### Table No. I. Table Of Pitches Produced By A Set Of 15 Change Wheels With Main Screw Of 10 Threads Per Inch

Wheel to be placed on the Mandrel in Top Horizontal line. Wheel to be placed on the Screw in Left Vertical column.

 144 120 96 72 60 53 50 48 36 144 12. 15. 20. 24 27.16 28.8 30. 40. 120 8.33. 12.5 16.66 20 22.64 24. 25. 33.33 96 6.66 8. -. 13.33 16 18.11 19.2 20. 26.66 72 5. 6. 7.5 12 13.58 14.4 15. 20. 60 4166 5. 6.25 8.33 10 11.32 12. 12.5 16.66 53 3.68 4.416 5.52 7.361 8.833 10.6 11.04 14.72 50 3.472 4166 5.208 6.944 8.33 9.433 10.417 13.88 48 3.33 4. 5. 6.66 8 9056 9.6 13.33 36 2.5 3. 3.75 5. 6 6.792 7.2 75 1 24 1.66 2. 2 5 3.33 4 4.528 4.8 5. 6.66 20 1.388 1.66 2.0833 2.77 3.33 3.773 4. 4.166 5.55 18 1.25 1.5 1.875 2.5 3 3.396 3.6 3.75 5. 16 1.11 1.33 1.66 2.22 2.66 3.018 3.2 3.33 4.44 15 1.0416 1.25 1.5625 2.0833 2.5 2.83 3. 3.125 4.166

In the foregoing table, every intersection shows the screw resulting from the several combinations; thus,

 60 on Mandrel with 60 on Screw gives 10 threads per inch. 60 " " 15 " " 2½ " " 36 " " 144 " " 40 " "

In describing the method of attaining screws of any required pitch from a guide or copy screw, by the system of change wheels, page 625, Vol. II., to which the reader is referred; it is stated, inter alia, that the value of any combination of wheels may be calculated as vulgar fractions, by multiplying together all the driving wheels as the numerators, and all the driven wheels as denominators, together with the fractional value or pitch of the guide screw. Thus from the foregoing table, neglecting the intermediate wheel altogether, the fraction of the wheels 96 and 48, shows that the screw resulting, will have an interval of one-fifth of an inch between every thread, or be five threads per inch.

Mandrel or driver 96 x 1 = 96 = 1/5. -- -- ---

Screw or driven 48 10 480

Any other pair of wheels of the same ratio, two to one, such as 36 and 18, 48 and 24, 60 and 30, 72 and 36, 96 and 48, 144 and 72, etc.; will produce the same result of five threads to the inch. The particular pair selected for use therefore is unimportant, but the choice is influenced by the distance of the rest from the axis of the lathe, which is determined by the diameter of the work and other circumstances. When the position of any of these pairs of wheels is exchanged, the smaller being placed on the mandrel and the larger on the screw, all will in like manner produce a screw having twenty threads to the inch, or twice the number of threads in the guide screw, instead of half the number as previously; analogous results given in the table, arise from the similar employment of single pairs of wheels of other ratios.

The value of the train of wheels is entirely different, when in place of a single wheel, merely to change the direction of the thread produced, the intermediate arbor carries two wheels, fixed and revolving together, one with the other. These two wheels, cannot both be in one plane with the rest of the train; but one of the pair will be driven by the wheel on the mandrel, and carry the other with it, which latter in turn will drive the wheel on the screw. The results previously obtained, will then be multiplied or divided by the fraction, the value of the pair of wheels on the intermediate arbor. At the same time, as these two wheels turn both together in the same direction, they fulfil the purpose of the single wheel previously employed on the intermediate arbor, and cause the thread to be right handed.

Thus, the pair of wheels 20 and 60, in their lowest terms a fraction of one third, when used upon the intermediate arbor, either multiply the previous value of the train by three, making the screw three times as coarse, or divide it by three, making it three times as fine; according to their relative position being 20/50 , or 60/20, as the one or the other is the driver or the driven wheel. The screw of five threads to the inch, produced by 96 on the mandrel and 48 on the screw would thus be rendered three times as coarse, or 1.66 by the combination A. and three times as fine or fifteen threads to the inch, by the combination B.

