Let the angle A=32° 15′, angle B=114° 24', and the side AB=98 perches. Hence the side AC=162.34 perches. To find the Area of a Triangle. Multiply any two sides and the natural sine of their contained angle continually together, and half the resulting product will be the required area. Then 162.34×98×.53361=8489.3722452. 8489.3722452+2=4244.6861226 square perches. To survey a Polygon by the Chain and Theodolite. If the polygon have five sides, measure four of the angles by the theodolite, and three of the sides by the chain; then find the sides and angles of every one of the three triangles into which the polygon may be divided, thence the area of the triangles separately; and the sum of the several areas will give the contents of the polygon. In the four-sided figure, E D the polygon. For, joining AC, AD, the figure is divided into B C three triangles, the sides and angles of each of which can be found by trigonometry; and having the sides and angles of every triangle in the figure, the area of each may be found respectively. The aggregate of all the triangles is the required area. It is necessary to know that when the polygon has n sides, to find the area it is only requisite to measure (n-2) sides, together with (n-1) angles; (n-1) sides, together with (n-2) angles; or n sides, together with (n-3) angles. It may, however, be advisable to measure as many of the sides and angles as are accessible, that the admeasurement might serve as a proof of the accuracy of the results by calculation. If all the internal angles of the polygon be measured, their sum ought to be equal to twice as many right angles as the figure has sides, wanting four right angles. When there is an angle that bends inwards, and when you measure the external angle, which is less than two right angles, deduct it from four right angles, and the remainder will be the internal angle. In measuring the sides, due allowance should be made in chaining hilly ground, otherwise the polygon would not close. In taking the angles round the polygon, it is not necessary to set the instrument to 360°, except in the first angle; but the preceding reading of the instrument must be always deducted from the subsequent one, and the difference will be the measure of the last angle. This practice has two advantages to recommend itthe time lost in setting the instrument to 360° at every angle is saved; and after having gone round the figure, the instrument itself affords a proof of the correctness of the angular part of the work-zero, on one of the verniers, always coinciding with 360°, if the angles are correctly taken. In making a trigonometrical survey of a large estate, it is sometimes difficult to find stations so as to do all by triangles. When such a difficulty presents itself, survey that particular part by going round it, which will be sufficiently accurate, provided the sides of the polygon occupy even ground, which can be correctly chained; and let these sides serve as the bases of so many new triangles. This method is never practised in a very extensive survey, such as that of a kingdom, barony, or even a parish-all should be triangulated when the survey is very extensive. On the Protracting and Plotting of Surveys. Every surveyor should calculate and plot his work daily, if possible; otherwise, if left undone for a long time, and an error should have been committed in any of the preceding days' work, it might be a task of great labour and difficulty to correct it: but when calculated and plotted daily, any error that may have been committed is easily detected and corrected. In a very extensive trigonometrical survey, the sides of every triangle should be calculated and plotted as soon as possible. We would recommend to lay down every triangle from its sides, using the compasses, as being much less liable to error than the protractor. When a triangle is delineated from its angles, by the use of the protractor, the smallest error in the angle will cause a very considerable error in the length of the sides. The method of laying down a survey from the sides has been already given; therefore we shall confine ourselves here to the use of the protractor, which can be employed only when the angles of the survey are taken by means of some angular instrument, such as the theodolite or sextant. To protract with any degree of accuracy, a circular protractor, divided with the same degree of minuteness as the instrument employed in the field, is indispensable. The most convenient is one of about five or six inches diameter, divided on silver, with two verniers and a tangent screw. To use this instrument, it must be so placed on the paper that its centre, which is the intersection of two lines passing through a circular disc of glass, covers or exactly coincides with the angular point. The divisions at 360° and 180° must be on the line passing through the point over which is the centre of the protractor. When the instrument is placed in this position, press it slightly against the paper, so as to prevent it from moving; then by means of a tangent screw, or rack and pinion (the former is better), the vernier is brought to the required angle. This instrument sometimes goes out of adjustment. When this happens, it is corrected by means of two small screws, on which two branches attached to the instrument, play. The centre and two pins at the extremities of the arms must, when joined, form one right line, otherwise some derangement exists, which requires correction before laying down the angle. When extreme accuracy is not required, a semicircular protractor would be found very convenient and expeditious. To lay down angles with it, set its centre over the angular point, and make the line, with which the angle is to be made, pass through the two points on the protractor marked 180°; then count off the required angle on the edge of the protractor, and join where the reckoning ends with the centre Angles may be very correctly protracted by means of a table of natural sines. Thus, if the given angle be denoted by A, look in the table for the natural sine of A; then double this will be the chord of the angle A, to the radius 1: hence 200 times this chord, or 400 times the sine of A, will be the chord of the angle A, to the radius 200. Then measure the distance 200 from the angular point, and on this line describe a triangle, having 200 for the side terminating at the angular point, and 400 × sin. A for the side opposite the angle. The triangle thus constructed will have an angle equal to the given one. Before you begin the actual survey, it would be well to travel over the property to be surveyed, in order to make yourself well acquainted with the ground, and ascertain the most favourable position for your base line, which should be always laid out on the most level part of the survey, so as to admit of correct measurement. Having determined the position of your base line, measure it correctly, and at its extremities measure, with the theodolite, the angles it forms with two lines joining its extreme points and a station taken at as great a distance from the base as the nature of the ground will admit.* Draw a sketch of the triangle thus formed in your field book, marking the angular * Let these lines be as nearly equal to the base as the ground will permit. |