vol- XVI,] fHILOSOPHrCAL TRANSACTION*. 26g 



tions. For, as c is to f :: so is ?^ = GB, by the 6 prop, to^ = GF, that 



is, as radius to sine of elevation, so is the parameter to the line GF ; so that 

 the lines GF are as the sines of elevation, and the times are proportional to 

 those lines ; therefore the times are as the sines of elevation. — Ergo, &c. 



Prop. IX. Problem I. A projection being made at pleasure ; having the distance 

 and altitude or descent of an object, through which the project passes, together 

 with the angle of elevation of the line of direction ; to find the parameter and 

 velocity; that is, in fig, 2, having the angle FGB, GM, and MX. — Solution. 

 As radius to secant of FGB, so GM the distance given, to GL ; and as radius 

 to tangent of FGB, so GM to LM. Then LM — MX in heights, or + 

 MX in descents, or else MX — ML, if the direction be below the horizontal 

 line, is the fall in the time that the direct impulse given at G, would have car- 

 ried the project from G to L, = LX = GY ; then, by reason of the parabola, 

 as LX or GY is to GL or YX, :: so is GL to the parameter sought. To find 

 the velocity of the impulse by prop. 1 and 4, find the time in seconds that 9 

 body would fall through the space LX, and by that dividing the line GL, the 

 quotient will be the velocity, or space moved in a second sought, which is always 

 a mean proportional between the parameter and 16 feet I inch. 



Prop. X. Prob. 2. Having the parameter, the horizontal distance, and height 



or descent of an object ; to find the elevations of the line of direction necessary 



to hit the given object ; that is, having given GM, MX, and the greatest 



random equal to half the parameter ; to find the angles FGB. — Let the tangent 



of the angle sought be = t, the horizontal distance GM = b, the altitude of 



the object MX = h, the parameter = p, and radius = r; then it will be, as r 



., J . fb T.,T .tb — , fin ascents > ^v yvtb 



tot8oao-=ML; and- + A|j^j^^^^^^J = LX, and^- T ph ^ 



GL^ = XY^* by the parabola ; but ^* + '-^*- = GL' (47, 1 Euclid). There- 

 fore^ — T ph = bb -{- ; which equation transposed, is ='' T ph 



— bb; and divided by bb, is — = ^ + |^ — 1. this equation shows that 



the question has two answers, and its roots are - = -^ X // PP + *ph _ 

 ^ r 26 ^ 4,bb » 



from which the following rule is derived ; divide half the parameter by the hori^ 

 zontal distance, and reserve the quotient, viz. ^ ; then say, as square of the 

 distance given is to the half parameter, so is the half parameter + double 

 1, descent \ ^° ^^^ square of a secant = ^^^/ >tbe tangent answering to that 



