( 52.) 53 Dat. • 2 Dat. PROB.-To find the ratio of two rectilineal figures E, F, given in species and described upon the straight lines Ab, cd, which have a given ratio to one another. Take a straight line H given in magnitude; and because the rectilineal figures E, AG, given in species, are described upon the same straight line AB, find their ratio by the 53d Dat. and make the ratio of H to K the same, K is therefore given. And because the similar rectilineal figures AG, F are described upon the straight lines AB, CD, which have a given ratio, find their ratio by the 54th Dat. and make the ratio of K to L the same: the figure E has to F the same ratio which H has to L for by the construction, as E is to AG, so is H to K; and as AG to F, so is K to L: therefore, ex æquali, as E to F, so is H to L. PROPOSITION LVI. If a rectilineal figure given in species be described upon a straight line given in magnitude, the figure is given in magnitude. Let the rectilineal figure ABCDE given in species, be described upon the straight line AB given in magnitude: the figure ABCDE is given in magnitude. Upon AB let the square AF be described; therefore AF is given in species and magnitude, and because the rectilineal figures ABCDE, AF, given in species, are described upon the same straight line AB, the ratio of ABCDE to AF is given*: but the square AF is given in magnitude, therefore* also the figure ABCDE is given in magnitude. D E B PROB. To find the magnitude of a rectilineal figure given in species described upon a straight line given in magnitude. Take the straight line GH equal to the given straight line AB, and by the 53d Dat. find the ratio which the square AF upon AB has to the figure ABCDE; and make the ratio of GH to HK the same; and upon GH describe the square GL and complete the parallelogram LHKM; the figure ABCDE is equal to LHKM. Because AF is to ABCDE, as the straight line GH to HK, that is, as the figure GL to HM; and AF is equal to GL; therefore ABCDE is equal to HM *. PROPOSITION LVII. If two rectilineal figures be given in species, and if a side of one of them have a given ratio to a side of the other, the ratios of the remaining sides to the remaining sides shall be given. Let AC, DF be two rectilineal figures given in species, and let the ratio of the side AB to the side DE be given the ratios of the remaining sides to the remaining sides are also given. Because the ratio of AB to DE is given, as also* the ratios of AB to BC, and of DE to EF, the ratio of BC to EF is given *. In the same manner the ratios of the other sides to the other sides are given. A B GH KL The ratio which BC has to EF may be found thus: take a straight line G given in magnitude, and because the ratio of BC to BA is given, make the ratio of G to H the same; and because the ratio of AB to DE is given, make the ratio of H to K the same; and make the ratio of K to L the same with the given ratio of DE to EF. Since therefore as BC to BA, so is G to H; and as BA to DE, so is H to K; and as DE to EF, so is K to L; ex æquali, BC is to EF, as G to L; therefore the ratio of G to L has been found, which is the same with the ratio of BC to EF. PROPOSITION LVIII. If two similar rectilineal figures have a given ratio to one another, their homologous sides have also a given ratio to one another. Let the two similar rectilineal figures A, B have a given ratio to one another: their homologous sides have also a given ratio. Let the side CD be homologous to EF; and to CD, EF, let the straight line & be a third proportional. As therefore* CD to G, so is the figure A to B; and the B EFG HL K ratio of A to B is given, therefore the ratio of CD to G is 14. 5. (53.) 3 Def. 10 Dat. (G.) See N. * 2 Cor. 20.6. given; and CD, EF, G are proportionals; wherefore the ratio of CD to EF is 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 same with it and, as the 13th Dat. directs to be done, find a mean proportional L between H and K; the ratio of CD to EF is the same with that of H to L. Let G be a third proportional to CD, EF; therefore as CD to G, so is (A to B, and so is) H to K; and as CD to EF, so is H to L, as is shewn in the 13th Dat. PROPOSITION LIX. If two rectilineal figures given in species have a given ratio to one another, their sides shall likewise have given ratios to one another. Let the two rectilineal figures A, B, given in species, have a given ratio to one another: their sides shall also have given ratios to one another. If the figure A be similar to B, their homologous sides shall have a given