自然哲学的数学原理-30

uniformly resisting medium approaches nearerto these hyperbolas than to a parabola. Thatline is certainly of the hyperbolic kind, butabout the vertex it is more distant from theasymptotes, and in the parts remote from thevertex draws nearer to them than these hy- M JL BD Kperbolas here described. The difference, however, is not so great betweenthe one and the other but that these latter may be commod^ously enoughused in practice instead of the former. And perhaps these may prove moreuseful than an hyperbola that is more accurate, and at the same time morecompounded. They may be made use of, then, in this manner.Complete the parallelogram XYGT, and the right line GT will touchthe hyperbola in G, and therefore the density of the medium in G is re-GT 2ciprocally as the tangent GT, and the velocity there as ^ -^=- ; and theresistance is to the force of gravity as GT toTherefore if a body projected from theplace A, in the direction of the right lineAH, describes the hyperbola AGK andAH produced meets the asymptote NX inH, arid AI drawrri parallel to it meets theother asymptote MX in I; the density ofthe mediu.n in A will be reciprocally asAH. and the velocity of the body as -JAH1. . . and the resis an^e there to the forceAl2nnn +2^- X GV.. TT 2nn + 2nof gravity rs AH to ZiTo~ X AI. Her,ce the following rules a ededuced.RULE 1. If the density of the medium at A, and the velocity with whichthe body is projected remain the same, and the angle NAH be changed ,the lengths AH, AI, HX will remain. Therefore if those lengths, in any276 THE MATHEMATICAL PRINCIPLES [BOOK II.one case, are found, the hyperbola may afterwards be easily determinedfrom any given angle NAH.RULE 2. If the angle NAH, and the density of the medium at A, remain the same, and the velocity with which the body is projected bechanged, the length AH will continue the same ; and AI will be changedin a duplicate ratio of the velocity reciprocally.RULE 3. If the angle NAH, the velocity of the body at A, and the accelerativegravity remain the same, and the proportion of the resistance atA to the motive gravity be augmented in any ratio ; the proportion of AHto A I will be augmented in the same ratio, the latus rectum of the above-AH2mentioned parabola remaining the same, and also the length AI proportionalto it; and therefore AH will be diminished in the same ratio, andAI will be diminished in the duplicate of that ratio. But the proportionof the resistance to the weight is augmented, when either the specific gravity is made less, the magnitude remaining equal, or when the density ofthe medium is made greater, or when, by diminishing the magnitude, theresistance becomes diminished in a less ratio than the weight.RULE 4. Because the density of the medium is greater near the vertexof the hyperbola than it is in the place A, that a mean density may bepreserved, the ratio of the least of the tangents GT to the tangent AHought to be found, and the density in A augmented in a ratio a littlegreater than that of half the sum of those tangents to the least of thetangents GT.RULE 5. If the lengths AH, AI are given, and the figure AGK is to bedescribed, produce HN to X, so that HX may be to AI as n -。- 1 to 1; andwith the centre X, and the asymptotes MX, NX, describe an hyperbolathrough the point A, such that AI may be to any of the lines VG as XV"to xr.RULE 6. By how much the greater the number n is, so much the moreaccurate are these hyperbolas in the ascent of the body from A, and lessaccurate in its descent to K ; and the contrary. The conic hyperbolakeeps a mean ratio between these, and is more simple than the rest. Therefore if the hyperbola be of this kind, and you are to find the point K,where the projected body falls upon any right line AN passing throughthe point A, let AN produced meet the asymptotes MX, NX in M and N,and take NK equal to AM.RULE 7. And hence appears an expeditious method of determining thishyperbola from the phenomena. Let two similar and equal bodies be projected with the same velocity, in different angles HAK, hAk, and let themfall upon the plane of the horizon in K and k ; and note the proportionof AK to AA". Let it be as d to e. Then erecting a perpendicular AI ofuny length, assume any how the length AH or Ah, and thence graphically,SEC. II. OF NATURAL PHILOSOPHY. 27?or by scale and compass, collect the lengths AK, A/>* (by Rule 6). If theratio of AK to A/.