The determination of areas between known boundaries is accomplished in several ways, but the one commonly used consists in determining the bearings and lengths of the outside boundary lines. This is done by beginning at one corner and running around the area to the starting point, and observing the bearing and measuring the distance of each boundary line in the order in which it is encountered. To illustrate, reference will be made to the diagram, in which the lines A B C D E represent the boundary of a field of land whose area is required. Beginning at A, the surveyor proceeds around the field, ob taining by his compass the bearings, and by measurement with a surveyor's chain, the lengths, of the lines A B, B C, C D, D E, and E A, one after the other in the order named. This being done lie states these bearings and distances on paper, giving the figure ABODE shown by the sketch. Ile next draws a meridian line NS through the westernmost corner of his boundary, and draws the various perpendiculars shown by the broken lines. These perpendiculars represent what are known as the latitudes of the several lines; thus for line DC, dl) is the latitude, De is the departure, and (fD dC) is the double meridian distance. Knowing the bearings and lengths of the various courses by simple trigo nometrical calculation, the surveyor is enabled to calculate the latitudes, departures, and double meridian distances. This being known, the fol lowing rule is used for calculating the area: Twice the area of the figure is equal to the algebraic sum of the products of the double me ridian distances of the several courses into the corresponding latitudes, north latitude being reckoned positively and south latitude negative ly. Expressed symbolically, for the figure shown,
this rule is as follows: 2A = (bB cC) Be + (al fD) Dd— (b11) Ab — (aE) Aa— (aE+ fD) Eg. It may often happen that one boundary of a field is not a right line, but is an irregular curved line such as the course of a stream C F B instead of by the right line B C. The area of this field is then the area of A B C D E plus the area of C F B; the surveyor when measuring the line B C measures two or more perpendiculars such as yy, zz, and ww, known as offsets and extend ing from the line B C to the stream. These per pendiculars divide the area C F B into a number of small areas each of which approximates close ly a simple trapezoid or triangle whose area is easily determined and the sum of whose areas is the area of the total tract. There is of course a slight error in this method, but it can be made so small by an experienced surveyor as to be practically negligible. Systematic methods have been devised for making field notes so as to in clude a register of the topographical features incident to the chain lines, the relation of these lines one to another, and a description of monu ments sufficient to identify the lines in the field. A column is usually taken in the centre of each page, of the field-book to represent the chain line, and the necessary notes are entered on either side. These notes if properly kept enable the surveyor to execute a topographical map of the territory surveyed, and to calculate its area.