The handle h, gives rotation to the work; and at the same time, by means of the rack r r, and the pinion fixed on its axis, the handle traverses a slide which carries on its upper surface a bar i; the latter moves on a center, and may be set at any inclination by the adjusting screw and divisions; it is then fixed by its clamping screws. The slide s, carries the tool, and the end of this slide rests against the inclined plane i, through the intervention of a saddle or swing piece; the slide and tool are drawn to the left hand by the chain which is coiled round the barrel b, by means of a spiral spring contained within it.

Supposing the bar i i, to stand square or at zero, no motion would be impressed on the tool during its traverse, which we will suppose to require 10 revolutions of the pinion. But if the bar were inclined to its utmost extent, so that we may suppose the one end to project exactly one inch beyond the other, in reference to the zero line or the path of the slide, then during the 10 revolutions of the screw, the tool would traverse one inch, or the difference between the ends of the inclined bar i; and it would thereby cut a screw of the length of one inch, or the total inclination of the bar, and containing ten coils or threads.

But the inclination of the bar is arbitrary, and may be any quantity less than one inch, and it may lean either to the right or left; consequently the instrument may be employed in cutting all right or left hand screws, not exceeding 10 turns in length, nor measuring in their total extent above one inch, or the maximum inclination of the bar.

The principle of this machine may be considered faultless; but in action it will depend upon several niceties of construction, particularly the straightness of the slide and inclined bar, the equality of the rack and pinion, and the exact contact between the tool slide and the inclined plane. These difficulties augment very rapidly with the increase of dimensions; and probably the machine made by Mr. Adam Reid exclusively for cutting screws, is as large as can be safely adopted; the inclined plane is 44 inches long, but the work cannot exceed 1 6/16 inch diam., 2 1/4 inch long, or ten threads in total length. The application of the inclined plane to cutting screws is therefore too contracted for the ordinary wants of the engineer, which are now admirably supplied by the screw-cutting lathes with guide screws and change wheels.

* The drawing is the half size of fig. 1, plate xvii. of Ferdinand Berthoud's Essai sur L'Horlogerie, Paris, 1763. M. Berthoud says, "The instrument is the most perfect with which I am acquainted; it is the invention of M. le Lievre, and it has been reconstructed and improved by M. Gideon Duval." The templet or shaper plate determines the hyperbolical section of the fusee. Plate 37 of Rees's Cyclopedia contains an engraving of a different modification of the fusee engine, also with an inclined plane, which is ascribed to Hindley of York.

The accuracy of screws has always been closely associated with the successful performance of engines for graduating circles and right lines, and the next examples will be extracted from the published accounts of the dividing engines made by Mr. Rainsden.*

* This eminent individual received a reward from the Board of Longitude, upon the condition that he would furnish, for the benefit of the public, a full account of the methods of constructing and using his dividing machines, and which duly appeared in the following tracts: - "Description of an Engine for dividing Mathematical Instruments, by Ramsden, 4 to, 1777." Also, "Description of an Engine for Dividing Straight Lines, by Ramsdcn, 4to, 1779," from which the following particulars are extracted: -

The circular dividing engine consisted of a large wheel moved by a tangent screw; the wheel was 45 inches diameter, and had 2160 teeth, so that six turns of the tangent screw moved the circle one degree; the screw had a micrometer, and also a ratchet wheel of 60 teeth, therefore one tooth equalled one-tenth of a minute of a degree. The screw could be moved a quantity equal to one single tooth, or several turns and parts, by means of a cord and treadle, so that the circular works attached to the dividing wheel could be readily graduated into the required numbers, by setting the tangent screw to move the appropriate quantities; the dividing knife or diamond point always moved on one fixed radial line, by means of a swing frame.

In retching or cutting the wheel, says Mr. Ramsden, "the circle was divided with the greatest exactness I was capable of, first into 5 parts, and each of these into 3; these parts were then bisected 4 times; "this divided the wheel into 240 divisions, each intended to contain 9 teeth. The retching was commenced at each of the 240 divisions, by setting the screw each time to aero by its micrometer, and the cutter frame to one of the great divisions by the index; the cutter was then pressed into the wheel by a screw, and the cutting process was interrupted at the ninth revolution of the screw. It was resumed at the next 240th division (or nine degrees off), as at first, and so on.

This process was repeated three times round the circle, after which the retching was continued uninterruptedly around the wheel about 300 times; this completed the teeth with satisfactory accuracy. The tangent screw was subsequently made, as explained in the text

The first application of the tangent screw and ratchet to the purposes of graduation, appears to have been in the machine for cutting clock and watch wheels, by Pierre Fardoil; see plate 23 of Thiout's Traite Horlogerie, etc. Paris, 1741. At page 55 is given a table of ratchets and settings for wheels from 102 to 800 teeth.

In Mr. Ramsden,s description of his dividing engine for circles, he says: "Having measured the circumference of the dividing wheel, I found it would require a screw about one thread in a hundred coarser than the guide screw." He goes on to explain that the guide-screw moved a tool fixed in a slide carefully fitted on a triangular bar, an arrangement equivalent to a slide-rest and fixed tool; the screw to be cut was placed parallel with the slide and the guide-screw and copy were connected by two change wheels of 198 and 200 teeth (numbers in the proportion required between the guide and copy), with an intermediate wheel to make the threads on the two screws in the same direction. As no account is given of the mode in which the guide-screw was itself formed, it is to be presumed it was the most correct screw that could be obtained, and was produced by some of the means described in the beginning of the present section.