7. When a submultiple of the first is contained in the second oftener than an equisubmultiple of the third is contained in the fourth, then the first is said to have to the second a less ratio than the third has to the fourth and on the other hand, the third is said to have to the fourth a greater ratio than the first has to

the second.

Note. For example, 4 has a less ratio to 12 than 10 has to 20, for any submultiple of 4, suppose 2, is contained in 12 oftener than an equisubmultiple of 10, i. e. 5 is contained in 20. If then there be any two magnitudes, A and B, and two others of the same kind, C and D, and if the half of A be contained in B six times, and the half of C be contained in D only four times, then A has a less ratio to B than C has to D; for A: 3 A:: C: 3C (or A:A:: C: C); but 3 C is greater than D .·. C has a greater ratio to D than C has to 3 C, i. e. than A has to 3 A, which is a quantity not greater than B.. &c.

6

8. Proportion is the similitude of ratios.

9. Proportion consists in three terms at least.

Note. When proportion exists only between three terms, they are said to be in continued proportion, the first consequent being the second antecedent. When this is the case the middle term is a mean proportional between the extremes.

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

11. When there are four magnitudes in continued proportion (A: B:: B: Cand P: C::C: D), then the first is said to have to the fourth (A: D) a triplicate ratio of that which it has to the second (i. e. of the ratio of A: B).

12. If there be any number of magnitudes of the same

kind (A, D, C, F) the first is said to have to the last (A: F) a ratio compounded of the ratios which the first has to the second, the second to the third, the third to the fourth (A: D and D: C and C: F) and so on to the last.

Note. By the duplicate ratio of A to B is meant the ratio of A to a third proportional to A and B, if C be the third proportional, the ratio of A to C is the one that is meant, thus the duplicate ratio of 1 to 3 is the ratio of 1 to 9, and the triplicate ratio of 1 to 3 is the ratio of 1

to 27.

There are other kinds of ratios sometimes mentioned by mathematicians, namely subduplicate, subtriplicate, &c. sesquialterate or sesquiplicate.

If several magnitudes be in continued proportion, the ratio of the first to the second is said to be subduplicate of the ratio of the first to the third, and so on.

If there be three magnitudes proportional (2: 16:128) and four others also proportional (1: 4: 16: 64), and if the first be to the last in the first series (2: 128) as the first to the last in the second (as 1 : 64), the ratio which the first has to the second in the first series (216) is said to be sesquialterate or sesquiplicate of the ratio which the first has to the second in the last series (1: 4).

It is not necessary that the magnitudes mentioned in the twelfth def. should be proportionals, for if there be any numbers 2, 9, 5, 15, then 2 will be to 15 in a ratio compounded of the ratios of 2: 9, 9: 5 and 5: 15; but a ratio may be compounded of several ratios, whose terms are not taken in continuation, as the ratio compounded of A: B and C D would not be the ratio of A: D, but of A to another quantity, suppose X, which is to B as D: C, for if B: X: C: D, then the ratio of A: X is compounded of the ratios of A: B and B : X; i. e. of the ratios of A: B and C: D.

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

:

Thus if A: B: C: D, then A is homologous to C, and B to D.

Geometers make use of the following terms, to express

certain modes of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.

14. By permutando or alternando; when it is concluded, that if there be four magnitudes of the same kind proportionals, the first is to the third as the second to the fourth.

Note. In this the antecedents are compared with one another, and the consequents with one another; thus if A: B : : C : D, by permutando A: C: : B : D.If 1:2::3: 6, permutando 1: 3 :: 2 : 6.

This definition differs from those following; in it it is necessary that the quantities should be of the same kind, but in the others it is not. It is called alternate proportion, and is proved by the 35th Prop. of the 5th Book.

15. By invertendo; when it is concluded that if there be four magnitudes proportional, the second is to the first as the fourth to the third.

Note. Here the antecedent is made consequent, and the consequent antecedent. If A : B::C: D then inA: vertendo, B: A:: D: C; this is called inverse ratio. The def. is proved by Prop. 20. Book 5.

16. Componendo ; when it is concluded, that if there be four magnitudes proportional, the first together with the second is to the second, as the third together with the fourth is to the fourth.

Note. Hence A+B :B : : C+D : D. Thus if 2: 16: 3, then 2+1:1::6+3: 3. This def. is proved by Cor. Prop. 21. Book 5.

17. Dividendo; when it is concluded, that if there be four magnitudes proportional, the difference between the first and second is to the second, as the difference between the third and fourth is to the fourth.

Note. If A: B:: C: D then A-B: B:: C-D: D. This def. is proved by Cor. 2. Prop. 25. Book 5.

18. Convertendo; when it is concluded, 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.

Note. If A: B:: C: D, then A: A+B:: C: C±D. This def. is proved by Prop. 21 & 25 and by Cor. 1. Prop. 25. Book 5.

19. Ex æquali, or ex æquo; when we infer that if there

be any number of magnitudes more than two, and as many others, which taken two by two of each series are in the same ratio, the first is to the last in the first series, as the first to the last in the second series.

Of this there are two species.

20. Ex æquo ordinate. When the first magnitude is to the second in the first series as the first to the second in the second series, and the second to the third in the first series as the second to the third in the second series, and so on, then it is concluded as in the preceding definition that the first is to the last in the first series, as the first to the last in the second.

Note.-If there be three magnitudes, A, B, C.
And three others, D, E, F.

Let A: B:: D: E,

And BC::E:F

Ex æquo ordinate A: C:: D: F. This is proved by Prop. 34, Book 5.

21. Ex æquo perturbate; when the first magnitude is to second in the first series, as the penultimate is to the last in the second series, and the second is to the third in the first series, as the antepenultimate is to the penultimate in the latter series, and so on, then it is concluded as above that the first is to the last in the first series, as the first is to the last in the second.

Note. As above, let there be three magnitudes A, B, C, And three others D, E, F,

Then if A : B::E:F

And B: C:: D: E

Ex æquo perturbate A: C:: D: F.

This definition is proved by Prop. 28, Book 5.

END OF DEFINITION'S.

THE

ELEMENTS OF GEOMETRY.

BOOK VI.

DEFINITIONS.

1. SIMILAR rectilineal figures are those which have all their angles respectively equal, and the sides about the equal angles proportional.

2. A parallelogram described on a right line is said to be applied to that line.

3. The altitude of any figure is the right line from the vertex perpendicular to the base.

4. A right line is said to be cut in extreme and mean ratio, when the whole line is to the greater segment as the greater segment is to the less.

5. A right line is said to be cut harmonically when it is divided into three segments, such that the whole line is to one extreme as the other extreme is to the middle part.

Three quantities are said to be harmonically proportional when the 1st: 3d :: difference between the first and second: the difference between the second and third.

Taking any right line b a, (Fig. 1) let it be cut so that bc:cd::ba: a d, the three lines b a, ca and da, are in harmonic proportion, since they fulfil the above definition. There is also an exemplification of this in Note 3 to Prop. 3 of this Book.