If the first Mag-
nitude AB has the
Same Proportion to
the Second C, as the
H third DE to the

fourth F; and if the
fifth BG be to the

Second C, as the fixth EH to the fourth F; then shall the first compounded with the fifth, viz. AG, be to the Second C, as the third compounded with the fixth viz. DH, is to the fourth F.

bypa

For because AB: C:: DE: F; but from the Hyp. and Inversely C: BG :: F: EH; therefore by equality AB: BG :: DE: EH. and fo by b 22.5. compounding AG: BG :: DH: EH. also BG: hyp. C:: EH: F. therefore again by equality AG: C:: DH: F. Q. E. D.

PROP. XXV.

19.5.

Make AG=E, and CH = F. • Because hyp. AB:CD::E:F:: AG: CH. therefore is 7.5. AB: CD :: GB: HD. but AB CD. whence byp * GBHD. but AG+FE+CH. there- & fchol. 1 fore AGF+GB-E+CH + HD, that 5. is, AB +FE+CD.

End of the Fifth Book.

K

EU

EUCLI D's

ELEMENTS.

I.

S

BOOK VI.

DEFINITIONS.

Imilar right-lined Figures, as ABC,
DCE, are such that have each
Angle of the one equal to each

Angle of the other, and the Sides

about the equal Angles proportional.

1

III. A right Line AC B C is faid to be divided into extreme and mean

when the whole AC is to the greatAB, as the greater Segment AC effer one CB. that is AC:AB::,

Triangles ABC, FCD, and Parallelograms BCAE, CDFA, that have the same altitude, are to one another as their Bases BC, CD.

E A F

Hence the T rallelograms AC Jes BC, KM, AJ, HF.

G

Take a any Number of Lines on BC, as BG, HG, each equal to BC, also DI = CD, and join AG, AH, FI.

The Triangles ACB, ABG, AGH are equal; also the Triang. FCD = FDI. Therefore the Triangle ACH is the fame Multiple of the Triangle ACB, as the Base HC is of the Base BC; and the Triangle FCI is the same Multiple of the Triangle FCD, as the Base CI is of the Base CD. But if HC be, =, or than CI, in like manner shall the Triang. AHC be, =, or FCI; and therefore d BC: CD:: Triang. ABC: Triang. FCD :: Pgr. CE: CF. Q. E. D.

t

SCHOL.

Hence the Triangles ABC, HKM, and the Parallelograms AGBC, DKHM, that have equal Ba Ses BC, KM, are to one another as their Altitudes AI, HF.

a

7.5.

* Take IL = CB, and EF = KM ; and join a 3. 1. LA, LG, ED, EH; it is manifest that the Triang. ABC: KHM::ALI: HEF :: AI: HF:: Pgr. d AGBC: DKHM.

A

PROP. II.

A

D

E

If any right Line DE be drawn parallel to one Side BC of a Triangle ABC; this shall cut or divide the Sides of that Triang. proportionally, viz. AD: BD:: AF: EC. And if the Sides of a Triangle be cut or divided proportionally, viz. AD: BD:: CAE: EC. a right Line DE joining the Points D, E, of Divifions, will be parallel to the other Side BC of the Triangle.

B

Draw CD, BE.

d

1.6.

41. Ι. 15. 5.

1 Hyp. Because the Triang. DEB = DEC; * 37. I. then shall the Triang. ADE: DBE :: ADE: 7.5. ECD; but the Triang. ADE: DBE:: AD: 8 1. 6.