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. length such 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, making CD=BC', the dis A B E D tance BD is then equal to either of these measured lines. 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 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 B C on a large scale, and taking the distance Bo from the plot. A 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 AX 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. A 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 Memoire.") 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 ad-AD evident. AD} d 9 D B D F a 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 Df=FD. Produce AD and ec till they meet in a, and also BD and fc till they meet in b and 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 AB. A right angle can often be laid off when no means of measuring other divisions of the circle are at hand. The distance A B can then be thus obtained :-- BC and DE are both perpendicular to AD, and the points E and C are marked in a line with A, then Of course with a sextant, or other means of observing the angle A perpendicular can always be thus laid off with the chain: suppose a the point the chain will stretch, holding by the centre fifty link brass ring as at c-the sides of ACB, AB becomes simply the tangent of that angle to the radius BC, and a table of natural sines and tangents engraved on the lid of any portable reflecting instrument is often of great service, particularly in sketching ground without any previous triangulation, and in obtaining the distance, to an enemy's batteries, &c., on a military reconnaissance. The height of a point on an inaccessible hill may also be obtained without the use of instruments, thus: Drive a picket three or four feet long at H, and another at L, where the top of a long rod FD is in a line with the object S from the point A (the heads of these pickets being on the same level); mark also the point C, where the head of the rod is in the same line with S, from the top of any other picket B; and measure AF and BC; lay off the distance BC from F to b, and the two triangles ADb and ASB are evidently similar, whence Ps H1 PS AB HI fore-DF. and AP-AF. HI HO Но and H. PS there AP AB HI Ab HO A few other methods of ascertaining distances and heights, more particularly connected with military reconnaissances, will be found in the next chapter. Where angles can be taken between three inaccessible objects, the relative positions of which are known, and can be laid down on paper, the place of the observer can be ascertained either by calculation, by con the triangle are then in the proportion of three, four, and five, and consequently cab must be a right-angle. An angle equal to any other angle can also be marked on the ground, with the chain only, by measuring equal distances on the sides containing it, and then taking the length of the chord: the same distances, or aliquot parts thereof, will, of course, measure the same angle. H struction, or by means of an instrument used for that purpose, called a "station pointer;" or, what is better still, a piece of thin tracing paper, with the observed angles plotted upon it, can be shifted about until the point falls into the only spot from whence the lines containing these angles pass through the three fixed stations. The case is a very common one in maritime surveying, where the two first methods of solution, calculation and construction, are seldom thought of; and the last, which is the most simple, and sufficiently correct for the purpose, generally adopted. In a trigonometrical survey, of course, this method would never be thought of for fixing a station, but the calculations for the different cases that may occur of the three points being in one line, or forming a triangle within, or without which the observer may happen to be, will be found, with a mass of other information on such subjects, in "Adam's Geometrical Essays," pp. 169 to 177. The following is the mode of obtaining the position of the observer by construction, in the case that most commonly occurs, viz., when the three points form a triangle, without which the place of observation lies. O, P, and Q represent the three points on shore whose positions have been determined by interior triangulation, and S a rock or anchorage, whose place is to be determined with relation to the stations above mentioned. Suppose the angle QSP is observed 35o, and PSO=40°, describe a circle passing through Q, S, and P, which is thus done. Double the angle QSP which=70°; subtract this from 180, leaving 110°; lay off half of this, or 55° at PQR and QPR, and the angle at R is evidently=70°, or double QSP; now the angle at the centre being double |