146 



" The furface of an oblique Cylinder is equal to a reElangle contained by the 

 " diameter of its bafe and the circumference of an ellipfe, the axes of -which 

 " are the kngth and perpendicular height of the cylinder." 



This Theorem is more readily inveftigated from fluxional principles than 

 from confiderations purely geometrical ; yet I believe no author has hi- 

 therto communicated it even fo derived. The geometrical demonftration is 

 according to the method of the ancients by means of circumfcribed and in- 

 fcribed prifras. 



As a neceffary fupplement for applying this theorem in praftice, fome 

 obfervations on the methods of obtaining the circumference of a very ex- 

 centric ellipfe are given. The circumference of an ellipfe approaching to a 

 circle is readily found, but that of an ellipfe very excentric requires much 

 greater refearch. The bed method perhaps of obtaining the circumference 

 of fuch an ellipfe is by the affiftance of the theorem of Count Fagnani. 

 This remarkable theorem has been inveftigated heretofore by the application 

 of algebra and fluxions. But by help of a curious, and I believe, new pro- 

 perty of the ellipfe, it admits of a fimple geometrical deraonftration ; and I 

 am enabled to derive the following theorem. 



If that femi-diameter of an ellipfe he taken, which is a mean proportional be- 

 tween the femi-axes , and be produced to meet the circumfcribing circle ; then 

 the point, where the ordinate to the circle drawn from the point of interferon 

 cuts the ellipfe, divides the quadrantal arc of the ellipfe into two parts, 

 the difference of which is equal to the difference of the femi-axes. 



THEOREM. 



The furface of an oblique cylinder is equal to a reftangle contabed by 



the diameter of its bafe and the circumference of an ellipfe, the axes of 



which are the flant fide and height of the cylinder. 



Demon- 



