By means of rods, universal joint, &c. it is often made to act within the carriage, so that the person may, at any moment, without the trouble of getting out, see the number of the revolutions of the wheel. If the instrument is to be always applied to one wheel, a table may easily be constructed to shew the distance in miles and its parts by inspection only.

The pedometer is exactly the same kind of instrument as the way-wiser. The box containing the wheels is made of the size of a watch-case, and goes into the fob, or breeches pocket; and, by means of a string and hook fastened to the waistband or at the knee, the number of steps a man takes in his regular paces are registered, from the action of the string upon the internal wheel-work, at every step, to the amount of 30,000. It is necessary to ascertain the distance walked, that the average length of one pace be previously known, and that multiplied by the number of steps registered on the dial-plate. This instrument requires a person to be very uniform in his paces, and the acting distance invariable, or a false step or pull will be made.

OF THE SURVEYING CROSS, fig. 2, plate 14.

The cross consists of two pair of sights, placed at right angles to each other: these sights are sometimes pierced out in the circumference of a thick tube of brass about 24 inches diameter, see fig. 3, plate 14. Sometimes it consists of four sights strongly fixed upon a brass cross; this is, when in use, screwed on a staff having a sharp point at the bottom to stick in the ground; one of this kind is represented at fig. 2, plate 14. The four sights screw off to make the instrument convenient for the pocket, and the staff which is about 4 or five feet in length (for both the crosses) unscrews into three parts to go in a portmanteau, &c.

The surveying cross is a very useful instrument for

placing of offsets, or even for measuring small pieces of ground; its accuracy depends on the sights being exactly at right angles to each other. It may be proved by looking at one object through two of the sights, and observing at the same time, without moving the instrument, another object through the other two sights; then turning the cross upon the staff, look at the same objects through the opposite sights; if they are accurately in the direction of the sights, the instrument is correct.*

It is usual in order to ascertain a crooked line by. offsets, first to measure a base or station line in the longest direction of the piece of ground, and while measuring, to find by the cross the places where perpendiculars would fall from the several corners and bends of the boundary; this is done by trials, fixing the instrument so, that by one pair of sights both ends of the line may be seen; and by the other pair, the corresponding bend or corner; then measuring the length of the said perpendicular. To be more particular, let A, h, i, k, l, m, fig. 35, plate 9, be a crooked hedge or river; measure a straight line, as AB, along the side of the foregoing line, and while measuring, observe when you are opposite to any bend or corner of the hedge, as at c, d, e; from thence measure the perpendicular offsets, as at ch, di, &c. with the offset staff, if they are not too long; if so, with the chain. The situation of the offsets are readily found, as above directed, by the cross, or King's surveying quadrant; they are to be registered in the field-book.

Of surveying with the chain and cross. What has been denominated by many writers, surveying by the

* I have made some additions to the box cross staff, which have been found useful and convenient for the pocket, where great accuracy is not required. See fig. 6. A compass and needle at the top A, to give the bearings, and a moveable graduated base at B, by rack-work and pinion C, to give an angle to 5′ of a degree by the nonius divided on the box above. Thus the surveyor may have a small theodolite, circumferentor, and cross staff all in one instrument. EDIT.

chain only, is in fact surveying by the cross and chain, for it is necessary to use the cross, or optical square, for determining their perpendicular lines, so that all that has been said, even by these men, in favour of the chain alone, is founded in fallacy. To survey the triangular field ABC, fig. 22, plate 9, by the chain and cross: 1. Set up marks at the corners of the field. 2. Beginning, suppose at A, measure on in a right line till you are arrived near the point D, where a perpendicular will fall from the angle, let the chain lie in the direction or line AB. 3. Fix the cross over AB, so as to see through one pair of sights the mark at A or B, and through the other, the mark at C; if it does not coincide at C with the mark, the cross must be moved backwards or forwards, till by trials one pair of the sights exactly coincide with the mark at C, and the other with A or B. 4. Observe how many chains and links the point D is from A, suppose 3.03. which must be entered in the fieldbook. 5. Measure the perpendicular DC, 643.7, finish the measure of the base line, and the work is done. This mode is used at present by many surveyors, probably because there is no check whereby to discover their errors, which must be very great, if the survey is of any extent.

