At present, we are about to consider this subject merely as a particular case of motions regulated by gravitation, reserving the particular consideration of the modifications of these motions by the resistance of the air, till we shall have made ourselves acquainted with the general laws of such resistance.
Let a body (Plate CCLXXXVI. Fig. 1.) be projected in any direction AB, which deviates from the vertical ASV. Then it would move on in this direction, and in equal suc ceeding moments would describe the equal spaces AB, BH, HI, 1K, KL, Ste. But suppose, that when the body is at B it receives an instantaneous impulse in the direction of the vertical BB', such that by this impulse it would de scribe the line B b uniformly in the same time that it would have continued its motion along Or, to speak more accurately, let the motion or velocity B b be com pounded with the motion 13H. The body must describe the diagonal BC of a parallelogram B b CH, and, at the end of this second moment, it must be in C, in the verti cal line HCC', and moving with the velocity BC. There fore, in the third moment it would describe CN, equal to BC. But let another impulse in the direction of the ver tical CC' generate the velocity C c, equal to 13 6. By the composition of this with the motion CN, the body will de scribe the diagonal CD of the parallelogram C c DN, and at the end of the third moment must he in D, moving in the direction and with the velocity C]). It would describe 1)0 equal to CD in the fourth moment. Another impulse of gravity D d, in the vertical, and equal to either of the former impulses, will make the body describe DE; and an equal impulse E c will deflect the body into IT; and another impulse Ff will deflect it into FG, &cc.
Thus it is plain that the body, by the composition of these equal and parallel impulses, will describe the poly gonal figure ABCDEFG, all in one vertical plane, and in every instant or point, such as E, will be found in the ver tical line KE, drawn from the point at which it would have arrived in that instant by the primitive projection.
Now, let the interval between these impulses be dimi nished, and their number be increased without end. It is evident that this polygonal motion will ultimately coincide with the motion in a path of continued curyation, by the continual and unvaried action of gravity.
The line described by the body has evidently the follow ing properties.
1st, If a number of equidistant vertical lines BB', HCC', IDD', KEE', Exc. be drawn, cutting the curve in B, C, D, E, Sze. and if the chords AB, BC, CD, DE, Sze.
drawn through the points of intersection, be produced till they cut the verticals in H, N, 0, P, Szc. the intercepted portions IIC, ND, OE, PF, Sze. are all equal.
2d, The curve is a parabola, in which the verticals BB', CC', Sze. are diameters. The property mentioned in the last paragraph belongs exclusively to the parabola. As the circle is the curve of uniform deflection in the direc tion of the radius, so the parabola is the curve of uniform deflection in the direction of the diameter. That the curve in which the chords drawn through the intersection of equidistant verticals cut off equal portions of these ver ticals is a parabola, is easily proved in a variety of ways. Since B b, C c, D d, E e, are all equal, and the verticals arc equidistant, B c d E must be a straight line. So must C de F; BE must he parallel to CD, and CF to DE. Therefore BF and CE are parallel, and are bisected in m and n by the vertical DD'. Also, if FC be produced till it meet the next vertical in 1, i B is equal to D m. All this is very plain. Hence B, or Dm:dm= BF : m F = m : o E; but d : D o — 712 0 E ; therefore D in: D o= : o ; and D, E, F are in a parabola, of which D in'is a diame ter, and o E, m F are semiordinates. We should prove, in the same manner, that BG is parallel to CF, and AG to BF, and D nz : DD'=m : and the points D, F, G, in the same parabola.
Thus we have demonstrated, that the equal and parallel impulse of gravity produces a motion in a parabola, whose diameters are perpendicular to the horizon. This was the great discovery of Galileo, and the finest example of his genius. His discoveries in the heavens have indeed at tracted more notice, and he is oftener spoken of as the first person who sheaved the mountains in the moon, the phases of Venus, the satellites of Jupiter, Scc. But in all these he was obliged to his telescope ; and another per son who had common curiosity would have seen the same things. But, in the present discovery, every step was an effort of judgment and reasoning, and the whole investi gation was altogether novel. No attempt had been made, since the first dawn of mechanical science, to explain a curvilineal motion of any kind; and even the law of the composition of motion, though faintly seen by the ancients, had never been applied to any WC (except by Stevinus) tili this sagacious philosopher saw its immense import ance, and brought it into constant service.