* bo the same with that of d to e, the length of AH wasrightly assumed. If not, take on the indefinite right line SM, the lengthSM equal to the assumed AH ; and erect a perpendicular MN equal to theAK ddifference -r-r of the ratios drawn into any given right line. By thelike method, from several assumed lengths AH, you may find several pointsN ; and draw througli them all a regular curve NNXN, cutting tr.e rightline SMMM in X. Lastly, assume AH equal to the abscissa SX, andthence find again the length AK ; and the lengths, which are to the assumed length AI, and this last AH, as the length AK known by experiment, to the length AK last found, will be the true lengths AI and AH,which were to be found. But these being given, there will be given alsothe resisting force of the medium in the place A, it being to the force ofgravity as AH to JAI. Let the density of the medium be increased byRule 4, and if the resisting force just found be increased in the same ratio,it will become still more accurate.RULE 8. The lengths AH, HX being found ;let there be now required the position of the line AH, according to which a projectile thrownwith that given velocity shall fall upon any point K. At the joints Aand K, erect the lines AC, KF perpendicular to the horizon : whereof letAC be drawn downwards, and be equal to AI or ^HX. With the asymptotes AK, KF, describe an hyperbola, whose conjugate shall pass throughthe point C ; and from the centre A, with the interval AH. describe a circle cutting that hyperbola in the point H ; then the projectile thrown inthe direction of the right line AH will fall upon the point K. Q.E.I. Forthe point H, because of the given length AH, must be somewhere in thecircumference of the described circle. Draw CH meeting AK and KF inE and F: and because CH, MX are parallel, and AC, AI equal, AE willbe equal to AM, and therefore also equal to KN. But CE is to AE asFH to KN. and therefore CE and FH are equal. Therefore the point Hfalls upon the hyperbolic curve described with the asymptotes AK,.KFwhose conjugate passes through the point C ; and is therefore found in the27S THE MATHEMATICAL PRINCIPLES [BOOK 11common intersection of this hyperboliccurve and the circumference of the described circle. Q.E.D. It is to be observed that this operation is the same,whether the right line AKN be parallel tothe horizon, or inclined thereto in any angle : and that from two intersections H,//., there arise two angles NAH, NAA ;and that in mechanical practice it is sufficient once to describe a circle, then toapply a ruler CH, of an indeterminate length, HO to the point C, that itspart PH, intercepted between the circle and the right line FK, may boequal to its part CE placed between the point C and the right line AKWhat has been said of hyperbolas may be easilyapplied to pir i >;>l.i3. For if a parabola be represented by XAGK, touched by a right line XVin the vertex X, and the ordinates IA, YG be asany powers XI", XV"; of the abscissas XI, XV ;draw XT, GT, AH, whereof let XT be parallelto VG, and let GT, AH touch the parabola inG and A : and a body projected from any placeA, in the direction of the right line AH, with adue velocity, will describe this parabola, if the density of the medium ineach of the places G be reciprocally as the tangent GT. In that case thevelocity in G will be the same as would cause a body, moving in a nonresistingspace, to describe a conic parabola, having G for its vertex, VG2GT2produced downwards for its diameter, and -. for its latusnn n X VGrectum. And the resisting force in G will be to the force of gravity as GT to2nti 2tt~2~ VG. Therefore if NAK represent an horizontal line, and botlithe density of the medium at A, and the velocity with which the body isprojected, remaining the same, the angle NAH be any how altered, thelengths AH, AI, HX will remain; and thence will be given the vertex Xof the parabola, and the position of the right line XI ; and by taking VGto IA as XVn to XI", there will be given all the points G of the parabola,through which the projectile will pass.SEC. IILJ OF NATURAL PHILOSOPHY. 279SECTION III.Of the motions of bodies which are resisted partly in the ratio of the velocities, and partly in the duplicate of the same ratio.PROPOSITION XI. THEOREM VIII.If a body be resisted partly in the ratio and partly in the duplicate ratioof its velocity, and moves in a similar medium by its innate forceonly; and the times be taken in arithmetical progression; thenquantities reciprocally proportional to the velocities, increased by a certain given quantity, will be in geometrical progression.With the centre C, and the rectangular asymptotes ^OADd and CH, describe an hyperbola BEe, and let| 。