自然哲学的数学原理-11

COR. 6. If the periodic times are in the sesquiplicate ratio of the radii,and therefore the velocities reciprocally in the subduplicate ratio of theradii, the centripetal forces will be in the duplicate ratio of the radii inversely : and the contrary.COR. 7. And universally, if the periodic time is as any power Rn of theradius R, and therefore the velocity reciprocally as the power Rn ] ofthe radius, the centripetal force will be reciprocally as the power R2n 1 ofthe radius; and the contrary.COR. 8. The same things all hold concerning the times, the velocities,and forces by which bodies describe the similar parts of any similar figuresthat have their centres in a similar position with those figures ; as appearsby applying the demonstration of the preceding cases to those. And theapplication is easy, by only substituting the equable description of areas inthe place of equable motion, and using the distances of the bodies from thecentres instead of the radii.COR. 9. From the same demonstration it likewise follows, that the arcwhich a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time, is a mean proportional between thediameter of the circle, and the space which the same body falling by thesame given force would descend through in the same given time.SCHOLIUM.The case of the 6th Corollary obtains in the celestial bodies (as SirChristopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ;and therefore in what follows, I intend to treat more at large of thosethings which relate to centripetal force decreasing in a duplicate ratioof the distances from the centres.Moreover, by means of the preceding Proposition and its Corollaries, weSEC. II.] OF NATURAL PHILOSOPHY. 109may discover the proportion of a centripetal force to any other knownforce, such as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetalforce of that body. But from the descent of heavy bodies, the time of oneentire revolution, as well as the arc described in any given time, is given(by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in hisexcellent book De Horologio Oscillatorio, has compared the force ofgravity with the centrifugal forces of revolving bodies.The preceding Proposition may be likewise demonstrated after thismanner. In any circle suppose a polygon to be inscribed of any numberof sides. And if a body, moved with a given velocity along the sides of thepolygon, is reflected from the circle at the several angular points, the force,with which at every reflection it strikes the circle, will be as its velocity :and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly ; that is (if the species ofthe polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of thecircle ; that is, as the square of that length applied to the radius ; andtherefore the polygon, by having its sides diminished in inftnitum, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the bodyimpels the circle; and to which the contrary force, wherewith the circlecontinually repels the body towards the centre, is equal.PROPOSITION V. PROBLEM I.There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some commoncentre : to find that centre.Let the three right lines PT, TQV, VRtouch the figure described in as many points,P, Q, R, and meet in T and V. On the tangents erect the perpendiculars PA, QB, RC,reciprocally proportional to the velocities of thebody in the points P, Q, R, from which theperpendiculars were raised ; that is, so that PAmay be to QB as the velocity in Q to the velocity in P, and QB to RCas the velocity in R to the velocity in Q. Through the ends A, B, C, ofthe perpendiculars draw AD, DBE, EC, at right angles, meeting in D andE : and the right lines TD, VE produced, will meet in S, the centre required.For the perpendiculars let fall from the centre S on the tangents PT.QT. are reciprocally as the velocities of the bodies in the points P and Q110 THE MATHEMATICAL PRINCIPLES [BOOK 1(by Cor. 1, Prop. I.), and therefore, by construction, as the perpendicularsAP, BQ, directly ;that is, as the perpendiculars let fall from the point Don the tangents. Whence it is easy to infer that the points S, D, T, arein one right line. And by the like argument the points S, E, V are alsoin one right line ; and therefore the centre S is in the point where theright lines TD; YE meet. Q.E.D.PROPOSITION VL THEOREM