec Curva ut
binarum quarumlibet ordinatarum intervallum.
Menſura ſit Rationis inter ipſas Ordinatas, quae
eſt notiſſima proprietas Logiftica : Eſt ergo Curyi
B g Logiſtica, Dico et eandem eſſe cum priore
B G, ſed inverſo ſitu; quod ſic oſtendo. Accedere
intelligantur Puncta F ad Punctum A in utraque
Figura, ita tamen ut Longitudines A F, AF,
ſemper ſint invicem aequales, et coeuntibus F et A

fiet Lineola AF aequalis ipſis A F, etenim cum fit

SF ad SA ut SA ad Sf, et dividendo A F ad
S A ut A ad Sf, erit viciſſim A F ad Af ut S A
ad Sf; b. e. ultimo in Ratione aequalitatis;

unde cum Subtangens Curvae B G fit RNAS

ABxXAFf

et Subtangens Curvae B g ſit XB

erit Sub-

4 | : : – tangens

BFG.

tan

OT

2

1

De Areis comparandis THEOREMATA duo. 223
tangens S B G ad SR Curvae Bg.

r A BFU ad KEF. 2 |
Porro, preſſio Aëris et Vis Gravitatis in Loco
A eaedem ponuntur in utraque Hyp. et cocuntibus

A et F Vis Gravitatis inter A et F in ſecunda

Hypotheſi pro uniformi tuto haberi poteſt ; quare
ndus Aeris inter A et F, idem eſt in utraque
Hypotheſi; unde et horum ponderum, exponentes,

numerum ABF G in prima Figura et A Bf

in ſecunda ſunt aequales; ergo Curvarum Subtan-
gentes etiam aequales ſunt ; quare Figura B g,.
eadem omnino eſt cum Figura BG; harum enim
Conſtructiones ex ſolis Subtangentibus pendere,
conſtat ex praecedentibus. Q. E. DP).

Theoremata duo pro comparandis Areis curvi-
lineis for mae ſimplicioris.

Sir P Area Curvae cujus Abſciſſa eſt ⁊, et
ordinatim, applicata 2˙¹wv x e+f 2%, ſit etiam
T vel T’ Area Curvae 1 eadem eſt A bſciſſa zz

ordinatim vero Applicata z8:= N f=,

poſito v pro Numero quolibet integro; quaeritur

Relatio inter P et T, vel inter P et T.

8OLlUur Tro.
an 9＋ . N (AA ll en, e En

Fiat . == 2 8 2 7, 2
— TAT „
7 me”

_ © SB. + AIG P GI
Wear wad 1 ——

n 7

224 De Areis comparandisTu FOREMAT A duo

modo fit s ultimus Terminus Seriei „„

quorum tot ſumendi ſunt quot ſunt Unitates in c,
fiat etiam Regreſſus a Termino p, ad Terminos
PD 5 2 , 5, , fed mutatis omnium Signis; B. e.

. e e 2
——— x FEI.

_ can to A flat eti: 25 9
1 *. =P. Rune 2 95
— e OT — 4 e “Ss, E

z
„ „et — & -A, fi modo

ſit 5′ ultimus Terminus Seriei 97% % quorum

etiam tot ſumendi ſunt quot ſunt unitates in .
His poſitis eritP=p g Ar ö Fa- D

Zh D* ＋ T, ſi modo habeatur — in Valore

1 ſin habeatur — c, erit P e þ 7 2 — .

7 r r 58 15 XE\

N. B. Si fiat 0 A4 ett endnas

= * AF ee Xe AT
Concinnari etiam poteſt Theorema ſecundum

I „ — e
in modum ſe uentem; fiat — =
| 1 6 * —1 5 7
9—1 : i 9—2 .
7, — AN

Ea TEE “7

timus Terminus Seriei 25 4 T, 5, quorum tot
ſumendi

SE =2″, fi modo fit 5 ul-

= „ „

De Areis comparandis THROR EMA A duo. 22 5

ſumendi ſunt quot ſunt Unitates in , et erit

Py 3 3 * P
PSN 2 292 20 —35 * and ＋ T’.
u. :
Demonſtratur Theorema primum per Aequati-
ones a nobis alibi traditas et demonſtratas; nam fi

ſcribantur Q, R, S, pro Areis intermediis inter P et
T, et y pro , erit primo Xe PIN
„n · x&. ſecundo Aix Q+8+a+ix

FRM zh box? tertio 6 2 XeR +3 +Þa+Þ2x

TS Na XZ, quarto 0+ 3 x eS4+6FXF 3X
1

fFT=2) +5 x” : harum Aequationum ope exter-

minenturFluentes intermediae Q. R, S, et elicietur
Aequatio exhibens Relationem inter P et T ut
ſupra. Porro demonſtrari poteſt Theorema ſe-
cundum, vel ex priore vel ad modum prioris.

8 COR OI. I. |
Si Series alterutra abrumpatur propter Terminos
evaneſcentes vel infinitos, dabitur Valor Quanti-
tatis alterius P vel T, vel Quantitatis alterius P
vel T’ abſque ulla relatione ad alteram, praeſertim
poſt diviſionem Aequationis totius . eee

infinitam ſi qua in Acquatione exiſtit.
Cano
Quod ſi Quantitas T, vel T’ abeat in infinitum,
prodibit Series Newtoniana pro inveniendo Valore

Quantitatis P, accurate quidem, fi abrumpatur; vel
per Approximationem, ſi procedat in infinitum.

. APR o-

226 QvarrATURE of @

A PROBLEM.

To meaſure the Quantity of Surface generated by an
Arc of the Rectangular Hy p t & BOL a revolving

about one of its 3 See Harmon. Men- |

nad p. 94.

SOLUTION.
12 1m* be the Power of the Hyperbola ferred

by perpendicular Ordinates to either of its Aſymp-

totes, and let y repreſent indifferently any one of
theſe Ordinates whoſe Abſciſs or Diſtance from
the Center let be x; and laſtly, let the Diameter
of a Circle be to its Periphery as 1 to p, then will

2 DyXx/x+y be the Fluxion of the Surface

ſought; or, dropping p till the Concluſion, the
Fiuxion will be 25 x i z, out of which either

x or y muſt be expunged by the Nature of the

Hyperbola. Let x be expunged thus, According
to the Nature of the HYPO, r there-

_
fore x? D -, 7 ” and x2 = N — 5 , there

fore N TN. 5 1 = 7. Ge.

becauſe y flows . to our Surface, whole
Fluxion was put affirmative: Therefore 29 x

• EN = —29 N nN. But this is a Fluxion
  of the third Form, where bo, y=4, d= —2,
  g 222 f= I, PN 9, R = mm,

T=

in che

Lill be EDA BC|

HYPERBOLIC SURFACE, 227

1 Nahe D; therefore 2 1p—24R
RT
8

Vu D n [= _ ——_ : But from the

Nature of the Hyperbola, * , and mi=yx3,
and -T pe, and N
ſubſtitute therefore y x V inſtead of Vn
in the foregoing Fluent, as alſo x y inſtead of m*

=o m + 5,

„ or the whole Fluent wake! will be

„and the Fluent will

ſtand thus, —y X TIT XITIXN x2 TP

v5

• wrote? | — _ .

Now to conſtruct this Fluent, let A be the

Center, and A B the Aſymptote referred to (See

Cotes, Page 94.) let B C be a fixt Ordinate, and
DE a moveable one, comprehending between
them the generating Arc C E, and the Line AE

when drawn will be YK An, ſince A D=x and

D E); therefore XV – is the Rectangle

AED, or AE xXxE D; moreover the Modulus ma
will in this Cale be the Rectangle A BC, therefore
according to this Conſtructions the general Fluent
DA-+AE E.
| DE

D coincided with B, this Fluent was = &CR

But when

| 4 ABC B 2 – . therefore ſo much of the

Q 2 Fluent

22:8 QvaDrATURE, &c.
Fluent as makes for our Purpoſe will be AC B

| DATA E B B A+AC 5
AE PPA BC DE —ABC __ 7
Now, for a more convenient Subtraction, let the

Rectangles AED and ACB be . to a
WER E

— Si one common “MV _

common —_ as well as s the Ratio *
and B 2

Let hs 1 AE cut BC in F, and we ſhall
have, from the Nature of the Hyperbola, B C to
D E as AD to A B, or as A E to AF; therefore
by multiplying Extremes and Means we have
AE DAF XB C; therefore AC B- AED, or
the rational Part of the Fluent, =AC—AF x BC.

Again, draw C G parallel to A E, and the Ratio

DA+AE | B -O
—_ E will be equal to the Ratio .

and the Exceſs of the Ratio — 5 ©: above

the Ratio 8 will now be the Exceſs of

B AAC

1
B G4
BA-AC’

the R atio B oe — L the Ratio

which Exceſs is equal to the Ratio

and the Fluent will ſtand thus, A C—A FxBC

B GG C
+A BC b AAC
BG4GC _
BAFAC

4 ü ing

Make A C—AF4A B

= A; and then the F luen (reſtor-

om. ²˙ inn ond ꝗ ‚—ͤAͤṼ es. be Ss ws

tn th. OO

ASYMPTOTIC SURFACE, | 229
ing p, that was ſet aſide) will be pxBCx AH,

e —
or 9x BC X BC but px B C? is the Area of a

Circle whoſe Semi-diameter is B C, therefore the
Surface ſought is to the Area of a Circle whoſe

Semi-diameter is BC, as AHtoBC. Q.E.D,

A THEOREM. Fig. 54.

If the Space between any Curve and its Aſymptote be
ſuppoſed to turn round that Aſymptote, ſo as to
generate a Tapering Solid of an infinite Length;
T ſay, that the Surface of this Solid will be finite or
infinite in Quantity, according as the Quantity of
the Space, that generates this Solid, is ſo.

Let the ſtreight Line ACDF be an Aſymptote
to the Curve B E, and let the Space betwixt them
be terminated at one End by the Line AB per-
pendicular to the Aſymptote A F, let DE and d e
be any two neighbouring Ordinates parallel to
AB, and let BC and E F be Tangents to the
Curve in the Points B and E; laſtly, let g be the
Circumference of a Circle whoſe Diameter is DE :

Then if we ſuppoſe the Ordinate de to approach
nearer and nearer to the Ordinate DE, and at laſt

to coincide with it, the Fluxion of the Surface at
E will be to a correſpondent Fluxion of the Space

ADEB in the ultimate Ratio of g X Ee to

DE x Dd; but this Ratio is compounded of two

others, viz. that of g to D E and that of Ee to
Da; che firſt of theſe Ratio’s, viz. that of g

Q3

to

|
F
|
t

230 When finite, when infinite.
to DE will always be finite and the ſame, being
the Ratio of the Circumference of a Circle to the
Radius, the other component Ratio, viz. that of
E e to Dd tends to a Ratio of Equality; becauſe

the nearer the Curve approaches to its Aſymptote,

the nearer will the Lines Ee and Dd approach
towards a State of Paralleliſm and Equality :

Therefore the Ratio of Ee to D d or of EF to DF
will always be leſs than that of B C to AC; but
the Ratio of BC to AC will always be Wer
when the Ordinate A B cuts and does not touch
the Curve; therefore, a fortiori, the Ratio of Ee
to D 4 muſt be finite; therefore every Fluxion of
the Surface will be to a like Fluxion of the Aſymp-
totic Space in a finite Ratio, ſince this Ratio will
always be the Sum of two others which are both
finite ; therefore the whole Surface will be to the
whole Space in a finite Ratio; therefore the Surface
muſt be finite or infinite in Quantity, according
as the Quantity of the Space that ne, the
Solid, is ſo. . D. .

„

In PRHIL Os. NEVUTON. SCHOLIA. 231

..

V. C..
NICOLAI SAUNDERSON

I N
PHILOSOPHIAM NEUTONIANAM _

S CH O

PROBLEMA ad illuſtrandum CoRor. II. ad
| Leges Motis. F ig. 58.

X Punctis M et N i in Plano quovis ad TEEN
Zontem normal: ſuſpendantur Corpora A et p,
ita tamen ut Corpus A liberè dependeat, et Cor-
pus p incumbat Plano dato G; guaeritur Ratio
Ponderum A et p ita comparatorum ut ſe mutuò in
Aequilibrio ſu ſuſtineant. Oportet autem ut innoteſcat
Ratio Diſtantiarum minimarum O K, ox, Filorum
MA, Np a Centro O, et Ratio Rectarum N p,
pH; poſità HN ad 2 G normali.

Filo NP ſupponatur Trochlea Z, dein tollatur
Planum p G, ita ut Fili Pars N Z eandem ſervet
Poſitionem quam prius; pars autem ZP libere

dependeat; et Corpus P jam toto ſuo Pondere
tendet Filum NZ. Sit Corpus A 20 Librarum,
ſitque minima Diſtantia Centri O a Filo M A ad

Q 4 minimam

N

232 In Pu11.0s. NzUToON.

minimam Diſtantiam ejuſdem Centri O a Filo
NZ ut 3 et 4, et fi Pondus P ſuſtineat Pondus

A in Aecquilibrio, erit hujus Pondus 15 Librarum,

propterea quod fit 20 ad 15 ut 4 ad 3. Reſtitua- |

tur jam Corpus P in Locum ſuum priorem, ni-

mirum ut incumbat Plano p G, et Tenſio Fili

minuetur hac Tranſlatione in Ratione p H ad pN.
Sitp H ad ↄ N ut 6 ad 5, et Tenſio Fili Ny jam
minuetur in hàc Ratione; ergo Pondus P jam non
ampliùs in Aequilibrio ſuſtinebit Pondus A, niſi

tanto augeatur Pondus abſolutum Corporis P,
quantd diminuta erat Tenſio Fili per Corporis p

Tranſlationem : Quare ut Corpus P ſuſtineat Cor-
pus A in Aequilibrio, oportet ut Pondus ejus
abſolutum 15 Librarum augeatur in Ratione

ad 6, vel in Ratione 15 ad 18. Quare fi Pondus

A fit 20 Librarum, oportet ut Pondus Corporis p
Plano p G incumbentis fit 18 Librarum; hic
enim Ratione Corpora A et p in Aequilibrio con-
ſiſtent ; et Ratio 20 ad 18 componitur ex Ratio-
nibus 20 ad 15 et 15 ad 18, hoc eſt, ex Ratio-
nibus 4 ad 3, et 5 ad 6.

Proprietates

Ny:

SCHOLIA 233

Proprietates quaedam Centri GRAvITATIS.

THEOREM A, Fig. 56.
Si Corpora quotcunquèe (utcunque ſita inter ſe) A, B, C,
Viribus quibuscunque de Locis ſuis A, B, C, fimul
impellantur, commune eorum Gravitatis Centrum
perinde movebitur, tum quoad Metis ſui Deter-
minationem, tum quoad Velocitatem, atque Corpus
Unicum AB A-C ex omnibus Compoſitum et in
commune omnium Gravitatis Centro conſtitutum
i modo Corpus illud compoſitum iiſdem quibus

componentia, Viribus ſimul impreſſis, et fimiliter
determinatis, impellatur. –

Eſto enim S commune omnium Gravitatis
Centrum, et excluſo Corpore A, fit T commune
Gravitatis Centrum reliquorum B, et C; et Cor-
poribus in Locis A, B, C, primò quieſcentibus,
ut et Centro S in Loco E; transferatur poſtea |
( reliquis quieſcentibus) Corpus A in Locum a,
atque adeo Centrum S de Loco E, in Locum
quendam F, ubi rurſus quieſcant, et Punctum F
erit in Recta Ta, ex Natura Centri Gravitatis ;
erit etiam Recta EF, Rectae A a parallela; ex eo
quod Rectae T A, T a ſimiliter ſecantur in Punctis
E et F; Denique cum ſit ex Natura Centri Gra-
vitatis, T E ad E A, ut A ad B+C; erit com-
ponendo TE ad TA vel E F ad Aa ut A ad
A4+B+C. Transferatur jam Corpus B in Lo-
cum b, et ſimul Centrum S de Loco F in Locum G,

ubi

234 In PHIL OS. NRUTOx.

ubi rurſus quieſcant; et Recta FG parallela erit
Rectae B 5, eritque ad B þ ut Corpus B ad ſum-
mam Corporum ABC. Poſtremo transfe-
ratur Corpus C in Locum c, et ſimu] Centrum S
de Loco G in Locum H; et Recta G H parallela
erit Rectae Cc, eritque ad C c ut Corpus C ad

ſummam Corporum AB C; Reſtituantur
jam Corpora in Loca ſua prima A, B, C, et Viri-
bus quibuſcunque de his Locis ſecundum direc-

tiones, et cum Velocitatibus Aa, Bb, Cc, ſimul

impellantur, nempe ut ad Loca a, 5, c, eodem
Tempore 7? ſimul appellant; et Corporum in his

Locis ſimul exiſtentium, commune Gravitatis

Centrum, erit in Loco H ut ſupra definito.
Fingatur jam Corpus Unicum D quod aequale
ſit omnibus A, B, C, ſimul ſumptis, er ſtatuatur
hujus Centrum in Loco E; et fi buic Corpori
(hoc eſt Centro ejus) imprimeretur ſola vis motrix
Corporis A, ob ſuppoſitam Virium Aequalitatem,
et ſimilem Determinationem, Corpus illud D

moveretur in Recta EF cum Velocitate quae

foret ad Velocitatem Corporis A, ut Corpus A
ad Corpus D, vel ut EF ad Aa; ergo Vis

motrix Corporis A Corpori D impreſſa efficeret ut

hocce percurreret Longitudinem E F, quo Tem-
pore Corpus A percurrebat Longitudinem A 4
(id eſt Tempore t); et pari Ratione ſola Vis mo-
trix Corporis B ipfi D impreſſa, efficeret ut Cor-
pus illud ex E decedens percurreret Longitudinem
aequalem et parallelam ipſi FG eodem Tempore t;
et ſola vis motrix Corporis C impelleret CorpusD,

per

‘»

W ˙ T Ver WOT

SCHOLIA, | 235
per Longitudinem aequalem et parallelam ipſi GH
Tempore praefinito ?: Quaxe Vires omnes ſimul .
et ſemel impreſſae, efficient ut Corpus D percurrat
Longitudinem E H Tempore t, ita ut in fine
Temporis illius reperiatur in H Loco communis
Centri S; erat autem Tempus f pro Arbitrio
aſſumptum: Quare ſi ſtatuatur Centrum Corporis
D in Centro communi totius Syſtematis A, B, C,
et Corpus illud iiſdem quibus Partes Syſtematis
Viribus ſimul impreſſis et ſimiliter applicatis, im-
pellatur, erit hujus Centrum ſemper in Centro
communi et idem erit motus utriuſque. Q. E. D.

