One very important application of the screw, is to the gradua-tion of mathematical scales, the screw is then employed to move a platform, which slides very freely, and carries the scale to be graduated; and the swing frame for the knife or diamond point is attached to some fixed part of the framing of the machine. Supposing the screw to be absolutely perfect, and to have fifty threads per inch, successive movements of fifty revolutions, would move the platform and graduate the scale exactly into true inches; but on close examination, some of the graduations will be found to exceed, and others to fall short of the true inch.

The scales assume, of course, the relative degree of accuracy of the screw employed. No test is more severe; and when these scales are examined by means of two microscopes under a magnifying power of ten or twenty times, the most minute errors become abundantly obvious, from the divisions of the scales, failing to intersect the cross wires of the instrument; the result clearly indicates, corresponding irregularities in the coarseness of the screw at the respective parts of its length. An accustomed eye can thus detect, with the microscope, differences not exceeding the one thirty-thousandth part of an inch, the twenty-five-thousandth part being comparatively of easy observation.

From Mr. Donkin's investigation of the subject, he was led to conclude that it is quite impossible to produce a screw which shall be absolutely free from error, when micrometrically proved; and in 1823, he was in consequence led to consider that as Mr. Maudslay's method of the bent lever and inclined straight bar, would compensate the error of total length in a nearly perfect screw, a similar mode might be applied to all the intermediate errors, by the employment of a curve, experimentally obtained by the method of continual bisection employed inn hand dividing.

It having been explained in reference to the diagram on page 644. that the inclination given to the bar i i, would reduce the effective length of a screw, and the reverse inclination would increase it, Mr. Donkin considered that from the observed fact of one half of the screw, (as estimated by counting the number of threads,) being generally too coarse, and the other half too fine, the compensation would require the one half of the bar i i, to be inclined to the right as in the diagram, and the other half to the left, in fact thus bending the right line into an obtuse angle.

Extending this mode, upon the presumption that the quarters, eighths, or sixteenths, of the screw were also dissimilar, the bar would require many flexures instead of the one only, giving to it a more or less zig-zag character, or rather that of a gently undulating line. The undulations being proportioned experimentally, to effect such compensations, as should add to the movement of the upper platform or supplementary table, where the screw was too fine, and subtract from its motion, where the screw was too coarse; so as, from a screw known to be slightly irregular, to produce the divisions of a scale, or the thread of another screw, considerably nearer to equality.

He carried out this project in 1826, and he has satisfactorily proved the existence of a correctional method, which is within reach of any clever workman who will devote sufficient patience to the adjustment of the engine, and which latter will be now briefly explained.

Mr. Donkin's dividing machine consists first of a table or platform moving on a railway, the platform being supported by four or any greater number of wheels, that may be required for preventing flexure and for diminishing friction. The upper edges of the two rails on which the wheels turn, are made as perfectly straight as possible, the rails lie in the same horizontal plane; and they are placed at any convenient distance from each other. The table or platform is guided laterally in its course upon the rails, by four wheels, of which two arc placed on each side of one of the rails; two wheels turn on fixed axles on one side of the rail, whilst the two on the other side are held tight to the rail by means of springs, thus preventing any deviation from the rectilinear course in which the platform ought to travel. To the under side of the platform is attached a clasp-nut, the two parts of which are so constructed, as to be applied to, or separated from the main screw, which lies below the platform, and is exactly parallel with the rails, or with the line in which the platform is made to move.

To effect the compensation, the platform or table consists of an Upper and lower plate, which arc capable of a small inde-pendent motion. The lower plate carries the fulcrum of the bent lever,whose arms are at right angles and as fifty to one, the lever moves in the vertical plane, so that its longer arm lies by gravity alone on the curvilinear edge of the compensation bar; upper platform is pressed endlong against the shorter arm of the bent lever, by a spring which always keeps them in close contact. The attachment of the two platforms is peculiar; the upper rides upon four rollers or rather sectors, and the two plates are connected by two slight rods placed transversely between them the ends of the rods are fixed over the one rail to the lower, and Offer the other rail to the upper platform; the bars consequently fulfil the office of the radius bars of a parallel rule, and suffice by their flexure alone, for the very limited and exact motion required in the upper table.

The compensating bar which is of the length of the screw, or

24 inches, has 48 narrow slips of metal placed like the keys of a piano-forte, each having an appropriate adjusting and fixing screw, by which the ends of the pieces may be placed in a con-tinuous line, or any of them may be placed above or below the line as required in the following mode of compensation. For change of total length and adjustment for temperature, the cur bar is more or less inclined, as in the former example, except that it is placed edgeways or vertically; it is attached to the outside of one of the rails, by a pivot which intersects the one end of its curvilinear edge, and the other end is raised or depressed by a screw, which effects the adjustment for temperature. Conceiving the length of the guide-screw divided into 48 equal parts, denoted by the figures 0 to 48, it would be first ascertained by two fixed microscopes, if the halves of the screw, measured from 0 to 24, and from 24 to 48, were absolutely equal quantities; if not, the central slip or finger would be raised or lowered until on repeated trials the due correctional movement was applied to the table. The two halves would he similarly bisected and corrected in the points 12 and 36, and the quarters again bisected in 6, 18, 80, and 42; and the eighths when also bisected, would extend the examination to the points 0, 3, 6, 9, etc., to 48. The easiest method is to compare the path of the slide, with the divisions of a superior scale, fixed upon the slide or platform of the machine.