SIM

IMILAR polygons may be divided into the fame number of fimilar triangles, having the fame ratio to one another that the polygons have; and the polygons have to one another the duplicate ratio of that which their homologous fides have.

BOOK VI.

Let ABCDE, FGHKL be fimilar polygons, and let AB be the homologous fide to FG: The polygons ABCDE, FGHKL may be divided into the fame number of fimilar triangles, whereof each to each has the fame ratio which the polygons have; and the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the fide AB has to the fide FG. Join BE, EC, GL, LH: And because the polygon ABCDE is fimilar to FGHKL, the angle BAE is equal to GFL, and a 1. Def.6. BA is to AE, as GF to FL : Wherefore, because the triangles ABE, FGL have an angle in one equal to an angle in the other, and their fides about these angles proportionals, the triangle ABE is equiangular, and therefore fimilar to FGL; wherefore the angle ABE is equal to FGL: And, because the polygons are fimilar, the whole angle ABC is equal to FGH; therefore the remaining angle EBC is equal to LGH: And because the triangles ABE, FGL are fimilar, EB is to BA, as LG to GF ; and alfo, because the polygons are fimilar, AB is to BC, as FG to GH; therefore, by equality, EB is to BC, as d 22. 5° LG to GH; that is, the fides about the equal angles EBC, LGH are proportionals; therefore the triangle EBC is equiangu

a

lar to LGH, and

fimilar to it .

For the fame rea

fon, the triangle ECD is fimilar to LHK: Therefore the fimilar polygons ABCDE, FGHKL are di

E

a

b6.6.

€4.6.

vided into the

same number of fimilar triangles.

Alfo thefe triangles have, each to each, the fame ratio which the polygons have to one another, the antecedents being ABE, EBC, ECD, and the confequents FGL, LGH, LHK: And the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which the fide AB has to the homologous fide FG.

U

Because

Book VI.

f II. 5.

Because the triangle ABE is fimilar to FGL, ARE has to FGL the duplicate ratio of that which the fide BE has to e 19. 6. GL: For the fame reason, the triangle BEC has to GLH the duplicate ratio of that which BE has to GL: Therefore, as the triangle ABE to FGL, fo is the triangle BEC to GLH. Again, because the triangle EBC is fimilar to LGH, EBC has to LGH the duplicate ratio of that which EC has to LH: For the fame reason, the triangle ECD has to LHK the duplicate ratio of that which EC has to LH: As, theref 11. 5. fore, the triangle EBC to LGH, fo is f ECD to LHK: But it has been proved, that the triangle EBC is to LGH, as ABE to FGL; therefore, as ABE to FGL, fo is EBC to LGH, and ECD to LHK: and as one of the antecedents to 12. 5. its confequent, fo are all the antecedents to all the confequents. Wherefore, as the triangle ABE to FGL, fo is the polygon ABCDE to FGHKL: But the triangle ABE has to FGL the duplicate ratio of that which the fide AB has to the homologous fide FG. Therefore alfo the polygon ABCDE has to the polygon FGHKL the duplicate ratio of that which AB has to the homologous fide FG. Wherefore, fimilar polygons, &c. Q. E. D.

COR. 1. In like manner, it may be proved, that fimilar fourfided figures, or of any number of fides, are one to another in the duplicate ratio of their homologous fides; and it has already been proved in triangles. Therefore, univerfally fimilar rectilineal figures are to one another in the duplicate ratio of their homologous fides.

COR. 2. And if to AB, FG, two of the homologous fides, hio.Def.5. a third proportional M be taken, AB has to M the duplicate ratio of that which AB has to FG: But the four-fided figure or polygon upon AB has to the four-fided figure or polygon upon FG likewife the duplicate ratio of that which AB has to FG: Therefore, as AB is to M, fo is the figure upon AB to the i Cor.19.6. figure upon FG; which was alfo proved in triangles. Therefore, univerfally, it is manifeft, that if three ftraight lines be proportionals, as the firft is to the third, fo is any rectilineal figure upon the firft, to a fimilar and fimilarly defcribed figure upon the fecond.

R

PROP. XXI. THEOR.

ECTILINEAL figures which are fimilar to the fame rectilineal figure, are alfo fimilar to one another.

Let

Let each of the rectilineal figures A, B be fimilar to the rec- Book VI. tilineal figure C: The figure A is fimilar to the figure B.

Because A is fimilar to C, they are equiangular, and also

have their fides about the equal angles proportionals. Again, a 1. Def. 6. because B is fimilar to C, they

are equiangular, and have their fides about the equal angles proportionals Therefore the fi

a

gures A, B are each of them

equiangular to C, and have the

fides about the equal angles of each of them and of C

propor

tionals. Wherefore the rectilineal figures A and B are equian

gular, and have their fides about the equal angles propor- b 1.Ax. 1. tionals C. Therefore A is fimilar to B. Q. E. D.

