THE PROPOSITIONS OF EUCLID,

BOOK V.

[The Student is recommended to take the Propositions in the following order.

§ 1. 1, 5, 3, 2 (without Corollary), 6, Corollary of 2.

§ 2. C, D, 15.

§ 3. 11, 13.

§ 4. 7, 9, 8, 10.

§ 5. 16, B, 18, 17, 17*, E, 12, 19, 24, 22, 23.

§ 6. 4.

§ 7. A, 14, 20, 21, 25.]

PROP. I.

If any number of magnitudes be equimultiples of as many: whatever multiple any one of them is of its part, the same multiple are all the first magnitudes of all the others.

[If any number of magnitudes (A, B, C, &c.) be equimultiples of as many (a, b, c, &c.), each of each; then, whatever multiple any one of them (4) is of its part (a), the same multiple are all the first magnitudes (A+B+C+ &c.) of all the others (a+b+c+&c.)].

PROP. II.

If the first magnitude be the same multiple of the second as the third is of the fourth, and if the fifth be the same multiple of the second as the sixth is of the fourth; the first and fifth together are the same multiple of the second as the third and sixth together are of the fourth.

[If the first magnitude (a) be the same multiple of the second (b) as the third (c) is of the fourth (d), and if the fifth (e) be the same multiple of the second (b) as the sixth (ƒ) is of the fourth (d); then the first and fifth together (a+e) are the same multiple of the second (b) as the third and sixth together (c+f) are of the fourth (d).]

The Proposition holds true of two ranks of magnitudes, of which each of the first rank is the same multiple of a single magnitude as each of the second rank is of another single magnitude.

If

[This Corollary is usually stated as follows.]

any number of magnitudes (A, B, C, &c.) be multiples of another (X), and as many others (a, b, c, &c.)

the same multiples of (x): all the first (A+B+C+ &c.) are the same multiple of (X) as all the others (a+b+c+&c.) are of (x).

If the first be the same multiple of the second as the third is of the fourth, and if of the first and the third there be taken equimultiples: these are equimultiples of the second and the fourth.

[If the first (a) be the same multiple of the second (b) as the third (c) is of the fourth (d), and if of the first (a) and the third (c) there be taken equimultiples (A, C): these are equimultiples of the second (b) and the fourth (d).]