equal. (Euc. 37. 1.) Taking from each the common part DBH, there remains BGH equal to DCH. If then the triangle DCH be thrown out of the plot, and BGH be added, we shall have the five-sided figure AGDEF equal to the sixsided figure ABCDEF. In the same manner, the line EL may be substituted for the two sides AF and EF; and then DM, for EL and ED. This will reduce the whole to the triangle MGD, which is equal to the original figure. The area of the triangle may then be found by multiplying its base into half its height; and this will be the contents of the field. In practice, it will not be necessary actually to draw the parallel lines BD, GC, &c. It will be sufficient to lay the edge of a rule on C, so as to be parallel to a line supposed to pass through B and D, and to mark the point of intersection G. 126. If after a field has been surveyed, and the area computed, the chain is found to be too long or too short; the true contents may be found, upon the principle that similar figures are to each other as the squares of their homologous sides. (Euc. 20. 6.) The proportion may be stated thus ; As the square of the true chain, to the square of that by which the survey was made; So is the computed area of the field, to the true area. Ex. If the area of a field measured by a chain 66.4 feet long, be computed to be 32.6036 acres; what is the area as measured by the true chain 66 feet long? Ans. 33 acres. 127. A plot of a field may be changed to a different scale, that is, it may be enlarged or diminished in any given ratio, by drawing lines parallel to each of the sides of the original plan. To enlarge the perimeter of the figure ABCDE (Fig. 36.) in the ratio of aG to AG; draw lines from G through each of the angular points. Then beginning at a, draw ab parallel to AB, bc parallel to BC, &c. It is evident that the angles are the same in the enlarged figure, as in the original one. And by similar triangles, AG: aG:: BG : bG::CG: cG:: &c. And AG: aG::AB: ab :: BC: bc:: &c. Therefore ABCDE and abcde are similar figures. (Euc. Def. 1. 6.) In the same manner, the smaller figure a'b'c'd'e' may be drawn, so as to have its perimeter proportioned to ABCDE as a'G to AG. SECTION II. METHODS OF SURVEYING IN PARTICULAR CASES. ART. 128. MEASURING round a field, in the manner explained in the preceding section, is by far the most common method of surveying. The following problems are sometimes useful. They may serve to verify or correct the surveys which are made by the usual method. PROBLEM I. To survey a field from TWO STATIONS. 129. FIND THE DISTANCE OF THE TWO STATIONS, AND THEIR BEARINGS FROM EACH OTHER; THEN TAKE THE BEARINGS OF THE SEVERAL CORNERS OF THE FIELD FROM EACH OF THE STATIONS. In the field ABCDE, (Fig. 37.) let the distance of the two stations S and T be given, and their bearings from each other. By taking the bearing of A from S and T, or the angles AST and ATS, we have the direction of the lines drawn from the two stations to one of the corners of the field. The point A is determined by the intersection of these lines. In the same manner, the point B is determined by the intersection of SB and TB; the point C, by the intersection of SC and TC; &c. &c. The sides of the field are then laid down, by connecting the points ABCD, &c. The area is obtained, by finding the areas of the several triangles into which the field is divided by lines drawn from one of the stations. Thus the area of ABCDE (Fig. 37.) is equal to ABT+BCT+CDT+DET+EAT or to ABS+BCS+CDS+DES+EAS. Now we have the base line ST given and the angles, in the triangle AST, to find AS and AT; in the triangle BST, to find BS and BT, &c. After these are found, we have two sides and the included angle in the triangles ABT, BCT, &c. from which the areas may be calculated. (Mens. 9.) Example. Let the station T (Fig. 37.) be N. 80° E. from S, the distance ST 27 chains, and the bearings of the several corners of the field from S and T as follows; TA N. 30° W. TC S. 53 E. TD S. 55 W. SA N. 17° E. SB N. 55 E. SC S. 73 E. TE N. 70 W. These will give the following angles; ATS 70° AST 63° SE N. 26 W. ATB= 45° From which, with the base line ST, are calculated the fol Contents of the field, 230.22 acres. The course and length of each of the sides of the field may be found, if necessary. After the parts mentioned above are calculated, there will be given two sides and the included angle, in the triangle ATB, to find AB, in BTC to find BC, &c. If the base line between the two stations be too short, compared with the sides of the field and their distances, the survey will be liable to inaccuracy. It should not generally be less than one tenth of the longest straight line which can be drawn on the ground to be measured. 130. It is not necessary that the base line, from the extremities of which the bearings are taken, should be within the field. It may be one of the sides, or it may be entirely without the field. Let S and T (Fig. 38.) be two stations from which all the corners of a field ABCDE may be seen. If the direction and length of the base line be measured, and the bearings of the points A, B, C, D, and E, be taken at each of the stations, the areas of the several triangles may be found. The figure ABCTDE is equal to DET+EAT+ABT+BCT From this subtracting DCT, we have the area of the field ABCDE. In this manner, a piece of ground may be measured which, from natural or artificial obstructions, is inaccessible. Thus an island may be measured from the opposite bank, or an enemy's camp, from a neighboring eminence. 131. The method of surveying by making observations from two stations, is particularly adapted to the measurement of a bay or harbor. The survey may be made on the water, by anchoring two vessels at a distance from each other, and observing from each the bearings of the several remarkable objects near the shore. Or the observations may be made from such elevated situations on the land, as are favorable for viewing the figure of the harbor. If all the parts of the shore cannot be seen from two stations, three or more may be taken. In this case the direction and distance of each from one of the others should be measured. PROBLEM II. To survey a field by measuring from ONE STATION, 132. TAKE THE BEARINGS OF THE SEVERAL CORNERS OF THE FIELD, AND MEASURE THE DISTANCE OF EACH FROM THE GIVEN STATION. If the length and direction of the several lines AT, BT, CT, DT, and ET, (Fig. 37.) be ascertained; there will be given two sides and the included angle of each of the triangles ABT, BCT, CDT, DET, and EAT; from which their areas may be calculated, (Mens. 9.) and the sum of these will be the contents of the whole figure. |