pAB, DE, de. be parallel to the asymptote CH. In|the asymptote CD let A, G be given points ; and ifthe time be expounded by the hyperbolic area ABEDuniformly increasing, I say, that the velocity may ~rbe expressed by the length DF, whose reciprocalGD, together with the given line CG, compose thelength CD increasing in a geometrical progression.For let the areola DEec/ be the least given increment of the time, andDd will be reciprocally as DE, and therefore directly as CD. Thereforethe decrement of ^TR, which (by Lem. II, Book II) is ^ no , will be also asDtfCD CG + GD 1 CGGO* r GD2~fc1S>aS GD +GJD 2* * nerefore tne timcuniformly increasing by the addition of the given particles EDcfe, it follows that r decreases in the same ratio with the velocity. For the decrement of the velocity is as the resistance, that is (by the supposition), asthe sum of two quantities, whereof one is as the velocity, and the other asthe square of the velocity ; and the decrement of ~~ is as the sum of the1 C^(^ 1quantities ~-^=randpfp,>whereof the first is ^^r itself, and the lasti iis a* /-TFT; therefore T^-R is as tne velocity, the decrements of both- CilJbeing analogous. And if the quantity GD reciprocally proportional toT, be augmented by the given quantity CG ; the sum CD, the timeABED uniformly increasing, will increase !n a geometrical progression.Q.E.D.THE MATHEMATICAL PRINCIPLES [BOOK IICOR. 1. Therefore, if, having the points A and G given, the time boexpounded by the hyperbolic area ABED, the velocity may be expoundedby -r the reciprocal of GD.COR. 2. And by taking GA to GD as the reciprocal of the velocity atthe beginning to the reciprocal of the velocity at the end of any timeABED, the point G will be found. And that point being found the velocity may be found from any other time given.PROPOSITION XII. THEOREM IX.The same things being supposed, I say, that if the spaces described aretaken in arithmetical progression, the velocities augmented by a certain given quantity will be in geometrical progression.In the asymptote CD let there be given thepoint R, and, erecting the perpendicular RSmeeting the hyperbola in S, let the space described be expounded by the hyperbolic areaIRSED ; and the velocity will be as the lengthJ GD, which, together with the given line CG,**composes a length CD decreasing in a geometrical progression, while the space RSED increases in an arithmetical[(regression.For, because the incre nent EDde of the space is given, the lineola DC?,which is the decrement of GD, will be reciprocally as ED, and thereforedirectly as CD ; that is, as the sum of the same GD and the given lengthCG. But the decrement of the velocity, in a time reciprocally proportional thereto, in which the given particle of space D^/eE is described, isas the resistance and the time conjunctly, that is. directly as the sum oftwo quantities, whereof one is as the velocity, the other as the square ofthe velocity, and inversely as the veh city ; and therefore directly as thesum of two quantities, one of which is given, the other is- as the velocity.Therefore the decrement both of the velocity and the line GD is as a givenquantity and a decreasing quantity conjunctly; and, because the decrements are analogous, the decreasing quantities will always be analogous ;viz., the velocity, and the line GD. U.E.D.COR. 1. If the velocity be expounded by the length GD, the space described will be as the hyperbolic area DESR.COR. 2. And if the point be assumed any how, the point G will befound, by taking GR to GD as the velocity at the beginning to the velocity after any space RSED is described. The point G being given, thespace is given from the given velocity : and the contrary.Cotw 3. Whence since (by Prop. XI) the velocity is given from the givenSEC. Ilt.1 Or NATURAL PHILOSOPHY. 281time, and (by this Prop.) the space is given from the given velocity ; thespace will be given from the given time : and the contrary.PROPOSITION XKI. THEOREM X.Supposing that a body attracted downwards by an uniform gravity ascends or descends in a right line; and that the same is resistedpartly in the ratio of its velocity, and partly in the duplicate ratiothereof: I say, that, if right lines parallel to the diameters of a circleand an hyperbola, be drawn through the ends of the, conjugate diameters, and the velocities be as some segments of those parallels drawnfrom a given point, the times will be as the sectors of the, areas cutoff by right lines drawnfrom the centre to the ends of the segments ;and the contrary.CASE 1. Suppose first that the body is ascending,and from the centre D, with any semi-diameter DB,describe a quadrant BETF of a circle, and throughthe end B of the semi-diameter DB draw the indefinite line BAP, parallel to the semi-diameter DF. Inchat line let there be given the point A, and take thesegment AP proportional to the velocity. And sinceone part of the resistance is as the velocity, andanother part as the square of the velocity, let thewhole resistance be as AP 2-f 2BAP. Join DA, DP, cutting the circlein E and T, and let the gravity be expounded by DA2, so that the gravityshall be to the resistance in P as DA2 to AP2+2BAP ; and the time of thewhole ascent will be as the sector EDT of the circle.For draw DVQ,, cutting off the moment PQ, of the velocity AP, and themoment DTV of the