hedge and ditch A B, in fig. 61., belong to the enclosure x; but on the other three sides to adjacent grounds. Fig. 61. B X PROBLEM VII. To set out small Allotments of Land. The gardener or bailiff may occasionally be called upon to divide and set out small allotments of land for cottagers to cultivate; and, although sufficient has already been taught to enable the student to accomplish such operations, yet an example or two may help him to manage these matters more readily. Ex. 1. Suppose a rectangular piece of ground (fig. 62.), whose length A B is 800 links, and breadth a C 500 links (4 acres), is to be divided into sixteen allotments, four of which shall contain one rood and ten poles each; four more B one rood each; and the remaining eight plots, thirty-five poles each; what will be the dimensions and proportions as laid down to a scale? Here the first four divisions will be of links in the line AC; viz. 500, will produce 200 { 350 to be set off on the line AB; as Ab=250, bc=200, cв= And these spaces again subdivided will show the proportions of each allotment, which will stand thus: Ex. 2. Let it be proposed to divide into ten equal portions an irregular plot of ground, of the shape of fig. 63., Fig. 63. B and the dimensions as follows: A B 1180 links the perpendiculars to the three angular points B, C, E, are 250, 190, and 80. Now, in cases like this, the proper way is to make an accurate plan of the ground to be divided, to any scale you please, the larger the better, and from the same scale plot in each allotment (found as below), which will then be easily transferred from the paper to the ground itself. Here, from the above dimensions, the area will be found to be 291,000 square links, which, being divided by 10, will give the area of each division equal to 29,100 square links. Let the several allotments run at right angles with the line AB: then, by the application of the scale, cỏ will be found to measure 300; therefore the half of this, to divide 29,100 by, will produce 194, the distance a b, consequently the triangle a b c is one allotment sought. Now for the next, the average height will be found to be 310; consequently, dividing by this number the area as before, the distance be will be found to be 97 nearly. Proceeding as before, the average height will by the scale be found to be 330, therefore ce will be found (as near as can be laid down) 88, for the third allotment. And for all the rest, the process is the same. Also, the quantities, reduced from square links, will each be 1 rood 6.56 poles. In cases where the land varies in quality, and it is desired at the same time that the cottagers should one with another have an allotment of equal value, the intelligent gardener or bailiff will of course give an additional quantity where there is a defect in value. An allowance too should be made for paths which may be common to two or more allotments in setting out, all of which should be properly considered. Ex. 3. Fig. 64. represents an unequal-sided piece of ground, measuring 1-882 acre, which it is required to lay off in five allotments. The first thing to be done is to ascertain the area of the piece. In doing this, let the measurer holding the back end of the chain start from A, noting the offsets on that line in his field-book as nearly as he can to the lines in the figure. When he has marked all the offsets on that line to B, with his cross staff let him set off the line B C, marking the offsets in the same manner. Then set off the line C D also at right angles. If this has been correctly done, the line DA will also be found to be at right angles with the other lines. The offsets, being all either triangles or trapezoids, must be calculated by the rules applicable to such figures (as in page 102.); and these, added to the area of the interior rectangle or parallelogram, will be found to amount to 1 acre 88,200 links, which, divided by 5, gives 37,640 square links to each allotment, or 1 rood, 20 poles, 7 yards. Supposing the lines of division to run in the direction of and parallel to the line A D, we find the offsets on that line to measure To the first offset 1500 on the line A B With ten times ten links at the corner And 100 links added to 250 in the line CD links 5600 1500 100 350 7550 500 7050 37640 30590 From which deduct 1000 links, taking the half Which deduct from the square links in one allotment and divide by 300, which gives within a small fraction of 102 links from the line A D for that division. On the same principle proceed with the rest of the four divisions, first finding the offsets in square links, and setting off from the last line accordingly. A rectangular piece of ground or parallelogram will be easily set off in the same manner, without any reference to offsets. 169 CHAP. IX. LEVELLING. LEVELLING is the art of finding a line parallel to the horizon at one or more stations, in order to determine the height of one place with regard to another. A truly level surface is a segment of a spherical surface, which is concentric with the globe of the earth. A true line of level is an arc of a great circle concentric with the globe of the earth. Hence, two or more places are on a true level, when they are equally distant from the centre of the earth. Also, one place is higher than another, or out of level with it, when it is further from the centre of the earth; thus, taking the surface of the ocean as an elastic band covering the lower part of the shell of the earth, and yielding to the lunar attraction, we should estimate all heights in reference to this datum. The apparent level is a straight line drawn tangent to an arc or line of true level. Every point of the apparent level, except the point of contact, is higher than the true level. Thus, let E A G, in fig. 65., be an arc of a great circle drawn upon the earth. To a person who stands upon the earth at A, the line HD is the apparent level to his rational ho D Fig. 65. B A H G E rizon; but this line the farther it is extended R R from his station A, the farther it recedes from the centre; for B C is longer than A C, and D C is longert han B C. Hence, we discover that the line of sight given by the operations of levels, is a tangent, or a right line perpendicular to the semi-dia |