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PRO P. XXV. THE OR,
IF four magnitudes be proportional, the greates 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.
Τ Η Ε
E L E M E N T S
E U U CL I D.
BO O K VI.
D E F INI TI O'N S.
1. Similar right-lined figures are such as have each of their se. Book VI. veral angles equal to one another, and the sides about the equal angles proportional to each other.
II. Figures are reciprocally proportional to each other, when the antecedent and consequent 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 segment as the greater segment is to the leffer.
IV. The altitude of any figure, is a line drawn from the vertex perpendicular to the base,
V. Ratio is said to be compounded of ratios, when the ratio of the first term to the last is produced from the quantities of the ratios of the intermediate terms, either by multiplication, divifion, or both.