Mifcellanea Cur to fa. g i 5 



Let the Arch B C (in Fig, 1. Tab. 5- J be double 

 the Arch B F, and A the Center ; draw the H«- 

 dii A B, AF, AC, and the Chord BD Q and 

 let fall BE perpendicular to AC, and the An- 

 gle EBC, will be equal to the Angle ABD, 

 and the Triangle B C E, will be like to the Tri- 

 angle BDA; wherefore it will be as A B to 

 A D, fo B C or twice B D, to B E ; that is, is 

 Radius to Co-fine, fo twice Sine to Sine of the 

 double Arch. And as A B to BD, fo twice 

 B D or B C to EC, that is, as Radius to Sine, 

 fo twice that Sine, to the Verfed Sine of the 

 double Arch which two Analogies refolved into 

 Equations, are the Propsfitions contained in the 

 Lemma to be proved. 



Prop. VI. The Horizontal Diftances of Pro- 

 jeBions made with the fame Velocity, at feveral 

 Elevations of the Line of Direction, are as the 

 Sims of the doubled Angles of Elevation. 



Let G B i ) the Horizontal Diftancc 



be = the Sine of the ^wg/<? of Elevation, 

 F GB, be = /, its Co- fine = c, Radius ~ r, and 

 the Parameter =: p* It will be as c to j • fb y to 



^= F B= G C, and by reafon of the Parabola 



III— to the %^ of C B, or G F • Now as c 

 c 



to r, fo is ^ to ~ = G F, and its 5 ? r f 



will be therefore == to P -L* . Which Equation 



reduced will be But by the former 



r r 



.Lewm* is equal to the Sine of the dou- 

 ble 



