necessary to move the protractor, held in this position, slightly up and down upon this line, until its bevelled edge touches the point D; D F is then at once drawn in the required direction. The distances may also be set off from a scale graduated on the edge of the protractor, by merely moving it along this line, DF, until some defined division corresponds with the station D. By observing with a sextant the angles between three or more known stations, the place of the observer can be ascertained, both instrumentally and by calculation, but not so readily as with the compass. The method of thus determining the position of any point will be explained hereafter. The plane table is perhaps the best contrivance for sketching in the interior detail of a survey with accuracy, but its size renders it too inconvenient to be termed portable, and its use is now almost universally superseded by the portfolio and compass. The little reflecting semicircle invented by Sir Howard Douglas, is so far an improvement on the sextant that it protracts the angles it observes by means of a contrivance by which the reflected angle is doubled instrumentally, and the angle is protracted upon the paper by means of a bevelled projection of the radius. Other varieties of small reflecting instruments have also been contrived for the same purpose. the The process of sketching between the fixed points plotted on paper is similar to surveying with the chain and theodolite as far as the natural and artificial boundaries are concerned; the distances being obtained by pacing; the offsets (if small) by estimation; and the bearings of the lines by the compass or sextant*. Everything is, however, here drawn at once upon the paper, instead of being entered in a field-book. The features of the ground are sketched at the same time as the boundaries and other details; and this part of the operation, being less mechanical * A straight walking-stick will be found very useful in sketching, not only for the purpose of getting in line between two objects, which is easily done by laying the stick on the ground, in the direction of one of them, and observing by looking from the other end to which side of the opposite station it cuts, but also for prolonging a line directed on any known point to the rear. A bush or any other mark, observed in the line of the stick, answers as well as another known point for pacing on. than the preceding, requires far more practice before anything like facility of execution can be acquired; it is, however, more particularly connected with the subject of the next chapter, where the different methods of delineating ground in the field will be explained. The following are the best practical methods of passing obstacles met with in surveying, and of determining distances which do not admit of measurement, by means adapted for use in the field, most of them requiring no trigonometrical calculation. Some of these problems are solved without the assistance of any instrument for observing angles; but as a general rule (subject of course to some few exceptions), it is always better to make use of the theodolite, sextant, or other portable instrument, than to endeavour by any circuitous process to manage without angular measurement. The measurement of the line AD, supposed to be run for the determination of a boundary, is stopped at B by a river or other obstacle. The point F is taken up in the line at about A F' F B E the estimated breadth of the obstacle from B; and a mark set up at E at right angles to AD from the point B, and about the same distance as BF. The theodolite being adjusted at E, the angle BEC is made equal to BEF, and a mark put up at C in the line AD; BC is then evidently equal to the measured distance FB. If the required termination of the line should be at any point C', its distance from B can be determined by merely reversing the order of the operation, and making the angle BEF' equal to BEC', the distance BF' being subsequently measured. There is no occasion in either case to read the angles. The instrument being levelled and clamped at zero, or any other marked division of the limb, is set on B; the upper plate is then unclamped, and the telescope pointed at F, when being again clamped, it is a second time made to bisect B; releasing the plate, the telescope is moved towards D till the vernier indicates zero, or whatever number of degrees it was first adjusted to; and the mark at C has then only to be placed in the line AD, and bisected by the intersection of the cross wires of the telescope. If it is impossible to measure a right angle at B, from some local obstruction, lay off any convenient angle AB E, and set up the theodolite at E. Make the angle BEC equal to one-half of ABE, and a mark being set up at C in the prolongation of A B, BC is evidently equal to BE, which must be measured, and which may at the same time be made subservient to the purpose of delineating the boundary of the river. as a house or barn, is by turning off to the right or left at right angles till it is passed, and then returning in the same manner to the original line. But perhaps a more convenient method is to measure on a line making an angle of 60° with the original direction a distance sufficient to clear the obstacle, and to return to the line at the same angle, A B E D making CD=BC'; the distance BD is then equal to either of these measured lines. C The distance from B on the line Ao, to the trigonometrical point o, which is inaccessible, is determined in the manner explained in the first method in the last page; the point C is taken at right angles to BA from the point B, and the angles B E A OCB and BCD being made equal, BD is equivalent to the distance Bo required. The same object is attained by laying down the plan of the building on a large scale, and taking the distance Bo from the plot. To find the point of intersection of two lines meeting in a lake or river, and the distance DB to the point of meeting :-From any point F on the line A X draw FD, and from any other point E draw ED, produce both these lines to H and G, making the prolongations either equal to the lines themselves, or any aliquot part of their length, suppose one-half; join HG, and produce it to O, where it meets the line CB, then OH is one half of EB, and OD equal to half of DB; which results give the point of intersection B, and the distance to it from D. To find the distance to any inaccessible point, on the other side of a river for instance, without the use of any instrument to measure angles.-(This and the two following are taken from the " Aide Mémoire.") A is any inaccessible point the distance of which from B is required: produce AB to any point D; draw Dd in any direction bisected in C; join BC and produce it to b, Cb being equal to BC; join db and produce it to a, the intersection of the prolongation of AC, then ab AB > The proof is and a d=AD evident. Another method Prolong AB to any point D, making BC equal to CD; lay off the same distances in any direction Dc cb; mark the intersection E of the line joining Bc and cb; mark also F the intersection of DE produced, and of Ab; produce Db, and BF, till they meet in a, and ab=AB ac=AC aD=AD To measure the distance between A and B, both being inaccessible :-From any point C draw any line Cc bisected in D; take any point E in the prolongation of AC, and join ED, producing the line to De ED; in like manner take any point F in the prolongation of BC, and make DfFD. Produce AD and ec till they meet in a, and also BD and fc till they meet in b; then ab AB. If AB cannot be measured, but the points A and B are accessible, their distances from any point O are determined; and by producing these lines any aliquot part of their length, as OP, OQ, the distance PQ will bear the same proportion to A B. |