144 A Specimen of a Method 



on tangencies may be conftructed on the fphere j for infhnce, having three 

 circles given upon a fpher'e, a fourth may be found to touch them ; for their 

 politions on the fphere being given, their proje&ions will alfo be given on a 

 plane itereographically j and as a circle may be found in Vi eta's method 

 to touch them on that plane, the fituation of that circle may be found 

 upon the fphere, and hence properties may be found for conftructing the 

 problem independent of the jt-ereographic projection ; and if we fuppofe 

 the centre of projection to be the centre or focus, &c. of a fpheroid or 

 other folid, innumerable properties may be found relative to their tangents,, 

 curvatures, &c. regard being had to the pofition of the plane, &c. 



To give a fpecimen of the aforefaid method in fortification, let h (fee 

 pp, 22, 23, 24, and 25 of Dejdier's Per/eel French Engineer) reprefent 

 the height of a wall j then according to Vauban's meafures, if five feet 

 be the thicknefs at the top, yh + 5, will be the thicknefs at the bottom; 

 and according to Belidor's method ~ $1+3,5, will be the thicknefs at the 

 top, arid yh + 3,5, that at the bottom. The length of the counterfort (ac- 

 cording to Vauban), will be-|h + 2; alfo ~ h + 2 is the thicknefs next the 

 wall, and, (-Jh + 4) the thicknefs at the other end of the counterfort. If 

 part of the wall is gazoned, let e be the height of that part and h'that of 

 the wall j then -§-(h + e) .+ 5 is the thicknefs at the bottom j fe + 5, is the 

 thicknefs at the topj -§-(h4-e) + 2, is the length of the counterfort j 

 ~ (h + e) + a its thicknefs next the wall, and f(K n + e ) +4) its thicknefs 

 fartheft from the walk When there are cavaliers, let c be their height 

 in feet j then -—(ae-nr+^o) is the thicknefs of the revetement at the top, 

 and ■- (ah + ae + c + $ti) is the thicknefs .at the bottom. 



