54. See N. a 3. def. b 9. dat. 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; therefore 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. PROP. LIX. F two rectilineal 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 fpecies, have a given ratio to one another, their fides fhall 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 preceding propofition; and because the figures are given in fpecies, the fides of each of them have given ratios to one another; there fore each fide of one of them has to each side of the other a given ratio. A G DE BF 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, so that CD, EF be homologous fides; therefore EG is given in fpecies; and the figure B is given 53. dat. in fpecies; wherefore the ratio H of B to EG is given; and the ratio of A to B is given, therefore the ratio of the fi C K M gure A to EG is given; and L d 58. dat. A is fimilar to EG; therefore the ratio of the fide CD to EF is given; 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 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 aequali, as A to EG, fo is H to L: And the figures A, EG are fimilar, and M is a mean proportional between H and L; therefore, as was fhewn in the preceding propofition, CD is to EF as H to M. PROP. LX. [F a rectilineal figure be given in fpecies and magnitude, the fides of it fhall be given in magnitude. IF 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, 55. and upon BC defcribe the figure D fimilar, and fimilarly a 18. 6. placed, to the figure A, G and let EF be the fide of b b 56. dat. 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, wherefore d EF is given; And e 58. dat. the ratio of EF to EG is given, therefore EG is given. And, d 2. dal. in the fame manner, each of the other fides of the figure A can be fhewn to be given. PROBLEM. e 3. def. I. 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. Because each of the figures D, H is given, their ratio is gi ven, which may be found by making upon the given ftraight f cor. 45line BC the 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 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 58th 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 I 4 fide of the figure to be defcribed, homologous to BC the fide of D, and the figure itfelf can be defcribed by the 18th prop. b. 6. which, by the conftruction, is fimilar to D; and because 8 2. Cor. D is to A, as BC to CL, that is as the figure BK to KL; and that D is equal to BK, therefore A is equal to KL, that is, to 20. 6. h 14. 5. 57. See N. H. IF PROP. LXI. Fa parallelogram given in magnitude has one of its fides and one of its angles given in magnitude, the other fide alfo is given. Let the parallelogram ABDC given in magnitude, have the fide AB and the angle BAC given in magnitude, the other fide AC is given. C B Take a ftraight line EF given in position and magnitude; and because the parallelogram AD is given in magnitude, a rectilineal a 1. def. figure equal to it can be found 2. And a parallelogram equal to this b Cor. 45. figure can be applied b to the given ftraight line EF in an angle equal to the given angle BAC. Let this be the parallelogram EFHG having the angle FEG equal to the angle BAC. And because the parallelograms AD, EH are equal, and have I. C 14. 6. d 12. 6. H. 出站 See N. E D F H the angles at A and E equal; the fides about them are reciprocally proportional; therefore as AB to EF, fo is EG to AC; and AB, EF, EG are given, therefore alfo AC is given 4. Whence the way of finding AC is manifeft. PROP. LXII. IF a parallelogram has a given angle, the rectangle contained by the fides about that angle has a given ratio to the parallelogram. Let the parallelogram ABCD have the given angle ABC, the rectangle AB, BC has a given ratio to the parallelogram AC. From the point A draw AE perpendicular to BC; "because the angle ABC is given, as alfo the angle AEB, the triangle 43. dat. ABE is given in fpecies; therefore the ratio of BA to AE is given. But as BA to AE, fo is the rectangle AB, BC to the rectangle AE, BC; therefore the ratio of b 1. 6. B A D the rectangle AB, BC to AE, BC, that is, to the parallelo- c 35. x. gram AC, is given. And it is evident how the ratio of the rectangle to the pa rallelogram may be found, by making the angle FGH equal to the given angle ABC, and drawing, from any point F in one of its fides, FK perpendicular to the other GH; for GF is to FK, as BA to AE, that is, as the rectangle AB, BC, to the parallelogram AC. COR. And if a triangle ABC has a given angle ABC, the rectangle AB, BC contained by the fides about that angle, fhall have a given ratio to the triangle ABC. 66. Complete the parallelogram ABCD; therefore, by this propofition, the rectangle AB, BC has a given ratio to the parallelogram AC; and AC has a given ratio to its half the triangle ABC; therefore the rectangle AB, BC has a given ra- d 14. 1. tio to the triangle ABC. с And the ratio of the rectangle to the triangle is found thus: Make the triangle FGK, as was shown in the propofition; the ratio of GF to the half of the perpendicular FK is the fame with the ratio of the rectangle AB, BC to the triangle ABC. Because, as was shown, GF is to FK, as AB, BC to the paralle logram AC; and FK is to its half, as AC is to its half, which is the triangle ABC; therefore, ex aequali, GF is to the half of FK, as AB, BC rectangle is to the triangle ABC. PROP. LXIII. IF two parallelograms be equiangular, as a fide of the firft to a fide of the fecond, fo is the other fide of the second to the straight line to which the other fide of the first has the fame ratio which the first parallelogram has to the fecond. And confequently, if the ratio of the first parallelogram to the fecond be given, the ratio of the other fide of the first to that straight line is given; and if the ratio of the other fide of the first to that straight line be given, the ratio of the first parallelogram to the fecond is given. Let AC, DF be two equiangular parallelograms, as BC, a fide of the firft, is to EF, a fide of the fecond, fo is DE, the other fide of the fecond, to the ftraight line to which AB, the o ther e 9. dat. 56. a 14. 6. 74.73. See N. a 35. I. A B H ther fide of the firft has the fame ratio which AC has to DF. G D F And if the ratio of the parallelogram AC to DF be given, then the ratio of the ftraight line AB to BG is given; and if the ratio of AB to the ftraight line BG be given, the ratio of the parallelogram AC to DF is given. IF PROP. LXIV. two parallelograms have unequal, but given angles, and if as a fide of the first to a fide of the fecond, fo the other fide of the fecond be made to a certain straight line; if the ratio of the firft parallelogram to the fecond be given, the ratio of the other fide of the first to that ftraight line fhall be given. And if the ratio of the other fide of the first to that ftraight line be given, the ratio of the first parallelogram to the fecond fhall be given. Let ABCD, EFGH be two parallelograms which have the unequal, but given, angles ABC, EFG; and as BC to FG, fo make EF to the ftraight line M. If the ratio of the parallelogram AC to EG be given, the ratio of AB to M is given. At the point B of the straight line BC make the angle CBK equal to the angle EFG, and complete the parallelogram KBCL. And becaule the ratio of AC to EG is given, and that AC is equal to the parallelogram KC, therefore the ratio of KC to EG is given; and KC, EG are equiangular; thereb 63. dat, fore as BC to FG, fo is EF to the ftraight line to which KB has a given ratio, viz. the fame which the parallelogram KC has to EG: But as BC to FG, fo is EF to the straight line M; therefore KB has a given ratio to M; and the ratio of |