an instrument for laying down and measuring angles upon paper with accuracy and dispatch; and by which the use of the line of chords is superseded. This instrument is variously formed, as semicircular, rectangular, or circular; and constructed of different materials, as brass, ivory, &c. It is necessary in laying down those surveys or other plans where angles are concerned.

The rectangular protractor is constructed in form of a right-angled parallelogram, which, when applied to a case of mathematical instruments, is substituted in place of the semicircular protractor and scale of equal parts. Fig. 1. is a representation of it: the manner of using it is exactly similar to that of the semicircular one.

The circular protractor, as its name implies, is a complete circle, and is superior by far to either of the former, both in point of accuracy and dispatch, especially when several angles are to be formed at the same point. The limb of this instrument is divided into 360 degrees, and each degree in some protractors is halved: it has a subdividing scale or vernier, by which an angle may be laid down measured to a single minute. In the centre of the protractor is a fine mark, which, when an angle is to be protracted or measured, is to be laid upon the angular point, and 0° or zero on the limb, upon the given line forming one side of the angle.

Fig. 2. represents a circular protractor whose limb is divided as above described, and the dividing scale on the index, which moves round the limb of the protractor on a conical centre, gives every minute of a degree. That part of the index beyond the limb has a steel point fixed at the end, in a direct line with the centre of the protractor, and whose use is to prick off the proposed angles.

Fig. 3. is another circular protractor, a little differently constructed from the former. The central point is formed by the intersection of two lines crossing each other at right angles, which are cut on a piece of glass. The limb is divided into degrees and half degrees, having an index with a vernier graduated to count to a single minute, and is furnished with a tooth and pinion, by means of which the index is moved round by turning a small nut. It has two pointers, one at each end of the index, furnished with springs for keeping them suspended while they are bringing to any angle; and being being brought, applying a finger to the top of the pointer, and pressing it down, pricks off the angle. There is this advantage in having two pointers, that all the bearings round a circuit may be laid or pricked off, although the index traverses but one-half of the protractor.

Another circular protractor, different from either of the former, is represented at fig. 4. The centre is also formed by the intersection of two lines at right angles to each other, which are cut on glass, that all parallax may thereby be avoided. The index is moved round by a tooth and pinion. The limb is divided into degrees and half degrees, and subdivided to every minute by the vernier. The pointer may be set at any convenient distance from the centre, as the socket which carries it moves upon the bar BC, and is fixed thereto by the nut D, at right angles to the bar BC, and moveable with it. There is another bar EF; on this bar different scales of equal parts are placed; so that by moving a square against the inner edge thereof, angles may be transferred to any distance within the limits, from the centre containing the same number of degrees marked out by the index.

It would indeed be superfluous to describe any more of these circular protractors, especially as the little alterations in them depend very much upon the fancy of the artist. Suffice it however to say, that we have seen others still differently constructed, one of which we shall briefly describe. The divisions upon the limb of this instrument are similar to those already described; but the index is a straight bar continued to some considerable distance each way beyond the limb of the instrument, and has a vernier to show minutes as usual; a mark upon one of the edges of the index, always coincides with the centre of the instrument. Instead, therefore, of pricking down the angle as in the former, part of the line containing the angle may be drawn, which, although perhaps not so accurate as a point, is more conspicuous, and the line is easily completed upon removal of the protractor. The common dimensions of the circular part of these instruments is from five to ten inches diameter; and they are made of brass.