æquali, as BF, that is, BH, to BG, so is LM to Q: but BH is to BG, as KB to BC; as therefore KB to BC, so is LM to Q; and because BE is to BK, as LO to LM; and as BK to BC, so is LM to Q; and as BC to BA, so Q was made to R; therefore, ex æquali, as BE to BA, so is LO to R, that is, to LN; and the angles ABE, NLO, are equal; therefore the parallelogram BF is similar to LS. PROP. LXX. Ir two straight lines have a given ratio to one an- See N other, and upon one of them be described a rectilineal figure given in species, 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 species. Let the two straight lines AB, CD, have a given ratio to one another, and upon AB let the figure AEB given in species 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 species. Upon the straight line AB, conceive the parallelogram AG to be described, similar and similarly placed to FD; and because the ratio of AB to CD is given, and upon them are described the similar rectilineal figures AG, FD; the ratio of AG to FD is given; and A the ratio of FD to AEB is given; therefore the ratio of AEB to AG is given; and the angle ABG is given, because it is equal to the angle FCD; be cause therefore the parallelo gram AG, which has a given E B • 51 Dat. G 9 Dat. M N H K L angle ABG, is applied to a side AB of the figure AEB C given in species, and the ratio of AEB to AG is given, the parallelogramı AG is given in species: but FD is similar © 69 Dat. to AG; therefore FD is given in species. A parallelogram similar to FD may be found thus: Take 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 same with it: Also, because the ratio of the straight line CD to AB is given, find by the 54th dat. the ratio which the figure FD described upon CD has to the figure AG described upon AB similar to FD; and make the ratio of K to L the same with this ratio: And because the ratios of H to K, and of K to L, are given, the ratio of 69 Dat. H to L is given; because, therefore, as AEB to FD, so is H to K; and as FD to AG, so is K to L; ex æquali, as AEB to AG, so is H to L; therefore the ratio of AEB to AG is given; and the figure AEB is given in species, and to its side AB the parallelogram AG is applied in the given angle ABG; therefore by the 69th dat. a parallelogram may be found similar to AG: Let this be the parallelogram MN; MN also is similar to FD; for, by the construction, MN is similar to AG, and AG is similar to FD; therefore -the parallelogram FD is similar to MN. If the extremes of three proportional straight lines have given ratios to the extremes of other three proportional straight lines; the means shall 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 shall have to the other a given ratio. Let A, B, C, be three proportional straight 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 also given. Because the ratio of A to D, as also of C to E, is given, the ratio of the rectangle A, C, to the rectangle D, F, is * 67 Dat. given; but the square of B is equal to the rectangle A, C; 17. 6. and the square of E to the rectangle b D, F; therefore the с ratio of the square of B to the square of E is given; where* 58 Dat. forec also the ratio of the straight line B to E is given. Next, let the ratio of A to D, and of B to E, be given; then the ratio of C to F is also given. Because the ratio of B to E is given, the ratio 54 Dat. of the square of B to the square of E is given; AB C therefore the ratio of the rectangle A, C, to the DEF rectangle D, F, is given; and the ratio of the side A to the side D is given; therefore the ratio of * 65 Dat. the other side C to the other F is given". 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 shall have a given ratio to the other mean, as may be shown in the same manner as in the foregoing proposition. t PROP. LXXII. IF four straight lines be proportionals; as the first is to the straight line to which the second has a given ratio, so is the third to a straight line to which the fourth has a given ratio. Let A, B, C, D, be four proportional straight lines, viz. as A to B, so 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. A BE Let E be the straight line to which B has a given ratio, and as B to E, so 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, so is C to D; and as B to E, so D to F: therefore, ex æquali, as A to E, so is C to F; and E is the straight line C D F to which B has a given ratio, and F that to which D has a given ratio; therefore as A is to the straight line to which B has a given ratio, so is C to a line to which D has a given ratio. 82. Hyp IF four straight lines be proportionals; as the first See N. is to the straight line to which the second has a given ratio, so is a straight line to which the third has a given ratio to the fourth. Let the straight line A be to B, as C to D; as A to the straight line to which B has a given ratio, so is a straight 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, so make F to C; because the ratio of B to E is given, the ratio of C to F is given: And because A is to B, as C to D; and as B to E, so F to C; therefore, ex æquali in proportione perturbato a, 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. E * 23.5. Ir a triangle has a given obtuse angle; the excess of the square of the side which subtends the obtuse angle, above the squares of the sides which contain it, shall have a given ratio to the triangle. Let the triangle ABC have a given obtuse angle ABC; and produce the straight line CB, and from the point A draw AD perpendicular to BC: The excess of the square 12. 2. of AC above the squares 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 43 Dat. ABD is given in species; and therefore the ratio of AD 1. 6. to DB is given: And as AD to DB, so is the rectangle AD, BC, to the rectangle DB, BC; wherefore the ratio of the rectangle AD, BC, to the rectangle DB, BC, is given, as also the ratio of twice the rectangle DB, BC, to the rectangle AD, BC: But the ratio of the rectangle AD, BC, to the triangle ABC, is given, because 41. 1. it is doubled of the triangle; therefore the ratio of twice the rectangle DB, BC, 9 Dat. to the triangle ABC, is given and D a E FG B twice the rectangle DB, BC, is the excess of the square of AC above the squares of AB, BC; therefore this excess has a given ratio to the triangle ABC. And the ratio of this excess to the triangle ABC may be found thus: Take a straight line EF given in position and magnitude; and because the angle ABC is given, at the point F of the straight line EF make the angle EFG equal to the angle ABC; produce GF, and draw EH perpendicular to FG; then the ratio of the excess of the square of AC above the squares of AB, BC, to the triangle ABC, is the same with the ratio of quadruple the straight lice 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 tri14. 6. angle ADB is equiangular to EHF; therefore as BD to * Cor. 4. 5. DA, so FH to HE; and as quadruple of BD to DA, so ist quadruple of FH to HE: But as twice BD is to DA, so is twice the rectangle DB, BC, to the rectangle AD, BC; C. 5. and as DA to the half of it, so ish the rectangle AD, BC, to its half the triangle ABC; therefore, ex æquali, as twice BD is to the half of DA, that is, as quadruple of BD is to DA, that is, as quadruple of FH to HE, so is twice the rectangle DB, BC, to the triangle ABC. Ir a triangle has a given acute angle, the space by which the square of the side subtending the acute angle is less than the squares of the sides which contain it, shall have a given ratio to the triangle. Let the triangle ABC have a given acute angle ABC, and draw AD perpendicular to BC, the space by which the of AC is less than the squares of AB, BC, that is a, 13. 2. square the double of the rectangle contained by CB, BD, has a given ratio to the triangle ABC. Because the angles ABD, ADB, are each of them given, the triangle ABD is given in species; and therefore the ratio of BD to DA is given: And as BD DC a to DA, so is the rectangle CB, BD, to the rectangle CB, AD: Therefore the ratio of these rectangles is given, as also the ratio of twice the rectangle CB, BD, to the rectangle CB, AD: But the rectangle CB, AD, has a given ratio to its half the triangle ABC: Therefore the ratio of twice the rectangle CB, 9 Dat. BD, to the triangle ABC is given; and twice the rectangle CB, BD, is a the space by which the square of AC is less than the squares of AB, BC; therefore the ratio of this space to the triangle ABC is given: And the ratio may be found as in the preceding proposition. LEMMA. IF from the vertex A of an isosceles triangle ABC, any A a 86. 1. 47. 1. € 5. 2. square of EB is equal to the rectangle BD, DC, together with the square BDE of DE; therefore the square of AB is D |