ensure that the cutting edge, A, shall not be in advance of the axis of the straight part of the shaft of the tool. The object of this arrangement is, that any deflection which the tool-shaft may undergo through excessive resistance to the cut, may tend to move the cutting edge, A, away from the work, and not to make it dig into it.

In tools for rough work, and having very stiff shafts, the cranked shape is unnecessary; and then the upper side of the shaft is in the plane BA F; the proper position of the upper plane of the cutting edge being given by means of a hollow or flute in the upper side of the tool.

Fig. 286 represents a paring tool designed on the same principles with that shown in fig. 284, but with two cutting edges, and a three-sided pyramidal

point. The upper part of the figure, marked with capital letters, is an elevation; the lower part, marked with italic letters, is a plan, or horizontal projection; and corresponding letters mark corresponding points. The cutting edges in the plan are marked a band a c; abed and a cƒ d are the projections of the two faceplanes; ad is the projection of the front edge; and abg hc that of the upper plane. In the elevation are shown one of the cutting edges, A B; one of the face-planes, A BED; and the front edge, A D. Sometimes the front edge is rounded, together with

the angles c a b and ƒ de; thus connecting the two straight cutting edges by means of a short curved cutting edge. Sometimes the whole cutting edge is a curve; and then, instead of two face-planes, there is a cylindrical front surface.

The relations amongst the angles made by the planes and edges of such a tool as that shown in fig. 286 may be determined either by spherical trigonometry, or by the rules in descriptive geometry given in this book, Articles 24 to 30, pages 9 to 12. They are treated of also in an essay by Mr. Willis, already mentioned as

having been published in the appendix to the second volume of Holtzapffel On Mechanical Manipulation.

A tool with one curved or two straight cutting edges may be used to cut a groove, or to pare a layer by successive shavings off the surface of a piece of work. In the latter case the shaft of the tool is to be so formed and held, that one of the straight cutting edges (for example, a b), or one side of the curved edge, touches the pared face of the work, and the other (for example, a c) cuts into the side of the unpared part of the layer that is being removed; and according as the tool is shaped and placed so as to lie to the right or to the left of the unpared part of the layer, it is called a right-side or a left-side tool. Thus, in a right-side tool, a b touches the pared face; a c, the side of the unpared layer; and in a leftside tool, a c touches the pared face, and ab the side of the unpared layer. A tool with one cutting edge which is parallel to the face of the work, as in fig. 284, or a tool with two cutting edges, as in fig. 286, making equal angles with the face, is called a face-tool. Fig. 287 shows how the principle of having a small angle of relief,

with a sufficiently acute cutting angle, is applied to drills or boring bits. The figure shows a front view, lettered A, B, C, &c.; an edge view, lettered A', B', C', &c.; and an end view, lettered in italics. A A is the axis of rotation; B C, B C, the cutting edges; and the requisite acuteness is given to the cutting angle (marked D'C'E' in the edge view) by means of a flute or curved hollow parallel to each of the cutting edges.

481. Cutting Angles of Tools. -The best cutting angles for paring tools suited to different materials have been ascertained by practical experience. The following are the principal results:

Cutting Angles. from 20° to 45° from 60° to 70°

(The smaller angles for the softer kinds;

the greater for the harder.)

Brass and Bronze,.......

80° and upwards.

In the case of scraping tools, the size of the cutting angle is a question mainly of convenience and strength; for the same tool which is a paring tool, when the working angle is only a little greater than the cutting angle, becomes a scraping tool when the working angle is sufficiently increased.

It may be considered, however, that in order to give sufficient strength to the tool, the least cutting angle for a scraping tool should not be less than the cutting angle of a paring tool suited to the same material.

The cutting angle of scraping tools for iron ranges from 60° to nearly 135; the former value being met with in triangular scrapers for finishing plane surfaces; the latter, in octagonal broaches for enlarging and correcting conical holes.

482. Speed of Cutting Tools.-The speed of the cutting motion of tools is limited by the heat produced by their action: it must not be so great as to cause that heat to affect the temper of the steel. Hence it is less, the harder the material of the work. following are examples.

MATERIAL.

White Cast Iron,

Speed of Cutting Motion.

Feet per Minute.

The

Higher speeds may be used for planing, and for ordinary turning, where the tool and the cut surface are freely exposed to the air, than for drilling and boring.

