ftance, in the equilateral Pyramid ABCD, whofe Axe is AX, fuppofing that the lateral furface. of the Pyramid confifts of Peri-. meters of triangles, parallel to, the bafe BCD, these can neither be computed by the altitude, AX, nor by the fide AB, (for by the former, the thing requir'd, would be wanting of the true Dimenfion, and by the latter 'twould exceed it,) but by the line AE drawn from the vertex A perpendicular to the fide BC of the bafe; the reafon of which is, that every plane fide of a Pyramid as ABC, confifts of parallel right lines computed by the altitude AE. After the fame manner, fuppofing that the fuperficies of the Hemisphere M X C BAD, confift, of peripheriesof circles parallel to the bafe BCD, the number of them is not to be computed by the Axis AX, but by the Quadrantal Arc AB, because that every point of the Arc AD in revolving produces a circumference. And fo any fuperficies, whether plane of curv'd, which is conceived to confift of equidiftant right or curv'd lines, is to be computed by a line cutting thofe equidiftant lines perpendicularly. For fince thofe equidiftant lines, in this Method of Indivifibles, are not confider'd abfolutely as lines having an infinitely fmall breadth, which is the fame with the breadth or thickness of the point defcribing thofe equidiftant lines in their Circumvolution, and fince the fame equidiftant lines divide the line cutting them perpendicularly into parts measuring its breadth, those parts are to be look'd upon as fuch fort of points, and confequently the number of equidi ftant lines, or the fum of those breadths is to be computed by the number of points in the line.. cutting cutting them perpendicularly, that is, by the length of that line, and not by a line of any other length, for that will confift of more or lefs points. Hence therefore in the fpeculation of the fu perficies of folids, the Method of Indivifibles is not unufeful, but rather very commodious, provided it be rightly understood, and applied according to the Rule prefcrib'd. For by the help of it even thefe fuperficies may be found, if fo be we have fome convenient Data prefuppos'd, on which the reafoning may be founded. For instance, we might by the help of it, inveftigate the fuperficies of a Cone, by reafoning after this manner. If the fuperficies of the cone ABC (fig.pag.362.) be divided into innumerable Peripheries of circles 6x parallel to the bafe BCD, the breadth of thofe Peripheries taken together, make up the fide 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 exprefs'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 fuperficies will be made out of thofe perpendiculars equal to the fuperficies of the Cone. But that fuperficies will be a triangle whofe heighth is AB, and bafe equal to the greateft Periphery BDC, and fo the fuperficies of the Cone will be ABx Periph. BDC, which conclufion agrees with the things laid down and demonftrated by Archimedes. After the fame manner, if you take any right α e line 3 equal to points pel 1 perpendiculars ur be erected equal to the Radii MN of parallel circles MOM paffing thro' the correfponding points M of that quadrantal Arc, the greatest of which let be equal to the Radius BX of the bafe of the Hemisphere: The figure aß will contain the Radii of all the circles of whofe Peripheries the fuperficies of the Hemisphere confifts. And if the perpendicular , B be erected equal to the Peripheries MOM, BDB, there will be made the figure a equal to the fuperficies of the Hemifphere. The dimenfion of which figure if you can by any means find (as in this cafe you are to find the Area of the figure a) thence you will eafily deduce the content of the fegment of the sphere, agreeing to what you may gather by any lawful realoning. Which Obfervation, I think will not be unuseful in Geometry. FINIS. A T the Hand and Pen in Barbican are Taught, Writing, Common and Merchants Accounts, after that well approved Method of Mr. Normanfel, algeba, Geometry, Dealuring, Surveying, Gauging,Javigation and Dialing, with other parts of the athematicks, also the Ufe of the Globes, Maps, and other Inftruments, by me Robert Arnold. |