50 to KL the fame with it. Find alfo the ratio of the triangle Because the triangle EAD is to E G D B F compofition as often as the number MNO 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, becaufe 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 æquali, as the rectilineal ABCDE to ABFG, fo is the ftraight line HM to MO. PROP. LIV. IF 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 fi gures 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 Gis ap. dat. alfo given a, But as AB to G, fo is E G F B C D H K L the figure E to the figure b F. Therefore the ratio of E to kis given PRO. PROBLEM. To find the ratio of two fimilar rectilineal figures, E, F fimilarly described 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 His 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 æquali, as AB to G, fo is H to L: But the figure E is to b the figure F, as AB to G, b2 cor. 20, that is, as H to L. PRO P. LV. IF 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. 6. E C D B F G -K 1 a 53. dat. 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 ipecies: Therefore, fince the figures E, AG which are given in fpe. Ar cies, are described upon the fame ftraight line AB, the ratio of E to AG is given a, and because the ratio of AB to 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 b;, and the ratio of AG to E is given; therefore the ratio of E to F is given c. H PROBLEM. To find the ratio of two rectilineal figures E, F given in species, 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 described 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 becaufe the fimilar rectilineal figures AG, Faie defcribed b 54 dat. cg. dat. defcribed 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 described upon a straight line given in magnitude; the figure is given in magnitude, Let the rectilineal figure ABCDE given in fpecies be described 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 fi. gures ABCDE, AF given in fpecies are defcribed upon the fame ftraight line AB, a 53. dat. the ratio of ABCDE, to AF is given a : But the fquare AF is given in magnitude, b. dat. therefore b alfo the figure ABCDE is given in magnitude. C 14 5. 53. PROB. To find the magnitude of a rectilineal figure given in fpecies defcribed upon a ftraight line given in magnitude. Take the ftraight line GH equal to the given straight line AB, and by the 53d dat. find the ratio which the fquare AF upon AB has to the figure ABCDE; B 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 ta HM c. 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 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 alfo given. Because the ratio of AB to DE is given, as alfo a the ratios a 3. def. of AB to BC, and of DE to EF; the ratio of BC to EF is gi ven b. In the fame manner, the ra tios of the other fides to the other fides are given. A D C E GH KL The ratio which BC has to EF may be found thus; take a ftraight B 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 EE, fo is K to L; ex æquali, BC is to EF, as G to L; therefore the ratio of G to Lhas been found, which is the fame with the ratio of BC to EF. PROP. LVII. bro. dat. G. IF two fimilar rectilineal figures have a given ratio to see N one another; their homologous fides have also a gi ven 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 b the ratio of CD to EF is given. A B DE FGb 13. dat. HL 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 10 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 54. See N. between H and K; the ratio of CD to EF is the fame with that of H to L. Let G be a third proportional to CD, EF; there fore as CD to G, fo 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 13th dat. IF PROP. LIX. F two rectilineal figures given in fpecies have a given ratio to one another; their fides fhall likewife have given ratios to one another. Let the two rectilineal figures A, B given in fpecies, have a given ratio to one another; their fides fhall also have given ra tios to one another. If the figure A be fimilar to B, their homologous fides fhall have a given ratio to one another, by the preceding propofition; and because the figures are given in fpecies, the a 3. def. fides of each of them have given ratios a to one another; thereb 9. fore each fide of one of them has b to each fide of the other a given ratio. dat. But if the figure A be not fimilar to B, let CD, EF be any two of their fides; and upon EF conceive the figure EG to be defcribed fimilar and fimilarly placed to the figure A, fo that CD, EF be homologous fides; therefore EG is given in fpe- C cies; and the figure B is given c 53. dat. in fpecies; wherefore the ratio d H of B to EG is given; and the K gure A to EG is given; and M G A DE BF A is fimilar to EG; therefore a the ratio of the fide CD to EF 58. dat. is given; and confequently b the ratios of the remaining fides to the remaining fides 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 fame with it. And by the 53d 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 æquali, as A |