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logous side to fg: the polygons abcdefghkl may be divided into the same number of similar triangles, whereof each to each has the same ratio which the polygons have; and the polygon abcde has to the polygon fghkl the duplicate ratio of that which the side a b has to the side fg.

Join be, ec, gl, 1h and because the polygon abcde is similar to the polygon fghkl, the angle bae is equal to the angle gfl (vi. def. 1). and ba is to a e, as gf to fl (vi. def. 1): wherefore, because the triangles a be, fgl have an angle in one equal to an angle in the other, and their sides about these equal angles proportionals, the triangle abe is equiangular (vi. 6), and therefore similar, to the triangle fgl (vi. 4); wherefore the angle abe is equal to the angle fg1: and, because the polygons are similar, the whole angle abc is equal (vi. def. 1) to the whole angle fgh; therefore the remaining angle ebc is equal to the remaining angle 1gh and because the triangles abe, fgl are similar, eb is to ba, as 1g to gf (vi. def. 1); and also, because the polygons are similar, a b is to bc, as fg to gh (vi. def. 1); therefore, ex æquali (v. 22) eb is to bc, as 1g to gh: that is, the sides about the equal angles ebc, lgh are proportionals; therefore (v. 22) the triangle ebc is equiangular to the triangle 1gh, and similar to it (vi. 4).

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For the same reason, the triangle ecd likewise is similar to the triangle lhk therefore the similar polygons abcde, fghkl are divided into the same number of similar triangles.

Also these triangles have each to each the same ratio which the polygons have to one another, the antecedents being abe, ebc, ecd, and the consequents fgl, lgh, lhk: and the polygon abcde has to the polygon fghkl the duplicate ratio of that which the side ab has to the homologous side fg.

Because the triangle a be is similar to the triangle fgl, abe has to fgl, the duplicate ratio (vi. 19) of that which the side be has to the side gl: for the same reason, the triangle bec has to glh the duplicate ratio of that which be has to g1: therefore, as the triangle a be to the triangle fgl, so (v. 11) is the triangle bec to the triangle g1h. Again, because the triangle ebc is similar to the triangle 1gh, ebc has to lgh the duplicate ratio of that which the side ec has to the side 1h: for the same reason, the triangle ecd has to the triangle 1hk the duplicate ratio of that which ec has to lh as therefore the triangle ebc to the triangle 1gh, so is (v. 11) the triangle e c d to the triangle 1hk: but it has been proved that the triangle ebc is likewise to the triangle 1gh, as the triangle a be to the triangle fg 1. Therefore, as the triangle abe is to the triangle fg so is triangle ebc to triangle 1g h, and triangle ecd

to triangle 1hk: and therefore, as one of the antecedents to one of the consequents, so are all the antecedents to all the consequents (v. 12). Wherefore, as the triangle abe to the triangle fgl, so is the polygon.

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abcde to the polygon fghkl: but the triangle a be has to the triangle fgl, the duplicate ratio of that which the side ab has to the homologous side fg. Therefore also the polygon abcde has to the polygon fghkl the duplicate ratio of that which ab has to the homologous side fg. Wherefore similar polygons, &c. Q. E. D.

COR. 1. In like manner, it may be proved, that similar four-sided figures, or of any number of sides, are one to another in the duplicate ratio of their homologous sides, and it has already been proved in triangles. Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.

COR. 2. And if to a b, fg, two of the homologous sides, a third proportional m be taken, a b has (v. def. 10) to m the duplicate ratio of that which ab has to fg; but the four-sided figure or polygon upon a b has to the four-sided figure or polygon upon fg likewise the duplicate ratio of that which ab has to fg: therefore, as a b is to m, so is the figure upon ab to the figure upon fg, which was also proved in triangles (vi. 19, cor.). Therefore, universally, it is manifest that if three straight lines be proportionals, as the first is to the third, so is any rectilineal figure upon the first to a similar and similarly described rectilineal figure upon the second.

PROPOSITION XXI.-THEOREM.

Rectilineal figures which are similar to the same rectilineal figure are also similar to one another.

LET each of the rectilineal figures a, b be similar to the rectilineal figure c: the figure a is similar to the figure b.

Because a is similar to c, they are equiangular, and also have their sides about the equal angles proportionals (vi. def. 1). Again, because b

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is similar to c, they are equiangular, and have their sides about the equal angles proportionals (vi. def. 1). Therefore the figures a, b are each of

them equiangular to c, and have the sides about the equal angles of each of them and of c proportionals. Wherefore the rectilineal figures a and b are equiangular (i. ax. 1), and have their sides about the equal angles proportionals (v. 11). Therefore a is similar (vi. def. 1) to b. Q. E. D.

PROPOSITION XXII.-THEOREM.

If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals; and if the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals.

LET the four straight lines a b, c d, ef, gh be proportionals, viz. a b to cd, as ef to gh, and upon a b, c d let the similar rectilineal figures kab, lcd be similarly described; and upon ef, gh the similar rectilineal figures mf, nh, in like manner. The rectilineal figure kab is to lcd as m f to n h.

To ab, cd take a third proportional (vi. 11) x; and to ef, gha third proportional o: and because a b is to cd as ef to gh, and that cd is (v. 11) to x as g h to o; wherefore, ex æquali (v. 22), as ab to x, so is ef to o but as ab to x, so is (vi. 20, cor. 2) the rectilineal k a b to the rectilineal lcd, and as ef to o, so is (vi. 20, cor. 2) the rectilineal m f to the rectilineal n h. Therefore, as kab to lcd, so (v. 11) is mf to nh.

