GRADUATION. There are few advances or refinements in the exact sciences which have not depended considerably on corresponding refinement in linear or angular measurement.
Graduation, or "dividing," as it is usually called, is the art of dividing straight lines, circular arcs, or whole circumferences into any required number of equal parts. It is the most important and difficult part of the work of the mathematical instrument maker.
According to Shuckburgh, from the time of Hipparchus and Ptolemy to that of Copernicus in the beginning of the 16th cen tury, few astronomical observations can be depended on to within less than 5 to io minutes of arc, those of Tycho Brahe (1546 16o1) being reliable to within one minute. The errors of Hevel ius's 6-foot sextant (middle 17th century) might amount to 15 or 20 seconds of arc, Flamsteed's sextant to 10 or 12 seconds, and Graham's 8-foot mural quadrant, used by Bradley from 1742, to 7 or 8 seconds.
There is still no such thing as a perfectly graduated circle in existence. A fair indication of the present state of progress towards this ideal may be found in the fact that it is possible to produce circles of 12 inches diameter divided into very many nearly equal parts by divisions none of which is out of its true position by more than two seconds of arc. Graduation may be considered under three heads, viz., original graduation, hand copy ing, and machine graduation performed by a dividing engine.
The foundation of the Royal Observatory at Greenwich, and the increasing demand for more accurate determination of the positions of the heavenly bodies, induced great advances in the art of graduation. In England, outstanding amongst those who made definite contributions to these advances from the late 17th to the early 19th century, are Abraham Sharp (1651-1742), Thomas Tompion (1639-1713), George Graham (1673-1751), Jonathan Sisson (c. 1690-1747), Jeremiah Sisson (c. 1715–C. 1780), John Bird (17o9-1776), John Smeaton (1724-1792), Jesse Ramsden John Troughton (c. 1747–c. 1790), Edward Trough ton (1753-1835), Thomas Jones (c. 1780–c. 1840), and William Simms (1793-1860).
The first example in which the method of performing the gradua tion is described in detail is the 8-ft. mural circle graduated by George Graham for Greenwich Observatory in 1725. In this two concentric arcs of radii 96.85 and 95.8 in. respectively were first described by the beam-compass. On the inner of these the arc of 9o° 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 obtained by continual bisection alone (which in Graham's opinion was the only accurate method), and would thus serve as a check upon the accuracy of the divisions of the inner arc. With the same distance on the beam-compass as was used to describe the inner arc, laid off from o°, the point 6o° was determined. With the points o° and 6o° as centres successively, and a distance on the beam-compass very nearly bisecting the arc 6o°, two slight marks were made on the arc ; the distance between these marks was divided by the hand aided by a lens, and this gave the point 30°. The chord of 6o° laid off from the point 3o° 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, and a table was constructed by which readings of the one could be converted into those of the other.
After the dots indicating the required divisions were obtained small arcs were drawn through them by the beam-compass having its fixed point somewhere on the line which was tangent to the quadrantal arc at the point where a division was to be marked.
The next important example of graduation was performed by Bird in 1767. His quadrant, which was also of 8-ft. radius, was divided into degrees and 12th parts of a degree. Bird computed the chords of certain arcs, so that when taken from an accurate scale of equal parts (previously constructed by him, employing continual bisection), and marked on the quadrant in their proper order, he obtained the point 85° 20'=1,024X 5'. As 1,024 is equal to the tenth power of 2, he was able to obtain 5' by continual bisection of this arc.
The Duc de Chaulnes published in 1768 a method of dividing in which greater accuracy was obtainable by replacing the points of a pair of beam compasses by two micrometer microscopes. These microscopes, having cross wires in the foci of their eye-glasses were fixed to a frame, and several pieces of brass with divisions on them could be temporarily secured by wax as required, to the circle to be divided; these were used as trial divisions. The microscopes were first fixed as nearly as possible at opposite ends of a diameter of the circle to be divided, and a trial division placed under each, so that the intersection of the cross wires in each case was coin cident with the middle of the division when viewed through the microscope. By repeatedly turning the circle half round, and by slight adjustment of the position of one microscope and one trial division, positions were obtained which were exactly diamet rically opposite. A cutting point was then placed over one division and a fixed microscope over the other, so that when any division was brought to bisect the cross wires in the microscope, the cutting point made one diametrically opposite. By a process of trial and adjustment with bisections and trisections, the circle was divided into spaces of ion, then by obtaining the arc of 9°, by trial on the arc of 180°, the circle could be divided into spaces of I °, or by similar means into smaller spaces.
