FIFTH BOOK.

DEFINITIONS.

1. A less magnitude is called an aliquot part, or a submultiple of a greater, when the less measures the greater.

2. A greater is called a multiple of a less, when the greater is measured by the less.

3. Ratio is a mutual relation of two magnitudes of the same kind, in respect of quantity.

4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the greater.

5. Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when any submultiple whatsoever of the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth.

6. Magnitudes which have the same ratio are called proportionals.

7. When a submultiple of the first is contained oftener in the second, than an equi-submultiple of the third is contained in the fourth, then the first is said to have a less ratio to the second, than the third has to the fourth; and conversely, the third is said to have a greater ratio to the fourth, than the first to the second. 8. Proportion is the similitude of ratios.

9. And proportion consists of three terms, at least.

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10 When three magnitudes are proportional (A to B as B to C), the first is said to have to the third (A to C) a duplicate ratio of that which it has to the second, namely, of the ratio of A to B.

11. When four magnitudes are proportional, one after the other (A to B as B to C, and B to C as C to D), the first is said to have to the fourth (A to D) a triplicate ratio of that which it has to the second (namely, of the ratio of A to B). And so on, increasing by unity, whatsoever be the number of proportionals.

12. If there be any number of magnitudes of the same kind (A, D, C, and F), the first is said to have to the last (A to F) a ratio compounded of the ratios which the first has to the second, and the second to the third, and the third to the fourth (A to D, D to C, and C to F), and so on to the last.

13. In proportionals, the antecedents are called homologous to the antecedents, and the consequents to the consequents.

Either the order or the magnitude of proportionals can be changed, so that they remain still proportionals; the various modes of changing, which are made use of by Geometers, are designated by the following names :—

14. By Permutation, when it is inferred that if there be four magnitudes, of the same kind, proportional, the first is to the third as the second to the fourth; as is shown in Prop. 33, B. 5. Ex. gr.-A to B as C to Do, becomes A to C as B to Do.

15. By Inversion, when it is inferred that if four magnitudes be proportional, the second is to the first as the fourth to the third; as is demonstrated in Prop. 20, namely-A to B as CD to E, becomes B to A as E to CD.

16. By composition, when it is inferred, if four magnitudes be proportional, that the first together with the second is to the second, as the third together with

the fourth is to the fourth; as shown in Cor. Prop. 21. Thus-A to B as C to D, will be A + B to B as C + D to D.

17. By Division, when it is inferred, if there be four magnitudes proportional, that the difference between the first and the second is to the second as the difference between the third and the fourth to the fourth; as is proved in Cor. 2, Prop. 25. Ex. gr.-A to B as C to D, becomes A - B to B as C-D to D.

18. By Conversion, when it is inferred, that if there be four magnitudes proportional, the first is to the sum or difference of the first and second, as the third is to the sum or difference of the third and fourth. See Props. 21 and 25, also Cor. 1, Prop. 25. Thus-A is to B as C is to D, becomes A to (A+B) as C to (C + D), or A to (A — B) as C to (C — D).

19. Ex æquali vel ex æquo (or from equality), when it is inferred, that if there be many magnitudes, and an equal number of others, which, if taken two by two, are in the same ratio, the first is to the last in the first series, as the first to the last in the next series.

Of this there are the two following species :—

20. Ex æquo ordinate (from orderly equality), when the first magnitude will be to the second in the first series, as the first to the second in the next series, and the second to the third in the first series, as the second to the third in the next, and so on; and it is inferred, as in Def. 19, that the first is to the last in the first series, as the first to the last in the next series; as shown in Prop. 34. Ex. gr.-Let A, B, and CO, be three magnitudes in the first series, and D, E, and F, be three in the second series; then, according to the Definition, we have

A to B as D to E, and B to CO as E to F,

and A to CO as D to F,

And so on, if there were four or more magnitudes.

21. Ex æquo perturbate (or disorderly proportion), when the first magnitude will be to the second in the first series, as the last but one to the last in the next; and the second to the third in the first series, as the last but two to the last but one in the next, and so on; and as above, that the first is to the last in the first series, as the first to the last in the next series; as shown in Prop. 38. Thus,

1. Equi-multiples, or equi-submultiples of the same, or equal magnitudes, are equal.

2. Magnitudes of which the same is an equi-multiple, or an equi-submultiple, or of magnitudes of which equal magnitudes are equi-multiples or equi-submultiples, are also equal to each other.

3. A multiple or a sub-multiple of a greater magnitude is greater than an equi-multiple or equi-submultiple of a less.

4. A magnitude whose multiple or sub-multiple is greater than the equi-multiple or equi-submultiple of that other, is greater than that other.

PROPOSITION I. THEOREM.

If there be given two equal magnitudes (BC and DE), so often as any third magnitude (A) is contained in the one, so often it is contained in the other.

First let one of the given magnitudes, BC, be a multiple of A, and A is not oftener contained in the one than the other. For, if it be possible, let

A be contained oftener in DE than in BC; and as often as A is contained in BC, so often take away from DE, and let any part mE remain, therefore BC and Dm are equi-multiples of the same, A, and therefore BC is equal to Dm (by Ax. 1, B. 5), but BC is equal to DE (by Hypoth.), and therefore Dm is equal to DE (by Ax. 3, B. 1), which is absurd. Now, if it be possible, let A be contained oftener in BC than in DE, and take away A as often as possible from DE, letting a part mE remain less than A, take it away so often from BC, and since it is contained oftener in BC than in DE, a part nC remains greater than A, or equal to it, and therefore is greater than mE; but Bn and Dm are equi-multiples of the same A (by Const.), and therefore equal (by Ax. 1, B. 5), BC and DE are also equal (by Hypoth.), and therefore nC is equal to mE (by Ax. 3, B. 1), but also greater than it, which is absurd.

Secondly, let neither of the given magnitudes be a multiple of A, and A is not oftener contained in one than in the other.

For, if it be possible, let A be contained oftener in BC than in DE, and take it away from DE as often as it can be done, and mE shall remain less than A; take it away as often from BC, and nC will remain greate than A, and therefore greater than mE; but Bn an Dm are equi-multiples of the same A, and therefore equal (by Ax. 1, B. 5), and BC and DE are also equa. (by Hypoth.), therefore nC is equal to mE (by Ax. 3, B. 1), but it is also greater than it, which is absurd.

So that in no case is A contained oftener in either of the given magnitudes, BC or DE, than in the other.

COR. 1.-Hence it appears, if one of the given magnitudes be a multiple of A, the other will be an equi-multiple of A.

COR. 2.-If two magnitudes be equal, as often as the first is contained in any third, so often is the secon contained in the same third..