The area found by calculation from the offsets is Oa. 2r. 17p. and the perpendicular of the triangle, found by the casting of the parallel ruler, is 161 links. Hence 751 x 161 = 0·60455 square links, or somewhat more than Oa. 2r. 161p. It hence appears that the method of casting by the ruler, gives the content very near the truth. In fact this method is mathematically accurate; but the danger of error, as already said, arises from the accidental slipping of the ruler during the operation. 2. Lay down and find the area of a curve-lined offset-piece, from the following field notes; and find its area both by calculation, and by the casting of the parallel ruler. By both the methods the area is found to be Oa. 2r. 15p. nearly, the perpendicular on A B being found 199 links by the casting operation. PROBLEM III. TO REDUCE THE IRREGULAR FIELD ABCDEFGHK TO A TRAPEZIUM BY THE PARALLEL RULER. Prolong the line AK at pleasure. Lay the ruler from K to G; move it parellel to, and mark A K prolonged at 1. Lay the ruler from 1 to F; move it parallel to G, and mark A K B at 2. Lay the ruler from 2 to E; move it parallel to F, and mark A1 at 3. Draw a line 3 to E and prolong from E. Lay the ruler from E to C; move it parallel to D, and mark 1 E at 4. Lay the ruler from 4 to B; move it parallel to C, and mark 1 E prolonged at 5. Draw a line from 5 to B; then shall AB53 be a trapezium, equal in area to the irregular figure ABCDEFG K; the area of which may be found by multiplying the diagonal B 3 by half the sum of the perpendiculars thereon from A and 5. NOTE. In this manner the crooked sides of a field may be successively reduced to straight ones. Thus, if the side A B had been crooked, the operation of straightening might be continued by prolonging the dotted line 5 B, and find successive points therein, corresponding to the assumed angles, till the last angle was brought thereon, and so on with respect to the side A K, had it also been crooked. When the sides of a field are curved, the method of reducing them to straight lines is the same as shewn in Problem II. A PROBLEM IV. 2 K 3 H I TO DRAW AN EQUALIZING LINE THROUGH THE CROOKED FENCE abcde, so THAT THE TWO FIELDS A Bea, a DC e MAY quantities shall be the same as those before separated by the crooked fence a b c d e. It is scarcely necessary to add, that had the fence abcde been curved, the equalizing line might have been found as in Prob. II. NOTE. The owners of adjoining estates sometimes agree to straighten fences or boundaries, by an equalizing line of fence, or, as they term it, " by giving and taking equal quantities of land." When this is required to be done, the crooked or curved boundary fence must be first measured on the ground, and then plotted on a large scale; when the equalizing line may be drawn on the plan, as shewn in the last Problem, and the distance from D to 3, or from a to 3, must be correctly measured by the scale: this distance must next be measured in the field, accordingly as it is taken from D or a; and the new line of boundary e 3 may now be ranged and marked out, preparatory to making the required fence. Moreover, to guard against errors, in the preceding operation, it is advisable to measure, both on the plan and on the ground, the parts cut off, on each side of the new boundary line, thus proving the work; as an error is of serious consequence in these matters: and, if one is found to exist, it must be corrected before the boundary fence is made. SURVEYING LARGE ESTATES OR PARISHES BY THE CHAIN ONLY. 1. Having perambulated the boundary of the estate, parish, or lordship to be surveyed, if you find that its boundary approaches somewhat near to that of a four-sided figure, or trapezium, the system of fundamental lines, adopted by order of the Tithe Commissioners of England and Wales, is to be preferred. These fundamental lines are six in number, of which four must run close by, or as nearly as possible to, the boundary in question, thus forming a trapezium, four lofty station poles being placed at each angle, as objects for running the lines; the other two lines must form the diagonals of this trapezium, and therefore pass through the central parts of the survey, intersecting each other, the.points of intersection being noted on measuring each line, so that when the system of lines are laid down on the plan, the proof of the accuracy of the work may be fully established, before the minor operations, or filling up, as it is called, is commenced. It will be necessary, moreever, in almost every case, to range the lines between every two of the main stations with long slender ranging poles, as the intervention of hills, fences, trees, buildings, &c., will frequently interrupt the view of even the loftiest station poles that can be obtained; and more especially so, when the main stations are at a great distance, which depends on the magnitude of the survey, and is sometimes as much as ten miles. In measuring these main lines, every fence, road, stream, building, &c., which is passed or crossed must be noted in the field book, the several crossings and bends being sketched therein, to the latter of which offsets must be taken. Stations must also be left on these main lines, at convenient situations for taking the interior fences, &c., of the survey, and their distances carefully noted in the field book. From and to the stations, thus left, or from and to points near them, secondary lines must be run, as near the interior parts of the survey as possible, the crossings, offsets, and other remarks being made in the field book, as already directed for the measurement of the main lines. These secondary lines will accurately fit between the points from and to which they have been measured, when laid down on the plan; thus forming a net work of small triangles within the four large triangles, into which the survey is divided by the six fundamental lines. This principle of proof is founded on the obvious property of triangles having a common angle always fitting one within another, the common angle of both being coincident. The lines marked with the figures 1 to 6, represent the system in question, those without figures are the secondary lines. The main lines are numbered with the figures 1, 2, 3, &c., in small circles, as the most convenient method of reference to the field book: the secondary lines must have these numbers continued on them, for the same purpose, but this is not done in the diagram, to avoid confusing it. It will be seen that the secondary lines m n, r s are prolonged beyond the system of main lines to give stability to the parts of the survey that protrudes beyond line 2. It will also be seen that the positions of several straight fences are determined by triple intersections of the main and secondary lines, or by direction points taken in them, as shewn in Prob. IX. In the author's practice, as a surveyor, he adopted this method, in the survey of the Parish of Tillington, Sussex, with the exception of bringing one of the diagonals to one of the angles of the trapezium, which, of course terminated in one of the side lines; thus giving a system of lines equally perfect, the end of the diagonal being only a few chains from the angle; there being a considerable practical difficulty in laying down this system of lines, where the ground is hilly and woody, as was the case on this occasion. 2. The system of main lines adopted by the author in the survey of the Parish of Woolbedding, in the same county, was the following, A B the base line AB being about five miles in length. To this survey, it may be clearly seen that the Tithe Commissioners' system of lines would be not at all adapted; though some surveyors, in compliance with their orders, adopted their system, whatever might be the shape of the survey, thus wasting much valuable time, in running lines over grounds at a great distance, in some parts, from the parish to be surveyed, and incurring the ridicule of both the scientific and the ignorant to boot. Such a course would evidently be required, in using their system in this survey: meanwhile a system of lines adapted to the shape of the survey, and constituting proof among themselves, as those shewn above do, are evidently the best. In this figure the interior fences and secondary lines of the survey are not drawn, as their great number would confuse the student; the author's object being to present a proper system of fundamental lines for the survey in question. |