自然哲学的数学原理-17

SPy13, are severally to the sev eral areas CSD, CBED,SDEB, in the given ratio of the heights CP, CD, andthe area SP/B is proportional to the time in whichthe body P will move through the arc P/B. the areaSDEB will be also proportional to that time. Letthe latus rectum of the hyperbola RPB be diminishedin infitiitum, the latus transversum remaining thesame; and the arc PB will come to coincide with theright line CB, and the focus S, with the vertex B, Aandthe right line SD with the right line BD. And therefore the areaBDEB will be proportional to the time in which the body C, by its perpendicular descent, describes the line CB. Q.E.I.CASE 3. And by the like argument, if the figureRPB is a parabola, and to the same principal vertex B another parabola BED is described, thatmay always remain given while the former parabola in whose perimeter the body P moves, byhaving its latus rectum diminished and reducedto nothing, comes to coincide with the line CB,the parabolic segment BDEB will be proportional ifto the time in which that body P or C will descend to the centre S or BQ.K.Tfl.l OF NATURAL PHILOSOPHY.PROPOSITION XXXIII. THEOREM IX.The tilings above found being supposed. I say, thai the, velocity of a Jailing body in any place C is to the velocity of a body, describing acircle about the centre B at the distance BC, in, the subduplicate ratioof AC, the distance of the body from the remoter vertex A of the circleor rectangular hyperbola, to iAB, the principal semi-diameter of theLet AB, the common diameter of both figures RPB,DEB, be bisected in O; anddraw the right line PT thatmay touch the figure RPBin P, and likewise cut thatcommon diameter AB (produced, if need be) in T; andlet SY be perpendicular tothis line, and BQ to this diameter, and suppose the latusrectum of the figure RPB tobe L. Prom Cor. 9, Prop.XVI, it is manifest that thevelocity of a body, movingin the line RPB about thecentre S, in any place P, isto the velocity of a body describing a circle about the same centre, at thedistance SP, in the subduplicate ratio of the rectangle L X SP to SY2Por by the properties of the conic sections ACB is to CP2 as 2AO to L.2CP5 X AOand therefore rrrr; is equal to L. Therefore those, velocities ano--ACBto each other in the subduplicate ratio ofCP3 X AO X SPACB toSY~. Moreover, by the properties of the conic sections, CO is to BO as BO to TV.?and (by composition or division) as CB to BT. Whence (by division cscomposition) BO or + CO will be to BO as CT to BT, that is, ACCP2 X AO X SPACB"will be to AO as CP to BQ; and therefore is equal to~AO X BC * ^ W suPPose GV, tne breadth of the figure RPB, tobe diminished in infinitum, so as the point P may come to coincide withthe point C, and the point S with the point B. and the line SP with theline BC, and the line SY with the line BQ; and the velocity of the bodynow descending perpendicularly in the line CB will be to the velocity of11162 THE MATHEMATICAL PRINCIPLES [BOOK 1a body describing a circle about the centre B, at the distance BC, in thrBQ2 X AC X SPsubduplicate ratio of--r-^-^- to SY2, that is (neglecting the ra- X Jotios of equality of SP to BC, and BQ,2 to SY2), in the subduplicate ratioof AC to AO, or iAB. Q.E.D.COR. 1 . When the points B and S come to coincide, TC will become toTS as AC to AO.COR. 2. A body revolving in any circle at a given distance from thecentre, by its motion converted upwards, will ascend to double its distancefrom the centre.PROPOSITION XXXIV. THEOREM X.If the. figure BED is a parabola, I say, that the velocity of a fallingbody in any place C is equal to the velocity by which a body mayuniformly describe a circle about the centre B at half the interval BCFor (by Cor. 7, Prop. XVI) the velocity of abody describing a parabola RPB about the centre S, in any place P, is equal to the velocity ofa body uniformly describing a circle about the csame centre S at half the interval SP. Let thebreadth CP of the parabola be diminished initifiiiitirni, so as the parabolic arc P/B may cometo coincide with the right line CB, the centre S swith the vertex B, and the interval SP with theinterval BC, and the proposition will be manifest. Q.E.D.PROPOSITION XXXV. THEOREM XLThe same things supposed, I say, that the area of the figure DES, described by the indefinite radius SD, is equal to the area which a bodywith a radius equal to h df the latus rectum of the figure DES, byuniformly revolving about the centre S, may describe in the same tijiw.1 JD/AJSEC. ni: OF NATURAL PHILOSOPHY.For suppose a body C in the smallest moment of time describes in falling the infinitely little line Cc. while another body K, uniformly revolving about the centre S in the circle OK/r, describes the arc KA:. Erect theperpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/.SK. SA*; and draw Del meeting the axis AS in T, and thereon let fall theperpendicular SY.CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisectits transverse diameter AS in O, and SO will be half the latus rectum.And because TC is to TD as Cc to Dd, and TD to TS as CD to SY ;ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1,Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescenceof the points D, d, the ultimate ratios of the lines are taken. WhereforeAC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity ofthe descending body in C IF, to the velocity of a body describing a circleabout the centre S, at the interval SC, in the subduplicate ratio of AC toAO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of abody describing the circle OKA: in the subduplicate ratio of SK to SC(by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is,the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC,that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to ACX KA*, and consequently AC to SK as AC X KA: to SY X IW. andthence SK X KA: equal to SY X Drf, and iSK X KA: equal to SY X DC/,that is, the area KSA* equal to the area SDrf. Therefore in every momentof time two equal particles, KSA" and SDrf, of areas are generated, which,if their magnitude is diminished, and their number increased in iiifinif t-w,obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the wholeareas together generated are always equal. Q..E.D.CASE 2. But if the figure DES is aparabola, we shall find, as above. CD XCc to SY X Df/ as TC to TS, that is,as 2 to 1; and that therefore |CD X Ccis equal to i SY X Vd. But the velocity of the falling body in C is equal tothe velocity writh which a circle may beuniformly described at the interval 4SC(by Prop" XXXIV). And this velocityto the velocity with which a circle maybe described with the radius SK, that is,the little line Cc to the arc KA,is (byCor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in theratio of SK to *CD. Wherefore iSK X KA: is equal to 4CD X Cc, andtherefore equal to SY X T)d ; that is, the area KSA* is equal to the areaSIW, as above. Q.E.D.164 THE MATHEMATICAL PRINCIPLES [BOOK 1.PROPOSITION XXXVI. PROBLEM XXV.To determine the times of the descent of a body falling fromplace A.Upon the diameter AS, the distance of the body from thecentre at the beginning, describe the semi-circle ADS, aslikewise the semi-circle OKH equal thereto, about the centreS. From any place C of the body erect the ordinate CD.Join SD, and make the sector OSK equal to the area ASD.It is evident (by Prop. XXXV) that the body in falling willdescribe the space AC in the same time in which another body,uniformly revolving about the centre S, may describe the arcOK. Q.E.F. Ma givenPROPOSITION XXXVII. PROBLEM XXVI.To define the times of the ascent or descent of a body projected upwardsor downwards from a given place.Suppose the body to go oif from the given place G, in the direction ofthe line GS, with any velocity. In the duplicate ratio of this velocity tothe uniform velocity in a circle, with which the body may revolve about。HDthe centre S at the given interval SG, take GA to AS. If that ratio isthe same as of the number 2 to 1, the point A is infinitely remote ; inwhich case a parabola is to be described with any latus rectum to the vertex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio isless or greater than the ratio of 2 to 1, in the former case a circle, in thelatter a rectangular hyperbola, is to be described on the diameter SA; asappears by Prop. XXXIII. Then about the centre S, with an intervalequal to half the latus rectum, describe the circle H/vK ; and at the placeG of the ascending or descending body, and at any other place C, erect theperpendiculars GI, CD, meeting the conic section or circle in I and D.Then joining SI, SD, let the sectors HSK, HS& be made equal to thesegments SEIS, SEDS. and (by Prop. XXXV) the body G will describeSEC. VII.] OF NATURAL PHILOSOPHY. 165the space GO in the same time in which the body K may describe t*he arcKk. Q.E.F.PROPOSITION XXXVIII. THEOREM XII.Supposing that the centripetal force is proportional to the altitude ordistance ofplaces from the centre, I say, that the times and velocitiesoffalling bodies, and the spaces which they describe, are respectivelyproportional to the arcs, and the right and versed sines of the arcs.Suppose the body to fall from any place A in the A.right line AS ; and about the centre of force S, withthe interval AS, describe the quadrant of a circle AE ;and let CD be the right sine of any arc AD ; and thebody A will in the time AD in falling describe thespace AC, and in the place C will acquire the velocity CD.This is demonstrated the same way from Prop. X, as Prop. XXX11 wasdemonstrated from Prop. XI.COR. 1. Hence the times are equal in which one body falling from theplace A arrives at the centre S, and another body revolving describes thequadrantal arc ADE.COR. 2. Wherefore all the times are equal in which bodies falling fromwhatsoever places arrive at the centre. For all the periodic times of revolving bodies are equal (by Cor. 3; Prop. IV).PROPOSITION XXXIX. PROBLEM XXVIT.Supposing a centripetal force of any kind, and granting the quadratnresof curvilinear figures ; it is required to find the velocity of a bod)/,ascending or descending in a right line, in the several