284 THE MATHEMATICAL PRINCIPLES [BOOK II2APQ, + 2BA X PUII of this Book) the moment KL of AK will be equal to2BPQor Z -, and the moment KLON of the area ANK will be equal to2BPQ X LO BPQ, X BD 3~~Z~~ >r 2Z X OK x~AB"CASE 1. Now if the body ascends, and the gravity be as AB2 + BD 9BET being a circle, the line AC, which is proportional to the gravityAW2 i RF)2will be - ~--, and DP 2 or AP 2 + 2BAP + AB 2 + BD2 will beAK XZ + AC X Z or CK X Z : and therefore the area DTV will be tothe area DPQ, as DT2 or I)B 2 to CK X Z.CASE 2. If the body ascends, and the gravity be as AB 2 BD 2, theA r>2 _ HI) 2line AC will be"--^---, and DT2 will be to DP 2 as DF 2 or DB 2Zto BP2 BD 2 or AP 2 + 2BAP + AB 2 BD 2, that is, to AK X Z +HAC X Z or CK X Z. And therefore the area DTV will be to the areaDPQ as DB2 to CK X Z.CASE 3. And by the same reasoning, if the body descends, and thereforethe gravity is as BD 2 AB 2, and the line AC becomes equal toor) 2 AB 25T r ; the area DTV will be to the area DPQ as DB2 to CK Z XZ : as above.Since, therefore, these areas are always in this ratio, if for the areaSEC. 111^ OF NATURAL PHILOSOPHY. 285DTY, by which the moment of the time, always equal to itself, is expressed, there be put any determinate rectangle, as BD X m, the area DPQ,,that is, |BD X PQ, will be to BD X mas CK X Z to BD 2. And thencePQ X BD 3 becomes equal to 2BD XmX CK X Z, and the moment KLONBP X BD X tnof the area A6NK, found before, becomes-.-^--. From the areaDET subduct its moment DTV or BD X m, and there will remain----Pp . Therefore the difference of the moments, that is, theAP X BD X mnt of the difference of tne areas, is equal to --7-5---; andtherefore (because of the given quantity---T-~ ) as the velocity AP ;that is, as the moment of the space which the body describes in its ascentor descent. And therefore the difference of the areas, and that space, increasing or decreasing by proportional moments, and beginning together orvanishing together, are proportional. Q.E.D.COR. If the length, which arises by applying the area DET to the lineBD, be called M ; and another length V be taken in that ratio to the lengthM, which the line DA has to the line DE; the space which a body, in aresisting medium, describes in its whole ascent or descent, will be to thespace which a body, in a non-resisting medium, falling from rest, can describe in the same time, as the difference of the aforesaid areas toBD X V2--TO"""" jan(^ therefore is given from the time given. For the space in aA.LJnon-resisting medium is in a duplicate ratio of the time, or as V2; and.BD X V 2because BD and AB are given, as ----TTT- . This area is equal to theDA 2 X BD x M2area --fvGr*~~~T~R "~ anc* ^ne moment Of M is m ; and therefore theDA2 X BD X 2M X mmoment of this area is ---=----^5 -. But this moment is to"" X .A Dthe moment of the difference of the aforesaid areas DET and A6NK, viz., toAP X BD X m DA2 X BD X M x DA2- -- . ^^ , as - -r- - to |BD X AP, or as into DETto DAP ; and, therefore, when the areas DKT and DAP are least, in theBD X V 2ratio of equality. Therefore the area r^-- and the difference of theareas DET and A&NK, when all these areas are least, have equal moments ;and { re therefore equal. Therefore since the velocities, and therefore alsothe s] aces in both mediums described together, in the beginning of the descent or the end of the ascent, approach to equality, and therefore are then286 THE MATHEMATICAL PRINCIPLES [BOOK IIBD X V2one to another as the area Ar-D^ , and the difference of the areas DETand A6NK ; and moreover since the space, in a non-resisting medium, isBD X V 2perpetually as Tu~~> an(^ tne sP ace>in a resisting medium, is perpetually as the difference of the areas DET and A&NK ;it necessarily follows,that the spaces, in both mediums, described in any equal times, are one toBD X V 2another as that area 7-5 an(^ ^he difference of the areas DET andA6NK. Q.E.D.SCHOLIUM.The resistance of spherical bodies in fluids arises partly from the tenacity, partly from the attrition, and partly from the density of the medium.And that part of the resistance which arises from the density of the fluidis, as I said, in a duplicate ratio of the velocity ; the other part, whicharises from the tenacity of the fluid, is uniform, or as the moment of thetime ; and, therefore, we might now proceed to the motion of bodies, whicliare resisted partly by an uniform force, or in the ratio of the moments ofthe time, and partly in the duplicate ratio of the velocity. But it is sufficient to