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PROPOSITION XVIII. THEOREM.
The terms of an analogy are proportional by composition.

Given A:B ::C:D; to prove that by composition, A + B :B::C +D:D.

(Prep.) Take m(A + B) and nB any multiples whatever of A + B and B; and first, let m be greater than n. (Dem.) Then, because A + B is also greater than B, m(A + B) 7 n.B.

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PROPOSITION XX. THEOREM. If there be three magnitudes, and other three, which, taken two and two in order, have the same ratio; if the first be

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PROPOSITION XXII. THEOREM. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.

First, let there be given three magnitudes, A, B, and C, and other three, D, E, and E, which, taken two and two in order, have the same ratio; namely, A:B::D:E, and B:C::E:F; to prove that A:C:: D:F. (Prep.) Take of A and D any equimultiples whatever, mA, mD;

of B and E any whatever, nB, nE; and of C and F any whatever, qC, F.

(Dem.) Because A:B::D: E, MA :nB :: mD:nE (V. 4); and for the same reason, nB : qC :: nE : qF. - Therefore (V. 20), according as mA is greater than 9C, equal to it, or less, MD is greater

mD, NE than gF, equal to it, or less; but mA, mD are any equimultiples of A and D; and qC, F are any equimultiples of C and F; therefore (V. Def. 10) A:C::D:F.

Again, let there be given four magnitudes, and other four, which, taken two and two, have the same ratio ; namely, A:B::E:F; B:C::F:G; C:D::G:H; to prove that A:D::E:H.

(Dem.) For since A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio, by the foregoing case A:C::E: G. And because also C:D | A, B, C, D, :: G:H, by that same case, A:D::E:H. E, F, G, H.

In the same manner the demonstration is extended to any number of magnitudes.

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

First, let there be given three magnitudes, A, B, C, and other three, D, E, and F, which, taken two and two in a cross order, have the same ratio; namely, A:B :: E:F, and B: C ::D:E; to prove that A:C::D:F. (Prep.) Take of A, B, and D any equimultiples mA, mB, mD;

and of C, E, F any equimultiples nC, nE, NF.

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PROPOSITION E. THEOREM. Ratios which are compounded of the same ratios are the same with one another.

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