Let the rectilineal figure ABCDE be given in species; ABCDE may be divided into triangles given in species. B A E Join BE, BD; and because ABCDE is given in species, the angle BAE is given (3. def.), and the ratio of BA to AE is given (3. def.); wherefore the triangle BAE is given in species (44. dat.), and the angle AEB is therefore given (3. def.). But the whole angle AED is given, and therefore the remaining angle BED is given, and the ratio of AE to EB is given, as also the ratio of AE to ED; therefore the ratio of BE to ED is given (9. dat.). And the angle BED is given, wherefore the triangle BED is given (44. dat.) in species. In the same manner, the triangle BDC is given in species: therefore rectilineal figures which are given in species are divided into triangles given in species. PROP. LII. C D 48. Ir two triangles given in species be described upon the same straight line, they shall have a given ratio to one another. Let the triangles ABC, ABD, given in species, be described upon the same straight line AB; the ratio of the triangle ABC to the triangle ABD is given. Through the point C draw CE parallel to AB, and let it meet DA produced in E, and join BE. Because the triangle ABC is given in species, the angle BAC, that is, the angle ACE, is given; and because the triangle ABD is given in species, the angle DAB, that is, the angle AEC E C L is given. Therefore the triangle ACE is given in species; wherefore the ratio of EA to AC is given (3. def.), and the ratio of CA to AB is given, as also the ratio of A BA to AD; therefore the ratio of (9. dat.) EA to AD is given, and the triangle ACB is equal (37. 1.) to the triangle AEB, and as the triangle AEB, or ACB, is to the triangle ADB, so is (1. 6.) the straight line EA to AD. But the ratio of EA to AD is given, therefore the ratio of the triangle ACB to the triangle ADB is given. PROBLEM. To find the ratio of two triangles ABC, ABD given in species, and which are described upon the same straight line AB. Take a straight line FG given in position and magnitude, and because the angles of the triangles ABC, ABD are given, at the points F, G of the straight line FG, make the angles GFH, GFK (23. 1.) equal to the angles BAC, BAD: and the angles FGH, FGK equal to the angles ABC, ABD, each to each. Therefore the triangles ABC, ABD are equiangular to the triangles FGH. FGK, each to each. Through the point H draw HL parallel to FG, meeting KF produced in L. And because the angles BAC, BAD are equal to the angles GFH, GFK, each to each; therefore the angles ACE, AEC are equal to FHL, FLH, each to each, and the triangle AEC equiangular to the triangle FLH. Therefore as EA to AC, so is LF to FH; and as CA to AB, so HF to FG ; and as BA to AD, so is GF to FK; wherefore, ex æquali, as EA to AD, so is LF to FK. But, as was shown, the triangle ABC is to the triangle ABD, as the straight line EA to AD, that is, as LF to FK. The ratio therefore of LF to FK has been found, which is the same with the ratio of the triangle ABC to the triangle ABD. If two rectilineal figures given in species be described upon the same straight line, they shall have a given ratio to one another.* Let any two rectilineal figures ABCDE, ABFG, which are given in species, be described upon the same straight line AB; the ratio of them to one another is given. Join AC, AD, AF: each of the triangles AED, ADC, ACB, AGF, ABF is given (51. dat.) in species. And because the trian gles ADE, ADC given in species are E G A D B described upon the same straight line AD, the ratio of EAD to DAC is given (52. dat.); and, by composition, the ratio of EACD to DAC is given (7. dat.). And the ratio of DAC to CAB is given (52. dat.) because they are described upon the same straight line AC; therefore the ratio of EACD to ACB is given (9. dat.): and, by composition, the ratio of ABCDE to ABC is given. In the same manner, the ratio of ABFG to ABF is given. But the ratio of the triangle ABC to the triangle ABF is given; wherefore (52. dat.), because the ratio of ABCDE to ABC is given, as also the ratio of ABC to ABF, and the ratio of ABF to ABFG; the ratio of the rectilineal ABCDE to the rectilineal ABFG is given (9. dat.). PROBLEM. H K L M N ----|-0 To find the ratio of two rectilineal figures given in species, and described upon the same straight line. Let ABCDE, ABFG be two rectilineal figures given in species, and described upon the same straight line AB, and join AC, AD, AF. Take a straight line HK given in position and