46b rHILOSOPHICAL TRANSACTIONS. [aNNO 1685-6. 



the angle EBC will be equal to the angle BAD, and the triangle BCE will be 

 similar to the triangle ABD ; therefore it will be as AB to AD, so BC, or twice 

 IBD, to BE, that is, as radius to cosine, so twice sine to sine of the double 

 arch. And as AB to BD, so twice BD or BC to EC ; that is, as radius to sine, 

 so twice that sine to the versed sine of the double arch ; which two analogies re- 

 solved into equations are the propositions contained in the lemma to be proved. 

 Prop. VI. The horizontal distances of projections made with the same velo- 

 city, at several elevations of the line of direction, are as the sines of the dou- 

 bled angles of elevation. — Let GB, fig. 2, the horizontal distance be = z, the 

 sine of the angle of elevation, FGB, be = s, its cosine = c, radius = r, and 



the parameter = p. It will be as c to j, so z to - = FB = GC; and by reason 

 of the parabola, ^ =r to the square of CB, or of GF. Now as c to r, so is 

 2 to — = GF, and its square ^^^ will be therefore = to^; which equation 



an ft f , i? i C 



reduced, will be^— = z. But, by the former lemma, — is equal to the sine 



of the double angle of which s is the sine, therefore it will be as radius to sine 

 of double the angle FGB, so is half the parameter to the horizontal range or 

 distance sought ; and at the several elevations, the ranges are as the sines of 

 the double angles of elevation. Q. E. D. 



Carol. Hence it follows, that half the parameter is the greatest random, and 

 which happens at the elevation of 45°, the sine of its double being radius. Also, 

 that the ranges equally distant above and below 45 are equal, as are the sines of 

 all doubled arches to the sines of their doubled complements. 



Prop. VII. The altitudes of projections made with the same velocity at se- 

 vei'al elevations, are as the versed sines of the doubled angles of elevation. 



For, as c is ta* :: so is ^ = GB, to ^ = BF, and VK = VR = 4- BF, 



the altitude of the projection = ^*-. Now, by the foregoing lemma, -^ = to 



the versed sine of the double angle, and therefore it will be as radius to versed 

 sine nf double the angle FGB, so is the height of the parameter, to the height 

 of the projection VK ; and so these heights, at several elevations, are as the 

 said versed sine. Q. E. D. 



Carol. From hence it is plain, that the greatest altitude of the perpendicular 

 projection is a 4th of the parameter, or half the greatest horizontal range ; the 

 versed sine of 180° being = 2 r. 



Prop. Fill. The lines GF, or times of the flight of a project, cast with the 

 same degree of velocity, at different elevations, are as the sines of the eleva- 