have cleared the way to this speculation in Prop. VIII and IXforegoing, and their Corollaries. For in those Propositions, instead of theuniform resistance made to an ascending body arising from its gravity,one may substitute the uniform resistance which arises from the tenacityof the medium, when the body moves by its vis insita alone ; and when thebody ascends in a right line, add this uniform resistance to the force ofgravity, and subduct it when the body descends in a right line. Onemight also go on to the motion of bodies which are resisted in part uniformly, in part in the ratio of the velocity, and in part in the duplicateratio of the same velocity. And I have opened a way to this in Prop.XIII and XIV foregoing, in which the uniform resistance arising from thetenacity of the medium may be substituted for the force of gravity, or becompounded with it as before. But I hasten to other things.SKC. -IV.] OF NATUEAL PHILOSOPHY. 2S?SECTION IV.Of the circular motion of bodies in resisting mediums.LEMMA III.Let PQR be a spiral rutting all the radii SP, SO, SR, <J*c.,in equalangles. Draw tfie right line PT touching the spiral in any point P,and cutting the radius SQ in T ;cfo er?0 PO, QO perpendicular tothe spiral, and meeting- in, O, andjoin SO. .1 say, that if Hie pointsP a/*(/ Q approach and coincide, the angle PSO vri/Z become a rightangle, and the ultimate ratio of the rectangle TQ, X 2PS to P^3//>i///>e /ie ya/io o/" equality.For from the right angles OPQ, OQR, subduct the equal angles SPQ, SQR, and therewill remain the equal angles OPS, OQS.Therefore a circle which passes through thepoints OSP will pass also through the pointQ. Let the points P and Q, coincide, andthis circle will touch the spiral in the placeof coincidence PQ, and will therefore cut theright line OP perpendicularly. Therefore OP will become a diameter ofthis circle, and the angle OSP, being in a semi-circle, becomes a rightone. Q.E.1).Draw Q,D, SE perpendicular to OP, and the ultimate ratios of the lineswill be as follows : TO to PD as TS or PS to PE, or 2PO to 2PS andPD to PO as PO to 2PO ; and, ex cequo pertorbatt, to TO to PO as POto 2PS. Whence PO2 becomes equal to TO X 2PS. O.E.D.PROPOSITION XV. THEOREM XII.Tf the density of a medium in each place thereof be reciproniJly as thedistance of the places from an immovable centre, aud the centripetalforce be in the duplicate ratio of the density ; I say, that a body mnyrevolve in a spiral which cuts all the radii drawn from that centrein a given angle.Suppose every thing to be as in the foregoing Lemma, and produce SO to V so that SVmay be equal to SP. In any time let a body,in a resisting medium, describe the least arcPO, and in double the time the least arc PR :and the decrements of those arcs arising fromthe resistance, or their differences from thearcs which would be described in a non-resisting medium in the same times, will be to eachother as the squares of the times in which theyare generated ; therefore the decrement of the288 THE MATHEMATICAL PRINCIPLES [_BoOK 11arc PQ is the fourth part of the decrement of the arc PR. Whence alsoif the area QSr be taken equal to the area PSQ, the decrement of the arcPQ will be equal to half the lineola Rr ; and therefore the force of resistance and the centripetal force are to each other as the lineola jRrandTQwhich they generate in the same time. Because the centripetal force withwhich the body is urged in P is reciprocally as SP 2, and (by Lem. X,Book I) the lineola TQ, which is generated by that force, is in a ratiocompounded of the ratio of this force and the duplicate ratio of the timein which the arc PQ, is described (for in this case I neglect the resistance,as being infinitely less than the centripetal force), it follows that TQ XSP 2, that is (by the last Lemma), fPQ2 X SP, will be in a duplicate ratio of the time, and therefore the time is as PQ, X v/SP ; and the velocity of the body, with which the arc PQ is described in that time, asPQ 1-p or , that is, in the subduplicate ratio of SP reciprocally.And, by a like reasoning, the velocity with which the arc QR is described,is in the subduplicate ratio of SQ reciprocally. Now those arcs PQ andQR are as the describing velocities to each other; that is, in the subduplicate ratio of SQ to SP, or as SQ to x/SP X SQ; and, because of theequal angles SPQ, SQ? , and the equal areas PSQ, QSr, the arc PQ is tothe arc Qr as SQ to SP. Take the differences of the proportional consequents, and the arc PQ will be to the arc Rr as SQ to SP VSP X ~SQ~,or ^VQ. For the points P and Q coinciding, the ultimate ratio of SPX SQ to |VQ is the ratio of equality. Because the decrement ofthe arc PQ arising from the resistance, or its double Rr, is as the resistanceand the square of the time conjunctly, the resistance will be &Sp-^r*1 op. XBut PQ was to Rr as SQ to |VQ, and thence SSaTXToD becomes asJr vst X oJr-VQ -OSpWxsvxSQ, or nsETp^TsPForthe poillts p and a coincidin&SP and SQ coincide also, and the angle PVQ becomes a right one; and,because of the similar triangles PVQ, PSO, PQ. becomes to -VQ as OPOSto | OS. Therefore : y-- is as the resistance, that is, in the ratio of。