for each turn of the mandril, it follows that the cutting tool will move longitudinally a distance equal to the pitch of the guiding-screw while the bar placed in the lathe makes one turn. Thus the point of the cutter will form on the bar a screw having the same pitch as the guiding-screw of the lathe. Here we have a striking illustration of the copying principle, for the lathe thus produces an exact copy of the screw which it contains. The screw-thread is traced out on the cylindrical bar, which is operated upon by the combination of the circular motion of the mandril with the longitudinal movement of the slide-rest. By modifying the relative amounts of these movements, screw-threads of any desired pitch can be made, and it is for this purpose that the _change wheels_ are provided. If the thread of the guiding-screw makes two turns in one inch, one revolution of the wheel C will advance the cutter half an inch along the length of the bar. If the numbers of teeth in the wheels be such that the wheel D makes ten revolutions while C is making one, then in the length of half an inch the thread of the screw produced by the cutter will go round the core ten times, or, in technical language, the screw will be of 1/20 inch pitch. [Illustration: FIG. 30.—_Whitworth’s Measuring Machine._ ] Since a screw turning in a nut advances only its pitch distance at each revolution, a finely-cut screw furnishes an instrument well adapted to impart a slow motion, or to measure minute spaces. Suppose a screw is cut so as to have fifty threads in an inch, then each turn will advance it 1/50 in.; half a turn 1/100 in.; a quarter of a turn, 1/200, and so on. It is quite easy to attach a graduated circle to the head of the screw, so that, by a fixed pointer at the circumference, any required fraction of a revolution may be read off. Thus if the circle had two hundred equal parts, we could, by turning the screw so that one division passed the index, cause the screw to advance through 1/200 of 1/50 inch, or 1/10000 part of an inch. This is the method adopted for moving the cross-wires of the instruments for measuring very small spaces under the microscope. Sir Joseph Whitworth, who has done so many great things in mechanical art, was the first mechanician to perceive the importance of extreme accuracy of workmanship, and he invented many beautiful instruments and processes by which this accuracy might be attained. Fig. 30 represents one of his measuring machines, intended for practical use in the workshop, to test the dimensions of pieces of metal where great precision is required. The base of the machine is constructed of a rigid cast iron bed bearing a fixed headstock, A, and a movable one, B, the latter sliding along the bed, C, with a slow movement, when the handle, D, is turned. This slow motion is produced by a screw on the axis, _a_, working in the lower part of the headstock, just as the slide-rest is moved along the bed of the lathe.