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 G be a third proportional. As therefore CD to G, so 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* the ratio of CD to EF is given. A B C DEFG H TK 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 18th Dat. If two rectilineal figures given in species have a given See N. 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* * 3 Def. to one another; therefore each side of one of them has to each side of the other a given ratio. C H G A DE 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. K M L 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. * 9 Dat. * 53 Dat. 9 Dat. 58 Dat. * 9 Dat. 55. * 18.6. * 56 Dat. PROP. 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, and upon BC describe the figure D' similar, and similarly placed, to the figure A, and 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 simi lar to D: therefore the ratio of the side EF to the ho 58 Dat. mologous side BC is given*; and BC is given, where-a fore* 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 a shewn to be given. * 2 Dat. * 3 Def. To describe a rectilineal figure A similar to a given a figure D, and equal to another given figure H... It is Prop. 25. B. 6. Elements. * 7 A Because each of the figures D, H, is given, their ray Cor.45.1. tio is given, which may be found by making upon the given straight line BC the parallelogram BK equal to * Cor.45.1. D, and upon its side CK making 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 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; by the 58th Dat. the ratio of the homologous sides BC, EF, is the same with 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 to A, as BC to CL, that is, as the figure BK to KL; and 2 Cor. that D is equal to BK, therefore A* is equal to KL, that is, to H. PROP. LXI. 20.6. * 14.5. 57. If a parallelogram given in magnitude have one of its sides See N. 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. A B * 1 Def. Cor. 45.1. C D F E 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 parallelogram 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 pa- G rallelograms AD, EH, are H 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. If a parallelogram have a given angle, the rectangle con- See N. tained by the sides about that angle has a given ratio to the parallelogram. C C Let the parallelogram ABCD have the given angle B E FA GK H From the point A draw AE perpendicular to BC; because the angle ABC is given, as also the angle * 43 Dat. AEB, the triangle ABE is given* in species; therefore the ratio of BA to AE is given. But as BA to AE, so is the rectangle AB, RC, to the rectangle AE, BC, therefore the ratio of the rectangle AB, BC, to AE, BC, that is*, to the parallelogram AC, is given. 1. 6. * 35. 1. 66. 41. 1. * 9 Dat. 56. And it is evident how the ratio of the rectangle to the parallelogram may be found, by making the angle FGH equal to the given angle ABC, and drawing from any point F in one of its sides, 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 sides about that angle, shall have a given ratio to the triangle ABC. : Complete the parallelogram ABCD; therefore by this proposition, 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* ratio to the triangle ABC. And the ratio of the rectangle to the triangle is found thus make the triangle FGK as was shewn in the proposition: the ratio of GF to the half of the perpendicular FK, is the same with the ratio of the rectangle AB, BC, to the triangle ABC. Because, as was shewn, GF is to FK, as AB, BC, to the parallelogram AC; and FK is to its half, as AC is to its half, which is the triangle ABC: therefore, ex æquali, GF is to the half of FK, as the rectangle AB, BC, is to the triangle ABC. PROP. LXIII. If two parallelograms be equiangular, as a side of the first to a side of the second, so is the other side of the second to the straight line to which the other side of the first has the same ratio which the first parallelogram |