A taper gauge may then be made as in Fig. 1229, the line a representing the bore of the hole, and line b the diameter of the internal piece, the distance between the two being the amount found by trial to be necessary for the forcing or driving. The same gauge obviously serves for testing the taper of the holes reamed.

Chucked or Face Plate Work.—This class of work requires the most skillful manipulation, because the order in which the work may most advantageously proceed and the method of chucking are often matters for mature consideration.

In a piece of work driven between the lathe centres, the truth of any one part may be perceived at any time while operating upon the others, but in chucked work, such is not always the case, and truth in the work is then only to be obtained by holding it truly. Again, the work is apt to be sprung or deflected by the pressure of the devices holding it, and furthermore the removal of the skin or surface will in light work sometimes throw it out of true as the work proceeds, the reason being already given, when referring to turning plain cylindrical work.

To Turn a Gland.—There are three methods of turning a gland: first, the hole and the face on the outside of the flange may be turned first, the subsequent turning being done on a mandrel; second, the hole only may be bored at the first chucking, all the remaining work being done on a mandrel; and, third, the hole, hub, and one radial face may be turned at one chucking, and the remaining face turned at a separate chucking.

If the first plan be adopted, any error in the truth of the mandrel will throw the hole out of true with the hub, which would be a serious defect, causing the gland to jamb against one side of the piston rod, and also of the gland bore. The same evil is liable to result from the second method; it is best, therefore, to chuck the gland by the hub in a universal chuck, and simply face the outer face of the flange, and also its edge. The gland may then be turned end for end, and the hole, the hub, the inside radial flange face, and the hub radial face, may then all be turned at one chucking; there is but one disadvantage in this method, which is that the gland must be unchucked to try its fit in the gland hole, but if standard gauges are used such trial will not be necessary, while if such is not the case and an error of measurement should occur, the gland may still be put on a mandrel and reduced if necessary.

In either method of chucking, the fit of the hole to the rod it is intended for cannot be tested without removing the gland from the chuck.

To Turn a Plain Cylindrical Ring all over in a Universal Chuck.—Three methods may be pursued in doing this simple job: first, the hole may be bored at one chucking, and the two radial faces and the circumference turned at a second chucking; second, the diameter may be turned, first on the hole and two radial faces turned at a second chucking; and third the hole and one radial face may be turned at one chucking, and the diameter and second radial face at a second chucking. The last method is best for the following reasons. The tool can pass clear over the surfaces at each chucking without danger of coming into contact with the chuck jaws, which would cause damage to both; second, at the last chucking, the chuck jaws being inside the ring, the latter may be tested for truth with a pointer fixed in the tool rest, and therefore set quite true.

It is obvious that at neither chucking should the ring be set so far within the chuck jaws that there will be danger of the tool touching them when turning the radial face.

In the case of a ring too thin to permit this, and of too large a bore to warrant making a mandrel for it, the ring may be held on the outside and bored, and both radial faces turned to within a short distance of the chuck jaws; at the second chucking, the chuck jaws being within the ring bore, the work may be set true with a pointer, as before, and finished.

If, however, a number of such rings were to be turned, it would pay to turn up another and thicker ring, and use it as a mandrel after the bore and one radial face of the ring had been turned.

To Turn an Eccentric Strap and Eccentric.—The eccentric strap should be turned first, because it can then be taken apart and its fit to the eccentric tried while the latter is in the lathe, which is not the case with the eccentric. The strap should first be held in a universal chuck bolted to the face plate, or held in dogs such as shown in Fig. 893 at c, and one face should be turned. It should then be turned round on the chuck to bore it, and face the other side.

If the shape of the strap will admit it, it is best chucked by plates and bolts holding the face first turned to the face plate, because in this case there will be no pressure tending to spring the straps out of their natural shape; otherwise, however, it may be held in a universal or independent jaw chuck, or if too large for insertion in chucks of this kind (which are rarely made for large lathes) it may be held in dogs such as shown in Fig. 893 at c.