V.In a space void of resistance, if a body revolves in any orbit about an immovable centre, and in the least time describes any arc just then, nascent ; and the versed sine of that arc is supposed to be drawn bisecting the chord, and produced passing through the centre offorce: thecentripetal force in the middle of the arc will be as the versed sine directly and the square of the time inversely.For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ;and augmenting the time in any ratio, because the arc will be augmentedin the same ratio, the versed sine will be augmented in the duplicate ofthat ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and thesquare of the time. Subduct on both sides the duplicate ratio of thetime, and the force will be as the versed sine directly, arid the square ofthe time inversely. Q.E.D.And the same thing may also be easily demonstrated by Corol. 4 ?T,em. X.COR. 1. If a body P revolving about thecentre S describes a curve line APQ,, which aright line ZPR touches in any point P ; andfrom any other point Q, of the curve, QJl isdrawn parallel to the distance SP, meetingthe tangent in R ; and QT is drawn perpen-(licular to the distance SP ; the centripetal force will be reciprocally as thesp2 x Q/r2solid- :, if the solid be taken of that magnitude which it ultimatelyacquires when the points P and Q, coincide. For Q,R is equal tothe versed sine of double the arc QP, whose middle is P : and double thetriangle SQP, or SP X Q,T is proportional to the time in which thatdouble arc is described ; and therefore may be used for the exponent ofthe time.COR. 2. By a like reasoning, the centripetal force is reciprocally as theSY2 X QJP2solid-7^5-;if SY is a perpendicular from the centre of force onPR the tangent of the orbit. For the rectangles SY X QP and SP X Q,Tare equal.SEC. II.] OF NATURAL PHILOSOPHY. IllCOR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncentrically,that is, contains with a circle the least angle of contact or section, having the same curvature rnd the same radius of curvature at thepoint P : and if PV be a chord of this circle, drawn from the body throughthe centre of force ;the centripetal force will be reciprocally as the solidQP2SY2 X PV. For PV is -.COR. 4. The same things being supposed, the centripetal force is as thesquare of the velocity directly, and that chord inversely. For the velocityis reciprocally as the perpendicular SY, by Cor. 1. Prop. I.COR. 5. Hence if any curvilinear figure APQ, is given, and therein apoint S is also given, to which a centripetal force is perpetually directed.that law of centripetal force may be found, by which the body P will bcjcontinually drawn back from a rectilinear course, and. being detained inthe perimeter of that figure, will describe the same by a perpetual revolu-SP2 x QT2tion. That is, we are to find, by computation, either the solid -----or the solid SY2 X PV, reciprocally proportional to this force. Example:of this we shall give in the following Problems.PROPOSITION VII. PROBLEM II.Tf a body revolves in the circumference of a circle; it is proposed to finiithe law of centripetal force directed to any given, point.Let VQPA be the circumference of thecircle ; S the given point to which as toa centre the force tends : P the body moving in the circumference ; Q the nextplace into which it is to move; and PRZthe tangent of the circle at the precedingplace. Through the point S draw thevchord PV, and the diameter VA of thecircle : join AP, and draw Q,T perpendicular to SP, which produced, may meetthe tangent PR in Z ; and lastly, throughthe point Q, draw LR parallel to SP, meeting the circle" in L, and thetangent PZ in R. And, because of the similar triangles ZQR, ZTP.VPA, we shall have RP2, that is. QRL to QT2 as AV2 to PV2. AndQRlj x PV2 SI3 -therefore - TS--is equal to QT2. Multiply those equals by -.and the points P and Q, coinciding, for RL write PV ; then we shall haveSP- X PV5 SP2 x QT2And therefore flr Cor 1 and 5. Prop. VI.)112 THE MATHEMATICAL PRINCIPLES [BOOK I,SP2 X PV3the centripetal force is reciprocally as -ry^~ J that is (because AV2ia given), reciprocally as the square of the distance or altitude SP, and the3ube of the chord PV conjunctly. Q.E.LThe same otherwise.On the tangent PR produced let fall the perpendicular SY ; and (because of the similar triangles