自然哲学的数学原理-27

DF, and the lines ER, ES to FB, FD respectively, and therefore QS tobe always equal to CK; and (by Cor. 2, Prop. XCVII) PD will be to QDas M to N, and therefore as DL to DK, or FB to FK ; and by division asDL FB or PH PD FB to FD or FQ QD ; arid by compositionas PH FB to FQ, that is (because PH and CG, QS and CE, are equal),as CE + BG FR to CE FS. But (because BG is to CE as MN to N) it. comes to pass also that CE + BG is to CE as M to N; andtherefore, by division, FR is to FS as M to N ; and therefore (by Cor. 2,Prop XCVI1) the superficies EF compels a body, falling upon it in thedirection DF, to go on in the line FR to the place B. Q.E.D.SCHOLIUM.,In the same manner one may go on to three or more superficies. Butof all figures the sphserical is the most proper for optical uses. If the object glasses of telescopes were made of two glasses of a sphaerical figure,containing water between them, it is not unlikely that the errors of therefractions made in the extreme parts of the superficies of the glasses maybe accurately enough corrected by the refractions of the water. Such object glasses are to be preferred before elliptic and hyperbolic glasses, not onlybecause they may be formed with more ease and accuracy, but because thepencils of rays situate without the axis of the glass would be more accurately refracted by them. But the different refrangibility of different rayais the real obstacle that hinders optics from being made perfect by sphaerical or any other figures. Unless the errors thence arising can be corrected,all the labour spent in correcting the others is quite thrown away.BOOK IIBOOK II.OF THE MOTION OF BODIES.SECTION I.Of the motion of bodies that are resisted in the ratio of the velocity.PROPOSITION I. THEOREM I.Tf a body is resisted in the ratio of its velocity, the motion lost by resistance is as the space gone over in its motion.For since the motion lost in each equal particle of time is as the velocity,that is, as the particle of space gone over, then, by composition, the motionlost in the whole time will he as the whole space gone over. Q.E.D.COR. Therefore if the body, destitute of all gravity, move by its innateforce only in free spaces, and there be given both its whole motion at thebeginning, and also the motion remaining after some part of the way isgone over, there will be given also the whole space which the body can describe in an infinite time. For that space will be to the space now described as the whole motion at the beginning is to the part lost of thatmotion.LEMMA I.Quantities proportional to their differences are continually proportional.Let A be to A B as B to B C and C to C D, (fee., and, by conversion, A will be to B as B to C and C to D, &c. Q.E.D.PROPOSITION II. THEOREM II.If a body is resisted in the ratio of its velocity, and moves, by its vis insitaonly, through a similar medium, and the times be taken equal,the velocities in the beginning of each of the times are in a geometrical progression, and the spaces described in each of the times are asthe velocities.CASE 1. Let the time be divided into equal particles ; and if at the verybeginning of each particle we suppose the resistance to act with one singleimpulse which is as the velocity, the decrement of the velocity in each ofTHE MATHEMATICAL PRINCIPLES [BOOK II.the particles of time will be as the same velocity. Therefore the velocities are proportional to their differences, and therefore (by Lem. 1, BookII) continually proportional. Therefore if out of an equal number of particles there be compounded any equal portions of time, the velocities at thebeginning of those times will be as terms in a continued progression, whichare taken by intervals, omitting every where an equal number of intermediate terms. But the ratios of these terms are compounded of the equajratios of the intermediate terms equally repeated, and therefore are equalTherefore the velocities, being proportional to those terms, are in geometrical progression. Let those equal particles of time be diminished, andtheir number increased in infinitum, so that the impulse of resistance maybecome continual; and the velocities at the beginnings of equal times, always continually proportional, will be also in this case continually proportional. Q.E.D.CASE 2. And, by division, the differences of the velocities, that is, theparts of the velocities lost in each of the times, are as the wholes ;but thespaces described in each of the times are as the lost parts of the velocities(by Prop. 1, Book I), and therefore are also as the wholes. Q.E.D.TT COROL. Hence if to the rectangular asymptotes AC, CH,the hyperbola BG is described, and AB, DG be drawn perpendicular to the asymptote AC, and both the velocity of. the body, and the resistance of the medium, at the very beginning of the motion, be expressed by any given line AC,and, after some time is elapsed, by the indefinite line DC ; the time maybe expressed by the area ABGD, and the space described in that time bythe line AD. For if that area, by the motion of the point D, be uniformly increased in the same manner as the time, the right line DC will decrease in a geometrical ratio in the same manner as the velocity ; and theparts of the right line AC, described in equal times, will decrease in thesame ratio.PROPOSITION III. PROBLEM I.To define the motion of a body which, in a similar medium, ascends ordescends in a right line, and is resisted in the ratio of its velocity, andacted upon by an uniform force of gravity.The body ascending, let the gravity be expounded by any given rectangle BACH ; and the resistance of the medium, at the beginning of the ascent,by the rectangle BADE, taken on the contrary sideJfl e B^l | L- of the right line AB. Through the point B, withthe rectangular asymptotes AC, CH, describe anhyperbola, cutting the perpendiculars DE, de, IDSEC. I.j OF NATURAL PHILOSOPHY. 