between two dots, any point of the sector must pass over 16 times that interval, that is to say, over an angle represented by ?^L x 16 = 22 30 . This interval was therefore divided by 16g, 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 1 20 ; and, in order to provide for the necessary gths 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 Avas 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 ^th of a division on the section which by construction is 1 20 . By proceeding in this Avay 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 pi oper division before commencing the third course. By pro ceeding in this manner through the whole circle, the microscope H was finally found with its Avire at zero, and the sector Avith 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 vork, 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 kuife 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 bs 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 lass 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 re quired division on the scale. Call this ratio a. Then, if the required divisions are longer than those of the pattern, the angle is cos~ l a, but, if shorter, the angle is sec~ J a. In the former case two operations are required before the divi sions are cut : first, the square is laid on the pattern, and the corresponding divisions merely notched very faintly on the edge of the vork; 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^a, 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 index is laid on the plate, with the notch grasping the cen tral pin, the straight edge of the steel bar lies exactly along a radius. The Avork to be graduated is laid flat on the divid ing plate, and fixed by two clamps in a position exactly con centric with it. The index is now laid on, with its edge coinciding with any required division on the dividing plate, and the corresponding division on the work is cut by draw ing the dividing knife along the straight edge of the index.
Machine Graduation.—The first dividing engine was probably that of Henry Hindley of York, constructed in. 1740, and used for the most part by him for cutting the teeth of clock wheels. This was followed shortly after by an engine devised by the Due de Chaulnes ; but the first engine which obtained distinct notoriety was that made by Ramsden, of which an account was published by the Board of Longitude in 1777. He was rewarded by that board with a sum of 300, and a further sum of 315 was given to him on condition that he would divide, at a certain fixed rate, the instruments of other makers. The essential principles of Ramsden s machine have been repeated in almost all succeeding engines for dividing circles, and it Avill be well, therefore, to give a brief description of it. It consisted of a large brass plate 45 inches in diameter, carefully turned, and movable on a vertical axis. The edge of the plate was ratched with 2160 teeth, into which a tangent screw Avorked, by means of Avhich the plate could be made to turn through any required angle. Thus six turns of the screw moved the plate through 1, and ^th O f a turn through -^-^th of a degree. On the axis of the tangent screAV was placed a cylinder having a spiral groove cut on its surface. A ratchet-wheel containing 60 teeth Avas attached to this cylinder, and Avas so arranged that, when the cylinder moved in one direction, it carried the tangent screAV with it, and so turned the plate, but Avhen it moved in the opposite direction, it left the tangent screw, and with it the plate, stationary. Round the spiral groove of the cylinder a catgut band was wound, one end of A T hich Avas attached to a treadle and the other to a counterpoise Aveight. When the treadle Avas depressed the tangent screw turned round, and when the pressure was removed it returned, in obedience to the Aveight, to its former posi tion without affecting the screw. Provision Avas also made Avhereby certain stops could be placed in the Avay of the screAV, which only allowed it the requisite amount of turning according to the gradua tion required. The Avork to be divided Avas firmly fixed on the plate, and made concentric Avith it. The divisions were cut, while the screw Avas stationary, by means of a dividing knife attached to a swing frame, Avhich alloAved it to have only a radial motion. In this Avay the artist could divide very rapidly by alternately depress ing the treadle and working the dividing knife. Ramsden also constructed a linear dividing engine on essentially the same principle. If we imagine the rim of the circular plate with its notches stretched out into a straight line and made movable in a straight slot, the screw, treadle, &c., remaining as before, we shall get a very good idea of the linear engine. In 1793 Edward Troughton finished a circular dividing engine, of which the plate was smaller than in Ramsden s, and which differed considerably otherwise in simplifying matters of detail. The plate was originally divided by Troughton s own ingenious method, already described, and the divisions so obtained were employed to ratch the edge of the plate for receiving the tangent screw with great accuracy. In the Transactions of the /Society of Arts for 1830-31 there is a full description, with illustrative figures, of a dividing engine, constructed by Andrew Ross, which differs considerably from those of Ramsden and Troughton. The essential point of difference is that, in Eoss s engine, the tangent screw does not turn the engine plate ; that is done by an independent apparatus, and the function of the tangent screw is only to stop the plate after it has passed through the required angular interval betAveentAvo divisions on the work to be graduated. Round the circumference of the plate are fixed 48 projections which just look as if the circumference had been divided into as many- deep and somewhat peculiarly shaped notches or teeth. Through each of these teeth a hole is bored parallel to the plane of the plate