Con ot.

Si Corpora A, B, C, moveantur uniformiter in
direQum, five motus illi fiant in eodem Plano, five
non, Centrum Corporis D, et proinde etiam com-
mune Gravitatis Centrum vel quieſcet vel move-
bitur uniformiter in Directum.

Corort” It

Longitudines EF, FG, GH, HE, ſunt ut
motus Corporum A, B, C, D: nam quo Tempore
Corpus A percurrebat Longitudinem A a, Corpus
D percurrebat L,ongitudinem E H; ergo Velo-
citas Corporis D, eſt ad Velocitatem Corporis A
ut EHad Aa; et motus ad motum, ut DXE H
ad Ax A a; erat autem E F ad Aaut A ad D,
quare factum Dx E F aequatur facto Ax A a; et
motus Corporis D, jam erit ad motum Corporis
A ut DX E H ad Dx E F, ſeu ut E H ad EF;

unde

236 In Pnitos Nx ToN.

unde fi Longitudo E H referat motum Corporis
D, Longitudines E F, F G, G , pariter referent
motus Corporum A, B, C.

Coror. III.

Hinc datis Motibus Corporum quotcunque
A, B, C, dabitur Motus communis omnium Cen-
tri Gravitatis; incipiendo nempe à Puncto quovis
E et ducendo rectas EF, F G, G , exhibentes
Motuum illorum et Quantitates et determinationes;
nam recta E H pariter exhibebit Motum totius
Syſtematis, ſimpliciter cum Velocitate communis
Centri Gravitatis lati, et quo Tempore Corpus A
percurrit Longitudinem, quae ſit ad E F ut D ad
A, commune Centrum Gravitatis percurret Lon-
gitudinem aequalem et parallelam ipſi E H.

CoR ol. *

Si Punctum ultimum H cum primo E coinci-
derit, Indicium erit commune Gravitatis Centrum

quieſcere.

| c aok. V.
Quemadmodum Longitudo E H refert Motum

Centri communis totius Syſtematis A. B, E, Lon-

gitudo E G referet Motum Centri communis

partium A et B, et Longitudo FH Motum Centri

communis partium B et C; et ſi compleatur Pa-
rallelogrammum F G St I, Longitudo E] pariter
exhibebit Motum Centri communis partium A
ct _ – Haec ita ſunt ex Hypotheſi quod Lon-
| 9

Ad = £4ADO

as Ss AHA oi,

SCHOLIA. 237

gitudines E F, F G, GH, referant Motus ipſorum
FA ,. reſpective.

• Corot. VI.
  Actiones extrinſecùs in partes Syſtematis A, B, C,
  non ſecus afficiunt Motum vel quietem Centri
  communis, quam ſimiles et aequales Actiones in

Centrum Corporis D exercitae afficiunt hujus

Motum vel Quietem ; neque refert quicquam in
quas partes Syſtematis impetus fant, modo virium
Quantitates et Determinationes eaedem maneant.

S VII.

Quare vires aequales et contrariae, in partes
Syſtematis ſimul impreſſae, nihil perturbabunt
Motum vel Quietem Centri communis, ſive vires

illae in eaſdem ſive in diverſas partes Syſtematis
imprimantur.

Cox or. VH.

Unde cum Actioni ſemper aequali et contraria
ſit Reactio, ſtatus Centri communis vel quieſcendi
vel movendi uniformiter in Directum ab Actio-

nibus partium Syſtematis inter ſe nullatenus per-
turbabitur aut mutabitur.

Cakot 1K

Eft itaque eadem lex Motus Corporum duorum
vel plurium quae Corporis ſolitarii; ſiquidemMotus
Syſtematis pariter atque Corporis ſolitarit ex Motu
Centri Gravitatis aeſtimetur, perſeverabit hoc in

ſatu Jos quieſcendi vel movendi uniformiter in

Directum

2
Fre *
_————

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238 In PHILOS. Neu ToON,

Directum, niſi quatenus a Viribus extrinſecus im-
preſſis cogitur ſtatum illum mutare. |

His adjungere liceat (ne alias pereat) Solutionem

non invenuſtam ſequentis Problematis, cujus qui-

dem Demonſtrationem ut ſatis obviam, vel ſaltem

ex praeccdentibus haud diffculter deducendam
praetereo.

PROBLEMA.

Poſitis omnibus ut in praecedenti propoſitione,
Corporibus A, B, C, in redtis Aa, B, Cc, uni-
formiter delatis, imprimantur Vires novae, quarum

et Determinationes et Quantitates exponantur per

Rectas LI, Mm, Nu, ita ut Vis L / ſit ad Vim
inſitam quo Corpus A prius movebatur ut Recta
LI ad Rectam EF; et lic de caeteris, quaeruntur
Motus reſultantes?

898 1 vr
Conſtructà ut in Corol. tertio ſuperioris Pro-
poſitionis, Figurà E F GH fit P locus Puncti E;
quo notato transferatur Figura tota, ſecundum
Determinationem LI, ita ut ſingula ejus Puncta
percurrant Longitudines aequales et parallelas
Rectae LI, et conſtituta Figura in hoc ſitu, notetur
Q locus Puncti F: ex hoc loco transferantur rurſus

Figura ſimiliter ſecundum Determinationem Mn;
dein ſiſtatur ut notetur R locus Puncti G; hin
denique transferatur Figura ſecundum Determina-
tionem Nx, et notetur 8 locus Puncti H, et jun-
_ pa PQ, QR, RS, SP; dico has
Rectas +

SCHOLIA. 239
Reftas rite exhibere et Determinationes et Quan-
titates Motuum reſultantium: h. e. Corpus A poſt
Actionem Vis L movebitur ſecundum Determi-
nationem P Q cum Motu et Velocitate quae erunt
ad Motum et Velocitatem priorem ut PQ adEF;
et fic de reliquis. Movebitur etiam commune
omnium Gravitatis Centrum, jam ſecundum De-
terminationem PS; et cum Velocitate quae erit

ad Velocitatem priorem ut P’S ad E H. 2 E. I.

CO RO. L:

Si Impreſſiones omnes eandem habuerint Deter-
minationem vel contrarias, Figura EFG H nunc
prorſum, nunc retrorſum, ſecundum haſce determi-
nationes mota inter eaſdem ſemper parallelas con-
tinebitur. Quod ſi Actiones omnes ex una Parte
ſimul ſumptae aequales fuerint Actionibus omnibus
ex altera, Figura illa E FG H, utcunque hac et
illac inter eaſdem ſemper parallelas mota, redibit
tamen ultimo in priſtinum ſitum et locum; et
Figura E QR H, exhibebit Motus reſultantes.
Unde rurſus facile colligitur commune Gravitatis
Centrum totius Syſtematis, a Viribus aequalibus
et contrariis ſimul impreſſis nihil omnino paſſurum

quoad ſtatum ſuum vel quieſcendi vel movendi
uniformiter in directum.

3 II.

Si Motus illi, quos Vires LI Mm, Nx, ſimpli-
citer et ſeorſim generarent, ſint ut Corpora A, B, C
reſpective, et ſimul aequales Motui E, fiantque

in

— gue 2 HIDE

wo OIL FO

• . p — — —
  . — Pots — — — 2
  — * a
  f Ln ͤͤ̃ ⁰ üA— ˙ eas DD
  p — SOLLADT 09 Ae eb 02.4 ane — 88
• ap 2 — 1

4 8
4

1%
I
N
vt
23 3

1. 1.

ö
l

mers;

240 In PHIT LOS. NEv Ton.
in eandem Plagam, h. e. in Plagam H E, Punctum

H dum Figura tota E F G, transfertur ſecun-

dum Determinationem Virium, locabitur ſemper
in Recta HE, cadetque ultimo in Punctum P locum
primum Puncti E. Unde conſequens eſt, Motum
omnem communis Centri Gravitatis his Actionibus

ſublatum eſſe, et Motus reliquos P Q. QR, RP,

ex Centro illo jam quieſcente ſpectatos omnino
proinde ſe habere ac Motus priores ex eodem Cen-
tro uniformiter progrediente. Nam cum Vires
novae ſint ut Corpora, Velocitates quas ſolae gene-
rarent aequales erunt, et praeterea aequales et con-
trariae Velocitati qua Centrum Gravitatis movetur

cum omnem hunc Motum tollant. Unde Motus

reliqui poſt ſubductum Motum Centri Gravitatis

iidem omnino erunt quoad hoc Centrum qui prius.

Coror. III.
Hinc et ex Corol. quinto praecedentis mani-
feſtum eſt, Motus Corporis unius cujuſque et

Centri Gravitatis reliquorum utcunque ſe habeant

abſolute, five in eodem fiant Plano, ſive non, ex
communi omnium Gravitatis Centro viſos, fieri
ſemper in Plagas contrarias. Sic Motus Corporis
A poſt ſubductum Motum Centri Gravicatis, fit in
Plagam PQ cum tamen (per modo citatum Corol.)
commune Gravitatis Centrum reliquorum B et C

feratur in Plagam QP, et par eſt Ratio reliquo-

rum, ut ex n Figurae P QRP ſatis

liquebit.

5 Co RO

yp 79

yy
.

we

CJ RY 2 9 woe PY OF OF

— *
wa

80H oi. 241

” Conn. If

| Qua Ratione retulimus Motus partium Syſte-
matis ad commune Centrum Gravitatis ſubducendo

Motum Centri ex Motibus partium, eàdem inve-
niri poſſunt Motus reſpectivi alterius cujuscunque

Centri uniformiter et in directum progredientis;

nimirum quaerendo Vires L Mn, Nx, in pla-
gam contrariam applicandas, quae ſemel et ſimpli-

citer agentes generare poſſint in Corporibus ſingulis

Velocitatem Centri; et notando Motus reliquos
juxta praeſcriptum Theorematis.

In e ad Lacks Mir

Demonſtrationem Neutonianam, quod Gravium
Deſcenſus ſint ut Quadrata Temporum, ſic explico.

Cadant e Quiete duo Corpora, ſintque Tem-
pora deſcenſuum (Fig. 57.) AB et ab: Accipi-
antur in his Temporibus duo Momenta P’Q et p +
totis proportionalia, et ſimiliter ſita, hoc eſt, fit
AP ad a p ut A B adas, et Velocitas in Inſtanti
P erit ad Velocitatem in Inſtanti p ut Tempus

AP ad Tempus ap, hoc eſt, ut Tempus AB ad

Tempus @ þ; ergo Velocitas in Inſtanti Pet Mo-
mentum P Q ſimul, erit ad Velocitatem in In-
ſtanti p, et Momentum pg ſimul, ut A B* ad 4h69.

Sed Spatiolum a Corpore cadente, Momento P Q

emenſum, eſt ad Spatiolum emenſum Motnento p q

ut Velocitas in P ad Velocitatem in p, atque ut
Momentum PQ ad Momentum p 9 conjunctim:
5 | Ergo

9
— ee e e Tor ee eee eee

}
3
|
1
‘
1

|
4
;

W
— IeEanneIe . ye
— ro ee ann Lt who I
=. — — — he — ——

2 —

rere
—— —ͤ—

23K» y —
r 0

—

—

242 H PHIILOS. Nev TON.

Ergo Spatium emenſum Momento ? Q eſt ad

Spatium emenſum Momento pq ut KB ad ad.
Dividantur jam Tempora AB et ab in Momenta
quam minima, numero quidem aequalia fed Mag-

‘ nitudine totis proportionalia ; et cum Spatia Mo-

mentis ſingulis ſimiliter ſitis deſcripta ſint ut Qua-
drata Temporum, erunt it Spatia tota in hac Ra-
tione *.

-— Fe Demonſtrationem tradit Cl. Auctor Regular Neutonianae

de Motibus ex collifione Corporum Aeſtimandis per Experimenta tan-
quam in Vacuo inflitita ; ſed prolixam ni mis, quum paucis ea res
abhſalvi poterat.

Arcus ſcilicet R V quatuor Refflentias compledti cenſendus eft,
fere arithmetice proportionales, ſed quarum duabus duae ſunt Situ
diſſimiles: nam Arcus ille, quem pro menſura Reſiftentiae per dimidiam
Oſcillationem genitae uſurpare licet, infra puncrum unde demittitur

_ Pendulum locandus eft, et ſupra punctum ad quod aſcenditur. Dus
igitur utriuſque Situs ratio habeatur, Arcum S T per quem Re-

feftentiarum pars guarta mixtim exponitur, nec totum infra, nec

totum ſupra medium Arcus RV punttum flatuere convenit, ſed in
10 PPS medio: Sic enim, neque alias, Refiflentiam utramwis
quam patitur Corpus deſcendendo ab S ad A, wel aſcendendo ab
A ad T, aeguo jure metietur eft enim 8 T utrique Arcui, S A et
AT, communis ; dum interea horum Arcuum ſemi ſumma guartae
e e 5 manet.

Quadratura

Soni, OA. — — ere A d i FRET ©”

ht A

4

Sc HOL I A. 243

Quadratura Parabolae Apolloniae,ofendens uſum

Lemmatum quatuor primorum Libri primi
Principiorum, ut et uſum Rationum pri marum
et ultimarum in Ratiociniis Geometricis.

Sit A Mm (Fig. 58.) Parabola cujus Parame-
ter t, Axis A Pp, ordinatim applicatae M P, m wp:
Quaeritur Spatium parabolicum A P M.

Compleantur Parallelogramma A P M Q,
Ap mg, occurrantque M Q, MP ipſis mp, n 120 in
R et 8 7 0 eritque primo X AP = P > M>

etixXAp =pm”; qi quare / xP p=mp’—M PPS

—PM*=PSFPMxPS—PM=—PSFÞ-PM x MS,

hoc eſt, c eſt, Rectangulum txP p aequatur Rectangulo
P MAPS S Ms; et Solidum AP x 7 x Pp Solido
APP M-EP 5 S M 8, vel Solidum P M* NE
Solido P M+PSxX APXxXMS. Sed PMxPp

aequatur Rectangulo PR, et PM x Pe aequatur
Rectangulo PR ducto in Rectam PM. Similiter
PMP SNA Px Ms aequatur Rectangulo O8
ducto in Rectam P MPS ; quare Rectangulum
PR eſt ad Rectangulum QS ut PMP S ad PM.
Manentibus MP, MQ, minuantur Latitudines
Rectangulorum PR, QS, ad infinitum, et Ratio

ultima Parallelogrammorum evaneſcentium erit
2PM ad PM ſive 2 ad 1: Exit ergo ſumma ultima
Parallelogrammorum omnium PR ad ſummam ulti-
mam Parallelogrammorum omnium QS ut 2 ad 1,
hoc eſt, Spatium i internum APM duplum erit Spatii
externi A QM, adeoque Spatium internum APM
ſubſeſquialterum erit totius Spatii PQ. Q. E. I.

244 NPRHII OS. NEU TOR.

De Rationibus pri mis et ultimis.

Quanquam Magnitudines variabiles conſiderari

poſſunt, vel tanquam ex nihilo naſcentes, vel tan-

quam in nihilum evaneſcentes; fieri tamen poteſt,
imo ſaepenumero fit, ut harum Rationes neque
naſcantur neque evaneſcant, neque ultra certam
ac determinatam Magnitudinem excrefcant, ſed
finitae exiſtant; etiam tum cum Magnitudines,
quarum illae ſunt Rationes vel in nihilum abiere,
vel in immenſum excrevere; et hae Rationes eae
ipſae ſunt, quas primas naſcentium, vel ultimas
evaneſcentium ſub diverſo conſiderandi modo nun-

cupamus, vel etiam ultimas Rationes Magnitudi-

num auctarum in infinitum. Sunt itaque Rati-
ones pri mae et ultimae, non Rationes Magnitudi-
num pri marum et ultimarum, quales non dantur
in rerum Natura, non Rationes Magnitudinum

finitarum, aut vere exiſtentium; ſed Rationum

illarum Limites, ad quos nimirum Rationes Mag-

nitudinum ſine Limite creſcentium vel decreſcen-

tium propius accedere poſſunt quam pro data
quavis differentiaz nunquam autem tranſgredi,
neque attingere quidem, niſi tunc cum Magnitu-
dines, quarum ſunt Rationes vel immenſae evaſe-
rint, vel nullae. Haec omnia utcunque conceptu
difficilia prima fronte videantur, unico tamen ex-
emplo inter alia quae paſſim plurima occurrunt,
ſatis illuſtrari poſſe exiſtimo.

