Book V.

a 10,

PRO P. XXV. THE OR.

IF four magnitudes be proportional, the greatest and least wil 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 greateft, 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; 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

THE

ELEMENT S

O F

EUCLID.

BOOK VI.

DEFINITIONS.

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 base.

V.

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

Book VI.

a 38. 1.

PROP. I. THE OR.

RIANGLES and parallelograms that have the fame altitude, are to each other as their bafes.

TR

Let the triangles ABC, ACD, and the parallelograms EB, FD have the fame altitude, they are to one another as their bafes; viz. as BC to CD; for, produce BC both ways to H, M; take BG, GH each equal to BC; and DK, KL, LM each equal to CD; and join AG, AH, AK, AL, AM; then the triangles ABC, ABG, AGH are equal to one another; and ACD, ADK, AKL, ALM equal to one another; then, because HC is taken any multiple of BC, and CM any other multiple of CD, the triangle AHC is the fame multiple of the triangle ABC that HC is of BC; and the triangle CAM the fame multiple of CAD that CM is of CD: If HC be equal to CM, the triangle AHC will be equal to the triangle ACM; if greater, greater; and if less, lefs; therefore ABC b Def. 5.5. is to ACD as BC is to CD; but the parallelogram EB is double the triangle ABC ©, and FD double ACD; therefore the parallelogram EB is to the parallelogram DF, as the triangle ABC is to the triangle ACD; therefore the parallelograms EB, DF are to each other as their bafes BC, CD. Wherefore, &c.

C 41. I.

d 15. 5.

e II, 5.

PRO P. II. THE O R.

F a right line be drawn parallel to one of the fides of a triangle, it will cut the other hides proportionally; and if a line cut the two fides of a triangle proportionally, that right line fhall be parallel to the other fide of the triangle.

Let DE be drawn parallel to BC, one fide of the triangle ABC, then AD will be to DB as AE is to EC; for join DC, BE, then the triangles BDE, DEC are equal; and ADE is fome other triangle; therefore BDE is to ADE as DEC is to ADE b: But BDE is to ADE as BD is to AD; and DEC is to ADE as EC is to AE; therefore BD is to DA as CE is to EAd; and if BD is to DA as CE is to EA, then DE is parallel to BC.

For the fame construction remains: As BD is to DA, fo is BDE to ADE; and CE is to EA as DEC is to ADE; therefore the triangle BDE is to the triangle ADE as DEC is to ADE; therefore the triangles BDE, CDE are equal; therefole DE is parallel to BC f. Wherefore, &c.

PROP.