H7 



Demonstration. 



Let the circle L R M L reprefent the bafe of the cyhnder, Q_R its flant Fig. i 



Cde, and OS its perpend, height. Draw the diameter L M perpendicular 



to R S alfo S P perpendicular to the tang. R P, and then QJ* will be per- 



III 



pendicular to P R. Let the tangent m R n be the fide of a polygon 



which is the bafe of a prifm circumfcribing the cylinder. 



Fig. 2 



Let L R A be a circle, the diameter of which is equal to (^R the flant 

 fide. Conceive this circle inclined to the plane of its orthographical projec- 



tion L B A in an angle equal R C^S the complement of the inclina- 



III 

 tion of the cylinder. Take the angle L C R = L C R. Let ffi n be the 



fide of a polygon circumfcribing the circle, Cmilar and fimilarly fituate to the 



polygon of which m n n z fide, and let p q be the projeftion of m n. 

 Draw L V parallel to the fide of the polygon mn, alfo draw L<u the pro- 

 jeftion of L V, Vw and R N perpendicular to C L, and join vw. 



Then the right angled triangles V v w and R Q^S are fimilar, becaufe p; , ^^^ ^ 



R <^S = V w -y (by conftr.) : and alfo the right angled triangles P R S 

 and R N C are equi-angular.. 



Hence Vt; : Vw : : RS : RQ ^ ^ 



Vw: RN = LV : : CV = CN : RC : : PR : RS 



Therefore Va; : LV : : PR : R(^ 



( T 2 ) whence 



