1 68. If four quantities be proportionals, the like pow ers or roots of these quantities will be proportionals. 69. A ratio compounded of several ratios is indicated by the continued product of the quantities which denote the component ratios. Thus, the ratio compounded of the ratios of a: b, of c: d, and of e: f, is indicated by The ratio which the first of a series of quantities, of like kind, has to the last, is the same as the ratio compounded of the ratios of the first to the second, of the second to the third, &c. to the last. Let a, b, c, d, e, f, be quantities of a like kind, then 70. In two ranks of proportionals, if the corresponding terms be multiplied together, the products will be proportionals. Leta:b::c:d, e:f::g: h. The demonstration may be easily applied to any number of proportions. 71. A ratio compounded of two, three, four, &c. equal ratios, is called the duplicate, triplicate, quadruplicate, &c. of one of the component ratios. The ratio compounded of any number of equal ratios is the same as the ratio of such power of the first term, as is indicated by the number of component ratios, to a like power of the second. a ab α b .. (art. 67,) ==, or a : b : : a: c. Again, let a : b :: b : c :: c:d::d: e, &c. n equations; and multiplying the first and second mem bers respectively together, 72. When the ratio of the first of three quantities to the second, is the same as the ratio of the second to the third, the ratio of the first to the second is termed the sub-duplicate of the ratio of the first to the third. When four quantities are continued proportionals, the ratio of the first to the second is called the sub-triplicate of the ratio of the first to the fourth. A ratio compounded of a simple and sub-duplicate ratio, is called a sesquiplicate ratio. 73. The sub-duplicate ratio is equivalent to the ratio of the square roots; the sub-triplicate, to the ratio of the cube roots; and the sesquiplicate, to the ratio of the square roots of the cubes. Let a:b::b:c::c:d. Also, b:e::e:c. Also, since b:e::e:c, and a:b::c:d. e=bc=ad, (art. 62.) But a : b :: a : e, in which the ratio of a : e, is compounded of the ratios of a: b, and of b: e, or of the ratio of a : b, and the sub-duplicate of the ratio of b:c. ON THE VARIATIONS OF QUANTITIES. 74. In the investigation of the relation which varying and dependent quantities bear to each other, the conclusions are more readily obtained, by expressing only two terms in each proportion, than by retaining the four. But though, in considering the variation of such quantities, two terms only are expressed, it must be remembered, that four are supposed; and that the operations, by which our conclusions are obtained, are in reality the operations of proportionals. 75. One quantity is said to vary directly, as another, when one is such a function* of the other, that if the former be changed, the latter will be changed in the same ratio. Thus, if B be such a function of A, that by changing A to a, B shall be changed to b; making A:a:: B: b, * The function of any variable quantity x, is an algebraic expression, in which x, combined with invariable quantities, is involved. Thus, 1+x, (1+x), ax, xn, ax, &c. are functions of x. Analysts sometimes use the Greek letter, to denote a function. Thus, ox may represent any function of x. A is said to vary directly as B. This relation is designated thus, A&B. 76. One quantity is said to vary inversely as another, when the latter is such a function of the former, that the one being increased or diminished, the other will be diminished or increased in the same ratio. Thus, if B be such a function of A, that by changing A to a, B becomes changed to b; making A:a::b: B. Then A is said to vary inversely as B. Indicated thus, Ac 1 B 77. One quantity is said to vary as two others jointly, when the two last are such functions of the first, that, the ratio which any two values of the first bear to each other, shall be the same as the ratio compounded of the ratios of the corresponding values of the other two. Thus, A varies as B and C jointly, (ABC) when A being changed to a, B changes to b, and C to c, so that 78. One quantity is said to vary directly as a second, and inversely as a third, when the ratio which any two values of the first bear to each other, is the same as the ratio compounded of the direct ratio of the corresponding values of the second, and the inverse ratio of those of the third. Thus, A varies directly as B, and inversely as C, when A, B, C; a, b, c, being corresponding values, In the following articles, A, B, C, &c. represent corresponding values of any quantities, and a, b, c, &c. any other corresponding values of the same quantities. 79. If AB, and m, n denote any given numbers, I 80. If ACB, then A" &B"; and Anx Br. See art. 68. 81. If ACB, and CD, then ACBD. See art. 70. From the three preceding articles it appears, that quantities connected by the sign cc, may be treated as the members of an equation, as far as multiplication, division, involution or evolution is concerned. 83. If ACC, and BCC, A and B being quantities of the same kind, then (A+B)CC, and A a ABC. A+B a+b ABBC a+b And±1=±1, or B Again, since ACC, and BCC, (art. 81,) ABCC. 84. If while A and B vary AB= a constant quantity, |