PROP. VII. IN any triangle, as the rectangle contained by two fides, is to the rectangle contained by half the fum, and half the difference of the bafe and the difference of the fides, fo is the fquare of the radius to the fquare of the fine of half the angle oppofite to the bafe. Let ABC be a triangle, of which the fide AC is greater than AB; and make AD equal to AC; therefore BD is the difference of the fides AC, AB: The rectangle BA, AC is to the rectangle contained by half the fum, and half the difference of CB, BD, as the fquare of the radius to the fquare of the fine of half the angle BAC. Join CD, and draw a AE, BF perpendicular to CD, and from the centre B, at the diftance BC, defcribe a circle, meeting AD in G, H, and CD in K: and because AD is equal to AC, the angle ACD 5.1. is equal to ADC, and AED, AEC are right angles, and the fide AE is common to the triangles AED, AEC; € 26. 1. therefore DE is equal to EC, and the angle DAE to EAC: and because c BF drawn through the centre cuts CK d 3. 3. at right angles, it bifects it ; therefore CK is double of CF: and CD is double of CE; therefore the remainder DK is double of the remain der EF and because the ftraight lines CK, GH in the circle, 35. 3. cut one another, the rectangle CD, DK is equal to the rectangle GD, DH; and the rectangles contained by their halves are equal; therefore the rectangle DE, EF is equal to the rectangle contained by half of DG, the fum of CB, BD, and half of DH their difference: but becaufe BF is parallel to AE, DA is to AB, as DE to EF; therefore the rectangles DA, AB, and 81.Def. 6. DE, EF, are fimilar figures & upon AB, EF: and the fquares of AD, DE are fimilar; therefore the rectangle DA, AB, or CA, h 22.6. AB, is the rectangle DE, EF, as the fquare of AD to the fquare of DE; that is, as the fquare of the radius to the fquare of the Therefore the rectangle CA, f 2.6. 1. P. T. fine of DAE, half of BAC. AB is to the rectangle contained by half the fum, and half the difference of CB, BD as the fquare of the radius to the fquare of the fine of half the angle BAC. TRIGONOMETRY. 239 PROP. VIII. N any triangle, the rectangle contained by two fides, is to the rectangle contained by half the fum of the three fides, and its excefs above the base, as the fquare of the radius to the fquare of the cofine of half the angle oppofite to the base. Let ABC be any triangle, the rectangle BA, AC is to the rectangle contained by half the fum of the three fides and its excess above the base BC, as the fquare of the radius to the fquare of the cofine of half the angle BAC. : e d D In BA produced, take AD equal to AC; and join DC, and draw a AE, BF perpendicular to DC; and from the centre B, at the distance BC, defcribe a circle, cutting DC again in K, and BD in H, G: and bifect DG in L: and because AD is equal to AC, the angles ADC, ACD are equal: and AED, AEC are right angles; therefore DE is equal to EC: and the angle BAC is equal to ACD, ADC, it is therefore double of ADC and because BF, from the centre B, is at right angles to CK, it bifects it ; therefore CK is double of CF; and CD is double of CE; therefore the remainder DK is double of the remainder EF: For the fame reason, DH is double of BL: and because DG, DC cut the circle, the rectangle GD, DH is equal & to the rectangle CD, DK; and therefore the rectangles A contained by their halves are equal; that is, the rectangle DE, EF is equal to the rectangle contained by DL half of DG the fum of the three fides, and BL the excess of LG that half fum above BG or BC the bafe: but because AE is parallel to BF, DA is to AB, as DE to EF; therefore the rectangle DA, AB is fimi- 'G lark to the rectangle DE, EF; and the fquares Pl. Trig. a 12. 2. K b 10. 1. c . . d 26. i. e 32. 1. £ 3.3. £ g Cor. 36. 3. of DA, DE are fimilar; therefore the rectangle DA, AB, or CA, AB, is to the rectangle DE, EF, as the fquare of DA to h 16. 