It is obvious that the first graduated instrument must have been done by the hand and eye alone, whether it was in the form of a straight-edge with equal divisions, or a screw, or a divided plate; but, once in the possession of one such divided instrument, it was a comparatively easy matter to employ it as a standard, and copy its divisions on any other article that might be desired. Hence graduation naturally divides itself into two distinct branches, original graduation and copying, which latter may be done either by the hand or by a machine called a dividing engine. We may thus speak of graduation under the three heads of original graduation, copying, and machine graduation. Original Graduation.— This is by far the most difficult part of the art—so difficult, indeed, and requiring such accuracy of hand and eye, that but few in any generation have been completely competent, for the task. The earlier astronomers graduated their own instruments, and, from the examples that have come down to us, it must have been very roughly done as compared with modern work.

In regard to the graduation of straight scales, we have, by elementary geometry, the means, theoretically, of dividing a straight line into any number of equal parts; but the practical carrying out of the geometrical construction is so beset with difficulties as to render the method untrustworthy. This method, which employs the common diagonal scale, was used in dividing a quadrant of 3 feet radius, which belonged to Napier of Merchiston, and which only read to minutes—a result, say Thomson and Tait {Nat. Γhilf f “giving no greater accuracy than is now attainable by the pocket sextants of Troughton and Simms, the radius of whose arc is little more than an inch.”

The original graduation of a straight line is, in practice, done either by the method of continual bisection or by stepping. In continual bisection the entire length of the line is first laid down. Then, as nearly as possible, half that distance is taken in the beam-compass and marked off by faint arcs from each end of the line. Should these marks coincide the exact middle point of the liue is obtained. If not, as will almost always be the case, the distance between the marks is carefully bisected by hand with the aid of a magnifying glass. The same process is again applied to the halves thus obtained, and so on in succession, dividing the line into parts represented by 2, 4, 8, 16, <fcc., till the desired divisions are reached. In the method of stepping the smallest division required is first taken, as accurately as possible, by spring dividers, and that distance is then laid off, by successive steps, from one end of the line. It is evident that, in this method, any error at starting will be multiplied at each division by the 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. On 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 w⅛s 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 indicating 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 beam-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. With the beamcompass 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 computed, 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 0° 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' × 1024 = 5' × 2 10 , the final divisions of 85° 20' were found by continual bisections. For the remainder of the quadrant beyond 85° 20', containing 56 divisions 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 rneansmf 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 0’ 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 gave 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' lΛstly, the arc between 85° and 85° 25' being quinquesected, and four-fifths taken beyond 85°, gave 85° 20', which as before is = 5' × 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 Philosophical Transactions for 1809, as employed by himself to divide a meridian circlé 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 frictional 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 bylines parallel to its axis. While 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 rollerdivisions. 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 consequence of the error of each dot being ascertained on the supposition of its neighbours being all correct. For this purpose two microscopes (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 0 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 0 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 + or - according as the arcual distance between 0 and 128 was found to exceed or fall short of the remaining part of the circumference. The microscope B was now shifted, A remaining opposite dot 0 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 0 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. With 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 :—

⅜(α α + z e ) + z=the real error of dot b, where x a is the real error of dot a, x c the real error of dot c, and x 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 0 to dots a and c respectively; and let b be the number of similar divisions from 0 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 - J (c - α) = b - ⅜(a + c).

Now the real positions of a and c are a-x a and c-x e respectively.

Therefore the real position of b, the point midway between a and and c, is =a-x a + ⅛{c-Xc-(a-x t i)}

= 4(α + c)-⅛(z α + a⅛).

Therefore the real error of b = b- real position of b

= δ - 4 (α + c) + ⅜(ic α + αr c )=a + ⅛ (a;« + ¾)∙

Having obtained the 256 dots and their real errors, the third and last part of Troughton’s process was to employ them to cut the final divisions of the circle, which were to be spaces of 5' each. Now the mean interval between any two dots is = 5' × 16j, 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 augular space equal to the mean interval between two dots, any point of the sector must pass over 16 times that interval, that is to say, over an angle represented by

× 16 = 22° 30'. Tlιis interval was therefore divided by I65, and 256

a space equal to 16 of the parts taken. This was laid off on the arc of the sector and carefully divided into 16 equal parts, each equal to l o 20'; and, in order to provide for the necessary ⅞ths of a division, there was laid off at each end of the sector, and beyond the 16 equal parts, two of these parts each subdivided into 8 equal parts. A microscope with cross wires, which we shall call I was placed on the main frame, so as to command a view of the sector divisions, just as the microscope H viewed the final divisions of the circle. Before the first or zero mark was cut, the zero of the sector was brought under I and then the division cut at the point on the circle indicated by H, which also coincided with the dot 0. The frame was then slipped along the circle by the slow screw motion provided for the purpose, till the first sector-division, by the action of the roller, was brought under I. The second mark was then cut on the circle at the point indicated by H. That the marks thus obtained are 5' apart, as they should be, is evident when we reflect that the distance between them must be ⅛th of a division on the section which by construction is 1° 20'. By proceeding in this way the first 16 divisions were cut; but before cutting the 17th it was necessary to adjust the micrometer wires of H to the real error of dot 1, as indicated by the table, and bring back the sector, not to zero, but to ∣th short of zero. Starting from this position the divisions between dots 1 and 2 were next filled in, and then H was adjusted to the real error of dot 2, and the sector brought back to its proper division before commencing the third course. By proceeding in this manner through the whole circle, the microscope H was finally found with its wire at zero, and the sector with its 16th division under its microscope indicating that the circle had been accurately divided.

Copying.— In graduation by copying the first requisite is a pattern, which must be either an accurately divided straight scale, or an accurately divided circle, commonly called a dividing plate.

In copying a straight scale the pattern and scale to be divided, usually called the work, are first fixed side by side, with their upper faces in the same plane. The dividing square, which closely resembles an ordinary joiner’s square, is then laid across both, and the point of the dividing knife dropped into the zero division of the pattern. The square is now moved up close to the point of the knife; and, while it is held firmly in this position by the left hand, the first division on the work is made by drawing the knife along the edge of the square with the right hand. Great care must be taken that the knife is held exactly in the same position in cutting the division and in setting the square.

It frequently happens that the divisions required on a scale are either greater or less than those on the pattern. To meet this case, and still use the same pattern, the work must be fixed at a certain angle of inclination with the pattern. This angle is easily found in the following way. Take the exact ratio of a division on the pattern to the required division on the scale. Call this ratio α. Then, if the required divisions are longer than those of the pattern, the angle is co3 -1 α, but, if shorter, the angle is sec -1 α. In the former case two operations are required before the divisions are cut: first, the square is laid on the pattern, and the corresponding divisions merely notched very faintly on the edge of the work; and, secondly, the square is applied to the work and the final divisions drawn opposite each faint notch. In the second case, that is, when the angle is sec^ 1 α, the dividing square is applied to the work, and the divisions cut when the edge of the square coincides with the end of each division on the pattern.

In copying circles use is made of the dividing plate. This is a circular plate of brass, of 36 inches or more in diameter, carefully graduated near its outer edge. It is turned quite flat, and has a steel pin fixed exactly in its centre, and at right angles to its plane. For guiding the dividing knife an instrument called an index is employed, This consists of a straight bar of thin steel of length equal to the radius of the plate. A piece of metal, having a V notch with its angle a right angle, is riveted to one end of the bar in such a position that the vertex of the notch is exactly in a line with the edge of the steel bar. In this way, when the...