APPENDIX.

This—described and drawn first in "Holtzapffel's Mechanical Manipulation," to which work the author, and, indeed, most authors of books of the nature of the present, are indebted for much of their information—is now become very general, and from its perfect action ought to be universally used in all factories in which the lathe bears a part. It permits the tool to be set at any required angle upon the bed of the slide rest, and holds it securely when placed in position. It is likewise so constructed as to be easily removed from the table of the rest, so that other forms of apparatus may be attached if desired. One nut only has to be turned to fix the tool, this nut turning on a strong central screw, A, in the figure, the lower part of which, as far as the shoulder, is screwed into the top plate of the rest. This shoulder is directed to be made with flattened sides, so as to be capable of being unscrewed by the application of a wrench. The actual clamp is a triangular piece of cast or wrought iron, B, in the centre of which is a hole to allow this piece to go easily over the screw. The hole is hollowed out into a cup-shaped cavity, into which fits a hemispherical washer, shown at C in the section. The clamping nut, D, acts upon this washer, which permits the triangle to take up a position notnecessarilyquite parallel to the bed of the slide rest, and thus a tool whose upper and lower surfaces may not be strictly parallel will be securely grasped. The piece called triangular is not precisely of that form, but of the shape shown in the second figure, in which E, E, represent two hard steel pins, slightly projecting—one of these, E, appearing in the first figure. These pins rest upon the upper surface of the tool. At the third angle theclamping piece is drilled and tapped to receive a screw, which must work stiffly in this hole. Thus when a tool is placed in position, as shown, the clamping nut maintains a pressure upon the three points beneath the apices of the triangle. As thus arranged the tool would be stiffly and securely held; but Professor Willis has added a second triangular piece, nearly similar to the first, except that it is provided with a boss, in which a notch or groove is cut, K, in both figures, into which the point of the small screw falls. This lower triangle, which is free to revolve round the central screw, is also cut away at the line L, L, of the second figure, so as to form a guide or rest for the side of the tool, which is thus kept at the same distance from the central screw, and placed in a moment exactly under the studs or points of the upper plate. A careful inspection of the two drawings will make the precise arrangement clear. Inmakingit, which is not very difficult, care must be taken to make the triangle of such size and so to place it, that no angle can overhang the top plate of the rest, in whatever position it may be. The hole in the upper triangle or clamp must be tolerably large and slightly conical—the base of the cone upwards, to allow this piece to take up a bearing, as described. The hemispherical washer is always in a horizontal position, and the hole through it may be only of sufficient diameter to allow it to pass freely over the central screw.

In theEnglish Mechanicof Nov. 2, 1866, a brief notice was given of the above. The author of the present work having carefully inspected the machine and seen it in operation, considers it of such great value to the amateur mechanic, as well as to the professional turner of metal work, that he has had an engraving of the machine carefully made from a photograph, and has here appended it to illustrate the description given.

MUNRO'S MACHINE TURNING LATHE FOR PLANING, ETC.

It is a lathe for planing, cutting key-grooves in wheels, collars, &c., and cutting racks on the teeth of wheels. The lathe is of the usual construction, but outside the right hand standard is fixed a vertical spindle, which is made to rotate by a pair of bevel wheels, the pinion being fast to the end of the crank shaft, and in contact with a wheel of double the number of teeth on the vertical spindle. On the top of the latter is a crank-plate, which will give a stroke of ten inches or less at pleasure. The planing-machine is fixed by two bolts to the lathe-bed, and a connecting rod is attached to the sliding plate or bed of the planing machine, the other end of which is made fast to the pin of the crank-plate. The work is clamped by simple means to this sliding bed, and thus passes to and fro under the tool which, by self-acting gear, is made to traverse sidewaysafter each stroke as in the large planing machines. The whole works almost noiselessly and with the greatest ease, each part being accurately fitted, and the whole well finished. For such purposes as planing the face of the slide valve and its bed in small engines, or shaping the guide bars of eccentric and other chucks, facing the frames of slide-rests, &c., it is exactly what is needed by the amateur, rendering the workshop complete for all purposes without the necessity for adding a large and separate planing machine, which takes up room that cannot always be conveniently spared. With such a lathe as that in the frontispiece, fitted with one of these planing machines, there is scarcely a model of machinery that could not be made. Any of our readers interested in mechanics would be wise to trip over to Lambeth and view the machine in operation; and the writer will guarantee, not only the most civil and obliging attention from the inventor, but the greatest pleasure and satisfaction from the working of the machine itself. There is asimple arrangementfor key-grooving and slotting, by attaching the upper slide of the ordinary rest to the crank plate of this machine, in which case most of the apparatus is removed.