6. k1. Deté. the fquare of DE; that is, as the fquare of the radius to the 1 22. ú. fquare of the cofine of ADC, half of the angle BAC". Where- m 1. P. T. fore, as the rectangle contained by the fides BA, AC is to the, rectangle contained by half the fum of the three fides, and its excefs above the base BC, fo is the fquare of the radius to the fquare of the cofine of half the angle BAC. COR. Pl. Trig. COR. Because the rectangle contained by half the fum of the three fides, and its excefs above the base, is to the rectangle BA, AC, as the fquare of the cofine of half BAC to the square n S. P. T. of the radius and that the rectangle BA, AC is to the rectangle contained by half the fum, and half the difference of the bale and difference of the fides, as the fquare of the radius to 0 7. P. T. the square of the fine of half BAC; therefore, by equality P, P 22. 6. the rectangle contained by half the fum of the three fides, and its excess above the bafe, is to the rectangle contained by half the fum, and half the difference of the bafe and difference of the fides, as the fquare of the cofine of half BAC to the square of its fine; that is, as the fquare of the radius to the fquare 42. Cor. of the tangent of half the angle BAC oppofite to the 10. Def. bafe. SOLUTION of the CASES of OBLIQUE-ANGLED TRIANGLES. GENERAL PROPOSITION. N any triangle, of the three fides, and any two of the angles, any three being given, the other two IN and the remaining angle may be found. But if the three angles be given, the ratios only of the fides are given, being the fame with thofe of the angles oppofite to them and in this cafe, the fides cannot be found. A4 B D D If two angles of a triangle be given, the third is also given, being the fupplement of their fum: and, on the contrary, if one of the angles be given, the fum of the other two is also given. The The four cafes of this propofition may be refolved by the help Pl. Trig. of fome of the preceding propofitions; as in the following Table. |AC: AB :: fin. B : fin. C. P. 3- The fum of BA, AC: diff. of BA, Whence B and C are found, by AB, AC, BC, the A, B,C, Let AD be perp. to BC. BC: fum of BA, AC: diff. of BA, If BD be greater than CB, C is three fides. The rect. BA, Pl. Trig. a 30. 3. b12. 1. c 33.6. d 20. 3. e 31. 3. f 4. 6. To conftruct a Table of Sines, Tangents, &c. B E Let ABC be a circle, of which the diameter is AC, and the centre D and let AB be any arch of it; and bifect AB in F; and join AB, BC, BD, DF: and let DF meet AB in E; and drawb BG prpendicular to AC: and because the arch AB is double of AF, the angle ADB is double of ADF: but ADB is double of ACB 4, therefore the angle ADF is equal to ACB; and DE is parallel to BC, or at right angles to AB; therefore AC is to CB, as AD to DE; and AC is double of AD; therefore CB is C g 14. 5. double of DE 8: and because the triangles ADE, BCG are equiangular, AD is to DE, as BC to CG f; therefore the rectangle AD, CG is equal to the rectangle BC, DE, that is, to double of the fquare of DE and DE is the cofine of AF 1, and CG the fum of the radius CD, and DG 11. Cor. the cofine of AB. Therefore, the rectangle contained by the Def. 10. radius, and the fum of the radius and the cofine of any arch, Pl. T. is double of the fquare of the cofine of half that arch. h 16. 6. k 1. 2. 4. DGA : If AB be equal to the radius AD, it is the fide of a hexagon m Cor. 15. infcribed in a circle ; and the arch AB is the fixth part of the circumference; that is, it is an arch of 60°; and becaufe then n 47. 1. the triangle ABD is equilateral, AG is equal to GD "; that is, it is the half of the radius AD. If, therefore, the radius be reprefented by unity, and the decimal notation be ufed, the fum of the radius and the cofine DG is represented by 1.5, and the rectangle contained by the radius and this fum is also represented by 1.5, because the radius is I; therefore half of this rectangle is .75, which is therefore the fquare of DE the cofine of AF; that is, of 30. Wherefore the cofine of 30° is the fquare-root of .75; that is, .8660254039344+. In the fame manner, if the radius 1 be added to this number, half the fum .9330127019672+ is the fquare of the cofine of 15°; and therefore its fquare-root .96592582632789 + is the cofine of 15o. In like manner, the fquare-root of half the fum of 1 and the cofine of 15° is .99144485936634 + the cofine of 7°30'. In the fame manner are found the cofine of 3° 45', of 1° 57' 30', and fo on till at length, after twelve bifections of the arch of 6c, the cofine of 52" 44" 03" 45""" is found to be •9999999673176965713. And if the fquare of the cofine be fubtracted from 1, which is the fquare of the radius; the remainder is the fquare of the fine of the fame arch ". Thus, the fine |