If after an eccentric strap is bored, and the bolts that hold its two halves together have been slackened, its diameter at a and at c, Fig. 1230, be measured, it will be found that a is less than c. The cause of this is partly explained under the head of tension of castings; but it is necessary to add that the diameters at a and at c in the figure are equal while the strap is in the lathe, or until the bolts holding the two halves of the strap together are released, yet so soon as this is done the diameter at a will reduce, the bore becoming an oval.[18]

Now, it is obvious that the eccentric must be turned to the diameter at c, or otherwise it will have lost motion in the strap. If, however, the eccentric be turned to the diameter of c, the strap cannot be tried on, as it will bind at the corners, as shown in Fig. 1231. To remedy this evil it is usual to put a piece of sheet tin or metal between the joint faces of the two halves of the eccentric straps before they are chucked to turn them, and to bore them too large to the amount of the thickness of sheet metal so employed. After the straps are bored these pieces of metal are removed, and the strap halves bolted together as in Fig. 1230, the diameter at c being that to which the eccentric must be turned.

If the sheet metal so inserted were thick enough, the strap bore will measure the same at a as at c, Fig. 1230. If it were too thick the diameter at a will be greatest, while if too thin the diameter at a will be the least. There is no rule whereby the necessary thickness for a given size of strap may be known, and the workman is usually governed by his experience on castings of similar metal, or from the same moulding shop.

He prefers, however, to be on the safe side by not putting in too great a thickness, because it is easier to scrape away the bore at the joint than it is to file away the joint faces. The following thicknesses for the respective diameters may be considered safe for castings that have not been reheated after casting.

In turning a new strap for an old eccentric, it will be necessary, when taking the diameter of the eccentric, to take a piece of tin of the same thickness as that placed between the eccentric lugs or jaws, and place it between the caliper leg and the eccentric, so that the diameter of the strap across c, Fig. 1230, may be made equal when the tin is removed to the diameter of the eccentric.

In turning up the eccentric, the plain face should be faced first, setting it true, or nearly so, with the circumference of the eccentric, as will be the case if the circumference is held in a universal chuck, but if the hub is so long that this cannot be done because the chuck jaws cannot reach the circumference, the hub itself may be held in an independent jaw chuck.

The face turned may then be turned round, so as to meet the face of the chuck against which it should bed fairly, so as to run true. At this chucking the hole bore, the hub, and the radial faces should be turned, all these surfaces being roughed out before any one surface is finished.

The eccentric must then be again reversed, so that the face of the hub meets, the face plate being held by bolts as shown for a crank in figure, when the work being set to the lines marked (so as to give it the correct amount of throw) may be turned to fit the bore of the strap, the strap being taken apart so as to try it on, which this method of chucking will readily permit.

Now, in an eccentric, the surfaces requiring to be most true one with the other are those of the bore and of the circumference where the strap fits, and since the latter was turned with the hub face to the chuck, and that hub face was turned at the same chucking as the hole was bored (and must, therefore, be true to the bore), the bore and circumference will be as true as it is practicable to get them, because upon the truth of the last chucking alone will the truth of the work depend.

Small eccentrics may be held for all their chuckings in jaw chucks, but not so truly as if chucked on a face plate, because of the difficulty of keeping the radial faces of such jaws true, which occurs by reason of the causes explained with reference to Figs. 848 and 849.

Eccentrics having so much throw upon them as to render it difficult to hold them for the last chucking by the method above given (by bolts through the bore), usually have openings through them on the throw side, and in this case parallel pieces may be placed behind the radial face (on the hub side of the eccentric), such parallel pieces being thick enough to keep the hub face clear of the chuck face, and bolts may be passed through the said opening to hold the eccentric. Another method would be as follows:—

