as the squares of the times in which they are generated ;if so be theseerrors are generated by any equal forces similarly applied to the bodies,and measured by the distances of the bodies from those places of the similar figures, at which, without the action of those forces, the bodies wouldhave arrived in those proportional times.COR. 2. But the errors that are generated by proportional forces, similarly applied to the bodies at similar parts of the similar figures, are asthe forces and the squares of the times conjuiu tly.COR. 3. The same thing is to be understood of any spaces whatsoeverdescribed by bodies urged with different forces ;all which, in the very begnning of the motion, are as the forces and the squares of the times conjunctly.100 THE MATHEMATICAL PRINCIPLES I SEC. 1COR. 4. And therefore the forces are as the spaces described in the verybeginning of the motion directly, and the squares of the times inversely.COR. 5. And the squares of the times are as the spaces described directly, und the forces inversely.SCHOLIUM.If in comparing indetermined quantities of different sorts one withanother, any one is said to be as any other directly or inversely, the meaning is, that the former is augmented or diminished in the same ratio withthe latter, or with its reciprocal. And if any one is said to be as any othertwo or more directly or inversely, the meaning is, that the first is augmented or diminished in the ratio compounded of the ratios in which theothers, or the reciprocals of the others, are augmented or diminished. Asif A is said to be as B directly, and C directly, and D inversely, the meaning is, that A is augmented or diminished in the same ratio with B X CX -jj-,that is to say, that A and - arc one to the other in a given ratio.LEMMA XLThe evanescent subtense of the angle of contact, in all curves which atthe point of contact have a finite curvature, is ultimately in the duplicate rati 1) of the subtense of the conterminate arc.CASE 1. Let AB be that arc, AD its tangent, BDthe subtense of the angle of contact perpendicular onthe tangent, AB the subtense of the arc. Draw BGperpendicular to the subtense AB, and AG to the tangent AD, meeting in G ;then let the points D, B, andG. approach to the points d, b, and g, and suppose Jto be the ultimate intersection of the lines BG, AG,when the points D, B, have come to A. It is evidentthat the distance GJ may be less than any assignable.But (from the nature of the circles passing throughthe points A, B, G, A, b, g,) AE2= AG X BD, andA62=Ag X bd ; and therefore the ratio of AB2 to Ab2is compounded oithe ratios of AG to Ag, and of Ed to bd. But because GJ may be assumed of less length than any assignable, the ratio of AG to Ag may besuch as to differ from the ratio of equality by less than any assignabledifference ; and therefore the ratio of AB2 to Ab2 may be such as to differfrom the ratio of BD to bd by less than any assignable difference. Therefore, by Lem. I, the ultimate ratio of AB2 to Ab2is the same with tho ultimate ratio of BD to bd. Q.E.D.CASE 2. Now let BD be inclined to AD in any given an*r1 r, and theultimate ratio of BD to bd will always be the same as before, and therefore the same with the ratio of AB2 to Ab2. Q.E-PBOOK I.] OF NATURAL PHILOSOPHY. 101CASE 3. And if we suppose the angle D not to be given, but that theright line BD converges to a given point, or is determined by any othercondition whatever ;nevertheless the angles D, d, being determined by thesame law, will always draw nearer to equality, arid approach nearer toeach other than by any assigned difference, and therefore, by Lem. I, will atlust be* equal ; and therefore the lines BD; bd arc in the same ratio to eachother as before. Q.E.D.COR. 1. Therefore since the tangents AD, Ad, the arcs AB, Ab, andtheir sines, BC, be, become ultimately equal to the chords AB, Ab} theirsquares will ultimately become 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 versedsines are as the subtenses BD, bd.COR. 3. And therefore the versed sine is in the duplicate ratio of thetime in which a body will describe the arc with a given velocity.COR. 4. The rectilinear triangles ADB, Adb areultimately in the triplicate ratio of the sides AD, Ad, cand in a sesquiplicate ratio of the sides DB, db ; asbeing in the ratio compounded of the sides AD to DB,and of Ad to db. So also the triangles ABC, Abeare ultimately in the triplicate ratio of the sides BC, be.What I call the sesquiplicate ratio is the subduplicateof the triplicate, as being compounded of the simpleand subduplicate ratio. jCOR. 5. And because DB, db are ultimately paral- glei and in the duplicate ratio of the lines AD, Ad, theultimate curvilinear areas ADB, Adb will be (by the nature of the parabola) two thirds of the rectilinear triangles ADB, Adb and the segmentsAB, Ab will be one third of the same triangles. And thence those areasand those segments will be in the triplicite ratio as well of the tangentsAD, Ad, as of the chords and arcs AB, AB.SCHOLIUM.But we have all along supposed the angle of contact to be neither infinitely greater nor infinitely less than the angles of contact made by circles and their tangents ; that is, that the curvature at the point A is neitherinfinitely small nor infinitely