£^3] 
BDC, and let fall BE perpendicular to and the Jngl^ 
EBQ will be equal to the Jngle ABD, and the Triangle BCE> 
will be like to the Triangle BDA ; wherefore it will be as 
ABto AD,fbBC or twice BD • to BE, that is as Raiim to 
Co-fme^ fo twice Sine^ to Sine of the double Arch. And as AB 
to BD, fb twice BD or BC, to EC, that is as Radius to Sine; 
fb twice that Sine to the Ver fed-fine of the double Arch\ which 
two Analogies refblved into Equations^ are the Profcftions 
contained in the Lemma to be proved. 
Prop. VI. The Horizontal diftances of Project ions made 
with the fame Velocityy at feveral Elevations of the Line of 
direction, are as the Sines of the AoMhled Angles of Elevation. 
LetGB (/^.av* the Horizontal diftance be = ^, the fine 
of the Angle of Elevation, FGB, be =: j-, its Ct3-/-??<? = 
zzrr, and the Parameter =p. It will be as c tos'^ fb ^ to 
S Z/ p s z> 
^=^B=:GCj and by reafbn of the Parabola-^ =:tothG 
c ^ c 
fquare of CB, or GF,. Now as t to r, fo is z to zrj=^ GF, and 
e 
z z r r ' "P s z 
itsfquare — — will be therefore=to^ : which Eo[mtionvt- 
c c c 
P S C Q. S C ' 
duced, will be ^ — But by the former Lemma is 
equal to the Sine of the double Angle ^ whereof s is the Sine : 
wherefore 'twil be as Kadiu<s to Sine of double the Angle FGB^ 
fb is half the Parameter ^ to the Horizontal rang or diUance 
fought ; and at the feveral Elevations^ the ranges are as the 
^nes of the double of Elevation ^E.D. 
Corollary. 
Hence it follows, that half the Parameter is the greateft 
'Kandony and that that happens at the Elevation of 45 de- 
grees, the fine of whofe double is Radiusy Like wife that 
die Ranges equally diftant above and below 45 are .qual, as 
B 2 are 