Figs. 1, 2, 3.

Mention of this has been made in the body of the work. It is used for turning rings and washers, and various sizes of these can be turned upon the same mandrel, so that a set of three will suffice for all the work likely to be met with even in the largest factories.Fig. 1represents the mandrel complete. F, F is the central part, with a conical boss, A, cast upon it, and the whole turned with great accuracy. Four longitudinal dovetailed slots, seen plainly inFig. 3,are then planed in the conical part, and into these are fitted steel wedges, Fig. 2, A and B, and B, Fig. 3. C, Fig. 1, is a hollow conical washer, which can be advanced over the central part when driven forward by the nut D. This washer, acting on the ends of the sliding wedges, causes them to move towards the large end of the cone A, and, from the form of these and of the cone, any washer or ring will be held tightly when placed outside these wedges, and will also be mounted concentrically.

It is but right to state that the above method has been objected to by a practical workman, whose business has led him to study the matter closely. He states that it is impossible in this way to effect the desired object. As the writer has not been able to test the working of the apparatus on his own lathe, he felt inclined, at first, to withdraw the whole chapter. The objections offered, however, were not, to his mind, entirely satisfactory; and the opinion of other equally scientific and practical men being favourable, the chapter has been retained. It is possible, nevertheless, that there may be a mathematical reason which the writer is not competent to work out, and the objector being a man of great mechanical knowledge and experience, his remarks are worthy of consideration. The practical (not insuperable) difficulty appears to be the production of a proper tool for this work.

This chuck is put in motion by an entirely new method; none of its parts being attached to the lathe head, the whole can be put in motion or released in an instant, and without stopping the lathe.

The whole of its work is executed by the continuous motion of the lathe, so that, when the chuck is adjusted, any figure (no matter how complex) may be begun and completed without once stopping the lathe.

By the different arrangements and adjustments of the chuck and slide rest, an infinite variety of the most beautiful geometrical figures may be produced; and some of them of so strange and fortuitous a nature as to bid defiance to any imitation.

Description of the Drawings.

Fig. 1is a front view, and

Fig. 2a view of the back of the chuck.

Fig. 1.

Fig. 2.

FRONT ELEVATION

A A. The foundation plate screwing on the plate of the mandrel and carrying the whole of the other parts of the chuck.

B, C. The two driving wheels giving an independent motion to the chuck.

D. Angular wheel moving freely on the wheel C for the angular adjustment of the figures.

E. Pinion of any number of teeth fitting on the shaft carrying D and C.

H. Large wheel of 120 teeth, forming the foundation of the second part, and driven from the pinion E by the wheels F, G, and T.

L. Large wheel of 96 teeth driven by the pinions and wheels U, I, J, K, and forming the foundation plate of the third part, M, which carries the nose of the chuck.

N, N. Self-adjusting radius plates for carrying the various change wheels.

O, P. The eccentric slides of the first and second parts.

Fig. 2shows the arrangement of the driving wheels and pinions on the back of the chuck.

The working of the chuck is as follows:—

If the pinion E has 20 teeth, and is geared direct into the wheel H, by means of an intermediate wheel, it will give six loops inwards if the motions are similar, and outward loops if the motions are contrary.

If the wheel H is driven from the pinion G it will give 12, 24, or 48 loops.

Pinion of 24 teeth will give 5, 10, 20, or 40 loops.

Pinion of 30 teeth will give 4, 8, 16, or 32 loops.

