Let a : b :: c : d; then ma : nb :: mc : nd. Dem.—We have a : b :: c : d (hyp.); therefore = . Hence, multiplying each fraction by , we get = ; therefore ma : nb :: mc : nd. PROP. V.—Theorem. If two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a′, b′) are of another (c′), then the difference of the two first is the same multiple of their submultiple (c), which the difference of their corresponding magnitudes is of their submultiple (c′) (compare Proposition ii.). Dem.—Let m and n be the multiples which a and b are of c. Then we have a = mc, and a′ = mc′, b = nc, and b′ = nc′. Therefore (a−b) = (m−n)c, and (a′−b′) = (m−n)c′. Hence a−b is the same multiple of c that a′− b′ is of c′. Cor.—If a − b = c, a′− b′ = c′; for if a − b = c, m − n = 1. PROP. VI.—Theorem. If a magnitude (a) be the same multiple of another (b), which a magnitude (a′) taken from the first is of a magnitude (b′) taken from the second, the remainder is the same multiple of the remainder that the whole is of the whole (compare Proposition i.). Dem.—Let m denote the multiples which the magnitudes a, a′ are of b, b′; then we have a = mb, a′ = mb′. Hence (a − a′) = m(b − b′). Prop. A.—Theorem (Simson). If two ratios be equal, then according as the antecedent of the first ratio is greater than, equal to, or less than its consequent, the antecedent of the second ratio is greater than, equal to, or less than its consequent. Dem.—Let a : b :: c : d; then = ; and if a be greater than b, is greater than unity; therefore is greater than unity, and c is greater than d. In like manner, if a be equal to b, c is equal to d, and if less, less. Prop. B.—Theorem (Simson). If two ratios are equal their reciprocals are equal (invertendo). Let a : b :: c : d, then b : a :: d : c. Dem.—Since a : b :: c : d; then = ; therefore 1 ÷ = 1 ÷, or = Hence b : a :: d : c. Prop. C.—Theorem (Simson). If the first of four magnitudes be the same multiple of the second which the third is of the fourth, the first is to the second as the third is to the fourth. Let a = mb, c = md; then a : b :: c : d. Dem.—Since a = mb, we have = m. In like manner, = m; therefore = . Hence a : b :: c : d. Prop. D.—Theorem (Simson). If the first be to the second as to the third is to the fourth, and if the first be a multiple or submultiple of the second, the third is the same multiple or submultiple of the fourth.