In the first stages of an original graduation, Ramsden used beam compasses as employed by Bird. Micrometer microscopes were then used as in the method of the Duc de Chaulnes to measure the errors in the positions of the dots. Corrections were made by pressing the dots backwards or forwards by hand, with a fine conical point. This method, known as "coaxing," is capable of a great degree of accuracy, but is extremely tedious.
Ramsden's original graduation of the wheel of his dividing engine (described later) was as follows:—It was divided with the greatest exactness of which he was capable, first into 5 parts, and each of these into 3 ; these parts were then bisected 4 times. Supposing the whole circumference of the wheel to contain 2,160 teeth, this gave successively spaces corresponding to 36, 18 and 9 teeth. To check the accuracy of these, he divided another circle (one-tenth of an inch within the first one), by con tinual bisections, giving i,o8o, 540, 270, 135, 671 and 33i. Not finding any sensible differences as observed by means of a fixed radial thin silver wire and magnifying lens, he used the former set of divisions for reference in ratching the edge of the wheel.
The method of original graduation adopted by Edward Trough ton is fully described in the Philosophical Transactions for 1809, as employed by himself to divide a meridian circle of 4 ft. diameter, made for Stephen Groombridge, and now preserved in the Science Museum. The circle was first accurately turned both on its face and its inner and outer edges. A roller was next pro vided, of such diameter that it revolved 16 times on its own axis while made to roll once round the outer edge of the circle. The roller, of ter having been properly adjusted as to size, was divided as accurately as possible into 16 equal parts by radial lines near the edge. While the frame carrying the roller was moved once round along the circle, the points of coincidence of the roller divisions with the circle were accurately observed by two micro scopes attached to the frame, one of which commanded the ring on the circle near its edge, which was to receive the divisions, and the other viewed the roller-divisions. The points of contact thus ascertained were marked with faint dots, and the circle thereby divided into 256 very nearly equal parts.
From observations by means of two microscopes a table of errors in the positions of these dots was prepared. The last part of Troughton's process was to employ them in cutting the final divisions of the circle, which were to be spaces of 5' each. The mean interval between any two dots is 36o°/256= 5'X 16i, and in the final division, this interval must be divided into 16i parts. This was accomplished by means of an instrument called a sub dividing sector.
Troughton estimated (1809 paper) that 13 days of eight hours each would be well employed in dividing a 4-ft. circle by his own method, and 52 days by Bird's method; whereas the method by adjustment supposing every dot to be tried, and that two thirds of them wanted adjusting, would require approximately 15o days.
Copying.—In copying a linear scale the pattern and scale to be divided are first fixed side by side, with their upper faces in the same plane. The dividing square, resembling an ordinary joiner's square, is then laid across both, and the point of the dividing knife dropped into the 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 correspond ing division on the work is made by drawing the knife along the edge of the square with the right hand.
In copying circles use is made of the dividing plate. This is a circular plate of brass, three feet or more in diameter, carefully graduated near its outer edge. The work to be graduated is centred and clamped to the dividing plate, and by setting a radial straight-edge to any required division on the dividing plate, the corresponding division on the work is cut by drawing the dividing knife along the straight-edge.
Machine Graduation.—Henry Hindley of York, about constructed a small engine for cutting the teeth in clock wheels, and for dividing instruments. In this he used the roller method for the original division of the dividing plate, which was actuated by an endless screw.
