as CA to AB, fo HF to FG; and as BA to AD, fo is GF to FK; wherefore, ex aequali, as EA to AD, fo is LF to FK. But, as was shown, the triangle ABC is to the triangle ABD, as the straight line EA to AD, that is, as LF to FK. The ratio therefore of LF to FK has been found, which is the fame with the ratio of the triangle ABC to the triangle ABD. IF F two rectilineal figures given in fpecies be defcribed See N. upon the fame ftraight line; they fhall have a given ratio to one another. Let any two rectilineal figures ABCDE, ABFG which are given in fpecies, be defcribed upon the fame ftraight line AB; the ratio of them to one another is given. E A D C b 52. dat. B C 7. dat. F KL MN O d 9. dat. Join AC, AD, AF; each of the triangles AED, ADC, ACB, AGF, ABF is given in fpecies. And because the tri- a 51. dat. angles ADE, ADC given in fpecies are defcribed upon the fame ftraight line AD, the ratio of EAD to DAC is given; and, by compofition, the ratio of EACD to DAC is given. And the ratio of DAC to CAB is given b, because they are defcribed upon the fame ftraight line AC; therefore the ratio of EACD to ACB is given; and, by compofition, the ratio of ABCDÉ to ABC is given. In the fame manner, the ratio of ABFG to ABF is given. But the ratio of the triangle ABC to the triangle ABF is given; wherefore, because the ratio of ABCDE to ABC is given, as alfo the ratio of ABC to ABF, and the ratio of ABF to ABFG; the ratio of the rectilineal ABCDE to the rectilineal ABFG is given ". G H PROBLEM. To find the ratio of two rectilineal figures given in species, and defcribed upon the fame straight line. Let ABCDE, ABFG be two rectilineal figures given in fpecies, and defcribed upon the fame ftraight line AB, and join AC, AD, AF. Take a ftraight line HK given in pofition and magnitude, and by the 52d dat. find the ratio of the triangle ADE to the triangle ADC, and make the ratio of HK C c 4 to . 50. a 9. dat. to KL the fame with it. Find alfo the ratio of the triangle E A G H C B F KL MN O Because the triangle EAD is to the triangle DAC, as the ftraight line HK to KL; and as the triangle DAC to CAB, fo is the ftraight line KL to LM; therefore, by using compofition as often as the number of triangles requires, the rectilineal ABCDE is to the triangle ABC, as the ftraight line HM to ML. In like manner, because the triangle GAF is to FAB, as ON to NM, by compofition, the rectilineal ABFG is to the triangle ABF, as MO to MN; and, by inverfion, as ABF to ABFG, fo is NM to MO. And the triangle ABC is to ABF, as LM to MN. Wherefore, because as ABCDE to ABC, fo is HM to ML; and as ABC to ABF, fo is LM to MN; and as ABF to ABFG, fo is MN to MO; ex aequali, as the rectilineal ABCDE to ABFG, fo is the ftraight line HM to MO. IF PROP. LIV. two ftraight lines have a given ratio to one another; the fimilar rectilineal figures defcribed upon them fimilarly, fhall have a given ratio to one another. Let the ftraight lines AB, CD, have a given ratio to one another, and let the fimilar and fimilarly placed rectilineal figures E, F be defcribed upon them; the ratio of E to F is given. To AB, CD, let G be a third proportional; therefore as AB to CD, fo is CD to G. And the ratio of AB to CD is given, wherefore the ratio of CD to G is given; and A confequently the ratio of AB to G is alfo given. But as AB to G, so is G E F BC D HKL b2. cor. 20. the figure E to the figure F. Therefore the ratio of E to Fis 6. given. PROBLEM. PROBLEM. To find the ratio of two fimilar rectilineal figures, E, F, fimilarly defcribed upon ftraight 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 fame with it; and because H is given, K is given. As H is to K, fo make K to L; then the ratio of E to F is the fame 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 aequali, as AB to G, fo is H to L: But the figure E is to the figure F, as AB to G, that is, as H to L. IF two ftraight lines have a given ratio to one another; the rectilineal figures given in fpecies defcrited upon them, fhall have to one another a given ratio. Let AB, CD be two ftraight lines which have a given ratio to one another; the rectilineal figures E, F given in fpecies and defcribed upon them, have a given ratio to one another. Upon the ftraight line AB, defcribe the figure AG fimilar and fimilarly placed to the figure F; and because F is given in fpecies, AG is alfo given in fpecies: Therefore, fince the figures E, AG which are given in fpe- A cies, are defcribed upon the fame ftraight line AB, the ratio of E to AG is given, and because the ratio of AB to CD is given, H E C D B F b2.cor.20. 