Fig.144. Fig.145. base, is a circle equal to its base. Any plane section * These methods of defining a cylinder are preferable to the description given in Art. 274 ; for they will apply to all cylinders, if, instead of the term circle, we substitute a plane surface embraced by a curve line; as it is not an essential property of a cylinder, that any section of it should be a circle, Nor is it an essential property of a cone, that any section of it should be a circle. inder belongs to the family of prisms; and therefore to Fig.146. Fig.147. proof that—The volume of a cylinder has for its measure the product of the area of its base multiplied by its height. Remark. It is manifest that this last proposition applies as well to oblique as to right cylinders, there being nothing in the reasoning which, supposes the axis of the cylinder perpendicular to the base. 281. If the semicircle ACB (fig. 147) revolve about its diameter AB, it will generate a sphere ; the semicircumference generates the spherical surface. Every point in this semi-circumference, as D, generates the circumference of a circle whose centre is in the diameter A.B (here called the axis) and whose plane is perpendicular to this axis. The extremities A, B, of the axis, are, in reference to this revolution, called poles. They are particularly the poles of all the circles whose circumferences are generated by the several points in the arc ACB of the revolving semicircle. As every point of the arc of the semicircle is at the same distance from its centre, and as this centre does not move during the process of revolution ; it follows that every point of the spherical surface is equally distant from the point O, the centre of the generating semicircle and also the centre of the sphere. It is also evident that any radius of the semicircle, in any part of its revolution, is also radius of the sphere; and that these radii are all equal. 282. If we suppose the sphere to be cut by a plane passing through its centre, it is perfectly manifest that the section will be a circle, whose radius will be the radius of the sphere. But suppose the plane to pass through the sphere on one side of the centre; and let the section be DG FH ; from O the centre of the sphere, draw OE perpendicular to this section, and draw radii to the several points in the periphery of this section ; they will be equal oblique lines, and will therefore meet the plane DGFH, at equal distances from the perpendicular OE (206). Every point in this periphery is therefore at the same distance from E; the section is consequently a circle whose centre is E. And as this reasoning applies to any plane section on either side of the centre, we infer that—Every plane section of a sphere is a circle ; and also that—Radius perpendicular to any plane section of the sphere, passes Fig.147. through its centre. 2-3. Remark. As DE is always one side of a rightangled triangle whose hypothen use is radius of the sphere, DE inust be less than radius; the plane section whose radius is radius of the sphere, is called a great circle ; other plane sections are called less circles. 284. As two great circles are planes, their intersection must be a straight line; and as each passes through the Centre of the sphere, the line of intersection must pass through the centre of each of them, that is—Two great circles mutually bisect each other. 2S5. Any portion of a spherical surface embraced by the arcs of three great circles, is called a spherical triangle. As two circumferences of great circles bisect each other, if each angle of a spherical triangle is salient, each of the arcs must be less than a semi-circumference. 286, Let ('M I be a spherical triangle, and draw the radii OC, OM, OI; it is evident that they determine a triedral angle whose vertex is at O, and whose plane angles are measured by the arcs CI, CM, I.M. And as the sum of any two of these plane angles is greater than the third (220) the sum of the arcs which measure them must be greater than the third arc (113); that is – The sum of any two sides of a spherical triangle, is greater than the third. Whence we infer that—The shortest way from one point to another, on the surface of a sphere, is in the arc of a great circle passing through these points. For if we suppose it shorter to pass through any point out of this arc, that point and the two points proposed may be made the vertices of a spherical triangle, one side of which will be the arc first proposed and less than the sum of the other two ; the inference is therefore manifestly true. 287. Suppose the two semi-circumferences ACB and AIB which meet each other in A and B, to be cut by the arc IM of another great circle; we shall have M I < MB + BI ; and consequently AM -- AI + IM 3 AM B + AIB ; that is—The sum of the three sides of any spherical triangle, is less than the circumference of a great circle. Remark. We might have deduced this truth from the limit of the magnitude of the plane angles which form the triedral angle at the centre of the sphere (222); this sum being less than four right angles, the sum of the Fig.147. arcs which measure these angles. must be less than an |