62 Motion of jected downwards. became an object of attention; and Mr Muller, through inadvertency, or want of discernment, has fallen into this mistake, and with that arrogance and self-conceit which mark all his writings, has made this mistake a fundamental principle, because it led him to establish a novel set of doctrines on this subject. He was fretted at the superior knowledge and talents of Mr Simpson, his inferior in the academy, and was guilty of several mean attempts to hurt his reputation. But they were unsuccessful. We might proceed to consider the motion of a body body pro-projected downwards. While the velocity of projection is less than the terminal velocity, the motion is determined by what we have already said: for we must compute the height necessary for acquiring this velocity in the air, and suppose the motion to have begun there. But if the velocity of projection be greater, this method fails. We pass it over (though not in the least more difficult than what has gone before), because it is of mere curiosity, and never occurs in any interesting case. We may just observe, that since the motion is swifter than the terminal velocity, the resistance must be greater than the weight, and the motion will be retarded. The very same process will give us for the space describe M: A Χλ น ย +v น :t, and we have 0,43429: 0,22122 น-บ g =21", 542 10",973 the time required. This is by far the most distinct way of conceiving the subject; and we should always keep in mind that the numbers or symbols which we call logarithms are really parts of the line MI in the figure of the logistic curve, and that the motion of a point in this line is precisely similar to that of the body. The Marquis Poleni, in a dissertation published at Padua in 1725, has with great ingenuity constructed logarithmics suited to all the cases which can occur. Herman, in his Phoronomia, has borrowed much of Poleni's methods, but has obscured them by an affectation of language geometrically precise, but involving the very obscure notion of abstract ratios. It is easy to see that น is the cotangent of the complement of an arch, whose radius is 1, and whose sine is: For let KC (fig. 6.) be=u, and Fig. 6. BE=v; then KD≈u+v, and DA=u-v. Join KB and BA, and draw CG parallel to KB. Now GA is the tangent of BA, complement of HB. Then, by similarity of triangles, GA : AC=AB=: BK= √AD: √DK=√uv: √u+v and AC BA)= cotan. GA Therefore tural sines, or for log. among the logarithmic sines, This fluent needs no constant quantity to complete it, or rather Co; fort must be when vo This will evidently be the case: for then L√/" + is L√ LI, =0. บ H u+v น-บ น But how does this quantity Χ λι signify a Mg time? Observe, that in whatever numbers, or by whatever units of space and time, u and g are expressed, ย น g L locity is communicated or extinguished by gravity; of one in vacuo. In the next place, let a body, whose terminal velothe ascent city is u, be projected perpendicularly upwards, with of a body velocity V. It is required to determine the height to which it ascends, so as to have any remaining velocity, and the time of its ascent; as also the height and time in which its whole motion will be extinguished. projected perpendicularly. 26 terminal velocity, or V=u; then v= If V= We have now We must in the last place ascertain the relation of g(u2+v3) for the expression of f; the space and the time. g( Here £(u*+v3) ;=~ (art. FLUXIONS) is an arch whose tangent i=—v, and i—— = g u S= -v v, and s ยย +C, = v v and t = แข g with that with which the ground and the complete fluent will be s≈ u2 + V3 rise. Then sh when vo; and # + Va u u L = X λ h=- X u ent is tx (arc. tan. u tities within the brackets express a portion of the arch of a circle whose radius is unity; and are therefore ab We learn from this expression of the time, that however great the velocity of projection, and the height, to which this body will rise, may be, the time of its mited. ascent is limited. It never can exceed the time of falling from the height a in vacuo in a greater proportion than that of a quadrantal arch to the radius, nearly the proportion of 8 to 5. A 24 pound iron ball cannot continue rising above 14 seconds, even if the resistance to quick motions did not increase faster than the square of the velocity. It probably will attain its greatest height in less than 12 seconds, let its velocity be ever so great. In the preceding example of the whole ascent, e=0, and 68 This time compared in bodies projected in air and in vacuo. Fig. 6. Cor. 1. The time in which a body, projected in the air with any velocity V, will attain its greatest height, is to that in which it would attain its greatest height in vacuo, as the arch whose tangent expresses the velocity is to the tangent; for the time of the ascent in the air V บ is x arch; the time of the ascent in vacuo is g g V X tan. g Now istan. and Vux tan. and If therefore a body be projected upwards with the 2 2. The height H to which a body will rise in a void, rate notions of the air's resistance. Mr Robins's me- and a considerable deviation from their intended direc tion does not cause any sensible error in the conse- 70 71 But we must now proceed to the general problem, of obto determine the motion of a body projected in any di-lique prorection, and with any velocity. Our readers will be-jection. lieve beforehand that this must be a difficult subject, when they see the simplest cases of rectilineal motion abundantly abstruse: it is indeed so difficult, that Sir Isaac Newton has not given a solution of it, and has This prothought himself well employed in making several ap-blem not proximations, in which the fertility of his genius appears solved by in great lustre. In the tenth and subsequent proposi- Newton. tions of the second book of the Principia, he shows what state of density in the air will comport with the motion of a body in any curve whatever: and then, by applying this discovery to several curves which have some similarity to the path of a projectile, he finds one which is not very different from what we may suppose to obtain in our atmosphere. But even this approximation was involved in such intricate calculations, that it seemed impossible to make any use of it. In the second edition of