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 sin (Es—b) . sin (3s—c), sin b. sin c sin AV Or, log sin A log sin (s—b)+log sin (sc)+ar. comp. log sin bar. comp. log sin c: where s a+b+c. 9.8415749 log sin (sb) = log sin 43°58' = 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′ = 0'0409771 19-2274413 Sum of the four logs = 9.6137206 Then, by the common analogy, · • · sin 56°40′ log = 9.9219401 9.8745679 sin 83°13′ · • • ... Aş, sin a To, sin A .. log Sois, sin b log To, sin B sin 62°56′ log = 9.9495770 And so is, sin c sin 114°30′ log = 9.9590229 To, sin c sin 125°19′ 9-9116507. ..log 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 (b+a) . tan †(b—a) tan have tan diff. seg. of c = tan c This in logarithms, is log tan (b+-a) log tan (b-a) = = log tan 69°56′ = 10·4375601 Their sum 198103420 log tan 57°15′ =101916394 = Subtract log tan c =3 9.6187026 Therefore, Therefore, since cos A = tan 79°49' X cot b: 9.8209820 log cos A = log cos 48°32' The other two angles may be found as before. The preference 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° 36' 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. Ex. 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, 62°42' ; to find the other parts. Ans. The third angle is either 130°56' or 156°14'. Side betw. giv. angles, either 119°4' or 152° 14'. either 79°12' or 100°48'. 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 7957_miles? Ans. 76.75299, or nearly 76% square miles. E.r. 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 l. Ans. Plane angle between each two lateral faces 126°52'11"1. between the base and each face 66°35'12" Solid angle at the vertex 114.497687 The max. angle Each ditto at the base 222.34298 S being 1000. 58 8', 2. CHAPTER CHAPTER V. 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 second 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 geographical positions of the principal places, whether on the coast or inland, in an island or kingdom; with a view to give greater accuracy to inaps, 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 the best county maps being so erroneous, as in some cases to exhibit blunders of three miles in distances of less than twenty. maps, 3, The a 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 correctness in the results, the instruments employed, the operations performed, and the computations required, must each have the greatest possible degree of accuracy. Of these, the 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 degree, the whole line intervening between two extreme points is not absolutely measured ; for this, on account of the inequalities 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 suecessively from one station to another. In such a series of triangles, care being always taken that one side is common to two of them, all the angles are known from the observations at the several 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 trigonometry. 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 extremnity 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 ail the operations made in the series between the two bases, w R 3. Now, in 'every series of triangles, where each angle is to be ascertained with the same instrument, they should, as nearly as circumstances will permit, be equilateral. For; if it were possible to choose the stations in such manner, that each angle should be exactly 60 degrees; then, the half number of triangles in the series, multiplied into the length of one side of either triangle, would, as in the annexed figure, give at once the total distance; and then also, not only the sides of the scale or ladder, constituted by this series of triangles, would be perfectly parallel, but the diagonal steps, marking the progress from one extremity to the other, would be alternately parallel throughout the whole length. Here too, the first side might be found by a base crossing it perpendicularly of about half its length, as at h; and the last side verified by another such base, R, at the opposite extremity. If the respective sides of the series of triangles were 12 or 18 miles, these bases might advantageously be between 6 and 7, or between 9 and 10 miles respectively; according to circumstances. It may also be remarked, (and the reason of it will be seen in the next section) that whenever only two angles of a triangle can be actually observed, each of them should be as nearly as possible 45°, or the sum of them about 90°; for the less the third or computed angle differs from 90°, the less probability there will be of any considerable error. See prob. 1 sect. 2, of this chapter. 4. The student may obtain a general notion of the method, employed in measuring an arc of the meridian, from the following brief sketch and introductory illustrations. The earth, it is well known, is nearly spherical. It any be either an ellipsoid of revolution, that is, a body formed by the rotation of an ellipse, the raiio of whose axes is nearly that of equality, on one of those axes; or it may approach nearly to the form of such an ellipsoid or spheroid, while its deviations from that form, though small relatively, may still be sufficiently great in themselves, to prevent its being called a spheroid with much more propriety than it is called a sphere. One of the methods made use of to determine this point, is by means of extensive Geodesic operations. The earth however, be its exact form what it may, is a planet, which not only revolves in an orbit, but turns upon an axis. Now, if we conceive a plane to pass through the axis of rotation of the earth, and through the zenith of any place on its surface, this plane, if prolonged to the limits of VOL. III. I the |