SYP, VPA), we shall have AV to PV as SPSP X PV SP2>< PV3to SY, and therefore--^~- = SY, and - ^- = SY2 A V A X PV. VAnd therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip-SP2 X PV3rocally as -~~ry~~~ I*na* *s (because AV is given), reciprocally as SP"X PV3. Q.E.I.Con. 1. Hence if the given point S, to which the centripetal force always tends, is placed in the circumference of the circle, as at V, the centripetal force will be reciprocally as the quadrato-cube (or fifth power) ofthe altitude SP.COR. 2. The force by which the body P in thecircle APTV revolves about the centre of force Sis to the force by which the same body P may revolve in the same circle, and in the same periodictime, about any other centre of force R, as RP2 XSP to the cube of the right line SG, which, fromthe first centre of force S is drawn parallel to thedistance PR of the body from the second centre of force R, meeting thetangent PG of the orbit in G. For by the construction of this Proposition,the former force is to the latter as RP2 X PT3 to SP2 X PV3; that is, asSP3 X PV3SP X RP2 to --p ; or (because of the similar triangles PSG, TPV)to SGS.COR. 3. The force by which the body P in any orbit revolves about thecentre of force S, is to the force by which the same body may revolve inthe same orbit, and the same periodic time, about any other centre of forceR. as the solid SP X RP2, contained under the distance of the body fromthe first centre of force S, and the square of its distance from the second centre of force R, to the cube of the right line SG, drawn from thefirst centre of the force S, parallel to the distance RP of the body fromfch*3 second centre of force R, meeting the tangent PG of the orbit in G.For the force in this orbit at any point P is the same as in a circle of thesame curvature.SJSG. IL] OF NATURAL PHILOSOPHY. 113PROPOSITION VIII. PROBLEM III.If a body mi ues in the semi-circuwferencePQA: it is proposed to findthe law of the centripetal force tending to a point S, so remote, that allthe lines PS. RS drawn thereto, may be taken for parallels.From C, the centre of the semi-circle, letthe semi-diameter CA he drawn, cutting theparallels at right angles in M and N, andjoin CP. Because of the similar trianglesCPM, PZT, and RZQ, we shall have CP2to PM2 as PR2 to QT2; and, from the nature of the circle, PR2is equal to the rectangle QR X RN + QN, or, the points P, Q coinciding, to the rectangleQR x 2PM. Therefore CP2is to PM2 as QR X 2PM to QT2; andQT2 2PM3 QT2 X SP2 2PM3 X SP2QR therefore (byCorol.8PM3 X SP2, and QR And1 and 5, Prop. VI.), the centripetal force is reciprocally as2SP2.that is (neglecting the given ratio-ppr)> reciprocally asPM3. Q.E.LAnd the same thing is likewise easily inferred from the preceding Proposition.SCHOLIUM.And by a like reasoning, a body will be moved in an ellipsis, or even iaan hyperbola, or parabola, by a centripetal force which is reciprocally aethe cube of the ordinate directed to an infinitely remote centre of force.PROPOSITION IX. PROBLEM IV.If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc.,in a given angle; it is proposed to find thelaio of the centripetal forcetending to tJie centre of that spiral.Suppose the indefinitely small angle AYPSQ to be given ; because, then, all theangles are given, thefigure SPRQT will ,_ be given in specie.vQT Q,T2Therefore the ratio-7^- is also given, and is as QT, that is (belot IX QKcause the figure is given in specie), as SP. But if the angle PSQ is anyway changed, the right line QR, subtending the angle of contact QPUtU THE MATHEMATICAL PRINCIPLES [BOOK J(by Lemma XI) will be changed in the duplicate ratio of PR or QTQT2Therefore the ratio ~TVD~remains the same as before, that is, as SP. AndQT2 x SP2-^ is as SP3, and therefore (by Corol. 1 and 5, Prop. YI) thecentripetal force is reciprocally as the cube of the distance SP. Q.E.I.The same otherwise.The perpendicular SY let fall upon the tangent, and the chord PY ofthe circle concentrically cutting the spiral, are in given ratios to the heightSP ; and therefore SP3is as SY2 X PY, that is (by Corol. 