y
GIFT OP
JULIUS WAMGEN1H1E11M 87
Mathematics Dept
COMMENTARY
NEWTON S PRINCIPIA.
A SUPPLEMENTARY VOLUME.
DESIGNED FOR THE USE OF STUDENTS AT THE UNIVERSITIES,
BY
J. M. F. WRIGHT, A. B.
LATE SCHOLAR OF TltlXITV COLLEGF, CAM15ftID R, AUTHOR OF
OF THE CA.Mnr.IPGK PROBLEMS, & C . &C.
IX TWO VOLUMES.
VOL. I.
LONDON:
PRINTED FOR T. T. & J. TEGG, 73, CHEAPSIDE?
AND RICHARD GRIFFIN & CO., GLASGOW.
MDCCCXXXIII.
f\ SO:
\V\cod;
GLASGOW :
GEORGE BROOKMAN, PRINTER, VILLAP1ELD.
TO THE TUTORS
OF THE SEVERAL COLLEGES AT CAMBRIDGE,
THESE PAGES,
WHICH WERE COMPOSED WITH THE VIEW
OF PROMOTING THE STUDIES
OVER WHICH THEY SO ABLY PRESIDE,
ARE RESPECTFULLY INSCRIBED
BY THEIR DEVOTED SERVANT,
THE AUTHOR.
609740
PREFACE.
THE flattering manner in which the Glasgow Edition of Newton s Prin
cipia has been received, a second impression being already on the verge
of publication, has induced the projectors and editor of that work, to
render, as they humbly conceive, their labours still more acceptable, by
presenting these additional volumes to the public. From amongst the
several testimonies of the esteem in which their former endeavours have
been held, it may suffice, to avoid the charge of selfeulogy, to select the
following, which, coming from the high authority of French mathematical
criticism, roust be considered at once as the more decisive and impartial.
It has been said by one of the first geometers of France, that " L edition
de Glasgow fait honneur aux presses de cette ville industrieuse. On peut
affirmer que jamais 1 art typographique ne rendit un plus bel hommage
a la memoire de Newton. Le merite de 1 impression, quoique tresremar
quable, n est pas ce que les editeurs ont recherche avec le plus de soin,
pour tant le materiel de leur travail, ils pouvaient s en rapporter a 1 habi
Lite de leur artistes : mais le choix des meilleures editions, la revision la
plus scrupuleuse du texte et des epreuves, la recherche attentive des fautes
qui pourraient echapper rneme au lecteur studieux, et passer inapercues
ce travail consciencieux de 1 intelligence et du savoir, voila ce qui eleve
cette edition audessus de toutes celles qui 1 ont precedee.
" Les editeurs de Glasgow ne s etaient charges que d un travail de re
vision. S ils avaient concu le projet cTamcliorer et complcter I ccuvre des
a3
vl PREFACE.
commentators, Us auraient sans Joule employe, comme eux$ les travaux des
successeurs de Newton sur les questions traitees dans le livre des Principes.
" Les descendans de Newton sont nombreux, et leur genealogie est
prouvee par des titres incontestibles; ceux qui vivent aujourd hui verraient
sans doute avec satisfaction que Ton format un tableau de leur famille, en
reunissant les productions les plus remarquables dont Pouvrage de Newton
a fourni le germe : que ce livre immortel soit entoure de tout ce Ton peut
regarder comme ses developpemens : voila son meilleur commentaire.
L? edition de Glasgow pourrait done etre continuee, et prodigieusement
enrichie"
The same philosopher takes occasion again to remark, that " Le plus
beau monument que Von puisse clever a la glcire de Newton, c est une
bonne edition de ses ouvrages : et il est etonnant que les Anglais en aient
laisse ce soin aux nations etrangeres. Les presses de Glasgow viennent
de reparer, en partie, le tort de la nation Anglaise : la nouvelle edition
des Principes est effectivement la plus belle, la plus correcte et la plus com
mode qui ait parujusqu tci. La collation des anciennes editions, la revi
sion des calculs, &c. ont ete confiees a un habile mathematicien et rien
n a ete neglige pour eviter toutes les erreurs et toutes les omissions.
" II faut esperer que les editeurs continueront leur belle entreprise, et
qu ih y seront assez encourages pour nous donner, non seulement tous les
ouvrages de Newton, mais ceux des savans qui ont complete ses travaux."
The encouragement here anticipated has not been withheld, nor has
the idea of improving and completing the cpmments of " The Jesuits",
contained in the Glasgow Newton, escaped us, inasmuch as long before
these hints were promulgated, had the following work, which is composed
principally as a succedaneum to the former, been planned, and partly writ
ten. It is at least, however, a pleasing confirmation of the justness of our
own conceptions, to have encountered even at a ny time with these after
suggestions. The plan of the work is, nevertheless, in several respects,
a deviation from that here so forcibly recommended.
The object of the first volume is, to make the text of the Principia, by
PREFACE. VII
supplying numerous steps in the very concise demonstrations of the pro
positions, and illustrating them by every conceivable device, as easy as
can be desired by students even of but moderate capacities. It is univer
sally known, that Newton composed this wonderful work in a very hasty
manner, merely selecting from a huge mass of papers such discoveries as
would succeed each other as the connecting links of one vast chain, but
without giving himself the trouble of explaining to the world the mode of
fabricating those links. His comprehensive mind could, by the feeblest
exertion of its powers, condense into one view many syllogisms of a pro
position even heretofore uncontemplated. What difficulties, then, ta him
would seem his own discoveries? Surely none; and the modesty for
which he is proverbially remarkable, gave him in his own estimation so
little the advantage of the rest of created beings, that he deemed these
difficulties as easy to others as to himself: the lamentable consequence of
which humility has been, that he himself is scarcely comprehended at this
day a century from the birth of the Principia.
We have had, in the first place, the Lectures of Whiston, who des
cants not even respectably in his lectures delivered at Cambridge, upon
the discoveries of his master. Then there follow even lower and less
competent interpreters of this great prophet of science for such Newton
must have been held in those dark days of knowledge whom it would be
time misspent to dwell upon. But the first, it would seem, who properly
estimated the Principia, was Clairaut. After a lapse of nearly half a cen
tury, this distinguished geometer not only acknowledged the truths of the
Principia, but even extended the domain of Newton and of Mathematical
Science. But even Clairaut did not condescend to explain his views and
perceptions to the rest of mankind, farther than by publishing his own
discoveries. For these we owe a vast debt of gratitude, but should have
been still more highly benefited, had he bestowed upon us a sort of run
ning Commentary on the Principia. It is genemlly supposed, indeed,
that the greater portion of the Commentary called Madame Chastellet s,
was due to Clairaut. The best things, however, of that work are alto
Vlll PREFACE
gether unworthy of so great a master ; at the most, showing the perform
ance was not one of his own seeking. At any rate, this work does not
deserve the name of a Commentary on the Principia. The same may
safely be affirmed of many other productions intended to facilitate New
ton. Pemberton s View, although a bulky tome, is little more than
a eulogy. Maclaurin s speculations also do but little, elucidate the
dark passages of the Principia, although written more immediately for
that purpose. This is also a heavy unreadable performance, and not
worthy a place on the same shelf with the other works of that great
geometer. Another great mathematician, scarcely inferior to Maclaurin,
has also laboured unprofitably in the same field. Emerson s Comments
is a book as small in value as it is in bulk, affording no helps worth the
perusal to the student. Thorpe s notes to the First Book of the Princi
pia, however, are of a higher character, and in many instances do really
facilitate the reading of Newton. Jebb s notes upon certain sections deserve
the same commendation ; and praise ought not to be withheld from several
other commentators, who have more or less succeeded in making small
portions of the Principia more accessible to the student such as the Rev.
Mr. Newton s work, Mr. Carr s, Mr. Wilkinson s, Mr. Lardner s, &c.
It must be confessed, however, that all these fall far short in value of the
very learned labours, contained in the Glasgow Newton, of the Jesuits
Le Seur and Jacquier, and their great coadjutor. Much remained, how
ever, to be added even to this erudite production, and subsequently to its
first appearance much has been excogitated, principally by the mathema
ticians of Cambridge, that focus of science, and native land of the Princi
pia, of which, in the composition of the following pages, the author has
liberally availed himself. The most valuable matter thus afforded are the
Tutorial MSS. in circulation at Cambridge. Of these, which are used in
explaining Newton to the students by the Private Tutors there, the author
confesses to have had abundance, and also to have used them so far as seem
ed auxiliary to his own resources. But at the same time it must be remark
ed, that little has been the assistance hence derived, or, indeed, from all
PREFACE. IX
other known sources, which from the first have been constantly at com
mand.
The plan of the work being to make those parts of Newton easy which
are required to be read at Cambridge and Dublin, that portion of the
Principia which is better read in the elementary works on Mechanics,
viz. the preliminary Definitions, Laws of Motion, and their Corollaries,
has been disregarded. For like reasons the fourth and fifth sections have
been but little dwelt upon. The eleventh section and third book have
not met with the attention their importance and intricacy would seem to
demand, partly from the circumstance of an excellent Treatise on Physics,
by Mr. Airey, having superseded the necessity of such labours; and
partly because in the second volume the reader will find the same subjects
treated after the easier and more comprehensive methods of Laplace.
The first section of the first book has been explained at great length,
and it is presumed that, for the first time, the true principles of what has
been so long a subject of contention in the scientific world, have there
been fully established. It is humbly thought (for in these intricate specu
lations it is folly to be proudly confident), that what has been considered
in so many lights and so variously denominated Fluxions, Ultimate Ratios,
Differential Calculus, Calculus of Derivations, &c. &c. is here laid down
on a basis too firm to be shaken by future controversy. It is also hoped
that the text of this section, hitherto held almost impenetrably obscure, is
now laid open to the view of most students. The same merit it is with some
confidence anticipated will be awarded to the illustrations of the 2nd, 3rd,
6th, 7th, 8th, and 9th sections, which, although not so recondite, require
much explanation, and many of the steps to be supplied in the demon
stration of almost every proposition. Many of the things in the first
volume are new to the author, but very probably not original in reality
so vast and various are the results of science already accumulated. Suffice
it to observe, that if they prove useful in unlocking the treasures of the
Principia, the author will rest satisfied with the meed of approbation,
which he will to that extent have earned from a discriminating and im
partial public.
X PREFACE.
The second volume is designed to form a sort of Appendix or Supple
ment to the Principia. It gives the principal discoveries of Laplace, and,
indeed, will be found of great service, as an introduction to the entire
perusal of the immortal work of that author the Mecanique Celeste.
This volume is prefaced by much useful matter relative to the Integra
tion of Partial Differences and other difficult branches of Abstract Ma
thematics, those powerful auxiliaries in the higher departments of Physical
Astronomy, and which appear in almost every page of the Mecanique
Celeste. These and other preparations, designed to facilitate the com
prehension of the Newton of these days, will, it is presumed, be found
fully acceptable to the more advanced readers, who may be prosecuting
researches even in the remotest and most hidden receptacles of science ;
and, indeed, the author trusts he is by no means unreasonably exorbitant
in his expectations, when he predicates of himself that throughout the
undertaking he has proved himself a labourer not unworthy of reward.
THE AUTHOE.
A COMMENTARY
NEWTON S PRINCIPIA.
SECTION I. BOOK I.
1. THIS section is introductory to the succeeding part of the work. It
comprehends the substance of the method of Exhaustions of the Ancients,
and also of the Modern Theories, variously denominated Fluxions, Dif
ferential Calculus, Calculus of Derivations, Functions, &c. &c. Like
them it treats of the relations which Indefinite quantities bear to one ano
ther, and conducts in general by a nearer route to precisely the same
results.
2. In what precedes this section, Jinite quantities only are considered,
such as the spaces described by bodies moving uniformly in Jinite times
with Jinite velocities ; or at most, those described by bodies whose mo
tions are uniformly accelerated. But what follows relates to the motions
of bodies accelerated according to various hypotheses, and requires the
consideration of quantities indefinitely small or great, or of such whose
Ratios, by their decrease or increase, continually approximate to certain
Limiting Values, but which they cannot reach be the quantities ever so
much diminished or augmented. These Limiting Ratios are called by
Newton, " Prime and Ultimate Ratios," Prime Ratio meaning the Limit
from which the Ratio of two quantities diverges, and Ultimate Ratio that
towards which the Ratio converges. To prevent ambiguity, the term Li
miting Ratio will subsequently be used throughout this Commentary.
A COMMENTARY ON [SECT. I.
LEMMA I.
3. QUANTITIES AND THE RATIOS OF QUANTITIES.] Hereby Newton
would infer the truth of the Lemma not only for quantities mensurable
by Integers, but also for such as may be denoted by Vulgar Fractions.
The necessity or use of the distinction is none ; there being just as much
reason for specifying all other sorts of quantities. The truth of the LEMMA
does not depend upon the species of quantities, but upon their confor
mity with the following conditions, viz.
4. That they tend continually to equality, and approach nearer to each
other than by any given difference. They must tend continually to equa
lity, that is, every Ratio of their successive corresponding values must be
nearer and nearer a Ratio of Equality, the number of these convergen
cies being without end. By given difference is merely meant any that can
be assigned or proposed.
5. FINITE TIME.] Newton obviously introduces the idea of time in this
enunciation, to show illustratively that he supposes the quantities to con
verge continually to equality, without ever actually reaching or passing that
state ; and since to fix such an idea, he says, " before the end of that
time," it was moreover necessary to consider the time Finite. Hence
our author would avoid the charge of " Fallacia Suppositionis" or of
" shifting the hypothesis" For it is contended that if you frame certain
relations between actual quantities, and afterwards deduce conclusions
from such relations on the supposition of the quantities having vanished,
such conclusions are illogically deduced, and ought no more to subsist
than the quantities themselves.
In the Scholium at the end of this Section he is more explicit. He
says, The ultimate Ratios, in which quantities vanish, are not in reality the
Ratios of Ultimate quantities ; but the Limits to which the Ratios of quan
tities continually decreasing always approach ; which they never can pass
beyond or arrive at, unless the quantities are continually and indejinitely
diminished. After all, however, neither our Author himself nor any of
his Commentators, though much has been advanced upon the subject, has
obviated this objection. Bishop Berkeley s ingenious criticisms in the
Analyst remain to this day unanswered. He therein facetiously denomi
nates the results, obtained from the supposition that the quantities, before
BOOK L] NEWTON S PRINCIPIA. 3
considered finite and real, have vanished, the " Ghosts of Departed
Quantities / and it must be admitted there is reason as well as wit in the
appellation. The fact is, Newton himself, if we may judge from his own
words in the above cited Scholium, where he says, " If two quantities,
whose DIFFERENCE is GIVEN are augmented continually, their Ultimate
Ratio will be a Ratio of Equality," had no knowledge of the true nature
of his Method of Prime and Ultimate Ratios. If there be meanino in
O
words, he plainly supposes in this passage, a mere Approximation to be
the same with an Ultimate Ratio. He loses sight of the condition ex
pressed in Lemma I. namely, that the quantities tend to equality nearer
than by any assignable difference, by supposing the difference of the quan
tities continually augmented to be given, or always the same. In this
sense the whole Earth, compared with the whole Earth minus a grain of
sand, would constitute an Ultimate Ratio of equality ; whereas so long as
any, the minutest difference exists between two quantities, they cannot be
said to be more than nearly equal. But it is now to be shown, that
6. If two quantities tend continually to equality, and approach to one
another nearer than by any assignable difference, their Ratio is ULTIMATE
LY a Ratio of ABSOLUTE equality. This may be demonstrated as fol
lows, even without supposing the quantities ultimately evanescent.
It is acknowledged by all writers on Algebra, and indeed selfevident, that
if in any equation put = 0, there be quantities absolutely different in kind,
the aggregate of each species is separately equal to 0. For example, if
since A + a is rational, (B + b) V~2 surd and C V 1 imaginary,
they cannot in any way destroy one another by the opposition of signs,
and therefore
A + a = 0, B + b = 0, C = 0.
In the same manner, if logarithms, exponentials, or any other quantities
differing essentially from one another constitute an equation like the above,
they must separately be equal to 0. This being premised, let L, L de
note the Limits, whatever they are, towards which the quantities L + I,
L 7 + 1 continually converge, and suppose their difference, in any state of
the convergence, to be D. Then
L + 1 L V = D,
or L L + 1 1 D = 0,
and since L, L are fixed and definite, and 1, 1", D always variable, the
former are independent of the latter, and we have
A2
4 A COMMENTARY ON [SECT. I.
L
L L = 0, or j> = 1, accurately. Q. e. d.
This way of considering the question, it is presumed, will be deemed
free from every objection. The principle upon which it rests depending
upon the nature of the variable quantities, and not upon their evanescence,
(as it is equally true even for constant quantities provided they be of dif
ferent natures), it is hoped we have at length hit upon the true and lo
gical method of expounding the doctrine of Prime and Ultimate Ratios,
or of Fluxions, or of the Differential Calculus, &c.
It may be here remarked, in passing, that the Method of Indeterminate
Coefficients, which is at bottom the same as that of Prime and Ultimate
Ratios, is treated illogically in most books of Algebra. Instead of
" shifting the hypothesis," as is done in Wood, Bonnycastle and others,
by making x = 0, in the equation
a + bx + cx 2 +dx 3 + = 0,
it is sufficient to know that each term x being indefinitely variable, is he
terogeneous compared with the rest, and consequently that each term
must equal 0.
7. Having established the truth of LEMMA I. on incontestable princi
ples, we proceed to make such applications as may produce results useful
to our subsequent comments. As these applications relate to the Limits
of the Ratios of the Differences of Quantities, we shall term, after Leib
nitz, the Method of Prime and Ultimate Ratios,
THE DIFFERENTIAL CALCULUS.
8. According to the established notation, let a, b, c, &c ., denote con
stant quantities, and z, y, x, &c., variable ones. Also let A z, A y, A x,
&c., represent the difference between any two values of z, y, x, &c., re
spectively.
9. Required the Limiting or Ultimate Ratio of A (a x) and A x, i. e.
the Limit of the Difference of a Rectangle having one side (a) constant, and
the other (x) variable, and of the Difference of the variable side.
Let L be the Limit sought, and L + 1 any value whatever of the va
rying Ratio. Then
A (a x) a (x + A x) ax T
L + = } AIT =  ir ; = a  b * No  7
L = a.
BOOK I.] NEWTON S PRINCIPIA. 5
In this instance the Ratio is the same for all values of x. But if in the
Limit we change the characteristic A into d, we have
d(ax)
* A ; A
ax) = a d x^
or
d (ax)
d (a x), d x being called the Differentials of a x and x respectively.
A (x 2 )
10. Required the Limit of .
Let L be the Limit required, and L + 1 the value of the Ratio gene
rally. Then
A (x 2 ) (x + A x) 2 x 2
L + * = AX = AX
2 X A X + A X 2
"AIT"  = 2 x + A x.
. . L 2 X + 1 A X =r
and since L 2 x and 1 Ax are heterogeneous
L 2 x = 0,
or
L = x2
and ..
or
d (x ~) = 2 x d x (c)
A (x n )
1 1. Generally, required the Limit of A x .
Let L and L + 1 be the Limit of the Ratio and the Ratio itself re
spectively. Then
T A ( X ) (X + AX) X n
L + 1 = ^~ =  AX
n. (n 1)
= n x n  + j . x n ~ 2 A x + &c.
and L n x n l being essentially different from the other terms of
the series and from 1, we have
d (x n )
d x = L = n x n ~ l or d (x ") = n x  l d x (d)
or in words,
A s
6 A COMMENTARY ON [SECT. I.
The Differential of any potsoer or root of a variable quantity is equal to
the product of the Differential of the quantity itself^ the same power or
root MINUS one of the quantity, and the index of the power or root.
We have here supposed the Binomial Theorem as fully established by
Algebra. It may, however, easily be demonstrated by the general prin
ciple explained in (7).
12. From 9 and 1 1 we get
d(ax n )=nax n ~ 1 dx ...... (e)
A(a+x n + cx m + exP + &c.)
13. Required the Limit of  A x
Let L be the Limit sought, and L + 1 the variable Ratio of the finite
differences; then
A(a + bx D + cx m + &c.)
*+!=,. AX
+ &c. a bx n cx m &c.
_
A X
= nbx n ~ 1 + mcx m ~ l + &c. + P AX + Q(A x) 2 f &c.
P, Q, &c. being the coefficients of A x, A x 2 + &c. And equating the
homogeneous determinate quantities, we have
d(a + bx n + cx m + &c.)
  
A(a + bx n + cx m + &c.) r
14. Required the Limit of  ~~A~X~~
By 1 1 we have
d. (a + bx n + cx m + &c.) r
d(a + bx + &c.) =r(a + bxn + C x + &c.) 
and by 13
d(a+bx n + cx m + &c.) = (nbx 11 " 1 f mcx 1 * 1 + &c.) dx
r
= r(nbx n  1 + mcx 1 " 1 + &c.)(a + bx n + &c.) r  ! .. (g)
the Limiting Ratio of the Finite Differences A(a + bx n fcx m + &c.),
A x, that is the Ratio of the Differentials ofa + bx n + cx m + &c.,
and x.
A + Bx u fCx m + &c.
15. Required the Ratio of the Differentials gf a .i.b x 4.Cx^J&c
and x, or the Limiting Ratio of their Finite Differences.
Let L be the Limit required, and L + 1 the varying Ratio. Then
__ A + B (x + A x)? 1 + C (x + A x) m + &c. A + B x n + &c.
L + a + b(x + AX) + c(x + Ax)^+ &c. ~ a + bx + &c.~
BOOK L] NEWTON S PRINC1PIA. 7
which being expanded by the Binomial Theorem, and properly reduced
gives
L X ( a + b x + &c.) 2 + L X P. AX + Q (A x) 2 +&c. + 1 X {a+bx + &c.
+ P. A x + Q (A x) 2 + &c.} =(a+bx + cx^+ &c.) X (nBx"" 1
+ mCx 1 + &c.) (A+Bx n +Cx m + &c.) X (^bx 1
+ ft c x / l + &c.) + P . A x + Q (A x) 2 + &c.
P, Q, P , Q &c. being coefficients of A x, (A x) 2 &c. and independent of
them.
Now equating those homogeneous terms which are independent of the
powers of A x, we get
(A + Bx n + Cx m + &c.) (vbx  l + / ucx^ + &c.)
A+Bx n + Cx m + &c.
and putting u = a ^b ^TVx^^&cT we have finall y
du du
 = L, and therefore  =
(a + b x + c x <" + &c.) *
the Ratio required.
16. Hence and from 1 1 we have the Ratio of the Differentials of
(A + Bx+Cx + & c .) *
(a + b x + cx^ + &c.) i and x and ln S rt
ready been delivered it is easy to obtain the Ratio of the Differentials of
any Algebraic Function whatever of one variable and of that variable.
N. B. By Function of a variable is meant a quantity anyhow involving
that variable. The term was first used to denote the Powers of a quan
tity, as x 2 , x 3 , &c. But it is now used in the general sense.
The quantities next to Algebraical ones, in point of simplicity, are Ex
ponential Functions; and we therefore proceed to the investigation of
their Differentials.
17. Required the Ratio of the Differentials of a x and x ; or the Limit
ing Ratio of their Differences.
Let L be the required Limit and L + 1 the varying Ratio ; then
A(a x ) a * + Ai a x
L + l =
= a x X
AX AX
a ** 1
A X
8 A COMMENTARY ON [SECT. I.
But since
ay = (1 + a l)y
2.3 (a l) 3 + &c.,
it is easily seen that the coefficient of y in the expansion is
(a I) 2 (a I) 3
g + v__  __ &c<
Hence
a* (a I) 2 (a _ I) 3
L + ] = Z1Z Ha 1 2 + V  3 ~ &c.) A x + P (AX) 2 + &c.J
and equating homogeneous quantities, we have
d  (a *) (a I) 2 (a I) 3
~d^~= L = fc 1  g  + L_J__ &c> j a*
= A a* ........ (h)
or the Ratio of the Differentials of any Exponential and its exponent is
equal to the product of the Exponential and a constant Quantity.
Hence and from the preceding articles, the Eatio of the Differentials of
any Algebraic Function of Exponentials having the same variable index,
may be found. The Student may find abundance of practice in the Col
lection of Examples of the Differential and Integral Calculus, by Messrs.
Peacock, Herschel and Babbage.
Before we proceed farther in Differentiation of quantities, let us inves
tigate the nature of the constant A which enters the equation (h).
For that purpose, let (the two first terms have been already found)
a x = l+Ax+Px 2 +Qx
Then, by 13,
d (a x )
3
But by equation (h)
d (a*)
~ also = A
and equating homogeneous quantities, we get
2 P = A 2 , 3 Q = A P, 4 R = A Q, &c. = &c.
BOOK I.] NEWTON S PR1NCIPIA.
whence
P _A_! o^ il n AQ A4
2 y  3  2. 3 K = 4 = 2. 3.4 <
Therefore,
A_ 2 A 3 A 4
Again, put A x = 1, then
A 1 l 
a = 1+1 + 2+ + 27374 + &c.
= 2.718281828459 as is easily calculated
= e
by supposition. Hence
loff. a
(a 1) 2 (a 1) 3 log. a
a " ~2~ ~ + " ~3~"  &c = To^Te = L a
for the system whose base is e, 1 being the characteristic of that system.
This system being that which gives
(e I) 2 (e I) 3
e 1 s 2  +  3  &c. = 1
is called Natural from being the most simple.
Hence the equation (h) becomes
d(a x )
^rrlaXa* ........ (1)
17 a. Required the Ratio of the Differentials of 1 (x) and x.
Let 1 x = u. Then e u = x
.. d x = d (e u ) = 1 e X e u d u = e u d u, by 16
d (1 x) _1_ 1
~~d!T = e u = x ........ ( m )
Ix
In any other system whose base is a, we have log. (x) = y^.
d lo, x 1 1
n x x
We are now prepared to differentiate any Algebraicj or Exponential
Functions of Logarithmic Functions, provided there be involved but
one variable.
Before we differentiate circular functions, viz. the sines, cosines, tan
gents, &c., of circular arcs, we shall proceed with our comments on the
text as far as LEMMA VIII.
10 A COMMENTARY ON [SECT. I.
LEMMA II.
18. In No. 6, calling L and L Limits of the circumscribed and inscribed
rectilinear figures, and L + 1, L + 1 any other values of them, whose
variable difference is D, the absolute equality of L and L is clearly de
monstrated, without the supposition of the bases A B, B C, C D, D E,
being infinitely diminished in number and augmented in magnitude. In
the view there taken of the subject, it is necessary merely to suppose them
variable.
LEMMA III.
19. This LEMMA is also demonstrable by the same process in No. 6,
as LEMMA II.
Cor. 1. The rectilinear figures cannot possibly coincide with the curvi
linear figure, because the rectilinear boundaries albmcndoE,
aKbLcMdDE cut the curve a b E in the points a, b, c, d, E in
finite angles. The learned Jesuits, Jacquier and Le Setir, in endeavour
ing to remove this difficulty, suppose the four points a, 1, b, K to coincide,
and thus to form a small element of the curve. But this is the language
of Indivisibles, and quite inadmissible. It is plain that no straight line,
or combination of straight lines, can form a curve line, so long as we un
derstand by a straight line " that which lies evenly between its extreme
points," and by a curve line, " that which does not lie evenly between its
extreme points ;" for otherwise it would be possible for a line to be
straight and not straight at the same time. The truth is manifestly this.
The Limiting Ratio of the inscribed and circumscribed figures is that of
equality, because they continually tend to a fixed area, viz, that of the
given intermediate curve. But although this intermediate curvilinear
area, is the Limit towards which the rectilinear areas continually tend and
approach nearer than by any difference ; yet it does not fallow that the
rectilinear boundaries also tend to the curvilinear one as a limit. The
rectilinear boundaries are, in fact, entirely heterogeneous with the interme
diate one, and consequently cannot be equal to it, nor coincide therewith.
We will now clear up the above, and at the same time introduce a strik
ing illustration of the necessity there exists, of taking into consideration
the nature of quantities, rather than their evanescence or infinitesimality.
BOOK I.]
NEWTON S PRINCIPIA.
11
B
D
E
Take the simplest example of LEMMA II., in the case of the right
angled triangle a E A, having its two legs A a, A E equal.
The figure being constructed as in the text of LEMMA II, it fol
lows from that Lemma, that the Ultimate Ratio of the inscribed and cir
cumscribed figures is a ratio of equality ; and moreover it would also
follow from Cor. 1. that either of these
coincided ultimately with the triangle a 1
a E A. Hence then the exterior boundary
albmcndoE coincides exactly with _,
a E ultimately, and they are consequently
equal in the Limit. As we have only
straight lines to deal with in this example,
let us try to ascertain the exact ratio of
a E to the exterior boundary.
If n be the indefinite number of equal
bases A B, B C, &c., it is evident, since
A a ~ A E, that the whole length of
a * b "i c n d o E = 2 n X A B. Also since a b = b c = &c.
= V~al r ~+ b 1 * = V 2. A B, we have a E = n V 2. A B.
Consequently,
albmcndoE:aE::2: V 2 : : V~2 : 1.
Hence it is plain the exterior boundary cannot possibly coincide with
a E. Other examples might be adduced, but it must now be sufficiently
clear, that Newton confounded the ultimate equality of the inscribed and
circumscribed figin es, to the intermediate one, with their actual coinci
dence, merely from deducing their Ratios on principles of approximation
or rather of Exhaustion, instead of those, as explained in No. 6 ; which
relate to the homogeneity of the quantities. In the above example the
boundaries being heterogeneous inasmuch as they are incommensurable,
cannot be compared as to magnitude, and unless lines are absolutely equal,
it is not easy to believe in their coincidence.
Profound as our veneration is, and ought to be, for the Great Father
of Mathematical Science, we must occasionally perhaps find fault with
his obscurities. But it shall be done with great caution, and only with
the view of removing them, in order to render accessible to students in
general, the comprehension of " This greatest monument of human ge
nius."
20. Cor. 2. 3. and 4. will be explained under LEMMA VII, which re
lates to the Limits of the Ratios of the chord, tangent and the arc.
12
A COMMENTARY ON
[SECT. I.
LEMMA IV.
21. Let the areas of the parallelograms inscribed in the two figures be
denoted by
P, Q, R, &c.
p, q, r, &c.
respectively ; and let them be such that
P : p : : Q : q : : R : r, &c. : : m : n.
Then by compounding these equal ratios, we get
P+Q+R + : p + q + r + ; ; m:n
But P + Q + R . . . . and p + q + r + . . . . have with the curvili
near areas an ultimate ratio of equality. Consequently these curvilinear
areas are in the given ratio of m : n.
Hence may be found the areas of certain curves, by comparing their
incremental rectangles with those of a known area.
Ex. 1. Required the area of the common Apollonian parabola comprised
between its vertex and a given ordinate.
Let a c E be the parabola,
whose vertex is E, axis E A and
LatusRectum = a. Then A A
being its circumscribing rectan
gle, let any number of rectan
gles vertically opposite to one
another be inscribed in the areas
a E A, a E A , viz. A b, b A ;
B c, c B , &c.
And since
A b = A K. A B
A b = A 1. A B = ^. A B
a
from the equation to the parabola.
A b a. A B
Also
A b ~ A K. A B
or
(A a) 2 Bb 2 = axAE a
(A a + B b) X A B = a X A B
BOOK I.]
a X A
NEWTON S PRINCIPIA.
!3
= Aa f Bb
A B
A b Aa + B b 2 Bb + K a Ka.
AMb = ~ETb~~ B b + B b
A b
Hence, since in the Limit Trr becomes fixed or of the same nature with
the first term, we have
A b
A b
= 2
ultimately.
And the same may be shown of all other corresponding pairs of rec
tangles ; consequently by LEMMA IV.
a E A : a E A : : 2 : 1
.. a E A : rectangle A A : : 2 : 3.
or the area of a parabola is equal to two thirds of its circumscribing rec
tangle.
Ex. 2. To compare the area of a, semiellipse with that of a semicircle
described on the same diameter.
J>
Taking any two corresponding inscribed rectangles P N, P N ; we
have
P N : F N : : P M : P M : : a : b
a and b being the semiaxes major and minor of the ellipse ; and all other
corresponding pairs of inscribed rectangles have the same constant ratio ;
consequently by LEMMA IV, the semicircle has to the semiellipse the ratio
of the major to the minor axis.
As another example, the student may compare the area of a cycloid
with that of its circumscribing rectangle, in a manner very similar to
Ex. 1.
This method of squaring curves is very limited in its application. In
the progress of our remarks upon this section, we shall have to exhibit a
general way of attaining that object.
14 A COMMENTARY ON [SECT. I.
LEMMA V.
22. For the definition of similar rectilinear figures, and the truth of this
LEMMA as it applies to them, see Euclid s Elements B. VI, Prop. 4, 19
and 20.
The farther consideration of this LEMMA must be deferred to the ex
planation of LEMMA VII.
LEMMA VI.
23. In the demonstration of this LEMMA, " Continued Curvature" at
any point, is tacitly defined to be such, that the arc does not make with the
tangent at that point, an angle equal to ajinite rectilinear angle.
In a Commentary on this LEMMA if the demonstration be admitted,
any other definition than this is plainly inadmissible, and yet several of
the Annotators have stretched their ingenuity to substitute notions of
continued curvature, wholly inconsistent with the above. The fact is,
this LEMMA is so exceedingly obscure, that it is difficult to make any
thing of it. In the enunciation, Newton speaks of the angle between the
chord and tangent ultimately vanishing, and in the demonstration, it is
the angle between the arc and tangent that must vanish ultimately. So
that in the Limit, it would seem, the arc and chord actually coincide.
This has not yet been established. In LEMMA III, Cor. 2, the coinci
dence ultimately of a chord and its arc is implied ; but this conclusion by
no means follows from the LEMMA itself, as may easily be gathered from
No. 19. The very thing to be proved by aid of this LEMMA is, that the
Ultimate Ratio of the chord to the arc is a ratio of equality, it being
merely subsidiary to LEMMA VII. But if it be already considered that
they coincide, of course they are equal, and LEMMA VII becomes nothing
less than " argumentum in circulo"
Newton introduces the idea of curves of " continued curvature" or
such as make no angle with the tangent, to intimate that this LEMMA does
not apply to curves of noncontinued curvature, or to such as do make a
Jinite angle with the tangent. At least this is the plain meaning of his
words. But it may be asked, are there any curves whose tangents are
inclined to them ? The question can only be resolved, by again admitting
BOOK I.]
NEWTON S PRINCIPIA.
15
the arc to be ultimately coincident with the chord ; and by then showing,
that curves may be imagined whose chord and tangent ultimately shall be
inclined at a finite angle. The Ellipse, for instance, whose minor axis
is indefinitely less than its major axis, is a curve of that kind ; for taking
the tangent at the vertex, and putting a, b, for the semiaxes, and y, x, for
the ordinate and abscissa, we have
b 2
, X (2axx 2 )
and
y 2 =
2 a
X 1 =
,
= V 2 a x
.. since b is indefinitely smaller than a V x, x is indefinitely greater than
y, and supposing y to be the tangent cut off by the secant x parallel to
the axis, x and y are sides of a right angled A, whose hypothenuse is the
chord. Hence it is plain the z_ opposite x is ultimately indefinitely
greater than the L. opposite to y. But they are together equal to a right
angle. Consequently the angle opposite x, or that between the chord and
tangent, is ultimately finite. Other cases might be adduced, but enough
has been said upon what it appears impossible to explain and establish as
logical and direct demonstration. We confess our inability to do this,
an(J feel pretty confident the critics will not accomplish it.
24. Having exposed the fallacy of Newton s reasoning in the proof of
this LEMMA, we shall now attempt something by way of substitute.
Let A D be the tangent to the curve at the
point A, and A B its chord. Then if B be
supposed to move indefinitely near to A, the
angle BAD shall indejinitely decrease,, pro
vided the curvature be not indefinitely great.
Draw R D passing through B at right an
gles to AB, and meeting the tangent AD and
normal A R in the points D and R respective
ly. Then since the angle BAD equals the
angle A R B, if A R B decrease indefinitely
when B approaches A ; that is, if A R be
come indefinitely greater than A B ; or
which is the same thing, if the curvature at A, be not indefinitely great ;
the angle BAD also decreases indefinitely. Q. e. d.
We have already explained, by an example in the last article, what is
16 A COMMENTARY ON [SECT. I.
meant by curvature indefinitely great. It is the same with Newton s ex
pression " continued curvature." The subject will be discussed at length
under LEMMA XI.
As vanishing quantities are objectionable on account of their nothing
ness as it has already been hinted, and it being sufficient to consider va
riable quantities, to get their limiting ratios, as capable of indefinite diminu
tion, the above enunciation has been somewhat modified to suit those
views.
LEMMA VII.
25. This LEMMA, supposing the two preceding ones to have been fully esta
blished, would have been a masterpiece of ingenuity and elegance. By
the aid of the proportionality of the homologous sides of similar curves,
our author has exhibited quantities evanescent by others of any finite
magnitude whatever, apparently a most ingenious device, and calculated
to obviate all objections. But in the course of our remarks, it will be
shown that LEMMA V cannot be demonstrated without the aid of this
LEMMA.
First, by supposing A d, A b always finite, the angles at d and b and
therefore those at D and B which are equal to the former are virtually
considered finite, or R D cuts the chord and tangent at finite angles.
Hence the elaborate note upon this subject of Le Seur and Jacquier is
rendered valueless as a direct comment.
Secondly. In the construction of the figure in this LEMMA, the de
scription of a figure similar to any given one, is taken for granted. But
the student would perhaps like to know how this can be effected.
LEMMA V, which is only enunciated, from being supposed to be a mere
corollary to LEMMA III and LEMMA IV, would afford the means immedi
ately, were it thence legitimately deduced. But we have clearly shown
(Art. 19.) that rectilinear boundaries, consisting of lines cutting the inter
mediate curve ultimately atjinite angles, cannot be equal ultimately to the
curvilinear one, and thence we show that the boundaries formed by the
chords or tangents, as stated in LEMMA III, Cor. 2 and 3, are not ulti
mately equal, by consequence of that LEMMA, to the curvilinear one.
Newton in Cor. 1, LEMMA III, asserts the ultimate coincidence, and
therefore equality of the rectilinear boundary whose component lines cut
the curve at finite angles, and thence would establish the succeeding cor
ROOK 1.1 NEWTON S PRINCIPIA. 17
ollaries a fortiori. But the truth is that the curvilinear boundary is the
limit, as to magnitude, or length, of the tangential and chordal bounda
ries ; although in the other case, it is a limit merely in respect of area.
Yet, we repeat it, that LEMMA V cannot be made to follow from the
LEMMAS preceding it. According to Newton s implied definition of simi
lar curves, as explained in the note of Le Seur and Jacquier, they are the
curvilinear limits of similar rectilinear Jigures. So they might be consi
dered, if it were already demonstrated that the limiting ratio of the chord
and arc is a ratio of equality ; but this belongs to LEMMA VII. Newton
himself and all the commentators whom we have perused, have thus
committed a solecism. Even the best Cambridge MSS. and we have
seen many belonging to the most celebrated private as well as college tu
tors in that learned university, have the same error. Nay most of them
are still more inconsistent. They give definitions of similar curves wholly
different from Newton s notion of them, and yet endeavour to prove
LEMMA V, by aid of LEMMA VII. For the verification of these asser
tions, which may else appear presumptuously gratuitous, let the Cantabs
peruse their MSS. The origin of all this may be traced to the falsely
deduced ultimate coincidence of the curvilinear and rectilinear boundaries,
in the corollaries of LEMMA III. See Art. 19.
We now give a demonstration of the LEMMA without the assistance of
similar curves, and yet independently of quantities actually evanescent.
By hypothesis the secant R D cuts the chord and tangent at finite an
gles. Hence, since
A + B + D = 180
.. B + D  180 A
orL + lfL +l = 180 A
L and L being the limits of B and D and 1, V their variable parts as in
Art. 6 ; and since by LEMMA VI, or rather by Art. 24, A is indefinitely
diminutive, we have, by collecting homogeneous quantities
L + L = 180
But A B, A D being ultimately not indefinitely great, it might easily
be shown from Euclid that L = L , and .. A B = A D ultimately, (see
Art. 6 ) and the intermediate arc is equal to either of them.
18
A COMMENTARY ON
[SECT. 1.
OTHERWISE,
If we refer the curve to its axis,
A a, B b being ordinates, &c. as
in the annexed diagram. Then,
by Euclid, we have
$ ... AD; I , + B + D . Bt>t
. Bd
2Bd
Now, since by Art. 24 or LEMMA VI, the L. B A D is indefinitely less
than either of the angles B or D, .. B D is indefinite compared with A B
A D
or A D. Hence L being the limit of .  and 1 its variable part, if we
extract the root of both sides of the equation and compare homogeneous
terms, we get,
L = 1 or &c. &c.
26. Having thus demonstrated that the limiting Ratio of the chord, arc
and tangent, is a ratio of equality, when the secant cuts the chord and tangent
at FINITE angles, we must again digress from the main object of this work,
to take up the subject of Article 17. By thus deriving the limits of the rati
os of the finite differences of functions and their variables, directly from the
LEMMAS of this Section, and giving to such limits a convenient algorithm
or notation, we shall not only clear up the doctrine of limits by nume
rous examples, but also prepare the way for understanding the abstruser
parts of the Principia. This has been before observed.
Required to find the Limit of the Finite Differences of the sine of a cir
cular arc and of the arc itself, or the Ratio of their Differentials.
Let x be the arc, and A x its finite variable increment. Then L being
the limit required and L + 1 the variable ratio, we have
L + 1 = A s *" x _ sm  (x + A x) sin, x
A X A~X
_ sin, x. cos. (A x) + cos, x. sin. (A x) sin, x
A X
sin. (A x) sin.x. cos. AX sin. x
COS. X.
A :c
AX
AX
Now by LEMMA VII, as demonstrated in the preceding Article, the li
, cos. (A x) sin. x ,
l, and  5  t  7^7 have no definite limits.
. ,. sm. A x .
nut or is
A X
A X
A X
BOOK L] NEWTON S PRINCIPIA. 19
Consequently putting
sin. (A x)
cos. x. 3 = cos. x + 1 ,
A X
we have
L + 1 = cos. x + 1 + 
AX AX
and equating homogeneous terms
L = cos. x
or adopting the differential symbols
d. sin. x
~d~
or
d sin.
27. Hence and from the rules for the differentiation of algebraic, expo
nential, &c. functions, we can differentiate all other circular functions of
one variable, viz. cosines, tangents, cotangents, secants, &c.
Thus,
d sin. (  x^
\8 / f^ \ 
= cos. ( x ) = sin. x
(i*)
or
d. cos. x
sin. x ~\
.. _^_. pnQ v M
lx f
>in. x = d x. cos. x J
dx
or
d. cos. x
= sin. x
or
d
1. COS. X . ^V
j = sin. x J
d x
. cos. x = d x. sin. x J
(b)
Again, since for radius 1, which is generally used as being the most simple,
1 + tan. 2 x = sec. 2 x =
COS. X
1 2 cos. x. d. cos. x
.*. 2 tan. x. d. tan. x = d. = 1
cos. 2 x cos. x
See 12 (d). Hence and from (b) immediately above, we have
d x. sin. x
tan. x. d. tan. x =
cos. 3 x
. . d. tan. x = d x.  (c)
cos.  x
Again,
cot. x =
tan. x
B 2
SO A COMMENTARY ON [SECT. I.
Therefore,
, 1 d. tan. x
cot x = d ta,,. X  5S?nr
tan. 2 x. cos. 2 x sin. 2 x
Again,
1
(d)
sec. x =
.. d. sec. x = d.
COS. X
d
d cos. x
/]9 A}
COS. X
d x. sin. x
COS. 2 X
(* u )
(e)
cos. 2 x
v c /
we have
and lastly since cosec. x = sec. (
d. (
\2
d. cosec. x = d. sec. ( x) =
d x. cos. x
sn
(0
sin. x
Any function of sines, cosines, &c. may hence be differentiated.
28. In articles 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26 and 27, are to
be found forms for the differentiation of any function of one variable,
whether it be algebraic, exponential, logarithmic, or circular.
In those Articles we have found in short, the limit of the ratio of the
first difference of a function, and of the first difference of its variable.
Now suppose in this first difference of the function, the variable x should
be increased again by A x, then taking the difference between the first
difference and what it becomes when x is thus increased, we have the dif
ference of the first difference of a function, or the second difference of a
function, and so on through all the orders of differences, making A x al
ways the same, merely for the sake of simplicity. Thus,
A (x 3 ) = (x + Ax) 3 x 3
= 3X 2 AX + 3XAX 2 + AX 3
and A 4 (x) 3 = 3 (x + AX) ~ A x +<8 (x + AX) Ax 2 + A x 3 3 x 2 AX
3XAX 2 AX 3
= 3. 2xAX 2 3Ax 3
BOOK L] NEWTON S PRINCIPIA. 21
denoting by A 2 the second difference.
Hence,
~A~x~ 2 = 3  2  x + 3 A x
and if the limiting ratio of A 2 (x 3 ) and Ax 2 , or the ratio of the second
differential of x 3 , and the square of the differential of its variable x, be
required, we should have
L + 1 = 3. 2. x + 3 A x
and equating homogeneous terms
d 2 (x 3 )
j , = L = 3. 2. x
d x 2
In a word, without considering the difference, we may obtain the se
cond, third, &c. differentials d 2 u, d 3 u, &c. of any function u of x im
mediately, if we observe that r is always a function itself of x, and
(I X
make d x constant. For example, let
u = ax n + bx m + &c.
Then, from Art. 13. we have
j = nax nI + m b x m ~ + &c.
CL X
, /d u\
VcTx/ d (d u) d 2 u
j = ~r 9 = j o (by notation)
d x d x 2 d x 2 v J
= n. (n l)ax n  2 + m(m 1) b x m ~ 2 + &c.
Similarly,
d 3 u
j 5 = n. (n 1). (n 2) a x "  3 + &c.
&c. = &c.
Having thus explained the method of ascertaining the limits of the ra
tios of all orders of finite differences of a function, and the corresponding
powers of the invariable first difference of the variable, or the ratios of the
differentials of all orders of a function, and of the corresponding power
of the first differential of its variable, we proceed to explain the use of
these limiting ratios, or ratios of differentials, by the following
B3
22
A COMMENTARY ON
[SECT. I.
APPLICATIONS
29. Let it be required to draw a tangent to a given curve at any given
point of it.
Let P be the given point, and A M
being the axis of the curve, let P M
= y, A M = x be the ordinate and
abscissa. Also let P be any other
point; draw P N meeting the ordi
nate P M in N, and join P P . Now
let T P R meeting M P and M A in
R and T be the tangent required.
Then since by similar triangles
P N : P N : : P M : M T
.. M T = M T + T T = y.
A X
Now y being supposed, as it always is in curves, a function of x, we have
seen that whether that function be algebraic, exponential, &c.
 in the limit, or ^ is always a definite function of x. Hence putting
J *
AX

Ay
d x
j
dy
we have
M T + T T = y (~ + l)
and equating homogeneous terms,
which being found from the equation to the curve, the point T will be
known, and therefore the position of the tangent P T. M T is called
the subtangent.
Ex. 1. In the common parabola,
y 2 = a x
BOOK I.] NEWTON S PRINCIPIA. 23
Therefore,
d x 2 y
d y ~~ a
and
2 v 2
MT = ^ = 2x
or the subtangent M T is equal to twice the abscissa.
Ex. 2. In the ellipse,
b 2
y 2 = ^(a 2 x*)
and it will be found by differentiating, &o. that
/A 2 v Z\
MT = ~
X
*
Ex. 3. In the logarithmic curve,
y = a*
dy
* cfx  * a x y ( see 17 )
MT = r a "v^C ^, ^ r, ^"^i ",i
which is therefore the same for all points.
The above method of deducing the expression for the subtangent is
strictly logical, and obviates at once the objections of Bishop Berkeley
relative to the compensation of errors in the denominator. The fact is,
these supposed errors being different in their very essence or nature from
the other quantities with which they are connected, must in their aggre
gate be equal to nothing, as it has been shown in Art. G! This ingenious
critic calls F R = z ; then, says he, (see fig. above)
y. d x
M T = d y + z accurately ;
whereas it ought to have been
y A x y
MT =
A y f z Ay
AX A X

