306. If four quantities are proportional, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. Let a:b::c:d. By composition, Art. 304, a+6:6::c+d:d. By alternation, Art. 301, a+b:c+d::b:d. Also by division, Art. 305, a-b:b::c-d:d; by alternation, a-6:0-d::b:d. By equality of ratios, Art. 303, a+b:ctd::2–6:c-d, or a C 307. If four quantities are in proportion, any equimultiples of the first couplet will be proportional to any equimultiples of the second couplet. Let a:b::c:d; then ъ đ Multiply both terms of the first fraction by m, and both terms of the second fraction by n, and we have nc mb nd' or ma: mb:: nc: nd. ma 308. If four quantities are in proportion, any equimultiples of the antecedents will be proportional to any equimultiples of the consequents. Let a:b::c:d ; then 7 ď Multiply each of these equals by m, and we have a с Divide each of these equals by ni ma mc no nd' or ma: nb :: mc:nd. 309. If any number of quantities are proportional, any one antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents, Let a:6::c:d::e:f; then, since a:b::c:d, (1.) and, since a:b:: e: fi (2.) also ab=ba. (3.) Adding (1), (2), and (3), ab+ad+af=ba+bc+be; that is, a(b+d+f)=b(a+c+e). Hence, Art. 300, a:0:: a+c+e:6+d+f. a с 310. If there are two sets of proportional quantities, the products of the corresponding terms will be proportional. Let ca:b::c:d, and e:f::g:h; then ae : bf :: cg: dh. For d' and 우 = 1 cg ae or 311. If four quantities are in proportion, like powers or roots of these quantities will also be in proportion. a с Let a:0::c:d; then õ=ă ъ * Raising each of these equals to the nth power, we obtain an сп brani 312. If three quantities are in continued proportion, the product of the extremes is equal to the square of the mean. If a:b:: b:c, then, by Art. 299, ac=bb=02. a с 313. If three quantities are in continued proportion, the first is to the third in the duplicate ratio of the first to the second. Let a:b::b:c; b then b Multiply each of these equals by, and we have 6 3^2 a? that is, 724 a a a 810 = or 314. If four quantities are in continued proportion, the first is to the fourth in the triplicate ratio of the first to the second. Let a:b::b:c:: c:d; then ūc (1.) a ale alala (2.) also (3.) Multiplying together (1), (2), and (3), we have a abc 73 hence a:d:: a:b. VARIATION. 315. Proportions are often expressed in an abridged form. Thus, if A and B represent two sums of money put out for one year at the same rate of interest, then A:B :: interest of A: interest of B. This is briefly expressed by saying that the interest varies as the principal. A peculiar character (cc) is used to denote this relation. Thus interest a principal denotes that the interest varies as the principal. 316. One quantity is said to vary directly as another when the two quantities increase or decrease together in the same ratio. Thus, in the above example, A varies directly as the interest of A. In such a case, either quantity is equal to the other multiplied by some constant number. Thus, if the interest varies as the principal, then the interest equals the product of the principal by some constant number, which is the rate of interest. If A a B, then A= mB. If the space (S) described by a falling body varies as the square of the time (T), then S=mT, where m represents a constant multiplier. 317. One quantity may vary directly as the product of several others. Thus, if a body moves with uniform velocity, the space described is measured by the product of the time by the velocity. If we put s to represent the space described, T the time of motion, and V the uniform velocity, then we shall have Sa TxV. Also the area of a rectangular figure varies as the product of its length and breadth. The weight of a stick of timber varies as the product of its length x its breadth xits depth x its density. 318. One quantity is said to vary inversely as another when the first varies as the reciprocal of the second. Thus, if the area of a triangle be invariable, the altitude varies inversely as the base. If the product of two quantities is constant, then one varies inversely as the other. In uniform motion the space described is measured by the product of the time by the velocity ; that is, Sa TxV; S whence V Τα Ta yi that is, the time required to travel a given distance varies inversely as the velocity. Conversely, if one quantity varies inversely as another, the product of the two quantities is constant. 1 Thus, if V then the product of T by V is equal to a constant quantity. To 319. One quantity is said to vary directly as a second, and inversely as a third, when it varies as the product of the second by the reciprocal of the third. Thus, according to the Newtonian law of gravitation, the attraction (G) of any heavenly body varies directly as the quantity of matter (Q), and inversely as the square of the distance (D). Q That is, D2 |