Principia Mathematica (Motte Translation, 1848) — part 7 of 45

So those weights are of equal force to move the arms of a balance ; which during the play of the balance are reciprocally as their velocities upwards and downwards ; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis of the balance ; but if they are turned aside by the interposition of oblique planes, or other ob- stacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are reciprocally as the heights of their ascent and de- jscent taken according to the perpendicular ; and that on account of the determination of gravity downwards.

94 THE MATHEMATICAL PRINCIPLES

And. in like manner in the pully, or in a combination of puUies, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight.

In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other.

The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body.

The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the pro- gress of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines.

The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary : from whence, in all sorts of proper machines, we have the solution of this problem ; To move a given weight loith a given poiver, or with a given force to over- come any other given resistance. For if machines are so contrived that the velocities of the agent and resistant are reciprocally as their forces, the agent will just sustain the resistant, but with a greater disparity of ve- locity will overcome it. So that if the disparity of velocities is so great a^s to overcome all that resistance which commonly arises either from the attrition of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those re- sistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of the machine as in the resisting body. But to treat of mechanics is not my present business. I was only willing to show by those examples the great extent and certainty of the third Law of motion. For if we estimate the action of the agent from its force and velocity conjunctly, and likewise the reaction of the impediment conjunctly from the velocities of its several parts, and from the forces of resistance arising fro^n the attrition, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last impressed upon the resisting body, the ultimate determination of the action will be always contrary to the determination of the reaction.

OF NATURAL PHILOSOPHY.

95

BOOK I.

OF THE MOTION OF BODIES.

•  SECTION  I. 
  

Of the method of first and last ratios of quantities^ by the help whereof loe demonstrate the propositions that follow.

LEMMA I.

Quantities, and the ratios of quantities, vjhich in any finite time convers^e continually to equality, and before the end of that tim^e approach nearer the one to the other than by any given difference, become ultimately equal.

If you deny it, suppose them to be ultimately unequal; and let D be tbeir ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D ; which is against the supposition.

LEMMA IL

If in any figure AacE, tei^minated by the right lines Aa, AE, and the curve acE, there be in- scribed any mimher of parallelograms Ah, Be, Cd, iSfc, com^prehended under equal bases AB, BC, CD, Sfc, and the sides, Bb, Cc, Dd, (^c, parallel to one side Aa of the figure ; and the parallelogram's aKbl, bLcm, cMdn, ^c, are co??i- pleted. Then if the breadth of those parallelo- grams be supposed to be diminished, and their a BP C D E num,ber to be augmented in infinitum ; / say, that the idtimate ratios which the inscribed figure AKbLcMdD, the circumscribed figure AalbmcndoE, and curvilinear figure AabcdE, will have to one another^ are ratios of equality.

For the difference of the inscribed and circumscribed figures is the sum of the parallelograms lil, L«z, Mvz, Do, that is (from the equality of all their bases), the rectangle under one of their bases lih and the sum of their altitudes Aa, that is, the rectangle KEla. But this rectangle, because

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THE MATHEMATICAL PRINCIPLES

[Book I.

its breadtli AB is supposed diminished m infinitum^ becomes less than any given space. And therefore (by Lem. I) the figures inscribed and circumscribed become ultimately equal one to the other ; and much more will the intermediate curvilinear figure be ultimately equal to either.

aE.D.

LEMMA m.

The same ultimate ratios are also ratios of equality ^ when the breadths, AB; BC, DC, ^c, of the parallelograms are unequal, and are all di- minished in infinitum. For suppose AF equal to the greatest breadth, and complete the parallelogram ¥Aaf This parallelo- gram will be greater than the difference of the in- scribed and circumscribed figures ; but, because its breadth AF is diminished in infinitwm, it will be- come less than any given rectangle. Q,.E.D.

Cor. 1. Hence the ultimate sum of those evanes- cent parallelograms will in all parts coincide with the curvilinear figure.

Cor. 2. Much more will the rectilinear figure comprehended under the chords of the evanescent arcs ah, be, cd, &c., ultimately coincide with the curvilinear figure.

Cor. 3. And also the circumscribed rectilinear figure comprehended under the tangents of the same arcs.

