Book VI. C; but B is equal to D; wherefore as A to B, fo B to C. Therefore, if three ftraight lines, &c. Q. E. D.

a 23. I.

32. I.

c 4.6.

d 22. 5.

UPO

PON a given ftraight line to defcribe a rectilineal figure fimilar, and fimilarly fituated to a given rectilineal figure.

Let AB be the given ftraight line, and CDEF the given rectilineal figure of four fides; it is required upon the given ftraight line AB to defcribe a rectilineal figure fimilar, and fimilarly fituated to CDEF.

G

H

Join DF, and at the points A, B in the ftraight line AB, make the angle BAG equal to the angle at C, and the angle ABG equal to the angle CDF; therefore the remaining angle CFD is equal to the remaining angle AGB: Wherefore the triangle FCD is equiangular to the triangle GAB: Again, at the points G, B in the straight line GB make the angle BGH equal to the angle DFE, and the angle GBH equal to FDE; therefore the

A

E

L

K

B C

D

remaining angle FED is equal to the remaining angle GHB, and the triangle FDE equiangular to the triangle GBH: Then, because the angle AGB is equal to the angle CFD, and BGH to DFE, the whole angle AGH is equal to the whole CFE: For the fame reason, the angle ABH is equal to the angle CDE; alfo the angle at A is equal to the angle at C, and the angle GHB to FED: Therefore the rectilineal figure ABHG is equiangular to CDEF: But likewife thefe figures have their fides about the equal angles proportionals: Because the triangles GAB, FCD being equiangular, BA is c to AG, as DC to CF; and because AG is to GB, as CF to FD; and as GB to GH, fo, by reafon of the equiangular triangles BGH, DFE, is FD to FE; therefore, ex aequali, AG is to GH, as CF to FE: In the fame manner it may be proved that AB is to BH as CD to DE: And GH is to HB, as FE to ED c. Wherefore, because

the

C

1

the rectilineal figures ABHG, CDEF are equiangular, and Book VI.
have their fides about the equal angles proportionals, they are
fimilar to one another *.

Next, Let it be required to defcribe upon a given straight line
AB, a rectilineal figure fimilar, and fimilarly fituated to the
rectilineal figure CDKEF.

c I. def. 6.

Join DE, and upon the given straight line AB defcribe the rectilineal figure ABHG fimilar, and fimilarly fituated to the quadrilateral figure CDEF, by the former cafe; and at the points B, H in the ftraight line BH, make the angle HBL equal to the angle EDK, and the angle BHL equal to the angle DEK; therefore the remaining angle at K is equal to the remaining angle at L: And because the figures ABHG, CDEF are fimilar, the angle GHB is equal to the angle FED, and BHL is equal to DEK; wherefore the whole angle GHL is equal to the whole angle FEK: For the fame reason the angle ABL is equal to the angle CDK: Therefore the five fided figures AGHLB, CFEKD are equiangular: And because the figures AGHB, CFED are fimilar, GH is to HB, as FE to ED; and as HB to HL, fo is ED to EK; therefore, ex aequali d, c 4. 6. GH is to HL, as FE to EK: For the fame reason, AB is to BL, as CD to DK: And BI. is to LH, as DK to KE, because the triangles BLH, DKE are equiangular: Therefore, because the five fided figures AGHLB, CFEKD are equiangular, and have their fides about the equal angles proportionals, they are fimilar to one another: And in the fame manner a rectilineal figure of fix or more fides may be described upon a given straight line fimilar to one given, and fo on. Which was to be done.

d 22. S.

STH

IMILAR triangles are to one another in the duplicate
ratio of their homologous fides.

Let ABC, DEF be fimilar triangles, having the angle B equal to the angle E, and let AB be to BC, as DE to EF, fo that the fide BC is homologous to EF: The triangle ABC has to the a 12. def. 5.. triangle DEF, the duplicate ratio of that which BC has to EF.

Take BG a third proportional to BC, EF, fo that BC is to b 11. 6.