The outside diameter of the eccentric may be gripped in a dog chuck, if the dogs of the chuck project out far enough to reach it (otherwise the dogs may grip the hub of the eccentric), while the hole is bored and the plain face of the eccentric turned. The eccentric must then be reversed in the lathe, and the hub and the radial face on that side must be turned. Then the plain face of the eccentric must be bolted to the face plate by plates placed across the spaces which are made to lighten the eccentric, and by a plate across the face of the hub. The eccentric, being set true to the lines, may then be turned on its outside diameter to fit the strap; to facilitate which fitting, thin parallel strips may be placed between the face plate and the plain face of the eccentric at this last chucking. It will be observed that, in either method of chucking, the outside diameter of the eccentric (that is to say, the part on which the strap fits) is turned with the face which was turned at the same chucking at which the hole was bored, clamped to the face plate. In cases where a number of eccentrics having the same size of bore and the same amount of throw are turned, there may be fitted to the face plate of the lathe a disk (such as shown in Fig. 888), of sufficient diameter to fit the hole of the eccentric, the said disk being fastened to the face plate at the required distance from the centre of the lathe to give the necessary amount of throw to the eccentric. The best method of fastening such a disk to the face plate is to provide it with a plain pin turned true with the disk, and let it fit a hole (bored in the face plate to receive it) sufficiently tightly to be just able to be taken in and out by the hand, the pin being provided with a screw at the end, so that it can be screwed tight by a nut to the face plate. The last chucking of the eccentric is then performed by placing the hole of the eccentric on the disk, which will insure the correctness of the throw without the aid of any lines on the eccentric which may be set as true as the diameter of the casting will permit, and then turned to fit the strap.

To Turn a Cylinder Cover.—A cylinder cover affords an example of chucking in which the work done at one chucking requires to be very true with that done at a subsequent chucking, thus the gland hole which is on one side requires to be quite true with the diameter that fits into the cylinder bore, this diameter being on the opposite side.

If the polished or gland side of the cover be turned first, the hole for the packing ring and that for the gland may be bored with the assurance that one will be true with the other, while the polished outside face may be turned at the same chucking.

But when the cover is turned round in the lathe to turn the straight face, though the hole may be set true as far as can be ascertained in its short length, yet that length is too short to be an accurate guide, and the hole for the packing ring may appear true, while that for the gland, being longer, will have any error in the setting, multiplied by reason of its greater length. It is better, therefore, to turn the plain face first, gripping the cover by the gland flange so that the plain radial face, the step that fits the cylinder bore, and the outer edge of the cover flange may be turned at one chucking; then when the cover is turned round in the chuck, the flat face may be set true by resting against the radial surface of the chuck jaws, and the concentric truth may be set by the outer edge of the flange, which, being of the extreme diameter of the cover, will most readily show any want of truth in the setting. If in this case a universal chuck be used, and the work does not run quite true, it may be corrected by slacking the necessary dog or jaw on one side, and tightening up again from the screw of the necessary jaw on the other.

This occurs because from the wear, &c., there is always some small amount of play or lost motion in the jaw screws, and in the mechanism operating them, and by the above means this is taken advantage of to true the work.

If from any cause the work cannot be held for the first chucking by means of the gland hole flange, it must be held by the circumferential edge of the cover, letting the jaws envelop as small a distance over that edge as possible, the protruding part of it may then be turned up as close to the chuck jaws as possible, and this turned part may still be used to set the cover concentrically true at the second chucking.

In a very small cover the gland hole may have a mandrel fitted to it and be turned therefrom on both radial faces, or on one face only, the other being turned at the chucking at which the holes were bored.

In a cover too large to be held in a jaw chuck, the cover may be held in chucking dogs such as shown at c in Fig. 893, the edge protruding as much as possible from the dog screws, and being turned half way across at one chucking, and finished at the second chucking. To set the radial face at the second chucking, the surface gauge, applied as shown in Fig. 894, may be employed. If the bore of the packing ring or piston rod hole is large enough to permit it, that hole and the gland hole may be bored at the same chucking as that at which the plain face and step that fits in the cylinder bore is turned, thus ensuring truth in all the essential parts of the cover.

But in this case these operations should be performed at the last of the two chuckings, so as to eliminate any error that might arise from the casting altering its shape by reason of the removal of the metal on the radial face of the gland hole side of the cover.

To Turn a Pulley.—A pulley affords an excellent example of lathe work, because it may be operated upon by several different methods: thus, for boring it may be held, if small, in a dog chuck, with the jaws inside the rim; in a dog chuck with the jaws outside the rim; in a dog chuck by the hub itself (if the hub is long enough). A larger pulley may be chucked for boring by the rim held in a jaw chuck; by the rim held by bolts and plates, or by the rim held by dogs, such as shown in Fig. 893, or by the arms rested on pieces placed between them and the chuck, and then bolts and plates applied to those arms.