Pinion of 40 teeth will give 3, 6, 18, or 36 loops.

Pinion of 60 teeth will give two loops inwards, if the motions are similar, but, if the motions are contrary, it will produce an ellipse of any proportion from a straight line to a circle.

Other combinations will give circulating or overlaying loops.

By the different arrangements of wheels and pinions on the plates N, N any number of loops can be produced up to 2,592 in the circle.

On the opposite page we illustrate some work executed with this chuck by Mr. Plant.

Fig. 3is a side elevation of the chuck full size.

A, A, the foundation plate screwing on the nose of the mandrel, and carrying the whole of the other parts of the chuck.

B, C, the two driving wheels giving an independent motion to the chuck.

D, D, angular wheel moving freely on the wheel C, for the angular adjustment of the figures.

E, E, pinion of any number of teeth fitting on the shaft carrying C and D.

H, H, large wheel of 120 teeth, forming foundation of the second part, and driven from the pinion E by the wheels F, G, and T.

L, large wheel of 96 teeth driven by the wheels and pinions I, J, K, and forming the foundation of the third part, M, which carries the nose of the chuck.

N, N, self-adjusting radius plates for carrying the various change wheels, &c.

O, P, the eccentric slides of the first and second parts.

Q, R, the screws working the eccentric slides.

"The formation of the tools used for turning and planing the metals is a subject of very great importance to the practical engineer, and it is indeed only when the mathematical principles upon which such tools act are closely followed by the workman that they produce their best effects."—Holtzapffel, vol. 2, p. 983.

As the best lathe can do no more than place the work in the most favourable position for the operation of the tool, and the best tool can only do good work whenappliedas well asconstructedon true principles, no argument is needed to prove the truth of the statement taken as the text of this paper.

But while many of our most eminent practical authorities, such as Nasmyth, Holtzapffel, Babbage, Prof. Willis, and others, have contributed valuable papers on the subject, no single writer can be said to have embodied all that should be known upon it as a whole.

Principle may be looked upon as the essence of practice, and in connection with this particular subject, the reduction of practice to principle is of comparatively modern growth. This will account for the fragmentary character and occasional difference of opinion, which marks the treatises of the above-named eminent authorities when compared with each other. As a step towards some more concise and perfect code of principle, I have endeavoured to collate and arrange in consecutive order, all those laws which govern the action of acute edged turning tools.

The object of this paper is not to supply patterns of tools, as the best form will be no better than the worst unless properly applied; but to set forth those general principles, which may enable the workman to distinguish between forms which are accidental and those which are essential, and thus to make the shape of any tool his servant rather than his guide.

Whatever the shape or purpose of any acute-edged tool may be, its action will always depend on the manner in which the extremeedge is applied to the surface acted upon; and as the same laws govern the action of every acute edge, whether formed on a razor or a tool for cast iron, it will assist a clear comprehension of this subject to consider first the action of edges generally, without reference to any particular tool.

The same edge may be made to act in four different ways, viz.: to cut, dig, chatter or scrape. Digging and chattering are intermediate stages between cutting and scraping, and are fatal to good work. Thuscuttingandscrapingremain the two standard principles, on one of which every tool should be made to act; and while cutting depends on the penetration of the edge, scraping results from using an edge so that it cannot penetrate. Consequently, the conditions most favourable to cutting will give the key to both principles of action.

Every cutting edge is simply a wedge, keen enough to guide its own path without depending on the grain or other accidental line of separation in the material on which it is employed; and when such a wedge is forced into any substance, it will show a constant tendency to penetrate in a line with that face which receives most opposition. The comparative amount of opposition which each face receives, will be determined either by one having more of its surface in contact with the material than the other as inFig. 2, or by the material giving way on one side, as inFig. 1and3. These last two figures illustrate the action of allparingtools, to which class cutting lathe tools belong. The dotted lines are added inFig. 2, to show that the action of the edge is the same, whether it be formed by one or two bevels.