EF, as EF to BG, and join GA: Then, because as AB to BC,

fo DE to EF; alternately c, AB is to DE, as BC to EF: But c 16. 5.

as

一

a 11. 5.

Book VI. as BC to EF, fo is EF to BG; therefored as AB to DE, fo is EF to BG: Wherefore the fides of the triangles ABG, DEF which are about the equal angles, are reciprocally proportional: But triangles which have the fides about two equal angles reciprocally proportional

are equal to one ane 15. 6. other: Therefore the triangle ABG is equal to the triangle DEF: And because as BC is to EF, fo EF to BG;

A

D

portionals, the first is

g 1. 6.

f1o. def. 5. faid to have to the third the duplicate ratio of that which it has to the fecond; BC therefore has to BG the duplicate ratio of that which BC has to EF: But as BC to BG, fo is the triangle ABC to the triangle ABG. Therefore the triangle ABC has to the triangle ABG the duplicate ratio of that which BC has to EF: But the triangle ABG is equal to the triangle DEF; wherefore also the triangle ABC has to the triangle DEF the duplicate ratio of that which BC has to EF. Therefore fimilar triangles, &c. Q. E. D.

COR. From this it is manifeft, that if three ftraight lines be proportionals, as the firft is to the third, fo is any triangle upon the first to a fimilar, and fimilarly defcribed triangle upoя

the fecond.

PROP. XX.

THEOR.

SIMILAR polygons may be divided into the same num

ber 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.

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

a 1. def. 6.

Gimilar to the polygon FGHKL, the angle BAE is equal to the Book VI. angle GFL, and 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 thefe equal angles proportionals, the triangle ABE is equiangular, and there-b 6. 6. fore fimilar to the triangle FGL; wherefore the angle ABE c 4. 6. is equal to the angle FGL: And, because the polygons are fimilar, the whole angle ABC is equal to the whole angle FGH; therefore the remaining angle EBC is equal to the remaining angle 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, ex aequali 4, EB is to BC, as LG to GH; that is, the fides about d 22. 5. the equal angles EBC, LGH are proportionals; therefore b the triangle EBC is equiangular to the triangle LGH, and

fimilar to it.

For the fame

reafon, the tri

angie ECD

likewife is fi- E

milar to the

triangle LHK:

Therefore the

fimilar poly

gons ABCDÉ,

FGHKL are divided into the fame 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.

1

Because the triangle ABE is fimilar to the triangle FGL, ABE has to FGL the duplicate ratio of that which the fide € 19. 6.} BE has to the fide GL: For the fame reafon, the triangle BEC has to GLH the duplicate ratio of that which BE has to GL: Therefore, as the triangle ABE to the triangle FGL, fof is the frus triangle BEC to the triangle GLH. Again, because the triangle EBC is fimilar to the triangle LGH, EBC has to LGH the duplicate ratio of that which the fide EC has to the fide LH: For the fame reafon, the triangle ECD has to the triangle

M

LHK,

11. 5.

Book VI. LHK, the duplicate ratio of that which EC has to LH: As therefore the triangle EBC to the triangle LGH, fo is the triangle ECD to the triangle LHK: But it has been proved that the triangle EBC is likewife to the triangle LGH, as the triangle ABE to the triangle FGL. Therefore, as the triangle ABE is to the triangle FGL, so is triangle EBC to triangle LGH, and triangle ECD to triangle LHK: And therefore, as one of the antecedents to one of the confequents, fo are all the antecedents to all the confequents. Wherefore, as the triangle ABE to the triangle FGL, fo is the polygon ABCDE to the polygon FGHKL: But the triangle ABE has to the triangle 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.

g 12. 5.

COR. 1. In like manner, it may be proved, that fimilar four fided 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, h 10. 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 figure i Cor. 19 6. upon FG, which was alfo proved in triangles i. Therefore, univerfally, it is manifeft, that if three ftraight lines be proportionals, as the firft is to the third, fo is any rectilineal figure uppon the first, to a fimilar and fimilarly defcribed rectilineal fi gure upon the fecond.

PROP.