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PROPOSITION 3. Invertendo or Inversely. If two ratios are equal, their reciprocal ratios are equal. Let

A:B:: 0 :D, then shall

B: A:: D:C. For, by hypothesis, the multiples of A are distributed among those of B in the same manner as the multiples of C are among those of D. therefore also, the multiples of B are distributed among those of A in the same manner as the multiples of D are among those of C. That is,

B : A :: D : C.

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Note. This proposition is sometimes enunciated thus :

If four magnitudes are proportionals, they are also proportionals when taken inversely.

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Equal magnitudes have the same ratio to the same magnitude; and the same magnitude has the same ratio to equal magnitudes.

Let A, B, C be three magnitudes of the same kind, and let A be equal to B; then shall

A:C::B:C and

C:A::C: B. Since A = B, their multiples are identical and therefore are distributed in the same way among the multiples of C. :: A:C::B:C,

Def. 5. :: also, invertendo, C:A::C: B.

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V. 3.

PROPOSITION 5.

Of two unequal magnitudes, the greater has a greater ratio to a third magnitude than the less has; and the same magnitude has a greater ratio to the less of two magnitudes than it has to the greater. First,

let A be > B; then shall

A:C be > B:C. Since A > B, it will be possible to find m such that mA exceeds mB by a magnitude greater than C; hence if mA lies between nC and (n+1)Con B < nC:

and if mA=nC, then mB < nC;
.. A:0>B:C.

Def. 7. Secondly,

let B be < A; then shall

C: B be > C: A. For taking m and n as before,

nC > mB, while nC is not > mA;

:: C:B>C: A.

m

Def. 7.

PROPOSITION 6.

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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:C::B:C; then shall

A=B.
For if A > B, then A:0>B:C,
and if B > A, then B:C> A:C,

v. 5. which contradict the hypothesis ;

:: A=B. Secondly,

let C:A :: C:B; then shall

A=B.

Because C : A ::C: B, :: invertendo, A:C::B:C,

V. 3. :: A=B, by the first part of the proof.

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PROPOSITION 7.

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 :C be > B:C; then shall

A be > B. For if A= B, then A:C:: B:C,

V. 4. which is contrary to the hypothesis.

And if A <B, then A:C<B:C; which is contrary to the hypothesis ;

.. A > B.

V. 5.

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PROPOSITION 8. Magnitudes have the same ratio to one another which their equimultiples have.

Let A, B be two magnitudes; then shall

A:B :: mA : mB.
If p, q be any two whole numbers,

then m.pA >, =, or <m.qB
according as pA >, =, or <qB.
But m.pA=p.mA, and m.qB=q.mB;
:. p.mA >, =,

or <q.mB
according as pA >, =, or <qB;
.: A:B :: mA : MB.

Def. 5. Cor.

Let A:B::C: D.
Then since A:B :: mA : mB,

and C :D:: nC : nD;
.. mA : mB :: nc :nD.

v. 1.

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PROPOSITION 9.

Let p, a

If two ratios are equal, and any equimultiples of the antecedents and also of the consequents are taken, the multiple of the first ante. cedent has to that of its consequent the same ratio as the multiple of the other antecedent has to that of its consequent.

Let A:B::C:D; then shall

mA:nB :: mC:nD.

be any two whole numbers; then because A:B ::C:D,

pm.C >, =, or <qn. D according as pm. A >, =, or <qn.B,

Def. 5. that is, p.mC>, =, or <q.nD, according as p.mA >, =, or <q.1B; .. mA : nB :: mC : nD.

Def. 5.

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PROPOSITION 10.

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If four magnitudes of the same kind are proportionals, the first is greater than, equal to, or less than the third, according as the second is greater than, equal to, or less than the fourth. Let A, B, C, D be four magnitudes of the same kind such that

A :B::C:D;

then A >, =, or <C
according as B >, =, or < D.
If B > D, then A :B < A :D;

but A:B::C:D;
.: C:D<A:D;
.. A:D>C:D;
.. A >C.

v. 7. Similarly it may be shewn that

if B < D, then A < C, and if B D, then A C.

V. 5

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PROPOSITION 11.

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Alternando or Alternately. If four magnitudes of the same kind are p portionals, they are also proportionals when taken alt ately. Let A, B, C, D be four magnitudes of the same kind such that

A:B::C:D; then shall

A:C::B: D.
Because A:B :: MA : mB,

V. 8. and C:D:: nC : nD;

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i. mA : mB:: nC : nD.

v. 1. .. mA >, =, or < nc according as mB>, =, or < nD.

v. 10. And m and n are any whole numbers ; .. A:C::B: D.

Def. 5.

PROPOSITION 12.

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Addendo. If any number of magnitudes of the same kind are proportionals, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents.

Let A, B, C, D, E, F, ... be magnitudes of the same kind such that

A :B::C:D :: E:F:: ...... ;
then shall A:B :: A+C+E+ :B+D+F+....
Because A :B::C:D:: E:F:: ...,
:: according as mA >, =, or <nB,

so is mC>, =, or <nD,
and mE>, =,

or <nF,

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so is mA +mC+mE+ ... >, =, or <nB+nD+nF +...

or m(A + C + E +...)>, =, or <n(B+D+F+ ...); and m and n are any whole numbers ;

:. A:B :: A+C+E+... :B+D+F+.... Def. 5.

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PROPOSITION 13.

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Componendo. If four magnitudes are proportionals, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth.

Let A:B ::C:D; then shall

A+B:B::C+D:D. If m be any whole number, it is possible to find another number n such that mA =nB, or lies between nB and (n+1) B, ;. mA = mB=mB+nB, or lies between mB+nB and mB+ (n +1)B.

But mA +mB=m(A + B), and mB+nB=(m + n)B; :. m(A + B)=(m + n)B, or lies between (m + n)B and (m +n +1) B.

Also because A:B::C:D,

:: mC=nD, or lies between nD and (n+1)D; Def. 5. .. m(C+D)=(m + n) D or lies between (m + n) D and (m+n+1)D; that is, the multiples of C + D are distributed among those of D in the same way as the multiples of A + B among those of B;

.: A + B :B:: C+D:D. Dividendo. In the same way it may be proved that

A - B:B:: C-D :D, or B - A:B :: D-C :D, according as A is > or < B.

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