253G, g ; and the body ascending will in the time DGgd describe the spaceEG-e; in the time DGBA, the space of the whole ascent EGB ;in thetime ABK1, the space of descent BFK ; and in the time IKki the space ofdescent KFfk; and the velocities of the bodies (proportional to the resistance of the medium) in these periods of time will be ABED, ABed, O,ABFI, AB/z respectively ; and the greatest velocity which the body canacquire by descending will be BACH.For let the rectangle BACH be resolved into innumerable rectangles AA , K/, Lm, M//, *fea, whichshall be as the increments of the velocities producedin so many equal times; then will 0, AAr, AL Am, An,ifec., be as the whole velocities, and therefore (by supposition) as the resistances of the medium in the beginningof each of the equal times. Make AC toAJLLBAK, or ABHC to AB/vK, as the force of gravity to the resistance in thebeginning of the second time ; then from the force of gravity subduct theresistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the absolute forces with which the body is acted upon in the beginning of each ofthe times, and therefore (by Law I) as the increments of the velocities, thatis, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, BookII) in a geometrical progression. Therefore, if the right lines K, LIM/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee..the areas AB^K, K</rL, LrsM, MsJN, (fee., will be equal, and therefore analogous to the equal times and equal gravitating forces. But thearea AB^K (by Corol. 3, Lem. VII and VIII, Book I) is to the area Bkqas K^ to 。kq, or AC to |AK, that is, as the force of gravity to the resistance in the middle of the first time. And by the like reasoning, the areas<?KLr, rLMs, sMN/, (fee., are to the areas qklr, rims, smnt, (fee., as thegravitating forces to the resistances in the middle of the second, third, fourthtime, and so on. Therefore since the equal areas BAKq, </KLr, rLMs,sMN/, (fee., are analogous to the gravitating forces, the areas Bkq, qklr,rims, smut, (fee., will be analogous to the resistances in the middle ofeach of the times, that is (by supposition), to the velocities, and so to thespaces described. Take the sums of the analogous quantities, and the areasBkq, Elr, Ems, But, (fee., will be analogous to the whole spaces described ;and also the areas AB<?K, ABrL, ABsM, AB^N, (fee., to the times. Therefore the body, in descending, will in any time ABrL describe the space Blr,and in the time Lr^N the space rlnt. Q,.E.D. And the like demonstration holds in ascending motion.COROL. 1. Therefore the greatest velocity that the body can acquire byfalling is to the velocity acquired in any given time as the iven force olgravity which perpetually acts upon it to the resisting force which opposesit at the end of that time.854 THE MATHEMATICAL PRINCIPLES [BOOK ILCOROL. 2. But the time being augmented in an arithmetical progression,the sum of that greatest velocity and the velocity in the ascent, and alsotheir difference in the descent, decreases in a geometrical progression.COROL. 3. Also the differences of the spaces, which are described in equaldifferences of the times, decrease in the same geometrical progression.COROL. 4. The space described by the body is the difference of twospaces, whereof one is as the time taken from the beginning of the descent,and the other as the velocity; which [spaces] also at the beginning of thedescent are equal among themselves.PROPOSITION IV. PROBLEM II.Supposing the force of gravity in any similar medium to be uniform,and to tend perpendicularly to the plane of the horizon ; to define themotion of a projectile therein, which suffers resistance proportional toits velocity.Let the projectile go from any place D inthe direction of any right line DP, and letits velocity at the beginning of the motionbe expounded by the length DP. From thepoint P let fall the perpendicular PC on thehorizontal line DC, and cut DC in A, sothat DA may be to AC as the resistanceof the medium arising from the motion upwards at the beginning to the force of gravity; or (which comes to the same) so thatt ie rectangle under DA and DP may be tothat under AC and CP as the whole resistance at the beginning of the motion to theforce of gravity. With the asymptotesDC, CP describe any hyperbola GTBS cutting the perpendiculars DG, AB in G andB ; complete the parallelogram DGKC, andlet its side GK cut AB in Q,. TakeN in the same ratio to QB as DC is in to CP ; and from any point R of theright line DC erect RT perpendicular to it, meeting the hy] erbola in T,and the right lines EH, GK, DP in I, t, and V ; in that perpendiculartake Vr equal to ~- , or which is the same thing, take Rr equal to(""""PIT?