1. Let a : b :: c : d, and let a be a multiple of b, then c is the same multiple of d. Dem.—Let a = mb, then = m;but = ; therefore = m, and c = md. 2. Let a = , then = ; therefore = , Hence c = . PROP. VII.—Theorem. 1. Equal magnitudes have equal ratios to the same magnitude. 2. The same magnitude has equal ratios to equal magnitudes. Let a and b be equal magnitudes, and c any other magnitude. Then 1. a : c :: b : c, 2. c : a :: c : b. Dem.—Since a = b, dividing each by c, we have therefore a : c :: b : c. Again, since a = b, dividing c by each, we have therefore c : a :: c : b. Observation.—2 follows at once from 1 by Proposition B. PROP. VIII.—Theorem. 1. Of two unequal magnitudes, the greater has a greater ratio to any third magnitude than the less has; 2. any third magnitude has a greater ratio to the less of two unequal magnitudes than it has to the greater. 1. Let a be greater than b, and let c be any other magnitude of the same kind, then the ratio a : c is greater than the ratio b : c. Dem.—Since a is greater than b, dividing each by c, therefore the ratio a : c is greater than the ratio b : c. 2. To prove that the ratio c : b is greater than the ratio c : a. Dem.—Since b is less than a, the quotient which is the result of dividing any magnitude by b is greater than the quotient which is got by dividing the same magnitude by a; therefore is greater than . Hence the ratio c : b is greater than the ratio c : a. PROP. IX.—Theorem. Magnitudes which have equal ratios to the same magnitude are equal to one another; 2. magnitudes to which the same magnitude has equal ratios are equal to one another. 1. If a : c :: b : c, to prove a = b. Dem.—Sincea : c :: b : c, = . Hence, multiplying each by c, we get a = b. 2. If c : a :: c : b, to prove a = b. Dem.—Since c : a :: c : b, by inversion, a : c :: b : c; therefore a = b. [1]. PROP. X.—Theorem. Of two unequal magnitudes, that which has the greater ratio to any third is the greater of the two; and that to which any third has the greater ratio is the less of the two. 1. If the ratio a : c be greater than the ratio b : c, to prove a greater than b. Dem.—Since the ratio a : c is greater than the ratio b : c, Hence, multiplying each by c, we get a greater than b. 2. If the ratio c : b is greater than the ratio c : a, to prove b is less than a. Dem.—Since the ratio c : b is greater than the ratio c : a, Hence 1 ÷ is less than 1 ÷, that is, is less than . Hence, multiplying each by c, we get PROP. XI.—Theorem. Ratios that are equal to the same ratio are equal to one another.