50. triangle ACD to the triangle ACB. And make the ratio of KL to LM the same. Also, find the ratio of the triangle ABC to the triangle ABF, and make the ratio of LM to MN the same. And, lastly, find the ratio of the triangle AFB to the triangle AFG, and make the ratio of MN to NO the same. Then the ratio of ABCDE to ABFG is the same with the ratio of HM to MO. C Because the triangle EAD is to the triangle DAC, as the straight F line HK to KL; and as the triangle DAC to CAB, so is the straight line KL to LM; therefore by using A composition as often as the number of triangles requires, the rectilineal G ABCDE is to the triangle ABC, as the straight line HM to ML. In like H manner, because the triangle GAF is to FAB, as ON to NM, by composition, the rectilineal ABFG is to the triangle ABF as MO to NM, and by inversion, as ABF to ABFG, so is NM to MO. And the triangle ABC is to ABF, as LM to MN. Wherefore, because as ABCDE to ABC, so is HM to ML; and as ABC, to ABF, so is LM to MN; and as ABF to ABFG, so is MN to MO; ex æquali, as the rectilineal ABCDE to ABFG, so is the straight line HM to MO. PROP. LIV. Ir two straight lines have a given ratio to one another; the similar rectilineal figures described upon them similarly, shall have a given ratio to one another. Let the straight lines AB, CD, have a given ratio to one another, and let the similar and similarly placed rectilineal figures E, F, be described upon them; the ratio of E to Fis given. To AB, CD, let G be a third fore the ratio of CD to G is given; a 9 Dat. to G is also givena. But as AB E G F B D H K L 2 Cor. to G, so is the figure E to the figureb F. Therefore the ra b 20. 6. tio of E to F is given. PROBLEM. To find the ratio of two similar rectilineal figures E, F, similarly described upon straight lines AB, CD, which have a given ratio to one another: Let G be a third proportional to AB, CD. Take a straight line H given in magnitude; and because the ratio of AB to CD is given, make the ratio of H to K the same with it; and because H is given, K is given. As H is to K, so make K to L: then the ratio of E to F is the same with the ratio of H to L; for AB is to CD, as H to K, wherefore CD is to G, as K to L; and, ex æquali, as AB to G, so is H to L: But the figure E is to the fi- 2 Cor. gure F, as AB to G, that is, as H to L. PROP. LV. Ir two straight lines have a given ratio to oné another; the rectilineal figures given in species, described upon them, shall have to one another a given ratio. Let AB, CD, be two straight lines which have a given ratio to one another; the rectilineal figures E, F, given in species and described upon them, have a given ratio to one another. Upon the straight line AB, describe the figure AG similar and similarly placed to the figure F; and because F is given in species, AG is also given in species: Therefore, since the figures E, AG, which are given in species, are described upon same straight line AB, the 51. 20. 6. E A the H KL ratio of E to AG is givena, and because the ratio of AB to CD is given, and upon them are described the si milar and similarly placed rectilineal figures AG, F, the ra tio of AG to F is given; and the ratio of AG to E is ' 54 Dat. given; therefore the ratio of E to F is given. PROBLEM. To find the ratio of two rectilineal figures E, F, given in species and described upon the straight lines AB, CD, which have a given ratio to one another. Take a straight line H given in magnitude; and because the rectilineal figures E, AG, given in species, are described upon the same straight line AB, find their ratio by the 53d dat. and make the ratio of H to K the same, K 9 Dat. 2 52. is therefore given. And because the similar rectilineal fi gures AG, F, are described upon the straight lines AB, CD, which have a given ratio, find their ratio by the 54th dat. and make the ratio of K to L the same: The figure E has to F the same ratio which H has to L: For, by the construction, as E is to AG, so is H to K; and as AG to F, so is K to L: Therefore, ex æquali, as E to F, so is H to L. PROP. LVI. Ir a rectilineal figure given in species be described upon a straight line given in magnitude; the figure is given in magnitude. Let the rectilineal figure ABCDE given in species, be described upon the straight line AB given in magnitude; the figure ABCDE is given in magnitude. Upon AB let the square AF be described; therefore AF is given in species and magnitude, and because the rectili'neal figures ABCDE, AF, given in species, are described upon the same straight line AB, the 53 Dat. ratio of ABCDE to AF is given a: But the square AF is given in mag b2 Dat. nitude, therefore also the figure D ABCDE is given in magnitude. C Take the straight line GH equal C B F M H K by the 53d dat. find the ratio which the square AF upon AB has to the figure ABCDE; and make the ratio of GH to HK the same; and upon GH describe the square GL and complete the parallelogram LHKM; the figure ABCDE is equal to LHKM; because AF is to ABCDE, as the straight line GH to HK, that is, as the figure GL to HM and AF is equal to GL; therefore ABCDE is equal 14. 