as the triangle APD to the hyperbolic sector ATD. For the velocity ina non-resisting medium Avould be as the time ATD, and in a resisting medium is as AP, that is, as the triangle APD. And those velocities, at thebeginning of the descent, are equal among themselves, as well as thoseareas ATD, APD.COR. 4. By the same argument, the velocity in the ascent is to the velocity with which the body in the same time, in a non-resisting space, wouldlose all its motion of ascent, as the triangle ApD to the circular sectorAtD ;or as the right line Ap to the arc At.COR. 5. Therefore the time in which a body, by falling in a resistingmedium, would acquire the velocity AP, is to the time in which it wouldacquire its greatest velocity AC, by falling in a non-resisting space, as thesector ADT to the triangle ADC : and the time in which it would lose itsvelocity Ap, by ascending in a resisting medium, is to the time in whichit would lose the same velocity by ascending in a non-resisting space, asthe arc At to its tangent Ap.COR. 6. Hence from the given time there is given the space described inthe ascent or descent. For the greatest velocity of a body descending inwfinitum is given (by Corol. 2 and 3, Theor. VI, of this Book) ; and thencethe time is given in which a body would acquire that velocity by fallingin a non-resisting space. And taking the sector ADT or ADt to the triangle ADC in the ratio of the given time to the time just now found,there will be given both the velocity AP or Ap, and the area ABNK orAB//&, which is to the sector ADT, or AD/, as the space sought to thespace which would, in the given time, be uniformly described with thatgreatest velocity found just before.COR. 7. And by going backward, from the given space of ascent or descent AB?A: or ABNK, there will be given the time AD* or ADT.268 THE MATHEMATICAL PRINCIPLES [BOOK iiPROPOSITION X. PROBLEM III.Suppose the uniform force of gravity to tend directly to the plane of thehorizon, and the resistance to be as the density of the medium and thesquare of the velocity coiijunctly : it is proposed to find the density ofthe medium in each place, which shall make the body move in anygiven carve line ; the velocity of the body and the resistance of themedium in each place.Let PQ be a plane perpendicular tothe plane of the scheme itself; PFHQa curve line meeting that plane in thepoints P and Q ; G, H, I, K fourplaces of the body going on in this。 curve from F to Q ; and GB, HC, ID,KE four parallel ordinates let fallp A. 33 c^D E Q from these points to the horizon, andstanding on the horizontal line PQ at the points B, C, D, E ; and let thedistances BC, CD, DE, of the ordinates be equal among themselves. Fromthe points G and H let the right lines GL, HN, be drawn touching thecurve in G and H, and meeting the ordinates CH, DI, produced upwards,in L and N : and complete the parallelogram HCDM. And the times inwhich the body describes the- arcs GH, HI, will be in a subduplicate ratioof the altitudes LH, NI; which the bodies would describe in those times,by falling from the tangents; and the velocities will be as the lengths described GH, HI directly, and the times inversely. Let the times be ex-C*TT TTTpounded by T and t, and the velocities by =- and ---; and the decrementJ_ L/-^TT TTTof the velocity produced in the time t will be expounded by -7^.This decrement arises from the resistance which retards the body, and fromthe gravity which accelerates it. Gravity, in a falling body, which in itsfall describes the space NI, produces a velocity with which it would be ableto describe twice that space in the same time, as Galileo has demonstrated ;2NIthat is, the velocity : but if the body describes the arc HI, it augmentsMIxNlthat arc only by the length HI ; and therefore generates HI HN oronly the velocity iff- Let this velocity be added to the beforetX H.JLmentioned decrement, and we shall have the decrement of the velocityGH HI SMI X Nlarising from the resistance alone, that is, -^ : h TSEC. II.J OF NATURAL PHILOSOPHY. 