circumscribed square; if the semi- The altitude AD of the cylinder is equal to the diameter PQ; the A base of the cylinder is equal to a great circle, since its diameter AB is equal to MN; hence, the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter (P. 1). This measure is the same as that of the surface of the sphere (P. 10); hence, the surface of the sphere, is equal to the convex surface of the circumscribed cylinder. But the surface of the sphere is equal to four great circles; hence, the convex surface of the cylinder is also equal to four great circles: and adding the two bases, each equal to a great circle, the total surface of the circumscribed cylinder is equal to six great circles; hence, the surface of the sphere is to the total surface of the circumscribed cylinder, as 4 is to 6, or as 2 is to 3; which is the first branch of the proposition. In the next place, since the base of the circumscribed cylinder is equal to a great circle of the sphere, and its altitude to the diameter, the solidity of the cylinder is equal to a great circle multiplied by its diameter (P. 2). But the solidity of the sphere is equal to four great circles multiplied by a third of the radius (P. 14); in other terms, to one great circle multiplied by of the radius, or by of the diameter; hence, the sphere is to the circumscribed cylinder as 2 to 3, and consequently, the solidities of these two bodies are as their surfaces. Scholium 1. Conceive a polyedron, all of whose faces touch the sphere; this polyedron may be considered as composed of pyramids, each pyramid having for its vertex the centre of the sphere, and for its base one of the poly edron's faces. Now, it is evident that all these pyramids have the radius of the sphere for their common altitude: so that the solidity of each pyramid will be equal to one face of the polyedron multiplied by a third of the radius: hence, the whole polyedron is equal to its surface multiplied by a third of the radius of the inscribed sphere. It is therefore manifest, that the solidities of polyedrons circumscribed about the sphere, are to each other as their surfaces. Thus, the property, which we have shown to be true with regard to the circumscribed cylinder, is also true with regard to an infinite number of other solids. We might likewise have observed, that the surfaces of polygons, circumscribed about a circle, are to each other as their perimeters. PROPOSITION XVI. THEOREM. If a circular segment is revolved about a diameter exterior to it, the solid generated is measured by one-sixth of into the square of the chord, into the distance between two perpendiculars let fall from the extremities of the arc on the axis. Let DMB be a circular segment, and AC the axis about which it is revolved. On the axis, let fall the perpendiculars BE, DF; from the centre C, draw CI perpendicular to the chord BD; also draw the radii CB, CD. The solid generated by the sector CDMB is measured by × CB3× EF (P. 14, s. 1). The solid generated by the isosceles triangle CDB has for A B E M DA F its measure TX CIXEF (P. 12, c.); hence, the solid gen. erated by the segment DMB, is measured by But in the right-angled triangle CBI, we have (B. IV. P. 8, C.), CB2- CI2=BI2=BD3 : hence, the solid generated by the segment DMB, has for its measure }«×EF×}BD2=¿«>BD3×EF. Scholium. The solid generated by the segment BMD is to the sphere which has BD for a diameter, as X BD3× EF is to XBD, or as EF to BD. PROPOSITION XVII. THEOREM. Every segment of a sphere is measured by half the sum of its bases multiplied by its altitude, plus the solidity of a sphere whose diameter is this same altitude. Let DMB be the arc of a circle, and DF, BE, perpendiculars let fall on the radius CA: then, if the area FDMBE be revolved about the radius CA it will generate a spherical segment. It is required to find the measure of this segment. The solid generated by the circular segment DMB is measured by (P. 16) TX BD XEF: the frustum of the cone described by the trapezoid FDBE is measured by (P. 6) }«×EF×(BE2+DF2+BE×DF): D B M hence, the segment of the sphere, which is the sum of these two solids, is measured by }«×EF ×(2BE2+2DF2+2BE×DF+BD"). But by drawing BO parallel to EF, we have, DO=DF-BE and DO2=DF2—2DF×BE+BE2 and, BD3=B0'+D02=EF2+DF2−2DF×BE+BE3. Substituting this value for BD in the expression for the solidity of the segment, we have, ¿T×EF ×(2BE2+2DF2+2BE\DF+EF2+DF2—2DF×BE+BE'), equal to } ̃×EF×(3BE2+3DF2+EF3); an expression which may be written in two parts, viz., EF×(*×BE2+*×DF2 and XEF3; 2 and these parts correspond with the enunciation. Cor. If the radius of either base is nothing, the segment becomes a spherical segment with a single base; hence, any spherical segment, with a single base, is equivalent to half the cylinder having the same base and the same altitude, plus the sphere of which this altitude is the diameter. GENERAL SCHOLIUMS. 1. Let R be the radius of a cylinder's base, H its altıtude: the solidity of the cylinder is XR2XH. 2. Let R be the radius of a cone's base, H its altitude: the solidity of the cone is 3. Let A and B be the radii of the bases of a frustum of a cone, H its altitude: the solidity of the frustum is } ̃×H×(A2+B2+A×B). 4. Let R be the radius of a sphere; its solidity is 5. Let R be the radius of a spherical sector, H the altitude of a zone, which forms its base: the solidity of the sector is 6. Let P and Q be the two bases of a spherical segment, H its altitude: the solidity of the segment is BOOK IX. SPHERICAL GEOMETRY. DEFINITIONS. 1. A SPHERICAL TRIANGLE is a portion of the surface of a sphere, bounded by three arcs of great circles. These arcs are named the sides of the triangle, and each is less than a semicircumference. The angles which the planes of the circles make with each other, are the angles of the triangle. 2. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a rectilineal triangle. 3. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by three or more arcs of great circles. 4. A LUNE is a portion of the surface of a sphere included between two semi-circles intersecting in a common diameter of the sphere. 5. A SPHERICAL WEDGE, or UNGULA, is that portion of a solid sphere, included between two planes passing through the centre, and the lune which forms its base. 6. A SPHERICAL PYRAMID is a portion of the solid sphere, included between three or more planes. The base of the pyramid is the spherical polygon intercepted by the same planes. These planes bound a polyedral angle, whose vertex is at the centre of the sphere. 7. The POLE OF A CIRCLE is a point on the surface of the sphere, equally distant from every point in the circum 'ference. |