The rim may be turned by placing the pulley on a mandrel and driving that mandrel by a dog or carrier; by placing it on a mandrel and driving it by a Clements driver such as shown in Fig. 753, and having two diametrically opposite driving pins, placed to bear against diametrically opposite arms; by holding the arms to the chuck as before described, and performing the boring and facing at one chucking; or by holding the rim on its inside by the chuck jaws, so as to turn and bore the pulley at one chucking, which can be done when the inside of the rim is parallel, or not sufficiently coned to cause it to slip off the jaws, or when the jaws will reach to the centre of the rim width.

The advantages and disadvantages of these various methods are as follows:—

From the weakness of the pulley rim it is apt to distort when held with sufficient chuck-jaw pressure to enable the turning of the rim face and edge. But this would not affect the truth of the hole; hence the rim may be gripped in a chuck to bore the hole and face the hub. If so held it should be held true to the inside face of the rim, so that the bore will be true to the same, and then in turning the outside diameter it will be made as true as possible with the rim, which will preserve the balance of the pulley as much as possible. For these reasons the inside of the rim should be the part set to run true, whatever method of chucking be employed; hence, if the circumstances will permit of holding the hub to bore it, an independent jaw chuck should be employed (that is, of course, a chuck capable of independent jaw movement).

If the pulley be chucked by the arms, it is well-nigh impossible to avoid springing those arms from the pressure of the bolts, &c., holding them, and as a result the pulley face, though turned true, will not be true of itself, nor true with the hole, when the arms are released from such pressure.

If the pulley is of such a large size that its rim must be held by bolts and plates while the boring is progressing, such bolts, &c., must be placed on the outside of the rim, so as not to be in the way when setting the pulley true to the inside of the rim.

A small pulley may be turned on a mandrel driven by a dog, which is the truest method of turning, because the rim is in this case strained by the pressure of the cut only. But a dog will not drive a cut at such a leverage as exists at the rim of a pulley above about 18 inches in diameter; furthermore, in a large wheel there would not be sufficient friction between a mandrel and the pulley bore to drive the roughing cut on the pulley face.

It is necessary, therefore, to drive the pulley from the arms, while holding it on a mandrel, but if it be driven by one arm the whole strain due to driving will fall on that one arm, and on one side of the pulley only, and this will have a tendency to cause the rim at and near its junction with that arm to spring or deflect from its natural position, and, therefore, to be not quite true; all that can be done, therefore, is to drive by two arms with a Clements driver, so as to equalize the pressure on them.

An excellent method of chucking a pulley, and one that with care avoids the disadvantages mentioned in the foregoing methods, is shown in Figs. 1232 and 1233. It consists of a clamping dog, Fig. 1234, that fastens to the lathe face plate, and secures the pulley by its arms, while supporting the rim and preventing it from chattering, if it is weak or slight.

This dog is bolted to the face plate by the two studs a and b. At c is a set screw for clamping the pulley arms against the screw d, and at f is a screw that steadies the pulley rim between the arms.

Cutting Screws in the Lathe with Slide Rest Tools.—In order to cut a thread in the lathe with a slide rest tool, it is necessary that the gear-wheels which transmit motion from the cone spindle to the feed screw shall be of the proportions necessary to give to the lathe carriage and slide rest sufficient lateral movement or traverse for lathe revolution to cut a thread of the desired pitch.

Suppose now that the feed screw makes a revolution in the same time that the cone spindle does, and it is evident that the thread cut by the slide rest tool will be of the same pitch as is the pitch of the lathe feed screw. If the feed screw gear-wheels of the lathe are what is called single geared (which means that no one stud in the change gearing carries more than one gear-wheel), it does not matter what are the sizes or how many teeth there are in the wheels used to convey or transmit motion from the cone spindle to the feed screw, for so long as the number of teeth on the cone spindle gear and that on the feed screw are equal, the feed screw will make one revolution in the same time as the cone spindle makes a revolution, and the cutting tool will travel a lateral distance equal to the pitch of the lead screw.

