to EG is given, and that AC is equal b to BL; therefore the ratio b. 35. 1. of BL to EG is given. and becaufe BL is equiangular to EG, and c by the hypothefis, the ratio of BC to FG is given; therefore the c. 65. Dat, ratio of KB to EF is given. and the ratio of KB to BA is given; the ratio therefore of AB to EF is given. E F A K D L d. 9. Dat, B C M H N P The ratio of AB to EF may be found thus; take the flraight line MN given in position and magnitude; and make the angle NMO equal to the given angle BAK, and the angle MNO equal to the given angle EFG or AKB. and becaufe the parallelogram BL is equiangular to EG, and has a given ratio to it, and that the ratio of BC to FG is given; find by the 65. Dat. the ratio of KB to EF; and make the ratio of NO to OP the fame with it. then the ratio of AB to EF is the fame with the ratio of MO to OP. for fince the triangle ABK is equiangular to MON, as AB to BK, fo is MO to ON; and as KB to EF, fo is NO to OP; therefore, ex aequrli, as AB to EF, fo is MO to OP. PROP. LXVII. 70. IF F the fides of two equiangular parallelograms have gi- See N. ven ratios to one another; the parallelograms fhall have a given ratio to one another. Let ABCD, EFGH be two equiangular parallelograms, and let the ratio of AB to EF, as alfo the ratio of BC to FG be given; the ratio of the parallelogram AC to EG is given. Take a straight line K given in magnitude, and because the ratio of AB to EF is given, make the ratio of K to L the fame with it; A therefore L is given 2. and because the ratio of BC to FG is B given, make the ratio of L to M Kthe fame. therefore M is given2; L and K is given, wherefore the M DE H a. 2. Dat. C the ratio of K to M is given. but the parallelogram AC is to the parallelogram EG, as the straight line K to the straight line M, as Dat. 70. See N. is demonftrated in the 23. Prop. of B. 6. Elem. therefore the ratio of AC to EG is given. From this it is plain how the ratio of two equiangular parallelograms may be found when the ratios of their fides are given. IF PRO P. LXVIII. TF the fides of two parallelograms which have unequal, but given, angles, have given ratios to one another; the parallelograms fhall have a given ratio to one another. Let two parallelograms ABCD, EFGH which have the given unequal angles ABC, EFG have the ratios of their fides, viz. of AB to EF, and of BC to FG given; the ratio of the parallelogram AC to EG is given. At the point B of the ftraight line BC make the angle CBK equal to the given angle EFG, and complete the parallelogram KBCL. and because each of the angles BAK, BKA is given, the tria. 43. Dat. angle ABK is given in fpecies. therefore the ratio of AB to BK b. 9. Dat. is given; and the ratio of AB to EF is given, wherefore the ratio of BK to EF is given. and KA the ratio of BC to FG is gi ven; and the angle KBC is equal to the angle EFG; there c. 67. Dat. fore the ratio of the parallelogram KC to EG is given. d. 35. 1. but KC is equal to AC; B LDEH therefore the ratio of AC to EG is given. The ratio of the parallelogram AC to EG may be found thus; take the ftraight line MN given in pofition and magnitude, and make the angle MNO equal to the given angle KAB, and the angle NMO equal to the given angle AKB or FEH. and because the ratio of AB to EF is given, make the ratio of NO to P the fame; alfo make the ratio of P to Q the fame with the given ratio of BC to FG. the parallelogram AC is to EG, as MO to Q. Because the angle KAB is equal to the angle MNO, and the angle AKB equal to the angle NMO; the triangle AKB is equiangular to NMO. therefore as KB to BA, fo is MO to ON; and as BA to EF, fo is NO to P; wherefore, ex aequali, as KB to EF, so is MO to P. and BC is to FG, as P to Q, and the parallelograms KC, EG are equiangular; therefore, as was fhewn in Prop. 67. the parallelogram KC, that is AC, is to EG, as MO to Q. COR. 1. If two triangles ABC, DEF have two equal angles, or 71. two unequal, but given angles ABC, DEF, and if the ratios of the fides about thefe angles, viz. the ratios of AB to DE, and of BC to EF be given; the triangles fhall have a given ratio to one another. A GD H N N Complete the parallelograms BG, B EH; the ratio of BG to EH is given"; CE F and therefore the triangles which are the halves of them have a given ratio to one another. c COR. 2. If the bases BC, EF of two triangles ABC, DEF have a given ratio to one another, and if alfo the straight lines AG, DH which are drawn to the bafes from the oppofite angles, either in equal angles, or unequal, but given, angles AGC, DHF have a given ratio to one another; the triangles K A fhall have a given ratio to one LD CEH F a. 67. or 68. Dat. b. 34. I. C. 15.5. 72. grams KC, LF. and because the angles AGC, DHF, or their equals the angles KBC, LEF are either equal, or unequal, but given; and that the ratio of AG to DH, that is of KB to LE is given, as alfo the ratio of BC to EF; therefore the ratio of the parallelogram KC. 67. or 68. to LF is given. wherefore alfo the ratio of the triangle ABC to DEF is given b. IF a PROP. LXIX. F a parallelogram which has a given angle be applied to one side of a rectilineal figure given in species; if the figure have a given ratio to the parallelogram, the parallelogram is given in fpecies. Let ABCD be a rectilineal figure given in fpecies, and to one fide of it AB let the parallelogram ABEF having the given angle ABE be applied; if the figure ABCD has a given ratio to the parallelogram BF, the parallelogram BF is given in fpecies. Thro' the point A draw AG parallel to BC, and thro' the point C draw CG parallel to AB, and produce GA, CB to the points H, K. be Dat. b. S41.1. 215.5. 61. a. 3. Def. 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 pa c. I. 6. rallelogram BG is given in fpecies. and becaufe upon the fame ftraight line AB the two rectilineal figures BD, BG given in fpecies b. 53. Dat. are defcribed, the ratio of BD to BG is given b. and, by hypothec. 9. Dat. fis, the ratio of BD to the parallelogram BF is given; wherefore © d. 35. 1. the ratio of BF, that is of the parallelogram BH, to BG is given, and therefore the ratio of the straight 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 becaufe 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 ratio 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; D A G C N C Ο M B take a ftraight line LM gi HF KEP ven in pofition and magni- .. tude; 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 53. 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, thro' 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 KBE 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 because as the figure BF to BD, fo is the straight line LM to P; and as BD to BG, fo is P to Q; ex aequali, as BF, that is d BH, to BG, fo is LM to Q. but BH is to BG, as KB to BC; as therefore KB to BC, fo is LM to Q. and becaufe 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 LO to R, that is to LN. and the angles ABE, NLO are equal; therefore the parallelogram BF is fimilar to LS. I PROP. LXX. 62.78. F 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 straight lines AB, CD have a given ratio to one another, and upon AB let the figure AEB given in fpecies be described, 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 defcribed the fimilar rectilineal figures AG, FD; the ratio of E AG to FD is given. and the ratio of FD to AEB is given; therefore the A c the ratio of AEB to AG is given, the parallelogram AG is given in c. 69. Dat. fpecies. but FD is fimilar to AG; therefore FD is given in fpecies. A parallelogram fimilar to FD may be found thus; take a straight 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 54. 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 to L are given, the ratio of H to L is given b because |