PROBLEM. To find the ratio of two fimilar rectilineal figures E, F fimilarly 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 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 figure F, as AB to G, that is as H to L. PROP. LV. b the IF two straight lines have a given ratio to one another; the rectilineal figures given in fpecies defcribed 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 fpecics 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 b. 2. Což 20.6. 516 fince the figures E, AG which are given in fpecies, are defcribed upon A the fame ftraight line AB, the ratio lar and fimilarly placed rectilineal figures AG, F, the ratio of AG to F is given b. and the ratio of AG to E is given; therefore the ra- b. 54. Dat. tio 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 53. Dat, and make the ratio of H to K the fame; K is therefore given. and because the similar recti C c c. 9. Dat 52. rectilineal figures AG, F are defcribed upon the ftraight lines AB, CD which have a given ratio, find their ratio by the 54. 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. I PRO P. LVI. Fa rectilineal figure given in fpecies be described upon a ftraight line given in magnitude; the figure is given in magnitude. Let the rectilineal figure ABCDE given in fpecies be described upon the ftraight 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 defcri bed upon the fame ftraight line AB, the raa. 53. Dat. tio of ABCDE to AF is given. but the fquare AF is given in magnitude, there- D b. 2. Dat. foreb alfo the figure ABCDE is given in magnitude. PROB. To find the magnitude of a rectilineal figure given in fpecies defcribed upon a C B E A ftraight line given in magnitude. Take the ftraight line GH cqual to the given ftraight line AB, and by the 53. Dat. G H K find the ratio which the fquare AF upon AB has to the figure ABCDE; and make the ratio of GH to HK the fame; and upon GH defcribe the fquare GL, and complete the parallelogram LHKM; the figure ABCDE is equal to LHKM. because AF is to ABCDE, as the figure GL to HM; c. 14. §. and AF is equal to GL, therefore ABCDE is equal to HM *. 53. as the ftraight line CH to HK, that is, PROP. LVII. IF two are in 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 shall be given. Let Let AC, DF be two rectilineal figures given in fpecies, 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. Def. Because the ratio of AB to DE is given, as also the ratios of a. 3. AB to BC, and of DE to EF; the ratio of BC to EF is given b. in b. 10. Dat. the fame manner, the ratios of the other fides to the other fides are gi D GH KL and because the ratio of AB to DE is IF F two fimilar rectilineal figures have a given ratio to See N one another; their homologous fides have alfo 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 alfo a given ratio. 15.6. Let the fide CD be homologous to EF, and to CD, EF let the ftraight line G be a third proportional. as therefore CD to G, fo a. 2. Coř. 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. The ratio of CD to EF may be found thus; take a ftraight line H gi AA H DEFGb. 13. Dati L K ven in magnitude; and becaufe the ratio of the figure A to B is given, make the ratio of H to K the fame with it. and, as the 13. Dat. directs to be done, find a mean proportional L between H and K; the 54. See N. a. 3. Def. b. 9. Dat. ratio of CD to EF is the fame with that of H to L. let G be d third proportional to CD, EF; therefore as CD to G, so is (A to B, and fo is) H to K. and as CD to EF, fo is H to L, as is fhewn in the 13. Dat. IF PRO P. LIX. F two rectilincal figures given in fpecies have a given ratio to one another; their fides fhall 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 fides shall also have given ratios to one another. If the figure A be fimilar to B, their homologous fides fhall have a given ratio to one another, by the preceeding Propofition; and because the figures are given in fpecies, the fides of each of them have given ratios to one another. therefore each fide of one of them has b to each fide of the other a given ratio. But if the figure A be not similar to B, let CD, EF be any two of their fides; and upon EF conceive the figure EG to be deferized fimilar and fimilarly placed to the figure A, fo that CD, EF be homologous fides; therefore EG is given in fpecics. and the fi- C gure is given in fpecies, where c. 53. Dat. fore the ratio of B to EG is gi ven; and the ratio of A to B is figure A to EG is given. and I K M L G A DE B F 58. Dat. A is fimilar to EG, therefore the ratio of the fide CD to EF is gi ven; and confequently the ratios of the remaining fides to the remaining fides are given. 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 by the 53. Dat. find the ratio of the figure B to EG, and make the ratio of K to L the fame; between H and L find a mean proportional M; the ratio of CD to EF is the fame with the ratio of H to M. because the figure A is to B, as H to K; and as B to EG, fo is K to L; ex acquali, as A to EC, fo is H to and the figures A, EG are fimi L. lar, lar, and M is a mean proportional between H and L; therefore, as was fhewn in the preceeding Propofition, CD is to EF, as H to M. IF PRO P. LX. a rectilineal figure be given in fpecies and magnitude, Let the rectilineal figure A be given in fpecies and magnitude; its fides are given in magnitude. Take a ftraight line BC given in pofition and magnitude; and 55. upon BC defcribe the figure D fimilar, and fimilarly placed, to the a. 18.6. figure A, and let EF be the M b 56. Dar. ven in magnitude. and the figure A is given in magnitude, therefore the ratio of A to D is given. and the figure A is fimilar to D; therefore the ratio of the fide EF to the homologous fide BC is given . and BC is given, c. 53 Dit. wherefore EF is given. and the ratio of EF to EG is given", therefore EG is given. and, in the fame manner, each of the other fides of the figure A can be fhewn to be given. PROBLEM. To defcribe a rectilineal figure A fimilar to a given figure D, and equal to another given figure H. It is Prop. 25. B. 6. Elem. d. 2. D.. 4.3. Dat Because each of the figures D, H is given, their ratio is given, which may be found by making f upon the given ftraight line BC the f. Cor. 45-1. parallelogram BK equal to D, and upon its fide CK making f the parallelogram KL equal to H in the angle KCL equal to the angle MBC. therefore the ratio of D to H, that is of BK to KL is the fame with the ratio of BC to CL. and becaufe the figures D, A are fimilar, and that the ratio of D to A, or H, is the fame with the ratio of BC to CL; by the 58. Dat. the ratio of the homologous fides BC, EF is the fame with the ratio of BC to the mean proportional between BC and CL. find EF the mean proportional; then EF is the fide of the figure to be defcribed, homologous to BC the |