2692NI.Therefore since, in the same time, the action of gravity generates, in a falling body, the velocity , the resistance will be to the gravity as 7^HI 2MI X NI 2NI t X GH 2MI X NI+ TTT- to or as ^ HI -fNow for the abscissas CB, CD,CE, put o, o, 2o. For the ordinateCH put P ; and for MI put any seriesQo + Ro 2 + So 3 +, &c. And allthe terms of the series after the first,that is, Ro 2 + So 3 +, (fee., will beNI ; and the ordinates DI, EK, andBGwill be P QoRo2 So 3 p A B c T> E(fee., P 2Qo 4Ro 2 SSo 3, (fee., and P -。- Qo Ro 2 + So 3,(fee., respectively. And by squaring the differences of the ordinates BGCH and CH DI, and to the squares thence produced adding the squaresof BC and CD themselves, you will have oo -f- QQoo 2QRo 3 +, (fee.,and oo -f QQoo -f 2QRo 3 +, (fee., the squares of the arcs GH, HI ; whoseQRoo QRooroots o y -, and o</! 4- QQ 4- are the1 + QQ v/1 + QQ s/1 -f QQarcs GH and HI. Moreover, if from the ordinate CH there be subductedhalf the sum of the ordinates BG and DI, and from the ordinate DI therebe subducted half the sum of the ordinates CH and EK, there will remainRoo and Roo + 3So 3, the versed sines of the arcs GI and HK. And theseare proportional to the lineolae LH and NI, and therefore in the duplicateratio of the infinitely small times T and t : and thence the ratio ~, is ^R + SSo R -f^ orSo , t X GH TTT 2MI X NI ,R : and T^ HI H HTTIT , by substitutingthe values of , GH, HI, MI and NI just found, becomes -^- J- /w-Lt/I + QQ. Arid since 2NI is 2Roo, the resistance will be now to theOOgravity as -- TT Q to 2Roo>that is>as 3S r to 4RR.And the velocity will be such, that a body going off therewith from anyplace H, in the direction of the tangent HN, would describe, in vacuo, aparabola, whose diameter is HC, and its latus rectum NT or --^----.And the resistance is as the density of the medium and the square ofthe velocity conjunctly ; and therefore the density of the medium is as theresistance directly, and the square of the velocity inversely ; that is, as270 THE MATHEMATICAL PRINCIPLES [BOOK II.QQ __4RRQ.E.I.COR. 1. If the tangent HN be produced both ways, so as to meet any HTordinatc AF in T - will be equal toV/T+ QQ; and therefore in whathas gone before may be put for ^ 。 -。- QQ. By this means the resistancewill be to the gravity as 3S X HT to 4RR X AC ; the velocity will be a*r-pj --^, and the density of the medium will be as -AO TT-n. -v/ i Jti X H 1COR. 2. And hence, if the curve line PFHQ be denned by the relationbetween the base or abscissa AC and the ordinate CH, as is usual, and thevalue of the ordinate be resolved into a converging series, the Problemwill be expeditiously solved by the first terms of the series ; as in the following examples.EXAMPLE 1. Let the line PFHQ, be a semi-circle described upon thediameter PQ, to find the density of the medium that shall make a projectile move in that line.Bisect the diameter PQ in A ; and call AQ, n ; AC, a ; CH, e ; andCD, o ; then DI2 or AQ, 2 AD 2 = nn aa 2ao oo, or ec. 2aooo ; and the root being extracted by our method, will give DI= eao oo aaoo ao 3 a 3 o 3~e~~~2e 2e?~~~W~2?&C* Here put nn f r ee + aa> andao nnoo anno 3DI will become = e , &c.e 2e 3 2e 5Such series I distinguish into successive terms after this manner : I callthat the first term in which the infinitely small quantity o is not found ;the second, in which that quantity is of one dimension only ; the third, inwhich it arises to two dimensions ; the fourth, in which it is of three ; andso ad infinitum. And the first term, which here is e, will always denotethe length of the ordinate CH, standing at the beginning of the indefinitequantity o. The second term, which here is , will denote the differencebetween CH and DN ; that is, the lineola MN which is cut off by completing the parallelogram HCDM; and therefore always determines theCM?position of the tangent HN ; as, in this case, by taking MN to HM asGto o, or a to e. The third term, which here is -, will represent the lineola IN, which lies between the tangent and the curve ; and thereforedetermines the angle of contact IHN, or the curvature which the curve lineSEC. II.] OF NATURAL PHILOSOPHY. 