Questions for Exercise in Spherical Trigonometry. Ex. 1. In the right-angled spherical triangle BAC, rightangled at a, the hypothenuse a = 78°20′, and one leg c = 76°52′, are given; to find the angles B, and c, and the other leg b. sin c Here, by table i case 1, sin c = ; Or, log sin c = log sin c-log sin a + 10. log cos B = log tan c log tan a + 10. log cos b = log cosa - log cos c + 10. Hence, 10 + log sin c = 10 + log sin 76°52′ = 19-9884894 Remains, log sin a = log sin c = log sin 78°20′ = 9-9909338 log sin 83°56′ = 9-9975556 Here c is acute, because the given leg is less than 90°. Again, 10 + log tan c = 10 + log tan 76°52′ = 20-6320468 log tan 78 20′ = 10-6851149 log cos 27°45′ = 9-9469319 B is here acute, because a and care of like affection. Lastly, 10 + log cos a = 10 + log cos 7820' = 19-3058189 Remains, log cos c = log cos 76°52′ = 9-3564426 log cos b = log cos 27° 8' = 9-9493763 where b is less than 90°, because a and e both are so. Ex. 2. In a right-angled spherical triangle, denoted as above, are given a = 78°20′, в = 27°45′; to find the other sides and angle. Ans. b = 27°8′, c = 76°52′, c = 83°56'. Ex. 3. In a spherical triangle, with a a right angle, given b = 117°34′, c = 31°51'; to find the other parts. Ans. a = 113°55′, c = 28°51′, в = 104°8′. Ex. 4. Given b = 27°6', c = 76°52′; to find the other parts. Ans. a = 78°20′, в = 27°45′, c = 83°56′. Ex. 5. Given b = 42°12′, в = 48°; to find the other parts. Ans. a = 64°40'1⁄2, or its supplement, c = 54°44′, or its supplement, c = 64°35′, or its supplement. Ех. 6. Given в = 48°, с = 64°35'; required the other Ans. b = 42°12', c = 54°44′, a = 64°40′. Ex. 7. In the quadrantal triangle ABC, given the quadrantal side a = 90°, an adjacent angle c = 42°12′, and the opposite angle A = 64°40'; required the other parts of the triangle ? Ex. 8. In an oblique-angled spherical triangle are given the three sides, viz. a = 56°40', b = 83°13′, c = 114°30'; to find the angles. Here, by the fifth case of table 2, we have Or, 2 log sin a=log sin(8-6)+log sin (s-c)+ar. comp. log sin b+ar. comp.log.sinc: where s = a + b + c. log sin (8-6) = log sin 43584 = 9-8415749 log sin (sc) = log sin 12°41′ = 9-3418385 A.c.log sin b = a.c.log sin 83°13′ = 0.0030508 A.c.log sin c = a.c.log sin 114°30′ Sum of the four logs = 0.0409771 19-2274413 Half sum = log sin A = log sin 24°15′ = 9 6137206 Consequently the angle a is 48'31'. Then, by common analogy, As, sin a... To, sin A.. So is, sin b To, sin B ... sin 56°40′ .. log = 9-9219401 sin 48'31' ...log = 9.8745679 sin 83°13′ ...log = 9.9969492 sin 62'56' ...log = 9.9495770 ...log = 9.9590229 ...log = 9.9116507 And so is, sin c... sin 114'30' So that the remaining angles are, B=62°56′, and c=125°19′. 2dly. By way of comparison of methods, let us find the angle A, by the analogies of Napier, according to case 5 table 3. In order to which, suppose a perpendicular demitted from the angle c on the opposite side c. Then shall we have tan diff. seg of c = This, in logarithms, is tan (b+a).tan(b-a) 3 log tan (b+a) = log tan 69°56′ = 10-4375601 Subtract log tan c = log tan 57°15′ = 10-1916394 Therefore, since cos A tan 79°49′ × cotb: The sum, rejecting 10 from the index) = = = 9-8209-20 log cos A log cos 48°32′ The other two angles may be found as before. The pre. ference is, in this case, manifestly due to the former method. Ex. 9. In an oblique-angled spherical triangle, are given two sides, equal to 114°30′ and 56°40′ respectively, and the angle opposite the former equal to 125°20'; to find the other parts. Ans. Angles 48°30′ and 62°55′; side, 83°12′. Ex. 10. Given, in a spherical triangle, two angles, equal to 48°30′ and 125°20', and the side opposite the latter; to find the other parts. Ans. Side opposite first angle, 56°40'; other side, 83°12′; third angle, 62°54'. Ex. 11. Given two sides, equal 114°30′ and 56°40'; and their included angle 62°54: to find the rest. Ex. 12. Given two angles, 125°20′ and 48°30', and the side comprehended between them 83°12': to find the other parts. Er. 13. In a spherical triangle, the angles are 48°31′, 62°56′, and 125°20'; required the sides ? Ex. 14. Given two angles, 50°12′, and 58°8′; and a side opposite the former, 62242; to find the other parts. Ans. The third angle is either 130°54′33′′ or 156°16′32′′. Side betw. giv. angles, either 119°3′32′′ or 152214′14′′. Side opp. 58°8′, 72°12′13′′ or 100°47 37". either Ex. 15. The excess of the three angles of a triangle, measured on the earth's surface, above two right angles, is 1 second; what is its area, taking the earth's diameter at 79574 miles? Ans. 76-75299, or nearly 764 square miles. Ex. 16. Determine the solid angles of a regular pyramid with hexagonal base, the altitude of the pyramid being to each side of the base, as 2 to 1. Ans. Plane angle between each two lateral