483. Resistance and Work of Paring Tools.—The following estimate of the resistance to the motion of a tool paring iron, and of the work done, is based on that given by Weisbach :

Let R denote the resistance to the cutting motion of the tool; b the breadth, and t the thickness of the shaving which it pares off; so that bt is the sectional area of the shaving; then

in which ƒ is a co-efficient depending on the hardness and toughness of the material. The value of ƒ for iron is estimated by Weisbach at 50,000 lbs. on the square inch, or 35 kilogrammes on the square millimètre. For steel, it is probably from once-and-half to twice as great. Let / be a given length of shaving; then the work done in paring that length off is

Let w be the heaviness of the material; then wbt is the

weight of material pared off; and the work done in that process is evidently equal to the work of lifting that weight to the height whose value for iron is

15,000 feet, or
4,570 mètres.

The counter-efficiency of the machines in which paring tools are used may be estimated at from 13 to 15; or say, on an average, 14; so that the total work of a machine in paring away a given weight of iron may be estimated as being equal to that of lifting the same weight to a height of

21,000 feet, or

6,400 mètres.

The work done by cutting tools produces heat, which, unless abstracted, tends to injure the tools by raising their temperature. In order to abstract the heat and keep the temperature moderate, the point of the tool, in cutting wrought iron and steel, is moistened with a suitable liquid, such as oil, or a solution of carbonate of soda in water. Pure water should not be used, as it causes iron and steel to rust. Cast iron, brass, and bronze are cut with dry tools.

484. Combinations of Cutting Tools.-The boring bit, already mentioned in Article 480, page 566, is in fact a combination of two cutters or paring tools. A combination of several paring tools, working side by side, is seen in the tool sometimes called a comb, used in screw-cutting lathes, for cutting several turns of the thread at the same time. Cutters following each other in succession occur in taps and dies, for cutting internal and external screws by hand. A circular cutter, or rose-cutter, used in shaping the teeth of wheels, is itself a toothed wheel, each of its teeth being a paring tool. The teeth of a saw form a series of small paring tools or scraping tools, according to their working angles; and by the process of setting them-that is, bending them alternately to the right and left they are made alternately into left-side and rightside tools, so as to cut the two sides of the saw-kerf.

485. Motions of Machine-Tools in General.—In most examples of machine-tools, the relative motion of the tool and the work is the resultant of three component motions, usually at right angles to each other, or of two out of those three: the cutting motion, the traversing motion, or transverse feed motion, and the advancing feed motion; the first two taking place parallel to the face of the work, and the third in a direction normal to it.

The cutting motion is the most rapid of the three, being that by which the tool acts on the face of the work, leaving a narrow strip or band from which a portion of the material has been pared or

scraped away. In many instances, the cutting motion is effected by a motion of the work, the tool remaining fixed, and such is the case especially in turning and screw-cutting lathes, and in almost all planing machines. There are other operations in which the cut is made by a motion of the tool; such as drilling, boring, slotting, and shaping. The speed of the cutting motion has been already mentioned in Article 482, page 567.

The transverse feed motion takes place parallel to the face of the work, and at right angles to the cutting motion: it is that motion by which the tool is made to shift its position relatively to the work, so as to make a series of parallel cuts side by side, leaving a series of strips or bands which compose a surface of any required extent. It is sometimes continuous, and sometimes intermittent. The general nature of transverse feed motions has already been described in Article 258, page 293.

The rate at which the traversing motion takes place in paring a continuous surface depends on the breadth of the cut; which, in iron, ranges from 0·01 to 0·05 inch (= from 0·25 to 1.25 millimètre). In screw-cutting, the traverse at each revolution is equal to the pitch of the screw.

The advancing feed-motion is that by which, after a certain depth of material has been cut away from the face of the work, the tool is advanced so as to cut away an additional depth. This is very often an intermittent motion; and in turning and planing machines it is usually an adjustment, made from time to time by hand. Its extent, at each adjustment, is equal to the depth of the cut; which, in iron, ranges from the smallest appreciable quantity up to 0.75 inch (= 19 millimètres) in ordinary cases.

486. Making Ruled Surfaces-Planing — Slotting — Shaping.—A ruled surface is one in which every point is traversed by a straight line lying wholly in the surface. Such a straight line is called a generating line of the surface; and the surface may be regarded as generated by the motion of the straight line. A ruled surface may be cut to any required degree of precision, by the successive strokes of a tool, each stroke being made along a straight generating line of the surface.

The class of ruled surfaces comprehends the following different kinds, amongst others:

I. All straight surfaces: that is, surfaces in which the straight generating lines are all parallel to each other. Such surfaces may be either plane or cylindrical; the term cylindrical surfaces embracing not only those whose transverse sections are circular, but those whose transverse sections are curves of any figure, such as the acting surfaces of the teeth of spur-wheels (Article 130, page 120). It has already been stated, in Article 38, page 17, that the bearing surfaces of sliding primary pieces must all be straight.