[519]
cutting them perpendicularly, that is, by the length of that line, and not by a line of any other length, for that will conſiſt of more or leſs points.
Hence therefore in the ſpeculation of the ſuperficies of ſolids, the Method of Indiviſibles is not unuſeful, but rather very commodious, provided it be rightly underſtood, and applied according to the Rule preſcrib'd. For by the help of it even theſ ſuperficies may be found, if ſo be we have ſome convenient Data preſuppos'd, on which the reaſoning may be founded. For inſtarice, we might by the help of it, inveſtigate the ſuperflcies of a Cone, by reaſoning after this manner.
If the ſuperficies of the cone ABC (fig. pag. 362.) be divided into innumerable Peripheries of circles βχδ parallel to the baſe BCD, the breadth of thoſe Peripheries taken together, make up the ſide 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 expreſs'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 ſuperficies will be made out of thoſe perpendiculars equal to the ſuperficies of the Gone. But that ſuperficies will be a triangle whoſe heighth is AB, and baſe equal to the greateſt Periphery BDC, and ſo the ſuperficies of the Cone will be = 12 AB × Periph. BDC, which concluſion agrees with the things laid down and demonſtrated by Archimedes.
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After the ſame manner, if you take any right line αβ equal to the quadrantal Arc AB of the Hemiſphere (in pag.364.) and to each of its points μ let the
