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

V. 4.

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..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. D.

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 nC;

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, q▲ and qB, that each of them may be greater than D.

Then A is greater than B,

..qA is greater than qB.

Let qA = qB, R together.

Note 3, p. 216.

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 q▲ and qB, mq▲ 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.

QE. D.

PROPOSITION VIII. (Eucl. v. 9.)

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.

Let A and B have the same ratio to C.

Then must A = B.

For if A were greater than B,

A would have a greater ratio to C than B has to C; V. 7. which is not the case.

And if A were less than B,

B would have a greater ratio to C than A has to C;

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Next, let C have the same ratio to A that C has to B.

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V. 7.

For we can show, as before, that A cannot be greater or less than B.

.. A = B.

Q. E. D.

PROPOSITION IX. (Eucl. v. 10.)

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.

Let A have to C a greater ratio than B has to C.
Then must A be greater than B.

For if A were equal to B, then would A have the same ratio to C that B has to C; which is not the case. V. 8. And if A were less than B, then would A have to Ca ratio less than that which B has to C; which is not the case. V. 7. .. A is greater than B.

Next, let C have a greater ratio to B than it has to A.

Then must B be less than A.

For if B were equal to A, then would C have the same ratio to B which it has to A; which is not the case.

V. 8.

And if B were greater than A, then C would have to B a ratio less than that which C has to A; which is not the

case.

.. B is less than A.

V. 7.

Q. E. D.

PROPOSITION X. (Eucl. v. 12.)

If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so must all the antecedents taken together be to all the consequents.

Let any number of magnitudes A, B,C,D,E,F...be proportionals, that is, A to B as C to D and as E is to F...

Then must A be to B as A, C, E...together is to B, D, F...together. 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 and as E is to F... .. if mA be greater than nB, mC is greater than nD, and mE is greater than nF...; and if equal, equal; if less, less.

V. 4.

.. if mA be greater than nB, mA, mC, mE...together are greater than nB, nD, nF...together; and if equal, equal; if less, less.

Now mA and mA, mC, mE...together are equimultiples of A and A, C, E...together. V. 1. And nB and nB, nD, nF...together are equimultiples of B and B, D, F...together.

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

PROPOSITION XI. (Eucl. v. 15.)

V. Def. 5.

Q. E. D.

Magnitudes have the same ratio to one another which their equimultiples have.

Let A be the same multiple of C that B is of D.

Then must C be to D as A to B.

Divide A into magnitudes E, F, G,...each equal to C,

and B into magnitudes H, K, L,...each equal to D, the number of the magnitudes being the same in both cases, because A and B are equimultiples of C and D.

Then E, F, G.........are all equal,

and H, K, L.......................are all equal.

.. E is to H, as F to K, as G to L...

.. E is to together,

V. 6

H as E, F, G...together is to H, K, L...

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SECTION IV.

On Proportion by Inversion, Alternation, and Separation.

PROPOSITION XII. (Eucl. v. B.)

If four magnitudes be proportionals, they must also be proportionals when taken inversely.

Let A be to B as C is to D.

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

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

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.

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

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

V. Def. 5.

Q. E. D.

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