In 1766 Jesse Ramsden had made his first dividing engine, with a dividing plate 3o inches in diameter. Though this engine gave more accurate results than the ordinary dividing plate method, and was good enough for dividing the circle of the com mon surveying instruments, it was not sufficiently accurate for nautical instruments used in the determination of position. In Ramsden had completed his second and very much better engine. A sextant divided by it was examined by Bird, who reported favourably on it. For his invention Ramsden received L615 from the Commissioners of Longitude on condition that he would divide sextants and octants for the trade at the rate of 6s. per sextant and 3s. per octant, also that he should instruct a certain number of persons (not exceeding ten) in the method of making and using the engine, during the period 28th October to 28th October 1777, the engine to become the property of the Commissioners.
Jesse Ramsden's engine consists of a horizontal wheel or plate 45 inches in diameter, which turns on a vertical axis; its outer edge is ratched or cut into 2,16o teeth, into which an endless screw gears. The downward stroke of a treadle turns the screw through any portion of a revolution as fixed by the setting of suitable mechanism. By means of a free wheel on the worm axis, the upward stroke of the treadle leaves the worm stationary. The circle to be divided is centred and fixed securely to the horizontal plate of the engine, and after each downward stroke of the treadle a division is cut by hand, the cutting point being carried in a frame (invented by Hindley) which allows only a radial to-and-fro motion of the point. One forward revolution of the screw advances the wheel through io minutes of arc. A brass plate on the screw arbor is divided into 6o parts, so that one division of this corresponds to 1 o seconds of arc on the wheel. In cutting the teeth on the wheel, the first light marks were made for the whole circle by a series of 240 operations, in each of which the space corresponding to 9 teeth was dealt with by 9 turns of the endless screw, the divided circle being referred to at the commencement of each operation so as to eliminate any slight errors which may have occurred during the previous oper ation. The whole series of operations was repeated three times round, to make the impression of the screw deeper. The wheel was then ratched round continuously about 30o times, until the teeth were finished. As the screw in ratching had continually hold of several teeth at the same time, and these continually changing, the inequalities of the teeth soon corrected themselves, and the teeth were reduced to what Ramsden described as "a perfect equality." This engine was used continuously by Ramsden until his death in 1800. Since that time it has been in the possession of several dividers, and is now in the U.S. National Museum at Washington, together with his original machine by which the endless screw of the dividing engine was cut. Ramsden also constructed a linear dividing engine on essentially the same principle, a straight rack taking the place of the notched rim of the circular plate.
In 1778 John Troughton completed an engine which had occu pied him three years. It was in general construction like that of Ramsden, but according to Edward Troughton it was thought to be superior in point of accuracy. Writing in 1830, he states : "The excellent engine of my late brother being fully four feet in diameter gave the operator, when at work near the centre, a position so painful, that it had done no good to either his health or my own, and has materially injured that of a worthy young man then my assistant; it was evident that, by making one of smaller dimensions, this evil would in a great measure be removed, and I foresaw that by employing my own method of original dividing from which to rack the plate, a considerable reduction might be effected without any sacrifice to accuracy. I also per ceived, that by contriving the parts with more simplicity than Ramsden had done, I could get through the work at less than two-thirds of the labour and expense. Such were my motives for making an engine and the work was accomplished in the year The description of the engine was published only in 1830. The engine was driven by a treadle and the divisions were cut by hand, as in Ramsden's engine. The circle is 34 inches in diameter, the worm has 20 threads to the inch, and the edge of the plate is ratched into 2,160 teeth.
In Troughton's dividing engine and those of Ramsden's con struction the operator could cut about 24 divisions per minute and could continue at this rate for hours, allowing for slight interruptions. In one minute as many as 3o divisions could be cut, but this rate could not be maintained.