Cor. 4 And therefore these ultimate figures (as to their perimeters acE), are not rectilinear, but curvilinear limits of rectilinear figures.

LEMMA IV.

If in two figures AacE, PprT, you inscribe {as before

tiDO ranks of parallelograms, an equal number in

each rank, and, wlien their breadths are di?ninished

in infinitum, the ultimate ratios of the parallelograms

in one figure to those in the other, each to each respec- tively, are the same; I say, that those two figures

AacE, PprT, are to one another in that sam£ ratio.

For as the parallelograms in the one are severally to p the parallelograms in the other, so (by composition) is the ^ sum of all in the one to the sum of all in the other ; and so is the one figure to the other ; because (by Lem. Ill) the former figure to the former sum, and the latter figure to the latter sum, are both in the ratio of equality. Q.E.D.

Cor. Hence if two quantities of any kind are any how divided into an equal number of parts, and those a

Sec. L] of natural philosophy. 9/

partS; when their number is augmented, and their magnitude diminished in wfinitum, have a given ratio one to the other, the first to the first, the second to the second, and so on in order, the whole quantities will be one to the other in that same given ratio. For if, in the figures of this Lemma, the parallelograms are taken one to the other in the ratio of the parts, the sum of the parts will always be as the sum of the parallelograms ; and therefore supposing the number of the parallelograms and parts to be aug- mented, and their magnitudes diminished in infinitum, those sums will be in the altimate ratio of the parallelogram in the one figure to the corres- pondent parallelogram in the other ; that is (by the supposition), in the ultimate ratio of any part of the one quantity to the correspondent part of the other.

LEMMA v.. In similar figures, all sorts of homologous sides, whether curvilinear or rectilinear, are proportional ; and the areas are in the duplicate ratio ■ of the homologous sides.

LEMMA VL If any arc ACB, given in position is sub- tended by its chord AB, and in any point A, in the middle of the continued curva- ture, is touched by a right line AD, pro- duced both ivays ; then if the points A and B approach one another and meet, I say, the angle BAD, contained betiveen the chord and the tangent, will be dimin- ished in infinitum, and ultimately vnll vanish.

For if that angle does not vanish^ the arc A OB will contain with the tangent AD an angle equal to a rectilinear angle; and therefore the cur- vature at the point A will not be continued, which is against the supposi- tion.

LEMMA VII. The same things being supposed, I say that the ultimate ratio of the arc^

chord, and tangent, any one to any other, is tlie ratio of equality.

For while the point B approaches towards the point A, consider always AB and AD as produced to the remote points b and d, and parallel to the pecant BD draw bd : and let the arc Acb be always similar to the arc ACB. Then, supposing the points A and B to coincide, the angle dAb will vanish, by the preceding Lemma ; and therefore the right lines A^., Ad (which are always finite), and the intermediate arc Acb, will coincide, and become equal among themselves. Wherefore, the right lines AB, AD,

98 THE MATHEMATICAL PRINCIPLES [SeC. I.

and the intermediate arc ACB (^Thich are always proportional to tlie former), will vanish, and ultimately acquire the ratio of equality. Q..E.D.

Cor. 1. Whence if through B we draw . -pl /b

BF parallel to the tangent, always cutting ^ a --^::::;;^^^ \ / "

any right line AF passing through A in / r/_L^__^^^

F, this line BF will be ultimately in the ^^ ^^

ratio of equality with the evanescent arc ACB ; because, completing the parallelogram AFBD, it is always in a ratio of equality with AD.

Cor. 2. And if through B and A more right lines are drawn, as BE, BD, AF, KQf^ cutting the tangent AD and its parallel BF : the ultimate ratio of all the abscissas AD, AE, BF, BG, and of the chord and arc AB, any one to any other, will be the ratio of equality.

Cor. 3. And therefore in all our reasoning about ultimate ratios, we may freely use any one of those lines for any other.

LEMMA YIII.

If the right lines AR, BR, with the arc ACB, the chord AB, and the tangent AD, constitute three triangles RAB. RACB, RAD, and the points A a?id B approach and meet : I say, that the ultimate form of these evaiiesc&nt triaiigles is that of similitude^ and their ultimate ratio that of equality.

