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

D

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 AB, 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

C

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

D

A

E

o CB 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 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 E B, 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

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 Be 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 Df FD.

Produce AD and ec till they meet in a, and also BD and fe 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 AB.

*

A right angle can often be laid off when no

means of

measuring other divisions of the circle are at hand. The distance AB can then be thus obtained :

:

Of course with a sextant, or other means of observing the angle ACB, AB becomes simply the tangent of that angle to the radius BC: 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:—

* A perpendicular can always be thus laid off with the chain :-suppose a the point at

50

30

which it is required to erect a right-angle: fix an arrow into the ground at a, through the ring of the chain, marking twenty links; measure forty links on the line ab, and pin down the end of the chain firmly at that spot, then draw out the remaining eighty links as far as the chain wil stretch, holding by the centre fifty-link brass ring as at c; the sides of the triangle are then in the proportion of three, four, and five, and consequently cab must be a right angle.

20

a

40

b

An angle equal to any other angle can also be marked on the ground, with the chain only, by measuring cqual 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.