VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 633 



twelfth book of Euclid, there follows a summary of what is most material in the 

 treatises of Archimedes, concerning the sphere and cylinder, conoids and 

 spheroids, the quadrature of the parabola and the spiral lines. The demon- 

 strations are not precisely in the same form as those of Archimedes, but are 

 often illustrated from the elementary propositions concerning the cone, or 

 corollaries from them, after the example of Pappus, (Coll. Math. Prop. 21st, 

 lib, 4) from whom a proposition is demonstrated, and rendered more general, 

 concerning the area of the spiral generated on a spherical surface by the com- 

 position of two uniform motions, analogous to those by which the spiral of 

 Archimedes is described on a plane. This area, though a portion of a curve 

 surface, is found to admit of a perfect quadrature, and this proposition con- 

 cludes the abstract. He takes occasion from these theorems to demonstrate 

 some properties of the conic sections, that are not mentioned by the writers on 

 that subject; and there are more of this kind described in the llth and 14th 

 chapters of the first book. 



It is known, that if a parallelogram, circumscribed about a given ellipse, have 

 its sides parallel to the conjugate diameters, then shall its area be of an invaria- 

 ble or given magnitude, and equal to the rectangle contained by the axes of the 

 figure ; but this is only a case of a more general proposition. For if, upon any 

 diameter produced without the ellipse, you take two points, one on each side of 

 the centre at equal distances from it, and the four tangents be drawn from these 

 points to the ellipse, those tangents shall form a parallelogram, which is always 

 of a given or invariable magnitude, when the ellipse is given, if the ratio of 

 those distances to the diameter be given ; and when the ratio of those distances 

 to the semidiameter is that of the diagonal of a square to the side, (or of 

 n/l to l) the parallelogram has its sides parallel to conjugate diameters. It is 

 likewise shown here, how the triangles, trapezia, or polygons of any kind are 

 determined, which, circumscribed about a given ellipse, are always of a given 

 mag 



nit 



itude. 

 There is also a general theorem concerning the frustum of a sphere, cone, 

 spheroid, or conoid, terminated by parallel planes, when compared with a 

 cylinder of the same altitude on a base equal to the middle section of the 

 frustum made by a parallel plane. The difference between the frustum and the 

 cylinder, is always the same in different parts of the same, or of similar solids, 

 when the inclination of the planes to the axis, and the altitude of the frustum, 

 are given. This difference vanishes in the parabolic conoid. It is the same in 

 all spheres ; being equal to half the content of a sphere of a diameter equal to 

 the altitude of the frustum. In the cone it is -j-th of the content of a similar 



VOL. VIII. 4 M 