For while the point B approaches towards a. D ^

the point A, consider always AB, AD, AR, as produced to the remote points h, d, and r, and rhd as drawn parallel to RD, and let the arc Ach be always similar to the arc ACB. Then supposing the points A and B to coincide, the angle bKd will vanish ; and therefore the three triangles rA6, rAch, rAd (which are always finite), will coincide, and on that account become both similar and equal. And therefore the triangles RAB, RACB, RAD which are always similar and proportional to these, will ultimately be- come both similar and equal among themselves. Q,.E.D.

Coil And hence in all reasonings about ultimate ratios, we may indif- ferently u^e any one of those triangles for any other.

LEMMA IX.

If a right lirm AE, and a curve line ABC, both given by position, cut each other in a given angle, A ; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C : and the points B and C together approach towards and Tneet in the point A : / say, that the areas of the tria?igles ABD, ACE, will ultimately be one to the other in the duplicate ratio of the sides.

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OF NATURAL PHILOSOPHY.

99

For while the points B, C, approach towards the point A, suppose always AD to be produced to the remote points d and . e, so as Ad, Ae may be proportional to AD, AE ; and the ordinates db, ec, to be drawn parallel to the ordinates DB and ^ EC, and meeting AB and AC produced D in b and c. Let the curve Ai>c be similar to the curve ABC, and draw the right line A^ so as to touch both curves in A, and cut the ordinates DB, EC, db ec, in F, G, f, g. Then, supposing the length Ae to remain the same, let the points B and C meet in the point A ; and the angle cKg vanishing, the curvilinear areas Abd, A.ce will coincide with the rectilinear areas Afd, Age ; and therefore (by Lem. Y) will be one to the other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas ; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately one to the other in the duplicate ratio of the sides AD, AE. aE.D.

LEMMA X. The spaces ivhich a body describes by any finite force urging it, ivhether

that force is determined and immutable, or is continually augmented

or continually diminished, are in the very beginning of the m^otion one

to the other in the duplicate ratio of the tim^es.

Let the times be represented by the lines AD, AE, and the velocities generated in those times by the ordinates DB, EC. The spaces described with these velocities will be as the areas ABD, ACE, described by those ordinates, that is, at the very beginning of the motion (by Lem. IX), in the duplicate ratio of the times AD, AE. Q.E.D.

Cor. 1. And hence one may easily infer, that the errors of bodies des- cribing similar parts of similar figures in proportional times, are nearly as the squares of the times in which they are generated ; if so be these errors are generated by any equal forces similarly applied to the bodies, and measured by the distances of the bodies from those places of the sim- ilar figures, at which, without the action of those forces, the bodies would have arrived in those proportional times.

Cor. 2. But the errors that are generated by proportional forces, sim- ilarly applied to the bodies at similar parts of the similar figures, are as the forces and the squares of the times conjunctly.

Cor. 3, The same thing is to be understood of any spaces whatsoever described by bodies urged with different forces ; all which, in the very be- ginning of the motion, are as the forces and the squares of the times conjunctly.

100

THE MATHEMATICAL PRINCIPLES

[Sec. I.

CoH. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversely.

Cor. 5. And the squares of the times are as the spaces described direct- ly, and the forces inversely.

SCHOLIUM.

If in comparing indetermined quantities of diiferent sorts one with another, any one is said to be as any other directly or inversel}^, the mean- ing is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. And if any one is said to be as any other two or more directly or inversely, the meaning is, that the first is aug- mented or diminished in the ratio compounded of the ratios in which the others, or the reciprocals of the others, are augmented or diminished. As if A is said to be as B directly, and C directly, and D inversely, the mean- ing is, that A is augmented or diminished in the same ratio with B X C X -^, that is to say, that A and -^ are one to the other in a given ratio.

LEMMA XI.