the finite differences being here considered. Now in the limit,  becomes a
A X
dy
definite function of x represented by g^r Consequently if 1 be put for
A y
the variable part of ~ , we have
24 A COMMENTARY ON [SECT. I.
_ __
dx + * + A X
and it is evident from LEMMA VII and Art. 25, that z is indefinite com
z d y
pared with A x. .. ^^ is indefinite compared with M T, 5 , and y ;
and 1 is also so ; hence
" .. M T.  + (l + ^j) M T = y
gives
y. d x z
M T = ^y, and 1 + =
which proves generally for all curves, what Berkeley established in the
case of the common parabola ; and at the same time demonstrates, as had
been already done by using T T instead of P R, incontestably the ac
curacy of the equation for the subtangent.
30. If it were required to draw a tangent to any point of a curve, re
ferred to a center by a radiusvector and the L. 6 which describes by
revolving round the fixed point, instead of the rectangular coordinates
x, y ; then the mode of getting the subtangent will be somewhat different.
Supposing x to originate in this center, it is plain that
x g cos. 6 1
y = g sin. 6 )
and substituting for x, y, d x, d y, hence derived in the expression (29.
e.) we have
d P cos. 6 P d 6 sin. 6
MT = g sin.Jx d ; 8in ., + ; d , cog ., ..... (f)
Ex. In the parabola
2a
"" 1 cos. 6
where a is the distance between the focus and vertex, or the value of g at
the vertex. Then substituting we get, after proper reductions
and the distance from the focus to the extremity of the subtangent is
cos. 6 cos. 6
 cos . 6 = 2 a
BOOK I.] NEWTON S PRINCIPIA. 25
2 a
= 1 cos. a ~ Si
as is well known.
30. a. The expression (f ) being too complicated in practice, the following
one may be substituted for it.
Let P T be a tangent to the
curve, refei red to the center S,
at the point P, meeting S T
drawn at right angles to S P,
ill T ; and let P be any other
point. Join P P and produce
it to T , and let T P be pro
duced to meet S P produced in
R, &c. Then drawing P N parallel to S T, we have
PN
x SF
But
P N = g tan. A 6, S P = f + A g
and
Therefore, substituting and equating homogeneous terms, after having
applied LEMMA VII to ascertain their limits, we get
Ex. 1. In the spiral of Archimedes we have
S = ad;
.. S T = S 
Ex. 2. In the hyperbolic spiral
a
g = y;
.. S T = a
31. It is sometimes useful to know the angle between the tangent and
axis.
P M dy
Tan. T = 5rT = d ^ (h)
See fig. to Art 29.
26 A COMMENTARY ON [SECT. I.
Again, in fig. Art. 30 a.
SP dg
Tan. T = ^ = ^ d , (k)
32. It is frequently of great use, in the theory of curves and in many
other collateral subjects, to be able to expand or develope any given func
tion of a variable into an infinite series, proceeding according to the
powers of that variable. We have already seen one use of such develop
ments in Art. 17. This may be effected in a general manner by aid of
successive differentiations, as follows.
If u = f (x) where f (x) means any function of x, or any expression
involving x and constants ; then, as it has been seen,
d u = u d x
(u being a new function of x)
Similarly
d u = u" d x
d u"  u " d x
&c. = &c.
But
. /d u\ d 2 u X d x d 2 x X d u
d u = d (tt = ~ d^~ ( 6 k )
. &c. = &c.
denoting d. (d u), d. (d x) by d ~ u, d 2 x, and (d x) 2 by d x 2 ,
according to the received notation ;
Or, (to abridge these expressions) supposing dx constant, and .. d 2 x = 0,
(a;
which give the various orders of fluxions required.
Ex. 1. Let u = x n
Then
du
5 n ~x a i
d X ~
d*u
3, = n. (n l)x" z
BOOK I.] NEWTON S PRINCIPIA. 27
d 3 u
j^ 3 = n. (n 1). (n 2)x n ~
&c. = &c.
d n u
J^TE = n. (n 1). (n 2) ..... 3. 2. 1.
Ex. 2. Let u = A + B x + C x 2 + D x 3 + E x 4 + &c.
Then,
du
^ = B + 2Cx + 3Dx 2 + 4Ex 3 + &c.
d 2 u
j^, = 2C + 2. 3Dx + 3. 4 E x 2 + &c.
d 3 u
d^~ 3 = 2. 3 D + 2, 3. 4 E x + &c.
&c. = &c.
Hence, if u be known, and the coefficients A, B, C, D, &c. be un
known, the latter may be found ; for if U, U , U", U" , &c. denote the
d u d 2 u d 3 u
when x = 0, then
A = U, B = 17, C = jg U", D = ~ U ", E = ~ U"",
&c. = &c.
and by substitution,
u = U + U x + U" l + U " ^ + &c. . ^ ^ ; . . (b)
This method of discovering the coefficients is named (after its inventor),
MACLAURIN S THEOREM.
The uses of this Theorem in the expansion of functions into series are
many and obvious.
For instance, let it be required to develope sin. x, or cos. x, or tan. x,
or 1. (1 f x ) into series according to the powers of x. Here
u = sin. x, or = cos. x, or = tan. x, or = 1. (1 + x ),
. d x = cos. x, or =  sin. x, or = ~g or = f j x
llH 2 sin. x 1
dx2   sin. x, or = _ cos. x, or = ,, or =   ,
28 A COMMENTARY ON [SECT. I.
d 3 u 2 + 4 sin. 2 x 2
H 3 = COSt x or = sm * x O1 = 4 ~> or = /V" i x \s
&c. = &c.
.. U =0, or = 1, or = 0, or =
U = 1, or = 0, or = 1, or = 1
U" =0, or = 1, or = 0, or = 1
U" = 1, or = 0, or = 2, or = 2
&c. = &c.
Hence
x 3 x 5
sin. x = x 273 + 2 . 3. 4. 5 ~ &c 
x 2 x 4
cos. x = 1 g + 2 3 4, & c 
x 3 2x 5 17x 7
tan. x = x + g + gy + 3 z 57 + &c.
X 2 X 3
1. (1 + X) = X g + g &C.
Hence may also be derived
TAYLOR S THEOREM.
For let
f (x) = A + Bx + Cx 2 + Dx 3 + Ex 4 + &c.
Then
f (x + h) = A + B. (x + h) + C. (x + h) 2 + D . (x + h) 3 + &c.
+ (B + 2Cx + 3Dx 2 )h
+ (C + 3Dx + 6Ex 2 )h 2
+ (D + 4 Ex + 10 Fx 2 ) h 3
+ &c.
d. f (x) d^XW !l!
d 3 f(x) h 3
the theorem in question, which is also of use in the expansion of series.
For the extension of these theorems to functions of two or more varia
bles, and for the still more effective theorems of Lagrange and Laplace,
the reader is referred to the elaborate work of Lacroix. 4to.
Having shown the method of finding the differentials of any quanti
BOOK I.] NEWTON S PRINCIPIA. 29
ties, and moreover, entered in a small degree upon the practical applica
tion of such differentials, we shall continue for a short space to explain
their farther utility.
33. To find the MAXIMA and MINIMA of quantities.
If a quantity increase to a certain magnitude and then decrease, the
state between its increase and decrease is its maximum. If it decrease
to a certain limit, and then increase, the intermediate state is its mi
nimum. Now it is evident that in the change from increasing to decreas
ing, or vice versa, which the quantity undergoes, its differential must have
changed signs from positive to negative, or vice versa, and therefore (since
moreover this change is continued) have passed through zero. Hence
When a quantity is a MAXIMUM or MINIMUM, its differential = 0. . . (a)
Since a quantity may have several different maxima and minima, (as for
instance the ordinate of an undulating kind of curve) it is useful to have
some means of distinguishing between them.
34. To distinguish between MAXIMA and MINIMA.
LEMMA. To show that in Taylor s Theorem (32. c.) any one term can
be rendered greater than the sum of the succeeding ones, supposing the
coefficients of the powers of h to be finite.
Let Q h n  l be any term of the theorem, and P the greatest coefficient
of the succeeding terms. Then, supposing h less than unity,
P h n (1 + h + h 2 + . . . . in infin.) = P h n X *
is greater than the sum ( S) of the succeeding terms. But supposing h to
decrease in infin.
1
P n> " l h = P h n ultimately. Hence ultimately
P h > 8
Now
Q h "  . ; p h n ; : Q : p h,
and since Q and P are finite, and h infinitely small ; therefore Q is > P h,
Hence Q h n ~ l is > P h n , and a fortiori > S.
Having established this point, let
u = f(x)
be the function whose maxima and minima are to be determined ; also
when u = max. or min. let x = a. Then by Taylor s Theorem
fl du. du h 2 d 3 u h 3
f.b = f._  h . . &c.
30 A COMMENTARY ON [SECT. I.
and
and since by the LEMMA, the sign of each term is the sign of the sum of
that and the subsequent terms,
.. f (a h) = f (a) ^?. M
d a
f(a + h) = f(a) + ^?. N
d a
Now since f (a) = max. or min. f (a) is > or < than both f (a h)
and f (a + h), which cannot be unless
d  u o
da 
Hence
d 2 u
f(ah) =f(a)
f(a + h) = f(a)
da 2
and f (a) is max. or min. or neither, according as f (a) is >, < or = to
both f (a h) and f (a + h), or according as
d 2 u .
T  is negative, positive, or zero
If it be zero as well as , , we have
d a
f(a + h) =f(a) +
O. i I
and f (a) cannot = max. or min. unless
d 3 u
d^~ :
which being the case we have
d 4 u
f(a h) = fa + M" )
da f
f(a + h)=fa + ^l?. N)
d a
and as before,
BOOK I.] NEWTON S PRINCIPIA. 31
d T" " *"" """ *
1 4
f (a) is max . or min. or neither, according as , " is negative, positive, or
zero, and so on continually.
Hence the following criterion.
If in u f (x), = 0, the resulting value of x shall give u MAX.
or MIN. or NEITHER, according as j 2 w negative, positive, or aero.
r d u d 2 u d^ u
J ~dx = dTx~ 2 ~ dlT 3 = tlien the Tesultin S value of u
shall be a MAX., MIN. or NEITHER according as  " is NEGATIVE, PO
11 X.
SITIVE, or ZERO ; and so on continually.
Ex. 1. To find the MAX. and MIN. of the ordinate of a common para
bola.
y = V a x
d y _ J^ V~SL
d x 2 y~^
which cannot = 0, unless x = oc .
Hence the parabola has no maxima or minima ordinates.
Ex. 2. To Jind the MAXIMA and MINIMA of y in the equation
y 2 2axy + x 2 = b 2 .
Here
d y /d y
d_y _ ay x d 2 y __ * ^jJ^
dx y_ax dx 2 ~ y_ ax
d V
and putting ^ = 0, we get
 _____ _
V (l_a 2 ) " V (1 a 2 ) dx 2 ~ b~Vjl a 8 )
which indicate and determine both a maximum and a minimum.
Ex. 3. 7b rfiw/Wc a in such a manner that the product of the m lh power
of the one part, and the power of the other shall be a maximum.
Let x be one part, then a x = the other, and by the question
u = x m . (a _ x )n  max .
du _
* a^ ~ ~ ( a x ) " ~ X (m a x. m + n)
32 A COMMENTARY ON [SECT. I.
and
d x
_ x m  2 ( a _ x ) n  2 x ( m + n 1. m + n. x 2 &c.)
Put r = ; then
d x
m a
x 0, or x = a, or x
in + n
the two former of which when m and n are even numbers give minima,
and the last the required maximum.
j]
Ex. 4. Let u = x x .
Here
d u 1 1.x 7 7 u
j = u. i = 0, . . 1. x = 1, and x e the hyperbolic base
Cl A. A.
= 2.71828, &c.
Innumerable other examples occur in researches in the doctrine of
curves, optics, astronomy, and in short, every branch of both abstract and
applied mathematics. Enough has been said, however, fully to demon
strate the general principle, when applied to functions of one independent
variable only.
For the MAXIMA and MINIMA of functions of two or more variables, see
Lacroix, 4to.
35. If in the expression (30 a. g) S T should be finite when s is infinite,
then the corresponding tangent is called an Asymptote to the curve, and
since g and this Asymptote are both infinite they are parallel. Hence
To Jind the Asymptotes to a curve,
d 6
In S T = f 2 j , make g = oc , then each finite value of S T gives an
Asymptote ; which may be drawn, by finding from the equation to the
curve the values of 6 for g = a, (which will determine the positions of g),
then by drawing through S at right angles to g, S T, S T , S T", &c. the
several values of the subtangent of the asymptotes, and finally through
T, T , 1", &c. perpendiculars to S T, S T , S T", &c. These perpen
diculars will be the asymptotes required.
Ex. In the hyperbola
_ _ _ _
^ ~~ a (1 e cos. 6)
1
Here g = a , gives 1 e cos. 6 = 0, .. cos. 6 =
/. + 6 = L. whose cos. is 
e
BOOK I.]
Also S T =
NEWTON S PRINCIPIA.
33
b 2 b 2
\T~A T^o 7 = b > whence it will be seen that
a e sin. 6 a V e z 1
the asymptotes are equally inclined (viz. by L. 6) to the axis, and pass
through the center.
The expression (29. e) will also lead to the discovery and construction
of asymptotes.
Since the tangent is the nearest straight line that can be drawn to the
curve at the point of contact, it affords the means of ascertaining the in
clination of the curve to any line given in position ; also whether at any
point the curve be inflected, or from concave become convex and vice ver
sa ; also whether at any point two or more branches of the curve meet,
i. e. whether that point be double, triple, &c.
36. To Jind the inclination of a curve at any point of it to a given line ,
Jmd that of the tangent at that given point, which will be the inclination
required.
Hence if the inclination of the tangent to the axis of a curve be zero,
the ordinate will then be a maximum or minimum ; for then
tan. T = =
dx
(31. h)
37. To Jind the points of Inflexion of a curve.
A B A B
Let y = f (x) be the equation to the curve a b ; then A a, B b being
any two ordinates, and ana tangent at the point a, if we put A a = y,
and A B = h, we get
A a = f x
dx
i> r h 4. lly
But B n =
mn = y. f. .
dy
d
dT* 172 + &c & 2 c]
Consequently B b is < or > B n
d 2 y.
according as r is negative or positive, i. e. the curve is
concave or con
34 A COMMENTARY ON [SECT. I.
d 2 y .
vex towards its axis according as ^  z is negative or positive.
Hence also, since a quantity in passing from positive to negative, and
vice vwsa, must become zero or infinity, at a point of inflexion
d 2 y
i or a
d x 2
Ex. In the Conchoid of Nicomedes
x y = (a + y ) V (b 2 y 2 )
which gives, by making d y constant,
d 2 x _ 2 b 4 a b 2 y 3 3 b 2 a y *
and putting this = 0, and reducing, there results
which will give y and then x.
These points of inflexion are those which the Theory of (34) indicates
as belonging to neither maxima nor minima , and pursuing this subject
still farther, it will be found, in like manner, that in some curves
d 4 v d 6 v
^ 4 = or a , , 4 = or a , &c. = &c.
d x 4 d x 6
also determine Points of Inflexion.
38. Tojind DOUBLE, TRIPLE, Sfc. points of a curve.
If the branches of the curve cut one another, there will evidently be as
many tangents as branches, and consequently either of the expressions,
Tan. T = ^ (31. h)
d x
M T = 2ji (29. e)
as derived from the equation of the curve, will have as many values as
there are branches, and thus the nature and position of the point will be
ascertained.
If the branches of the curve touch, then the tangents coincide, and the
method of discovering such multiple points becomes too intricate to be in
troduced in a brief sketch like the present. For the entire Theory of
Curves the reader is referred to Cramer s express treatise on that subject,
or to Lacroix s Different, and Integ. Calculus, 4to. edit.
39. We once more return to the text, and resume our comments. We
pass by LEMMA VIII as containing no difficulty which has not been al
ready explained.
As similar figures and their properties are required for the demonstra
BOOK I.] NEWTON S PRJNCIPIA. 35
tion of LEMMA IX, we shall now use LEMMA VII in establishing LEMMA
V, and shall thence proceed to show what figures are similar and how to
construct them.
According to Newton s notion of similar curvilinear figures, we may
define two curvilinear figures to be similar when any rectilinear polygon
being inscribed in one of them, a rectilinear polygon similar to the former,
may always be inscribed in the other.
Hence, increasing the number of the sides of the polygons, and dimi
nishing their lengths indefinitely, the lengths and areas of the curvilinear
figures will be the limits by LEMMAS VII and III, of those of the recti
linear polygons, and we shall, therefore, have by Euclid these lengths
and areas in direct and duplicate proportions of the homologous sides
respectively.
40. To construct curves similar to given ones.
If y, x be the ordinate and abscissa, and x the corresponding abscissa
of the required curve, we have
x:y: :x :Z x x = y \ .* . . . < a j ^
the ordinate of the required curve, which gives that point in it which
corresponds to the point in the given curve whose coordinates are x, y;
and in the same manner may as many other points as we please be de
termined.
In such curves, however, as admit a practical or mechanical construc
tion, it will frequently be sufficient to determine but one or two values of y .
Ex. 1. In the circle let x, measured along the diameter from its extre
mity, be r (the radius) ; then y r, and we have
y = 2 x x = x
X
where x may be of any magnitude whatever. Hence, all semicircles, and
therefore circles, are similar Jfgures.
Ex. 2. In a circular arc (2 a) let x be measured along the chord (2 b),
and suppose x = r sin. a ; then y = r . vers. a
, vers. a
y = HI x x
sin. a
which gives the greatest ordinate to any semichord as an abscissa, of the
required arc, and thence since
y = r V r 2 x *
it will be easy to find the radius r and centre, and to describe the arc
required.
36 A COMMENTARY ON [SECT. I.
But since
y r vers. a! vers. vers. a
x r sin. a! sin. a sin. a
therefore
a a
2 sin. * ,2 sin. ~
1 cos. a 21 cos. a
sin. a a . a sin. a a . a
2 cos. sin. 2 cos. sin.
& 22
or
a a
tan.  = tan. ^
and
. .a = a
which accords with Euclid, and shows that similar arcs of circles subtend
equal angles.
Ex. 3. Given an arc of a parabola, "whose latusrectiim is p, to Jind a
similar one, whose latusrectum shall be p .
In the first place, since the arc is given, the coordinates at its extremi
ties are ; whence may be determined its axis and vertex ; and by the usual
mode of describing the parabola it may be completed to the vertex.
Now, since
y 2 = p x
x, x being measured along the axis, and when
v P
/. v = . x = . x = 2 x
x y
which shows that all semiparabolas, and therefore parabolas, are similar
figures. Hence, having described upon the axis of the given parabola,
any other having the same vertex, the arc of this latter intercepted be
tween the points whose coordinates correspond to those of the extremi
ties of the given arc will be the arc required.
Ex. 4. In the ellipse whose semidiameters are a, b, if x be measured
along the axis, when x = a, y = b. Hence
b
y = _. x
and x or the semiaxis major being assumed any whatever, this value of
y will give the semiaxis minor, whence the ellipse may be described.
This being accomplished, let (a, (3) (a , 8 } be the coordinates at the
BOOK L] NEWTON S PRINCIPIA. 37
extremities of any given arc of the given ellipse, then the similar one of
the ellipse described will be that intercepted between the points whose
coordinates, (x , y ) (x", y") are given by
y = V (2 a x x 2 )
: 3 : : x : y l J a
a : /3 : : x : y"J an b" , /2 ,
y z V (2 a x x 2 )
B
In like manner it may be found, that
All cycloids are similar.
Epicycloids are so, when the radii of their wheels a radii of the spheres.
Catenaries are similar when the bases <x tensions, Sfc. Sfc.
40. If it were required to describe the curve A c b (fig. to LEMMA
VII) not only similar to A C B, but also such that its chord should be of
the given length (c) ; then having found, as in the last example, the co
ordinates (x , y ) (x", y") in terms of the assumed value of the abscissa
(as a in Ex. 4), and (, /3), (a , /3 ) the coordinates at the extremities of
the given arc, we have
a function of a : whence a may be found.
Ex. In the case of a parabola whose equation is y 2 = a x, it will be
found that (y 2 = a x being the equation of the required parabola)
whence (a ) is known, or the latusrectum of the required parabola is so
determined, that the arc similar to the given one shall have a chord = c.
41. It is also assumed in the construction both to LEMMA VII and
LEMMA IX, that, If in similar Jigures, originating in the same point., the
chords or axes coincide, the tangents at that origin will coincide also.
Since the chords A B, A b (fig. to LEMMA VII), the parallel secants
B D, b d, and the tangents A D, A d are corresponding sides, each to
each, to the similar figures, we have (by LEMMA V)
A B : B D : : A b : b d
and L B = L. b. Consequently, by Euclid the L. B A D = L b A d,
or the tangents coincide.
38 A COMMENTARY ON
To make this still clearer. Let
M B, M B be two similar curves,
and A B, A B similar parts of them.
Let fall from A, B, A , B , the or
dinates A a, B b, A a , B b cut
ting off the corresponding abscissas
M a, M b, M a , M b , and draw
the chords A B, A B ; also draw
A C, A C at right angles to B b, B C .
Then, since (by LEMMA V)
M
a :
Mb : :
A a
;.
Bb
M
a
: Mb :
: A
a
: B b
M
a :
ab : :
Aa :
B C
M
a
: a b :
: A a
: B
C
AC :
B C : :
Ma
*
A a
A
C
: B C :
: M
a
: A
a
But
M a : A a : : M a : A a
.. A C : B C : : A C : B C
and the L. C = L. C
.. the triangles A B C, A B C are similar, and the L. B A C =
^ B A C , i. e. A B is parallel to A B .
Hence if B, B move up to A, the chords A B, A B shall ultimately
be parallel, i. e. the tangents (see LEMMA III, Cor. 2 and 3, or LEMMA
VI,) at A, A are parallel.
Hence, if the chords coincide, as in fig. to LEMMA VII, the tangents
coincide also.
The student is now prepared for the demonstration of the LEMMA.
He will perceive that as B approaches A, new curves, or parts of curves,
A c b similar to the parts A C B are supposed continually to be described,
the point b also approaching d, which may not only be at ajinite distance
from A, but absolutely fixed. It is also apparent, that as the ratio be
tween A B and A b decreases, the curve A c b approachas to the straight
line A b as its limit.
42. LEMMA XL The construction will be better understood when
thus effected.
Take A e of any given magnitude and draw the ordinate e c meeting
A C productd in c, and upon A c describe the curve Abe (see 39)
BOOK I.] NEWTON S PRINCIPIA. 39
A D
similar to A B C. Take A d = A e X r ^ and erect the ordinate d b
A &
meeting A b c in b. Then, since A d, A e are the abscissae corre
O *
spending to A D, A E, the ordinates d b, e c also correspond to the
ordinates D B, E C, and by LEMMA V we have
d b : D B : : e c : E C : : A e : A E
: : A d : A C (by construction)
and the / D = L. d. Hence
b is in the straight line A B produced, &c. &c.
43. This LEMMA may be proved, without the aid of similar curves, as
follows :
A B D = ^^ . (D F + F B)
AD 2 tan> " i A D B F
~ 2 2
and
where a = /L D A F.
A BD _ AD 2 , tan, a + A D . B F
" A C E " A E 2 . tan. a + A E . C G
Now by LEMMA VII, since L. B A F is indefinite compared with F or B ;
therefore B F, C G are indefinite compared with A D or A E. Hence
if L be the limit of . ., , and L + 1 its varying value, we have
A C Jti
AD 2 , tan, a + A D . B F
= A E 8 . tan. a + A E . C G
and multiplying by the denominator and equating homogeneous terms
we get
L . A E 2 . tan. a = AD", tan.
 f A BD _ AD 2
)! AlTE ~ A~E~ 2
44. LEMMA X. " Continually increased or diminished." The word
" continually" is here introduced for the same reason as continued
curvature" in LEMMA VI.
If the force, moreover, be not "Jinite" neither will its effects be ; or
the velocity, space described, and time will not admit of comparison.
40 A COMMENTARY ON [SECT. I.
45. Let the time A D be divided into several portions, such as D d,
A b B being the locus of the extremities of the ordinates which D repre
sent, the velocities acquired D B, d b, B
&c. Then upon these lines D d, &c.
\B
as bases, there being inscribed rect
angles in the figure A D B, and when
their number is increased and bases
diminished indefinitely, their ultimate
sum shall = the curvilinear area D d D A
A B D (LEMMA III.) But each of these rectangles represents the space
described in the time denoted by its base ; for during an instant the ve
locity may be considered constant, and by mechanics we have for constant
velocities S = T X V. Hence the area A B D represents the whole
space described in the time A D.
In the same manner, ACE (see fig. LEMMA X) represents the tune
A E. But by LEMMA IX these areas are " ipso motus initio," as A D *
and A E 2 Hence, in the very beginning of the motion, the spaces de
scribed are also in the duplicate ratio of the times.
46. Hence may be derived the differential expressions for the space
described^ velocity acquired, &c.
Let the velocity B D acquired in the time t (A D) be denoted by v,
and the space described, by s.
Then, ultimately >, we have
Dd = dt,Bn = dv,
and
Dnbd = ds = Ddxdb = dtXv.
Hence
d s . d s
v = ,ds = vdt, d t = (a)
Again, if D d =r d D , the spaces described in these successive instants,
are D b, D m, and therefore ultimately the fluxion of the space repre
sented by the ultimate state of D m is b n r m or 2 b m B . Hence
d (d s) = 2 X b m B ultimately,
and supposing B to move up to A, since in the limit at A, B coincides
with A, arid B m with A D, and therefore b m B or d (d s) represents
the space described " in the very beginning of the motion."
Hence by the LEMMA,
d (ds) a 2 d t 2 a d t 2
or with the same accelerating force
d 2 s a d t 2 (b)
BOOK I.] NEWTON S PRINCIPIA. 41
With different accelerating forces d 2 s must be proportionably increased
or diminished, and .*. (see Wood s Mechanics)
d 2 s a Fdt 2
Hence we have, after properly adjusting the units of force, &c.
d 2 s = Fdt 2 .
and . . I
d 2 s f  .*vtfv.. f  ( c )
F:= dT 2 3
Hence also and by means of (a) considering d t constant,
F = , v d v = F d s (d)
all of which expressions will be of the utmost use in our subsequent
comments.
47. LEMMA X. COR. I. To make this corollary intelligible it will be
useful to prove the general principle, that
If a body, moving in a curve, be acted upon by any new accelerating
force, the distance between the points at which it would arrive WITHOUT
and WITH the new force in the same time, or " error," is equal to the space
that the new force, acting solely, would cause it to describe in that same
time.
Let a body move in the curve ABC, and when at B, let an additional
force act upon.it in the direction B b. Also let B D, D E, E C ;
B F, F G, G b be spaces that would be described in equal times by the
body moving in the curve, and when moved by the sole action of the new
force. Then draw tangents at the points B, D, E meeting D d, E e, C c,
each parallel to B b, in P, Q, R. Also draw F M, G R, b d parallel to
B P; M S, R N, d e parallel to D Qj and S V, N T, e c parallel to
ER.
42 A COMMENTARY ON [SECT. I.
Now since the body at B is acted upon by forces which separately
would cause it to move through B D, B F, or, when the number of
the spaces is increased and their magnitude diminished in infinitum,
through B P, B F in same time, therefore by LAW III, Cor. 1, when
these forces act together, the body will move in that time through the
diagonal up to M. In the same manner it may be shown to move from
M to N, and from N to C in the succeeding times. Hence, if the num
ber of the times be increased and their duration indefinitely diminished,
the body will have moved through an indefinite number of points M, N,
&c. up to C, describing a curve B C. Also since b d, d e, e c are each
parallel to the tangents at B, D, E, or ultimately to the curve B D E C ;
.. b d e c ultimately assimilates itself to a curve equal and parallel to
B D E C ; moreover C c is parallel to B b. Hence C c is also equal
to Bb.
Hence, then, The Error caused by any disturbing force acting upon a
body moving in a curve, is equal to the space that would be described by
means of the sole action of that force, and moreover it is parallel to the
direction of that force. Wherefore, if the disturbing force be constant, it is
easily inferred from LEMMAS X and IX, and indeed is shown in all books
on Mechanics, that the errors are as the squares of the times in which they
are generated. Also, if the disturbing forces be nearly constant, then the
errors areas the squares of the times quamproxime. But these conclusions,
the same as those which Note 118 of the Jesuits, Le Seur and Jacquier,
(see Glasgow edit. 1822.) leads to, do not prove the assertion of Newton
in the corollary under consideration, inasmuch as they are general for all
curves, and apply not to similar curves in particular.
48. Now let a curve similar to the above be constructed, and completing
the figure, let the points corresponding to A, B, &c. be denoted by
A , B , &c. and let the times in which the similar parts of these curves,
viz. B D, B D ; D E, D E ; E C, E C are described, be in the ratio
t : t . Then the times in which, by the same disturbing force, the spaces
B F, B F ; F G, F G ; G b, G b are described, are in the ratio of
t : t . Hence, " in ipso motus initio" (by LEMMA X) we have
B F : B F : : t 2 : t 2
F G : FG : : t 2 : t 2
&c. &c.
and therefore,
B F + F G + &c. : B F + F 7 G + &c. : : t 2 : t 2
BOOK I.] NEWTON S PRINCIPJA. 43
But, (by 15,)
B F + F G + &c. = the error C c,
and
B F + F G + &c. = the error C c ,
and the times in which B C, B 7 C are described, are in the ratio t : t .
Hence then
C c : a c : : t 2 : t
or The ERRORS arising from equal forces, applied at corresponding points,
disturbing the motions of bodies in similar curves, which describe similar
parts of those curves in proportional times, are as the squares of the times
in which they are generated EXACTLY, and not " quam proxime"
Hence Newton appears to have neglected to investigate this corollary.
The corollary indeed did not merit any great attention, being limited by
several restrictions to very particular cases.
It would seem from this and the last No. that Newton s meaning in
the forces being " similarly applied," is merely that they are to be applied
at corresponding points, and do not necessarily act in directions similarly
situated with respect to the curves.
For explanation with regard to the other corollaries, see 46.
49. LEMMA XI. " Finite Curvature" Before we can form any precise
notion as to the curvature at any point of a curve s being Finite, Infinite or
Infinitesimal, some method of measuring curvature in general must be de
vised. This measure evidently depends on the ultimate angle contained by
the chord and tangent ( A B, A D) or on the angle of contact. Now, although
this angle can have no finite value when singly considered, yet when two
such angles are compared, their ratio may be finite, and if any known
curvature be assumed of a standard magnitude, we shall have, by the
equality between the ratios of the angles of contact and the curvatures, the
curvature at any point in any curve whatever. In practice, however, it
is more commodious to compare the subtenses of the angles of contact
(which may be considered circular arcs, see LEM.MA VII, having; radii in
^
a ratio of equality, and therefore are accurate measures of them), than the
angles themselves.
50. Ex. 1. Let the circumference of a circle be divided into any num
ber of equal parts and the points of division being joined, let there be ?
tangent drawn at every such point meeting a perpendicular let fall from
the next point ; then it may easily be shown that these perpendiculars or
subtenses are all equal, and if the number of parts be increased, and their
A COMMENTARY ON
[SECT. I.
magnitude diminished, in infinitum, they will have a ratio of equality.
Hence, the CIRCLE has the same curvature at every point, or it is a curve
of uniform curvature.
51. Ex. 2. Let two circles touch one
another in the point A, having the
common tangent A D. Also let B D
be perpendicular to A D and cut the
circle A D in B . Join A B, A B .
Then since A "B, A B are ultimately
equal to A D (LEMMA VII) they are
equal to one another, and consequently
the limiting ratio of B D and B D, is
that of the curvatures of the respective
circles A C, A D (by 17.)
But, by the nature of the circle,
A D 2 = 2 R X D B D B 2 = 2 r X
R and r being the radii of the circles.
Therefore
D B 2 R D B
L+ DB ~2r DB
and equating homogeneous terms we have
D B D B
i. e. The curvatures of circles are inversely as their radii.
52. Hence, if the curvature of the circle whose radius is 1, (inch, foot,
or any other measure,) be denoted by C, that of any other circle whose
radius is r, is
r
53. Hence, if the radius r of a circle compared with 1, definite, its
curvature compared with C, is finite ; if r be infinite the curvature is
infinitesimal , if r be infinitesimal the curvature is infinite, and so 011 through
all the higher orders of infinites and infinitesimals. By infinites and in
finitesimals are understood quantities indefinitely great or small.
The above sufficiently explains why curvature, compared with a given
standard (as C), can be said to definite or indefinite. We are yet to show
the reason of the restriction to curves of finite curvature, in the enuncia
tion of the LEMMA.
54. The circles which pass through A, B, G ; a, b, g, (fig. LEMMA XI)
BOOK I.] NEWTON S PRINCIPIA. 45
have the same tangent A D with the curve and the same subtenses. Hence
(49. and 52.) these circles ultimately have the same curvature as the curve,
i. e. A I is the diameter of that circle which has the same curvature as the
curve at A. Hence, according as A I is finite or indefinite, the curvature
at A is so likewise, compared with that of circles of finite radius.
Now A G ultimately, or
AB"~
: ^~D
whether A I be finite or not. If finite, B D a A B 2 , as we also learn
from the text.
A B 
55. If the curvature be infinitesimal or A I infinite ; then since gjj
is infinite, B D must be infinitely less than A B ~, or, A B being
always considered in its ultimate state an infinitesimal of the first order,
B D is that of the third order, i. e. B D oc A B 3 . The converse is
also true.
Ex. In the cubical parabola, the abscissa tx as the cube of the or
dinate ; hence at its vertex the curvature is infinitely small. At other
points, however, of this curve, as we shall see hereafter, the curvature is
finite.
To show at once the different proportions between the subtenses of the
angles of contact and the conterminous arcs, corresponding to the differ
ent orders of infinitesimal or infinite curvatures, and to make intelligible
this intricate subject, let A B ultimately considered be indefinitely small
A B 2
compared with 1 ; then since ^^ = A B, A B 2 is infinitesimal com
A B n
pared with A B ; and generally . p nI = A B, shows that A B n is
infinitely small compared with A B n  * so that the different orders of in
Jinitesimals may be correctly denoted by
AB, AB 2 , AB 3 , A B*, &c.
Also since 1 is infinite compared with the infinitesimal A B, and A B
compared with A B \ &c. the different orders of infinites may be repre
sented by
1 * 1 1 &c
A~B A~B~ 2 AB 3 AB 4
56. Hence if the curvature at any point of a curve be infinitesimal in
the second degree
46 A COMMENTARY ON [SECT. ]
A B 2 1
v> TV Qt . , and B D oc A B 4 , and conversely.
D L) A 15
And generally, if the curvature be infinitesimal in the n th degree,
A B 2 1
JVfT a A R n? an( ^ ^^ a A B n + 2 , and conversely.
Again, if the curvature be infinite in the n th degree,
A B 2
^ cc A B n , and B D oc A B 2  n , and conversely.
The parabolas of the different orders will afford examples to the above
conclusions.
57. The above is sufficient to explain the first case of the LEMMA.
Case 2. presents no difficulty ; for b d, B D being inclined at any equal
angles to A D, they will be parallel and form, with the perpendiculars let
fall from b, B upon A D, similar triangles, whose sides being propor
tional, the ratio between B D, b d will be the same as in Case 1.
Case 3. If B D converge, i. e. pass through when produced to a given
point, b d will also, and ultimately when d and D move up to A, the
difference between the angles A d b, A D B will be less than any
that can be assigned, i. e. B D and b d will be ultimately parallel;
which reduces this case to Case 2. (See Note 125. of PP. Le Seur and
Jacquier.)
Instead of passing through a given point, B D, b d may be supposed
to touch perpetually any given curve, as a circle for instance, and B D
will still a AD 2 ; for the angles D, d are ultimately equal, inasmuch as
from the same point A there can evidently be but one line drawn touch
ing the circle or curve.
Many other laws determining B D might be devised, but the above
will be sufficient to illustrate Newton s expression, " or let B D be deter
mined by any other law whatever." It may, however, be farther observed
that this law must be definite or such as will^r B D. For instance, the
LEMMA would not be true if this law were that B D should cut instead of
touch the given circle.
58. LEMMA XL Cou. II. It may be thus explained. Let P be
the given point towards which the sagittae S G, s g, bisecting the chords
A B, A b, converge. S G, s g shall ultimately be as the squares of
A B, A b, &c.
BOOK I.]
NEWTON S PRINCIPIA.
47
For join P B, P b and produce
them, as also P G, P g, to meet the
tangent in D, d, T. t. Then if B
and b move up to A, the angles
T P D, t P d, or the differences be
tween the angles ATP and A D P,
and between A t P and. A d P, may
be diminished without limit; that is,
(LEMMA I), the angles at T, D and
at t, d are ultimately equal. Hence
the triangles ATS, A D B are
similar, as likewise are A t s, A d b.
Consequently
ST : D B
and
s t : d b
and
.. S T : s t :
Also by LEMMA VII,
S T : st :
and by LEMMA XI, Case 3,
D B : d b
.. S G : s s :
S G :
AB 2
AB 2
A S : A B
Ab
D B : db
S S
Ab 2
Ab 2
Q. e. d.
Moreover, it hence appears, that the sagittte which cut the chords, in
ANY GIVEN RATIO WHATEVER., and tend to a given point, have ultimately
the same ratio as the subtenses of the angles of contact, and are as the squares
of the corresponding arcs, chords, or tangents.
59. LEMMA XL COR. III. If the velocity of a body be constant or
"given," the space described is proportional to the time t. Hence
A B a t, and .. S G a A B 2 a t 2 .
60. LEMMA XL COR. IV. Supposing B D, b d at right angles to
A D (and they have the same proportion when inclined at a given angle
to A D, and also when tending to a given point, &c.) we have
A COMMENTARY ON [SECT. I.
AADB: AAdb::^^? A d x db
2 2
:: ?T XAD: Ad
A D 2
r^* AD: Ad
: : A D 3 : A d 3 .
Also
A d
: : (D B) ^ : (d b) *
It may be observed here, that the tyro, on reverting to LEMMA IX,
usually infers from it that
A A D B a A D 2 and does not a. AD 3 ,
but then he does not consider that A D, in LEMMA IX, cuts or makes a
Jinite angle with the curve, whereas in LEMMA XI it touches the curve.
61. LEMMA XI. COR. V. Since in the common parabola the ab
scissa a square of the ordinate, and likewise B D or A C a A D 2 or
C D 2 , it is evident that the curve may ultimately be considered a
parabola.
This being admitted, we learn from Ex. 1, No. 4, that the curvilinear
area A C B = f of the rectangle C D. Whence the curvilinear area
A B D =  of C D = f of the triangle A B D, or the area A B D a
triangle A B D a A D 3 , &c. (by Cor. 4.) So far B D, b d have been
considered at right angles to A D. Let them now be inclined to it at a
given angle, or let them tend to a given point, or " be determined by any
other law;" then (LEMMA, Case 3, and No. 25) B D, b d will ultimately
be parallel. Hence, B D , b d (fig. No. 26) being the corresponding
subtenses perpendicular to A D, it is plain enough that the ultimate dif
ferences between the curvilinear areas A B D, A B D and between
A b d, A b d are the similar triangles B D D , b d d , which
differences are therefore as B D 2 , b d 2 , or as A B 4 , A b 4 , i. e.
BDD a A B 4 .
But we have shown that A B D a A B 3 .
BOOK I.] NEWTON S PRINCIPIA. 49
Consequently
ABD = ABD+BDD / = axAB 3 + bxAB 4 =AB 3 (a + bxAB)
and b X A B being indefinite compared with a, (see Art. 6,)
ABD = axAB 3 a A B 3 .
Q. e. d.
SCHOLIUM TO SECTION I.
62. What Newton asserts in the Scholium, and his commentators Le
Seur and Jacquier endeavour (unsuccessfully) to elucidate, with regard to
the different orders of the angles of contact or curvatures, may be briefly
explained, thus.
Let D B ex A D m . Then the diameter of curvature, which equals
A D 2
jjg (see No. 22 and 24), a AD 2 " 1 . Similarly if D B ot AD", the
diameter of curvature <x A D 2 ~ n . Hence D and D represents these
diameters, we have
D a X A D 2 ~ m a
D 7 = a X AD 2  n = "a 7 D m ( a and a bein S finite )
and if n = 2 or D definite, then D will bejinite, infinitesimal, or infinite,
according as m = 2, or is any number, (whole, fractional, or even transcen
dental) less than 2, or any number greater than 2. Again, if m = n
then D compared with D is finite, since D : D : : a : a . If m be less
than n in any finite degree, then n m is positive, and D is always in
finitely less than D . If m be greater than n, then
D a 1
= 7 X
D " a AD
and m n being positive, D is always infinite compared with D .
Hence then, there is no limit to the orders of diameters of curvature,
with regard to infinite and infinitesimal, and consequently not to the
curvatures.
63. In this Scholium Newton says, that " Those things which have
been demonstrated of curve lines and the surfaces which they comprehend
are easily applied to the curve surfaces and contents of solids." Let us
attempt this application, or rather to show,
1st, That if any number of parallelepipeds of equal bases be inscribed in
any solid, and the same number having the same bases be also circumscribed
VOL. I. D
50
A COMMENTARY ON
[SECT. I.
about it ; then the number of these parallelepipeds being increased and their
magnitude diminished IN INFINITUM, the ultimate ratios which the aggre
gates of the inscribed and circumscribed parallelopipeds have to one another
and to the solid, are ratios of equality.
A
Let A S T U V Z Y X W A be any portion of a solid cut off by three
planes A A V, A A Z and Z A V, passing through the same point A ,
and perpendicular to one another. Also let the intersections of these
planes with one another be A A , A V, A Z, arid with the surface of the
solid be A U V, A Y Z and Z 1 V. Moreover let A V, A Z be each
divided into any number of equal parts in the points B , T , U ; D , X , Y ,
and through them let planes, parallel to A A" Z and A A V respectively,
be supposed to pass, whose intersections with the planes A A V, A A Z
BOOK I.] NEWTON S PRINCIPIA. 51
shall be S B 7 , T T, U U ; W D 7 , X X , Y Y 7 , and with the plane
A 7 Z V, 1 W, m T 7 , n U ; t D 7 , s X , o Y , respectively. Again, let the
intersections of these planes with the curve surface be S P 1, T Q m,
U R n ; WPt, XQs, YRo respectively. Also suppose their several
mutual intersections to be P C 7 , P E 7 , P" x, P " G 7 , Q F 7 , Q H 7 , Q" K 7 ,
&c. ; those of these planes taken in pairs and of the plane A Z V, being
the points C 7 , E 7 , x, G 7 , F 7 , H 7 , K , F, &c. and those of these pairs of
planes and of the curve surface, the points P, P , P", P" , Q, Q , Q", R, &c.
Now the planes, passing through B 7 , T 7 , U 7 , being all parallel to
A A Z, are parallel to one another and perpendicular to A A V. Also
because the planes passing through D 7 , X 7 , Y are parallel to A A 7 V,
they are parallel to one another, and perpendicular to A A? Z. Hence
(Euc. B. XL) S B , T T 7 , U U 7 , W D , X X 7 , Y Y 7 , as also P C 7 , P 7 E 7 ,
P 77 x, P 777 G 7 , Q F 7 , Q 7 H 7 , Q 77 K 7 , &c. &c. are parallel to A A 7 and to
one another. It is also evident, for the same reasons, that B 7 1, T 7 m, U 7 n,
are parallel to A 7 Z and to one another, as also are D 7 1, X 7 s, Y 7 o to
A 7 V and to one another. Hence also it follows that A 7 B 7 C 7 D 7 ,
B 7 C 7 E 7 T , &c. are rectangles, which rectangles, having their sides equal,
are themselves equal.
Again, from the points A, P, Q, R in the curve surface, draw A B,
A D; P E, P G; Q H, Q K; R L, R N parallel to A 7 B 7 , A 7 D 7 ;
C 7 E 7 , C 7 G 7 ; F 7 H 7 , T 7 K 7 , I 7 o, I 7 n and meeting B 7 S, D 7 W; E 7 P ,
G 7 P 777 ; H 7 Q 7 , K 7 Q 77 produced in the points B, D; E, G; H, K, re
spectively. Then complete the rectangles A C, P F, Q I which, being
equal and parallel to A 7 C 7 , C 7 F 7 , F 7 I 7 , will evidently, when C 7 P, F 7 Q,
I 7 R are produced to C, F, I, complete the rectangular parallelepipeds
A C 7 , P F 7 , Q I 7 . Moreover, supposing F 7 I 7 the last rectangle wholly
within the curve Z V produce K 7 I 7 , H 7 I 7 and make I 7 L 7 , I 7 N 7 equal
K I 7 , H 7 I 7 , and complete the rectangle I M 7 . Also complete the
parallelepiped R M 7 .
Again, produce E P, G P, H Q, K Q; L R, N R to the points d, b ;
g, e; k, h, and complete the rectangles Pa, Q p, R q thereby dividing
the parallelepipeds A C 7 , P F 7 , Q I 7 , each into two others, viz. A P,
aC 7 ; PQ, pF 7 ; Q R, q I .
Now the difference between the sum of the inscribed parallelepipeds
a C 7 , p F 7 , q P, and that of the circumscribed ones A C 7 , P F 7 , Q P, R M 7 ,
is evidently the sum of the parallelepipeds A P, P Q, Q R, R M 7 ; that
is, since their bases are equal and the altitudes P R 7 , R I, Q F, PC
are together equal to A A 7 , this difference is equal to the parallelepiped
A C 7 . In the same manner if a series of inscribed and circumscribed
D2
52 A COMMENTARY ON [SECT. I.
rectangular parallelepipeds, having the bases B E , E H , H L , be
constructed, the difference between their aggregates will equal the paral
lelepiped whose base is B E and altitude S B , and so on with every
series that can be constructed on bases succeeding each other diagonally.
Hence then the difference between the sums of all the parallelepipeds
that can be inscribed in the curve surface A Z V and circumscribed about
it, is the sum of the parallelepipeds whose bases are each equal to A C
and altitudes are A A , S B , T T ; , U U , W D , X X , Y Y . Let
now the number of the parts A B , B T , T U , U V, and of the parts
A D , D X , X? Y , Y 7 Z be increased, and their magnitude diminished
in infinitum, and it is evident the aforesaid sum of the parallelepipeds,
which are comprised between the planes A A Z, S B 1 and between the
planes A A! V, W D t, will also be diminished without limit ; that is, the
difference between the inscribed and circumscribed whole solid is ulti
mately less than any that can be assigned, and these solids are ultimately
equal, and a fortiori is the intermediate curvesurfaced solid equal to either
of them (see LEMMA I and Art. 6.) Q. e. d.
Hitherto only such portions of solids as are bounded by three planes
perpendicular to one another, and passing through the same point, have
been considered. But since a complete curve surfaced solid will consist of
four such portions, it is evident that what has been demonstrated of any
one portion must hold with regard to the whole. Moreover, if the solid
should not be curvesurfaced throughout, but have one, two, or three plane
faces, there will be no difficulty in modifying the above to suit any parti
cular case.
2dly, If in two curvesurfaced solids there be inscribed two series of paral
lelepipeds, each of the same number , and ultimately these parattelopipeds
have to each other a given ratio., the solids themselves have to one another
that same ratio.
This follows at once from the above and the composition of ratios.
3dly, All the corresponding edges or sides, rectilinear or curvilinear, of
similar solids are proportionals ; also the corresponding surfaces, plane or
curved, are in the duplicate ratio of the sides ; and the volumes or contents
are in the triplicate ratio of the sides.
When the solids have plane surfaces only, the above is shown to be
true by Euclid.
When, however, the solids are curvesurfaced, wholly or in part, we
must define them to be similar when any plane surf aced solid whatever
being inscribed in any one of them, similar ones may also be inscribed in the
BOOK I.] NEWTON S PRINCIPIA. 53
others. Hence it is evident that the corresponding plane surfaces are
similar, and consequently, by LEMMA V, the corresponding edges are
proportional, and the corresponding plane surfaces are in the duplicate
ratio of these edges or sides. Moreover, if the same number of similar
parallelepipeds be inscribed in the solids, and that number be indefinitely
increased, it follows from 63. 1 and the composition of ratios, that the
curved surfaces are proportional to the corresponding plane surfaces, and
therefore in the duplicate ratio of the corresponding edges ; and also that
the contents are proportional to the corresponding inscribed parallelepi
peds, or (by Euclid) in the triplicate ratio of the edges.
These three cases will enable the student of himself to pursue the ana
logy as far as he may wish. We shall " leave him to his own devices,"
after cautioning him against supposing that a curvedsurface, at any point
of it, has a certain fixed degree of curvature or deflection from the tangent
plane, and therefore that there is a sphere, touching the tangentplane at
that point, whose diameter shall be the limit of the diameters of all the
spheres that can be made to touch the tangentplane or curvedsurface
analogously to A I in LEMMA XI. Every curvilinear section of a curved
surface, made by a plane passing through a given point, has at that point
a different curvature, the curvedsurface being taken in the general sense;
and it is a problem of Maxima and Minima To determine those sections
which present the greatest and least degrees of curvature.
The other points of this Scholium require no particular remarks. If
the student be desirous of knowing in what consists the distinction be
tween the obsolete methods of Exhaustions, Indivisibles, &c. and that of
PRIME AND ULTIMATE RATIOS, let him go to the original sources to the
works of Archimedes, Cavalerius, &c.
64. Before we close our comments upon this very important part of the
Principia, we may be excused, perhaps, if we enter into the detail of the
Principle delivered in Art. 6, which has already afforded us so much
illustration of the text, and, as we shall see hereafter, so many valuable
results. We have thence obtained a number of the ordinary rules for
deducing indefinite forms from given definite functions of one variable ;
and it will be confessed, by competent and candid judges, that these ap
plications of the principle strongly confirm it. Enough has indeed been
already developed of the principle, to prove it clearly divested of all the
metaphysical obscurities and inconsistencies, which render the methods of
Fluxions, Differential Calculus, &c. &c. so objectionable as to their logic,
and which have given rise to so many theories, all tending to establish
n 3
54, A COMMENTARY ON [SECT. I.
the same rules. It is incredible that the great men, who successively in
troduced their several theories, should have been satisfied with the
reasonings by which they attempted to establish them. So many con
flicting opinions, as to the principles of the science, go only to show that
all were founded in error. Although it is generally difficult, and often
impossible, for even the most sharpsighted of men, to discern truth
through the clouds of error in which she is usually enveloped, yet, when
she does break through, it is with such distinct beauty and simplicity that
she is instantly recognized by all. In the murkiness around her there are
indeed false lights innumerable, and each passing meteor is in turn, by
many observers, mistaken for the real presence ; but these instantly vanish
when exposed to the refulgent brightness of truth herself. Thus we have
seen the various systems of the world, as devised by Ptolemy, Tycho
Brahe, and Descartes, give way, by the unanimous consent of philoso
phers, to the demonstrative one of Newton. It is true, the principle of
gravitation was received at first with caution, from its nonaccordance
with astronomical observations ; but the moment the cause of this discre
pancy, viz. the erroneous admeasurement of an arc of the meridian, was
removed, it was hailed universally as truth, and will doubtless be coeval
with time itself. The Theories relative to quantities indefinitely variable,
present an argument from which may be drawn conclusions directly op
posite to the above. Newton himself, dissatisfied with his Fluxions, pro
duces PRIME AND ULTIMATE RATIOS, and again, dissatisfied with these, he
introduces the idea of Moments in the second volume of the Principia.
He is every where constrained to apologize for his obscurities, first in his
Fluxions for the use of time and velocities, and then again in the Scholium,
at the end of Sect. I of the Principia, (and in this instance we have shown
how little it avails him) for reasoning upon nothings. After Newton comes
Leibnitz, his great though dishonest rival, (we may so designate him, such
being evidently the sentiments of Newton himself), who, bent upon oblite
rating all traces of his spoil, melts it down into another form, but yet falls
into greater errors, as to the true nature of the thing, than the discoverer
himself. From his Infinitesimals, considered as absolute nothings of the dif
ferent orders, nothing can be logically deduced, unless by Him (we speak
with reverence) who made all things from nothing. Suchjiats we mortals
cannot issue with the same effect, nor do we therefore admit in science, finite
and tangible consequences deduced from the arithmetic of absolute no
things, be they ever so many. Then we have a number of theories pro
mulgated by D Alembert, Euler, Simpson, Marquise L Hopital, &c. &c.
BOOK L] NEWTON S PRINCIPIA. 55
all more or less modifications of the others all struggling to establish
and illustrate what the great inventor, with all his almost supernatural
genius, failed to accomplish. All these diversities in the views of philo
sophers make, as it has been already observed, a strong argument that
truth had not then unveiled herself to any of them. Newton strove most
of any to have a full view, but he caught only a glimpse, as we may per
ceive by his remaining dissatisfied with it. Hence then it appears, to us
at least, that the true metaphysics of the doctrine of quantities indefinitely
variable, remain to this day undiscovered. But it may be asked, after
this sweeping conclusion, how comes it that the results and rules thence
obtained all agree in form, and in their application to physics produce
consequences exactly in conformity with experience and observation ?
The answer is easy. These forms and results are accurately true, al
though illogically deduced, from a mere compensation of errors. This has
been clearly shown in the general expression for the subtangent (Art. 29),
and all the methods, not even Lagrange s Calcul des Fonctions excepted,
are liable to the paralogism. Innumerable other instances might be
adduced, but this one we deem amply sufficient to warrant the above
assertion.
After these preliminary observations upon the state of darkness and
error, which prevails to this day over the scientific horizon, it may per
haps be expected of us to shine forth to dispel the fog. But we arrogate
to ourselves no such extraordinary powers. All we pretend to is self
satisfaction as to the removal of the difficulties of the science. Having
engaged to write a Commentary upon the Principia, we naturally sought
to be satisfied as to the correctness of the method of Prime and Ultimate
Ratios. The more we endeavoured to remove objections, the more they
continually presented themselves ; so that after spending many months in
the fruitless attempt, we had nearly abandoned the work altogether ;
when suddenly, in examining the method of Indeterminate Coefficients in
Dr. Wood s Algebra, it occurred that the aggregates of the coefficients of
the like powers of the indefinite variable, must be separately equal to zero,
not because the variable might be assumed equal to zero, (which it never
is, although it is capable of indefinite diminution,) but because of the
different powers being essentially different from, and forming no part of
one another.
From this a train of reflections followed, relative to the treatment of
homogeneous definite quantities in other branches of Algebra. It was
soon perceptible that any equation put = 0, consisting of an aggregate of
D 4
56 A COMMENTARY ON [SECT. I.
different quantities incapable of amalgamation by the opposition of plus
and minus, must give each of these quantities equal to zero. Reverting to
indefinites, it then appeared that their whole theory might be developed
on the same principles, and making trial as in Art. 6, and the subsequent
parts of the preceding commentary, we have satisfied ourselves most fully
of having thus hit upon a method of clearing up all the difficulties of
what we shall henceforth, contrary to the intention expressed in Art. 7,
entitle
THE CALCULUS
INDEFINITE DIFFERENCES.
65. A constant quantity is such, that from its very nature it cannot be
made less or greater.
Constants, as such quantities may briefly be called, are denoted generally
by the first letters of the alphabet,
a, b, c, d, &c.
A definite quantity is a GIVEN VALUE of a quantity essentially variable.
Definite quantities are denoted by the last letters of the alphabet, as
z, y, x, w, &c.
An INDEFINITE quantity is a quantity essentially variable through all
degrees of diminution or of augmentation short of absolute NOTHINGNESS or
INFINITUDE.
Thus the ordinate of a curve, considered generally, is an indefinite,
being capable of every degree of diminution. But if any particular value,
as that which to a given abscissa, for instance, be fixed upon, this value is
definite. All abstract numbers, as 1, 2, 3, &c. and quantities absolutely
fixed, are constants.
66. The difference between two definite values of the same quantity (y) is
a definite quantity, and may be represented by
adopting the notation of the Calculus of Finite (or definite} Differences.
In the same manner the difference between two definite values of A y is
a definite quantity, and is denoted by
A (A y)
BOOK L] NEWTON S PRINCIPIA.
or more simply by
and so on to
57
(b)
67. The difference between a Definite value and the Indefinite value of
any quantity y is Indefinite, and we call it the Indefinite Difference of y, and
denote it, agreeably to the received algorithm, by
dy (c)
In the same manner
d(dy)
or
the Indefinite Difference of the Indefinite Difference of y, or the second in
definite difference of y.
Proceeding thus we arrive at
d n y (d)
which means the n th indefinite difference of y.
68. Definite and Indejinite Differences admit of being also represented
by lines, as follows :
t"
Let P P = y be any fixed or definite ordinate of the curve A U, and
taking P Q = Q R = R S = &c. let ordinates be erected meeting
the curve in Q, R, S, T, &c. Join P Q, Q R, R S, &c. and produce
them to meet the ordinates produced in r, s, t, &c. Also draw r s , s t ,
58 A COMMENTARY ON [SECT. I.
&c. parallel to R S, S T, &c. and draw s t", &c. parallel to s t , &c. ; and
finally draw P m, Q n, R o, &c. perpendicular to the ordinates.
Now supposing not only P P but also Q Q , R R , &c. fixed or defi
nite ; then
Q m = Q Q PP = APP = Ay
Rr = n r n R r: Q m R n = A Q m
= A(AP P) = A 2 PF = A 2 y
ss =Ss Ss / = Ss Rr = A R r
= A 3 y
t t" = t t t t" = t t S S = A S S
= A(A 3 y) = A 4 y.
and so on to any extent.
But if the equal parts P Q , Q R , &c. be arbitrary or indefinite, then
Q m, R r, s s , 1 1", &c. become so, and they represent the several Inde
Jinite Differences of y, viz.
dy, d 2 y, d 3 y, d 4 y, c.
69. The reader will henceforth know the distinction between Definite
and Indefinite Differences. We now proceed to establish, of Indefinite
Differences., the
FUNDAMENTAL PRINCIPLE.
It is evidently a truth perfectly axiomatic, that No aggregate of INDEFI
NITE quantities can be a definite quantity, or aggregate of definite quanti
ties^ unless these aggregates are equal to zero.
It may be said that (a x) + ( a + x ) = 2 a, in which (x) is indefinite,
and (a) constant or definite, is an instance to the contrary ; but then the
reply is, a x and a + x are not indefinites in the sense of Art. 65.
70. Hence if in any equation
A + B x + C x 2 + D x 3 + &c. =
A, B, C, &c. be definite quantities and x an indefinite quantity ; then we
have
A = 0, B = 0, C = 0, &c.
For Bx+ Cx 2 + Dx 3 + &c. cannot equal A unless A = 0.
But by transposing A to the other side of the equation, it does = A.
Therefore A = and consequently
B x + C x 2 + D x 3 + &c. =
or
x(B + Cx + Dx 2 + &c.) =
BOOK I.] NEWTON S PRINCIPIA. 59
But x being indefinite cannot be equal to ; ..
B + Cx + Dx 2 + &c. =
Hence, as before, it may be shown that B =: 0, and therefore
x (C + D x + &c.) =
Hence C = 0, and so on throughout.
71. Again, if in the equation
A, B, B , C, C , C", D, &c. be definite quantities, and x, y INDEFINITES ;
then
A = 0\
B x + B y = \ivhen y is a function ofx.
C x 2 + C xy + C"y 2 = Oj
&c. =
For, let y = z x, then substituting
A + x (B + B z) + x 2 (C + C z + C" z 2 )
+ x 3 (D + D z + D" z 2 + D " z 3 ) + &c. =
Hence by 70,
A = 0, B + B z = 0, C + C z + C" z 2 = 0, &c.
y
and substituting for z and reducing we get
X.
A = 0, B x + B y = 0, &c.
In the same manner, if we have an equation involving three or more
indefinites, it may be shown that the aggregates of the homogeneous terms
must each equal zero.
This general principle, which is that of Indeterminate Coefficients
legitimately established and generalized, (the ordinary proofs divide
B x + C x 2 + &c. = by x, which gives B + Cx+Dx 2 + &c. =
z
and not ; x is then put = 0, and thence truly results B = , which
instead of being 0, may be any quantity whatever, as we know from alge
bra ; whereas in 70, by considering the nature of x, and the absurdity of
making it = we avoid the paralogism) conducts us by a near route to
the Indefinite Differences of functions of one or MORE variables.
72. To fold the Indefinite Difference of any function ofx.
Let u = f x denote the function.
Then d u and d x being the indefinite differences of the function and
of x itself, v;e have
u + d u = f (x + d x)
Assume
f (x + d x) = A + B d x + C d x " + &c.
60 A COMMENTARY ON [SECT. I.
A, 13, &c. being independent of d x or definite quantities involving x and
constants ; then
u + du=A + B d x + C d x 2 j &c.
and by 71, we have
u = A, d u = B . d x
Hence then this general rule,
The INDEFINITE DIFFERENCE of any function of x, f x, is the second
term in the devclopcmcnt off (x + d x) according to the increasing powers
Ex. Let u = x n . Then it may easily be shown independently of the
Binomial Theorem that
(x + dx) n = x n + n . x n  d x + Pdx 2
.. d (x n ) = n . x " 1 d x
The student may deduce the results also of Art. 9, 1 0, &c. from this general
rule.
73. To find the indefinite difference of the product of two variables.
Let u = x y. Then
u + du=(x + dx).(y + cly) = xy+x dy + y dx + dx dy
.. d u = x dy+y dx + dx dy
and by 71, or directly from the homogeneity of the quantities, we have
d u = x d y + y d x ........ (a)
Hence
d (x y z) = x d (y z) + y z d x
= xzdy + xydz + yzdx . . . (b)
and so on for any number of variables.
Again, required d . .
7
j
Let = u. Then
y
x = y u, and d x u dy + y d u
x d x u
. . d d u =  d y
y y y
_y dx x dy }
y 2 u
Hence, and from rules already delivered, may be found the Indefinite
Differences of any functions whatever of two or more variables. We
refer the student to Peacock s Examples of the Differential Calculus for
practice.
The result (a) may be deduced geometrically from the fig. in Art. 21.
The sum of the indefinite rectangles A b, b A makes the Indefinite
Difference.
BOOK I.]
NEWTON S PRINCIPIA.
61
We might, in this place, investigate the second, third, &c. Indefinite
Differences, and give rules for the maxima and minima of functions of two
or more variables, and extend the Theorems of Maclaurin and Taylor to
such cases. Much might also be said upon various other applications,
but the complete discussion of the science we reserve for an express
Treatise on the subject. We shall hasten to deduce such results as we
shall obviously want in the course of our subsequent remarks ; beginning
with the research of a general expression for the radius of curvature of a
given curve, or for the radius of that circle whose deflection from the
tangent is the same as that of the curve at the point of contact.
74. Required the radius of curvature for any point of a given curve.
Let A P Q R be the given
curve, referred to the axis A O
by the ordinate and abscissa
P M, A M or y and x. P M
being fixed let Q N, O R be
any other ordinates taken at
equal indefinite intervals M N,
N O. Join P Q and produce
it to meet O R in r ; and let
P t be the tangent at P drawn
by Art. 29, meeting Q N, O R
in q and t respectively. Again
draw a circle (as in construc
tion of LEMMA XI, or other
wise) passing through P and Q and touching the tangent P t, and there
fore touching the curve ; and let B D be its diameter parallel to A O.
Now
Q n = d y, P 11 = d x, P q = P Q (LEMMA VII) =
V (d x 2 f d y 2 ) or d s, if s = arc A P.
Moreover let
P M = y ;
then it readily appears (see Art. 27) that d s = , R being the ra
dius of the circle.
Again
Pq 2 = Qq X (Qq + 2 Q N )
= Q q (Q q + 2 d y + 2 /)
62 A COMMENTARY ON [SECT. I.
or
But since
R t : Q q : : P r 2 : P Q 2 : : 4 : i (LEMMA XI)
and
Q q : t r : : 1 : 2
.. R t = 2 t r, or R r = t r = 2 Q q
 Q q = ^ = ^ (by Art. 68.)
Consequently
(d 2 y) 2 , R dx d 2 y
 
and equating Homogeneous Indefinites
R dx d 2
d s z =
,
d s
R  ds3 _ (dx 2 + dy
= dxd 2 y = dx d 2 y
dx 2
the general expression for the radius of curvatui e.
Ex. 1. In the parabola y 2 = a x.
d y a
die ~~ !2y
and since when the curve is concave to the axis d 2 y is negative,
d y a dy a 2
~ die 2 = ~ 2~ 2 oTx = ~ 4~ =
3.
Hence at the vertex R = , and at the extremity of the latus rectum,
R = a = a V 2.
BOOK I.] NEWTON S PRINCIPIA. 63
Ex. 2. If p be the parameter or the double ordinate passing through
the focus and 2 a the axismajor of any conic section, its equation is
Hence
2 y d y = pdxHh^xdx
and
2 d y 2 + 2yd 2 y = +. d x s
d_y _
" d X " ~2
and
d X s 4 y
.. R =
which reduces to
R =
2p 2
Ex. 3. In the cycloid it is easy to show that
Aj_ _ j 2r y
dx *v y
r being the radius of the generating circle, and x, y referred to the base
or path of the circle.
d g y _ _r_
* cTx" 2 = " y~*
. . R = 2v 2ry=2 the normal.
Hence it is an easy problem iojind the equation to the locus of the centres
of curvature for the several points of a given curve.
If y and x be the coordinates of the given curve, and Y and X those of
the required locus, all referred to the same origin and axis, then the stu
dent will easily prove that
A COMMENTARY ON [SECT. I.
^d x
and
Y = v
dx 2
which will give the equation required, by substituting by means of the
equation to the given curve.
In the cycloid for instance
X = x + V (2 r y y *)
Y = y
whence it easily appears that the locus required is the same cycloid, only
differing in position from the given one.
75. Required to express the radius of curvature in terms of the polar co
ordinates of a curve, viz. in terms of the radius vector g and traced
angle 6.
x = g cos. 6 \
and >
y = g sin. 6 J
.. taking the indefinite differences, and substituting in equation (d) of Art.
74, we get
2 dr d_
which by means of the equation to the curve will give the radius of curva
ture required.
Ex. 1. In the logarithmic spiral
d = la Xa
. R _ (g 2 + (la) 2 ^) g 3 (^j! a) 2)

I
BOOK L] NEWTON S PRINCIPIA. 05
Ex. 2. In the spiral of Archimedes
f = a ^
and
Ex. 3. /w the hyperbolic spiral
a
Ex. 4. In the Lituus
. R _ _
4a 4 P 4
Ex. 5. / Me 1 Epicycloid
g = (r + r ) " 2 r (r + r ) cos. d
r and r being the radius of the wheel and globe respectively.
Here
R  ( r + r ) (3 r 2 2 r r r * + 2 g)*
2 (3 r 2 r r r 2 ) f 3 g
Having already given those results of the Calculus of Indefinite Differ
ences which are most useful, we proceed to the reverse of the calculus,
which consists in the investigation of the Indefinites themselves from their
indefinite differences. In the direct method we seek the Indefinite Differ
ence of a given function. In the inverse method we have given the Inde
finite Difference to find the function whose Indefinite Difference it is. This
inverse method we call
THE INTEGRAL CALCULUS
OF
INDEFINITE DIFFERENCES.
76. The integral of d x is evidently x + C, since the indefinite differ
ence of x + C is d x.
77. Required the integral of a. d x ?
By Art. 9, we have
d (a x) = a d x.
Vor. I. E
06 A COMMENTARY ON [SECT. I.
Hence reversely the integral of a d x is a x. This is only one of the in
numerable integrals which there are of a d x. We have not only d (a x)
= a d x but also
d(ax+C) = adx
in which C is any constant whatever.
.. ax + C =/adx = a/d x . . . (a) (see 76)
generally, /being the characteristic of an integral.
78. Required the integral of
a x P d x.
By Art. 12
d(ax n fC)= n a x "  * d x
..ax" f C =/n a x^ d x
= n X/ax n ~ 1 dx (77)
/(i ** X V
a x n ~ l d x = 1 .
n n
r*
But since C is any constant whatever may be written C.
. ./ax 1 dx = + C
n
Hence it is plain that
ft Y P +
Or To find the integral of the product of a constant the p th power of the
variable and the Indefinite Difference of that variable, let the index of the
power be increased by \, suppress the Indefinite Difference, multiply by the
constant, divide by the increased index, and add an arbitrary constant.
79. Hence
/(a x P d x + b x * d x + &c.) =
a XP+ 1 bx"* 1
F+T + q +r + &c  + c
80. Hence also
/ax n dx =   + C.
(n 1) x n ~ l n
81. Required the integral of
ax m ~ 1 dx(b + ex m )P.
Let
u = b + e x m
. . d u = mex m ~ l dx
. . a x m ~ d x = . d u
m e
../ax m ~ 1 dx(b+ ex m )P = /* u^du
v / m ft
m e
BOOK I.] NEWTON S PRINCIPIA. 67
= /* TTV u + + C (78)
m e . (p + 1)
= , a , .. . (b X e x m ) P+ 1 + C.
m e (p + 1)
d x
82. Required the integral of .
By 80 it would seem that
f ci x i r~
and if when
x p /d x 110
^ ~J~T ~o~~"o~ : :
But by Art. 17 a. we know that
d x
d . 1 x =
x
Therefore
J x
Here it may be convenient to make the arbitrary constant of the form 1 C
Therefore %
/* = 1 x + 1 C = 1 C x
x
Hence the integral of a fraction whose numerator is the Indefinite Differ
ence of the denominator, is the hyperbolic logarithm of the denominator PLUS
an arbitrary constant.
83. Hence
/ax m ~ 1 dx a /" mx m "~ 1 dx
bx m + e bm , xra
f
/
= JL.l.( x m + 
b m \ b
s .i.v^iA f r I ,
b m ^ , b/
and so on for more complicated forms.
84. Required the integral of&* d x.
By Art. 17
d.a x = la.a x dx
E 2
68 A COMMENTARY ON [SECT. I.
85. If y, x, t, s denote the sine, cosine, tangent, and secant of an angle
6 ; then we have, Art. 26, 27.
d y d x d t __ ds
: =: ~ "
= tan. t+C
/" ds _. = + C = se^ s + C
J s V2s s 2
sin. ~~ l y, cos. ~ l x, &c. being symbols for the arc whose sine is y, cosine is
x, &c. respectively.
86. Hence, more generally,
du _ _i_ f v T du
bu ). Vb/ vfl _b u
V
< a >
or = TT X angle whose sine is u ^J to rad. 1 + C.
Also
/ du 1 / b /i\
lm  r^v = TT cos.  1 u / + C . . (b)
J V (a bu 2 ) V b V a
Again
^du
, . f V d 11
/d u __ 1 / a
a + b u 2 " V ab J b
and
. da _ J_ f V^ u
/u V(bu 2 a) " V a / " /b, //b .
UX U  1
BOOK I.] NEWTON S PRINCIPIA. 69
Moreover, if u be the versed sine of an angb 6, then the sine
= V (2 u u s ) and
d u = d (1 cos. 6) = d 6 . sin. 6 (Art. 27.)
= cU. V (2u u 2 )
. dtf du
" V(2u u 2 )
Hence
/_ du  6 , r
./V(2u u 2 )"
= vers. ~ J u + C
and generally
2b j
du T du
a \ a
2b
87. Required the integrals of
dx dx d x
a + bx a bx a bx 2
f dx = 2. /d ( a +
^a + bx b / a + bx
dx
_ ___ L /*d(abx)
x~ b a bx
see Art. 17 a.
Hence,
A
/
a + bx^a bxj ~/a s b 2 x
}_ af
E3
c (f)
and
bx
(S)
70 A COMMENTARY ON [SECT. I.
Hence we easily get by analogy
/ d x 1 , V a + V b . x
J a bx 2 ~ vTb V a b x 1 ^
1 i ^ a + ^ b . x
2V r aT>* " V a v b. x
88. Required the integral of
dx
ax 2 + bx + c"
In the first place
ax2 , f b V (b 2 4 a c) \
a t 5 h 2a 2a ~J X
f , _b_ V (b 2 4 ac)\ f/ J)_x 2 _ b 2 4ac
1 2a~ 2a ) & \\** + 2a/ 2a }
Hence, putting
we have
^ i f\ ~~ ^
2 a
d x = d u
and
d x d u
ax* + bx + c a(u._ b J=ii<)
which presents the following cases.
Case 1. Let a be negative and c be positive ; then
d x d u "
f d x _ \7~2 / 2 a p,
* ^ ^x"Tb^" _ / 2 tan " b 2 4ac" f
+4ac) V b 2 +4ac
(see Art. 86) =  / ^ tan.Yx + ^) ^/ r ^~ +C . . . (i)
V ab 2 4ac V 2a/A b* 4ac
a(b 2 +4ac)
Case 2. Let c fo negative and a positive ; then
r d x _ / d u
4 a c
du
2a(b 2 +4ac)
b 2 + 4 ac
^ U
/
b 2 +4jic b
~"~
/
x
2 a 2a
see Art. 87.
BOOK I.] NEWTON S PRINCIPIA.
Case 3. Let b 2 be > 4 a c and a, c be both positive , then
d_x r d u
ax 2 + bx + c~ ~~ b 2 4 a
du
a / b 2 4 a c
/ T5 u
I x I 
f
A* 2a(b 2 4ac) b 2 4ac b
v / x
V 2 a 2a
Case 4. Let b 2 be < 4 a c and a, c be both positive ;
Then
d x 1 r d u
a/ 4ac b 2 2
"~
Case 5. 7^b 2 ^>4ac a;rf a, c both negative ;
Then
/d x _ 1 /" d u
ax 2 +bx c~^a / b 2 4ac 8
Case 6. Ifb 2 be < 4 a c awrf a awrf c both negative ;
Then
d x 1 f d u
c a/ 4ac b 2 2
/ 4ac b 2 h_
~ V 2a(4ac b 2 ) 1 , 4ac^b^ IZ +C "" ^
N " 2 a 2 a
89. Required the integral of any rational function whatever of one
variable, multiplied by. the indefinite difference of that variable.
Every rational function of x is comprised under the general form
AY ro _ i TJ v ni l ^i C^ V* m ~~ 2 i ft*/^ T^" v ^L
A "y* J_> A. y ~ \_^ A p <Xi IV A. "^ J_j
a x n + b x n ~ + c x n ~ 2 + &c. k x +1
E 1
a v m + 1 m
Cx m ~
72 A COMMENTARY ON [SECT. I
in which A, B, C, &c. a, b, c, &c. and m, n are any constants whatever.
If
n = 0,
then we have (Art. 77)
/(A x m + B x m  f c.)
^ f &c.) f constant.
/ a
Again, if m be > n the above can always be reduced by actual division
to the form
A x m ~ n + B x m  n f. &c. 4 =  *
a x " + bx"" 1 + &c.
and if the whole be multiplied by d x its integral will consist of two parts,
one of which is found to be (by 77)
A B . x m ~ n
m n + 1 X m + ln~^n~ + &C>
and the other
r A"x 1 + B // x n ~ 2 + &c.
d
X.
9 a x n + b x " ~ + c.
Hence then it is necessary to consider only functions of the general
form
x" 1 + A x p  2 + Bx u ~ 3 + &c. U
x"+ ax"~ J + bx u ~ 2 + &c. " V
in order to integrate an indefinite difference, whose definite part is any
rational function whatever.
Case 1. Let the denominator V consist ofn unequal real factors, x a,
x (S, c. according to the theory of algebraic equations. Assume
U _ P Q R
V ~ X a + x /3 + x ;
and reducing to a common denominator we shall have
&C>
U = P.x /3 . x 7 ... to (n 1) terms
+ Q.X a.x 7
+ R.X a.x f3
 (P + Q + R 4. fcc^x" 1
P.(S a) + Q. (S 8) + &C.} X n 
+ P.(S a.S a) + Q.(S /3.S
1. S 3 1.2 1
&c.
where S, S &c. denote the sum of a, /3, y &c. the sum of the products of
1 1.2
every two of them and so on.
BOOK I.] NEWTON S PRINCIPIA. 73
But by the theory of equations
S= a
S= b
1.1
&c. = &c.
... u = (P + Q + R + &c.)x n 
+ {a(P + Q + R + &c.) + P a +Q/3 + Ry + &c.} X x B ~ 2
+ {b (P + Q + R + &c.) + a(P a + Q8 + &c.) +
(P 2 + Q/3 2 + Ry 2 + &c.)} x n  3 + &c.
Hence equating like quantities (6)
P + Q 4. R + & c . = 1
a + Pa + Q/3+R 7 + &c. = A
b + a (A a) + P a 2 + Q /3 2 + R 7 2 + &c. = B
&c. = &c.
giving n independent equations to determine P, Q, R, &c.
F T U x 2 + 6 x + 3
1 
Here
P+ Q + R = 1j
6+P+2Q+3R=6 Vwhence
11 + P + 4Q + 9R = 3J
P = 1, Q = 5andR = 3
Hence
U d x r d x r 5 dx / 3 d x
/

= C 1. (x + 1) + 5 1. (x + 2) 3 1. (x + 3).
P, Q, R, &c. may be more easily found as follows :
Since
x" 1 + Ax n ~ 2 &c. = P (x /3). (x 7). &c.
+ Q ( X ). (x 7). &c.
+ R (x a), (x 8). &c.
+ &c.
let x = a, j8, 7, &c. successively ; we shall then have
a n ~ 1 + Aa tt ~ 2 + &C. = P . (a /3) . (a 7) &C. \
(8 n  l + A j8 n  2 + &c. = Q . ( ) . (p 7) &c. V. . . ( A)
&c. = &c.
In the above example we have
a = 1, J3 = 2, 7 = 3 and n = 3
A = 6 and B = 3.
. . P =
A COMMENTARY ON [SECT. I.
6 + 3 *
1. 2.
O 4 6. 2 + 3
y = = r = 5
as before.
Hence then the factors of V being supposed all unequal, either of the
above methods will give the coefficients P, Q, R, &c. and therefore
enable us to analyze the general expression ^ into the partial fractions
as expressed by
" P + Q +& ,
V ~~ x a ^ x
and we then have
Udx
= F . 1 (x
dx a 4 b / d x a + bydx
/iLp = P.l(xa) + Ql. (x/3) + 8cc. + C.
b / d x a + D /_c
f~/ a^TlE 2 /a
~ J x 2 ^ a x 2 J a+ x
+^l(a_x)_ a + b l.( a + x) + C
= a 1 x (a + b) 1 V a 2 x 2 + C
by the nature of logarithms.
TT. /3x 5 /*d
Ex  9  dx = ~
= I . 1 ( X _ 4) i 1. (x 2) + C.
Ex 4 f xdx /* pt1 ^ rQ dx
+ Q 1 . (x + j8) + C
where
and
p  _J*_  2 a + V (4 a 2 + b 2 )
j8 \/(4a 2 + b 2 ) 2a
= P 1 (x + )
a j8 ~~2 V (4 a 2 + b 2 )
Case 2. irf a// #7^ factors of V be real and equal, or suppose a = (3
= y = &c.
Then
U ._ x n ~ + A_x_  2 + &Q.
V = ~ X "a" n
BOOK I.] NEWTON S PRINCIPIA. 75
and since
a /3 = 0, a 7 = &c.
the forms marked (A) will not give us P, Q, R, &c. In this case we
must assume
U P Q R
V " (x a) n "*" (x a) n  l + (x a) n ~ 2 "*
to n 1 terms, and reducing to a common denominator, we get
U = P + Q . (x ) + R (x a) 2 + &c.
now let x = a, and we have
a n  1 + A a n  2 + &c. = P.
Also
^ = Q + 2 R . (x a) + 3 S . (x a) ~ + &c.
1 1 X.
d2J =2R+3.2.S.(x a)+4.3.T(x ) 2 +&c.
dx 2
d 3 U
= 2 . 3 . S + 4 . 3 . 2 T (x a) + &c.
dx
&c. = &c.
and if in each of these x be put = a, we have by Maclaurin s theorem
the values of Q, R, S, &c.
TT 1 T , U X 2 3X+ 2
Ex.1. Let =
(x _ 4)3
Then
U = x 8 3x + 2

dx
d*U
:
dx
.. P = 6
Q = 8 3 = 5
R = i. 2 = 1
/U d x * 6 d x / 5 d x / d x
~V~ ~ J (x 4) 3 + J (x 4) 2 + y x^T
Let U 
.Let 
(x _ 3)6
76 A COMMENTARY ON [SECT. I.
Here
U = x s + x 3
i5 = 5x+Sx
dx
d x
.. P = 3 5 + 3 3 = 27 X 10 = 270
Q = 27 X 16 = 432
R = 20 X 27 + 6 X 3 =
IB
9X_60 + 6
2X3
_ 360 _
27374.
W 12 1
" 2.3.4.5 ~
Hence
l8.(^93. F ! 5p .^i gr8  1 5. x i 5 + I.(x_3)
which admits of farther reduction.
x 2 + x U
Ex. 3. Let  _ yy = y .
Here
U = x 2 +x
i
dx
and
Ii5
d5
X 2
NEWTON S PRINCIPIA. 77
(x I) 2
2(x I) 4
// appears from this example, and indeed is otherwise evident, that the
number of partial fractions into which it is necessary to split the function
exceeds the dimension ofx. in U, by unity.
This is the first time, unless we mistake, that Maclaurin s Theorem
has been used to analyze rational fractions into partial rational fractions.
It produces them with less labour than any other method that has fallen
under our notice.
Case 3. Let the factors of the denominator V be all imaginary and un
equal.
We know then if in V, which is real, there is an imaginary factor of
the form x + h + kV 1, then there is also another of the form
x + h k V ]. Hence V must be of an even number of dimensions,
and must consist of quadratic real factors of the form arising from
(x + h + k V 1) ( X + h k V 1)
or of the form
(x + h) 2 + k 2 .
Hence, assuming
U P + Qx P + Q x
V ir H 7^ or~l T & c 
and reducing to a common denominator, we have
U = (P + Qx) J(x + a ) 2 + I 8 8 } H x + a// ) 2 + I 3 " 2 } x &c 
}. (P _f Q x ) ( x + a) 2 + (3 2 ] {(x + a") 2 + 13" *} X &c.
+ (P" + Q" x) J(x + a) 2 + 8 2 } J( x + " ) 2 + ^ 2 J X &C
+ &C.
Now for x substitute successively
a + 3 V 1, + fy / ], a! + j3" V 1, &C.
then U will become for each partly real and partly imaginary, and we
have as many equations containing respectively P, Q ; P , Q ; P", Q", &c.
as there are pairs of these coefficients ; whence by equating homogeneous
quantities, viz. real and imaginary ones, we shall obtain P, Q ; P , Q , &c.
78 A COMMENTARY ON [SECT. 1.
Ex. 1 . Required the integral of
x 3 d x
x 4 + 3x 2 + 2
Here the quadratic factors of V are x 2 + 1> x 2 + 2
.. a = 0, = 0, /3 = 1, and /3 = V~2 .
Consequently
x = (P + Qx)(x 2 + 2)
+ (P + Q x)(x 2 + l)
Let x = \/^n. Then
V 1 = (P + Q V 1) . ( 1 + 2)
_.. P=0, Q =  1
Again, let x = V 2. V 1, and we have
3
_2? V 1 = (P + Q V 2. V l) (2+ 1)
= P Qf V~2 . V~^l
.. P = 0, and Q = 2
Hence
* x s dx / x d x /*2 x d x
X y O X "y f *^ X ^^ A X "^ ^
^ /""i j I /. 2 i 1 \  1 /xr 2 t O\
Ex. 2. Required the integral of
dx
To find the quadratic factors of
1 +x 2n
we assume
x 2 n + 1 = 0,
and then we have
X 2n = 1 = cos. (2p+ 1) *r+ V 1 sin. (2 p + ])T
T being 180 of the circle whose diameter is 1, and p any integer what
ever.
Hence by Demoivre s Theorem
2p+l .   . 2p+l
x = cos.  it + v 1 . sin. j  w
2 n 2 n
But since imaginary roots of an equation enter it by pairs of the form
A. + V 1 . B, we have also
2 p + 1   . 2 p + 1
x = cos. \. ff V 1 . sin. *s  v
2 n 2 n
BOOK I.] NEWTON S PRINCIPIA. 79
and
2 n
which is the general quadratic factor of x 2 n + 1. Hence putting
p = 0, 1, 2 ...... n 1 successively,
x 2 +l = (x 2 2xcos. ^ + l) . (x 2 2xcos. ~ + 1 ) X
Hence to get the values of P and Q corresponding to the general factor,
assume
P+Qx N
M
2n
Then
_ 2xcos. 
But
TV/T l+X 2n
M =  ?
x 2 2 x cos.  TT f 1
2 n
and becomes of the form when for x we put cos. P it + V lx
2 p+ 1
sin. ^ ; its value however may thus be found
*w XI
T , 2 p + 1 . 2 p + 1
Let cos. 2 ic + V 1 sin. ^ ^ ff = r
2 n 2 n
then
2 p + 1 . 2 p + 1 1
cos. T V 1 . sin. ^ r =
2 n 2 n r
and
M= 1+x " .
Again let x r = y ; then
M = l + y 2n + 2n y
80 A COMMENTARY ON [SECT. I.
But
r 2 n = cos. 2 p + 1 . * + V 1 sin. 2p+l.w
M  y* n ~ + 2n y 2n  2 .r+ . . . . 2n r 2 "" 1
X
r
Hence when for x we put r, y = 0, and
and from the above equation we have
O n v 2 n 1
2
or
_ , . 2 p+ 1 D 2 p + 1 . 2 n 1 .
2V 1 sin. ~r v = 2 n P . cos. ^ . it + 2 n P V IX
2 n 2 n
2p+1.2ii 1 _ ^ i 2n ,\
sin. r  <r 2 n Q (since r 2n =: 1)
ii
.. equating homogeneous quantities we get
. 2p+l
sin. *1 ff=n. sin.
. .
2 n 2 n
and
2p+1.2n 1
P . cos. it Q.
fw n
But
2 n
Hence the above equations become
. 2 p + 1 r, 2 p +1
.. sin. ^r T = n P sin. T 
2n 2 n
2 p+ 1 ^

1 iri 1 2p+ 1
.. P = , andQ=  . cos. ^ 
n n 2 n
Hence the general partial integral of
dx
BOOK L] NEWTON S PRINCIPIA. 81
. r (\ x cos.  tt\ d x
1 / \ 2 n /
V
. n / x 
2 x cos. ~ v l l * + 1
n
cos. v. TT / 2 x d x 2 cos. ^~ it . d x
2 n / 2 n
2n (
/ v*
2 n 4 1
2 x cos. + 1
dx
2 p 4 1
2 X COS. ^~ r + 1
2 n
2p+ 1
S> 2n ,/ , 2p+ 1 .
_ . 1 f x 2 g x cos< _ri T + 1 )
2 n V 2 n /
2p+ 1 ,
sin. ^ v / x cos.
2n
see Art 88. Case 4.
d x
Hence then the integral of y   , which is the aggregate of the results
l j x
obtained from the above general form by substituting for p = 0, 1, 2 . . .
n 1, may readily be ascertained.
r d x
As a particular instance let J ^  6 be required.
Here
n = 3
and the general term is
2p+ 1
cos.  ft * _
1 2 x cos.
. 2p + 1 2
sin. r x cos.
/% * ^~ vv/o. ff
.tan. 1  6
sin ?JL_i
6
Letp = 0, 1, 2, collect the terms, and reduce them ; and it will appear that
dx _lj^3 , x 2 +xV3 + l _,8x(l x
By proceeding according to the above method it will be found, that the
general partial fractions to be integrated in the integrals of
VOL. I. F
A COMMENTARY ON
[SECT. I.
dx , x r d x
~ r and 
are respectively
v "
A
cos. 2 p it 1
* _ n , v _,
JW  X
x 2 2 x cos.
Q r\ fjf
.dx
and
2 cos
X
n
2pw 2 r p *
E x cos.
2 p it
x 2 2 cos. x + 1
dx.
and when these partial integrals are obtained, the entire ones will be
found by putting p = 0, 1 .... or according as p is even or
odd.
Ex. 3. Required the integral of
x r dx
x sn 2ax n + 1
"where a is < 1.
First let us find the quadratic factors ofx 211 2ax n + 1. For that
purpose put
x 2n 2ax n = 1
Then
= a + V 1. V 1 a 2
since a is <1 1.
Novr put a = cos. d , then
x n = cos. d + V 1 sin. 5
= cos. (2 p * + 5) + V^l sin. (2 p T + a)
2 p 9 + ^ , , ; . 2 p r + 3
. . x = cos. * ^ + V 1 sin. :
and the general quadratic factor of
is
2 x cos.
2 p <r f d
where p may be any number from 0, 1, &c. to n 1.
Hence to find the general partial integral of the given indefinite differ
ence, we assume
x P + Qx u N
x iTZ 2 a x " +1  2.+  M
BOOK I.] NEWTON S PRINCIPIA. 83
and proceeding as in the last example, we get
. (r 1 + 1) (2pcr + 3) 1
Q = sin. ^   v v L
=
n n sin. 5
and
. (n r) . (2 p r f 6) 1
P = sin.    v " x _  _____
n n sin. 3
whence the remainder of the process is easy.
Case 4. Let the factors of the denominator be all imaginary and equal in
pairs.
In this Case, we have the form
u_ _ y
V" {(x+)+0 f a ~
and assuming as in Case 2.
u P + Qx P + Q x
, &c
u  1 H
K + Lx K + L x
and reducing to a common denominator,
U = P + Qx + (F + Q x) (x~+~^ 2 + /3 2 ) + &c.
and substituting for x one of its imaginary values, and equating homoge
neous terms, in the result we get P and Q. Deriving from hence the
values of : , =  , &c. and in each of these values substituting for x
d x d x 2
one of the quantities which makes x + a] 2 + /S 2 = 0, and equating ho
mogeneous terms we shall successively obtain
P , Q ; P", Q", &c.
This method, however, not being very commodious in practice, for the
present case, we shall recommend either the actual developement of the
above expression according to the powers of x, and the comparison of the
coefficients of the like powers (by art. 6), or the following method.
Having determined P and Q as above, make
U  U  (P + Q x)
x~+^r + /3<
_ U  (F + Q! x)
2 T+^p + /3*
_. U"  (P" + Q! 1 x)
&c. = &c.
Then since U , U", U" , &c. have the same form as U, or have an
F2
84 A COMMENTARY ON [SECT. I.
integer form, if we put for x that value which makes (x + a) 2 f /3 2 =
0, and afterwards in the several results, equate homogeneous quantities
we shall obtain the several coefficients.
P , Q ; P", Q",&c.
Case 5. If the denominator V consist of one set of Factors simple and
unequal of the form
x ax a , &c. ;
of several sets of equal simple Factors, as
(x e) P, (x e ) % &c.
and of equal and unequal sets of quadratic factors of the forms
x 2 + a x + b, x s + a x + b , See.
(x 2 + 1 x + r) ^, (x 2 + I x + r ) , &c.
then the general assumption for obtaining the partial fractions must be
U M M
V == x^=~^ + xTZT H
I E A F i a
E
1 F/ 1
1 (xe)p 1 (xe)P >
P + Q x F + Q
(x_e )<
1 (xer
x G 4 H x x
L x 2 + ax + b + x 2 + a x
R + Sx R + S x
+ b 4
G + H
and the several coefficients may be found by applying the foregoing rules
for each corresponding set. They may also be had at once by reducing
to a common denominator both sides of the equation, and arranging the
numerators according to the powers of x, and then equating homogeneous
quantities.
We have thus shown that every rational fraction, whose denominator
can be decomposed into simple or quadratic factors, may be itself analyzed
into as many partial fractions as there are factors, and hence it is clear
that the integral of the general function
&c. Kx + L
ax n + bx u  l + &c. kx + 1
may, under these restrictions, always be obtained. It is always reducible,
in short, to one or other or a combination of the forms
r / d x f d x
/xdx,/^^, /^r+T
Having disposed of rational forms we next consider irrational ones.
Already (see Art. 86, &c.)
/+dx ,* d x /* d x _
V(a bx 2 ) /xV(bx 2 a) / V (ax bx 2 )
BOOK I.] NEWTON S PRINCIPIA. 85
have been found in terms of circular arcs. We now proceed to treat of
Irrationals generally ; and the most natural and obvious way of so doing
is to investigate such forms as admit of being rationalized.
90. Required the integral of
I 1 1 J. 1 7
dx X F Jx, x m , x n , x*, x S &c. S
where F denotes any rational function of the quantities between the brackets.
Let
x _ u m n p q f &c>
Then
i
x m = u npqr ....
JL
X n =U mpqr ....
1
( x p _ u mnq r> t m %
&C. = &C.
and
d x = mnpq . ... x u mnp< *  1 xdu
and substituting for these quantities in the above expression, it becomes
rational, and consequently integrable by the preceding article.
,,
Ex.
b + cx*
Here
x = u 69
3 180
and
dx = 6u 59 du.
Hence the expression is transformed to
n "9^ u l60 +2au 4 +l
60 u 9 d u  1* 1S
b + c u 15
whose integral may be found by Art. 89, Case 3, Ex. 2.
91. Required the integral of
d x X F x, (a + b x) ", (a + b x) s , &c.^
where F, as before, means any rational function.
Put ajbx = u ntnp then substitute, and we get
nmp . /u nm a \
^  . u nmp  1 duXF(  ^  , u m P ,u n r,& c .)
which is rational.
86 A COMMENTARY ON
Examples to this general result are
x 4 dx
[SECT. I.
,
and
c x 5 + (a + b x) * x + c (a + b x) ^
which are easily resolved.
92. Required the integral of
f /a 4 b x\  /a 4 b x\ 1
dx F Sx, ( ) u>(f )q, Sec. >
Vf + gx/ vf + gx/
Assume
a + bx
and then by substituting, the expression becomes rational and integrable.
93. Required the integral of
d x F fx, V (a + b x + c x 2 )}
Case 1. When c is positive, let
a + b x + c x 2 = c (x + u) 2 .
Then
a cu"
2cu b
(2cu b)
cu
and substituting, the expression becomes rational.
Case 2. When c is negative, if r, r be the roots of the equation
a + bx cx 2 =
Then assume
V c (x r) (r x) (x r) c u
and we have
cru 2 4r , (r r )2cudu
V . _ r\ v v _
~~ o T J u A / a i l \ I
cu 2 + 1 (cu 2 + I) 2
and by substitution, the expression becomes rational.
94. Required the integral of
d x F Jx, (a + b x) S (a? + b x) *}
Make
a + b x = (at + b x) u " ;
Then
a a u
b x =
(a b b a) 2u d u
,  a b) _V(ab a b)
BOOK I.] NEWTON S PRINCIPIA. 87
Hence, substituting, the above expression becomes of the form
duF fu, V(b u 2 b)J
F denoting a rational function different from that represented by F.
But this form may be rationalized by 93 ; whence the expression becomes
integrable.
95. Required the integral of
p
x m i dx(a + b x n )T~.
. . m m p .
This form may be rationalized when either , or  1 is an integer.
Case 1. Leta+bx B =u q ; then(a+bx n )T = UP, x^^", x m =
u q
Hence the expression becomes
which is rational and integrable when is an integer.
Case 2. Let a + bx n = x n u q ; then substituting as before, we get the
transformed expression
u
whicli is rational and integrable when + is an integer.
Examples are
x g dx x 2m dx
(a a + x 2 )^ (a 2 + x 2 )^
2m + * X 6 d X
~~
96. Required the integral of
x ra  d x (a + b x n ) q X F (x n ).
This expression becomes rational in the same cases, and by the same sub
stitutions, as that of 95. To this form belongs
x m+ n i dx(a + bx n )?
and the more general one
88 A COMMENTARY ON [SECT. I.
where
and
Q = A + B x n + C x 2n + &c.
97. Required the integral of
x m 1 dx X F{x m , x n , (a + bx n )^
Make a + bx n =u q ; then
x m  l d x = S . ( U u a )" ~ l d u
n b V. b /
and in the cases where is an integer, the whole expression becomes ra
tional and inte^rable.
tegral of
Xdx
"o
98. Required the integral of
X + X" + V(a + bx + cx 2 )
where X, X , X" denote any rational functions 0/*x.
Multiply and divide by
X + X" V(a + bx + x 2 )
and the result is, after reduction,
XX dx XX"dx Va
_ __
X /2 X" 2 (a + bxf ex 2 ) X /2 X //2 (a +bx + cx 2 )
consisting of a rational and an irrational part. The irrational part, in
many cases, may also be rationalized, and thus the whole made integrable.
99. Required the integral of
x m dxF x n , V (a + bx n f ex 2 ")}
Let x n = u ; then the expression may be transformed into
1 m + 1 ,
u n " diiF u, V (a + bu + cu 2 )]
n
which may be rationalized by Art. 93, when is an integer.
100. Required the integral of
x m dxFx n , V (a + b 2 x 2 "), bx n + V (a + b 2 x 2n )}.
Let
bx u + V (a + b 2 x 2n ) = u;
then
and the whole expression evidently becomes rational when  is an
integer.
Many other general expressions may be rationalized, and much might
BOOK I.] NEWTON S PRINCIPIA. 89
be said further upon the subject ; but the foregoing cases will exhibit the
general method of such reductions. If the reader be not satisfied let him
consult a paper in the Philosophical Transactions for 1816, by E. Ffrench
Bromhead, Esq. which is decidedly the best production upon the Integrals
of Irrational Functions, which has ever appeared.
Perfect as is the theory of Rational Functions, yet the like has not been
attained with regard to Irrational Functions. The above and similar arti
fices will lead to the integration of a vast number of forms, and to that of
many which really occur in the resolution of philosophical and other
problems ; but a method universally applicable has not yet been discover
ed, and probably never will be.
Hitherto the integrals of algebraic forms have been investigated. We
now proceed to Transcendental Functions.
101. Required the integral of
a x dx.
By Art. 17,
d.a x = l.a X a x dx
1 x
Hence
/a m x d x = i a m x + C (b)
102. Required the integral of
Xa x dx
where X is an algebraic Junction of*.
By the form (see 73)
d (u v) = u d v + v d u
we have
f u d v = u v f\ d u.
Hence
11 ^ /*l ^  .7
"* "" 1 a J 1 a
/dX a*dx _ dX a* f a* d 2 X
J dx la = dx (la) 2 /(la) 2 dx
a x dx d X a* f a x d 3 X
> ax_
dx 2 la 2 ""
(la) 2 "" dx 2 (la)
&c. = &c.
the law of continuation being manifest.
90 A COMMENTARY ON [SECT. I
Hence, by substitution,
/v x i _ v a * dX * x d 2 X a x
*n~~dx (la) 2 + dx 2 (la) 3 ~
which will terminate when X is of the form
A + Bx
Ex /*x 3 a x dx aXx3 3aX * 2 3.2a x x 3.2a x
X ** * 1 a " (1 a) 2 (la) 3 "(la) 4 "
OTHERWISE
/a x Xdx = a x /Xdx /la.a x dx/Xdx
= a x X / la/a x X / dx
putting
X =/Xdx.
Hence
/a x X dx = a x X" la/a x X"dx
&c. = &c.
and substituting, we get
/a x Xdx= a x X la.a x X" + (la) 2 a x X " &c.
X , X", X ", &c. being equal to /X d x, /X d x, /X" d x, &c. re
spectively.
T? r x dx vi , xla x 2 (la) 2 x 3 (la) 3
&/.  = a l* + _ + L.2 .+ _A_L + 4,, + C .
which does not terminate.
By this last example we see how an Indefinite Difference may be in
tegrated in an infinite series. If in that example x be supposed less
than 1, the terms of the integral become less and less or the series is con
vergent. Hence then by taking a few of the first terms we get an ap
proximate value of the integral, which in the absence of an exact one, will
frequently suffice in practice.
The general formula for obtaining the integral in an infinite or finite
series, corresponding to that of Taylor in the Calculus of Indefinite
Differences, is the following one, ascribed to John Bernoulli, and usually
termed
JOHN BERNOULLI S THEOREM.
/Xdx = Xx /xdX
rdX _ dX x 2 r x 2 dx d 2 X
J dx ~^ ~2~J~^ dlF
x 3 dx d 3 X
2.3 dx 3
&c. = &c.
/d^X x g dx _ d 2 X x 3 f
J dx 2 2  "31* !""/
BOOK L] NEWTON S PRINCIPIA. 91
Hence .
f*r i XT X  Cl A. X ~,
/Xdx = X*^. 5 + ^ . gjSc. + C
the theorem in question.
Ex.l.x m dx = x m + 1 x m + 1 + ^ n ! 1 x m + 1 &c. + C
But since
m
as in Art. 78.
1 02. Required the integral of
Xdx(lx)"
where X is any Algebraic Function ofx, \ x the Hyperbolic logarithm of x,
and n a positive integer.
By the formula
f u d v = u v f v d u
we have
dx
x n /Yl x) B  /"X d x
x "
= (lx)"X n/(lx) ^ X X
J x " v x
&c. = &c.
where X , X", X ", &c. are put for/Xdx, /"^ dx, / d x, &c. re
 / X ^ X
spectively.
Hence
/Xdx(lx)" = X (lx)"
] 93. Required the integral of
inhere U w any function of I x.
x
92 A COMMENTARY ON [SECT. I.
Let u = 1 x.
Then
A dx
d u = .
x
and substituting, the expression becomes algebraic, and therefore integra
ble in many cases.
104. Required the integral of
Xdx (lx) n
"where n is negative.
Integrating by Parts, as it is termed, or by the formula
/u d v = u v f\ d u
we get, since
X d x d x .
/Xdx __ Xx 1 rdx_ dJXxj
^(lx) n (n 1) (lx)" 1+ n lV(l x )ni dx
and pursuing the method, and writing
y, _ d (X x)
dx
we have
X Xx
x// _ d (X x)
~dT~
&c. = &c.
(n 1) . . . 2/1(5)
or
Xx  x^ 1 / dx
__ __ _  _ x __
(n 1 ) (1 x) ~ ~ J (n 1) . (n 2) . . . . (n m) (Uj^^
according as n is or is not an integer, m being in the latter case the
greatest integer in n.
/x"dx__ x + f 1 m+1
X V (1 X )" n __ ! t(lx)" 1 + (n SH rxp^ 4 C
(m + I)" 1 /*x m d x
__ (m + I)" 1 /*
n 1) n 2 ____ I/
(n 1) (n 2) ____ I Ix
when m is an integer.
105. Required tlie integrals of
i A i * d I . d f . . d I
d . cos. 0, d d . sin. <L d 6 . tan. 0. d 6 . sec. 0,   , = A , 
cos. I sin. ^ tan.
By Art. 26, &c.
d sin. 6 = d 6 . cos. 0, and d cos. 6 = d 6 sin. 6
/./d 6 cos. r= sin. 6 + >C ......... (a)
and
sin. = C cos. 6 ........ (b)
BOOK L] NEWTON S PRINCIPIA. 93
Again let tan. 6 = t ; then
dt
d 6 =
1 + t 2
and
t d t
t 2 )
= C 1. cos. 6 . ;i (c)
since
1
1 + t 2 = sec. 2 6 =
d 6 sec. 6 =
cos. z 6
Again
d 6 d 6 cos. 6
cos. 6 ~ 1 sin. 2 1
d (sin. 6}
"1 sin. 2 6
l d (sin. d) d sin.
1 sin. 6 * 1 + sin.
/.yd 6 sec. 4 = l.(lfsin.0) % 1 (1 sin.0)fC
rrl.tan. (45+^) + C. . . (d)
which is the same as f   .
* cos. 6
Again
C.   = fd d cosec. 6
J sin. 6
= /d * sec. (!*)= /d . (  tf sec. 
= 1. (tan. I) + C . . . . ...... (e)
Again
f ~
= Icos. 0) + C(byc)
= 1 . sin. 6 + C . . . . ........ (f)
106. Required the integral of
sin. m cos. n 6 . d 0.
m and n fomg positive or negative integers.
94 A COMMENTARY ON [SECT. I.
Let sin. 6 = u ; then d 6 cos. d = d u and the above expression becomes
u m du(l u 2 ) ^
which is integrable when either or J f n ~ = m "*""
2
is an integer (see 95.) If n be odd, the radical disappears ; if n be even
and m even also, then ^ = an integer ; if n be even and m odd, then
m + I . ,. T _ T ,
^ is an integer. Whence
u m d u (1 u ! ) n "a
is integrable by 95.
OTHERWISE,
Integrating by Parts, we have
ClTl HI ~ 1 A vyi
/d 6 sin. m 6 cos. n 6= cos." + 1 6+ */cos. n + 2 6. sin. m  2 Q X d 6
n + 1 m j 1
sin." 1  1 6 m 1 ,
= cos. n + 1 ^ H / dx sin. m ~ 2 ^ cos. "tf
m 4 n m + n j
and continuing the process m is diminished by 2 each time.
In the same way we find
and so on.
107. Required the integrals of
d u = d 6 sin, (a 6 + b) cos. (a 6 + b )
dv = d*sin.(af + b) sin. (a 6 + b )
and
dw = d 6 cos. (a 6 + b) cos. (a 6 + b )
By the known forms of Trigonometry we have
du = d 6 [sin. (a + a . 0+b + b x ) + sin. (a a . + b b )}
i d v = dd {cos. (a~+T .0+b + b ) cos. (a a .<J+b b )}
d w = d d {cos. (a + a". 0+b + b ) + cos. (a^a 7 . 6+b b )}
Hence by 105 we have
r t / cos. (a^Fa 7 . ^ + b + bQ , cos, (a a 7 . + b b ) \
~*t~ a + a a a ~J
sin. (a+ a . + b + b ) sin. (a a 7 ".* + b b 7 )
 ~ "
 p i i/ sin  (a + a  + b + bQ . sin, (a^^a 7 . tf + b I/)
rt  a + a a a
These integrals are very useful.
BOOK I.]
NEWTON S PRINCIPIA.
95
108. Required the integrals of
6 n d 6 sin. 0, and d n d 6 cos. d.
Integrating by Parts we get
/ <) n xd0sin. 0=C n cos. 0fn d" 1 sin. 0+n . (n 1) n  8 cos.0 &c.
and
/tJ"xd0cos.d:=:C + n sin. 6 + nd ll ~ l cos.d n. (n 1) n  2 sinJ +&c.
109. Required the integrals of
X d x sin. l x
X d x tan. l x
X d x sec. ~ x
&c.
Integrating by Parts we have
/Xdxsin.>x = sin.ix/Xdx
/ V 1 i r v 1 / d X /" X d X
J X d x tan.  l x = tan.  l x/ X d x / r^  j
. ~^
x V(x 2 1)
/X d x sec.  l x = sec.  l xA X d x /* r5.
^ x x 2
&c. = &c.
see Art. 86.
110. Required the integral of
d u  ( f + g cos, tf) d 6
(a + b cos. 6) n
Integrating by Parts and reducing, we have
(ag bf)sin. 6 _ _
(n l)(a 2 b 2 )(a
/(n 1) (at* bg) + (n^2) (a g
(n I)(a 2
cos.^
(a + b cos. 6) n ~
which repeated, will finally produce, when n is an integer, the integral
required.
^
(a b) tan.
Ex./ d " 2
r a
+ b cos. d V (a z b 2 )
=r . tan.
V (a 2 b 2 )
1 , b+acos. 6+ sin. 6 V (b 2 a ) _
V(b z a 2 ) a + b cos. 6
Notwithstanding the numerous forms which are integrable by the pre
ceding methods, there are innumerable others which have hitherto resisted
all the ingenuity that has been employed to resolve them. If any such
appear in the resolution of problems, they must be expanded into con
96 A COMMENTARY ON [SECT. I
verging series, by some such method as that already delivered in Art. 101 j
or with greater certainty of attaining the requisite degree of convergency,
by the following
METHOD OF APPROXIMATION.
111. Required to integrate between x = b, x = a, any given Indefinite
Difference,, in a convergent series.
Let f (x) denote the exact integral of f X d x; then by Taylor s
Theorem
and making
h = b a
f (x + ba)fx = X. (ba) + d d . 1^1!+ &c.
Again, make
x = a
then
dX d 2 X
dT "d^ 2 &C<
become constants
A, A , &c.
and we obtain
f(b)f(a) = A(ba) + . (ba) + ^ (ba) 3
which, when b a is small compared with unity, is sufficiently conver
gent for all practical purposes.
If b a be not small, assume
b a = p./3
p being the number of equal parts 3, into which the interval b a is sup
posed to be divided, in order to make /3 small compared with unity. Then
taking the integral between the several limits
a, a + /3
a, a + 2 18
&c.
a, a + p /3
BOOK I.] NEWTON S PRINCIPIA. 97
we get
f. ( a + /3) f( a ) = A/3+ ^. 0* + ^. . 0s + &c.
A.O
f (a + 2jg) f (a + flrrBjS+S . /3 2 + l/3 3 + &c.
/i <c. o
&c. = &c.
f (a + pj8) f (a+J=I./8) = P/3 + /3 2 +
A, A 7 , &c. B, B , &c ....... P, P , & c .
being the values of
v dx
Xj dT &c "
when for x we put
a, a + ft a + 2 ft &c.
Hence
f(b)f(a) =(A + B + ..;..
+ (A + B + . . . . F)
+ (A" + B" + . . . . F ) 1^3
+ &c.
the integral required, the convergency of the series being of any degree
that may be demanded.
If j3 be taken very small, then
f (b) f (a) = (A + B +  P) nearly.
Ex. Required the approximate value of
/X^^dx X (1 x n )f
between the limits of x = and x = 1, when neither 9 n r  f
is an integer.
Here
X = x 1 " 1 (l_x n )T
and
d X p JL n p
jY rr (m + n l)x n  2 (] ^x) _ ^x n  2 (l X
b^a  ]  1.
Assume ] = 10 X ft and we have for limits
1 2
Tb ; To ; &c 
YOI. I. G
98 A COMMENTARY ON fSEcr. I.
Hence m being > 1,
A =
"
&c. = &c.
Hence, between the limits x = 1 and x =
1 f P H
/Xdx =  X (10 n l)q + (10 n 2 n )q
_j_ (10 " 3 n ) "? + &c. + (10 n 9 n )"!r  nearly.
W T e shall meet with more particular instances in the course of our
comments upon the text.
Hitherto the use of the Integral Calculus of Indefinite Differences has
not been very apparent. We have contented ourselves so far with
making as rapid a sketch as possible of the leading principles on which
the Inverse Method depends ; but we now come to its
APPLICATIONS.
112. Required to Jind the area of any curve, comprised between two
given values of its ordinate.
Let E c C (fig. to LEMMA II of the text) be a given or definite area
comprised between and C c, or and y. Then C c being fixed or De
finite, let B b be considered Indefinite, or let L b = d y. Hence the
Indefinite Difference of the area E c C is the Indefinite area
B Ccb.
Hence if E C = x, and S denote the area E c C ; then
d S = B Ccb = CL + Lcb
y d x + L c b.
But L c b is heterogeneous (see Art. 60) compared with C L or y d x.
... d S = y d x
BOOK I.] NEWTON S PRINCIPIA. 99
Hence
S=/ydx,
the area required.
Ex. 1. Required the area of the common parabola.
Here
y 2 = a x.
2y dy
.. d x = * J
a
and
s _ "

