entering their distances in the field-book; then measure the distance between the two marks for a proof line:-or, one mark only may be left in one of the lines, which may be connected with its opposite angle for a proof line. EXAMPLES. 1. Required the construction and area of a field from the following dimensions. NOTE. If the proof line measured from the plan, does not exactly, or very nearly, agree with that measured in the field, some error has been made, and the work must be repeated. TO FIND THE AREA OF A TRIANGLE FROM THE THREE SIDES. RULE. From half the sum of the three sides subtract each side severally and reserve the three remainders; multiply the half sum continually by the three remainders, and the square root of the product will be the area. NOTE. By this rule the area of a triangle may be found without laying it down, or finding the perpendicular. Adopting the preceding example, we have by the rule, 1338+852 + 1244 half sum of the three sides. Then 1717 865 2 1717 1338 = 379 = 1st remainder; 1717 852= 2nd remainder; 1717 1244 = 473 = 3rd remainder; whence (1717 x 379 × 865 x 473) 5·15022 5a. Or. 24p. the sum as the area already found by measuring the perpendicular from the plan. NOTE. This method of finding the areas of triangles is very little used in practice, on account of its requiring a tedious calculation, which may, however, be more readily performed by logarithms, as shall hereafter be shewn. 2. It is required to lay down a survey and find its content from the following field notes. Having drawn the figure, the proof line m n will be found to measure 351 links, as in the field notes; and the perpendicular Bb to be 1056 links. Double areas. 2644224 Triangle A B C 655676 Offsets on A B and A C 3299900 Sum 200616 Insets on B C 2)3099284 Difference 15.49642 = 15a. 2r. Op. nearly, the area required. COMPUTATION OF THE AREA BY CASTING, THAT IS, BY REDUCING THE CROOKED SIDES TO STRAIGHT ONES. The offsets in the last example have been computed, in manner already shewn in Chap. I.; but by this method straight lines are drawn on the plotted figure so as to include as much space in the area to be measured, as they exclude, as nearly as can be judged by the eye, the area to be measured is thus reduced to a figure bounded by right lines only, which may thence be much more expeditiously reduced to triangles, trapeziums, &c. The method of drawing these lines is usually by a straight edged ruler of transparent horn, or by a silken thread stretched with a bow; the ruler or thread being moved over the crooked fence, till it appear to the eye to enclose as much of the adjoining ground as is left out, a line is then drawn in this position; and so on for other crooked fences. Thus the trouble of calculating numerous offsets is completely avoided, and with proper care equal accuracy is obtained. Ъ B 2. We shall adopt the last example for this method of casting, that it may be seen how near the two methods agree. The figure being constructed, and the boundary drawn carefully with ink, the chain-lines must then be rubbed out, and the three dotted lines A B, BC, CA must now be drawn, in such a manner, that the parts excluded by them may be equal to the parts included, as nearly as can be judged by the eye. The base AC will be found to be 2584 links, and the perpendicular B = 1200 links. Whence 2584 X 1200 2 =15·50400=15a. 2r. 1p. nearly, the area by this method. If the area found by the true method be taken from the area, just found by casting, it will be seen that they differ by little more than one pole out of 25 acres, or little more than 1 in 4000: thus 15.50400 15.49642 758 square links, or little more than one pole. NOTE. It will hence be seen at once that a great deal of trouble is saved by this method, which is therefore generally adopted by practical surveyors; although it is certainly less correct than by calculating from the offsets, the former method depending chiefly on the accuracy of the casting lines for the truth of its results; but practice will soon render it easy to draw the lines so as to obtain almost perfect accuracy. 3. Required the plans and contents of two fields by both the methods of calculation, viz. offsets and by castings, from the following field-notes. NOTE. It will be seen that in the main triangles of these two surveys, the proof lines have been taken from a side of each to its opposite angle; which is the best method of proof, when convenient to make it; but it may be performed with equal accuracy by taking a proof line from one side to another, at a short distance from one of the angles of the triangle. PROBLEM VI. FOUR SIDED FIELDS. When a field has four sides, straight or crooked, measure the four sides, or lines near them, if crooked, taking the offsets: also measure one or both the diagonals, one of which will serve as a base in plotting the work, and the other for a proof-line; or the proof-line may be measured in any other direction that may be most convenient. Sometimes the measurement of both the diagonals is prevented by obstructions, in such cases it will be sufficient to measure tie-lines across two of the angles of the trapezium, at the distance of from two to five chains from each angle, according to the size of the field. These tie-lines with their distances from the angles on the main-lines will be found sufficient for planning the lines and proving them. A EXAMPLES. 1. In the annexed figure the lines A B, BC, CD, DA are measured, marks being left at p, q, and r, and their respective distances on the lines noted in the field-book, thus furnishing the follow method of laying down the plan. On A B, as a base, take A p = given distance, and with the distances Ar, pr, and centers A and p describe arcs cutting in r; then prolong Ar, and lay off thereon the given length A D. In the same manner construct the triangle p B q, and make B C its given length. Lastly, join DC, which must be of the length shewn in the field-book, otherwise there has been some mistake either in the measurement, or in laying it down. Should this be the case the whole of the work, firstly on the plan, and secondly in the field, must be gone over again till the error be discovered. NOTE. When the main lines that include the chief part of the ground to be |