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THEOREM (m). Parallelograms which are equal and equiangular have the sides about their equal angles reciprocally proportional.

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BH Let QH, QK be equal and equiangular parallelograms having the angles BQC, AQD equal to one another.

Then QA : QB as QC : QD.

Let the Os be placed so that AQ, QB may be in the same straight line, and also DQ, QC in the same straight line;

produce KA, HC to meet in X: then QX is a O.

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THEOREM (n). Equal triangles having one angle of the one equal to one angle of the other, have the sides about those angles reciprocally proportional.

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Euclid's test of proportion is the following:

The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any equimultiples whatever of the first and third being taken and any equimultiples whatever of the second and fourth, if the multiple of the first be equal to that of the second, then the multiple of the third is equal to that of the fourth, if greater, greater, and if less, less.

Prop. I. Book V., and Prop. XIII. Book VI., may be established by the direct application of the above test.

THEOREM (6). PROP. I. Book V. Straight lines have to one another the same ratio as the rectangles of equal altitude described upon them.

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BH RLK Let AD, BF be rectangles of equal altitude on the straight lines AG, BH.

Then shall AG have to BH the same ratio as AD to BF.

From AG produced cut off any number of parts GP, PQ, &c. each = AG, and from BH produced any number of parts HR, RL, &c. each = BH.

Complete the rectangles PD, FR, &c. Thus any equimultiples whatever have been taken of AG and rectangle AD; namely, AS and rectangle AY;

and any equimultiples whatever have been taken of BH and rectangle BF; namely, BK and rectangle B2. Also if AS be = BK, then rectangle AY is = rectangle B2,

if greater, greater, and if less, less;
... AG : BH as AD : BF.

THEOREM (D). PROP. XIII. Book VI.

In equal circles, angles at the centre have the same ratio as the arcs on which they stand; so also have the sectors.

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Let AKG, BLH be angles at the centres K, L of equal circles.

Then shall AKG have the same ratio to . BLH as arc AG to arc BH.

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Similarly it may be shewn that
sector AKG : sector BLH as arc AG : arc BH.,

COMPOUND RATIOS.

When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratios of the ist to the 2nd, the 2nd to the 3rd, and so on to the last (or of ratios the same as these ratios).

Thus, if there are three magnitudes of the same kind, the ist has to the 3rd the ratio compounded of the ratios of

the ist to the 2nd, and the and to the 3rd. Moreover, if these three magnitudes are proportional, then the ratio of the ist to the 2nd is the same as the ratio of the 2nd to the 3rd. Accordingly, when three magnitudes are proportional,

the ist is said to have to the 3rd
the duplicate ratio of that which it has to the 2nd,

The antecedent terms of proportionals are said to be homologous to one another; so also are the consequents.

Thus, the corresponding sides of similar figures are omologous to one another.

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