for uniting any detached underground survey to a surface survey, for the purpose of ascertaining the extent of workings in regard to adjoining property; also to determine the height and distance of objects relatively to one another. In ordinary practice, where minutes and seconds of an angle are recorded as well as degrees, a table of logarithms is indispensable for the solution of trigonometrical problems. The signs of the trigonometrical expressions are also necessary to be understood. Suppose NC H and LC M to be two lines at right angles to one another, and fixed in position, and that the radius C H traverses the circle in the contrary direction to the hands of a watch, as indicated by the arrows; in doing so it passes through the sectors of the circle marked 1, 2, 3, 4, 5, 6, 7, 8 in the figure (page 42). Assuming H to be a point upon the circumference of the circle at zero on a scale of degrees, if radius C H = a, and C F = x and F H = b, then x-a-b. Now, so long as b<a, x is +, and Flies to right of C; but if b>a, x is and lies to the left of C. Hence any line measured along N H or parallel to it, is said to be a positive line on the right of C, cr a negative line on the left of C, that is along C N or parallel to it, and the symbol + or represents the direction. The radius lying in the direction of neither the vertical nor horizontal direction cannot change its sign, and is always reckoned positive. Hence the sign of the trigonometrical expressions will be as follows:In 1, 2, 3, 4 the sine is + In 5, 6, 7, 8 the sine is and these signs determine the signs of the remaining trigonometrical ratios of tangent and cotangent, secant and cosecant. In dealing with an angle a, the expression cos a, &c., &c., will be independent of the magnitude of the revolving line and depend only on the absolute inclination of the two lines. Since cos a is never greater than unity, vers a is always + and its greatest value is when a becomes 180° when cos I and vers a = 2. a= Dividing the circle into eight parts as shown in the figure It is useful to remember also that the 2 sin of larger angle (cos of smaller angle) 2 sin of smaller angle (cos of larger angle) 2 cos of one angle (cos of another angle) = cos of sum + cos of diff. 2 sin of one angle (sin of another angle) If A < 45° cos A> sin A If A> 45° and <90° sin A> cos A. The student desirous of working out one or two practical examples in the application of plane trigonometry to surveying, will find the following questions useful to test his proficiency. (1.) Having measured B A up a slope, and taken the angles D BA, BAD, C BA, CA B, show how to calculate the lengths of B C, B D, and C D. (2.) Two men are surveying when each is at a distance of 300 yards from a flagstaff; the one finds the angle subtended by the position of his companion and the staff to be 32° 45'. Find how far they are apart. (3.) In walking towards a flagstaff, a surveyor found the angle of elevation of its top to be 2° 19′ 13′′ at one milestone, and after proceeding to the next milestone, the angle of elevation was 3° 28′ 49′′. How much farther should he have to walk before he reached it, assuming they were all on the same level? (4.) Wishing to know the height of a certain house standing on the summit of a hill of uniform slope, a surveyor descended the hill 40 feet and then found that the height subtended an angle of 34° 18′ 19′′, but upon descending a farther distance of 20 feet, he found this angle had become 19° 14' 32". Find the height of the house. The student who can work out these problems, and also is able to follow the trigonometry involved in the formulæ for the setting out of curves, given on pages 203-225, will be able to deal with any problem in plane trigonometry likely to come before him as a surveyor. CHAPTER VI. FIELD-BOOK. THE field-book of a survey should contain upon its first page the surveyor's name conspicuously written, to provide for its chance of recovery, if lost, an index to the base lines, not necessarily drawn to scale, but sketched so as to indicate the relative position of the main base lines, and the direction in which they are chained. Thus the index sketch of the base lines used in plotting a survey made from chain measurements only is represented in the accompanying diagram (pages 47-49). The base lines are numbered, and an arrow upon each line shows the direction in which they were measured, in the field or upon the ground. Pencil entries are preferable to the use of ink in the field, because the ink will run in wet weather. No note should be left to memory, but should be written or booked at once. The surveyor should always carry a piece of india-rubber in his pocket. Few can sketch correctly, as well as clearly, without the use of india-rubber. Such corrections would mean erasing if pen and ink were employed. Measurements should be written small but clear. There is no occasion to chain a base line twice in order to prove its accuracy. The tie-line is the best proof of correct measurement, and is founded upon a proposition of Euclid (book 1., prop. 7), that upon the same line, and upon the same side of it, there cannot be two different triangles, having their sides terminating at the same extremity of the base, equal to one another. Thus the line marked 7 in the diagram, proceeding from a point in the line A B to the intersection of lines marked 3 and 6, forms a tie to the triangle formed by the lines marked 1, 3, and 6 respectively. Scarcely any two surveyors set down their field notes exactly in the same manner. Usually, however, the main terminal stations are lettered in the field-book, but not the |