Suppose, for example, that Fig. 1235 represents the screw cutting gear or change wheels of a lathe, wheel d being the driver, i an intermediate wheel for transmitting motion from the driver d to the lead-screw wheel s. Suppose, also, that d has 32, i 80, and s 32 teeth, and we have a simple or single-geared lathe. In this case it may first be proved that we need not concern ourselves with the number of teeth in the intermediate i, because its number of teeth is of no consequence. For example, the 32 teeth in d will in a revolution move 32 of the teeth in i past the line of centres, and it is obvious that i will move the 32 teeth in s past the line of centres, causing it to make one revolution the same as d. If any other size of wheel be used for an intermediate, the effect will be precisely the same, the revolutions of d and of s remaining equal. Under these conditions the lathe would cut a thread whose pitch would be the same as that of the thread on the lead screw.

Now let us turn to Fig. 1236, representing an arrangement of gearing common in American practice, and we have within the lathe-head three gears, a, b, and c, which cannot be changed. Of these, b and c are simply intermediate wheels, the respective diameters of which have no effect upon the revolutions of the lead screw, except that they convey the motion to d. To demonstrate this, suppose the wheels to have the number of teeth marked respectively against them in the end view of the figure, c and d having each 20 teeth, and the one revolution of the live spindle wheel a will cause the lead-screw wheel to make one revolution, because a and s contain the same number of teeth. This may be made plain as follows: The 20 teeth in a will in one revolution cause b to make two revolutions, because b has but half as many teeth as a. The two revolutions of b will cause c to make but one revolution, because c has twice as many teeth as b has. Now, c and d are fast on the same shaft r; hence they revolve together, the one revolution of c simply being conveyed by the shaft r to d, and it is clear that the one revolution of a has been conveyed without change to d, and that, therefore, d may be considered to have simply taken the place of a, unaffected by the wheels b, c. Wheel i is again an intermediate, so that, whatever its diameter or number of teeth, one revolution of d will cause one revolution of s. Thus in this arrangement the lead screw will again revolve at the same speed as the live spindle, and the thread cut will be of the same pitch as the pitch of the lead screw. Practically, then, all the wheels between a and s, as thus arranged, act as simple intermediates, the same as though it were a single-geared lathe, which occurs because c and d have the same number of teeth, and we have, therefore, made no use of the shaft r to compound the gearing.

The term “compounded” as applied to the change gears of a lathe, means that there exists in it a shaft or some equivalent means by which the velocity of the wheels may be changed. Such a shaft is shown at r in Fig. 1236, and it affords a means of compounding by placing on its outer end, as at d, a wheel that has a different number of teeth to that in wheel c. In Fig. 1237 this change is made, wheel d having 40 teeth instead of the 20 it had before. As in the former case, however, it will make one revolution to one of c or one of a, but having 40 teeth it will move 40 of the teeth in i past the line of centres, and this will cause the lead screw wheel s to make two revolutions, because it has 20 teeth only. Thus, the compounding of c and d on shaft r has caused s to make two revolutions to one of a, or, what is the same thing, one revolution of a will in this case cause s to make two revolutions, and the thread cut would be twice as coarse as the lead-screw thread. In the case of a lathe geared as in either Fig. 1235 or 1236, all the wheels that we require to consider in calculating the change wheels are d and s. Now, the shaft r is called the “mandrel,” the “stud,” or the “spindle,” all three terms being used, and the wheel d is the wheel on the stud, mandrel, or spindle, while in every case s is that on the lead screw, and the revolutions of this wheel d and those of the lead screw will be in the same proportion as exists between their numbers of teeth. In considering their revolutions it is to be borne in mind that when d has more teeth than s the speed of the lead screw is increased, and the lathe will cut a thread coarser than that of its lead screw, or when d has less teeth than s the speed of the lead screw is diminished, and the pitch of thread cut will be finer than that of the lead screw.

