h a common multiple. They are seldom or
never removed and their diameters are made dissimilar only for the
purpose of giving to the guide screw a slower rate of motion than
that of the mandrel whereby it may be made of coarser pitch than
that of the screw to be cut and its wear materially lessened.

The introduction of gearing between the spindle and the lead screw, for
whatever purpose, could not help but introduce variable factors caused
by inaccuracies in the gears themselves and in their mounting. These
were of little consequence for common work, particularly when coupled to
a screw which, itself, was of questionable accuracy. The increasing
refinements demanded in scientific instruments and in machine tools
themselves after they had reached a relatively stable form dictated that
attention be dedicated to improved accuracy of the threaded components.

[Illustration: Figure 15.--MAUDSLAY'S WELL-KNOWN screw-cutting lathe of
1797-1800, showing the method of mounting and driving changeable master
screws. (_Photo courtesy of The Science Museum, London._)]

[Illustration: Figure 16.--HEADSTOCK OF A GERMAN INSTRUMENT-MAKER'S
LATHE, typical of the mid-19th century, showing the traverse spindle,
interchangeable lead screws, and semicircumferential nut containing
several leads. The nut may be brought into engagement by the lever at
top rear of the headstock. This releases the end thrust control on the
spindle simultaneously with engagement of the nut. (Smithsonian photo
49839.)]

[Illustration: Figure 17.--DAVID WILKINSON'S SCREW-CUTTING LATHE,
patented in the United States in 1798. Note the ready facility with
which the lead screw may be exchanged for another and the same means of
supporting and driving as in figure 15. (U.S. National Archives photo.)]

An attack on this problem, which interestingly reverts to the
fundamental principle of motion derived from a master screw without the
intervention of other mechanism (fig. 19), is covered by a patent[5]
issued to Charles Vander Woerd, one-time superintendent of the Waltham
Watch Company. The problem is well stated in the patent:

This invention relates to the manufacture of leading screws to be
used for purposes requiring the highest attainable degree of
correctness in the cutting of the screw-threads of said screw ...
as, for example, in machines for ruling lines in glass plates to
produce refraction [sic] gratings for the re