reafon the angle ABL is equal to the angle CDK: therefore Book VI. 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 ; © 4. 6. therefore, ex æquali d, GH is to HL, as FE to EK: for the d 22, 5. fame reason, AB is to BL, as CD to DK: and BL is to LH, as c 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 defcribed upon a given ftraight line fimilar to one given, and so on. Which was to be done. S IMILAR triangles are to one another in the du- Let ABC, DEF be fimilar triangles, having the angle B to EF. CE F Take BG a third proportional to BC, EF b, or fuch that b 11. 6. AB: DE:: BC: EF; but BC: EF:: EF: BG; therefore d AB: DE:: EF: BG: wherefore the fides of the triangles ABG, DEF, which are about the equal angles, are reciprocally proportional: but triangles, which have the fides a N bout C 16.5 Book VI. bout two equal angles reciprocally proportional, are equal to one e 15. 6. anothere: therefore the triangle ABG is A D be proportionals, the firft has to the third the duplicate ratio of B G CE F 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, f 1.6. 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: and the triangle ABG is equal to the triangle DEF; wherefore alfo 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 first is to the third, fo is any triangle upon the first to a fimilar, and fimilarly described triangle upon the fecond. S PROP. XX. THEOR. 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. 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 has to each 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 becaufe the polygon ABCDE is fimilar to the polygon FGHKL, the angle BAE is equal a 1. def. 6. to the angle GFL, and BA: AE::GF: FL a: where fore, because the triangles ABE, FGL have an angle in one equal to an angle in the other, and their fides about thefe thefe equal angles proportionals, the triangle ABE is equiangular, and therefore fimilar, to the triangle FGL; wherefore the angle ABE is equal to the angle FGL: and, because the polygons are fimilar, the whole angle ABC is equal a to the whole angle FGH; therefore the remaining angle EBC is equal to the remaining angle LGH: now because the triangles ABE, FGL are fimilar, EB: BA:: LG : GF; and alfo because the polygons are fimilar, AB: BC:: FG: GH a; therefore, ex æquali d; EB: BC:: LG: GH; that is, the fides about the equal angles EBC, LGH are proportionals; therefore b the triangle EBC is equiangular to the triangle LGH, and fimilar to it c. For the fame reason, the triangle ECD is likewise fimilar to the triangle LHK; therefore the fimilar polygons ABCDE, FGHKL are divided into the fame number of fimilar triangles. Also these 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. Because the triangle ABE is fimilar to the triangle FGL, ABE has to FGL the duplicate ratio e of that which the er. 6. fide 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 triangle BEC to the triangle GLH. Again, be- f11. 5. cause 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 reason, the triangle ECD has to the triangle LHK, the duplicate ratio of that which EC has to Li: therefore, as the triangle EBC to the triangle LGH, fo is f the triangle ECD to the triangle LHK: but it has been proved, that the triangle EBC is likewife to the N 2 triangle Book VI. triangle LGH, as the triangle ABE to the triangle FGL. Therefore, as the triangle ABE is to the triangle FGL, so is the triangle EBC to the triangle LGH, and the triangle ECD to the triangle LHK: and therefore, as one of the antecedents to one A M of the confequents, fo are all the antecedents to all the con12. 5. fequents 5. 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. 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, a b11. def. 5. third proportional M be taken, AB hash 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 i Cor.19.6. the figure upon FG, which was alfo proved in triangles i, Therefore, univerfally, it is manifeft, that if three straight lines be proportionals, as the first is to the third, fo is any rectilineal figure upon the firft, to a fimilar and fimilarly defcribed rectilineal figure upon the second. COR. 3. Because all fquares are fimilar figures, the ratio of any two fquares to one another is the fame with the duplicate ratio of their fides; and hence, alfo, any two fimilar rectilineal figures are to one another as the fquares of their ho mologous fides. PROP. Book VI. R PROP. XXI. THEOR. ECTILINEAL figures which are fimilar to the fame rectilineal figure, are alfo fimilar to one another. Let each of the rectilineal figures A, B be fimilar to the rectilineal figure C: The figure A is fimilar to the figure B. Because A is fimilar to C, they are equiangular. and alfo have their fides about the equal angles proportionals a. Again, a 1. def. G. because B is fimilar to C, they are equiangular, and have their fides about the equal angles proportionalsa: therefore the figures A, B are each of them equiangular to C, and have the fides about C A B the equal angles of each of them, and of C, proportionals. Wherefore the rectilineal figures A and B are equiangular b. br. Ax. 1. and have their fides about the equal angles proportionals c, c 11. 5. Therefore A is fimilar a to B. QE. D. 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 straight lines be proportionals, thofe ftraight lines fhall be proportionals. Let the four straight lines AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD N 3 let |