6. 51. a 53. dat. and upon them are defcribed the fimilar and fimilarly placed rectilineal figures AG, F, the ratio of AG to F is given; b 54. dat. and the ratio of AG to E is given; therefore the ratio of E to F is given. PROBLEM. To find the ratio of two rectilineal figures E, F given in fpecies, and defcribed upon the ftraight lines AB, CD which have a given ratio to one another. Take a ftraight line H given in magnitude; and because the rectilineal figures E, AG given in fpecies are defcribed upon the fame ftraight line AB, find their ratio by the 53d dat. and make the ratio of H to K the fame; K is therefore given : And because the fimilar rectilineal figures AG, F are defcribed upon c 9. dat. 52. ■ 53. dat. 2. dat. upon the ftraight lines AB, CD, which have a given ratio, find their ratio by the 54th dat. and make the ratio of K to L the fame: The figure E has to F the fame ratio which H has to L: For, by the conftruction, as E is to AG, fo is H to K; and as AG to F, fo is K to L; therefore, ex aequali, as E to F; fo is H to L. PROP. LVI. IF a rectilineal figure given in fpecies be defcribed upon a ftraight line given in magnitude; the figure is given in magnitude. Let the rectilineal figure ABCDE given in species be defcribed upon the straight line AB given in magnitude; the figure ABCDE is given in magnitude. Upon AB let the fquare AF be defcribed; therefore AF is given in fpecies and magnitude, and because the rectilineal figures ABCDE, AF given in fpecies are defcribed upon the fame ftraight line AB, the ratio of ABCDE, to AF is given: But the fquare AF is given in magnitude, therefore alfo the figure ABCDE is gi- D ven in magnitude. PROB. To find the magnitude of a rectilineal figure given in fpecies described upon a ftraight line given in magnitude, F Take the ftraight line GH equal to the given ftraight line AB, and by the 53d dat. find the ratio which the fquare € 14. 5. 53. AF upon AB has to the figure ABCDE; and make the ratio of GH to HK the fame; and upon GH defcribe the square GL, and conplete the parallelogram LHKM; the figure ABCDE is equal to LHKM: Becaufe AF is to ABCDE, as the ftraight line GH to HK, that is, as the figure GL to HM; and AF is equal to GL; therefore ABCDE is equal to HM. PROP. LVII. F two rectilineal figures are given in fpecies, and if a fide of one of them has a given ratio to a fide of the other; the ratios of the remaining fides to the remaining fides fhall be given. Let Let AC, DF be two rectilineal figures given in species, and let the ratio of the fide AB to the fide DE be given, the ratios of the remaining fides to the remaining fides are also given. Because the ratio of AB to DE is given, as also the ratios a 3. def. of AB to BC, and of DE to EF, the ratio of BC to EF is gi ven. In the fame manner, the ratios of the other fides to the other fides are given. D A C E GH KL The ratio which BC has to EF may be found thus: Take a straight line G given in magnitude, and because the ratio of BC to BA is given, make the ratio of G to H the fame; and because the ratio of AB to DE is given, make the ratio of H to K the fame; and make the ratio of K to L the fame with the given ratio of DE to EF. Since therefore as BC to BA, fo is G to H; and as BA to DE, fo is H to K; and as DE to EF, fo is K to L; ex aequali, BC is to EF, as G to L; therefore the ratio of G to L has been found, which is the fame with the ratio of BC to EF. IF PRO P. LVIII. , b 10. dat. G. two fimilar rectilineal figures have a given ratio to See N. one another, their homologous fides have also a given ratio to one another. Let the two fimilar rectilineal figures A, B have a given ratio to one another, their homologous fides have also a given ratio. 20. 6. Let the fide CD be homologous to EF, and to CD, EF let the straight line G be a third proportional. As therefore a CD a 2. Cor. to G, fo 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 C is given. b A B DEFG b 13. dat. H L K The ratio of CD to EF may be found thus: Take a ftraight 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 fame with it: And, as the 13th dat. directs to be done, find a mean proportional L between |