The First Six Books of the Elements of Euclid

by Euclid; John Casey

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