Exponantur

. „„ kt A OA oy _ ÞxX>% > ay

So AOL IA. ; 245

Exponantur itaque duae Quantitates A et B,
quarum fit Az4 xx+-3 x et B=2 xxx, varia-
biles quidem propter variabilem x, et immenſae
vel nullae prout Radix earum x infinita fuerit, vel
nulla; dico jam ultimam Rationem evaneſcentis A
ad evaneſcentem B, eam ipſam fore quam habet
3 ad 1; ultimam autem Rationem infinitae & ad
infinitam B, eam fore quam habet 2 ad 1. Etenim
cum fit A=q4xx+3 x, et B=2 xxx, erit A
ad B (dividendo per &) ut 4x+3 ad 2x+1:;
Eſt autem haec Ratio major quam 4×2 ad 2x;
vel 2 ad 1, et minor quam Ratio 6×3 ad 2x-k1
vel 3 ad 1. Evaneſcat x et ſimul evaneſcentibus
4x et 2x erit A ad B ultimo ut 3 ad 1. Rurſus
cum fit A ad B univerſaliter, ut 4x–3 ad 2×1,
vel (dividendo per x) ut 44-2 ad 2+; augeatur
x in infinitum, et evaneſcentibus – et – „ exit A

ad B ultimo ut 4 ad 2, vel 2 ad 1; nunquam ergo

erit A ad B, vel in Ratione 3 ad 1, vel in Ratione

2 ad 1, ſed in Ratione intermedia; et hae Rationes,
nempe 3 ad 1 et 2 ad 1, ſunt Limites certi, et jam
determinati, Rationum omnium quas habere poteſt
Quantitas A ad Quantitatem B; inter quos quidem
Limites, Rationes illae indeterminatae tanquam
inter cancellos cohibentur, et ad quos propius ac-
cedere poſſunt quam pro data quavis differentia.
(Vide Neutonum ſub finem Sectionis primae.)

—— 1 ²-VÄ⁵Ü— ˙ . —˙ꝛe ̃

— . — —

246 un Pnirtos, NEVvToON.

De Curvatura et Angulo Contactus.

DrxrxyFiniTIoO.

Si ad Curvae alicujus particulam ſeu Arcum
quam minimum, et ad eaſdem partes, applicetur
Arcus quam minimus Circularis, qui cum priore,
quantum fieri poteſt, congruat; horum Curvatura

eadem eſſe dicitur : Sin alter alterum in ſinu ſuo

complectatur, exterioris Curvatura minor eſſe di-
citur, interioris major: Et univerſaliter, Quantitas
Curvaturae in loco quovis ex reciproca Ratione diametri
Circuli aeque curvi ſemper aeſtimatur; unde Curva-
tura finita eſſe dicitur, ubi Diameter Curvaturae,
hoc eſt, Diameter Circuli aeque curvi eſt finita.

THEOREM A I. Fig. 59.

Si per Curvae alicujus tria Punta A, B, C, ad

invicem continuo accedentia, et in loco B ultimo
coeuntia, tranſeat ſemper Circulus D; dico Cir-
culum illum D eandem ultimo Curvaturam habi-
turum quam Curva in loco B.

DEMONSTRATIO.
Habeat enim (fi negas) alius Circulus E eandem
Curvaturam, congruatque quantum fieri poteſt,
cum Curva in loco B, et Circulus iſte E vel tranſ-
ibit per tria illa Punta A, B, C, vel non;
tranſeat ergo, et duo Circuli D et E ſe mutuo ſeca-
bunt in pluribus quam duobus Punctis, contra quod

demonſtravit Euclides in decima Propeſitione Libri

tertii Elementorum. Si non tranſeat, ergo Cir-
| culus

3

1

—— ww, — 8 — Wy DD ©

;

Senor 3

culus D magis congruet cum Curva in loco B
quam E Circulus ejuſdem Curvaturae (hoc eſt)

quami Circulus maxime congruus QE. A. Habet

itaque Circulus D Curvaturam Praefinitam.

Q. p.

„

Si ex tribus Punctis A, B, C, duo tantum a
et B cojerint, immoto manente tertio C, Circulus
quidem D tanget Curvam in loco B, ſed eandem
Curvaturam non habebit neceſſario, niſi et tertium
C cum reliquis duobus A et B coaluerit.

SCHOLIUM.
Quemadmodum dantur Circuli numero infiniti,

qui per duo data Punta tranſire poſſunt, unicus

autem qui per tria; ita dantur Circuli numero

infiniti qui eandem Curvam in eodem loco con-

tingere poſſunt, unicus autem qui eandem habebit
Curvaturam, congruetque cum Curva in loco
Contactus. Porro ex jam dictis et demonſtratis,
facile conſtabit, particulas quam minimasCurvarum

omnium ubi Curvatura eſt finita, iiſdem omnino

affectionibus gaudere quoad ipſarum Chordas,
Tangentes, Sagittas, Sc. quibus Arcus quam mi-
nimi Circulares ejuſdem Curvaturae; harum itaque
nonnullas ex natura Circuli, tanquam ex proprio
fonte deductas, praeſertim quas in Philoſophia
Neutoniana maxime uſui fore praevidero in medium
proferre et explicare non pigebit. Sunt autem
hujusmodi.

i I. Si

248 In Pruilos Nev Ton.

_ ns
Si Circulum ABC (Fig. 60.) tetigerit Recta

quaepiam AP, et per hujus Circumferentiam

moveatur Punctum B verſus Punctum immotum
A, coeatque ultimo cum eodem, et ſi per immetum
Punctum A, et motum B, tranſire ſemper intelli-

gatur Chorda A B; dico Angulum B A D inter

Chordam A B, et Tangentem A D, ultimo mino-
rem fore dato quovis Rectilinco. Nam Angulus

iſte B A D ſemper aequalis erit Angulo A CB in

alterno Segmento, qui coeuntibus A et B evaneſcit.

| * 5
Si a Puncto B agatur utcunque ſubtenſa B D,
efficiens Angulum quemlibet finitum BD A cum
Tangente A D, dico Chordam AB, tangentem
AD, et Arcym AB fore ultimo in Ratione
aequalitatis. Poſita enim A C ſubtenſae B D pa-

rallela, ſimilia erunt Triangula C A B, AB D;
unde AB et AD et ADD B erunt ad invicem

ut C A et CB et CBB A. Coeat jam Punctum

B cum Puncto A, eruntque CA et CB et C BBA
ultimo in Ratione acqualitatis ; 3 quare ABetAD
et AD DB erunt etiam ultimo in Ratione aequa-

litatis; et à fortiori A B et AD et Arcus AB erunt

ultimo in Ratione aequalitatis.

8 .
Quapropter, in omni de Rationibus ultimis
computatione, hae lineae pro ſe invicem uſurpari

_

IV.

oi. ts oe ©

. _ Ly A. bh 249

| N |

Si ducatur Chorda BF G, Tangenti AD paral-
lela, ſecans Chordam A C in E, Sagitta A F ultimo
biſecabit et Arcum BAG et Chordam BFG;
nam ob parallelas AD, B G, eſt Punctum A in
medio Arcus BAG, et ft AF et BD etiam

parallelae ponantur, erit BF vel AD aequalis

Arcui AB; et pari Ratione erit F G n
Arcui AG.

a

| Subtenſa BD, et huic aequalis et parallela
Sagitta A F, erunt ut Quadratum Arcus B A, vel
hujus dupli BAG directe, atque ut Chorda A C
inverſe; ſi modo Chorda illa A C ſit parallela
ſubtenſae B D, et Chorda B G, Tangenti AD.
Nam ob dicta Triangula ſimilia C Al B, ABD,
eſt CA ad AB ut AB ad BD: Ergo vniverſallter
eſt BD vel AF aequalis Quadrato Chordae A B.
applicato ad Chordam A C; et evaneſcens Chorda
aequalis eſt Arcui evaneſcenti,

=

Adee in Figura quavis curvilinea, fi detur
et Angulus ad D et Curvatura ad A, ſubtenſa
evaneſcens B D et Sagitta AF erunt ut Quadratum
Arcus B A vel Arcus B A G.

f VII. |
Quod fi plures fuerint Arcus quam minimi, et
viciniſſimi, erunt horum Quadrata ut Sagittae illae
1 | =

250 In PRILTLOS. NEU Ton.

quae Chordas biſecant, et ad datum Punctum
convergunt. Nam ob parvitatem et vicinitatem
Arcuum, erit eadem omnium Curvatura, Tan-

gentes quaſi in eadem ſunt Recta, et omnium
Sagittae quaſi parallelae. |

VIII.

Si ex Curvae alicujus Punctis A et B, ducantur
utcunque Rectae AF, BF, ita tamen ut B F
parallela fit Tangenti in A, et fi producatur A F
ad C, ita ut Longitudo A C aequalis fit ei quae
ultimo fit, applicando Quadratum vel Arcus
evaneſcentis A B, vel Chordae A B, vel Ordinatae
B F ad interceptam A F, Circulus qui tangit Cur-
vam in A tranſitque per C, eandem habebit
Curvaturam cum Curva in loco Contactus.

IX.

Vel fi tangat Recta AD Curvam aliquam in
loco A quam ſecat ſub:enſa BD in B, et fi ex
Punctis A et B ducantur Rectae A C, B C effici-
entes Angulos ABC, BAC Angulis ADB,
A BD aequales reſpective, Rectae illae AC, BC
ultimo concurrent in peripheria Circuli aeque
Curvi, et cum altera illa Curva congruentis in A.

X.

Poſtremo, ſi in Figura quavis curvilinea, Cur-
vatura ad A non detur, fi tamen Recta A C
ſubtenſae BD parallela terminetur ad peripheriam
Circuli aeque Curvi, hoc eſt, fi Arcus quam
minimus A B compleatur in Circulum aeque
| Curvum

SCHOLIA, 251

Curvum ABC, ſubtenſa evaneſcens BD aequalis
erit Quadrato Arcus evaneſcentis A E B applicato
ad Rectam illam A C.

Theorema ſequens reſpicit Arcus quam mini-
mos Curvarum, quorum Curvaturae infinite ma-
jores ſunt, vel infinite minores cireularibus. Vide
Scholium ad finem Sectionis primae Principiorum.

TrnroremMa II. Fig. 61.

si Curvae A B, AC, et Recta AD ſe mutuo
contingant in loco A, actàque ſecundum quamcun-
que legem Recta D BC, fuerit DB, vel univer-
faliter, vel ſaltem evaneſcens, ut A D®, et CD ut
A D’, exiſtente m majore quam »: Dico Angulum
Contactus BAD infinite minorem fore Angulo
CAD, et idcirco nullam Lineam Curvam ejuſdem
generis cum A C duci poſſe inter Curvam A B et
Tangentem ſuam AD.
DrMONSTRAT IO.

Eſto enim BD= AD , et 90 phe poſitis
7

N et q finitis et conſtantibus, erit BD ad CD ut
AD . AÞD*

—— ad

: vel ut AD” ad AD” xc, vel ut

4A5.— ad : Evaneſcat jam AD et ſimul

AD==”, et BD jam erit ad C D ut Quantitas
7

evaneſcens AD” ad Quantitatem finitam , vel

ut finitum ad infinitum : Eſt itaque ſubtenſa evane-
ſcens B D infinite minor ſubtenſa evaneſcente C D,
et N Angulus BAD “AN CAD, Q.E.D.

Demon-

252 In PHIL Os. NEUToON,

Demonſiratio Corollarii oftavi Propoſitionis
| _ quartae,

Si duo Corpora in Orbibus ſimilibus revolvantur,
Viribus centripetis ad Centra ſimiliter poſita ten-
dentibus : dico Vires haſce centripetas in locis ſimi.
libus eſſe ut Arcuum quam minimorum et ſimul de-

ſcriptorum Quadrata applicata ad Radios ab his
locis ad Centra Virium ductos. F 1g. 62.

Sint enim S et 5 Centra Virium, B et 5 Orbi—
tarum loca ſimilia, ABC et abc Arcus quam
minimi ſimu] deſcripti, quorum media Puncta
ſunt B et þ; et compleantur hi Arcus in Circulos
aeque curvos A BCE et abce, ſecantes Radios
B S et 55 (ſi opus productos) in E et e; et cum
Arcus AB Cet abc ſimul deſcribantur, erit Vis
centripeta in B ad Vim centripetam in &, ut Arcus
ABC Sagitta, quae Chordam biſecat, et per
Centrum Virium tranſit, ad ſimilem Sagittam
Arcus abc : Sed hae Sagittae ſunt ad invicem ut

A A BC* —_— ee abc

BE |
ſcripſimus; ergo Vis nen -i in B eſt ad Vim

centripetam in 6 ut ABC C ad abe . Sed in Fi-

BE be

guris ſimilibus, lineamenta omnia ſimilia ſunt
proportionalia 3 adeoque BE eſt ad be ut BS ad bs:
Subſtituantur ergo BS et hs pro B E et be, et Vis
centripeta in B erit ad Vim centripetam in b ut

1 ad 25 E.: D.
2 Q

Per ea quae fupra de Curvatpre

dor

• • 
    4
    e

.

— ar apr le EY PIG ar SEED ABD 9s —— re

oScCHOLIA. 253

Cotok.::

Vires centripetae in locis ſimilibus ſunt ut
Quadrata Velocitatum in his locis directe, atque
ut Radii ad Centra Virium ducti inverſe. Nam
Velocitates ſunt ut Arcus ſimul deſcripti.

Conor.

Sunt etiam Vires centripetae ſimilibus in locis ut
Radii ad Centra Virium ducti directe, atque ut
Quadrata temporum periodicorum inverſe. Re-
ferant enim AB Cet a hc jam non amplius Arcus
quam minimos ſimul deſcriptos, ſed ſimiles ſimi-
lium Figurarum particulas, temporibus quam
minimis T et ? deſcriptas; et Velocitas in B erit
ABC 6 : 0

ad Velocitatem in h̊ ut ad nempe ut

T
Spatia deſcripta directe, et tempora a Sed
Arcus ABC eſt ad Arcum ſibi ſimilem ac ut
Longitudo BS ad ſimilem Longitudinem 55;

quare Velocitas in B eſt ad Velocitatem in þ ut

BS b
F ad 5 et vadratum Via in B ad

| BS, 4 &
Quadraturn Velocitatis in 5 ut 222 ad ©. Sed
per ultimum Corollarium, Vires centripetae in
3 et 6 ſunt ut Quadrata Velocitatum in his locis
directe, atque ut Radii ab jiſdem locis ad Centra
Virium ducti inverſe: Ergo Vis centripeta in B

BS

bs
eſt ad Vim centripetam in 4 ut T7 ee . Verum

enim vero, cum ſimiles ſimilium 2 partes
S ABC,

r A. a ns ES nn, EE es as La ”
_— * 5 = . — —— —— —

254 TM PHIL OS. Nevron.
S ABC, sabc ſint ad Areas totas in eadem
Ratiohe, erunt et tempora T et ? in eadem Ratione

ad tempora tota periodica: Quare Vires centri-

petae in locis ſimilibus B et 5 erunt ut Radii ab
iiſdem locis ad Centra Virium ducti directe, atque

ut Quadrata Temporum periodicorum inverſe.
Reliqua n , ut ex jiſdem demonſtratis

in Circulo.

Demonfiratio n noni Fete
quartae.

Fig. 63.

Si Corpus revolvatur uni formiter in Orbe circulari,
Vi centripeta ad Centrum tendente percurrens Arcum
gquemlibet AB tempore t: Dico Arcum illum A B

medium fore Proportionalem inter A E Diametrum
Circuli et s deſcenſum rectilineum eodem tempore t
confectum, ſi Corpus revolvens Motu omni circulari
privatum retta verſus Centrum dimitteretur; eadem
urgente Vi centripeta qua prius in Orbe circulari
coercebatur. |

Nam data Vi centripeta, Deſcenſus s erit ut
Quadratum temporis , hoc eſt, ob ſuppoſitum
uniformem Corporis | Motum, ut Quadratum Ar-

cus A B, vel ut

A E ; unde fi in caſu aliquo
ARB
AE
aequales eſſe, ſemper „ erunt. Tangat

Recta A D Circulum in A, agaturque Subtenſa

particulari,

BD

ene
BD Dlametro AE parallela, et ſi Arcus AB

ponatur infinite Parvus, erit Subtenſa B DA.
ut ex us conſtat quae de Curvatura demonſtravi-
mus. Sed in hoc caſu, eſt etiam BD aequalis
Spatio 5: nam tempore quam minimo / conſide-
rari poteſt Vis centripeta tanquam agens ſecun-
dum Rectas Diametro A E parallelas : Quare in

Illo caſu, ubi Arcus A B infinite parvus eſt, Lon-

gitudines 40 E aequales ſunt ; 3 ergo 3

erunt univerſaliter, Q. E. D.

COROLLARIU M.

Et hinc rurſus patet, Vires centripetas in Or-
bibus circularibus, quae ad horum Orbium Centra
tendunt, eſſe ut Quadrata Arcuum ſimul deſcrip-
torum directe, atque ut Circulorum Diametri vel
Semidiametri inverſe. Nam hae Vires erunt ut
Deſcenſus rectilinei quos eodem N efficiunt.

Ad illuſtrandum Corollarium nonum Propof tions

quartae.

PROBLEMA.

Referat s Spatium @ Gravi e Quiete demiſſo, tempore
unius minuti ſecundi deſcriptum, d Diametrum
Terrae, et ſeponatur omnis ex Aere Reſiſtentia;
quaeritur quanta cum Velocitate Projectile aliquod
ex editiore in Terrae Superficie loco, et ſecundum
KNeltam Horizonti parallelam emittendum fit ut fiat

Planeta

—
———— ::: — ˙

1 h PRI os. NEU TO x.

Planeta ſecundarius, boc eſt, ut inſtar Lunae noſtrae,

Circulum Terrae concentricum uniformi cum Motu
Percurrat.

