Let A and B be equal magnitudes, and C any other magnitude: each of the magnitudes A and B shall have the same ratio to C; and C shall have the same ratio to cach of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E; and of C any multiple whatever F.

Then, because D is the same multiple of A that E is of B, [Construction.

and that A is equal to B;

[Hypothesis.

therefore D is equal to E. [V. Axiom 1. Therefore if D be greater than F, E is greater than F; and if equal, equal; and if less, less.

But D and E are any equimultiples whatever of A and B, and F is any multiple whatever of C; [Construction. therefore A is to Cas B is to C. [V. Def. 5.

E B

Also C shall have the same ratio to A that it has to B. For the same construction being made, it may be shewn, as before, that D is equal to E.

Therefore if F be greater than D, F is greater than E; and if equal, equal; and if less, less.

But F is any multiple whatever of C, and D and E are any equimultiples whatever of A and B; therefore Cis to A as C is to B.

Wherefore, equal magnitudes &c.

PROPOSITION 8.

Q.E.D.

[Construction. [V. Definition 5.

THEOREM.

Of unequal magnitudes, the_greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater.

Let AB and BC be unequal magnitudes, of which AB is the greater; and let D be any other magnitude whatever: AB shall have a greater ratio to D than BC has to D; and D shall have a greater ratio to BC than it has to AB.

If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG the doubles of AC, CB (Figure 1).

But if that which is not the
greater of the two AC, CB, be
less than D (Figures 2 and 3),
this magnitude can be multiplied,
so as to become greater than D,
whether it be AC or CB.

Let it be multiplied until it be-
comes greater than D, and let the
other be multiplied as often.
Let EF be the multiple thus taken
of AC, and FG the same multiple
of CB;

therefore EF and FG are each
of them greater than D.

And in all the cases, take H the double of D, Kits triple, and so on, until the multiple of D taken is the first which is greater than FG. Let L be that multiple of D, namely, the first which is greater than FG; and let K be the multiple of D which is next less than L.

Then, because L is the first multiple of D which is greater than FG,

[Construction.

F

Fig. 1.

A

B

M

H

Fig. 2.

Fig. 3.

E

E

F

F

G

LK HD G

A

JAM

therefore EG is the same multiple of AB that FG is of CB;

that is, EG and FG are equimultiples of AB and CB.

[V. 1.

And it was shewn that FG is not less than K,

and EF is greater than D;

[Construction.

therefore the whole EG is greater than K and D together.

But K and D together are equal to Z; therefore EG is greater than L.

But FG is not greater than L.

And EG and FG were shewn to be equimultiples of AB and BC;

and L is a multiple of D.

[Construction.

Therefore AB has to D a greater ratio than BC has to D.

[V. Definition 7.

Also, D shall have to BC a greater ratio than it has to AB.

For, the same construction being made, it may be shewn, that L is greater than FG but not greater than EG.

And Z is a multiple of D,

[Construction.

and EG and FG were shewn to be equi

multiples of AB and CB.

[Construction.

Therefore D has to BC a greater ratio than it has to AB.

[V. Definition 7. Wherefore, of unequal magnitudes &c. Q.E.D.

PROPOSITION 9. THEOREM.

Magnitudes which have the same ratio to the same magnitude, are equal to one another; and those to which the same magnitude has the same ratio, are equal to one another.

First, let A and B have the same ratio to C: A shall be equal to B.

For, if A is not equal to B, one of them must be greater than the other; let A be the greater.

Then, by what was shewn in Proposition 8, there are

some equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the multiple of C.

Let such multiples be taken; and let D and E be the equimultiples of A and B, and F the multiple of C; so that D is greater than F, but E is not greater than F.

Then, because A is to C as B is to C; and of A and B are taken equimultiples D and E, and of C is taken a multiple F;

and that D is greater than F; therefore E is also greater than F.

But E is not greater than F;

which is impossible.

[Construction.

[V. Definition 5. [Construction

Therefore A and B are not unequal; that is, they are equal.

Next, let C have the same ratio to A and B: A shall be equal to B.

For, if A is not equal to B, one of them must be greater than the other; let A be the greater.

Then, by what was shewn in Proposition 8, there is some multiple F of C, and some equimultiples E and D of B and A, such that F is greater than E, but not greater than D.

And, because C is to B as C is to A,

[Hypothesis. and that F the multiple of the first is greater than E the multiple of the second,

[Construction. therefore F the multiple of the third is greater than D the multiple of the fourth.

But Fis not greater than D;

which is impossible.

[V. Definition 5.

[Construction.

Therefore A and B are not unequal; that is, they are equal.

Wherefore, magnitudes which &c. Q.E.D.

That magnitude which has a greater ratio than another has to the same magnitude is the greater of the two; and that magnitude to which the same has a greater ratio than it has to another magnitude is the less of the two.

First, let A have to C a greater ratio than B has to C: A shall be greater than B.

For, because A has a greater ratio to C than B has to C, there are some equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the multiple of C. [V. Def. 7. Let such multiples be taken; let D and E be the equimultiples of A and B, and F the multiple of C'; so that D is greater than F, but E is not greater than F;

therefore D is greater than E.

and

이

B

And because D and E are equimultiples of A and B, and that D is greater than E,

therefore A is greater than B.

[V. Axiom 4.

Next, let C have to B a greater ratio than it has to A: B shall be less than A.

For there is some multiple F of C, and some equimultiples E and D of B and A, such that F is greater than E, but not greater than D; [V. Definition 7.

therefore E is less than D.

And because E and D are equimultiples of B and A, and that E is less than D,

therefore B is less than A.

[V. Axiom 4.

.Wherefore, that magnitude &c. Q.E.D.