The evanescent subtense of the angle of contact, hi all curves which at the point of contact have a finite curvature, is ultimately in the dupli- cate ratio of the subtense of the conterininate arc. Case 1. Let AB be that arc, AD its tangent, BD the subtense of the angle of contact perpendicular on the tangent, AB the subtense of the arc. Draw BG perpendicular to the subtense AB, and AG to the tan- gent AD, meeting in G ; then let the points D, B, and G, approach to the points d, b, and g, and suppose J to be the ultimate intersection of the lines BG, AG, when the points D, B, have come to A. It is evident that the distance GJ may be less than any assignable. But (from the nature of the circles passing through the points A, B, G, A, b, §;.) AB^'= AG X BD, and Ab^ = Ag X bd : and therefore the ratio of AB^ to .A¥ is compounded of the ratios of AG to Ag, and of B<i to bd. But because GJ may be as- sumed of less length than any assignable, the ratio of AG to Ag may be such as to differ from the ratio of equality by less than any assignable difference ; and therefore the ratio of AB^ to A¥ may be such as to differ from the ratio of BD to hd by less than any assignable difference. 7.'here- fore, by Lem. I, the ultimate ratio of AB^ to Ab^ is the same with the ul- timate ratio of BD to bd. Q.E.D.

Case 2. Now let BD be inclined to AD in any given angle, and the ultimate ratio of BD to bd will always be the same as before, and there- fore the same with the ratio of AB^ to Ail Q,.E.D.

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OF NATURAL PHILOSOPHY.

101

Case 3. And if we suppose tlie angle D not to be given, but that the right line BD converges to a given point, or is determined by any other condition whatever ; nevertheless the angles D, d, being determined by the same law, will always draw nearer to equality, and approach nearer to each other than by any assigned difference, and therefore, by Lem. I, will at last be equal : and therefore the lines BD, bd are in the same ratio to each other as before. Q.E.D.

Cor. 1. Therefore since the tangents AD, Ac/, the arcs AB, Kh, and their sines, BC, be, become ultimately equal to the chords AB, A6, their squares will ultimately becom^e as the subtenses BD, bd.

Cor. 2. Their squares are also ultimately as the versed sines of the arcs, bisecting the chords, and converging to a given point. For those versed sines are as the subtenses -BD, bd.

Cor. 3. And therefore the versed sine is in the duplicate ra.tio of the time in which a body will describe the arc with a given velocity.

Cor. 4. The rectilinear triangles ADB, Adb are ultimately in the triplicate ratio of the sides AD, Kd, and in a sesquiplicate ratio of the sides DB, db ; as being in the ratio compounded of the sides AD to DB, and of Ad to db. So also the triangles ABC, Ahc are ultimately in the triplicate ratio of the sides BC, be. Yv'hat I call the sesquiplicate ratio is the subduplicate of the triplicate, as being compounded of the simple and subduplicate ratio.

Cor. 5. And because DB, db are ultimately paral- ^ lei and in the duplicate ratio of the lines AD, Ao?, the ultimate curvilinear areas ADB, Adh will be (by the nature bola) two thirds of the rectilinear triangles xiDB, Adb ; and AB, Ab will be one third of the same triangles. And thence those areas and those segments will be in the triplicate ratio as well of the tangents AD, xic/, as of the chords and arcs AB, AB.

SCHOLIUM.

But we have all along supposed the angle of contact to be neither infi- nitely greater nor infinitely less than the angles of contact made by cir- cles and their tangents ; that is, that the curvature at the point A is neither infinitely small nor infinitely great, or that the interval A J is of a finite mag- nitude. For DB may be taken as AD^ : in which case no circle can be drawn through the point A, between ihQ tangent AD and the curve AB, and therefore the angle of contact will be infinitely less than those of circles. And by a like reasoning, if DB be made successfully as AD'^, AD^, KD^, Ajy, (fee, we shall have a series of angles of contact, proceeding in ivftni- turn, wherein every succeeding term is infinitely less than the pre-

G

of the para- the seo-ments

102 THE MATHEMATICAL PRINCIPLES [BoOK I.

ceding. And if DB be made successively as AD^^ AD| AD^ AD^ AD|,

AD3, &c., we shall have another infinite series of angles of contact, the first of which is of the same sort with those of circles, the second infinitely greater, and every succeeding one infinitely greater than the preceding. But between any two of these angles another series of intermediate angles of contact may be interposed, proceeding both ways in wfinitum, wherein every succeeding angle shall be infinitely greater or infinitely less than the preceding. As if between the terms AD^ and AD^ there were interposed the series ADf , KD% AD|, AD|, AD| AD|, AD-^ AD^, ADf , (fee. And again, between any two angles of this series, a new series of intermediate angles may be interposed, differing from one another by infinite intervals. Nor is nature confined to any bounds.