5. to HMS. IF two rectilineal figures are given in species, and if a side of one of them has a given ratio to a side of the other; the ratios of the remaining sides to the remaining sides shall be given. Let AC, DF, be two rectilineal figures given in species, and let the ratio of the side AB to the side DE be given, the ratios of the remaining sides to the remaining sides are also given. Because the ratio of AB to DE is given, as also the ra- * 3 Def. tios of AB to BC, and of DE to EF, the ratio of BC to EF is givenb. In the same manner the ratios of the other sides ↳ 10 Dat. to the other sides are given. The ratio which BC has to A EF may be found thus: Take a straight line G given in mag- B same; and because the ratio of AB to DE is given, make the ratio of H to GH KL D K the same; and make the ratio of K to L the same with the given ratio of DE to EF. Since therefore as BC to BA, so is G to H; and as BA to DE, so is H to K; and as DE to EF, so is K to L; ex æquali, BC is to EF, as G to L; therefore the ratio of G to L has been found, which is the same with the ratio of BC to EF. PROP. LVIII. b G. If two similar rectilineal figures have a given ratio See N. to one another, their homologous sides have also a given ratio to one another. Let the two similar rectilineal figures A, B, have a given ratio to one another, their homologous sides have also a given ratio. Let the side CD be homologous to EF; and to CD, EF, let the straight line G be a third proportional. As thereforea CD to G, so is the figure A to B; and the ratio of A to B is given, therefore the ratio of CD to G is given; and CD, EF, G, are proportionals; wherefore the ratio of CD to EF is given. a 2 Cor. 20. 6. A DE B b 13 Dat. H L K The ratio of CD to EF may be found thus: Take a straight line H given in magnitude; and because the ratio of the figure A to B is given, make the ratio of H to K the same with it: And, as the 13th dat. directs to be done, find a mean proportional L between H and K; the ratio of CD to EF is the same with that of H to L. Let G be a third proportional to CD, EF; therefore as CD to G, so is (A to B, and so is) H to K; and as CD to EF, so is H to L, as is shown in the 13th dat. See N. IF two rectilineal figures given in species have a given ratio to one another, their sides shall likewise have given ratios to one another. Let the two rectilineal figures A, B, given in species, have a given ratio to one another, their sides shall also have given ratios to one another. If the figure A be similar to B, their homologous sides shall have a given ratio to one another, by the preceding proposition; and because the figures are given in species, 3 Def. the sides of each of them have given ratiosa to one another; b9 Dat. therefore each side of two of them has to each side of the other a given ratio. A But if the figure A be not similar to B, let CD, EF, be any two of their sides; and upon EF conceive the figure EG to be described similar and similarly placed to the figure A, so that CD, EF, C be homologous sides: therefore EG is given in species: and the figure B H is given in species; where- K 53 Dat. forec the ratio of B to EG is M given and the ratio of AL Ꭰ E B F to B is given, therefore the ratio of the figure A to EG is given and A is similar to EG; therefored the ratio of the a 58 Dat. side CD to EF is given; and consequently the ratios of the remaining sides to the remaining sides are given. The ratio of CD to EF may be found thus: Take a straight line H given in magnitude, and because the ratio of the figure A to B is given, make the ratio of H to K the same with it. And by the 53d dat, find the ratio of the fi gure B to EG, and make the ratio of K to L the same: Between H and L find a mean proportional M, the ratio of CD to EF is the same with the ratio of H to M; because the figure A is to B as H to K: and as B. to EG, so is K |