Principia Mathematica (Motte Translation, 1848) — part 19 of 45

the particles of time will be as the same velocity. Therefore the veloci- ties are proportional to their differences, and therefore (by Lem. 1, Book II) continually proportional. Therefore if out of an equal number of par- ticles there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progression, which are taken by intervals, omitting every wliere an equal number of interme- diate terms. But the ratios of these terms are compounded of the equal ratios of the intermediate terms equally repeated, and therefore are equal. Therefore the velocities, being proportional to those terms, are in geomet- rical progression. Let those equal particles of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual ; and the velocities at the beginnings of equal times, al- ways continually proportional, will be also in this case continually pro- portional. Q.E.D.

Case 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes ; but the spaces described in each of the times are as the lost parts of the velocities (by Prop. 1; Book I), and therefore are also as the wholes. Q.E.D. jj CoROL. Hence if to the rectangular asymptotes AC, CH,

the hyperbola BG is described, and AB, DG be drawn per- V^^^_^;B pendicular to the asymptote AC, and both the velocity of j7— -^1 the body, and the resistance of the medium, at the very be- ginning of the motion, be expressed by any given line AC, and, after some time is elapsed, by the indefinite line DC ; the time may be expressed by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniform- ly increased in the same manner as the time, the right line DC will de- crease in a geometrical ratio in the same manner as the velocity ; and the parts of the right line AC, described in equal times, will decrease in the same ratio.

PROPOSITION III. PROBLEM I.

7o define the motion of a body which, in a similar medium^, ascends or descends in a right line, a7id is resisted in the ratio of its velocity, and acted upon by an uniform, force of gravity.

J The body ascending, let the gravity be expound-

ed by any given rectangle BACH ; and the resist- ance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side ,-^ of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an

o I I I

y> d^ J^ X / C hyperbola, cutting the perpendiculars DE, de, in

Sec. I.] OF NATURAL PHILOSOPHY. 253

G, g ; and the body ascending will in the time BGgd describe the space EG^e; in the time DGBA, the space of the whole ascent EGB ; in the time ABKI, the space of descent BFK ; and in the time IKJd the space of descent KFfk ; and the velocities of the bodies (proportional to the re- sistance of the medium) in these periods of time will be ABED, ABed, O, ABFI, ABj^ respectively ; and the greatest velocity which the body can acquire by descending will be BACH.

For let the rectangle BACH be resolved into in- numerable rectangles Ak, Kl, Lm, Mn, &c., which shall be as the increments of the velocities produced in so many equal times; then will 0, Ak, AL Am, An, &c., be as the whole velocities, and therefore (by suppo- sition) as the resistances of the medium in the be-

ginning of each of the equal times. Make AC to

AK, or ABHG to ABA:K, as the force of gravity to the resistance in the beginning of the second time ; then from the force of gravity subduct the resistances, and ABHC, KA'HC, LZHC, MmHC, (fee, will be as the abso- lute forces with which the body is acted upon in the beginning of each of the times, and therefore (by Law I) as the increments of the velocities, that is, as the rectangles Ak, K/, hm, M«, (fcc, and therefore (by Lem. 1, Book II) in a geometrical progression. Therefore, if the right lines K^', U, Mm, 'Nn, &c., are produced so as to meet the hyperbola in q, r, s, t, &c., the areas ABqK, Kqrh, hrsM, M^^N, &c., will be equal, and there- fore analogous to the equal times and equal gravitating forces. But the area AB^-K (by Corel. 3, Lem. VII and VHI, Book I) is to the area BA:^' as K^' to ^kq, or AC to |AK, that is, as the force of gravity to the resist- ance in the middle of the first time. And by the like reasoning, the areas qK.hr, rLMs, ^MN^, (fee, are to the areas qklr, 7^lm,s, snmt, (fee, as the gravitating forces to the resistances in the middle of the second, third, fourth time, and so on. Therefore since the equal areas BAKq, qKhr, rLMs, sMW, (fee, are analogous to the gravitating forces, the areas Bkq, qklr, rims, smnt, (fee, will be analogous to the resistances in the middle of each of the times, that is (by supposition), to the velocities, and so to the spaces described. Take the sums of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, (fee, will be analogous to the whole spaces described ; and also the areas AB^'K, ABrL, AB^M, ABifN, (fee, to the times. There- fore the body, in descending, will in any time ABrL describe the space Blr, and in the time hrtN the space rlnt. Q.E.D. And the like demonstra- tion holds in ascending motion.

