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and fourth, according to any multiplication whatsoever, either together exceed, or are together equal, or are together deficient to each other.* 6. Magnitudes having the same ratio are called proportionals.

7. But when of equimultiples, the multiple of the first exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second, than the third has to the fourth.+

8. Proportion is a similitude of ratios.

9. Proportion consists of three terms at least. 10. If three magnitudes be proportionals, the first is said to have to the third a duplicate ratio of that which it has to the second.

11. If four magnitudes be proportionals, the first is said to have to the fourth a triplicate ratio of that which it has to the second, and so forwards, always more by one, as long as the proportion

continues.

12. Magnitudes are called homologous when the antecedents are to the antecedents, as the consequents to the consequents.

13. Alternate ratio is the assumption of the antecedent to the antecedent, and of the consequent to the consequent. ‡

14. Inverse ratio is an assumption of the consequent as the antecedent, and so compared with the antecedent to the consequent. §

* See Euclid's other definition of proportion in the Seventh Book. Such as the ratios 3 1 and 10: 7, for if the first and third be multiplied by 2 and the second and fourth by 4, there will result 6:4; 20: 28; where the first 6 is greater than the second 4, whilst the third 20 is less than the fourth 28.

If A: C: D

3:46:3} then alternately

ACB: D

3:6::4:8

In alternate proportion, it is necessary that the four magnitudes be of the same kind. For if a line A be to a line B, as a number c is to a number D, it does not follow that the line A will be to the number c as the line в to the

number D, since no ratio between a line and number can be assigned.

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15. Composition of ratio is the assumption of the antecedent together with the consequent taken as one, to that consequent. *

16. Division of ratio is the assumption of the excess, by which the antecedent exceeds the consequent, to that consequent. †

17. Conversion of ratio is the assumption of the antecedent to the excess, by which the antecedent exceeds that consequent. ‡

18. Ratio of equality is when there are several magnitudes, and as many others, so that the first of the first magnitudes shall be to the last, as the first in the second magnitudes to the last. Or otherwise, the assumption of the extremes by subtracting the means. §

19. Ordinate proportion is, when it shall be as antecedent to a consequent, so is an antecedent to a consequent, and as the consequent is to any other, so is the consequent to any other. || 20. Perturbate proportion is, when there are three magnitudes and others equal to them in number, it shall be as an antecedent in the first magnitudes is to a consequent, so is an antecedent in the second magnitudes to a consequent. And as a consequent in the first magnitudes to another, so is some other in the second magnitudes to an antecedent.

If A : B :: C: D
3:4::6:8

If A B:: C: D
3:4::6 8

If A : B :: C: D
3:4::68

If A: B:: D: E

B: C:: E: F

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then from equality {

: 4-3::6:8-6

AC: D: F

On the supposition that A, B, and c, are magnitudes in one order, and D, E, and F, in another.

If A: B::a: b then if the ratios are taken equal in a direct order,
B:C:: b : C and that the extremes are proportional, viz. A : D
C: D::C: d :: ad, it is called ordinate proportion.

** As suppose the magnitude A is to the magnitude в as the magnitude c is to the magnitude D; and again, suppose the consequent в is to some other magnitude E as some other magnitude F is to the antecedent c; then is this proportion called perturbate. For further elucidation, consult Fenn's Euclid, page 167.

AXIOMS.

1. "Equimultiples of the same, or of equal magnitudes, are equal to one another."

2. "Those magnitudes of which the same, or equal magnitudes, are equimultiples, are equal to one another."

3. "A multiple of a greater magnitude is greater than the same multiple of the less."

4. "That magnitude of which a multiple is greater than the same multiple of another, is greater than that other magnitude.'

PROPOSITION I.

THEOREM.

If there be any number of magnitudes equimultiples of as many other magnitudes, each of each; whatsoever mul tiple one magnitude is of one, the same multiple shall all be of all.

Let AB, CD, be any number of magnitudes, equimultiples of as many other magnitudes E, F, each of each; whatsoever multiple AB is of E, the same multiple AB, CD, together, is of E and F together.

For because AB is an equimultiple of E, and CD of F; as many magnitudes as are in AB equal to E, so many will there be in CD equal to F. Divide

a

And

H

D

AB into parts equal to E, which let be AG, GB; also CD into parts equal to F, namely, CH, HD. Therefore the multi- G tude of parts CH, HD, will be equal to B the multitude of them AG, GB. because AG is equal to E, also CH equal to F; AG, CH, Ax. 2. 1. will be equal to E and F. For the same reason GB is equal to E, and HD to F; therefore GB, HD, will be equal to E and F: whence as many magnitudes as are in AB equal to E, so many are there in AB, CD, equal to E, F. Wherefore what multiple AB is of E, the same multiple will AB, CD, be of E, F. Therefore, if there be any magnitudes, &c. Q. E. D.*

The same by Algebra.

Let there be any number of magnitudes a m, a n, equimultiples of as many others m, n; then shall a m be the same multiple of m as a m + a n is of m + n. For a m is contained a times in m, and a man is also contained a times in m + n. Q. E. D.

*This is only a particular case of proposition 12.

PROPOSITION II.

THEOREM.

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

For let the first magnitude AB be the same multiple of the second c, as the third DE is of the fourth F; and a fifth magnitude BG be the same multiple of the second c, as the sixth EH is of the fourth F: then is AG the first and fifth taken together the same multiple of the second c, as DH the third and sixth together is of the fourth F.

H

For because AB is the same multiple of c as DE is of F; as many magnitudes as are in AB equal to c, so many will there be in DE equal to F. And for the same reason as many magnitudes as are in BG equal to c, so many will there be in EH equal to F: therefore as many magnitudes as are in the whole AG equal to C, so many will there be in B the whole DH equal to F. Wherefore

a

whatever multiple AG is of c, the same multiple is DH of F: therefore AG the first and fifth taken together shall be the

A

same multiple of the second c, as DH the third and sixth, is of F the fourth. Wherefore if the first be the same multiple, &c. Q. E. D.

The same by Algebra.

Let a m and a n be equimultiples of the magnitudes m, n; also b m, bn, equimultiples of the same magnitudes m, n; then shall a m+bm be the same multiple of m, as a n + b n is of n. For m is contained a+b times in a m + bm; and times in a n + b n. Q. E. D.

n is contained a + b

a 1. 5.

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