a 3a
and between the limits of y = r and y = r becomes
If m and m be the corresponding values of x, we have
2
S = (r m r m )
2
= of the circumscribing rectangle.
Let r = 0, then
2
S = r m (see Art. 21.)
Ex. 2. Take the general Parabola whose equation is
y m __ <! x n
Here it will be found in like manner that
m
m + n* "
between the limits of n = y = 0, and x = a, y = /3.
Hence all PARABOLAS may be squared, as it is termed ; or a square may
be found whose area shall be equal to that of any Parabola.
Ex. 3. Required the area of an HYPERBOLA comprised by its asymptote,
and one infinite branch.
If x, y be parallel to the asymptotes, and originate in the center
x y = a b
is the equation to the curve.
Hence
d x = abd y
y 8
G2
100 A COMMENTARY ON [SECT. I.
and
Q f abdy
s =/ IT* = C a b 1 y.
Let at the vertex y = /3, and x == ; then the area is and
C = a b . 1 8.
Hence
S = a b . 1 . P .
y
1 13. If the curve be referred to a fixed center by the radiusvector and
tracedangle 6; then
ds = l ^ "{. . : . !; . : : ,,, ? ;.
For d S the Indefinite Area contained by f , and f + d = (gfdg) ^5
2
d 6 . g d e d
+ s  (Art. 26) and equating homogeneous quantities we
have
d S =
Ex. 1. In the Spiral of Archimedes
= a
n 2 _ 2
. Q f AZ A A 43 I p
. . 5 _ j e a. d  . + \^.
Ex. 2. In the Trisectrix
S = 2 cos. d + 1
.. d S = i/(2cos. tf l) 2 d^
which may easily be integrated.
Hence then the area of every curve could be found, if all integrations
were possible. By such as are possible, and the general method of ap
proximation (Art. Ill) the quadrature of a curve may be effected either
exactly or to any required degree of accuracy. In Section VII and many
other parts of the Principia our author integrates Functions by means of
curves ; that is, he reduces them to areas, and takes it for granted that
such areas can be investigated.
114. To find the length of any curve comprised within given values of the
ordinate ; or To RECTIFY any curve.
Let s be the length required. Then d s = its Indefinite Chord, by
Art. 25 and LEMMA VII.
.. d s = V (d x 2 + d y 2 )
and
s =fV (dx* + dy 2 ) (a)
BOOK L] NEWTON S PRINCIPIA.
Ex. 1. In the general parabola
y m = ax".
Hence
and
dx 2 =
2 2 m
.dy
n a n
n a
which is integrable by Art. 95 when either
1_ 1 1_
O TV* O ryi "* O
that is, when either
In
or
1m
101
2 m n 2 m n
is an integer ; that is when either m or n is even.
The common parabola is Rectifiable, because then m = 2. In this case
ds= dy V(l + y 2 )
Hence assuming according to Case 2 of Art. 95,
(r)
we get the Rational Form
ds =
Hence by Art. 89, Case 2,
 + V u
l 4 8
Butu = +/ * zy .
cessary reductions
Hence by substituting and making the ne
G3
** ^ * *
102
s =
A COMMENTARY ON
.
a 1 .
[SECT. I.
Let y = ; then s = and we get C =
and .. between the Limits of y = and y = /3
+ a 1 .
In the Second Cubical Parabola
and
y 3 = ax 2
d s = d y
which gives at once (Art. 91)
Ex. 2. In the circle (Art. 26)
ds =
C.
)
4;
V(ly^)
which admits of Integration in a series only. Expanding (1 v z )~"i
by the Binomial Theorem, we have
Hence,
and
and between the limits of y = and y =  or for an arc of 30 we have
s  j ___ _ 4. .
2 h 2. 3. 2 3 + 2.4.5.2 5
1
1 4. _L_ 4, 3 , 5 5. 7
2 + 3. 2 4 "*" 5. 2 8 + 7. 2 11 + 9. 2 16
f.5
! .0208333333
+ &c.
= <! .0023437500
I .0003487720
L.0000593390
Sec.
= .5235851943 nearly.
BOOK I.] NEWTON S PRINCIPIA. 103
Hence 180 of the circle whose radius is 1 or the whole circumference
it of the circle whose diameter is 1 is
cr = . 5235851943 ... X 6 nearly
= 3.1415111658
which is true to the fourth decimal place ; or the defect is less than .
By taking more terms any required approximation to the value of T may
be obtained.
Ex. 3. In the Ellipse
a 2 e 2 x 2
s = /dx. N / a2 _ x2
where x is the abscissa referred to the center, a the semiaxis major and
ae the eccentricity (see Solutions to Cambridge Problems, Vol. II. p. 144.)
115. If the curve be referred to polar coordinates, and 8; then
s =fV fe*d* +dg 2 ) (b)
For
y = g sin. 6
x = m + cos. d
and if d x 2 , d y 2 be thence found and substituted in the expression
(114. a) the result will be as above.
Ex. 1. In the Spiral of Archimedes
P a $
. 8 a ! t+ * (s + O  c
2 a
see the value for s in the common parabola, Art. 1 14.
Ex. 2. In the logarithmic Spiral
e
S  e
or
and we find
s = V~2fd g = g V 2 + C.
116. Required the Volume or solid Content of any solid formed by the
revolution of a curve round its axis.
Let V be the volume between the values and y of the ordinate of the
generating curve. Then d V = a cylinder whose base is T y 2 and alti
tude d x + a quantity Indefinite or heterogeneous compared with either
d V or the cylinder.
G4
104 A COMMENTARY ON [SECT. I.
But the cylinder = v y ~ d x. Hence equating homogeneous terms, we
have
d V = cry 2 dx
and
V = cr/y*dx (c)
Ex. 1. In the sphere (rad. = r)
y 2 = r 2 x 2
.. V = ff/r 2 d x w/x 2 d x
/ x 3
and between the limits x = and r
which gives the Hemisphere.
Hence for the whole sphere
Ex. 2. In the Paraboloid.
y z = ax
.. V = ,r/a x d x
<x a 2
: ~2~ : C;
and between the limits x = and a
Ex. 3. In the Ellipsoid.
,.V: ^./(a 2 dx~x 2 dx)
V2
a^
and between the limits x = and a
..
Hence for the whole Ellipsoid
V = jUab 2 .
O
The formula (c) may be transformed to
y (d)
BOOK I.]
NEWTON S PRINCIPIA.
105
where S = f y d x or the area of the generating curve, which is a singular
expression, f S d y being also an area.
In philosophical inquiries solids of revolution are the only ones almost
that we meet with. Thus the Sun, Planets and Secondaries are Ellip
soids of different eccentricities, or approximately such. Hence then in
preparation for such inquiry it would not be of great use to investigate
the Volumes of Bodies in general.
If x, y, z, denote the rectangular coordinates, or the perpendiculars let
fall from any point of a curved surface upon three planes passing through
a point given in position at right angles to one another, then it may easily
be shown by the principles upon which we have all along proceeded,
that
d V = d y/z d x"l
or
= d z/y d x \ . ..... (e)
or
= dx/zdyj
according as we take the base of d V in the planes to which z, y, or x is
respectively perpendicular
For let the Volume V be cut off by a plane passing through the point
in the surface and parallel to any of the coordinate planes ; then the area
of the plane section thus made will be
/z d
or
or
fy d x
/ z d y
see Art. 112.
Then another section, parallel to^z d x, orfy d x, or J z d y and at
the Indefinite distance d y, or d z, or d x from the former being made,
ic Indefinite Difference of the Volume will be the portion comprised by
icse two sections ; and the only thing then to be proved is that this por
tion is = d yyz d x or d zy*y d x, or d x JT* d y. But this is easily to
proved by LEMMA VII.
This, which is an easier and more comprehensible method of deducing
V than the one usually given by means of Taylor s Theorem, we have
lerely sketched ; it being incompatible with our limits to enter into de
1. To conclude we may remark that in Integrating both y z d x, and
d y y z d x must be taken within the prescribed limits, first considering
Definite and then .r.
106 A COMMENTARY ON [SECT. I.
117. To find the curved surface of a Solid of Revolution.
Let the curved surface taken as far as the value y of the ordinate re
ferred to the axis of revolution be a, and s the length of the generating
curve to that point; then d a = the surface of a cylinder the radius of
whose base is y and circumference 2 <r y, and altitude d s, by LEMMA VII.
and like considerations. Hence
d * = 9 *r y d a
and
. = 2<r/yds . . . . . . . .: . . (a)
or
= 2vrys 2cr/sdy ...... (b)
which latter form may be used when s is known in terms of y ; this will
not often be the case however.
Ex. In the common Paraboloid.
y 2 = a x
and
Let y = and S, then a between these limits is expressed by
If the surface of any solid whatever were required, by considerations
similar to those by which (116. e) is established, we shall have
d a = V (dy 2 + dz 2 )/ V (d x 2 + d z 2 ) . . . . (c)
and substituting for dzin V d x 2 + dz 2 its value deduced from z = f .
(x, y) on the supposition that y is Definite ; and in V (d y 2 + d z 2 ) its
value supposing x Definite. Integrate first V (d x 2 + d z 2 ) between the
prescribed limits supposing y Definite and then Integrate V (d y 2 + d z 2 )
f V (d x 2 + d z 2 ) between its limits making x Definite. This last result
will be the surface required.
We must now close our Introduction as it relates to the Integration of
Functions of one Independent variable.
It remains for us to give a brief notice of the artifices by which Func
tions of two Independent Variables may be Integrated.
118. Required the Integral of
Xdx + Ydy = 0,
where X is any function o/ x, and Y a function ofy the same or different.
BOOK L] NEWTON S PRINCIPIA. 107
When each of the terms can be Integrated separately by the preceding
methods for functions of one variable, the above form may be Integrated,
and we have
/Xdx+/Ydy = C.
This is so plain as to need no illustration from examples. We shah 1 ,
nowever, give some to show how Integrals apparently Transcendental
may in particular cases, be rendered algebraic.
Ex. 1. + ^ = 0.
.. 1 x + 1 y  C  1 . C
and
.. x y  C or = C.
d x d y _
ii(X. . i TJ 3V ~f~ ~i" i , _. o\ "
Here
sin.  l x + sin. ~ y C = sin.  C
.. C = sin. sin.  1 x sin.  1 y}
=. sin. (sin. ~ x) . cos. (sin. l y ) + cos. (sin. ~ l x) sin. (sin. "~ l y)
 x . V (1 y 2 ) + y V (1 x 2 )
which is algebraic.
Generally if the Integral be of the form
f (x) + f. (y) = C
Then assume
C = f. (C)
and take the inverse function off" 1 (C) and we have
which when expanded will be algebraic.
119. Required the Integral of
Ydx + Xdy = 0.
Dividing by X Y we get
which is Integrable by art. 118.
120. Required the Integral of
inhere P and Q are each such Junctions of\ and y that the sum of the expo
nents ofx. and y in every term of the equation is the same.
108 A COMMENTARY ON [SECT. I.
Let x = u y. Then if m be the constant sum of the exponents, P and
Q will be of the forms
U X y m U y m
U and U being functions of w.
Hence, since dx = udy + ydu, we have
U.(udy + ydu) + U dy =
and
(UufU )dy + Uydu =
d y , U d u 
y y + tnr+U = ..... ^
which is Integrable by art. 118.
Ex. 1. (a x + b y) d y + (f x 4 g y) d x = 0.
Here
P = f x + g y, Q = a x + b y
U= fu+ g, U = a u + b
. ( ll 4. (fu + g) du _
y h fu 2 + (g + a)u + b 
which being rational is Integrable by art. (88, 89)
Ex. 2. x d y y d x d x V (x 2 + y 2 )
Here
Q = x, P = y V (x 2 +y z )
U = u, U = 1 V (1 + u 2 )
d_y 2
u V (1 + u 2 )
or
ll  1^ 4. du
y u r U V (1 + U 2 ) "
which is Integrable by art. (82, 85.)
These Forms are called Homogeneous.
121. To Integrate
(ax4by4c)dy + (mx+ny4p)dx = 0.
By assuming
= vj
and
m x + n y + P
we get
m d u adv bdv ndu
d y =  1 , and d x  T 
J mb na mb ria
and therefore
(m u n v) d u + (b v a u) d u =
which being Homogeneous is Integrable by Art. 120.
BOOK L]
NEWTON S PRINCIPIA.
109
We now come to that class of Integrals which is of the greatest use in
Natural Philosophy to
LINEAR EQUATIONS.
122. Required to Integrate
dy + yXdx^X dx,
where X, X are functions of X.
Let
y = u v.
Then
udv+vdu+Xuvdx= X dx
Hence assuming
d v + v X d x = (a)
we have also
v d u = X d x (b)
Hence
.. Iv +/Xdx = C
or
v = e c/xdx
= e c X e~
= C X e xdx .
Substituting for v in (b) we therefore get
1 /Xdx
du = p,.e X dx
\s
which may be Integrated in many cases by Art. 118.
Ex. dy + ydx = ax 3 dx.
Here
X = 1, X = a x 3
/X d x = x
and
/X d x e Xdx = a/x 3 e * d x
= a e x (x 3 3 x 2 +
see Art. (102)
Hence
y = Ce~* + a (x 3 3x 2 + 6x 6)
6 x 6)
110 A COMMENTARY ON [SECT. I.
122. Required to Integrate the LINEAR Equation of the second order
dx 2 .dapT
"where X, X are functions o/*x.
Let y = e/" 11 ^; then y^ = ue /udx
d x
d x 2 ~ MX /
and .*. by substitution,
d~x "*
which is an equation of the first order and in certain cases may be Integ
rable by some one of the preceding methods. When for instance X and
X are constants and a, b roots of the equation
u 2 + Xu+ X =
then it will be found that
y = C e a x + C e b x .
123. Required the Integral of
_
d x 2 d x
where X" is a new function ofx.
Let y = t z ; then Differencing, and substituting, we may assume the
result
^+X^+X Z = .... ... . (a)
and
d z\ , X /x ,.
E) dx:= ir ( b >
Hence (by 122) deriving z from (a) and substituting in (b) we have a
Linear Equation of the first order in terms of fjrrJJ whence (g
be found ; and we shall thus finally obtain
dx).
Here
X v ~\ ft
, ov j , vv
BOOK L] NEWTON S PRINCIPIA.
Equat. (a) becomes
d 2 z dji 1 z
dx~ 2 + dx x " x 2
whence
wherein z = e /udx ; which becomes homogeneous when for u we put v~ \
Next the variables are separated by putting (see 120)
X = V S
and
we have
dv
1 S2 +
s 1
V
s(s 2
 1)
and
1
/s + 1
s Vs 1
Hence
x 2 + 1 , , x 2 1
TTi fx/ udx = ! 
and
x 2 1
X
Again
f> /Xd x Ix v
^ C A
and
/X" e/ Xdx z d x =/a d x = a x + C
and
x 2 1 r (a x + C) xd x
r _\ _ !___
y x J (x 2 I) 2
which being Rational may be farther integrated, and it is found that
finally
A v 4 C v 2 1 ,, x _
C* A. J \*J A. Jl I / f~^i "
2 x 4 x \ x +
Here we shall terminate our long digression. We have exposed both
the Direct and Inverse Calculus sufficiently to make it easy for the
reader to comprehend the uses we may hereafter make of them, which
was the main object we had in view. Without the Integral Calculus, in
some shape or other, it is impossible to prosecute researches in the higher
branches of philosophy with any chance of success ; and we accordingly
see Newton, partial as he seems to have been of Geometrical Synthesis,
frequently have recourse to its assistance. His Commentators, especially
112 A COMMENTARY ON [SECT. II.
the Jesuits Le Seur and Jacquier, and Madame Chastellet (or rather
Clairaut), have availed themselves on all occasions of its powers. The
reader may anticipate, from the trouble we have given ourselves in establish
ing its rules and formulae, that we also shall not be very scrupulous in that
respect. Our design is, however, not perhaps exactly as he may suspect.
As far as the Geometrical Methods will suffice for the comments we may
have to offer, so far shall we use them. But if by the use of the Algo
rithmic Formulae any additional truths can be elicited, or any illustrations
given to the text, we shall adopt them without hesitation.
SECTION II. PROP. I.
124. This Proposition is a generalization of the Law discovered by Kepler
from the observations of Tycho Brahe upon the motions of the planets
and the satellites.
" When the body has arrived at B," says Newton, " let a centripetal
force act at once with a strong impulse, #c."] But were the force acting
incessantly the body will arrive in the next instant at the same point C.
For supposing the centripetal force
incessant, the path of the body will
evidently be a curve such as A B C.
Again, if the body move in the chord
A B, and A B, B C be chords de
scribed in equal times, the deflection
from A B, produced by an impulsive
force acting only at B and communi ^
eating a velocity which wouldhavebeen
generated by the incessant force in the time through A B, is C c. But
if the force had been incessant instead of impulsive, the body would have
been moving in the tangent B T at B, and in this case the deflection at the
end of the time through B C would have been half the space describ
ed with the whole velocity generated through B C (Wood s Mech.)
But
C T = i C c
.. the body would still be at C.
BOOK I.] NEWTON S PRINCIPIA. 113
AN ANALYTICAL PROOF.
Let F denote the central force tending constantly to S (see Newton s
figure), which take as the origin of the rectangular coordinates (x, y)
which determine the place the body is in at the end of the time t. Also
let g be the distance of the body at that time from S, and 6 the angular
distance of g from the axis of x. Then F being resolved parallel to the
axis of x, y, its components are
F. and F. 3L
S S
and (see Art. 46) we .. have
d 2 x _ F x ^ d 2 y _ F JT
Hence
yd 2 x _ p,xy__xd 2 y
d t 2 ~J~ ~dT 2
< y d 2 x x d g y
d t
But
yd 2 x xd 2 y = dydx + yd 2 x dxdy xd 2 y
= d . (y d x x d y)
.. integrating
y d x x d v
j = constant = c.
d t
Again,
x = g cos. d, y = g sin <?, x 2 + y 2 = g 2
. . d x = g d 6 sin. + d g cos. 6
d y = g d 6 cos. 6 + d g sin. S;
whence by substitution we get
ydx xdy = f 2 d
But (see Art. 1 13)
g 2 = d . (Area of the curve) = d . A
.. d t = s  r = . d A.
c c
Vor.. I. H
114 A COMMENTARY ON [SECT. II.
Now since the time and area commence together in the integration
there is no constant to be added.
2
.. t = X A oc A.
c
Q. e. d.
J25. COR. 1. PROP. II. By the comment upon LEMMA X, it appears
that generally
ds
v = dt
and here, since the times of describing A B, B C, &c. are the same by
hypothesis, d t is given. Consequently
v oc d s
that is the velocities at the points A, B, C, &c. ai*e as the elemental spaces
described A B, B C, C D, &c. respectively. But since the area of a A
generally = semibase X perpendicular, we have, in symbols,
d . A = p X d s
d. A
. . V OC d S CC  ;
p
and since the A A B S, B C S, C D S, &c. are all equal, d A is constant,
and we finally get
1 c
v a or =
P P
the constant being determinable, as will be shown presently, from the
nature of the curve described and the absolute attracting force of S.
126. COR. 2. The parallelogram C A being constructed, C V is equal and
parallel to A B. But A B = B c by construction and they are in the
same line. Therefore C V is equal and parallel to B c. Hence B V is
parallel to C c. But S B is also parallel to C c by construction, and
B V, B S have one point in common, viz. B. They therefore coincide.
That is B V, when produced passes through S.
127. COR. 3. The body when at B is acted on by two forces ; one in
the direction B c, the momentum which is measured by the product of its
mass and velocity, and the other the attracting single impulse in the di
rection B S. These acting for an instant produce by composition the
momentum in the direction B C measurable by the actual velocity X mass.
Now these component and compound momentums being each propor
tional to the product of the mass and the initial velocity of the body in
the directions B c, B V, and B C respectively, will be also proportional
to their initial velocities simply, and therefore by (125) to B V, B c, B C.
BOOK I.]
NEWTON S PRINCIPIA.
115
C
Hence B V measures the force which attracts the body towards S when
the body is at B and so on for every other position of the body.
128. COR. 1. PROP. II. In the annexed
figure B c = A B, C c is parallel to
S B, and C c is parallel to S B. Now
A S C B = S c B = S A B, and if the
body by an impulse of S have deflected
from its rectilinear course so as to be
in C, by the proposition the direction
in which the centripetal force acts is that
of C c or S B. But if, the body having
arrived at C , the ASBC be>SAB
(the times of description are equal by
hypothesis) and .. > S B C, the vertex
C falls without the A S B C, and the
direction of the force along c C or B S ,
has clearly declined from the course
B S in consequentia.
The other case is readily understood
from this other diagi am.
129. To prove that a body cannot de
scribe areas proportional to the times round
two centers.
If possible let
AS AB = A S BC
and
S A B = S B C.
Then
A S B C ( = S A B) = S B c
and C c is parallel to S B. But it is
also parallel to S B by construction.
Therefore S B and S B coincide, which
is contrary to hypothesis.
130. PROP. III. The demonstration of this proposition, although strictly
rigorous, is rather puzzling to those who read it for the first time. At least
so I have found it in instruction. It will perhaps be clearer when stated
symbolically thus :
Let the central body be called T and the revolving one L. Also lef
the whole force on L be F, its centripetal force be f, and the force ac
H2
116 A COMMENTARY ON [SECT. II.
celerating T be f . Then supposing a force equal to f to be applied to
L and T in a direction opposite to that of f , by COR. 6. of the Laws,
the force f will cause the body L to revolve as before, and we have
remaining
f = F f
or
F = f + f .
Q. e. d.
ILLUSTRATION.
Suppose on the deck of a vessel in motion, you whirl a body round in a
vertical or other plane by means of a string, it is evident the centrifugal
force or tension of the string or the power of the hand which counteracts
that centrifugal force i. e. the centripetal force will not be altered by the
force which impels the vessel. Now the motion of the vessel gives an
equal one to the hand and body and in the same direction ; therefore the
force on the body = force on the hand + centripetal power of the hand.
131. PROP. IV. Since the motion of the body in a circle is uniform by
supposition, the arcs described are proportional to the times. Hence
., , arc X radius
t a arc described oc 5
oc area of the sector.
Consequently by PROP. II. the force tends to the center of the circle.
Again the motion being equable and the body always at the same dis
tance from the center of attraction, the centripetal force (F) will clearly
be every where the same in the same circle (see COR. 3. PROP. I.) But
the absolute value of the force is thus obtained.
Let the arc A B (fig. in the Glasgow edit.) be described in the time T.
Then by the centripetal force F, (which supposing A B indefinitely small,
may be considered constant,), the sagitta D B (S) will be described in
that time, and (Wood s Mechanics) comparing this force with gravity as
the unit of force put = 1, we have
S = fFT<
g being = 32 feet.
But by similar triangles A B D, A B G
BOOK I.]
(LEMMA VII.)
If T be given
If T = arc second
NEWTON S PRINCIPIA.
2S (arc A B)
~ jfT 2 = g R T 2
117
Fa (arcA^
_ (arcAB)
r 7^
132. COR. 1. Since the motion is uniform, the velocity is
arc
:
F  oc
* * TJ "* T>
g R R
133. COR. 2. The Periodic Time is
circumference 2 it R
p _
velocity
* 2 R
gRP 2 gP
134. COR. 3, 4, 5, 6, ^. Generally let
P = k x R n ,
k being a constant.
Then
2 T R 2 if
~P~~ = k R
and
*
^
P 2
ex
F =
4r
gk 2 R
oc
Conversely. If F a Rgn1 ; P will a R
For (133)
Pcc
135. COR. 8. A B, a b are similar
arcs, and A B, a h contemporaneous
ly described and indefinitely small. M
Now ultimately
an: am:: ah :ab*
and
a m : A M : : a b : A B
(LEMMA V)
.. an : A M : : ah 2 : ab. A B
118
A COMMENTARY ON
[SECT. II.
or
f : F
ah* e A B
a b
a s
A B 2
. ,% (LEMMA V)
A Ib v
a s
or
A B
V 2
A~S
_V 2
* A S*
And if the whole similar curves A D, a d be divided into an equal
number of indefinitely small equal areas A B S, B C S, &c. ; a b s, b c s,
&c. these will be similar, and, by composition of ratios, (P and p being
the whole times)
P : p : : time through A B : time through a b
A B ab AS
Hence
v V
PCX A S
V
V 2
a s
v
A S
rp 2
AC
RB
; RB (131)
AB 2
136. COR. 9. Let A C be uniformly described,
and with the force considered constant, suppose
the body would fall to L in the same time in
which it would revolve to C. Then A B being
indefinitely small, the force down R B may be
considered constant, and we have (131)
A C 2 : A B 2 : : T 2 :
A B
. . TT 2 .
A L
: : AL
: . AL
Hence
AB 2 =ALxAD.
PROP. VI. Sagitta ex F when time is given. Also sag. a (arc) 2 by
LEMMA XI, ex t 2 when F is given
.. when neither force nor time is given
sag. ex F X t 2 ;
BOOK I.] NEWTON S PRINCIPIA. 119
OTHERWISE.
By LEMMA X, COR. 4,
space ipso motus initio
* C a 7g
sag.
<*=?.
To generalize this expression, let ^ be the space described in I" at
the surface of the Earth by Gravity. Also let the unit of force be Gravi
ty. Then
1 : : 1^" : 2X I 772
2_sag. _2_ ^ }
gt 2 g t 2
by hypothesis.
137. COR. 1. F oc 5J 1 a Q ^
t 2 (area S P Q) 2
QR
S P 2 x Q T 2
To generalize this, let a be the area described in I . Then the area
described in t" a X t = .
&
SP x QT
~2~a~
and substituting in (a) we get
8 a 2 _QR b)
~Y < SP 2 x QT 2
Again, if the Trajectories turn into themselves, there must be
a : I" : : A (whole Area) : T (Period. Time)
a = A
Hence by (b) we have
8A 2 QR ,^
r = _. X
gT 2 S P 2 x QT 2
which, in practice, is the most convenient expression.
8 A 2 Q R
138. COR. 2. F = g^ 2 X gy^x Q P 2 ^
139. Cor. 3. F = A X SY x PV (e)
120 A COMMENTARY ON [SECT. II.
Hence is got a differential expression for the force. Since
P v  ~P d *
A. T i
dp
__.8 A 2 i
.". T lFf\H X
gT 2 2p 2 pdg
dp
 1A! x dp m
gT 2 Vdg
Another is the following in terms of the reciprocal of the Radius Vector
o and the tracedangle 6.
Because
1 . dg 2 + g 2 d
p" 2 = g 4 d d z
d 2 1
Let
Then
, du
d g = 5
u 2
also
1 du
H
dp 2dud 2 u
~^
dp d 2 u 2 3
p^d~g == "d^ L
and substituting in f we have
F  1AI v ( u M u
 ^F 2 \d ^ 2
140. COR. 4. F a _ x V 2 X
* FV*
This is generalized thus. Since
v _ space __ P Q
Time "" t
and
F2
BOOK L]
NEWTON S PRINCIPIA.
121
aXt(=7~Xt) = area described
P Qx S Y
Hence
and by COR. 3.
.v PQ  2A
* t T
2
S~Y
S Y
4 A
v 2 V
I =  X
(h)
g" PV
From this formula we get
V 2 = X F X P V
n
 2 F
4
But by Mechanics, if s denote the space moved through by a body
urged by a constant force F
V 2 = 2gF x s
P V
s = 4
that is, the space through which a body must fall when acted on by the force
continued constant to acquire the velocity it has at any point of the Trajec
tory, is % of the chord of curvature at that point.
Also
dp dp
The next four propositions are merely examples to the preceding formulae.
141. PROP. VII.
R P 2 (= Q R x R L) : Q T 2 : : A V 2 : P V*
. QR * RL x PV 2 _, n T
A ir 9 V^ 1
A V
SP
W J
and multiplying both sides by ^^ and putting P V for R L, we have
Also by (137 c.)
S P 2 x P V 3 _ SP* x QT S
AV 2 QR
A V 2 1
C SP 2 x P V 3<X S P" 2 x P V 3
_ 8 A 2
: x
A V
1
P V
S P 2 X P V 3 *
122 A COMMENTARY ON [SECT. II
OTHERWISE.
From similar triangles we get
A V : P V : : S P : S Y
SP x PV
.. SY =
AV
S P 2 x P V 2
S Y  X P V = A yf X P V
S P 2 x PV 3
^A V 2
Foe 1
SY* X PV SP 2 x PV
as before.
OTHERWISE.
r 2 a 2 + e 2
P ~ 2~
is the equation to the circle ; whence
_dp _ j_
df ~" r
_ 4A 2 dp _ 4 A 8
= gT~ 2X p T dl~gT 2
4?rr 8 r 3
3 2crr 4
OTHERWISE.
The polar equation to the circle is
2 a cos.
~ 1 + cos. 2 6
cos.
. /_ U 1
\ p / 2 a cos.
d_u 1 / sin. ^
" d 6 2 a \cos. 2 ^
I 2 a
sin.
1 sin.
~ ~ X
2a cos. 2
d u _ J_ /3 sin. 2 tf 2 sin 4 tf \
d 6 "" 2~a \ cos. 6 cos7 3 1/
= 0^ X^V^X(3sin
2 a cos. 3 6 v
BOOK I.]
NEWTON S PRINCIPIA.
123
Hence
d 2 u
sin. z 6 . 2 _J cos. I
2 a cos. 3 <T (A ~ 6) + 2 a cos.^ * ~2~a~
1
2 a cos. 3 d
2 a cos. 3 6
1
X (3 sin * 6 sin. 4 6 + cos. 2 6 + cos. 4
X (2 sin. 2 sin. 4 0+ 1 + 12 sin.
a cos. 3 6
which by (139) gives
4A 2 u
a cos. d 6
4 A 2 ( 1 + cos. 2 6) z
v \ * __
g T 2 4 a 3 cos. 5 ^
cos. 5 6
142. COR. 1. F oc 
But in this case
.. Foe
S P = P V.
1 _32gr 4
or ^ m X
SP 5 gT 2 ^ SP S
COR. 2. F: F:: RP 2 x PT 3 : SP 2 x PV S
: : S P x R P
SP 3 x PV
PT 3
:: S P x R P 2 : SG 3 ,
by similar triangles.
This is true when the periodic times are the same. When they are
different we have
T
F: F::SPxRP 2 4xSG 3 ,
S K 1
R
where the notation explains itself.
143. PROP. VIII.
C P 2 : PM 2 :: PR 2 : QT 2
and
PR 2 = QRx(RN + QN)=QR X 2PM
.. C P 2 : P M 2 : : Q R x 2 P M : Q T*
Q T " _ 2PM 3
* Q R : C P 2
124 A COMMENTARY ON [SECT. II.
and
QT Z X SP 2 2PM 3 X SP 2
QR CP 2
CP 2
".
2PM 3 x SP 2 PM 3
Also by 137,
CP 2
g SP 2 x PM 3
But
_ S P X velocity __ S P X V
V 2 CP 2
.. F = X
g PM 3
OTHERWISE.
By PROP. VII,
F oc
SP 2 X P V 3
But S P is infinite and P V = 2 P M.
1
.. F
P M 3
OTHERWISE.
The equation to the circle from any point without it is
c 2  r 2 g 2
P ~ " 2r
where c is the distance of the point from the center, and r the radius,
Moreover in this case
g = c + PM = c + y
c 2 r 2 c 2 2cy y 8
.. P =  jj
c y
r
. dp _ c + y x
i"
r c y
BOOK I.] NEWTON S PRINCIPIA. 125
Hence (139)
_ 4a*r 2 1 ._ V 2 r 2 1
* "^ ~* " o ^> ~^ ^\ q
c 2 g y 3 g y 3
SCHOLIUM.
144. Generally we have
P R 2 : QT 2 : : P C 2 : P M*
PR 2 O T
J 1V ^ J
. . p p 2 . p TV/I 2
 Q~R J
But
PR 2
P V

QR
and
P C : P M : : 2 R (R = rad. of curvature) : P V
QT 2 _ PM 2 _2RxPM
QR : PC 2 : P C
2 R X P M 3
PC 3
But
QT 2 _2AC 2
QR B C 4 *
and
From the expression (g. 139) we get
,, 4a 2 d 2 u
F = X j r X u 2 .
& U a
But
a dd dx
4a 2 V
a X t = 5g = a X ^
d d 2 " d x 2
Also
j UP
.. d u = ~
g
126
and
A COMMENTARY ON
Hence
= r ( see 69 )
V 2 ? 4 d 2 e 1
T _ V/ VX
g ax^
This is moreover to be obtained at once from (see 48)
T
F =  x
For
d t =
.. F =
g dt"
a4
v
V 2
g d x :
145. PROP. IX. Another demonstration is the following:
[SECT. II.
gdx 2
V 2
f 2
, d 2 f
r
g
V 2
dx 2
d 2 y
X^
(1).
S T P
Let ZPSQ = /pSq. Then from the nature of the spiral the
angles at P, Q, p, q being all equal, the triangles S P Q, S p q are simi
lar. Also we have the triangles R P Q> r p q similar, as likewise Q P T,
qpt.
Hence
QT 2
OR
li!
qr
S P : S p
and by LEMMA IX.
q r : q r : : p r 2 : p r 2 : : q t /2 : q t 4
BOOK I.]
Hence
nut!
q r
NEWTON S PRINCIPIA.
q t 2 _ q_t_ 8
q r
qr
QT
,, . c p
a S P
QR
.. F
S P
127
OTHERWISE.
The equation to the logarithmic spiral is
b
p =  X S
d_p _ b
d a a
and by (f. 139) we have
F =
gb
Using the polar equation, viz.
b
4 a "
dp
4 a 2
b
XV
a 3
S
4 a 2 .a 2
ftd*
g
a
bV
X
/ / 2 2\
v (a 2 b 2 ) a
the force may also be found by the formula (g).
146. PROP. X.
P v x v G : Q v 2 : : P C 2 : C D 2
I
Q v 2 : QT 2 : : P C 2 : P F
.. P v x v G : QT 2 : : PC 4 : CD 2 x P F 2
.. v G :
P v
CD 2 X P F
PC 2
But
P V = Q R, and C D x P F = (by Conies) B C X C A
also
ult. v G = 2 P C.
B C 2 X C A 2
..2 PC :
PC 8
128 A COMMENTARY ON [SECT. II.
Q R PC p r
*QT 2 X CP 2BC 2 x C A 2
Also by expression (c. 137) we get
_ 8A 2 PC
r = Fi=n; X
gT 2 2B C 2 X C A 2
But
A= * X BC X C A
The additional figure represents an Hyberbola. The same reasoning
shows that the force, being in the center and repulsive, also in this curve,
a C P.
ALITER.
Take
Tu = TV
and
u V : v G : : D C 2 : P C*
Then since
Q v 2 : Pv x vG : : D C 2 : P C 2
.. u V : v G : : Q v 2 : P v x v G
.. Q v 2 = P v x u V
..Qv 2 + uPxPv= PvX (uV + uP)
= P v x V P.
But
Qv J = QT 2 + Tv 2 = QT 2 + Tu 2
= p Q 2 P T 2 + T u 2
= p Q2_ (PT Tu 2 )
= PQ2_P U X Pv
(chord PQ) 2 = Pv x VP.
Now suppose a circle touching P R in P and passing through Q to
cut P G in some point V . Then if Q V be joined we have
z.PQv=zQPR = ^QV P
and in the A Q P v, Q V P the L. Q P V is common. They are there
fore similar, and we have
P v : P Q : : P Q : P V
.. P Q 2 = P v x V P = PvxVP
.. V P = V P
or the circle in question passes through V ;
.. P V is the chord of curvature passing through C.
BOOK I.] NEWTON S PRINCIPIA. 129
Again, since
DC 2
u V = v G x p ^ a = C x v G
or
p V P u = C (P G P v)
and
P V, PG
being homogeneous
2 D^C 2 2CD*
PC 2 PC
.. (Cor. 3, PHOP. VI.)
F PC
2 PF a x CD 2
But since by Conies the parallelogram described about an Ellipse is
equal to the rectangle under its principal axes, it is constant. .. P F x
CD is.
and
F p C.
OTHERWISE.
By (f. 139) we have
g T
But in the ellipse referred to its center
_
 a * + b 8 ~
._ a 2 + b 2
and differentiating, and dividing by 2, there results
dp _ g
p s dg a 2 b i!
which gives
4A 2 P 4^r 2
TT _ _ v _  \( a.
~gT 2 a 2 b 2 ~ g T*
In like manner may the force be found from the polar equation to die
ellipse, viz.
b
" 1 _ e 2 cos. 2 6
by means of substituting in equat. (g. 131). )
130 A COMMENTARY ON [SECT. II.
147. COR. 1. For a geometrical proof of this converse, see the Jesuits
notes, or Thorpe s Commentary. An analytical one is the following.
Let the body at the distance R from the center be projected with the
velocity Vin a direction whose distance from the center of attraction is P.
Also let
F = *<
IL being the force at the distance 1. Then (by f )
4 A 2 ^ dp
= ^r**P**l = ^
which gives by integration, and reduction
1 _^gT 2
p" 2 "^^ h
R and P being corresponding values of g and p.
But in the ellipse referred to its center we have
_!_ _ a 2 +b 2 g 2
p 2 ~ a 2 b 2 a 2 b 2
which shows that the orbit is also an ellipse with the force tending to its
center, and equating homogeneous quantities, we get
J_
2
a 2 b 2 4 A 2 P
and
a 2 b 2 4 A 2
But
A = cr a b
... T = 2 * (1)
Vug
which gives the value of the periodic time, and also shows it to be con
stant. (See Cor. 2 to this Proposition.)
Having discovered that the orbit is an ellipse with the force tending to
tne center, from the data, we can find the actual orbit by determining its
semiaxes a and b.
By 140, we have
v  L
V T 1 P
a 2 + b g _ R 2 l_
* ~~ a^b 2 " ^g x V 2 P 2 "*~P 2
and
1 _L
aTp^S x V 2 P 2
BOOK L] NEWTON S PRINCIPIA. 131
V 2
l 2 ih 2 R 2 _L
. . cl ] U IV }
and
2 VP
2 a b = =r
\
. . a + b = ^
and
g v>g
which, by addition and subtraction, give a and b.
OTHERWISE.
By formula (g. 139,) we have
...4. u _
d tf 2 ^ TT 2" x u~ 3 ~ (
and multiplying by 2 d u, integrating and putting 8 * T 2 _ ^
4< A
d u 2 M
dT* + u2 + ^2 + ^ =
To determine C, we have
d u 2 _ J_ dj 8
d0 2 ~ f * dl" 8
and in all curves it is easily found that
. I 1 ," 2 _ e 2 p 8 _ _i
d^ g p s = p~"
Hence, when ^ = R, and p = P,
which gives the constant C.
Again from (2) we get
_ _
 V ( M Cu u 4 )
which being integrated (see Hersch s Tables, p. ItiO.^Englished edit
published by Baynes & Son, Paternoster Row) and the constants properly
determined will finally give g in terms of * whence from the equation to
the ellipse will be recognised the orbit and its dimensions.
I 2
A COMMENTARY ON
[SECT. II.
148. COR. 2. This Cor. has already been demonstrated see (1).
Newton s Proof may thus be rendered a little easier.
By Cor. 3 and 8 of Prop. IV, in similar ellipses
T is constant.
Again for Ellipses having the same axismajor, we have
/
(
V
cr a b b
oc a
But since the forces are the same at the principal vertexes, the sagittae
are equal, and ultimately the arcs, which measure the velocities, are equal
to the ordinates, and these are as the axesminores. Hence, a (which
v X S Yx ,
 ) oc b.
.*. T oc T <x ] or is constant.
Again, generally if A and B be any two ellipses whatever, and C a third
one similar to A, and having the same axismajor as B ; then, by what
has just been shown,
T in B = T in C
and
T in C = T in A
.. T in B = T in A.
149. SCHOL. See the Jesuits Notes. Also take this proof of, " If one
curve be related to another on the same axis by having its ordinates in a
given ratio, and inclined at a given angle, the forces by which bodies are
made to describe these curves in the same time about the same center in
the axis are, in corresponding points, as the distances from the center."
The construction being intelligible from the figure, we have
P N : Q N : : p O : q O
.. P N : p O : : Q N q O
: : N T : O T ultimately.
BOOK L]
NEWTON S PRINCIPIA.
133
.. Tangents meet in T,
the triangles C P T, C Q T are in the ratio of P N : Q h or of parallelo
grams PNOp, QNOq ultimately, i. e. in the given ratio, and
C p P : C P T : : p P : P T ultimately.
: : NO: NT
: : qQ:QT
: :CQq: CQT
.. C p P : C q Q in a given ratio.
. bodies describing equal areas in equal times, are in corresponding
points at the same times.
.. P p, Q q are described in the same time, and m p and k q are as the
forces.
Draw C R, C S parallel to P T, Q T; then
p O : q O : : P N : Q N : : n O : 1 O
.. n O : p O : : 1 O : q O
1 O : O S)
and
n p
: n O
: Iq
1 Q)
but
f
nO
: n R
: IQ
1 S j
(since n O
: O R
TO
O C:
* P
: n R
1 q
1 S
.. n p
: p R
1 q
qS
and
n p
1 q
: p R
: q S
m p
k q
pC
q C)
.. mp
: P C
k q
qC
or
Fatp
: F at q
P C
qC,
Q. e. d.
SECTION III.
150. PROP. XI. This proposition we shall simplify by arranging the pro
portions one under another as follows :
But
LxQR
( = Px) : L xPv :
: PE
P C
A C
P C
LxPv
: G v x P v
L
G v
G v x P v
: Q v 2
PC 2
CD
Qv 2
:Qx
1
1
Qx 2
:QT =
PE 2
P F 2
/
CA 2
P F
CD 2
C B
134 A COMMENTARY ON [SECT. III.
..Lx QR: QT 2 : : A CxLx P C 2 xCD 2 : PC xGvxCD 2 xCB 2
and
QR A Cx PC A C x PC AC
QT 2 ~ G v x C B* " 2 PC X C B 2 2 C B 2
T? QR (_ AC x 1
* * a Q T 2 x S P A 2 C B 2 x S P 2 ) SP 2
Q. e. d.
Hence, by expression (c) Art. 137, we have
8 A 2 AC
T? v
 rr* o ^
gT 2 2 CB 2 x S P 2
" 2 b 2 x e 2
X^ (a]
where the elements a and T are determinable by observation.
OTHERWISE.
A general expression for the force (g. 139) is
4 A 2 9 /d 2 u , x
F = gT 2 x u (dT 2 + u )
But the equation to the Ellipse gives
_ 1 _ 1 f e cos. 6
=  = ^(T^^eY
where a is the semiaxis major and a e the eccentricity.
d u e sin. 6
dT : ~a(l e 2 )
and
d 2 u e cos. 8
d 2 u
dT 2 + u =
and
F =
.
T 2 X a(l e 2 )
But
A s = w 2 a 2 b 2 = * 2 a 2 (a 2 a 2 e 2 )
. F _ ill^ 3 x u ,
gT 2
the same as before.
BOOK L] NEWTON S PRINCIPIA. 135
OTHERWISE.
Another expression is (k. 140)
_ 4A 2 dp
F = 77^ X ,f .
g 1 p a g
Another equation to the Ellipse is also
j. _ 2 a g _ 2 a J_
dp a
P 5 ^! "" ^V
T? _
 b 2 g
u a v
X i o 5  rfi ij ^N " o *
b 2 P^ G L ~
^ b O
151. PROP. XII. The same order of the proportions, which are also let
tered in the same manner, as in the case of the ellipse is preserved here.
Moreover the equations to the Hyperbola are
a (e 2 I)
1 + e cos. 6
and
P 2 = ^i
which will give the same values of F as before excepting that it becomes
negative and thereby indicates the force to be repulsive.
152. PROP. XIII. By Conies
4SP.Pv = Qv 2 = Qx 2 ultimately.
But
.. 4 SP. Q R : Qx 2 : : 1 : 1
and
Qx 2 : QT 2 : : SP 2 : S N 2
:: S P : S A
.. 4SP.QR : QT 2 :: S P : S A
QR 1 . !_
" Q T 8 ~ 4 S A " L
L being the latus rectum.
.. F
Q~T 2 X S P 2 S P 2
or
F = ! X g^QHj T , (b.
136 A COMMENTARY ON [SECT. III.
8a 2 1 2 P 2 V 2 1
X ] X
~ g L S P 2 gL
a being the area described by the radiusvector in a second, or P the per
pendicular upon the tangent and V the corresponding velocity.
OTHERWISE.
In the parabola we have
and
which give
12 22
u =  = j (1 + cos. 0) = J + T cos.
1^ 4 1
P~ 2 ~ L X J
d 2 u 2
d7^ + u = X
and
d p _2 _1
and these give, when substituted in
F  ?2 V2 u 2 (
g \p~}
or
P V a dp
o; p a d P
O Jt 9
the same result, viz.
x
g
Newton observes that the two latter propositions may easily be deduced
from PROP. XL
In that we have found (Art. 150)
4 A 2 a
= JTT 5 x
P 2 V 2
g r
Now when the section becomes an Hyperbola the force must be repul
sive the trajectory being convex towards the force, and the expression re
mains the same.
BOOK I.] NEWTON S PRINCIPIA. 137
Again by the property of the ellipse
b : : ax a: g a 
4 4 \
which gives
a_ _2 _ JL
b 2 ~ L ~~4 a
and if c be the eccentricity
b 2 = a 2 c 2 = (a + c) X (a c)
" _ a _ _ 2 1
(a + c) X (a c) "" L ~ 4 a
Now when the ellipse becomes a parabola a and c are infinite, a c is
finite, and a + c is of the same order of infinites as a. Consequently r 9
nJ
\sjinite, and equating like quantities, we have
_a _J2
b~ 2 ~"L
which being substituted above gives
2P 2 V 2 1
the same as before.
Again, let the Ellipse merge into a circle; then b = a and
v P 2 V 2 a
TV _ _ v
A A
K 2 2
5 D S
a V 2 ^
g X ? 2
V s
(c)
g X
153. PROP. XIII. COR. 1. For the focus , point of contact, and position of
the tangent being given, a conic section can be described having at that point
a given curvature. ]
For a geometrical construction see Jesuits note, No. 268.
The elements of the Conic Section may also be thus found.
The expression for R in Art. 75 may easily be transformed to
R =
p 2 d 6* * d <T
for
p z ~dl~ ~7~ a ** 
138 A COMMENTARY ON [SECT. III.
Now the general equation to conic sections being
b 2 1
s = x
a 1 + e cos. 6
the denominator of the value of R is easily found to be
which gives
R  *
a p 3
Hence
b 2 p 3
 =  x R
a f J
is known.
Again, by the equation to conic sections we have
b 2 g
P = 2 a ipe
which, by aid of the above, gives
a = 2g 2 ~p R
And
p 8 R
~ 2 g 2 p R*
Whence the construction is easy.
154. The Curvature is given from the Centripetal Force and Velocity being
given.~\
If the circle of curvature be described passing through P, Q, V, and O
(P V being the chord of curvature passing through the center of force,
and P O the diameter of curvature) ; then from the similar triangles
P Q R, P V Q, we get
P Q 2
QR  .
iv  P V
Also from the triangles P Q T and P S Y (S Y being the perpendicu
lar upon the tangent) we have
SPx QT
^^~ SY
and from P S Y, P V O,
^i**SY
BOOK I.I NEWTON S PRINCIPIA. 139
whence by substitution, &c.
Q R SP
QT 2 xSP*~2Rx SY 2
_ 2P 2 V 2 QR V 2 x SP
g * QT 2 x SP 2 " Rx SY
which gives
SP_ V 2
 > ~
Hence, S P, S Y and g being given quantities, R is also given if V and
F are.
155. Two orbits which touch one another and have the same centripetal
force and velocity cannot be described. ]
This is clear from the " Principle of sufficient Reason." For it is a
truth axiomatic that any number of causes acting simultaneously under
given circumstances, viz. the absolute force, law of force, velocity, direc
tion, and distance, can produce but one effect. In the present case that
one effect is the motion of the body in some one of the Conic Sections.
OTHERWISE.
Let the given law of force be denoted generally by f g, where f g means
any function; then (139)
F _ P 2 V 2 dp
g *p a de
and since P and V are given
pv _ . ^ P .
g p 3 d P
But if A be the value of F at the given distance (r) from the center to
the point of contact ; then
F : A : :fg :fr
and
F: A: :f :fr
and
~ f r S
HO A COMMENTARY ON [SECT. III.
Hence
1Z! jL.p_ A r
3 d ~ fr X
and
P
g P
P 2 V 2 dp A
r Via
/ Q i / r* ^. * C
s p 3 d / f r
and integrating, we have
P 2 V 2 f r
and
Po TT o n i i
2 V 2 f r /!_ JL \ _ / i e
Nowyd g f g and yd g f g are evidently the same functions of g and g t
which therefore assume
a P and P 5
and adding the constant by referring to the point of contact of the two
orbits, and putting
p? v*fr
2 g A = M,
we get
M X (p^ r,) = <p g <f> r
MX (1 L)=^ ?r.
. J_  L _L
" p =: "M "" P*~
p 7 " 2 =: 31 + F 2 ~
in which equations the constants being the same, and those with which
g and f are also involved, the curves which are thence descriptible are
identical. Q. e. d.
These explanations are sufficient to clear up the converse proposition
contained in this corollary.
156. It may be demonstrated generally and at once as follows :
By the question
BOOK I.] NEWTON S PRINCIPIA. Ml
then
and
/d g ^
and substituting in (d) we have
1 1 1 1
p 8 = r M + P 2 H " M g
But the general equation to Conic Sections is
_L 2a _L
p 2 ~ b 3 g b 2 "
Whence the orbit is a Conic Section whose axes are determinable from
2a. 1 2 g A r 2
b"2 M = pa V 2
and
 _L 1 1
+ b 2 ~ ~ r M. " P 2
1 2 g A r
^0^2 " p 2 "y 2"
and the section is an Ellipse, Parabola or Hyperbola according as
V 2 is >, or = or < 2 g Ar.
Before this subject is quitted it may not be amiss by these forms also to
demonstrate the converse of PROP. X, or Cor. 1, PROP. X.
Here
ff = <
f r = r
VV hence
1 r 2 J_ g 2
p 2 ~ 2 M H " P 2 2 M
But in the Conic Sections referred to the center, we have
which shows the orbit to be an Ellipse or Hyperbola and its axes may be
found as before.
142
A COMMENTARY ON
[SECT. III.
In the case of the Ellipse take the following geometrical solution and
construction
C, the center of force and distance C P are given. The body is projected
at P with the given velocity V. Hence P V is given, (for V 2 = . F . P V.)
Also the position of the tangent is given, .. position of D C is given, and
SCO 2
P V = .p, n . Hence C D is given in magnitude. Draw P F per
P C
pendicular to C D. Produce and take P f = CD. Join C f and bisect
in g. Join P g, and take g C, g f, g p, g q, all equal. Draw C p, C q.
These are the positions of the major and minor axis. Also \ major axis
= P q, \ minor axis = P p.
For from g describe a circle through C, f, p, q, and since C F f is a
right Zj it will pass through F.
..Pp.Pq= PF.Pf= PF.CD
Also
PC 2 +Pf 2 = Pg 2 f g C 2 + Pg 2 + gf 2 , (since base of A bisected in g)
or
= Pq 2 2Pg.gq+Pp 2 + SPg.gp
= Pq 2 + Pp 2
... Pp. Pq PF.CD \ But a and b are determined by the same
Pp 2 + Pq 2 = PC 2 + CD 2 / equations. . . P q = a, P p = b.
Also since p and F are right angles, the circle on x y will pass through p
and F, and APpx = Cpq=CFq = xFp, because ^xFC = pFq.
.. L. Pp x = /in alternate segment. .. P p is tangent.
Pp = PF.Px .. P F. Px = b 2 .
But if in the Ellipse C x be the major axis, P F . P x = b 2 .
BOOK I.] NEWTON S PRINC1PIA. 143
.. C x is the major axis, and . . C q is the minor axis.
. . the Ellipse is constructed.
PROP. XIII, COR. 2. See Jesuits note. The case of the body s
descent in a straight line to the center is here omitted by Newton, be
cause it is possible in most laws of force, and is moreover reserved for a
full discussion in Section VII.
The value of the force is however easily obtained from 140.
O T 2 O T 2
157. PROP. XIV. L = ^jL ^5
W tt
a Q T 2 X S P 2 by hypoth.
OTHERWISE.
By Art. 150,
F  4 A a 8 A 2 \
 g T* X b V " LgT 2 * p
for the circle, ellipse, and hyperbola, and by 152.
O P 2 V2 1
F= r x
T ^ *>
Lg e 2
for the parabola.
Now if /& be the value of F at distance 1, we have
Whence in the former case
8 A 2 2 P 2 x V 2
T T 2 P r
gL
and in the latter
2 P 2 x V 2
But
~8 * : 1 2 : A 2 : T 2
4
Aj; S P 2 x QT 2 P^x V 2
" * T 2 ~ 4 ~ "4 ~
158. PROP. XIV. COR. I. By the form (a) we have
A(= crab) = JtJl X V L X 1.
T V L.
144 A COMMENTARY ON [SECT. III.
159. PROP. XV. From the preceding Art.
T= / X .
~ V AC, g V L
But in the ellipse
L =
a
T X a 2 ... (e)
V /*g
160. PROP. XVI. For explanations of the text see Jesuits notes.
OTHERWISE.
By Art. 157 we get
for the circle, ellipse, hyperbola, and parabola.
But in the circle, L = 2 P.
.. V = V^X^p  A/Y^x ^ . . . (g)
r being its radius.
In the ellipse and hyperbola
L  2b2
161. PROP. XVI, COR. I. By 157,
L = X P 2 X V 2 .
g/*
162. COR. 2. V = / X ,
D being the max. or min. distance.
163. COR. 3. By Art. 160, and the preceding one,
* X : X
: : V L : V 2 D.
164. COR. 4. By Art. 160,
BOOK I.]
But
NEWTON S PRINCIPIA.
2 b 2
L = , P = b, and r = a
145
.*. v : v : :
b V a Va*
165. COR. 6. By the equations to the parabola, ellipse, and hyper
bola, viz.
n * . O
the Cor. is manifest.
166. COR. 7. By Art. 160 we have
/2 1 L 1
v 2 . y 2 . . .
2 P 2 r
which by aid of the above equations to the curves proves the Cor.
OTHERWISE.
By Art. 140 generally for all curves
pv 
JL. T 
i
dp
P V = 2 g (racl. = e)
But generally
and in the circle
An analogy which will give the comparison between v and v ; for any
curve whose equation is given.
167. COR. 9. By Cor. 8,
and
.: ex equo
v : v : : = : p
sv::
VOL. ].
K
146 A COMMENTARY ON [SECT. III.
1GB. PROP. XVII. The " absolute quantity of the force" must be
known, viz. the value of ^ or else the actual value of V in the assumed
orbit will not be determinable ; i. e.
L : L : : P 2 V 2 : P /2 V /2
will not give L .
It must be observed that it has already been shown (Cor. 1, Prop.
XIII) that the orbit is a conic section.
See Jesuits notes, and also Art. 153 of this Commentary.
169. PROP. XVII, COR. 3. The two motions being compounded, the
position of the tangent to the new orbit will thence be given and therefore
the perpendicular upon it from the center. Also the new velocity.
Whence, as in Prop. XVII, the new orbit may be constructed.
OTHERWISE.
Let the velocity be augmented by the impulse m times.
Now, if /* be the force at the distance 1, and P and V the perpendicu
lar and velocity at distance (R) of projection, by 156 the general equation
to the new orbit is such that its semiaxes are
R R
or =
 2 m 2 " rn 2 2
and
=
2 m 2
according as the orbit is an ellipse or hyperbola. Moreover it also
thence appears that when m 2 = 2, the orbit is a parabola, and that the
equations corresponding to these cases are
g
R
2 m
or
m 2 P
or
= P X
BOOK I.]
NEWTON S PRINCIPIA.
147
DEDUCTIONS AND ADDITIONS
TO
SECTIONS II AND III.
170. In the parabola theforce acting in lines parallel to the axis, required F,
4SP.QR:QT 2 ::Qv 2 :QT 2 ::YE 2 :YA 2 ::SE:SA::SP:SA
Q R 1
" QT 2 ~~ 4 S A a ^ ^ * s constant * F i g constant.
S
Let u be the velocity icsolved parallel to P M then since the force acts
perpendicular to P M, u at any point must be same as at A. .. if P Q be
the velocity in the curve, Q T = u = constant quantity, and a = S/ P Q T
S P.u
2
. F _ 8a*.QR 2u 2
" gS P .QT* ~ g~L" ^ S
which avoids the consideration of S P being infinite ; and
. . body must fall through  to acquire the velocity at vertex, which agrees
with Mechanics. (At any point V = u / .
^/ S A
171. In the cycloid required the force when acting parallel to the axis.
148
A COMMENTARY ON
[SECT. III.
RP 2 : QT 2 :: Z P 2 : ZT 2 :: VF 2 : E F* : : VB: BE
and since the chord of curvature (C. c) = 4 P M, R P 2 = 4 P M. R Q,
/. 4 P M. R Q : Q T 2 : : V B : (B E =) P M
" QT 2 ~~ 4
p M
F =
S P constant )
* a = Yelocit > parallel to A B 
/* / B V
(At any point v = u . ^/ p^
172. In the cycloid the force is parallel to the base
R P 2 : QT 2 : : Z P e : ZT 8 : : V E 2 : V M 2 : : VB: VM
and since C . c = 4 E M
R P 2 = 4 E M.RQ,
.. 4EM.RQ:QT :: VB: V M,
QR V B 1
[f V M = y, F =
gy
4 E M. V M a E M. V M
u g r / VBx
2 r y v " V  2 >/
u = velocity parallel to V B.
BOOK I.] NEWTON S PRINCIPIA.
8a 2 QR 2u 2 .QJR __""JL V ^__ \
= ^TS P^Q T 2 = g . Q T 2 ~ 2 g . E M . V M V
149
(At any point v = u
B V
)
173. Find F in a parabola tending to the vertex.
TAN
T P : P N : : T A : A E
or
V 4 x 2 + y 2 : y : : x :
1
V 4 x 2 + y
4 x 2 + ax 4 x + a
cl X
= p, (A E),
p ax
2 dp __ 4dx.ax g 2axdx(4x
~ a 2 x 4
ax
dp _ 2 x + a
p 3 ax 3
2 a x , 2 2x + a ,, v
. d x = . 5 d x,
a x 3
Also
= V x* + y 2 ,
 x d x + y d y
d x
V x
dp _ 2 x + a 2 V x 2 +
* p *~d~p a x 3 2 x + a
A P
A N
y 2 V X 2 + tl X
^~x 2 V x a + ax
a x
K3
150 A COMMENTARY ON [SECT. III.
174. Geometrically. Let P Q O be the circle of curvature,
but
but
P v (C. c through the vertex of the parabola) =
PQ 2 PO.Az
PP. Az
AP
QR
PQ 2
AP
A P 2
QT 2 ~
QR
Az 2
A P 3
QT 2 *
. T?
PO.Az 3
8a 2 .Q R
8a 8 . A P
g. A P 2 .QT 2 ~ g.PO. A z s
P O.Az 3 = 2 A S.
.. F =
S P 3 SY 3 AT
gy,
4a l . A P
SP ;
= 2 A S.A N 3
g.A S.AN 3
175. If the centripetal be changed into a repelling force, and the body
revolve in the opposite hyperbola, F ot p g .
BOOK I.]
NEWTON S PRINCIPIA.
151
The body is projected in direction P 11 ; R Q is the deflection from the
Tangent due to repelling force H P, find the force.
L . S P
176. In any Conic Section the chord of curvature = ^^r
for
pv QP 2 QT 2 .S P 2 L.SP 2
QR " Q RS Y 2 v*
L.SP 3
S Y
177. Radius of curvature = ~ o~
for
P W =
PV.SP L.SP
S"V SY 3
o .. 2
178. Hence in any curve F = s~Y*~p\r
_8ji*_ 4a a .SP
~g.SY .2RTSY~
SP
K 4
see Art.
152
A COMMENTARY ON
[SECT. Ill
179. Hence in Conic Sections
8a 2 8a 2
F =
g.SY 2 .PV~g.SY 2 .L.SP 2
S Y 2 "
8 a 2 1
2(X
gTL.S P 2 SP~ 2
L S P 2
180. If the chord of curvature be proved = vs  independently of
f~\ T 1
he proof that ~ = L, this general proof of the variation of force in
tonic sections might supersede Newton s; otherwise not.
181. A body attached to a string, whose length = b, is whirled round so as
to describe a circle whose center is the Jixed extremity of the string parallel
to the horizon in T" ; required the ratio of the tension to the weight.
Gravity = 1, .*. v of the revolving body = V g F b, if b be the length of
the string ; t
V 2
.. F (= centripetal force = tension) = T (131)
arid
_ circumference 2 v b V b
i =?^ ===== __ 2 T . ,
4b
V g F b
 gT 2
4 T 2 b
.. F : Gravity : : ^ ,, : 1, or Tension : weight : : 4
O
If Tension = 3 weight; required T.
4* 2 b:gT 2 : :3: 1,
b : g T z .
T 2 =   
If T be given, and the tension = 3 weight, required the length of the string.
"jp 2 __ _ ff
.. b =
 
