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PROP. XXV. THE OR.

IF four magnitudes be proportional, the greatesi and least will be greater than the other two.

Let four magnitudes AB, CD, E, F, be proportional, viz. AB to CD as E to F; of which let AB be the greatest, and F the leaft; then AB and F together, will be greater than CD and E; for, cut off AG equal to E, and CH to F; then AB is to CD as AG is to CH; therefore the remainder BG, will be to the remainder DH, as the whole AB is to the whole DC a; but AB is greater than CD; therefore GB is greater than HD; and, because AG is equal to E, and CH to F, then AG and F are equal to CH and E; but BG is greater than HD; therefore AB and F are greater than DC and E. Wherefore, &c.

THE

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ELEMENTS

O F

EUCL I D.

BOOK VI.

DEFINITION S.

1.

Similar right-lined figures are fuch as have each of their fe- Book VI. veral angles equal to one another, and the fides about the equal angles proportional to each other.

II.

Figures are reciprocally proportional to each other, when the antecedent and confequent terms of the ratio are in each figure. III.

A right line is cut into extreme and mean ratio, when the whole is to the greater fegment as the greater fegment is to the

leffer.

IV.

The altitude of any figure, is a line drawn from the vertex perpendicular to the bafe.

V.

Ratio is faid to be compounded of ratios, when the ratio of the firft term to the laft is produced from the quantities of the ratios of the intermediate terms, either by multiplication, divifion, or both.

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