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PROPOSITION XXVI. (Eucl. v. 19.)

If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder must be to the remainder as the whole is to the whole.

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

For

Then must A be to C as A, B together is to C, D together.
A, B together is to C, D together as B is to D,
.. A, B together is to B as C, D together is to D,
and.. A is to B as C is to D,

Hence A is to C as B is to D.

But A, B together is to C, D together as B is to D. .. A is to Ɑ as A, B together is to C, D together.

PROPOSITION XXVII. (Eucl. v. 21.)

V. 15.

V. 25.

V. 15.

Hyp.

V. 5.

Q. E. D.

If there be three magnitudes, and other three, which have the same ratio, taken two and two, but in a cross order, then if the first be greater than the third, the fourth must be greater than the sixth; and if equal, equal; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, and let A be to B as E is to F,

and B be to C as D is to E.

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

First, if A be greater than C,

A has to B a greater ratio than C has to B, and.. E has to F a greater ratio than C has to B.

Now B is to C as D is to E,

.. C is to B as E is to D.

Hence E has to F a greater ratio than E has to D.

..D is greater than F.

Similarly the other cases may be proved.

V. 7.

V. 13.

Hyp.

V. 12.

V. 9.

Q. E. D.

PROPOSITION XXVIII. (Eucl. v. 23.)

If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first must have to the last of the first magnitudes the same ratio which the first of the others has to the last of these.

Let A, B, C be three magnitudes, and D, E, F other three, and let A be to B as E is to F,

[blocks in formation]

Of A, B, D take any equimultiples mA, mB, mD, and of C, E, F take any equimultiples nC, nE, nF.

[blocks in formation]

V. 17.

.. mB is to nC as mD is to nE.

Hence, if mA be greater than nC, mD is greater than nF;

and if equal, equal; and if less, less.

.. A is to C as D is to F.

V. 27.

V. Def. 5.

The proposition may be easily extended to any number of magnitudes.

QE. D.

PROPOSITION XXIX. (Eucl. v. 25.)

If four magnitudes of the same kind be proportionals, the greatest and least of them together must be greater than the other two together.

Let A be to B as C is to D,

and let A be the greatest of the four magnitudes, and consequently D the least.

V. 18, and V. 14.

Then must A, D together be greater than B, C together.

B, P together is to B as D, Q together is to D,

Let AB, P together, and C

=

D, Q together.

Then

.. P is to B as Q is to D,

and B is greater than D.

.. P is greater than Q.

V. 25.

V. 14.

Hence P, B, D together are greater than Q, B, D

together.

.. A, D together are greater than B, C together.

I. Ax. 4.

Q. E. D.

PROPOSITION XXX. (Eucl. v. C.)

If the first be the same multiple of the second, or the same submultiple of it, that the third is of the fourth, the first must be to the second as the third is to the fourth.

First, let A be the same multiple of B, that C is of D.
Then must A be to B as C is to D.

[blocks in formation]

Take of A and C any equimultiples mA, mC,
and of B and Dany equimultiples nB, nD.

Then mA =

mpB and mC = mpD.

Now if mpB be greater than nB,
mpD is greater than nD ;

and if equal, equal; if less, less.

V. 3.

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

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

V. Def. 5.

Next, let A be the same submultiple of B, that C is of D. Then must A be to B as C is to D.

For ... A is the same submultiple of B, that C is of D, .. B is the same multiple of A, that D is of C,

and

.. B is to A as D is to C, by the first case,

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

V. 12. Q. E. D.

PROPOSITION XXXI. (Eucl. v. E.)

If four magnitudes be proportionals, they must also be proportionals by conversion; that is, the first must be to its excess above the second as the third is to its excess above the fourth.

Let A, B together be to B as C, D together is to D.
Then must A, B together be to A as C, D together is to C.

For A, B together is to B as C, D together is to D,

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

V. 25.

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

V. 12.

and ..

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

V. 16. Q. E. D.

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