CoROL. 1. Therefore the greatest velocity that the body can acquire by falling is to the velocity acquired in any given time as the given force of gravity which perpetually acts upon it to the resisting force which opposes it at the end of that time.

254

THE MATHEMATICAL PRINCIPLES

[Book IL

CoROL. 2. But tlie time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the descent, decreases in a geometrical progression.

CoROL. 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the same geometrical progression.

CoROL. 4. The space described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity ; which [spaces] also at the beginning of the descent are equal among themselves.

PROPOSITION IV. PROBLEM II.

Supposing the force of gravity in any similar medium to be uniform j and to tend perpendicularly to the plane of the horizon ; to define the motion of a projectile therein, lohich suffers resistance proportional to its velocity.

Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. Prom the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, so that DA may be to AC as the resistance of the medium arising from the motion up- wards at the beginning to the force of grav- ity; or (which comes to the same) so that the rectangle under DA. and DP may be to that under AC and CP as the whole resist- ance at the beginning of the motion to the force of gravity. With the asymptotes DC, CP describe any hyperbola GTBS cut- ting the perpendiculars DG, AB in G and B ; complete the parallelogram DGKC, and D BA. 3? ^ let its side GK cut AB in Q. Take a line

N in the same ratio to QB as DC is in to CP ; and from any point R of the right line DC erect RT perpendicular to it, meeting the hyperbola in T, and the right lines EH, GK, DP in I, t, and Y ; in that perpendicular

/f^T take Vr equal to -^f^ , or which is the same thing, take Rr equal to

N

GTIE

N

and the projectile in the time DRTG will arrive at the point ?',

describing the curve line DraF, the locus of the point r ; thence it will come to its greatest height a in the perpendicular AB ; and afterwards

Sec. I.] OF NATURAL PHILOSOPHY. 255

ever approach to the asymptote PC. And its velocity in any point r will be as the tangent rh to the curve. Q.E.I.

For N is to QB as DC to CP or DR to RV, and therefore RV is equal to

^ , and Kr (that is, Rv — vr, or z-- ) is equal to

DRXAB — RDGT ^^ ^ . .^ .- ^. .1. .

^^ . JNow let the time be expounded by the area

RDGT and (by Laws, Cor. 2), distinguish the motion of the body into two others, one of ascent, the other lateral. And since the resistance is as the motion, let that also be distinguished into two parts proportional and contrary to the parts of the motion : and therefore the length described by the lateral motion will be (by Prop. II, Book II) as the line DR, and the height (by Prop. Ill, Book II) as the area DR X AB — RDGT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR X AQ, and therefore that line Rr (or

—- ^^^ — ) will then be to DR as AB — AQ or QB to N,

that is, as CP to DC ; and therefore as the motion upwards to the motion lengthwise at the beginning. Since, therefore, Rr is always as the height, and DR always as the length, and Rr is to DR at the beginning as the height to the Itogth, it follows, that Rr is always to DR as the height to the length ; and therefore that the body will move in the line DraF, which is the locus of the point r. Q.E.D.

n 1 n^^ ^ T? • 1 DRxAB RDGT ., ^

Cor. 1. Iiiereiore Rr is equal to ^j^: =^^ — . and therefore

DR V AR

if RT be produced to X so that RX may be equal to ^ , that iS;

if the parallelogram ACPY be completed, and DY cutting CP in Z be drawn, and RT be produced till it meets DY in X ; Xr will be equal to

I^DGT , , , . , . .

— -^^: — , and therefore proportional to the time.