Another method of compounding is shown in Fig. 1238, the compounded pair c d being on a stud carried in the swing frame f. Now, suppose a has 32, c 64, d 32, and s 64 teeth, the revolution being in the same proportion as the numbers of teeth, c will make one-half a revolution to one revolution of a, and d, being fast to the same stud as c, will also make one-half revolution to one revolution of a. This one-half revolution of d will cause s to make one-quarter of a revolution; hence the thread cut will be four times as fine as the pitch of the thread on the lead screw, because while the lathe makes one turn the lead screw makes one-quarter of a turn. In this arrangement we are enabled to change wheel c as well as wheel d (which could not be done in the arrangement shown in Fig. 1236), and for this reason more changes can be made with the same number of wheels. When the wheel c makes either more or less revolutions than the driver a, it must be taken into account in calculating the change wheels. As arranged in Fig. 1236, it makes the same number as a, which is a very common, arrangement, but in Fig. 1238 it is shown to have twice as many teeth as a; hence it makes half as many revolutions. In the latter case we have two pairs of wheels, in each of which the driven wheel is twice the size of the driver; hence the revolutions are reduced four times.

Suppose it is required to cut a thread of eight to an inch on a lathe such as shown in Fig. 1235, the lead screw pitch being four per inch, and for such simple trains of gearing we have a very simple rule, as follows:—

Rule.—Put down the pitch of the lead screw as the numerator, and the pitch of thread you want to cut as the denominator of a vulgar fraction, and multiply both by the pitch of the lead screw, thus:

There are three things to be noted in this rule; and the first is, that when the pitch of the lead screw and the pitch of thread you want to cut is put down as a fraction, the numerator at once represents the wheel to go on the stud, and the denominator represents the wheel to go on the lead screw, and no figuring would require to be done providing there were gear-wheels having as few teeth as there are threads per inch in the lead screw, and that there was a gear-wheel having as many teeth as the threads per inch required to be cut. For example, suppose the lathe in Fig. 1236 to have a lead screw of 20 per inch, and that the change wheels are required to cut a pitch 40, then we have ²⁰⁄₄₀, the 20 to go on at d in Fig. 1236 and the 40 to go on the lead screw. But since lead screws are not made of such fine pitch, but vary from two threads to about six per inch, we simply multiply the fraction by any number we choose that will give us numbers corresponding to the teeth in the change wheels. Suppose, for example, the pitch of lead screw is 2, and we wish to cut 6, then we have ²⁄₆, and as the smallest change wheel has, say, 12 teeth we multiply the fraction by 6, thus: ²⁄₆ × ⁶⁄₆ = ¹²⁄₃₆. If we have not a 12 and a 36 wheel, we may multiply the fraction by any other number, as, say, 8; thus: ²⁄₆ × ⁸⁄₈ = ¹⁶⁄₄₈ giving us a 16 wheel for d, Fig. 1236, and a 48 wheel for the lead screw.

The second notable feature in this rule is that it applies just the same whether the pitch to be cut is coarser or finer than the lead screw; thus: Suppose the pitch of the lead screw is 4, and we want to cut 2. We put these figures down as before ⁴⁄₂, and proceed to multiply, say, by 8; thus: ⁴⁄₂ × ⁸⁄₈ = ³²⁄₁₆, giving a 32 and a 16 as the necessary wheels.

The third feature is, that no matter whether the pitch to be cut is coarser or finer than the lead screw, the wheels go on the lathe just as they stand in the fraction; the top figure goes on top in the lathe, as, for example, on the driving stud, and the bottom figures of the fraction are for the teeth in the wheel that goes on the bottom of the lathe or on the lead screw. No rule can possibly be simpler than this. Suppose now that the pitch of the lead screw is 4 per inch and we want to cut 1¹⁄₂ per inch. As the required pitch is expressed in half inches, we express the pitch of the lead in half inches, and employ the rule precisely as before. Thus, in four there are eight halves; hence, we put down 8 as the numerator, and in 1¹⁄₂ there are three halves, so we put down 3 and get the fraction ⁸⁄₃. This will multiply by any number, as, say, 6; thus: (⁸⁄₃) ×  (⁶⁄₆) = (⁴⁸⁄₁₈), giving us 48 teeth for the wheel d in Fig. 1236, and 18 for the lead screw wheel s.

