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 PRQ P. LV. F two ftraight 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 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. there b 2. Cor. 2.2.0. 51. CD is given, and upon them are defcribed the fimilar and fimilarly placed rectilineal figures AG, F, the ratio of AG to F is given . and the ratio of AG to E is given; b. 54. Dat. 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 becaufe the rectihineal figures E, AG given in fpecies are defcribed upon the fine 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 furikat Сс 9. Dat. 52. rectilineal figures AG, F are defcribed upon the ftraight lines AP, CD which have a given ratio, find their ratio by the 54. Da 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. IF PROP. LVI. TF a 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 defcribed upon the ftraight line AB given in magnitude; the figure ABCDE is given in magnitude. Upon AB let the square AF be defçribed; therefore AF is g ven in fpecies and magnitude. and becaufe the rectilineal figures ABCDE, AF given in fpecies are defcribed upon the fame ftraight line AB, the ratio of 2. 53. Dat. ABCDE to AF is given. but the fquare AF is given in magnitude, therefore b alfo the figure ABCDE is given in magnitude. b. 2. Dat. C. 14. 5. 53. PROB. To find the magnitude of a rectilineal figure given in fpecies defcribed upon a ftraight line given in magnitude. D C B G H K Take the ftraight line GH equal to the given ftraight line AB, and by the 53. Dat. 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. 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 . IF 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 AC, DF be two reftilineal figures given in fpecies, and let the ratio of the fide AB to the file DE be given; the ratios of . the remaining fides to the remaining fides are alfo given. Because the ratio of AB to DE is given, as alfo the ratios of a. 3. Def. AB to BC, and of DE to EF; the ratio of BC to EF is givenb. in b 10 Dat, the fame manner, the ratios of the other fides to the other fides are given. The ratio which BC has to EF may be found thus; take a straight ine G given in magnitude, and beaufe the ratio of BC to BA is given, hake the ratio of G to H the fune; nd because the ratio of AB to DE ; given, make the ratio of H to K A E !!!! GMKL D ༢ e fame; and make the ratio of K to L the fame with the given. tio of DE to EF. fince therefore as BC to BA, fo is G to H; and › BA to DE, fo is H to K; and as DE to EF, fo is K to L; ex equali, BC is to EF, as G to L. therefore the ratio of G to L is been found which is the fame with the ratio of BC to EF. PROP. LVIII. F two fimilar rectilineal figures have a given ratio to one another; their homologous fides have alfo a ven ratio to one another. Let the two fimilar rectilineal figures A, B have a given ratio one another; their homologous fides have alfo a given ratio. Let the fide CD be homologous to EF, and to CD, EF let the aight line G be a third proportional. as therefore CD to G, is the figure A to B; and the ratio A to B is given, therefore the raof CD to G is given; and CD, G are proportionals, wherefore he ratio of CD to EF is given. The ratio of CD to EF may be and thus; take a ftraight line H gi a. 2. Cor 20 6. A C DE FG b. 13. Dat. H I K in magnitude; and becaufe. the ratio of the figure A to B is gii, make the ratio of H to K tife fame with it. and, as the 13. Dat." ects to be done, find a mean proportional L between II and K; the See N. 54. a. 3. Def. b. 9. Dat. ratio of CD to EF is the fame with that of H to L. let Gla PROP. LIX. IF two rectilineal figures given in fpecies have a gis ratio to one another; their fides fhall likewife ha given ratios to one another. Let the two rectilineal figures A, B given in fpecies have age ratio to one another; their fides fhall alfo have given ratios to another. If the figure A be fimilar to B, their homologous fides have a given ratio to one another, by the preceding Propoút”, and because the figures are given in fpecies, the fides of ea them have given ratios to one another. therefore each one of them has to each fide of the other a given ratio. But if the figure A be not fimilar to B, let CD, EF be any of their fides; and upon EF conceive the figure EG to be der fimilar and fimilarly placed to the figure A, fo that CD, EF be homologous fides; therefore EG is given in fpecies. and the figure B is given in fpecies, c. 53. Dat. wherefore the ratio of B to EG is given; and the ratio of C H K A to B is given, therefore the M A DEBA Τ 458. Dat. given. and A is fimilar to EG, therefore the ratio of the fide to EF is given; and confequently the ratios of the rema fides to the remaining fides are given. The ratio of CD to EF may be found thus ; take a straighte H given in magnitude, and because the ratio of the figure Atoas given, make the ratio of H to K the fame with it. and by the Dat. find the ratio of the figure B to EG, and make the ratin to L the fame; between H and L find a mean proportional M; ratio of CD to EF is the fame with the ratio of H to M. bec the figure A is to B, as H to K; and as B to EG, fo is K to 1 aequali, as A to EG, fo is H to L. and the figures A, EG are , and M is a mean proportional between H and L; therefore, as as hewn in the preceding Propofition, CD is to EF, as H to M. PROP. LX. Fa rectilineal figure be given in fpecies and magnitude, the fides of it fhall be given in magnitude. Let the rectilineal figure A be given in fpecies and magnitude; : fides are given in magnitude. a 55. Take a straight line BC given in pofition and magnitude; and Don BC defcribe the figure D fimilar, and fimilarly placed, to the a. 18.6. gure A, and let EF be the ae figure A is given in magnitude, therefore the ratio of A to D given. and the figure A is fimilar to D; therefore the ratio of he fide EF to the homologous fide BC is given . and BC is gi- . 58. Dat. en, wherefore EF is given. and the ratio of EF to EG is gien, therefore EG is given. and, in the fame manner, each of e. 3. Der. he 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. Elein. Because each of the figures D, H is given, their ratio is given, d. 2. Dat. which may be found by making fupon the given ftraight line BC the f.Cor.45.1 parallelogram BK equal to D, and upon its fide CK making 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 because 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 proportion:1; then EF is the fide of the figure to be defcribed, homologous to BC the С с 3 |