great, or that the interval AJ is of a finite magnitude. For DB may be taken as AD3: in which case no circle can be drawnthrough the point A, between the tangent AD and the curve AB, andtherefore the angle of contact will be infinitely less than those of circles.And by a like reasoning, if DB be made successfully as AD4, AD5, AD8,AD7, etc., we shall have a series of angles of contact, proceeding in itifinitum,wherein every succeeding term is infinitely less than the pre102 THE MATHEMATICAL PRINCIPLES [BOOK 1ceding. And if DB be made successively as AD2, AD|, AD^, AD], AD|AD7, &c., we shall have another infinite series of angles of contact, the firstof which is of the same sort with those of circles, the second infinitelygreater, and every succeeding one infinitely greater than the preceding.But between any two of these angles another series of intermediate anglesof contact may be interposed, proceeding both ways in infinitum. whereinevery succeeding angle shall be infinitely greater or infinitely less than thepreceding. As if between the terms AD2 and AD3 there were interposedthe series AD f, ADy, AD49, AD|, AD?, AD|, AD^1, AD^, AD^7, &c. Andagain, between any two angles of this series, a new series of intermediateangles 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 theeuperfices which they comprehend, may be easily applied to the curve superficesand contents of solids. These Lemmas are premised to avoid thetediousness of deducing perplexed demonstrations ad absurdnm, accordingto the method of the ancient geometers. For demonstrations are morecontracted by the method of indivisibles : but because the hypothesis ofindivisibles seems somewhat harsh, and therefore that method is reckonedless geometrical, I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios ; and so topremise, as short as I could, the demonstrations of those limits. For herebythe same thing is performed as by the method of indivisibles ; and nowthose principles being demonstrated, we may use them with more safety.Therefore if hereafter I should happen to consider quantities as made up ofparticles, or should use little curve lines for right ones, I would not be un-(lerstood to mean indivisibles, but evanescent divisible quantities : not thesums and ratios of determinate parts, but always the limits of sums andratios ; and that the force of such demonstrations always depends on themethod laid down in the foregoing Lemmas.Perhaps it may be objected, that there is no ultimate proportion, ofevanescent quantities ; because the proportion, before the quantities havevanished, is not the ultimate, and when they are vanished, is none. Butby the same argument, it may be alledged, that a body arriving at a certain place, and there stopping has no ultimate velocity : because the velocity, before the body comes to the place, is not its ultimate velocity ; whenit has arrived, is none i ut the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrivesat its last place and the motion ceases, nor after, but at the very instant itarrives ; 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 ratio of evanescent quantities is to Le understood the ratio of the ijuantitieaSEC. II.] OF NATURAL PHILOSOPHY. 103not before they vanish, nor afterwards, but with which they vanish. Inlike manner the first ratio of nascent quantities is that with which they beginto be. And the first or last sum is that with which they begin and ceaseto be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is theultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain anddefinite, to determine the same is a problem strictly geometrical. Butwhatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical.It may also be objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will be also given : and so allquantities will consist of indivisibles, which is contrary to what Euclidhas demonstrated concerning incommensurables, in the 10th Book of hisElements. But this objection is founded on a false supposition. Forthose ultimate ratios with which quantities vanish are not truly the ratiosof ultimate quantities, but limits towards which the ratios of quantitiesdecreasing without limit do always converge ; and to which they approachnearer than by any given difference, but never go beyond, nor in effect attainto, till the quantities are diminished in wfinitum. This thing will appearmore evident in quantities infinitely great. If two quantities, whose difference is given, be augmented in infin&um, the ultimate ratio of thesequantities will be given, to wit, the ratio of equality ; but it does not fromthence follow, that the ultimate or greatest quantities themselves, whoseratio that is, will be given. Therefore if in what follows, for the sake ofbeing more easily understood, I should happen to mention quantities asleast, or evanescent, or ultimate, you are not to suppose that quantities ofany determinate magnitude are meant, but such as are conceived to be always diminished without end.SECTION II.Of the Invention of Centripetal Forces.PROPOSITION I. THEOREM 1.The areas, which revolving bodies describe by radii drawn to an ^mmovablecentra offorce do lie in tJ:e same immovable planes, and are proportionalto the times in which they are described.For