sector DET answering to a given moment of time ;and that decrement PQ, of the velocity will be as the sum of the forces ofgravity DA2 and of resistance AP 2 + 2BAP, that is (by Prop. XIIBookII,Elem.),asDP*. Then the arsa DPQ, which is proportional to PQ:is as DP2, and the area DTV, which is to the area DPQ, as DT2 to DP 2, itas the given quantity DT2. Therefore the area EDT decreases uniformlyaccording to the rate of the future time, by subduction of given particles DTV,and is therefore proportional to the time of the whole ascent. Q..E.D.CASE 2. If the velocity in the ascentof the body be expounded by the lengthAP as before, and the resistance be madeas AP2-f- 2BAP,and if the force of gravity be less than can be expressed by DA2;take BD of such a length, that AB2BD 2 maybe proportional to the gravity,and let DF be perpendicular and equalF OS2 THE MATHEMATICAL PRINCIPLES [BOOK ll.to DB, and through the vertex F describe the hyperbola FTVE, whose conjugate semi -diameters are DB and DF; and which cuts DA in E, and DP,DQ in T and V ; and the time of the whole ascent will be as the hyperbolic sector TDE.For the decrement PQ of the velocity, produced in a given particle oftime, is as the sum of the resistance AP2 -f 2BAP and of the gravityAB2 BD2, that is, as BP 2 BD 2. But the area DTV is to the areaDPQ as DT2 to DP 2; and, therefore, if GT be drawn perpendicular toDF. as GT2 or GD 2 DF2 to BD 2, and as GD2 to BP 2, and, by division, as DF2 to BP 2 BD 2. Therefore since the area DPQ is as PQ,that is, as BP 2 BD 2, the area DTV will be as the given quantity DF 2.Therefore the area EDT decreases uniformly in each of the equal particlesof time, by the subduction of so many given particles DTV, and thereforeis proportional to the time. Q.E.D.CASE 3. Let AP be the velocity in the descent of""" the body, and AP 2 + 2BAP the force of resistance,and BD 2 AB 2 the force of gravity, the angle DBAbeing a right one. And if with the centre D, and theprincipal vertex B, there be described a rectangularhyperbola BETV cutting DA, DP, and DQ producedin E, T, and V : the sector DET of this hyperbola willD be as the whole time of descent.For the increment PQ of the velocity, and the area DPQ proportionalto it, is as the excess of the gravity above the resistance, that is, asm)2?_ AB 2 _2BAP AP2 or BD 2 BP 2. And the area DTVis to the area DPQ as DT 3 to DP 2; and therefore as GT2 or GD" -BD 2 to BP 2, and as GD 2 to BD 2, and, by division, as BD 2 to BD2 -BP2. Therefore since the ami DPQ is as BD2 BP2, the area DTVwill be as the given quantity BD 2. Therefore the area EDT increasesuniformly in the several equal particles of time by the addition of asmany given particles DTV, and therefore is proportional to the time ofthe descent. Q.E.D.Con. If with the centre D and the semi-diameter DA there be drawnthrough the vertex A an arc A/ similar to the arc ET, and similarly subtendino^the angle A DT, the velocity AP will be to the velocity which thebody in the time EDT, in a non-resisting space, can lose in its ascent, oracquire in its descent, as the area of the triangle DAP to the area of theBector DA/ ; and therefore is given from the time given. For the velocityir a non-resistin^ medium is proportional to the time, and therefore to thissector : in a resisting medium, it is as the triangle ; and in both mediums,where it is least, it approaches to the ratio of equality, as the sector andtriangle doSEC. III.] OF NATURAL PHILOSOPHY. 283SCHOLIUM.One may demonstrate also that case in the ascent of the body, where theforce of gravity is less than can be expressed by DA2 or AB 2 + BD 2, andgreater than can be expressed by AB 2 DB 2, and must be expressed byAB2. But I hasten to other thingsPROPOSITION XIV. THEOREM XLThe same things being supposed, 1 say, that the space described in theascent or descent is as the difference of the area by which the time isexpressed, and of some other area which is augmented or diminishedin an arithmetical progression ; if the forces compounded of the resistance and the gravity be taken, in a geometrical progression.Take AC (in these three figures) proportional to the gravity, and AKto the resistance ; but take them on the same side of the point A, if the。*"1。B A K QPbody is descending, otherwise on the contrary. Erect A b, which make toDB as DB 2 to 4BAC : and to the rectangular asymptotes CK, CH, describe the hyperbola 6N ; and, erecting KN perpendicular to CK, the areaA/AK will be augmented or diminished in an arithmetical progression,while the forces CK are taken in a geometrical progression. I say, therefore, that the distance of the body from its greatest altitude is as the excessof the area A6NK above the area DET.For since AK is as the resistance, that is, as AP 2 X 2BAP ; assumeany given quantity Z, and put AK equal to then (by Lem,