This method of determining the side c, requires the previous determination of one of the angles A, B, and is therefore, when it is only required to calculate the side c, less convenient than the following method, by which the side c is calculated immediately from the two sides and angle which are given. (57.) By Art. (55.) we have c = √ a2 + b2 This expression, however, is not adapted to logarithmic computation; we must, therefore, give it a more convenient form for this purpose. c = √ a2 2 ab + b2 + 2 ab (1-cos. C) *This is an instance in which the angle is too small to be determined with accuracy from its logarithmic tangent by means of a table calculated for minutes only. The above value is found from a table calculated for each second. Where is determined from the expression C 2 log, tan. = log; 4 + log; a log b + 2 log, sin. 2 And we have or + 2. A. C. log, (a - b) — 20 log Clog (a - b) + log, sec. 0 We will solve the last example by this method. To find the angle ✪ we have log, 4 0.6020600 (58.) In the first of these solutions we have to determine a small angle, and in the second an angle nearly 90° from their logarithmic tangents. With a table in which these tangents are calculated for every second, this may be done with accuracy; but if our table be computed for minutes only, this cannot be done conveniently, because the method of proportional parts will not apply. It may, therefore, sometimes be desirable to avoid this difficulty, which may be done as follows: log, clog, a + b + log; cos. 0-10 By using this formula we should avoid, in the above example, the objection above-mentioned. Any of the above formulæ would be theoretically sufficient for the complete solution of the case we have been considering; but the observation we have made on the last example, will show that different formulæ will be preferable in different cases. It is the business of the analyst to investigate these formulæ, and of the computist to select that which is most applicable to the particular circumstances of his problem. In this selection he will be guided by such considerations as the above, or others which experience will suggest. SECTION VIII. General Account of the Method of observing Angles.-Principles on which Observations are corrected.—Examples of the Measurement of Heights and Distances. In this section we shall explain the methods of ascertaining heights and distances by means of trigonometry. In the operations necessary for this purpose, the length of at least one line must be ascertained by actual admeasurement, and the magnitude of certain angles by actual observation. (59.) The former may be effected without difficulty, when no great accuracy is required, by means of a string, chain, or pole of given length; when extreme nicety is essential, as in extensive trigonometrical surveys, many precautions must be taken, which, however, it is not now our object particularly to point out. We shall proceed to show generally how the angular distance of two points, as seen from any proposed station, may be observed. (60.) Let P Q be the two points, C the position of the eye of the observer, the plane passing through these points being supposed to coincide with the plane of the paper. Suppose a circular rim divided into degrees, minutes, &c., to have its centre at C, and its plane in that passing through P and Q; then let a line C D, moveable round C, be made to coincide with C A, a fixed radius of the instrument (A being the point where the graduation begins), and then the rim turned round an axis through C, and perpendicular to its plane, till C D be directed to P. Let CD be then moved till it be directed to Q, the rim remaining fixed; the graduation at K (supposed to proceed in direction A K) will give the number of degrees, minutes, &c., in the angle P C Q. (61.) If, instead of moving C D while the rim remains fixed, CD be supposed to remain fixed, and the rim be made to revolve till C A be directed to Q, and C K be directed to P, the graduation of A K will determine the magnitude of the angle P C Q. Nothing can appear more simple than this operation; but in cases of great nicety, where, perhaps, an error of a few seconds would be fatal to the required accuracy of our calculations, it will be easily conceived that many precautions become necessary to avoid such errors. The different instruments for observing angles are, in fact, so many contrivances for performing the operation we have described with greater facility and accuracy than the simple apparatus above-mentioned would admit of. They all possess the graduated rim, the perfect circularity and exact graduation of which are most essential, and though this graduating is done in the present day with wonderful precision, it is a matter of too much practical difficulty not to admit of some correction by methods the principles of which are independent of mere mechanical skill. (62.) The most important is the principle of repetition, by which any error in the graduation and in the reading off of the numbers of degrees, &c., to which a single observation is liable, is divided among many repeated observations, so that by a sufficient number of repetitions the required angle, as far as it depends on the above circumstances, can be obtained to any assignable degree of accuracy. This is easily explained. When the moveable line C D has been first directed to P and then to Q, as above described, let it be fixed to the rim at K, and then be made to move by the motion of the rim about its axis through C till it comes again into the direction C P. A will then have come to K,, and L into the direction C Q. Let C D be unfastened from the rim, and brought again into the direction C Q, and therefore passing through L; the graduation corresponding to this position of C D will denote twice the required angle PCQ; and if s" be the error arising from the graduation or reading off after the second observation, and N denote the angle read off in degrees, minutes, &c. 2.PCQN+8" N or the error in the observed value of PCQ will be let n be the number of observations, N the degrees, &c., read off, and suppose the circular rim to have made besides m complete revolutions. Then the error being s", n.PCQ m 360° +N+ N 360° + + and the error in the observed value of PC Q will be only ገ. It will be remarked that the reading off is only once necessary, i.e., after the last observation. It is in the application of this principle that the peculiarity of the repeating circle consists. (63.) Another principle of correction for erroneous graduation and reading off, consists in taking the mean of several readings off on different parts of the rim as the true reading. In this case the observation must be made as described in (61.) Thus suppose the point A of the rim to revolve till it come to K, A' coming to K', the angle PC Q may then be read off both at C' and K'. In the same manner it may be read off at any other parts of the rim. Suppose N, N,... N, to be the several readings off, the true value of the angle is taken to be N+N+-+N, n N2 (64.) The most common instruments for measuring angles in terrestrial observation are the quadrant, repeating circle, and theodolite. A full description of these instruments does not come within our immediate object, which is merely to give a general notion of the manner in which the magnitude of angles may be observed, and to point out the principles of the methods by which observations may be corrected. We shall merely observe at present that the quadrant is simply a quarter of such a circle as that above described, bounded by the two fixed radii C A, C B, and gra duated from 0 to 90°. It has the disadvantage of offering no practical facility for the application of the above principles of correction. The repeating circle consists of a circle like the above, mounted in a manner convenient for its use. In the theodolite, the graduated circle is horizontal when properly adjusted, the instrument being intended for the observation of horizontal angles only. With the quadrant, or repeating circle, angles in any plane may be observed. (65.) In order to ascertain the height of an object, it is generally much the most convenient to observe its angular altitude, or the angle which a line from the object to the observer's eye makes with a horizontal line. This horizontal line being seldom exactly known, it is usual to make use of a plumb line, which determines the position of a line perpendicular to the horizon. Thus, if P be the object, let the side A C of the quadrant be directed to P; then if CD be the plumb line, the graduation of B K will give the angle PCG, CG being horizontal. K B Having thus given the student a general notion of the methods by which the magnitude of angles may be observed, we shall lay before him a few examples which may enable him to judge of the best method of proceeding in any particular cases that may present themselves. Examples. To find the height and distance of an inaccessible object on a horizontal plane. Let the angle CAP be observed at the station A = α. Measure A B in the direction of the object, and let it =a, and then observe the angle CBP = B. Let the height CPy and the distance B C = a. Now PB sin. PAB = sin. a AB sin. APB sin. (6-a) B P and y sin. (B-a)' PB. sin.ẞ sin. a. sin. B sin. (B-a) log, y = log a + log, sin. a + log, sin. Blog, sin. (ẞ—a) — 10; also = PB cos. B sin. (B-a) .. log; x = log; a + log, sin. a + log, cos. Blog, sin. (6-a) — 10. |