Since DBE and dbe represent the two successive positions of the tool, at the beginning and end of one rotation of the cylinder, a little consideration will show that the shaded space between them, namely mbn, is the section of the spiral shaving which runs off the work during the process of turning. In this diagram bn is the breadth of the shaving, and bm its thickness; but by varying the position or angles of the tool, and its relative motion to the work, it may happen that the reverse may be the case; that is to say, that bm, may be the breadth, and bn the thickness. In all cases, however, the two cutting edges of the tool are employed in detaching the shaving, the one, (as BE in this figure,) separating its breadth from the solid, and the other, (as DB,) separating its thickness, or vice versa.

In adjusting the position and angles of a tool for turning or planing a given piece of work, it appears to me essential that its action, as shown by such a diagram as this, should be well foreseen and investigated, and I can only regret that the narrow limits within which I am at present circumscribed, prevent me from explaining the consequences of this principle by a variety of figures.

For example. In practice, if a tool were used in the position of fig. 997, the motion would be slow, and the space Bb or mb, which is the thickness of the shaving, would be much less than in the diagram. It would be usually argued, that BE was the real cutting edge, and that the shaving would come off without the assistance of the other edge BD. Nevertheless, the action of this edge BD is the only one which left upon the surface of the work, and if the shaving be torn off edgewise by neglecting the action of this edge the surface will necessarily be left rough.

* In fig. 997 the thread of the screw, is inadvertently drawn, so as to incline in the wrong direction. In fact the figure now shows the lower surface of the work seen transparently, instead of the upper as it ought to have done.

By placing the edge BD still more nearly, or even exactly parallel to the axis of rotation, and rounding off the corner D * to prevent it from catching the surface. screw form may be wholly obliterated, and if the edge BO be carefully sharpened a finished surface will result; for it is clear, that thus the edge BE is wholly occupied with the hard work of separating the breadth of the shaving, and that the surface which it leaves at each turn is wholly removed in the next, whereat the edge BD has the lighter work of separating the shaving edgewise, and the surface which it leaves, is in fact the visible surface of the work when completed.

Let us now examine the angles that may be given to the tool edges. Fig. 998 shows the pointed tool in its simplest form, AB and AC are its cutting edges. The stem of the tool may be of various shapes for convenience, but the cutting portion of the instrument is bounded by three planes, namely, two tide planes, one of which only, S, is shown in the figure, and a third or upper plans U. The intersection of this upper plane, with the two side planes respectively, produces the cutting edges AB, AC, and the intersection of the two side planes produces the line of the front angle AD.

By a proper management of the inclination of these planes to each other, we obtain the desired form of the point of the tool, and the proper acuteness of the cutting edges. This is the subject to which I wish in the next place to direct attention.

The front angle upon AD determines the form of the point of the tool in plan, and also the section of the shaving, as already explained. As to the cutting edges, a greater or less inclination of the upper plane U of the tool to the horizon (always supposing the tool to rest on a horizontal bed), will produce a greater or less degree of acuteness in these cutting edges.

If the upper plane be horizontal the cutting edges will plainly be square, whatever be the front angle of the tool. But if not, then the angle of the edges will vary conjointly with the front angle of the tool, and the inclination of the upper plane.

Different metals, and qualities of metal, require different degrees of acuteness in the cutting edges, which have not been as far an I know exactly determined. In the present case I will assume, that wrought iron requires an edge of 60°, cast iron of 70°; that brass may be roughed out with an edge of 80°, and finished with one of 90°. These angles I believe to be very near to the best; probably a variation of a degree or two is of little consequence. But as the finishing of some kinds of work requires that the edge of the tool should endure through a long process, without giving way and requiring fresh grinding, it is of some importance that the angles of the edged should bo carefully investigated.

In grinding a tool of this form it is convenient to consider only the angle which the upper plane U makes with the line of the front angle A D. In other words the angles of the cutting edges AB AC being equal, if we suppose a vertical plane to pass through AD and make equal angles with the side planes S, it will clearly intersect the upper plane in a line AK, bisecting the angle BAC, and the upper piano will be perpendicular to this vertical plane. A rough goniometer† will enable us to grind this upper plane at any desired angle KAD, and thus to ensure that the cutting edges shall be alike.

* In fact, there is no occasion to round off the corner D, because the edge BD is, in most cases, inclined downwards, and the corner D carried thereby clear of the surface of the work, except in face turning.

† By this tens is meant a frame attached to a grinding machine, capable of being set at different angles, so as to ensure that the tool, which rests upon it during the process, shall receive its proper form. A goniostat would be a better name.