In a lathe geared as in Fig. 1235 the top wheel d could not be readily changed, and it would be more convenient to change the lead screw wheel s only. Suppose, then, that the lead screw pitch is 2 per inch, and we want to cut 8. Putting down the fraction as before, we have ²⁄₈, and to get the wheel s for the lead screw we may multiply the number of teeth in d by 8 and divide it by 2; thus: 32 × 8 = 256, and 256 ÷ 2 = 128; hence all we have to do is to put on the lead screw a wheel having 128 teeth. But suppose the pitch to be cut is 4¹⁄₄, the pitch of the lead screw being 2. Then we put both numbers into quarters, thus: In 2 there are 8 quarters, and in 4¹⁄₄ there are 17 quarters; hence the fraction is ⁸⁄₁₇. If now we multiply both terms of this ⁸⁄₁₇ by 4 we get ³²⁄₆₈, and all we have to do is to put on the lead screw a wheel having 68 teeth.

When we have to deal with a lathe compounded as in Fig. 1238, in which the combination can be altered in two places—that is, between a and c and between d and s—the wheel a remaining fixed, and the pitch of the lead screw is 2 per inch, and it is required to cut 8 per inch—this gives us the fraction ²⁄₈, which is at once the proportion that must exist between the revolutions of the wheel a and the wheel s. But in this case the fraction gives us the number of revolutions that wheel s must make while the wheel a is making two revolutions, and it is more convenient to obtain the number that s requires to make while a is making one revolution, which we may do by simply dividing the pitch required to be cut by the pitch of the lead screw, as follows: Pitch of thread required, 8; pitch of lead screw, 2; 8 ÷ 2 = 4 = the revolutions s must make while a makes one. We have then to reduce the revolutions four times, which we may do by putting on at c a wheel with twice as many teeth in it as there are in a, and as a has 32, therefore c must have 64 teeth. When we come to the second pair of wheels, d and s, we may put any wheel we like in place of d, providing we put on s a wheel having twice as many.

But suppose we require to cut a fractional pitch, as, say, 4¹⁄₈ per inch, the pitch of lead screw being 2, all we have to do is to put the pitch of the lead screw into eighths, and also put the number of teeth in a into eighths; thus: In two there are 16 eighths, and in the pitch required there are 33 eighths; hence for the pitch of the lead screw we use the 16, and for the thread required we use the 33, and proceed as before; thus:

The simplest method of doing this would be to put on at c a wheel having 2¹⁄₁₆ times as many teeth as there are in a. Suppose then that a has 32 teeth, and one sixteenth of 32 = 2, because 32 ÷ 16 = 2. Then twice 32 is 64, and if we add the 2 to this we get 66; hence, if we give wheel c 66 teeth, we have reduced the motion the 2¹⁄₁₆ times, and we may put on d and s wheels having an equal number of teeth. Or we may put on a wheel at c having the same number as a has, and then put on any two wheels at d and c, so long as that at s has 2¹⁄₁₆ times as many teeth as that at d.

Again, suppose that the pitch of a lead screw is 4 threads per inch, and that it be required to find what wheels to use to cut a thread of ¹¹⁄₁₆ inch pitch, that is to say, a thread that measures ¹¹⁄₁₆ inch from one thread to the other, and not a pitch of ¹¹⁄₁₆ threads per inch: First we must bring the pitch of the lead screw and the pitch to be cut to the same terms, and as the pitch to be cut is expressed in sixteenths we must bring the lead screw pitch to sixteenths also. Thus, in an inch of the length of the lead screw there are 16 sixteenths, and in this inch there are 4 threads; hence each thread is ⁴⁄₁₆ pitch, because 16 ÷ 4 = 4. Our pitch of lead screw expressed in sixteenths is, therefore, 4, and as the pitch to be cut is ¹¹⁄₁₆ it is expressed in sixteenths by 11; hence we have the fraction ⁴⁄₁₁, which is the proportion that must exist between the wheels, or in other words, while the lathe spindle (or what is the same thing, the work) makes 4 revolutions the lead screw must make 11.

Suppose the lathe to be single geared, and not compounded, and we multiply this fraction and get—

But suppose the lathe to be compounded as in Fig. 1235, and we may arrange the wheels in several ways, and in order to make the problem more practical, we may suppose the lathe to have wheels with the following numbers of teeth, 18, 24, 36, 36, 48, 60, 66, 72, 84, 90, 96, 102, 108, and 132.