Book VI. so are all the antecedents to all the consequents g.

Where

fore, as the triangle ABE to the triangle FGL, so is the polygon g 12. 5. ABCDE to the polygon FGHKL: but the triangle ABE has to the triangle FGL the duplicate ratio of that which the side AB has to the homologous side FG. Therefore also the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous side FG. Wherefore, similar polygons, &c. Q. E. D.

5.

COR. 1. In like manner, it may be proved, that similar four sided figures, or of any number of sides, are one to another in the duplicate ratio of their homologous sides, and it has already been proved in triangles. Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.

COR. 2. And if to AB, FG, two of the homologous sides, a h 10. def. third proportional M be taken, AB hash to M the duplicate ratio of that which AB has to FG: but the four sided figure or polygon upon AB has to the four sided figure or polygon upon FG likewise the duplicate ratio of that which AB has to FG; therefore, as AB is to M, so is the figure upon AB to the figure upon FG, iCor. 19. which was also proved in triangles. Therefore, universally, it is manifest, that if three straight lines be proportionals, as the first is to the third, so is any rectilineal figure upon the first, to a similar and similarly described rectilineal figure upon the second

6.

Book VI,

PROP. XXI. THEOR.

RECTILINEAL figures which are similar to the same rectilineal figure, are also similar to one another.

Let each of the rectilineal figures A, B be similar to the rectilineal figure C: the figure A is similar to the figure B.

Because A is similar to C, they are equiangular, and also have their sides about the equal angles proportionalsa. Again, be- a 1. def.6. cause B is similar to

figures A, B are each

of them equiangular to C, and have the sides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal figures A and B are equiangularb, and have their sides b 1. Ax. about the equal angles proportionals. Therefore A is similar a to B. Q. E. D.

PROP. XXII. THEOR.

IF four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals; and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be propor, tionals.

Let the four straight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD let the similar rectilineal figures KAB, LCD be similarly described; and upon EF, GH the similar rectilineal figures MF, NH in like manner: the rectilineal figure KAB is to LCD, as MF to NH.

1.

c 11.5.

To AB, CD take a third proportionala X; and to EF, GHa 11.6. a third proportional O: and because AB is to CD, as EF to b 11. 5. GH, and that CD is to X, as GH to 0; wherefore, ex æqualis, c 22. 5. as AB to X, so EF to O: but as AB to X, so is d the rectilineal d 2. Cor, Book VI. KAB to the rectilineal LCD, and as EF to O, so is d the rectilineal MF to the rectilineal NH: therefore, as KAB to LCD, sob is MF to NH.

d 2 Cor. 20.6.

b 11. 5.

e 12. 6.

f 18. 6.

And if the rectilineal KAB be to LCD, as MF to NH; the straight line AB is to CD, as EF to GH.

Make e as AB to CD, so EF to PR, and upon PR describe f the rectilineal figure SR similar and similarly situated to either

g 9. 5.

of the figures MF, NH: then because as AB to CD, so is EF to PR, and that upon AB, CD are described the similar and similarly situated rectilineals KAB, LCD, and upon EF, PR, in like manner, the similar rectilineals MF, SR; KAB is to LCD, as MF to SR; but, by the hypothesis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the same ratio to each of the two NH, SR, these are equals to one another: they are also similar, and similarly situated; therefore GH is equal to PR: and because as AB to CD, so is EF to PR, and that PR is equal to GH; AB is to CD, as EF to GH, If, therefore, four straight lines, &c. Q. E. D.

See N.

PROP. XXIII. THEOR.

EQUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Let AC CF be equiangular parallelograms, having the angle BCD equal to the angle ECG: the ratio of the parallelogram AC to the parallelogram CF is the same with the ratio which is compounded of the ratios of their sides.

Let BC, CG be placed in a straight line; therefore DC and Book VI. CE are also in a straight linea; and complete the parallelogram DG; and, taking any straight line K, makeb as BC to CG, a 14. 1. so K to L, and as DC to CE, so makeb L to M: therefore b 12. 6., the ratios of K to L, and L to M, are the same with the ratios of the sides, viz. of BC to CG, and DC to CE. But the ratio of K to Mis that which is said to be compounded of the cA.def.5: ratios of K to L, and L to M: wherefore also K has to M the ratio compounded of the ratios of the A

D H

sides; and because as BC to CG, so is the parallelogram AC to the parallelogram CHd; but as BC to CG, so is K to L; therefore Kise to L, as the parallelogram AC to the parallelogram CH: again, because as DC to CE, so is the parallelogram CH to the parallelogram CF; but as DC to CE, so is L to M; wherefore Lise to M, as the

parallelogram CH to the parallelogram CF: therefore, since it has been proved

KLM

Gd 1.6. e 11. 5.

EF

that as K to L, so is the parallelogram AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æqualif, K is to M, as the parallelogram AC to the f 22. 5.. parallelogram CF: but K has to M the ratio which is compounded of the ratios of the sides; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore, equiangular parallelqgrams, &c. Q. E. D.

PROP. XXIV. THEOR.

THE parallelograms about the diameter of any See N. parallelogram, are similar to the whole, and to one another.

Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK the parallelograms about the diameter: the parallelograms EG, HK are similar both to the whole parallelogram ABCD, and to one another.

Because DC, GF are parallels, the angle ADC is equal to a 29. 1. the angle AGF: for the same reason, because BC, EF are pa

Book VI. rallels, the angle ABC is equal to the angle AEF: and each of the angles BCD, EFG is equal to the opposite angle DAB, b 34. 1. and therefore are equal to one another; wherefore the parallelograms ABCD, AEFG are equiangular: and because the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, EAF, they are equiangu

с 4. 6.

lar to one another; therefore cas AB

d 7.5.

to BC, so is AE to EF: and because
the opposite sides of parallelograms
are equal to one another b, AB is d to
AD, as AE to AG; and DC to CB,
as GF to FE; and also CD to DA,
as FG to GA: therefore the sides of
the parallelograms ABCD, AEFG
about the equal angles are proportion- D
als; and they are therefore similar to

F

G

e1. def. 6. one anothere: for the same reason, the parallelogram ABCD is similar to the parallelogram FHCK. Wherefore each of the parallelograms GE, KH is similar to DB: but rectilineal figures which are similar to the same rectilineal figure, are also similar f 21.6. to one anotherf; therefore the parallelogram GE is similar to KH. Wherefore, the parallelograms, &c. Q. E. D.

PROP. XXV. PROB.

See N.

TO describe a rectilineal figure which shall be si. milar to one, and equal to another given rectilineal figure.

Let ABC be the given rectilineal figure, to which the figure to be described is required to be similar, and D that to which it must be equal. It is required to describe a rectilineal figure similar to ABC, and equal to D.

Upon the straight line BC describe a the parallelogram BE equal to the figure ABC; also upon CE describe the parallelogram CM equal to D, and having the angle FCE equal b29. 1. to the angle CBL: therefore BC and CF are in a straight 14. 1. line b, as also LE and EM: between BC and CF find a mean c 13. 6. proportional GH, and upon GH described the rectilineal fid 18. 6. gure KGH similar and similarly situated to the figure ABC: e 2. Cor. and because BC is GH as GH to CF, and if three straight 20. 6. lines be proportionals, as the first is to the third, so is the