VOL. XXXI.] PHILOSOPHICAL TRANSACTIONS. 511 



in |3. Then since the space cc, and therefore also (3c, is supposed very small, 

 the velocities with which they are described will be nearly equable; therefore 

 the spaces cc, Bb or cj3, being described in the same time, will be as the ve- 

 locities with which they are described ; and again the velocities will be as 

 the spaces. Let the points c and c coincide, and then these ratios will be 

 accurate. But in that case, because of the similar triangles cj3c, pbc, it is 

 cjS : j3c :: fb : bc; therefore the velocities with which Bb and (3c are described, 

 are as fb and bc, that is, they are equal. But the velocity by which Bb is 

 described, is that with which the projectile moves at the point a, and the other 

 velocity by which (3c is described, is that which a body acquires by falling 

 through the altitude bc, the 4th part of the parameter at the point a. There- 

 fore the velocity of the projectile in any point a, is equal to the velocity which 

 a body can acquire by falling from a height equal to a 4th part of the parameter 

 at that point, a. e. d. 



Prop. III. Having given the Velocity and Direction of Projection ; to Jind 

 the Trajectory of the projected Body. — 1. Let the body be projected from the 

 place A, fig. 3, in the direction ab. Draw ac in the direction of gravit}{, 

 viz. perpendicular to the horizon, and of that length which is due to the velo- 

 city of projection at a. Draw ap = ac, making the angle bap = the angle 

 BAC. Draw cd perp. to ac, that is parallel to the horizon, meeting pd, pa- 

 rallel to AC, in D. Bisect pd in e : then ep will be the axis, and e the prin- 

 cipal vertex of the parabola described by the projectile. Hence the trajectory 

 will be described by the known properties of the parabola, a. e. p. 



For AC is the 4th part of the parameter at the point a. Hence the rest 

 appears from conies. 



1. At any point g of the trajectory, draw gh parallel to ac, and meeting 

 CD in h; then hg is the altitude due to the velocity of the projectile at g, that 

 is, through which a body by falling can gain that velocity, a. e. p. 



This also appears from Prop. 2, and from conies. 



Scholium. If to the points a and c, fig. 2, there be drawn the tangents ab 

 and CG, meeting the perpendiculars to the horizon, cb and ag, in b and g; 

 then the velocities at a and c will be to each other, as the intercepted parts, 

 ab, cg, of the tangents. 



Prop. IV. From One Experiment made, to find the Velocity of Projection. 

 — Let the body be projected from the point a, fig. 2, in any direction ab, and 

 observe the point struck c. In the direction of gravity draw cb, meeting ab 

 in B, and take l a 3d proportional to cb and ab. Then will the 4th part of l 

 be the altitude due to the projectile velocity at a. a. e. i. 