places throughwhich it passes ; as also the time in which it will arrive at any place :and vice versa.Suppose the body E to fall from any placeA in the right line ADEC ; and from its placeE imagine a perpendicular EG always erectedproportional to the centripetal force in thatplace tending to the centre C ; and let BFGbe a curve line, the locus of the point G. And Din the beginning of the motion suppose EG tocoincide with the perpendicular AB ; and thevelocity of the body in any place E will be asa right line whose square is equal to the curvilinear area ABGE. Q.E.I.In EG take EM reciprocally proportional toE366 THE MATHEMATICAL PRINCIPLES [BOOK 1a right line whose square is equal to the area ABGE, and let VLM be acurve line wherein the point M is always placed, and to which the rightline AB produced is an asymptote; and the time in which the body infalling- describes the line AE, will be as the curvilinear area ABTVME.Q.E.I.For in the right line AE let there be taken the very small line DE ofa given length, and let DLF be the place of the line EMG, when thebody was in D ; and if the centripetal force be such, that a right line,whose square is equal to the area ABGE; is as the velocity of the descending body, the area itself will be as the square of that velocity ; that is, iffor the velocities in D and E we write V and V + I, the area ABFD willbe as VY, and the area ABGE as YY + 2VI -f II; and by division, thearea DFGE as 2VI -f LI, and therefore ^ will be as--^rthat is. if we take the first ratios of those quantities when just nascent, the2YIlength DF is as the quantity -|yrran(i therefore also as half that quantity1 X YBut the time in which the body in falling describes the veryline DE, is as that line directly and the velocity Y inversely ; andthe force will be as the increment I of the velocity directly and the timeinversely ; and therefore if we take the first ratios when those quantitiesI X Vare just nascent, as-jy==r-.that is, as the length DF. Therefore a forceproportional to DF or EG will cause the body to descend with a velocitythat is as the right line whose square is equal to the area ABGE. Q.E.D.Moreover, since the time in which a very small line DE of a givenlength may be described is as the velocity inversely, and therefore alsoinversely as a right line whose square is equal to the area ABFD ; andsince the line DL. and by consequence the nascent area DLME, will be as(he same right line inversely, the time will be as the area DLME, andthe sum of all the times will be as the sum of all the areas : that is (byCor. Lern. IV), the whole time in which the line AE is described will beas the whole area ATYME. Q.E.D.COR. 1. Let P be the place from whence a body ought to fall, so asthat, when urged by any known uniform centripetal force (such asgravity is vulgarly supposed to be), it may acquire in the place D avelocity equal to the velocity which another body, falling by any forcewhatever, hath acquired in that place D. In the perpendicular DF letthere be taken DR., which may be o DF as that uniform force tothe other force in the place D. Complete the rectangle PDRQ,, and cutiff the area. ABFD equal to that rectangle. Then A will be the placeSEC. VII. I OF NATURAL PHILOSOPHY. 10;from whence the other body fell. For completing the rectangle DRSE, since the areaABFD is to the area DFGE as VV to 2VI,and therefore as 4V to I, that is, as half thewhole velocity to the increment of the velocityof the body falling by the unequable force; andin like manner the area PQRD to the areaDRSE as half the whole velocity to the increment of the velocity of the body falling by theuniform force ; and since those increments (byreason of the equality of the nascent times)are as the generating forces, that is, as the ordinatesDF, DR, and consequently as the nascent areas DFGE, DRSE :therefore, ex aq-uo, the whole areas ABFD, PQRD will be to one anotheras the halves of the whole velocities ; and therefore, because the velocitiesare equal, they become equal also.COR. 2. Whence if any body be projected either upwards or downwardswith a given velocity from any place D, and there be given the law ofcentripetal force acting on it, its velocity will be found in any other place,as e, by erecting the ordinate eg, and taking that velocity to the velocityin the place D as a right line whose square is equal to the rectanglePQRD, either increased by the curvilinear area DFge, if the place e isbelow the place D, or diminished by the same area DFg-e, if it be higher,is to the right line whose square is equal to the rectangle PQRD alone.COR. 3. The time is also known by erecting the ordinate em reciprocally proportional to the square root of PQRD -f- or T)Fge, and takingthe time in which the body has described the line De to the time in whichanother body has fallen with an uniform force from P, and in falling arrived at D in the proportion of the curvilinear area DLme to the rectangle 2PD X DL. For the time in which a body falling with an uniformforce hath described the line PD, is to the time in which the same bodyhas described