the lead-screw, in a given time, it is evident that the
tool will cut six threads per inch. If the spindle revolved twice as
fast as the lead-screw, it would make twelve turns while the tool moved
one inch, and, consequently, twelve threads per inch would be cut; but
to get this difference in speeds it is necessary to use a combination of
gearing that will cause the lead-screw to revolve once while the lathe
spindle and work make two revolutions.

[Illustration: Fig. 29. (A) Lathe with Simple Gearing for Thread
Cutting. (B) Compound Geared Lathe]

Suppose that nine threads to the inch are to be cut and the lead-screw
has six threads per inch. In this case the work must make nine
revolutions while the lead-screw makes six and causes the carriage and
thread tool to move one inch, or in other words, one revolution of the
lead-screw corresponds to one and one-half revolution of the spindle;
therefore, if the lead-screw gear _c_ has 36 teeth, the gear _a_ on the
spindle stud should have 24 teeth. The spindle will then revolve one and
one-half times faster than the lead-screw, provided the stud rotates at
the same rate of speed as the main lathe spindle. The number of teeth in
the change gears that is required for a certain pitch can be found by
multiplying the number of threads per inch of the lead-screw, and the
number of threads per inch to be cut, by the same trial multiplier. The
formula which expresses the relation between threads per inch of
lead-screw, threads per inch to be cut, and the number of teeth in the
change gears, is as follows:

threads per inch of lead-screw teeth in gear on spindle stud
------------------------------ = -----------------------------
threads per inch to be cut teeth in gear on lead-screw

Applying this to the example given, we have 6/9 = 24/36. The values of
36 and 24 are obtained by multiplying 6 and 9, respectively, by 4,
which, of course, does not change the proportion. Any other number could
be used as a multiplier, and if gears having 24 and 36 teeth were not
available, this might be necessary. For example, if there were no gears
of this size, some other multiplier as 5 or 6 might be used.

Suppose the number of teeth in the change gears supplied with the lathe
are 24, 28, 32, 36, etc., increasing by four teeth up to 100, and assume
that the lead-screw has 6 threads per inch and that 10 threads per inch
are to be cut. Then,

6 6 x 4 24
-- = ------ = --