And if the rectilineal kab be to lcd as mf to nh; the straight line ab is to cd as ef to g h

Make (vi. 12) as ab to cd, so efto pr, and upon pr describe (vi. 18) the rectilineal figure sr similar and similarly situated to either of the

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figures mf, nh: then, because as ab to cd, so is ef to pr, and that upon a b, c d are described the similar and similarly situated rectilineals kab, lcd, and upon ef, pr, in like manner, the similar rectilineals m f, sr; kab is to lcd, as mf to sr; but, by the hypothesis, kab is to lcd as mf to nh; and therefore the rectilineal mf having the same ratio to each of the two nh, sr, these are equal (v. 9) to one another: they are also similar and similarly situated; therefore gh is equal to pr. And because as ab to cd, so is ef to pr, and that pr is equal to gh; ab is to c d as ef to gh. If therefore four straight lines, &c. Q. E. D.

PROPOSITION XXIII.-THEOREM.

Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

LET a c, cf be equiangular parallelograms, having the angle b c d equal to the angle e cg: the ratio of the parallelogram ac to the parallelogram cf is the same with the ratio which is compounded of the ratios of their sides.

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Let bc, cg be placed in a straight line; therefore dc and ce are also in a straight line (i. 14); and complete the parallelogram dg; and taking any straight line k, make (vi. 12) as bc to cg, so k to 1; and as dc to ce, so make (vi. 12) 1 to m: therefore the ratios of k to 1, and 1 to m, are the same with the ratios of the sides. viz. of But the ratio of k bc to cg, and dc to ce. to m is that which is said to be compounded (v. def. A.) of the ratios of k to 1, and 1 to m wherefore also k has to m the ratio compounded of the ratios of the sides. And because as bc to cg, so is the parallelogram ac to the parallelogram ch (vi. 1); but as bc to cg, so is k to 1; therefore k is (v. 11) to 1, as the parallelogram a c to the parallelogram c h. Again, because as dc to ce, so is the parallelogram ch to the parallelogram cf; but as dc to ce, so is 1 to m; k l m wherefore 1 is (v. 11) to m, as the parallelogram

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ch to the parallelogram cf: therefore since it has been proved, that as k to 1, so is the parallelogram a c to the parallelogram ch; and as 1 to m, so the parallelogram ch to the parallelogram cf: ex æquali (v. 22), 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 sides; therefore also the parallelogram ac has to the parallelogram cf the ratio which is compounded of the ratios of the sides. Wherefore, equiangular parallelograms, &c. Q. E. D.

PROPOSITION XXIV.-THEOREM.

The parallelograms about the diameter of any parallelogram are similar to the whole and to one another,

LET abcd be a parallelogram, of which the diameter is a c; and eg, hk the parallelograms about the diameter. The parallelograms eg, h k are similar both to the whole parallelogram a bcd, and to one another.

Because de, gf are parallels, the angle a dc is equal (i. 29) to the angle a gf. For the same reason, because b c, e f are parallels, the angle abc is equal to the angle a ef: and each of the angles bc d e f g is equal to the opposite angle da b (i. 34), and therefore are equal to one another wherefore the parallelograms a bcd, a efg, are equiangular and because the angle a bc is equal to the angle aef, and the angle bac

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common to the two triangles bac, eaf, they are equiangular to one another; therefore (vi. 4) as ab to bc, so is ae to ef and because the opposite sides of parallelograms are equal to one another (i. 34), ab (v. 7) is to ad as ae to ag; and dc to cb as gf to fe; and also cd to da as fg to ga: therefore the sides of the parallelograms abcd, aefg about the equal angles are proportionals; and they are therefore similar to one another (vi. def. 1); for the same reason the parallelogram abcd is similar to the parallelogram fhck. Wherefore each of the parallelograms ge, kh is similar to db: but rectilineal figures which are similar to the same rectilineal figure are also similar to one another (vi. 21); therefore the parallelogram ge is similar to kh. Wherefore the parallelogram, &c. Q. E. D.

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PROPOSITION XXV.-PROBLEM.

To describe a rectilineal figure which shall be similar to one and equal to another given rectilineal figure.

LET a b c be the given rectilineal figure to which the figure to be described is required to be similar, and d that to which it must be equal. It is required to describe a rectilineal figure similar to a b c, and equal to d.

Upon the straight line bc describe (i. 45, cor.) the parallelogram be equal to the figure abc; also upon ce describe (i. 45, cor.) the parallelogram cm equal to d, and having the angle fce equal to the angle cbl: therefore bc and c f are in a straight line (i. 29, and i. 14), as also le and em: between bc and cf find (vi. 13) a mean proportional gh, and upon gh describe (vi. 18) the rectilineal figure kgh similar and similarly

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situated to the figure abc: and because bc is to gh as gh to cf; and if three straight lines be proportionals, as the first is to the third, so is (vi. 20, cor. 2) the figure upon the first to the similar and similarly described figure upon the second; therefore, as b c to cf, so is the rectilineal figure abc to kgh: but as bc to cf, so is (vi. 1) the parallelogram be to the parallelogram ef: therefore as the rectilineal figure abc is to kgh, so is the parallelogram be to the parallelogram ef (v. 11): and the rectilineal figure a bc is equal to the parallelogram be; therefore the rec

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