The general rule which is applicable to all screw-cutting should be thus stated: - As the step of the lathe-screw is to the step of the screw to be cut, so is the number of teeth on the driving wheel or wheels to the number of teeth on the driven wheel or wheels.

We now first apply this rule to the cutting of a screw with simple-gear wheels. The screw required is to have 4 per inch, and the lathe-screw to be used has 2 per inch, the wheel on the mandril has 20 teeth ; consequently, by applying the ordinary rule of three to these three terms, the number of teeth on the desired screw-wheel is obtained thus :

2 : 4 :: 20 _20 2)80 40. A 40-teeth wheel is therefore the one required. Again, if the screw to be cut is to have l 1/2 steps per inch, the numbers appear thus:

2: 1|::20: 15. Or, if the 20 and 15 wheels are too small to engage with a connecting-wheel, a 40-wheel is put upon the mandril, and the figures appear thus:

2 : 1 1/2 :: 40 : 30. It is now presumed that the fractional pitch 2.3 per inch, is to be cut with a lathe-screw having 1.8 per inch, and that a 20-wheel must be used on the mandril. In this case, the figures appear as follow:

1 8/10 : 2 3/10 :: 20 : 25 5555/10000

This fourth term correctly represents the teeth of a screw-wheel which, together with a 20-wheel on the mandril would cut the desired pitch, supposing that a wheel with 25 and the fraction of teeth which is noted could be obtained. Therefore, in order to cut the thread with wheels in ordinary use, it is requisite to employ some of them in such a manner that they possess the same relation as that existing between 20 and 25 5555/10000. It will be seen presently that four wheels can be easily selected for the purpose; but it is requisite to here give one or two more examples of simple-gear.

If a thread to be cut is required to have only half a step per inch, with a lathe-screw of 2 per inch, the two terms are placed in the same order as for the shorter steps before given. Thus:

2: 5/10, and when placed, the operator immediately sees that the step of the one is four times as great as that of the other, so that he must select some comparative large wheel for the mandril, in order that the one on the screw may not be too small. Either a 100-wheel, 80, or 120, can therefore be selected, and the screw-wheel ascertained thus:

2 : 5/10 :: 80 : 20 or, 2 : 5/10 :: 100 : 25 or, 2 : 5 /10 :: 120 : 30. Either 20, 25, or 30, on the screw, will therefore cut the desired thread with half a step per inch.. Suppose now that a small thread to have 15 an inch is required, and to be cut with the same screw of 2 per inch. By placing the two terms, the operator sees that the intended pitch is seven and a half times smaller than that of the lathe-screw, and therefore selects about the smallest wheel he has for the primary, such as 20, or 25, and obtains the following:

2 : 15 :: 25 : 187 1/2 or, 2 : 15 :: 20 : 150. Consequently, the 150-wheel must be put upon the screw, the term 187 1/2 not having any single wheel to represent it.

In all these examples of simple-gear screw-cutting, it is to be remembered that the two first terms of any proposition contain the representatives of the required wheels, independently of the third and fourth terms. For this reason, both of the two simple-gear wheels required for a stated pitch can be ascertained by inspecting, multiplying, or dividing the two terms only. In these operations the thing to be observed is, the numbers of teeth on the wheels which the operator has at command. For instance, suppose a pitch of 15 per inch is to be cut as in the last example with a screw having 2 an inch. Place the two terms according to the rule in page 374, thus:

2 : 15. These very symbols alone indicate two wheels which would produce the thread - a 2-cog wheel, and a 15-cog wheel, which is proved by inspecting the four terms annexed:

2 : 15 :: 2 : 15. But because such wheels are not in use, the multiplication or division referred to is necessary in order that ordinary wheels can be employed. For this purpose the two terms, when requisite, are reduced to their lowest names, and next multiplied or divided by 2 1/2, 5, 4, 10, 20, 25, 100, or any number which suits the wheels of the lathe. In the example now presented, no reduction is needful, it being only necessary to multiply each term by 10, as here shown:

2x10 = 20, and 15 x 10 = 150. They can be also multiplied by the other multipliers given, and other wheels indicated :

2 x 15 = 30, and 15x15 = 225, or, 2 x 5 = 10, and 15x5 = 75, or, 2x2 1/2 = 5, and 15x2 1/2 = 37 1/2. By these and similar operations, it is seen that 20 and 150 are the only suitable wheels which will cut such a small pitch with simple-gear only.

It is now intended to cut a pitch of 2 1/4 per inch with a lathe-screw having 2 per inch. This is a case in which reduction is employed, the 2 and the 2| being shown as improper fractions. The two terms appear thus :

8/4:9/4

Having now reduced both terms to one name, which in this case is fourths, it is not necessary to regard them further when calculating for two wheels only; and the only thing which need be done is to take away the two numerators or upper symbols, and multiply them, because the same relation exists between 8 and 9 as between 2 and 2 1/4. The 8 and 9 are therefore multiplied by 5 :

8 x 5 = 40, and 9 x 5 = 45. 40 and 45 are thus seen to be the wheels desired. And because the second term 9 represents the intended screw to be cut, therefore the fourth term 45 represents the screw-wheel, according to the rule (page 374), the first and second terms of any proposition always denoting the lathe-screw and required screw, while the third and fourth terms denote the mandril-wheel and screw-wheel.

The next example consists in cutting a thread of 2 3/4 per inch with a lathe-screw of 1 per inch. After changing the terms to fractions as directed, they are seen in this form:

4 /4 : 11/4. By now taking away the two upper ones, which of themselves exactly indicate the relation between the desired wheels, and multiplying them by 5, Ave have 20 and 55 as the wheels which will produce the pitch.

A few instances in tabular form are now subjoined, in which the reduction of the two terms are involved, as will be observed by the student.

In the first and second of these examples it is not requisite to use the improper fractions 2/1,6/1, and 9/1; the two first terms can be merely multiplied by 10, as before stated ; but the fractional forms are given to show that the method is applicable to all. To obtain the improper fractions, the first and second terms of the examples are multiplied according to the ordinary rules of simple arithmetic.

Sufficient has now been stated to illustrate the selection of simple-gear wheels, and to show that they will cut a great number of fractional threads, in addition to those in every-day use.