Let a : b :: e : f, and c : d :: e : f, to prove a : b :: c : d. Dem.—Since a : b :: e : f, = . In like manner, = . Hence = [I., Axiom i.], and a : b :: c : d. PROP. XII.—Theorem. If any number of ratios be equal to one another, any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents. Let the ratios a : b, c : d, e : f, be all equal to one another; it is required to prove that any of these ratios is equal to the ratio a + c + e : b + d + f. Dem.—By hypotheses, Since these fractions are all equal, let their common value be r; then we have = r, = r, = r; therefore a = br, c = dr, e = fr; therefore a + c + e = (b + d + f)r. Hence = r; therefore = , and a : b :: a + c + e : b + d + f. Cor.—With the same hypotheses, if l, m, n be any three multipliers, a : b :: la + mc + ne : lb + md + nf. PROP. XIII.—Theorem. If two ratios are equal, and if one of them be greater than any third ratio, then the other is also greater than that third ratio. If a : b :: c : d, but the ratio of c : d greater than the ratio of e : f; then the ratio of a : b is greater than the ratio of e : f. Dem.—Since the ratio of c : d is greater than the ratio of e : f, Again, since a : b :: c : d, = ; therefore is greater than . or the ratio of a : b is greater than the ratio of e : f. PROP. XIV.—Theorem. If two ratios be equal, then, according as the antecedent of the first ratio is greater than, equal to, or less than the antecedent of the second, the consequent of the first is greater than, equal to, or less than the consequent of the second. Let a : b :: c : d; then if a be greater than c, b is greater than d; if equal, equal; if less, less. Dem.—Since a : b :: c : d. we have = , and multiplying each by we get × = ×, or = ; therefore a : c :: b : d. Hence, Proposition [A], if a be greater than c, b is greater than d; if equal, equal; and if less, less. PROP. XV.—Theorem. Magnitudes have the same ratio which all equimultiples of them have. Let a, b be two magnitudes, then the ratio a : b is equal to the ratio ma : mb. Dem.—The ratio a : b = , and the ratio of ma : mb = ; but since the value of a fraction is not altered by multiplying its numerator and denominator by the same number, = ; therefore a : b :: ma : mb. PROP. XVI—Theorem. If four magnitudes of the same kind be proportionals they are also proportionals by alternation (alternando). Let a : b :: c : d, then a : c :: b : d. Dem.—Since a : b :: c : d, and multiplying each by , we get . = .,