As to the second solution, x = a + a2, it is inapplicable to the present question; we shall see that it belongs, as well as the first, to this other question, of an abstract nature, furnished by the equation a2, or 3 ax The known line AN (fig. 56), being divided into three equal parts Fig. 56. at the points B and D, to find in the direction of this line a point P, such that the part AD shall be a mean proportional between the distances of the point P from the extremities A and N. Indeed, if we designate the third AD of the line AN by a, and AP by x, we shall have PN 3 ax; and the conditions of the question give this proportion whence = as above found. We shall have both also by the same construction, except that for the second root, or 3 a + √ a2, we apply MO from M to P toward N, and then AP and. AP will be the two values of x. Other Applications of Algebra. 96. In order to resolve the last question, we were obliged to calculate the algebraic expression of a spherical sector, and of the cone, which makes a part of it. The bodies, which are the subject of consideration in geometry, being the elements of all others, often present themselves in our inquiries, and especially in physico-mathematical questions. If we represent by rc the ratio of the radius to the circumference of a circle, a ratio which is known with a degree of exactness sufficient for practical purposes (Geom. 292), the circum ference of any other circle whose radius is a will be a c r (Geom. or 220 (Geom. 289). From this 2 r it will be evident, that the surfaces of circles increase as the increases only in proportion to the increase of a2. If h be the altitude of a cylinder, the radius of whose base is d, we shall have a2ch for the solidity of this cylinder (Geom. 2 r for the same reason we shall have a2 c 2 r 516); Xh' for the solidity of another cylinder, whose altitude is h' and the radius of whose base is a'; consequently the solidity of the two cylinders will be to each other that is, the solidities of cylinders are to each other as the altitudes multiplied by the squares of the radii of the bases. If the altitudes are proportional to the radii of the bases, we shall have h: h'::a: a', that is, the solidities are as the cubes of the radii of the bases (Geom. 518). We have seen in the Elements of Geometry, that surfaces depend upon the product of two dimensions, and solids upon the product of three dimensions; so that, if the several dimensions of one of two solids, or two surfaces, which we would compare, have to the several dimensions of the other, each the same ratio, the two surfaces will be to each other as the squares, and the two solids as the cubes, of the homologous dimensions; and more generally still, if any two quantities of the same nature are expressed each by the same number of factors, and if the several factors of the one have to the several factors of the other, each the same ratio, the two quantities will be to each other as their homologous factors, raised to a power whose exponent is equal to the number of factors. If, for example, the two quantities were a b c d, a' b' c' d', and we had What is here said is true not only of simple quantities; the same may be shown with respect to compound quantities. Let the quantities whose dimensions are proportional be a/2 b α abcd: a'b' + c' d' :: a b + cd: + :: a b + cd: a12 c d a2 a'2 a b + a22 cd a2 It follows, from what is here demonstrated, that the surfaces of similar figures are as the squares of their homologous dimensions, and that the solidities of similar solids are as the cubes of their homologous dimensions; for, whatever these figures and these solids may be, the former may always be considered as composed of similar triangles, having their altitudes and bases proportional (Geom. 219), and the latter as composed of similar pyramids, having their three dimensions also proportional (Geom. 433). It will hence be perceived, that quantities may be readily compared, when they are expressed algebraically; and this may be done, whether the quantities be of the same or of a different species, as a cone and a sphere, a prism and a cylinder, provided only that they are of the same nature, that is, both solids, or both surfaces, or both &c. 97. We are taught in the Elements of Geometry, how to find the solidity of the frustum of a pyramid, and the frustum of a cone (Geom. 422, 527). If now we designate by h the altitude. of the entire pyramid, and by h' the altitude of the pyramid cut off, by s the surface of the inferior base, and by s the surface of the superior base, we shall have But, if we designate by k the altitude of the frustum, we shall whence we deduce, by the common rules of algebra, Now the solidity of the entire pyramid is s X and the solidity of the pyramid cut off is s' × or, putting for h' its value, hs hs' s h or 3 putting for h its value, found above, we shall have or, the whole being divided by vs — √5, We see, therefore, that the frustum of a pyramid, or of a cone, is composed of three pyramids of the same altitude, of which one has for its base the inferior base s of the frustum, another the superior bases', and the third the mean proportional between these, which agrees with the propositions above referred to. 98. If a represent the radius of a sphere, face of a great circle of this sphere, and a2 c 2 r 4 a2 c 2 r will be the sur the entire surface (Geom. 536); consequently, If we designate by a the altitude of any segment, we shall have, as in art. 95, (2 a x a2 cx 3 r for the solidity of the sector, and C 2 r x2) for the solidity of the cone, which makes a ax part of it; hence the solidity of the segment will be from which it will be seen, that the solidity of the segment is equal to the product of the circle, whose radius is the altitude of this segment, multiplied by the radius of the sphere minus the third of this altitude. When we have the algebraic expression of quantities, it is easy to resolve several questions that may be raised respecting these quantities. If, for example, it were asked, what must be the altitude of a cone which shall be equal in solidity to a given sphere, and which shall have for the radius of its base, the radius of the sphere; designating this altitude by h, and the radius of the base by a, we shall have for the solidity of the cone a2 h X ; 2 r 3 |