Pone factum, et Aon ſingulis Minutis ſecundis
deſcriptus, medius erit Proportionalis inter Dia-
metrum d et Spatium g. Innoteſcunt autem et d
et 5 ex notis Terrae Magnitudine, et Longitudine

Penduli ad Minuta ſecunda oſcillantis. Emittatur
iraque Projectile tanta cum Velocitate, quanta |

ſufficiat ad Longitudinem 2 Spatio unius Minuti
ſecundi uniformiter percurrendam, et ſolvetur
Problema. :

CoROLLARIUM.
Si deſideretur tempus periodicum, dicendum eſt,

ut ds ad totum Terrae Ambitum ita tempus
unius Minuti ſecundi ad tempus quaeſitum.

SCHOLIUM,
Ambitus Telluris eſt 123249600 pedum Pari-
ſienſium; unde Diameter ejus d eſt eorundum
pedum fere 39231600 ; fed et Spatium 5 ad

eandem Menſuram exactum, eſt 1577 pedum;
ergo prodit tandem 4/4 pedum Gallicorum 24326,

Anglicorum?2 5982, hoc eſt, milliariumAnglicorum
4.9208 3. Tanta itaque eſſe debet Velocitas pro-

jectilis, quanta ſufficiat ad Spatium 5 fere milli-

arium tempore Minuti unius ſecundi uniformiter
percurrendum 3 unde tempus periodicum pro-
jectilis erit unius horae, 24 minutorum primorum,

27 ſecundorum. . Ex his autem manifeſtum eſt,
quod

SCH OLIA. 257

quod ſi Terra noſtra Revolutiones ſingulas circa
proprium Axem Spatio unius horae, 24 minutorum
primorum, 27 ſecundorum perageret, Partes
omnes Juxta Aequatorem ſitae, Gravitatem ſuam
omnino amitterent. Quod ſi Terra incitatiore
adhuc provolveretur Motu, Aer omnis et Aqua
et Partes omnes juxta Aequatorem, Terrae non
adhaerentes, in Spatia circumjacentia penitus avo-
larent, et ibi orbitas ellipticas umbilicos alteros
in centro telluris habentes, motu ſuo deſcriberent,
ut in ſequentibus luculentius conſtabit.

THEOREMA.

Si Corpus in Circuli peripheria, cujus Diameter eſt d,
uniformiter Motum, ope Fili teneatur, ne ulterius
excurrat, abſolvatque Arcum quemlibet p, quo tem-
Pore idem vel aliud grave Quiete demiſſum de-
ſcenderet per Spatium s: dico Vim centrifugam
Corporis revolventis fore ad Vim Gravitatis ut p p
Quadratum Arcus deſcripti ad Rectangulum ds,

ſeu (quod eodem recidit) dico Tenſionem Fili ex Vi
centrifuga Corporis oriundam fore ad Tenſionem ej uſ-
dem ex eodem pondere libere ſuſpenſo in bac Ratione.

Nam Vis centrifuga Corporis revolventis aequalis

eſt Vi centripetae qua verſus Centrum urgetur,
quaque uniformiter urgente, ſi nihil impediret,

Corpus deſcenderet per Spatium = 2 , quo tempore

deſcendit grave per Spatium s ex Hs : Ergo
Vis centripeta vel centrifuga Coporis revolventis

eſt ad Vim Gravitatis ut = ad 5 vel ut pa ad ds.
Q. E. BD. 8 | Jn

— ——
. © Or IA SS *

• — * — –

258 I: PHILOS. Nev TON:

In eguſdem Propoſitionis S c HOLIUM.

Si Polygonum detur Magnitudine, et Velocitas
non detur, Vis ictum ſingulorum erit ut Velocitas,
hoc eſt, ut Arcus rectilineus dato Tempore de-
ſcriptus; z et numerus ictuum dato Tempore erit
etiam ut Arcus rectilineus: Quare Vis et numerus
ictuum fimul, hoc eſt, ſumma Virium ſive Vis
centrifuga, erit ut Quadratum Arcus reCtilinei dato
Tempore deſcripti. Haec ita ſunt ex Hypotheſi,
quod Polygonum detur Magnitudine, et Velocitas
non detur: Detur jam Velocitas, et etiam Poly-
gonum ſpecie, non autem Magnitudine; et Vis
ictuum dabitur : Sed numerus ictuum jam erit
reciproce ut Semidiameter Circuli cui inſcribitur
Polygonum: Nam quo major eſt Semidiameter
Circuli, eo minor eſt numerus Angulorum in Arcu
rectilineo datae Longitudinis; ergo quo major eſt
Diameter Circuli, eo minor erit numerus Reflec-
tionum dato Tempore confectarum; quare ſi de-
tur Velocitas, numerus Reflectionum erit reciproce
ut Semidiameter Circuli: Quare ſi neque Poly-
gonum detur Magnitudine, neque Velocitas,
ſumma Virium in dato Tempore erit ut Quadra-
tum Arcus rectilinei dato Tempore deſcripti ap-
plicatum ad Circuli circumſcripti Semidiametrum.

PA o-

SCHOLIA. – 259

PROBLEM A.

Efficere | ut Corpus aliqQuod in Peripheria dati Grreul |

uniformiter moveatur, et Vim determinare quae

ſufficiat ad iſtiuſmodi Motum conſervandum.

AER Fig. 64.

Moveatur Corpus primo in perimetro Polygoni

eujuſcunque acquilateri ABC DEFG HI dato

Circulo inſcripti, cujus Centrum fit S, et produ-

catur Latus quodvis A B ad c, ita ut aequales ſint
AB, Be; et Corpus ubi ad B pervenerit, (ſi nihil

impediret) Longitudinem Be eodem Tempore

deſcriberet quo Latus AB antea deſcripſit: Verum

quo minus hoc fiat, agat Vis quaedam impulſu

unico ea Quantitate act Determinatione ut Corpus
hac Vi impulſum de Recta Be in Rectam BC
detorqueatur, eamque praeterea eodem Tempore
percurrat, quo antea Latus A B percurrebat: Et

fi Corporis in Latere A B moti, exponatur Vis

inſita per Longitudinem illam AB vel Bc, et

ſimul compleatur Parallelogrammum Be CK,

Longitudo B K vel c C, novi hujus impulſus tum

Quantitatem tum Determinationem exhibebit: Et

ſi ad Angulos ſingulos Corpus hoc modo de Latere
in Latus deflectatur, perget idem in Polygoni
hujus perimetro unif.rmi cum Velocitate moveri.
Connectantur jam SA, SB, et cum Anguli SAB,
ASB ſimul ee Angulo S Be; (per prop.

xvi. I. 1. El.) ſi auferantur degusles Anguli SAB,

SB C, 28 Angulus C Bc acqualis Angulo

‘S 2 ASB;

260 In PHIL OS. NEU TOR.

ASB; unde cum aequales ponantur B C, B A
ſimilia erunt Triangula iſoſcelia AS B, CBc;
unde Recta Ce parallela erit ipſi S B, eritque ad
CB vel AB ut AB ad AS: Tendet itaque Vis
BK ad Centrum S, et aequalis erit Longitudini
A B* | |

Supponamus uſdem impulſibus accelerantibus,
quibus hoc Corpus in perimetro Polygoni retine-
tur, et ſimiliter agentibus, urgeri Corpus aliud in
directum, Motu aequabiliter accelerato; et conſtat
dum Corpus primum percurrit Longitudinem
A B, Corpus illud alterum primo ſuo impulſu
conficere Longitudinem AS wel ck vel

re aka
z Similiter dum Corpus primum

percurrit Longitudinem AB C, Corpus alterum

duobus jam impulſibus ſucceſſive agentibus feretur
per Longitudinem AB T2AB va = vel
ae l

2 KS et porro dum Corpus primum

Longitudinem A B CD conficit, Corpus alterum
AB* T2 AB ＋ AB

conficiet Longitudinem Te :
ns Ki, , ARCONLAREDE
yel N vel —— R Et uni-

verſaliter dum Corpus primum Arcum quemvis
tectilineum abſolvit, Corpus alterum impelletur
per L.ongitudinem quae fit ad hunc Arcum ut

• 5 Arcus

kk. 22 — 5 — va

OD OM lp pd en

3
1

W

2 8 Hy

11

2 92

BCwoL tk 261
Arcus iſte unico Polygoni Latere auctus, ad Circuli
Diametrum. Augeatur jam numerus et minuatur
Longitudo Laterum i in infinitum, ita nempe, ut
Polygonum coincidat omni ex parte cum Circulo
circumſcripto, et Vis acceleratrix jam continue
efficiet ut Longitudo a Corpore aequabiliter acce-
lerato confecta, fir ad Arcum circularem a Corpore
uniformiter gyrante interea deſcriptum, ultimo ut

Arcus iſte ad Circuli Diametrum. Q. E. I.

Seien.

Hinc e converſo, ſi detur Vis centripeta Cor-
poris in dato Circulo revolventis, facile invenietur
Motus Corporis in hoc Circulo, Nam Arcus dato
quovis Tempore deſcriptus, medius erit propor-

tionalis inter Circuli Diametrum, et Longitudinem

quae Motu ab hac Vi aequabiliter accelerato pari
Tempore conficitur.

COnot It

Spatia Motibus aequabiliter acceleratis, et e

Quiete inchoatis, deſcripta ſunt ut Quadrata Tem-
porum ab initio Motuum, et Vires acceleratrices
conjunctim. Conſtat enim ex Demonſtratione
Propoſitionis, Longitudines a Corpore aequabiliter
accelerato confectas, fuiſſe ut Quadrata Arcuum

a Corpore revolvente interea deſcriptorum, et

proinde etiam ut Quadrata Temporum ſiſdem
Arcubus proportionalium. Quod ſi e contra den-

tur Tempora, Spatia percurſa erunt ut F

adeoque ut Vires generantes.
8 3 con OL,

R

262 N PRHIILOS. NeEvToON.

Coro. MI.

Sunt etiam Spatia illa ut Vires ſimul et Qua-
drata Velocitatum quae dictis Temporibus gene-
rantur. Nam Velocitates, caeteris paribus ſunt ut
Tempora. |

Conor. IV.

Spatia Motibus aequabiliter retardatis, ac tandem

evaneſcentibus deſcripta, ſunt ut Quadrata Velo-
citatum ſub initio Motuum.

CoroL. V.

$i Tempora in partes acquales dividantur,
Spatia Motibus aequabiliter acceleratis 1iſdem
Temporibus deſcripta, erunt ut numeri impares
I, 3» 5, 7» 9, Ce.

Cool. VI.

Velocitas Corporis Vi quavis centripeta, in

Circuli peripheria revoluti, eſt ad Velocitatem

aequali Vi uniformiter agente, dato quovis Tem-
poor genitam, ut Circuli Semidiameter ad Arcum
eodem Tempore deſcriptum. Nam Velocitas
Corporis in Orbe rectilineo erat ad Velocitatem

alterius in directum poſt primum impulſum ut

B Cad Cc, vel ut S B ad AB; et haec Velocitas
erat ad Velocitatem poſtea, ut unitas ad numerum
ictuum accelerantium, hoc eſt, ut AB ad Arcum
rectilineum toto Accelerationis Tempore emenſum.
Quare ex aequo Velocitas prima eſt ad Velocitatem
ultimam ut Semidiameter Circuli circumſcripti ad
Arcum praefinitum: et par eſt Ratio in Orbe cir-

n ot.

ScuoLlLia 267

| Cone. VIE.

Spatium Motu aequabiliter accelerato emenſum,
dimidium tantum eſt ejus quod uniformiter, Ve-
locitate ultimo acquiſita eodem Tempore percurri
poteſt. Nam Spatium Motu aequabiliter accele-

rato emenſum eſt ad Arcum circularem, eadem

Vi centripeta, eodemque Tempore deſcriptum,

ut Arcus iſte ad Circuli Diametrum, (per Corol. 1.)

vel ut Semi- arcus ad Semidiametrum, Verum
Spatium, Velocitate ultimo acquiſita percurſum
erit ad eundem Arcum interea deſcriptum, ut
Velocitas ad Velocitatem, hoc eſt, (per ultimum
Corol.) ut Arcus totus ad Circuli Semidiametrum.

CoR OL. VIII.

Vires centripetae 1iſque aequales etiam Vires
centrifugae Corporum in Circulis gyrantium ſunt
ut Arcuum ſimul deſcriptorum Quadrata ad Cir-
culorum Diametros vel Semidiametros applicata.

Cook. I
unt etiam hae Vires ut Quadrata Velocitatum
itidem ad Circulorum Diametros vel Semidiame-

tros applicata : Nam Velocitates ſunt ut Arcus
ſimul deſcripti.

Co ROT. X.

Poſtremo ſunt hae Vires ut Circulorum Semi-
diametri directe, et Quadrata Temporum periodi-
corum inverſe. Nam ſi ſublato omni Motu cir-
culari Corpora in directum iiſdem Viribus cen-

84 tripetis

264 NJ Prilos Nev Tox:
tripetis uniformiter agentibus impellerentur, Vires
per Corol. 2. eſſent ut Spatia deſcripta directe, et
Quadrata Temporum inverſe. Verum fi Tem-
pora ea ſumantur quibus ſimiles Arcus peragun-
tur, Spatia quae ſunt ut Arcuum illorum Qua-
drata ad Circulorum Semidiametros applicata, jam
erunt ut eaedem Semidiametri. Sunt itaque vires
contripetæ vel centrifuge, ut Semidiametri illi di-
recte, et Quadrata Temporum periodicorum in-
verſe, ſeu (quod perinde eſt) ſunt hae Vires reci-
proce ut Quadrata Temporum periodicorum ad
Circulorum Semidiametros applicata.

Cool. XI.

HFinc fi Tempora periodica ſint ut dignitas ali-

Semidiametrorum vel Diſtantiarum a Centro,
cujus Index eſt 2, Vires centripetae erunt reciproce
ut dignitas 22—1, vel directe ut dignitas 1—22
earundem Diſtantiarum. Nam Quadrata Tem-
porum periodicorum erunt ut harum Diſtantiarum
dignitas 27, adeoque horum Quadrata ad haſce
Diſtantias applicata, quibus nempe reciprocantur
Vires centripetae, erunt ut Diſtantiarum dignitas
2N—1. F 5
COo ROL. XII.

Velocitates etiam erunt reciproce ut Diſtantia-
rum dignitas 2-1. Nam Velocitatis illae (per
Corol. .) ſunt in ſubduplicata Ratione Diſtantia-
rum et Virium centripetarum, conjunctim, hoc eſt,
(per Corol. ult.) ut Diſtantiarum dignitas 1—#
directe vel 8 reciproce. Q. E. D. |

THEOREMA.

Senor: 265

THEOREM A,

Motus circularis uniformis erit, frve ab Afione Vis
alicujus ad Centrum tendentis, ſve a Reactione Fili
alicujus non elaſtici, vel Superficiei Cylindricae

politiſſimae, Corpus motum à curſu rectilineo per-
petuo deflectentis, oriatur. |

Patet pars prior ex Aequalitate Arearum Tem-
poribus aequalibus circa Centrum deſcriptarum:
Et pars poſterior ſic patebit.

Sunto AB, BC, (Fig. 65.) Latera contigua
| Polygoni cujusvis acquilateri, et Circulo inſcripti
cujus Centrum ſit D, et jungantur AD, BD, CD,
ut et AC ſecans Semidiametrum BD in E.
Producatur C BF cui perpendicularis fit AF;
et ob ſimilia Triangula ABF, ADE, erit AB ad
BF ut AD ad DE. His praemiſſis, feratur
Corpus aliquod uniformiter in Latere A B, cum
Vi vel Momento AB, et impinget hoc ſeſe in
Latus BC cum Vi A F, et poſt impactum retine-
bit Momentum BF : Unde Motus Corporis in
AB erit ad Motum in B Cut AB ad B F, vel ut
AD ad D E; quare Motus in AB erit ad Motum
amiſſum ad Angulum B ut AD ad BE, vel ut

AD ad – vel ut 2 AU* ad AB*, Vocetur

perimeter Polyg goni p; et ſi Motus amiſſi Quan-
titatem ad Angulos ſingulos eandem eſſe ponamus,
Motus ſub initio erit ad Motus Decrementum poſt

primans Revolutionem, | ut 2 AD* ad px AB.
Atqui

266 In PHIL OS. NEUT ON.

Atqui Motus decrementum ad Angulos ſingulos
non eſt idem, ſed perpetuo diminuitur, et in eadem
Ratione qua Motus ipſe diminuitur: Nam decre-
mentum Motus ad Angulum quemvis eſt ad Mo-

tum decieſcentem, in data Ratione B E ad BD;

quare decrementum Motus poſt unicam Revolu-

tionem factam, minus eſt eo quod exponitur per

Quantitatem PX AB. Augeatur j Jam numerus, et

minuatur Longitudo Laterum A B, BC, et Quan-

titas px AB evaneſcet, et Matus jam fiet unifor-

mis; et par eſt Ratio in omnibus Curvaturis quae
non ſunt infinite majores circularibus.

Tn SCHoLIUM Prep. 8. Fig. 66.

Revolvatur Corpus aliguod P in perimetro Ellipſeos
AP B Vi centripeta tendente ad Punfum R, ades
longinguum ut Refiae omnes RP pro parallelis
baberi paſſint: Quaeritur lex Vis centripetae tenden-
tis ad Punctum R.

