| = π.AB.(AE + DE)(AE − DE); | |||||||||||
| but | AE + DE | = 2OG, and AE − DE = AD. |
Hence volume described by the rectangle ABCD
= 2π.OG.AB.AD.
= rectangle ABCD multiplied by the circumference of the circle described by its middle point O.
Observation.—This Cor. is a simple case of Guldinus’s celebrated theorem. By its assistance we give in the two following corollaries original methods of finding the volumes of the cone and sphere, and it may be applied with equal facility to the solution of several other problems which are usually done by the Integral Calculus.
Cor. 3.—The volume of a cone is one-third the volume of a cylinder having the same base and altitude.
Dem.—Let ABCD be a rectangle whose diagonal is AC. The triangle ABC will describe a cone, and the rectangle a cylinder by revolving round AB. Take two points E, F infinitely near each other in AC, and form two rectangles, EH, EK, by drawing lines parallel to AD, AB. Now if O, O′ be the middle points of these rectangles, it is evident that, when the whole figure revolves round AB, the circumference of the circle described by O′ will ultimately be twice the circumference of the circle described by O; and since the parallelogram EK is equal to EH [I. xliii.], the solid described by EK (Cor. 1) will be equal to twice the solid described by EH. Hence, if AC be divided into an indefinite number of equal parts, and rectangles corresponding to EH, EK be inscribed in the triangles ABC, ADC, the sum of the solids described by the rectangles in the triangle ADC is equal to twice the sum of the solids described by the rectangles in the triangle ABC—that is, the difference between the cylinder and cone is equal to twice the cone. Hence the cylinder is equal to three times the cone.
Or thus: We may regard the cone and the cylinder as limiting cases of a pyramid and prism having the same base and altitude; and since (v. Cor. 2) the volume of a pyramid is one-third of the volume of a prism, having the same base and altitude, the volume of the cone is one-third of the volume of the cylinder.
Cor. 4.—If r be the radius of the base of a cone, and h its height,