wise the duplicate ratio of that which AB has to FG: therefore, as AB is to M, so is the figure upon AB to the figure upon FG, which was also proved in triangles (Cor. 19. 6.). 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. COR. 3. Because all squares are similar figures, the ratio of any two squares to one another is the same with the duplicate ratio of their sides; and hence, also, any two similar rectilineal figures are to one another as the squares of their homologous sides. PROP. XXI. THEOR. 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 (def. 1.6.). Again, because B is similar to C, they are equiangular, and have their sides about the equal angles proportionals (def. 1. 6.): therefore the figures A, B, AAA 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 (1. Ax. 1.), and have their sides about the equal angles proportionals (11. 5.). Therefore A is similar (def. 1. 6.) to B. Q. E. D. PROP. XXII. THEOR. 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, AB, CD, EF, GH be proportionals, viz. AB to CD, as EF to GH, and upon AB, CD let the similar reetilineal 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 MF to NH. To AB, CD take a third proportional (11. 6.) X; and to EF, GH, a third proportional O: and because AB: CD: EF: GH, and CD:X:: GH: (11.5.) O, ex æquali (22. 5.) AB: X (2. Cor. 20. 6.) :: KAB: LCD: and KAB: LCD (2. Cor. 20. 6.) :: MF: NH. And if the figure KAB be to the figure LCD, as the figure MF to the figure NH, AB is to CD, as EF to GH. Make (12. 6.) as AB to CD, so EF to PR, and upon PR describe (18.6.) the rectilineal figure SR similar, and similarly situated to ei ther of the figures MF, NH: then, because that as AB to CD, so is EF, to PR, and upon AB, CD are described the similar and similarly situated rectilineals KAB, LCD, and upon EF, PR, in like manner, the similar rectilineals MF, 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 two are equal (9. 5.) to one another: they are also similar, and similarly situated; therefore GH is equal to PR: and because as AB te CD, so is EF to PR, and because PR is equal to GH, AB is to CD, as EF to GH. If therefore four straight lines, &c. Q. E. D. → PROP. XXIII. THEOR, Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms having the angle BCD equal to the angle ECG; 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. B D H G Let BC, CG be placed in a straight line; therefore DC and CE are also in a straight, line (14. 1.); complete the parallelogram DG; and, taking any straight line K, make (12. 6.) as BC to CG, so K to L; and as DC to CE, so make (12. 6.) L to M: therefore the ratios of K to L, and L to M, are the same with the ratios of the sides, viz. of BC to CG, and of DC to CE. But the ratio of K to M is that which is said to be compounded (def. 10. 5.) of the ratios of K to L, and L. to M; wherefore also K has to M the ratio compounded of the ratios of the A sides of the parallelograms. Now, because as BC to CG, so is the paralJelogram AC to the parallelogram CH (1. 6.); and as BC to CG, so is K to L; therefore K is (11. 5.) to L, as the parallelogram AC to the parallelogram CH: again, because as DC to CE, so is the parallelogram CH to the parallelogram CF: and as DC to CE, so is L to M; therefore L is (11. 5.) to M, as the parallelogram CH to the parallelogram CF: therefore, since it has been proved, that as K to L, so is the parallelogram AC to the parallelogram CH; and as L to M, so the parallelogram CH to the parallelogram CF; ex æquali (22. 5.), 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. E F 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 AC; and EG, HK the parallelograms about the diameter: the parallelograms EG, HK are similar, both to the whole parallelogram ABCD, and to one another. Because DC, GF are parallels, the angle ADC is equal (29. 1.) to the angle AGF: for the same reason, because BC, EF are parallels, the angle ABC is equal to the angle AEF: and each of the angles BCD, EFG is equal to the opposite angle DAB (34. 1.), and therefore are equal to one another, wherefore the parallelograms ABCD, AEFG are equiangular. And because the angle ABC is equal to the angle G A K E B H AEF, and the angle BAC common to the two triangles BAC, EAF, they are equiangular to one another; therefore (4. 6.) as AB to BC, so is AE to EF; and because the opposite sides of parallelograms are equal to one another (34 1.), AB is (7. 5.) 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 parallelo- D grams ABCD, AEFG about the equal angles are proportionals; and they are therefore similar to one another (def. 1. 6.): 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 (21.6.); therefore the parallelogram GE is similar to KH. Wherefore the parallelograms, &c. Q. E. D. PROP. XXV. PROB. To describe a rectilineal figure which shall be similar to one, and equal to another given rectilineal figure. Let ABC 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 ABC, and equal to D. Upon the straight line BC describe (cor. 45.1.) the parallelogram BE equal to the figure ABC; also upon CE describe (cor.45.1.) the parallelogram CM equal to D, and having the angle FCE equal to the angle CBL: therefore BC and CF are in a straight line (29. 1. 14. 1.), as also LE and EM; between BC and CF find (13. 6.) a mean proportional GH, and upon GH describe (18. 6.) the rectilineal figure KGH simi lar, and similarly 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 (2. Cor. 20. 6.) the figure upon the first to the si T milar and similarly described figure upon the second; therefore as BC to CF, so is the figure ABC to the figure KGH: but as BC to CF, so is (1.6.) the parallelogram BE to the parallelogram EF: therefore as the figure ABC is to the figure KGH, so is the parallelogram BE to the parallelogram EF (11. 5.): but the rectilineal figure ABC is equal to the parallelogram BE; therefore the rectilineal figure KGH is equal (14.5.) to the parallelogram EF: but EF is equal to the figure D; wherefore also KGH is equal to D; and it is similar to ABC. Therefore the rectilineal figure KGH has been described similar to the figure ABC, and equal to D. Which was to be done. PROP. XXVI. THEOR. If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter. Let the parallelograms ABCD, AEFG be similar and similarly situ. ated, and have the angle DAB common: ABCD and AEFG are about the same diameter. For, if not, let, if possible, the parallelogram BD have its diameter AHC in a different straight line from AF, the diameter of the parallelogram EG and let GF meet K AHC in H; and through H draw HK parallel to AD or BC; therefore the para!lelograms ABCD, AKHG being about the same diameter, are similar to one another (24. 6.): wherefore, as DA to AB, so is B (def. 1. 6.) GA to AK; but because ABCD G D H and AEFG are similar parallelograms, as DA is to AB, so is GA to AE; therefore (11. 5.) as GA to AE, so GA to AK; wherefore GA has the same ratio to each of the straight lines AE, AK; aud consequently AK is equal (9. 5.) to AE, the less to the greater, which is impossible; therefore ABCD and AKHG are not about the same diameter; wherefore ABCD and AEFG must he about the same diameter. Therefore, if two similar, &c. Q. E. D. PROP. XXVII. THEOR. Of all the rectangles contained by the segments of a given straight line, the greatest is the square which is described on half the line. Let AB be a given straight line, which is bisected in C; and let D be any point in it, the square on AC is greater than the rectangle AD, DB. A C D B For, since the straight line AB is divided in two equal parts in C, and into two unequal parts in D, the rectangle contained by AD and DB, together with the square of CD, is equal to the square of AC |