24 X 20 X 1 = 480 or reduced to its lowest terms 1 60 X 72 X 6 = 25920 54

The fraction denotes that 1/34 of an inch is the pitch of the screw, or the interval from thread to thread; also that it has 54 threads in each inch, and which is called the rate of the screw.

And in C, the numbers in which example were selected at random, the screw would be found to possess rather more than 85 threads per inch.*

 27 X 39 X 1 1114 or reduced to its lowest terms 1 53 X 107 X 6 39026 35 71/1113

In imitation of the method of change-wheels, the slide-rest screw is sometimes moved by an arrangement of catgut bands, resembling that represented in Besson's screw lathe, page 616.

One band proceeds from the pulley on the mandrel to a spindle overhead having two pulleys, and a second cord descends from this spindle to a pulley on the slide-rest.f The method offers facility in cutting screws of various pitches, by changing the pulleys, and also either right or left hand screws, by crossing or uncrossing one of the bands.

* The fractions should be reduced to their lowest terms before calculation, to avoid the necessity for multiplying such high numbers. Thus the first example would become reduced to 1/3x1/3x1/6 = 1/54 and would be multiplied by inspection alone, as the numerators and denominators may be taken crossways if more convenient; thus 32/12 is equal to 1/3, and 20/60 is also equal to 1/2, fractions which are smaller than | and ft, the lowest terms respectively of 24/60 and 20/72; the second case could not be thus treated, and the whole numbers must there be multiplied, as they will not admit of reduction. Other details will be advanced, and tables of the combinations of the change-wheels will be also given, in treating of the practice of cutting screws.

† This apparatus has been applied to cutting the expanding horn snakes. See Manuel du Tourneur, first edit, 1796, vol. il, plate 21; and second edit., 1816, vol. il, plate 16; see also page 1245 of the first volume of this work.

The plan is unexceptionable, when applied for traversing the tool slowly for the purpose of turning smooth cylinders, or surfaces (which is virtually cutting a screw or spiral of about 100 coils in the inch); and in the absence of better means, pulleys and bands are sometimes used in matching screws of unknown or irregular pitches, by the tedious method of repeated trials; as on slightly reducing, with the turning tool, the diameter of either of the driving pulleys, the screw or the work becomes gradually finer; and reducing either of the driven pulleys makes it coarser; but the mode is scarcely trustworthy, and is decidedly far inferior to its descendant, or the method of change wheels.

The screw tools, or chasing tools, employed in the traversing lathes for cutting external and internal screws, resemble the fixed tools generally, except as regards their cutting edges; the following figures 604 to 606 refer to angular threads, and 607 and 608 to square threads.

Angular screws are sometimes cut with the single point, fig. 604, a form which is easily and correctly made; the general angle of the point is about 55° to 60°, and when it is only allowed to cut on one of its sides or bevels, it may be used fearlessly, as the shavings easily curl out of the way and escape. But when both sides of the single point tool are allowed to cut, it requires very much more cautious management; as in the latter case, the duplex shavings being disposed to curl over opposite ways, they pucker up as an angular film, and in fine threads they are liable to break the point of the tool, or to cause it to dig into, and tear, the work. Sometimes, also, a fragment of the shaving is wedged so forcibly into the screw by the end of the tool, that it can only be extricated by a sharp chisel and hammer.

In cutting angular screws, it is very much more usual and expeditious to employ screw tools with many points, which are made in the lathe by means of a revolving cutter or hob, figs. 550 and 551, page 591. Screw tools with many points, are always required for those angular threads which are rounded at the top and bottom, and which arc thence called rounded or round threads.*

Mr. Clement gives to the screw tool for rounded threads the profile of fig. 605, which const ruction allows the tool to be inverted, so that the edges may be alternately used for the purpose of equalizing the section of the thread. In making the tool 605, the hob (whinch is dotted), is put between centers in the traversing lathe, and those wheels are applied which would serve to cut a screw of the same pitch as the hob; the bar of steel is then fixed in the slide rest, so that the dotted line or the axis of the tool intersects the center of the hob. The tool is afterwards hollowed on both sides with the file, to facilitate the sharpening, and it is then hardened. In using the tool, it is depressed until cither edge comes down to the radius, proceeding from (black) circle, which is supposed to represent the screw to be cut; the depression gives the required penetration to the upper angle, and removes the lower out of contact.†

Mr. Bodmcr's patent chasing tool is represented in fig. 606; the cutter, c, is made as a ring of steel which is screwed internally to the diameter of the bolt, and turned externally with an undercut groove, for the small screw and nut by which it is held in an iron stock, s, formed of a corresponding sweep; for distinctness the cutter and screw are also shown detached. The center of curvature of the tool is placed a little below the center of the lathe, to give the angle of separation or penetration; and after the tool has been ground away in the act of being sharpened, it is raised up, until its points touch a straight edge applied on the line a a of the stock; this denotes the proper height of center, and also the angle to which the tool is intended to be hooked, namely 10 degrees: each ring makes four or five cutters, and one stock may be used for several diameters of threads.