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 mC 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 introduction 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 m▲ 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.

Then A is to B as C is to D,

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

V. 4.

Again, ·.· C is to D as E is to F,

.. if mC be greater than nD, mE is greater than nF; and if equal, equal; if less, less.

V. 4.

Hence, if mA be greater than nB, mE is greater than nF;

and if equal, equal; if less, less.

.. A is to B as E is to F.

V. Def. 5.

Q. E. Di

PROPOSITION VI. (Eucl. v. 7.)

Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other magnitude.

Then must A be to C as B is to C,

and C must be to A as C is to B.

Take mA and mB any equimultiples of A and B, and nC any multiple of C.

Then ... A

=

B,...
.. MA =

mB.

V. Ax. 1.

..if mA be greater than nC, mB is greater than nⱭ;

and if equal, equal; if less, less.

.. A is to C as B is to C.

V. Def. 5.

Again, if nC be greater than mA, nC is greater than mB;

and if equal, equal; if less, less.

.. C is to A as C is to B.

V. Def. 5.

Q. E. D.

PROPOSITION VII. (Eucl. v. 8.)

Of two unequal magnitudes, the greater has a greater ratio to any other magnitude than the less has; and the same magnitude has a greater ratio to the less, of two other magnitudes, than it has to the greater.

Let A and B be any two magnitudes, of which A is the greater, and let D be any other magnitude.

Then must the ratio of A to D be greater than the ratio of B to D.

Take such equimultiples of A and B, qA and qB, that each of them may be greater than D.

Then A is greater than B,

Note 3, p. 216.

..qA is greater than qB.

Let qAqB, R together.

V. Ax. 3.

Then, however small R may be, we can find a multiple of R, suppose mR, such that mR is greater than qB.

Note 3.

Take equimultiples of qA and qB, mqA and mqB, and take a multiple of D, nD, such that nD is not less than mqB and not greater than (mq+q) B.

Then

mqA

=

mqB, mR together, and mR is greater than qB,

.. mqA is greater than (mq + q) B,

and, a fortiori, mqA is greater than nD.

But mqB is not greater than nD,

Note 3.
V. 1.

.. the ratio of A to D is greater than the ratio of B to D.

V. Def. 7.

Also, the ratio of D to B must be greater than the ratio of D to A.

For, the same multiples being taken as before,

⚫nD is not less than mqB,

and nD is less than mqA,

.. D has to B a greater ratio than D has to A.

V. Def. 7.

Q. E. D.