J i X olthe density of the medium in P and the duplicate ratio of the velocityconjunc-tly. Subduct the duplicate ratio of the velocity, namely, the ratio1 OS^5, and there will remain the density of the medium in P. as 7^5-= OA Ur X feiLet the spiral be given, and; because of the given ratio of OS to OP, thedensity of the medium in P will be as~-p.Therefore in a medium whoseSEC. IV,] OF NATURAL PHILOSOPHY. 2S9density is reciprocally as SP the distance from the centre, a body will revolve in this spiral. Q.E.D.COR. 1. The velocity in any place P, is always the same wherewith abody in a non-resisting medium with the same centripetal force would revolve in a circle, at the same distance SP from the centre.COR. 2. The density of the medium, if the distance SP be given, is asOS OSTTp,but if that distance is not given, as ^ ^5. And thence a spiralmay be fitted to any density of the medium.COR. 3. The force of the resistance in any place P is to the centripetalforce in the same place as |OS to OP. For those forces are to each other^VQ x PQ iPQ2as iRr and TQ, or as 1 ^-^~- and ^-, that is, as iVQ and PQ,ol%, olor |OS and OP. The spiral therefore being given, there is given the proportion of the resistance to the centripetal force ; and, vice versa, from thatproportion given the spiral is given.COR. 4. Therefore the body cannot revolve in this spiral, except wherethe force of resistance is less than half the centripetal force. Let the resistance be made equal to half the centripetal force, and the spiral will coincide with the right line PS, and in that right line the body will descendto the centre with a velocity that is to the velocity, with which it wasproved before, in the case of the parabola (Theor. X, Book I), the descentwould be made in a non-resisting medium, in the subduplicate ratio ofunity to the number two. And the times of the descent will be here reciprocally as the velocities, and therefore given.COR. 5. And because at equal distancesfrom the centre the velocity is the same in thespiral PQ,R as it is in the right line SP, andthe length of the spiral is to the length of theright line PS in a given ratio, namely, in theratio of OP to OS ; the time of the descent inthe spiral will be to the time of the descent inthe right line SP in the same given ratio, andtherefore given.COR. 6. If from the centre S, with any twogiven intervals, two circles are described ; andthese circles remaining, the angle which the spiral makes with the radius"PS be any how changed ; the number of revolutions which the body cancomplete in the space between the circumferences of those circles, goingPSround in the spiral from one circumference to another, will be as 7^, or asOk5ths tangent of the angle which the spiral makes with the radius PS ; and19290 THE MATHEMATICAL PRINCIPLES [BOOK IIOPthe time of the same revolutions will be as -^, that is, as the secant of the Uosame angle, or reciprocally as the density of the medium.COR. 7. If a body, in a medium whose density is reciprocally as the distances of places from the centre, revolves in any curve AEB about thatcentre, and cuts the first radius AS in the sameangle in B as it did before in A, and that with avelocity that shall be to its first velocity in A reciprocally in a subduplicate ratio of the distancesfrom the centre (that is, as AS to a mean proportional between AS and BS) that body will continue to describe innumerable similar revolution?BFC, CGD, &c., and by its intersections willdistinguish the radius AS into parts AS, BS, CS, DS, &c., that are continually proportional. But the times of the revolutions will be as theperimeters of the orbits AEB, BFC, CGD, &c., directly, and the velocities3 3at the beginnings A, B, C of those orbits inversely ;that is as AS % BS %CS"2". And the whole time in which the body will arrive at the centre,will be to the time of the first revolution as the