Cor. 2. Whence if innumerable lines CR, or, which is the same, innu- merable lines ZX, be taken in a geometrical progression, there, will be as many lines Xr in an arithmetical progression. ^ And hence the curve DraF is easily delineated by the table of logarithms.

Cor. 3. If a parabola be constructed to the vertex D, and the diameter DG produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the motion to the gravitating force, the ve- locity with which the body ought to go from the place D, in the direction of the right line DP, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the same place D, in the direction of the same right line DP, so as to describe

256

THE MATHEMATICAL PRINCIPLES

[Book II.

p a parabola in a non-resisting medium. For the latus rectum of this parabola, at the very

beginning of the motion, is -^^^ ; and Vr is

tGT BRxTt ^ . , ,

"1^ ^^ 2^" — * ^ nght line, which,

if drawn, would touch the hyperbola GTS in

^ G, is parallel to DK, and therefore T^ is

CK X DR QR V DO

And there-

fore Vr is equal to

DC DR2 X CK X CP

CK X DR ^ ^^ . QB X DC

, and N is gp .

and DP are proportionals), to

2DC=^ X QB

DV2 X CK X CP

, that is (because DR and DC, DV

and the latus rectum

DV2

• comes out

2DP2 X QB

CKx 2DP2 X DA

2DP^ X QB -~P, that is (because QB and CK, DA and AC

and therefore is to 2DP as DP X DA to

Vr

are proportional), ^^ ^ ^^

CP X AC ; that is, as the resistance to the gravity.

Q.E.D.

Cor. 4. Hence if a body be projected from any place D with a given velocity, in the direction of a right line DP given by posi- tion, and the resistance of the medium, at the beginning of the motion, be given, the curve DrrtF, which that body will describe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given. Then cutting DC in A, so that CP X AC may be to DP X DA in the same ratio of the gravity to the resistance, the point A |-j- will be given. And hence the curve DraF ^ is also given.

.^ Cor. 5. And, on the contrary, if the curve DraF be given, there will be given both the velocity of the body and the resistance of the medium in each of the places r. For the ratio of CP X AC to DP X DA being given, there is given both the resistance of the medium at the beginning of the motion, and the latus rectum of the parabola ; and thence the velocity at the be- ginning of the motion is given also. Then from the length of the tangent

Sec. L] / OF natural philosophy. 257

rL there is given both the velocity proportional to it, and the resistance proportional to the velocity in any place f.

Cor. 6. But since the length 2DP is to the latus rectum of the para- bola as the gravity to the resistance in D ; and, from the velocity aug- mented, the resistance is augmented in the same ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio, it is plain that the length 2DP is augmented in that simple ratio only ; and is therefore always proportional to the velocity ; nor will it be augmented or dimin- ished by the change of the angle CDP, unless the velocity be also changed.

Cor. 7. Hence appears the method of deter- /^

mining the curve DraP nearly from the phe- / y^

nomena, and thence collecting the resistance ,and velocity with which the body is projected. Let two similar and equal bodies be projected with the same velocity, from the place D, in differ- ent angles CDP, CDj? ; and let the places F,

/", where they fall upon the horizontal plane /^ \ \

DC, be known. Then taking any length for ^ J^ ^ O

DP or DjD suppose the resistance in D to be to

the gravity in any ratio whatsoever, and let that -^^

ratio be expounded by any length SM. Then, g , I >-qCl!^r —

by computation, from that assumed length DP, ' t^

find the lengths DF, D/; and from the ratio

F/

^^, found by calculation, subduct the same ratio as found by experiment ;

and let the difference be expounded by the perpendicular MN. Repeat the same a second and a third time, by assuming always a new ratio SM of the resistance to the gravity, and collecting a new difference MN. Draw the affirmative differences on one side of the right line SM, and the negative on the other side ; and through the points N, N, N, draw a regular curve NNN, cutting the right line SMMM in X, and SX will be the true ratio of the resistance to the gravity, which was to be found. From this ratio the length DF is to be collected by calculation ; and a length, which is to the assumed length DP as the length DF known by experiment to the length DP just now found, will be the true length DP. This being known, you will have both the curve line Dr«F which the body describes, and also the velocity and resistance of the body in each place.