suppose the time to be divided into equal parts, and in the first partof that time let the body by its innate force describe the right line ABIn the second part of that time, the same would (by Law I.), if not hindered,proceel directly to c, alo ILJ; the line Be equal to AB ; so that by the radiiAS, BS, cS, draw. i to the centre, the equal areas ASB, BSc, would be de104 THE MATHEMATICAL PRINCIPLES [BOOK Iscribed. But when the bodyis arrived at B, supposethat a centripetal force actsat once with a great impulse, and, turning aside thebody from the right line Be,compels it afterwards to continue its motion along theright line BC. Draw cCparallel to BS meeting BCin C ; and at the end of thesecond part of the time, thebody (by Cor. I. of the Laws)will be found in C, in thesame plane with the triangleA SB. Join SC, and, because sSB 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 thecentripetal force acts successively in C, D, E. &c., and makes the body, ineach single particle of time, to describe the right lines CD, DE, EF7 &c.,they will all lie in the same plane : and the triangle SCD will be equal tothe triangle SBC, and SDE to SCD, and SEF to SDE. And therefore,in equal times, equal areas are described in one immovable plane : and, bycomposition, any sums SADS, SAFS, of those areas, are one to the otheras the times in which they are described. Now let the number of thosetriangles be augmented, and their breadth diminished in wjinitum ; and(by Cor. 4, Lem. III.) their ultimate perimeter ADF will be a curve line :and therefore the centripetal force, by which the body is perpetually drawnback from the tangent of this curve, will act continually ; and any describedareas SADS, SAFS, which are always proportional to the times of description, 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 fallfrom that centre on the right line that touches the orbit. For the velocities 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 perpendicularslet fall upon them.COR. 2. If the chords AB, BC of two arcs, successively described inequal times by the same body, in spaces void of resistance, are completedinto a parallelogram ABCV, and the diagonal BV of this parallelogram;in the position which it ultimately acquires when those arcs are diminishedin irifinitum, is produced both ways, it will pass through the centre of force.COR. 3. If the chords AB, BC, and DE, EF, cf arcs described in equalSEC. II.] OF NATURAL PHILOSOPHY. 105times, in spaces void of resistance, are completed into the parallelogramsABCV, DEFZ : the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when those arcs are diminished ininfinitum. For the motions BC and EF of the body (by Cor. 1 of theLaws) are compounded of the motions Be, BV, and E/", EZ : but BV andEZ, which are equal to Cc and F/, in the demonstration of this Proposition, 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, aredrawn back from rectilinear motions, and turned into curvilinear orbits,are one to another as the versed sines of arcs described in equal times ; whichversed sines tend to the centre of force, and bisect the chords when thosearcs are diminished to infinity. For such versed sines are the halves ofthe diagonals mentioned in Cor. 3.COR. 5. And therefore those forces are to the force of gravity as the saidversed sines to the versed sines perpendicular to the horizon of those parabolic 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 centresof force which are placed in those planes, are not at rest, but move uniformly forward in right lines.PROPOSITION II. THEOREM II.Every body that moves in any curve line described in a plane, and by aradius, drawn to a point either immovable, or moving forward withan uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to thatpointCASE. 1. For every bodythat moves in a curve line,is (by Law 1) turned asidefrom its rectilinear courseby the action of some forcethat impels it. And that forceby which the body is turnedofffrom its rectilinear course,and is made to describe, inequal times, the equal leasttriangles SAB, SBC, SCD,&c., about the immovablepoint S (by Prop. XL. Book1, Elem. and Law II), actsin the place B, according tothe direction of a line par1U6 THE MATHEMATICAL PRINCIPLES [BOOK f.allel K cC. that is, in the direction of the line BS. and in the place C,accordii g to the direction of a line parallel to dD, that is, in the directionof the line CS, (fee.; and therefore acts always in the direction of linestending to the immovable point S. Q.E.I).CASE. 2. And (by Cor. 5 of the Laws) it is indifferent whether the superficesin which a body describes a curvilinear figure be quiescent, or movestogether with the body, the figure described, and its point S, uniformlyforward in right lines.COR. 1. In non-resisting spaces or mediums, if the areas are not proportional to the times, the forces are not directed to the point in which theradii meet ; but deviate therefrom in. consequently or towards the parts towhich the motion is directed, if the description of the areas is accelerated ;but in antecedentia, if retarded.COR. 2. And even in resisting mediums, if the description of the areasis accelerated, the directions of the forces deviate from the point in whichthe radii meet, towards the parts