faces 125°22′35′′. between the base and each face 66°35′12′′. Solid angle at the vertex 89-60648) The max. angle 218-19367} being 1000. (60) ON GEODESIC OPERATIONS, AND THE FIGURE OF THE EARTH. SECTION I. General Account of this kind of Surveying. ART. 1. In the treatise on Land Surveying in the first volume of this Course of Mathematics, the directions were restricted to the necessary operations for surveying fields, farms, lordships, or at most counties; these being the only operations in which the generality of persons, who practise this kind of measurement, are likely to be engaged: but there are especial occasions when it is requisite to apply the principles of plane and spherical geometry, and the practices of surveying, to much more extensive portions of the earth's surface; and when of course much care and judgment are called into exercise, both with regard to the direction of the practical operations, and the management of the computations. The extensive processes which we are now about to consider, and which are characterised by the terms Geodesic Operations and Trigonometrical Surveying, are usually undertaken for the accomplishment of one of these three objects. 1. The finding the difference of longitude, between two moderately distant and noted meridians; as the meridians of the observatories at Greenwich and Oxford, or of those at Greenwich and Paris. 2. The accurate determination of the geogra. phical positions of the principal places, whether on the coast or inland, in an island or kingdom; with a view to give greater accuracy to maps, and to accommodate the navigator with the actual position, as to latitude and longitude, of the principal promontories, havens, and ports. These have, till lately, been desiderata, even in this country: the position of some important points, as the Lizard, not being known within seven minutes of a degree; and, until the publication of the Board of Ordnance maps, the best county maps being so erroneous, as in some cases to exhibit blunders of three miles in distances of less than twenty. 3. The measurement of a degree in various situations; and thence the determination of the figure and magnitude of the earth. When objects so important as these are to be attained, it is manifest that, in order to ensure the desirable degree of cor. rectness in the results, the instruments employed, the opera. tions performed, and the computations required, must each have the greatest possible degree of accuracy. Of these, he first depend on the artist; the second on the surveyor, or engineer, who conducts them; and the latter on the theorist and calculator: they are these last which will chiefly engage our attention in the present chapter. 2. In the determination of distances of many miles, whether for the survey of a kingdom, or for the measurement of a de. gree, the whole line intervening between two extreme points is not absolutely measured; for this, on account of the in. equalities of the earth's surface, would be always very difficult, and often impossible. But, a line of a few miles in length is very carefully measured on some plain, heath, or marsh, which is so nearly level as to facilitate the measurement of an actually horizontal line; and this line being assumed as the base of the operations, a variety of hills and elevated spots are selected, at which signals can be placed, suitably distant and visible one from another: the straight lines joining these points constitute a double series of triangles, of which the assumed base forms the first side; the angles of these, that is, the angles made at each station or signal staff, by two other signal staffs, are carefully measured by a theodolite, which is carried suc. cessively from one station to another. In such a series of tri. angles, care being always taken that one side is common to two of them, all the angles are known from the observations at the severa! stations; and a side of one of them being given, namely, that of the base measured, the sides of all the rest, as well as the distance from the first angle of the first triangle, to any part of the last triangle, may be found by the rules of trigo. nometry. And so, again, the bearing of any one of the sides, with respect to the meridian, being determined by observation, the bearings of any of the rest, with respect to the same meridian, will be known by computation. In these operations, it is always advisable, when circumstances will admit of it, to measure another base (called a base of verification) at or near the ulterior extremity of the series: for the length of this base, computed as one of the sides of the chain of triangles, compared with its length determined by actual admeasurement, will be a test of the accuracy of all the operations made in the series between the two bases. |