62. The formulæ requisite for the solution of right-angled triangles may be obtained from the preceding chapter by supposing one of the angles a right angle, as C for example. They may also be obtained very easily in an independent manner, as we will now shew. Let ABC be a spherical triangle having a right angle at C; let O be the centre of the sphere. From any point P in OA draw PM perpendicular to OC, and from M draw MN perpendicular to OB, and join PN. Then PM is perpendicular to MN, because the plane AOC is perpendicular to the plane BOC; hence PN2 = PM2 + MN2 – OP2 – OM2 + OM2 – ON2 – OP2 – ON2 ; = therefore PNO is a right angle. And = cosa cos b 1 = by (1); COS C (5). Crosswise the second formula in (2) and the first sacos B tan c tan a sin A sin c; sin A cos c ços a sin A cos b by (1). cos B-sin A cos b) cos 4= sin B cos af .(6). 1 six formule comprise ten equations; and thus we can As we have stated, the above six formulæ may be obed from those given in the preceding chapter by supposing Ca angle. Thus (1) follows from Art. 39, (2) from Art. 41, from the fourth and fifth equations of Art. 44, (4) from the and second equations of Art. 44, (5) from the third equation Art. 47, (6) from the first and second equations of Art. 47. Since the six formulæ may be obtained from those given in the preceding chapter which have been proved to be universally true, we do not stop to shew that the demonstration of Art. 62 may be applied to every case which can occur; the student may for exercise investigate the modifications which will be necessary when we suppose one or more of the quantities a, b, c, A, B equal to a right angle or greater than a right angle. 64. Certain properties of right-angled triangles are deducible from the formulæ of Art. 62. From (1) it follows that cos c has the same sign as the product cos a cos b; hence either all the cosines are positive, or else only one is positive. Therefore in a right-angled triangle either all the three sides are less than quadrants, or else one side is less than a quadrant and the other two sides are greater than quadrants. From (4) it follows that tan a has the same sign as tan A. Therefore A and a are either both greater than or both less π П 2 than; this is expressed by saying that ▲ and a are of the same affection. Similarly B and b are of the same affection, 65. The formulæ of Art. 62 are comprised in the following enunciations, which the student will find it useful to remember; the results are distinguished by the same numbers as have been already applied to them in Art. 62; the side opposite the right angle is called the hypothenuse:* Sine side = sine of opposite angle × sine hyp................. .(2), .(3), Tan side = tan opposite angle - sine of other side..............(4), Cos angle cos opposite side × sine of other angle..............(6). 66. Napier's Rules. The formulæ of Art. 62 are comprised in two rules, which are called, from their inventor, Napier's Rules of Circular Parts. Napier was also the inventor of Logarithms, and the Rules of Circular Parts were first published by him in a work entitled Mirifici Logarithmorum Canonis Descriptio...... Edinburgh, 1614. These rules we will now explain. Multiply together the two formulæ (4); thus, Multiply crosswise the second formula in (2) and the first in (3); thus sin a cos B tan c = tan a sin A sin c ; These six formulæ comprise ten equations; and thus we can solve every case of right-angled triangles. For every one of these ten equations is a distinct combination involving three out of the five quantities a, b, c, A, B; and out of five quantities only ten combinations of three can be formed. Thus any two of the five quantities being given and a third required, some one of the preceding ten equations will serve to determine that third quantity. 63. As we have stated, the above six formulæ may be obtained from those given in the preceding chapter by supposing Ca right angle. Thus (1) follows from Art. 39, (2) from Art. 41, (3) from the fourth and fifth equations of Art. 44, (4) from the first and second equations of Art. 44, (5) from the third equation of Art. 47, (6) from the first and second equations of Art. 47. Since the six formulæ may be obtained from those given in the preceding chapter which have been proved to be universally true, we do not stop to shew that the demonstration of Art. 62 may be applied to every case which can occur; the student may for exercise investigate the modifications which will be necessary when we suppose one or more of the quantities a, b, c, A, B equal to a right angle or greater than a right angle. 64. Certain properties of right-angled triangles are deducible from the formulæ of Art. 62. From (1) it follows that cos c has the same sign as the product cos a cos b; hence either all the cosines are positive, or else only one is positive. Therefore in a right-angled triangle either all the three sides are less than quadrants, or else one side is less than a quadrant and the other two sides are greater than quadrants. From (4) it follows that tan a has the same sign as tan A. Therefore A and a are either both greater than or both less π 2' than; this is expressed by saying that A and a are of the same affection. Similarly B and b are of the same affection, 65. The formulæ of Art. 62 are comprised in the following enunciations, which the student will find it useful to remember; the results are distinguished by the same numbers as have been already applied to them in Art. 62; the side opposite the right angle is called the hypothenuse:* Cos hyp product of cosines of sides = Cos hyp product of cotangents of angles = Sine side = sine of opposite angle × sine hyp..... Tan side tan hyp × cos included angle = .(1), ..(5), (2), .(3), Tan side 66. Napier's Rules. The formula of Art. 62 are comprised in two rules, which are called, from their inventor, Napier's Rules of Circular Parts. Napier was also the inventor of Logarithms, and the Rules of Circular Parts were first published by him in a work entitled Mirifici Logarithmorum Canonis Descriptio... Edinburgh, 1614. These rules we will now explain. ...... |