PROP. XX. THEOR. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons have to one another the duplicate ratio of that which their homologous sides have. Let ABCDE, FGHKL, be similar polygons, and let AB be the homoogous side to FG: the polygons ABCDE, FGHKL, may be divided into the same number of similar triangles, whereof each has to each the same ratio which the polygons have; and the polygon ABCDE has to the polygon FGHKL a ratio duplicate of that which the side AB has to the side FG. Join BE, EC, GL, LH: and because the polygon ABCDE is similar to the polygon FGHKL, the angle BAE is equal to the angle GFL (def. 1. 6.), and BA: AE::GF: FL (def. 1.6.): wherefore, because the tri angles ABE, 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 (6. 6.), and therefore similar, to the triangle FGL (4. 6.): wherefore the angle ABE is equal to the angle FGL: and, because the polygons are similar, the whole angle ABC is equal (def. 1. 6.) to the whole angle FGH; therefore the remaining angle EBC is equal to the remaining angle LGH: now because the triangles ABE, FGL are similar, EB: BA:: LG: GF; and also because the polygons are similar, AB: BC:: FG: GH (def. 1.6.); therefore, ex æquali (22. 5.) EB: BC:: LG: GH, that is, the sides about the equal angles EBC, LGH are proportionals; therefore (6. 6.) the triangle EBC is equiangular to the triangle LGH, and similar to it (4. 6.). For the same reason, the triangle ECD is likewise 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 ABE is similar to the triangle FGL, ABE has to FGL the duplicate ratio (19. 6.) 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 GL: therefore, as the triangle ABE to the triangle FGL, so (11.5.) is the triangle BEC to the triangle GLH. Again, because the triangle EBC is similar to the triangle LGH, EBC has to LGH the duplicate ratio of that which the side EC has to the side LII : for the same reason, the triangle ECD has to the triangle LHK, the duplicate ratio of that which EC has to LH: therefore, as the triangle EBC to the triangle LGH, so is (11.5.) the triangle ECD to the triangle LHK : but it has been proved, that the triangle EBC is likewise to the 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 .HK: and therefore, as one of the antecedents to one of the consequents, so are all the antecedents to all the consequents (12.5.). Wherefore, as the triangle ABE to the tri angle FGL, so is the polygon ABCDE to the polygon FGHKL: but the triangle ABE 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. COR. 1. In like manner it may be proved, that similar figures of four sides, or of any number of sides, are one to another in the duplicate ratio of their homologous sides, and the same has already been proved of triangles : therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. COR. 2. And if to AB, FG, two of the homologous sides, a third proportional M be taken, AB has (def. 11. 5.) to M the duplicate ratio of that which AB has to FG: but the four-sided figure, or polygon, upon AB has to the four-sided figure, or polygon, upon FG likewise 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 he proportionals, as the first 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. SCHOLIUM. If two polygons are composed of the same number of triangles similar, and similarly situated, those two polygons will be similar. For the similarity of the two triangles will give the angles EAB=LFG ABE=FGL, EBC=LGH: hence, ABC=FGH, likewise BCD=GHK &c. Moreover, we shall have, EA: LF::AB:FG::EB: LG :: BC : GH, &c.; hence the two polygons have their angles equal and their sides proportional; consequently they are similar. 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, 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 Bare equiangular (1. Ax. 1.), and have their sides about the equal angles proportionals (11.5.). Therefore A is similar (def. 1. 6.) to B. 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 rectilineal figures KAB, LCD be similarly described; and upon EF, GH the similar recti lineal 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: 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 either 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 to CD, so is EF to PR, and because PR is equal to GH, AB is to CD, as EF to GH. 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. 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 sides of the parallelograms. Now, because as BC to CG, so is the parallelogram 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 Lis (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. Cor. Hence, any two rectangles are to each other as the products of their bases multiplied by their altitudes. SCHOLIUM. Hence the product of the base by the altitude may be assumed as the neasure of a rectangle, provided we understand by this product the product of two numbers, one of which is the number of linear units contained in the base, the other the number of linear units contained in the altitude. Still this measure is not absolute but relative: it supposes that the area of any other rectangle is computed in a similar manner, by measuring its sides with the same linear unit; a second product is thus obtained, and the ratio of the two products is the same as that of the two rectangles, agreeably to the proposition just demonstrated. For example, if the base of the rectangle A contained three units, and its altitude ten, that rectangle will be represented by the number 3×10, or 30, a number which signifies nothing while thus isolated; but if there is a second rectangle B, the base of which contains twelve units, and the altitude seven, this rectangle would be represented by the number 12×7=84; and we shall hence be entitled to conclude that the two rectangles are to each other as 30 is to 84; and therefore, if the rectangle A were to be assumed as the unit of measurement in surfaces, the rectangle B would then have for its absolute measure; or, which amounts to the same thing, it would be equal to of a superficial unit. It is more common and more simple to assume the squares as the unit of surface; and to select that square whose side is the unit of length. In this case, the measurement which we have regarded merely as relative, becomes absolute: the number 30, for instance, by which the rectangle A was measured, now represents 30 superficial units, or 30 of those squares, which have each of their sides equal to unity. |