To plot this, make AB equal 11.41, A D equal to 3.03; on the point D erect the perpendicular DC, and make it equal 6.43; then draw A C, B C2 and the triangle is formed.

OF THE OPTICAL SQUARE, fig. 4, plate 14.

This instrument has the two principal glasses of Hadley's quadrant, and was contrived by my father; it is in most, if not in all respects superior to the common surveying cross, because it requires no staff, may be used in the hand, and is of course of great use to a military officer. It consists of two plain mirrors, so disposed, that an object seen by reflexion from both, will appear to coincide with another ob

ject seen by direct vision whenever the two objects subtend a right angle from the centre of the instrument, and serves therefore to raise or let fall perpendiculars on the ground, as a square does on paper, of which we shall give some examples. Its application to the purposes of surveying will be evident from these, and what has been already said concerning the cross.

Fig. 4, plate 14, is a representation of the instrument without its cover, in order to render the construction more evident. There is a cover with a slit or sight for viewing the objects; the object seen directly, always coincides with the object seen by reflexion, when they are at right angles to each other.*

From a given point in a given line, to raise a perpendicular. 1. The observer is to stand with this instrument over the given point, causing a person to stand with a mark, or fixing one at some convenient distance on the given line. 2. An assistant must be placed at a convenient distance, with a mark somewhere near the line in which it is supposed the perpendicular will fall; then if on looking at one of the objects, the other be seen in a line with it, the place where the mark of the assistant is fixed is the required point.

From a given point over a given line to let fall a perpendicular. Every straight line is limited and determined by two points, through which it is supposed to pass; so in a field the line is determined by two fixed objects, as steeples, trees, marks erected for the purpose, &c. For the present operation, the two objects that determine must be on one side the point where the perpendicular falls; or, in other words, the observer must not be between the objects, he must place himself over the line, in which he will always be when the two objects coincide; he must move himself backwards on the line, till the mark,

See the description of a considerable improvement upon it, after the description of the Hadley's sextant, plate 19. EDIT.

from whence the perpendicular is to be let fall, seen by direct vision, coincides with one of the objects which determine the given line seen by reflexion, and the instrument will be over the required point. To measure inaccessible distances by the optical

square.

Required the distances from the steeple A, fig. 20, plate 9, to B. Let the observer stand with his instrument at B, and direct an assistant to move about C with a staff as a mark, until he sees it coincide by direct vision with the object at A; let him fix the staff there at C; then let the observer walk along the line A C, until, when at D, A and B coincide in the instrument, and BD will be perpendicular to A C; measure the three lines BC, DC, BD, and then the following proportion will give the required distance, for as DC is to DB, so is BC to AB.

Second method. 1. Make BC, fig. 21, plate 9, perpendicular to BA. 2. Divide BC into four equal parts. 3. Make CD perpendicular to C B. 4. Bring FEA into one line, and the distance from C to F will be of the distance from B to A.

OF THE CIRCUMFERENTOR, fig. 1, plate 15.

This instrument consists of a compass box, a magnetic needle, and two plain sights, perpendicular to the meridian line in the box, by which the bearings. of objects are taken from one station to another. It is not much used in England where land is valuable; but in America where land is not so dear, and where it is necessary to survey large tracts of ground, overstocked with wood, in a little time, and where the surveyor must take a multitude of angles, in which the sight of the two lines forming the angle may be hindered by underwood, the circumferentor is chiefly

used.

The circumferentor, see fig. 1, plate 13, consists of a brass arm, about 14 or 15 inches long, with