but when of different names, their difference is a meridian distance of the same name with the greater. Prob. 18. Having the bearing and distance of all the lines in a survey given, except one, to find the bearing and distance of that line. Suppose, in the last example, we want the bearing and distance of the last line from the rest. Add up the four first latitudes, and take the differance between the northings and southings, which must be northing or southing of the last line as 17. 62. In like manner you get the easting, 0. 76. Now, per 47. 1, you may find the chain line. The bearing can be found by plain trigonometry. Cor. From this you may find the bearing and distance of a line which joins two given points, without chaining the line or taking the bearing. Prob. 19. Required to find the position of a line, to run from the station Q, fig. 10, to cut off any part required. Suppose the division line to fall on DE, as QO. Find the latitude and departure of QD by the last problem and then the area of DEAQ. Now, if this area is the required division, all is right. But if not, take the difference between the required division and this part; then you see which side of QD the division line must fall. Allow it to falls as QO. We have the area of QO D, and angle D, because we have the Having the area Hence, if we lay and then run the bearing of DE and DQ given; therefore, we get the perpendicular QO let fall on the line DE. and perpendicular we get the base DO. off the distance DO from the angle D, line OQ, it will cut off the required area. the bearing of DO and set it off from the cut off the part required. Or, if you find point Q, it will On Local Attraction.-Where local attraction exists, the variation will differ at different stations, and, consequently, much effect the area of the survey, unless corrected, and allowed on the corresponding bearings. The Surveyor ought at each station see if the compass will truly reverse; if not found to do so, after taking it once or twice, the needle must be effected by local attraction. To find how much this attraction is, take the difference between last bearing, which we suppose to be true, and the present one. This difference is the attraction at the present station. This difference must be applied to the following bearing according to its name. If this variation is met with at the second station, another station must be taken, either inside or outside the land, where it must be proved there is no attraction. From this station to the stations attracted, take bearings back and forward, so as to find the vale of the attraction and the station where the attraction commences at. It would be much better to use the theodolite, or chain only, where there is local attraction. We can find the angles of the field truly, in case of local attraction, from the back and forward bearings. Then reduce these to bearings and find the area as before. Prob. 20. To trace out the true position of a defaced bounds from the map or deed. Let fig. 10 represent the Let DE be the defaced bounds. From the map take off the length of DE, and the angle CDE; both of which must be taken off with the greatest care. Now go map. on the land to the point D; from this point, with a good circumferentor or theodolite set off the angle found from the map, which gives the position of the required boundary on the land. Chain off also the line found from the map, then will the bounds be truly traced out on the land. If the point D is not a given point on the land, you have only to start from some other given point, as, suppose C, and trace out CD and DE. In consequence of the variation of the needle and local attraction, it is much better to use the angles than the bearings. But we may use either. The bounds may be traced by the chain only, but not so correct as with the instrument. In the year 1802 the following method of surveying the public lands was adopted by Colonel J. Mansfield, Surveyor-General of the North-Western Territory. The part of the country to be surveyed he first divided by parallel meridians, six miles from each other, and again by east and west lines, also six miles from each other. After this manner the whole country is divided into equal squares, called townships. Therefore, each township is a square, six miles on a side, and contains 36 square miles. Each township is divided into squares, by meridians one mile from each other, and by east and west lines at the same distance from each other. Hence, each township is divided into 36 square miles, each square called a section, it is evident each section contains 640 acres. The sections are divided into half, quarter, and eighths of sections. The principal meridians and east and west lines, have been established by astronomical observations, and the sub-division lines run with the compass. There are occasions when it is requisite to apply the principles of plain and spherical geometry to much more extensive portions of the earth's surface, as in the determination of distances of many miles, whether for the sur vey of a kingdom, or for the measurement of a degree in the survey of a kingdom, there must be a line of a few miles measured, actually horizontal, as near as possible. This must be taken on the best possible level, the line so measured is assumed the base of the whole operation. Then a variety of hills and elevated spots are selected, at which signals can be placed, suitably distant and visible one from the other. A good theodolite is used, and each angle in every triangle observed. Then having one side and all angles given, we can calculate for all other sides in each triangle through the operation. I have before said, in order to insure accuracy in surveys of this kind, it is proper to find every angle of the triangles by observation, if the situation of the triangle will permit it; as the difference of their sum from 180 will enable us to judge of the accuracy of the work. As many angles are taken from each station as possible. After carrying on a series of triangles in the manner here described to some distance, it is customary to find, by actual measurement, the interval of two objects whose distance has been previously obtained by calculation, in order to determine the error of the calculated distance, and this line is called the base of verification. The rules for calculating such work we have already given. SECTION V. OF THE THEODOLITE. In surveying where angles are required to be taken, whether horizontal or vertical, no angular instrument is so well adapted for the purpose as the telescopic theodolite, for the angles deduced from this far exceeds in accuracy those obtained from any other instrument. There are various forms of this instrument, but the one we refer to is the five-inch theodolite, being the most convenient for land surveying. A vertical angle is any angle in a plane perpendicular to the horizon. Adjustment of the Theodolite. The first adjustment is the line of collimation, that is, to make the intersection of the cross-wires coincide with the axis of the rings on which the telescope turns it is known to be correct, when an eye looking through the telescope observes their intersection continue on the same point of a distant object during an entire revolution of the telescope. The following is the method of making this adjustment: First, make the centre of the horizontal wire coincide with some well-defined part of a distant object; then turn the telescope half round in its (Y's,) till the level lies above it, and observe if the point is again cut by the centre of the wire; if not, move the wire one half the quantity of deviation, by turning two of the screws at the extremity of said wire, releasing one, before tightening the other, and correct the other half by elevating or depressing the telescope; now, if the coincidence of the wire and object remain perfect in both positions of the telescope, the line of collimation is correct, but if not, the operation must be repeated carefully, until the adjustment is perfect. After the same manner, the vertical line is put correct, or, the point of intersection, when there are two oblique lines instead of a vertical one. The second adjustment, is that which puts the level attached to the telescope, parallel to the rectified line of collimation. The clips being open, and the vertical arc clamped, bring the air-bubble of the level, to the centre of its glass tube, by turning the tangent-screw of the vertical arc, |