4 cr
1 82. If a body suspended by a string from
any point describe a circle, the string describes
a cone , required the time of one revolution or
of one oscillation.
Let A C = 1, B C = b,
The body is kept at rest by 3 forces, gra
vity in the direction of A B, tension in the
direction C A, and the centripetal force in
the direction C B.
BOOK I.]
As before, centripetal force =
NEWTON S PRINCIPIA.
z b
153
2 >
and centripetal force : gravity : b : V 1 s b 2 , (from A)
4 vr 2 V 1  b 2
. rPZ _ ^
= 2 if = a constant quantity if V 1
2 n 2
c
c n
2 n ;
be given.
.. the time of oscillation is the same for all conical pendulums having a
common altitude.
183. v in the Ellipse at the perihelion : v in the circle e. d. : : n : 1, Jind
the major axis, eccentricity, and compare its T with that in the circle, and
Jind the limits ofn.
Let S A = c,
v in the Ellipse : that in the circle e. d. : : V H P : V A C
: : V H A : VAC in this case
: : n : 1 by supposition,
..2 AC AS = n 2 AC,
... A C = C
Excentricity =AC A S = 5
s
T : T in the circle : : A C^ : A S * : :
(2n 2 )
Also n must be <C V 2,
for if n = V 2, the orbit is a parabola
if n > V 2, the orbit is an hyperbola.
184. Suppose of the quantity of
matter ofto be taken away. How
much would T of D be increased, and
what the eccentricity of her new orbit ?
the D s present orbit being considered
circular.
At any point A her direction is
perpendicular to S A,
.. if the forcer be altered at any
point A, her v in the new orbit will
c 2 : : 1 : (2 n s )
154 A COMMENTARY ON [SECT. III.
= her v in the circle, since v = y > an d S Y = S A, and a is the
same at A.
Let A S = c, P V at A = L, and F = * ^ oc _.
in this case,
2 b z
/. 3:4::2c(=Lin the circle) : ( = L in the ellipse)
3b 2 3(a 2 a c) 3(2ac c 2 ) _ 3c 2
. . 4 c ~~  _ N 7 * ft f
a a a
3c 2
.. = 2c,
3c
And T in the circle : T 7 in the ellipse : : ^7 : (~~]
V 4 v 2 /
V 3
V~3 f 3 \ ^ 1 3
V4 V2/ V 2 2
: : V 2 : 3.
And the excentricity = a
.
W
185. What quantity must be destroyed that D s T may be doubled, and
what the excentricity of her new orbit ?
Let F of : f (new force) : : n : 1
.*. v = / s. F . P V, and v is given,
1
P V
2b 2 n a 2 a c 2ac c 2
.. n : 1 : : : 2 c : : : c : :  : c : : 2 a c : a
a a a
.*. n a =r 2 a c,
c
2 n
Also T in the circle : T in the ellipse : : 1 : 2
<)
: : (2 n) * : n *
/. 1 : 4 : : (2 n) 3 : n .. n =r 4 (2 n) 3 , whence n.
BOOK I.] NEWTON S PRINCIPIA.
And the excentricity
155
c =
c c (2 c nc) __ c (n 1)
2 n~ 2 n 2 n
186. What quantity must be destroyed that D s orbit may become a
parabola ?
L = 4 c,
.. F : / : : 4 c : 2 c : : 2 : 1,
.. \ the force must be destroyed.
187. F a =TJ, a body is projected at \ given D, v = v in the circle,
L. with S B = 45,yrcc7 axis major, excentricity, and T.
Since v = v in the circle, .*. the body is projected from B,
and L. S B Y= 45 ;
.. L. S B C, or B S C = 45,
S B
.. S C = S B. cos. 45 =
But
V 2
S B = D
axis major
~~
.. axis major and excentricity are found.
And T may be found from Art. 159.
Y
P
188. Prove that the angular v round H : that round S : : S P : H P.
This is called Seth Ward s Hypothesis.
In the ellipse. Let P m, p n, be perpendicular to S p, H P,
.. p in = Increment of S P = Decrement of H P = P n
.. triangles P m p, P n p, are equal,
.. P m = p n, and angular v p
distance
189. Similarly in the hyperbola.
Angular v of S P : angular vofSY::PV:2SP:: $.* S P
C D 2
: : HP : A C.
156 A COMMENTARY ON [SECT. III.
190. Compare the times of fatting to the center of the logarithmic spiral
from different points.
The times are as the areas.
P
c s
, 2
d . area = ( 6  ^_ C S P), ford, area =  .
fit fit
f~\ r r* i A
Also Tp^p =  , = tan. L. Y P T = tan. , (a being constant) = a
a p
.. area = s  g 2 , (for when P, = 0, area = 0, /, Cor. = 0)
.*. if P, p, be points given,
T from P to center : t from p to center : : S P 2 : S p 2 .
191. Compare v in a logarithmic spiral with that in a circle, e. d.
2 V 2
F =
.. if F be given, V oc
.. v in spiral : v in the circle : : V P V in spiral : V 2 S P ::!:!.
192. Compare T in a logarithmic spiral with that in a circle., e. d.
rr . ! whole area a e 2 a P 2
J in spiral =  5 ?
 :   X7   _ 
area in 1 4 . v . S Y 2 v . . sin. a
T in cirrlp  whole area  g g* _ 2 or g 2 _ 2 cr
A ill Clinic  ;  _  . __   
area in 1" v . ib Y v . v
ap* 2^p a
 ^_  .  i . .  : 2 T : : a : 4 * . sin. a.
v . g . sin. a v 2 sin. a
: : tan. a : 4 T . sin. a : : 1 : 4 T cos. .
BOOK I.]
NEWTON S PRINCIPIA.
157
192. In the Ellipse compare the time from the mean distance to the Aphe
lion, with the time from the mean distance to the Perihelion. Also given the
Excentricity, tojind the difference of the times, and conversely.
A D V is   described on A V.
G
T of passing through Aphelion : t through Perihelion
: : S B V : S B A
: : S D V : S D A
Let Q = quadrant C D V,
. rp (^ a . a e a.ae
^ " 2 : ^ " 2
.. (T + t =) P : T t : : 2 Q : a . a e
P a.ae
.. T t =
2 Q
whence T t, or, if T t be given, a e may be found.
193. If the perihelion distance of a comet in a parabola = 64, s mean
distance = 100, compare its velocity at the extremity of It with s velocity
at mean distance.
Since moves in an ellipse, v at the mean distance = that in the circle
e . d . and v in the parabola at the extremity of L
: v in the circle rad. 2 S A : : V 2 : 1
v in the circle rad. 2 S A
: v in the circle rad. AC
. v in the parabola at L
: v in the ellipse at B
V A C : V S A
V 2 . AC : V S~A.~ 2
: : 10 V 2 : 8 V 2
: : 5 : 4
194. What is the difference between L of a parabola and ellipse, having
the same < st distance = 1, and axis major of the ellipse = 300? Compare
the \ at the extremity ofL> and <" distances.
In the parabola L = 4 A S = 4.
158
A COMMENTARY ON
[SECT. III.
In the ellipse L =
2B C
A C
300
1
T50
(A C 2 A C SA")
(2 AC. AS AS 2 ) =
600 J
150
V 2 : 1
V AC: VH P
\/50 : V299
V 300 : V 299.
v in the parabola at A : v in the circle rad. S A
v in the circle rad. S A : v in the ellipse e. d.
: : V AC : V 2AC SA
/. v in the parabola at A : v in the ellipse e. d. :
Similarly compare v s . at the extremity of Lat. R.
1 95. Suppose a body to oscillate in a
whole cycloidal arc, compare the tension
of the string at the lowest point with
the weight of the body.
The tension of the string arises
from two causes, the weight of the
body, and the centrifugal force. At
V we may consider the body revolving
in the circular arc rad. D V, . . the
centrifugal = centripetal force. Now
the velocity at V = that down C V by the force of grav.
= that with which the body revolves in the circle rad.
2 C V.
.. grav. : centrifugal force
.. tension : grav.
196. Suppose the body to oscillate
through the quadrant A B, compare the
tension at B with the weight.
At B the string will be in the direction of
gravity; . . the whole weight will stretch
the string; /.the tension will = centrifugal \
force + weight. Now the centrifugal
force = centripetal force with which the
body would revolve in the circle e. d.
!:!,
2 : 1
And v in the circle = V 2 g . F .
BOOK I.]
NEWTON S PRINCIPIA.
159
/. F =
gRgCB
in this case,
also v at B from grav. = V 2 g . C B, grav. = 1 .
/. grav. = 1 =
.*. F : grav. : :
2g C B
v 2 v
2gCB g C B
since v = v .
/. tension : grav. : : 3 : 1.
197. A body vibrates in a circular arc
from the center C ; through what arc must
it vibrate so that at the lowest point the
tension of the string = 2 X weight?
v from grav. = v d . N V, (if P
be the point required) v of revo
, CV
lution m the circle = v d . ^r .
2: 1,
C
/QV _
.. centrifugal force : grav. : : v : v : : / : V N V
.. centrifugal force + grav. (= tension) : grav.
_ _
1 V N V : V N V
2 : 1 by supposition.
C V
C V
.. N V =
= V N V,
C V
198. There is a hollow vessel in form
of an inverted paraboloid down which
a body descends,, the pressure at lowest
point = n . weight, find from what point
it must descend.
At any point P, the body is in the
same situation as if suspended from G,
P G being normal, and revolving in the
circle whose rad. G P. Now P G =
V 4 A S . S P, .*. at A, P G =
]60 A COMMENTARY ON [SECT. 111.
V 4 A S 2 = 2 A S. Also.,* 2 at A with which the body revolves =
.. centrifugal force =
2g A S
v
and grav. =  p , if h = height fallen from.
o
But the whole pressure arises from grav. + centrifugal force, and = n . grav.
. . centrifugal force + grav. : grav. : : n : 1
or
1 1 1
AS+ h : ::n:1
1 1
A~S : h ::n ~ l : J
... h  n 1 . A S.
199. Compare the time (T ; ) in which a body de
scribes 90 of anomaly in a parabola with T in the
circle rad. = S A.
Time through A L : 1 : : area A S L : a in 1"
 A S. SL 4 A S 2
a 3 a
T in the circle rad. S A : 1 : : whole circle : a in 1"
w A Q 2
a A ij
A S a
.. T =
ar
. T 1 T
"
and
a: a :: VL: V2AS:: V4AS: V2AS::
4
2:1
.. T : T : :
: r : : 2 V 2 : 3 r.
3 V 2
Compare the time of describing 90 in the parabola A L voith that hi the
parabola A 1, (fig. same.)
t : T in the circle rad. S A : : 4 : 3 V 2 . T
T in the circle S A : T in the circle rad. o A : : S A.% : a A*
(since T 2 R 3 )
T : t through 1 A : : 3 V 2 . v : 4
frj
~ e .. ~~ z . k % i. VU g.i v ;.*. . . ^ i* : <7 A ff .
See Sect. VI. Prop. XXX.
BOOK L]
NEWTON S PRINCIPIA.
161
200. Draw the diameter P p such that the time through P V p : time
through p A P : : n : 1, force oc gp^
Describe the circle on A V.
Let t = time through P V p, and T the periodic time
n _ PVpS _ QVq S circle + A Q q S
n + 1 ellipse " circle ;i*
circle
circle S R. 2 C Q
2 2
circle
j a e . sm. u . a
_ , (u = excentric anomaly)
= + e . sin. u
.. n it n + 1 . {= + e sin. u)
sn. u
sin. u =
n 1
r
1
*
5
2 e
which determines u, &c.
201. The Moon revolves round the Earth in 30 days, the mean distance
from the Earth = 240,000 miles. Jupiter s Moon revolves in day, the
mean distance from Jupiter = 240,000 miles. Compare the absolute Jorces
of Jupiter and the Earth.
VOL. I. L
162 A COMMENTARY ON [SECT. III.
if
T oc  , A being the major axis of the ellipse,
. . If A be given, ^ ^ >
Mass of Jupiter _ T 2 of the Earth s Moon _ 30j _ 14,400
* Mass of the Earth ~~ T 7 2 of Jupiter s Moon 1 1
4 2
202. A Comet at perihelion is 400 times as near to the Sun as the Earth
at its mean distance. Compare their velocities at those points.
Velocity 2 of the Comet F . 4 A S F 4 _ _L
Velocity 2 of the Earth " F. 2 B S = F 2~." 400 F 200
400" 2 1
1 2 200
= 800
V V 2 . 20 30 ,
=r nearly.
v
203. Compare the Masses of the Sun and Earth, having the mean distance
of the Earth from the Sun = 400, the distance of the Moon from the Earth,
and Earth s P d . = 13. the Moon s P d .
T 2 oc ,
ft
a
" i" 1 a T^Z
Mass of the Sun 400 3 J^ _ 64,000,000 _ 40Q Q0
* Mass of the Earth = I 3 13 2 " 169
204. If the force  2 z , where x is the distance from the center
of force, it will be centripetal whilst 2 > ~ , or x > a ; there will be
\. A
a jKMTtf o/ 7 contrary flexure in the orbit when  = ^ , or x = a, and
afterwards when x < a, the force will be repulsive, and the curve change
its direction.
BOOK I.]
NEWTON S PRINCIPIA.
163
205. The body revolving in an ellipse, at
B the force becomes n times as great. Find
the new orbit, and under what values ofn it
will be a parabola, ellipse, or hyperbola.
S being one focus since the force
the other focus must lie
distance
in B H produced both ways, since
S B, H B, make equal angles with
the tangent. V 2 =  F.PV = ^F.2ACin the original ellipse, or
* &
=  n F . P V in the new orbit.
2AC = n.PV = n.
2 SB. h B
S B"+"h B*
2
.. (S B + h B) A C = 2 n . S B . h B,
.. A C 2 + h B . A C = 2 n A C . h B,
hB f=r
If 2 n 1 = 0, or n = $, the orbit is a parabola ; if n > , the orbit
is an ellipse; if n < , the orbit is an hyperbola.
Let S C in the original ellipse be given B C,
.. S B H = right angle, and S B or A C = B h . cot. B S h
whence the direction of a a , the new major axis ; also
Sh
a a = S B + B h, and S c =
VBh 2 SB 2
2
If the orbit in the parabola a a be parallel to B h, and L . R = 2 S B,
since S B h = right angle.
206. Suppose a Comet in its or
bit to impel the Earth from a cir
cular orbit in a direction making
an acute angle with the Earth s
distance from the Sun, the velo
city after impact being to the velo
city before : : V~B : V~2. Find
the alteration in the length of the
year.
Since V 3 : V 2 < ratio than V 2 : 1, .*. the new orbit will be an
ellipse.
L2
164
X!
V 2
A COMMENTARY ON
2 S P. H P HP
[SECT. III.
p y
2 SP
A C. 2 S P " A C
C S P
A C
.. 3AC=4AC 2SP
.. 2 S P = A C
T in ellipse _ 2^ S P? 8
T r in circle = gp f : ~3 l
207. A body revolves in an ellipse, at any given point the force becomes
diminished by ^ th part. Find the new orbit.
\~ F. P V
in this case P V = ,
r
P V in ellipse 1 n n 1
pv in new orbit ~ I n
But
if
in conic section pv n 1 PV
 _* _ . _ _ of M
2 in circle e. d. " 2 S P " 2 S P
n HP
" n 1" A C
. H P = AC, the new orbit is a Circle
= 2 A C, Parabola }>
n 1
< 2 A C,
> 2 A C,
Ellipse
Hyperbola J
If  ^ = 2, or n = 2, then when the orbit is a circle or an ellipse, P
11 J.
must be between a, B ; when the orbit is a parabola, P must be at B ;
when the orbit is an hyperbola, Pmust be between B, A.
BOOK I.] NEWTON S PRINCIPIA. 165
208. If the curvature and inclination of tlie tangent to the radius be the
same at two points in the curve, the forces at those points are inversely as the
radii z .
8 a 2 __ 8 a 2 _ 8 a 2 1
g.SY 2 .PV~g.SY.S P.R~g.sin.0SP 2 .R SP 2
This applies to the extremities of major axis in an ellipse (or circle) in
the center offeree in the axis.
209. Required the angular velocity of %.
By 46, 6 being the tracedangle,
dd
W rr r; 
d t
But by Prop. I. or Art. 124,
d t : T : : d A : A
d 6 _ 2 A 1
dl  T~ x ~
or
P x V
(a)
210. Required the Centrifugal Force (<p) in any orbit.
When the revolving body is at any distance g from the center of force,
the Centrifugal Force, which arises from its inertia or tendency to persevere
in the direction of the tangent (most authors erroneously attribute this force
to the angular motion, see Vince s Flux. p. 283) is clearly the same as it
would be were the body with the same Centripetal Force revolving in a
circle whose radius is g. Moreover, since in a circle the body is always
at the same distance from the center, the Centrifugal Force must always
be equal to the Centripetal Force.
But in tlie circle
or
g e 3
P and V belonging to the orbit.
L3
A COMMENTARY ON [SECT. III.
P 2 V 2 1
g
Hence also and by 209,
g
And 139,
F : <* IE If
(b)
P S 3
211. Required the angular velocity of the perpendicular upon the
tangent.
If two consecutive points in the curve be taken ; tangents, perpendiculars
and the circle of curvature be described as in Art 74, it will readily ap
pear that the incremental angle (d ^} described by p = that described
by the radius of curvature. It will also be seen that
But from similar triangles
P V : 2 R : : p : g.
.. d 6 : d 4> : : P V : 2 g
P V being the chord of curvature.
Hence
d ^\ d 6 2
or
or
Ex. 1. In the circle P V = 2 ; whence
PxV
~ ^ "
Ex. 2. In the other Conic Sections, we have
2g
w X
(di
; PV
2P X V
(e)
g X P V
P X V dp
ftt
p =
BOOK I.] NEWTON S PRINC1PIA. 167
which gives by taking the logarithms
2 lp = lb 2 + lg l(2a + g)
and (17 a.)
2 d p _ dj , d g _ 2 a d g
P "T~2aqp^~ f (2a + g)
whence
aP X V
=
212. Required the Paracentric Velocity in an orbit.
It readily appears from the fig. that
d s : d g : : g : V g 2 p 2 .
.. If u denote the velocity towards the center, we have
/ d g\ ds V e p
u ( = i ! ) = 3~i X 
\ d t/ d t g
x eP (125)
or
2 A //I IN ,.
~T~ X V \p~ 2 ~~ g z ) *
Also since
p 2
= PV
213. Tojind where in an orbit the Paracentric Velocity is a maximum.
From the equation to the curve substitute in the expression (212. g)
for p*, then put d u = 0, and the resulting value off will give the posi
tion required.
Thus in the ellipse
 2a g
and
v s  = ) = max.
2 a 1 1
_
" b =
L4
168 A COMMENTARY ON [SECT. III.
8ad g , 2 d g _
"" ~~
and
b s LatusRectum
e = T =  g
or the point required is the extremity of the LatusRectum.
OTHERWISE.
Generally, It neither increases nor decreases when F = p. Hence
when u = max. (see 210)
d p _ d g
? :: 7
which is also got from putting
d (u 2 ) =
in the expression 212. h.
214. Tojlnd where the angular velocity increases fastest.
By Art. 209 and 125,
 ,
"l   < JT V J\  ; 7\ o 1 .  4 <" 1 f, >
d t 3 f 2 d 6 f 4 g d
But from similar triangles
t / r> 9\ f~\ T 1 ~D T* .  ^1 A . A
p: V (g 2 p } : y 1 : " l ::gae:ag
j ,. o pa V 2
... "I = l^j X V ( f 2  p 2 ) = max.
,.!*?*= I i^max (b)
either of which equations, by aid of that to the curve, will give the point
required.
Ex. In the ellipse
_  = TO ax. = m
i d m _ .
and , = gives
4 .
b
BOOK I.] NEWTON S PRINCIPIA. 169
which gives
P = a + V (49 a 2 48 b 2 )
6 6
for the maxima or minima positions.
If the equation
_ b_ 8 1
a 1 + e cos. d
and the first form be used, we have
d e a e .
T* = TT * s sln 
d d b 2
and
sin. 6
 3  = max. = m.
Whence and from d m = 0, we get finally
215. To fold where the Linear Velocity increases fastest.
Here
d v
p = max.
d t
But (125)
P x V
P
and
g 2 d 6 p d g
: p^v~ P x v ): v s z p l
d v py V(g 8 p 8 ) d_p_
Tt = ~~i~ < p^dl
V ^ 2 D 2 ^
 g F X ".
I
, V (?* P 2 ) v d P
or ? max. = m.
(
and
d m =
will give the point required.
170 A COMMENTARY ON [SECT. IV.
Thus in the ellipse
p 2 1 b 2
= 4 H j = max.
which gives
d m _ _4_ 10 a b 8 g 4 6 b 2 g 5
d s " g 5 " (2 a g)V
whence the maxima and minima positions.
In the case of the parabola, a is indefinitely great and the equation
becomes
4 a 2 1 4 a b 2 =
IB
5 b 2 5
f = o x T^ * Latus Rectum.
o a ID
Many other problems respecting velocities, &c. might be here added.
But instead of dwelling longer upon such matters, which are rather
curious than useful, and at best only calculated to exercise the student,
I shall refer him to my Solutions of the Cambridge Problems, where he
will find a great number of them as well as of problems of great and
essential importance.
SECTION IV.
216. PROP. XVIII. If the two points P, p, be given, then circles whose
centers are P, p, and radii AB+SP, AB+Sp, might be described
intersecting in H.
If the positions of two tangents T R, t r be given, then perpendiculars
S T, S t must be let fall and doubled, and from V and v with radii each
= A B, circles must be described intersecting in H.
Having thus in either of the three cases determined the other focus H,
the ellipse may be described mechanically, by taking a thread = A B in
length, fixing its ends in S and H, and running the pen all round so as to
stretch the string.
BOOK I.] NEWTON S PRINCIPIA. 171
This proposition may thus be demonstrated analytically.
1st. Let the focus S, the tangent T R, and the point P be given in
position ; and the axismajor be given in length, viz. 2 a. Then the per
pendicular S T ( = p), and the radius vector S P ( = f) are known.
But the equation to Conic Sections is
whence b is found.
Also the distance (2 c) between the foci is got by making p = g, thence
finding and therefore c = a Ijl .
This gives the other focus ; and the two foci being known, and the axis
major, the curve is easily constructed.
217. 2d. Let two tangents T R, t r, and the focus S be given in position.
Then making S the origin of coordinates, the equations to the trajectory
are
b 2 g b 2 1
P " 3 :
a being the inclination of the axismajor to that of the abscissae.
Now calling the angles which the tangents make with the axis of the ab
scissae T and T , by 31 we have
tan * T = dV
But
x = cos. 6, y = g sin. 6
whence
tan. T = d g sm  6 + g d 6 cos. 6
d cos. d 6 sin. 6
ed
Vl tan. 6 + I
= ^  (b)
g d 6
Also from equations (a) we easily get
tan.
(0
V 2
COS. (d a) = g (2)
ae g
sin. (d a) = ^X V (2ag g b) . . (3)
and
_ Sap*
172 A COMMENTARY ON [SECT. IV.
and putting
R = V (2ae e b 2 ) ... (5)
we have
R , tan. 6 tan. a
{ = tan. (6 a) = . (6)
b 2 ag 1 + tan. . tan. 6
which gives tan. 6 in terms of a, b, , and tan. a.
Hence by successive substitutions by means of these several expres
sions tan. T may be found in terms of a, b, p, tan. a, all of which are given
except b and tan. . Let, therefore,
tan. T = f (a, p, b, tan. a).
In like manner we also get
tan. T = f (a, p , b, tan. a)
p belonging to the tangent whose inclination to the axis is T.
From these two equations b and tan. a may be found, which give
c = V a 2 b 2 and a, or the distance between the foci and the position
of the axismajor; which being known the Trajectory is easily con
structed.
218. 3d. Let the focus and two points in the curve be given in posi
tion, &c.
Then the corresponding radii g, f , and traced angles 6, tf, in the
equations
+ a (1 e 2 )
1 + e cos. (6 a)
a(le)
1 + e cos. (& a)
are given ; and by the formula
cos. (d a) = cos. 6 . cos. a f sin. & sin. a
2 a e and a or the distance between the foci and the position of the axis
major may hence be found.
This is much less concise than Newton s geometrical method. But it
may still be useful to students to know both of them.
219. PROP. XIX. To make this clearer we will state the three cases
separately.
Case 1. Let a point P and tangent T R be given.
Then the figure in the text being taken, we double the perpendicular
S T, describe the circle F G, and draw F I touching the circle in F and
passing through V. But this last step. is thus effected. Join V P, sup
pose it to cut the circle in M (not shown in the fig.), and take
V F 2 = V M x (V P + P M).
The rest is easy.
BOOK I.]
NEWTON S PRINCIPIA.
173
Case 2. Let two tangents be given. Then V and v being determined
the locus of them is the directrix. Whence the rest is plain.
Case 3. Let two points (P, p) be given. Describe from P and p the
circles F G, f g intersecting in the focus S. Then draw F f a common
tangent to them, &c.
But this is done by describing from P with a radius = S P S p, a
circle F G , by drawing from p the tangent p F as in the other case (or
by describing a semicircle upon P p, so as to intersect F G in F ) by
producing P F to F, and drawing F f parallel to F p.
See my Solutions of the Cambridge Problems, vol. I. Geometry, where
tangencies are fully treated.
174 A COMMENTARY ON [SECT. IV
These three cases may easily be deduced analytically from the general
solution above ; or in the same way may more simply be done at once,
from the equations
L L 1
P 2 = T ft S = Z X
2 1 + cos. (6 a)
220. PROP. XX. Case 1. Given in species] means the same as " simi
lar" in the 5th LEMMA.
Since the Trajectory is given in species, &c.] From p. 36 it seems that
the ratio of the axes 2 a, 2 b is given in similar ellipses, and thence the
same is easily shown of hyperbolas. Hence, since
c 2 = a
2 c being the distance between the foci, if = m, a given quantity, we
a
have
V a
which is also given.
With the centers B, C, &c.]
The common tangent L K is drawn as in 219.
Cases 2. 3. See Jesuits Notes.
OTHERWISE.
221. Case 1. Let the two points B, C and the focus S be given.
Then
+ a (1 e 2 )
1 + e cos. (6 a)
 a(l e 2 ) f
S I + e cos. (d f a))
a being the inclination of the axis of abscissas to the axis major.
But since the trajectory is given in species
e = is known,
a
and in equations (1), g, 6; g , tf, are given.
Hence, therefore, by the form
cos. (6 a) = cos. 6 . cos. a + sin. 6 sin. a,
a and , or the semi axismajor and its position are found;
also c = a e is known ;
which gives the construction.
BOOK I.j NEWTON S PRINCIPIA. 175
Case 2. By proceeding as in 220, in which expressions (e) will be
known, both a, a e, and a may be found.
Case 3. In this case
will give a. Hence c a e is known and
+ a(l e 2 )
S ~ I + e cos. (0 a)
gives a.
Case 4. Since the trajectory to be described must be similar to a given
one whose a and c are given,
c c
G = a = 17
is known (217).
Also g and 6 belonging to the given point are known.
Hence we have
1 + e cos. (d a)
And by means of the condition of touching the given line, another
equation involving a, a may be found (see 217) which with the former
will give a and a.
222. SCHOLIUM TO PROP. XXI.
Given three points in the Trajectory and the focus to construct it.
ANOTHER SOLUTION.
Let the coordinates to the three points be g, 6 ; g, Of ; g", 6", and a the
angle between the major axis and that of the abscissae. Then
+ a.(le 2 ) "I
1 + e cos. (d a)
4 a H P 2 ^
/ ~ H J. I , i x
~ 1 + e cos. (<f a)
/= + a (l e 2 )
1 + e cos. (d" ) ^
and eliminating + a (1 e 2 ) we get
g g = e . cos. (& a) e cos. (6 a)
g = e . cos. (d" a) e cos. (6 ) J
176 A COMMENTARY ON [SECT. V.
from which eliminating e, there results
__ gg __ gg"
__ _ _
g . COS. (^ a) g COS. (0 a) g" COS. (0" a) COS. (<J a)
Hence by the formula
cos. (P Q) = cos. P . cos. Q + sin. P . sin. Q
gfV cos. J" (g  g") g cos. f + s (f  g Qcos.d
_
(S gO f" sin. <?" (g e" ) /sin. + ^ f <")sinJ
which gives a.
Hence by means of equations (B) e will be known ; and then by substi
tution in eq. (A), a is known.
SECTION V.
The preliminary LEMMAS of this section are rendered sufficiently intel
ligible by the Commentary of the Jesuits P.P. Le Seur, &c.
Moreover we shall be brief in our comments upon it (as we have been
upon the former section) for the reason that at Cambridge, the focus of
mathematical learning, the students scarcely even touch upon these sub
jects, but pass at once from the third to the sixth section.
223. PROP. XXII.
This proposition may be analytically resolved as follows :
The general equation to a conic section is that of two dimensions (see
Wood s Alg. Part IV.) viz.
y 2 + Axy+Bx 2 + Cy + Dx + E =
in which if A, B, C, D, E were given the curve could be constructed.
Now since five points are given by the question, let their coordinates be
a, j3 ; a, j3 ; a, j3 ; a, (3 ; a, /3 .
11 22 33 44
These being substituted for x, y, in the above equation will give us five
simple equations, involving the five unknown quantities A, B, C, D, E,
which may therefore be easily determined: and then the trajectory is
easily constructed by the ordinary rules (see Wood s Alg. Lacroix s Diff.
Cal. &c.)
224. PROP. XXIII. The analytical determination of the trajectory
from these conditions is also easy.
Let
a, /3 ; a, ; a, ; a, /3
11 22 33
BOOK I.] NEWTON S PRINCIPIA. 177
be the coordinates of the given point. Also let the tangent given in posi
tion be determinable from the equation
y = m x + n ......... (a)
in which m, n are given.
Then first substituting the above given values of the coordinates in
y 2 + Axy + Bx 2 + Cy+Dx+ E = . . . (b)
we get four simple equations involving the five unknown quantities
A, B, C, D, E ; and secondly since the inclination of the curve to the axis
of abscissas is the same at the point of contact as that of the tangent,
d y _ d y
cfx dx
y = y
x = x
. Ay+ 2Bx + D
2y+ Ax + C
and substituting in this and the general equation for y its value
y = m x + n
we have
A(mx + n)+2Bx +
2(mx + n) + Ax+C
and
from the former of which
, _ n A + m C + D
2(m 2
and from the latter
2(m 2 +nA+B)
and equating these and reducing the result we get
4,m 2 n 2 = (nA + mC + D+ 2 m n) 2 (n 2 + n C + E) (m 2 +m A+B)
and this again reduces to
+ 2mCD nBCmAE BE+ 3mn 2 A
+ 3nm 2 C + 4mnD n B m 2 E n a m 2 =
which is a fifth equation involving A, B, C, D, E.
From these five equations let the five unknown quantities be determined,
and then construct eq. (b) by the customary methods.
M
178 A COMMENTARY ON [SECT. V.
225. PROP. XXIV.
OTHERWISE.
Let
,<3; a , /3 ; a", 0"
be the coordinates of the three given points, and
y = m x + n
y"= m x" + n
the equations to the two tangents. Then substituting in the general
equation for Conic Sections these pairs of values of x, y, we get three
simple equations involving the unknown coefficients A, B, C, D, E ; and
from the conditions of contact, viz.
dy _ dy _
dx ~~ dx
x = x
We also have two other equations (see 224) involving the same five un
knowns, whence by the usual methods they may be found, and then the
trajectory constructed.
226. PROP. XXV.
Proceeding as in the last two articles, we shall get two simple equations
and three quadratics involving A, B, C, D, E, from whence to find them
and construct the trajectory.
227. PROP. XXVI.
In this case we shall have one simple equation and four quadratics to
find A, B, C, D, E, with, and wherewith to describe the orbit.
228. PROP. XXVII.
In the last case of the five tangents we shall have five quadratics,
wherewith to determine the coefficients of the general equation, and to
construct.
BOOK 1.] NEWTON S PRINCIPIA. 179
SECTION VI.
229. PROP. XXX.
OTHERWISE.
After a body has moved t" from the vertex of the parabola, let it be re
quired to find its position.
If A be the area described in that time by the radius vector, and P, V
the perpendicular or the tangent and velocity at any point, by 124 and
125 we have
c P V
A = ~2 * * = 3 X t
and by 157,
L being the latusrectum.
* A = 1 vV x L
But
ASP = A OP SOP=f AOxPO SOxPO
= ixy_ Mx __ r)y
where r = A S, &c. (see 21) and
y 2 = 4rx
. .y 3 + 12r z y = 12 rt V g^r
by the resolution of which y may be found and therefore the position of P.
OTHERWISE.
230. By 46 and 125,
V " C
Also
i ?d o
U S = r2
M 9
180 A COMMENTARY ON [SECT. VI.
...dt=
which is an expression of general use in determining the time in terms of
the radius vector, &c.
In the parabola
P 2 = re,
whence
c V(g r)
and integrating by parts
2 V r . . 2 V r ,, .. .
t = rj* S V (S ~ r) fd g^ (g
2 V r
But
c= PV= VlT^r (229)
which gives
whence we have g and the point required.
liy the last Article the value of M in Newton s Assumption is easily
obtained, and is
1VA ~~ ___ ~* /\ / _ *
4 r 4 V 2r
231. COR. 1. This readily appears upon drawing S Q the semilatus
rectura and by drawing through its point of bisection a perpendicular to
GH.
232. COR. 2. This proportion can easily be obtained as in the note of
the Jesuits, by taking the ratio of the increments of G H and of the curve
at the vertex ; or the absolute value of the velocity of H is directly got
thus.
d.GH3dM * /
dt dt 4^ 2r
Also the velocity in the curve is given by (see 140)
BOOK I.] NEWTON S PRINCIPIA. 181
and at the vertex = r,
.. v : v : : 3 : 8.
233. COR. 3. Either A P, or S P being bisected, &c. will determine
the point H and therefore
4 2 r
X GH.
234. LEMMA XXVIII. That an oval cannot be squared is differently
demonstrated by several authors. See Vince s Fluxions, p. 356; also
Waring.
235. PROP. XXXI. This is rendered somewhat easier by the follow
ing arrangement of the proportions :
If G is taken so that
OG:OA::OA:OS
or
and
GK: 2*OG:: t: T
or
n v 2^x OA 2 t
Then, &c. &c. For
ASP= ASQx
a
= ~ X(OQA OQS)
= 27 (OQx AQ OQx SR)
= ^(AQ SR).
But
S R : sin. A Q : : S O : O A
: : O A : O G : : A Q : F G
S R A Q sin. A Q
FG
and
M3
182 A COMMENTARY ON SECT. VI.
= x(FGsin.AQ)
(FG _ sin<AQ)
(see the Jesuits note q.) which is identical with (a), since
_1 A S _^
T ~~ Ellipse
_ ASP
v ab
OTHERWISE.
236. By 230 we have
But in the ellipse
2a
r\ t
. . U I  . . , j j%
and putting
g a zz u
it becomes
. b . (a + u) d u
2 a e being the excentricity.
Hence
(b)
b a p du _b / udu
T^ V(a 2 e 2 u 2 ) + ~IJ V(a 8 e 8 u
L? s in. . U  V(a 2 e 2 u 2 ) + C.
c a e c
Let t zz 0, when u zz a e ; then
and we get
_ba *
~ V X 2
BOOK L] NEWTON S PRINCIPIA. 183
which is the known form of the equation to the Trochoid, t being the ab
scissa, &e.
Hence by approximation or by construction u and therefore may be
found, which will give the place of the body in the trajectory.
It need hardly be observed that (157)
OTHERWISE.
237. dt =
but in the ellipse
b 2 1
= x
a 1 j e cos. 6
b
.. d t =  x
a 2 c ( 1 + e cos. 6} 2
and (see Hirsch s Tables, or art. 110)
a 2 (l e 2 ) f 1 . e + cos. 6 e sin. d
t = x < cos. ~ .
c I M* e ) l+ecos.0 1 + ecos.
which also indicates the Trochoid.
To simplify this expression let
. e 4 cos. d
V/VJ
1 + e cos. i
1 ~
then
e + cos. d
*^~ /*OC 11
1 + e cos. 6
CUp* II
and
rr\<i A
e <
e cos. u 1
Hence
A/ { 1 2\
sin. 6 =
1 e cos. u
and
e sin, d e sin. u
1 + ecos. d ~ V (1 e 2 )
a 2 V (I e 2 )
.. t = * X ^u esm. u}
But (157)
184 A COMMENTARY ON [SECT. VI.
5
a 2
.*. t = == x (u e sin. u)
V cr/jb
a* 1
Let = _ .
V g(* n
Then
nt=.u esin. u (1)
Again, 6 may be better expressed in terms of u, thus
2 _0 1 cos. 6 _ 1+e 1 cos. u _ 1 + e 2 u
2 ~ 1 + cos. 6 ~ T^Ti X 1 + cos. u ~ T^> tan> "2~
u
Moreover g is expressible in terms of u, for
g= I a i 1 ~ e2) , = a(l ecos.u) ..... (3)
1 + e cos. 6
In these three equations, n t is called the Mean Anomaly , u the
Excentric Anomaly, (because it = the angle at the center of the ellipse
subtended by the orduiate of the circle described upon the axismajor
corresponding to that of the ellipse) ; and 6 the True Anomaly.
238. SCHOLIUM.
Newton says that " the approximation is founded on the Theorem that
The area APSocAQ SF, SF being the perpendicular let fall from
S upon O Q."]
First we have
APR=AQRx
S P R = S Q R x
a
..ASP = ASQx
a
But
ASQ=AOQ SOQ
zrJAQxAO SFxOQ
= i AO x (AQ SF).
.. A S P = ~ x (A Q S F)
= X (a u a e sin. u ) ..... (1)
J9
u being the L A O Q.
BOOK I.] NEWTON S PRINCIPIA. 185
(Hence is suggested this easy determination of eq. 1. 237. 4 .
For 5 (a  >
t = T x A Sp 2 * flg v 2
Ellipse V ^ g
= X (u e sin. u). )
V i
* 
Again, supposing u an approximate value of u, let
u = u H
a
Then, by the Theorem, we have
2 A b Sp = A q S O X sin. A q
= AQ + Qq + SOx sin. (A Q + Q q)
to radius 1.
But A Q being an approximate value of A q, Q q is small compared
with A O, and we have
sin. ( A Q + Q q) = sin. A Q cos. Q q + cos. A Q sin. Q q
= sin. A Q + Q q cos. A Q nearly.
J_
.*. Q q = ( ^ ? A Q + SO sin. A Q) x = nearly
^=r + cos. A Q
which points out the use of these assumptions
XT/ 2 A Sp 2 t , ,
N = 1 ^ = r?f, X area of the Ellipse
and
D = S O. sin. A Q = B sin. A Q
I/ a I
 so
Then
Q q = (N A Q + D ) X , , _
, , _ .
\J + cos. A Q
in which it is easily seen B , N x , D , I/
are identical with B, N, D, L.
Hence
E = Qq= (NAQ + D) T _ L _.
L + cos. A Q
186
A COMMENTARY ON
[SECT. VII.
Having augmented or diminished the assumed arc A Q by E, then re
peat the process, and thus find successively
G, I, &c.
For a developement of the other mode of approximation in this
Scholium, see the Jesuits note 386. Also see Woodhouse s Plane Astro
nomy for other methods.
SECTION VII.
1 A
239. PROP. XXXII. F ^ 2 Determine the spaces which a
distance
body descending from A in a straight line towards the center of
force describes in a given time. P
If the body did not fall in a straight line to the center, it would
describe some conic section round the center of force, as focus
C ellipse ~\
(which would be < parabola > if the velocity at any point were to S
(_ hyperbola j
the velocity in the circle, the same distance and force, in R.J = V
V 2 : 1.)
(I) Let the Conic Section be an Ellipse A R P B.
Describe a circle on Major Axis A B, draw
C P D through the place of the body perpendi
cular to A B.
The time of describing A P a area A S P a
area A S D, whatever may be the excentricity
of the ellipse.
Let the Axis Minor of the ellipse be diminish
ed sine limite and the ellipse becomes a straight
line ultimately, A B being constant, and since
A S . S B = (Minor Axis) 2 = 0, and A S finite
/.SB = 0, or B ultimately comes to S, and
time d . A C a area A D B. .. if A D B be taken proportional to time,
C is found by the ordinate D C.
(T . A C a area ADBaADO + ODBaarcAD + CD
/. take 6 + sin. & proportional to time, and D and C are determined.)
D
BOOK I.]
(Hence
the time down A O
T.O B
NEWTON S PRINCIPIA.
187
18
T
N. B. The time in this case is the time
from the beginning of the fall, or the time
from A.
(II) Let the conic section be the hyperbola
B F P. Describe a rectangular hyperbola on
Major Axis A B.
T a area S B F P or area SEED.
Let the Minor Axis be diminished sine
limite, and the hyperbola becomes a straight
line, and T or area B D E.
N. The time in this case is the time from
the end of motion or time to S.
Let the conic section be the parabola B F P.
Describe any fixed parabola BED.
T or area S B F P a area SEED.
Let L . R. of B F P be diminished sine
limite the parabola becomes a straight line,
and T a area B D E.
N. The time in this case is the time from
the end of motion, or time to S.
Objection to Newton s method. If a
straight line be considered as an evanescent
conic section, when the body comes to peri
helion i. e. to the center it ought to return to aphelion i. e. to the original
point, whereas it will go through the center to the distance below the
center r= the original point.
240. We shall find by Prop. XXXIX, that the distance from a center from
which the body must fall, acted on by a blc force, to acquire the velocity such
as to make it describe an ellipse = A B (finite distance), for the hyperbola
= A B, for the parabola = a .
241. Case 1. v d v = g ^ d x, f = force distance 1,
x 2
v =
if a be the original point
a ~ x \
ax/
dx Va
Ut =  =
v V 2
T
dx .
V ax x
188
A COMMENTARY ON [SECT. VII.
r(_x)dx fdx 
Vax x
. . t = / ^ .. /
V 2g/z \
/ a
^ . / rt
Va. x x !
+ C, when t = 0, x = a,
V ax x 2 + /circumference
V 2~
vers
rad
rs. ~ l x ~)
1 1
vers. ~ x,
rad. = 
^ (C D + A D)
if the circle be described on B A = a,
/"~a~ 4 /CD.OB AD.ODx
"~ S 2fir/ct aA 2 2 ~/
.BAD.
Case 2. v 2 = 2 g ^ .
a x
, if a be an original point,
/__
tf/V V
xdx
x +
2
for t in this case is the time to the center, not the time from the original
point,
, . d x d x
.*. d t = , or d t = .
v v
Now if with the Major Axis A B = a, we describe the rectangular
hyperbola,
D
we have
B
A
E
BOOK I.] NEWTON S PRINCIPIA.
d.BED=d.BEDC d.ABDC=ydx
189
Vax + x
a x d x
i /
.*. t from B = / .BED, for they begin or end together at 15.
B
1),
T7I
Jli
Case 3. v 2 = 2 g p , if a be a ,
, . dx Vxdx
.*. a t = = _
v V 2 2 u
. . . T,
, t being time to B,
+ C, when t = 0, x = 0, .. C = 0.
V" 2 g /A 3
Describe a parabola on the line of fall, vertex B, L . R. = any fixed
distance a,
.
. v/ x . x =
2 V 2
 2 V 2 T> ^ T.
a x . x = ===== .BED.
2 V 2
Hence in general, in Newton Prop. XXXII, t = _. . curvili
V a g ft
near area, a being L . R. of the figure described.
T a, T D 2 (Ax. Min.) 2 . f A
In the evanescent conic sections, L . R. = \ ^ , . . if Ax,
Min. be indefinitely small, L. R. will be indefinitely small with respect to the
Ax. Min. The chord of curvature at the finite distance from A to B is ulti
mately finite, for P V = LL*_^L?j but at A or B, P V = L, = in
~2~
finitesimal of the second order. Hence S B is also ultimately of the second
order, for at B, S B = L. 4^
2 AS
PROP. XXXIII. Force a
Vat C
1
(distance) *
VAC
r = i JTT ?T7s = in the ellipse and hyperbola.
v in the circle distance S C V SA
Ma
A COMMENTARY ON [SECT. VII.
HP VAC
~Ajr "" ~T^T whentheconiGsectlonbecomesa straight line 1 )
2 V 2
NEWTON S METHOD.
V 2 SY 2 L SP
v 2 ~" 2SP " 2 S Y 2
A C.CB AO 2 2 AO 2AO
C P 2 /Min. AX.N 8  2/Min.Ax.x * ~ L
V 2 / V 2 /
AO
. L _ AO.CP
2 : A C.C B
* v 2 = ATCTC~BTS Y 2
but
CO BO
B O "" TO
C O __ C B comp. in the ellipse
B O ~~ B T div. in the hyperbola,
. A C _ C_T div. in the ellipse C P
B O " B T comp. in the hyperbola ~~ BQ
A C 2 _ C P 2
* AO 2 ~ BQ 1
. BQ 2 .A C _ AQ.CP 2
A O AC
V 2 _ BQ 2 . A C.S P
*v 2 ~AO.BC.SY 2
but ultimately
B/~k__C!\7 Q"D T> C*
v^ o l,Oirr o *~/f
, . A , V 2 in a straight line A C
.. ultimately ^. r 5^ = TTS;*
v 2 in the circle A O
AC
Y  /
v ~ V
COR, 1. It appeared in the proof that
A.
A O
AC C T
BOOK I.] NEWTON S PRINCIPIA. 191
AC CT
.. ultimately ^^ = ^^ .
(This will be used to prove next Prop.)
COR. 2. Let C come to O, then A C = A O and V = v,
.. the velocity in the circle = the velocity acquired by falling externally
through distance = rad. towards the center of the force a
ILL 11 Ul3lt.ll.ld ~ 1 till* i,\J W til Ho Lilt/ V.,^11 lv>i W* H.1W JWJ. WV> A o
distance z
242. I Vwrf actual Velocity at C.
V 2 atC = AC
v 2 in the circle distance B C B A*
. vz _. ^ " ^ T7 2 _ 2A C g /* R p
"FA" 1 "BT Fc 5
if At = the force at distance 1,
. V2 AC
~ g ^B A.B C
V a x
.. V = V 2 g y, . r " , if B A = a, B C = x.
V a x
Tr . 1,. V space described
If a is given, V a r
V space to be described
In descents from different points,
, T V space described
V a * .
V space to be described X initial height ;
In descents from different points to different centers,
V space described X absolute force
V or .
V space to be described X initial height
243. Otherwise. vdv = ^dx,
ft T X
.\v 2 = 2 g (A. ^^ > wn en a is positive, as in the ellipse
o ^L Y
= 2 g ^ . when a is negative as in the hyperbola
= 2 g fi . , when a is a , as in parabola
(when x = 0, v is infinite)
V 2 in the circle radius x (in the ellipse and hyperbola)
v 2 2 a x . , a
. . yj = in the ellipse, =
192
A COMMENTARY ON
[SECT. VII.
v 2 2 a + x . , , a + x
^^ = in the hyperbola, =
V a ()
f /* X
 z . =
iC
2 ff fJ>
V 2 in the circle radius = (in the parabola) =
m
V 2 1
.*. ^2 = in the parabola.
244. In the hyperbola not evanescent
Velocity at the infinite distance _ S A
velocity at A " S Y
finite R., but when the hyperbola van
ishes, S Y ultimately = Min. Ax. for
S Y S C
rr = rp 5 and ultimately S C =
A C, and b C = A C, .. ultimately S Y = A b = C B, .. ultimately
S Y _ infinitesimal of the first order
S A " of the 2d order
velocity at A
velocity at cc distance
AS
245. PROP. XXXIV.
Velocity at C  _L f
velocity in the circle, distance S C ~~ T (
2
S P
what
the parabola.
For the velocity in the parabola at P = velocity in the circle
ever be L . R . of the parabola.
246. PROP. XXXV. Force oc rr ^ .
(distance) 2
The same things being assumed, the area swept out by the indefinite
T T?
radius S D in fig. D E S = area of a circular sector (rad. =
of fig.) uniformly described about the center S in the same time.
Whilst the falling body describes C c indefinitely small, let K k be the
arc described by the body uniformly revolving in the circle.
Case ] . If D E S be an ellipse or rectangular hyperbola,  = ^ ,
CT
Cc
Dd
CJD
S Y
DT
DT
T S
BOOK I.] NEWTON S PRINCIPLE
.Cc.CD CT AC
D~~A c~v ~ Tf*~S ~r~r\ ultimately.
ti.oY IS AO J
(Cor. Prop. XXXIII.)
But
velocity at C VAC
v in the circle rad. S C "" v^L^
v in the circle rad. S C S K / A~O
v in the circle rad. S K ~~ ** S C = +*> ~S~C
193
velocity at C x __ Cc _ A^C _ A C
in the circle rad. S K/ ~" K k ~ V ~S~C ~ (Tl)
.. Cc.CD = Kk.AC
. Kk. A C _ A C
D d . S Y r A~O
.. A O. Kk = Dd. S Y,
. . the area S K k = the area S D d,
.. the nascent areas traced out by S D and S K are equal
.*. the sums of these areas are equal.
Case 2. If D E S be a parabola S K = L ; R .
iO
As above
Cc. CD _ CT 2
D d . S Y ~ T~S = T
also ^ p
Cc = _ velocity at C _ ^locity in the circle ^
Kk "" velocity in the circle L . R ~~ velocity in the circle L . R
_ 2 2"
_ VS K SK
_
v~$Tc CD
2 2
..Cc.CD=r2.Kk.SK
.. K k . S K = D d . S Y.
247. PROP. XXXVI. Force a _ L_
(distance)
To determine the times of descent of a body falling from the given (and
. .finite] altitude A S
On A S describe a circle and an equal circle round the center S.
From any point of descent C erect the ordinate C D, join S D. Make
the sector O S K = the area A D S (O K = A D + D C) the body
will fall from A to C in the time of describing O K about the center S
Vet. J. N
194 A COMMENTARY ON [SECT. VII.
uniformly, the force oc _ . Also S K being given, the period
in the circle may be found, (P =^/ . r . S K *), and the time through
O
O K = P .  ? . the time through O K is known. .. the time
circumference
through A C is known.
248. Find the time in which a Planet would fall from any point in its
orbit to the Sun.
H
circle S P
Time of fall = time of describing ^ O K H, S O = g ,
5.
period in the circle O K H _ period in the circle rad. S O _ S O g
period in the ellipse " period in the circle rad. AC A C ^
.. the time of fall = i . P .
be considered a circle
, P = period of the planet. If the orbit
AC
and the time of fall
4 V 2
p
.
vs
= P. nearly.
= nearly.
BOOK I.]
NEWTON S PRIMCIPIA.
195
249. The time down A C a (arc
= A D + C D), a C L, if the cy
cloid be described on A S. Hence,
having given the place of a body at a
given time, we can determine the
place at another given time.
..
time d. A C
Draw the ordinate m 1 ; 1 c will deter
mine c the place of the body.
250. PROP. XXXVII. To determine the times of ascent and descent of a
body projected upwards or downwards from a given point, F a .
distance
Let the body move off from the point G with a given velocity. Let
V 2 at G m 2
== TV i i = 7, (V and v known, . . m known),
v 2 m the circle e. d. 1 v
To determine the point A, take
G A
m
2
G A
m z
G A + G S
G A
2
m ~
" G S
2 m 2
.. if m 2 = 2, G A is + and 00 , /. the parabola \ must be des
if m 2 < 2, G A is + and fin. .. the circle Vended on the
if m s > 2, G A is andfin. .. the rectangular hyperbola/ axis S A.
With the center S and rad. =   of the conic section, describe the
m
circle k K H, and erecting the ordinates G I, C D, c d, from any places
of the body, the body will describe G C, G c, in times of describing the
areas S K k, S K k , which are respectively = S I D, S I d.
251. PROP. XXXVIII. Force ex distance.
Let a body fall from A to any point C,
by a force tending to S, and a g . as the
distance. Time a arc A D, and V acquir
ed a C D. Conceive a body to fall in an
evanescent ellipse about S as the center.
.*. the time down A P or A C
aASPaASDocAD.
40
a A D for the same descent, i. e. when
A is given.
NT 2
196
A COMMENTARY ON
[SECT. VII.
The velocity at any point P
oc V F. P V
S P.
a CD.
COR. 1. T. from A to S = period in an evanescent ellipse.
= % period in the circle A D E.
= T. through A E.
COR. 2. T. from different altitudes to
S a time of describing different quadrants
about S as the center oc 1.
N. In the common cycloid A C S it is
proved in Mechanics that ifSca=SCA
and the circle be described on 2 . Sea,
and if a c = A C, the space fallen through,
then the time through A C a arc a d,
and V acquired a c d, which is analogous
to Newton s Prop.
Newton s Prop, might be proved in the
same way that the properties of the cycloid
are proved.
OTHERWISE.
252. vdv = g/AX.dx,
..v 2 = 2 g A* (a 2 x 2 ), if a = the height fallen from
.. v = V 2o/i . V a 2 x 2 = v/2gAt . C D.
d x d x 1
v V Spy* V a 2 x 2
.. t = +
arc
COS. =
. = *N
.. = a/
.AD.
a V2g/j,
.: velocity oc sine of the arc whose versed sine = space, and the arc
a time, (rad. = original distance.)
253. The velocity is velocity from ajinite altitude.
If the velocity had been that from infinity, it would have been infinite
BOOK I.] NEWTON S PRINCIPIA. 197
cl x x
and constant. .. d t = , and t = ., ._ j C, when t = 0,
v . a. V g p.
a, a = a .
1
x = a, .*. c is finite, . . t = C =
v g (i
Similarly if the velocity had been > velocity from infinity, it would
have been infinite.
254. PROP. XXXIX. Force a (distance}*, or any function of distance.
Assuming any oc n . of the centripetal force, and also that quadratures of
all curves can be determined (i. e. that all fluents can be taken) ; Re
quired the velocity of a body, when ascending or descending perpendicu
larly, at different points, and the time in which a body will arrive at any
point.
(The proof of the Prop, is inverse. Newton assumes the area A B F D
to oc V 2 and A D to oc space described, whence he shows that the force
D F the ordinate. Conversely, he concludes, if F oc D F, A B F D
a V 2 .)
v 2 a/v d v oc/F. ds.
Let D E be a small given increment of space, and I a corresponding
increment of velocity. By hypothesis
A BFD V_ 2 _ V 2
AB G E " v 2 =: V 2 + 2V.I+ I 2
ABFD V 2 V 2
* TPP~P~F = 9~V 1~ t" 2 = 9~V T u " iniate y
But
ABFDocV 2 .. D F G E 2 V . I
.. D E . D F ultimately, a 2 . V . I
2V. I I.V
.". JJ r a ex ^ .
But in motions where the forces are constant if I be the velocity gene
rated in T, F oc _ , f F oc = \ and if S be the space described with uni
form velocity V in T, ~ = ^ , (d t = ) . Also when the force is
fe JL v /
I V
a ble , the same holds for nascent spaces. .. F , and D E re
o
presents S. . . D F represents F.
N 3
198
A COMMENTARY ON
[SECT. VII.
Let D L a , = , .. D L M E ultimately = D L . D E
V A B F D v
D F
=^ <x time through D E ultimately.
.. Increment of the area A T V M E increment of the time down A D.
.. A T V M E oc T.
/dt
/d s
V
(Since A B F D vanishes at A, .. A T is an asymptote to the time
curve. And since E M becomes indefinitely small when A B F D is in
finite, .. A E is also an asymptote.)
255. COR. 1. Let a body fall from P, and be acted on by a constant
force given. If the velocity at D = the velocity of a body falling by the
action of a ble force, then A, the point of fall, will be found by making
ABFD = PQRD.
For
ABFD
D FGE
D FGE
= ~ by Prop.
D R S E " D R ~ i
if i be the increment of the velocity generated through D E by a constant
force.
DRSE
PQTTD
ABFD
* PQRD
2i
V
256. COR. 2. If a body be projected up or down in a straight line
from the center of force with a given velocity, and the law of force given ;
Find the velocity at any other point E . Take E g / for the force at E r .
BOOK I.] NEWTON S PRINCIPIA. 199
velocity at E = velocity at D.  ^^^^~ + if pr "
jected down, if projected up.
y P Q R U D F g E V A Bg j/x
( V P QRD" " V A B F D
257. COR. 3. Find the time through D E . _ _
Take E m inversely proportional to V P Q R D + D F g E (or
to the velocity at E ).
T.PD _ V"P D _ _VTD__ = _ Vir P* ( D E small)
T77E~ VTHE~ V(PD+DE) .
2V PD
PD
T.PD _ 2 PD _ 2 P D . D L
T.DE " DE DLME
also
T.D Eby ble force _ DJL^M E
T.DE by do7~ = D L m E /J
but T . D E by a constant force = T . D E by a ble force since the velo
/ d s\
cities at D are equal ( d t = 1
T. PD _ 2 PD. D L
T.DE X " D L m E x
d v
258. It is taken for granted in Prop. XXXIX, that F a ^ (46),
and that v = ^ , whence it follows that ifc.F = ^,dv = c.F.dt,
d t
and vdv = cF.ds.
.. v 2 = 2c/Fds
Newton represents/ F d s by the area A B F D, whose ordinate D I
always = F.
d ds
,.=/
v " V2c./Fds ;
d s
s
N4
200 A COMMENTARY ON [SECT. VII.
Newton represents / .  by the area A B T U M E, whose or
J V f d s
dinate D L always = 
. A BTF D
sg
In COR. 1. If F be a constant force V 2 = 2 g F . P D, by Mechanics
but
V 2 = 2c./Fds
And F 7 . P D or P Q R D is proved =/F d s or A B F D,
.. c = g
and
v* = 2g./Fds.
In COR 2 velocit y at E/ _ V/Fd s when s = A E 7
velocity at D "" VfF d s when s = A D
V A B g E
V AB FD
In COR. 3. t = time through D E X =/A? f __
v V 2 gy F d s
T = time through P D = =
VatD
= 2 P D. D L
P D . D L
t D L m E
259. The force a x n .
.. v d v = g ^ x n d x, fjt, the force distance 1.
if a be the original height.
Let n be positive.
V from a finite distance to the center is finite 1
V from x to a finite distance is infinite. /
Let n be negative but less than 1.
V from a finite distance to the center is finite 1
V from co to a finite distance is infinite. J
Let n = 1 the above Integral fails, because x disappears, which
cannot be.
BOOK 1.1 NEWTON S PRINCIPIA. 201
dx
v d v = g p 
.. V from a finite distance to the center is infinite 1
V from x to a finite distance is infinite. f
1 x
But the log. of an infinite quantity is x ly less than the quantity itself  when
X
x is infinite =  . Diff, and it becomes * = = .
x x
Tx
Let n be negative and greater than 1.
V from a finite distance to the center is infinite ")
V from oo to a finite distance is finite. /
260. If the force be constant, the velocitycurve is a straight line parallel
to the line of fall, as Q R in Prop. XXXIX.
DEDUCTIONS.
261. To find under what laws of force the velocity from x to a finite
distance will be infinite or finite, and from a finite distance to the center
will be finite or infinite.
If (1) F a x 2 , V a V~^ 3 ~~^
(2)
, v
. */ n 2 TT 2
(3)
1
1
/ a
(4)
x
~ V x
1
Va x
X 2
ax
(6)
1
/a 2 x 2
X 3
V a z x 2
1
n 1 x n
(7)
y
x n
V o n 1 v n I
^ cli A
In the former cases, or in all cases where F cc some direct power of
distance, the velocity acquired in falling from co to a finite distance or to
the center will be infinite, and from a finite distance to the center will be
finite.
202
A COMMENTARY ON
[SECT. VII,
In the 4th case, the velocity from oo to a finite, and from a finite dis
tance to the center will be infinite.
In the following cases, when the force a as some inverse power of
distance, the velocity from CD to a finite distance will be finite, for
a" 1 x n ~ l _ /_L_
V a n 1 x n 1 ^/ ^n 1
when a is infinite. And the velocity from a finite distance to the center
will be infinite, for
 a n  l x n 
when x = 0.
262. On the Velocity and TimeCurves.
B A B
n D
/
C
(4)
H
( 1 ) Let F a D, the area which represents V 2 becomes a A.
For D F a D C.
(2) Let F a V D, /. D F 2 a D C and Vcurve is a parabola.
(3) Let F a D 2 , . . D F a DC 2 , and Vcurve is a parabola the
axis parallel to A B.
(4) Let F a yr, /. D F a yxfo * Vcurve is an hyperbola referred
to the asymptotes A C, C H.
(5) If F a D, and be repulsive, V 2 aDC.DFDC 2 j
/. V a D C, . . the ordinate of the time curve a ^ a T\ n >
.. Tcurve is an hyperbola between asymptotes.
(6) If a body fall from co distance, and F a =p, V a ^,
.. the ordinate of the timecurve D, . . Tcurve is a straight line.
(7) If a body fall from , and F <x jp , V a  ,
.. the ordinate of Tcurve V D C, . . Tcurve is a parabola.
(8) If a body fall from x, and F a
V a ,
.. the ordinate of Tcurve a D C 2 , .. Tcurve is a parabola as in case 3.
BOOK I.]
NEWTON S PRINCIPIA.
203
EXTERNAL AND INTERNAL FALLS.
263. Find the external fall in the ellipse, the force in the focus.
Let x P be the space required to acquire the velocity in the curve at P.
V 2 downPx Px
V 2 in the circle distance S P
S x
2
V * in the circle distance S P A a
V 2 in the ellipse at P " 2. H P
, V 2 down Px A a . P x
V 3 in the ellipse at P ~~ S x. H P
Sx " A a
P x _ HP
S P " S P
.. P x = H P
.. S x = SPfPx = Aa, and the locus of x is the circle on 2 A a,
the center S.
264. Find the internal fall in the ellipse, the force in the focus.
V z down P x
V * in the circle S x
V 8 in the circle S x
SP
2
_SP I
V 2 in the circle S P ~ S x tOrCe a distance 1
204
A COMMENTARY ON
V 2 in the circle S P A a
[SECT. VII.
V 2 in the ellipseatfP
V 2 down P x
V z in the ellipse at P
S x
Pc
2 H P
Px. Aa
S x . H P
HP
A a
H P
SP~Aa + HP
Describe a circle from H with the radius A a. Produce P H to the
circumference in F. Join F S. Draw H x parallel to F S.
265. Generally.
For external falls.
V 2 down P x Sg.areaAB FD Newton s fig.
V 2 in the circle distance S P ~ g F . S P F = force at distance S P
V 2 in the circle S P 2 S P
V 2 in the curve at P
V 2 down P x
P V
4. A B FD
* V 2 in the curve " F. P V
.. 4 . A B F D = F . P V
. , , . , fordinate = F I
K md the area in general < , >
t abscissa = space J
In the general expression make the distance from the center = S P,
and a the original height, S x will be found.
266. For internal falls.
V 2 down P x 2g. AB F D Newton s fig.
"2 g F . ST F = force at P
2 SP
V 2 in the circle S P
V 2 in the circle S P
V 2 in the curve at P
V 2 down P x _
"V 2 in the curve at P ~ F. P V
.. if the velocities are equal, 4 A B F D =
BOOK I.] NEWTON S PRINCIPIA.
267. Ex. For internal and external falls.
205
In the ellipse the force tending to the center.
In this case, D F a D S. Take A B for the force at A. Join B S.
A D
.. D F is the force at D, and the area A B F D =  (A B + D F)
= AS _ S D A B + D F. Let /^ equal the absolute force at the dis
tance 1. Let S A = a, S D = x, .. A B = a p.
D F =
.. A B F D = ft .
x. a f x
and
cr
4ABFD = F.PV,
CD 2
x 2 = C P .
 in the ellipse,
or
a 2 x 2 = C D 2 .
For the external fall, make x = C P, then a = C x, and C x 2 C P 2 = C D 2 ,
or Cx 2 = C P 2 + CD 2
= A C 2 + B C 2
= AB 2
.. C x = A B.
For the internal fall, make a = C P, then x = C x , and
C P 2 Cx /2 = CD 2 ,
or
Cx 2 = CP 2 CD 2 ,
.. Cx = V C P 2 CD 2 .
268. Similarly, in all cases where the velocity in the curve is quadrable,
without the Integral Calculus we may find internal and external falls.
But generally the process must be by that method.
206
A COMMENTARY ON
[SECT. VII.
Thus in the above Ex.
vdv = g/ux.dx
gfi (a 2 x 2 )
.. v
269. And in general,
n+1
, as above, &c.
(a n + 1 x 1 ^ 1 ), if the force a
^
Also
dp
+  x +) = ,. P.
And to find the external fall, make x = , and from the equation find a,
the distance required.
And to find the internal fall make a = r, and from the equation find x,
the distance required.
270. Find the external fall in the hyperbola, the force oc from the focus.
V 2 down O P : V in the circle rad. S P : : O P :
SO
V * in the circle S P : V 2 in the hyperbola at P : : A C : H P
BOOK L] NEWTON S PRINCIPIA. 207
.. V 2 down OP: V 2 in the hyperbola : A C. O P : SQ ^ H P
.. 2 A C. O P = SO. HP
.. 2AC.SO 2AC.SP = SO. HP

To find what this denotes, find the actual velocity in the hyperbola.
Let the force = /3, at a distance = r, . . the force at the distance
Also
V 2 in the circle S P jS. r* x /3 x
2 g x 2 2 2 x
V 2 in the hyperbola _ (2 a + x) j3 r 2
2 g a . 2 x
 /3r , $Ll
x "2 a
V 2 B r z V 
But ^ when the body has been projected from oo =  1 $ of
g x ^g
V s 8. r z V 2
projection from oo , ..  of projection from oo = ^ = down 2 a,
O O
r
F being constant and =  5 , or = V 2 from GO to O , when S O = 2 A C.
.*. V in the hyperbola is such as would be acquired by the body ascend
ing from the distance x to CD by the action of force considered as repul
sive, and then being projected from co back to O , S O being = 2 A C.
In the opposite hyperbola the velocity is found in the same way, the
c , . ,, 2 H C . S P
torce repulsive, p externally = \T\>
A \^i  AT. L
271. Internal fall
V 2 down P O : V 2 in the circle rad. SO: : P O :
V 2 in the circle S O : V 2 in the circle S P : : S P : S O
V 2 in the circle S P : V 2 in the hyperbola at P : : A C : H P
.. V down P O : V 2 in the hyperbola : : A C. PO : S H P
.. 2 AC. PO = S O. H P
or
2AC(SP SO) = SO.HP
2 A C. SP
208
A COMMENTARY ON
[SECT. VII.
and
Hence make H E = 2 A C, join S E, and draw H O parallel to S E.
Hence the external and internal falls are found, by making V acquired
down a certain space p with a ble force equal that down i . P V by a
constant force, P V being known from the curve.
272. Find how far the body must fall externally to the cir
cumference to acquire V in the circle, F distance towards the
center of the circle.
Let OC = p,OB = x,QA=ta,C being the point re
quired from which a body falls.
Let the force at A = 1, .*. the force at B =
x A
~* "I
v d v g.F.dx, (for the velocity increases as x decreases)
* i
= g . d x
fo a
and when v = 0, x = p,
v z =
and when x = a,
at A =
(P 
But
v 2 at A = 2g. g
the force at A being constant, and
= ga
p a s = a
p 2 = 2 a 2 , .. p = V 2 . a.
273. Find how far the body must fall internally from the circumference to
acquire V in the circle, F a distance towards the center of the circle.
Let P be the point to which the body must fall, O A = a, O P=p,
O Q = x, F at A = 1, .. the force at Q = .
BOOK 1.1 NEWTON S PRINCIPIA. 209
i X
.. v d V = ft . . d x
a
... v 2 =  .x 2 + C,
and when v = 0, x = a,
a
.. v 2 = (a 2 x 2 )
a
and when x = p,
v 2 = (a 2 p s )from a ble force
a
and
v 2 = g . a, from the constant force 1 at A.
..a 2 p 2 = a 2 , .. p = 0, .. the body falls from the circumference
to the center.
274. Similarly, when F oc . .
distance
O C, or p externally = a V~~e, (e = base of hyp. log,)
and
OP, or p internally = ^ .
275. When F a .
distance 2
p externally = 2 a
2 a
p internally =  .
o
276, When F oc r. J
distance 3
p externally = x .
p internally = ~^
277. When Fa 1
distance n + l
n
p externally = a ^J 
n
p internally = a / ;
V 2 + n
If the force be repulsive, the velocity increases as the distance increases,
. . v d v = g F . d x
Vor.. I. O
210
A COMMENTARY ON
[.SECT. VI J.
278. Find how far a body must fall externally to any point P in the
parabola, to acquire v in the curve. F a =^2 towards the focus.
P V = 4 S P = c, S Q = p, S B = x, S P = a, force at P = 1,
a 2
but
FatB =
\o
. yl _ s! 4. r
o v
when v = 0, x = p
, r _ g a2
= 2ga 2 (  ) = 2ga 2 f ) at P,
Vx p/ \ a p/
= 2g.~ = 2ga,
279. Similarly, internally, p =
280. In the ellipse, F a
xternally =
p internally =
towards a focus
p externally = PH + P S. (. . describe a circle with the center S, rad. = 2 A C)
PH. PS
(Hence V at P = V in the circle e. d.)
281. In the hyperbola, F a ^p towards focus
pexternally 2 A C (Hence V at P = V in the circle e. d.)
P H PS
p internally = p ^^j . (Hence V at P= V in the circle e. d., p. 190)
<5 A \~/ "} Jr Jrl
282. In the ellipse F cc D from the center
pexternally= V A C 2 f B C 2 , (= A B)} (Hence construction)
or (= V C D 2 + C P 2 )
(Hence also V at P = V in the circle radius C P, when C D = C P)
p internally = V G P 2 CD 2 .
BOOK I.]
NEWTON S PRINCIPIA.
211
(Hence if C P = C D s p = 0, and V at P = V in the circle e. d, as
was deduced before)
(If C P < C D, p impossible,. .. the body cannot fall from any distance
to C and thus acquire the V in the curve)
283. In the ellipse, F a D from the center.
External fall.
The velocitycurve is a straight line, (since D F a C D, also
since F = 0, when C P = 0, this straight line comes to C, as
C d b, V a V TTO b a C O, O being the point fallen from, to acquire
Vat P.
.. V from O to C : V from P to C : : O C : P C
Also since vdv = gF.dx, and if the force at the distance 1 = 1,
the force at x = x. .. v d v = g x d x, and integrating and correct
ing, v 2 = g (p * x 2 ), where p = the distance fallen from.
.; v a V p 2 x 2 , and if a circle be described, with center C, rad. C O
a P N (the right sine of the arc whose versed P O is the space fallen
through).
.. V from O to P : V from O to C : : P N : (C M =) O C
and
V from P to C : V in the circle rad. C P : : 1 : 1
(for if P v = i P C, v d = C d P) and
V in the circle C P : V in the ellipse : : C P : C D.
Compounding the 4 ratios,
V down O P : V in the ellipse : : P N : C D
.. Take P N = C D, and
V down O P = V in the ellipse,
.. C O = C N = V C P 2 + C D .
02
212
Internal fall.
A COMMENTARY ON
[SECT. II.
V in the ellipse : V in the circle rad. C P : : CD: C P
V in the circle : V down C P : : 1 : ]
V down C P : V down PO: : (CM=)CP: ON
.. V in the ellipse : V down P O : : C D : O N
.. Take O N = C D, and V in the curve = V down P O, and C O
= V C P * C D 2 .
284. Find the point in the ellipse.! the force in the center, where V = the
velocity in the circle, e. d.
D
In this case C P = C D, whence the construction.
Join A B, describe
circle
on it, bisect the circumference in D , join
B D , A D . From C with A D cut the ellipse in P.
2AD /2 (=2PC 2 ) = AB 2 =AC 2 + BC s (=CP e + CD 2 )
.. 2 C P 2 = C P 2 + CD 2
... C P 2 = C D 2 . (C P will pass through E.)
A simpler construction is to bisect A B in E, B M in F, then C P is
the diameter to the ordinate A B, and from the triangles C E B, C F B,
C F is parallel to A B, .. C D is a conjugate to C P and = C P.
p externally (to which body must
285. In the hyperbola,
force repulsive, a D, from the center
rise from P,)= V C D 2 + C P 2
p internally (to which body must
rise from the center) = VCP CD*
(Hence if the hyperbola be rectangular p internally = 0, or the body must
rise through C P.)
BOOK I.]
NEWTON S PRINCIPIA.
213
286. In any curve, F oc jqri ,Jind p externally.
where a = S P, c = P V.
287. If the curve be a logarithmic spiral, c = 2 a,
/a \ i
.. p = a I
n a
o F a jp, ( ..
.. n = 2 j
p = a  n = cc
288. In any curve, F a p. n + ][ t jind p internally.
f a \ I . / 4a + 1 x
p = a /  \ V. (p *      )
n c v 1 4 a + n c/
\ a + I
4> J
289. If the curve be a logarithmic spiral, c = 2 a, n = 2,
290. If the curve be a circle, F in the circumference, c = a, and n = 4,
/a \
.*. p externally = a ( ) * = x
* cl tl/
/ 1 \ *L
and p internally = a ( ) * = ~ .
\a + a/ * f
291. In the ellipse, F a =^ from focus. External fall.
V 2 down O P : V 2 in the circle radius S P : : O P : , Sect. VII.
iO
V ! in the circle S P : V 2 in the ellipse at P : : A C : II P,
03
214, A COMMENTARY ON
.. V 2 down O P : V 2 in the ellipse : : A C . O P :
[SECT. VII.
SO.H P
.. s o =
.. 2 AC.O P = SO. HP
SAC. OP 2AC.SO 2AC.SP
H P
H P
Internal fall.
2AC.SP
U  2 A C H P  ^ A
V 2 down P O : V 2 in the circle radius S O : : P O :   ,
<w
V 2 in the circle S O : V 2 in the circle S P : : S P : S O
V * in the circle S P : V 2 in the ellipse at P : : A C : H P
.. V 2 down P O : V 2 in the ellipse : : P O . A C :
.. 2PO.AC = SO.HP
.. 2SP.AC 2SO.AC = SO.HP
2 A C.S P
SO.H P
.. S O =
F a
2 A C + H P
Hence, make.H E = 2 A C, join S E, and draw H O parallel to E S.
292. External fall in the parabola,
^ from focus.
E
V 2 d . O P : V 2 in the circle radius S P
SO
:: OP:
, Sect. VII.
V 2 in the circle S P : V 2 in the parabola
atP:: 1 : 2,
BOOK I.J NEWTON S PRINCIPIA. 215
.. V 2 down O P : V " in the parabola : : O P : S O
.. O P = S O, .. S O = a
Internal fall.
Vdown OP : V 2 in the circle S O : : O P : ~
fit
V 2 in the circle S O : V 2 in the circle S P : : S P : S O
V 2 in the circle S P : V 2 in the parabola at P: : 1 : 2
.. V 2 down OP: V 2 in the parabola : : O P : S O,
.. O P = S O,
.. S O  ~ .
V = V down  =r V down S P = V . down E P = V of a body describ
T*
ing the parabola by a constant vertical force = force at P. / x
293. Find the external fall so that the velocity* ac
quired = n . velocity in the curve, Fax".
v d v = g ,a . x n . d x, (/ = force distance 1),
.. v 2 = ~~ (a n + l x tt + l ) a = original height, /x
\TI .1 P d P g ., 2 p d P /
V " in the curve = a u, . ^ = ^  u. . c, if c = , =, /
dp 2 dp
* 2 ~~n + 1 ^ ~" n+1* "
Make x = S P = g, and from the equation we get a, which = S x.
For the internal fall, make a = S P = g, and from the equation we get
x, which = S x .
294. Find the external fall in a LEMNISCATA.
(x 2 + y 2 ) 2 = a 2 (x 2 y 2 )
is a rectangular equation whence we must get a polar one
Let L. N S P = 6,
.*. y = g. sin. 6 t x = g. cos. &, g 2 = (x 2 + y 2 )
.. g 4 = a *" . (g 2 (cos. 8 d sin. z 6}} = a 2 g 2 . cos. 2 0,
.. g 2 = a 2 . cos. 2 d
r e\
.: 2 6 = L. (cos. = a;),
V av
2gdg 2 g d g
a 2 : Va 4 g 4
/ 1 L!
v ""
O4
216
A COMMENTARY ON [SECT. VII.
but in general
..p =
r d~
p*
= in this case 
P 6
D 8 i
" a 4
4
_ ?J_P  _ SL
.*. force to S a

r
v d v = *f . d x,
Also
P V =
dp
*  g^ ^l  2 g^ JL

^ 3 6
Make x in the formula above = j,
. . ^ = 0, ,*. a is infinite.
rt D *
BOOK I.]
NEWTON S PRINCIPIA.
217
295. Find the force and external fall in an EPICYCLOID
CY 2 =CP 2 YP 2 = CP 2 CA 2 .
Let
CY = p,
YB !
CB
= g, CB = c, CA=b,
C 2 p 2
.. c 2 p 2 =
b 2 c
b 2 p
c 2 b 2
JL c 2 b 2
* * 2 "~ ""^ /""" 2 V\ gV
2 dp _ c 2 b 2 (
p 3 "" c 2
.. force
b 2 )
oc i.
(as in the Involute of the circle which is an Epicycloid, when the radius
of the rota becomes infinite.)
Having got <x a of force, we can easily get the external (or internal) fall.
296. Find in what cases we can integrate for the Velocity and Time.
Case 1. Let force a x a ,
.. v d v = g (i . x n d x,
1
... t  /*"~ dx = / n + 1 /*
J v > 2s //, / Vfa
dx
Now in general we can integrate x m dx.(a + bx n1 ) , when
mf1. , . m+lp ,,
is whole or \ whole.
n n q
. . in this case, we can integrate, when
Let
1
or
= p any whole number
= p
.. n = " , (p being positive), (a)
A COMMENTARY ON [SECT. Vll.
<0
.*. these formulas admit only and 1 for integer positive values of n, and
no positive fractional values. . . we can integrate when F a x, or F a 1.
297 oc 1
297. Case 2. Let force oc 1 ,
x"
, d x
.*. v d v = g .
fo x n
2rv n /Q n 1 ^^ ,,. n i
... v* = ^ ^ (? : x _>
/ dx /n l.a n1 r dx.x
. . t = I = *J n / ; ; ,
J V ** 2 g (Jj J V/a n ~
n 1
2
,n I
in which case we can integrate, when  ^ , or ^ , whole.
i. e. if  \ or ^ , be whole.
2 n 1, n 1
Let r =: p, any whole positive No,,
1 _ 2 p 1
n 1 2
2
2p V
. . these formulae admit any values of n, in which the numerator ex
ceeds the denominator by 1, or in which the numerator and denominator
are any two successive odd numbers, the numerator being the greater.
1111
.. we can integrate, when F 5, 7= , ,  , &c.
X X g" ^3" ^ X
or
1 1 J_ l *
 ~ 5 5 <v 9 n 9 *^
v 3 v*? v 7 ~v y
X A * A ^ A 7
BOOK I.] NEWTON S PRINCIPIA. 219
298. Case 3. The formulas ( ) (ft ), in which p is positive, cannot be
come negative. But the formulas (a) and (/3) may. From which we can
integrate, when F oc ___.____ & c .
299. When the force a x. n ,Jind a n . of times from different altitudes
to the center of force. Find the same, force a s  .
X
Fa x n , .. v d v = g / u.x n dx,
dx
v
d x n+ 1
a 7 , , , ; which is of ^ dimensions.
^/ <j n + 1 X n ~^~ 2
. . t will be of dimensions.
and when x = 0, t will
on~
x a _ 2 
dx
+.
a 2
n + 1 x n + i a
1
! *" + ,1.3 x ! "
+ lr r r+au.. + a74 8n + 3
when t = 0, x = a,
.. C a I. + 1 . __ . U _?_ . &c I
I 2 n + 8+ 2. 4" 8l , + 3 i
. . when x zr 0, t a  " a ^
220 A COMMENTARY ON [SECT. VII.
1 n + l
when n is negative t a _ n _ 1 a a 2 .
a _
COR. If n be positive and greater than 1, the greater the altitude, the
less the time to the center.
300. A body is projected up P A with the velocity V from the given
po nt A, force in S K^jind the height to which the body will rise.
vdv = g ,& x n d x,
for the velocity decreases as x increases, A
.
when v = V, x = a,
8
. x  /v g .n + i
V
COR. Let n = 2, and V = the velocity down , force at A con
slant, = velocity in the circle distance S A.
= 2 a.
BOOK I.] NEWTON S PRINCIPIA. 221
SECTION VIII.
301. PROP. XLI. Resolving the centripetal force I N or D E (F)
into the tangential one IT (F ) and the perpendicular one T N, we
have (46)
I N : I T : : F : F : : ^ :
d t d t
.. d v : d v : : d t x I N : d t x I T.
But since (46)
v v
and by hypothesis
v v 7
.. d t : d t : : d s : d s : : I N : I K
.. d v : d v : : I N 2 : I K x I T
: : 1 : 1
or
d v = d v ,
&c. &c.
OTHERWISE.
302. By 46, we have generally
vdv gFds
s being the direction of the force F. Hence if s be the straight line and
s the trajectory, &c. we have
vdv =. gFds
v d v = g F d &
.. v 2 V 2  2g/Fds
v"_ V 1 = 2g/Fd s
V and V being the given values of v and v at given distances by which
the integrals are corrected.
Now since the central body is the same at the same distance the central
force must be the same in both curve and line. Therefore, resolving F
222 A COMMENTARY ON [SECT. VIII.
when at the distance s into the tangential and perpendicular forces, we
have
_,
= F x
_
IN  I K
d s
d s
.. F d s = F d s
and
v/a _ V /2 = 2 g/F d s = v 2 V 2
which shows that if the velocities be the same at any two equal distances,
they are equal at all equal distances i. e. if
V = V
then
v = v .
303. COR. 2. By Prop. XXXIX,
v 2 A B G E.
But in the curve
y a Fa A n  l
.. y d x a A" 1 d A
Therefore (112)
ABGE =/ydxa ~ + C
P n A n
a
n
Hence
v 2 a P n A".
304. Generally (46)
v
OTHERWISE.
d v = gFds
and if
F =
then
v.^ll^
n
But when v = 0, let s = P ; then
and
C = P".
BOOK I.] NEWTON S PRINCIPIA. 223
,2 _ jLfaJ 6 /pn _
n
in which s is any quantity whatever and may therefore be the radius vector
of the Trajectory A ; that is
v 2 = i^(pn_ A") or = ?g^(D n e n )
n n v
in more convenient notation.
N. B. From this formula may be found the spaces through which a
body must fall externally to acquire the velocity in the curve (286, &c.)
305. PROP. XLI. Given the centripetal Jbjce to construct the Trajec
tory, and to find the time of describing any portion of it.
By Prop. XXXIX,
v = V~2~g. V A B F D = ^ (46) = ^
But
T /i yr v , Time T _, XT Time
d t = I C K X T = I CxK NX
A .m. vx ^s *** .L^ ^\ _. j
Area 2 Area
= p TT~ (P being the perpendicular upon the
tangent when the velocity is V. See 125, &c.)
Moreover, if V be the velocity at V, by Prop. XXXIX,
V = V~2~~. V A B L V.
Whence
P VABLV
V A B F D  x
v

