Hyp For because BG is to C, as EH is to F, it fhall be (in verfely) as C is to BG, fo is F to EH. Then fince AB is to C, as DE is to F, and as C is to BG, fo is F to *22 of this. EH; it fhall be by Equality as A B is to BG, fo is DE to EH. And because Magnitudes, being divided, +18 of this. are proportional, they fhall alfo bet proportional when compounded. Therefore, as AG is to GB, fo is DH to HE: But as GB is to C, fo alfo is HE to F. Wherefore, by Equality *, it fhall be as AG is to C, fo is DH to F. Therefore, if the firft Magnitude has the fame Proportion to the fecond, as the third to the fourth; and if the fifth has the fame Proportion to the fecond, as the fixth has to the fourth; then shall the firft, compounded with the fifth, have the fame Proportion to the fecond, as the third compounded with the fixth has to the fourth; which was to be demonftrated. PROPOSITION XXV. THE ORE M. If four Magnitudes be proportional, the greatest and the leaft of them, will be greater than the other two. ET four Magnitudes A B, CD, E, F, be propor tional, whereof AB is to CD, as E is to F; let AB be the greatest of them, and F the leaft. I fay AB, and B For let AG be equal to E, * 19 of this. alfo be* as the Refidue GB to the Refidue HD; fo is the G D H A CE F But AB is greater than CD; therefore alfo GB fhall be greater than HD. And fince AG is equal to E and and CH to F, AG and F will be equal to CH and E. But if equal things are added to unequal things, the wholes fhall be unequal. Therefore GB, HD being unequal, for GB is the greater: If A G, and F, are added to GB, and CH, and E, to HD; AB and F will neceffarily be greater than CD and E. Wherefore, if four Magnitudes be proportional, the greatest, and the leaft of them, will be greater than the other two; which was to be demonftrated. The END of the FIFTH BOOK, L EUCLID's $48 EUCLID's ELEMENTS. BOOK VI. 1. S DEFINITION S. IMILAR Right-lined Figures, are fuck as bave each of their feveral Angles equal to one another, and the Sides about the equal Angles proportional to each other. II. Figures are faid to be reciprocal, when the Antecedent and Confequent Terms of the Ratio's are in each Figure. III. A Right Line is faid to be cut into mean and IV. The Altitude of any Figure, is a perpendicular PRO PROPOSITION I. THEOREM. Triangles and Parallelograms that have the fame L ET the Triangles ABC, ACD, and the Parallelograms EC, CF, have the fame Altitude, viz. the Perpendicular drawn from the Point A to BD. I fáy, as the Base BC, is to the Bafe CD, fo is the Triangle ABC, to the Triangle ACD; and fo is the Parallelogram EC to the Parallelogram CF. * For produce BD both ways to the Points H and L, and take GB, GH, any Number of Times equal to the Bafe BC; and DK, KL, any Number of Times equal to the Bafe CD, and join AG, AH, AK, AL. Then because CB, BG, GH, are equal to one another, the Triangles AHG, AGB, ABC, also will be equal to one another: Therefore the fame * 38. tà Multiple that the Bafe HC is of BC, fhall the Triangle AHC be of the Triangle ABC. By the fame Reafon, the fame Multiple that the Bafe LC is of the Bafe CD, fhall the Triangle ALC be of the Triangle ACD. And if HC, be equal to the Bafe CL, the Triangle AHC is alfo equal to the Triangle ALC And if the Base HC, exceeds the Bafe CL, then the Triangle AHC, will exceed the Triangle ALC. And if HC be lefs, then the Triangle AHC will be lefs. Therefore fince there are four Magnitudes, viz. the two Bafes BC, CD, and the two Triangles ABC, ACD; and fince the Bafe HC, and the Triangle AHC, are Equimultiples of the Bafe BC, and the Triangle ABC: And the Base CL, and the Triangle ALC, are Equimultiples of the Bafe CD, and the Triangle ADC. And it has been proved, that if the Bafe HC, exceeds the Bafe CL, the Triangle AHC, will exceed the Triangle ALC; and if equal, equal; if lefs, lefs. Then as the Bafe BC, is to the Bafe CD, fot is the Trian-t Def. 5i 56 gle ABC, to the Triangle ACD, † 41. I. 11. 5. † 37. I. † 7. 5. And because the Parallelogram EC, is † double to the Triangle ABC; and the Parallelogram F C, double to the Triangle ACD; and Parts have the fame Proportion as their like Multiples. Therefore as the Triangle ABC is to the Triangle ACD, fo is the Parallelogram EC to the Parallelogram CF. And fo fince it has been proved, that the Base BC is to the Base CD, as the Triangle ABC, is to the Triangle ACD; and the Triangle ABC is to the Triangle A CD, as the Parallelogram EC is to the Parallelogram CF; it fhall be as the Bafe BC is to the Base CD, fo is the Parallelogram EC to the Parallelogram FC. Wherefore Triangles, and Parallelograms, that have the fame Altitude, are to each other as their Bafes; which was to be demonstrated. PROPOSITION II. THEORE M. If a Right Line be drawn Parallel to one of the Sides of a Triangle, it shall cut the Sides of the Triangle proportionally; and if the Sides of the Triangle be cut proportionally, then a Right Line joining the Points of Section, fhall be parallel to the other Side of the Triangle. LET DE be drawn parallel to BC, a Side of the Triangle ABC. I fay, DB is to DA, as CE is to EA. For let BE, CD, be joined. Then the Triangle BDE is * equal to the Triangle CDE, for they ftand upon the fame Bafe DE, and are between the fame Parallels DE and BC; and ADE is fome other Triangle. But equal Magnitudes have the fame Proportion to one and the fame Magnitude. Therefore as the Triangle BDE is to the Triangle ADE, fo is the Triangle CDE to the Triangle ADE. But as the Triangle BDE, is to the Triangle I of this. ADE, fot is BD to DA; for fince they have the fame Altitude, viz. a Perpendicular drawn from the Point E to AB, they are to each other as their Bases. And for the fame Reason, as the Triangle CD E, is te |