Illustration No. 1.(1, 2, 3)

Thus in all cases,—except when an edge is applied so that the pressure is equal on both faces,—one face will guide the course of the edge, and in paring tools this will always be the lower face, or that next the surface of the work.

The first consideration in placing any paring tool must therefore always be that,the lower face of the edge should lie as nearly as possible in a line with the direction the cut is intended to follow, so as to placethe whole edge in its natural wedge-like position: for when any edge is compelled to act in a manner contrary to this, it will assuredly assert its natural tendency by digging and chattering in the direction of its lower face. But when the action of the tool is continuous as in turning, planing, or boring, care must be taken that this face of the edge does not actually rub against that of the work; and, to avoid this, Nasmyth recommends that the face of the edge should be inclined from the surface of the work at an angle of 3°. Babbage calls this angle "the angle of relief," because it relieves the friction; and to show how little variation is admissible in this angle, Holtzapffel places its maximum at 6°. In cylindrical work the angle of relief is estimated from a tangent to the circumference. Thus, in Figs.4 and 7, the lines C, D, may represent plane surfaces or tangents at pleasure, and in either case the lower face of each edge is supposed to make an angle of 3° with these lines respectively.

Illustration No. 2.(4, 5, 6, 7)

An examination of the nature of the force required to separate any shaving will show the importance of close attention to the above rule. Babbage has pointed out that this process involves two forces, which, though simultaneous in their action, are distinct in the nature of their operation. The first is that necessary to divide the material atom from atom, and depends on the kind of edge employed. The second force is that required to wedge back the shaving, so as to make way for the further progress of the edge, and depends on the manner in which it is applied to the work. Now in fibrous and cohesive materials, the amount of force required to wedge back the shaving is usually greater than that required to effect the initial penetration, and must always depend on the angle which theupper surfaceof the edge makes with the face of the work; while it is obvious that, whatever the acuteness of the particular edge employed may be, this angle will be reduced to the minimum obtainable with suchan edge, by keeping its lower face as close as possible to the surface from which the shaving is being wedged off.[25]A comparison of Figs. 44 and 5will illustrate this. Both edges are supposed to be of the same acuteness, viz., 60°, and inFig. 4, where the angle of relief is only 3°, the edge of 60° will wedge off the shaving at the smallest available angle, viz., 63°, while the position of the same edge inFig. 5increases this angle to 90°.

[25]In adopting Mr. Babbage's arguments I have varied their form. Mr. Babbage takes the square of 90° and divides it into three parts, viz.:Angle of relief 3°Angle of edge 60°Angle of escape 27°big right bracket90°The angle of escape is thus estimated from the horizontal line perpendicular to a base line presented by the surface of the work or by a tangent to it. But as the value of this angle depends directly on its relation to the base line, and has only a complementary relation to the horizontal line, I have thought it better to confine the illustration to the same base as being more directly connected with the wedge-like action of the edge.

[25]In adopting Mr. Babbage's arguments I have varied their form. Mr. Babbage takes the square of 90° and divides it into three parts, viz.:

Angle of relief 3°Angle of edge 60°Angle of escape 27°

The angle of escape is thus estimated from the horizontal line perpendicular to a base line presented by the surface of the work or by a tangent to it. But as the value of this angle depends directly on its relation to the base line, and has only a complementary relation to the horizontal line, I have thought it better to confine the illustration to the same base as being more directly connected with the wedge-like action of the edge.