the line PE in the subduplicate ratio of PD to PE ; that is(the very small line DE being just nascent), in the ratio of PD to PD -f^DE; or 2PD to 2PD -f- DE, and, by division, to the time in which thebody hath described the small line DE, as 2PD to DE, and therefore asthe rectangle 2PD X DL to the area DLME ; and the time in whichboth the bodies described the very small line DE is to the time in whichthe body moving unequably hath described the line De as the area DLMEto the area DLme ; and, ex aquo, the first mentioned of these times is tothe last as the rectangle 2PD X DL to the area DLrae.163 THE MATHEMATICAL PRINCIPLES [BoOK ISECTION VIII.Of the invention of orbits wherein bodies will revolve, being acted uponby any sort of centripetal force.PROPOSITION XL. THEOREM XIII.// a body, acted upon by any centripetal force, is any how moved, andanother body ascends or descends in a right line, and their velocitiesbe equal in amj one case of equal altitudes, t/ieir velocities will be alsoequal at all equal altitudes.Let a body descend from A through D and E, to the centre(j : and let another body move from V in the curve line VIK&.From the centre C, with any distances, describe the concentriccircles DI, EK, meeting the right line AC in I) and E; andthe curve VIK in I and K. Draw 1C meeting KE in N, andon IK let fall the perpendicular NT and let the interval DEor IN between the circumferences of the circles be very small ;K/and imagine the bodies in D and I to have equal velocities.Then because the distances CD and CI are equal, the centripetal forces in D and I will be also equal. Let those forces be k)expressed by the equal lineoke DE and IN ; and let the forceIN (by Cor. 2 of the Laws of Motion) be resolved into twoothers, NT and IT. rl hen the force NT acting in the directionline NT perpendicular to the path ITK of the body will not at all affector change the velocity of the body in that path, but only draw it asidefrom a rectilinear course, and make it deflect perpetually from the tangentof the orbit, and proceed in the curvilinear path ITK/j. That wholeforce, therefore, will be spent in producing this effect: but the other forceIT, acting in the direction of the course of the body, will be all employedin accelerating it, and in the least given time will produce an accelerationproportional to itself. Therefore the accelerations of the bodies in D andI, produced in equal times, are as the lines DE, IT (if we take the firstratios of the nascent lines DE, IN, IK, IT, NT) ; and in unequal times asthose lines and the times conjunctly. But the times in which DE and IKare described, are, by reason of the equal velocities (in D and I) as thespaces described DE and IK, and therefore the accelerations in the courseof the bodies through the lines DE and IK are as DE and IT, and DEand IK conjunctly ; that is, as the square of DE to the rectangle IT intoIK. But the rectangle IT X IK is equal to the square of IN, that is,equal to the square of DE ; and therefore the accelerations generated inthe passage of the bodies from D and I to E and K are equal. Thereforethe velocities of the holies in E and K are also equal, and by the samereasoning they will always be found equal in any subsequent equal distances. Q..E.D.SEC. VI11.J OF NATURAL PHILOSOPHY. 169By the same reasoning, bodies of equal velocities and equal distancesfrom the centre will be equally retarded in their ascent to equal distances.Q.E.D.COR. 1. Therefore if a body either oscillates by hanging to a string, orby any polished and perfectly smooth impediment is forced to move in acurve line ; and another body ascends or descends in a right line, and theirvelocities be equal at any one equal altitude, their velocities will be alsoequal at all other equal altitudes. For by the string of the pendulousbody, or by the impediment of a vessel perfectly smooth, the same thingwill be effected as by the transverse force NT. The body is neitheraccelerated nor retarded by it, but only is obliged to leave its rectilinearcourse.COR. 2. Suppose the quantity P to be the greatest distance from thecentre to which a body can ascend, whether it be oscillating, or revolvingin a trajectory, and so the same projected upwards from any point of atrajectory with the velocity it has in that point. Let the quantity A bethe distance of the body from the centre in any other point of the orbit ; andlet the centripetal force be always as the power An, of the quantity A, theindex of which power n 1 is any number n diminished by unity. Thenthe velocity in every altitude A will be as v/ P11A", and therefore willbe given. For by Prop. XXXIX, the velocity of a body ascending anddescending in a right line is in that very ratio.PROPOSITION XLI. PROBLEM XXVTII.Supposing a centripetal force of any kind, and granting the quadratures of curvilinear figures, it is required to find as well the trajectories in which bodies will move, as the times of their motions in thetrajectories found.Let any centripetal force tend tothe centre C, and let it be requiredto find the trajectory VIKAr. Let R,