(159) The field notes from which the area is to be computed may be imperfect. There may be obstacles which prevent the measuring of one side, or the notes may be defaced so as to render some of the numbers illegible. If the bearings and lengths of all the sides of a field except one are given, the remaining side may easily be found by calculation. For the difference between the sum of the northings and the sum of the southings of the given sides will be the northing or southing of the remaining side; and the difference between the sum of the eastings and the sum of the westings of the given sides will be the easting or westing of the remaining side. Having, then, the difference of latitude and departure of the required side, its length and direction are easily found by Trigonometry (Art. 47). Ex. Given the bearings and lengths of the sides of a tract of land as follows: Required the bearing and distance of the fourth side. Ans., S. 15° 33' E., distance 8.62 chains. B (160.) There is another method of finding the area of a field which may be practiced when great accuracy is not required It consists in first drawing a plan of the field, as in Art. 153 then dividing the field into triangles by diagonal lines, and measuring the bases and perpendiculars of the triangles upon the same scale of equal parts by which the plot was drawn. Thas, if we take Ex. 1, and draw the diagonals AC, AD, the field will be divided into three triangles, whose area is easily found when we know A E H F G: the diagonals AC, AD, and the perpendiculars BF, DG, EH. The diagonal AC is found by measurement upon the scale of equal parts to be 16.87; the diagonal AD is 15.67; the perpenlicular BF is 6.30; DG is 4.92; and EH is 6.42. the triangle ABC=16.87×3.15= 53.14 66 ADC 16.87×2.46= 41.50 66 Hence =144.94 sq. chains. This method of finding the area of a field is very expeditious, and when the plot is carefully drawn, may afford results sufficiently precise for many purposes. (161.) To survey an irregular boundary by means of off sets. When the boundaries of a field are very irregular, like a river or lake shore, it is generally best to run a straight line, coming as near as is convenient to the true boundary, and measure the perpendicular distances of the prominent points of the boundary from this line. C Let ABCD be a piece of land to be surveyed; the land being bounded on the east by a lake, and on the west by a creek We select stations A, B, C, D, so as to form a polygon which shall embrace most of the proposed field, and find its area. We then measure perpendiculars aa', bb', cc', &c., as also the distances Aa, ab, bc, &c. Then, considering the spaces Aaa', abb'a', &c., as triangles or trapezoids, their area may be computed; and, adding these areas to the figure ABCD, we shall obtain the area of the proposed field nearly. C C' b Ъ α D (162.) To determine the bearing and distance from one point to another by means of a series of triangles. When it is required to find the distance between two points remote from each other, we form a series of triangles such that the first and second triangles may have one side in common; the second and third, also, one side in common; the third and fourth, &c. We then measure one side of the first triangle for a base line, and all the angles in each of the triangles. These data are sufficient to determine the length of the sides of each triangle; for in the first triangle we have one side and the angles to find the other sides. When these are found, we shall have one side and all the angles of a second triangle to find the other sides. In the same manner we may calculate the dimensions of the third triangle, the fourth, and so on. We shall illustrate this method by an example taken from the Coast Survey of the United States. F B The object here is to make a survey of Chesapeake Bay and its vicinity; to determine with the utmost precision the position of the most prominent points of the country, to which subordinate points may be referred, and thus a perfect map of the country be obtained. Accordingly, a level spot of ground was selected on the eastern side of the bay, on Kent Island, where a base line, AB, of more than five miles in length, was measured with every precaution. A station, C, was also selected upon the other side of the bay, near Annapolis, so situated that it was visi ble from A and B. The three angles. of the triangle ABC were then measured with a large theodolite, after which the length of BC may be computed. A fourth station, D, is now taken on the western shore of the bay, visible from C and B, and all the angles of the triangle BCD are measured, when the line BD can be computed. A fifth station, E, is now taken on an island near the eastern shore, visible both from B and D, and all the angles of the triangle BDE are measured, when DE can be computed. Also, all the angles of the triangle DEF are measured, and EF is computed. Then all the angles of the triangle EFG are meas I H G ured, and FG is computed. So, also, all the angles of the tri. angle FGH are measured, and GH is computed; and thus a chain of triangles may be extended along the entire coast of the United States. To test the accuracy of the work, it is common to measure a side in one of the triangles remote from the first base, and compare its measured length with that deduced by computation from the entire series of triangles. This line is called a base of verification. Such a base has been measured on Long Island; and, indeed, several bases have been measured on different points of the coast. These are all connected by a triangulation, and thus the length of a side in any triangle may be deduced from more than one base line, and the agreement of these results is a test of the accuracy of the entire work. Thus the length of one of the sides of a triangle which was twelve miles, as deduced from the Kent Island base, differed only twenty inches from that derived from the Long Island base, distant two hundred miles. The superiority of this method of surveying arises from the circumstance that it is necessary to measure but a small number of base lines along a coast of a thousand or more miles in extent; and for these the most favorable ground may be selected any where in the vicinity of the system of triangles. All the other quantities measured are angles; and the precision of these measurements is not at all impaired by the inequalities of the surface of the ground. Indeed, mountainous countries afford peculiar facilities for a trigonometrical survey, since they present heights of ground visible to a great distance, and thus permit the formation of triangles of very large dimensions. (163.) To divide an irregular piece of land into any two given parts. We first run a line, by estimation, as near as may be to th required division line, and compute the area thus cut off. It this is found too large or too small, we add or subtract a triangle, or some other figure, as the case may require. Suppose it is required to divide the field ABCDEFGHI into two equal parts, by a line IL, running from the corner I to the opposite side CD. We first draw a line from I to D, and compute the area of the part DEFGHI; and, knowing the area B K L D of the entire field, we learn the area which must be contained in the triangle DIL, in order that IL may divide the field into two equal parts. Having the bearings and distances of the sides DE, EF, &c., we can compute the bearing and distance of DI. Thus the an- A gle IDK is known; and, having the hypothenuse ID, we can compute the length of the perpendicular IK let fall on CD. Now the base of a triangle must be equal to its area divided by half the altitude. Hence, if we divide the I E H G area of the triangle DIL by half of IK, it will give DL. F In a similar manner we might proceed if it was required to divide a tract of land into any two given parts. Variation of the Needle. (164.) The line indicated by a magnetic needle, when freely supported and allowed to come to a state of rest, is called the magnetic meridian. This does not generally coincide with the astronomical meridian, which is a true north and south line. The angle which the magnetic meridian makes with the true meridian is called the variation of the needle, and is said to be east or west, according as the north end of the needle points east or west of the north pole of the earth. The variation of the needle is different in different parts of the earth. In some parts of the United States it is 10° west, and in others 10° east, while at other places the variation has every intermediate value. Even at the same place, the variation does not remain constant for any length of time. Hence it is necessary frequently to determine the amount of the variation, which is easily done when we know the position of the true meridian. The latter can only be determined by astronornical observations. The best method is by observations of the pole star. If this star were exactly at the pole, it would always be on the meridian; but, being at a distance of about |