SPHEROID, in geometry, a solid, ap proaching to the figure of a sphere.
The spheroid is generated by the en tire revolution of a semi-ellipsis about its axis. Thus, if the semi-ellinsis AUF11 (Plate rm. Miscel. fig. 6.) be supposed to revolve round its transverse axis AB, it will generate the oblong spheroidAHFBC. Now as all circles are as the squares scribed upon their radii, that is, the cle of the radius EH is to the circle of the radius EG, as CF' to CD', because EH : EG : : CF : CD, and since it is so every where, all the circles described with the respective radii EH, (that is, the spheroid made by the rotation of the semi-ellipsis AFB about the axis AB) will be to all the circles described by the respective radii EG, (that is, the sphere described by the rotation of the semicircle ABD on the axis AB,) as FC' to that is, as the roid is to the sphere on the same axis, so is the other axis of the generating ellipsis to the square of the diameter or axis of the sphere : and this holds, whether the spheroid be formed by a revolution around the greater or lesser axis.
Hence it appears, that the half of the spheroid, formed by the rotation of the space AHFC, around the axis AC, is double of the cone generated by the triangle AFC, about the same axis. Hence, also, is evident the measure of segments of the spheroid, cut by planes perpendicular to the axis: for the seg ment of the spheroid, made by the ro tation of the space ANNE round the axis AE, is to the segment of the sphere, having the same axis AC, and made by the rotation of the segment of the cir cle AMGE, as CF' to CD'. But the measure of this solid may be found with less trouble by this analogy ; vi:. as
BE: AC EB: : so is the cone genera ted by the rotation of the triangle AUE round the axis AE; to the segment of the sphere made by the rotation of the space AMIE round the same axis AE, as is demonstrated by Archimedes of colloids and spheroids, prop. 34. This agrees as well to the oblate as to the oblong spheroid. A spheroid is also equal to two thirds of its circumscribing cylinder. As to the superficies of a sphe roid, M. Huygens gives the two follow ing constructions in his Horolog. Oscill. For describing a circle equal to the superficies of an oblong and prolate spheroid: 1. Let an oblong spheroid be generated by the rotation of the ellipsis ADRE, (fig. 7.) about its transverse axis AB, and let DE be its conjugate ; make DF equal to CB, or let F be one of the foci, and draw BG parallel to FD, and about the point G, with the radius BG, describe an arch, BHA, of a circle ; then between the semi-conjugate CD, and a right line equal to DE + the arch AHB, find a mean proportional, and that will be the radius of a circle equal to the superficies of the oblong spheroid. 2. Let a prolate spheroid be generated by the rotation of the ellipsis ADBE (fig. 8.) about its conjugate axis AB. Let F be one of the foci, and bisect CF in G, and let AGB be the curve of the common parabola, whose base is the conjugate diameter AB, and axis CG. Then if between the transverse axis DE, and a right line equal to the curve AGB of the parabola, a mean proportional be taken, the same will be the radius of a circle equal to the surface of that prolate spheroid.