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PROPOSITION XIII. (Eucl. v. 13.)

If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first must also have to the second a greater ratio than the fifth has to the sixth.

Let A have to B the same ratio that C has to D, but C to D a greater ratio than E has to F.

Then must A have to B a greater ratio than E has to F.

For C has to D a greater ratio than E has to F, we can find such equimultiples of C and E, suppose mCand mE, and such equimultiples of D and F, suppose nD and nF, that mC is greater than nD, but mE not greater than nF.

Then A is to B as C is to D,
and mC is greater than nD,
.. mA is greater than nB.

And mE is not greater than nF.

..A has to B a greater ratio than E has to F.

V. Def. 7.
Hyp.

V. 4.

V. Def. 7.

Q. E. D.

PROPOSITION XIV. (Eucl. v. 14.)

If the first has to the second the same ratio which the third has to the fourth; then, if the first be greater than the third the second must be greater than the fourth; and if equal, equal ; and if less, less.

Let A have the same ratio to B that C has to D.

Then if A be greater than C, B must be greater than D.
For A is greater than C,

and B is any other magnitude,

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

V. 7.

But A is to B as C is to D.

V. 13.

V. 9.

.. C has a greater ratio to D, than C has to B. .:B is greater than D. Similarly it may be shown that if A be less than C, B must be less than D; and that if A be equal to C, B must be equal to D.

Q. E. D.

PROPOSITION XV. (Eucl. v. 16.)

If four magnitudes of the same kind be proportionals, they must also be proportionals when taken alternately.

Let A, B, C, D be four magnitudes of the same kind, and let A be to B as C is to D.

Then alternately A must be to C as B is to D.

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

Then mA is to mB as A is to B,
and C is to D as A is to B,

V. 11.

Hyp.

... mA is to mB as C is to D.

V. 5.

But nC is to nD as C is to D;

and .. mA is to mB as nC is to nD.

If... mA be greater than nC, mB is greater than nD;

V. 11.

V. 5.

V. 14.

and if equal, equal; if less, less.

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

V. Def. 5.

Q. E D.

PROPOSITION XVI. (Eucl. v. 18.)

If magnitudes taken separately be proportionals, they must be proportionals also when taken jointly.

Let A have the same ratio to B that C has to D.

Then must A, B together have the same ratio to B, that C, D together has to D.

First, when all the magnitudes are of the same kind,
A is to B as C is to D,

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

V. 15.

V. 10.

V. 15.

.. A, B together is to C, D together as B is to D, and .. A, B together is to B as C, D together is to D. Next, when all the magnitudes are not of the same kind, we may employ a method of proof which includes the former case: thus

Take of A, B, C, D any equimultiples mA, mB, mC, mD, and of B and D take any equimultiples nB, nD.

Then. A is to B as C is to D,

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

V. 4.

If then mA, mB together be greater than mB, nB together, mC, mD together is greater than mC, nD together;

and if equal, equal; if less, less.

I. Ax. 2, 4.

Now mA, mB together is the same multiple of A, B together that mC, mD together is of C, D together;

V. 1.

and mB, nB together is the same multiple of B that mD, nD together is of D.

V. 2.

.. A, B together is to B as C, D together is to D. V. Def. 5.

Q. E. D.

SECTION V.

Containing the Propositions occasionally referred to in Book VI.

PROPOSITION XVII. (Eucl. v. 4.)

If the first of four magnitudes has to the second the same ratio which the third has to the fourth, and any equimultiples of the first and third be taken, and also any equimultiples of the second and fourth, then must the multiple of the first have the same ratio to the multiple of the second which the multiple of the third has to that of the fourth.

If A be to B as C is to D,

and mA, mC be taken equimultiples of A and C,
and nB, nD............

then must m▲ be to nВ as mC is to nD.

of B and D,

Take of mA, mC any equimultiples pmA, pmC, and of nB, nD.....

qnB, qnD.

Then pmA, pmC are equimultiples of A and C, and qnB, qnD........

V. 3.

of B and D.

V. 3.

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Then

and qnB, qnD..........

pmA, pmO are equimultiples of mA, mC,

.. mA is to nB as mC is to nD.

of nB, nD,

V. Def. 5.

Q. E. D.

PROPOSITION XVIII. (Eucl. v. A.)

If the first of four magnitudes have the same ratio to the second that the third has to the fourth, then, if the first be greater than the second, the third must be greater than the fourth; and if equal, equal; and if less, less.

Let A be to B as C is to D.

Then if A be greater than B, C must be greater than D; and if equal, equal; and if less, less.

Take any equimultiples of each, mA, mB, mC, mD.

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..if mA be greater than mB, mC is greater than mD; and if equal, equal; and if less, less.

First, suppose A greater than B,

V. 4.

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PROPOSITION XIX. (Eucl. v. D.)

If the first be to the second as the third is to the fourth, and if the first be a multiple, or a submultiple, of the second, the third must be the same multiple, or the same submultiple, of the fourth.

Let A be to B as C is to D,

and, first, let A be a multiple of B.

Then must C be the same multiple of D.

Let A=mB, and take mƊ the same multiple of D that A is of B.

[blocks in formation]
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