ca; also it will be observed that BC" = 180° BC= 180° BC= 180° - a', as previously appeared from the calculation. If BC is less than BA, then C' would fall between B and A, and the above construction would be no longer possible. Hence if c> a, the case is not ambiguous. (15.) To solve the Sixth Case of Oblique-Angled Triangles. In this case we have given A,B,C. We shall find a from formula (11), and b and c from similar formulas. This case, however, never occurs in any of the practical applications of spherical trigonometry. On the Solution of Quadrantal Triangles. A quadrantal triangle (Spherical Geometry, def. xiv. p. 260) has one side of 90°, and .. the corresponding polar triangle is a right-angled triangle. From this consideration it would be easy to modify Napier's Rule to suit the case of the quadrantal triangle. In practice, however, it is better to treat them as oblique triangles--on doing so it will be found in practice that the circumstance of one side being equal to 90° will introduce important simplifications. ON THE FORMULAS PECULIAR TO GEODETICAL OPERATIONS. We have already stated in general terms that the science of spherical trigonometry finds one of its applications in Geodesy. It is to be observed that this application possesses some peculiarities in consequence of the sides of the triangles employed in a survey, on even the largest scale, being small compared with the radius of the earth, and consequently small when estimated in degrees or minutes; whereas, in astronomy, there is no limitation imposed on the magnitudes of the sides of the triangles employed in that science; our object in the present article is to explain concisely the results of this limitation, and to deduce certain formulas depending on it. (16.) To state the Object of a Trigonometrical Survey of a Country. F B The object of the survey is, (1) to fix accurately the relative positions of certain chief points in the country, so as to lay them down on a map; and (2), having fixed these chief points, then by means of subsidiary operations to lay down in detail all the minor features of the country, its roads, rivers, towns, hamlets, &c. The accompanying figure will be sufficient to illustrate this matter for our present purpose, which is from an actual survey. A, is a place called Ruckinge, B, High Nook, C, Allington, D, Lydd, E, Fairlight Down, and F, Tenterden. The line AB, is measured very accurately, and is called the base line; and then the angles CAB, ABC, are measured; from these data, AC, E Fig. 8. and CB, can be calculated; then CB being known, the angles DCB, and CBD, can be measured, and thus CD, and DB, be determined; and this operation continued for any number of triangles whatever. It is usual in the larger triangles to measure all the three angles of any triangle, and not merely the two at the base; this is done with a view of keeping a check upon the various errors to which all observations are liable. When the triangulating has been continued for some distance, it is necessary to compare the calculated length of a line that has been fixed upon, and then measure it; the coincidence of the two results is a verification of all previous measurements and calculations; hence such a line is called a base of verification. It is usual to choose stations that are from ten to twenty miles apart; also it is usual to choose for a base line a line of about four or five miles long. In late French surveys only two bases of verification have been used. The accuracy attainable in practice will be appreciated when the fact is stated that, in some English bases of verification, of four or five miles long, the computed and measured lengths have differed only by one or two inches. The operations of a trigonometrical survey are then two,—(1) the measurement of base lines, (2) the measurements of angles. We will proceed to consider each of these. (17.) The Measurement of a Base Line. A space of open ground which is nearly level must be chosen, the line to be measured being indicated by stations and stages erected, if necessary, to secure the horizontality of the base; the measure may be made by rods of glass, or steel, proper corrections being applied for temperature; a more convenient contrivance for securing accuracy in the measures has been devised of late for the Irish Survey, it is of the following kind : A B different expansions for the same temperature; suppose then that AB, for a change in temperature, becomes ab, while CD becomes cd, then Ap will assume the position Ар δα ap, and let Aa = δα Cc = db, then if Cp is taken so long that then p Cp sb will not be changed by the change of temperature; since within very large limits the δα expansion of a metal is proportional to the increase of temperature, and .. is бо constant; the same arrangement being made at the end, BD, the distance pq, will not be affected by change of temperature. (18.) To Correct for want of Straightness in the Base Line. The nature of the ground may be such as to render it impossible to measure a perfectly straight base line of sufficient length. This was the case in some of the French surveys, where the actual measurement was of two straight lines inclined to each other at an angle of very nearly 180°. C Fig. 10. b. CB a. and BCN B = a2 + b2 + 2 ab cos. 02 1 if we omit 0" 2 (a + b)2 { (a + b) { 1 − 2 (a + ab02 if we omit ef .. AB = a + b ab02 The correction to be applied to the sum of a and b, to obtain the true distance, AB. (19.) To measure the Angles of the Triangles of a Survey. Any two of the three angular points of one of these triangles is rarely in a horizontal plane passing through the third; the angle required is, of course, such a horizontal angle. Now the angles are measured either by a theodolite, or by a repeating circle-in the case of the former instrument, the vertical elevation of each object is observed, and the horizontal angle between them-that is to say, if ABC Fig. (D) are the stations, ANM the horizontal plane through A, BN and CM perpendiculars from B and C on AMN, then by the theodolite we observe the angles BAN, CAM, and MAN; the last is the horizontal angle required. The theodolite is the instrument that has been used in the English surveys. But if a repeating circle is employed, the angle BAC is the one observed; and it is necessary to deduce from this the horizontal angle MAN. The repeating circle is the instrument used in the French surveys. (20). To determine the Correction for reducing an Angle to the Horizon. Let ABC M N be the same as in last article; with centre A and any radius describe a sphere, which meets the lines AB AC, AN, AM in p, q, n, m, respectively. pn and mq, if joined by great circles, clearly meet in Z vertically over A, since the circles must be perpendicular to the horizontal plane. Then mn, or the angle mZn (Spherical Geometry, prop. v. cor. 4) is the angle required, and for its determination we have given pq = A, pn h, qm determine SA. = =h'. Suppose mZn = A + SA, then our object is to Now we will suppose dA so small that we can omit SA2... and h and l' so small that we can omit every power and product higher than h2, hh', and h". Hence, whether we use the theodolite, or the repeating circle, we obtain the same end, namely, the determination of the horizontal angle. It is plain that the MATHEMATICAL SCIENCES.-No. XIV. 2 E horizontal angle being determined at each of the stations, these are the true angles, supposing the triangle spherical; and the triangle so determined is spherical. We may proceed with the triangle thus obtained in either of the following ways: (21). To Explain the Methods of Treating the Triangles thus Obtained. (a.) Delambre's Method. It is supposed that we have tables of the kind described in Art. 72, Plane Trigonometry. Now we know a, the length of a side in feet, hence sin. a α subtended at the centre by a is known, and hence log. suppose the table to give us this for every value of a adding log. a we obtain log. sin. a. Now from formula (1). L sin. b L sin. a + L sin. Bar: c: L sin. A obtained. (b.) By the Method of the Chordat Triangle. in α the angle sin. A hence 10, from whence b can be Let ABC be a given triangle, the sides of which are a, b, c, and angles A, B, C, join AB, BC, CD, by straight lines, then the plane triangle formed by these chords is called the chordal triangle. Let CSC be the angle of the chordal triangle corresponding to the angle C of the spherical triangle. Now, by Plane Trigon. art. 37, 2 (chord BC) (chord CA) cos. (C — 8C) = (chord BC)2 + (chord CA)2 (chord AB)2 |