KN
. . putting
A V 2 g
we have
ABFD : Z 2 : : I K 2 : KN 2
.. A B FD Z 2 : Z 2 : : I K 2 K N 2 : K N 2
and
V A B F D Z" 2 : Z =  : : I N : K N
A.
. A x K N  Q X IN
V (AB FD Z 2 )
Also since similar triangles are to one another in the duplicate ratio 01
icir homologous sides
YXxXC = AxKNx ^ 2
224 A COMMENTARY ON [SECT. VIII.
_ Q x CX a x I N
= A 2 V (A B FD Z 2 )
and putting
y = ]>b = 2 V (A B FD Z )
and
/  n Q x CX*
y ~ 2 A 2 V (A B FD Z 2 )
Then
Area V C I =/ y d x = V D b a\ ( ^
AreaVC X = /y dx = V D caj
Now (124)
2 VCI _ 2 V_D>ji
P X V : P X V
or
2 V D b a
" <v/2g.Px VABLV
the time of describing V I.
Also, if^.VC 1=6, we have
_ _ XV X CV = <_x CV*
_ 2VDca
P 2
which gives the Trajectory.
306. To express equations (5) and (6) in terms ofg and 6, ( = A).
First
V 2
ABLV =
and
Q 2 _P 2 xV 8
" "
v 2 P 2 X V*
BOOK I.] NEWTON S PRINCIPIA.
Hence
225
P X Vg
and
P 3 x V
P 2 V 2 )
and
2 J V (e 2 v 2 P 2 V 2 )
P 3 v r de
x /
l = 7 v<**v 2 P 2 V 2 )
P 2 V 2 )
But by Prop. XL.
the integral being taken from v = 0, or from f =D, D being the same as
P in 304.
fs d e
V
P 8 V r )
, or =/
V 2 V 2_
/.
J
Px Vdg
P S V
. . (7)
ix ( 8 )
307. Tojtnd t awrf ^ m terms of % and p.
Since (125)
/
" J
P 2 V 2 p
and
(10)
But previous to using these forms we must find the equation to the tra
jectory, thus (139)
P 2 V 2 d D
X 4 = F = f(j)
f denoting the law of force.
VOL. I.
22G
or
A COMMENTARY ON
pays
P<
~~ 17 2 o . r A 4
[SECT. VIII.
(U)
308. To these different methods the following are examples :
1st. Let F a s = p s . Then (see 304)
and if P and V belong to an apse or when P = g ;
V 2 = g ft (D 2 P 2 )
A/ , .. J
P 2 (D 2 P 2 )}
?_ 2
2
Let g 2 = u. Then we easily get
du
2 t V or u =
pa  1  7
2
and making t = at an apse or when g = P, we find
D 2
C = sin. . ^psr = sin. ~ l 1
2
2 V
sin.
1)2 )
T r(
D 2 "" 2 f
"a" 3
Also
/dt__ 1
J^~9 A/
du
" 2 ^/( u+ W{( p ^V" ! }
and assuming
2
we get
PV " 2
D
2V 5 2
BOOK I.] NEWTON S PRINCIPIA. 227
and making 6 = 0, when g = P we find
C =_ sin.l = 1..
Also
V= V gp. V (D 2 P 2 )
= sn.
 sin. ( , + i
= cos. 20=2 cos. 2 1
which gives
P2 /T)2 _ p2\
r ^ r
Now the equation to the ellipse, g and 6 being referred to its center, is
b 2
o nn? 
1 e 2 cos. 8 d
Therefore the trajectory is an ellipse the center of force being in its
center, and we have its semiaxes from
b 2 = D 2 P 2
c 2 a 2 b 2 2P Z D
C
a 2 a 2 P 2
viz.
b = V(D 2 P 2 )}
and V (3)
a = P J
which latter value was already assumed.
Tojind the Periodic time.
From (3) it appears that when
,
and substituting in (1) we have
L=X
P2
228 A COMMENTARY QN [SECT. Vlll
But
sm. l ( 1) = .
4 " 2 V gf*
and
T = 2ff , . . (4)
V gi*
which has already been found otherwise (see 147).
To apply (9) and (10) of 307 to this example we must first integrate
(11) where f $ = /tig; that is since
~ 2 * 2
we have
P2 YT2
P 2 =
V 8
But
V 2 = g^*(D 2 P 2 )
D 2  g s
which also indicates an ellipse referred to its center, the equation being
generally
2 _ a 2 b 2
 a 2 + b 2 g 2
Hence
g2 g2( D 2_g2)_p2(D2_ P 2)
p P 2 (D 2 P 2 )
. t _ i
the same as before.
With regard to 6 t the axes of the ellipse being known from (5) we have
the polar equation, viz.
,2 r .
"1 e 2 cos. 2 6
309. Ex. 2. Let F = 4 . Then (304)
s
s
V 2_?J
BOOK I.]
and
NEWTON S PRINCIPIA.
= 2ffA6 X ^~ S
V2*X^lI
^ DP
229
P and V belonging to an apse.
\
g* DP + P 2 )
D 8
which, adding and subtracting , transforms to
t V D r
A i D
and making g = u
t =
VD
(see 86).
Let t = 0, when
= P. Then
D
D
Also
J_ . r dt 
PVJT? 
D
But assuming
tlie above becomes rationalized, and we readily find
P3
230
A COMMENTARY ON
du
[SECT. VI11.
VP.(D P)
X < tan. ~ l
D
and making 6 = 0, when g = P, or when u = P   , we get
Hence, since moreover
Dx D
or
2 8
= sm.
= sin. (0 4. *2~] = cos 6
P 2
2 P . (D P)
D
2P>
!+(!_) cos..
. (2)
But the equation to the ellipse referred to its focus is
b 2 1
S= T x
a 1 + e cos.
b_ 2 P (D P)
a " D
and
e" =
D
BOOK I.] NEWTON S PRINCIPIA. 231
.b*_4P 4P*_4P
a" 2 "  TJ " " ~D^ ET* 
b 2 2
_
~a~ X D
and
(3)
b= VP x (D
To find the Periodic Time ; let 6 = *. Then g = 2 a P = JD P,
and equation (1) gives
T
tf IL ^2 2 /
see 159,
OTHERWISE.
First find the Trajectory by fonnula (11. 307) ; then substitute for
in 9 and 10, &c.
310. Required the Time and Trajectory when F= ^
By 304,
V 2=_g ;u ,x (D 2 P 2 )
O \ 9 *
, 2
~D 2 X f 2
.. if V and P belong to an apse we have
g tt D 2 P 2
= JJ2 X p 2
232 A COMMENTARY ON [SECT. VJH.
D
V L .
and taking t = at an apse or when g = P, C = 0,
V g/i
also
6 r dt D
X (C+ VP 8 %*)
= 0,
. . . . (1)
But

(P 2 f 2 ) ~ +P
and
X
11 _ , ^ (P 2 g 2 )+P
Tk 2 T>2\ * I
V (D s P 2 )
and making ^ = at the apse or where g = P,
C = l. =
 V D
,
e\/(D 2
wliich gives
pa
311. Required the Trajectory and circumstances of motion when
or for any inverse law of the distance.
The readiest method is this ; By (11) 307, if r, and P be the values of
g and p for the given velocity V (P is no longer an apsidal distance)
p2Af2 O,, r n 1 __ p n 1
v _, v 2 4 ^ v g n)
p 2 h (n l)r n  J g" 1
the equation to the Trajectory.
Also since
vdv = gFdf
I.]
NEWTON S PRINCIPIA.
233
Hence
and if we put
to
(n l)r
in which m may be > = or < 1 we easily get
D / m  PeV
~ V m 1
P
n
m= 1
/ ni
J r X
N m 1
n 1
2
P =
w (r=^n"~ e
To Jind 6 on this hypothesis.
We have (307)
. m< 1
n 1
which gives by substitution
n 5 ,
r P 2 d
n 3
(2)
m= 1
d
= +
m
^
X PX
n_3
a d
lm
the positive or negative sign being used according as the body ascends or
descends.
Ex. If n = 2, we get
. . . . m> 1
234
A COMMENTARY ON
[SECT. VIII.
P =
P =
m = 1
1 m
P.
the equations to the ellipse, parabola and hyperbola respectively.
Also we have correspondingly
I m l
= +r P.
m
m
which are easily integrated.
Ex. 2. Let n = 3. Then we get
Vm _,
5T3n>< p x
. . m > 1
P
P = T g
. . m = 1
. m < 1
cH =
Pr
mP 2
C V (r 2 P 2 )
+ / m v
 V 1 m X
mP
. m> 1
m= 1
. m< 1
312. In the first of these values of 6, m P 8 may be > = or < r 2 .
(1). Let m P 2 > r 2 . Then (see 86)
/ m / / m 1 / m 1 N
/ i n I / ^S \f t* I c**r* " I / Cfi/" "~* * 1* / ,
^V mP r zXj \ se W nTP r 2 ~ V mP rV
at an apse or when r = P
6 = + J m , X P X sec. 1 4 . . . (a)
>r m 1 i
BOOK I.]
for
NEWTON S PRINCIPIA.
235
/ m 1 1 1
/ , Ol* ~~*
V m P 2 r 2 P " r
(2) Let m P ~ r 2 . Then we have
r
V (m l) f
P = 
/ r
* ( + HT^l
(b)
a= +
V (m 1)
dj
1 X J 2 ~
j.
= i
V m 1
X (
>
X
e r
 V m _ 1
which indicates the Reciprocal or Hyperbolic Spiral.
(3) LetmP 2 be < r 2 . Then
/_L
V i
p =
ni
+ r m *"S
m i y
L
m r
^mP Xl V
at an apse r = P ; and then
6 = + .
+c
i^ir 2 mP) V(i 2 mP 2 )
^i 2 P) V(r z mP :
(e)
1.
V (r 8 g 8 ) r
Thus the first of the values of 6 has been split into three, and integrat
ing the other two we also get
Pr
tf a = +
= +
V (r s P 2 )
Pr
V(r e P !
s ; * (1 ~ 1 )
x 1.^
236
A COMMENTARY ON
[SECT: VIII,
m
^ 1
X
* mP g V(m.r 2 ^ 2 ) V (r 8 mP 2 )
and if rf is measured from an apse or r = P it reduces to
= + P /SLi.
N 1 m
313. Hence recapitulating we have these pairs of equations, viz.
or
=*v
_ x sec. l.
m 1 P
Jb construct the Trajectory,
put = 0, then
g = P= SA;
let f = CD, then
and
m
m
1
and taking A S B, A S B for these values of 0,
and S B, S B for those of p and drawing B Z,
B Z 7 at right angles we have two asymptotes ; S C being found by put
ting 6 = it. Thus and by the rules in (35, 36, 37, 38.) the curve may
be traced in all its ramifications.
2. p =
V (m 1)
//
V \ S
and
V (m 1)
X
BOOK I.]
NEWTON S PRINCIPIA.
237
This equation becomes more simple when
we make 6 originate from = oo ; for then
it is
1
V (m
and following the above hinted method the
curve, viz. the Reciprocal Spiral, may easily be B
described as in the annexed diagram.
I
. p = p /i X
^V 1 m
and
> = +rP /
*j f2
2 \/"nT(F 2 P 2 )
and when 6 is measured from an apse or when P = r
^Igg+rz mP) V(r 2 mP
V (r 2 mP 2 )~
Whence may easily be traced this figure.*
A j *
Z
From which may be described the Logarithmic Spiral.^
m
^=+rP /
 Vr 2

mP
xl
(m.r 8  2 ) V (r 2 m P )
238
or
= r
m
A COMMENTARY ON
i r V(g 2 r 2 )
[SECT. VIIF
i ^ / ,
V 1 m g
when P = r.
Whence this spiral.
These several spirals are called Cotes Spirals,
because he was the first to construct them as
Trajectories.
314. If n = 4. Then the Trajectory, &c.
are had by the following equations, viz.
p=
d 6 = r P
315. Ifn = 5. Then
p = P V in
d6 = r P
m
I m
m 1
X
!(,* 5_
V V m 1
V (m 1 .g 2 +r 4 )
m d
m
m 1 m
which as well as the former expression is not integrable by the usual
methods.
When
"1 T* O O .
m 1
is a perfect square, or when
m 1
m 2 P
m _ 1  4 (m 1)
then we have
Therefore (87)
" 2 (m 1)
/ m P *
m
1) , N2(m 1)
BOOK L]
NEWTON S PRINCIPIA.
239
V (2.m l.g 2 m P 2 )
V(mP 2 2.m l.
and these being constructed will be as subjoined.
316. COR. 1. OTHERWISE.
To find tlie apses of an orbit where F = ^ .
Let
Then
P = f
S
m
n 1
f n 3
~
,P>!>+
 ;
m 1
= m > 1
, m = 1
and
nl +
. n ]
. . . m < 1
1 ni l_m
which being resolved all the possible values off will be discovered in each
case, and thence by substituting in 6, we get the position as well as the
number of apses.
Ex. 1. Let n = 2. Then
,* + .JL mpt o
r m 1* m 1
L
PJ _ 4" r J^
r 4
mP !
r 1
240 A COMMENTARY ON [SECT. VIII.
which give
r r 2 + 4m P g .(m 1)
? g( m _ 1)~ 4(m I) 2
^L
S = : 4
and
r /r 2 4 m P 2 .(l m)
K 2 (1 m) V 4.(1 m) 2
Whence in the ellipse and hyperbola there are two apses (force in the
focus) ; in the former lying on different sides of the focus ; in the latter
both lying on the same side of the focus, as is seen by substituting the
values of g in those of 6. Also there is but one in the parabola.
Ex. 2. Let n = 3. Then eq. (A) become
2 _ m P 2 + r 2
which indicate two apses in the same straight line, and on different sides
of the center, whose position will be given by hence finding 6 ;
2
r o
(2) S = = <
pi
because r is > P,
whence there is no apse.
r 2 mP 2
(3) g =  l __ m
which gives two apses, r 2 being > m P 2 because m is < 1 and P < r ;
and their position is found from 6.
317. COR. 2. This is done also by the equation
P
p = g. sin. <p, or sin. <f> = ~
<p being the L. required.
Ex. When n = 3, and m = 1, we have (4. 313)
P
p = T e
p
.. sin. <p = y
.. v is constant, a known property of the logarithmic spiral.
318. To find when the body reaches the center of force.
Put in the equations to the Trajectory involving p, g ; or g, 6
Ex. 1. When n = 3, in all the five cases it is found that
p =
BocfK I.] NEWTON S PRINCIPLE. 241
and
6 =r x .
Ex. 2. When n = 5 in the particular case of 315, the former value of
6 becomes impossible, being the logarithm of a negative quantity, and the
latter is = co .
319. Tojind when the Trajectory has an asymptotic circle.
If at an apse for & = cc the velocity be the same as that in a circle at
the same distance (166), or if when
6 co
and
P = f
we also have
p dp
then it is clear there is an asymptotic circle.
Examples are in hypothesis of 315.
320. Tojind the number of revolutions from an apse to = co .
Let & be the value of d a when g = p or at an apse, and (/ when
f = co . Then
6"
= the number of revolutions required.
2 >,
Ex. By 313, we have
/
1
= P sec.
>r m
m ft
.*. v =
 .
m 1
321. COR. 3. First let V R S be an hyperbola whose equation, x being
measured from C, is
Then
VCR = y^ X
But
/ydx = ^/dx V
a a J V (x * a 2 )
VOL. I. Q,
242 A COMMENTARY ON [SECT. VIII.
b ,/o ox b ,, , ,, ,x b /* a 2 dx
=xWx 2 a 8 )  /dxWx 2 a 2 ) / r  ^
a a 17 a* 7 V(x 2 a 2 )
.*. 2/y d x = xV(x 2 a ^) abl. X +V ( x2 ~ a *)
a a
and
VCR=^l. X+ *( *> .... (1)
Again
g=CP=CT=x subtangent
= x
dy
x 2 a 2 _ a^
x " x
and substituting for x in (1) we have
VCR = ~.l.
2 a
+ V ^~ e} .... (2)
N being a constant quantity.
322. Hence conversely
and differentiating (17) we get
x ^2. _L\
N 2 V u aV
dd 2 ~ a 2 b 2
and again differentiating (d 6 being constant)
dT 2 = a 2 b 2 N 8 X ]
Hence (139)
P*V 2 / 4
g
322. By the text it would appear that the body must proceed from V
in a direction perpendicular to C V i. e. that V is an apse.
From (1) 322, we easily get
BOOK L] NEWTON S PRINCIPIA. 243
and since generally
de 2
4
a b N  
P 2 =    .... (i)
which is another equation to the trajectory involving the perpendicular
upon the tangent.
Now at an apse
P = g
and substituting in equation (1) we get easily
g = a
which shows V to be an apse.
*._.,
OTHERWISE.
Put d g = 0, for g is then = max. or min.
324. With a proper velocity. ~]
The velocity with which the body must be projected from V is found
from
vdvrr gFdf.
325. Descend to the center}. When
s = 0, p = (1. 323) and = oo (2. 321).
326. Secondly, let V R S be an ellipse, whose equation referred to the
center C is
Then
and as above, integrating by parts,
x v (ti z v ^ a z A v
/dxV(ax)=i*Jl J4i "
^ V (a 2 x 2 )
Q2
A COMMENTARY ON [SECT. VIII.
x y ( a a _x 2 ) a]_ / . , x r
o o
A
Also
a 8 x ;
~~ X "*" x
and
rr Sin.~*
a / w 2 \ 20
 =. sin ( rxf] = cos. , XT
j \2 ablN/ abJN
and
2 tf ..... (2)
Conversely by the expression for F in 139, we have
Foe 1
327. Tojind when the body is at an apse, either proceed as in 323,
or put
d x . sin. x
By (27) d . sec. x =
sin. 6
cos. 2 6
or
=
6=
that is the point V is an apse.
328. The proper velocity of projection is easily found as indicated
in 324.
329. Will ascend perpetually and go ojfto infinity. }
From (2) 327, we learn that when
2 6 *
a~FN " 2
g is ce>;
also that g can never = 0.
BOOK L] NEWTON S PR1NCIPIA. 245
330. When the force is changed from centripetal to centrifugal, the
sign of its expression (139) must be changed.
331. PROP. XLII. The preceding comments together with the Jesuits
notes will render this proposition easily intelligible.
The expression (139)
F _P 2 V* dp
L XN q 1
g P d
or rather (307)
pz y 2
in which P and V are given, will lead to a more direct and convenient
resolution of the problem.
It must, however, be remarked, that the difference between the first
part of Prop. XLI. and this, is that the force itself is given in the former
and only the law of force in the latter. That is, if for instance F = p n  *,
in the former /^ is given, in the latter not. But since V is given in the
latter, we have //. from 304.
SECTION IX.
332. PROP. XLIII. To mafce a body move in an oibit revolving about
the center of force, in the same way as in the same orbit quiescenf]
that is, To adjust the angular velocity of the orbit, and centripetal force
so that the body may be at any time at the same point in the revolving
orbit as in the orbit at rest, and tend to the same center.
That it may tend to the same center (see Prop. II), the area of the new
orbit in a fixed plane (V C p) must a time a area in the given orbit
(V C P); and since these areas begin together their increments must also
be proportional, that is (see fig. next Prop.)
CPxKRocCpxkr
But
KR = CK x ^KCP
kr=Ckxz.kCp
and C P = C p, and C K = C k
.. L K C P a k C p
and the angles V C P, V C p begin together
/.^.VCP a /LVCp.
Q3
A COMMENTARY ON [SECT. IX.
Hence in order that the centripetal force in the new orbit may tend to
C, it is necessary that
. V C p a ,L V C P.
Again, taking always
CP= Cp
and
VCp: VCP:: G: F
G : F being an invariable ratio, the equation to the locus of p or the orbit
in fixed space can be determined; and thence (by 137, 139, or by Cor.
1, 2, 3 of Prop. VI) may be found the centripetal force in that locus.
333. Tojind the orbit infixed space or the locus of p.
Let the equation to the given orbit V C P be
where = C P, and 6 = V C P, and f means any function; then that of
the locus is
which will give the orbit required.
OTHERWISE.
Let p be the perpendicular upon the tangent in the given orbit, and p
that in the locus ; then it is easily got by drawing the incremental figures
and similar triangles (see fig. Prop. XLIV) that
K R : k r : : F : G
k r : pr :: p : V (f 2 p 2 )
pr :PR:: 1 : 1
PR :KR:: V ( z p 2 ) : p
whence
1 : 1 : : F . p V (^ p 2 ) : G p V ( s 9 p s )
and
" F 2 g 2 + (G 2 F 2 )p /2
334. Ex. 1. Lei the given Trajectory be the ellipse with the force in
its focus; then
K2 a M P 2 1
p> 2 = JLi, and g = * ^ ecos ,
and therefore
b 2 G 2 (2ag)g 2
/ " b 2 (G 8 F 2 ) H F 2 (2ag g 2 )
BOOK I.] NEWTON S PRINCJPIA. 24,1
and
 a.(Ie 2 )
/ F
1 + e co
Hence since the force is ( 139)
and here we have
a(l e *) u r= 1 J e cos.
Z
2F 2 F 2

and again differentiating, &c. we have
d 2 u F 2 G 2 F 2
d~^ H = Ga(l e*) H ~G^~
But if 2 R = latusrectum we have
/. the force in the new orbit is
p V 2 ( F 2 R G R F 2
X 1 ~T2 T ~5
gRG 2 ^ s g j
335, Ex. 2. Generally let the equations to the given trajectory be
g = f (0 )~)
and
Then since
d e u F 2 d u
d fl 2 H = G 2 d^ J H
F 8 v /d 2 u . N F 2
= G~ 2 X VdT 2 + u ) + u ~ G^ u
and if the centripetal forces in the given trajectory and locus be named
X, X , by 139 we have
gX . FJ gX G 2 F v 1
p? yi  G t A p ,j y /t 1 ^ x yy
Qi
248 A COMMENTARY ON [SECT. IX.
_ p*y* , F 2 X G 2 F 2 _1
/4 X
Also from (2. 333) we have
JL  L s _I_ G 8 F _i
p 2 G 2 X p 2 ~ G 2 X ? 2
dp Fj dp / G 2 F * 1
p 3 d f ~G 2X p 3 d f 4 ~^ X "p"
.. by 139
gX .._F gX GF 1
p 2 y 2 ~ p/ 2 y/ 2 "t (j 2 "P
the same as before.
This second general example includes the first, as well as Prop. XLIV,
&c. of the text.
836. Another determination of the force tending to C and which shall
make the body describe the locus of p.
First, as before, we must show that in order to make the force X tend
to C, the ratio L. V C P : L. V C p must be constant or = F : G.
Next, since they begin together the corresponding angular velocities
u, u f of C P, C p are in th^t same ratio ; i. e.
: : : F : G.
Now in order to exactly counteract the centrifugal force which arises
from the angular motion of the orbit, we must add the same quantity to
the centripetal force. Hence if p, p denote the centrifugal forces in the
given orbit and the locus, we have
X = X +. p p
X being the force in the given orbit.
But (210)
P 2 V 1
P =  x 3
g ?
and
a w 2
when o is given.
a/ ! G 2 P 2 V G 2 1
<?/ 0; v _ ffi V  __ V V
a 2 ~ * F 1 ~ g X F f j 8
p*Y 2 G 2 F 2 1
(D
v
BOOK I.] NEWTON S PRINCIPIA. 2t9
or
pz y 2
g
or
P 2 V 2 / dp , G 2 F 2 X ._.
= T~~ * V dl + FV ) (8)
O ib b
337. PROP. XLIV. Take u p, u k similar and equal to V P and V K ;
also
m r : k r : : . V C p : V C P.
Then since always C P = p c, we have
p r = P R.
Resolve the motions P K, p k into P R, R K and p r, r k. Then
and therefore when the centripetal forces PR, p r are equal, the body
would be at m. But if
P Cn:pCk::VCp:VCP
and
C n = C k
the body will really be in n.
Hence the difference of the forces is
m k X m s (m r k r) . (m r + k r)
m n = = * * ~ .
m t m t
But since the triangles p C k, p C n are given,
1
K r a m r a
Cp
1 1
.. m n cc = , X  .
C p 2 m t
Again since
p C k : p C n : : P C K : p C n : : V C P : V C p
: : k r : m r by construction
: : p C k : p C m ultimately
. . p C n = p C m
and m n ultimately passes through the center. Consequently
m t = 2 C p ultimately
and
1
in n a ^ 3
Cp
250 A COMMENTARY ON [SECT IX.
OTHERWISE.
338. By 336,
X X = / p
p 2 y 2 G 2 F 2 1
v v
g F 2 3
ex  .
S 3
339. To trace the variations of sign qfmn. .
If the orbit move in consequentia, that is in the same direction as C P,
the new centrifugal force would throw the body farther from the center,
that is
Cmis>CnorCk
or m n is positive.
Again, when the orbit is projected in antecedentia with a velocity <
than twice that of C P, the velocity of C p is less than that of C P.
Therefore
C m is < C n
or m n is negative.
Again, when the orbit is projected in antecedentia with a velocity =
twice that of C P, the angular velocity of the orbit just counteracts the
velocity of C P, and
mn = 0.
And finally, when the orbit is projected in antecedentia with a velocity
> 2 vel. of C P, the velocity of C p is > vel. of C P or C m is > C n, or
m n is positive.
OTHERWISE.
By 338,
m n oc <p p
oc u * 2
But
= 01 + W
W being the angular velocity of the orbit.
.. m n oc + 2 uW+ W 2
+ 2 + W
j or being used according as W is in consequentia or antecedentia.
BOOK I.]
NEWTON S PRINCIPIA.
251
Hence m n is positive or negative according as W is positive, and nega
tive and > 2 ; or negative and < 2 u. That is, &c. &c.
Also when W is negative and = 2 <w, m = 0. Therefore, &c.
340. COR. 1. Let D be the difference of the forces in the orbit and in
the locus, and f the force in the circle K R, we have
D: f : : m n : z r
.ink X m s . r k 2
m t 2~k~c
(m r + r k) (m r r k) r k *
2 k c 2kc
::mr 2 rk 2 :rk 2
:: G 2 F 2 : F 2 .
341. COR. 2. In the ellipse with the force in the focus, we have
F 2 R G 2 R F 2
x/a i 2 +  ^ 
For (C V being put = T)
v 2 y 2
Force at V in Ellipse : Do. in circle : : = jwrr TVTT/
chord P V : P V
1 1
Also F in Circle : m n at V
m n at V : m n at p :
.*. F at V in ellipse : m n at p ;
Hence
2 R 2 T
T: R
F 2 : G 2 F 2
J_ JL
T 3 A *
TF 2 RG 2 RF 2
we have
x F2
x 
F in ellipse at V = ~^
and
RG 2
m n =
and
X = X + m n
F 2 RG 2 RF
see 834.
252 A COMMENTARY ON [SECT. IX.
OTHERWISE.
342. By 336,
P 2 v 2 n 2
But
X  ^
2
and
P 2 V 2 L
 = ^ ft = R p (157)
g 2
p r F 2 G 2 F 2 1
= F 2 x i 7^ + ~p J
343. COR. 3. /w the ellipse with the force in the center.
X FZ A . R G 2 R F 2
T 3 A 3
v 2
For here X a A and the force generally oc ^^ (140)
/Force in ellipse at V : Force in circle at V : : T : R
J F in circle : m n at V : : F 2 : G z F 2
(.m n at V : m n at p : : 7 ^r 3 : r 3
1 A
T? I. /W F 2 ^ RG 2 RF Z
.. F in ellipse at V : m n at p : : 7^3 . T :
7^3 .  T
1 A
F 2 A
assuming F in ellipse at P = .3 , we have
and
F 2
F in ellipse at V = =r 3 x T
RG 2 R F 2
.. m n =  A3
.. X 7 a X + m n a , &c.
OTHERWISE.
. P 2 V 2 4 (Area of Ellipse)
344. X = p P, and = T^ . i 
g g( Period) 2
_ 4?r 8 a 2 b 2 _ 2 ,
g( Period) 2
BOOK I.] NEWTON S PRINCIPIA. 253
Therefore by 886
ft2 p 2 1
X ^ + ^a b X^i X
^a 3 fF 2 g , b 2 x (G 2 F 2 ))
F 2 \ a 3 ag 3 /
RG 2 RF*1
S * J
a
845. COR. 4. Generally let X &? /Ac >rce <tf P, V ~ at V, R the
radius of curvature in V, C V = T, &c.
V R ft 2 V R F 2
X a X 4 AS
A 3
For
f F in orbit at V : F in circle at V : : T : R
jl* : m n at V ::F 2 :G 2 F 2
Im n at V : m n : : A 3 : T 3
V TT 2 ft 2 TT *
.. F in orbit at V : m n : : 8 : V R . !2^l
. . since by the assumption
F in orbit at V =
T 2
VR(G 2 F 8 )
A 3
and
OTHERWISE.
This may better be done after 336, where it must be observed V is not
the same as the indeterminate quantity V in this corollary.
346. COR. 5. The equation to the new orbit is (333)
2 _
1
G*p /2 g 2
""
g 2 + (G" 2 " F 2 )p /8
p belonging to the given orbit.
Ex. 1. Let the given orbit be a common parabola.
Then
s _ G 2 rg s
: F 2 g + (G 2 F 8 )r
and the new force is obtained from 836.
254 A COMMENTARY ON [SECT. IX.
Ex. 2. Let the given orbit be any one of Cotes Spirals, whose general
equation is
D"  
 2
Then the equation of 333 becomes
G 2
b 2 ? 2
u
which being of the same form as the former shows the locus to be similar
in each case to the given spiral.
This is also evident from the law of force being in each case the same
(see 336) viz.
/  U
~~
~~ ~ " ~ /~*
f g
Ex. 3. If the given orbit be a circle, the new one is also.
Ex. 4. Let the given trajectory be a straight line.
Here p is constant. Therefore
2 _ G 2 p /2 x g g
P T? 2 n>2 __ T2
the equation to the elliptic spiral, &c. &c.
Ex. 5. Let the given orbit be a circle with the force in its circumference.
Here
2/^2 2\
P (4 1 " P )
P 2 = 47*
and we have from 333
" 4r 2 F 2 + (G 2 F 2 )g 2
Ex. 6. Let the given orbit be an ellipse with force in the focus.
Here
2a g
and this gives
P 8 = F * g (2 a ? ) + b s (G J Y
BOOK I.] NEWTON S PRINCIPIA. 255
347. To find the points of contrary Jlexure, in the locus put
dp = 0;
and this gives in the case of the ellipse
b 2 F 2 G*
OTHERWISE.
In passing from convex to concave towards the center, the force in the
locus must have changed signs. That is, at the point of contrary flexure,
the force equals nothing or in this same case
F 2 A + RG 2 RF 2 =
.. A = S s x (F 8 G 2 )
 k! F 2 G 3
: T* F 1
And generally by (336) we have in the case of a contrary flexure
which will give all the points of that nature in the locus.
348. To Jind the points where the locus and given Trajectory intersect
one another.
It is clear that at such points
g = g , and tf = 2 W + 6
W being any integer whatever. But
f = ~ 6 = m 6
2 W*
= "nT+T
This is independent of either the Trajectory or Locus.
349. Tojtnd the number of such intersections during an entire revolution
ofCP.
Since 6 cannot be > 2 *
W is < m + 1 and also < m 1
.. 2 W is < 2 m.
Or the number required is the greatest integer in 2 m or  .
F
This is also independent of either Trajectory or Locus.
25G
A COMMENTARY ON
[SECT. IX.
350. Tojlnd the number and position of the double points of the Locus,
i. e. of those points where it cuts or touches itself.
Having obtained the equation to the Locus find its singular points
whether double, triple, &c. by the usual methods ; or more simply,
consider the double points which are owing to apses and pairs of equal
values of C P, one on one side of C V and the other on the other, thus :
The given Trajectory V W being V
symmetrical on either side of V W, let
W 7 be the point in the locus correspond
ing to W. Join C W 7 and produce
it indefinitely both ways. Then it is
clear that W is an apse; also that the
angle subtended by V v x 7 W 7 is
/~i
= r X it = \v r
L. V C y , w being
f~\
the greatest whole number in ^
I.
supposes the motion to be in consequentia). Hence it appears that where
ever the Locus cuts the line C W 7 there is a double point or an apse, and
also that there are w + 1 such points.
s~*
Ex. 1. Let T=T = 2 ; i. e. let the orbit move in conse
b
quentia with a velocity = the velocity of C P. Then L.
V C y 7 = 0, y 7 coincides with V, and the double points
are y 7 V, x 7 and W 7 .
The course of the Locus is indicated by the order of
the figures 1, 2, 3, 4.
Ex. 2. Let p = 3.
Then the Locus resembles this figure, 1, 2, 3,
4, 5, 6. showing the course of the curve in which
V, x 7 , A, W 7 are double points and also apses.
/
Ex. 3. Let ^ = 4.
Then this figure sufficiently traces the Locus.
Its five double points, viz. V, x 7 , A, B, W 7 are
also apses.
G
Higher integer values of p will give the Locus
BOOK I.]
NEWTON S PRJNCIPIA.
257
still more complicated. If ^ be not integer, the
figure will be as in the first of this article, the
double points, lying out of the line C V. More
s1
over if ^ be less than 1, or if the orbit move in
r
antecedentia this method must be somewhat
varied, but not greatly. These and other curio
sities hence deducible, we leave to the student.
351. To investigate the motion of (p) when the
ellipse, the force being in the focus, moves in ante
cedentia with a velocity = velocity of C P in
consequentia.
Since in this case
G =
.. (333) also
p =
ov the Locus is the straiht line C V.
Also (342)
/F 2
^
= it x
eR
Hence
i Y/ i
v d v ex X d oc
Rd
. . V z OC
R
.
1 a
1 e 2
, , axis major , , , , .
(where  ^ = 1 ;) and the body stops when
or when
g  1 e.
Hence then the body moves in a straight line C V, the force increasing
3
to of the latusrectum from the center, when it = max. Then it
4
decreases until the distance = or R. Here the centrifugal force pre
vails, but the velocity being then = max. the body goes forward till the
VOL. I. R
258 A COMMENTARY ON [SECT. IX.
distance = the least distance when v = 0, and afterwards it is repelled
and so on in infinitum. .
352. Tojind "when the velocity in the Locus = max. or min.
Since in either case %
d.v 2 = 2vdv =
and
v d v = X d f
.. X =
.. (336)
pz v 2 G 2 _ F 2 1
X+^ x UpA x = o
Ex. In the ellipse with the force in the focus, we have (342)
v <" f FZ j_ RG R F S
= F^p + p y
F 2 R G 2 R F 2
o
S
u
.. = R x
F
b 2 F 2 G 2
: a X ~ F 2
b 2 L
If G = 0, v = max. when g = , or when P is at the extre
a &
mity of the latusrectum.
If F = 2 G, v = max. when e = R . ~^ = R =  
4 \j ~ 4 o
lat. rectum.
353. To find when the force X in the Locus = max, or min.
Put d X = 0, which gives (see 336)
3 p 2 v* G 2 F 2 1
d X = r X FZ X p
Ex. In the ellipse
~ T 2
and (157)
pa v 2
r v , . R
p" lit
g
2 F 2 d g 3 R G 2 d g 3 RF 2 dg _
which gives
3 R F 2 G 2
Q , y __^ , .
C r / ^ 1^ O
BOOK I.] NEWTON S PRINCIPIA. 259
Hence when
G =
X = max. when = ~ .
&
When g = R, and G = 0. Then
Y F 2 RF 2
X = R 2 ~ RT =
When F = 2 G, or the ellipse moves in consequentia with the velo
city of C p ; then
X = max. when
3 J^ 4G 2 G 2 j)
2 4, G 2 : 8
354. COR. 6. Since the given trajectory is a straight line and the center
offeree C not in it, this force cannot act at all upon the body, or (336)
X = 0.
Hence in this case
x/ _ P 2 V 2 v G 2 F 2 1
~F^ 73
where P = C V and V the given uniform velocity along V P.
In this case the Locus is found as in 346.
355. If the given Trajectory is a circle, it is clear that the Locus of p
is likewise a circle, the radiusvector being in both cases invariable.
356. PHOP. XLV. The orbits (round the same center of force) acquire
the same form, if the centripetal forces by which they are described at equal
altitudes be rendered proportional. ]
Let f and f be two forces, then if at all equal altitudes
f a f
the orbits are of the same form.
For (46)
dt 2 dt 2 S P 2 x QT
1 1
a
or
QT* SP 2 x
1
1
d 6 2 d V ~
and
d C a d V.
R 2
260 A COMMENTARY ON [SECT. IX.
But they begin together and therefore
6 a (f
and
p f/.
Hence it is clear the orbits have the same form, and hence is also sug
gested the necessity for making the angles 0, 6 proportional.
rfi
Hence then X , and X being given, we can find ^ such as shall ren
der the Trajectory traced by p, very nearly a circle. This is done ap
proximately by considering the given fixed orbit nearly a circle, and
equating as in 336.
357. Ex. 1. To Jind the angle between the apsides when X is constant.
In this case (342)
X a 1 a ^ a jt j^ .
Now making = T x, where x is indefinitely diminishable, and
equating, we have
(T x) 3 = F 2 T F 2 x + RG 2 RF 2
= T 3 3T 2 x + 3Tx 2 x 3
and equating homologous terms (6)
T 3 F 2 T+RG 2 RF 2 = F 2 x (T R) + RG 2
and
F 2 = 3T 2
G_ 2 T 3 T R
* F 2 ~ R F 2 R
T 3 _J_ Jl
= 3 R T 2 R
T T R _ 3 R 2T
~ 3 R ~ R 3 R
= nearly
9
since R is = T nearly.
Hence when F = 180 = it
the angle between the apsides of the Locus in which the force is constant.
358. Ex. 2. Let X a g n ~ 3 . Then as before
(T x) n = F 2 (T x) + RG 2 RF 2
and expanding and equating homologous terms
T n = F 2 T + RG 2 RF 2
BOOK I.] NEWTON S PRINCIPIA.
and
But since T nearly = R
T n_l = G 2
.*. !
* F 2 ~ n
and when F = <s
T
y (JT i .
V n
Thus when n 3 = 1, we have
261
When n 3 = 1, n = 2, and y = ^ = 127. 16 . 45".
When n 3 = ^ , n = ] , and 7 = 2 ?r = 360.
4 4
359. Let X oc
l> n m _j_ f, n
Pg Cg
. Then
b.(T x) m + c(T x) n = F 2 .(T x)+ R.(G 2 F 2 )
and expanding and equating homologous terms we get
bT m + cT n = F 2 (T R) + RG 2
and
bm T 111  1 ^ en T 11  1 = F 2 .
But R being nearly = T, we have
bT mi4. cT ni _ G 2
G 2 bT m  1 + cT n ~ 1 b T m + c T n
F 2 " bm T m  1 + cnT n ~ 1 == mbT m + ncT n
which is more simply expressed by putting T = 1. Then we have
G* b+ c
F 2 ~ mb + nc
and when F = it
b + c
360. COR. 1. Given the L. between the apsides to Jind the force.
Let n : m : : 360 : 2 7
: : 180 K \ y
m
.*. y = r
n
ButifX oc e p
y *~~
113
262 A COMMENTARY ON [SECT. IX
n 1
p = t
. . X 7 gib" 3
Ex. 1. If n : m : : 1 : 1,
X oc L
as in the ellipse about the focus.
2. If n : m : : 363 : 360
3. Ifn : m : : 1 : 2
1
X
And so on.
IL
4
Again if X _ _

and the body having reached one apse can never reach another.
IfX oc
+ q
.. the body never reaches another apse, and since the centrifugal force
 , if the body depart from an apse and centrifugal force be > centri
petal force, then centrifugal is always > centripetal force and the body
will continue to ascend in infinitum.
Again if at an apse the centrifugal be < the centripetal force, the centri
fugal is always < centripetal force and the body will descend to the center.
The same is true if X a and in all these cases, if
centrifugal = centripetal
the body describes a circle.
361. COR. 2. First let us compare the force ^ c A, belonging to
the moon s orbit, with
Fo T> /"I o "D TJ^ 2
rv \jf ri J;
A? + A 3
Since the moon s apse proceeds, (n m) is positive.
BOOK I.] NEWTON S P1UNCIP1A. 263
.. c A does not correspond to n m and . . ^ does not correspond
F_*
Now
j^ A c A 4 a b A m c A p
A 2 ~ C "A 3 A 3
l*c Ft
.. X oc A i a A 02
1 _ 4 c _ F_ 2
* l 2 = G 2
F2 RG 2 RF 2 1 4c , 3cR
+
A 3 A 3
1 4 c , 1
and
3 c R
mn = ~A^~
Hence also
y =T / _ . . &C. &C. &C.
*V 1 4 c
362. To determine the angle between the apsides generally.
Let
f (A) meaning any function whatever of A. Then for Trajectories which
are nearly circular, put
f(A) F 2 A + R.(G 2 F 2 )
IT A 3
... f. A = F 2 A + R(G 8 F 2 )
or
f.(T x) = F 2 (T x) + R(G 2 F 2 )
But expanding f (T x) by Maclaurin s Theorem (32)
u = f (T x) =U U x + U"^ 2 &c.
t J, U &c. being the values of u, T , T &c.
(1 X. Cl. X
when x = 0, and therefore independent of x. Hence comparing
homologous terms (6) we have
U = F 2 T+R(G 2 F z )
U = F 2
R4
264
A COMMENTARY ON
Also since R = T nearly
U = TG 2
1 U
F ~~ T . U 7
Hence when F =r v, the angle between the apsides is
or
N U
making T = 1.
Ex. 1. Let f (A) = b A m + c A n = u
Then
du
dx
= mbA m  1 +ncA n  1 .
Hence since A = T when x =
U = fT = b T ra + c T n
U = mbT" 1  1 + n c T 11  1
G 2 b T m f c T n
F 2 " mbT
or
G_ 8 b+ c
F 2  m b + n c
and
7 =
b + c
m b + n c
[SECT. IX.
as in 359.
Ex. 2. Let f . (A) = b A m + c A n + e A r + &c.
j^ = mbA m  1 + ncA n  1 + reA r  1
.. U = bT m cTeT r &c.
and
T X U = m b T m + n c T " + r e T r + &c.
 b Tm + c T " + e Tr + &c 
F 2 mbT
or
when T = 1.
Also
&c.
7 =
b + c + e
m b f n c f r e + . .
(1)
&c.
BOOK I.] NEWTON S PRINCIPIA.
Ex. 3. Let *4* = a A = u.
A.
Here (17)
du
j^ = A 2 a A x(3 + Ala)
Hence
U = Ta T x (3 f Tla)
T X U = T 3 a T (3 + Tla)
G 2 1
F 1 =: T X (3 + T 1 a)
and when T = 1
G 2 1
265
F 2 ~ 3 +la
.*, <y sr r / _
V3 + la
Hence if a = e the hyperbolic base, since 1 e = 1, we have
Ex. 4. Let f (A) = e A = u.
Then
du
j~ e
d x
.. U = e T
and
T . U = T e T
.*. 7 = T.
Ex. 5. Let ti^i = sin. A.
u = f(A) = A 3 sin. A
.. U = T 3 sln. T
and
^ = 3A 8 sin. A + A 3 cos. A
.. T U = 3 T 3 sin. T + T 4 cos. T
. G_ 2 _ _ sin. T
F 2 ~3sin.T+ t cos. T
sin. T
sin.T + Tcos.T*
266 A COMMENTARY ON [SECT. IX.
" 4"
ry
363. To prove that
bA m +cA n _ 1 mb + nc_ 3
~K~ 3 ~b~+~c
= b + c (mb + nc)x+ &c,
1 /, mb+nc
= f i I 1 CT x + &c
b + c v b + c
1 mb + n c
~b + c v
1 mb + nc
= b + c*
364. To Jind the apsides when the eccentricity is infinitely great.
Make
2 q : V (n + 1) : *. velocity in the curve : velocity in the circle of the
same distance a.
Then (306) it easily appears that when F g n
n + 3
!
g V (a n + 1 f n + 1 )g 2 q 2 a n + 1 (a 2
and
gives the equation to the apsides, viz.
(a + 1 g n + l )g z q a"* 1 (a 2 ? 2 ) =
whose roots are
a (and a when n is odd) and a positive and negative quantity (and when
n is odd another negative quantity).
Now when q =
two of whose roots are 0, 0, and the roots abovementioned consequently
arise from q, which will be very small when q is.
Again since
when q and are both very small
BOOK I.] NEWTON S PIUNCIPIA. 267
and
s = q
.. the lower apsidal distance is a q.
A nearer approximation is
g= + . 1SL_.
Hence
n + 3
g v/(^_a 2 q 2 + /3) X Q
where /S contains q 4 &c. &c., and this must be integrated from g = b to
g = a (b = a q).
But since in the variation off from b to c, Q may be considered con
stant, we get
6 = sec.  . J + C = sec.  . .
aq a q
and
if 3 it 5 it a , .
7  Q g > g > &c  ultimately
the apsidal distances required.
Next let
Then again, make
v : v in a circle of the same distance : : q V 2 : V (n 1)
and we get (306)
and for the apsidal distances
which gives (n > 1 and < 3)
2
fsaqfrr
Hence
*=/=
a tj f n 3n
a q d
2 a 3  " + /?) x Q
V Q J ^ */ t n 3 n r"g~3^TiT
268 A COMMENTARY ON SECT. IX.
and
3 n
g 3
7 3 n 3 ~" 3 n 3 n
qa 2
Hence, the orbit being indefinitely excentric, when
F oc g . ... we have . . . . *y = ^
for
1 T
Foe
any number < 1 2
Foe . . . 7 =
r oc y ^
g number between 1 and 2 2
FpWs ?>*
But by the principles of this 9th Section when the excentricity is inde
finitely small, and F a n
y = V (n + 3)
(see 358), and when
1
V (3 n)
Wherefore when n is > 1
7 increases as the excentricity from
V (3 + n) t0 2
When F oc g
7 is the same for all excentricities.
When F a g 1 "?
7 decreases as the excentricity increases from
tol
n) 2
which is also true for F oc .
S
BOOK i.j NEWTON S PRINCIPIA. 269
decreases as the excentricity increases from
T
; tO
V(3 n) 3 n
When F oc L
When F oc
2 <3
7 increases with the excentricity from
to
V(3_n) 3 n*
If the above concise view of the method of rinding the apsides in this
particular case, the opposite of the one in the text, should prove obscure ;
the student is referred to the original paper from which it is drawn, viz.
a very able one in the Cambridge Philosophical Transactions, Vol. I,
Part I, p. 179, by Mr. Whewell.
365. We shall terminate our remarks upon this Section by a brief dis
cussion of the general apsidal equations, or rather a recapitulation of re
sults the details being developed in Leybourne s Mathematical Repository,
by Mr. Dawson of Sedburgh.
It will have been seen that the equation of the apsides is of the form
x n Ax m B = (1)
the equation of Limits to which is (see Wood s Algeb.)
nx ni mAx m  1 = (2)
and gives
/" m A \ n
x = ( A)
V n /
i
m
If n and m are even and A positive, * has two values, and the number
of real roots cannot exceed 4 in that case.
Multiply (1) by n and (2) by x and then we have
(m ^ n) A x m nB =
which gives
B v "in"
and this will give two other limits if A, B be positive and m even ; and if
(1) have two real roots they must each = x.
270 A COMMENTARY ON [SECT. X.
If m, n be even and B, A positive, there will be two pairs of equal roots.
Make them so and we get
n _ /nx
\ni/
n nm
which will give the number of real roots.
(1). If n be even and B positive there are two real roots.
(2). If n be even, m odd, and B negative and (M), the coefficient to
A n , negative, there are two ; otherwise none.
(3). If n, m, be even, A, B, negative, there are no real roots.
(4). If m, n be even, B negative, and A positive, and (M) positive there
are four real roots ; otherwise none.
(5). If m, n be odd, and (M) positive there will be three or one real.
(6). If m be even, n odd, and A, B have the same sign, there will be
but one.
(7). If m be even, n odd, and A, B have different signs, and M s sign
differs from B s, there will be three or only one.
(8). If
x n _j_ An m B =
then
is positive, and it must be > B, and the whole must be positive.
If
x n Ax m + B =
the result is negative.
SECTION X.
366. PROP. XLVI. The shortest line that can be drawn to a plane
from a given point is the perpendicular let fall upon it. For since
Q C S = right L^ any line Q S which subtends it must be > than either
of the others in the same triangle, or S C is < than any other S C.
A familiar application of this proposition is this :
367. Let S Q be a sling with a body Q at the end of it, and by the hand
S let it be whirled so as to describe a right cone whose altitude is S C, and
base the circle whose radius is Q C; required the time of a revolution.
Let S C = h, S Q = 1, Q C = r = VI 2 h 2 .
BOOKI.] NEWTON S PRINCIPIA. 271
Then if F denote the resolved part of the tension S Q in the direction
Q C, or that part which would cause the body to describe the circle P Q,
and gravity be denoted by 1, we have
F : 1 : : r : h
But by 134, or Prop. IV,
p x p _ ^^ * \ A. v p 2
J. /\ JL i S\ A
g h
A**.7/i (1)
<\ or
the time required.
If the time of revolution (P) be observed, then h may be hence obtained.
If a body were to revolve round a circle in a paraboloidal surface, whose
axis is vertical, then the reaction of the surface in the direction of the
normal will correspond to the tension of the string, and the subnormal,
which is constant, will represent h. Consequently the times of all such
revolutions is constant for every such circle.
368. PROP. XLVII. When the excentricity of the ellipse is indefi
nitely diminished it becomes a straight line in the limit, &c. &c. &c.
369. SCHOLIUM. In these cases it is sufficient to consider the motion
in the generating curves.]
Since the surface is supposed perfectly smooth, whilst the body moves
through the generating curve, the surface, always in contact with the
body, may revolve about the axis of the curve with any velocity whatever,
without deranging in the least the motion of the body ; and thus by ad
justing the angular velocity of the surface, the body may be made to trace
any proposed path on the surface.
If the surface were not perfectly smooth the friction would give the
body a tangential velocity, and thence a centrifugal force, which would
cause a departure from both the curve and surface, unless opposed by
their material ; and even then in consequence of the resolved pressure a
rise or fall in the surface.
Hence it is clear that the time of describing any portion of a path in a
surface of revolution, is equal to the time of describing the corresponding
portion of the generating curve.
Thus when the force is in the center of a sphere, and whilst this force
causes the body to describe a fixed greatcircle, the sphere itself revolves
with a uniform angular velocity, the path described by the body on the
surface of the sphere will be the Spiral of Pappus.
272 A COMMENTARY ON [SECT. X.
370. PROP. XLVIII and XLIX. In the Epicycloid and Hypocycloid,
f
s : 2 vers. : : 2 (R + r) : R
where s is any arc of the curve, s the corresponding one of the wheel, and R
the radius of the globe and r that of the wheel, the + sign being used for
the former and in the Hypocycloid. (See Jesuits notes.)
OTHERWISE.
If p be the perpendicular let fall from C upon the tangent V P, we
have from similar triangles in the Epicycloid and Hypocycloid
PY: CB:: VY: VC
or
p 2 n 2 R 2 . . /T> JL 9 ,N 2 n 2 . / U _i_ O \2
J JL v I L I 1 * 1 j Y* * \ /
which gives
n 2_ / R J O r \ 2 v S ** / 1 \
JJ \i\,^TA IJ * / r> J_ Q \ 2 T> 2 \ l l
Now from the incremental figure of a curve we have generally
d s
But
d]
p 2 )
(2)
R
40 r \ 2 _ .2?
,
,.ds =
X
and integrating from
s = 0, wheng = R
we get _
V (R + 2r) 2
+ 2r ) 8 R 8 V(R2 r) 8
which is easily transformed to the proportion enunciated.
The subsequent propositions of this section shall now be headed by a
succinct view of the analytical method of treating the same subject.
371. Generally, A body being constrained to move along a given curve by
known forces, required its velocity.
Let the body P move along the curve
P A, referred to the coordinates x, y
originating in A ; and let the forces be
resolved into others which shall act
parallel to x, y and call the respective
aggregates X, Y. Besides these we
have to consider the reaction (R) of the
BOOK L] NEWTON S PRINCIPIA. 273
curve along the normal P K, which being resolved into the same direc
tions gives (d s, being the element of the curve)
n dx d y
It 3 , and R r* .
ds ds
Hence the whole forces along x and y are (see 46)
d l x
dt
2 =X +
LV ds
Again, eliminating R, we get
2dxd 2 x + 2dyd 2 y
jji * l = 2Xdx + 2Ydy
and
d x z + dy 2
But
. . v*= 2/(Xdx+ Ydy) (])
Hence it appears that The velocity is independent of the reaction of the
curve.
372. If the force be constant and in parallel lines, such as gravity, and
x be vertical ; then
and
Y =
and we have
v 2 = 2/gdx
= 2g(c x)
= 2g(h x)
h being the value of x, when v = ; and the height from which it begins to
fall.
373. To determine the motion in a common cycloid, when the force is gravity.
The equation to the curve A P is
!2rK
x
r being the radius of the generating circle.
/2r
.. ds = dx I
V x
VOL. I.
A COMMENTARY ON [SECT. X.
and
ds
 / r
x ~ V
. A/(hxj ~ g V (hxx 2 )
t being = 0, when x = h.
Hence the whole time of descent to the lowest point is
T , r
F^VTg
which also gives the time of an oscillation.
374. Required the time of an oscillation in a small circular arc.
Here
y = V (2rx x 2 )
r being the radius of the circle, and
__ rd x
S  V (2rx x 2 )
ds
* dt  v 2g V (h x)
dx
x)(2rx x 2 )}
r dx
X V {(hx x z )(2r x)}
to integrate which, put
x
dx
d 9  2 A/Oix x*j
and since
= 2r(l a 2 sin.
/
d * = V
Now since the circular arc is small, h is small ; and therefore 3 is so.
And by expanding the denominator we get
sn.
BOOK I.] NEWTON S PRINCIPIA. 275
and integrating by parts or by the formula
fd & . sin. m = cos. 6 sin. m  J 6 A fdd sin. m %Q
m m J
and taking it from
d = to 6 = J
we get
rr I
f. d 6 sin. m 6 = ^ f,dd sin. m ~ 2 &
m Jl
the accentedy denoting the Definite Integration from 6 = 0, to 6 = .
In like manner
f. d 6 sin. ra  2 = ~ /~ d sin. m  *
m 2 <//
and so on to
Hence
(m 1) (m 3) 1 r
7, d Sin. m = V T ^rr 1 X 5
m (m 2) 2 2
and
/. d d f = 0")
I , , .. r? 5~; I rom I
/ x V (1 5 2 sin. 2 6 & _ _* >
is the same as
V (1 3 s sin. 2 0) from
whence then
CJ* 
= oi
and taking the first term only as an approximate value
t = ~2"tJ ~~^
fi
which equals the time down a cycloidal arc whose radius is 7
TJ
If we take two terms we have
 I?(14
2 V g V 4
S2
276 A COMMENTARY ON [SECT. X.
375. To determine the velocity and time in a Hypocycloid, the force
tending to the center of the globe and g.
By (370)
the equation to the Hypocycloid is
R 2 P Z
by hypothesis.
Now calling the force tending to the center F, we have
X = F x  ,Y = F
...v 2 = C 2/Fd f
But by the supposition
Hence
ds_
Cl L ^
To integrate it, put
2 D = u 8
and
V R 2 D 2 d u
dt =  ;
V^a V (h 2 D 2 u 2 )
V (R 2 D 2 ) /^ _ D 2
< v : pcm 1 / =
RVp Vh 2 D 2
Hence making = D, we have
Oscill. cr /R 2 D
2
376. Since h does not enter the above expression the descents are
Isochronous.
We also have it in another form, viz.
I /rJi J__A
2 " A/ VR^ RV/
BOOK I.]
NEWTON S PRINCIPIA.
If R ^ = g or force of gravity and R be large compared with b,
T
277
the same as in the common cycloid.
377. Required to Jind the value of the reaction R, when a body is con
strained to move along a given curve.
As before (46)
d 2 x r. d y
_ Y R
 i
dx
Hence
. Xdy Ydx , dyd 2 x dxd 8 y
"> j~I T j * 2 J
ds
But if r be the radius of curvature, we have (74)
ds 3
~ dyd x dxd 2 y
Hence
dt 2 ds
B Ydx Xdy^ ds 2
rt i r ..j i2
d s
rdt !
Another expression is
Y d x X d y v^
or
Ydx Xdy
,
+ P
<p being the centrifugal force.
If the body be acted on by gravity only
ds + rdt !
or
or
(1)
(2)
(3)
ds
If the body be moved by a constant force in the origin of x, y, we have
xdy y d x
\ d x X d y = F
J *
= F i d 6.
83
278
A COMMENTARY ON
[SECT. X.
for
xdy y d x = g 2 d
i? . ,
.. R =
,
ds
d s r d t
or
or
_ FjjH
d s
378, To Jind the tension of the string in the oscillation of a common
cycloid.
Here
but
dy =
d s =
d s ^ r d t
;2 a x
2_a
x
and
d_y _ 2 a x
d s ~ V WIT
r  2 V 2 a V (2 a x)
~~ = 2 g (h x)
^R = g J*Z
& V 2 a
= g
__ _
\/2a V(2 a x)
2 a + h 2x
When x =r h
When x =
2 ax)
R = .
_
a 2 2 a h)
(2 a)
2 a + h
rr
K rr
J.V .
2 a
When moreover h = 2 a, the pressure at A the lowest point is = 2 g.
379. To Jind the tension "when the body oscillates in a circular arc by
gravity.
BOOK I.] NEWTON S PRINCIPIA. 279
Here
dv  (C ~ x) d X
y A/(2cx x*)
c d x
d s =
V (2 ex x 2 )
d y _ c x
d x c
r = c
= g
When x =
c c
c + 2 h 3 x
v c + 2 h
rv
e>
c
r= 3 g or h = c.
If it fall through the whole semicircle from the highest point
h = 2c,
and
R = 5g
or the tension at the lowest point is five times the weight.
When this tension = 0,
c+2h 3x = 0, orx = ^
A body moving along a curve whose plane is vertical will quit it when
R =
that is when
c + 2h
x 
3
and then proceed to describe a parabola.
380. To Jind the motion of a body upon a surface of revolution, when
acted on by forces in a plane passing through the axis.
Referring the surface to three rectangular axes x, y, z, one of which (z)
is the axis of revolution, another is also situated in the plane of forces, and
the third perpendicular to the other two.
Let the forces which act in the plane be resolved into two, one parallel
to the axis of revolution Z, and the other E, into the direction of the
radiusvector, projected upon the plane perpendicular to this axis. Then,
S4,
280
A COMMENTARY ON
[SECT. X.
calling this projected radius g, and resolving the reaction R (which also
takes place in the same plane as the forces) into the same directions, these
components are
dz
R ds
d s
supposing ds= V / (dz
of jf is
F + R
and the whole force in the direction
d_z
ds
and resolving this again parallel to x and y, we have
d 2 x _
~ ds
F j
~ "
and
7
= 
Hence we get
xd g y y d 2 x _ _ d xdy y dx
dt 2
and
dxd 2 x + dy d 2 y+dzd 2 z
Which, since
dt
x d x+ y dy
.
d s
d s
xdx + y dy _ d
e
Again
dz 2 _dz 2 dg 2
dt 2 ~" d * d t 2
(D
(2)
and from the nature of the section of the surface made by a plane passing
through the axis and body, ~ is known in terms of g. Let therefore
dz
BOOK I.] NEWTON S PRINCIPIA. 281
and we have
d_z_ 2 2 dg 2
d t 2 ~ dt 2 *
Also let the angle corresponding to g be 0, then
xdy ydx = g 2 d
and
dx 2 + dy 2 = dg 2 fg 2 d0 2 ,
and substituting the equations (2) and (3) become
Integrating the first we have
P 2 d = h d t
h being the arbitrary constant.
or
The second can be integrated when
2 Fdg 2Zdz
is integrable. Now if for F, Z, z we substitute their values in terms of e,
the expression will become a function of and its integral will be also a
function of g. Let therefore
/(F d g + Z d z) = Q
and we get
dp 2 p 2 d 6 Z dp 2
2
which gives, putting for d t its value
~o~vv\ i i jj . (6)
Hence also
Ol L 7 f. c\~~s~\\ o I o^ ( )
V \ (c 2 Q) g 2 h *} v
If the force be always parallel to the axis, we have
F =
and if also Z be a constant force, or if
we then have
Q = /Z d z = g z (8)
282 A COMMENTARY ON [SECT. X.
Z being to be expressed in terms of g.
381. Tojind under what circumstances a body will describe a circle on a
surface of revolution.
For this purpose it must always move in a plane perpendicular to the
axis of revolution ; g, z will be constant; also (Prop. IV)
cos. 6 = x
d 2 x _ g cos. 6 d 6 z
dT 2 " : dt 2
Also
d*x
dt 2
Hence as in the last art.
. 2
If the force be gravity acting vertically along z, we have
yj _ d z
Hence may be found the time of revolution of a Conical Pendulum.
(See also 367.)
382. To determine the motion of a body moving so as not to describe a
circle, when acted on by gravity.
Here
Q = gz
and
C 2 Q = 2g. (k z)
k being an arbitrary quantity.
Also
g 2 = 2 r z z 2
z being measured from the surface.
.. cl g = (r z) d z
and
_ * r ~
+ P  + (r z) z ~ (r z)
BOOK I.] NEWTON S PRINCIPIA. 283
Hence (380)
*

In order that
the denominator of the above must be put = ; i. e.
2 g (k z) (2 r z z 2 ) h 2 =
or
h 2
z 3 (k + 2r)z 2 + 2krz  =
g
which has two possible roots ; because as the body moves, it will reach
one highest and one lowest point, and therefore two places when
Hence the equation has also a third root. Suppose these roots to be
", ft 7
where a is the greatest value of z, and j3 the least, which occur during the
body s motion.
Hence
___
(2g) V {(_ z ).(z j8)(y z)
To integrate which let
Then
dz
d 6 =
_ cfrz
= 2V {( z) (z 
Also
.. z = 13 + (a jg) sin. l 6
and
y z = 7 {3 + ( /S) sin. *
= (7 P) U a sin. ^,
if
284 A COMMENTARY ON [SECT. X.
.. d t =
V2g. (y /3).
which is to be integrated from z = /3, to z = a ; that is from
6 = to 6 = ~
this expanded in the same way as in 374 gives
t= va 2r _^:
which is the time of a whole oscillation from the least to the greatest
distance.
Also
h d t h d t
= 2 ~~ = 2 r z z 2
and & is hence known in terms of z.
383. A body acted on by gravity moves on a surface of revolution whose
axis is vertical : when its path is nearly circular, it is required to find the
angle between the apsides of the path projected in the plane o/ x, y.
In this case
and if at an apse
o = a, z = k
we have
(C 2gk)a z h 2 :=
... C = ^ + 2 g k.
Hence (380)
\j
d 6 =
il=
V(l +p)~,"*
Let = +
? a
(1+
BOOK I.] NEWTON S PRINCIPIA. 285
<i a 2g (k z) h
Q U
It is requisite to express the righthand side of this equation in terms
of w
Now since at an apse we have
w = 0, z = k, and g = a
we have generally
dz , d 2 z w 2
z = k + d. w + dV 2 i72 + &c 
the values of the differential coefficients being taken for
w = (see 32)
And
d z = p d f z= p 2 d w
d 2 z = 2p^dgdw g d w d p
or, making
d p = q d g
d 2 z = (2p + qg)dgdw = (2 p + q g) f 3 d 2 .
And if p/ and q/ be the values which p and q assume when w = 0,
=r a, we have for that case,
^f,= (2p,+ q ,a)a>
Z = k p 7 a 2 w + (2p + q,a) a 3 . ^ Sac.
Also
wV = + + w 2
/ "a 2 a
Hence
1 _ /I , \ 2
e 2 " v a ;
2g(k_z)h 2 (l_^)
becomes
2 g (p, a 2 co _ (2 p, + q, a) a 3 . ^ + &c.) h *(^ + **).
But when a body moves in a circle of radius = a, we have
h2 = Sf s P = ga p,
in this case. And when the body moves nearly in a circle, h 2 will have
nearly this value. If we put
h 2 = (1 + a)ga 3 P/
we shall finally have to put
5 =
280 A COMMENTARY ON [SECT. X.
in order to get the ultimate angle when the orbit becomes indefinitely near
a circle. Hence we may put
and
becomes
in which the higher powers of u may be neglected in comparison of u 2 ;
. d " 2 _ _ ga 3 (3 P/ + q/ a)q. 2 _ (3 P/ + q/ a)^_ 2
d^ 2 " h 2 (1 + p 2 ) P/ (1 + P 2 )
_ (3 p,+ q/ a) a 2
P/(l +P/ 2 )
again omitting powers above u z : for p = p / + A u + &c.
Differentiate and divide by 2 d ca, and we have
suppose ; of which the integral is taken so that
6 = 0, when u =
is
u = C sin. 6 V N.
And 01 passes from to its greatest value, and consequently g passes
from the value a. to another maximum or minimum, while the arc 6 V N
passes from to <r. Hence, for the angle A between the apsides we have
A V N v or A = r^j
V N
where
N  3 P/ + q/ a .
384. Let the surface be a sphere and let the path described be nearly a
circle : to Jind the horizontal angle between the apsides.
Supposing the origin to be at the lowest point of the surface, we have
z = r V (r 2 e ")
d z g
" P  V( r _ a J
BOOK 1.1 NEWTON S PRINCIPIA. 287
4 r 2 3 a
/. N = =
Hence the angle between the apsides is
A 