Ducatur Semidiameter C A ReQis R P parallela,
cui productae occurrat in G Recta PG tangens
be in loco P, et ſi Vis centripeta ad Centrum
C tenderet, foret haec Vis ut CP Diſtantia Cor-
| poris a Centro, (per hujus Prop. 10.) et proinde
per Sol:dum RP*xCP recte exponitur, cum

. > detur RP: Verum fi Vis centripeta ad C tendens

exponatur per R CP, Vis centripeta ad Punc-

tum R tendens exponetur per CG (per Corol. 3.

praecedentis.) Ergo Vis centripeta tendens ad R
4 eſt

PPP ANNIE 9 Oe OO

SCHOLIA, „

eſt ut CG3. Ducatur Semidiameter C B Semi-
diametro C A conjugata, ut et ordinatim appli-
catae PQ, PM; P Qiph CA, et PM ipſi C B;
et ex natura Tangentis erit CG ad C A ut CA nl
CQ velPM; quare C G erit reciproce ut PM;
ergo Vis centripeta in loco P erit reciproce ut
Cubus Ordinatae PM. Q. E. I. |

Abeat jam Figura Elliptica in Probing et
Ordinata PM jam pro invariabili haberi poteſt :
Evadet enim PM in hoc caſu infinita ; et Varia-
tiones finitae Quantitatum infinitarum non plus
valent quam Variationes infinite parvae Magnitu-
dinum finitarum : Ergo in Parabola Vis centripeta
ſecundum Rectas parallelas agens, eſt invariabilis
ſi Vis centripeta ſecundum Rectas parallelas agens,
ſit invariabilis, Figura deſcripta erit Farabola,
quod eſt Theorema Galilaei.

Applicari poteſt haec Demonſtratio etiam ad
Hyperbolam, nominando et conſtruendo duntaant
ann pro Ellipſi. |

Ad Prop. 9. LEMMA. Fig. 67.

Conſtituatur Series Triangulorum conſimilium

8 AB, SBC, SCD, S DE, S EF, Sc. aequales

Angulos habentium ad Punctum 8, et Figurae ſic
conſtitutae Proprietates erunt quae ſequuntur.

1. Anguli omnes S AB, SBC, S CD, S DE,

SEF erunt aequales.

2. Sectores omnes rectilinei ſub aequalibus An-

gulis ad Punctum S erunt Figurae ſimiles.

: oo

268 ” to PHILOos. NEU TON.

Sic Sector rectilineus S AB C ſimilis erit
Sectori rectilineo 8S DE F, cum ex partibus
ſimilibus conſtituantur.

3. Lineae omnes in Sectore S A B C erunt ad
   Lineas omnes ſimiliter ſitas in Sectore SD E F
   in data Ratione SA ad SD.
4. Augeatur jam numerus et minuatur Longi-

tudo Rectarum A B, B C, CD, DE, EF,

et Figura rectilinea S AB CDE F migrabit

in Spiralem aequiangulum ſecantem Radios

omnes S A, SB, &c. in Angulis aequalibus;
et Sectores omnes jam mixtilinei ſub aequa-

libus Angulis ad Punctum S contenti erunt

adhuc Figurae ſimiles. |
Haec eſt natura Spiralis aequiangulae.

Ad Corol. 2. Prop. 10. Fig. 68.
Sunto A Pa, Apa Ellipſes, communem habentes

Axem majorem A a; et fi ducatur Ordinata com-

munis QpP, erit QP ad Qp in data Ratione;

nimirum in Ratione Axis minoris Figurae A Pa ad
Arem minorem Figurae A p a.

Sit enim CB Axis minor Fi igurae A Pa, et Ch

Axis minor Figurae A Pa, et erit PQ Q ad Rectan-

gulum AQs ut CB* ad CA, ex conicis; ergo

viciſim P Q* erit ad C B* ut Rectangulum AQa
ad C 2 et pari Ra Ratione p C eritad ut AQ
ad TA ; ergo T C erit ad CB ut y Q ad C 5

et viciſſim PQ’ ad p »ÞQ ut CB ad C, et PQ

exit ad 7 Qin data Ratione C B ad Co. Q. E. D.
Revolvantur

ͤ—y— ——— er er rr rn nn non —

Sc HOL IA. 2369
Revolvantur j jam in his Ellipſibus Corpora duo
P et p, Vi centripeta tendente ad Centrum com-
mune : Dico Velocitatem Corporis P in Vertice
principali A fore ad Velocitatem Corporis p in
| eodem loco in data Ratione Q ad Qp. Moveatur
enim Punctum Q una cum Ordinata communi
QP uſque ad Verticem A; et fi Corpora P et p
ſimul exeant de loco A, ſimul pervenient ad Or-
dinatam communem QPP,; id adeo ob Vires ac-
celeratrices utriuſque Corporis in loco A aequales:
Erunt ergo Arcus AP, Ap Arcus ſimul deſcriptiz
et Velocitas Corporis P erit ad Velocitatem Cor-
poris p ut AP ad Ap, vel Q ad Q. Reſti-
tuatur jam Ordinata QpP in locum priorem, et
manebit etiamnum Ratio Q ad Q p.

f In idem Corollarium.

Ex hoc Corollario facile elicitur T HEOREMA
ſequens ; nimirum. Si Vis centripeta ad Centrum
quodvis tendens fuerit ut Diſtantia locorum ab hoc
Centro, Corpora omnia, five propiora ſive remo- –
tiora ; Temporibus aequalibus e Quiete ad Cen-
trum deſcendent. Deſcribant enim Circulos vel
Ellipſes, et omnium Tempora periodica erunt
aequalia, quaecunque fuerint Ellipſeon Longitu-
dines vel Latitudines. Minuantur Latitudines in
infinitum, et deſcenſus obliqui jam vertentur in
rectilineos; et deſcenſuum Tempora omnia aequalia
erunt £ Temporis periodici Corporis circa idem
Centrum uniformiter in Circulo revolventis,

= | Tn

270 In Pruirtros NEU TON.

In SCHOLIUM Prop. 10.

Sunto (in eadem Fig.) AP, Ap duae Lineae
curvae communem habentes Verticem A, abſciſ-
ſamque communem AQ ; ea vero fit harum Cur.
varum indoles, ut acta communi Ordinata Qp P,
Ordinata QP fit ſemper ad Ordinatam Q- in data

Ratione. Revolvantur jam in his Figuris duo

Corpora P et p Vi centripeta tendente ad Punctum

quodvis S in abſciſſa communi A Q eave producta

conſtitutum; ſintque Corporum P et p Tempora

periodica aequalia: Dico Vim centripetam in loco P

fore ad Vim centripetam in loco p ut Diſtantia S P ad
Diſtantiam S p. | |

Primo enim cum Tempora periodica aequentur,
Areae ſimul deſcriptae erunt ut Areae totae harum
Figurarum, hoc eſt, in Ratione Q ad Qp. Jam

vero Figura mixtilinea APQ erit ad Figuram

mixtilineam ApQutQPad Qp; et Triangulum
S PQ eſt ad Triangulum S p Qetiam in Ratione

QP ad Qp; ergo (componendo) erit Area mix- |

tilinea ASPad Aream ASp ut QP ad Q7?p:
Quare quo tempore Radius SP percurrit Aream
ASP, Radius Sp percurret Aream A Sp; adeo-
que (fi Corpora Pet p ſimul exeant de loco A)
_ fimul pervenient ad Ordinatam communem QpP.
Quare Motus Corporum P et p ſecundum Rectas
ipſt A’S parallelas agentes, aequales erunt. Porro
cum Q ſemper fit ad QP in data Ratione, motus

et motuum mutationes, et Vires mutantes ſecun-

dum

SCHOLIA. 271
dum Rectas ipſi PQ parallelas agentes, ſemper
erunt in data Ratione QP ad Qp. Referat 3j Jam
SP Vim centripetam Corporis P in loco quovis P,
et reſolvi poteſt Vis SP in Vires PQ, QS; ergo
Vis centripeta Corporis p reſolvi poteſt in Vires
2 Q, QS: Ergo Vis centripeta Corporis p exprimi

debet per Ream Sp; quare Vis centripeta in
loco P eft ad Vim centripetam in loco p, ut Di-
ſtantia S P ad Diſtantiam Sp. Q. E. D.
Servatis Ordinatarum QP, Q Longitudini-
bus, mutetur jam Angulus SQ p, ne Ordinatae
QP, Qp amplius jaceant in eadem Recta; et
applicari poteſt praecedens Demonſtratio ad hunc
caſum, ſimiliter atque ad priorem ; hoc eſt, Cor-
pora P et p ſimul pervenient ad Ordinatas QP,
Q, et Vis centripeta in loco P erit ad Vim cen-
tripetam in loco p adhuc ut Diſtantia 8 P ad
Piſtantiam 8 P. Q. E. D.

8 @ My . © . C
Meorema ad Demonſlrationem Corollarii prims

Propofitionis 13. apprime neceſſarium.
| Fig. 69,

Detur Figura quaevis rectilinea AB CDE FG.
et Punctum quodvis S intra hujus Cavitatem
conſtitutum ; et in Latere AB feratur Corpus
quacunque cum Velocitate: Dico ita modulari ac
temperari poſſe impulſus centripetos ad Centrum 8

tendentes, ut Corpus illud in dictae Figurae peri-
metro perpetuo moveatur.

DEMON-

|
|

272 In PHIL OS. NEUTON.

DRMONSTRAT TO.
Acta enim Cc ipſi S B parallela, et occurrente

productae A B in c, completoque Parallelogrammo
Be C K, exponatur Motus Corporis in Latere AB

per Rectam Be, et appulſo Corpore ad Angulum
B, agat Vis centripeta verſus S impulſu unico BK ;
et haec Vis BK una cum Vi Corporis inſita Be

efficiet ut Corpus deinceps moveatur in Latere

BC cum Velocitate quae eſt ad Velocitatem in
AB ut BC ad Bc. Similiter ubi Corpus acceſ-
ſerit ad Angulum proximum C, agat rurſus Vis
centripeta impulſu ſimplici verſus S, detorquens
Corpus de Latere BC in Latus CD, idemque
ſemper fieri intelligatur ad ſingulos reliquos An-

gulos D, E, F, G, A, et movebitur Corpus in

perimetro Figurae propoſitae. Q. E. F.

CoROLLARIUM.
Valebunt etiam praecedentia in Figuris curvi-

lineis, cum in his Figurae rectilineae inſcriptae et
circumſcriptae ultimo terminentur; adeoque nulla

extat Figura curvilinea verſus Centrum Virium
cava, in cujus perimetro Corpus juſta Vi centripeta
ad Centrum illud tendente moveri non poteſt,
etiamſi Velocitas in loco aliquo particulari fit data,
et Mathematicorum utique eſt harum Virium leges
exquirere, et Calculo determinare. Velocitates
autem atque adeo Vires centripetae in iiſdem locis
eaedem ſemper eſſe debent; ſiquidem Areae circa

Centrum Virium defcriptae my ee ſint
Temporibus. | Ad

ane,
Ad Prop. 1 5.

Sit @ Axis major Ellipſeos, b minor, / Latus :
rectum, Tempus periodicum; et erit a/==/?,
et a3/=4ab, Eſt autem ad ut vl, et ab ut .
Quare 451 erit ut 3), et H ut as, et f ut a3. Q.E.D.

27Þ.

Ad Corel. 4. Prep. 16.

Sit ? Semiaxis major, c Semiaxis minor, v Ve-
locitas in mediocri Diſtantia, et exponatur v* per

/
Quantitatem 3 et cum fit co M t, erit
A
—== _ — — Sit! jam V Velocitas in Circulo ad

Diſtantiam t, et cum exponatur Velocitas v2 per
1 : 2 t 2 |
F: Exponetur Velocitas V* per —_ vel 0 Ae-
quantur itaque V et v.

Ad Corol. 6, ejuſdem. Fig. 70.

Sit A a Axis Ellipſcos, A Vertex ab Umbilico
S remotior, @ propior, et Velocitas maxima erit
in a, minima in A; quare ut Velocitatis variatio
patefiat, conferamus Velocitatem in à cum Velo-
citate in A: Eſt ergo Velocitas in a ad Velocitatem
in A ut SA ad Sa; ſed SA ad s eſt in
minore Ratione quam in Ratione SA ad Sa,
propterea quod Ratio prior dimidium tantum fit
poſterioris; quare Perpendicula SA er Sa, et
proinde etiam Velocitas, magis variantur quam pro
reciproca ſubduplicata Ratione Diſtantiae,

p Sit

294 Ht Pnitos. Nevron;

Sit a Vertex Hyperbolae, Fig. 71. cujus Um-
bilicus eſt 8, (et non oppoſitae) et ducatur Sy ad
Aſymptoton perpendicularis, et Velocitas in à erit
ad Velocitatem ad Diſtantiam infinitam ut Sy ad
Sa: Sed Ratio Sy ad S a finita eſt, quando Ratio
Diſtantiarum, et proinde etiam ſubduplicata Ratio,
eſt infinita; quare in Hyperbola, perpendiculum
et Velocitas minus variantur quam pro reciproca

ſubduplicata Ratione Diſtantiae.

Ad Corol. 7.

Sit in ſiſdem Figuris, Aa Axis major Ellipſeos
vel Hyperbolae, B & minor, S A Diſtantia maxima,
86 minima, L Latus rectum, C Centrum, V

Velocitas in Ellipſi vel Hyperbola, v Velocitas in
Circulo ad eandem Diſtantium, et oſtendendum eſt
fore V ad v in minore vel majore quam ſubdupli-
cata Ratione 2 ad 1, prout Corpus P in Ellipfi vel
Hyperbola verſatur. Nam in Ellipſeos Vertice A
eſt V ad v in ſubduplicata Ratione Lad 2 S A (per
hujus Corol. 3.); eſt autem L ad 2 8 A ut Lx Aa

ad 28 Ax Aa; hoc eſt, ut B ad 2 S Ax Aa,
vel ut 48 Ax Sa ad 28 Ax Aa, vel ut 2 Sa ad
Aa: Quare V eſt ad v in ſubduplicata Ratione
2Saad Aa: Eſt autem 28 @ minor quam 2 Ca,
hoc eſt, quam A 4 Ergo in Vertice A, V minor
eſt quam v; tantum abeſt ut major ſit in ſubdu-
plicata Ratipne 2 ad 1. Similiter in Vertice a, eſt
V advan ſubduplicata Ratione 2S A ad Aa, hoc
eſt, in minore _ ſubduplicata Ratione 2 Ag
ad Ss vel 2. ad 1 Yo N
ari

SCHOLIA., 275

Pari Ratiocinio, fi fit à Vertex Hyperbolae
cujus Umbilicus eſt S, eſt V ad v in Vertice à in
fubduplicata Rarione 2 S A ad Aa, hoc eſt, in
majore quam ſubduplicata Ratione 2 Aa ad Aa
vel 2 ad 1. Abeat jam Corpus P ad Diſtantiam
infinitam, et acta Sy ad Afymptoton perpendicu-

lari, eſt V 4 v in 2 Ratione 2 ad

28

IP vel Sp IS 55, eſt Quantirs init, tgp
infinite parva : * in Hyperbola ad Diſtan-

tiam infinitam eſt V infinities major quam v, et

Propterea in majore quam ſubduplicata Ratione
2 ad 1.

LEMMA. |
Si Ellipſin vel Hyperbolam tangat recta quae –
vis, et ex umbilicis demittantur in tangentem
perpendiculares, erit ſub his perpendicularibus
rectangulum aequale quadrato ſemiaxis conju-
gati.

Exſtat demonſtratum ad calcem Operum yaw:
humorum Cotęſii.

Loco Propofitionts 17. Libri 1. PRINCIPIORUM,
et Loco Corollariorum ad Prop. 16. ejuſdem,
ſſequentem adbibe cum Corollarus your

PROBLEMA. Figg. 70 et 71.

Exeat Corpus aliquod de loco dato P data cum
Velocitate ſecundum datam poſitione Rectam P
2 T 2 k et

—— — —— is 4 + err.

276 In Pn 1 os. Nzurox.

et ſimul urgeatur a Vi centripeta tendente ad 5
trum datum 8, quae in loco quidem P innoteſcat,
aliis vero in locis ſit reciproce ut Quadrata Diſtan-
tiarum ab hoc Centro; Quaeritur SECTIO Conica
quam Motu ſuo deſcriptarum eft hoc Corpus.

Cum tangat Recta Z P Figuram quaeſitam in

loco P, et cum detur Poſitione Recta 8 P per
umbilicorum alterum S tranſiens, dabitur Poſitione
Recta PH quae per alterum Figurae umbilicum H
tranſibit, et cujus inventa Longitudo, ſolvet Pro-

blema.