IF

PROP. XXII. THEOR.

F four ftraight lines be proportionals, the fimilar rectilineal figures fimilarly defcribed upon them, fhall alfo be proportionals; and if the fimilar rectilineal figures fimilarly defcribed upon four ftraight lines be proportionals, thofe ftraight lines fhall be proportionals.

Let the four ftraight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD let the fimilar rectilineal figures KAB, LCD be fimilarly defcribed; and upon EF, GH the fimilar rectilineal figures MF, NH, in like manner: The rectilineal figure KAB is to LCD, as MF to NH.

a

To AB, CD take a third proportional X; and to EF, GH a third proportional 0: And because AB is to CD, as EF to GH, and that CD is to X, as GH to O; wherefore, by equa lity, as AB to X, fo EF to O: But as AB to X, fo is the rectilineal KAB to the rectilineal LCD, and as EF to O, fo is the rectilineal MF to the rectilineal NH: Therefore, as KAB to LCD, fob is MF to NH.

C II. 5.

a 11. 6.

b 11. 5.

C 22. 5.

d 2. Cor.

20.6.

d 2. Cor.

20.6.

And if the rectilineal KAB be to LCD, as MF to NH; the b11. 5. ftraight line AB is to CD, as EF to GH.

e 12. 6.

Make as AB to CD, fo EF to PR, and upon PR defcribe f the rectilineal figure SR fimilar and fimilarly fituated to either f 18. 6, of the figures MF, NH: Then, because as AB to CD, fo is EF to PR, and that upon AB, CD are defcribed the fimilar and fimilarly fituated rectilineals KAB, LCD, and upon EF, PR, in like manner, the fimilar rectilineals MF, SR; KAB is to LCD,

U 2

BOOK VI. LCD, as MF to SR; but, by hypothefis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the g 9. 5. fame ratio to each of the two NH, SR, these are equal to one

another: They are also fimilar, and fimilarly fituated; therefore GH is equal to PR: And because as AB to CD, fo is EF to PR, and that PR is equal to GH; AB is to CD, as EF to GH. If therefore four ftraight lines, &c. Q. E. D.

PROP. XXIII. THEOR.

QUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of

their fides.

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 fame with the ratio which is compounded of the ratios of their fides.

Let BC, CG be placed in a straight line; therefore DC and a 3. Cor. CE are also in a straight line 2; and complete the parallelogram DG; and, taking any ftraight line K, make as BC to CG, b 12.6. fo K to L; and as DC to CE, fo make ↳ L to M : Therefore

15. I.

b

C

the ratios of K to L, and L to M, are the fame with the ratios of the fides, viz. of BC to CG, and DC to CE. But the raCA.Def.5 tio of K to M is that which is faid to be compounded of the ratios of K to L, and L to M: Wherefore alfo K has to M, the ratio compounded of the ratios of the fides: And because as BC to CG, fo is the parallelogram AC to the parallelogram d 1. 6. CH; but as BC to CG, fo is K to L; therefore K is to L, as the parallelogram AC to the parallelogram CH: Again, because as DC to CE, fo is the parallelogram CH to the parallelogram CF; but as DC to CE, fo is L to M; wherefore L is to M, as the parallelogram CH to the parallelogram

e 11. 5.

e

CF:

G

C

f 22. 5.

CF: Therefore, fince it has been proved, that as K to L, fo is BOOK. VI. the parallelogram AC to the parallelogram CH; and as L to M, fo the parallelogram CH to the parallelogram CF; by equality f, B K is to M, as the parallelogram AC to the parallelogram CF: But K has to M the ratio which is compounded of the ratios of the fides; therefore alfo the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the fides. Wherefore, equiangular parallelograms, &c. Q. E. D.

TH

KLM

PROP. XXIV. THEOR.

E F

HE parallelograms about the diameter of any parallelogram, are fimilar 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 fimilar both to the whole parallelogram ABCD, and to one another.

C

a 29. I.

C 4. 6.

A E

B

F

d 7.5.

H

Because DC, GF are parallels, the angle ADC is equal a to the angle AGF: For the fame reafon, because BC, EF are parallels, the angle ABC is equal to the angle AEF: And each of the angles BCD, EFG is equal to the oppofite 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 equiangular to one another; therefore, as AB to BC, fo is AE to EF: And because the oppofite fides of parallelograms are equal to one another, AB is d to AD, as AE G to AG; and DC to CB, as GF to FE; and alfo CD to DA, as FG to GA: Therefore the fides of the parallelograms ABCD, AEFG about the equal angles are proportionals; and they are therefore fimilar to one another : For the fame reason, the parallelogram ABCD is fimilar to the parallelogram FHCK. Wherefore each of the parallelograms GE, KH is fimilar to DB: But rectilineal figures which are fimilar to the

D K

C

fame

e 1.Def. 6.