Page:Euclid's Elements 1714 Barrow translation.djvu/531

 cutting them perpendicularly, that is, by the length of that line, and not by a line of any other length, for that will conit of more or les points.

Hence therefore in the peculation of the uperficies of olids, the Method of Indiviibles is not unueful, but rather very commodious, provided it be rightly undertood, and applied according to the Rule precrib'd. For by the help of it even the uperficies may be found, if o be we have ome convenient Data preuppos'd, on which the reaoning may be founded. For intarice, we might by the help of it, invetigate the uperflcies of a Cone, by reaoning after this manner.

If the uperficies of the cone ABC (fig. pag. 362.) be divided into innumerable Peripheries of circles &beta;&chi;&delta; parallel to the bae BCD, the breadth of thoe Peripheries taken together, make up the ide AB cutting them perpendicularly, and confequently there will be as many Peripheries as there are points in the line AB, that is, their number may be expres'd by the number of points in AB, or by its length. Wherefore, if you draw perpendiculars equal to the Peripheries to every point of AB, a uperficies will be made out of thoe perpendiculars equal to the uperficies of the Gone. But that uperficies will be a triangle whoe heighth is AB, and bae equal to the greatet Periphery BDC, and o the uperficies of the Cone will be = $1⁄2$ AB &times; Periph. BDC, which concluion agrees with the things laid down and demontrated by Archimedes.

After the ame manner, if you take any right line &alpha;&beta; equal to the quadrantal Arc AB of the Hemiphere (in pag.364.) and to each of its points &mu; let the