quentiy, the Sides that are about the equal Angles of be proportional; then, as the firft is to the third, so THEOREM. gles, equal in Number, and bomologous to the gous Side bas to the otber. LET ABCDEFGHEL, be similar Poly. gons, and let the Side A B be homologous to the Side FG. I say, the Polygons ABCDE, FGHKL, are divided into equal Numbers of similar Triangles, and homologous to the Wholes; and the Polygon A B C D E, to the Polygon FGHKL, is ! in the duplicate Proportion of that which the Side A B has to the Side F G. För let B E, E C, GL, L H, be joined. Then, because the Polygon ABCDE is fimilar to the Polygon FGHK L, the Angle B A E is equal to the Angle G FL; and B A is to A E, as GF is to FL. Now, fince A BE, FGL, are two Triangles having one Angle of the one equal to one. An gle of the other, and the sides about the equal Angles 16 of this proportional; the Triangle A B E will be & equian gular to the Triangle F G L, and also fimilar to it. Therefore the Angle A B E is equal to the Angle : Defir of FGL ; but the whole Angle A B C is * equal to the whole Angle F GH, because of the Similarity of the Polygons; therefore the remaining Angle É BC is equal to the remaining Angle LGH: (And since by the Similarity of the Triangles ABE, FGL), as E B is to B A, fo is L G to GF; and fince, also, by the Similarity offthe Polygons, A B is to B Ç as FG, 22.5 is to GH; it shall be + by Equality of Proportion, as E B is to B C, fo is LG to GH; that is, the sides about the equal Angles E BC, LGH, are proportional. Wherefore the Triangle E B C is equiangular to the Triangles LGH, and, consequently, allo similar to it. For the same Reason, the Triangle ECD is likewise fimilar to the Triangle LHK; therefore the similar Polygons ABCDE, FGHKL, are divided into equal Numbers of similar Triangles. I say, they are also homologous to the wholes; that is, that the Triangles ate proporcional, and the Antecedents are A BE, EBC, ECD; and their Consequents F GL, LGH, LHK: And the Polygon ABCDE, to the Polygon F GHKL, is in the duplicate Proportion of an homologous Side of the one, to an homologous Side of the other; that is, as AB to F G. For, because the Triangle A B E is similar to the * 19 of tbis. Triangle F GL, the Triangle A B E shall be * to the Triangle F GL, in the duplicate Proportion of B E to - the Triangle G LH, is * in a duplicate Proportion of + 11.5 BE to GL: Therefore the Triangle A BE is + to the Triangle FGL, as the Triangle BEC is to the Triangle G LH. Again, because the Triangle E B C, is is fimilar to the Triangle LGH, the Triangle EBC, to the Triangle L GH, shall be in the duplicate Proportion of the Right Line C E, to the Right Line HL; and so, likewise, the Triangle ECD to the Triangle LHK, shall be in the duplicate Proportion of CE to HL. Therefore the Triangle B E C is to the Triangle L GH, as the Triangle CE D is to the Triangle LHK. But it has been proved, that the Triangle E B C is to the Triangle LGH, as the Triangle A BE is to the Triangle F GL. Therefore, as the Triangle A B E is to the Triangle F GL, lo is the Triangle BIE C to the Triangle Ğ HL; and to is the Triangle E C D to the Triangle LHK. But as one of the Antecedents is to one of the Consequents, fo are all the Antecedents to all the Confequents. I 12. 5. Wherefore, as the Triangle A B E is to the Triangle FG L, fo is the Polygon A B C D E to the Polygon FGHKL: But the Triangle A B E, to the TriangleF GL, * is in the duplicate Proportion of the 19 of this. homologous Side A B to the homologous Side FG; for (imilar Triangles are in the duplicate Proportion of the homologous Sides. Wherefore, the Polygon ABC D E, to the Polygon F GHKL, 'is in the duplicate Proportion of the homologous Side A B to the homologous Side FG. Therefore, fimilar Polygons are divided into fimilar Triangles, equal in Number, and homologous to the Wholes; and, Polygon to Polygon, is in the duplicate Proportion of that which one homologous Side has to the other ; which was to be demonstrated. It may be demonstrated, after the same manner, Figures are to one another in the duplicate Propor- BGX 2. There F |