3 and 5, Prop.YI) reciprocally as the centripetal force.LEMMA XII.All parallelograms circumscribed about any conjugate diameters of agiven ellipsis or hyperbola are equal among themselves.This is demonstrated by the writers on the conic sections.PROPOSITION X. PROBLEM Y.If a body revolves in an ellipsis ; it is proposed to find the law of thicentripetal force tending to the centre of the ellipsis.Suppose CA, CB tobe semi-axes of theellipsis; GP, DK, conjugate diameters ; PF,Q,T perpendiculars tothose diameters; Qvan^rdinate to the diameter GP ; and if theparallelogram QvPRbe completed, then (bythe properties of thejonic sections) the reclanglePvG will be toQv2 as PC2 to CD2;and (because of thesimilar triangles Q^T, PCF), Qi>2 to QT2 as PC2 to PF2; and, by composition, the ratio of PvG to QT2is compounded of the ratio of PC21<QT2CD2, and of the ratio of PC2 to PF2, that is, vG to-pas PC;to_92L^_P_]^_. Put QR for Pr, and (by Lem. XII) BC X CA for CDK PF ; also (the points P and Q coinciding) 2PC for rG; and multiplySEC. II.] OF NATURAL PHILOSOPHY. 115QT2 x PC2ing the extremes and means together, we shall have rfo~ equal to2BC2 X CA2pp . Therefore (by Cor. 5, Prop. VI), the centripetal force is2BC2 X CA2reciprocally as ry~ ; that is (because 2I3C2 X CA2is given), reciprocally as-r^v; that is, directly as the distance PC. QEI.I OTJie same otherwise.[n the right line PG on the other side of the point T, take the point uso that Tu may be equal to TV ; then take uV, such as shall be to vG asDC2 to PC2. And because Qr9is to PvG as DC2 to PC2(by the conicsections), we shall have Qv2 -= Pi X V. Add the rectangle n.Pv to bothsides, and the square of the chord of the arc PQ, will be equal to the rectangle VPv ; and therefore a circle which touches the conic section in P,and passes through the point Q,, will pass also through the point V. Nowlet the points P and Q, meet, and the ratio of nV to rG, which is the samewith the ratio of DC2 to PC2, will become the ratio of PV to PG, or PV2DC2to 2PC : and therefore PY will be equal to . And therefore theforce by which the body P revolves in the ellipsis will be reciprocally as2 DC2ry X PF2(by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2isI Ogiven) directly as PC. Q.E.I.COR. 1. And therefore the force is as the distance of the body from thecentre of the ellipsis ; and, vice versa, if the force is as the distance, thebody will move in an ellipsis whose centre coincides with the centre of force,or perhaps in a circle into which the ellipsis may degenerate.COR. 2. And the periodic times of the revolutions made in all ellipseswhatsoever about the same centre will be equal. For those times in similar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipsesthat have their greater axis common, they are one to another as the wholeareas of the ellipses directly, and the parts of the areas described in thesame time inversely: that is, as the lesser axes directly, and the velocitiesof the bodies in their principal vertices inversely ; :hat is, as those lesseraxes dirtily, and the ordinates to the same point% f the common axes inversely ; and therefore (because of the equality of the direct and inverseratios) in the ratio of equality.SCHOLIUM.If the ellipsis, by having its centre removed to an infinite distance, degenerates into a parabola, the body will move in tin s parabola ; and the116 THE MATHEMATICAL PRINCIPLES [BOOK Iforce, now tending to a centre infinitely remote, will become equable.Which is Galileo s theorem. And if the parabolic section of the cone (bychanging the inclination of the cutting plane to the cone) degenerates intoan hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in likemanner as in the circle, or in the ellipsis, if the forces are directed to thecentre of the figure placed in the abscissa, those forces by increasing or diminishing the ordinates in any given ratio, or even by changing the angleof the inclination of the ordinates to the abscissa, are always augmentedor diminished in the ratio of the distances from the centre ; provided theperiodic times remain equal ; so also in all figures whatsoever, if the ordinatesare augmented or diminished in any given ratio, or their inclinationis any way changed, the periodic time remaining the same, the forces directed to any centre placed in the abscissa are in the several ordinateeaugmented or diminished in the ratio of the distances from the centreSECTION III.Of the motion of bodies in eccentric conic sections.PROPOSITION XL PROBLEM VI.If a body revolves in an ellipsis ; it is required to find the law of the