magnitude, and by the 52d dat. find the ratio of the triangle ADE to the tri * See Note. angle ADC, and make the ratio of HK to KL the same with it. Find also the ratio of the triangle ABD to the triangle ACB. And make the ratio of KL to LM the same. Also, find the ratio of the triangle ABC to the triangle ABF and make the ratio of LM to MN the same. And, lastly, find the ratio of the triangle AFB to the triangle AFG, and make the ratio of MN to NO the same. Then the ratio of ABCDE to ABFG is the same with the ratio of HM to MO. D E C A B G F K L M N Because the triangle EAD is to the triangle DAC as the straight line HK to KL; and as the triangle DAC to CAB, so is the straight line KL to LM; therefore, by using composition as often as the number of triangles requires, the rectilineal ABCDE is to the triangle ABC, as the straight line HM to ML. In like manner, because the triangle GAF is to FAB, as ON 'to NM, by composition, the rectilineal ABFG is to the triangle ABF, as MO to NM; and, by inversion, as ABF to ABFG, so is NM to MO. And the triangle ABC is to ABF, as LM to MN. Wherefore, because as ABCDE to ABC, so is HM to ML; and as ABC to ABF, so is LM to MN; andas ABF to ABFG, so is MN to MO: ex æquali, as the rectilineal ABCDE to ABFG, so is the straight line HM to MO. Ir two straight lines have a given ratio to one another, the similar rectilineal figures described upon them similarly, shall have a given ratio to one another. Let the straight lines AB, CD have a given ratio to one another, and let the similar and similarly placed rectilineal figures E, F be described upon them; the ratio of E to F is given. To AB, CD, let G be a third proportional: therefore, as AB to CD, so is CD to G. And the ratio of AB to CD is given, wherefore the ratio of CD to G is given; and consequently the ratio of AB to G is also given A (9. dat.). But as AB to G, so is the figure E to the figure (2. cor. 20. 6.) F. fore the ratio of E to F is given. There G E F B C D H K L PROBLEM. To find the ratio of two similar rectilineal figures, E, F, similarly described upon straight lines AB, CD which have a given ratio to one another: let G be a third proportional to AB, CD. Take a straight line H given in magnitude; and because the ratio of AB to CD is given, make the ratio of H to K the same with it; and because H is given, K is given. As H is to K, so make K to L; then the ratio of E to F is the same with the ratio of H to L: for AB is to CD, as H to K, wherefore CD is to G, as K to L: and, ex æquali, as AB to G so is H to L: but the figure E is to (2. cor. 20. 6.) the figure F, as AB to G, that is, as H to L. Ir two straight lines have a given ratio to one another; the rectilineal figures given in species described upon them, shall have to one another a given ratio. Let AB, CD be two straight lines which have a given ratio to one another: the rectilineal figures E, F given in species and described upon them have a given ratio to one another. Upon the straight line AB, describe the figure AG similar and similarly placed to the figure F; and because F is given in species, AG is also given in species: therefore, since the figures E, AG, which are given in species, are described upon A C D E B the same straight line AB, the ratio of F E to AG is given (53. dat.), and be cause the ratio of AB to CD is given, G and upon them are described the si- HK milar and similarly placed rectilineal L figures AG, F, the ratio of AG to F is given (54. dat.): and the ratio of AG to E is given: therefore the ratio of E to F is given (9. dat.). PROBLEM. 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. Ir 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 D upon the same straight line AB, the ratio of ABCDE to AF is given (53. dat.): but the square AF is given in magnitude, therefore (2. dat.) also the figure ABCDE is given in magnitude. PROBLEM. To find the magnitude of a rectilineal figure given in species described upon a straight line given in magnitude. C A E B 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 G 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 (14. 5.). Ir two rectilineal figures are given in species, and if a side of one of them has 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 (3. def.) the ratios of AB to BC, and of DE to EF, the ratio of BC to EF is given (10. dat.). In the same manner, the ratios of the other sides to the other sides are given. A D B CE F 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. G H K L |