For practical purposes the cylinder of revolution (right circular cylinder) is the one most frequently used, and the important formula is therefore _l_ = 2[pi]_rh_ where _l_ = the lateral area, _r_ = the radius, and _h_ = the altitude. Applications of this formula are easily found. THEOREM. _The volume of a circular cylinder is equal to the product of its base by its altitude._ Here again the important case is that of the cylinder of revolution, where _v_ = [pi]_r_^2_h_. The number of applications of this proposition is, of course, very great. In architecture and in mechanics the cylinder is constantly seen, and the mensuration of the surface and the volume is important. A single illustration of this type of problem will suffice. A machinist is making a crank pin (a kind of bolt) for an engine, according to this drawing. He considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown, where 7 3/4" means 7 3/4 inches. He has this formula for the weight (_w_) of a steel cylinder where _d_ is the diameter and _l_ is the length: _w_ = 0.07[pi]_d_^2_l_. Taking [pi] = 3 1/7, find the weight of the pin. The most elaborate study of the cylinder, cone, and sphere (the "three round bodies") in the Greek literature is that of Archimedes of Syracuse (on the island of Sicily), who lived in the third century B.C. Archimedes tells us, however, that Eudoxus (born _ca._ 407 B.C.) discovered that any cone is one third of a cylinder of the same base and the same altitude. Tradition says that Archimedes requested that a sphere and a cylinder be carved upon his tomb, and that this was done. Cicero relates that he discovered the tomb by means of these symbols. The tomb now shown to visitors in ancient Syracuse as that of Archimedes cannot be his, for it bears no such figures, and is not "outside the gate of Agrigentum," as Cicero describes. The cone is now introduced. A conic surface is easily illustrated to a class by taking a piece of paper and rolling it up into a cornucopia, the space inclosed being a _conic space_, a term that is sometimes convenient. The generation of a conic surface may be shown by taking a blackboard pointer and swinging it around by its tip so that the other end moves in a curve. If we consider a straight line as the limit of a curve, then the pointer may swing in a plane, and so a plane is the limit of a conic surface. If we swing the pointer about a point in the middle, we shall generate the two nappes of the cone, the conic space now being double. In practice the right circular cone, or cone of revolution, is the important type, and special attention should be given to this form. THEOREM. _Every section of a cone made by a plane passing through its vertex is a triangle._