NOTE 4. To make Def. v. clearer we give the following illustration. Suppose A, B, C, D to be four magnitudes; the scales of their multiples will then be—

where mA, mC stand for any equimultiples of A and C, and nB, nD stand for any equimultiples of B and D: then the Definition may be stated more briefly thus:

A is said to have the same ratio to B which C has to D, when mA is found in the same position among the multiples of B, in which mC is found among the multiples of D; or, which is the same thing, when the order of the multiples of A and B in the Scale of Relation of A and B, is precisely the same as the order of the multiples of C and D in the Scale of Relation of C and D; or, when every multiple of A is found in the same position among the multiples of B, in which the same multiple of C is found among the multiples of D.

NOTE 5. The use of Def. v. will be better understood by the following application of it.

To show that rectangles of equal altitude are to one another as their bases.

Let AC, ac be two rectangles of equal altitude.

Let B, B' and R, R' stand for the bases and the areas of these rectangles respectively.

Take AD, DE, EF,....................... .m in number, and all equal,

And ad, de, ef, fg, gh,................

n in number, and all equal.

Now we can prove, by superposition, that if AF be greater than ah, AP will be greater than ap, and if equal, equal; and if less, less.

That is, if mB be greater than nB', mR is greater than nR'; and if equal, equal; and if less, less.

Hence, by Def. v.,

B is to B' as R is to R'.

Hence we deduce two Corollaries, which are the foundation of the proofs in Book vi.

COR. I. Parallelograms of equal altitude are to one another as their bases.

For the parallelograms are equal to rectangles, on the same bases and between the same parallels.

COR. II. Triangles of equal altitude are to one another as their bases.

For the triangles are equal to the halves of the rectangles, on the same bases and between the same parallels.

N.B.-These Corollaries are proved as a direct Proposition in Eucl. vi. 1. Cor. II. could not, consistently with Euclid's method, be introduced in this place, for it assumes Proposition XI. of Book v.

DEF. VI. Magnitudes which have the same ratio are called Proportionals.

If A, B, C, D be proportionals, it is usually expressed by saying, A is to B as C is to D.

The magnitudes A and C are called the Antecedents of the ratios. B and D.......... .Consequents.

The antecedents are said to be homologous to one another, that is, occupying the same position in the ratios (óμóλoyoı), and the consequents are said to be homologous to one another.

DEF. VII. When of the equimultiples of four magnitudes, taken as in Def. v., the multiple of the first is greater than [or is equal to] the multiple of the second, but the multiple of the third is not greater than [or is less than] the multiple of the fourth, then the first is said to have to the second a greater ratio, than the third has to the fourth.

NOTE 6. The meaning of Def. vII. may be expressed, after taking the scales of multiples as in the explanation of Def. v., thus:

A is said to have to B a greater ratio than C has to D, when two whole numbers m and n can be found, such that mA is greater than nB, but m°C not greater than nD; or, such that mA is equal to nB, but mC less than nD.

SECTION III.

Containing the Propositions most frequently referred to in Book VI.

NOTE 7. The Fifth Book of Euclid may be regarded in two aspects: first, as a Treatise on the Theory of Ratio and Proportion, complete in itself, and depending in no way on the preceding Books of the Elements; and secondly, as a necessary 'ntroduction to the Sixth Book.

If we make the number of references in Book VI. a test of the importance of particular Propositions in Book v., they will be arranged in the following order :

Propositions X., XI, XV., XVI., XIX., XXII., are referred to once.

It is desirable, then, that the student should observe that the three Propositions, which are of especial importance for Book vi., are included in this Section.

PROPOSITION IV.

If four magnitudes be proportionals, and any equimultiples be taken of the first and third, and also any equimultiples of the second and fourth, if the multiple of the first be greater than that of the second, the multiple of the third must be greater than that of the fourth; and if equal, equal; and if less, less.

Let A be to B as C is to D,

and let any equimultiples mA, mC be taken of A and C, and any equimultiples nB, nD............ of B and D.

Then if mA be greater than nB, mC must be greater than nD ; and if equal, equal; if less, less.

For if mA be greater than nB, but mC not greater than nD, then will A have to B a greater ratio than C has to D; which is not the case. V. Def. 7.

Hence if mA be greater than nB, mC must be greater than nD.

Similarly it may be shown that, if mA be equal to, or less than, nB, mC must also be equal to, or less than, nD.

Q. E. D.

N.B.-We have added this Proposition to meet an objection, which might be made to a reference to Definition v., when the converse of that Definition is wanted. This reference is of frequent occurrence in Simson's edition.

PROPOSITION V. (Eucl. v. 11.)

Ratios that are the same to the same ratio, are the same to one another.

Let A be to B as C is to D,

and E be to F as C is to D.

Then must A be to B as E is to F.

Take of A, C, E any equimultiples mA, mC, mE, and of B, D, F any equimultiples nB, nD, nF.