SCHOLIUM.

But, yet, that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis than a physical one. In mediums void of all te- nacity, the resistances made to bodies are in the duplicate ratio of the ve- locities. For by the action of a swifter body, a greater motion in propor-

17

25S THE MATHEMATICAL PKINCIPLES [BoOK II.

tion to a greater velocity is communicated to the same quantity of tlie medium in a less time ; and in an equal time, by reason of a greater quan- tity of the disturbed medium, a motion is communicated in the duplicate ratio greater ; and the resistance (by Law II and III) is as the motion communicated. Let us, therefore, see what motions arise from this law of resistance.

SECTION IL

Of the motion of bodies that are i^esisted in the duplicate ratio of their

velocities,

PROPOSITION V. THEOREM III.

If a body is resisted in tlm duplicate ratio of its velocity , and moves by its innate force only through a similar inedium ; and the times be taken in a geoinetrical progression, proceeding from less to greater terms : I say, that the velocities at the beginning of each of the times are in the sa?ne geometrical progression inversely ; and that the spaces are equal, ichich are described in each of the times. For since the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resist- ance : if the time be divided into innumerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the diiFerences of the same velocities. Let those particles of time be AK, KL, LM, &c., taken in the right line CD ; and erect the perpendiculars AB, Y.k, LZ, Mm, &c., meeting the hyperbola BA'ZmG, described with the centre C, and the rectangular asymptotes CD, CH, in B, k, I, m, &c. ; then AB will be to ILk as CK to CA, and, by division, AB — Kk to Kk as AK to CA, and alternately, AB — Kk to AK as Kk

C jmM.T D ^^ ^^ . ^^^ therefore as AB X Kk to AB X CA.

Therefore since AK and AB X C A are given, AB — Kk will be as AB X Kk ; and, lastly, when AB and Kk coincide, as AB^. And, by the like reasoning, Kk-l^l, LZ-Mwi, &c., will be as Kk\ LZ^ &c. Therefore the squares of the lines AB, Kk, L/, Mm, &c., are as their differences ; and, therefore, since the squares of the velocities were shewn above to be as their differences, the progression of both will be alike. This being demonstrated it follows also that the areas described by these lines are in a like progres- sion with the spaces described by these velocities. Therefore if the velo- city at the beginning of the first time AK be expounded by the line AB,

Sec. II.] OF NATURAL PHILOSOPHY. 259

and the velocity at the beginning of the second time KL by the line K^, and the length described in the hrst time by the area AKkB, all the fol- lowing velocities will be expounded by the following lines hi, Mm, <fec., and the lengths described, by the areas K^, Lm, (fcc. And, by compo- sition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AMmB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c., so that CAj'CK, CL, CM, &c. may be in a geometrical progression; and those parts will be in the same progression, and the velocities AB, Kk, Lit, Mm, &c., will be in the same progression inversely, and the spaces de- scribed Ak, Kl, hm, (fee, will be equal. Q.E.D.

Cor. 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time Avill be expounded by the ordinate DG; and the whole space described by the adjacent hyperbolic area ABGD ; and the space which any body can describe in the same time AD, with the first velocity AB, in a non-resisting medium, by the rectan- gle AB X AD.

Cor. 2. Hence the space described in a resisting medium is given, by taking it to the space described with the uniform velocity AB in a non- resisting medium, as the hyperbolic area ABGD to the rectangle AB X AD.

Cor. 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the ve- locity AB in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the asymptote in T, the right line AT will be equal to AC, and will express the time in which the first resistance, uniformly con- tinued, may take away the whole velocity AB.

Cor. 4. And thence is also given the proportion of this resistance to the force of gravity, or any other given centripetal force.

Cor. 5. And, vice versa, if there is given the proportion of the resist- ance to any given centripetal force, the time AC is also given, in which a centripetal force equal to the resistance may generate any velocity as AB ; and thence is given the point B, through which the hyperbola, having CH, CD for its asymptotes, is to be described ; as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD, in a similar resisting medium.

