VOL. II.] PHILOSOPHICAL TRANSACTIONS. 357 



tion of their velocities, and of the direct proportion of the sines of their angles 

 of incidence. 



Let us suppose also the following axiom, to wit, that percussions, aeteris 

 paribus^ are in the direct proportion of the velocities, by which the moving 

 body approaches the resisting plane. Suppose CF the plane, (fig. 7, pi. 7); 

 and let there be two moveable bodies, in every respect alike, which approach 

 with an equal motion the plane CF from the point A, in the right line* 

 AD, A F ; I say, the percussion at the point D is to the percussion at the point 

 F, in the ratio compounded of the ratio of the velocity in the right line A D, to 

 the velocity in A F, and of the ratio of the sine of the angle A D E, to the sine 

 of the angle A F E. From the point A to the plane C F draw the perpendicular; 

 also make the right line A C equal to the right line A F, A B equal to the right 

 line A D, and the plane B G H parallel to the plane C F. Let us suppose the 

 moving body, as before mentioned, alike in all respects, to be moved equally in 

 the right line A C, with the same velocity with which the body is moved in the 

 right line A D ; then because the planes B G H, C F are parallel, and the mo- 

 tion in the right line A C is equable, therefore the moving body approaches the 

 plane BH with the same velocity with which it approaches the plane CF, and 

 thence the percussions at the points B and C are equal ; also the percussion at 

 the point D, is to the percussion at the point B, as the right line A E to the 

 right line A H, or (because of the equals A B, AD) as the sine of the angle 

 A D E to the sine of the angle A B H, which I thus prove : the velocity of the 

 body in the straight line A D, is equal to the velocity in the right line A B, 

 which is equal to AD, and therefore both the right lines AD, AB, are passed 

 over in the same time ; and so in the same time the accessions to the resisting 

 planes A E, A H, are performed ; therefore the velocities of the accessions to 

 the resisting planes are in the direct ratio of A E to A H, and likewise the per- 

 cussion at the point D is to tlie percussion at the point C, in the same ratio 

 of A E to A H, namely, as the sine of the angle of incidence A D E to the 

 sine of the angle of incidence ACE, or A F E. But because the right lines 

 AC, A F, are equally inclined to the plane C F, the moving bodies in the 

 right lines A C, A F, approach to the plane C F, in the same ratio in which they 

 are moved in the right lines AC, A F ; and therefore the percussion at C is to 

 the percussion at F, in the ratio of the velocity of the motion at A C, or in 

 A D, to the velocity of the motion in A F. But since it is before demonstrated 

 that the percussion at the point D, is to the percussion at the point C, in the 

 ratio of the sine of the angle A D E to the sine of the angle A F E ; and now it 

 is demonstrated that the stroke at the point C, is to the percussion at the point 

 F, as the velocity of the motion in AD to the velocity in AF: therefore (by 



VOL. I. K K 