271has in H. If that lineola IN is of a finite magnitude, it will be expressedby the third term, together with those that follow in wfinitu:.:i. But ifthat lineola be diminished in. infinitnm,the terms following become infinitely less than the third term, andtherefore may be neglected. Thefourth term determines the variationof the curvature ; the fifth, the variation of the variation ; and so on.Whence, by the way, appears no con-p~" ~K B~C~D~E~temptible use of these series in the solution of problems that depend upontangents, and the curvature of curves.ao 77/700 anno 3Now compare the series e ^ ^~ &c., with thee Ze 3 Ze*series P Qo - Roo So 3&c., and for P, Q, II and S? put e, -, ^-^and ~ , and for ^ 1 + QQ put 1 H or -; and the density oi2e 5 ee ethe medium will come out as ; that is (because n is given), as - orlie e~Yj, that is, as that length of the tangent HT, which is terminated at theOH.semi-diameter AF standing perpendicularly on PQ : and the resistancewill be to the gravity as 3a to2>/, that is, as SAC to the diameter PQ ofthe circle; and the velocity will be as i/ CH. Therefore if the body goesfrom the place F, with a due velocity, in the direction of a line parallel toPQ, and the density of the medium in each of the places H is as the lengthof the tangent HT, and the resistance also in any place H is to the forceof gravity as SAC to PQ, that body will describe the quadrant FHQ of acircle. Q.E.I.But if the same body should go-frorn the place P, in the direction of aline perpendicular to PQ, and should begin to move in an arc of the semicircle PFQ, we must take AC or a on the contrary side of the centre A ;and therefore its sign must be changed, and we must put a for + a.Then the density of the medium would come out as . But naturedoes not admit of a negative density, that is, a density which acceleratesthe motion of bodies; and therefore it cannot naturally come to pass thata body by ascending from P should describe the quadrant PF of a circle.To produce such an effect, a body ought to be accelerated by an impellingmedium, and not impeded by a resisting one.EXAMPLE 2. Let the line PFQ be a parabola, having its axis AF per272 THE MATHEMATICAL PRINCIPLES [BOOK ILpendicular to the horizon PQ, to find the density of the medium, whichwill make a projectile move in that line.From the nature of the parabola, the rectangle PDQ,is equal to the rectangle under the ordinate DI and somegiven right line;that is, if that right line be called b ;PC, a; PQ, c; CH, e; and CD, o; the rectangle aA. CD ~Q + o into c a o or ac aa 2ao -{-co oo, iaac aaequal to the rectangle b into DI, and therefore DI is equal to --7--hc 2a oo c 2a-. o r. Now the second term -, o of this series is to be putb b boofor Q,o, and the third term -r for Roo. But since there are no moreterms, the co-efficient S of the fourth term will vanish ; and therefore theSouantitv - , to which the density of the medium is proper- R v itional, will be nothing. Therefore, where the medium is of no density,the projectile will move in a parabola ; as Galileo hath heretofore demonstrated. Q.E.I.EXAMPLE 3. Let the line AGK be an hyperbola, having its asymptoteNX perpendicular to the horizontal plane AK, to find the density of themedium that will make a projectile move in that line.Let MX be the other asymptote, meetingthe ordinate DG produced