Cor. 5.-Ratios, subduplicate, subtriplicate, &c. of equal ratios, are equal; for, if they were unequal, the ratios, which are duplicate, triplicate, &c. of them, would be unequal (by the preced. cor.), contrary to the supposition.

Cor. 6.-Ratios, sesquiplicate of equal ratios, are equal.

portional, and three others D, E, F in the same ratio; let G be a mean pro

G

H

portional between B and C, and H between E and F ; the ratios of A to G and of D to H, are sesquiplicate of the equal ratios of A to B and of D to E (Def. 14. 5); the ratios of A to G and of D to H are equal.

For the ratios of B to G and of E to H, being subduplicaté of the equal ratios of B to C and E to F (Def. 14. 5), are equal (by the preced. cor.); whence, the ratios of A to B and of D to E being equal (Hyp.), A is to G, as D is to H (22. 5).

PROP. XXIII. THEOR.

If there be any number of magnitudes, and as many others, which, taken two and two in perturbate order, are in the same ratio; the ratio of the first, of the first magnitudes, to the last, is the same, as the ratio of the first to the last, of the others.

First, let there be three magnitudes
AB, CD, EF, and as many others GH, A-
IK, LM, and let AB be to CD, as IK

is to LM, and CD to EF, as GH to C-
IK; AB is to EF, as GH to LM.

G

Q

--B

-D

N

-F

R

-S

-H

P

-K

For, if the ratios of AB to EF, and E. of GH to LM be not equal, let one of them, if possible, as of AB to EF be the greater, and let EN a submultiple of EF be contained oftener in AB, than LO a like submultiple of LM is in GH [Def. I-| 7. 5]; let IP be a like submultiple of IK, as LO is of LM, AQ the greatest L——M multiple of EN which is in AB, and RS

a like multiple of IP.

0

The ratio of RS to IK is greater than that of GH to LM [Theor. 2. 15. 5 and 8. 5], therefore alternating, the ratio of RS to GH is greater than that of IK to LM [Cor. 16. 5], but IK is to LM, as AB is to CD (Hyp.), therefore the ratio of RS to GH is greater than that of AB to CD (13. 5); whence, GH being to IK, as CD is to EF (Hyp.), the ratio of RS to IK is greater than that of AB to EF (Cor. 1. 22. 5), which is absurd (Theor. 2. 15. 5 and 7 and 8. 5); therefore the ratios of AB to EF, and of GH to LM are not unequal, they are therefore equal.

For, because there are three magnitudes A, B, C, and three others F, G, H, which, taken two and two in a perturbate order, are in the same ratio; A is to C, as F is to H (by the preced. part; and C is to D, as E is to F (Hyp.); therefore (by the same part) A is to D, as E is to H.

In like manner the proposition might be demonstrated, if there were ever so many magnitudes.

Schol. By a like reasoning, as is used in the latter part of this proposition, it might be demonstrated, "if there be ever so "many magnitudes, and others equal to them in number, which "taken two and two, in any order whatever, are in the same "ratio; that the ratio of the first, of the first magnitudes, to "the last, is the same, as the ratio of the first to the last of the "others;" and therefore, that, "ratios compounded of equal ❝ratios (see Def. 13. 5), however disposed, are equal.”

PROP. XXIV. THEOR.

If to the antecedents (AB, EF) of four proportionals (AB, CD, EF, GH), magnitudes (BK, FL, which have the same ratio to their respective consequents (CD, GH), be added; the compounds (AK, EL), and consequents (CD, GH) are proportional.

Because BK is to CD, as FL is to GH (Hyp.), by inverting, CD is to BK, as GH to FL (Theor. 3. 15. 5); whence, AB being to CD, as EF to GH (Hyp.), by ordinate equality, AB is to BK, as EF to FL (22. 5); therefore, by compounding, AK is to BK, as EL to FL (18. 5), and therefore, BK being to CD, as FL to GH (Hyp.), by ordinate equality, AK is to CD, as EL to GH (22. 5).

A

C

B

-K

-D

F

E

--L

G.

-H

PROP. XXV. THEOR.

If four magnitudes of the same kind (AB, CD, 'EF, GH) be proportional; the greatest (AB) and least (GH), together, are greater than the other two (CD, EF) together.

K

B

L

C-

-D

-F

-H

E

Take AK on AB equal to EF, and CL on CD equal to GH (Post. 1. 5); Aand, since AB is to CD, as EF is to GH (Hyp.), or, (Cor. 1. 7. 5 and 11. 5), as AK is to CL, AB is to CD, as KB is to LD (19. 5); and AB is greater than CD (Hyp.), therefore KB is greater than LD (Cor. 13. 5); and, because AK is equal to EF, and CL to GH, AK and GH together, are equal to CL and EF together (Ax. 2. 1); therefore, adding to them the unequals KB, LD, the magnitudes AB and GH together, are greater than CD and EF together (Ax. 4. 1).

G

BOOK VI.

DEFINITIONS.

1. Similar rectilineal figures, are such, as have all the angles of one, severally equal to those of the other, and the sides about the equal angles proportional.

2. A right line is said to be divided in extreme and mean ratio, when the whole is to the greater segment, as the greater segment to the less.

3. The altitude or height of any figure, is a perpendicular, let fall from the vertex or top on the base.

4. A parallelogram is said to be applied to a right line on which it is described.

5. A parallelogram, described on a part of any right line, is said to be applied to that right line, deficient by a parallelogram described on the residue of the right line, in the same angle, and of the same altitude.

6. And if a right line be produced, a parallelogram described on the compound of the right line and part produced, is said to be applied to the first mentioned right line, exceeding by the parallelogram described on the part produced, in the same angle, and of the same altitude.

PROPOSITION I. THEOREM.

Triangles (ABC, DEF), and parallelograms (BG, DH), which have the same altitude, are to each other, as their bases (AB, DE).

Part 1.-Divide DE G into any number of equal parts DK, KI, IE (Cor. 7. 34. 1), and on AB take partsAO, ON, &c. as often as can be done, each equal to DK (3. 1), un

N

A ONML B D K I E

til a part LB remain less than DK, and join FK, FI, CO, CN, CM, CL.

Because the right lines DK, KI, IE, AO, ON, NM and ML are equal, the triangles FDK, FKI, FIE, CAO, CON, CNM, CML constituted on them, and being of the same altitude, are equal (38. 1), and the triangle CLB, being of the same altitude, and having its base LB less than DK, is less than the triangle FDK, therefore the triangle DKF is a like submultiple of DEF, as DK is of DE, and the triangle DKF and the right line DK, are contained equally often in the triangle ABC and right line AB.

In like manner it may be proved, that any other equisubmultiples of the triangle DEF and right line DE, are contained equally often in the triangle ABC and right line AB; therefore the triangle ABC is to the triangle DEF, as AB is to DE [Def. 5. 5].

Part 2. The parallelograms BG, DH, being double the triangles ABC, DEF (41. 1), are to each other, as these triangles (15. 5), and therefore, these triangles being to each other, as the bases AB, DE (by part 1), the parallelograms BG, DH are to each other, as the same bases (11. 5).