to which the motion tends.SCHOLIUM.A body may be urged by a centripetal force compounded of severalforces ;in which case the meaning of the Proposition is, that the forcewhich results out of all tends to the point S. But if any force acts perpetually in the direction of lines perpendicular to the described surface,this force will make the body to deviate from the plane of its motion : butwill neither augment nor diminish the quantity of the described surfaceand is therefore to be neglected in the composition of forces.PROPOSITION III. THEOREM III.Every body, that by a radius drawn to the centre of another body, howsoever moved, describes areas about that centre proportional to iJie times,is urged by a force compounded out of the centripetal force Bending fothat other body, and of all the accelerative force by which that otherbody is impelled.Let L represent the one, and T the other body ; and (by Cor. of the Laws)if both bodies are urged in the direction of parallel lines, by a neT forceequal and contrary to that by which the second body T is tinned, the firstbody L will go on to describe about the other body T the same areas asbefore : but the force by which that other body T was urged will be nowdestroyed by an equal and contrary force; and therefore (by Law I.) thatother body T, now left to itself, will either rest, or move uniformly forwardin a right line : and the first body L impelled by the difference of theforces, that is, by the force remaining, will go on to describe about the otherbody T areas proportional to the times. And therefore (by Theor. II.) thedifference ;f the forces is directed to the other body T as its centre. Q.E.DSEC. IL] OF NATURAL PHILOSOPHY. 107Co.*. 1. Hence if the one body L, by a radius drawn to the other body T,describes areas proportional to the times ; and from the whole force, by whichthe firr.t body L is urged (whether that force is simple, or, according toCor. 2 of the Laws, compounded out of several forces), we subduct (by thesame Cor.) that whole accelerative force by which the other body is urged ;the who_e remaining force by which the first body is urged will tend to the( ther body T, as its centre.COR. 2. And, if these areas are proportional to the times nearly, the remaining force will tend to the other body T nearly.COR. 3. And vice versa, if the remaining force tends nearly to the otherbody T, those areas will be nearly proportional to the times.COR. 4. If the body L, by a radius drawn to the other body T, describesareas, which, compared with the times, are very unequal ; and that otherbody T be either at rest, or moves uniformly forward in a right line : theaction of the centripetal force tending to that other body T is either noneat all, or it is mixed and compounded with very powerful actions of otherforces : and the whole force compounded of them all, if they are many, isdirected to another (immovable or moveaJble) centre. The same thing obtains, when the other body is moved by any motion whatsoever ; providedthat centripetal force is taken, wrhich remains after subducting that wholeforce acting upon that other body T.SCHOLIUM.Because the equable description of areas indicates that a centre is respected by that force with which the body is most affected, and by which itis drawn back from its rectilinear motion, and retained in its orbit ; whymay we not be allowed, in the following discourse, to use the equable description of areas as an indication of a centre, about which all circularmotion is performed in free spaces ?PROPOSITION IV. THEOREM IV.The centripetal forces of bodies, which by equable motions describe different circles, tend to the centres of the same circles ; and are one to tJieother as the squares of t/ie arcs described in equal times applied to theradii of the circles.These forces tend to the centres of the circles (by Prop. II., and Cor. 2,Prop. L), and are one to another as the versed sines of the least arcs described in equal times (by Cor. 4, Prop. I.) ; that is, as the squares of thesame arcs applied to the diameters of the circles (by Lem. VII.) ; and therefore since those arcs are as arcs described in any equal times, and the diameers ace as the radii, the forces will be as the squares of any arcs descrbed in the same time applied to the radii of the circles. Q.E.D.^OR. 1. Therefore, since those arcs are as the velocities of the bodies.I OS THE MATHEMATICAL PRINCIPLES [BOOK .the centripetal forces are in a ratio compounded of the duplicate ra jio ofthe velocities directly, and of the simple ratio of the radii inversely.COR. 2. And since the periodic times are in a ratio compounded of theratio of the radii directly, and the ratio of the velocities inversely, the centripetal forces, are in a ratio compounded of the ratio of the radii directly,and the duplicate ratio of the periodic times inversely.COR, 3. Whence if the periodic times are equal, and the velocitiestherefore as the radii, the centripetal forces will be also as the radii; andtke contrary.COR. 4. If the periodic times and the velocities are both in the subduplicateratio of the radii, the centripetal forces will be equal among themselves ; and the contrary.COR. 5. If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii; and thecontrary.