P ejuſmodi, ut quo Tempore percurritur Longi-
tudo 4 uniformi cum Velocitate Corporis in P,
eodem Tempore, Vi uniformi accelerattice quae

• ubique aequalis eſt Vi centripetae in loco P, recta

deſcenderetur per Altitudinem inter quam et Al-
titudinem 28 P Longitudo quaedam 5 media eſt
Proportionalis: Atque his quidem datis, dabitur
et Forma et Poſitione Trajectoria quaeſita: Erit
enim ut 25 à ad aa ita SP ad P H, inter quas
Semiaxis tranſverſus medius eſt arithmeticus; et
fi in Tangentem Z P Y demittantur Perpendicu-
lares SY, H Z, eritetiam ut 26? —az ad à ita SY
ad H Z, inter quas Semiaxis conjugatus medius eſt
geometricus.
DEMONSTRATIO.
Nam quo Tempore percurritur Longitudo a,

M 1 Longitado . f
otu aequabili, vel Longitudo 255 otu

aequabiliter accelerato, eodem Tempore Corpus i in
Circulo

Sunto itaque Velocitas et Vis centripeta in loco

0
&

SCHoLIA. ; 277

Circulo revolvens ad Altitudinem SP percurreret
Arcum 5; uti ex Corol. 9, Prop. 4, Lib. 1, Prin-
cipiorum, manifeſtam eſt : Ergo Velocitas in Cir-
culo ad Altitudinem SP eſt ad Velocitatem in
conica Sectione ad eandem Altitudinem ut þ ad a,
vel in ſubduplicata Ratione #* ad 2. Eſto jam
umbilicus alter H, quaecunque tandem fuerit
Lengitudo PH; et cum Sectiones omnes conicae
ad Elligſin referri poſſunt, ponamus Sectionem
quaeſitam ellipticam eſſe; et Quadratum Semiaxis

3 ——
conjugati aequale erit et Rectangulo S PP H ya,

et Rectangulo S XH Z; ergo S PP Fx

aequale erit 8 Y x H Z: Sed ob ſimilia Triangula
5

SPY, HPZ,ert HZ=SY.x 5 et S XH Z

— 5 e eſt SP-PH . I
— X | \ uaree s 2 ICY 8
® & 8 2 9 do p . — X 4 ; S P

8 P | —
quare SP+P HX +5 5H duatur + 25 . Sed Ve-

locitas in conica Sectione ad Altitudinem quamvis
SE, lt | reciproce in ſubduplicata Ratione Longi-

tudinis 1 „ vel ES per Prop. 16, Li „

Principiorum : Ergo eadem Velocitas eſt etiam
Feciproce in ſubduplicata Ratione Longitudinis

S PP H x 1 : Ergo Velocitas Corporis in Cir-
culo revolventis ad Altitudinem primam SP eſt
| T 3 | ad

278 Tn PAHII OS. NevToN.
ad Velocitatem Corporis e loco ſuo — . exe-
untis, in 2 Jonas Ratione SPP H PII

SPY PS – vel in ſubduplicata — MW

SPTPH ad 2 PH. Eſt ergo 52 ad a2 ut
SP+PH ad 2 PH; 2 ad & ut 28 Pz PH
ad 2P H; vel ut SP–PH ad PH, et (divi-

dendo) 2 52-4 ad a? ut SP ad PH vel ut S 1 ad
HZ. Q. E. D.

Conor. 1.

| Orbita deſcripta Ellipfs erit vel Parabola vel
Hyperbola, prout Quantitas a* minor fuerit, vel
aequalis, vel major quam 233.

Coo: HM.
T2
Si Parabola ſit, erit hujus Latus rectum 4 oy

per Lemma 14, et etiam ex Solutione hujus Pro-
blematis; et Axis per 8 tranſiens erit Recta ipſi
PH *

Conor. III.

Si En ſit vel Hyperbola, erit Ellipfece Semi–
5²

2 52—42

, et Semiaxis Hyperbolac

eaſu ar erit S XxX Vo RA” in poſteriore
6
8 * 42.25.

Porro beg pagan +.

SCHOLIA, 279

Coro. IV. | =

Traque ſi detur Velocitas, et Vis centripeta ad
Altitudinem quamvis SP, idem erit Axis tranſ-
verſus, quicunque fuerit Angulus S PY ad eandem

Altitudinem ; et Axis conjugatus erit ut Sinus
Anguli iſtius S P Y.

Coke, V.

Si producgmur SPQ in Ellipſi, vel PSQ in
Hyperbola donec SQ aequalis fit Axi tranſverſo
Figurae, erit Velocitas in conica Sectione ad Ve-
locitatem in Circulo ad eandem quamvis Altitu-
dinem SP in ſubduplicata Ratione PQ ad 3SQ.

Ca..

In Ellipſi, Velocitas Corporis P in mediocri ſua
ab umbilico Diſtantia aequatur Velgeinatl 1 in Cir-
culo ad eandem Diſtantiam.

Coror. VII.

In Parabola Velocitas eſt ad Veloritstem in
Circulo ad eandem Diſtantiam in ſubduplicata |
Ratione numeri binarii ad unitatem.

CoroLt. VIII.

Cum Velocitas in Circulo fit 4 reciproce
in ſubduplicata Ratione Diſtantiae SP, erit Ve-
locitas in conica Sectione in ſubduplicata Ratione
| Longitudinis PQ directe, et in ſubduplicata

Rectanguli P’S Q reciproce. |

114k

289 In PEHILOS. New Ton:

Conor. IX.
1 in data Figura, Velocitas: erit ubique ut
Ver.
8 P.
| ESS er. te

Adeoque in data Figura, fi detur veſocitas ad
Altitudinem quamvis, dabitur illa ad aliam quam-

u in Conan Xt,

• Parabbla Velccitas eſt reciproce in ſubdu-
  . Ratione Altitudinis SP.

j Conor. XII.

3 Hyperbola Velocitas ad Altitudinem quam-
vis 8P eſt ad Velocitatem in Altitudine infinita i in
ſubduplicata Ratione P Q ad PS.

Co Rol. XIII.
Eadem omnia obtinebunt, ſive Latitudo Orbitac
deſcriptae major fit, five minor, five nulla.
Co Rl. XIV.
Adeoque ſi Velocitates Corporum directe vel
oblique aſcendentium vel deſcendentium in aliquo

4 « ;

• aegualium Altitudinum caſu aequales fuerint, et

ad omnes alias Altitudines aequales, Velocitates

aequales erunt, quaecunque fuerint Latitudines
Figurarum ee

Co ROI.

1 a ., . 4A lad _

.

• uy

SCHOLIA. 28x
Coror. XV.
Si Corpus ſurſum emittatur de loco P cum Ve-

locitate quae ſit ad Velocitatem in Circulo ad ean-

dem Altitudinem, in minori Ratione quam V ad
1, ſeu (quod. eodem redit) fi Velocitas et Vis cen-
tripeta in loco P ejuſmodi fuerint, ut fit a* minor

quam 2 b*, et producatur Recta SPQ, ita ut ſit

.

erit S Q Altitudo maxima Corporis in Recta illa
S8 aſcendentis; et Velocitas ad omnes Alti-
tudines inter aſcendendum et deſcendendum deter-
minabitur * hujus Corol. g et 10.

e e XVI.

Quod ſi tanta fuerit Velocitas in loco P, ut ſit
qe ba, aſcendet Corpus ad Altitudinem infini-

tam, neque unquam redibit; et Velocitas deter-

minabirur ubique per hujus Corol. 11.

Consol. |
Si major adhuc fuerit Velocitas in loco P, Cor-
pus non modo aſcendet ad Altitudinem infinitam,
fed et Motus ſui Partem aliquam etiamtum re-
tinebit. Ut ſi Producatur Recta PS Qs ita ut fit

a*

PQ = ae ** 8 S P, Puncto Qjam exiſtente ad

partes alteras PunRi 8, Velocitas prima in loco P

erit ad Velocitatem ultimam ad Altitudinem in-

finitam in ſubduplicata Ratione PQ ad PS, et in

omnibus aliis Altitudinibus, Velocitas erit ut in
hujus Corel. 12. Cool..

232 InParilos NevrToN,

. Coror. XVII.
Si Corpus de loco quovis Q demiſfum deſcendat
uſque ad Centrum S, Velocitas ad Altitudinem
quamvis minorem SP eft ad Velocitatem Corporis

in Circulo revolventis ad eandem Altitudinem in

ſubduplicata Ratione P Q ad 28 Q.

CoroLt, XIX.
Et Tempus totius deſcenſus, dimidium erit
Temporis periodici Corporis in Circulo ad Alti-
tudinem 4 S Q revolventis. Nam ſpectandus eſt

hic deſcenſus tanquam factus in Semiperimetro

Ellipſeos nullius Latitudinis, cujus tamen Longi-
tudo ſeu Axis tranſverſys eſt 8 .

Coror. XX.

Ergo Tempus totius deſcenſus erit in ſeſqui-
W Ratione Altitudinis totius S *

| SenoLIUM.

Poſteaquam haec conſcripſiſſem, incidi in “I |

| praecedentis Problematis Conſtruftionem paulo
ſimpliciorem quam hic ſubjungere viſum eſt.

1. Sunto Velocitas et Vis centripeta in loco P

ejuſmodi, ut quo Tempore percurritur Longitudo

a uniformi cum Velocitate Corporis in P, eodem

Tempore Vi uniformi acceleratrice quae ubique
aqualis eſt Vi centripetae in loco P, deſcenderetur

Spatium 5, fiatque ipſis s et a tertia proportio-
e eritque SP ad PH, vel S 1 ad HZ ut

S Pad

hy 3 FIC I Ion ET „ mn

S hOLIA. 283
SP—! ad I: Nam Velocitas Corporis in Circulo

revolventis ad Altitudinem SP eſt ad Velocitatem

in canica Sectione ad eandem Altitudinem ut
media Proportionalis inter 2SP ets, ad a, five
mediam Proportionalem inter 4 / et s, hoc eft, in
ſubduplicata Ratione S P ad 21. Sed in Demon-
ſtratione praecedentis Solutionis, Fig. 70, erat
Velocitas illa ad hanc in fubduplicata Ratione

S PNP H ad2PH; quare S P PH eſt ad 2p H

ut SP ad 21; unde erit S Pad PH. vel SY ad
HZ ut 8 Pi ad J. Q. E. D. |

Longitudo Axis tranſverſi eſt Fl

S Ew
jugati 2 SY „ * — et Lateris recti (quod

Axibus tertium eſt Proportionale) 41 al
SP

Porro ſi Longitudo a nunc major, nunc minor
accipiatur, erit Spatium s ut Quadratum Temporis
quo 8 illa ann ; — crit 3

cunque magna vel exigua ſumatur Si a ;
et propterea quaecunque fuerit lex vel Quantitas
Vis centripetae aliis in locis, fi in Recta F VS vel
PSV capiatur Longitudo PV aequalis Longitu-
dini 47 ad locum P pertinenti, Circulus qui tangit
Rectam PY in P tranſitque per V, Curvaturam

habebit Trajectoriae deſcriptae in loco illo P, et

Diameter Curvaturae major erit quam Chorda P
in Ratione SP ad SY, (Vide Prop. 7, Lib. 1,
Prin-

„Axis con-

284 In PRILOs. NEUTOR.

Principiorum) Et fi ſumatur Arcus quam minimu.
Pq in conſequentia, Angulus S P q perpetuo au-
gebitur vel minuetur, prout in Recta PSV Lon-
gitudo S V major fuerit vel minor quam SP, vel
PV quam 2 PS, vel prout Velocitas in loco P
major fuerit vel minor quam quae ſufficiat ad

Circulum deſcribendum ad Altitudinem SP,

quando Angulus S P Q rectus eſt.

Conſequuntur haec ex natura Circuli, pendet

itaque Inctementum vel Decrementum An-

guli S Pꝗ ex Altitudine S P, et ex Velocitate et
Vi centripeta-in loco P; acceſſus autem Corporis
ad Centrum M vel receſſus ab codem, ab his mi-
nime dependent, ſed ex ſola Quantitate Anguli
SP Accedit enim vel recedit, prout Angulus
iſto minor elit vel major Redo |

. Diximus in hoc Scholio augeri vel diminui
en S Pe in conſequentia, prouc Longitudo
PV major fuerit ve! minor quam 2 PS, vel prout
Veiocitas in loco P major fuerit vel minor quam
Velocitas Corporis ad eandem diftantiam S P
eademque Vi centripeta in P Circulum deſcribentis:
Nam Angulus SP gq aequalis eſt Angulo in alterno
ſegmento per Prop. 32, Lib. 3, Elem. et proinde
aequalis eſt Angulo ad Centrum cujus Menſura

ſemiſſis eſt Arcus PV. Augetur itaque vel di-

minuitur Angulus S 4 pro Ratione Ln vel
diminuti Arcus illius.

Feratur Corpus P, Fig. 72, in Orbe circulari
AB . exiſtente Cond S extra Centrum Circuli

&

SCHOLIA.’., Ta
et percurrente Corpore P Arcum CDEFG,

Arcus PqV, et Angulus S Pꝗ perpetuo augetur,

et Longitudo PV major eſt Longitudine 2 PS
per Prop. 7 et 8, Lib. 3, Elem. Percurrente vero
Corpore Arcum GHABC, Arcus PA V, et

Ayngulus S Pq perpetuo diminuitur, et Longitudo

PV minor eft quam 2 PS: par eſt Ratio in Arcu

quolibet Aeque-curvo. Porro, quo Tempore
Corpus P, urgente uniformi Vi centripeta in P
deſcenderet per Longitudinem , idem eodem
Tempore cum Velocitate ſua in P uniformiter
percurreret Longitudinem a, quae ex Hypotheſi
media eſt proportionalis inter 4/ et 5; et eodem
etiam Tempore Corpus ad diſtantiam S P, eadem

Vi centripeta in Circulo revolutum pexcurreret

Arcum qui medius eſt proportionalis inter Dia-
metrum 2 P’S et Spatium 5. Eſt ergo Velocitas
in P ad Velocitatem in Circulo ad diſtantiam S P

in ſubduplicata Ratione 4 /X’5 ad 2P SXs, vel 4]

ad 2PS, vel PV ad2PS. Eft ergo Velocitas

in P major vel minor quam Velocitas in Circulo

ad diſtantiam S P, prout Longitudo PV major
eſt vel minor quam Longitudo 2 PS. —

PROBLEM A. Fig. 70.
Sit S Centrum Terrae, SP Altitudo 10 Semi-
‘diametrorum terreſtrium, SP Angulus, cujus
Sinus eſt ad Radium ut 3 ad 4, et emittatur Corpus
de loco P ſecundum directionem PV, cum Velo-
citate qua uniformiter percurri poſſunt pedes Pa-

rifientes 4665 ua – Tempore unius Minuti ſecundi.

Quaeritur

286 1 PHIL os. N x v TON.
Quaeritur Forma et Pofitio Trajectoriae deſcribendae,
et, fi Ellipfis fuerit, Tempus periodicum.
SGCLUTH0.

1. Pone 4865 1 Da, et erit a: 3669732;

et cum ad Superficiem telluris noſtrae deſcendant

Corpora omina per Altitudinem pedum Parifien-

ſium 1 52 Tempore unius Minuti ſecundi, de-

kendene eadem ad Alticudinem SP per partes
aeg hujus pedis; unde erit 3 et = vel
41 = 156926400, et ] = 39231600.

2. Semidiameter terreſtris eſt pedum Pariſien-
   ſium 196158003 unde erit S P 196158000 eorun-
   dem pedum : Eſt itaque 8 P major quam „ et
   Sectio conica deſcribenda erit Elliꝑſs.
3. Eſt SP ad J ut g ad 1, et SP, ad J, vel
   S P ad PH vel S 1 ad HZ ut 4 ad 1: Eſt itaque
   P H=2 + Semidiametris terreſtribus, et S PP H

five Axis major Ellipſeos aequalis 12+ Semidia-

metris.

4. Longitudo perpendiculi S Y eſt 77 Semi-
   diametrorum, et Longitudo perpendiculi H Z 14

Semidiametrorum; unde eſt S V XH Z ſeu Qua-

. dratum Semiaxis minoris _ et Axis minor erit

7: Semidiametrorum terreſtrium; unde Longi-
tudo Ellipſeos erit ad Latitudinem ut 5 ad 3.

5. Si a Quadrato Ala. mos — ſubducatur

Quadratum Axis minoris — , reſtabir Quadratum
“En Longitudinis

SCHOLIA, 287
Longitudinis 8 H = vel 100; unde erit SH=10

Semidiametris, et * dimidium ; eſt diſtantia
umbilicorum a Centro Ellipſeos : Eft itaque diſ-
tantia Apogaei a Centro Terrae 115 Semidiame-
trorum terreſtrium, et diſtantia Perigaei 13.

6. Cum dentur omnia Latera Trianguli 8 PH,
   invenietur Angulus PS H 14 22“; unde dabitur
   poſitio Lineae Ap/idum in antecedentia, vel in con-
   ſequentia, C conſtituendae, prout 2 SPY
   eſt minor vel major Reco.
7. Jam vero ut inveniatur Tempus periodicum
   in hac Orbita, animadvertendum, Corporis juxta

Terrae Superficiem in Circulo revolventis Tempus
periodicum inventum fuiſſe alibi unius horae,
24 minutorum primorum, 27 ſecundorum, hoc eſt,
5067 ſecundorum. Dicendum itaque ut Cubus
Diametri Orbitae circularis ad Cubum Axis tranſ-
verſi Ellipſeos, ita Quadratum Temporis periodici
prioris ad Quadratum Temporis periodici poſte-

rioris. Sunt autem dicti Cubi ut 8 ad —
vel ut 64 ad 25 25 23, et horum numerorum
Radices quadraticae ſunt ut 8 ad 5X5X5 vel ut 8
ad 125. Dic itaque ut 8 ad 125 ita 5067 minuta
ſecunda, ſive Tempus periodicum in Circulo ad
79172 minuta ſecunda quae conſtituunt Tempus
periodicum quaeſitum. Abſolvit igitur novus hic
Planeta periodum ſuam circa Centrum telluris
Spatio horarum 21, minutorum f 59,

ſecundorum 32. 3 E

In |

NF * ® = –
— — — – – ATT “CH n — 1 2 er ee ee . x — — \ rr 6 4 ” > — X 44S wi — — — — — ea

x r . 1 > *
err 0
r

• * — py – – — of

|;
|
}
[

, ” 7 ORs

— — — API 27 eo > Ion

288 MPrilos NEevuToON.

In Prop. 44. Vide Fig. apud auctorem.
Centro C, Inter vallo C n deſcribatur Circulus ſecans
Lineam min productam in t.