Those things which have been demonstrated of curve lines, and the superfices which they comprehend, may be easily applied to the curve su- perfices and contents of solids. These Lemmas are premised to avoid the tediousness of deducing perplexed demonstrations ad absurduni, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : but because the hypothesis of indivisibles seems somewhat harsh, and therefore that method is reckoned less geometrical, I chose rather to reduce the demonstrations of the follow- ing propositions to the first and last sums and ratios of nascent and evane- scent quantities, that is, to the limits of those sums and ratios ; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles ; and now those principles being demonstrated, we may use them with more safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curve lines for right ones, I would not be un- derstood to mean indivisibles, but evanescent divisible quantities ; not the sums and ratios of determinate parts, but always the limits of sums and ratios ; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas.

Perhaps it may be objected, that there is no ultimate proportion, of evanescent quantities ; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a cer- tain place, and there stopping, has no ultimate velocity : because the velo- city, before the body comes to the place, is not its ultimate velocity : when it has arrived, is none. But the answer is easy ; for by the ultimate ve- locity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives ; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ra- tio of evanescent quantities is to be understood the ratio of the quantities

Sec. II.] OF NATURAL PHILOSOPHY. 103

not before they vanishj nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the ve- locity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and pro- portions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and de- monstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratios of evanescent quan- tities are given, their ultimate magnitudes will be also given : and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge ; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in ivfinitum^. This thing will appear more evident in quantities infinitely great. If two quantities, whose dif- ference is given, be augmented in infinitum, the ultimate ratio of these quantities will be given, to wit, the ratio of equality ; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be al- ways diminished without end.

SECTION 11.

Of the Invention of Centripetal Forces,

PROPOSITION I. THEOREM L

The areas, which revolving bodies describe by radii drawn to an immo- vable centra of force do lie in the same immovable planes, and are pro- portional to the times in which they are described. For suppose the time to be divided into equal parts, and in the first part of that time let the body by its innate force describe the right line AB., In the second part of that time, the same would (by Law L), if not hindered, proceei directly to c, along the line Be equal to AB ; so that by the radii AS, BS, cS, drawD to the centre, the equal areas ASB, BSc, would be de-

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[Book 1.

scribed. But wlien the body is arrived at B, suppose that a centripetal force acts at once with a great im- pulse, and, turning aside the body from the right line Be, compels it afterwards to con- tinue its motion along the right line BC. Draw cC parallel to BS meeting BC in C ; and at the end of the second part of the time, the body (by Cor. I. of the Laws) will be found in C, in the same plane with the triangle ASB. Join SC, and, because ® SB and Cc are parallel, the triangle SBC will be equal to the triangle SBc, and therefore also to the triangle SAB. By the like argument, if tlie centripetal force acts successively in C, D, E, &c., and makes the body, in each single particle of time, to describe the right lines CD, DE, EF, (fcc, they will all lie in the same plane.; and the triangle SCD will be equal to the triangle SBC, and SDE^to SCD, and SEP to SDE. And therefore, in equal times, equal areas are described in one immovable plane : and, by composition, any sums SADS, SAFS, of those areas, are one to the other as the times in which they are described. Now let the number of those triangles be augmented, and their breadth diminished in infinitum ; and (by Cor. 4, Lem. III.) their ultimate perimeter ADP^ will be' a curve line : and therefore the centripetal force, by which the body is perpetually drawn back from the tangent of this curve, will act continually ; and any described areas SADS, SAFS, which are always proportional to the times of de- scription, will, in this case also, be proportional to those times. Q.E.D.

Cor. 1. The velocity of a body attracted towards an immovable centre, in spaces void of resistance, is reciprocally as the perpendicular let fall from that centre on the right line that touches the orbit. For the veloci- ties in those places A, B, C, D, E, are as the bases AB, BC, CD, DE, EF, of equal triangles ; and these bases are reciprocally as the perpendiculars let fall upon them.

