The process employed by Galileo in this investigation, and which has been copied by almost all the writers on the subject, is considerably different from the one now gone through. Galileo supposes the heavy body to fall in the vertical 13 with a uniformly accelerated motion, describ ing spaces as the squares of the times. He supposes this motion to be compounded with the uniform motion in the direction of the tangent BR. Then, supposing that B t and BT are fallen through while B r and BR are described by the motion of projection, it follows, that because B r is to BR as the time of describing B r to the time of describ ing BR, we shall have B t : BT=B : Therefore, completing the parallelograms B t C r, BTSR, we have B t : BT=t C: '1'S, and the points B, C, S are in a para bola, whose diameter is BT, and has BR a tangent in B.
No doubt, the result of these suppositions agrees per fectly with the phenomena, and gives a very easy and ele gant solution of the question. But, in the first place, it is more difficult, or takes more discourse, to prove this con tinued composition of motion (almost peculiar to the case) than to demonstrate the parabolic figure: and, secondly, it is not a just narration of the fact of the procedure of nature. There is no composition of such motions as are here supposed. When the body is at C, there is not a mo tion in the direction parallel to B r, compounding itself with a motion in the vertical, having the velocity which the falling body would have as it passes through the point t. The body is really moving in the direction CS of the tangent to the parabola, and it there receives the same in finitesimal impulse of gravity that it received at B. Its deflection, therefore, from the line 'of its motion, does not make any finite angle with that motion. Therefore, al though Galileo's demonstration does very well for a mere mathematical process, like the navigator's calculation of the ship's place by tables of difference of latitude and de parture, it by no means answers the purpose of the philo sophical investigation of a natural phenomenon. The me thod we have followed is a bare narration of the facts ; con sidering the motion of the body in every instant as it real ly is, and stating the force then really affecting its motion.
We have not scrupled to make use of the method em ployed by Newton in the demonstration of his fundamen tal proposition on curvilineal motions, first conceiving the action of gravity to be subsultory, and the motion to be polygonal, and then inferring a similar result from the un interrupted action of gravity. But if any person is so fas
tidious as to object to this, (as John Bernoulli has done to Newton's method,) he may remark, that the motion 13 b, which we compared with BH, in order to produce the mo tion BC, is just double of the space B t, through which the body falls during the motion along BII. Therefore the figure will be such, that the curvilineal deflection will be one half of B b, or of IIC, and the tangent to the curve, whatever it is, will bisect HC. Then, during the next moment, since the deflective action of gravity is sup posed the same, the body will be as much deflected from its path in C, that is, from the new tangent CS, whatever direction that tangent may have, as it was in the preceding moment. This gives us 8 D equal to r C, and this obtains throughout. Without entering on any discussion on the progress of the deflection in the different points of the arch BC or CD, it is enough for our purpose to chew that the curve described is such, that when equidistant verti cals are drawn, and tangents drawn through their inter sections with the curve, the portions of the verticals cut off by the tangents are everywhere equal. This also is a property of the parabola exclusively. That BCD is a parabola, of which BT is a diameter, and BR a tangent, is easily seen. For, drawing D u parallel to BR, it is plain that v N=2 r C, and ND =2 a D,=2 r C. There fore v = 4r C, and B u B t, and 11 t: B u—t : u And we should prove, in the same manner, that y E=9 r C, &c.
Having thus ascertained the general nature of the path of a projectile, we must now examine its motion in this path, determining its velocity in the different points, and the time employed in the description of the arches. For this purpose we must first ascertain the precise parabola described under the conditions of the projection, that is, depending on its direction and velocity. To do this in a way naturally connected with the acting forces, we shall consider the velocity of projection as having been gene rated by falling through some determinate height.
Let us therefore suppose that the body is projected from B, Plate CCLXXXVI. Fig. 1. in the direction BR, with the velocity acquired by falling through the vertical VB. Make la equal to VB, and BR equal to VT or 2 VB, and, lastly, draw TS parallel to BR, meeting the parabola in S.