148 



whence, the angles V -y L and (^P R being right angles, it follows that the 

 triangles L ^' V and PQR are fimilar, 



and that PQ^: qR=AL : : Lv: LY : : pq : mn 



II II 



but AL : LM :: mn : mn by fimilar figures. 



therefore PQ^: LM : : /-^ : mn or reft. PQ^ m7i = LM. pq. or the fide of 



the prifm = reft. pq. LM and confequently the whole furface of the cir- 



cumfcribing prifm = reftangle LM . x the circumference of the polygon cir- 

 cumfcribing the ellipfe, which polygon is the projeftion of the polygon cir- 

 cumfcribing the circle. 



In the fame manner it may be proved that the furface of a prifm infcribed 



in the oblique cylinder is equal to the reftangle contained by LM and the 

 circumference of the polygon infcribed in the ellipfe. 



And as this is true of any whatfoever circumfcribed and infcribed prifms, 

 it follows that tlie furface of the cylinder is alfo equal to the reftangle con- 

 tained by the diameter of the bafe LM and the circumference of the ellipfe 

 which is the projeftion of the circle, for otherwife, as may be eafily fliewn, 

 an abfurdity might be deduced Q^E.D. 



APPENDIX. 



The quadrature of the furface of an oblique cylinder is reduced, as above, 

 to the reftjfication of an ellipfe. The circumference of an ellipfe is, as is 



well known, equal to the circumference of the circumfcribing circle 

 e"- y" 3.3.5^5 



X : I — * &c. e being the excentricity to 



2.2 2.2.4.4 2.2.4.4.6.6 



the 

 * A feries converging much more rapidly when e is fmall, is given in Mr. Ivory's very 

 ingenious Effay, Edinburgh Tranfadions, Vol. 4. 1 798. 



