Number
of
Teeth
for
Wheel
to
go
on
Lead
Screw.		Lead
Screw.		121824303642485460667278849096
12222222222222222
18333333333333333
24444444444444444
30555555555555555
36666666666666666
42777777777777777
48888888888888888
54999999999999999
60101010101010101010101010101010
Pitch
of
Lead
Screw.		2345678910111213141516

This table may be used for compound lathes by simply dividing the pitch of the lead screw by the ratio of the compounded pair of wheels. For example, for the wheels to cut 8 threads per inch, the pitch of lead screw being 4 and the compounded gears 2 to 1, as the ratio of the compounded pair is 2 to 1, we divide the pitch of lead screw by 2, which gives us 2, and we thus find the wheels in the column of pitch of lead screw 2, getting 12 and 48 as the required wheels, the 12 going on top of the lathe because it is at the top of the table, and the 48 on the lead screw because it is at the left-hand end of the table, and the lead screw gear is at the left-hand end of the lathe.

The table may be made for half threads as well as whole ones by simply advancing the left-hand column by two teeth, instead of by four, thus:—

Teeth
for
Wheel
on
Lead
Screw.		Teeth for Wheel on Stud.
121620242832364044
123 3 3 3 3 3 3 3 3
1431⁄231⁄231⁄231⁄231⁄231⁄231⁄231⁄231⁄2
164 4 4 4 4 4 4 4 4
1841⁄241⁄241⁄241⁄241⁄241⁄241⁄241⁄241⁄2
205 5 5 5 5 5 5 5 5
2251⁄251⁄251⁄251⁄251⁄251⁄251⁄251⁄251⁄2
246 6 6 6 6 6 6 6 6
2661⁄261⁄261⁄261⁄261⁄261⁄261⁄261⁄261⁄2
287 7 7 7 7 7 7 7 7
3071⁄271⁄271⁄271⁄271⁄271⁄271⁄271⁄271⁄2
328 8 8 8 8 8 8 8 8
3481⁄281⁄281⁄281⁄281⁄281⁄281⁄281⁄281⁄2
369 9 9 9 9 9 9 9 9
3891⁄291⁄291⁄291⁄291⁄291⁄291⁄291⁄291⁄2
4010 10 10 10 10 10 10 10 10
42101⁄2101⁄2101⁄2101⁄2101⁄2101⁄2101⁄2101⁄2101⁄2
Pitch
of
Lead
Screw.		34567891011

For quarter threads we advance the left-hand column by one tooth, or for thirds of threads by three teeth, and so on.

If we require to find what wheels to provide for a lathe, we take the pitch of the lead screw for the numerator, and the pitch required for the denominator, and multiply them first by 2, then by 3, then by 4, and so on, continuing until the numerator or denominator is as large as it can be to give the required proportion of teeth, and not exceed the greatest number that the largest wheel can contain.

For example: A lathe has single gear, and its lead screw pitch is 8 per inch, what wheels will cut 18, 17, 16, 15, 14, or 13 threads per inch?

Wheels.
Pitch of lead screw8 × 2 = 16 24 32
Pitch required1836 54 72
Pitch of lead screw8„16 24 32
Pitch required1734 51 68
Pitch of lead screw8„16 24 32
Pitch required1632 48 64
Pitch of lead screw8„16 24 32
Pitch required1530 45 60
Pitch of lead screw8„16 24 32
Pitch required1428 42 56
Pitch of lead screw8„16 24 32 40
Pitch required1326 39 52 65

If we suppose that the greatest number of teeth permissible in one wheel is not to exceed 100, then in this table we have all the combinations of wheels that can be used to cut the given pitches; and having made out such a table, comprising all the pitches to be cut, we may select therefrom the least number of wheels that will cut those pitches. The whole table being made out it will be found, of course, that the numerators of the fractions are the same in each case; that is, in this case, 16, 18, 24, 32, and so on as far as we choose to carry the multiplication of the numerator. We shall also find that the denominators diminish in a regular order: thus taking the fractions whose numerators are in each case 16, we find their denominators are, as we pass down the column, 36, 34, 32, 30, 28, and 26, respectively, thus decreasing by 2, which is the number we multiplied the left-hand column by to obtain them. Similarly in the fractions whose numerators are 24, the denominators diminish by 3, being respectively 54, 51, 48, 45, 42, and 39; hence the construction of such a table is a very simple matter so far as whole numbered threads are concerned, as no multiplication is necessary save for the first line representing the finest pitch to be cut.

For fractional threads, however, instead of using the pitch of the lead screw for the numerator, we must reduce it to terms of the fraction it is required to cut. For example, for 51⁄2 threads we proceed as follows. The pitch of the lead screw is 8, and in 8 there are 16 halves, hence we use 16 instead of 8, and as in the 51⁄2 there are 11 halves we use the fraction 16⁄11 and multiply it first by 2, then by 3, and then by 4, and so on, obtaining as follows: 16⁄11, 32⁄22, 48⁄33, 64⁄44, obtaining as before three sets of wheels either of which will cut the required pitch. In selecting from such a table the wheels to cut any required number of pitches, the set must, in order to cut a thread of the same pitch as the lead screw, contain two wheels having the same number of teeth.