Book VI. bout two equal angles reciprocally proportional are equal to one è 15. 6. f 1. 6. three ftraight lines G A CE D F be proportionals, B COR. From this it is manifeft, that if three ftraight lines be proportionals, as the the firft is to the third, fo is any triangle upon the firft to a fimilar, and fimilarly described triangle upon the fecond. ST 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 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 the polygon FGHKL, the angle BAE is equal a 1. def. 6. to the angle GFL a, and BA is to AE, as GF to FL a: wherefore, 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 equi- Book VI. angular b, and therefore fimilar to the triangle FGL; where- b 6. o. fore the angle ABE is equal to the angle FGL: and, be- c 4. 6. cause 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: and because the triangles ABE, FGL are fimilar, EB is to BA, as LG to GF a; and also, because the polygons are fimilar, AB is to BC, as FG to GH a; therefore, ex æquali d, EB is to BC d 22. 5. as LG to 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 likewife is 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 e 19 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, fo f is the triangle BEC to the triangle GLH. Again, be- f 11. 5. cause the triangle EBC is fimilar to the triangle LGH, EBC has to LGH the duplicate ratio ef that which the fide EC has to the fide LH: for the fame reafon, the triangle ECD has to the triangle LHK, the duplicate ratio of that which EC has to LH: as therefore the triangle EBC to the triangle LGH, so is f the triangle ECD to the triangle LHK: but it has been proved, that the triangle EBC is likewife to the triangle N 2 Book VI. triangle LGH, as the triangle ABE to the triangle FGL. Therefore, as the triangle ABE is to the triangle FGL, fo is triangle EBC to triangle LGH, and triangle ECD to triangle LHK: and therefore, as one of the antecedents to one II. of the confequents, fo are all the antecedents to all the con12. 5. fequents 8. 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, 11. def. 5. a third proportional M be taken, AB has h 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 upon FG, which was alfo proved in trii Cor.19. 6. angles i. Therefore, univerfally, it is manifeft, that if three ftraight 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 upen 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 homologous fides. PROP. R PRO P. XXI. THE O R. 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... Book VI. Because A is fimilar to C, they are equiangular, and alfo have their fides about the equal angles proportionals a. Again, a 1. def. 6. because B is fimilar to C, they are equiangular, and have their fides about the equal angles proportionals a: therefore the figures A, B are each of them equiangular to C, and have the fides about A A B the equal angles of each of them and of C proportionals. Wherefore the rectilineal figures A and B are equiangular b, b 1. Ax. I, and have their fides about the equal angles proportionals c.c 11. 5. Therefore A is fimilar a to B. Q. E. D. PROP. XXII. THEOR. F four ftraight lines be proportionals, the fimilar I 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 lincs AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD N 3 let Book VI 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 11. 6. c 22 5. To AB, CD take a third proportional a X; and to EF, GH a third proportional O: and because AB is to CD, as b 11. 5. EF to GH, and that CD is b to X, as GH to O; wherefore, ex æquali c. as AB to X, fo EF to O: but as AB to X, fo d 2. cor. is d the rectilineal KAB to the rectilineal LCD, and as EF to O, fo is d the rectilineal MF to the rectilineal NH: therefore, as KAB to LCD, fo b is MF to NH. 20. 6. d 2. cor. 20. 6. e 12. 6. And if the rectilineal KAB be to LCD, as MF to NH; the ftraight line AB is to CD, as EF to GH. Make e as AB to CD, fo EF to PR, and upon PR defcribe f 18. 6. f the rectilineal figure SR fimilar and fimilarly fituated to K either of the figures MF, NH: then, because as AB to CD, fo is EF to PR, and that upon AB, CD are described the fimilar and fimilarly fituated rectilineals KAB, LCD, and upon EF, PR, in like manner, the fimilar rectilineals MF, SR; KAB is to LCD, as MF to SR; but, by the hypothefis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the fame ratio to each of the two NH, SR, these are equal g 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. |