Thus, as far as regards the force required to bend back the shaving, the edge ofFig. 5might just as well be nearly square, or 87°, taking off 3° for the angle of relief. Indeed, this less acute edge would work better than one more acute but badly placed, as inFig. 5; for the lower face here points too muchintothe work, creating the tendency to dig explained above. The same arguments and illustrations apply with equal force to drills and boring tools, andFig. 5may be looked at as representing one edge of a common drill, in which the acuteness is obtained by bevelling the under sides only, leaving the upper face of each edge perpendicular to the surface acted upon. Nasmyth has pointed out that the less acute drills of this class are made the better and more smoothly they will cut; for, so long as the upper faces are left square to the surface of the work, increasing the bevel of the lower faces can only increase the tendency to dig and chatter. Thus, whenever acuteness is desired in any cutting edge, it should always be obtained from the upper face; and the dotted lines inFig. 4, suggesting a tool for metal in one case, and a common wood-turning chisel in the other, are added to illustrate this, by showing that the line of the lower face is common to both. No tools afford a better illustration of this principle in boring tools than the American twist drills, which owe the ease and beauty of their action to the spiral flutes being placed so as to give the necessary acuteness from the upper face of each edge, thus allowing the lower faces to be kept as close as possible to the surface of the work. There is yet one more important practical advantage to be gained from adopting the smallest possible angle of relief. The arrow in Figs.4 and 5shows the direction in which the strain of the cut will fall on the edges respectively. Ithas been shown that the position ofFig. 5increases the amount of strain on the edge, and yet it is apparent that it is less able to bear this increased strain; for while this falls on Fig. 4 in its strongest direction—viz., almost down the length of one face—it falls onFig. 5acrossthe end of the edge, thus rendering it far more liable to wear and fracture.

It is therefore evident that, in treating plane surfaces, the cutting action of any acute edge is most favoured when its lower face is placed nearly parallel with the surface acted on; and in treating cylindrical surfaces, when the same face occupies the same position with regard to some tangent of the circumference; or, in other words, when the lower face is almost at right angles to some radius of the circle, as inFig. 4: and it follows that the tendency to penetrate will be most effectually counteracted when a line at right angles to the surface, or a radius of the circle, as inFig. 6, bisects the edge, making each face equidistant from the surface which moves across it. Thus,Fig. 6represents thescrapingposition; and it is obvious that all bow-drills or other tools, which aresaid to cut both ways, must really act on the scraping principle.

Practical illustrations in support of the universal application of these principles might be multiplied indefinitely; but two very common operations will suffice to prove that the position of the edge determines the nature of its action. If a penknife be not held with its blade perpendicular to the paper, when used for scratching out, it will be sure to hang and chatter; and the flatter a razor is held to the skin in shaving the more free will the chin be from uncomfortable digs and chatters afterwards.

The conditions which next demand notice in the case of turning-tools are those which must be observed to preserve the proper position of the edge under the strain put upon it. These relate to the form of the tool, and, in the case of cylindrical work with fixed tools, to the part of the surface at which the edge of the tool should be applied. Drills and boring tools require little notice in this respect, for, as the strain is round their axis, it is only necessary that their shafts should be strong enough not to twist or bend. It must, however, be remembered that when common drills are required to be very acute, the edges should be thrown up a little or hollowed out so as to give the acuteness on the upper face as explained above.[26]

[26]The common form of drill is rendered far more efficient with wrought iron and materials that requirecutting, by twisting the flat shaft when hot, so as to reverse the position of each edge after the manner of a screw-auger. The lower faces can then be kept as close as possible to the face of the work while the twist will give a moderate degree of acuteness on the upper face.

Hand-turning is simply a matter of manual dexterity, and as any part of the same plane or the same circumference presents the same surface to the edge of the tool, the correct relation between the edgeand the surface can be obtained in many places, and therefore the particular point at which the edge should be applied is simply a matter of personal convenience, and may vary with the height of the lathe or that of the workman, or the shape and nature of the tool employed. The use of the graver affords a good illustration of this; and it may be remarked, in connection with this tool, that none is more simple in construction, more perfect in principle, or more convenient in application. When its use is once thoroughly mastered it will do anything from smoothing a pin to roughing out a cylinder four or five inches in diameter. The graver is simply a square bar of steel ground off obliquely at the end; and by varying the obliquity of this slope the act of grinding one plane face will give two cutting edges of any desired acuteness, and three heels from which to use these edges at choice. In hand-turning only one edge of the graver is used at a time, and the lozenge-shaped face is made the lower face common to each edge. Now, when the graver is used for roughing, the point is generally buried in the clean metalbelowthe central line of the work, and the lower face is placed against, and takes the shaving from, the little shoulder which it forms on the cylinder. When the graver is used for smoothing, the lower face is placed nearly flat against the face of the work, and the edge is generally made to bite on, or alittle above, the central line. But for very light finishing cuts the graver may be used from the heel at the bottom of its lozenge face, and in this position its point is over the top of the work, bringing the biting part of the edgestill more abovethe central line. Thus, the only three points to consider in placing the tool in hand-turning are—first, that the lower face of the edge should occupy the proper position with regard to the surface; secondly, that the handle of the tool should come up conveniently to the hands of the operator; and thirdly, that while these two conditions are observed, the heel of the tool should be able to take a firm bearing on the rest.

