multiplied at each division by tlie number of that division. Errors so made are usually adjusted by the dots being put either back or forward a little by means of the dividing punch guided by a magnifying glass. This is an extremely tedious process, as the dots, when so altered several times, are apt to get insufferably large and shapeless. The division of circular arcs is essentially the same in principle as the graduation of straight lines, and of this we shall now give some examples. The first example of note is the 8-feet mural circle which was graduated by Graham for the National Observatory in 1725. In this two concentric arcs of radii 96 85 and 95 - 8 inches respectively were first described by the beam-compass. Ou the inner of these the arc of 90 was to be divided into degrees and 12th parts of a degree, while the same on the outer was to be divided into 96 equal parts, and these again into 16th parts. The reason for adopting the latter was that, 96 and 16 being both powers of 2, the divisions could be got at by continual bisection alone, which, in Graham s opinion, who first employed it, is the only accurate method, and would thus serve as a check upon the accuracy of the divisions of the outer arc. With the same distance on the beam-compass as was used to describe the inner arc, laid off from 0, the point 60 was at once determined. With the points 0" and 60 as centres successively, and a distance on the beam-compass very nearly bisecting the arc of 60, two slight marks were made on the arc ; the distance between these marks was carefully divided by the hand aided by a lens, and this gave the point 30. The chord of 60 laid off from the point 30 gave the point 90, and the quadrant was now divided into three equal parts. Each of these parts was similarly bisected, and the resulting divisions again trisected, giving 18 parts of 5 each. Each of these quinquesected gave degrees, the 12th parts of which were arrived at by bisecting and trisecting as before. The outer arc was divided by continual bisection alone, and a table was constructed by which the readings of the one arc could be converted into those of the other. After the dots indi cating the required divisions were obtained, either straight strokes all directed towards the centre were drawn through them by the dividing knife, or sometimes small arcs were drawn through them by the beaui-compass having its fixed point somewhere on the line which was a tangent to the quadrantal arc at the point where a division was to be marked. The next important example of graduation was done by Bird in 1767. His quadrant, which was also 8 feet radius, was divided Into degrees and 12th parts of a degree. He employed the method of continual bisection aided by chords taken from an exact scale of equal parts, which could read to 001 of an inch, and which he had previously graduated by continual bisections. AVith the beam- compass an arc of radius 95 938 inches was first drawn. From this radius the chords of 30 , 15 , 10 20 , 4 40 , and 42 40 were com puted, and each of them by means of the scale of equal parts laid off on a separate beam-compass to be ready. The radius laid off from D gave the point 60 ; by the chord of 30 the arc of 60 was bisected ; from the point 30 the radius laid off gave the point 90 ; the chord of 15 laid off backwards from 90 gave the point 75 ; from 75 was laid off forwards the chord of 10 20 ; and from 90 was laid off backwards the chord of 4 40 ; and these were found to coincide in the point 85 20 . Now 85 20 being = 5 x 1024 = 5 x 2 10 , the final divisions of 85 20 were found by continual bisections. For the remainder of the quadrant beyond 85 20 , containing 56 divi sions of 5 each, the chord of 64 such divisions was laid off from the point 85 40 , and the corresponding arc divided by continual bisections as before. There was thus a severe check upon the accuracy of the points already found, viz., 15, 30, 60, 75, 90, which, however, were found to coincide with the corresponding points obtained by continual bisections. The short lines through the dots were drawn in the way already mentioned. The next eminent artists in original graduation are the brothers John and Edward Troughton. The former was the first to devise a means. of graduating the quadrant by continual bisection without the aid of such a scale of equal parts as was used by Bird. His method was as follows : The radius of the quadrant laid off from gave the point 60. This arc bisected and the half laid off from 60 gave the point 90. The arc between 60 and 90 bisected gave 75; the arc between 75 and 90 bisected gave the point 82 30 , and the arc between 82 30 and 90 bisected g,ve the point 86 15 . Further, the arc between 82 30 and 86" 15 trisected, and two- thirds of it taken beyond 82 30 , gave the point 85, while the arc between 85 and 86 15 also trisected, and one-third part laid off beyond 85, gave the point 85 25 lastly, the arc between 85 and 85 25 being quinquesected, and four-fifths taken beyond 85, gave 85 20 , which as before is = 5 x 2 10 , and so can be finally divided by continual bisection.
