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Prop. xI. Algebraically.

Let the ratio of A, a, be the same as the ratio of A, : ", and the ratio of A,: a, be the same as the ratio of A, a ̧. Then the ratio of A,: a, shall be the same as the ratio of As : α。. For since A1: α2 :: A ̧: ɑ„

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Prop. XII. Algebraically.

Let A1, A2, A3, α, A ̧, α be proportionals,
so that A1: a2 :: A3 : α4 :: Д: α.

Then shall A1: α1:: A1 + As + Az : α2+ α12 + Ag•
For since A, a2 :: As : a4 :: A¿: α ̧

A1 As As

.. =
аз a

11

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Hence A, (a,+ a1 + αs) = α, (A1 + A, + A ̧), by addition,
and dividing these equals by a, (a2 + as + αs),

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and A1: a,:: A1 + As + A ̧ : ɑg + α1 + α ̧•

Prop. x. Algebraically.

Let Â1, α, Ag, a4, A5, a, be six magnitudes, such that A, : a,:: A, : das but that the ratio of A, a is greater than the ratio of A,: a. Then the ratio of A, : a, shall be greater than the ratio of A,: a.

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That is, the ratio of A,: a, is greater than the ratio of A: ag

Prop. XIV. Algebraically.

Let A, 2, As, as be proportionals,

Then if A, > A,, then a, > a, and if equal, equal; and if less, less.

For since A, ag¦¦ A3 : am
A1 A

=

a

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A,, then a, must be equal to a, and if A, be < A,, a, must be less than a

Hence, therefore, if &c.

Prop. xv. Algebraically.

Let A1, a, be any magnitudes of the same kind,

Then A, a :: mA : ma,;

:

mA, and ma2 being any equimultiples of A, and a.

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and since the numerator and denominator of a fraction may be multiplied by the same number without altering the value of the fraction.

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Prop. xvi. Algebraically.

Let A1, A2, A3, a, be four magnitudes of the same kind, which are proportionals,

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Then these shall be proportionals when taken alternately, that is,

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and A, A, A2 : ɑ4•

Prop. XVII. Algebraically.

Let A+ a, a, Aз + a, a, be proportionals,

then A, a,, A,, a, shall be proportionals.
For since A1+ Ag: Ag :: Ag + as: as,

or

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Prop. XVIII, is the converse of Prop. XVII.

The following is Euclid's indirect demonstration.
Let AE, EB, CF, FD be proportionals,

that is, as AE to EB, so let CF be to FD:

then these shall be proportionals also when taken jointly;
that is, as AB to BE, so shall CD be to DF.

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For if the ratio of AB to BE be not the same as the ratio of CD to DF; the ratio of AB to BE is either greater than, or less than the ratio of CD to DF.

First, let AB have to BE a less ratio than CD has to DF; and let DQ be taken so that AB has to BE the same ratio as CD to DQ: and since magnitudes when taken jointly are proportionals, they are also proportionals when taken separately; (v. 17.)

therefore AE has to EB the same ratio as CQ to QD;

but, by the hypothesis, AE has to EB the same ratio as CF to FD; therefore the ratio of CQ to QD is the same as the ratio of CF to FD. (v. 11.). And when four magnitudes are proportionals, if the first be greater than the second, the third is greater than the fourth; and if equal, equal; and if less, less; (v. 14.) but CQ is less than CF,

therefore QD is less than FD; which is absurd. Wherefore the ratio of AB to BE is not less than the ratio of CD to DF; that is, AB has the same ratio to BE as CD has to DF.

Secondly. By a similar mode of reasoning, it may likewise be shewn, that AB has the same ratio to BE as CD has to DF, if AB be assumed to have to BE a greater ratio than CD has to DF.

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Prop. xix. Algebraically.

Let the whole A, have the same ratio to the whole A,,
as a taken from the first, is to a, taken from the second,

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Then A, – a,: A, − ɑ, :: ɑ1 : α, is found proved in the preceding

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Dividing the latter by the former of these equals,

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Prop. xx. Algebraically.

Let A1, A2, A, be three magnitudes, and a1, a2, ag, other three, such that A : A2 ¦¦ α1: a2,

and A2 A3 Az: Az:

if A1> As, then shall a1 > Az,

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

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A3

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and since the fraction is equal to

and that A1> Az:

It follows that a1 is > ag.

In the same way it may be shewn

that if A1 Prop. xxi.

Ag, then a1
Algebraically.

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a; and if A, be < A3, then a1 < ɑz.

Let A1, A2, A3, be three magnitudes,
and a1, a2, as three others,

such that A : A2 :: α2: Az,
and 42: A, A1 A2.

If A, A,, then shall a1 > a, and if equal, equal; and if less, less.

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