The best rule for hand-turning is, therefore, to apply the tool to the work, with these ends in view before fixing the rest, and then to bring that up to the necessary position.

When the heel of any hand tool has a firm bearing on the rest, and the edge is applied in the wedge-like position, the preservation of this during the progress of the work depends on delicacy of touch rather than muscular power. But when the edges are applied out of their natural line, it puzzles a strong wrist to keep them to their cut at all without digging into the work. This affords the best practical illustration of the necessity of careful attention to the position of slide-rest tools, which are deprived of all power of accommodation to the sense of touch, and which therefore require accurate adjustment in the first instance.

For the motion of the tool is now confined to that of the rest, and as this moves in horizontal planes, the edge of the tool must be applied to the work on that parallel plane which passes through the lathe centres. The reason for this rule will be at once apparent, if the edge be not placed on this central line in facing up a plate—for then it will lose its cut before reaching the centre, leaving a core untouched. Now although it would require an exaggerated error in the position of the edge to lose cut altogether in turning a cylinder, yet this example proves that, unless the edge be applied exactly on the central line the relative position between it and the surface of the work, on which the cutting action depends, will imperceptibly change with the reduction of the work; and supposing this to vary much in diameter, the same tool may cut beautifully on one part and badly on another.Fig. 4, which illustrates the cutting action of the edge, has been purposely placed on a part of the circle where a slide-rest tool could only act for a very short time, in order to draw attention to the difference between those conditions which govern the cutting action, and those which depend on the motion of the rest from which the tool is used. It is obvious that ifFig. 4were moved inwards on a horizontal line the edge would pass over the smaller circle without touching it. The illustration is of course exaggerated, but it proves thatFig. 7is the only position in which the tool will cut over varying diameters without some change in the relative positions of its lower face and that of the work. Hence the usual instructions to apply the edge about the centre of the work. But Babbage has observed, that however good this direction may be as far it goes, it is insufficient and liable to mislead when given alone. It is impossible to do away with elasticity when the tool is supported at some lateral distance from the line of strain, as in the slide rest or planing machine; and unless this elasticity is counteracted by the position of the tool, it may upset the best position of the edge. To meet this, Babbage gives the following rule—First, consider whereabouts the tool itself will bend under strain on its edge, when fixed in the rest; and then take care that this part of the tool, which Babbage calls "the centre of flexure,"—is placed above a line joining the centre of the work and the edge of the tool.Fig. 7will explain the reasons for this rule and the consequences of neglecting it when there is much strain put on the edge.

LetE, I, F,be the line joining the centre of the work and the edge of the tool. Then if G above this line be the centre of flexure, when the tool bends its edge must follow some part of the arc, H, I, J, from G as a centre, and will be thrown out of the work. But if K below the line, E, I, F, be the centre of flexure, then, under the same circumstances, the edge will follow some part of the arc L, I, M, from K as a centre, and must dig into the work. It isimportant to recognise this principle, because while it shows that every tool in which the top of the edge standsabovethe shaft must be liable to the evils resulting from elasticity, it shows also that even cranked tools may fail to obviate the danger, unless care is taken to place the weakest point in the shaft above the central line. Babbage remarks, that although it is not always possible to strengthen any part of a tool, it is always possible and sometimes desirable to make some particular point weaker than the rest, by cutting away a little where the weak point should be.Fig. 9shows that the crank principle may be applied in another form, and although the crank is upwards in this case, the same object is attained by making P the weakest point, and placing it above the central line, N, O. This form, however, is only used for light finishing cuts; for any unnecessary length of crank evidently adds elasticity, and Holtzapffel observes that, "in adopting the crank form tools the principle must not be carried to excess, as it must be remembered we can never expunge elasticity from our materials, whether viewed in relation to the machine, the tool, or the work." The crank, therefore, should only be just sufficient to give the edge the right direction if the tool should spring; and Holtzapffel remarks, that as a tool will generally bend somewhere in the central line of its shaft, it is sufficient if the top of the edge is kept on or just below this line, as inFig. 8. Referring again toFig. 7, and looking at the line C, D, as a plane surface, and I, F, as a line perpendicular to that surface, the same arguments and illustrations apply to the form of a tool in the planing machine. The point at which the tool is now applied ceases to be of moment.