PROPOSITION VI. THEOREM lY.

Homogeneous and equal spherical bodies, opposed hy resisfMnc.es that are in the duplicate ratio of the velocities, ajid moving on by their innate force only, will, in times lohich are reciprocally as the velocities at the

260 THE MATHExMATlCAL PRINCIPLES [BoOK II.

beginning, describe equal spaces, and lose parts of their velocities pro- portional to the wholes.

To the rectangular asymptotes CD, CH de- , scribe any hyperbola B6Ee, cutting the perpen-

\ diculars AB, «6, DE, de in B, b, E, e ; let the

^^^ initial velocities be expounded by the perpendicu-

lars AB, DE, and the times by the lines Aa, Dc?. Therefore as Aa is to Dc?, so (by the hypothesis) . is DE to AB, and so (from the nature of the hy-

^ perbola) is CA to CD ; and, by composition, so is

Ca to Cd. Therefore the areas ABZ;«, DEec?, that is, the spaces described, are equal among themselves, and the first velocities AB, DE are propor- tional to the last ab, de; and therefore, by division, proportional to the parts of the velocities lost, AB — ab, DE — de. Q,.E.D.

PROPOSITION YII. THEOREM Y.

If spherical bodies are resisted in the duplicate ratio of their velocities^ in times which are as the first motions directly, and the first resist- ances inversely, they loill lose parts of their m^otions proportional to tfie wholes, and will describe spaces proportional to those times and the first velocities conjunctly.

For the parts of the motions lost are as the resistances and times con- junctly. Therefore, that those parts may be proportional to the wholes, the resistance and time conjunctly ought to be as the motion. Therefore the time will be as the motion directly and the resistance inversely. Where- fore the particles of the times being taken in that ratio, the bodies will always lose parts of their motions proportional to the wholes, and there- fore will retain velocities always proportional to their first velocities. And because of the given ratio of the velocities, they will always describe spaces which are as the first velocities and the times conjunctly. Q..E.D. Cor. 1. Therefore if bodies equally swift are resisted in a duplicate ra- tio of their diameters, homogeneous globes moving with any velocities whatsoever, by describing spaces proportional to their diameters, will lose parts of their motions proportional to the wholes. For the motion of each globe will be as its velocity and mass conjunctly, that is, as the velocity and the cube of its diameter ; the resistance (by supposition) will be as the square of the diameter and the square of the velocity conjunctly ; and the time (by this proposition) is in the former ratio directly, and in the latter inversely, that is, as the diameter directly and the velocity inversely ; and therefore the space, which is proportional to the time and velocity is as the diameter.

Cor. 2. If bodies equally swift are resisted in a sesquiplicate ratio of their diameters, homogeneous globes, moring with any velocities whatso-

Sec. II.] OF NATURAL PHILOSOPHY. 261

ever, by describing spaces that are in a sesquiplicate ratio of tlie diameters, will lose parts of their motions proportional to the wholes.

Cor. 3. And uniyersally, if equally swift bodies are resisted in the ratio of any power of the diameters, the spaces, in which homogeneous globes, moving with any velocity whatsoever, will lose parts of their motions pro- portional to the wholes, will be as the cubes of the diameters applied to that power. Let those diameters be D and E ; and if the resistances, where the velocities are supposed equal, are as D" and E"; the spaces in which the globes, moving with any velocities whatsoever, will lose parts of their motions proportional to the wholes, Avill be as D ^ — " and E ^ — °. And therefore homogeneous globes, in describing spaces proportional to D ^ — " and E"^ — ", will retain their velocities in the same ratio to one another as at the beginning.

Cor. 4. Now if the globes are not homogeneous, the space described by the denser globe must be augmented in the ratio of the density. For the motion, with an equal velocity, is greater in the ratio of the density, and the time (by this Prop.) is augmented in the ratio of motion directly, and tlie space described in the ratio of the time.