V(4r 2 3 a 2 )
The motion of a point on a spherical surface is manifestly the same as
the motion of a simple pendulum or heavy body, suspended by an inex
tensible string from a fixed point ; the body being considered as a point
and the string without weight. If the pendulum begin to move in a ver
tical plane, it will go on oscillating in the same plane in the manner al
ready considered. But if the pendulum have any lateral motion it will
go on revolving about the lowest point, and generally alternately ap
proaching to it, and receding from it. By a proper adjustment of the velocity
and direction it may describe a circle (134) ; and if the velocity when it
is moving parallel to the horizon be nearly equal to the velocity in a cir
cle, it will describe a curve little differing from a circle. In this case we
can find the angle between the greatest and least distances, by the for
mula just deduced.
Since
. _ *_r
: V (4 r a 3 a 8 )
if a = 0, A = 7 , the apsides are 90 from each other, which also ap
fit
pears from observing that when the amplitude of the pendulum s revolu
tion is very small, the force is nearly as the distance ; and the body de
scribes ellipses nearly ; of which the lowest point is the center.
If a = r,
A = * = 180
this is when the pendulum string is horizontal ; and requires an infinite
velocity.
If a =  ; so that the string is inclined 30 to the vertical ;
A = J _ =99 50 .
V 13
298 A COMMENTARY ON [SECT. X.
r*
If a 2 = jr ; so that the string is inclined 45 to the vertical ;
A = *r J  = J13. 56 .
3 r 2
If a 2 _ . so th a t the string is inclined 60 to the vertical ;
4
A = ^z = 136 nearly.
385. Let the surface be an inverted cone, with its axis vertical : to find
the horizontal angle between the apsides when the orbit is nearly a circle.
Let r be the radius of the circle and 7 the angle which the slant side
makes with the horizon. Then
z = g tan. 7
p = tan. 7
__
tan. 7. sec. * 7
and
A =
cos. 7 V 3
If 7 = 60
A =
386. Let the surface be an inverted paraboloid whose parameter is c.
= c z
d z
2
c
6_a 2a
2~a
If a =  , or the body revolve at the extremity of the focal ordinate,
m
N = 2
and
BOOK I ] NEWTON S PRINCIPIA. 281)
387. When a body moves on a conical surface, acted on by a force tend
ing to the vertex ; its motion in the surface will be the same., as if the sur
face were unwrapped, and made plane, the force remaining at the vertex.
Measuring the radiusvector (g) from the vertex, let the force be F,
and the angle which the slant side makes with the base = 7 : then
z = g tan. 7
p = tan. 7
1 + p 2 = sec. 2 7
also
Q=/(Fdg + Zdz) =/F dg .
Hence (380)
i A _ sec  7 h d g
=
or putting
h cos. 7 for h
d tf sec. 7 for d 6
and
g cos. 7 for g
we have
h dg
Now d (f is the differential of the angle described along the conical sur
face, and it appears that the relation between V and / will be the same as
in a plane, where a body is acted upon by a central force F. For in that
case we have
h 2 d t> 2 h 2 1
a ? j. _ I 9 F d P
4 d ^ 2 ^ ? 2 J "
and integrating
h 2 d g 2 h 2
/ 4 J / 2 +7^ = ^
^which agrees with the equation just found.
388. When a body moves on a surface of revolution, to Jind the reac
tion R.
Take the three original equations (380) and multiply them by x d z,
y d 2, g d f ; and the two first become
x d 8 x d z ^ _ F x 2 d z _ R d z 2 x 2
dt 2 T~ " "dT 7
y d*y dz F y 2 d z R dz 2 "y 2
dt 2 ? l ds y
VOL. 1. T
290 A COMMENTARY ON [SECT. X.
add these, observing that
and we have
<xd 2 x+yd 2 y)dz _ _ _ d z*
"dT 2 " WUF
Also the third is
Subtract this, observing that dz 2 + d g 2 = ds 2 , and we have
(xd 2 x + yd 2 y)dz gdgd 2 z
dt 2
s (Z d s F d z) R  d s.
But
x 2 + y 2 = g 2
xdx+ydy = gdg
xd 2 x + yd 2 y + dx 2 + dy 2 = gd 2 g + dg 2 .
Hence
(dg 2 dx 2 dy 2 ) dz gd z d 2 g gdgd 2 z
dt 2 dt 2
g (Z d g F d z) R g d s
and
dg 2 = ds 2 dz 2 .
Hence
R _ Z d g F d z dgd 2 z dzd 2 g
ds dt 2 ds
(dx 2 + dy 2 + dz 2 ds 2 )dz
gdt 2 d s
Now if r be the radius of curvature, we have (74)
ds 3
: dgd 2 z_dzd 2 g
and
d x 2 + dy 2 + d z 2 = d tf 2
a being the arc described.
Hence
Z d g F d z d s 2
ds h FdT 1
dg 2 ds 2 dz
1 gdt 2 d~s
Here it is manifest that
d s 2
BOOK I.] NEWTON S PRINCIPIA. 291
is the square of the velocity resolved into the generating curve, and that
d 2 d s 2
dt 2
is the square of the velocity resolved perpendicular to g. The two last
terms which involve these quantities, form that part of the resistance
which is due to the centrifugal force ; the first term is that which arises
from the resolved part of the forces.
From this expression we know the value of R ; for we have, as before
Also
da 2 _ds 2 _ g*dd ~ __h*
~ dt 2 TT 2 " = r*
Hence
j=C2/(F dHZdz)
h 2
389. To find the tension of a pendulum moving in a spherical surface.
C 2/(F dg + Zdz) = 2g(k,z)
*  V (2rz z 2 )
d _ r z
<Tz ~~ V (2rz z 2 )
d s r
de r z
d s _ r r
d~z ~ V(2rz z 2 ) = 7
Hence
2g(k z)  ~ 2
R = g( r  z ) + _: __ : __ ii
r r
_g(r+2k 3z)
r
and hence it is the same as that of the pendulum oscillating in a vertical
plane with the same velocity at the same distances.
390. To find the Velocity , Reaction, and Motion of a body upon any
surface whatever.
Let R be the reaction of the surface, which is in the direction of a nor
mal to it at eacli point. Also let i, s , t" be the angles which this normal
T2
+ _ __ __ + . .
j 3 r
292
A COMMENTARY ON
[SECT. X.
makes with the axes of x, y, z respectively ; we shall then have, consider
ing the resolved parts of R among the forces which act on the point
d 2 x
 = X + R cos.
d 2 z
dT 2
= Z+R.cos./
Now the nature of the surface is expressed by an equation between
x, y, z: and if we "suppose that we have deduced from this equation
dz =pdx + qdy
, dz , d z
where p = ~j and q = . .
dx dy
p and q being taken on the supposition of y and x being constants respec
tively ; we have for the equations to the normal of the points whose co
ordinates are
x, y, z
x x + p (z z) =0
yy + q(z f z) =
x , y , T! being coordinates to any point in the normal (see Lacroix,
No. 143.)
Hence it appears that if P K be the normal,
P G, P H its projections on planes parallel to
x z, y z respectively.
The equation of P G is
x x + p . (z z) = 0,
and hence
GN+pPN=
and
G N = p . P N.
Similarly the equation of P H is
y y + q( z z ) =
whence
HN+ q.PN =
H N = q . P N.
And hence,
cos
:. t = cos. K P h =
JPh
PK
GN
NG e + HN*)
BOOK I.] NEWTON S PRINC1PIA. 293
P
V (\ +p + i*)
cos. t = cos. K P g = p
HN
V(PN 2 + NG 2 + HN 2 )
q
V (1 + p 2 + q 2 )
Whence, since
cos. 2 f cos. 2 E + cos. s t" = 1
COS. 2 ?" = V ( 1 COS. 2 e COS. 2 e )
__ 1
Substituting these values; multiplying by d x, d y, d z respectively, in
the three equations ; and observing that
dz pdx q d y =
we have
dxd 8 x + dyd g y + d z d * z ,
Tri = Xdx + Ydy + Zdz
Cl I *
and integrating
dx s + dy* + dz 2
grfi 2/(Xdx+ Ydyf Zdz)
L4. L * *
and if this can be integrated, we have the velocity.
If we take die three original equations, and multiply them respectively
ty P, q, and 1, and then add, we obtain
 P X
But
d z = p d x + q d y.
Hence
^ 5 = D ^ x _L a ^J y 4. dp dx + d q d y
d t 2 P d t " q d t 2 "*" "dT 8 ""
Substituting this on the first side of the above equation, and takinr
the value of II, we find
R P^ + q Y Z ^1 p d x + d q d y
If in the three original equations we eliminate R, we find two second
differential equations, involving the known forces
X,Y, Z
T 3
y!)i A COMMENTARY ON [SECT. X.
and p, q, which are also known when the surface is known, combining
with these the equation to the surface, by which z is known in terms of
x, y, we have equations from which we can find the relation between the
time and the three coordinates.
391. To find the path which a body mil describe upon a given surface,
when acted upon by no force.
In this case we must make
X, Y, Z each = 0.
Then, if we multiply the three equations of the last art. respectively by
(qdz + dy), pdz + dx, qdx pdy
and add them, we find,
(qdz + dy)d 2 x + (pdz+dx) d 2 y+ (qdx pdy) d 2 z
/ (q d z + d y) cos. s ~\
= R d t 2  + (pdz + dx) cos. t t
v. + (qdx pdy) cos. t")
or putting for cos. e, cos. t , cos. *" their values
Rdt 2
Hence, for the curve described in this case, we have
(p d z + d x) d 2 y = (p d y q d x) d 2 z+ (q d z + d y) d 2 x.
This equation expresses a relation between x, y, z, without any regard
to the time. Hence, we may suppose x the independent variable, and
d 2 x = ; whence we have
(pdz + dx) d 2 y = (pdy qd x)d 2 z. t
This equation, combined with
dz=rpdx + qdy,
gives the curve described, where the body is left to itself, and moves along
the surface.
The curve thus described is the shortest line which can be drawn from
one of its points to another, upon the surface.
The velocity is constant as appears from the equation
v = 2/(Xdx + Ydy + Zdz).
By methods somewhat similar we might determine the motion of a point
upon a given curve of double curvature, or such as lies not in one plane
when acted upon by given forces.
392. To fold the curve of equal pressure, or that on which a body descend
ing by the force of gravity, presses equally at all points.
BOOK I.]
NEWTON S PRINCIPJA.
Let A M be the vertical abscissa = x, M P the hori
zontal ordinate = y ; the arc of the curve s, the time t,
and the radius of curvature at P = r, r being positive
when the curve is concave to the axis ; then R being the
reaction at P, we have by what has preceded.
R = ~dT + r~TF (1)
But if H M be the height due to the velocity at P,
A H = h, we have
ds"
295
H
M
dt
= 2g(hx).
Also, if we suppose d s constant, we have (74)
d s d x
and if the constant value of R be k, equation ( 1 ) becomes
k = S d y _ 2g(hx)d 2 y
d s d s d x
k dx , , d 2 v dy
d x
W
The righthand side is obviously the differential of
V (h
hence, integrating
k d v
g * d s
d_y = k C
: g  V (h x)
_
ds
If C = 0, the curve becomes a straight line inclined to the
which obviously answers the condition. The sine of inclination
In other cases the curve is found by equation (2), putting
V(dx 2 + dy 2 ) for ds
and integrating.
If we differentiate equation (2), d s being constant, we have
d*y_ Cdx
d S n /!_
horizoij,
is 
And if C be positive, r is positive, and the curve is concave to the axis.
T4
296
A COMMENTARY ON
[SECT. X.
We have the curve parallel to the axis, as at C, when ^ = 0, that is,
u s
; when
, k C
when = .,
g V (h x
x =
When x increases beyond this, the curve approaches the axis, and r^
U V
is negative ; it can never become < 1 ; hence B the limit of x is
found by making
x)
or
x =h
C
g)
2
If k be < g, as the curve descends towards Z, it approximates perpe
k
tually to the inclination, the sine of which is .
o
If k be > g there will be a point at which the curve becomes horizontal.
C is known from (2), (3), if we knew the pressure or the radius of cur
vature at a given point.
If C be negative, the curve is convex to the axis. In this case the part
of the pressure arising from centrifugal force diminishes the part arising
from gravity, and k must be less than g.
393. To find the curve which cuts a given assemblage of curves, so as to
make them Synchronous, or descriptive by the force of gravity in the same
time.
Let A P, A P , A P", &c. be curves of the
same kind, referred to a common base A D,
and differing only in their parameters, (or the
constants in their equations, such as the radius
of a circle, the axes of an ellipse, &c.)
Let the vertical A M = x, M P (horizontal)
= y ; y and x being connected by an equation
involving a. The time down A P is
/dx
V(2gx)
the integral being taken between
x = and x = A M ;
and this must be the same for all curves, whatever (a) may be.
BOOK I.] NEWTON S PRINCIPIA. 297
Hence, we may put
.
k being a constant quantity, and in differentiating, we must suppose (a)
variable as well as x and s.
Let
d s = pdx
p being a function of x, and a which will be of dimensions, because d x,
and d s are quantities of the same dimensions. Hence
f P dx _ k
J V (2gx)~
and differentiating
Now, since p is of dimensions in x, and a, it is easily seen that
r p
J V2
is a function whose dimensions in x and a are , because the dimensions
of an expression are increased by 1 in integrating. Hence by a known
property of homogeneous functions, we have
k p V x
q ~2a~a V (2g)
substituting this in equation (2) it becomes
pdx k d a p d a V x __ ..
V (2gx) H " ~2l a V (2 g) ~
in which, if we put for (a) its value in x and y, we have an equation to the
curve P P P".
If the given time (k) be that of falling down a vertical height (h), we
have
* = J,
and hence, equation (3) becomes
p(adx xda) + da V (h x) = . . . . (4)
Ex. Let the curves A P, A P , A P" be all cycloids of which the bases
coincide with A D.
Let C D be the axis of any one of these cycloids and = 2 u, : t being
the radius of the generating circle. If C N = x , we shall have as before
2 a
298 A COMMENTARY ON [SECT. X.
and since
x = 2 a x
/ 2 a
N 2a x
Hence
2a
P ~
and equation (4) becomes
v(8a)(adxxda)
V (2 a x)
Let = u
a
so that
adx xda= a 2 du
x = au;
and substituting
a 2 du V2 d v h ,
V(2 u)" 1
du V 2 da Vh
V(2uu 2 ) a f
. V 2 X vers.  l u 2 J C (6)
When a is infinite, the portion A P of the cycloid becomes a vertical
line, and
x = h, .. u = 0, .. C = 0.
Hence
x , 2 h ..
= vers. . / [71
a >r a
From this equation (a) should be eliminated by the equation to the
cycloid, which is
y = a vers.  V (2 a x x 2 ) . . . . (8)
and we should have the equation to the curve required.
Substituting in (8) from (7), we have
y = V (2 ah) V (2 ax x 2 )
_dav h xda + adx x d \
V (2a) V (2 ax x*)
and eliminating d a by (5)
dy _ 2 a x / 2 a x
dx~ ~ V (2ax x 2 ) ~ "~**J x
BOOK I.] NEWTON S PRINCIPIA. 299
But differentiating (8) supposing (a) constant, we have in the cycloid
2 a
And hence (31) the curve P P P" cuts the cycloids all at right angles,
the subnormal of the former coinciding with the subtangent of the latter,
each being
2 a
AGO
The curve P P P" will meet A D in the point B, such that the given
time is that of describing the whole cycloid A B. It will meet the vertical
line in E, so that the body falls through A E in the given time.
394. If instead of supposing all the cycloids
to meet in the point A, we suppose them all to
pass through any point C, their bases still being
in the same line A D ; a curve P P drawn so
that the times down P C, P C, &c. are all
equal, will cut all the cycloids at right angles.
This may easily be demonstrated.
395. Tojind Tautochronous curves or those down which to a given Ji xed
point a body descending all distances shall move in the same time.
(1) let the force be constant and act in parallel lines.
Let A the lowest point be the fixed point, D that
from which the body falls, A B vertical, B D, M P
horizontal. A M = x, A P = s, A B = h, and the
constant force = g.
Then the velocity at P is
v = V (2g.h x)
and
dt = =
ds
V 2g V (h x)
and the whole time of descent will be found by integrating this from
x = h, to x = 0.
Now, since the time is to be the same, from whatever point D the body
falls, that is whatever be h, the integral just mentioned, taken between the
limits, must be independent of h. That is, if we take the integral so as
to vanish when
x =
and then put h for x, h will disappear altogether from the result. This
must manifestly arise from its being possible to put the result in a form
300 A COMMENTARY ON [SECT. X.
v v 2
involving only r , as r^ , &c. ; that is from its being of dimensions in
x and h.
Let
d s = p dx
where p depends only on the curve, and does not involve h. Then, we
have
t  f
J
1 / f p d x t 1 pxd x 1.3 pxMx
Ws) J I *TT*T T* 271 ~vT c
and from what has been said, it is evident, that each of the quantities
/*p d x /pxdx /px n dx
y i y JT } y~~gir+T
h ^ h 2 h T
must be of the form
CX 2
2 n + 1
that is
f p x " d x must = c x 2~~ ;
hence
. , 2 n + 1 Hi^ 1 .
p x n d x = ^ c x a d x ;
2n + 1 c
P =
or if
2 n + 1
and
which is a property of the cycloid.
Without expanding, the thing may thus be proved. If p be a function
of m dimensions in x, ,/ . is of m i dimensions : and as the
V (h x)
dimensions of an expression are increased by 1 in integrating
f. P dx
BOOK I.]
NEWTON S PKINCIPIA.
301
is of m j 1 dimensions in x, and when h is put for x, of m { ^ dimen
sions in h. But it ought to be independent of h or of dimensions
Hence
i = o
p = a^
as before.
396. (2) Let the force tend to a center and vary as any function of the
distance. Required the Tautochronous Curve.
Let S be the center of force, A the point to
which the body must descend ; D the point from
which it descends. Let also
S A = e, S D = f, S P = ^ A P = s
P being any point whatever,
= C 2/Fdf
Now we have
or if
the velocity being when f.
Hence the time of describing D A is
t=/: ds
taken from g = f, to g = e. And since the time must be the same what
ever is D, the integral so taken must be independent of f.
Let
<pf <p e = h
d s = p d z
p depending on the nature of the curve, and not involving f. Then
/p d z f
r7j r , from z = h to z =
V (h z)
= /* r~i r from z = to z = h.
J V (h z)
And this must be independent of f, and therefore of <p f, and of h
Hence, after taking the integral the result must be when z = 0, and
independent of h, when h is put for z. Therefore it must be of dimen
sions in z and h. But if p be of n dimensions in z, or if
p = cz n
V (h z)
will be of n \ dimensions,
302 A COMMENTARY ON [SECT. X.
and
/. TT  r of n + s dimensions.
V (h z)
Hence, n + \ = 0, n = , and
Therefore
/ C C
d s =r d z / = <f> o d g J  7 ;
>r z V p g <f> e
whence the curve is known.
If 6 be the angle A S O, we have
and
g 2
whence may be found a polar equation to the curve.
397. Ex. 1. Let the force vary as the distance, and be attractive.
Then
F = Mg,
z <p p
dz = 2 # i
4,
d s
when = e, r is infinite or the curve is perpendicular to S A at A.
If S Y, perpendicular upon the tangent P Y, be called p, we have
p2_ ds 2 dg 2
8 " ds 2
j _ e 2 (1
4 C/(A
If e = 0, or the body descend to the center, this gives the logarithmic
spiral.
In other cases let
BOOK I.]
and
NEWTON S PRINCIPIA.
a 2 e 2
303
a
a* e z
the equation to the Hypocycloid (370)
If 4 c p = 1, the curve becomes a straight line, to which S A is per
pendicular at A.
If 4 c ^ be > 1 the curve will be concave to the center and go off to
infinity.
398. Ex. 2. Let the force vary inversely as the square of the distance.
Then
r
and as before we shall find
g 5 (g e)
c e
399. A body being acted upon by a force in parallel lines, in its descent
from one point to another, to find the Brachystochron, or the curve of quick
est descent between them.
Let A, B be the given points, and A O P Q B
the required curve. Since the time down
A O P Q B is less than down any other curve, if
we take another as A O p Q B, which coincides
with the former, except for the arc O P Q, we
shall have
Time down A O : T. O P Q + T. Q B, less than
Time down A O+T. O p Q + T. Q B
and if the times down Q B be the same on the two suppositions, we shall
have
T. O P Q less than the time down any other arc O p Q.
The times down Q B will be the same in the two cases if the velocity
at Q be the same. But we know that the velocity acquired at Q is the
same, whether the body descend down
A O P Q, or A O p Q.
Hence it appears that if the time down A O P Q B be a minimum, the
time down any portion O P Q is also a minimum.
304 A COMMENTARY ON [SECT. X.
Let a vertical line of abscissas be taken in the direction of the force;
and perpendicular ordinates, O L, P M, Q N be drawn, it being sup
posed that
L M = M N.
Then, if L M, M N be taken indefinitely small, we may consider them
as representing the differential of x : On this supposition, O P, P Q, will
represent the differentials of the curve, and the velocity may be supposed
constant in O P, and in P Q. Let
AL = x, L O = y, OA = s,
and let d x, d y, d s be the differentials of the abscissa, ordinate, and
curve at Q, and v the velocity there ; and d x , d y , d s , v be the cor
responding quantities at P. Hence the time of describing O P Q will
be (46)
d s d s
v + V
which is a minimum ; and consequently its differential = 0. This dif
ferential is that which arises from supposing P to assume any position as
p out of the curve O P Q ; and as the differentials indicated by d arise
from supposing P to Vary its position along the curve O P Q, we shall
use d to indicate the differentiation, on hypothesis of passing from one
curve to another, or the variations of the quantities to which it is
prefixed.
We shall also suppose p to be in the line M P, so that d x is not sup
posed to vary. These considerations being introduced, we may pro
ceed thus,
d. d,, ........
And v, v are the same whether we take O P Q, or O p Q ; for the
velocity at p = velocity at P. Hence
d v = o, a v = o
and
8 d s d d s _
" i~ 7 "
v v
Now
ds 2 =dx 2 + dy 2
.. d s a d s = d y d d y,
(for d d x = 0).
Similarly
d s 6 d s = d y d d y .
BOOK I.] NEWTON S PRINCIPIA. 905
Substituting the value of d d s, d d s which these equations give,
we have
dyddy d y _
vds ~v ds
And since the points O, Q, remain fixed during the variation of P s
position, we have
d y + d y = const.
d d y = d d y.
Substituting, and omitting 8 d y,
d y. jiy.
vds v 7 d s 
Or, since the two terms belong to the successive points O, P, their
difference will be the differential indicated by d; hence,
d ~/ =
vds
dy
.*. j = const ....... . iv\
vds v
Which is the property of the curve; and v being known in terms of x,
we may determine its nature.
Let the force be gravity ; then
v = V(2gx);
dy
"
dy J_
d s V x V a
a being a constant.
d ?  /
ds ~ V a
which is a property of the cycloid, of which the axis is parallel to x,
and of which the base passes through the point from which the body
falls.
If the body fall from a given point to another given point, setting off
with the velocity acquired down a given height; the curve of quickest
descent is a cycloid, of which the base coincides with the horizontal line,
from which the body acquires its velocity.
400. If a body be acted on by gravity, the curve of its quickest descent
from a given point to a given curve, cuts the latter at right angles.
Let A be the given point, and B M the given curve; A B the curve of
quickest descent cuts B M at right angles.
VOL. I.
u
306
A COMMENTARY ON
[SECT. X.
It is manifest the curve A B must be a cycloid, for
otherwise a cycloid might be drawn from A to B, in A
which the descent would be shorter. If possible, let
A Q be the cycloid of quickest descent, the angle
A Q B being acute. Draw another cycloid A P, and
let P P be the curve which cuts A P, A Q so as to
make the arcs A P, A P synchronous. Then (394) P P
is perpendicular to A Q, and therefore manifestly P is
between A and Q, and the time down A P is less than the time down
A Q ; therefore, this latter is not the curve of quickest descent. Hence,
if A Q be not perpendicular to B M, it is not the curve of quickest
descent.
The cycloid which is perpendicular to B
longest descent from A to B M.
M may be the cycloid of
401. If a body be acted on by gravity, and if A B be the
curve of quickest descent from the curve A L to the point B ;
A T, the tangent of A L at A, is parallel to B V, a perpen
dicular to the curve A B at B.
If B V be not parallel to A T, draw B X parallel to
A T, and falling between B V and A. In the curve A L
take a point a near to A. Let a B be the cycloid of quick
est descent from the point a to the point B; and Bb being
taken equal and parallel to a A, let A b be a cycloid equal
and similar to a B. Since A B V is a right angle, the
curve B P, which cuts off A P synchronous to A B, has B V for a tan
gent. Also, ultimately A a coincides with A T, and therefore B b with
B X. Hence B is between A and P. Hence, the time down A b is less
than the time down A P, and therefore, than that down A B. And
hence the time down a B (which is the same as that down A b) is less
than that down A B. Hence, if B V be not parallel to A T, A B is not
the line of quickest descent from A L to B.
402. Supposing a body to be acted on by any forces whatever, to determinf
the Brachystochron.
Making the same notations and suppositions as before, A L, L O. (see
a preceding figure) being any rectangular coordinates ; since, as before,
the time down O P Q is a minimum, we have
BoOK !] NEWTON S PRINCIPIA.
4. IfLi __ d s 3 v d s 3 v _
v v v 2 v 2 "~ *
Now as before we also have
* i d y 3 d y
o Cl S =r 1_ 
ds
supposing 6 d x = 0, and
_ dy .a.dy
307
d s d s
dv =
for v is the velocity at O and does not vary by altering the curve.
v = v + d v
dv = dv + ddv = ddv.
Hence
vds v d s v 72 =
Also
v ~ v+"d~v ~~ v v" 2
for d v 2 , & c . must be omitted. Substituting this in the second term of
the above equation, we have
j> __ dy ady d y d v a d y d s 8 d v
vds vds v 2 d~s ~ ~~V 2 ~
or
M! s d s) v "*" d s . v 2 v 72 " Tdy =
Now as before
d y d y d y
d7 ~~d7 d dY
And in the other terms we may, since O, P, are indefinitely near, put
d s, d y, v for d s , d y , v :
if we do this, and multiply by v, we have
d dy dy.dvds adv
which will give the nature of the curve.
If the forces which act on the body at O, be equivalent to X in the
direction of x, and Y in the direction of y, we have (371)
vdv=Xdx+Ydy
d v = Xd *+ Ydy
v
* i Yddy
. . o d v = i.
v
U2
308
A COMMENTARY ON
[SECT. X.
because 5 v = 0, <5 d x = ; also X and Y are functions of A L, and L O,
and therefore not affected by d.
Substituting these values in the equation to the curve, we have
d dy dy Xdx+Ydy ds Y = Q
d s d s v 2 v v
or
, dy dx Xdy Ydx _ _
a . = j .
d s d s  2
v~
which will give the nature of the curve.
If r be the radius of curvature, and d s constant, we have (from 74)
d s d x
r =:
d 2 y
r being positive when the curve is convex to A M ;
l d y _ d x
d s r
and hence
v_ 2 _ Xdy Ydx
r d s
v 2 .
The quantity is the centrifugal force (210), and therefore that part
,., . e . ,Xdy Ydx..
of the pressure which arises from it. And ^  is the pressure
which arises from resolving the forces perpendicular to the axis. Hence,
it appears then in the Brachystochron for any given forces, the parts of
the pressure which arise from the given forces and from the centrifugal
force must be equal.
403. If we suppose the force to tend to a center S,
which may be assumed to be in the line A M, and F
to be the whole force ; also if
then we have
C ^ ~ = force in P S resolved parallel to
and
Y S = F x 
C
v 2 = C 2/g F d
2 g./ F d g _ F p
r s
also
r
dp
BOOK I.] NEWTON S PH1NCIP1A. 309
2dp_ 2Fdg
p = C
and integrating
whence the relation of p and is known.
If the body begin to descend from A
C2g/Fdg =
when = a.
404. Ex. 1. Let the force vary directly as the distance.
Here
p=C>(a g 2 )
which agrees with the equation to the Hypocycloid (370).
405. Ex. 2. Let the force vary inversely as the square of the distance $
then
by supposition.
S 2 _ S 3 + C *f CE
C
p d g
ci a ~ . .
c V (a g) . d g
~ f V (g 3 + c *g c*a)
_ _ cdg
When g = a, d J = ; when
g 3 + c g c 2 a =
d 6 is infinite, and the curve is perpendicular to the radius as at B. Tills
equation has only one real root.
If we have c = , S B = ~
2
B being an apse.
U3
310 A COMMENTARY ON [SECT. X.
If c =
,.
n 3 + n n 2 f 1
406. When a body moves on a given surface, to determine the Brachy
stochron. ,
Let x, y, z be rectangular coordinates, x being vertical ; and as before
let d s, d s be two successive elements of the curve ; and let
d x, d y, d z,
d x , d y 7 , d z
be the corresponding elements of x, y, z ; then since the minimum pro
perty will be true of the indefinitely small portion of the curve, we have
as before, supposing v, v the velocities,
ds d s
j = mm.
v
The variations indicated by a are those which arise, supposing d x, d x
to be equal and constant, and d y, d z, d y , d z to vary
Now
d s 2 = dx 2 + dy 2 + dz 2
.. d s a d s = d y a d y + d z a d z.
Similarly
d s a d s = d y a d y + d z a d z.
Also, the extremities of the arc
d s + d s
being fixed, we have
d y + d y = const.
.. ady + ady =
d z f d z const
.. a d z + a d z = 0.
Hence
ads 
rt c " ft c W
(2)
And the surface is defined by an equation between x, y, z, which we
may call
L = 0.
BOOK I.] NEWTON S PRINCIPLE 311
Let this differentiated give
....... (3)
Hence, since d x, p, q are not affected by 8
3dz = q.3dy ......... (4)
For the sake of simplicity, we will suppose the body to be acted on
only by a force in the direction of x, so that v, v will depend on x alone,
and will not be affected by the variation of d y, d z. Hence, we have by ( 1 )
6ds d d s _
T
which, by substituting from (2) becomes
_ ; 
d s v d s / \ v d s v d s
Therefore we shall have, as before
* adz=0;
. .
v d s v d s
and by equation (4), this becomes
d.^L + qd.4^ = ....... (5)
v d s v d s
whence the equation to the curve is known.
If we suppose the body not to be acted on by any force, v will be con
stant, and the path described will manifestly be the shortest line which
can be drawn on the given surface, and will be determined by
d.iZ+q.d.^ = ..... " . (6)
d s d s v
If we suppose d s to be constant, we have
d 2 y + qd 2 z=:0
which agrees with the equation there deduced for the path, when the
body is acted on by no forces.
Hence, it appears that when a body moves along a surface undisturbed,
it will describe the shortest line which can be drawn on that surface, be
tween any points of its path.
407. Let P and Q be two bodies, of which the Jirst hangs
from ajixedpoint and the second from the Jirst by means of
inextensible strings A P, P Q; it is required to determine the
small oscillations.
Let
A M = x, M P = y,
AN = x / ,NQ = y /
A P = a, P Q = a
mass of P = p, of Q = p
tension of A P =p,ofPQ=r p .
U4
312 A COMMENTARY ON [SECT. X.
Then resolving the forces p, p , we have
y .p g./ y_p_g
d t 2 p a ft
<* 2 y _ _P^g y y
y_\
a t
(
_
d t 2 (j! a
By combining these with the equations in x, x and with the two
x 2 +y = a 2 ,
(x x)*+(y _y) 2 :=a 2 ;
we should, by eliminating p, p find the motion. But when the oscilla
tions are small, we may approximate in a more simple manner.
Let /3, j3 be the initial values of y, y . Then manifestly, p, p will de
pend on the initial position of the bodies, and on their position at the time
t : and hence we may suppose
p= M + P/3 + Q/3 + Ry + Sy + &c.
and similarly for p .
Now, in the equations of motion above, p, p are multiplied by y, y y
which, since the oscillations are very small are also very small quantities,
(viz. of the order /S). Hence their products with 3 will be of the order
B\ and may be neglected, and we may suppose p reduced to its first
term M.
M is the tension of A P, when /3, /3 &c. are all = 0. Hence it is the
tension when P, Q, hang at rest from A, and consequently
M = ^ + til.
Similarly, the first term of p , which may be put for it is m . Substi
tuting these values and dividing by g, equations (1) become
\
9
(
v /
y
_
gdt 2 " a " a
Multiply the second of these equations by X and add it to the first, and
we have
_ _/j m x / j^x
V a " t* a af ) y \ a! /*a ) y
_
g d I 8 , V a t* a
and manifestly this can be solved if the second member can be put in
the form
k.(y + xy )
that is, if
BOOK L] NEWTON S PRINCIPIA. 313
k x = 
a [A &
or
/i A 6 a u, a
a k = 1 1
> (8)
 = (a k 1)X
ft
Eliminating X we have
Hence
(a k) 2 ^1 + )(l + ^ a k =  .... (4)
From this equation we obtain two values of k. Let these be de
noted by
k, 2 k
and let the corresponding values of X, be
x, 2 x .
Then, we have these equations.
and it is easily seen that the integrals of these equations are
y + x y = 1 C cos. t V ( k g) + D sin. t V ( l k g)
y + 2 Xy = 2 Ccos.t V ( 2 kg) + 2 Dsin.t V ( 2 kg)
C, 1 D, 2 C, 2 D being arbitrary constants. But we may suppose
1 C = E cos. >e
D = E sin. >e
*C = 2 E cos. 2 e
D 2 = 2 E sin. 2 e
By introducing these values we find
y + X y = E cos. {t V ( k g) + ej
y + 2 X y = 2 E cos. [i V ( 2 k g) + 2 c}
From these we easily find
The arbitrary quantities *, e, &c. depend on the initial position and
314 A COMMENTARY ON [SECT. X.
velocity of the points. If the velocities of P, Q = 0, when t = 0, we
shall have
% 2 e, each =
as appears by taking the Differentials of y, y .
If either of the two J E, 2 E be = 0, we shall have (supposing the latter
case and omitting l e)
y = 8  j cos. t V ( k g)
y =
Hence it appears that the oscillations in this case are symmetrical : that
is, the bodies P, Q come to the vertical line at the same time, have similar
and equal motions on the two sides of it, and reach their greatest dis
tances from it at the same time. It is easy to see that in this case, the
motion has the same law of time and velocity as in a cycloidal pendulum ;
and the time of an oscillation, in this case, extends from when t = to
when t V ( J k g) = ir. Also if /3, /3 be the greatest horizontal deviation
of P, Q, we shall have
y = j3 . cos. t V ( : k g)
y = /S .cos. t V ( kg).
In order to find the original relation of /3, 3 , (the oscillations will be
symmetrical if the forces which urge P, Q to the vertical be as P M, Q N,
as is easily seen. Hence the conditions for symmetrical oscillation might
be determined by finding the position of P, Q that this might originally
be the relation of the forces) that the oscillations may be of this kind, the
original velocities being 0, we must have by equation (5) since 2 E = 0.
)8 + 2 X /3  0.
Similarly, if we had
8 + X /3 =
we should have *E = 0, and the oscillations would be symmetrical, and
would employ a time
When neither of these relations obtains, the oscillations may be consi
dered as compounded of two in the following manner : Suppose that we
put
y = Hcos. t V ( kg) + Kcos. t V ( 2 kg) ... (7)
omitting *e, 2 e, and altering the constants in equation (6) ; and suppose
that we take
M p = H . cos. t V ( l k g) ;
BOOK I.] NEWTON S PRINCIPIA. 315
Then p will oscillate about M according to the law of a cycloidal pen
dulum (neglecting the vertical motion). Also
p P will  K . cos. t V ( 2 k g).
Hence, P oscillates about p according to a similar law, while p oscil
lates about M. And in the same way, we may have a point q so moved,
that Q shall oscillate about q in a time
while q oscillates about N in a time
V( kg)
And hence, the motion of the pendulum A P Q is compounded of the
motion A p q oscillating symmetrically about a vertical line, and of A P Q
oscillating symmetrically about A p q, as if that were a fixed vertical line.
When a pendulum oscillates in this manner it will never return exactly
to its original position if V *k, V 2 k are incommensurable.
If V l k, V 2 k are commensurable so that we have
m V : k = n V 2 k
m and ri being whole numbers, the pendulum will at certain intervals, re
turn to its original position. For let
t V ( k g) = 2 n r
then
t V ( 2 k g) = 2 m T
and by (7)
y = H cos. 2 n *  K . cos. 2 m T
= H + K,
which is the same as when
t = 0.
And similarly, after an interval such that
t V ( l k g) = 4 n T, 6 n T, &c.
the pendulum will return to its original position, having described in the
intermediate times, similar cycles of oscillations.
408. Ex. Let (if = p,
a = a
to determine the oscillations.
Here equation (4) becomes
a 2 k 2 4 ak = 2
and
a k = 2 + V 2.
316 A COMMENTARY ON
Also, by equation (3)
[SECT. X
a k = 3 X
.. x = 1 + V 2, 2 X = 1 V 2.
Hence, in order that the oscillations may be symmetrical, we must
either have
/3 + ( I + V 2) j3 = 0, whence /3 = ( V 2 I)
or
f3 ( V 2 1) (S = 0, whence /3 = ( V 2 + 1) 0.
The two arrangements indicated by these equations are thus repre
sented.
Q N Q
The first corresponds to
/3 = (V 2 + l)./3
or
In this case, the pendulum will oscillate into the position A P Q , simi
larly situated on the other side of the line ; and the time of this complete
oscillation will be
In the other case, corresponding to
P = (V 2 l)/3
Q is on the other side of the vertical line, and
QN=(V2 1)PM.
The pendulum oscillates into the position A P Q , the point O remain
ing always in the vertical line ; and the time of an oscillation is
<jt /a
V (2 + V~2)+J g~
The lengths of simple pendulums which would oscillate respectively in
these times would be
2 _ V 2 and 2 + V 2
BOOK I ]
NEWTON S PRINCIPIA.
317
or
1 .707 a and .293 a.
If neither of these arrangements exist originally, let 8, /3 be the origi
nal values of y, y when t is 0. Then making t = in equation (5), we
have
E = 8 + ( V 2 + 1) S
and
2 E = /3 (V 2 1) /3 .
And these being known, we have the motion by equation (6).
409. Any number of material points P 1} P 2 , P 3 . . . Q,
^awg &/ means of a string without weight^ from a point
A ; it is required to determine their small oscillations in
a vertical plane.
Let A N be a vertical abscissa, and PJ M,, P 2 M 2 ,
&c. horizontal ordinates ; so that
A M! = x l5 A M 2 = x 2 , &c.
PI M! =.y,, P 2 M 2 = y 2 , &c.
A P! = a l5 P! P 2 = a 2 , &c.
tension of A P! = p l9 of P 1 P 2 = p 2 , &c.
mass of P! = p lt of P 2 = /4 2 , &c.
Hence, we have three equations, by resolving the forces parallel to the
horizon.
d2 yi _ PI g yi .PS g y a 3
d t 2 " ~LL ~FL~ * T
d 2 y 2 p 2 g y 2 ;
i rj r>
d t * ~~ ~ ~ ~ " 7T~
\i L fj^ tl^ fJ^2
" y?  Pag ys ya , p* g
2 "~
Tt
y 3
d t 2
^n _ P n
y n y n _
. . . (1)
And as in the last, it will appear that p,, p 2 , &c. may, for these small
oscillations, be considered constant, and the same as in the state of rest.
Hence if
then
P! = M, p 2 = M A&I, p 3 = M ^,  /Kg, &c.
Also, dividing by g, and arranging, the above equations may be put in
this form :
S18
A COMMENTARY ON
r SECT. X.
gdt
PI
;)y> +
y*
^2 _ 2_yi __ / p 2
(;
Pa
a 2
y 2 +
PS y 3
lL  P2_Z* __ / Ps , P4 \ ,
i 2 " /. Q \ "T" _ I J 3 ~i
r^3 3 r^i a*} f^ \ **4*
(1)
u v D v i n v
" jn ^ h n Jn I Pn Jn
gdt 2 /* a n /i n a n
The first and last of these equations become symmetrical with the rest
if we observe that
y = o
and
Pn + i = 0.
Now if we multiply these equations respectively by
1, X, X , X", &c.
and add them, we have
f\ 2 TT I % A 2 T T I \t A 2 ,, i 5irr
PI
a 3
a 3
a 4
wn  1 a n /z. n a n
and this will be integrable, if the righthand side of the equation be redu
cible to this form
k (y, + X y 2 + X y 3 + &c.).
That is, if
k _ _Pj_
, _
kx =
(n  2) _ __
/ (n  3) n > / (n  2)
_ Pn
n a n
(3)
BOOK I.] NEWTON S PRINCIPIA. 319
If we now eliminate
X, X , X", &c.
from these n equations, it is easily seen that we shall have an equation of
n dimensions in k.
Let
% 2 k, 3 k ...... k
be the n values of k ; then for each of these there is a value of
X , X", X"
easily deducible from equations (3), which we may represent by
X, X , >X", &c.
2 X , 2 X", 2 X ", &C.
Hence we have these equations by taking corresponding values X and k,
_ t
and so on, making n equations.
Integrating each of these equations we get, as in the last problem
yi + * y 2 + * y 3 + &c. = E cos. ft V (>k g ) + e $ 1 , r .
yi + 2 * y 2 + 2 > J 3 + &c. = 2 E cos. {t V ( 2 k g) + 2 e] )
1 E, 2 E, &c. ! e, 2 e, &c. being arbitrary constants.
From these n simple equations, we can, without difficulty, obtain the n
quantities y l5 y g , &c. And it is manifest that the results will be of this
form
yi= I H 1 cos.{t V ( kg) + 1 e}+ 2 H 1 cos.{lV( 8 kg) + 2 e} + &c.j
y^^cos.Jt V Ckg) + 1 e}+ 2 H 2 cos.Jt\/( 2 kg) + 2 e] + &c. V . . . (6)
&c. = &c. )
where H^ Ha, &c. must be deduced from (Sj, /3 2 , &c. the original values
of yi, y 2 , &c.
If the points have no initial velocities (i. e. when t = 0) we shall have
E = 0, 2 E = 0, &c.
We may have symmetrical oscillations in the following manner. If,
of the quantities 1 E, *E, 3 E, &c. all vanish except one, for instance n E ; we
have
yi + ^ y 8 + ^ y 3 + &c. = o ^
yi + ^y 2 + 2 ^y3 + &c. = o
yi + 3 ^y a + 3 x y 3 + &c. = o k   (T)
._.__
yi+ n ^y a + n ^y3+&c. n Ecos.tV( n kg)J
omitting n E.
320 A COMMENTARY ON [SECT. X.
From the n 1 of these equations, it appears that y 2 , y 3 , &c. are in a
given ratio to y l ; and hence
n > y3 + &c.
is a given multiple of y t and = m yj suppose. Hence, we have
m y! s= n E cos. V ( n k g) ;
or, omitting the index n, which is now unnecessary,
m y l = E cos. t V (k g).
Also if y 2 = e 2 y lt
m y 2 = E e 2 cos. t V (k g)
and similarly for y 3 &c.
Hence, it appears that in this case the oscillations are symmetrical. All
the points come into the vertical line at the same time, and move similar
ly, and contemporaneously on the two sides of it. The relation among
the original ordinates ft, /3 2 , /3 3 , &c. which must subsist in order that the
oscillations may be of this kind, is given by the n 1 equations (7),
ft + 1 X& + X /3 3 +&c. =
ft + 2 */3 2 + 2 >//3 3 + &c. =
ft + 3 *& + 3 >//3 3 + &c. =
&c. = &c.
These give the proportion of ft /3 2 , &c; the arbitrary constant n E, in
the remaining equation, gives the actual quantity of the original displace
ment.
Also, we may take any one of the quantities L E, S E, 3 E, &c. for that
which does not vanish ; and hence obtain, in a different way, such a sys
tem of n 1 equations as has just been described. Hence, there are n
different relations among ft ft, &c. or n different modes of arrangement,
in which the points may be placed, so as to oscillate symmetrically.
( We might here also find these positions, which give symmetrical oscil
lations, by requiring the force in each of the ordinates Pj MI, P 2 M 2 to
be as the distance; in which case the points P M P 2 , &c. would all come
to the vertical at the same time.
If the quantities V l k, V 2 k have one common measure, there will be
a time after which the pendulum will come into its original position. And
it will describe similar successive cycles of vibrations. If these quantities
be not commensurable, no portion of its motion will be similar to any
preceding portion.)
The time of oscillation in each of these arrangements is easily known ;
the equation
m yi = n E cos. t V ( n k g)
BOOK L] NEWTON S PRINCIPIA. 321
shows that an oscillation employs a time
And hence, if all the roots : k, 2 k, 3 k, &c. be different, the time is dif
ferent for each different arrangement.
If the initial arrangement of the points be different from all those thus
obtained, the oscillations of the pendulum may always be considered as
compounded of n symmetrical oscillations. That is, if an imaginary pen
dulum oscillate symmetrically about the vertical line in a time
A/Ckg)
and a second imaginary pendulum oscillate about the place of the first,
considered as a fixed line, in the time
and a third about the second, in the same manner, in the thnc
x
and so on ; the n th pendulum may always be made to coincide per
petually with the real pendulum, by properly adjusting the amplitudes of
the imaginary oscillations. This appears by considering the equations
(6), viz.
yi = Hj cos. t V ( k g) + 2 H! cos. t V ( 2 k g) + &c.
&c. = &c.
This principle of the coexistence of vibrations is applicable in all cases
where the vibrations are indefinitely small. In all such cases each set of
symmetrical vibrations takes place, and affects the system as if that were
the only motion which it experienced.
A familiar instance of this principle is seen in the manner in which the
circular vibrations, produced by dropping stones into still water, spread
from their respective centers, and cross without disfiguring each other.
If the oscillations be not all made in one vertical plane, we may take a
horizontal ordinate z perpendicular to y. The oscillations in the direc
tion of y will be the same as before, and there will be similar results ob
tained with respect to the oscillations in the direction of z.
We have supposed that the motion in the direction of x, the vertical
axis, may be neglected, which is true when the oscillations arc very
small.
410. Ex. Let there be three bodies all equal (each = /a,), and also their
distances a 1} a. 2 , a 3 all equal (each = a).
VOL. I. X
322 A COMMENTARY ON [SECT. X.
Here
p = 3 (*, p. 2 = 2 ft, p 3 = a
and equations (3) become
a k = 5 2 X
a k X = 2 + 3 X X
a k >/ = X + X .
Eliminating k, we have
5 X 2 X 2 = 2 + 3 X X ,
5 x 2 X X = X + X ,
or
X = 2X 2 2 X 2,
4 X 2 X X = X
v
.. x =
2 X 4
... (2X 2 2X 2)(2X 4) = X
or
X 3_3X 2 + ^X + 2 =0,
4
which may be solved by Trigonometrical Tables. We shall find three
values of X.
Hence, we have a value of X corresponding to each value of X ; and
then by equations (7)
ft f x ft + X X ft = \ , 7 v
j3 + 2 x ft + 2 x ft = J
whence we find ft, ft in terms of ft.
We shall thus find
ft = 2. 295 &
or
ft = ]. 348 ft
or
ft = .643,3,
according as we take the different values of X.
And the times of oscillation in each case will be found by taking tiie
value of
a k = 5 2 X;
that value of X being taken which is not used in equation (7 ). For the
time of oscillation will be given by making
t V (k g) = cr.
If the values of ft, ft, ft have not this initial relation, the oscillations
BOOK I.I
NEWTON S PRINCIPIA.
323
will be compounded in a manner similar to that described in the example
for two bodies only.
411. A flexible chain, of uniform thidcness, hangs from a Jixed point :
to find its initial form, that its small oscillations may be symmetrical.
Let A M, the vertical abscissa x ; M P the hori
zontal ordinate = y; A P = s, and the whole length
A C = a;
.. A P = a s.
And as before, the tension at P, when the oscillations
are small, will be the weight of P C, and may be represent
ed by a s. This tension will act in the direction of a
tangent at P, and hence the part of it in the direction
P M will be
C
tension X
ds
or
(a  s) $* .
d s
Now, if we take any portion P Q = h, we shall find the horizontal
force at Q in the same manner. For the point Q, supposing d s constant
h d 3 " ^ 2
dy . dy , d 2
* becomes 3= + =
ds d s d s
(see 32).
Also, the tension will be a s f h.
the direction N Q, is
y L ,
1 ^ d s 3
IT2 &C 
Hence the horizontal force in
Subtracting from this the force in P M, we have the force on P Q
horizontally.
h <P h 2
+ Z + &c.)
s d s 2 1 /
and the mass of P Q being represented by h, the accelerating force
( =  *j is found. But since the different points of P Q move
* rnuss /
with different velocities, this expression is only applicable when h is inde
finitely small. Hence, supposing Q to approach to and coincide with P,
we have, when h vanishes
12 1
accelerating force on P = (a s) , \ ^ .
d s 2 d s
X2
324 A COMMENTARY ON [SECT. X.
But since the oscillations are indefinitely small, x coincides with s and
we have
d 2 v d v
accelerating force on P = (a x) j ^.
dx 2 dx
Now, in order that the oscillations may be symmetrical, this force must
be in the direction P M, and proportional to P M, in which case all the
points of A C, will come to the vertical A B at once. Hence, we must
have
(a x) , ^ ^~ = kdy (!)
dx 2 d x
k being some constant quantity to be determined.
This equation cannot be integrated in finite terms. To obtain a
series let
y = A + B. (a x)+ C(a x) 2 + &c.
..i = B 2C(a x) 3D (a x) 2
... <JlZ = 1. 2. C + 2. 3 D (a x) + &c.
(I *v
Hence
~ v " dx 2 dx *
gives
= 1. 2. C (a x) + 2. 3 D (a x) 2 + &c.
+ B + 2 C (a x) + 3 D (a x) 2 + &c.
+ kA + kB(a x) + k C (a x) 2 + &c.
Equating coefficients ; we have
B = _ k A,
2 2 C= k B
3 2 D = k C
&c. = &c.
.. B = k A
k 2 A
C =
D =
2 2
k 3 A
2 2 .3 2
&c. = &c.
and
..(2)
BOOK L] NEWTON S PRINCIPIA. 325
Here
A is B C, the value of y when x = a. When x = 0, y = ;
k 2 a 2 k 3 1 3
. 1 L. r, _L I . _i_ &T O f^\
 1 K a f g 2 ga 32 "T "~ W
From this equation (k) may be found. The equation has an mfinite
number of dimensions, and hence k will have an infinite number of values,
which we may call
l ]f 2 k n t l
2v, Iv, ... IV . . . J ,
and these give an infinite number of initial forms, for which the chain
may perform symmetrical oscillations.
The time of oscillation for each of these forms will be found thus. At
the distance y, the force is k g y : hence by what has preceded, the time
to the vertical is
T
sVTFg)
and the time of oscillation is
(The greatest value of k a is about 1.44 (Euler Com. Acad. Petrop.
torn. viii. p. 43). And the time of oscillation for this value is the same as
2
that of a simple pendulum, whose length is a nearly.)
o
The points where the curve cuts the axis will be found by putting y = 0.
Hence taking the value n k of k, we have
n^a/o _ Y \ 2 n 1, 3 / _ x \ *
0= ln k(a _ x)+ _ (*J=2 _ + k ^ 32 X) +&C.
which will manifestly be verified, if
n k (a x) = k a
or
n k ( a _ x ) = 2k a
or
*k(a x) = 3 ka
&c. = &c.
because l k a, 2 k a, &c. are roots of equation (3).
That is if
x = a l  or = a J ~ or = &c 
Suppose k, 2 k, 3 k, &c. to be the roots in the order of their magnitude
k being the least.
Then if for n k, we take k, all these values of x will be negative, and
the curve will never cut the vertical axis below A.
X3
326
A COMMENTARY ON
[SECT. X.
If for n k, we take *k, all the values of x will be negative except the
first; therefore, the curve will cut A B In one point. If we take 3 k, all
the values will be negative except the two first, and the curve cuts A B
in two points ; and so on.
Hence, the forms for which the oscillations will be
symmetrical, are of the kind thus represented.
And there are an infinite number of them, each
cutting the axis in a different number of points.
If we represent equation (2) in this manner
y = A <f> (k, x)
it is evident that
y = 1 A? (>k, x)
y = 2 A p ( 2 k, x)
&c. = &c.
will each satisfy equation (1). Hence as before, if we put
y = A p ( k, x) + 2 A p ( 2 k, x) + &c.
and if A, *A, &c. can be so assumed that this shall represent a given
initial form of the chain, its oscillations shall be compounded of as many
coexisting symmetrical ones, as there are terms A, 2 A, &c.
We shall now terminate this long digression upon constrained mo
tion. The reader who wishes for more complete information may con
sult Whewell s Dynamics, one of the most useful and elegant treatises
ever written, the various speculations of Euler in the work above quoted,
or rather the comprehensive methods of Lagrange in his Mecanique
Analytique.
We now proceed to simplify the text of this Xth Section.
412. PROP. L. First, S II Q is formed by an entire revolution of the
generating circle or wheel, whose diameter is O R, upon the globe
SOQ.
413. Secondly, by taking
C A : C O : : C O : C R
we have
CA: CO:: CA CO: CO CR
: : A O : O R
and therefore if C S be joined and produced to meet the exterior globe
in D, we have also
AD : SO(:: C A: CO) :: AO: OR.
But
S O = the semicireumference of the wheel O R = ^^ .
BOOK I.] NEWTON S PR1NCIPIA. 327
. .AD = ^ = the circumference of the wheel whose diameter is
it
A O. That is S is the vertex of the Hypocycloid A S, and A S is per
pendicular in S to C S. But O S is also perpendicular to C S. There
fore A S touches O S in S, &c.
414. The similar jigures A S, S R.]
By 39 it readily appears that Hypocycloids are similar when
R : r : : R : r
R and r being the radii of the globe and wheel : that is when
C A : AO ::CO : O R
or when
CA:CO::CO:CR
.. A S, S R are similar
415. V B, V W are equal to O A, O R.]
If B be not in the circumference AD let C V meet it in B . Then
V P being a tangent at P, and since the element of the curve A P is the
same as would be generated by the revolution of B P around B as a
center, and .. B P is perpendicular both to the curve and its tangent
P V, therefore P B, P B and .. B, B coincide. That is
V B = O A.
Also if the wheel O R describes O V whilst A O describes A B, the
angular velocity B P in each must be the same, although at first, viz. at
O and A, they are at right angles to each other. Hence when they shall
have arrived at V and B their distances from C B must be complements
of each other. But
.TVW = BVP=5 PBV
m
/. T V is a chord in the wheel O R, and
.. V W = O R.
See also the Jesuits note.
OTHERWISE.
416. Construct the curve S P, to which the radius of curvature to every
point of S R Q is a tangent ; or which is the same, find S A the Locus of
the Centers of Curvature to S R Q.
Hence is suggested the following generalization of the Problem, viz.
417. To make a body oscillate in any given curve.
Let S R Q (Newton s fig.) the given curve be symmetrical on both sides
X4
328 A COMMENTARY ON [SECT. X.
of R. Then if x, y be the rectangular coordinates referred to the vertex
R, and a, (3 those of the centers of curvature (P) we have
r 2 _ p T 2 _ (y /3) 2 + ( X a) 2 .
Hence, the contact being of the second order (74)
X + (y 0)^ = (1)
and
d v 2 d 2 v
!+ HI + &$< (a)
These two equations by means of that of the given curve, will give us
Q in terms of , or the equation to the Locus of the centers of curvature.
Let S A be the Locus corresponding to S R, and A Q the other half.
Then suspending a body from A attached to a string whose length is R A,
when this string shall be stretched into any position APT, it is evident
that P being the point where the string quits the locus is a tangent to it,
and that T is a point in S R Q.
Ex. 1. Let S R Q be the common parabola.
Here
y 2 = 2 a x
d y a
d x y
d 2 y ady a 2 a
dx 2 = " y* d~x = ~ y" 3 ~ ~ 2 x y
/. substituting we get
/S).! =
v
and
... x a + fl + ^} . ^ = = 3x afa
V V. 2 x/ a
or
and
But
y 8 = 2 a x
8x ;
BOOK I.] NEWTON S PRINCIPIA. 329
8 (a a) 3 8 ..
_ vx V / _ __ /~ o I 3 /Ql
s\ ~nri ~ "fjry * \ ~~~ I \ )
Now when /3 = 0, a = a; which shows that A R the length of the
string must equal a. Also making A the origin of abscissas, that is, aug
menting a, by a, we have
 x
the equation to the semicubical parabola A S, A Q, which may be traced
by the ordinary rules (35, &c.); and thereby the body be made to oscillate
in the common parabola S Q R.
Ex. 2. Let S R Q be an ellipse.
Then, referring to its center, instead of the vertex,
or
b 2 x 2 = a 2 b
a y
d y
d x
... a 2 y + b 2 x =
J
and
ii y TJ r B j q p  v.
J dx d x. 2
These give
d y b 2 X.
dx. ~ ~ a 2 y
and
d^y b 4
dx 2 = a 2 y 3
Hence
(a 2 b 2 ) x 3
a 3 i
a 4
and
^= (a "" b 4 )y3 
Hence substituting the values of y and x in
a 2 y
we get
* M r\ i^ x *< o v .*_
(a)
b \f / a a \
the equation to the Locus of the centers of curvature.
330
A COMMENTARY ON
[SECT. X.
In the annexed figure let
SC = b, CR = a
C M = x, T M = y.
Then
P N = ft C N = .
And to construct A S by points, first put s
whence by equation (a)
j_ a 2 b *
C6 ^ "~
a
the value of A C. Let
a. =
then
S = + =^
the value of S C or C Q .
Hence to make a body oscillate in the semiellipse S R Q we must
take a pendulum of the length A R, (part = A P S flexible, and part
= S S rigid ; because S S is horizontal, and no string however stretched
can be horizontal see Whewell s Mechanics,) and suspend it at A.
Then A P being in contact with the Locus AS , P T will also touch
A S in P, &c. &c.
Ex. 3. Let S R Q be the common cycloid ,
The equation to the cycloid is
1?  /
d x ~ V
 /f? r
V V
d*y
dx 2=
whence it is found that
y
1
Hence
and
d^ _ 2r y
dx ~ y
dv
dx
/.
2r y
y
. /^. 3
do" Ar y
which is also the equation of a cycloid, of which the generating circle is
BOOK L] NEWTON S PRINCIPIA. 331
precisely the same as the former, the only difference consisting in a change
of sign of the ordinate, and of the origin of the abscissae.
The rest of this section is rendered sufficiently intelligible by the
Notes of P. P. Le Seur and Jacquier ; and by the ample supplementary
matter we have inserted.
SECTION XL
417. PROP. LVII. Two bodies attracting one another, describe round
each other and round the center of gravity similar figures.
Q
Since the mutual actions will not affect the center of gravity, the bodies
will always lie in a straight line passing through C, and their distances
from C will always be in the same proportion.
.. S C : T C : : P C : Q C
and
zSCT = QCP.
.*. the figures described round each other are similar.
Also if T t be taken = S P, the figure which P seems to describe
round S will be t Q, and
Tt : TQ:: S P: TQ
: : C P : C Q
and
t 1 T Q = P C Q.
.. the figures t Q, P Q, are similar ; and the figure which S seems to
describe round P is similar, and equal to the figure which P seems to
describe round S.
418. PROP. LVIII. If S remained at rest, a figure might be de
scribed by P round S, similar and equal to the figures which P and S
seem to describe round each other, and by an equal force.
332
A COMMENTARY ON
[SECT. XI.
Q
R
8
Curves are supposed similar and Q R, q r indefinitely small. Let P and
p be projected in directions P R, p r (making equal angles C P R, s p r)
with such velocities that
V V~S
_ V CP _ V PQ
P v sp ^ pq
Then (si
since a t =
ds
v
PQ
pq VPQ
Vp q VPQ _ V QR
V q r
Vpq
C
But in the beginning of the motion f =
F _ QR jr l_
f : : ~qr Q R =: 1
The same thing takes place if the center of gravity and the whole system
move uniformly forward in a straight line in fixed space.
419. COB. 1. If F cc D, the bodies will describe round the common
center of gravity, and round each other, concentric ellipses, for such would
be described by P round S at rest with the same force.
Conversely, if the figures be ellipses concentric, F D.
420. COR. 2. If F <x  the figures will be conic sections, the foci in
the centers of force, and the converse.
421. COR. 3. Equal areas are described round the center of gravity,
and round each other, in equal times.
V
422. COR. 3. Otherwise. Since the curves are similar, the areas, bounded
by similar parts of the curves, are similar or proportional.
.. spq : C P Q : : sp 2 : C P 2 : : (S + P) 2 : s 2 in a given ratio;
BOOK 1.1 NEWTON S PRINCIPIA. 333
and T. through s p q : T. through CPQ:: VS + P: V S, in a given ratio
and .. : : T. through spv: T. through CPV
.. T. through C P Q : T. through CPV:: T. through spq : T. through spv
: : s p q : s p v (by Sect. II.)
::CPQ:CPV
.. the areas described round C are proportional to the times, and the
areas described round each other in the same times, which are similar to
the areas round C, are also proportional to the times.
423. PROP. LIX. The period in the figure described in4ast Prop.
: the period round C : : V S + P : V S ; for the times through similar
arcs p q, P Q, are in that proportion.
424. PROP. LX. The major axis of an ellipse which P seems to de
scribe round S in motion (Force <x jrzl major axis of an ellipse which
would be described by P in the same time round S at rest : : S + P :
of two mean proportionals between S + P and S.
Let A = major axis of an ellipse described (or seemed to be described)
round S in motion, and which is similar and equal to the ellipse de
scribed in Prop. LVIII.
Let x = major axis of an ellipse which would be described round S at
rest in the same time.
period in ellipse round S in motion V S /p T TV\
period in same ellipse round Sat rest "" ^/~s^f^P r ^
and by Sect. Ill,
period in ellipse round S at rest A *
period in required ellipse round S at rest ~ f
A,
period in ellipse round S in motion A* V~S
period in required ellipse round S at rest ~~ I v~S~^TP
but these periods are to be equal,
.. A 3 S = x 3 .S~+~p
3
.. A:x:: V S + P: V S::S+ P: first of two mean proportionals
(for if a, a r, a r 2 , a r 3 , be proportionals, V~o. : V a r 3 : : a : a r.)
425. At what mean distance from the earth would the moon revolve
round the earth at rest, in the same time as she now revolves round the
earth in motion ? This is easily resolved.
426. PROP. LXI. The bodies will move as if acted upon by bodies at
the center of gravity with the same force, and the law of force with re
334 A COMMENTARY ON [SECT. XI.
spect to the distances from the center of gravity will be the same as with
respect to the distances from each other.
For the force is always in the line of the center of gravity, and .. the
bodies will be acted upon as if it came from the center of gravity.
And the distance from the center of gravity is in a given ratio to the
distance from each other, .. the forces which are the same functions of
these distances will be proportional.
427. P$,OP. LXII. Problem of two bodies with no initial Velocities.
F oc _ . Two bodies are let fall towards each other. Determine the
motions.
The center of gravity will remain at rest, and the bodies will move as
if acted on by bodies placed at the center of gravity, (and exerting the
same force at any given distance that the real bodies exert),
.. the motions may be determined by the 7th Sect.
428. PROP. LXIII. Problem of two bodies with given initial Velo
cities.
F ex j . Two bodies are projected in given directions, with given
velocities. Determine the motions.
The motion of the center of gravity is known from the velocities and
directions of projection. Subtract the velocity of the center of gravity
from each of the given velocities, and the remainders will be the velocities
with which the bodies will move in respect of each other, and of the cen
ter of gravity, as if the center of gravity were at rest. Hence since they
are acted upon as if by bodies at the center of gravity, (whose magnitudes
are determined by the equality of the forces), the motions may be deter
mined by Prop. XVII, Sect. Ill, (velocities being supposed to be acquired
down the finite distance), if the directions of projection do not tend to the
center, or by Prop. XXXVII, Sect. VII, if they tend to or directly from
the center. Thus the motions of the bodies with respect to the center of
gravity will be determined, and these motions compounded with the uni
form motion of the center of gravity will determine the motions of the
bodies in absolute space.
429. PIIOP. LXIV. F oc D, determine the motions of any number of
bodies attracting each other.
BOOK I.] NEWTON S PRINCIPIA. 335
T and L will describe concentric
ellipses round D.
Now add a third body S.
Attraction of S on T may be re
presented by the distance T S, and
on L by L S, (attraction at distance
being 1) resolve T S, L S, into
T D, D S ; L D, D S, whereof the
parts T D, L D, being in given
ratios to the whole, T L, L T, will
only increase the forces with which
L and T act on each other, and
the bodies L and T will continue to describe ellipses (as far as respect
these new forces) but with accelerated velocities, (for in similar parts of
similar figures V 2 F.R Prop. IV. Cor. 1 and 8.) The remaining
forces D S, and D S, being equal and parallel, will not alter the relative
motions of the bodies L and T, .. they will continue to describe ellipses
round D, which will move towards the line I K, but will be impeded in
its approach by making the bodies S and D (D being T + L) describe
concentric ellipses round the center of gravity C, being projected with
proper velocities, in opposite and parallel directions. Now add a fourth
body V, and all the previous motions will continue the same, only accel
erated, and C and V will describe ellipses round B, being projected with
proper velocities.
And so on, for any number of bodies.
Also the periods in all the ellipses will be the same, for the accelerating
force onT = L.TL+ S . TD = (T+L) . TD+S. T D = (T+L +S).
T D, i. e. when a third body S is added, T is acted on as if by the sum
of the three bodies at the distance T D, and the accelerating force on D
towards C=S.SD = S.CS+S.DC = (T+L).DC+S.DC
 (T + L + S). D C.
. . accelerating force on T towards D : do. on D towards C : : T D : D C
o
. . the absolute accelerating forces on T and D are equal, or T and D
move as if they revolved round a common center, the absolute force the
same, and varying as the distance from the center, i. e. they describe el
lipses, in the same periods.
Similarly when a fourth body V is added, T, L, D, S, C, and V, move
as if the four bodies were placed at D, C, B, i. e. as if the absolute forces
were the same, and with forces proportional to their respective distances
from the centers of gravity, and .. in equal periods.
53G
A COMMENTARY ON
[SECT. XL
And so on, for any number of bodies.
430. PROP. LXVI. S and P revolve round T, S in the exterior orbit,
P in the interior,
F oc , find when P will describe round T an orbit nearest to the
ellipse, and areas most nearly proportional to the times.
(1st.) Let S, P, revolve round the greatest body T in the same plane.
Take K S for the force of S on P at the mean distance S K,
and
L S = S K .
= force at p >
resolve L S into L M, M S,
L M is parallel to P T, and .. tends to the center T, .. P will con
tinue to describe areas round T proportional to the times, as when acted
on only by P T, but since L M does not oc p~ff~z > tne sum of L M and
P T will not TZ , .. the form of the elliptic orbit P A B will be
disturbed by this force, L M, M S neither tends from P to the center
T, nor oc
. from the force M S both the proportionality of areas
P T 2
to times, and the elliptic form of the orbit, will be disturbed, and the
elliptic form on two accounts, because M S does not tend to C, and be
cause it does not p~q^i
. . the areas will be most proportional to the times, when the force
M S is least, and the elliptic form will be most complete, when the forces
M S, L M, but particularly L M, are least.
Now let the force of S on T = N S, then this first part of the force
M S being common to P and T will not affect their mutual motions, .. the
BOOK 1.]
NEWTON S PRINCIPIA.
337
disturbing forces will be least when L M, M N, are least, or L M remain
ing, when M N is least, i. e. when the forces of S on P and T are nearly
equal, or S N nearly = S K.
(2dly) Let S and P revolve round T in different planes.
Then L M will act as before.
But M N acting parallel to T S, when S is not in the line of the Nodes,
(and M N does not pass through T), will cause a disturbance not only
in the longitude as before, but also in the latitude, by deflecting P from
the plane of its orbit. And this disturbance will be least, when M N is
least, or S N nearly = S K.
431. COR. 1. If more bodies revolve round the greatest body T, the
motion of the inmost body P will be least disturbed when T is attracted
by the others equally, according to the distances, as they are attracted by
each other.
432. COR. 2. In the system of T, if the attractions of any two on the
third be as y^, , P will describe areas round T with greater velocity near
conjunction and opposition, than near the quadratures.
433. To prove this, the following investigation is necessary.
Take 1 S to represent the attraction of S on P,
n S T,
Then the disturbing forces are 1 m (parallel to P T) and m n.
Now 
SI?
O I c, TJ
S.R
SP~ (R 2 2Rrcos.
S.R
V R 2 2Rrcos. A
JL C0 ^: 4. l\ /i 2 r cos. A r 2
R h RVV ~~KT r R
VOL. I.
338 A COMMENTARY ON [SECT XT.
S / 2r
S /. 3/2r r 2 x 3. 5 /2rcos. A rVo N
= KA 1 + rw cos  A  RI) + 274 ( R wl &c )
S /. 3r /3 3.5
= RTA 1 + Tr cos  A ~ (2  2^ cos<
S /. 3 r. cos. A\
= R 2 ^ ~R /
where R is indefinitely great with respect to r.
Also
Q Q^ ^ /, 3 r cos. A\ S S.Srcos.
Sn= w (l+ _^__)_ R2= __
ultimately
and Ira = SI.  = ~ (R 2 2 R r cos. A + r 2 )
* (R 2 ~2 Rr cos. A + r 2 )
2 &c
.
" R 3 R
= p^Y ultimately.
434. Call 1 m the addititious force
and m n the ablatitious force 
and m n = 1 m 3 cos. A.
Resolve m n into m q, q n.
The part of the ablatitious force which acts in the direction m q
= m n . cos. A
3 . S . r. cos. 2 A
R
= central ablatitious force.
3 S r
The tangential part r= m n . sin. A = ~WT~ sm  ^ cos  A
o Q r
=  . j^ . sin. 2 A = tangential ablatitious force
& JLv
., i i r ., j Tjrr, i S.r 3.S.r.cos. 2 A
% . the whole force m the direction PT = lm mq = ~, ^r^
it it
= ^ (l _ 3 cos. 2 A) and the
R 3 v
3 S.r
, whole force in the direction of the Tangent = q n = . ~^j . sin. 2 A.
til At
435. Hence COR. 2. is manifest, for of the four forces acting on P, the
BOOK I.]
NEWTON S PRINCIPJA.
339
three first, namely, attraction of T, addititious force, and central ablatiti
ous force, do not disturb the equable description of areas, but the fourth
or tangential ablatitious force does, and this is f from A to B, from B
to C, + from C to D, from D to A. /. the velocity is accelerated from A
to B, and retarded from B to C, /. it is greatest at B. Similarly it is a
maximum at D. And it is a minimum at A and C. This is Cor. 3.
436. To otherwise calculate the central and tangential ablititious forces.
On account of the great distance of S, S M, P L may be considered
parallel, and
.. P T = L M, and S P = S K = S T.
.. the ablatitious force = 3 P T. sin. 6 = 3 P K.
Take P m = 3 P K, and resolve it into P n, n m.
P n =; P m . sin. 6 = 3 P T. sin. 2 6 = central ablatitious force
= 3 p T. ] cos  2 *
n m = P m . cos, 6 3 P T. sin. 6 cos. 6 = ^ . P T. sin. 2 6 = tangential
SB
ablatitious force.
The same conclusions may be got in terms of 1 m from the fig. in Art
433, which would be better.
437. Find the disturbing force on P in the direction P T.
This = (addititious + central ablatitious) force = 1 m f 3 1 m . sin. 2 6
"I cos. 2
i 01 /"
= lm_31m(
438. To Jind the mean disturbing force of S during a whole revolution
in the direction P T.
Let P T at the mean distance = m, then 1 m f
Y2
1 3 cos. 2
340 A COMMENTARY ON [SECT. XI.
1m m , !!
= ^ since cos. 2 6 is destroyed during a whole revo
 A
lution.
439. The disturbing forces on P are
(1) addititious = =^ = A.
(2) ablatitious = 3 . A . sin. 6
3
~2
3 . A
which is (1) tangential ablatitious force * . cos. 2 &
I cos> 2
and (2) central ablatitious force = 3 A . 
2
3 A 3 A
.*. whole disturbing force in the direction P T = A j  . cos. 2 &
A , 3 A
= Q + . cos. 2 6.
But in a whole revolution cos. 2 6 will destroy itself, .. the whole dis
turbing force in the direction P T in a complete revolution is ablatitious
and = addititious force.
The whole force in the direction P T = ^j (1 3 sin. 2 6) (Art. 433)
multiply this by d.0, and the integral = ^y (0 6 + . sin. 2 i\
= sum of the disturbing forces ; and this when 6 = ir becomes
i
This must be divided by T, and it gives the mean disturbing force act
CJ
ing on P in the direction of radius vector = & ~t> s
440. The 2d COR. will appear from Art. 433 and 434.
3
For the tangential ablatitious force = . sin 2 6 . X addititious force,
.. this force will accelerate the description of the areas from the quadra
tures to the syzygies and retard it from the syzygies to the quadratures,
since in the former case sin. 2 is +, and in the latter .
441. COR. 3 is contained in COR. 2. (Hence the Variation in as
tronomy.)
BOOK I.]
NEWTON S PRINCIPIA.
341
442. P V is equivalent to P T, T V, and accelerates the motion ;
p V is equivalent to p T, T V, and retards the motion.
443. COR. 4. Cast, par., the curve is of greater curvature in the quadra
tures than in the syzygies.
For since the velocity is greatest in the syzygies, (and the central abla
titious force being the greatest, the remaining force of Pto T is the least)
the body will be less deflected from a right line, and the orbit will be less
curved. The contrary takes place in the quadratures.
444. The whole force from S in the direction P T=~^ (1 3 sin. 2 6}
T
(see 433) and the force from T in the direction P T = 
rp O
. . the whole force in the direction P T = i + ^ (1 3 sin. 2 6)
T S r
and at A this becomes V + 
r 2
at B
at C
atD
r 2
_T
r 2
T
. 2
2 . S . r
R 3
S.r
R 3
2 S. r
r R 3
(for though sin. 270 is , yet its syzygy is +).
Thus it appears that on two accounts the orbit is more curved in the
quadratures than in the syzygies, and assumes the form of an ellipse at
the major axis A C.
Y3
342 A COMMENTARY ON [SECT. XL
.*. the body is at a greater distance from the center in the quadratures
than in the syzygies, which is Cor. 5.
445. COR. 5. Hence the body P, caet. par., will recede farther from
T in the quadratures than in the syzygies ; for since the orbit is less
curved in the syzygies than in the quadratures, it is evident that the body
must be farther from the center in the quadratures than in the syzygies.
446. Con. 6. The addititious central force is greater than the ablati
tious from Q to P, and from P 7 to Q, but less from P to P , and from
Q to Q , .". on the whole, the central attraction is diminished. But it
may be said, that the areas are accelerated towards B and D, and .*. the
time through P P may not exceed the time through P Q, or the time
through Q Q exceed that through Q P. But in all the corollories, since
the errors are very small, when we are seeking the quantity of an error,
and have ascertained it without taking into account some other error,
there will be an error in our error, but this error in the error will be an
error of the second order, and may .*. be neglected.
The attraction of P to T being diminished in the course of a revolution,
the absolute force towards T is diminished, (being diminished by the
S r r &
mean disturbing force i _ s , 439,) .. the period which . is
R 3 V f
increased, supposing r constant.
But as T approaches S (which it will do from its higher apse to the
lower) R is diminished, the disturbing force fwhich involves y] will be
increased, and the gravity of P to T still more diminished, and .. r will
be increased ; . . on both accounts (the diminution of f and increase of r)
the period will be increased.
(Thus the period of the moon round the earth is shorter in summer
than in winter. Hence the Annual equation in astronomy.)
When T recedes from S, R is increased, and the disturbing force di
minished and r diminished. . . the period will be diminished (not in com
parison with the period round T if there were no body S, but in compari
son with what the period was before, from the actual disturbance.)
T 1 C
447. COR. 6. The whole force of P to T in the quadratures = ^+, 1
r 2 R s
T 2Sr
 the syzygies = 
. . on the whole the attraction of P to T is diminished in a revolution.
For the ablatitious force in the syzygies equals twice the addititious force
in the quadratures.
UOOK I.] NEWTON S PRINCIPIA. 343
At a certain point the ablatitious force zr the addititious; when
1 = 3 sin. 2 6
or
sill  = V3
and
A = 55, &c.
P
(the whole force being then = j.
Up to this point from the quadratures the addititious force is greater
than the ablatitious force, and from this point to one equally distant from
the syzygies on the other side, the ablatitious is greater than the addititious ;
.. in a whole revolution P s gravity to T is diminished.
Again since T alternately approaches to and recedes from S, the radius
I* 
P T is increased when T approaches S, and the period cc _____ ___ __ .
V absolute force
and since f is diminished, and .*. r increased, . . the periodic time is in
creased on both accounts, (for f is diminished by the increase of the dis
turbing forces which involve w.J If the distance of S be diminished, the
absolute force of S on P will be increased, . .thedisturbingforces which QCyr^
from S are increased, and P s gravity to T diminished, and .*. the periodic
time is increased in a greater ratio than r 2 (because of the diminution of
r f
fin the expression ryyj and when the distance of S is increased, the dis
turbing force will be diminished, (but still the attraction of P to T will be
diminished by the disturbance of S) and r will be decreased, .*. the
5
period will be diminished in a less ratio than r ^.
448. COR. 7. To find the effect of the disturbing force on the motion
of the apsides of P s orbit during one whole revolution.
FT! CJ
Whole force in the direction P T = f TTT (1 ^ cos> * A)
= T +T.c.r, (if T.c = 3 (l3cos.A) = Tr + 3  c  r4 >
I i c
. . the L. between the apsides =180   by the IXth Sect, which
1 + 4 c
is less than 180 when c is positive, i. e. from Q to P and from P to P,
344 A COMMENTARY ON [SECT. XL
(fig. (446,)) and greater than 180 when c is negative, i. e. from P to P
and from Q to Q ,
.. upon the whole the apsides are progressive, (regressive in the quadra
tures and progressive in the syzygies) ;
, T 3Sr
force = 75  TTT" force m conjunction
T 3 S i 1
f~ z  ^ = force in opposition
Now
R 3 T 3Sr 3 A R 3 T 3Sr /3
r R r 2 R 3
differ most from and =$
r 2 r /z
when r is least with respect to r ,
which is the case when the Apsides are in the syzygies.
But
R 3 T+ Sr 3 R 3 T+ Sr 3
r 8 R 8 ~?~ r R T ~
differ least from 2 an( ^ ~ when r is most nearly equal to r ,
449. COR. 7. Ex. Find the angle from the quadratures, when the apses
are stationary.
Draw P m parallel to T S, and = 3 P K, m n perpendicular to T P,
resolve. P m into P n, n m, whereof n m neither increases nor diminishes
the accelerating force of P to T, but P n lessens that force, .. when P n
= P T, the accelerating force of P is neither increased nor diminished,
and the apses are quiescent,
by the triangles PT:PK::PM=3PK:PnPT
.*. in the required position 3 P K 2 = P T ~
or
P K = ~^= PT.sin. P,
v o
BOOK L] NEWTON S PRINCIPIA. 345
.. ,
or
6 = 35 26 .
The addititious force P T P n is a maximum in quadratures.
^ P
F or P T : P K : : 3 P K : P n =
.. P T P n = PT 3 pJP , which is a maximum when P K = 0,
or the body is in syzygy.
450. COR. 8. Since the progression or regression of the Apsides de
pends on the decrement of the force in a greater or less ratio than D 2 , from
the lower apse to the upper, and on a similar increment from the upper
to the lower, (by the IXth Sect.), and is .. greatest when the proportion
of the force in the upper apse to that in the lower, recedes the most from the
inverse square of D, it is manifest that the Apsides progress the fastest from
the ablatitious force, when they are in the syzygies, (because the whole forces
in conjunction and opposition, i. e. at the upper and lower apses being
T 2 S r
I  rTT" > when the apsides are in the syzygies and when r is greatest
T
at the upper apse, being least, and the negative part of the expression
2 S r
,, 3 being greatest, the whole expression is .. least, and when r is least,
T
at the lower apse, ^ being greatest, and the negative part least, .. the
whole expression is greatest, and .*. the disproportion between the forces at
the upper and lower apse is greatest), and that they regress the slowest
T S r
in that case from the addititious force, (for + ^^ , which is the whole
v r 2 R 3
force in the quadratures, both before and after conjunction, r being the
semi minor axis in each case, differs least from the inverse square) ; there
fore, on the whole the progression in the course of a revolution is greatest
when the apsides are in the syzygies.
Similarly the regression is greatest when the apsides are in the quadra
tures, but still it is not equal to the progression in the course of the re
Volution.
451. COR. 8. Let the apsides be in the syzygies, and let the force
at the upper apse : that at the lower, : : D E : A B, D A
346
A COMMENTARY ON
being the curve whose ordinate is inversely
as the distance 2 from C, . . these forces being
diminished, the force D E at the upper apse
2 r S
by the greatest quantity oT~ > anc ^ tne f rce
A B at the lower apse by the least quantity
2^8
J5J ; the curve a d which is the new force
curve has its ordinates decreasing in a
greater ratio than ^^ .
Let the apsides be in the quadratures, then the force E D will be increased
by the greatest quantity ^ , and the force A B by the least quantity
S r
^j , /. the curve a d which is the new force curve will have its
ordinates decreasing in a less ratio than = 2 .
451. COR. 9. Suppose the line of apsides to be in quadratures, then while the
body moves from a higher to a lower apse, it is acted on by a force which
1 R 3 T f S r 3
does not increase so fast as ^~ 9 (for the force = r^, , .. the
r^ R, 3
numerator decreases as the denominator increases), . . the orbit will be
exterior to the elliptic orbit and the excentricity will be decreased. Also as
the descent is caused by the force rr^ (1 3 cos. 2 A), the less this
T
force is with respect to  , the less will the excentricity be diminished.
Now while the line of the apsides moves from the line of quadratures, the
force p 3 ( 1 3 cos. 2 A) is diminished, and when it is inclined at z_ 35
JLX
16 the disturbing force = 0, and .. at those four points the excentricity
is unaltered. After this, it may be shown in the same manner that the
excentricity will be continually increased until the line of apsides coin
cides with the line of syzygies. Here it is a maximum, since the disturb
ing force is negative. Afterwards it will decrease as before it increased
until the line of apsides again coincides with the quadrature, and then the
excentricity = maximum.
(Hence Evection in Astron.)
BOOK I.] NEWTON S PRINCIPIA. 347
452. LEMMA. To calculate that part of the ablatitious force which is
employed in drawing P from the plane of its orbit.
Let A = angular distance from syzygy.
Q = angular distance of nodes from syzygy.
I = inclination of orbit to orbit of S and T.
3 S r
Then the force required = jp . cos. A . sin. Q . sin. I. (not quite
accurately.)
When P is in quadratures, this force vanishes, since oos. A = 0.
When nodes are in syzygy, since sin. Q = 0,
quadratures, this force (cast, par.) = maxi
mum, since sin. Q = sin. 90 = rad.
453. COR. 12. The effects produced by the disturbing forces are all
greater when P is in conjunction than when in opposition.
For they involve , .. when R is least, they are greatest.
454. COR. 13. Let S be supposed so great that the system P and T re
volve round S fixed. Then the disturbing forces will be of the same kind
as before, when we supposed S to revolve round T at rest.
The only difference will be in the magnitude of these forces, which will
be increased in the same ratio as S is increased.
455. COR. 14. If we suppose the different systems in which S and S T
oc, but P T and P and T remain the same, and the period (p) of P round
T remains the same, all the errors p~3 a ~^7 , if A = density of S,
and d its diameter,
a <3 3 , if A given, and 3 = apparent diam.
also
TTs a o~3 ^ P = period of T round S,
.. the errors oc .
These are the linear errors, and angular errors oc in the same ratio,
since P T is given.
456. COR. 15. If S and T be varied in the same ratio,
S T
Accelerating force of S : that of T : : rf z : the same ratio as before.
R 2 r
. . the disturbances remain the same as before.
(The same will hold if R and r be also varied proportionally.)
.. the linear errors described in P s orbit oc P T, (since they involve r),
if P T oc, the rest remaining constant.
3*8 A COMMENTARY ON [SECT. XI.
i ,1 i r T> /> m linear errors P T
also the angular errors of P as seen from T oc a __ oc i }
and are . . the same in the two systems.
The similar linear errors a f . T 2 , .. P T a f . T \ and f
P T P T
Fpg , but f a accelerating force of T on P oc , (p = period of P
round T,)
..Tap and .. oc P
c S P 3
\ o T 1 Jr /
COR. 14. In the systems
S, T, P, Radii R, r Periods P, p
S , T, P R , r P , p.
Linear errors dato t. in 1st : do. in second : : p 2 : p^
.*. angular errors in the period of P : : :  : pj^ .
COR. 15. In the systems
S T P F? r P r
J  1 J x J ">  1 " ~ * 9 P
S , T, P R , r F, p ,
s;
s
.%* =2..
, S T , R r
so that ^ = ;= and ^
o R r
P
Linear errors in a revolution of P in 1 st. : do. in second : : r : r
angular errors : ::!:!.
COR. 16. In the systems
S, T, P, R, r P, p
S, T , P, R, i j P, p .
Linear errors in a revolution of P in 1st. : do in second : : r p 2 : r p /s
angular errors in a revolution of P : : : p 2 : p 2 .
To compare the systems
(1) S, T, P R, r P, p
/O\ O/ TV T)/ T>/ ,,/ ID/ y/
I ^ I O j J j JL *"  JLV I ~" " "   "~ A , Li
Assume the system
(Q\ C/ T 1 p T} r p/ n
l I ^ J J. j JL ^^ At* 5 J. JL A U
.*. by (14) angular errors in P S revolution in (1) : in (3) . : 51 p7 Z
by (16) angular errors in (3) : in (2) : : p 2 : p /2
~ p/2
therefore errors in (1) : in (2) : : p 2 : ^ z
BOOK I.] NEWTON S PRINCIPIA. 349
Or assume the system (3) 2, T, P g, r n, p
2 T Q r
so that ~T = , , = ,
i
/. the errors in (1) : errors in (3) :
(3) : (2}
I I S
2 S R 3
P 2 n 2 " R 3
e 3 " 2 s 3
:: 1 : 1
. S S . R 3 R 3 .
S T R 3
r /3
"S 7 2 R 3 3 *
s , T R , 3 
r 3
S r 3 . S r 73
..P 2 .].
: : R 3 T : n f 3 * T
pa p/ 2 *
457. COR. 16. In the different systems the mean angular errors of
P a ^~ whether we consider the motion of apses or of nodes (or errors
in latitude and longitude.)
For first, suppose every thing in the two different systems to be the same
except P T, . . p will vary. Divide the whole times p, p , into the same
number of indefinitely small portions proportional to the wholes. Then if
the position of P be given, the disturbing forces all cc each other cc P T
and the space cc f . T 2 , .*. the linear errors generated in any two corre
sponding portions of tune cc P T . p 2 .
.. the angular errors generated in these portions, as seen from T, oc p 2 .
.. Cornp . the periodic angular errors as seen from T x p 2 .
Now by Cor. 14, if in two different systems P T and .. p be the same,
every thing else varying, the angular errors generated in a given time, as in
$*?!
.. neutris datis, in different systems the angular errors generated in the
P 2
time p oc jl^ .
Now
// . // . . P_ 2 . JL
pa
.*. the angular errors generated in 1" (or the mean angular errors) or .
Hence the mean motion of the nodes as seen from T cc mean motion
of the apses, for each oc
458. COR. 17.
Mean addititious force : mean force of P on T : : p z : P 2 .
For
mean addititious force : force of S on T : : P T : S T,
350
A COMMENTARY ON [SECT. XL
Sr S
PT
i
S T
force of S on T : mean force of T on P: : r : ^ { *"
p
force oc
P
rad.
P.
p
.. mean addititious force : mean force of T on P: : p 2 : P 2
.*. ablatitious force : mean force of T on P: : 3 cos. 6 . p
Similarly, the tangential and central ablatitious and all the forces may
be found in terms of the mean force of T on P.
459. PROP. LXVII. Things being as in Prop. LXVI, S describes
the areas more nearly proportional to the times, and the orbit more ellipti
cal round the center of gravity of P and T than round T.
P T
For the forces on S are
PS
and
TS
.. the direction of the compound force lies between S P, S T; and T
attracts S more than P.
.. it lies nearer T than P, and .. nearer C the center of gravity of T
and P.
. . the areas round C are more proportional to the times, than when
round T.
Also as S P increases or decreases, S C increases or decreases, but S T
remains the same ; .*. the compound force is more nearly proportional to
the inverse square of S C than of S T; . . also the orbit round C is move
nearly elliptic (having C in the focus) than the orbit round T.
SECOND COMMENTARY
ON
SECTION XI.
460. To find the axis major of an ellipse, whose periodic time round
S at rest would equal the periodic time of P round S in motion.
Let A equal the axis major of an ellipse described round P at rest
equal the axis major of P Q v.
Let x equal the axis major required,
P. T. of P round S in motion : p S at rest : : V S : V S + P
3 3
P. T. of p in the elliptic axis A : P. T. in the elliptic axis x : : A 2 : x *
.. P. T. of P round S in motion : P.T. in the el.ax. x : : V A 3 !* : Vx 3 (S+P).
By hyp. the 1st term equals the 2d,
.. A 3 S = x 3 . S + P
461. PROP. LXIII. Having given the velocity, places, and directions
of two bodies attracted to their common center of gravity, the forces vary
ing inversely as the distance 2 , to determine the actual motions of bodies in
fixed space.
Since the initial motions of the bodies are given, the motions of the center
of gravity are given. And the bodies describe the same moveable curve
round the center of gravity as if the center were at rest, while the center
moves uniformly in a right line.
* Take therefore the motion of the center proportional to the time,
i. e. proportional to the area described in moveable orbits
* Since a body describes some curve in fixed space, it describes areas in proportion to the times
in this curve, and since the center moves uniformly forward, the space described by it is in pro
portion to the time, therefore, &c.
352
A COMMENTARY ON
[SECT. XI.
462. Ex. 1. Let the body P describe a circle round C, while the center C
moves uniformly forward. Take C G : C P : : v of C : v of P, and with the
center C and rad. C G describe a circle G C N, and suppose it to move
round along G H, then P will describe the trochoid P L T, and when P
has described the semicircle P A B, P will be at the summit of the trochoid
.*. every point of the semicircumference G F N will have touched G H,
.. G H equals the semicircumference G F N,
.. v of P : v of C : : P A B semicircumference : C 11 = G F N semicircle
* : : C P : C G Q. e. d.
463. Ex. 2. Let the moveable curve
be a parabola, and let the center of gravity
move in the direction of its primitive
axis. When the body is at the vertex
A , let S be the position of the center
of gravity, and while S has described
uniformly S S, let A have described the
arc of the parabola A P.
Let A N = x, N P = y, be the abA S
scissa and ordinate of the curve A P in fixed space.
Let 4 p equal the parameter of the parabola A P.
.. A N = f , A S = S S = x _ =
4p 4p
SN = AN A S = A N p = ?
P
4p
4p
AreaASP=ANP SNP=ANx N P } N S X NP
= s . Xl __ i y 3 4 P y _ y 3 + I2p g y
3 4p ^ 4 p 24 p
By Prop. S S oo A S P ; therefore they are in some given ratio.
Let A S P : S S : : a : b : :
24 p
4 p
* If C P = C G the curve in fixed space becomes the common cycloid.
If C P > C G    the oblongated trochoid.
BOOK I.]
NEWTON S PRINCIPIA.
353
. . y 3 f 12p 2 y = 4pax ay*
.. y 3 + ay 2 + 12 p 8 y 4pax == 0.
Equation to the curve in fixed space.
464. Ex. 3. * Let B B be the orbit of the earth round the sun, M A
that of the moon round the earth, then the moon will, during a revolution,
trace out a contracted or protracted epicycloid according as A L has a
greater or less circumference than A M, and the orbit of the moon round
the sun will consist of twelve epicycloids, and it will be always concave to
the sun. For
R r
F of the earth to the sun : F of the moon to the earth : : pj : t
400 1
(365) 2 (27) 2
in a greater ratio than 2:1. But the force of the earth to the sun is
nearly equal to the force of the moon to the sun, . .the force of the moon
to the earth, .. the deflection to the sun will always be within the tan
gential or the curve is always concave towards the sun.
465. PROP. LXVI. If three bodies attract each other with forces