Cor. 2. If the chords AB, BC of two arcs, successively described in equal times by the same body, in spaces void of resistance, are completed into a parallelogram ABCY, and the diagonal BY of this parallelogram, in the position which it ultimately acquires when those arcs are diminished in iifinihfmi, is produced both ways, it will pass through the centre of force.

Cor. 3. If the chords AB, BC, and DE, EF, of arcs described in equal

Sec. IL]

OF NATURAL PHILOSOPHY.

105

times, in spaces void of resistance, are completed into the parallelograms ABCV, DEFZ : the forces in B and E are one to the other in the ulti- mate ratio of the diagonals BV, EZ, when those arcs are diminished in wfinilum. For the motions BC and EP of the body (by Cor. 1 of the Laws) are compounded of the motions Be, BY, and E/, EZ : but BV and EZ, which are equal to Cc and F/", in the demonstration of this Proposi- tion, were generated by the impulses of the centripetal force in B and E, and are therefore proportional to those impulses.

Cor. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed sines of arcs described in equal times ; which versed sines tend to the centre of force, and bisect the chords when those arcs are diminished to infinity. For such versed sines are the halves of the diagonals mentioned in Cor. 3.

Cor. 5. And therefore those forces are to the force of gravity as the said versed sines to the versed sines perpendicular to the horizon of those para- bolic arcs which projectiles describe in the same time.

Cor. 6. And the same things do all hold good (by Cor. 5 of the Laws), when the planes in which the bodies are moved, together with the centres of force which are placed in those planes, are not at rest, but move uni- formly forward in right lines.

PROPOSITION IL THEOREM IL

Every body that moves in any curve line described in a plane, and by a radiiiSj draion to a point either immovable, or moving forward -with an uniform rectilinear motion, describes about thai point areas propor- tional to the times, is urged by a centripetal force directed to that point.

Case. 1. For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it. And that force by which the body is turned off from its rectilinear course, and is made to describe, in equal times, the equal least triangles SAB, SBC, SCD, &c., about the immovable point S (by Prop. XL. Book 1, Elem. and Law II), acts in the place B, according to the direction of a line par-

S-

106 THE MATHEMATICAL PRINCIPLES [BoOK L

allel to cC, that is, in the direction of the line BS ; and in the place C, according to the direction of a line parallel to o?D, that is, in the direction of the line CS, (fee. : and therefore acts always in the direction of lines tending to the immovable point S. Q.E.D.

Case. 2. And (by Cor. 5 of the Laws) it is indifferent whether the su- perfices in which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly forward in right lines.

Cor. 1. In non-resisting spaces or mediums, if the areas are not propor- tional to the times, the forces are not directed to the point in which the radii meet ; but deviate therefrom in con sequent ia, or towards the parts to which the motion is directed, if the description of the areas is accelerated ; hut hi antecedentia, if retarded.

Cor. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

SCHOLIUM.

A body may be urged by a centripetal force compounded of several forces ; in which case the meaning of the Proposition is, that the force which results out of all tends to the point S. But if any force acts per- petually in the direction of lines perpendicular to the described surface, this force will make the body to deviate from the plane of its motion : but will neither augment nor diminish the quantity of the described surface, and is therefore to be neglected in the composition of forces.

PROPOSITION III. THEOREM III. Every body, that by a radius draicn to the centre of another body, how- soever moved, describes areas about that centre proportional to the times j is urged by a force compounded out of the centripetal force tending to that other body, and of all the accelerative force by lohich that other body is impelled.

Let L represent the one, and T the other body ; and (by Cor. 6 of the Laws) if both bodies are urged in the direction of parallel lines, by a new force equal and contrary to that by which the second body T is urged, the first body L will go on to describe about the other body T the same areas as before : but the force by which that other body T was urged will be now destroyed by an equal and contrary force : and therefore (by Law I.) that other body T, now left to itself, will either rest, or move uniformly forward in a right line : and the first body L impelled by the difference of the forces, that is, by the force remaining, will go on to describe about the other body T areas proportional to the times. And therefore (by Theor. II.) the difference of the forces is directed to the other body T as its centre. Q.E.D.

Sec. II.] OF NATURAL PHILOSOPHY. 107