a 3. def. c 9. dat. d 35. I. e I. 6. e the points H, K; because the angle ABC is given, and the ratio of AB to BC is given, the figure ABCD being given in fpecies; therefore, the parallelogram BG is given in fpecies. And because upon the fame ftraight line AB the two rectilineal figures BD, BG given in fpecies are described, the ratio of b 53. dat. BD to BG is given b; and, by hypothefis, the ratio of BD to the parallelogram BF is given; wherefore the ratio of BF, that is, of the parallelogram BH, to BG is given, and therefore the ratio of the ftraight line KB to BC is given; and the ratio of BC to BA is given, wherefore the ratio of KB to BA is given: And because the angle ABC is given, the adjacent angle ABK is given; and the angle ABE is given, therefore the remaining angle KBE is given. The angle EKB is alfo given, because it is equal to the angle ABK ; therefore the triangle BKE is given in fpecies, and confequently the ra tio of EB to BK is given; and the ratio of KB to BA is given, wherefore the ratio of EB to BA is given; and the angle ABE is given, therefore the parallelogram BF is given in fpecies. A parallelogram fimilar to BF may be found thus: Take a ftraight line a N G C S Ο M B LM given in pofition and HF KE P Q R magnitude; and because the angles ABK, ABE are given, make the angle NLM equal to ABK, and the angle NLO equal to ABE. And because the ratio of BF to BD is given, make the ratio of LM to P the fame with it; and because the ratio of the figure BD to BG is given, find this ratio by the 53d dat. and make the ratio of P to Q the fame. Alfo, because the ratio of CB to BA is given, make the ratio of Q to R the fame; and take LN equal to R; through the point M draw OM parallel to LN, and complete the parallelogram NLOS; then this is fimilar to the parallelogram BF. Because the angle ABK is equal to NLM, and the angle ABE to NLO the angle KEE is equal to MLO; and the angles BKE, LMO are equal, because the angle ABK is equal to NLM; therefore, the triangles BKE, LMO are equiangular to one another; wherefore as BE to BK, fo is LO to LM; and becaufe as the figure BF to BD, fo is the ftraight line LM to P; and as BD to BG, fo is P to Q; ex aequali, as BF, that is BH, to BG, fo is LM to Q: But BH is to BG, BG, as KB to BC; as therefore KB to BC, fo is LM to Q; and because BE is to BK as LO to LM; and as BK to BC, fo is LM to Q: And as BC to BA, fo Q was made to R ; therefore, ex aequali, as BE to BA, fo is LÒ to R, that is to LN; and the angles ABE, NLO are equal; therefore the paralle logram BF is fimilar to LS. PROP. LXX. 62. 78. IF two straight lines have a given ratio to one another, See N. and upon one of them be described a rectilineal figure given in fpecies, and upon the other a parallelogram having a given angle; if the figure have a given ratio to the parallelogram, the parallelogram is given in fpecies. Let the two ftraight lines AB, CD have a given ratio to one another, and upon AB let the figure AEB given in fpecies be defcribed, and upon CD the parallelogram DF having the given angle FCD; if the ratio of AEB to DF be given, the parallelogram DF is given in fpecies. Upon the ftraight line AB, conceive the parallelogram AG to be defcribed fimilar, and fimilarly placed to FD; and because the ratio of AB to CD is given, and upon them are described the fimilar rectilineal figures AG, FD; the ratio of AG to FD is gi ven; and the ratio of FD to ALB A is given; therefore the ratio of E F B a 54. dat. b 9. dat. AEB to AG is given; and the angle GC D ABG is given, because it is equal to the angle FCD; becaufe therefore M the parallelogram AG which has a N given angle ABG is applied to a fide AB of the figure AEB given in fpe HKL cies, and the ratio of AEB to AG is given, the parallelogram AG is given in fpecies; but FD is fimilar to AG; therefore c 69. date FD is given in fpecies. A parallelogram fimilar to FD may be found thus: Take a ftraight line H given in magnitude; and because the ratio of the figure AEB to FD is given, make the ratio of H to K the fame with it: Alfo, because the ratio of the ftraight line CD to AB is given, find by the 54th dat. the ratio which the figure FD defcribed upon CD has to the figure AG defcribed upon AB fimilar to FD; and make the ratio of K to L the fame with this ratio: And because the ratios of H to K, and of K bg. da' to L are given, the ratio of H to L is given b; because, there fore, as AEB to FD, fo is H to K; and as FD to AG, fo is K to L; ex aequali, as AEB to AG, fo is H to L; therefore the ratio of AEB to AG is given; and the figure AEB is given in fpecies, and to its fide AB the parallelogram AG is applied in the given angle ABG; therefore by the 69th dat. a parallelogram may be found fimilar to AG: Let this be the parallelogram MN; MN alfo is fimilar to FD; for, by the con ftruction, MN is fimilar to AG, and AG is fimilar to FD; therefore the parallelogram FD is fimilar to MN. 81. a 67. dat. b 17. 