gle BAD, and its double the angle BAC, are given; and the angle ABC is given; therefore the triangle ABC is given in species*. * 43 Dat. A triangle which shall have the things mentioned in the proposition to be given may be thus found. Let EFG be the given angle, and the ratio of H to K the given ratio; and by Prop. 44. find the triangle EFL, which has the angle EFG for one of its angles, and the ratio of the sides EF, FL, about this angle the same with the ratio of H to K; and make the angle LEM equal to the angle FEL. And because the ratio of H to K is the ratio which two sides of a triangle have to the third, H must be greater than K. And because EF is to FL, as H to K, therefore EF is greater than FL, and the angle FEL, that is, LEM, is therefore less than the angle ELF. Wherefore the angles LFE, FEM are less than two right angles, as was shewn in the foregoing proposition, and the straight lines FL, EM must meet if produced; let them meet in G, EFG is the triangle which was to be found: for EFG is one of its angles, and because the angle FEG is bisected by EL, the two sides FE, EG together, have to the third side FG, the ratio of EF to FL, that is, the given ratio of H to K. H M PROPOSITION L. If from the vertex of a triangle given in species, a straight line be drawn to the base in a given angle, it shall have a given ratio to the base. From the vertex A, of the triangle ABC which is given in species, let AD be drawn to the base BC in a given angle ADB: the ratio of AD to BC is given. Because the triangle ABC is given in species, the angle ABD is given, and the angle ADB is given, therefore the triangle ABD is given* in species: wherefore the ratio of AD to AB is given. And the ratio of AB to BC is given; and therefore* the ratio of AD to BC is given. PROPOSITION LI. "Rectilineal figures given in species, are divided into triangles which are given in species. (76.) * 43 Dat. 9 Dat. (47.) • 3 Def. • 3 Def. • 44 Dat. • 8 Def. 9 Dat. 44 Dat. ( 48.) * 3 Def. • 9 Dat. ⚫ 37. 1, . 1. 6. Let the rectilineal figure ABCDE be given in species: ABCDE may be divided into triangles given in species. Join BE, BD: and because ABCDE is given in species, the angle BAE is given *, and the ratio of BA to AE is given; wherefore the triangle BAE is given in species, and the angle AEB is therefore given *. 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 *. And the angle BED is given, wherefore the triangle BED is given 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. B E PROPOSITION LII. If 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 E ACE, is given; and because the triangle ABD is given in species, the angle DAB, that is, the angle AEC, is given. Therefore the triangle ACE is given in * species; wherefore the ratio of EA to AC is given *, and the ratio of CA to AB is given, as also the ratio of BA to AD; therefore the ratio of * EA to AD is given, and the triangle ACB is equal to the triangle AEB, and as the triangle AEB, or ACB, is to the triangle ADB, so is 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* equal to the angles BAC, BAD, and the angles 23. 1. 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 is 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 shewn, 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, 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, PROPOSITION LIII. ( 49.) If two rectilineal figures given in species be described upon See N. the same straight line, they shall have a given ratio to one another. E * AGF, ABF is given in species. And because the triangles 15 Dat. ADE, ADC, given in species, are described upon the same straight line AD, the ratio of EAD to DAC is given * ; and by composition, the ratio of EACD to DAC is given And the ratio of DAC to CAB is given*, because they are described upon the same straight line AC; therefore the ratio of EACD to ACB is given; 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 because the 52 Dat, ratio of ABCDE to ABC is given, as also the ratio of ABC to A G D HKL MN B F • 52 Dat. • 7 Dat. * 52 Dat. • 9 Dat. * 9 Dat. (50.) ABF, and the ratio of ABF to ABFG; the ratio of the rectilineal ABCDE to the rectilineal ABFG is given *. PROB.-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 triangle ADC, and make the ratio of HK to KL the same with it. Find also the ratio of the triangle ACD to the triangle ACB, and make the ratio 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. E A G H D KL MN B F 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 compo sition 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; and as ABF to ABFG, so is MN to MO; ex æquali, as the rectilineal ABCDE to ABFG, so is the straight line HM to MO. PROPOSITION LIV. If 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 *. But as AB to G, so is the figure E to the figure* F. Therefore the ratio of E to F is given. E G H K F PROB.-To find the ratio of two similar rectilineal figures E, 7 have a given ratio to one another. A L 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 the figure F, as AB to G, that is, as H to L. PROPOSITION LV. If 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. 4. E B G C D F 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 upon the given in species, are described same straight line AB, the ratio of E to AG is given *, and 53 Dat. because the ratio of AB to CD is given, and upon them are described the similar and similarly placed rectilineal figures AG, F, the ratio of AG to F is given; and the ratio of AG to E is given; therefore the ratio of E to F is given *. HK L_ ⚫ 2 Cor. 20. 6. (51.) 54 Dat. 9 Dat. |