[ 142 ] 
and thefe are as the Difiances perpendicularly de- 
fended (by the precedent ). Whence, univerfally , 
if both the Elevations and the Velocities differ, the 
Amplitudes will be to each other in a Ratio com- 
pounded of the Ratio s of the Sines of double the 
Angles of Elevation, and of the duplicate Ratio’s 
of the Velocities, or impelling Forces. 
Problem II. 
! The Angle of Elevation , and the great eft horizontal 
Amplitude , being given , to find at what T) iftance 
the Piece ought to be planted , to hit an Object , 
whofe Tiftance, above or below the Plane of the 
Horizon , is alfo given . 
Let AB {Fig. 2 and 3.) be the Plane of the Ho- 
rizon, BC the perpendicular Height or Deprefllon 
of the Objed, and AB the required Diftance : Alfo 
let BC be produced to meet the Line of Diredion 
AT) in T ), and let P be the Place where the Path 
of the Projedile would meet the Horizon ; more- 
over, let Pffbe, perpendicular to A'P, and CM 
parallel to AT). Then, by the preceding Problem, 
it will be as Radius: the Sine of iBAT) the 
given (or greateft) Amplitude : A P > which there- 
fore, is known. 
Moreover, the Areas of fimilar Triangles being 
as the Squares of their homologous Sides, we have 
APxPQgABxBT : : A $* : AT>*. But A£f :AT> 2 : : 
ABxBTJ: :QP :T)C (from Principles already ex- 
plained) therefore, by Equality, APxP^fAB'X. 
BT) : : QP : T)C s and confequently AP : AB : ; 
B T ) : CT) j but (becaufe of the parallel Lines 
CM and AT)) BD .CT) \ \AB\AM\ whence, 
again 