6. PROP. LXXI. IF the extremes of three proportional ftraight lines have given ratios to the extremes of other three propor, tional ftraight lines; the means fhall also have a given ratio to one another: And if one extreme has a given ratio to one extreme, and the mean to the mean; likewise the other extreme fhall have to the other a given ratio. Let A, B, C be three proportional ftraight lines, and D, E, F three other; and let the ratios of A to D, and of C to F be given; then the ratio of B to E is alfo given. Because the ratio of A to D, as alfo of C to F is given, the ratio of the rectangle A, C to the rectangle D, F is given"; but the fquare of B is equal to the rectangle A, C; and the fquare of E to the rectangle b D, F; therefore the ratio of the 58. dat. fquare of B to the fquare of E is given; wherefore alfo the ra tio of the ftraight line B to E is given. Next, let the ratio of A to D, and of B to E be gi- d 54. dat. the fquare of B to the fquare of E is given; there- € 65. dat. the other F is given. AB C COR. And if the extremes of four proportionals have to the extremes of four other proportionals given ratios, and one of the means a given ratio to one of the means; the other mean fhall have a given ratio to the other mean, as may be shown in the fame manner as in the foregoing propofition. PROP. IF four ftraight lines be proportionals; as the first is to the ftraight line to which the fecond has a given ratio, fo is the third to a straight line to which the fourth has a given ratio. Let A, B, C, D be four proportional ftraight lines, viz. as A to B, fo C to D; as A is to the straight line to which B has a given ratio, so is C to a straight line to which D has a given ratio. Let E be the ftraight line to which B has a given ratio, and as B to E, fo make D to F: The ratio of B to E is given, and therefore the ratio of D to F; and because as A to B, fo is C to D; and as B to E fo D to F; therefore, ex aequali, as A to E, fo is ABE C to F; and E is the ftraight line to which B has a C DF given ratio, and F that to which D has a given ratio; therefore as A is to the ftraight line to which B has a given ratio, so is C to a line to which D has a given ratio. PR O P. LXXIII. 82. a Hyp. 83. IF four ftraight lines be proportionals; as the first is to See N. the ftraight line to which the fecond has a given ra tio, fo is a ftraight line to which the third has a given ratio to the fourth. Let the ftraight line A be to B, as C to D; as A to the ftraight line to which B has a given ratio, fo is a ftraight line to which C has a given ratio to D. Let E be the straight line to which B has a given ratio, and as B to E, fo make F to C; because the ratio of B to E is given, the ratio of C to F is given: And becaufe A is to B, as C to D; and as B ABE to E, fo F to C; therefore, ex aequali in proportione FC D perturbata, A is to E, as F to D; that is, A is to E to which B has a given ratio, as F, to which C has a given ratio, is to D. a 23.5. a 12. 2. 64. IF. PRO P. LXXIV. Fa triangle has a given obtufe angle; the excess of the fquare of the fide which fubtends the obtufe angle, above the fquares of the fides which contain it, fhall have a given ratio to the triangle. Let the triangle ABC have a given obtufe angle ABC; and produce the ftraight line CB, and from the point A draw AD perpendicular to BC: The excess of the fquare of AC above the fquares of AB, BC, that is, the double of the rectangle contained by DB, BC, has a given ratio to the triangle ABC. Because the angle ABC is given, the angle ABD is also given; and the angle ADB is given; wherefore the triangle b 43. dat. ABD is given in fpecies; and therefore the ratio of AD to c 1. 6. E H FG DB is given: And as AD to DB, fo is the rectangle AD, a And the ratio of this excefs to the triangle ABC may be found thus: Take a ftraight line EF given in pofition and magnitude; and because the angle ABC is given, at the point F of the ftraight line EF, make the angle EFG equal to the angle ABC; produce GF, and draw EH perpendicular to FG; then the ratio of the excefs of the fquare of AC above the squares of AB, BC to the triangle ABC, is the fame with the ratio of quadruple the ftraight line HF to HE. Because the angle ABD is equal to the angle EFH, and the angle ADB to EHF, each being a right angle; the triangle ADB is equiangular to EHF; therefore f as BD to DA, fo FH to HE; and as quadruple of BD to DA, fo is & quadruple of FH to HE: But as twice BD is to DA, fo is twice the rectangle DB, BC to the rectangle AD, BC; and as DA to the half of it, fo is the rectangle AD, BC to its half the triangle |