ratio to one another, by the preceding proposition; and because the figures are given in species, the sides of each of them have given ratios * to one another; therefore each side of one of them has to each side of the other a given ratio. H K M A B F But if the figure A be not similar to B, let CD, EF be any two of their sides ; and upon EF, conceive the figure EG to be described similar and similarly placed to the figure A, so that CD, EF be homologous sides: therefore EG is given in species: and the figure B is given in species; wherefore the ratio of B to EG is given: and the ratio of A to B is given, therefore the ratio of the figure A to EG is given: and A is similar to EG; therefore* the ratio of the side CD to EF is given; and consequently' the ratios of the remaining sides to the remaining sides 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 same 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 same: between H and L find a mean proportional M, the ratio of CD to EF is the same with the ratio of H to M. Because the figure A is to B as H to K, and as B to EG, so is K to L; ex æquali, as A to EG, so is H to L: and the figures A, EG are similar, and M is a mean proportional between H and L; therefore, as was shewn in the preceding proposition, CD is to EF as H to M. PROPOSITION LX. If a rectilineal figure be given in species and magnitude, the sides of it shall be given in magnitude. Let the rectilineal figure A be given in species and magnitude: its sides are given in magnitude. Take a straight line BC given in position and magnitude, G D E H M ( 55. ) and upon BC describe the figure D similar, and similarly 18.6. placed, to the figure A, and let EF be the side of the figure A homologous to BC the side of D; therefore the figure D is given in species. And because upon the given straight line BC the figure D given in species is described, D is given in magnitude, and the figure A is given in magnitude, therefore the ratio of A to D is given and the figure A is similar to D: therefore the ratio of the side EF to the homologous side BC is given * ; and BC is given, wherefore* EF is given: and the ratio of EF to EG is given *, therefore EG is given. And, in the same manner, each of the other sides of the figure A can be shewn to be given. 56 Dat. 58 Dat. 2 Dat. * 3 Def. *This is Prop. 25. * PROB.-To describe a rectilineal figure A similar to a given figure D, and equal to another given figure H. Because each of the figures D, H is given, their ratio is given, which may be found by making upon the given Cor. 45.1. straight line BC the parallelogram BK equal to D, and upon * its side CK making the parallelogram KL equal to H in the Cor. 45. 1. angle KCL equal to the angle MBC; therefore the ratio of D to H, that is, of BK to KL, is the same with the ratio of BC to CL and because the figures D, A are similar, and that the ratio of D to A, or H, is the same with the ratio of BC to CL, the ratio of the homologous sides BC, EF, is the same with 58 Dat. 6. the ratio of BC to the mean proportional between BC and CL. Find EF the mean proportional; then EF is the side of the figure to be described, homologous to BC the side of D, and the figure itself can be described by the 18th Prop. B. 6., which, by the construction, is similar to D; and because D is 2 Cor. 20. 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 H. ⚫ 14. 5. (57.) See N. • 1 Def. PROPOSITION LXI. If a parallelogram given in magnitude have one of its sides and one of its angles given in magnitude, the other side also is given. Let the parallelogram ABDC given in magnitude, have the side AB and the angle BAC given in magnitude: the other side AC is given. * B Take a straight line EF given in position and magnitude; and because the parallelogram AD is given in magnitude, a rectilineal figure equal to it can be found. And a paralleloCor. 45. 1. gram equal to this figure can be applied to the given straight 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 the angles at A and E equal; the sides about them are reciprocally proportional*; therefore as AB to EF, so is EG to AC: and AB, EF, EG are given, therefore also AC is given*. Whence the way of finding AC is manifest. ⚫ 14. 6. * 12. 6, (H.) ⚫ 48 Dat. PROPOSITION LXII. E If a parallelogram have a given angle, the rectangle contained by the sides about that angle has a given ratio to the parallel ogram. 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 also the angle AEB, the triangle ABE is given * in species; therefore the ratio of BA to AE is |