in V ; and fromthe nature of the hyperbola, the rectangle ofXV into VG will be given. There is alsogiven the ratio of DN to VX, and thereforethe rectangle of DN into VG is given. Letthat be bb : and, completing the parallelogram DNXZ, let BN be called a; BD, o;NX, c; and let the given ratio of VZ toMA. BD K N ZX or DN be -. Then DN will be equalbbto a o} VG equal to , VZ equal to X a o. and GD or NXa o nm m-VZ VG equal to c a + o . Let the term - ben n a o a obb bb bb bb ,resolved into the converging series~^"+ ^ + ^l00 + ^4 > &c andm bb m bb bb bbGD will become equal to c - a - + -o ~ o ^ o 251SEC. II.] OF NATURAL PHILOSOPHY. 273&c. The second term o o of this series is to be u?ed for Qo; then aathird o 2, with its sign changed for Ro2; and the fourth o 3, with itsm bb bb bbsign changed also for So 3, and their coefficients , and are ton aa a abe pat for Q, R, and S in the former rule. Which being done, the denbba*sity of the medium will come out as , ,bbammnn2mbbnaaImm -, that is, if in VZ you take VY equal toaa aa1 m2VG, as YT7- For aa and ^ a 22mbb bnn n aa2mbb b 4H are the squares of XZn aaand ZY. But the ratio of the resistance to gravity is found to be that of3XY to 2YG ; and the velocity is that with which the body would de-XY2scribe a parabola, whose vertex is G, diameter DG, latus rectum^v . Suppose, therefore, that the densities of the medium in each of the places Gare reciprocally as the distances XY, and that the resistance in any placeG is to the gravity as 3XY to 2YG ; and a body let go from the place A,with a due velocity, will describe that hyperbola AGK. Q.E.I.EXAMPLE 4. Suppose, mdeMtely, the line AGK to be an hyperboladescribed with the centre X, and the asymptotes MX, NX, so that, havingconstructed the rectangle XZDN, whose side ZD cuts the hyperbola in Gand its asymptote in V, VG may be reciprocally as any power DNn of theline ZX or DN, whose index is the number n : to find the density of themedium in which a projected body will describe this curve.For BN, BD, NX, put A, O, C, respec- ^tively, and let VZ be to XZ or DN as d tobbe, and VG be equal tobe equal to A O, VG == ^=then DN willVZ =O, and GD or NX VZ VG equal274termTHE MATHEMATICAL PRINCIPLES [BOOK Hnbbnn -f- nbbU ! J x *=rr be resolved into an infinite series -r- +A Of A" A.n3 -- 3nn + 2/iX O +n" ~ x bb O 3 g^TT-T ,&c,,andGD will be equal X 00 O 2 +c bb d nbb + ?m -toC -A--T-+-O- -r O - ~ e A" e A" + l 2An-f+ Hi^T t "。bb 3> &c- The second tcrm - O - -T 6An + e An4- lseries is to be used for 0,0, the third^66O 2 for Roo, the fourth-。~r~3 bbO 5 for So 3. And thence the density of the mediumOof this-, in any place G7 will be2dnbb nub*and therefore if in VZ you take VY equal to n X VG, that density is renw IT- j ^ *2rf//66 /mfi 4ciprocally as XY. For A 2 and A 2 -- A + r are thetc/ o^x ./。_"squares of XZ and ZY. Hut the resistance in the same place G is to theforce of gravity as 3S X - to 4RR, that is, as XY toAnd the velocity there is the same wherewith the projected body wouldmove in a parabola, whose vertex is G, diameter GD, and latus rectum2XY 2or --------- --. Q.E.I. R nn VGACHTSCHOLIUM.In the same manner that the density of the medium comes out to be asS X AC .Tjr m ^ r- 1) if the resistancelx X HIis put as any power V" of the velocityV, the density of the medium willcome out to be asB C D E Q. xSAnd therefore if a curve can be found, such that the ratio of to4 oR iSEC. II.J OF NATURAL PHILOSOPHY, 275n 1, or ofgr^ to may be given ; the body, in an unizHTACform medium, whose resistance is as the power V" of the velocity V, willmove in this curve. But let us return to more simple curves.Because there can be no motion in a parabola except in a non-resisting medium, butin the hyperbolas here described it is producedby a perpetual resistance ;it is evident thatthe line which a projectile describes in an