varying inversely as the square of the distance, but the two leas: revolve
To determine the nature of the curve dwribed by the moon with respect to the sun.
VOL. I. Z
354 A COMMENTARY ON [SECT. XI.
about the greatest, the innermost of the two will more nearly describe the
areas proportional to the time, and a figure more nearly similar to an el
lipse, if the greatest body be attracted by the others, than if it were at rest,
or than if it were attracted much more or much less than the other bodies.
(L M : P T : : S L : S P,
PT
.. L M oc
.. L M =
SP 3 *
PTxSL SK 3 xPT
S P S P 3
.. S K 2 : SP* : : SL : S P).
Let P and S revolve in the same plane about the greatest body T, and
P describe the orbit P A B, and S, E S E. Take S K the mean distance
of P from S, and let S K represent the attraction of P to S at that dis
tance. Take SL : SK :: SR 2 : SP 2 , and SL will represent the
attraction of S on P at the distance S P. Resolve it into two S M, and
L M parallel to P T, and P will be acted upon by three forces P T, L M,
S M. The first force P T tends to T , and varies inversely as the dis
tance 2 , .. P ought by this force to describe an ellipse, whose focus is T.
The second, L M, being parallel to P T may be made to coincide with it
in this direction, and .. the body P will still, being acted upon by a centri
petal force to T, describe areas proportional to the time. But since L M
does not vary inversely as P T, it will make P describe a curve different
from an ellipse, and .*. the longer L M is compared with P T, the more
will the curves differ from an ellipse. The third force S M, being neither
in the direction P T, nor varying in the inverse square of the distance, will
make the body no longer describe areas in proportion to the limes, and the
curve differ more from the form of an ellipse. The body P will .*. describe
areas most nearly proportional to the times, when this third force is a
minimum, and P A B will approach nearest to the form of an ellipse, when
both second and third forces are minima. Now let S N represent the
attraction of S on T towards S, and if S N and S M were equal, P and
T being equally attracted in parallel directions would have relatively the
same situation, and if S N be greater or less then S M, their difference
M N is the disturbing force, and the body P will approach most nearly
the equable description of areas, and P A B to the form of an ellipse,
when M N is either nothing or a minimum.
Case 2. If the bodies P and S revolve about T in different planes, L M
being parallel to P S will have the same effect as before, and will not
BOOK I.]
NEWTON S PRINCIPIA.
355
tend to move P from its plane. But N M acting in a different plane,
will tend to draw P out of its plane, besides disturbing the equable des
cription of areas, &c. and as before this disturbing force is a minimum,
when M N is a, minimum, or when S N = nearly S K.
466. To estimate the magnitude of the disturbing forces on P, when P
moves in a circular orbit, and in the same plane with S and T.
Let the angle from the quadratures P C T = S,
S T = d, P T = r, F at the distance (a) = M,
F on P  Ma2
SP
. . From P in the direction S P :
. . F in the direction P T = ^^] x
But S P 2 = d 2 + r 2 2 d r sin. 0,
.. F in the direction P T =
P T : : S P : P T,
par
M a*r
(d 2 + r * 2 d r sin. 6} f
M a 2 r
Mar
2 d r sin. 6
d 3 r d
IM. u. ~ r
= 33 = A nearly, since d being indefinitely great compared with r
in the expansion, all the terms may be neglected except two. First i
vanishes when compared with r 3 , .. the addititious force in the direction
T = A. By proportion as before, force in the direction S T
Ma 2 ST Mad f
: SP* SP " d 3 (1 + (r 2dr sin. 6
d*
Mji^ / 3 r 8  2 d r sin.
d* \ ~ 2 d >
_M a 2 3 M a 8 r 8 3 M a*r sin, t)
d* 2 d 4 + " d 3
356
A COMMENTARY ON
[SECT. XL
.. force in the direction S T =
M
3 M a 2 r
sin. t nearly, since
it **
1 1 Ma 2
, vanishes when compared with , , and the force of S on T = rr ,
d * d 3 d *
Ma 2 3 M a r . Ma 2
.. ablatitious F = TV; } rz sin  d Tr~
d d* d*
= 3 A . sin. 0.
If P T equal the addititious force, then the ablatitious force equals 3 P K,
for P K: PT: : sin. 6 : (1 = r),
.. 3 P K = 3 P T . sin. 6 = 3 A . sin. 6.
To resolve the ablatitious force. Take
P m : P n : : P T : T K : : 1 : cos. 6,
3 A
.. P n = P in X cos. 6 = 3 A X sin. 6 cos. 6 = . sin. 2 6
m n = P m X PK = 3A. sin. 2 6 = 3 A . * ~ C S> 2 *,
it
.. the disturbing forces of S on P are
M a 2 r
1. The addititious force = p = A.
2. The ablatitious force which is resolved into the tangential part
q A J 2 COS A
=  . sin. 2 6, and that in the direction T P = 3 A . =  ,
*^ ^
.. whole disturbing force in the direction P T = A 3 A . ^ 
A 3 A 3 A ASA i*iii
= A j . cos. 2 6 = H ~ . cos. 2 6, and in the whole
22 22
revolution the positive cosine destroys the negative, therefore the whole
disturbing force in a complete revolution is ablatitious, and equal to one
half of the mean addititious force.
467. To compare N M and L M.
L M : P T : : (S L = ^) : S P,
.. L M = f p, X P T
BOOK I.]
MN =
NEWTON S PRINCIPIA.
3_SP 3
v S T _ S T
_
SP 3 S P
SK 3 (SKKP) 3
S P 3
S K 3 S K* + 3 S R 2 x K P
S P 3
3 S K* x P K
X ST
X ST nearly
S P
X S T nearly =
L.
= gpr X P T X sin. 6,
.. M N : L M : : 1 : 3 sin. 6.
S P
X P K
357
468. Next let S and P revolve about T in different planes, and let
N P N be P s orbit, N N the line of the nodes. Take T K in T S =
3 A . sin. 6. Pass a plane through T K and turn it round till it is per
pendicular to P s orbit. Let T e be the intersection of it with P s orbit.
Produce T E and draw K F perpendicular to it, .. K F is perpendicular
to the plane of P s orbit, and therefore perpendicular to every line meet
ing it in that orbit, T in the plane of S s orbit ; draw K H perpendicular
to N N produced ; join H F, then F H K equals the inclination of the
planes of the two orbits. For K H T, K F T, K F H being all right angles,
KT* = K H* + HT*
K F * + H = K F 2 + F H * + H T ,
* .. F T * = F H 2 + H T ,
.. F H is perpendicular to H T.
Since P T = A, T K = A x sin. 6
Let the angle K H T = T, II T K e= <p = angular distance of the line of the no<!c
from S y z.
7.3
359
A COMMENTARY ON
[SECT. XI.
PT
TK
KH
PT
TK
KH
KF
KF
3 sin. t)
sin. <p
sin. T,
3 sin. d . sin.
sn.
T,
/. ablatitious force perpendicular to P s orbit = K F
= 3 P T X sin. 6. sin. p x sin. T = 3 A X sin. 6. sin. <p X sin. T.
2d. Hence it appears that there are four forces acting on P.
C
1. Attraction of P to T a
2. Addititious F in the direction P T =
3. Ablatitious F in the direction P T =
M a*r
d 3 *
3 Ma 2 r
sin. z 6.
4. Tangential part of the ablatitious force = f .
Ma !
sin. 2 6.
Of these the three first acting in the direction of the radiusvector do
not disturb the equable description of areas, the fourth acting in the di
rection of a tangent at P does interrupt it.
Since the tangential part of F is formed by the revolution of P M = 3 A X
sin. & at C, 6 = 0, therefore P m = 0, and consequently the tangential
F = ; from C to A, P n is in consequentia, and therefore accelerates
the body P at A, it again equals 0, and from A to D is in antecedentia,
and therefore retards P ; from D to B it accelerates; from B to C it re
tards.
Therefore the velocity of P is greatest at A and B, because these are
the points at which the accelerations cease and retardations begin, and
the velocity is least at D and C. To find the velocity gained by the ac
tion of the tangential force.*
sin. 26d6
* F in the direction P T is a maximum at the quadrature, because die ablatitious F in tie
quadrature is 0, and at every other point it is something.
BOOK I.] NEWTON S PRINCIPIA. 359
sin. 2 6 X 20 = (cos. 2 0) ,
v 2
. . Z = = Cor.  A. cos. 2 6.
5
But when 6 = 0, the tangential F = 0, and no velocity is produced,
.. cos. 2 = R = ],
(1 cos. 2 6} =  A. 2 sin. 2 6,
4
.. v 2 = 3g A. sui. 2 0,
/. v = V 3 g A. sin. d,
. . v a (sin. 0) ,
. . whole f on the moon at the mean distance : f of S on T : :  2 : p^
* r
and the force of S on T : add. f at the mean distance (m) : : :"4*s
d d
.*. whole f at the mean distance : m : : P s : p 2 and ~ t X whole f &c. = in.
f or
Now f on the moon at any distance (r) = ^j 3 and at the mean
distance (1) = f ^L 3 = f ,
_ p 2 f m p 1
.*. m =
12
JL ** A
2p 2 f
2 P 8 + p
and therefore nearly = ~ ^ p4 ,
( p 2 2 p 4 )
.\ in r (which equals the addititious force) =  rfl BT" J
469. To compare the ablatitious and addititious forces upon the moon,
with the force of gravity upon the earth s surface. (Newton, Vol. III.
Prop. XXV.)
add. f : f of Son T :: PT : ST
S T P V P T
f of S on T : f of the earth on the moon : : . Tj  : ^ = ,
.. add. f : f of the earth on the moon : : p 2 : P 2
f of the earth on the moon : force of gravity :: 1 ; 60 5 ,
.% add. f : force of gravity :: p* : P 2 . GO 2 . . . (])
Also ablat. f : addititious force : : 3 P K : P T,
.. ablat. f : addititious force : : 3 P K . p * : 60 2 . P T. P s . (2)
470. Con. 2. In a system of three bodies S, P, T, force oc* , the
4
360 A COMMENTARY ON [SECT. XI.
body P will describe greater areas in a given time at the syzygies than at
the quadrature.
The tangent ablatitious f = f . P T . sin. 2 6 ; therefore this force will
accelerate the description of areas from quadratures to syzygies and retard
it from syzygies to quadratures, since in the former case sin. 2 6 is positive,
and in the latter negative.
COR. 3. is contained in Cor. 2.
The first quadrant d. sin. being positive the velocity increases,
in the second d. sin. negative the velocity decreases, &c. for the 1st Cor.
2d Cor. &c.
Also v is a maximum when sin. 8 is a maximum, i. e. at A and B.
471. COR. 4. The curvature of P s orbit is greater in quadratures than
in the syzygy.
Ma 2 Ma e r 3Mar.. B 
The whole F on P = ^7 \  p  ~2~[3~ ( l ~~ cos> ) X
/3 M a 3 r . sin. 2 6\
\ 2 d 3 /
In quadratures sin. 20=0,
F M a 2 M a * r
~^~ ~d^~
And in syz. 29= 180,
.. sin. 20 = 0, cos. 201
3Ma g r 3 M a g r
TTF""*" d 3
, , v . M a J 2 M a * r
.*. the whole if on P in the svz. =  r  ,
r 2 d
.. F is greater in the quadratures than in the syzygies; and the velocity
is greater in the syzygies than in the quadratures.
1 T*
But the curvature a p^r cc v 2 , .. is greatest in the quadratures and
least in the syzygies.
472. COR. 5 Since the curvature of P s orbit is greatest in the quadra
ture and least in the syzygy, the circular orbit must assume the form of an
ellipse whose major axis is C D and minor A B
.*. P recedes farther from T in the quadrature than in the syzygy.
473. COR. 6.
The whole F on P in the line PT =
.Ma 2
= in quad. 5
BOOK i.]
NEWTON S PRINCIPIA.
361
M a * 2 M a 2 r
and in syz. = o
let the ablatitious force on P equal the addititious, and
M a 2 r 3 M a 2 r
. sin. 2 6
1
/. sin. Q = = sin. 35 . 16.
V 3
Therefore up to this point from quadrature the ablatitious force is less
than the addititious, and from this to one equally distant from the other
point of quadrature, the ablatitious is greater than the addititious, therefore
in a whole revolution the gravity of P to T is diminutive from what it
R2
would be if the orbit were circular or if S did not act, and P a  ,  ~
\ abl. r
and since the action of S is alternately increased or diminished, therefore
P ex from what it would be were P T constant, both on account of the
variation, and of the absolute force.
474. COR. 7. * Let P revolve round T in an elliptic orbit, the force on
Ma 2 , Ma 2 r , b
P in the quad. = , H ^ +  2 + c r.
.. G + 180
and since the number is greater than the de
b + 4 c
nomination G is less than 180. . . the apsides are regressive if the same
effect is produced as long as the addititious force is greater than the abla
titious, i. e. through 35. 16 .
The force on P in the syz. =
Ma
2Ma l r
* Since P a
II
and in whiter the sun is nearer the earth than in summer,
V ablatitious force
R Js increased in winter, and A i diminished, therefore the lunar months are shorter in vrintor
than in summer.
362 A COMMENTARY ON [SECT. XI.
= 180 . /, ~ I C > 180
V b 8 c
m c
.*. in the syz. the apsides are progressive, and since / : will be
an improper fraction as long as the ablatitious force is greater than the
addititious, and when the disturbing forces are equal, m c = n c, therefore
G = 180, i. e. the line of apsides is at rest (or it lies in V C produced
9th.) .*. since they are regressive through 141. 4 and progressive
2 18. 56 they are on the whole progressive.
To find the effect produced by the tangential ablatitious force, on the
velocity of P in its orbit. Assume u = velocity of a body at the mean
distance 1, then = velocity at any other distance r nearly, the orbit
being nearly circular.
Let v be the true velocity of P at any distance (r), vdv = gFdx
( f = 16 12 For the tan S ent ablatitious f = f . P T . 2 0, and x = r
= 3 P T . m r . sin. 26.tf,
.. v 2 = 3 P T m r cos. 2 6 + C,
and
2
v 2   frr
V n C\A 
f "
Hence it appears that the velocity is greatest in syzygy and least in
quadrature, since in the former case, cos. 2 d is greatest and negative, and
in the latter, greatest and positive.
To find the increment of the moon s velocity by the tangential force
while she moves from quadrature to syzygy.
v 2 = 3 P T . m . r . cos. 2 6 + C,
but (v) the increment = 0, when 6 = 0,
.. C = 3 P T . m . r,
.. v 2 = 3 P T . m . r (1 cos. 2 6) = G P T. m. r. sin. 8 0,
and when d = 90, or the body is in syzygy v " = 6 P T rn . r.
475. COR. 6. Since the gravity of P to T is twice as much diminished
in syzygy as it is increased in quadrature, by the action of the disturbing
force S, the gravity of P to T during a whole revolution is diminished.
Now the disturbing forces depend on the proportion between P T and
T S, and therefore they become less or greater as T S becomes greater
BOOK I.] NEWTON S PRINCIPIA. 363
or less. If therefore T approach S, the gravity of P to T will be still
more diminished, and therefore P T will be the increment.
Now P. T cc  ; since, therefore, when S T is di
V absolute force
minished, R is increased and the absolute force diminished (for the ab
solute force to T is diminished by the increase of the disturbing force) the
P . T is increased. In the same way when S T is increased the P . T is
diminished, therefore P. T is increased or diminished according as S T
* O
is diminished or increased. Hence per. t of the moon is shorter in winter
than in summer.
OTHERWISE.
476. COR. 7. To find the effect of the disturbing force on the motion
of the apsides of P s orbit during a whole revolution.
f
Let f = gravity of P to T at the mean distance ( 1 ), then = gravity
of P at any other distance r.
f f
Now in quadrature the whole force of P to T =  a f add. f = 2 + r
= 3 and with this force the distance of the apsides = 180 / ^
which is less than 180, therefore the apsides are regressive when the
f
body is in quadrature. Now in syz. the whole force of P to T =
r 2
f r g r 4
2 r =  y , therefore the distance between the apsides = 180
, f 2
fij f 8 which is greater than 180, therefore the apsides are progressive
when the body is in syzygy.
But as the force (2 r) which causes the progression in syzygy is double
the force (r) which causes the regression in quadrature, the progressive
motion in syzygy is greater than the regressive motion in the quadrature.
Hence, upon the whole, the motion of the apsides will be progressive
during a whole revolution.
At any other point, the motion of the apsides will be progressive or
P T S P T
retrograde, according as the whole central force + ^ . cos. 2 i)
t &
is negative or positive.
364
A COMMENTARY ON
[SECT. XL
477. COR. 8. To calculate the disturbing force when P s orbit is ex
centric.
m
P T 3 P T
The whole central disturbing force = f cos. 26=.
\ s . cos. 2 6 (m is the mean add. f). Now r =
1 e
= by div. 1 e 2 + e . cos. u + e 2 . cos. 2 u, &c. neglecting terms in
e 2 e 2
volving e 3 , &c. = 1 + e . cos. u + . cos. 2 u ; therefore the
whole central disturbing force =
in
m e
m . e . cos. u
me 8 cos. 2 u , 3 3 in e  o
,  + ~ ni cos. 2 6 . cos. 2 6 + m e . cos. u . cos. 2 6
^r *w *
f  m e ". cos. 2 u . cos. 2 6.
478. COR. 8. It has been shown that the apsides are progressive in
syzygy in consequence of the ablatitious force, and that they are regres
sive in quadrature from the effect of the ablatitious force, and also, that
they are upon the whole progressive. It follows, therefore, that the
greater the excess of the ablatitious over the addititious force, the more will
the apsides be progressive in the course of a revolution. Now in any
position m M of the line of the apsides, the excess of the ablatitious in
conjunction 2 A T in opposition = T B, therefore the whole excess
= 2 A B. Again, the excess of the addititious above the ablatitious force
in quadrature = C D. Therefore the apsides in a whole revolution will
be retrograde if 2 A B be less than C D, and progressive if 2 A B be
greater than C D. Also their progression will be greater, the greater the
excess of 2 A B above C D ; but the excess is the greatest when M m is
in syzygy, for then A B is greatest and C D the least. Also, when M in
is in syzygy the apsides being progressive are moving in the same direc
tion with S, and therefore will remain for some length of time in syzygy.
Again, when the apsides are in quadrature A B = P p, and C D = M m,
m
BOOK I.] NEWTON S PRINCIPIA. 365
but if the orbit be nearly circular, 2 A B is greater than C D ; therefore
the apsides are still in a whole revolution progressive, though not so
much as in the former case.
F
In orbits nearly circular it follows from G = = when F a AP S ,
V r
that if the force vary in a greater ratio than the inverse square, the
apsides are progressive. If therefore in the inverse square they are sta
tionary, if in a less ratio they are regressive. Now from quadrature to
35 a force which <x the distance is added to one varying inversely as
the square, therefore the compound varies in a less ratio than the inverse
square, therefore the apsides are regressive up to this point. At this point
F <x T. ; , therefore they are stationary. From this to 35 from
distance 2
another Q a quantity varying as the distance is subtracted from one
varying inversely as the square, therefore the resulting quantity varies
in a greater ratio than the inverse square, therefore the apsides are
progressive through 218.
OTHERWISE.
479. COR. 8. It has been shown that the apsides are progressive in
syzygy in consequence of the ablatitious force, and that they are regressive
in the quadratures on account of the addititious force, and they are on the
whole progressive, because the ablatitious force is on the whole greater
than the addititious. . . the greater the excess of the ablatitious force
above the addititious the more will be the apsides progressive.
In any position of the line A B in conjunction the excess of the ablati
tious force above the addititious is 2 P T, in opposition 2 p t. .*. the whole
excess in the syzygies = 2 P p. In the quadratures at C the ablatitious
force vanishes. /. the excess of the addititious = additious = C T.
.*. the whole addititious in the quadratures = C D.
Now the apsides will, in the whole revolution, be progressive or regres
sive, according as 2 Pp is greater or less than C D, and then the progres
sion will be greatest in that position of the line of the apses when 2 P p
C D is the greatest, i. e. when A B is in the syzygy, for then 2 P p
2 A B, the greatest line in the ellipse, and C D = R r =r ordinate =
least through the focus. .*. 2 P p CD is a maximum. Also when
A B is in the syzygy, the line of apsides being progressive, will move the
same way as S. .. it will remain in the syzygy longer, and on this account
the apsides will be more progressive. But when the apsides are in the
quadratures S P = R r and C D = A B, and the orbit being nearly
circular, R r nearly equals A B. . . 2 P p C D is positive, and the
366 A COMMENTARY ON [SECT. XI.
apsides are progressive on the whole, though not so much as in the last
case ; and the apsides being regressive in the quadratures move in the op
posite direction to S, .*. are sooner out of the quadratures, .. the regres
sion in the quadrature is less than the progression in the syzygy.
480. COR. 9. LEMMA. If from a quantity which GC r^ any quantity
A
be subtracted which a A the remainder will vary in a higher ratio than
the inverse square of A, but if to a quantity varying as ^ z another be
A
added which a A, the sum will vary in a lower ratio than rg .
1 j c A 2
If be diminished C A = 7= . If A increases 1 c A *
A 2 A z
decreases, and rj increases. . . the quantity decreases, 1 c A increases
1
and rj increases. .. increases from both these accounts. . . the whole
A i
quantity varies in a higher ratio than ^ .
1 4 c A 2
If C A be added rg , as A is increased the numerator increases,
and jg decreases. . . the quantity does not decrease so fast as j s , and
A. A
if A be diminished 1 + c A 2 is diminished, and ^ increased. . . the
quantity is not increased as fast as r 2 . .. &c. Q. e. d.
OTHERWISE.
481. COR. 9. To find the effect of the disturbing force on the excen
l
tricity of P s orbit. If P were acted on by a force a j z , the excentricity
of its orbit would not be altered. But since P is acted on by a force vary
ing partly as r z and partly as the distance, the excentricity will continual
ly vary.
Suppose the line of the apsides to coincide with the quadrature, then
while the body moves from the higher to the lower apse, it is acted upon
by a force which does not increase so fast as p } for the force at the quad
f
rature = + m r, and /. the body will describe an orbit exterior to the
elliptic which would be described by the force a  . Hence the body
BOOK I.] NEWTON S PRINCIPIA. 367
will be farther from the focus at the lower apse than it would have been
had it moved in an elliptic orbit, or the excentricity is diminished. Also
as the decrease in excentricity is caused by the force (m r), the less this
f
force is with respect to z , the less will be the diminution of excentricity.
Now while the line of apsides moves from the line of quadratures, the force
(m r) is diminished, and when it is inclined at an angle of 35 16 the
disturbing force is nothing, and .*. at those four points the excentricity
remains unaltered. After this it may be shown in the same manner that
the excentricity will be continually increased, until the line of apsides
coincides with the syzygies. Hence it is a maximum, since the disturbing
force in these is negative. Afterwards it will decrease as before it in
creased, until the line of apsides again coincides with the line of quadra
ture, and the excentricity is a minimum.
COR. 14. Let P T = r, S T = d, f = force of T on P at the distance
1, g = force of S on T at the distance, then the ablatitious force
= ~n J if " the position of P be given, and d varies, the ablati
tious force cc p . But when the position of P is given, the ablatitious
: addititious : : in a given ratio, . . addititious force cc p , or the dis
turbing force cc ^ . Hence if the absolute force of S should x the dis
i if
turbing force cc r^  . Let P = the periodical time of T about S,
.. p cc j^ * . Let A = density, 8 = diameter of the sun, then the
A .X 3 1
absolute force <x A d 3 , then the disturbing force a j cc p^ cc A (ap
parent diameter) 3 of the sun. Or since P T is constant, the linear as well
as the angular errors oc in the same ratio.
483. Con. 15. If the bodies S and T either remain unchanged, or their
absolute forces are changed in any given ratio, and the magnitude of the
orbits described by S and P be so changed that they remain similar to
what they were before, and their inclination be unaltered, since the accel
c rrrr. i A  r r o absolute force of T
crating force ot P to i : accelerating force of S : : p :
absolute force of S , , r . ,
r r z , and the numerators and denominators of the last
o JL
terms are changed in the same given ratio, the accelerating forces remain
in the same ratio as before, and the linear or angular errors cc as before,
368
A COMMENTARY ON
[SECT. XI.
i. e. as the diameter of the orbits, and the times of those errors oc P T s
of the bodies.
COR. 16. Hence if the forms and inclinations of the orbits remain, and
the magnitude of the forces and the distances of the bodies be changed ; to
find the variation of the errors and the times of the errors. In Cor. 14.
it was shown, how that when P T remained constant, the errors oc ^ .
Now let P T also a , then since the addititious force in a given position
of P oc P T, and in a given position of P the addititious : ablatitious in
a given ratio.
COK. If a body in an ellipse be acted upon bv a force which varies
in a ratio greater than the inverse
square of the distance, it will in de
scending fromthe higher apse Bto the
lower apse A, be drawn nearer to the ^
center. .*. as S is fixed, the excen
tricity is increased, and from A to B
the excentricity will be increased
also, because the force decreases the faster the distance 2 increases.
484. (CoR. 10.) Let the plane of P s orbit be inclined to the plane of T s
orbit remaining fixed. Then the addititious force being parallel to P T,
is in the same plane with it, and . . does not alter the inclination of the
plane. But the ablatitious force acting from P to S may be resolved into
two, one parallel, and one perpendicular to the plane of P s orbit. The
force perpendicular to P s orbit = 3 A X sin. 6 X sin. Q X sin. T
when d perpendicular distance of P from the quadratures, Q = angular
distance of the line of the nodes from the syzygy, T = first inclination of
the planes.
Hence when the line of the nodes is in the syzygy, 6 0,
.. no force acts perpendicular to the plane,
and the inclination is not changed. When
the line of the nodes is in the quadratures,
d = 90, /. sin. is a maximum, . . force per
pendicular produces the greatest change /
in the inclination, and sin. & being posi
tive from C to D, the force to change the
inclination continually acts from C to D
pulling the plane down from D to C. Sin. d
is negative, .*. force which before was posi
sin. =
H
BOOK I.] NEWTON S PRINCIPIA. 369
tive pulling down to the plane of S s orbit (or to the plane of the paper)
now is negative, and . . pulls up to the plane of the paper. But P s orbit is
now below the plane of the paper, . . force still acts to change the inclina
tion. Now since the force from C to D continually draws P towards the
plane of S s orbit, P will arrive at that plane before it gets to D.
If the nodes be in the octants past the quadrature, that is between C
and A. Then from N to D, sin. 6 being positive, the inclination is di
minished, and from Dto N increased, .*. inclination is diminished through
270, and increased through 90, . . in this, as in the former case, it is
more diminished than increased. When the nodes are in the octants be
fore the quadratures, i. e. in G H, inclination is decreased from H to C,
diminished from C to N, (and at N the body having got to the highest
point) increased from N to D, diminished from D to N , and increased
from 2 N to H, . . inclination is increased through 270, and diminished
through 90, /. it is increased upon the whole. Now the inclination of
P s orbit is a maximum when the force perpendicular to it is a minimum,
i. e. when (by expression) the line of the nodes is in the syzygies. When
is the quadratures, and the body is in the syzygies, the least it is increased
when the apsides move from the syzygies to the quadratures ; it is dimin
ished and again increased as they return to the syzygies.
485. (CoR. 11.) While P moves from the quadrature in C, the nodes
being in the quadrature it is drawn towards S, and .*. comes to the plane
of S s orbit at a point nearer S than N or D, i. e. cuts the plane before it
arrives at the node. . . in this case the line of the nodes is regressive. In
the syzygies the nodes rest, and in the points between the syzygies and
quadratures, they are sometimes progressive and sometimes regressive,
but on the whole regressive; .. they are either retrograde or stationary.
486. (CoR. 12.) All the errors mentioned in the preceding corollaries are
greater in the syzygies than in any other points, because the disturbing
force is greater at the conjunction and opposition.
487. (CoR. 13.) And since in deducing the preceding corollaries, no re
gard was had to the magnitude of S, the principles are true if S be so
great that P and T revolve about it, and since S is increased, the disturbing
force is increased ; .. irregularities will be greater than they were before.
Ma 2 r ^ T VT ^ r 3 Ma
r
488. (CoR. 14.) L M = ^j^ = N N M = " ^ , sin. 6, .. in
a given position of P, if P T remain unaltered, the forces N M and L M
VOL. T. A a
370 A COMMENTARY ON [SECT. XI.
cc .. X absolute force cc ^  of T for (sect. 3 . P 2 oc
j 2
.. ^  . .
d j (Per. T) 2 absolute f.
whether the absolute force vary or be constant. Let D = diameter of S,
<3 = density of S, and attractive force of S cc magnitude or quantity of
matter oc D 3 3,
D 3 &
/. forces L M and N M cc TT.
d 3
But r = apparent diameter of S,
.. forces cc (apparent diameter) 3 3 another expression.
489. (CoR. 15.) Let another body as P revolve round T in an orbit
similar to the orbit of P round T, while T is carried round S in an orbit
similar to that of T round S, and let the orbit of P be equally inclined to
that of T" with the orbit P to that of T. Let A, a, be the absolute forces
of S, T, A , a , of S , T ,
A a
accelerating force of P by S : that of P by T : : ~ p 2 : p ^ ,
and the orbits being similar
A a
accelerating force of P by S : that of P by T : : ~ p/ 2 : p}^ ,
.. if A : a : : A : a, and the orbits being similar,
S P : P T * : : S F : F T ,
accelerating force of P by S : that of P by T
: : force on P by S : force on P by T ,
and the errors due to the disturbing forces in the case of P are as
A A
cT>t r 3 x r i n l ^ e case f f" an d S are as Q , ^3 X R,
o L o JL
.. linear errors in the first case : that in the second : : r : R.
sin. errors
Angular errors cc
XI
in the first cas
linear errors
angular errors in the first case : that in the second : : 1 : 1.
Now Cor. 2. Lem. X. T 2 a
T C
angular errors
.\ T 2 x angular errors,
.. angular errors : 360 : : T 2 : P 2 ,
.. T 2 a P 2 X angular errors,
.. T oc P for = angular errors.
BOOK I.] NEWTON S PRINCIPIA. 371
490. (Con. 16.) Suppose the forces of S, P T, ST to vary in any man
ner, it is required to compare the angular errors that P describes in simi
lar, and similarly situated orbits. Suppose the force of S and T to be
constant, . . addititious force x P T, . . if two bodies describe in similar
orbits = evanescent arcs. Linear errors x p 2 X P T.
.. angular errors x p 2 (p = per. time of P round T, P = that of T
round S). But by Cor. 14. if P T be given, the absolute force of A and
STx.
Angular errors x yyy
.*. if P T, S T and the absolute force alternately vary,
angular errors x ~ ,
/P = per. time of P round T\ f M a 2 r
Vp = per. time of T round S J ~dP~
, linear errors
angular errors x p .
radius
M a 2 r
.. lin. errors x force T ! a pr X P 2 by last Cor.
d 3
, rP 2 P\
/. angular errors x x r ) .
d 3 X r p 2 /
M a 2
Now the errors d t X p = whole angular errors x L ,
.*. error d t x ,y 2 thence the mean motion of the apsides x mean motion
of the nodes, for each x J , for each error is formed by forces varying as
proof of the preceding corollaries, both the disturbing forces, and .. the
errors produced by them in a given time will x P T. Let P describe an
indefinite small angle about T (in a given position of P), then the linear
errors generated in that time x force T P time 2 , but the time of describ
ing = angles about T x whole periodic time (p), . . linear errors x
P T p *, and as the same is true for every small portion, similar; the
linear errors during a whole revolution x P T p 2 . Angular errors
lincni* cr
x j .. oc p * . when S T, P T, and the absolute force vary, the
p 2 absolute p * p *
angular errors a p~ a g ^ 3 a g*,_ , (when the absolute force is
A a2
372 A COMMENTARY ON [SECT. XI.
given.) Now the error in any given timex p varies the whole errors during
P 2 P
a revolution a ~ z . .*. the tfrrors in any given time oc p 2 . Hence the
mean motion of the apsides of P s orbit varies the mean motion of the
nodes, and each will a ~ 3 the excentricities and inclination being small
and remaining the same.
491. (CoR. 17.) To compare the disturbing forces with the force of
PtoT.
FofSonT:FofPonT: absoluteF a
a
ST 2 T P !
absolute F . A. ST . aTP
axis major 3 SS 3 T P s
. S.T . TP . ^L
* * P 2 " D * " " p 2
mean add. F : F of S on T : : ^^ : ^ : :
.. mean add. F : F P on T : : p 2 : P .
492. To compare the densities of different planets.
Let P and P be the periodic times of A and B, r and r their distances
from the body round which they revolve.
F of A to S : F of B to S : : ~ : ^
quantity of matter in A do. in B m D 3 of A X density ^ D 3 of Bx density
distance 2 distance 2 distance 8 distance *
r r>
p 2 p/ 2
D 3 xd
D /3 xd 1 1
r 3
r 3 p e p/ 2
r 3 r /3
A V
..a. a > p 3 p s jj/ 3 p/ E
1 1
* S 3 P 2 3 p/ J
where S and S represent the apparent diameters of the two planets.
493. In what part of the moon s orbit is her gravity towards the earth
unaffected by the action of the sun.
Ma 2 r 3Ma 2 r U cos. 2 ^ + 3 Ma r^ ^
2 ~ 2 d 3
and when it is acted upon only by the force of gravity = for the
other forces then have no effect.
BOOK I.]
NEWTON S PRINCIPIA.
373
M a*r
3
M
a 2
T 1 COS
. 2
1 3
M a 2 r .
) n
d 3
d
3
2
1 2
d J
1
3.
1
cos. 2 6 t
2
"3 s
;in. 2 6
=
2
3
3
3
1
2
f
~2
cos. 2 6 +
2
sin. 2
a = o
3
i
3
1 sin.
2 C
3
2
1
2
2
+ 2
Let x = sin. 6,
(.. 1 f +   sin. * +  X 2 sin. x cos. d = 0)
and
3x 2
1
+ 3 x V 1 x
0.
An equation from which x may be found.
494. LEMMA. If a body moving towards a plane given in position, be
acted upon by a force perpendicular to its motion tending towards that plane,
the inclination of the orbit to the plane will be increased. Again, if the body
be moving from the plane, and the force acts from the plane, the inclina
tion is also increased. But if the body be moving towards the plane, and
the force tends from the plane, or if the body be moving from the plane,
and the force tends towards the plane, the inclination of the orbit to the
plane is diminished.
495. To calculate that part of the ablatitious tangential force which is
employed in drawing P from the plane of its orbit.
Let the dotted line upon the ecliptic N A P N be that part of P s orbit
which lies above it. Let C D be the intersection of a plane drawn per
pendicular to the ecliptic ; P K perpendicular to this plane, and there
Aa3
374
A COMMENTARY ON
[SECT. XI.
fore parallel to the ecliptic. Take T F = 3 P K ; join P F and it will
represent the disturbing force of the sun. Draw P i a tangent to, and
F i perpendicular to the plane of the orbit. Complete the rectangle i m,
and P F may be resolved into P m, P i, of which P m is the effective force
to alter the inclination. Draw the plane F G i perpendicular to N N ;
then F G is perpendicular to N N . Also F i G is a right angle. As
sume P T tabular rad. Then
::R:3g,..
: : R : s > .
: : R : i J *
PT: Pm :: R 3 : 3g. s. i
_ PT.3g. s. i
Pm =
R
PT: TF::
T F : F G
FG: Pm
g = sin. 6 =r sin. L dist. from quad.
s = sin. <p = sin. /_ dist. of nodes from syz.
i =r sin. F T i = sin. F G i = sin. inclination of orbit to ecliptic.
Hence the force to draw P from its orbit =
_,
R
when P is in
the quadratures. Since g vanishes this force vanishes. When the nodes
are in the syzygies s vanishes, and when in the quadratures this force is a
maximum. Since s = rad. cotan. parte.
496. To calculate the quantity of the forces.
Let S T = d, P T = r, the mean distance from T = i. The force
of T on P at the mean distance = f; the force of S on P at the mean
distance = g.
Then the force S T =  , and the force S T : f. P T : : d : r,
IT r f? r 3 " r
.. force P T = ^r, hence the add. f = >; ablat. f = j? 3 sin. 0, the
d 3 d 3 d 3
mean add. force at distance 1 = K , the central ablat. = , , sin. 2 6, the
d* d 3
3 rr r
tangential ablat. f = S"jnT * S " T> ^ "
BOOK I.] NEWTON S PRINCIPIA. 375
* ~ f? i* 3 0* r
The whole disturbing force of S on P = ~o~T 3 ^ 9 1 3 cos> ^ ^ > l ^ e
mean disturbing f = ^ 3  (since cos. 2 & vanishes) = by supposi
tion.
Hence we have the whole gravitation of P to T = *3t + oiT *
r ci cl i& ci
x*
cos. 2 0, and the mean = n _ (since cos. 2 vanishes).
r a 2 d* v
PROBLEM.
497. Required the whole effect, and also the mean effect of the sun to
diminish the lunar gravity; and show that if P and p be the periodic
times of the earth and moon, f the earth s attraction at the mean
distance of the moon, r the radiusvector of the moon s orbit ; the additi
fp 2 P 4 1
tious force will be nearly represented by the formula \ \.~ ^n4 \ f r 
v X A *
P n = 3 P T. sin. 8 6, and P T 3 P T . sin. 8 d = 2 + JL p T X
cos. 26 = whole diminution of gravity of the moon, and the mean di
P T g r,
mmution = f ^ 3 by supposition.
Again,
P 1 a d 3
.ab. f ^_
498. To find the central and ablatitious tangential forces.
C
Take Pm = 3PK = 3PT. sin. 6 = ablatitious force.
Then P n = P m . sin. 6 = 3 P T . sin. * Q = central force
m n = P m . cos. 6 = 3 P T . sin. 6 . cos. 6
=  . P T sin. 2 6 = tangential ablatitious force.
To find what is the disturbing force of S on P.
A a 1
S76 A COMMENTARY ON [SECT. XI.
The disturbing force = P T 3 P T . sin. * 6 = (~ ! + 3cos  2 *\ x
P T" ^
P T = Ai +  P T. cos. 2 6.
d ti
To find the mean disturbing force of S during a whole revolution.
P T 3
Let P T at the mean distance = m, then 1 . P T cos. 2 6
id ft
g = H since cos. 2 6 is destroyed during a whole revolution.
499. To find the disturbing force in syzygy.
SAT A T = 2 A T = disturbing force in syzygy;
the force in quadrature is wholly effective and equal P T,
/. force in quadrature : f in syzygy : : P T : 2 P T : : 1 : 2.
To find that point in P s orbit when the force of P to T is neither
increased nor diminished by the force of S to T.
In this point Pn= P T or 3 P T sin. 2 6 = P T,
.*. sin. 6 = ==
V 3
and
6 = 35 W."
To find when the central ablatitious force is a maximum.
P n = 3 P T . sin. 2 & = maximum,
. . d . (sin. 2 6) or 2 sin. 6 . cos. & d 6 = 0,
. . sin. 6 . cos. 6 = 0,
or
sin. 6 . V 1 sin. 2 6 = 0,
and
sin. 0=1,
or the body is in opposition.
Then (Prop. LVIII, LIX,)
T 2 : t 2 : : S P : C P : : S + P : S
and
T 2 : t 2 : : A 3 : x 3
. A 3 : x 3 :: S+ P : S
and
A : x :: (S+ P) ^ : S*.
500. PROB. Hence to correct for the axis major of the moon s orbit.
Let S be the earth, P the moon, and let per. t of a body moving in a
secondary at the earth s surface be found, and also the periodic time of
BOOK I.] NEWTON S PRINCIPIA. 377
the moon. Then we may find the axis major of the moon s orbit round
the earth supposed at rest = x, by supposition. Then the corrected axis
or axis major round the earth in motion : x : : (S + P) : S s
(S +
. . axis major round the earth in motion = x . S
= y.
Hence to compare the quantity of matter in the earth and moon,
y:x:: VS+P: V S
... y 3_ x 3 : X 3 :: p : s.
501. To define the addititious and ablatitious forces. Let S T repre
sent the attractive force of T to S. Take
SL: ST::
1
1
: : ST 8 : S P 1
S P 2 S T
and S L will represent the attractive force of P to S. Resolve this into
S M, and L M ; then L M, that part of the force in the direction P T
is called the addititious force, and S M S T = NMis the ablatitious
force.
502. To compare these forces.
Since SL:ST::ST 2 :SP 2 , .. SL = i = attractive force of
P to S in the direction S P, and S P : S T : : 
= attractive
force of P to S in the direction TS = ST 4 (ST PK)~ =ST
+ 3 P K = S M nearly,
.. 3PK = TM = PL = ablatitious force = 3 P T . sin. 6.
c T 3 e T 3
Also S P P T  
SP ST" 1
P T = attractive force of P to S in the direction L M = P T nearly.
Hence the addititious force : ablatitious force : : P T : 3 P T . sin. 6 : 1
. 3 sin. 6. Q. e. d.
BOOK III.
1. PROP. I. All secondaries are found to describe areas round the
primary proportional to the time, and these periodic times to be to each
other in the sesquiplicate ratio of their radii. Therefore the center of
force is in the primary, and the force cc =Y .
2. PROP. II. In the same way, it may be proved, that the sun is the
center of force to the primaries, and that the forces oc ,.  . Also the
Aphelion points are nearly at rest, which would not be the case if the
force varied in a greater or less ratio than the inverse square of the dis
tance, by principles of the 9th Section, Book 1st.
3. PROP. III. The foregoing applies to the moon. The motion of the
moon s apogee is very slow about 3 3 in a revolution, whence the force
will x vsjTg ~Az It was proved in the 9th Section, that if the ablatitious
force of the sun were to the centripetal force of the earth : : 1 : 357.45,
that the motion of the moon s apogee would be ^ the real motion.
.*. the ablatitious force of the sun : centripetal force : : 2 : 357.45
: : 1 : 178 f.
This being very small may be neglected, the remainder x ^ .
4. COR. The mean force of the earth on the moon : force of attraction
; : 177 f I : 178 fg.
The centripetal force at the distance of the moon : centripetal force at
the earth:: 1 : D*.
5. PROP. IV. By the best observations, the distance of the moon from
the earth equals about 60 semidiameters of the earth in syzygies. If the
moon or any heavy body at the same distance were deprived of motion in
the space of one minute, it would fall through a space = 16 /^feet. For the
380
A COMMENTARY ON
III.
deflexion from the tangent in the same time = 16 T L feet. Therefore the
space fallen through at the surface of the earth in I" = 16 T V feet.
For 60" : t : : D : 1,
60
thence the moon is retained in its orbit by the force of the earth s gravity
like heavy bodies on the earth s surface.
6. PROP. XIX. By the figure of the earth, the force of gravity at
the pole : force of gravity at the equator : : 289 : 288. Suppose A B Q q
a spheroid revolving, the lesser diameter P Q, and A C Q q c a a canal
filled with water. Then the weight of the arm Q q c C : ditto of
A a c C : : 288 : 289. The centrifugal force at the equator, therefore 1
suppose 2^g f tne weight.
Again, supposing the ratio of the diameters to be 100 : 101. By com
putation, the attraction to the earth at Q : attraction to a sphere whose
radius == Q C : : 126 : 125. And the attraction to a sphere whose ra
dius A C : attraction of a spheroid at A formed by the revolution of an
ellipse about its major axis : : 126 : 125.
The attraction to the earth at A is a mean proportional between the at
tractions to the sphere whose radius =r A C, and the oblong spheroid,
since the attraction varies as the quantity of matter, and the quantity of
matter in the oblate spheroid is a mean to the quantities of matter in the
oblong spheroid and the circumscribing sphere.
Hence the attraction to the sphere whose radius = A C : attraction to
the earth at A : : 126 : 125 $.
. . attraction to the earth at the pole : attraction to the earth at the equa
tor : : 501 : 500.
Now the weights in the canals cc whole weights a magnitudes X gra
BOOK III.] NEWTON S PRINCIPIA. 381
vity, therefore the weight of the equatorial arm : weight of the polar
: : 500 X 101 : 501 X 100
: : 505 : 501.
4
Therefore the centrifugal force at the equator supports ^^ to make an
equilibrium.
But the centrifugal force of the earth supports
41 11
* 505 : 289 : : 100 : 229 = the GXCeSS f the e< l uatorial over the P olar
radius.
Hence the equatorial radius : polar : : 1 + ^^r : 1
: : 230 : 229.
Again, since when the times of rotation and density are different the
yz
difference of the diameter oc ,  . and that the time of the earth s rota
dens.
tion = 23h. 56 .
The time of Jupiter s rotation = 9h. 56 .
The ratio of the squares of the velocity are as 29 : 5, and the density
of the earth : density of Jupiter : : 400 : 94.5.
d the difference of Jupiter s diameter is as  X jrjp X rr ,
5 94.5 229
.. d : Jupiter s least diameter : :  x ^jr X ^5 : : 29 X 80 : 94.5 X 229
> .Jr..) 
: : 2320 : 21640
: : 232 : 2164
:: 1 : 9
The polar diameter : equatorial diameter : : 9 : 10^
ON THE TIDES.
7. THE PHENOMENA OF THE TIDES.
1. The interval between two succeeding high waters is 12 hours 25
minutes. The diminution varies nearly as the squares of the times from
high water.
2. Twenty four hours 50 minutes may be called the lunar day. The
interval between two complete tides, the tide day. The first may be call
382
A COMMENTARY ON
[BOOK III.
ed the superior, the other inferior, and at the time of new moon, the
morning and evening,
3. The high water is when the moon is in S. W. to us. The highest tide
at Brest is a day and a half after full or change. The third full sea after
the high water at the full moon is the highest ; the third after quadrature
is the lowest or neap tide.
4. Also the highest spring tide is when the moon is in perigee, the next
spring tide is the lowest, since the moon is nearly in the apogee.
5. In winter the spring tides are greater than in summer, and from the
same reasoning the neap tides are lower.
6. In north latitude, when the moon s declination is north, that tide in
which the moon is above the horizon is greater than the other of the same
day in which the moon is below the horizon. The contrary will take
place if either the observer be in south latitude or the moon s declination
south.
7. PROP. I. Suppose P to be any
particle attracted towards a center E,
and let the gravity of E to S be repre
sented by E S. Draw B A perpendi
cular to E S, which will therefore re
present the diameter of the plane of il
lumination. Draw Q P N perpendicu
lar to B A, P M perpendicular to E C.
Then take E I = 3 P N, and join P I,
P I will represent the disturbing force
of P. PI may be resolved into the
two P E, P Q, of which P E is counter
balanced by an equal and opposite force,
P Q acts in the direction N P.
Hence if the whole body be supposed
to be fluid, the fluid in the canal N P
will lose its equilibrium, and therefore
cannot remain at rest. Now, the equi
librium may be restored by adding a
small portion P p to the canal, or by
supposing the water to subside round
the circle B A, and to be collected to
wards O and C, so that the earth may put on the form of a prolate sphe
roid, whose axis is in the line O C, and poles in O and C, which may be
BOOK III.] NEWTON S PRINCIPIA. 383
the case since the forces which are superadded a N P, or the distance
from B A, so that this mass may acquire such a protuberancy at O and C,
that the force at O shall be to the force at B : : E A : E C ; and by the
above formula
_ 5JC __ E C E A
r ~ 4 g ~ E A
8. PROP. II. Let W equal the terrestrial gravitation of C; G equal its
gravitation to the sun ; F the disturbing force of a particle acting at O and
C ; S and E the quantities of matter in the sun and earth.
3 S C
.. F : W : :
CS* X CG C E 3
Since the gravitation to the sun oc s
C S 2 : E S 2 : : ES: C G
/. CG X C S 2 = ES 3 .
. F . W . . ?A . _A
E S 3 C E 3
and
E : S : : 1 : 338343
E C:ES: : 1 : 23668
. 3 S . _]?_.  1 12773541 F W
" * E s s cnr^ * iw**** . . , .
.. 4 W : 5 F : : C E : E C E A.
4 d 3d
Attraction to the pole : attraction to the equator : : 1 : 1
O t)
Quantity of matter at the pole : do. at equator : : 1 : 1 d.
Weight of the polar arm : weight of the equatorial arm : : 1 : 1 
O D
. Excess of the polar = attractive force : weight of the equator
or
mean weight W : : : 1
5F
9. PROP. III. Let A E a Q be the spheroid, B E b Q the inscribed
334
A COMMENTARY ON
III.
sphere, A G a g the circumscribed sphere, and D F d f the sphere equal
(in capacity) to the spheroid.
K
H
Then since spheres and spheroids are equal to f of their circumscribing
cylinder, and that the spheroid = sphere D F d f.
CF 2 xCD = CE 2 xCA
CE: C F 2 : : CD: C A,
and make
but
Also
CE:CF::CF:Cx
..C E 2 : C F 2 : : C E : Cx
/. CD:CA::CE:Cx
.. CD:CE::CA:Cx
C D = C E nearly
.. C A = C x.
E x = 2 E F nearly
.. A D = 2 E F.*
Let C E = a, C F = a + x,
_ aM2a*+x a _ a* + 2 *
~~T~ a
= a + 2 x nearly
.% E x = 2 x nearly.
BOOK III.] NEWTON S PRINCIPIA. 385
PROP. IV. By the triangles p I L, C I N,
A B : I L : : r 2 : (cos.) z L. T C A
.. I L = A B x (cos.) 2 z_ I C A = S X (cos.) ? x
(if S = A B and x = angular distance from the sun s place.)
Again,
G E : K I : : r 2 : (sin.) L. T C A
.. K I = S X (sin.) 2 . K.
COR. 1. The elevation of a spheroid above the level of the undisturbed
c _
ocean = 1 i 1 m = S X (cos.) 2 x \ = S X (cos.) 2 x
9
The depression of the same = S X (sin.) z x S = S X (sin.) 2 x f.
COR. 2. The spheroid cuts the sphere equal in capacity to itself in a
S
point where S X (cos.) 2 x = = 0, or (cos.) " x = .
o
. . cos. x = .57734, &c.
= cos. 54. 44 .
10. PROP. V. The unequal gravitation of the earth to the moon is
(4000) 3 times greater than towards the sun.
Let M equal the elevation above the inscribed sphere at the pole of
the spheroid, 7 equal the angular distance from the pole.
/. the elevation above the equally capacious sphere = M X (cos.) z /
the depression   = M X (sin.) * 7 .
Hence the effect of the joint action of the sun and moon is equal to the
sum or difference of their separate actions.
.. the elevation at any place = S X (cos.) 3 x f M X (cos.) 2 7 S + 5T
the depression  = S X (sin.) " x + M X (sin.) 2 7 f S + M.
1. Suppose the sun and moon in the same place in the heavens.
Then the elevation at the pole = S + M S + M =  S + M, and
the depression at the equator = S + M S + M = f, S + M,
. . the elevation above the inscribed sphere = S + M.
2. Suppose the moon to be in the quadratures.
The elevation at S = S i S + M = f S $ M.
the depression at M = S S + M = $ S M,
the elevation at S above the inscribed sphere = S M,
the elevation at M (by the same reasoning) = M S.
But (by observation) it is found that it is high water under the moon
when it is in the quadratures, also that tlie depression at S is below the
natural level of the ocean ; hence M is more than twice S, and although
VOL. I. B b
386
A COMMENTARY ON
[BOOK III.
the high water is never directly under the sun or moon, when the moon is
in the quadratures high water is always 6 hours after the high water at
full or change.
Suppose the moon to be in neither of the former positions.
Then the place of high water is where the elevation =r maximum,
or when S X cos. 2 x + M X cos. 2 y = maximum,
and since
cos. * x = + \ cos. 2 x,
and
cos. y = + \ cos. 2 y,
elevation = maximum, when S X cos. 2 x + M X cos. 2 y = max
imum.
Therefore, let A B S D be a great circle of the earth passing through
S and M, (those places on its surface which have the sun and moon in the
zenith). Join C M, cutting the circle described on C S in (m). Make
S d : d a : : force of the moon : force of the sun (which force is supposed
BOOK III.] NEWTON S PRINCIPIA. 387
known). Join ma, m d, and let H be any point on the surface of the
ocean. Join C H cutting the circle C m S in (h) ; draw the diameter
h d h , and draw m t, a x perpendicular to h h , and a y parallel to it.
Then
M=Sd, S=ad
and
and
.. d t = M X cos. 2 y, d x = S X cos. 2 x,
.. elevation = maximum when t x = a y = maximum,
or when a y = a m, i. e. when h h is parallel to a m, hence
CONSTRUCTION.
Make
S d : d a : : M : S,
and join m a, draw h h parallel to a m, and from C draw C h H cutting
the surface of the ocean in H, which is the point of hi^h water.
Again, through h 7 draw L C h , meeting the circle in L, L ; these are
the points of low water. For let
L C S = u, L C M = z .
cos. L a d x = cos. A S d h = cos. 2 u S C h"!= cos. 2 u = d x
and
cos. 2 z = cos. 2 L C M = d t.
.. S X cos. 2 u + M X cos. 2 z = max.
COR. If d f be drawn perpendicular to a m, a m represents the whole
difference between high and low water, a f equals the point effected by the
sun, m f that by the moon.
For
sin. * u = cos. 2 x,
sin. 2 y = cos. 2 x.
.*. elevation + depression = S X : cos.  x + M X : cos. i y
+ S X cos. 2 x ^ $
+ M X : cos. y f = S X : 2 cos. x 1 + M X : 2cos. y 1
= S X cos. 2 x + M x cos. 2 y
and
d t = M X cos. 2 y
d x = S X cos. 2 x.
388 A COMMENTARY ON [BOOK III.
12. Conclusions deduced from the above (supposing that both the sun
and moon are in the equator.)
H
1. At new and full moon, high water will be at noon and midnight.
For in this case C M, a m, C S, d h, C H, all coincide.
2. When the moon is in the quadrature at B, the place of high water is
also at B under the moon, when the moon is on the meridian, for C M is
perpendicular to C S, (m) coincides with C, (a m) with (a C), d h with
d C.
3. While the moon passes from the syzygy to the quadrature the place
of high water follows the moon s place, and is to the westward of it, over
takes the moon at the quadratures, and is again overtaken at the next
syzygy. Hence in the first and third quadrants high water is after noon
or midnight, but before the moon s southing, and in second and fourth vice
versa.
4. ZL M C H = max. when S C H = 45. S d h = 90. and m a
perpendicular to S C, and a m d = max., and a m d m d h = 2y .
BOOK III.] NEWTON S PRINCIPIA. 389
Hence in the octants, the motion of the high water = moon s easterly
motion ; in syzygy it is slower ; in quadratures faster. Therefore the tide
day in the octants = 24h. 50 = the lunar day ; in syzygy it is less = 24h.
35 ; in quadratures = 25h. 25 .
For take any point (u) near (m), draw u a, u d, and d i parallel to a u
and with the center (a) and radius a u, describe an arc (u v) which may
be considered as a straight line perpendicular to am; u m and i h are
respectively equal to the motions of M and H, and by triangles u m v,
dmf.
u m : i h : : m a : m f.
Therefore the synodic motion of the moon s place : synodic motion ot
high water : : m a : m f.
COR. 1. At new or full moon, m a coincides with S a, and m f with S d ;
at the quadratures, m a coincides with C a, and m f with C d ; therefore
the retardation of the tides at new or full moon : retardation at quadra
tures ::Sa:Ca::M + S:M S.
Con. 2. In the octants, m a is perpendicular to S a, therefore m a, m f
coincide. Therefore the synodic motion of high water equals the synodic
motion of the moon.
COR. 3. The variation of the tide during a lunation is represented by
(m a) ; at S, m a = S a, at C = C a.
Therefore the spring tide : neap tide : : M f S : M S.
COR. 4. The sun contributes to the elevation, till the high water is in
the octants, after which (a f ) is v e, therefore the sun diminishes the
elevation.
COR. 5. Let m u be a given arc of the moon s synodic motion, in v is
the difference between the tides m a, u a corresponding to it.
Therefore by the triangles m u v, m d f.
m u : m v : : m d : d f.
.*. m v <x d f ;
and since
m d : d f : : r : sin. dmf:: r : sin. m d h : : r : sin. 2 M C H
m v oc sin. 2 arc M H.
13. PROP. VI. In the triangle m d a, m d, d a and Z. m d a are known
when the proportion M : S is known and the moon s elongation.
Let the angle m d a = a,
and make
M + S : M S : : tan. a : tan. b
Bb3
390 A COMMENTARY ON [BOOK III.
then
a b a + b
v
For
y = g,x:= 2
M+ S:M S::md + da:md da
mad+amd mad amd
: : tan. ^ : tan.
2 x + 2 y 2x 2y
: : tan. = ; : tan. ~
: : tan. x f y : tan. x y
: : tan. a : tan. b,
. . x + y : x y : : a : b,
.. 2 x = a + b, 2 y = a b,
. x 
and
~ b
2 *
14. PROP. VII. To find the proportion between the accelerating forces
of the moon and sun. 1st. By comparing the tide day at new and full
moon with the tide day at quadratures.
35 : 85 : : M : S,
. . . . . . . . .
Also, at the time of the greatest separation of high water from the moon
in the triangle m d a, m d : d a : : r : sin. 2 y : : M : S,
.. jgj = sin. 2 y,
at the octants y is found =12 SO ,
.. ^ = sin. 25,
M
. . M : S : : 5 : 2} nearly.
Hence taking this as the mean proportion at the mean distances of the
moon and sun (if the earth =1) the moon = .
COR. 1. If the disturbing forces were equal there would be no high or
low water at quadratures, but there would be an elevation above the in
scribed spheroid all round the circle, passing through the sun and moon
BOOK 111.] NEWTON S PRINCIPIA. 391
COR. The gravitation of the sun produces an elevation of 24 inches,
the gravitation of the moon produces an elevation of 58 inches.
.. the spring tide = 82 inches, and the neap tide = 33 inches.
15. COR. 3. Though M : S : : 5 : 2, this ratio varies nearly from (6:2)
to 4 : 2, for supposing the sun and moon s distance each = 1000.
In January, the distance of the sun = 983, perigee distance of the
moon = 945.
In July, the distance of the sun = 1017, apogee distance of the moon
= 1055.
1
Disturbing force oc ~ 3 ; hence
S
M
apogee
1.901
4.258
mean
2
5
perigee
2.105
5.925.*
rr.i 5 A 3 d 3
The general expression is M = S X pr 3 X 75
JL) o
To find the general expression above.
Disturbing force of different bodies (See Newton, Sect, llth, p. G6,
Cor. 14.) a jL,
.. disturbing force S : disturbing force at mean distance : : D 3 : A 3
disturbing force M : disturbing force at mean distance : : d 3 : 8 3 ,
M 5 d_*_ <P
S : 2 :: D 3 * A~ 3
M ^ Aj> (P
* ~S > : 2 X D 1 X ^
(or supposing that the absolute force of the sun and moon are the same).
16. PROP. VIII. Let N Q S E be the earth, N S its axis, E Q its equa.
tor, O its center ; let the moon be in the direction O M having the de
clination B Q.
* The solar force may be neglected, but the variation of the moon s distance, and proportion
ally the variation of its action, produces an effect on the times, and a much greater on the heights
of the tides.
332
[BooK III.
Let D be any point on the surface of the earth, D C L its parallel of
latitude, N D S its meridian ; and let B F b f be the elliptical spheroid
of the ocean, having its poles in O M, and its equator F O f.
F
As the point D is carried along its parallel of latitude, it will pass
through all the states of the tide, having high water at C and L, and low
water when it comes to (d) the intersection of its parallel of latitude with
the equator of the watery spheroid.
Draw the meridian N d G cutting the terrestrial equator in G. Then
the arc Q G (converted into lunar hours) will give the duration of the
ebb of the superior tide, G E in the same way the flood of the inferior.
N. B., the whole tide G Q C , consisting of the ebb Q G, and the flood
G Q is more than four times G O greater than the inferior tide.
COR. If the spheroid touch the sphere in F and f, C C is the height
at C, L I/ the height at L, hence if L q be a concentric circle C q will
be the difference of superior and inferior tides.
CONCLUSIONS DRAWN FROM PROP. VIII.
1. If the moon has no declination, the duration of the inferior and su
perior tides is equal for one day over all the earth.
2. If the moon has declination, the duration of the superior will be
longer or shorter than the duration of the inferior according as the
moon s declination and the latitude of the place are of the same or differ
ent denominations.
3. When the moon s declination equals the colatitude or exceeds it,
BOOK III.] NEWTON S PRINCIPIA. 393
there will only be a superior or inferior tide in the same day, (the paral
lel of latitude passing through f or between N and f.)
4. The sin. of arc G O = tan. of latitude X tan. declination.
For
rad. : cot. d O G : : tan. d G : sin. G O,
.. sin. G O = cot. d O G X tan. G d
= tan. declination X tan. latitude.
17. PROP. IX. With the center C and radius C Q (representing the
P
whole elevation of the lunar tide) describe a circle which may represent
the terrestrial meridian of any place, whose poles are P, p, and equator
E Q. Bisect P C in O, and round O describe a circle P B C D ; let M
be the place on the earth s surface which has the moon in its zenith, Z
the place of the observer. Draw M C m, cutting the small circle in A,
and Z C N cutting the small circle in B ; draw the diameter BOD and
A I parallel to E Q, draw A F, G H, IK perpendicular to B D, and
join I D, A B, A D, and through I draw C M cutting the meridian in
M . Then after a diurnal revolution the moon will come into the
situation M , and the angle M C N ( = the nadir distance) = supplement
the angle ICB = ^IDB.
Also the .ADB = BCA = zenith distance of the moon.
A COMMENTARY ON [BooK III.
Hence D F, D K cc cos. * of the zenith and nadir distances to rad. D B.
oc elevation of the superior and inferior tides.
CONCLUSIONS FROM PROP. IX.
1. The greatest tides are when the moon is in the zenith or nadir of the
observer. For in this case (when M approaches to Z) A and I move to
wards D, B, and F coincides with B ; but in this case, the medium tide
which is represented by D H (an arithmetic mean to D K, D F) is di
minished.
If Z approach to M, D and I separate ; and hence, the superior and
inferior and the medium tides all increase.
2. If the moon be in the equator, the inferior and superior tides are
equal, and equal M X (cos) 2 latitude. For since A and I coincide with
C, and F and K with (i) D i = D B X (cos.) 8 B D C = M X (cos.) *
latitude.
3. If the observer be in the equator, the superior and inferior tides are
equal every where, and = M X (cos.) 2 of the declination of the moon.
For B coincides with C, and F and K with G ; P G = P C X cos. 2 of
the moon s declination = M x (cos.) z of the moon s declination.
4. The superior tides are greater or less than the inferior, according as
the moon and place of the observer are on the same or different sides of
the equator.
5. If the colatitude of the place equal the moon s declination or is less
than it, there will be no superior or inferior tide, according as the latitude
and the declination have the same or different denominations. For when
P Z = M Q, D coincides with I, and if it be less than M Q, D falls between
I and C, so that Z will not pass through the equator of the watery spheroid.
6. At the pole there are no diurnal tides, but a rise and subsidence
of the water twice in the month, owing to the moon s declining to both
sides of the equator.
18. PROP. X. To find the value of the mean tide.
A G = sin. 2 declination (to rad. = O C.)
and
O G = cos. 2 declination (to the same radius).
M
..OH = cos. 2 declination X cos. 2 lat. X _ t
SB
.. D H = O D + O H
M / 1 + cos. 2 lat. X cos. 2 declination
._ J.VI X x,
BOOK III.] NEWTON S PRtNCIPIA. 395
Now as the moon s declination never exceeds 30, the cos. 2 declination
is always + v 2 , and never greater than ; if the latitude be less than 45,
the cos. 2 lat. is + v e, after which it becomes v e.
Hence
1. The mean tide is equally affected by north and south declination of
the moon.
2. If the latitude = 45, the mean tide M.
3. If the lat. be less than 45, the mean tide decreases as the declina
tion increases.
4. If the latitude be greater than 45, the mean tide decreases as the
declination diminishes.
. Tr . , , . , ,, 1 + cos. 2 declination
5. If the latitude = 0, the mean tide = M X 5
BOOK I.
SECTION XII.
503. PROP. LXX. To find the attraction on a particle placed within
a spherical surface, force ce g . y  g .
Let P be a particle, and through P draw H P K,
I P L making a very small angle, and let them
revolve and generate conical surfaces I P H,
L P K. Now since the angles at P are equal
and the angles at H and L are also equal (for
both are on the same segment of the circle),
therefore the triangles H I P, P L K, are similar.
.. HI:KL::HP:PL
Now since the surface of a cone GC (slant side) 2 ,
.. surface intercepted by revolution of I P H : that of L P K : : P H : P L
:: HI 2 : KL
and attractions of each particle in I P H : that of L P K
P2 r> T
Jti J. Jr \~
1
but the whole attraction of P oc the number of particles X attraction of
each,
HI* K L*
.. the whole attraction on P from H I : from K L : : ^rm ir T .
rl 1 T Jv L, *
:: J : 1;
and the same may be proved of any other part of the spherical surface ;
.*. P is at rest.
504. PROP. LXXL To find the attraction on a particle placed without
a spherical surface, force cc g . p  .
distance *
398
A COMMENTARY ON
[SECT. XII.
Let A B, a b, be two equal spherical surfaces, and let P, p be two
particles at any distances P S, p s from their centers; draw P H K,
K
P I L very near each other, and S F D, S E perpendicular upon them, and
from (p) draw p h k, p i 1, so that h k, i 1 may equal H K, I L respective
ly, and s f d, s e, i r perpendiculars upon them may equal S F D, S E,
I R respectively ; then ultimately PE = PF = pe = pf, and D F
= d f. Draw I Q, i q perpendicular upon P S, p s.
Now
PI: PF:: IR : DF")
V.. PI pf:pi.PF::IR:ir::IH:ih
: : d f : i r I
and
pf:p i : :
Again
PI: PS:: IQ: SF
and
/ps:pi::sf:iq
.. PP.pf. ps: (pi) 2 .PF. PS:: IQ.IH:iq.ih
: : circumfer. of circle rad. I Q X I H : circumfer. of circle rad i q X i h
: : annulus described by revolution of I Q : that by revolution of i q.
Now
. . PI.ps:pi.PS::IQ:iq
attraction on 1st annulus : attraction on 2d
And
attraction on the annulus : attraction in the direction P S
P F
.. attraction in direction PS = p f. p s. ^~
P ]
.. whole att n . of p to S : whole att n . of (p) to s : : p f . p s . p;
1st annulus ^ 2d annulus
distance 2 distance 2
PP.pf.ps (pi)*.PF. PS
PI 2 (pi) 2
:: pf. ps :PF.PS.
P I : P Q
P S : P F
: P F . P S . 
PS 2 ps 2
BOOK I.] NEWTON S PRINCIPIA. 399
and the same may be proved of all the annul! of which the surfaces are
^ 
composed, and therefore the attraction of P cc pqi cc ^  j from
the center.
COR. The attraction of the particles within the surface on P equals the
attraction of the particles without the surface.
For K L : I H : : P L : P I : : L N : I Q.
.*. annulus described by I H : annulus described by K L
:: IQ.IH: K L. L N : : P P : P L 2
.*. attraction on the annulus I H : attraction on the annulus K L
PI 2 PL 2
: P I s : P L 8
and so on for every other annulus, and one set of annuli equals the part
within the surface, and the other set equals the part without.
506. PROP. LXXII. To find the attraction on a particle placed with
out a solid sphere, force oc g  p  r.
distance 2
Let the sphere be supposed to be made up of spherical surfaces, and
the attraction of these surfaces upon P will x TT  r, and therefore
distance z
the whole attractions
number of surfaces content of sphere diameter 3
P~W~ PS 2 PS 2 "
and if P S bear a given ratio to the diameter, then
the whole attraction on P oc , ,<x diameter
diameter
507. PROP. LXXIIL To find the attraction on the particle placed
within
Let P be the particle ; with rad. S P describe
the interior sphere P Q ; then by Prop. LXX.
(considering the sphere to be made of spherical
surfaces,) the attraction of all the particles con
tained between the circumferences of the two
circles on P will be nothing, inasmuch as they
are equal on each side of P, and the attraction
PS 3
of the other part by the last Prop, oc rr^i a P S.
400
A COMMENTARY ON
[SECT. X1J.
508. PROP. LXXIV. If the attractions of the particles of a sphere
<x T . : =. and two similar spheres attract each other, then the spheres
distance z
1
will attract with a force x g as
distance 3
of their centers.
For the attraction of each particle cc ~ = from the center of the
distance 2
attracting sphere (A), and therefore with respect to the attracted particle
the attracting sphere is the same as if all its particles were concentrated
in its center. Hence the attraction of each particle in (A) upon the
whole of (B) will a ^ of each particle in B from the center of P,
distance 2
and if all the particles in B were concentrated in the center, the attraction
would be the same; and hence the attractions of A and B upon each other
will be the same as if each of them were concentrated in its center, and
1
therefore cc
distance 2
509. PROP. LXXVI. Let the spheres attract each other, and let
them not be homogeneous, but let them be homogeneous at correspond
ing distances from the center, then they attract each other with forces
1
ac*.
distance
G
Suppose any number of spheres C D and E F, I K and L M, &c. to
be concentric with the spheres A B, G H, respectively; and let C D and
I K, E F and L M be homogeneous respectively ; then each of these
spheres will attract each other with forces cc g . T ; Now suppose
distance *
the original spheres to be made up by the addition and subtraction of
similar and homogeneous spheres, each of these spheres attracting each
BOOK I.]
other with a force a *. 7:  a ,
distance a
each other in the same ratio.
NEWTON S PRINC1PIA.
1
401
then the sum or differences will attract
510. PROP. LXXVII. Let the force cc distance, to find the attraction
of a sphere on a particle placed without or within it.
Let P be the particle, S the center, draw two planes E F, e f, equally
distant from S ; let H be a particle in the plane E F, then the attraction
of H on P oc HP, .and therefore the attraction in the direction S P a
P G, and the attraction of the sum of the particles in E F on P towards
S a circle E F . P G, and the attraction of the sum of the particles in
(e f) on P towards S cc circle e f . P g, therefore the whole attraction of
E F, e f, a circle EF(PG+Pg)cc circle E F . 2 P S, therefore the
whole attraction of the sphere cc sphere X P S.
When P is within the sphere, the attraction of the circle E F on P to
wards S x circle E F . P G, and the attraction of the circle (e f ) towards
S cc circle e f . P g, and the difference of these attractions on the whole
attraction to S cc circle E F (P g P G) cc circle E F . 2 P S. There
fore the whole attraction of the sphere on P cc sphere X P S.
511. LEMMA XXIX. If any arc be described with the center S, rad.
S B, and with the center P, two circles be described very near each other
Vot. I. C c
402
A COMMENTARY ON
[SECT. XII.
cutting, first, the circle in E, e, and P S in F, f; and E D, e d, be drawn
perpendicular to P S, then ultimately,
Dd: Ff: : PE: PS.
For
Dd: Ee::
DT: ET:: DE: ES
and
Ee: Ff ::
Ee:er::SE:SG
D rl F f 
T> E : S G ; :
P E : P S.
512. PROP. LXXIX. Let a solid be generated by the revolutions of at.
evanescent lamina E F f e round the axis P S, then the force with which
the solid attracts PocDE 2 . Ffx force of each particle.
Draw E D, e d perpendiculars upon PS; let e d intersect E F in r;
draw r n perpendicular upon E D. Then E r : n r : : P E : ED, .*.
Er.ED = nr.PS = Dd.PE, . . the annular surface generated by
the revolution of Era Er.EDacDd.PE, and (P E remaining the
same) a D d. But the attraction of this annular surface on P ex D d .
P E, and the attraction in the direction P E : the attraction in the direc
tion P S : : P E : P D,
.. the attraction in the direction P S oc
PD
PE
.Dd. PE oc PD.Dd
and the whole attraction on P of the surface described by E F a sum of
the PD.Dd.
Let P E = r, D F = x,
.. P D = r x,
. P D . D d = r d x x < x,
..sumofPD.DdS=/rdx xdx =
D E 1 ,
and therefore the attraction of lamina cc D E *. Ff X force of each particle.
BOOK I.]
NEWTON S PRINCIPIA. 403
D E*. P S
513. PROP. LXXX. Take D N proportional to
X force
of each particle at the distance P E, or if ^ represent that force, let D N
~T) "p 2 p C
a P E V tnen tne area trace d out by D N will be proportional to
the whole attraction of the sphere.
For the attraction of lamina EFfeaDE 2 . Ffx force of each parti
ID E 2 P S
cle a (LEMMA XXIII)    . D d x force of each particle, or
D E 2 PS
d "
a attraction of lamina E F f e, and the
p E. V
sum of these areas or area A N B will represent the whole attraction of
the sphere on P.
514. PROP. LXXXI. To find the area A N B.
Draw the tangent P H and H I perpendicular on P S, and bisect P I
in L ; then
Cc3
404 A COMMENTARY ON [SECT. XII.
But
SE S = SH 2 = PS.SI,
.. PE 2 = P S 2 + PS.SI + 2PS.SD
= PS{PS + SI + 2SD}
= P S J(P I + I S) + S I + 2 S DJ
DE*= SE 2 SD 2 = SE 2 (LD LS) 2
= SE 2 LD 2 LS 2 + 2LD.LS
2LD.LS LD 2 (LS+SE)(LS SE)
= 2LD.LSLD S LB.LA,
DE 2 .PS 2LD.LS.PS
.*. D N oc
PE.V V 2 SD.P S. V
LD 2 .PS LB.LA. PS
V 2 L D . P S". V V2LD.PS.V
and hence if V be given, D N may be represented in terms of L D and
known quantities.
515. Ex. 1. Let the force a ^.  ; to find the area A N B.
distance
_ XT 2LS.LD.PS LD 2 .PS AL.LB.PS
D H 2LD.PS" ~2LD.PS 2LD.PS
, LD A L . L B
~2~* 2LD
, _ , L D . D d AL.LB. D d
.. D N . D d, or d . area GC L b . D d  ^ 2 ^ ^  >
., area AND between the values of L A and L B
n TAX LB 2 LA 2 AL.LB .LB
= LS.(LB~LA)  j g ILA
Now
L B 2 L A 2 = (L B + L A) . (LB LA)
= (LS + AS + LS AS)AB = 2LS.AB,
A XT r^ A B . AL.LB . L B
.. area AND = LS.AB  ^  2  1 j^
L S . A B AL.LB . L B
~~2~ 2 L~A*
BOOK I.j
NEWTON S PRINC1PIA.
405
516. To construct this area.
To the points L, A, B erect L 1, A a, B b,
perpendiculars, and let A a = L B, and B b
= L A, through the points (a), (b), de
scribe an hyperbola to which L 1, L B are
asymptotes. Then. by property of the hy
perbola, AL.Aa = LD.DF,
AL.Aa AL.LB
LD "LTD
A L . L B . D d
L A
.. D F =
.. DF.Dd =
.. area A a F D =/D F. D d = A L. L B/L D,
. . hyperbolic area Aaf b B= A L L B f^L^.
J L A*
The area A a B b == B b A B 4 A B> a n
2
.
+ L A
= ^^AB
AB = LS.AB,
. . area a f b a = area A a B b area A a f b B
517. Ex. 2. Let the force a
n TO
Let V =
, to find the
2 A S*
. .DN=
but
L.V
PE.V
V.PE = JLJ^ := !P_SjiI^_
AL.LB. PS
A K9
. . DN =^i^J[ AL.LB. SI i
LD 2 ~2T7IP = 2PS. g L.LD ,
N.x = SI.L S/LD ^JjiPj. AL.LB. SI
2 ~2TT)
. . area between the values of L A and L B
S /^? !iiyLzi_LAJ_ /L^SJ AL.SI
AJ A 2 v^ o ~"~ : S
Cc3
406
A COMMENTARY ON
[SECT. xir.
To construct this area.
1 a
Take S I = S s, and describe a hyperbola passing through a, s, b, to which
L 1, L B are asymptotes; then as in the former case, the area A a n b B
.. the area A N B = S I . L S SLAB.
518. PROF. LXXXII. Let I be a particle within the sphere, and P
the same particle without the sphere, and take
S P : S A : : S A : S I,
then will the attracting power of the sphere on I : attracting power of the
sphere on P _ _
: : V S I. V force on I : V S P. V forpe on P.
D N force on the point P : D N 7 force on the point I
D E 2 P S D E 2 I S
:: ~FE7V~ : TE.V
: : P S . I E . V : I S . P E . V.
Let
V : V :: P E n : I E",
BOOK L] NEWTON S PRINCIPIA.
then
DN : D N :: PS.IE.IE n :IS.PE.PE",
but
PS:SE::SE:SI,
and the angle at S is common,
.. triangles P S E, I S E are similar,
.. P E : I E : : P S : S E : : S E : S I,
.. D N : D N :: PS.SE.I E n : PS. SI. P E ",
: : SE.I E n : S I.P E n
407
: : A/S P.I E n : VSI.PE"
:: VSP : SI VSI.PS*.
519. PROP. LXXXIII. To find the attraction of a segment of a spheie
.upon a corpuscle placed within its centre.
Draw the circle F E G with the
center P, let R B S be the segment of
the sphere, and let the attraction of the
spherical lamina E F G upon P be
proportional to F N, then the area de
scribed by F N oc whole attraction of
the segment to P.
Now the surface of the segment
E F G a P F D F, and the content
of the lamina whose thickness is O a
PF D F O.
1
G
Let F oc
distance
 and the attraction on P of the particle in that
DE !
spherical lamina, cc (Prop. LXXIII.) p pr^
f2 P F F D F D 2 ) O
a
PF n
2FD O FD 2 O
P F "  ! P F n
.. if F N be taken proportional to p ,., n _,
out by F N will be the whole attraction on P.
F D *
 , the area traced
520. PROP. LXXXIV. To find the attraction when the body is placed
in the axis of the segment, but not in the center of the sphere.
408
A COMMENTARY ON
[SECT. XIII.
Describe a circle with the radius P E, and the segment cut off by the
revolution of this circle E F K round P B, will have P in its center, and
FIB
the attraction on P of this part may be found by the preceding Proposi
tion, and of the other part by PROP. LXXXI. and the sum of these at
tractions will be the whole attraction on P.
SECTION XIII.
521. PROP. LXXXV. If the attraction of a body on a particle placed
in contact with it, be much greater than if the particle were removed at
any the least distance from contact, the force of the attraction of the par
ticles a in a higher ratio than that ofp .
distance z
For if the force oc rr j , and the particle be placed at any distance
from the sphere, then the attraction oc ^ from the center of the
distance 2
sphere, .and .. is not sensibly increased by being placed in contact with
the sphere, and it is still less increased when the force a in a less ratio
, and it is indifferent whether the sphere be homo
than that of p
distance
geneous or not ; if it be homogeneous at equal distances, or whether the
body be placed within or without the sphei e, the attraction still varying in
the same ratio, or whether any parts of this orbit remote from the point of
contact be taken away, and be supplied by other parts, whether attractive
or not, . . so far as attraction is concerned, the attracting power of this
sphere, and of any other body will not sensibly differ ; . . if the pheno
BOOK I.] NEWTON S PRINCIPIA. 409
mena stated in the Proposition be observed, the force must vary in a higher
ratio than that of p .
distance 2
522. PROP. LXXXVI. If the attraction of the particles cc in a higher
ratio than T. , or oc 7: 5 . then the attraction of a body placed
distance 3 distance d
in contact with any body, is much greater than if they were separated
even by an evanescent distance.
For if the force of each particle of the sphere oc in a higher ratio than
that of j = , the attraction of the sphere on the particle is indefinitely
distance
increased by their being placed in contact, and the same is the case for
any meniscus of a sphere; and by the addition and subtraction of attrac
tive particles to a sphere, the body may assume any given figure, and
.*. the increase or decrease of the attraction of this body will not be sensi
bly different from the attraction of a sphere, if the body be placed in con
tact with it.
523. PROP. LXXXVII. Let two similar bodies, composed of particles
equally attractive, be placed at proportional distances from two particles
which are also proportional to the bodies themselves, then the accelerat
ing attractions of corpuscles to the attracting bodies will be proportional
to the whole bodies of which they are a part, and in which they are simi
larly situated.
For if the bodies be supposed to consist of particles which are propor
tional to the bodies themselves, then the attraction of each particle in one
body : the attraction of each particle in the other body, : : the attraction
of all the particles in the first body : the attraction of all the particles in
the second body, which is the Proposition.
COR. Let the attracting forces cc p , then the attraction of a
distance n
particle in a body whose side is A : B
A 3 B 3
distance n from A distance n from B
A 3 B^
A" : B"
1 1
A n  3 B n ~ 3
if the distances oc as A and B.
410
A COMMENTARY ON
[SECT. XIII.
524. Paop. LXXXVIII. If the particles of any body attract with a
force a distance, then the whole body will be acted upon by a particle
without it, in the same manner as if all the particles of which the body is
composed, were concentrated in its center of gravity.
Let R S T V be the body, Z the par
ticle without it, let A and B be any
two particles of the body, G their cen
ter of gravity, then A A G = B B G,
and then the forces of Z of these parti
cles oc A A Z, B B Z, and these
forces may be resolved into A A G +
A G Z, B B G + B G Z, and A A G
being = B B G and acting in opposite
directions, they will destroy each other,
and . . force of Z upon A and B will be
proportional to A Z G + B Z G, or to (A + B) Z G, .. particles A
and B will be equally acted upon by Z, whether they be at A and B, or
collected in their center of gravity. And if there be three bodies A, B,
C, the same may be proved of the center of gravity of A and B (G) and
C, and . . of A, B, and C, and so on for all the particles of which the
body is composed, or for the body itself.
525. PROP. LXXXIX. The same applies to any number of bodies
acting upon a particle, the force of each body being the same as if it
were collected in its center of gravity, and the force of the whole system
of bodies being the same as if the several centers of gravity were collected
in the common center of the whole.
526. PROP. XC. Let a body be placed in a perpendicular to the plane
of a given circle drawn from its center ; to find the attraction of the circu
lar area upon the body.
With the center A, radius = A D, let
a circle be supposed to be described, to
whose plane A P is perpendicular. From
any point E in this circle draw P E, in
P A or it produced take P F = P E, and
draw F K perpendicular to P F, and let
F K oc attracting force at E on P. Let
J K L be the curve described by the point
K, and let I K L meet A D in L, take
P H = P D, and draw H I perpendicular
BOOK I.] NEWTON S PRINCIPIA. 411
to P H meeting this curve in I, then the attraction on P of the circle
a A P the area A H I L.
For take E e an evanescent part of A D, and join P e, draw e C per
pendicular upon P E, .. E e : E C : : P E : A E, .. E e . A E = E C x
P E cc annulus described by A E, and the attraction of that annul us in
P A
the direction P A oc E C . P E . ^^ X force of each particle at E cc E C X
lr ilj
P A X force of each particle at E, but E C = F f, .. F K . F f cc E C x
the force of each particle at E, . . attraction of the annulus in the direction
PA cc PA.Ff. FK, and ..PA X sum of the areas F K . Ff or P A
the area A H I L is proportional to the attraction of the whole part de
scribed by the revolution of A E.
527. COR. 1. Let the force of each particle a T  3, at P F = x.
distance 2
let b = force at the distance a,
b a 2
. . F K the force at the distance x = r ,
F K Ff ka dx
. . r A., f i ,
.. attraction = PA.FK.Ff=PA /" 2
J x
cc PA  x A p p ,
and between the values of P A and P H, the attraction
P A _L l ? A
L PA~~PH ~PH*
528. COR. 2. Letthe force cc ^^  . , then T K = ^ ,
distance n x n
^ /.b a n , PA 1
. . attraction = P A / d x cc  r X  r+ Cor.,
/x n n 1 x "
and between the values of P A and P H,
PA r i 1
.* *
attraction = 
:
1 P A
529. COR. 3. Let the diameter of a circle become infinite, or P II
cc co, then the attraction cc . .
1 A
530. PROP. XCI. To find the attraction on a particle placed in the
axis produced of a regular solid.
412
A COMMENTARY ON
R E
[SrCT. XIII.
Let P be a body situated in the axis A B of the curve D E C G, by
the revolution of which the solid is generated. Let any circle 11 F S
perpendicular to the axis, cut the solid, and in the semidiameter F S of
the solid, take F K proportional to the attraction of the circle on P, then
F K . F f o: attraction of the solid whose base = circle R F S, and depth
= F f, let I K L be the curve traced out by F K, .. A L K F a at
traction of the solid.
COR. 1. Let the solid be a cylinder, the force varying as j
2 *
D
Then the attraction of the circle R F S, or F K which is proportional
PF
to that attraction a 1
PR
Let P F = x, F R = b,
. . F K a 1 
x x
. . FK. Ff ac dx
.. area cc x Vx 2 + b * .
BOOK I.]
NEWTON S PRINCIPIA.
413
Now if P A = x, attraction = 0,
.. Cor. = PD P A,
.. whole attraction = P B P E + PD PA
= AB PE + PD.
Let AB= oo = P E = P D,
. . atraction = A B.
531. COR. 3. Let the body P be placed
within a spheroid, let a spheroidical shell
be included between the two similar
spheroids DOG, K N I, and let the
spheroid be described round S which
will pass through P, and which is simi
lar to the original spheroid, draw D P E,
F P G, very near each other. Now P D
= BE, PF = CG, P H = B I, P K
= CL.
.. F K = L G, and D H = I E,
and the parts of the spheroidical shell which are intercepted between these
lines, are of equal thickness, as also the conical frustums intercepted by
the revolution of these lines, and
.*. attraction on P by the part D K : . . . . G I
number of particles in D K __ ... G 2
G
E
PD
PD ?
PG 2
PG
: : 1 : 1,
PD 2 PG !
arid the same may be proved of every other part of a spheroidical shell, and
,\ body is not at all attracted by it; and the same may be proved of all the
other spheroidical shells which are included between the spheroids, A O G,
and C P M, and . . P is not affected by the parts exteinal to C P M, and
.. (Prop. LXXIL),
attraction on P : attraction on A : : P S : A S.
532. PROP. XCIII. To find the attraction of a body placed without an
infinite solid, the force of each particle varying as T. , where n is
distance n
greater than 3.
Let C be the body, and let G L, H M, K O, &c. be the attractions
at the several infinite planes of which a solid is composed on the
414
A COMMENTARY ON
[SECT XIII.
body C; then the area G L O K equals the whole attraction of a solid
onC.
L
M
N
G
H
I
K
1
m
n
o
Now if the force cc T . n ,
distance
Then
H M a c ^ n _ a (Cor. 3. Prop. XC)
../H M . d x a /*4^i cc L_^ + Cor.
J .j V n * xr n d *
cc
C G n ~ 3 C H n ~
and. if H C = GO
then the area G L O K a ^^  .
(j (_, " ~ d
Case 2. Let a body be placed within the solid.
N
C
O
I K
o
G
Let C be the place of the body, and take C K = C G ; the part of
the solid between G and K will have no effect on the body C, and there
fore it is attracted to remain as if it were placed without it at the distance
CK.
.*. attraction cc
BOOK I.]
NEWTON S PRINCIPLE
415
SECTION XIV.
534. PROP. XCIV. Let a body move through a similar medium, ter
minated by parallel plane surfaces, and let the body, in its passage through
this medium, be attracted by a force varying according to any law of its
distance from the plane of incidence. Then will the sine of inclination be
to the sine of refraction in a given ratio.
R a
Let A a, B b be the planes which terminate the medium, and G H be
the direction of the body s incidence, and I R that of its emergence.
Case 1. Let the force to the plane A a be constant, then the body will
describe a parabola, the force acting parallel to I R, which will be a diameter
of the parabola described. H M will be a tangent to the parabola, and if
K I be produced I L will also be a tangent to the parabola at I. Let K I
produced meet G M in L with the center L, and distance L I describe
a circle cutting I R in N, and draw L O perpendicular to I R. Now by a
property of the parabola M I =. I v,
.. M L = H L, /. M O = O R, and .. M N = I R.
The angle L M I = the angle of incidence, and the angle M I L = sup
plement of M I K = supplemental angle of emergence.
Now
L . M I = M H 2 = 4 M L *
416 A COMMENTARY ON [SECT. XIV.
but
MN.MI = MI.IR = MQ.MP=ML + LQ.ML LQ
__ 1V/T T 2 T {^\
. M j = ML 2 ^LQ 2
.. L:IR::4ML 2 :ML 2 LQ 1
but L and I R are given
..4 ML 2 a ML 2 LQ 1
..ML 2 aLQ 2 a LI 2
.% M L a L I or sin. refraction : sin. inclination in a given ratio.
Case 2. Let the force vary according to T G/
H / a
any law of distance from A a. g j/ ~ b
Divide the medium by parallel planes A a, c K/ c
B b, C c, D d, &c. and let the planes be at ^_
evanescent distances from each other, and
let the force in passing from A a to B b,
from B b to C c, from C c to D d, &c. be
uniform.
. . sin. I at H : sin. R at H : : a : b
sin. R or I at I : sin. R at K : : c : d
sin. R or I at K : sin. R at R : : e : f, and so on.
. . sin. I at H : sin. RatR::a.c.e:b.d.f and in a constant pro
portion.
535. PROP. XCV. The velocity of a particle before incidence : velocity
after emergence : : sin. emergence : sin. incidence.
G
B
D
K
Take A H = I d, and draw A G, d K perpendicular upon A a, D d,
meeting the directions of incidence and emergence in G, K. Let the
motion of the body be resolved into the two G A, A H, Id, d k, the ve
BOOK L] NEWTON S PRINCIPIA. 417
locity perpendicular to A a cannot alter the motion in the direction A a ;
therefore the body will describe G H, I K in the same time as the spaces
A H, I d are described, that is, it will describe G H, I K in equal times
before the incidence and after the emergence.
Velocity before incidence : velocity after emergence : : G H : I K
A H Id
" sin/incidence sin. emergence
: : sin. emergence : sin. incidence.
536. PROP. XCVI. Let the velocity before incidence be greater than
the velocity after emergence, then, by inclining the direction of the inci
dent particle perpetually, the ray will be refracted back again in a similar
curve, and the angle of reflection will equal the angle of incidence.
ri 2
*>
A P hXa
B \P P/ b
C
D
E
Let the medium be separated by parallel planes A a, B b, C c, D d,
E e, &c. and since the velocity before incidence is greater than the
velocity after emergence. .*. sin. of emergence is greater than sin. of in
cidence. . . H P, P Q, Q R, &c. will continually make a less angle with
H a, P b, Q c, R d, &c. till at last it coincides with it as at R ; and after
this it will be reflected back again and describe the curve R q p h g simi
lar to R Q P H G, and the angle of emergence at h will equal the angle
of incidence at H.
537. PROP. XCVII. Let sin. incidence : sin. refraction in a given ra
tio, and let the rays diverge from a given point ; to find the surface of
medium so that they may be refracted to another given point.
C MM
Let A be the focus of incident, B of refracted rays, and let C D E
be the surface which it is required to determine. Take D E a small arc,
VOL. I. D d
418
A COMMENTARY ON
[SECT. XIV.
and draw E F, E G perpendiculars upon A D and D B; then D F, D G
are the sines of incidence and refraction ; or increment of A D : decrement
of B D : : sin. incidence : sin. refraction. Take .*. a point C in the axis
through which the curve ought to pass, and let C M : C N : : sin. inci
dence : sin. refraction, and points where the circles described with radii
A M, B N intersect each other will trace out the curve.
538. COR. 1. If A and B be either of them at an infinite distance or at
any assigned situation, all the curves, which are the loci of D in different
situations of A and B with respect to C, will be traced out by this
process.
K
A C B
539. COR. 2. Describe circles with radii A C and C B, meeting A D,
B D in P and Q ; then P D : D Q : : sin. incidence : sin. refraction, since
P D, D Q are the increments of B C and A C.
BOOK II.
SECTION I.
1. PROP. I. Suppose the resistance oc velocity, and supposing the whole
time to be divided into equal portions, the motion lost will velocity, and
oc space described. Therefore by composition, the whole decrement of the
velocity cc space described.
COR. Hence the whole velocity at the beginning of motion : that part
which is lost : : the whole space which the velocity can describe : space
already described.
2. PROP. II. Suppose the resistance oc velocity.
Case 1. Suppose the whole time to be divided into equal portions, and
at the beginning of each portion, the force of resistance to make a single
impulse which will oc velocity, and the decrement of the velocity
cc resistance in a given time, oc velocity. Therefore the velocities
at the beginning of the respective portions of time will be in a con
tinued progression. Now suppose the portions of time to be diminished
sine limite, and then the number increased ad injinitum, then the force of
resistance will act constantly, and the velocity at the beginning of equal
successive portions of time will be in geometric progression.
Case 2. The spaces described will be as the decrements of the velocity
oc velocity.
3. COR. 1. Hence if the time be represented by any line and be divid
ed into equal portions, and ordinates be drawn perpendicular to this
line in geometric progression, the ordinates will represent the velocities,
and the area of the curve which is the logarithmic curve, will be as the
spaces described.
Dd 2
420
A COMMENTARY ON
[SECT. 1.
Suppose L S T to be the logarithmic curve to the asymptote A Z.
A L, the velocity of the body at the beginning of the motion.
P Q
h H
K Z
The space described in the time A H with the first velocity continued
uniform : space described in the resisting medium, in the same time : :
A H P L : area A L S H : : rect. A L X P L : rect. A L X PS*
: : P L : P S (if A L = subtan. of the curve).
Also since H S, K T representing the velocities in the times A H, A K ;
P S, Q T are the velocities lost, and therefore oc spaces described.
4. COR. 1. Suppose the resistance as well as the velocity at the begin
ning of the motion to be represented by the line C A, and after any time by
the line C D. The area A B G D will be as the time, and A D as the
space described.
For if A B G D increase in arithmetical progression the areas being
the hyperbolic logarithms of the abscissas, the abscissa will decrease in
geometrical progression, and therefore A D will increase in the same
proportion.
5. PROP. III. Let the force of gravity be represented by the rectangle
* Let the subtangent = M. Then the whole area of the curve = M X A L.
.. the area ALSH = MXAL MXHS = MXPS=ALXPS.
BOOK II.]
NEWTON S PRINCIPIA.
421
BACH, and the force of resistance at the beginning of the motion by
the rectangle B A D E on the other side of A B.
D
K
F
H
Describe the hyperbola G B K between the asymptotes A C and C H
cutting the perpendiculars D E, d e, in G and g.
Then if the body ascend in the time represented by the area D G g d,
the body will describe a space proportional to the area E G g e, and the
whole space through which it can ascend will be proportional to the area
E G B.
If the body descend in the time A B K I, the area described is B F K.
For>euppose the whole area of the parallelogram B A C H to be di
A a K L M N I
vided into portions, which shall be as the increments of the velocity in
equal times, therefore A k, A 1, A m, A n, &c. will <x velocity, and there
fore cc resistances at the beginning of the respective times.
Let A C : A K : : force of gravity : resistance at the beginning of the
second portion of time, then the parallelograms B A C H, k K C H, &c.
will represent the absolute forces on the body, and will decrease in geome
trical progression. Hence if the lines K k, L 1, &c. be produced to meet
D d 3
422 A COMMENTARY ON [SECT. I.
the curve in q, r, &c. these hyperbolic areas being all equal will repre
sent the times, and also the force of gravity which is constant. But the
area B A K q : area Bqk::Kq:4kq::AC:AK:: force of
gravity : resistance in the middle of the first portion of time.
In the same way, the areas q K L r, r L M s, &c. are to the areas
q k 1 r, r 1 m s, &c. as the force of gravity to the force of resistance in the mid
dle of the second, third, &c. portions of time. And since the first term is
constant and proportional to the third, the second is proportional to the
fourth, similarly as to the velocities, and therefore to the spaces described.
. . by composition B k q, B r 1, B s m, &c. will be as the whole spaces
described, Q. e. d.
The same may be proved of the ascent of the body in the same way.
6. COR. 1. The greatest velocity which the body can acquire : the velo
city acquired in any given time : : force of gravity : force of resistance
at the end of the given time.
7. COR. 2. The times are logarithms of the velocities.
8. COR. 4. The space described by the body is the difference of the space
representing the time, and the area representing the velocity, which at the
beginning of the motion are mutually equal to each other.
Suppose the resistance to <x velocity.
r v ~
. . C E : v 2 : : r : j = retarding force corresponding with the velocity (v)
c
.. x = b X 1 v + C,
.. t = b X + Cor.
= X cc ,
.. the times being in geometiical progression, the velocities C, d, E, &c.
will be in the same inverse geometrical progression.
Also the spaces will be in arithmetical progression.
BOOK II.]
NEWTON S PRINCIPIA.
423
9. PROP. IV. Let D P be the direction of the projectile, and let it
represent the initial velocity ; draw C P perpendicular to C D, and
N
E
let D A : A C : : resistance : gravity. Also DP: C P : : resistance :
gravity, .. DAxDP:CPxCA::R:G. Between D C, C P de
scribe a hyperbola cutting D G and A B perpendicular to D C in G and B,
from R draw R V perpendicular cutting D P in V and the hyperbola in T,
complete the parallelogram G K C D and make N : Q B : : C D : C P.
Take
GTt GTE I
V r =  N or R r = ^ ,
for s*ince
R V =
N:QB::CD:CP::DR:RV,
D R X QB
fTT"
and
GTEI D R X QB GTt _
~~ ~"
in the time represented by D R T G the body will be at (r), and the great
est altitude = a, and the velocity cc r L.
For the motion may be resolved into two, ascending and lateral. The
lateral motion is represented by D R, and the motion in ascent by 11 r,
which
a D R x QB GTt,
or
D R X A B D G . R T
N
Dd4
424 A COMMENTARY ON [SECT. II.
or
D R X A B D R x AQ
~N~
D R : R r : : N : A B A Q, or Q B
: : C D : C P,
: : lateral motion . ascending motion at the beginning,
(r) will be the place of the body required.
SECTION II.
10. PROP. V. Suppose the resistance to vary as the velocity 2 .
Then as before, the decrement of velocity a resistance a velocity ! .
AKI.M T D
Let the whole time A D be divided into a great number of equal por
tions, and draw the ordinates A B, K k, L 1, M m, &c. to the hyperbola
described between the two rectangular asymptotes, C H, CD; then by the
property of the hyperbola,
AB:Kk::CK:CA,
/. ABKk:Kk::AK:CA
::ABxAK:ABxCA.
.. AB KkaABxKk.
In the same way
Kk LI a Kk 2 , &c.
or
A B 2 , K k , L 1 2 , &c.
are proportional to their differences.
. . velocities will decrease in the same proportion. Also the spaces de
scribed are represented by the areas described by the ordinates ; hence in
NEWTON S PRINCiPIA.
425
BOOK II.]
the time A M the space described may be represented by the whole area
A M mB.
Now suppose the lines C A, C K, &c. and similarly A K, K L, &c. in
geometrical progression, then the ordinates will decrease in the inverse
geometrical progression, and the spaces will be all equal to each other.
Q. e. d.
1 1. COR. 1. The space described in the resisting medium : the space de
scribed with the first velocity continued uniform for the time AD:: the
hyperbolic area A D G B : rectangle A B X AD.
12. COR. 3. The first resistance equals the centripetal force which would
generate the first velocity in the time A C, for if the tangent B T be drawn
to the hyperbola at B, since the hyperbola is rectangular AT = AC, and
with the first resistance continued uniform for the time A C the whole
velocity A B would be destroyed, which is the time in which the same ve
locity would be generated by a force equal the first resistance. For the
first decrement is A B K k, and in equal times there would be equal de
crements of velocity.
13. COR. 4. The first resistance : force of gravity : : velocity generated
by the force equal the first resistance in the time A C : velocity generated
by the force of gravity in the same time.
14. COR. 5. Vice versa, if this ratio is given, every thing else may be
found.
H IS
P L K I A
15. PROP. VIII. Let C A represent the force of gravity, A K the resis
tance, . . C K represents the absolute force at any time (if the body de
scend) ; A P, a mean proportional to A C and A K, represents the velo
city ; K L, P Q are contemporaneous increments of the resistance and
the velocity.
Then since
AP aAK, KLaSAPxPQxAPxKC,
426
A COMMENTARY ON
[SECT. II.
the increment of velocity cc force when the time is given,
.. K L x K N a A P x K C x K N,
.. ultimately K L O N (equal the increment of the hyperbolic area)
GC A P cc velocity, cc space described, and the whole hyperbolic area =
the sum of all the K L O Ns which are proportional to the velocity, and
.*. space described. .*. If the whole hyperbolic area be divided into equal
portions the absolute force C A, C I, C K, &c. are in geometrical pro
gression. Q. e. d.
16. COR. 1. Hence if the space described be represented by a hyper
bolic area, the force of gravity, velocity, and resistance, may be repre
sented by lines which are in continued proportion.
17. COR. 2. The greatest velocity = A C.
18. COR. 3. If the resistance is known for a given velocity, the greatest
velocity : given velocity : : V force of gravity : V given resistance.
19. PROP. IX. Let A C represent the greatest velocity, and A D be per
B
pendicular and equal to it. With the center D and radius A D describe
the quadrant A t E and the hyperbola A V Z. Draw the radii D P, D p.
Then
Case 1. If the body ascend ; draw D v q near to D p, .. since the sector
and the triangle are small,
D v t : Dp q: : D t* : D p*
q p
*
v t oc
D
BOOK H.] NEWTON S PRINCIPIA. 427
& A D X p q p q
* AD 2 + ADxAK a C K
cc increment of the time.
.. bv composition, the whole sector a whole time till the whole
V = 0.
Case 2. If the body descend ; as before
D VT: D P Q: : D T 2 : D P 2
: : DX 4 : D A*: : T X 2 : A P*
: : DX 2 TX 2 : DA 2 AP 1
: : A D 2 : A D 2 ADx AK
: : A D : C K.
By the property of the hyperbola,
T X 2 = D X 2 D A 2
.. D A 2 = DX 2 TX 2
VT DPQ p Q
DVTcc AD X CK a CT
oc increment of the time.
.. by composition, the whole time of descent till the body acquire its
greatest V = the whole hyperbolic sector DAT.
20. COR. 1. If A B = \ A C.
The space which the descending body describes in any time : space
which it would describe in a nonresisting medium to acquire the greatest
velocity : : area A B N K : A A T D, which represents the time. For
since AC:AP::AP:AK
KL:PQ::AP:iAC
and
KN: AC ::AB:CK
.. KLON:DTV::AP:AC
: : vol. of the body at any time : the greatest vel.
Hence the increments of the areas cc velocity cc spaces described.
.. by composition the whole A B N K : sector A T D : : space described
to acquire any velocity : space described in a nonresisting medium for
the same time.
21. COR. 2. In the same way, if the body ascend, the space described
till the velocity = A p : space through which a body would move : :
A B n k : A D t.
22. COR. 3. Also, the velocity of a body falling for the time A T D :
velocity which a body would acquire in a nonresisting medium in the
same time : : A A D P : sector T D A ; for since the force is constant,
428 A COMMENTARY ON [SECT. II.
the velocity in a nonresisting medium cc time, and the force in a resist
ing medium ocAPccAADP.
23. COR. 4. In the same way, the velocity in the ascent : velocity with which
a body should move, to lose its whole motion in the same time : : A A p D
: sector A t D : : A p : arc A t.
For let A Y be any other velocity acquired in a non resisting medium
in the same time with A P.
.. A P : A C : : A P D : this area
and
AP:AC::APD:ACD.
Therefore the area which represents the time of acquiring the greatest
velocity in a nonresisting medium = A C D.
In the same way, let Ay be velocity lost in a nonresisting medium in
the same time as A p in a resisting medium.
.*. Ap:Ay::AApD: area which represents the time of losing the
velocity A p.
.. time of losing the velocity A y = A A p D.
24. COR. 5. Hence the time in which a falling body would acquire the
velocity A P : time in which, in a nonresisting medium, it would acquire
the greatest velocity : : sector A D T : A C A D.
Also the time in which it would lose the velocity A p : time in which,
in a nonresisting medium, it would lose the same velocity : : arc A t :
tangent A p.
25. COR. 6. Hence the time being given, the space described in ascent
or descent may be known, for the greatest velocity which the body can
acquire is constant, therefore the time in which a body falling in a non
resisting medium, would acquire that velocity is also known. Then the
sector ADTorADtrAADC:: given time : time just foundj there
fore the velocity A P is known or A p.
Then the area ABNKorABnk:ADTorADt:: space sought
for : space which the body would describe uniformly with its greatest
velocity.
26. COR. 7. Hence vice versa, if the space be given, the time will be
known.
BOOK II.] NEWTON S PRINCIPIA. 429
27. PROP. X. Let P F Q be the curve meeting the plane P Q. Let
T
B C D E Q
G, H, I, K be the points in the curve, draw the ordinates ; let B C = C D
= D E, &c.
Draw H N, G L tangents at H and G, meeting the ordinates produced
in L and N, complete the parallelogram C H M D. Then the times
cc V L H and V N I, and the velocities oc G H and H I, and the times
G H TT T
cc ; let T and t = times, and the velocities a ~j and  , therefore
the decrement of the velocity arising from the retardation of resistance and
G H H T
the acceleration of gravity cc ~, , also the accelerating force of
gravity would cause a body to describe 2 I N in the same time, therefore
the increment of the velocity from G =
2NI
again the arc is increased
by the space = HI HN = RI =  =5=  , therefore the de
F ,u GHHISMIxNI
crement from the resistance alone = ~   1  == , ..
GHxt T
resistance : gravity : :   H 1
T
2 M I x N I v T
HI   : 2 N L
Again, let
A B, C D, C E, &c. be o + o, 2o, 3o, &c.
C H = P
and
.. D I = P
EK = P
Q o + &c.
2Qo 4Ro 2 &c.
Qo+&c.
430 A COMMENTARY ON [SECT. II.
(BG CH) 2 + B C 2 (= G H 2 ) = o 2 + Q*o 2 + 3QRo 3 f &c.
.. G H 2 = 1 + Q 2 x o 2 + 3 Q R o 3 ,
Q Ro 2
.. G H = V 1 + Q 2 x o +
i + Q
and
Subtract from C H the sum G B and D I, and R o 2 and R o 2 +
3 S o 3 will be the remainder, equal to the sagittae of the arcs, and which
are proportional to L H and N I, and therefore, in the subtracted num
ber of the times,
. . ^ a /
R + 3 S o R + f So ^ j 3 S o
R " a 2 R " * + 2 R
Q" 2 2 R
S o 3
HI = o. V 1 + Q 4 +
Q R o 2
M I x NI__ Ro 2 x_Qo_+ Ro 2 H &c.
HI = o. V l~+ Q* Q Ro 2
I ^ ;
l + Q 2
GHxt HT ,2MIxNI
.. resistance : gravity : : Fp H 1 H rrr ^~ : " N I
: : 3 S V 1 + Q 2 : 4 R 2 .
The velocity is equal to that in the parabola whose diameter rr H C,
H N 2 1 + Q *
and the lat. rect. = , or n The resistance oc density x V e ,
e , , resistance 3 S V 1 + Q 2 v R
therefore the density ^j^ oc o ^ directly oc _
V s * K " ^14Q S
c
directly oc
R V 1 + Q 2
28. Ex. 1 . Let it be a circular arc, C H = e, A Q =r n, A C a,
CD = o,
.. D I 2 = n 2 (a+o) 2 = n 2 a 2 2aoo 8 =e 2 Sao o*,
BOOK II.]
and therefore
NEWTON S PRINCIPIA.
2 3
_ e
a o n o
_   5
a n o
r /" t?
P = c, Q = , R = ^
S =
a n :
.. density oc
oc
R V 1 + Q 2 " 2e5
sin.
2 e 5
a n 2 2 e 3 e
^
n
a a sin.
a a a oc tangent.
n e e cos.
3 a n " n n 4
The resistance : gravity : : g Q . X : 5 : : 3 a : 2 n.
29. Ex. 2. Of the hyperbola.
P I X b  P D 2 ,
.. put P C = a, C D = o, Q P = c,
.. a + o X c a o ac a 2 2ao + co o 2
a c a
.. D I =
2 a + c o 2
r TT .""B 1
and since there is no fourth term,
S = 0,
.*. draw y = 0.
30. PROP. XIII. Suppose the resistance to V + V 2 .
A Q P
431
D F
Case 1. Suppose the body to ascend ; with the center D and rad. D B,
432
A COMMENTARY ON
[SECT. II.
describe the quadrant B T F; draw B P an indefinite line perpendicular
to B D, and parallel to D F. Let A P represent the velocity ; join D P,
D A, and draw D Q near D P.
.. resistance AP 2 + 2BAxAP, suppose gravity oc D A *,
. . decrement of V oc gravity + resistance AD 2 +AP 2 + 2BAxAP.
a D P 2
DPQ( PQ) :D T V::D P 2 :DT 2 ,
.. D T Va D T 2 <x l,
therefore the whole sector E T D, is proportional to the time.
Case 2. Suppose the force of gravity proportional to a less quantity
than DA 2 , draw B D perpendicular to B P, and let the force of gravity
P Q
FG
oc A B 2 B D 2 . Draw D F parallel to P B and = D B and with the
center D axismajor = axisminor = D B, describe a hyperbola
from the vertex F, cutting A D produced in E, and D P, D Q in T, V.
Now since the body is supposed to ascend.
2AB X AP + AB 2
B 2 + 2AB x BP).
Also, DTV:DPQ::DT 2 :DP 2 (by similar triangles)
::TG 2 :BD 2 (TG perpendicular to G)
: : D F 2 : PB 2 D B 2 .
Now D P Q cc decrement of velocity oc P B 2 D B 2 ,
.. DTVocDF 2 al a increment of the time, since the time flows uni
formly.
The decrement of the velocity ccAP
B D 2 oc BP 2 BD 2 BP 2 = AP 2
BOOK II.] NEWTON S PRINCIPIA.
Case 3. If the body descend ; let gravity oc B D J A B *.
With center D and vertex B, describe the rectangular hyperbola B T V,
cutting the lines D A, D P, D Q produced in E, T, V.
The increment of V B D 2 A B 2 2 A B x AP _ A P *
oc B D 2 (A B + AP) 2 oc B D 2 B P 1
DTV:DPQ(ocpQ)::DT 2 :DP 2
::GT S :BP*::GD 2 BD 2 :BP 2
:: GD 2 : BD 2 :: BD 2 : BD 2 EPS
.. DT Voc BD oc i,
.. the whole sector E D T oc time.
31. COR. With the center C and distance D A describe an arc similar
to B T.
Then the velocity A P : the velocity which in the time E D t a body
would lose or acquire in a nonresisting medium : : A D A P sector
AD t.
For V in a nonresisting medium oc time.
32. In the case of the ascent,
Let the force of gravity I. Resistance oc 2 a v f v
. . d v oe 1 + 2 a v + v*
d v
. . by Demoivre s first formula,
f. or time =
when
f.
VOL. I.
= ~l X cir. arc. rad. = g and
tangent = v + a
Ee
434 A COMMENTARY ON [SECT. Ilf.
The whole time .*. when v = = , X cir. arc rad. = a
8
and tangent z= a f C.
.*. cor r . time = ; X cir. arc rad. = g and tangent v f a cir. arc rad.
O
= g and tangent a.
.. the time of ascent = sector EDT g s = 1 a*.
33. In the case of descent,
dv cc 1 2 a v v*
let
v + a = x
.*. d v = d x
.. \ z f 2 a v f a 2 = x s
.. 1ra 2 x 2 = 1 2 a v v *
Time = 0, v = 0,
/. x = a,
.. Cor 1 , time X ft  ffi^ .
2 g J g x J g a
34 PROP. XIV. Take A C proportional to gravity, and A K to the
esistance on contrary sides if the body ascend, and vice versa.
Between the asymptotes describe a hyperbola, &c. &c.
Draw A b perpendicular to C A, and
Ab:DB::l)B 2 :4BA X A C.
The area A b N K increases or decreases in arithmetic progression it
the forces be taken in geometric progression.
Now
A K a resistance a 2 B A P + A P 2 .
Let
. 2BAP + AP 2
A K. r= y
2BA X PQ+2APX PQ
.*. Jv JL T/ >
BOOK II.]
NEWTON S PRINCIPIA.
43.5
I)
A KQ P
KL =
2 B P Q
N T ow
Ab:LO::CK:CA
DB:Ab::4BAx CAtDB*
BD 3
.. L O =
4 B A x C K
. KT nv 2PB x PQ x BD 3
. . IV JL \J IS =: 7~rr~ i >S i=i .
4B A x CK x Z
Case 1. Suppose the body to ascend,
gravity ex A B + B D 2 = A B + BD
4S6
A COMMENTARY ON
[SECT. IV.

