is D to E; and H, K have been taken equimultiples of B, D; and L, M any other equimultiples of C, E which may accidentally happen; therefore it is, (by 4. 5.) as H is to L fo is K to M: but it has been demonftrated as G is to H fo is M to N: wherefore because there are three magnitudes G, H, L and others K, M, N equal to them in multitude; in the fame ratio taken two and two; and their proportion is perturbate; therefore by equality (by 21.

5

DEFK MN

GHLAB C if G exceed L; K alfo exceeds N: and if equal, equal and if lefs, lefs: and G, K are equimultiples of A, D; and L, N any ples of C, F which may accidentally happen; therefore (by 5. def. 5.)

A is to C as D is to F.

other equimulti

Wherefore if there be three magnitudes, and others, equal to them in multitude; in the fame ratio taken two and two; and if their proportion be perturbate; they will also be in the fame ratio by equality. Which was to be demonftrated.

Book V.

If the first have the fame ratio to the second which the third bes to the fourth; and if the fifth have alfo the fame ratio to the second which the fixth has to the fourth; alfo the first and fifth together.. will have the fame ratio to the fecond which the third and fixth together has to the fourth.

For let the first AB have the fame ratio to the fecond C, which the third DE has to the fourth F: and alfo let the fifth BG have the fame ratio to the fecond C, which the fixth EH has to the fourth F: I say that alfo AG the first together with the fifth, will. have the fame ratio to C the fecond which DH the third together with the fixth bas to F the fourth.

For because it is, as BG is to C fo is EH to F: therefore by inverfion (by cor. to 4. 5.) C is to BG as. F is to EH, Wherefore because it is as AB is to C fo is DE to F; and as C is to BG fo is

F tor

1

Book V. F to EH: therefore by equality (by 22. 5.) it is, as AB is to BG fo is DE to EH: and because the divided magnitudes are proportionals, they will also (by 18. 5) be proportionals when compounded: wherefore as AG is to GB fo is DH to HE: but it is, as GB is to C fo is HE to F: therefore by equality (by 22. 5.) it is, as AG is to C fo is DH to F.

G

H

B

E

AC DF

Wherefore if the first have the fame ratio to the fecond which the third has to the fourth; and if the fifth have alfo the fame ratio to the fecond which the fixth has to the fourth; alfo the first and fifth together will have the fame ratio to the fecond which the third and fixth together has to the fourth. Which was to be demonftrated.

P.R O P. XXV.

If four magnitudes be proportionals; the greatest and the least of them are greater than the other two.

Let the four magnitudes AB, CD, E, F be proportionals; viz. as AB is to CD fo let E be to F: and let AB be the greatest of them and F the leaft; I say that AB and F are greater than CD and E. For let AG be made equal to E and let CH be made equal to F.

B..

G D

+H

ACE F

Wherefore because it is, as AB is to CD fo is E to F; and AG is equal to E and CH to F; therefore it is as AB is to CD fo is AG to CH: and because it is as the whole AB is to the whole CD fo is AG taken away to CH taken away; therefore (by 19. 5.) the remainder GB will be to the remainder HD as the whole AB is to the whole CD: but (by fupp.) AB is greater than CD; therefore GB is greater than HD: and because AG is equal to E and CH to F: therefore AG and F together are equal to CH and E together; and because if equals be added to unequals, the wholes are unequal wherefore GB and HD being unequal: and AG and F be added to the greater GB and CH and E be added to the less HD: AB and F together are greater than CD and E.

Wherefore if four magnitudes be proportionals, the greatest and the least of them are greater than the other two. Which was to be demonftrated.

1. SIMILAR rectilineal figures are those which have their Book VI. angles equal each to each; and the fides about the equal angles proportionals. 2. But those are reciprocal figures; when the antecedent and consequent terms are in each of the figures.

3. A ftraight line is faid to have been cut in extreme and mean ratio, when it is, as the whole line is to the greater fegment, fo is greater Segment to the less.

the

4. The altitude of any figure is the perpendicular drawn from the vertex to the base.

5. A ratio is faid to be compounded of ratios, when the quantities of the ratios being multiplied into one another do make fome ratio.