Cor. 5. And if the globes move in different mediums, the space, in a medium which, cceteris paribus, resists the most, must be diminished in the ratio of the greater resistance. For the time (by this Prop.) will be di- minished in the ratio of the augmented resistance, and the space in the ra- tio of the time.

LEMMA n.

The moment of any genitum is equal to the ^moments of each of the gen- erating sides draion into the indices of the powers of those sides, and into their co-efficients contimially,

I call any quantity a genitum which is not made by addition or sub- duction of divers parts, but is generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever ; in geometry by the invention of contents and sides, or of the extremes and means of proportionals. (Quantities of this kind are products, quotients, roots, rectangles, squares, cubes, square" and cubic sides, and the like. These quantities I here consider as variable and indetermined, and increas- ing or decreasing, as it were, by a perpetual motion or flux ; and I under- stand their momentaneous increments or decrements by the name of mo- ments ; so that the increments may be esteemed as added or affirmative moments; and the decrements as subducted or negative ones. But take care not to look upon finite particles as such. Finite particles are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes. Nor do we in this Lemma regard the magnitude of the moments, but their first

262 THE MATHEMATICAL PRINCIPLES [BoOK II

proportion, as nascent. It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities), or any finite quantities proportional to those velocities. The co-efficient of any generating side is the quantity which arises by applying the genitum to that side.

Wherefore tlie sense of the Lemma is, that if the moments of any quan- tities A, B, C, &c., increasing or decreasing by a perpetual flux, or the velocities of the mutations which are proportional to them, be called a, b, c, <fec., the moment or mutation of the generated rectangle AB will be «B -h bA ; the moment of the generated content ABC will be aBC + bAC +

All

cAB ; and the moments of the generated powers A^, A^, A*, A^, A^. A^, A^, A—\A—^, A— ^ will be 2aA, 3aA^, 4aA^ |aA — ^, faA^,

1 _3

laA — '^, f r/A — 3, — a A — ^, — 2g:A — ^, — |«A — ^ respectively; and, in general, that the moment of any power A^, will be ^ aA^-^^. Also, that the moment of the generated quantity A^B will be 2aAB -{- bA^ ; the moment of the generated quantity A ^B^C^ will be SaA^B^C^ -- AbA^

A'

B^C^ +2cA^B^C; and the moment of the generated quantity ^ or

A^B -^ -' will be SaA^B — - —2bA^B — ^ ; and so on. The Lemma is thus demonstrated.

Case 1. Any rectangle, as AB, augmented by a perpetual flux, when, as yet, there wanted of the sides A and B half their moments ^a and ^b, was A — ^a into B — ^b, or AB — |a B — |6 A + ^ab ; but as soon as the sides A and B are augmented by the other half moments, the rectangle be- comes A-- \a into B + \b, or AB -^ \a B -- \b A + }ab. From this rectangle subduct the former rectangle, and there will remain the excess aB -f bA. Therefore with the whole increments a and b of the sides, the increment aB + bA of the rectangle is generated. Q.E.D.

Case 2. Suppose AB always equal to G, and then the moment of the content ABC or GC (by Case 1) will be gC + cG, that is (putting AB and aB + bA for G and g), aBC + bAC 4- cAB. And the reasoning is the same for contents under ever so many sides. Q.E.D.

Case 3. Suppose the sides A, B, and C, to be always equal among them- selves; and the moment aB + bA, of A^, that is, of the rectangle AB, will be 2a A ; and the moment aBC + bAC + cAB of A^, that is, of the content ABC, will be SaA^. And by the same reasoning the moment of any power A" is ?zaA" — K Q.E.D

Case 4. Therefore since -r- into A is 1, the moment of -r drawn into A A

Sec. 11.] OF NATURAL PHILOSOPHY. 263

A, together with -j drawn into a, will be the moment of 1, that is, nothing.

1 a

Therefore the moment of -r, or of A — % is -^^. And generally since

-r- into A" is 1; the moment of j^drawn into A" together with -r- into

A"^ A A

1^

A"

naA" — * will be nothing. And, therefore, the moment of ti; or A

will be — -— -r. Q.E.D.