the Principia, published in 1713, Newton corrects some mistakes which he had committed in the first, and carries his approximations much farther, but still does not attempt a direct investigation of the path which. a body will describe in our atmosphere. This is somewhat surprising. In prop. 14. &c. he shows how a body, actuated by a centripetal force, in a medium of a V density varying according to certain laws, will describe an eccentric spiral, of which he assigns the properties, and the law of description. Had he supposed the density constant, and the difference between the greatest and least distances from the centre of centripetal force exceedingly small in comparison with the distances themselves, his spiral would have coincided with the path of a projectile in the air of uniform density, and the steps of his investigation would have led him immediately to the complete solution of the problem. For this is the real state of the case. A heavy body is not acted on by equal and parallel gravity, but by a gravity inversely proportional to the square of the distance from the centre of the earth, and in lines tending to that centre nearly; and it was with the view of simplifying the investigation, that mathematicians have adopted the other hypothesis. and the height to which it rises in the air is 22 + V1 ; therefore H: h= V1 2g Mg Mg =V1: X 2λ =M•V1 : u3×λ WE have been thus particular in treating of the perpendicular ascents and descents of heavy bodies through the air, in order that the reader may conceive distinctly the quantities which he is thus combining in his algebraic operations, and may see their connection in nature with each other. We shall also find that, in the present state of our mathematical knowledge, this simple state of the case contains almost all that we can deterNecessity mine with any confidence. On this account it were to of further be wished that the professional gentlemen would make experiments. many experiments on these motions. There is no way that promises so much for assisting us in forming accu 69 72 Soon after the publication of this second edition of Disputes the Principia, the dispute about the invention of the among fluxionary calculus became very violent, and the great British and promoters of that calculus upon the continent were in foreign the habit of proposing difficult problems to exercise the ticians. talents of the mathematician. Challenges of this kind frequently passed between the British and foreigners. Dr mathema 73 Bernoulli's solution. Fig. 7: would cause the body to describe uniformly in the time 2 gy the ascent will be r+ The same fluxionary symbol Dr Keill of Oxford had keenly espoused the claim of PROBLEM. To determine the trajectory, and all the will express the retardation during the descent, because in the descent the ordinates decrease, and y is a negative quantity. The diminution of velocity is . This is proportional to the retarding force and to the time of its action jointly, and therefore-vr+xi; but the time is as the space z divided by the velocity v; therefore B and บ gy. Because m N is the deflection by gravity, it is as the force g and the square of the time -- y= =g. The fluxion of gy; therefore ૪ 2 a 2, or a y=zy, for the fluxionary equation of the Newton into his investigation of the spiral motions. And the equation is evidently an equation connected a y with the logarithmic curve and the logarithmic spiral. But we must endeavour to reduce it to a lower order of fluxions, before we can establish a relation between x, x, and y. x Let p express the ratio of y to x, that is, let p be= , or p xy. It is evident that this expresses the inclination of the tangent at M to the horizon, and that p is the tangent of this inclination, radius being unity. Or it may be considered merely as a number, multiply ing x, so as to make it =y. We now have y = p2x2, and since 21= x2 + y2, we have x1 = x2 + p2 x2, = 1 +p ×3, and x=x1+p'. 3d, We get y by the area of a third curve whose ab卫 scissa is p, and the ordinate is 9 76 To com The problem of the trajectory is therefore completely solved, because we have determined the ordinate, abscissa, and arch of the curve for any given position of its tangent. It now only remains to compute the magnitudes of these ordinates and abscissæ, or to draw them pute the magnitude by a geometrical construction. But in this consists the of the ordilengths of x and y, can neither be computed nor exhi- abscissa. difficulty. The areas of these curves, which express the nate and bited geometrically, by any accurate method yet discovered, and we must content ourselves with approximations. These render the description of the trajectory exceedingly difficult and tedious, so that little advantage bas as yet been derived from the knowledge we have got of its properties. It will however greatly assist our conception of the subject to proceed some length in this construction; for it must be acknowledged that very few distinct notions accompany a mere algebraic operation, especially if in any degree complicated, which we confess is the case in the present question. fig. 8. Let B m NR (fig. 8.) be an equilateral hyperbola, of Plate which B is the vertex, BA the semitransverse axis, CCCCXLII. which we shall assume for the unity of length. Let AV be the semiconjugate axis BA, unity, and AS the assymptote, bisecting the right angle BAV. Let PN, pn be two ordinates to the conjugate axis, exceedingly near to each other. Join BP, AN, and draw Bß, N, perpendicular to the assymptote, and BC parallel to AP. It is well known that BP is equal to NP. Therefore PN1 =BA2 + AP'. Now since BA=1, if we make AP-p of our formulæ, PN is 1+p2, and Pp is= p, and the area BAPNB=ƒ, ¿ √1+p"]: That is to say, the number ƒɔp √1+p* (for it is a number) has {p√1+p2+ hyperbolic logarithm p+√√1+p3. Now let AMD be another curve, such that its ordinates V m, PD, &c. may be proportional to the areas AB m V, ABNP and may have the same proportion to AB, the unity of length, which these areas have to ABCV, the unity of surface. Then VM: VC= V m BA: VCBA, and PD: P >=PNBA: VCBA, &c. These ordinates will now representƒ‚¿√1+p with reference to a linear unit, as the areas to the hyperbola represented it in reference to a superficial unit. Again, |