VOL. I.

* D

PROP.

The triangles and parallelograms which are under the fame altitude are to one another as their bases.

Let ABC, ACD be triangles; and EC, CF parallelograms, which are under the fame altitude, viz. the perpendicular drawn from the point A to BD: I say that it is, as the bafe BC is to the bafe CD fo is the triangle ABC to the triangle ACD: also fo is the parallelogram EC to the parallelogram CF.

For let BD be produced towards both parts to the points H, L: and let any number of lines BG, GH be made equal to the base BC: and any number of lines DK, KL equal to the base CD: and let AG, AH, AK, AL be joined.

And because CB, BG, GH are equal to one another; alfo (by 38. 1.) the triangles AGH, AGB, ABC are equal to one another: therefore whatsoever multiple the base HC is of the base BC; the fame multiple also is the triangle AHC

E AF

of the triangle ABC. Certainly for the H G B CDKL fame reafon alfo, whatsoever multiple the

bafe CL is of the base CD the fame multiple is the triangle ACL of the triangle ACD. And if the base HC be equal to the base CL (by 38. 1.) the triangle AHC is alfo equal to the triangle ACL and if the base HC exceed the bafe CL; the triangle AHC alfo exceeds the triangle ACL: and if lefs, lefs. There being four magnitudes, the two bafes BC, CD; and the two triangles ABC, ACD and equimultiples of the bafe BC and of the triangle ABC have been taken viz. the base HC and the triangle AHC : and of the bafe CD and of the triangle ACD any other equimultiples which may accidentally happen viz. the bafe CL and the triangle ACL and it has been demonstrated that if the base HC exceed the base CL; the triangle AHC also exceeds the triangle ACL: and if equal, equal: and if lefs, lefs: therefore (by 5. def. 5.) it is, as, the bafe BC is to the bafe CD fo is the triangle ABC to the triangle ACD.

And

And because the parallelogram EC is double of the triangle ABC Book VI. (by 41. 1.): and the parallelogram FC is double of the triangle ACD; and parts (by 15. 5.) have the fame ratio to one another as their equimultiples; therefore it is, as the triangle ABC is to the triangle ACD so is the parallelogram EC to the parallelogram FC: wherefore because it has been demonftrated that as the base BC is to the base CD fo is the triangle ABC to the triangle ACD: and as the triangle ABC is to the triangle ACD so is the parallelogram EC to the parallelogram FC; therefore also (by 11. 5.) the base BC is to the base CD as the parallelogram EC is to the parallelogram FC.

Wherefore the triangles and parallelograms, which are under the fame altitude, are to one another as their bafes. Which was to be demonftrated.

If any straight line be drawn parallel to one of the fides of a triangle, it will cut the fides of the triangle proportionally and if the fides of the triangle be cut proportionally, the straight line joining the sections will be parallel to the remaining fide of the triangle.

For let DE be drawn parallel to BC one of the fides of the triangle ABC: I say that it is, as BD is to DA so is CE to EA. For let BE, CD be joined.

Therefore the triangle BDE is equal to the triangle CDE; for they are upon the fame base DE and between the fame parallel lines DE, BC: but ADE is fome other triangle; and (by 7. 5.) equal magnitudes have the fame ratio to the same magnitude: therefore it is, as the triangle BDE is to the triangle ADE so is the triangle CDE to the triangle ADE:. but as the triangle BDE is to the triangle ADE so is (by 1.6.) BD to DA: for being under the fame altitude, the perpendicular drawn from the point E to AB, they are to one another as their bafes. Certainly for the fame reafon also, as the triangle CDE is to the triangle ADE fo is CE to EA: Therefore (by 11. 5.) as BD is to DA so is CE to EA.

But let AB, AC, the fides of the triangle ABC be cut proportionally in the points D, E viz. as BD is to DA fo let CE be to EA:

*D2

and