Combination A. M. I. S.

96 ------- 20

60 ------- 48

Combination B. M. I. S.

96 ------ 60

20 ------ 48

The values calculated as vulgar fractions, with all the driving and all the driven wheels, respectively multiplied together, the fractions reduced to their lowest terms, give

2 3 1 = 6 or 1.66.

A 1 1 10 10 1

2 x 1 x 1 x =2 or 15.

3 1 10 30

Any other pair of wheels of the same ratio, such as 16/48 or 24/72, produce the same result; and the pair of wheels selected for use, as previously explained, again greatly depends upon the diameter of the work, or the space required for the tool, between it and the slide rest. The following pairs of wheels, are used to multiply or divide the terms of the table by 2, 3, 4, etc., and such pairs of wheels as 48/60 which give 4/5ths or 5/4ths are employed for fractional numbers.

 24 and 48 by 2 16 - 48 3 15 - 60 - 4 24 - 120 - 5
 16 and 96 by 6. 16 - 120 7-5. 15 - 120 - 8. 16 - 144 - 9.

The finest combination conveniently obtained in this manner, with the set of change wheels specified in the first table; results from interposing the pair of wheels 120/15 on the intermediate arbor, between 36 on the mandrel and 144 on the screw, the setting given in the table for 40 threads. Which, multiplied by 8 the value of the fraction, results in a screw of 320 threads to the inch upon the work, or a screw so fine as to appear a smooth cylinder.

The following table, gives the combinations of the same set of wheels for screws coarser than half inch pitch or rise. The first screw mentioned, that of half inch pitch, or a rise of 5/10ths of an inch in every thread, is produced when the 144 wheel is on the mandrel, gearing into 24 placed on the intermediate arbor, (called also the middle and the double arbor,) which arbor also carries a 60 wheel, that gears into a 72 wheel on the screw of the slide rest. The last and coarsest screw of this series, has a pitch or rise of 7.2 inches in each revolution; resulting from 144 on the mandrel, working into 16 on the middle arbor, and the 120 also on the middle arbor, working into the pinion of 15 on the slide rest screw.

### Table No. II. Table Of Pitches Coarser Than Half Inch Rise, Produced By A Set Of 15 Change Wheels, With Main Screw Of 10 Threads Per Inch

 Screw produced. Wheel on Mandrel. Wheels on MiddleArbor. Wheel on Screw. 0 144 24 60 72 •6 144 15 30 48 75 144 72 60 16 •8 144 15 30 36 •9 144 24 72 48 1' 144 36 60 24 12 144 15 60 48 135 144 16 72 48 1.5 144 72 120 16 1.6 144 15 60 36 1.8 144 15 30 16 2' 144 36 120 24 2.25 144 48 120 16 2.4 144 15 60 24
 Screw produced. Wheel on Mandrel. Wheels on MiddleArbor. Wheel on Screw. 2.7 144 16 60 20 3' 144 36 120 16 3.2 144 15 60 18 3 6 144 15 60 16 3.84 144 30 120 15 4. 144 24 120 18 4 5 144 24 120 16 4.8 144 24 120 15 5.4 144 20 120 16 5.76 144 20 120 15 6. 144 18 120 16 6.4 144 18 120 15 7.2 144 16 120 15

The two succeeding tables, give a series of settings for the slide foot lathe, fig. 114, which has a guide screw of 4 threads to the inch; the wheels rising by 5 as usual, from 20 to 120 teeth. The pitches of the screws produced by any of these settings, it is almost needless to repeat, necessarily vary with that of the guide screw employed; therefore with the third and fourth tables, if a guide screw of 2 threads to the inch be employed, it would double the pitches of the resulting screws, while one of eight threads to the inch would halve them, and, so on in proportion.