[ *45 ] 
required Elevations 5 whence the Elevations them- 
felves are known. Q. E. I. 
Corol. 1 . 
Hence, if the Elevation of the Piece, with the Di- 
ftance and the Height (or Depreftion) of the Objcd 
be given, the greateft horizontal Amplitude may be 
found : For it will be AB : BC : : Radius : Tang, of 
BAC 5 whence CAE) is alfo known. 
Then, £ CAE) : S.ACE {A HE) : \AE ( HE ) :• 
AE. 
And, S. AE)C : Radius : ; AB : AE. 
Therefore, by compounding thefe Proportions, 
we have S. CAE x S AEC : Radius xS. ACE : : 
AB: AE i which is equal to twice the required 
Amplitude, by Conftrudion. 
Corol. 2. 
Moreover, if the Elevation, and the greateft ho- 
rizontal Amplitude be given, the Amplitude of the 
Projedion on any afeending or defeending Plane AE y 
whofe Inclination FAE is alfo given, may from 
hence be derived. For, S.AHE (ACE): S-EAH 
(CAE) : ; AE (2BQ ) : EH (AE) and ACE : 
S. AEC : :AE : AC ; whence, by compounding the 
two Proportions, Scf. S . ACE : S. CAE xS.AEC 
; ; 2B£E AC ; from which AC is known. 
Corol. 3 . 
Since it appears, that the Triangles AEB and 
EH 1 are equal and alike in all refpeds, and, there- 
fore, the horizontal Diftance AB, univerfally , equal 
to the Perpendicular HI, it is manifeft, that, when 
HI 
