objects in that edge produced. In this position he notes where the plumb-line intersects the arc of the quadrant. He then brings the same edge of the board to the direction of the second object, and notes again the intersection of the plumb-line with the quadrant. The difference of the two graduations thus noted is the measure of the angle required. But since only a few graduations are marked on this simple instrument, and since the observer can mostly select his own position, he should endeavour so to place himself, that, when he takes the first observation, the plumb-line shall pass over an exact division of the quadrant. The instrument is especially useful in measuring the height of a lofty building, or tree, whose base is accessible. In this case a single observation only is needed. The observer takes up such a position that, when the instrument is rightly fixed, without moving the board at all, by placing his eye at B he sees the top of the building in BA produced. He then knows that the line BA, produced to meet the top of the building, makes an angle of 45° with the vertical; and therefore the height required is equal to the horizontal distance of B from the building (which is readily measured), with the dif ference of level between B and the foot of the building added or subtracted, as the case may be. THE VERNIER. 272. The VERNIER, so called from the name of its inventor, is an instrument for measuring very small quantities on a graduated scale, either straight, as in the Barometer, or circular, as in the Theodolite, or Graphometer, or in many astronomical instruments. Let AB be any convenient unit, as one inch, on the limb of an instrument, and divided into 10 equal parts. It is required to subdivide each of these 10 parts into 10 more, or the whole into 100 equal parts, without making 100 lines between A and B; for if they were made, no eye could count them. PART III. 8 30 9 8 B 10 D = For this purpose annex a small sliding instrument CD, called a vernier, equal to in., and divide it into 10 equal parts; then, since the whole CD is equal to in., each of its parts is equal to one-tenth of in. 180 in., orin. o in.; that is, each division of the vernier is less than one of the limb AB, by of an inch. Then, if the lowest points of AB and CD are made to coincide, the 2nd mark on the vernier is lower than the 2nd on the limb by 18 in., or '02 in.; the 6th on the vernier is lower than the 6th on the limb by 18 in.; or 06 in.; and so on. 7 9 6 8 6 7 4 6 3 16 2 14 The divisions of both the limb and the vernier are measured upwards. The mode of using the vernier will be seen from the accompanying diagram of the upper portion of a Barometer. Let the upper end of the column of mercury stand somewhere between the 7th and 8th divisions of the 30th inch AB. Slide the vernier so that its highest point may come exactly opposite the head of the column. It is required to know by how many hundredths the upper end of the vernier is above the number 7 on the limb. Observe which division of the vernier corresponds with some one division of AB, so that they run into one horizontal line; let it be 6: then since each division of the vernier is onehundredth shorter than the one in the instrument, the 7 on the vernier is one-hundredth below the division 5 on the limb nearest to it; 8 on the vernier is two-hundredths below 6 on the limb; and 10 is four-hundredths below 8 on the limb, or six above the 7; that is, the number 6, at the point where the divisions of the limb and vernier coincide, tells how many hundredths the top of the vernier is above the highest division reached by it on the limb. In this case, therefore, the reading of the height of the column will be 29 in. 7 tenths, 6 hundredths, or 29.76 in. We see that the degree of accuracy to which the vernier measures is of th of an inch, or 0 in., because both limb and vernier were divided into 10 equal parts. If they had each been divided into 20 equal parts, the accuracy would have been carried to of th of an inch, or of an inch. 273. The limb may be circular, as in the next diagram; and here let the limb be divided into degrees, and the vernier, which is equal in length to 9 of these degrees, be subdivided into ten equal parts, as before: these will therefore enable us to measure tenths of a degree, or portions of 6 minutes. Let the extremity of the vernier fall, suppose, between the 45th and 46th degrees; and the division marked 8 on the vernier coincide with some division on the limb, then the reading will be 45o, and 8 portions of 6 minutes, or 45° 48′. Instead of taking the vernier one-tenth less than an inch, or than 10 degrees, we might take it one-tenth more; then, as before, the difference between the divisions on the limb and the vernier would be one-hundredth; but the divisions on the vernier would be numbered from the top. This is usually the case in the older barometers. THE CIRCULAR PROTRACTOR. 274. It was mentioned in (236) that a more complete and accurate form of Protractor was used in actual practice than the one there described. It consists of a complete brass circle, crossed by a brass band. On each semicircumference is a vernier, described in (274), for reading twice any angle observed; (the importance of two verniers will be seen presently). The advantages of this instrument over the semicircular one seen in ordinary cases of instruments are as follows: I. In the semicircular instrument, [see fig. to Art. (233),] in order accurately to measure an angle already laid down on the paper, or to lay down an angle, it is necessary to make the straight edge AB coincide with the middle of the black stroke representing one of the given mathematical lines which bound the angle, Now it is very difficult to do this with accuracy; but the desired coincidence is effected in the circular instrument, by placing the Protractor over the given line, so that two of the fine lines which mark the divisions on the limb, and which are always parts of a diametral line, may coincide with it; when, of course, the centre of the instrument will also be in that line. II. If the centre of the instrument be not obtained with perfect accuracy, the error in any angle observed, and arising from this inaccurate position of the centre, called the eccentricity of the circle, is compensated for by measuring the vertical or opposite angle with the second vernier, which angle will of course give as much too large an arc, as the first did too small, or the converse; and the mean of the two observed angles is then taken. III. An angle also can be taken from the circumference instead of from the centre, as in [Prob. 16. (3) p. 252], by bringing the circumference of the Protractor over the angular point C, and observing the arc AB intercepted between the lines AC, BC, half of which will be the measure of the angle ACB. THE GRAPHOMETER. 275. This instrument consists of a semicircle, or, which is much better, a complete circle of brass*, whose rim is divided into 180°, in the one case, and 360° in the other, with two diametral bands, one fixed, and the other moveable about the centre. At right angles to the extremities of each of these diametral bands is placed a pinnule, or sight. This consists of a thin, oblong, flat piece of brass, as represented at A and B, about 12 inches high, and having a slit pierced lengthways, and in the middle. One half of this slit is very narrow; the other is much broader, and is bisected lengthways by a wire, which, if continued, would also bisect the narrow part of the slit. At one end of each diametral band the narrow slit is up * The advantages of the complete circle over the semi-circle are well known to practical observers. And where great accuracy of measurement is of importance, the semi-circular instrument ought never to be used. permost; at the other it is reversed. Each end of the moveable diametral band is furnished with a vernier for reading off angles. The whole is attached centrally to a pivot, which works by a universal joint in an upright pillar resting upon a tripod. A compass and a spiritlevel are attached, so that the diametral bands of the instrument can be placed in any required position, with respect to the points of the compass, and the plane of the circle be made horizontal. The graphometer is used for taking angles, most commonly, either in an horizontal, or vertical, plane; but it may be turned in any direction, so as to bring it into the plane of any two or more objects whose bearings are required. The observer looks through the narrow slit, and therefore has the larger opening of the opposite sight in the direction of the object observed. He finds the object through that latter opening, and brings the wire which bisects it into the same plane with the narrow slit close to the eye, so that the plane passing through the wires bisects the object. Let E be the place of observation; O, P, and Q, objects in an horizontal plane, whose relative positions the observer wishes to ascertain. Place the circle in the plane passing through O, P, and Q, and let the fixed diametral band be directed to P, so that to the observer at p the vertical plane, passing through the wires of A and its opposite sight, may bisect P. Direct the moveable diametral band DH, so |