Edward Troughton's Method.—The method of original graduation discovered by Edward Troughton is fully described in the Philo sophical Transactions for 1809, as employed by himself to divide a meridian circle of 4 feet radius. The circle was first accurately turned both on its face and its inner and outer edges. A roller was next provided, of such diameter that it revolved 16 times on its own axis while made to roll once round the outer edge of the circle. This roller, made movable on pivots, was attached to a frame-work, which could be slid freely, yet tightly, along the circle, the roller meanwhile revolving, by means of fiictioual contact, on the outer edge. The roller was also, after having been properly adjusted as to size, divided as accurately as possible into 16 equal parts by lines parallel to its axis. AVhile the frame carrying the roller was moved once round along the circle, the points of contact of the roller-divisions with the circle were accurately observed by two microscopes attached to the frame, one of which (which we shall call H) commanded the ring on the circle near its edge, which was to receive the divisions, and the other viewed the roller- divisions. The exact points of contact thus ascertained were marked with faint dots, and the meridian circle thereby divided into 256 very nearly equal parts. The next part of the operation was to find out and tabulate the errors of these dots, which are called apparent errors, in conse quence of the error of each dot being ascertained on the supposition of its neighbours being all correct. For this purpose two micro scopes (which we shall call A and B) were taken, with cross wires and micrometer adjustments, consisting of a screw and head divided into 100 divisions, 50 of which read in the one and 50 in the opposite direction. These microscopes, A and B, were fixed so that their cross-wires respectively bisected the dots and 128, which were supposed to be diametrically opposite. The circle was now turned half-way round on its axis, so that dot 128 coincided with the wire of A, and, should dot be found to coincide with B, then the two dots were sure to be 180 apart. If not, the cross wire of B was moved till it coincided with dot 0, and the number of divisions of the micrometer head noted. Half this number gave clearly the error of dot 128, and it was tabulated -for -according as the arcual distance between and 128 was found to exceed or fall short of the remaining part of the circumference. The micro scope B was now shifted, A remaining opposite dot as before, till its wire bisected dot 64, and, by giving the circle one quarter of a turn on its axis, the difference of the arcs between dots and 64 and between 64 and 128 was obtained. The half of this difference gave the apparent error of dot 64, which was tabulated with its proper sign. AVith the microscope A .still in the same position the error of dot 192 was obtained, and in the same way by shifting B to dot 32 the errors of dots 32, 96, 160, and 224 were successively ascertained. By proceeding in this way the apparent errors of all the 256 dots were tabulated. From this table of apparent errors a table of real errors was drawn up by employing the following formula: ^(x a + x e ) + z = te real error of dot b, where x a is the real error of dot a, x c the real error of dot c, and * the apparent error of dot b midway between a and c. Thus having got the real errors of any two dots, the table of apparent errors gives the means of finding the real errors of all the other dots. The above formula is easily, derived as follows : Let a and c be the number of micrometer divisions from to dots a and c respectively; and let b be the number of similar divisions from to the point b, supposed to be midway between a and c. Also let x a and x c be the real errors of dots a and c, and z the apparent error of b. Then we have z=b-a- ^(c- a) = b- ! 2 ( Now the real positions of a and c are a ,r and c a c respectively. Therefore the real position of b, the point midway between a and and c, is = a - x a + i { c - y c - (a - x a ) ] = 4( + e)-i(a!a + !,,). Therefore the real error of b = b - real position of b Having obtained the 256 dots and their real errors, the third and last part of Trough ton s process was to employ them to cut the final divisions of the circle, which were to be spaces of f/ each. Now the mean interval between any two dots is _ = 5 x 16|, 256 and hence, in the final division, this interval must be divided into 16| equal parts. In order to accomplish this a small instrument, called a subdividing sector, was provided. It was formed of thin brass and had a radius about four times that of the roller, but made adjustable as to length. The sector was placed concentrically on the axis, and rested on the upper end of the roller. It turned by frictional adhesion along with the roller, but at the same time was sufficiently loose to allow of its being moved back by hand to any position without affecting the roller. Now it is evident that, while
the roller passes over an angular space equal to the mean interval