Quamvis Recta 7 m infinita ſit, tamen Angulus
7 C m nunquam major erit Recto; adeoque Recta

7 merit ad Rectam & in majori Ratione quam eſt

Angulus r Cm ad Angulum r Ck: Eſt autem m
ad 7 k ut Angulus r C x ad Angulum TC &; quare
Angulus r C » major eſt Angulo rCm. Sed quo

propius Punctum m accedit ad Punctum r, eo
minus erit Diſcrimen Angulorum Cm, CA;

coeuntibus autem Punctis m et r, aequales erunt
Anguli rCm, 7 Cn, hoc eſt, Punctum » jam ſitum
erit in Recta C m; id adeo, propterea quod in hoc
caſu eſt rm ad 7k ut Angulus Ce, vel Cn

ad r CE.

In Corol. 2. ejuſdem Prop.
. Vis autem qua Corpus in Circulo, &c.

Adhibita Notatione Neutoniana, poſitoque i in-
ſuper quod Velocitas Corporis P in Vertice fit c,
ſic argumentor. Vis oentrirere Corporis P in

Vertice V exponitur per Sed Curvatura

SV
Ellipſeos in vertice V eadem eſt cum Curvatura
Circuli cujus Semidiameter eſt R, dimidium ni-
mirum Lateris recti, ut patet ex Conicis: Ergo ſi
Corpus Circulum deſcribat ad Diſtantiam R eum

Velocitate 6 erit Vis centriperac in hoc Circulo

aequalis

SCHOLIA. 1
aequalis Vi centripetae in Vertice V; ergo Vis
centripeta imhoc Circulo erit 8 Revolvatur

2
jam Corpus aliquod cum Velocitate c in Circulo
ad Altitudinem CV, et Vis centripeta in priore

Circulo erit ad Vim centripetam in Circulo poſte-
2 .

EC Of
riore ut N ad S ber Corol. 1, Prop. 4, Lib. 1,

1 | 2 131
Principiorum; hoc eſt, ut Rad Y * ergo, ut
The” ad ita L 3. _ erit Vis

C yi C V* CV 3 V3
centripeta Corporis Circulum deſeribentis cum
| c ad Diſtantiam C V.

Lemma ad Prop. 45.

Si Quantitates duae ax et þ-+y componantur
ex partibus datis à et h, et ex partibus non datis,
ſimul tamen naſcentibus, vel ſimul evaneſcentibus
x et y; fuerit autem a+x ut >y dico fore x

ad y ſemper ut à ad 3. Etenim cum ponatur |

ax ut by, erit a+x ad b-+y ſemper in data
Ratione: Naſcentibus autem vel evaneſcentibus
xety, eſt a+x ad by ut à ad 5; quare a+x
erit ad b-+-y ſemper ut a ad 5; adcoque’ x erit ad y
ſemper ut @ ad b. QE. D.

290 N PRHIITIOS. Nev ToN,

Ad Prop. 45.

Ut in Corollariis ſecundo et tertio.} dt Vis centri-
peta ut Altitudinis Dignitas quaelibet, cujus Index
eſt n—=3: Quaeritur Angulus inter Apſidem ſummam
et Apfidem imam, ſecundum Hypothe/im Corollarii tertii,
ubi nimirum Centrum Virium coincidit cum Centro com-
muni Ellipſeon. | |

Secundum hanc Hypotheſim, adhibita Notatione
Neutonians, Vis centripeta eſt ut == —.—
Scribatur 1 pro T vel R, et Vis centripeta jam erit

G6 -* F2A4þ-G?* . F
OT A3 „ hoc eſt ut AZ wow >
Eft autem A4=1i—4X, Sc. et F:A+—=F2z—4F2X,

: &c.quare Vis centripeta eſt ore . 25 =
G2—4 FX 22 1 r
— ponitur autem Vis centripeta

ut A3 vel I quare A“ eſt ut S :

Sed A. 1 X, Sc. quare G’—4F*X eſt ut
1 X; unde eſt G“ ad 1 ut 4F“ ad »; unde
| EO F

G*u=4F?, et G. , et G . Sed in hac
Hy potheſi eſt F go Grad. cum fit go Grad. An-
gulus inter Apſidem ſummam et Apſidem imam
in Ellipfi quieſcente : Quare Angulus inter Apſi-

dem imam, ubi Vis centripeta eſt ut A3, . erit
180 To

Js idem nimirum qui prodit ex Hypotheſi
Corollarii ſecundi. Ad

Ad Gon 10 Prop. 45.

nn
Quantitas 2, exponens dignitatem Alti-

14400
tudinis eſt in caſu propokito – = 7541 —3 Eſt etiam
. | ; 2

14042. r, et fi Seb utrobique .,
       464k 25 241 „
       manebit— =1— et —— = I
       14641 14641 m m
       241 241

• 3 2 _— z ergo Vis cen pes
  eſt reciproce ut Altitudinis dignitas þ 2+ =.
  Fractionis –

1 5
— divide tum Numeratorem tum
I4041 | #
Denominatorem per 241, et prodibit Gora
| | | 2 7 17

Sed 34+ non multum abeſt ab 223 vel + 3 quare
Vis * eſt reciproce ut Altirudinis digits

2+ – 2 TNT : mow ab Indice 2 differentia eſt A 8 IT :

60
Sin 2 ＋ 88 — ſubducatur de 3, vel de 2 +557

manebit 25 ; quare differentia polibengy eſt ad

priorem ut 59+ ad 1,

Ad Prop. 58,

Si Velocitates in P et p fuerint in ſubduplicata
Ratione CP ad 5p, Corpora P et p deſcribent
Arcus ſimiles in Temporibus quae erunt etiam in

G | ſub-

SCHOLIA. 291

292 AUMnPnilos NRUTON.
ſubduplicata illa Ratione; et fi Velocitates in Qet
7 fuerint ut C Q ad v/sg, vel ob ſimilitudinem
Arcuum ut /CP ad p, Corpora P et p per-
gent Arcus ſimiles Temporibus proportionalibus
deſcribere. Videamus ergo an Velocitates in Q et 3
ſint ad invicem in hac Ratione. Anguli CPQ etspy,
ob ſimilitudinem ſitus Corporum P et ↄ in Curvis
ſimilibus, aequantur; ergo Vires centripetae, quibus
haec Corpora in IL. ineas curvas detorquentur, ſimi-
liter applicantur: Sed Vires aequales ſimiliter ap-
plicatae, Velocitates generant Temporibus pro-
Portionales; ergo Incrementa vel Decrementa Ve-
locitatum in locis Q et q ſunt ut VC P ad Vp.
Addantur haec Incrementa vel ſubducantur De-
crementa Velocitatibus in P et p quae ſunt etiam
ut VCP ad , et prodibunt Velocitates in Qet?
etiam in eadem illa Ratione. Sunt ergo Veloci-
tates in locis omnibus ſimilibus ut VC P ad Vp.

2 Ad Prop. 60.
Minuatur Tempus periedicum in Ratione
SP ad /s; et ſimul Axis major Ellipſeos
deſcriptae in Ratione A ad a, et erit SP ad
s ut AZ ad 42 (nam Tempora periodica ſunt in
ſeſquiplicata Ratione Axium A et @) ; ergo erit
SP ad S ut As ad a3. Sunto m et n mediae
proportionales inter SP et 8, ita ut ſint SP,
m, u, 8, continue proportionales, et Ratio SP
ad S aequalis erit Rationi SEP ad m triplicatae,
hoc eſt, erit SEP ad 8 ut Sn ad m3 : Erat

autem

Seo
autem SP ad Set A3 ad 43; quare A3 erit ad
a3 ut S PIM ad ms, . A 4 4 ut S+P ad m.

E

Ad 3 61. partem alteram.

Moveantur duo Corpora S et P circa commune
Gravitatis Centrum C, et nominetur S P, x, et

C P, z; et ponantur Corpora S et P aequalia, ita
ut ſit x 2E; exponamus denique Vim centripe-
tam qua Corpus S trahit Corpus P per Quantitatem
aliquam compoſitam, qualis eſt Ax+Bx?, habitis
A et B pro Quantitatibus quibuſlibet datis, et ſi
exponatur lex A xJ-Bx: per poteſtates ipſius x,
evadet 2A K ABG vel poſitis a et & pro 2 A et
4B, lex Vis centripetae ad commune Centrum C
tendentis fiet @ z-b 22; et Quantitas a z-+b 2
ſemper aequalis erit Quantitati A x+B x. Statua- tur jam Corpus aliquod in communi Centro C, quod trahat Corpus P Vi centripeta az—-b2
Corpore S jam deſtrudts ; et Corpus P jam tra-
hetur ad Corpus i in Centro C conſtitutum omnino
pariter ac prius, tum quoad directionem, tum quoad
Quantitatem Vis centripetae.

| Ad Prop. 64.

LEMMA,

Si hn Corpus aliquod L circa Centrum D
cum Vi centripeta quae ſit ut A Xx DL, Tempus
periodicum erit reciproce in ſubduplicata Ratione
Quantitatis A, Nam (per Corel. 2. Prop. 10.)

1 Tempus

294 In PII OS. NEUTON.

Tempus periodicum idem erit ac fi Corpus L.
Circulum deſcriberet circa Centrum D ad Diſtan-
tiam quamvis DL: Deſcribat ergo; et ſi Tem-

9. 
   pus periodicum vocetur T, erit Ax DL ut T.

(per Corol. 2. Prop. 4); unde erit T2 ut — et T

ut IN E.. | |

Poſitis quae in Prop. 64, trahat Corpus unum-
quodque Corpora reliqua cum Vi acceleratrice,
quae ſit ut Corpus trahens et diſtantia Corporum
attractorum conjunctim; et exponetur Vis qua
Corpus T trahit Corpus L per Quantitatem
TXT L; cumque ſit L ad T ut I D ad DL, erit
(componerdo) TL ad Tut TL ad DL; unde
‘Tx TL aequalis erit 14-LxDL: Trahitur
ergo Corpus L ad Centrum commune PD, perinde
ac ſi Corpus T tolleretur, et aliud Corpus T+L
in communi Centro D immotum conſtitueretur;
et Tempus periodicum Corporis L circum D erit
ad Tempus periodicum Corporis L circum T im-
motum in ſubduplicata Ratione Corporis T ad
ſummam Corporum TL. Accedat jam Corpus
tertium S, et Vis qua Corpus S trahit Corpus T
exponetur per S x S T aequalis Viribus S%X TBA.
SxS D: Similiter Vis qua Corpus S trahit Corpus |
L, aequalis erit Viribus S LDA SSB: Quare
ſumma Virium qua Corpus S trahit Corpora T et
Leſt SXxXTDþSxXDL+SX25SD ; Sed Vis
SxX2SD minime ꝓerturbat Motum Syſtematis ;

| | | ut

/

S

. SCHOLIA.
ut demonſtrat Neutonus; et Vi TTL xD L qua

Corpus L trahebatur ad Centrum D ante Acceſſum

Corporis S, jam additur Vis SX DL, ita ut Vis
rota qua Corpus L trahitur ad Centrum D jam fit

STT L NDL] minuitur ergo Tempus periodi-

cum Corporum T et L circa commune ipſorum

Gravitatis Centrum D per Acceſſum Corporis S in

ſubduplicata Ratione S + T +L ad TL; et
Corpus alterutrum puta L trahitur ad Centrum D
perinde ac fi Corpora S et T ſubmoverentur, et
Corpus STL in communi Centro D collo-
caretur. Similiter ſi accedat Corpus quartum V,
trahetur Corpus L ad Centrum D, perinde ac
ſi Corpora V, 8, IT ſubmoverentur, et Corpus
V4-S–T-EL, in Centro D collocaretur, et Tem-

pus periodicum Acceſſy Corporis V minuetur in
ſubduplicata Ratione VS -T, L ad STL.

Haec ſunt Phaenomena motuum binorum quo-

rumcunque Corporum circa commune ipſorum

Gravitatis Centrum: Videamus jam Phaenomena

motuum reſpectu reliquorum Centrorum C et B.

Corpus S trahitur ad Corpus T Vi centripeta

TxTS aequali Viribus TXT DTxS D; |

ſimiliter Vis qua Corpus S trahitur ad Corpus I,

aequatur Viribus LXL D-|LxSD; quare ſumma

Virium quibus Corpus S trahitur a Corporibus T
et L, et TxTD+LxXLD+T+LxSD.
Sed Vires TX TD et LXL D aequales cum ſint

et contrariae, ex Natura Centri Gravitatis ſe muruo

V4. deſtruunt;

295

= – — — HEY
— * — K TI TY. wh» * 6
— by — oy — — — GUI IE VU dy by
„ 2 — 9
——ůͤů—— — _ *
— —

296 | In Pnitos. Nev Ton.

deſtruunt; manet itaque Vis T + L.x SD qua
Corpus S trahitur a Corporibus T et L, pariter ac
fi Corpora illa T et L in unum coaleſcerent, et in

communi Centro D locarentur. Porro cum ſint
ex Natura Centri Gravitatis S ad TAL ut C Dad
_ CS, et componendo STL ad T–L ut SD

ad S C, erit TL xSD=S4+T+L.xSC; ergo
Corpus S trahitur ad commune Centrum C cum Vi
SETFLxXSC. Accedat rurſus Corpus quar-
tum V, et Vis qua V trahitur ad S erit S x VS =
S&S C -S XxVC; et Vis qua Corpus V trahitur a
Corporibus T et E erit TEL x CD4-T-+LXVC
uti ante dictum eſt ; quare Vis tota qua Corpus *
trahitur a Corporibus S, T, L, eſt SxSC+
T+LxXCD+S+T+LXVC: Sed Vires
SXS Cet T+LxCD ſe mutuo deſtruunt, ut
prius ; quare Vis qua Corpus V trahitur a Cor-
poribus 8, T, L eft SET+LxVC: Eſt autem
VadS+T+LutBC ad VB, et componendo,
VS+T-EL eft ad SEIT IL ut VC ad VB:
Eſt ergo S+T+-L x V C=V+S+1+Lx VB.
Trahitur ergo Corpus unumquodque V ad com-
mune omnium Centrum B omnino ſimiliter ac ſi
Corpus VESETAL aequale nempe toti Syſte-
mati, in communi Centro B conſtitueretur ; et
Vires abſoſutae ad Centra B, C, D, tendentes
aequales ſunt inter ſe. © Sunt ergo Tempora peri-
odica tum Corporis unius cujuſque circa Centrum
B, tum trium quorumcunque circa Centrum C,
1 tum

S HOL IA. 297

tum duorum quorumcunque circa Centrum 1
aequalia inter ſe; et ad Tempus periodicum Cor-
poris L circa Centrum immotum T in ſubdu-
plicata Ratione Corporis T ad totum Syſtema
VS -T TL. Trahant jam Corpora T et L ſe
mutuo Viribus vel majoribus vel minoribus quam
quibus trahunt caetera, et Tempus periodicum
Syſtematis T-L circa Centrum D minuetur vel

augebitur in ſubduplicata Ratione Virium auctarum
vel diminutarum; ſed Motus reliquorum ex hac

Virium intenſione vel remiſſione minim Pertur-
babitur.

Prop. 66. Corol. 6. Fig. 73.

Revolvatur Corpus P circum T Tempore a,
exiſtente mediocri diſtantia TPS d; dein per
actionem Corporis S minuatur Vis centralis Cor-
poris T in Ratione 7 ad s, et per hanc diminutio-
nem efficietur ut Corpus P jam revolvatur circum
T ad majorem Diſtantiam; quam Diſtantiam in
mediocri ſua Quantitate vocabimus D, et majori
etiam Tempore periodico quod ſit c; quaeritur
Ratio inter à et c.

Fingamus aliud Corpus circum T revolvens ad
Diſtantiam D Tempore 6, inter quod et Corpus 8
nulla intercedit attractio: quo polito erit Tempus
a ad Tempus. + ut d ad D4. Sed per 4 4.

Corol. 2. – eſt ad = ut r ad 5; adeoque b erit ad

c ut /s ad wr. G hae Rationes, et

prodibit a ad c ut d A x, ad D XVr.

298 In PRHIL OS. NRERUTON.

F
FS

| In Coral. 10 et 11,
Orbitam PAB Linea Nodorum in duos Semi-

circulos dirimit, quorum uterque binas habet facies,

alteras plano ES T inclinatas, alteras reclinatas;
et quanquam Vis MN agat ſemper ſecundum
rectas plano EST parallelas, dicemus tamen in
ſequentibus agere eam ad planum vel à plano
EST, prout ab inclinata vel reclinata facie im-
mediate dirigatur. Dicemus etiam motum Cor-
poris P fieri ad planum vel d plano EST, prout
Corpus illud ad Ned:m proximum vel d Node
proximo tendit. His praemiſſis Effata quaedam
proferemus, quibus ad Corollaria ſupra memorata
aditus ampliſſimus pateat, quae alias vix aut ne
vix quidem intelligi poſſunt. Horum vero De-
monſtrationes ad Macbinulam quandam ſimpliciſ-
ſimam, et huic potiſſimum Negotio Confiractan,
referemus.

N. B. Canſtat “Ws Machina ex duobus i
circularibus plano cuidam planum ES T referenti
inclinatis, et adglutinatis, et ſe mutuo extra hoc
planum in Angulo quam fieri poteft minimo de-
cuſſantibus.

1. Si vis MN et Motus Corporis P ejuſdem
   fuerint affectionis, hoc eſt, ſi fiat uterque ad pla-
   num vel à plano E58 1 inclinatio Orbitae P A B,
   hoc eſt, Angulus inclinationis perpetuo augebitur: :
   Alias minuetur.
2. Vis M N, et motus Corporis P, ejuſdem fone
   | affectionis in locis Orbitae PAB oppoſitis, et

SCHOLI A, 299

propterea quicquid praeſtant in uno Semicirculo,
idem praeſtabunt in altero.