sum of all the continued 142 proportionals AS 2, BS 2, CS 2, going on ad itifinitum, to the first term* i 3AS 2;that is, as the first term AS 2 to the difference of the two first AS 2BS% or as f AS to AB very nearly. Whence the whole time may beeasily found.COR. 8. From hence also may be deduced, near enough, the motions ofbodies in mediums whose density is either uniform, or observes any otherassigned law. From the centre S, with intervals SA, SB, SC, &c., continually proportional, describe as many circles; and suppose the time ofthe revolutions between the perimeters of any two of those circles, in themedium whereof we treated, to be to the time of the revolutions betweenthe same in the medium proposed as the mean density of the proposed medium between those circles to the mean density of the medium whereof wetreated, between the same circles, nearly : and that the secant of the anglein which the spiral above determined, in the medium whereof we treated,cuts the radius AS, is in the same ratio to the secant of the angle in whichthe new spiral, in the proposed medium, cuts the same radius : and alsothat the number of all the revolutions between the same two circles is nearlyas the tangents of those angles. If this be done every where between everytwo circles, the motion will be continued through all the circles. And bythis means one may without difficulty conceive at what rate and in whattime bodies ought to revolve in any regular medium.SEC. IY.1 OF NATURAL PHILOSOPHY. 291COR. 9. And although these motions becoming eccentrical should beperformed in spirals approaching to an oval figure, yet, conceiving theseveral revolutions of those spirals to be at the same distances from eachother, and to approach to the centre by the same degrees as the spiral abovedescribed, we may also understand how the motions of bodies may be performed in spirals of that kind.PROPOSITION XVI. THEOREM XIII.If the density of the medium in each of the places be reciprocally as thedistance of the>, places from the immoveable centre, and the centripetalforce be reciprocally as any power of the same distance, I say, that thebody may revolve in a spiral intersecting all the radii drawn fromthat centre in a given, angle.This is demonstrated in the same manner asthe foregoing Proposition. For if the centripetal force in P be reciprocally as any powerSPn + 1 of the distance SP whose index is n+ 1;it will be collected, as above, that thetime in which the body describes any arc PQ,iwill be as PQ, X PS 2U; and the resistance ini!! x _n; raS"~ X SPPQ, X SP"XSQ,, , 1 in X OS . 1 X OS .therefore asQp"^~gpirqTTtliat 1S> (because -~~Qp~~1S a lvenquantity), reciprocally as SPn + !. And therefore, since the velocity is reciprocally as SP3", the density in P will be reciprocally as SP.COR. 1. The resistance is to the centripetal force as 1 ^//. X OSto OP.COR. 2. If the centripetal force be reciprocally as SP 3. 1 w will be===; and therefore the resistance and density of the medium will benothing, as in Prop. IX, Book I.COR. 3. If the centripetal force be reciprocally as any power of the radius SP, whose index is greater than the number 3, the affirmative resistance will be changed into a negative.SCHOLIUM.This Proposition and the former, which relate to mediums of unequaldensity, are to be understood of the motion of bodies that are so small, thatthe greater density of the medium on one side of the body above that onthe other is not to be considered. I suppose also the resistance, caterisparibus, to be proportional to its density. Whence, in mediums whose292 THE MATHEMATICAL PRINCIPLES | BoOK IIforce of resistance is not as the density, the density must be so much augmented or diminished, that either the excess of the resistance may be takenaway, or the defect supplied.PROPOSITION XVII. PROBLEM IVTofind the centripetal force and the resisting force of the medium, bywhich a body, the law of the velocity being given, shall revolve in agiven spiral.Let that spiral be PQR. From the velocity,with which the body goes over the very small arcPQ,, the time will be given : and from the altitudeTQ,, which is as the centripetal force, and thesquare of the time, that force will be given. Thenfrom the difference RSr of the areas PSQ, andQ,SR described in equal particles of time, the retardation of the body will be given ; and fromthe retardation will be found the resisting forceand density of the medium.PROPOSITION XVIII. PROBLEM V.The law of centripptal force being given, to find the density of the me