^ T ; and the projectile in the time DRTG will arrive at the point rdescribing the curve line DraF, the locus of the point r ; thence it willcome to its greatest height a in the perpendicular AB j and afterwardsSEC. 1.J OF NATURAL PHILOSOPHY. 255ever approach to the asymptote PC. And its velocity in any pjint r willbe as the tangent rL to the curve. Q.E.I.For N is to Q,B as DC to CP or DR to RV, and therefore RV is equal toPR X QB , -.."."v v DRXQB-/GT^r-, and R/ (that is, RV Vr, or - --^---) is equal toD-R X-Ap RDGT ~---. JNow let the time be expounded by the areaRDGT and (by Laws, Cor. 2), distinguish the motion of the body intotwo others, one of ascent, the other lateral. And since the resistance is asthe motion, let that also be distinguished into two parts proportional andcontrary to the parts of the motion : and therefore the length described bythe lateral motion will be (by Prop. II, Book II) as the line DR, and theheight (by Prop. Ill, Book II) as the area DR X AB RDGT, that is,as the line Rr. But in the very beginning of the motion the area RDGTis equal to the rectangle DR X AQ, and therefore that line Rr (orDRx ABthat is, as CP to DC ; and therefore as the motion upwards to the motionlengthwise at the beginning. Since, therefore, Rr is always as the height,and DR always as the length, and Rr is to DR at the beginning as theheight to the length, it follows, that Rr is always to DR as the height tothe length ; and therefore that the body will move in the line DraF. whichis the locus of the point r. QJE.D.DR X AB RDGTCOR. 1. Therefore Rr is equal to --^------^-. and thereforeif RT be produced to X so that RX may be equal to --^--; that is,if the parallelogram ACPY be completed, and DY cutting CP in Z bedrawn, and RT be produced till it meets DY in X ; Xr will be equal toRDGT^ , and therefore proportional to the time.COR. 2. Whence if innumerable lines CR, or, which is the same, innumerable lines ZX, be taken in a geometrical progression, there will be asmany lines Xr in an arithmetical progression. And hence the curve DraFis easily delineated by the table of logarithms.COR. 3. If a parabola be constructed to the vertex D, and the diameterDG produced downwards, and its latus rectum is to 2 DP as the wholeresistance at the beginning of the notion to the gravitating force, the velocity with which the body ought *o go from the place D, in the directionof the right line DP, so as in an uniform resisting medium to describe thecurve DraF, will be the same as that with which it ought to go from thesame place D in the direction of the same right line DP, so as to describe256 THE MATHEMATICAL PRINCIPLES ~ [BOOK IIIa parabola in a non-resisting medium. Forthe latus rectum of this parabola, at the veryDV2beginning of the motion, is -y- , andVristGT DR x T*-~JTor^T. But a right line, which,if drawn, would touch the hyperbola GTS inK G, is parallel to DK, and therefore T* isCKX DRcQBx DC ^ , and N is ~pp Ahd there- DCDR2 X CK x CPfore Vr is equal to 2DC 2 X QlT~; *^at *S (Because D^ an<* ^)C, DVDV2 x CK ~x OPand DP are proportionals), to ^T5 Fcrr J an<* tne ^atus reeturnDV2- comes out -2DP2 X QBare proportional),CK X CP2DP 2 X DAAC X CPCP X AC ;that is, as the resistance to the gravity.(becauseand therefore is to 2DP as DP X DA toQ.E.D.COR. 4. Hence if a body be projected fromany place D with a given velocity, in thedirection of a right line DP given by position, and the resistance of the medium, atthe beginning of the motion, be given, thecurve DraF, which that body will describe,may be found. For the velocity beinggiven, the latus rectum of the parabola isgiven, as is well known. And taking 2DPto that latus rectum, as the force of gravityto the resisting force, DP is also given.Then cutting DC in A, so that GP X ACmay be to DP X DA in the same ratio ofthe gravity to the resistance, the point Awill be given. And hence the curve DraFis also given.COR. 5. And, on the contrary, if thecurve DraF be given, there will be givenx>th the velocity of the body and the resistance of the medium in each ofthe places r. For the ratio of CP X AC to DP X DA being given, thereis given both the resistance of the medium at the beginning of the motionand the latus rectum of the parabola ; and thence the velocity at the beginning of the motion is given also. Then from the length of the tangentSEC. I.]OF NATURAL PHILOSOPHY. 