In 1826 William Simms went into partnership with Troughton, who retired from business in 1831 and died in 1835. In Simms had completed his own dividing engine, which he described before the Royal Astronomical Society in 1843. The plate was 46 inches in diameter and divided with extreme care on a ring of silver into 4,320 divisions, adopting Troughton's method with some modifications. A single cutter, mounted in the endless screw frame, was used for ratching the edge of the plate, and as each of the 4,320 divisions in order was brought to coincide with the wires of a powerful microscope, the cutter was entered, and three circulations of the engine plate completed the work. Mr. Simms, in his paper, states:—"I was not without hope that the teeth on the edge would by this means be cut as truly as the original divisions themselves, and this expectation has, I believe, been fully realised." A new, important feature in this engine was the mechanism by which the engine became self-acting. When first used, it was driven by means of a descending weight in an open court ad joining the room in which the engine was placed ; later on. it was driven from an overhead shaft by a belt in the usual way.
The original Troughton engine of 1793 with self-acting mecha nism added by William Simms, appeared probably not long after Simms' larger engine had been made. The wooden stand, the worm mechanism, and the large circle were made by Trough ton in 1793. The casting seen on the left, with its wheelwork, the tracelet frame, and cam mechanism for moving the cutting knife up and down, and for regulating the length of cut, are due to Simms. This engine was in use almost continuously from until recent years, though it has been superseded for more accurate division by later and better engines constructed by Troughton and Simms. During the latter part of its existence it has not been used for graduating instruments reading to a greater accuracy than one minute of arc.
The re-graduation by hand of the two circles 3o and 24 inches in diameter of the Westbury altitude and azimuth circle made by Edward Troughton, which was performed by Simms in 1823, occupied nearly twelve weeks of six eight-hour days a week. By means of his automatic dividing engine (1843). after some five hours necessary for setting up the circles on the engine, the actual graduation would have been performed in about five hours.
There are two sets of teeth on the edge of the wheel, one having 4,32o teeth and the other 2,160.
After the circle to be graduated has been exactly centred and clamped upon the revolving wheel, the action of the engine is entirely automatic. The main frame, in which the revolving table moves, is of cast iron; the revolving table and its spindle are of phosphor bronze, and weigh about 5 cwt., the whole engine weighing about 2 tons. The original graduation into 4,320 equal divisions, which occupied five months, was made on three rings of silver inlaid on the top face near its outer edge. The gradua tions, successively made on circular lines cut on these silver rings, were re-cut from tables of errors many times, until the maximum error of any one division did not exceed o.6 of a second. From the final graduations the teeth on the edge of the revolving table were cut one by one. To avoid undue stress of the metal, and to maintain the keen edge of the V cutter, several cuts were taken, and the accuracy was checked by taking the mean of seven micrometer readings before each tooth was cut. The whole process of cutting the teeth occupied five weeks, care being taken to keep the temperature constant and uniform throughout the specially constructed room in which the opera tions were carried out. The machine is capable of graduating circles from 3 inches to 4 ft. 6 in. in diameter.
Dividing engines of different designs for both circular and linear graduation are manufactured in considerable numbers at Geneva by the Societe Genevoise d'Instruments de Physique. The machines designed and made for laboratory use can engrave lines either fine grade, 0.05 to 0.15 mm. (0.002" to 0.006") in thickness, or microscopic grade, 0.002 to 0.005 mm. (0.0001" to 0.0006") in thickness to a guaranteed accuracy of 0•002 mm. (o•000i") for linear machines and ±'1 second of arc for circular machines. The largest and most accurate of the circular dividing engines are I metre in diameter and the guaranteed accuracy is second of arc. An automatic correcting device is provided to compensate the slight errors existing in the spacing of the teeth on the periphery of the dividing wheel. In a larger machine, 2 metres in diameter, specially designed for use in gun factories, the accuracy is seconds.
In the automatic machines for workshop use, lines may be ruled either of medium or coarse grade, 0.1 to 0.2 mm. (0.004" to 0.008") in thickness to a guaranteed accuracy of ±0.010 mm. (0.0004") for linear machines, and 71:15 to ±30 seconds of arc for circular machines, at a rate of So to 200 lines per minute.
The high precision linear dividing engines are provided with a temperature compensation device which produces the same effect as if the pitch of the leading screw were varied by the small amount necessary to produce the correct compensation.