2/0 FHILOSOPHICAL TRANSACTIONS. [ANNO 1 685-6; 



secant will be «/ ^-^nr ^ or '■'" '• so then the sum and difference of the 



quotient above found, and this tangent will be the roots of the equation, and 

 the tangents of the elevations sought. 



Note, That in descents, if the tangent exceed the quotient, as it does when 

 J!) A is more than bb, the direction of the lower elevation will be below the horizon; 

 and if ph = bb, it must be directed horizontal, and the tangent of the upper 



elevation will be y. Note also, that if 4 bb + 4ph'm ascents, or 4 bb — 4ph 



in descents, be equal to pp, there is but one elevation that can hit the object, 



and its tangent is|^ ; and \{ Abb -\- 4ph'm ascents, or 4 hb — 4ph'm descents, 



exceed pp, the object is without the reach of a project thrown with that velo- 

 city, and then the thing is impossible. 



From this equation, 4 bb + 4ph = pp, are determined the utmost limits of 

 the reach of any project, and the figure assigned, wherein are all the heights 

 on each horizontal distance, beyond which it cannot pass ; for by reduction of 



that equation h will be found = -j-j& in heights, and ^p in descents; 



from whence it follows, that all the points h are in the curve of the parabola, 

 whose focus is the point from whence the project is castj and whose latus rectum 

 OS parameter to the axis is = p. Likewise, from the same equation may the 

 least parameter or velocity be found, capable to reach the object proposed ; for 



bb=i\pp + /> A being reduced, 4./) will be = ^ bb + hh + k 4 1" descents I" ' 

 which is the horizontal range at 45 degrees, that would just reach the object, 

 and the elevation requisite will be easily had ; for dividing the semiparameter so 

 found by the given horizontal distance b, the quotient into radius will be the 

 tangent of the elevation sought. This rule may be of good use to all bombar- 

 diers and gunners, not only that they may use no more powder than is neces- 

 sary to throw their bombs into the place assigned, but that they may shoot with " 

 much more certainty, because a small error committed in the elevation of the 

 piece will produce no sensible difference in the fall of the shot ; for which rea- 

 sons, the French engineers, in their late sieges, have used mortar pieces in- 

 clined constantly to the elevation of 45°, proportioning their charge of powder 

 according to the distance of the object they intend to strike on the horizon. 



And this is all that need to be said concerning this problem, of shooting 

 upon heights and descents. But if a geometrical construction of it be required, 

 "I think the following is as easy as any can be expected, which I deduce from the 



foregoing analytical solution, viz. - = ^ + i^ * ^ ~bb "" ' " * Having made 