3. Linea Quadraturarum fertur in conſequentia
   eadem Velocitate cum Linea Syzygiarum, idque
   ſive motus iſte motui Corporis S circum T vel
   Corporis T circum S debeatur.
4. Inter Quadraturam et Nodum proximum,

Vis MN agit 4 plano EST, aliis in locis ad
planum. |

5. Si Vis MN agat ad planum E whe Nodi

regrediuntur, alias progrediuntur.

In Corol. 14.

Quo melius intelligantur ea quae i in hoc Corol-

loraria tradita ſunt, Longitudines L M, MN
quibus Vires omnes perturbatrices Corporis P ex-

ponuntur, primo ſunt inveſtigandae. Nam ſi detur

Diſtantia S T, accipi poteſt Longitudo SK aequa-
lis ipſi S T; quo in caſu erit etiam LM in me-
diocri ſua Quantitate ãequalis Diſtantiae PT; ſin
augeatur Diſtantia S T, neceſſe erit ut minuatur
Longitudo SK; adeoque neque Longitudines

SK, ST, neque Longitudines LM, PT am-
plius pro aequalibus habendae ſunt, ſed eruenda

eſt Longitudo LM ex ſimilibus Triangulis S LM,
SPT per ſequentem Analogiam, nempe SP eſt

SLXPT
ad PT ut SL ad LM, adeoque L Ma

| Verum quoniam S L nunc major eſt, nunc minor
quam SK, et ab eadem SK nunquam aberrat
ſenſibiliter, 3 SL per mednacrem ſaam

Quan:

1 .
P r ** TIC
— n . hy . l

300 In PRILOs. NRU rox. 5
Quantitatem 8 K, et ſimiliter S P per S T, et pro-

dibit Longitudo LM=2 LN ND
Jam vero ut exquiramus L ongitudinem MN,
jungatur K N, et ad Rectam S T. perpendiculares
ducantur LX, PI, et ſi Orbita P AB circularis
fit, vel propemodum circu:aris, aequales erunt
SK, SN, et Anguli SKN, SNK tantum non
erunt recti propter Angulum ad S tantum non
evaneſcentem : Quare Rectae K N et LX pro
parallelis habendae ſunt ut et Rectae K L, NX
pro aequalibus et parallelis: Sed et Rectae S P, SI
erunt etiam aequales ob Angulum ad I rectum ;
quare cum ſit SK ad SL ut SP ad ST}, erit
etiam SKad SLuUtSFadsT;, vel ut SI ad
SI–2IT, et dividendo SK erit ad K L ut SI ad
2IT : Prodit ergo K L vel NX CIT
SKx2IT . 81
51 + Rurſus, propter ſimilia Triangula
SLMet SPT, LMXet PT I, erit MX ad IT
ut LM ad PT, hoc eſt, ut SL ad SP; quare

SLXIT skxIT.
e n ——_ quare Lon-

MX aequatur

gitudo MN ſeu x IX N ert L. Et

univerſaliter LM erit ad MN ut Radius ad
Triplum Coſinus Anguli PTS; et in dato ſitu
Corporum 8, T, P, ad invicem, ubi eſt 3I Tut
PT, ob datum Specie Triangulum PT I, erit
N | MN

SCHoOLIA, | 301

Fi
MN ut 8 K x quare Vis L M univerſaliter,

et Vis MN in integra Revolutione Corporis P

circum T, «ft Ut SK ST.

His expeditis, fit V Vis abſoluta attractiva
pgs

V
Corporis S, et s K erit ut ==; “_ S KN NF,
ſeu Vires LM, MN erunt ut N z unde in

dato Syſtemate Corporum P, T, erunt Vires

LM, MN ut = Sit jam J Tempus peri-

odicum Corporis T circum 8, et per Corol. 2.

• ST V I

Prop. 4. hujus, erit ST ut Tr, et 5 T; uta;
ergo in dato Syſtemate Corporum P, T, Vires
LM, MN ſunt reciproce ut Quadratum Temporis
periodici Syſtematis. circum S revolventis. Poſtre-
mo, fit D Diameter-Corporis S, et Diameter ejus

D
apparens ex Centro T ſpectata erit ut , et Cubus
3

Diametri apparentis ut ST.. Sunto V et Pz
3
ſemper proportionales, hoc eſt, Magnitudo Cor-

poris S et Vis abſoluta attractiva vel maneant vel

V
mutentur — eritque T5 SITh

hoc eſt, Vires LM, MN erunt ut Calo Dia-

metri * Corporis S ex Centro T ſpec-
tatae.

In

0,

muroam————uo——————@w_@=_w———_———————————— — —C— *
” A LSE 12 * Vn .

302 In PHIL OS. Ntvros.

1 Corol. 1 5.

Sit V Vis attractiva Corporis S, v Vis attractiva
Corporis T, D Diameter Orbitae Corporis T cir-
cum 8 revolventis, d Diameter Orbitae P A B,
T Tempus periodicum Corporis T circum 8, .
Tempus periodicum Corporis P circum T, et per

Corol. 2. Prop. 4. erit >> ut V, et Jy ut T-; unde |

erit T ad t ut x ad 75 vel ut Dv ad Vd, hoc eſt;
T- erit ad 72 in Ratione compoſita ex Ratione D
ad det Ratione v ad V. Manente Orbium pro-
portione mutentur eorum Magnitudines, et vel
maneant vel mutentur proportionaliter Vires V
et v, et propter ſervatas Rationes componentes
inter D et d, et v et V, ſervabitur Ratio compoſita
inter T* er Y, eritque T ad t in eadem Ratione
qua prius. Tangat jam Recta PR Orbitam PAB
in loco P, et a loco proximo Q age QR Diſtantiae
P T parallelam, et quo Tempore Corpus P ſola
Vi infita percurreret Tangentem P R, vel addita
Vi v percurreret Arcum quam minimum PQ
eodem Tempore idem P totis Viribus L M, MN
et v percurrat Arcum P x, et’ Lineola Q x erit
Effectus Virium LM, MN.
MManentibus Orbium Forma, Proportione, et
Inchinatione ad invicem, mutentur eorum Magni
tudines, ut et Magnitudo Arcus P Q in eadem
Ratione, et Vires Vet v vel maneant, vel muten-
4 | rur

SCHOLIA;

tur etiam in data aliqua Ratione. Detur poſtremo
ſitus rum Orbitarum ad invicem, tum Corporum

S. T, P, in his Orbitis, atque ob datum et ſitu et

ſpecie Triangulum SP T, et datam Rationem Vi-
rium, Lineola Q ſemper erit ſimiliter ſita quoad

hoc Triangulum : Erunt etiam in hoc caſu Vires
L M. M N ut Vis v, et Lineola Q x ut Lineola
fimul genita QR, et Errores omnes ex Lineola
Qx oriundi, ut Qx vel QR, vel ut Diametri
Orbitarum; atque adeo reddentur proportionales.
Denique ob datam Rationem inter Arcum PQ et

totam Orbitae PA B perimetrum, Tempus quo

generatur Qx erit ut Tempus periodicum Corporis
P circum T, vel Corporis T circum S. Sunt itaque
Errores ſimiles lineares ut Orbitarum Diametri, et
propterea Errores angulares ex Centro T ſpectati
aequales; et Errorum linearium ſimilium vel An-

gulorum aequalium Tempora ſunt ut Tempora
Pee

In Corol. 16.

Sit T Tempus periodicum Telluris noſtrae circa
Solem, # Tempus periodicum Lunae circa Cen-
trum Telluris, R Tempus periodicum Jovis circa
Solem, a Tempus periodicum Satellitis alicujus
circum-jovialis circa Centrum Jovis, et propofitum
fit Errores omnes Satellitis ex analogis Erroribus
lunaribus Ope Corol. 14. et 15. derivare.

Manentibus Orbita et Tempore periodico Satel-
titis, deturbetur Jupiter de loco ſuo, et ſtatuatur

ad Diſtantiam a — quae fit ad Diſtantiam Satel-

litis

303

;

L

4

E

: £
1
|
|

7 |

304 I PHIL OS NevuTon.
litis a Centro Jovis ut Diſtantia Terrae noſtrae a
Sole ad Diſtantiam Lunae a Centro Terrae.

Sit S Tempus periodicum Jovis circa Solem ad
hanc Diſtantiam revolventis, eritque per Corel. 15.
rad S ut f ad T; unde 5 aequalis erit T, et
1 *

SSS TFT Jam vero ex py 15. manifeſtum
eſt Errores omnes in Motu Satellitis Tempore r
commiſſos, aequales eſſe Erroribus ſimilibus in

motu Lunae commiſſis Tempore 7. Referat ita-
— 7

que = Quantitates pens, periodicorum luna-
ET et huic acqualis 88 exhibebit Errores ana-

logos aequales i in motu Satellitis. Reſtituatur jam
Jupiter in locum proprium, et per Corol. 14.
mutabuntur Quantitates Errorum in reciproca
duplicata Ratione Temporis 3 Jovis circa

6 |
“_ hoc eſt, in Ratione — I = ad RR Quare
KR lm exponet Quantitates Errorum Satellitis,
hoc eſt, Errores periodici Satellitis erunt ad Errores
1 12 omnino ut
EK TT
in Corol. x6. 16. alia Ratione demonſtratum eſt.

Vires L M, MN, caeteris flantibus, ſunt ut P T.]

Nam Vires LM. MN ſunt ut — Sed
fi detur S T, dabitur SK. ob datam Vim abſolutam
Corporis

| analogos Lunares, ut —

SCHOLIA, TT

Corporis 8, quo in caſu, dabitur S. et Vires
LM, MN erunt ut | us

Et motus uterque erit ut Tempus periodicum Corporis
P directe, et Quadratum Temporis nn Cor-
poris T inverſe.

Motus periodici ſunt qui Spatio unius Revolu-
tionis peraguntur, adeoque ſunt ut motus con-
temporanei et Tempora periodica conjunctim;
unde fit ut motus contemporanei ſint ut motus
periodici directe et Tempora 1 inverſe.
Sed motus periodici ſunt ut 1 ergo motus

contemporanei erunt ut 7 T

In Corol. 17.

Sit V Vis attractiva Corporis 8, v Vis attrac-
tiva Corporis T; fit T Tempus periodicum Cor-
poris I circum 8, Tempus periodicum Corporis
P circum T, et Vis mediocris LM erit ad V ut
S =
PTadST: Eft autem Vad v ut 55 ad ——
quare Vis mediocris L M eſt ad Vim vin Rub

compoſita ex Ratione PT ad ST, et ex Ratione

• ad —— 2 E, Sed hae duae Rationes componunt

TPxST TPRSE
Rationem TY ad = 77

, hoc eſt, Ra-

tionem 11 ad 25 vel tt ad TT; quare Vis

mediocris LM eft ad Vim qua Corpus P retinetur
docris LM * in

206 In PRILOS. NEUTOR.
in Orbe ſuo ut Quadratum Temporis periodiei

Corporis P circum T ad Quadratum Temporis
periodici Corporis T circum S.

Ad Prop. 1 Lib. 3.

Periodus Lunaris reſpectu fixarum eſt 27 die-
rum, 7 horarum, et 43 minutorum primorum;
ſeu minutorum primorum 39343: Sed Vis qua
Luna in Orbe ſuo retinetur minor eſt quam ſi a
Terra ſolum manaret in Ratione 1772 ad 17822,
ſeu 7109 ad 71493 adeoque fi periodus Lunaris
deſideretur, qualis ab Attractione ſola Telluris
prodiret, minuenda eſt periodus 39343 minutorum
primorum in difta Ratione ſubduplicata, nempe in
Ratione 7139 ad 7119, et prodibit minutorum
primorum 39233. Sic itaque procedit calculus.

Ambitus Terrae 123249600 pedes Pariſienſes,
Peripheria Orbitae Lunaris 7394976000, Arcus
quem Luna medio Motu (ut ſupra correcto) Tem-
pore unius minuti primi deſcribit 188489, hujus
Quadratum 35528103121, Diameter Telluris
392 31600, Diameter Orbitae Lunaris 2353896000,
Quadratum Arcus Tempore unius Minuti primi
deſcriptt ad hanc Diametrum applicatum 1 5,094 ;

quae Longitudo eſt pets Pariſienſium 1551
Digiti, et 15 Lineae. |

5 \ Them aliter.

Periodus Lunaris abſque Diminutione $0343 ©
minutorum primorum, Orbita ben 7394796,
„ – | Arcus |

DCROLITA. 397
Arcus Tempore minuti unius primi deſcriptus
187962, Quadratum Arcus 35329713444, Dia-
meter Orbitae Lunaris 235389600, Spatium rec-
tilineum per quod deſcenditur Tempore Minuti
unius primi apud Lunam, priuſquam corrigatur

15,009037 pedum, Spatium correctum 1 5,092.

Vis qua Luna retinetur in Orbe ſuo, oritur
potiſſimum ab Actione Terrae, ſed nonnihil etiam
ab Actione Solis; eſtque haec Vis compoſita mi-
nor quam fi a Terra ſola manaret in Ratione 17722.

ad 1784; quare fi deſideretur Gravitas Lunae ad
ſolam Terram, augendum eſt Spatum 15, 09037
. 22; atque haec eſt

Correctio ſupra memorata. Vide Corol. I 3.
Lib. 3.

Actiones Solis et Lunae in Aeſtibus marinis
excitandis eſſe, caeteris paribus, ut Vires
eorum abſolutae directe, et ut Cubus Di-
flantiae eorum a Centro Terrae inverſe.

Fig. 74.

Deſignet S Centrum Solis vel Lunae, T Cen-
trum Telluris, ZN et HO Diametros terrreſtres ſe
mutuo decuſſantes ad Angulos rectos in Centro T;
ita ut locus Z habeat Punctum S pro Zenith:
Deſignet etiam V Vim abſolutam attractivam Cor-
poris 8, et Attractiones in locis Z, T et N erunt

Attractionis in Z ſupra Attractionem 1 et

X 3 At-

Vf |
ut et = reſpective: et exceſſus

308 In Purtrtos NEvTon.
Attractionis in T ſupra Attractionem in N, erunt
V V WW
7: F et 55; — SK: eſpective; ergo Vis
qua elevantur aquae in loco Z erit a :
v v S2. 8 Tz
vel S SN prout conftirutum tuerit Corpus
S in Zenith vel Nadir hujus loci: Cape harum
Virium ſemi-ſummam, et habebis Vim medio-
crem qua excitantur aquae in loco Z, abſque ulla
habita Ratione ſitus Corporis S, five in Zenith

V 1 1
verſetur, five i in Nadir, nimirum *

1 SN
RET V 9 iN yr NT xSN—SZ
28 Na Xx SZ 5 IS
ey Vx2STxZN VXSEXEN

SNS el SNS Z el
r TVTouzhN _ n
—— vel r nam cum Longi-
tudo S T permagna fit prae Longitudine Semi-
diametri T Z erit Rectangulum SN X SZ ad
Quadratum Diſtantiae S T in Ratione Aequalitatis
your proxime. |
Porro, fi Vis attractiva Corporis S in loco H
exponeretur per Longitudinem S H, Vis qua de-
primuntur Aquae in loco H exponenda eſſet per
| Longitudinem H T; nam Vis SH idem valet
atque e Vires HT, TS; quarum Vis TS, trahendo
particulas Aquarum ſecundum Rectas ipfi T8
parallelas, nihil confert ad depreſſionem Aquarum,
dum altera Vis H T tota impenditur in hoc ne-

gotium:

at the, end.

5

S Hö LA.
gSotium: Sed Vis attractiva Corporis 8 non ex-
ponenda eſt per Longitudinem S H, ſed per Quan-

309

= VxXSH VxSH

titatem 5 8 vel D B V = ; (nam
cum Longitudo TH quam minima fit quoad
Longitudinem S T, et ponatur Angulus STH
rectus, Ratio S H ad S T perexigua erit fi con-
feratur cum Ratione S T ad S Z, atque adeo pro
Ratione Aequalitatis tuto haberi poteſt); ergo
depreſſio Aquarum in loco H, non amplius ex-
ponenda eſt per Longitudinem H T, fed per

. VxaHT a

Quantitatem —8 f z unde et ex computo priore
facile perſpicitur, Elevationem Aquarum in loco
Z duplo majorem eſſe, quam eſt depreſſio earum
in loco H; et Actionem utramque (caeteris pari-
bus), ficut et hinc pendentes Aeſtus marinos, eſſe
ut Vis abſoluta attractiva Corporis S directe, et
Cubus Diſtantiae S T inverſe. Q. E. D.

5 1 N 1

— r
F DG — * D.
if MN. * Wir ry Y
4 F * Y — * A *
” 2 7 * 8 A

• 8 — I. * CS
  ; ” a4. , <* % & . * 9 : i >3 3 CL
  þ J> y = \ *
  . * a.
  3 ** — 2 .
  3
  nx; — th”
  — n A

Pag. 56. for Prob. X. &c. read Prob. XI. & c.

86. 1. 8. for B read A.

FO FO

109. I. 10. for g read C Ci.
     11. for 2PC rad PC:
         136.1. penult. for zx— read ..
         148.1. 17. for ghz” read g rn.
         Ibid. 1. 19. for fg-þeh read Age hb.
         152.1. 7. for —g4kread gk.
110. I. 5. for —z? read —a*.
111. for vii. read viii.

If any other Miſtakes have happened, we hope they are
ſuch as an intelligent and candid Reader will eaſily correct and

forgive.

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HE Elements of ALGEBRA, in ten Books,
| By NicyoLas SAUNDERSON, LL. D.
Late Lucaſian Profeſſor of the MarTHEMATITICS
in the Neri of Cambridge, and F. R. S.
| To which is prefixed, |
An Account of the Author’s Life and Character,
and his palpable Arithmetic decyphered,
By Joan CoLson, F. R. S.

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