257L there is given both the velocity proportional to it, and the resistanceproportional to the velocity in any place r.COR. 6. But since the length 2DP is to the latus rectum of the parabola as the gravity to the resistance in D ; and, from the velocity augmented, the resistance is ti gmented in the same ratio, but the latus rectumof the parabola is augmented in the duplicate of that ratio, it is plain thotthe length 2DP is augmented in that simple ratio only ; and is thereforealways proportional to the velocity ; nor will it be augmented or diminished by the change of the angle CDP, unless the velocity be also changed.COR. 7. Hence appears the method of determining the curve DraF nearly from the phenomena,and thence collecting the resistance andvelocity with which the body is projected. Lettwo similar and equal bodies be projected withthe same velocity, from the place D, in different angles CDP, CDp ; and let the places F,f. where they fall upon the horizontal planeDC, be known. Then taking any length for D */ FDP or Dp suppose the resistance in D to be tothe suavity in any ratio whatsoever, and let thatratio be expounded by any length SM. Then, , _by computation, from that assumed length DP, ^xfind the lengths DF, D/; and from the ratioF/-p^,found by calculation, subduct the same ratio as found by experiment ;and let the cKfference be expounded by the perpendicular MN. Repeat thesame a second and a third time, by assuming always a new ratio SM of theresistance to the gravity, and collecting a new difference MN. Draw theaffirmative differences on one side of the right line SM, and the negativeon the other side ; and through the points N, N, N, draw a regular curveNNN. cutting the right line SMMM in X, and SX will be the true ratioof the resistance to the gravity, which was to be found. From this ratiothe length DF is to be collected by calculation; and a length, which is tothe assumed length DP as the length DF known by experiment to thelength DF just now found, will be the true length DP. This being known,you will have both the curve line DraF which the body describes, and alsothe velocity and resistance of the body in each place.SCHOLIUM.But, yet, that the resistance of bodies is in the ratio of the velocity, is morea mathematical hypothesis than a physical one. In mediums void of all tenacity, the resistances made to bodies are in the duplicate ratio of the velocities. For by the action of a swifter body, a greater motion in propor-17THE MATHEMATICAL PRINCIPLES [BoOK ILtion to a greater velocity is communicated to the same quantity of themedium in a less time ; and in an equal time, by reason of a greater quantity of the disturbed medium, a motion is communicated in the duplicateratio greater ; and the resistance (by Law II and III) is as the motioncommunicated. Let us, therefore, see what motions arise from this law ofresistance.SECTION II.If the motion of bodies that are resisted in tfie duplicate ratio of theirvelocities.PROPOSITION V. THEOREM III.Ff a body is resisted in the duplicate ratio of its velocity, and moves byits innate force only through a similar medium; and the times betaken in a geometrical progression, proceeding from less to greaterterms : I say, that the velocities at the beginning of each of the timesare in the same geometrical progression inversely ; and that the spacesare equal, which are described in each of the times.For since the resistance of the medium is proportional to the square ofthe velocity, and the decrement of the velocity is proportional to the resistance : if the time be divided into innumerable equal particles, the squares ofthe velocities at the beginning of each of the times will be proportional tothe differences of the same velocities. Let those particles of time be AK,KL, LM, &c., taken in the right line CD; anderect the perpendiculars AB, Kk, L/, Mm, &c.,meeting the hyperbola BklmG, described with thecentre C, and the rectangular asymptotes CD, CH.in B, kj I, m, (fee.; then AB will be to Kk as CKto CA, and, by division, AB Kk to Kk as AKC ARIMT to ^A>an(1 alternate^ AB ^C to AK as Kkto CA ; and therefore as AB X Kk to AB X CA.Therefore since AK and AB X CA are given,* AB Kk will be as ABX Kk ; and, lastly, when AB and KA* coincide, as AB2. And, by the likereasoning, KAr-U, J J-M/??, (fee., will be as Kk2. LI2, (fee. Therefore thesquares of the lines AB, KA", L/, Mm, (fee., are as their differences ; and,therefore, since the squares of the velocities were shewn above to be as theirdifferences, the progression of both will be alike. This being demonstratedit follows also that the areas described by these lines are in a like progression with the spaces described by these velocities. Therefore if the velocity at the beginning of the first time AK be expounded by the line AB,SEC. II.] OF NATURAL PHILOSOPHY. 25CJand the velocity at the beginning of the second time KL by the line K&and the length described in the hrst time by the area AKA*B, all the following velocities will be expounded by the following lines 。J. Mm, .fee.and the lengths described, by the areas K/, I mi. &c. And, by composition, if the whole time be expounded by AM, the sum of its parts, thewhole length described will be expounded by AM/ftB the sum of its parts.Now conceive the time AM to be divided into the parts AK, KL, LM, (feeso that CA, CK, CL, CM, (fee. may be in a geometrical progression ; andthose parts will be in the same progression, and the velocities AB, K/r,L/, Mm, (fee., will be in the same progression inversely, and the spaces described Ak, K/, Lw, (fee., will be equal. Q,.E.D.