A° 4- * .

Case 5. And since A^ into A^ is A, the moment of A^ drawn into 2A- will be a (by Case 3) ; and, therefore, the moment of A^ will be p-r-p or

^aA — 1. And, generally, putting A^' equal to B, then A™ will be equal to B", and therefore m«A™ — ^ equal to wiB" ■ — \ and maA — ^ equal to

nbB — ^ , or 7ibA — „ ; and therefore „ «A —^ is equal to b, that is, equal

to the moment of A~^. Q.E.D.

Case 6. Therefore the moment of any generated quantity A^'B" is the moment of A"" drawn into B", together with the moment of B" drawn into A"", that is, maA™ — ^ B" + ?z6B" — ^ A'" ; and that whether the indices on and n of the powers be whole numbers or fractions, affirmative or neg- ative. And the reasoning is the same for contents under more powers. Q,.E.D.

Cor. 1. Hence in quantities continually proportional, if one term is given, the moments of the rest of the terms will be as the sam.e terms mul- tiplied by the number of intervals between them and the given term. Let A, B, C, D, E, F, be continually proportional ; then if the term C is given, the moments of the rest of the terms will be among themselves as — 2A, — B, D, 2E, 3F.

Cor. 2. And if in four proportionals the two means are given, the mo- ments of the extremes will be as those extremes. The same is to be un- derstood of the sides of any given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the mo- ments of the sides will be reciprocally as the sides.

SCHOLIUM. 1

In a letter of mine to Mr. /. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Sluskis's method, which at that time was not made public, I subjoined these words : This is one particular, or rather a Corollary, of a general me-

264 THE MATHEMATICAL PRINCIPLES [BoOK II.

thod, which extends itself, without any troublesome calculatio7i, not only to the drawing of tangents to any curve lines, whether geometrical or mechanical, or any how respecting right lines or other curves, but also to the resolving otJier abstruser kinds of problems about the crookedness, areas, lengths, centres of gravity of curves, &c. : nor is it [as Hudden's method de Maximis & Minimis) limited to equations which are free from surd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite series. So far that letter. And these last words relate to a treatise I composed on that , sub- ject in the year 1671. The foundation of that general method is contain- ed in the preceding Lemma.

PROPOSITION YIII. THEOREM VI.

If a body in an uniform medium, being uniformly acted upon by the force of gravity, ascends or descends in a right line ; and the whole space described be distinguished into equal parts, and in the beginning of each of the parts {by adding or subducting the resisting force of the m^edium to or from the force of gravity, when the body ascends or de- scends) you collect the absolute forces ; I say, that those absolute forces are in a geometrical progression. n ig For let the force of gravity be expounded by the

given line AC ; the force of resistance by the indefi- nite line AK ; the absolute force in the descent of the ~^rf — body by the difference KG ; the velocity of the body

QPLKIA/^/^/ ' l>y ^ liiie AP, which shall be a mean proportional be-

tween AK and AC, and therefore in a subduplicate ratio of the resistance : the increment of the resistance made in a given particle of time by the li- neola KL, and the contemporaneous increment of the velocity by the li- neola PQ, ; and with the centre C, and rectangular asymptotes CA, CH, describe any hyperbola BNS meeting the erected perpendiculars AB, KN, LO in B; N and 0. Because AK is as AP^, the moment KL of the one will be as the moment 2APQ, of the other, that is, as AP X KC ; for the in- crement PQ. of the velocity is (by Law 11) proportional to the generating force KC. Let the ratio of KL be compounded with the ratio KN, and the rectangle KL X KN will become as AP X KC X KN ; that is (because the rectangle KC X KN is given), as AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle KL X KN becomes, when the points K and L coincide, the ratio of equality. Therefore that hyperbolic evanescent area is as AP. Therefore the whole hyperbolic area ABOL is composed of particles KNOL which are always proportional to the velocity AP ; and therefore is itself proportional to the space described with that velocity. Let that area be now divided into equal parts,

Sec. II.]

OP NATURAL PHILOSOPHY.

265