Z
..DP l = CKx Z.
.. DT 2 :DP*::DB :CKxZ
and in the other two cases the same result will obtain.
Make
DTV = DBx ra.
..DBxm:iDBx PQ::DB*:CK X Z
.. BD 3 xPQ = 2BDxmxCKxZ.
...AbNKDTV=
it will represent the space.
A P.* velocity.
SECTION IV.
35. PROP. XV. LEMMA. The
. O P Q = a rectangle = L. O Q R
ami
L. S P Q = L. of the spiral = A. S Q R
.. L. O P S = L O Q S.
.. the circle which passes through the points P, S, O, also passes
through Q. Also when Q coincides with P, this ^^ touches the spiral.
.. L. P S O L. in a
whose diameter = P O.
BOOK II.] NEWTON S PRINCIPIA. 437
Also
TQ : PQ :: PQ : 2 PS.
.r. PQ 2PS x TO
which also follows from the general property of every curve.
PQ 2 = P V x Q R.
OR  PQZ
QK TW
36. Hence the resistance density X square of the velocity.
37. Density oc j centripetal force oc density 2 p .
J distance distance 2
Then produce S Q to V so that S V = S P, and let P Q be an arc
described in a small time, P R described in twice that time, .. the decre
ments of the arcs from what would be described in a nonresisting me
dium a T 2 .
.. decrement of the arc P Q = \ decrement of the arc P R
.. decrement of the arc P Q = R r (if Q S r = area P S Q).
For let P q, q v be arcs described (in the same time as P Q, Q R) in a
nonresisting medium,
PSq PSQ = QSq = qSv Q S r
rSv QSq
.. 2 Q S q = r S v
.. if S T ultimately = S t be the perpendicular on the tangents
STxQq = Stxrv
.. 2 Q q = r v
and
R v = 4 Q q.
. . 2 Q q = R r.
Hence
Resistance : centripetal force : : R r : T Q,
Also
T Q X S P 2 a time 2 , (Newt. Sect. II.)
..PQ 2 X S P a time 2
/. time a P Q x V fc> I
.. V a
also
V at Q oc
VSQ
Eg
C J
438 A COMMENTARY ON [SECT. IV.
::SQ:VSQxSP
PQ: Q r : : SQ: SP
since the areas are equal, and the angles at P and Q are equal.
.. P Q : R r : : S Q : S P V S Q x S P
: : S Q : f V Q
For
SQ = SP VQ
.. S Q x S P = S P 2 V Q x SP
.. V SQx SP = SP J V Q_Z^_&c.
.. V Q ultimately = S P V S P X S Q
T, . decrement of V R r
Resistance oc = 5 cc
time 2 P Q 2 X S P
PQxSQxSP
JVQ: PQ::OS:PO
and
1 Q O
S Q = S P cc QJ f x gp2
O S
.. density X square of the velocity cc resistance cc Typ q
OS
/. density a /Y15 o~
O S
and in the logarithmic spiral ^rp is constant
.*. density cc ^g . Q. e. d.
88. ("OR. 1. V in spiral = V in the circle in a non resisting medium at
he same distance.
39. COR 3. Resistance : centripetal force : :  R r : T Q
jVQx PQ ^PQ 2
SQ S P
::VQ:PQ
: : I O S : O P.
. . the ratio of resistance to the centripetal force is known if the spiral be
given, and vice versa.
40. Con 4. If the resistance exceed the centripetal force, the body
cannot move in this spiral. For if the resistance equal  the centripetal
BOOK II.] NEWTON S PRINCIPIA. 439
force, O S = O P, . .the body will descend to the center in a straight
line P S.
V of descent in a straight line : V in a nonresisting medium of de
scent in an evanescent parabola : : 1 : V 2 ; for V in the spiral = V in the
circle at the same distance, V in the parabola = V in the circle at
 distance.
Hence since time a v ,
time of descent in the 1st case : that in 2d : : V 2 : 1.
41. Con. 5. V in the spiral P Q R = V in the line P S at the same
distance. Also
P Q R : P S in a given ratio : : P S : P T : : O P : O S
.. time of descending P Q R : that of "P S : : O P : O S.*
Length of the spiral = T P = sector of the L. T P S.
a : b : : b : c : : c : d : : d : e
a + b + c + &c. : b + c + d + &c. : : a : b
.*. a f b + c } &c. : a : : a : a b.
42. COR. 6. If with the center S and any two given radii, two
circles be described, the number of revolutions which the body makes
between the two circumferences in the different spirals <x tangent ot the
angle of the spiral cc
The time of describing the revolution : time down the difference of the
radii : : length of the revolution : that difference.
2d ex 4th,
/. time cc length of the revolution oc secant of the angle of the spiral
OP
p q : p t : : S p : S y
p (1 x
d w : : : x : p.
r. . t
440 A COMMENTARY ON [SECT. IV
43. Con. 7. Suppose a body to revolve as in the proposition, and to cut
the radius in the points A, B, C, D, the intersections by the nature of the
spiral arc in continued proportion.
Times of revolution a Perimetersdescribed
and velocity oc
1
^distance
aA W T} Q ST ^ G 2
i*. O J.) O V^ O
. . the whole time : time of one revolution : : A
: : A
44. PROP. XVI. Suppose the centripetal force x J n ,
time a P Q x S P
i
and velocity ex
&c. : A S
: A S B S *.
SP*
PQ:QR::SQ*:SP
Qr:PQ::SP :SQ
Qr : Q R
. . Q r : R r
For
S Q $  l : S Q * > _ S P S ~
S Q : l^TfiT. V Q.
S P = S Q + V Q,
BOOK II.] NEWTON S PRINCIPIA. 441
* 1 + 1. VQ x SQ* 2 + &c.
... SQ2 _SP = 1 x VQ x SQ? 2 .
Then as before it may be proved, if the spiral be given, that the density
^rp. Q. e. d.
45. COR. 1.
_
Resistance : centripetal force : : 1 n . O S : O P,
fur the resistance : centripetal force : :  II r : T Q
X VQx PQ PQ !
Q " TiT
x VQ:PQ
: : l  x O S: OP.
ii
46. COR. 2. If n + 1 = 3, 1 ~ = 0,
ii
.. resistance = 0.
COR. 3. If n + 1 be greater than 3, the resistance is propelling.
SECTION VI.
47. PROP. XXIV. The distances of any bodies centers, of oscillation from
the axis of motion being the same, the quantities of matter CD weight
X squares of the times of oscillation in vacua.
T, t i i , force X time ,, ,.
1< or the velocity generated GC ,  . r orce on bodies at
quantities 01 matter
c(jual distances from the lowest points GO weights, times of describing
corresponding parts of the motion cc whole time of oscillation,
c force X time of oscil.
.. quantities of matter cc , .,
velocities
co weights X squares of the times,
since the velocities generated cc : for equal spaces.
times
48. COR. 1. Ilence the times being the same, the quantities of matter
co weights.
Cou. 2. If the weights be the same, the quantities of matter co time %
COR. 3. If the quantities of matter be the same, the weights cc : .
time *
442 A COMMENTARY ON [SECT. VI
49. Cou. 4. Generally the accelerating force cc  p .^ of matter
quantities
and L oo T T 2 ,
T WxT*
Q~~
W x T 2
.. Q cc
L
.\ if W and Q be given L co T 2 .
If T and Q be given L oo W.
CA ~ _ . ... c weight X time 2 of oscillation
50. COR. 5. generally the quantity or matter cc a 1 = .
51 PROP. XXV. Let A B be the arc which a body would describe in a
nonresisting medium in any time. Then the accelerating force at any
point D oc C D ; let C D represent it, and since the resistance cc time,
it may be represented by the arc C o.
.*. the accelerating force in a resisting medium of any body d, = o d.
Take
od:CD::oB:CB.
Therefore at the beginning of motion, the accelerating force will be in
this ratio, .*. the initial velocities and spaces described will be in the some
ratio, .. the spaces to be described will also be in the same ratio, and
vanish together, . . the bodies will arrive at the same time at the points
C and o.
In the same way when the bodies ascend, it may be proved that they
will arrive at their highest points at the same time. .. If A B : a B in
the ratio. C B : o B, the oscillations in a nonresisting and resisting me
dium will be isochronous. Q. e. d.
BOOK II.]
NEWTON S PRINCIPIA.
443
Con. The greatest velocity in a resisting medium is at the point o.
The expression for the J time of an oscillation in vacuo, or time of de
scent down to the lowest point a quadrant whose radius = 1. Now
suppose the body to move in a resisting medium when the resistance
: force of gravity : : r : 1.
Then vdv = gFdx + grdz = gd 2 x + grdz. Now by
a property of the cycloid, if  be the axis, d x : d z : : x : \ : : z : a,
&
z d z
. . d x =r ,
a
. . v d v = i  x z d z f grdz ~
= f X z 2 + grz,
. . v
2grz+C.
Now
r= d, V = o,
.. v 2 = . xd 2 z 2 _2grxd z
=  x T r 4 a r d + 2 a J r z z*,"
2 2 a r d f 2 a r z z s ,
a d z
Vd z 2 a r d + 2 a r z z *.
Assume
z a r = y,
... z  2 a r z + a * r * = y\
.*. 2 a r z z * = a s v * y f ,
d 2 a r d f 2 a r z z = (d a r) 2 y ! = (b * y 5
444 A COMMENTARY ON [SECT. VI.
and
d z =r d y
dy
i r u Y
.. d t = / X  ^ J
J g Vb 2 y 8
.. t = f x circular arc, radius = 1,
J %
and
z a r
cos. = , f C and C = o.
d a r
/. the whole time of descent to the lowest point = f  X circular arc
, a r . .
whose cos. = ; , .. time in vacuo : tune in resisting medium
d a r
 _.  o
: : quadrant : arc whose cos. :
d a
a r
,
r
Cou. 1. Time of descent to the point of greatest acceleration is constant,
for in that case z = a r,
t = f x quadrant, for d v = 0,
&
.. v d v = 0,
/. g z d z + g a r i = 0,
/. z = a r,
.. z : r : : a : 1.
COR. 2. To find the excess of arc in descent above that in ascent.
v d v f g T d x + g r d z,
z d z
. v d v =  s r d z
a
v 2 m z 2
.. v 2 = & (d z 2 ) (z d) x 2 a r
=  X (d * 2 a r d) (2 a r z z )
which when the body arrives to the highest point = 0,
d ~ 2 a r d 2 a r z z 2 = 0,
,1 " 9 a i* rl 7 2 I O .1 T r,
.. d 5 2ard = z*+2a r z,
.. z + a r = d a r,
.. z = d 2 a r,
.. d z = 2 a r.
BOOK II.]
NEWTON S PRINCIPIA.
445
52. PROP. XXVI. Since V oc arc, and resistance a V, resistance a arc.
.. Accelerating force in the resisting medium GC arcs.
Also the increments or decrements of V a accelerating force.
.. the V will always <x arc.
But in the beginning of the motion, the forces which oo arcs will generate
velocities which are proportional to the arcs to be described. .. the velo
cities will always co arcs to be described.
.*. the times of oscillation will be constant.
53. PROP XXVIII. Let C B be the arc described in the descent, C a
in the ascent.
.. A a = the difference (if A C = C B)
Force of gravity at D : resistance : : C D : C O.
C A = C B
Oa = O B
. . CA OaorAa eO = CB OB = CO
.. C O = A a
. . Force of gravity at D : resistance : : C D : A a
. . At the beginning of the motion,
Force of gravity : resistance : : 2 C B : A a
: : 2 length of pendulum : A a.
54. PROB. To find the resistance on a thread of a sensible thickness.
Resistance CD V * X D * of suspended globe.
.*. resistance on the whole thread : resistance on the globe C : :
446 A COMMENTARY ON [SECT. VI.
::2a s V. (a b) 8 : a 3 r 2 c 2 r 2 c 2 . (a 2 b) 3 , c = a + r.
:: a 3 b 2 . (a b) * : 3a 2 r 2 c 2 b bab 2 r*c 2 f4b 3 r 2 c*,
:: a b . (a b) 2 : 3a z r 2 c 2 ba b r * c 2 + 4 b * r 2 c\
.. resistance on the thread : whole resistance
::a 3 b. (a b) a : r 2 c*.(3a ! bab+ 4b 3 ).
COR. If the thickness (b) be small when compared with the length (a)
bab4 4b 2 =3a 2 bab + 3 b 2 (nearly) = 3. (a b) .
and
3 a
. . Resistance on the whole thread : resistance on the globe
: : a 3 b : 3r a c
Resistance on the thread : whole resistance to the pendulum
Suppose, instead of a globe, a cylinder be suspended whose ax. = 2 r.
Now by differentials
the resistance on the circumference : resistance on the base : : 2 : 3.
By composition the resistance to the cylinder : resistance on the square
= 2 r : : 2 : 3.
Resistance a x 2 x ,
. . resistance ax 3 ,
.. resistance to the whole thread ot x 3 .
Resistance on A E a (a 2 b) 3 if 2 b = E D.
. . Resistance on the thread : resistance of the globe
: : 16 . a 3 b 2 . (a b) * ; 3 p . a 3 (a 2 b) 3 x r s . (a + i) *.
55. PROP. XXIX. B a is the whole arc of oscillation. In the line OQ
take four points S, P, Q, R, so that if O K, S T, P I, Q E be erected
BOOK II.]
NEWTON S PRINCIPIA.
447
perpendiculars to O Q meeting a rectangular hyperbola between the
asymptotes O Q, O K in T, I, G, E, and through I, K F be drawn
O S P rRQ M
parallel to O Q, meeting Q E produced in F. The area P I E Q may
be : area P I S T : : C B : C a. Also I E F : I L T : : O R : O S.
Draw M N perpendicular to O Q meeting the hyperbola in N, so that
P L M N may be proportional to C Z, and P I G R to C D.
Then the resistance : gravity QQ X TEF IGH:PINM.
Now since the force cc distance, the arcs and forces are as the hyper
bolic areas. .*. D d is proportional to R r G g.
Now by taking the differentials the increment of (QQ T E F I G H)
P ,. T TTT? T V V
= G H g h 
H G I E I
OQ
OR
: R rX G R : : H G Q : G R : : O R X
O P x
O R
OQ
(ORxHG = ORxHR
= PIHR=PIRG+IGH):. PIRG+IGH
x IEF:OPx PI.
O R
NowifTs g^XIEF I GH, the increment Y a PIG R Y.
Let V = the whole from gravity. .*. V R = actual accelerating
force. . . Increment of the velocity a V R X increment of the time.
As the resistance cc V 2 the increment of resistance a V X increment of
, . , , . . increment of the space f
the velocity, and the velocity a  ?. t . .. Increment of
increment ot the time
resistance a V R if the space be given, co P I G R Z, if Z be the
area which represents the resistance R e.
Since the increment Y a PIGR Y, and the increment of Z
448
A COMMENTARY ON
[SECT. VIII.
ccPIGR Z. IfY and Z be equal at the beginning of the motion and
begin at the same time by the addition of equal increments, they will still
remain equal, and vanish at the same time.
Now both Z and Y begin and end when resistance = 0, i. e. when
O R
OQ
. I E F I G H =
or
xOR IGH = 0.
O R x I E F
I G H = Z
.. Resistance : gravity : : g . I E F I G H : P M N I.
SECTION VIII.
66. PROP. XLIV. The friction not being considered, suppose the mean
K
M
E
altitude of the water in the two arms of the vessel to be A B, C D. Then
when the water in the arm K L has ascended to E F, the water in the arm
M N will descend to G H, and the moving force of the water equals the
excess of the water in one arm above the water in the other, equals twice
A E F B. Let V P be a pendulum, R S a cycloid = length of the
canal, and P Q = A E. The accelerating force of the water : whole
weight : : A E or P Q : P R.
BOOK II.] NEWTON S PRINCIFIA. 449
Also, the accelerating force of P through the arc P Q : whole weight
of P : : P Q : P R ; therefore the accelerating force of the water and P
cc the weights. Therefore if P equal the weight of the water in the canal,
the vibration of the water in the canal will be similar and cotemporaneous
with the oscillations of P in the cycloid.
COR. 1. Hence the vibrations of the water are isochronous.
Con. 2. If the length of the canal equal twice the length of the
pendulum which oscillates in seconds; the vibrations will also be performed
in seconds.
COR. 3. The time of a vibration will <* V L.
Let the length = L, A E = a,
then the accelerating force : whole weight : : 2 a : L,
.*. accelerating force = y ;
JU
2 A
.. when the surface is at 0, the accelerating force = ^
Put E = x,
A = a x,
, .. c 2 a 2 x
.. accelerating torce = = .
g . 2 a d x 2 x d x
* _
X 2 a x x 2 ,
x adx
V 2 a x x s
*. t ss^J ^ 5 X cir. arc rad. =a, and vers. = x
~ ^ g a
cor n . and cor". =r 0,
v t = 0, x = 0,
.. ifp = 3. 14159, &c.
= J
V
(x) = (a)} = /L X  P 
ga 2 V2ga
X
. . time of one entire vibration = p X 75 = time of one entire vi
V 2 o
bration of a pendulum whose length = .
8
\ oi. I, Ff
450
A COMMENTARY ON
[SECT. VIII.
57. COR. 1. Since the distance (a) above the quiescent surface docs
not enter into the expression. The time will be the same, whatever be
the value of A E.
58. COR. 2. The greatest velocity is at A = ^ x a, a a / 
\ 1* L A V I
59. PROP. XLVIL Let E, F, G be three physical points in the lin
B C, which are equally distant; E e, F f,
G g the spaces through which they move
during the time of one vibration. Let e, <p, y
be their place at any time. Make P S =
E e, and bisect it in O, and with center O
and radius O P = O S, describe a circle.
Let the circumference of this circle repre H
sent the time of one vibration, so that in
the time P H or P H S h, if H L or h 1
be drawn perpendicular to P S and E be
taken =r P L or P 1, E s may be found in
E ; suppose this the nature of the medium.
Take in the circumference P H Sh, the arcs
H I, I K, h i, i k which may bear the
same ratio to the circumference of the circle as E F or F G to
B C. Draw I M, K N or i m, k n perpendicular to P S. Hence
PI, or P H S i will represent the motion of F . and P K or
P H S k that of G . E , Fp, G 7 = P L, P M, P N or P 1,
P m, P n respectively.
Hence s 7 or E G + G 7 E e = GE L N = expan
sion at s 7 ; or = E G + 1 n.
. . in going, expansion : mean expansion : : G E L N : E G
In returning,
: : : E G + In : E G
Now join I O, and draw K r perpendicular to H L, UK r,
I O M are similar triangles, since the^KHr = ^KOk=^
I O i = L. I O P and ^ at r and M = 90,
.. L N : K H : : I M : I O or O P, and by supposition K H :
EG:: circumference PSLP:BC::OP:V = radius of
the circle whose circumference = B C.
.. by composition L N : G E : : I M : V.
.% expansion : mean expansion : : V I M : V,
G
FJ
E
B
JA
PROBLEMS.
401
Let A C and E be the respective places of the object, eye, and reflector
at first, and B Q and F their places at any other time, or if K F = F Q
= C E, K may also be the place of the eye, and since K F always = C E,
and that B F is constant, K will trace out an ellipse by next problem.
Also by optics the angle K F H = H F Q, and from similar triangles,
K H : K F : : K D : K B,
EF:BF::KD:KB,
.. KH + DH:KF + BF::KD:KB,
or
D Q : D K : : K F + B F : K B in a given ratio,
*. Q traces an ellipse.
To determine the quantity of fluid issuing through an orifice of a
given form and magnitude, in the side of a cylindrical vessel, supposed to
be kept constantly full.
Let
S B = h, S A = h , A P = x,
P M = y,
.. velocity of the efflux in M N
= V g (h + x)
and the area of the lamina = 2 y x
and the time = t,
.*. the quantity of fluid through M N
in n = area of the section x vel. X t,
= 2 y x V g (h + x) X t,
. . the quantity effluent through the whole area A m S t A = sum of
all the portions effluent through M N = / 2 y x V g h ^f~xt 2t
^gy^h fxfC connected between the values x = 0, and
x =r h  h .
S
_ f/y x
[y~x ~~+~C p
mean height.
BOOK II.] NEWTON S PRINCIPIA. 453
.. (ra b*) 8 xdx s = 2 r a b dy b 4 d y *,
.. r* a z dx* = 2 rab dy 8
if (b) be small compared to (aj,
r ad x*
d y  VKT
V r a X d x /jr_ _adx
^ " " V 2 a x x 2 / N/a v V2ax x 2
.\ v = / x circular arc whose rad. = a, and vers. = x
\l a
C, and cor". = 0,
because when y = 0, x = 0,
.. arc = 0.
.. C D = x quadrant B N E,
and therefore
cp
/.
V a "
BN E
B N x k
BNE
60. PROP. XLIX. Put A = attraction of a homogeneous atmosphere
when the weight and density equal the weight and density of the medium
through which the physical line E G is supposed to vibrate. Then every
thing remaining as in Prop. XL VI I. the vibration of the line E G will
be performed in the same times as the vibrations in a cycloid, whose
length = P S, since in each case they would move according to the Fame
law, and through the same space. Also, if A be the length of a pendulum,
since T a V L
The time of a vibration : time of oscillation of a pendulum A
: : V~T~O : V ~A.
Also (PROP. XLVIL), the accelerating force of E G in medium : ac
celerating force in cycloid
::AxHK:VxEG;
since H K : G E : : P O : V.
: : PO X A : V 2 .
Ff 3
454 A COMMENTARY ON [SECT VIIT.
Now
T x ji ~ when L is given.
.. the time of" vibration : time of oscillation of the pendulum A
: : V : A
: : B C : circumference of a circle rad. = A.
Now B C = space described in the time of one vibration, therefore
the circumference of the circle of radius A = space described in the time
of the oscillation of a pendulum whose length = A.
Since the time of vibration : time of describing a space =r circum
ference of the circle whose rad. = A : : B C : that circumference.
COR. 1. The velocity equals that acquired down half the altitude of
A. For in the same time, with this velocity uniform, the body would de
scribe A ; and since the time down half A : time of an oscillation : : r :
circumference. In the time of an oscillation the body would describe the
circumference.
Con. 2. Since the comparative force or weight cc density X attraction
of a homogeneous atmosphere, A GO , , and the velocity 00 V A.
V elastic force
ff ,
*JU ^
V density
SCHOLIUM.
61. PROP. XLIX. Sound is produced by the pulses of air, which
theory is confirmed, 1st, from the vibrations of solid bodies opposed to it.
2d. from the Coincidence of theory with experiment, with respect to the
velocity of sound.
The specific gravity of air : that of mercury : : 1 : 11890.
Now since the alt. oc  , .. 1 : 11890 : : 30 inches : 29725 feet =
sp. gr.
altitude of the homogeneous atmosphere. Hence a pendulum whose
length = 29725, will perform an oscillation in 190", in which time by
Prop. XLIX, sound will move over 186768 feet, therefore in I" sound
will describe 979 feet. This computation does not take into considera
tion the solidity of the particles of air, through which sound is pro
pagated instantly. Now suppose the particles of air to have the same
density as the particles of water, then the diameter of each particle : dis
BOOK II.] NEWTON S PRINCIPIA, 451
.. elasticity : mean elasticity : : : . In the same way, for the
points E and G, the ratio will be ^7 ^^ : _ a \ . JL
V rlL, V V KN V
: : excess of elasticity of E : mean elasticity
H L K N 1
V 2 HLxV KNx V + HLx KN : T
: : H L K N : V.
Now
V a 1.
.. the excess of E s elasticity cc H L K N, and since H L K N
= H r : H K : : O M : O P,
.. H L K N a O M,
/. excess of E s elasticity cc O M.
Since E and G exert themselves in opposite directions by the arc s ten
dency to dilate, this excess is the accelerating force of e 7, .. accelerating
force co O M.*
ON THE HARMONIC CURVE.
Since the ordinates in the harmonic curve drawn perpendicular to the
axis are in a constant ratio, the subtenses of the angle of contact will be
in the same given ratio. Now the subtenses oc . , and when
rad. of curv.
the curve performs very small vibrations, the arcs are nearly equal.
Now the curv. cc , , .*. subtense cc curvature,
rad.
Hence the accelerating force on any point of the string cc curvature at
that point.
* Now bisect F f in n,
.. O M = n<p
For
O M = O T PM=nF F = n ?
i. e. the accelerating force cc distance from f) the middle point. Q. e. d.
452 A COMMENTARY ON [SECT. VIII.
To find the equation to the harmonic curve.
C S
E D
Let A C be the axis of the harmonic curve C B A, D the middle point,
draw B D perpendicular cutting the curve in B; draw P M perpendi
cular to B D cutting the curve in P, and cutting the quadrant described
with the center D and radius D B in N. Draw P S perpendicular to A C
Put
BD = a, PM = y, B M = x,
.. D M = a x = P S.
r = rad. of curv. at B, B P = z,
d z d x
. . rad. of curv. =
Now
BD: PS
or
Now
Put
d*y
(if d K be constant).
a : a x :
: curvature at B : curvature at P
: rad. of cur. at P : rad. at B
d z d x
d y
.*. r a d * y + adzdx xdxdz = 0,
.\ rady + adzx = + C.
x = 0, d y = d x,
radz = + C = C,
rady + axdz
dz
= r a d z.
. . r a d y = r a b 2 d z,
..r a dy ir (ra b 2 ) 2 X d x 2 + r a 8 dy 2r a b ! d y ! + b 4 d y %
BOOK II.] NEWTON S PRINCIPIA. 457
Also
L t
D d :   = rad. of curve : : the moving force of D d : P
.. the moving force of D d = P x D T d X ap *
L w
. . accelerating force = P X Dd X a p* L
L* Dd X w
P X a p *
Lw.
if D O = x, D C = a, O C =r a x,
.*. the accelerating force at O = T" .
_ g. Pp
v A *  v i  i
. . v a s _ j x a d x x d x
P r>*
... v z = fe ^ X 2ax
L w
g pl
T /> V 2 ax x .
L w
.. C and 1 = 0,
d x / L w d x
. . d t = . / 75 rX ;
v VgPp 1 V 2 a x 
L w
rs , X cir. arc rad. = 1
and
x
vers. sine = ,
a
when x = a,
t = 0.
^ r X quadrant ss ./ ^  X
g P p 9 V g P p * 2
= * x \J "^F*
.". time of a vibration = / rr I"
/V g L
. . number of vibrations in 1 " = ^ / $ .
V L w
COR. Time of vibration =r time of the oscillation of a pendulum whose
L w
458
A COMMENTARY, &c.
[SECT. IX.
For this time = ./
64. PROP. LI. Let A F be a cylinder moving in a fluid round a
fixed axis in S, and suppose the fluid divided into a great number of solid
orbs of the same thickness. Then the disturbing force cc translation of
parts X Surfaces. Now the disturbing forces are constant. . . Transla
tion of parts, from the defect of lubricity a T. . Now the differ
distance
r.i ! . translation 1 * ,
ence ot the angular motions cc p ex :  . On A Q draw
distance d:stance*
A a, B b, C c, &c. : :
a hyperbolic area.
. . periodic time cc
distance 
1
then the sum of the differences
cc
1
cc distance.
angular motion hyperbolic area
In the same way, if they were globes or spheres, the periodic time
would vary as the distance *.
END OF THE FIRST VOLUME.
BOOK I.] NEWTON S PRINCIPIA. 455
tance between their centers : : 1 : 9, or 1 : 10 nearly. (For if there are
two cubes of air and water equal to each other, D the diameter of the par
ticles, S the interval between them, S + D = the side of the cube, and if
N = N. N S + N D = N. in the side of the cube, N. in the cube
jo N 3 . Also, if M be the N. in the cube of water, M D the side of the
cube and the N. in the cube cc M 3 .
Put 1 : A : : N 3 : M 3 ,
.. M = A * N,
By Proposition
.. S = D X A a 1,
.. S: D: : A 3 1 : 1,
.. S + D : D : : A 3 : 1 : : 9 : 1 if A = 870
or 10: 1 if A = 1000).
Now the space described by sound : space which the air occupies : : 9 : 11,
979
.. space to be added = ^ = 108 or the velocity of sound is 1088
feet per 1".
Again, also the elasticity of air is increased by vapours. Hence since
the velocity oc e . a  ^ ; if the density remain the same the velocity
V density
oc V elasticity. Hence if the air be supposed to consist of 11 feet, 10 of
air, and 1 of vapour, the elasticity will be increased in the ratio of 1 1 : 10,
therefore the velocity will be increased in the ratio of 11 : 10 or 21 : 20,
therefore the velocity of sound will altogether be 1 142 feet per 1", which
is the same as found by experiment.
In summer the air being more elastic than in winter, sound will be
propagated with a greater velocity than in winter. The above calculation
relates to the mean elasticity of the air which is in spring and autumn.
Hence may be found the intervals of pulses of the air.
By experiment, a tube whose length is five Paris feet, was observed to
give the same sound as a chord which vibrated 100 times in 1", and in
the same time sound moves through 1070 feet, therefore the interval of
the pulses of air = 10.7 or about twice the length of the pipe.
Ff 4
456 A COMMENTARY ON [SECT; VJIT,
62. On the vibrations of a harmonic string.
The force with which a string tends to the center of the curve : force
which stretches the string : : length : radius of curvature. Let P p be a
small portion of the string, O the center of the curve ; join O P, O p, and
draw P t, p t, tangents at P and p meeting in t, complete the parallelo
gram P t p r. Join t r, then P t, p t represent the stretching force of
the string, which may be resolved into P x, t x and p x, t x of which
P x, p x destroy each other, and 2 t x = force with which the string
tends to the center O. Now the /LtPr= z. P O p, . . z. t P x = .
P O p, .*. t r : P t : : P p : O P, i. e. the force with which any particle
moves towards the center of the curve : force which stretches it : : length
: radius.
63. To find the times of vibration of a harmonic string.
B
Let w = weight of the string. L = length.
D d : L : : weight D d : w
1* TTAJ D d X W
. . weight or D d = _
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