Illustration No. 3.(8, 9)

Having considered the conditions necessary to insure the best cutting action of acute edges and the preservation of that action during the progress of the work, it remains to treat of the edges most suitable to particular materials, the method of giving them any desired angle, and the manner of applying slide-rest tools so as to obtain the best work with the least expenditure of force and time.

Willis observes that different metals and qualities of the same metal require to be treated with edges differing in their degree of acuteness, and all the standard authorities concur in giving the following code as near enough for all practical purposes. Themodification of these angles is ruled by the general principle that fibrous and cohesive materials require more acute edges than crystalline and granular substances, as will be apparent in the following code:—

Thus the edges available for the metals commonly treated in the lathe find their maximum at 90° and minimum at 60°. The maximum requires no explanation, as when any edge is larger it ceases to be an acute edge.

With regard to the minimum of 60°, Babbage has pointed out that this is dependent not only on the strength necessary to resist the strain of the cut, but further and chiefly on the temper which must be preserved in the edge; and if this be less than 60° the mass of metal composing the extreme edge will be too small to carry off the heat generated by the cut, consequently the extreme edge would soon lose temper and become useless.

But these different edges are formed in very different ways according to the purpose for which the tool is intended; and this will be best understood by comparing the action of a hand-turning chisel with that of a pointed slide-rest tool. In the first case the edge is applied at an oblique tangent to the surface, and removes the shaving by passing under its whole width, much after the manner in which an apple is pared, or a ribbon unwound from a stick, when the lower edge of one turn just overlaps the top edge of the turn below it, and so on. In this case the shaving can be cleanly detached by one straight edge. But the position and motion of the slide-rest tool being perpendicular to the axis of the work, its action becomes that of uncoiling rather than paring; and as a cord or wire wound round a stick touches the face of the stick in one direction, and the coil next to itself in another, so in this case the width and thickness of the shaving lie in opposite directions, as illustrated by the dark band inFig. 10. Consequently, unless the shaving be cut simultaneously in these two directions—viz., from the face of the work on one side, and from the matter under removal on the other, it is obvious that it must betornfrom the work in one direction, thus increasing the labour and spoiling the appearance of the work if the tearing should be from its face. Now, in practice, at any rate in the rough cut, it is usual to take the width of the shaving from the superfluous matter; and if the tool be placed, as inFig. 11, it can only cut on one edge; thus the edge of the shaving will be torn from the face of the work, while the point of the tool will trace a fine thread in its progress along it, leaving the face with a rough unfinished appearance.

But if the edges be formed so that they can be placed as in Figs.10 and 12, then both can cut simultaneously, and the screw-like trace of the point may be obliterated. This method of using the tool will leave the work with a good face from the first rough cut, leaving very little for the finishing cut to do; in addition to which the labour will be reduced to a minimum, thereby permitting a much heavier cut from the same amount of force. In turning any plane surface the corner of the edge should be sufficiently relieved from it to avoid the danger of catching; but, in turning cylindrical surfaces, if the tool be carefully made and placed, the slope of the upper surface will carry the corner out of cut. Experiment must decide the exact adjustment; but the great aim should be to keep the face of the tool next the work as nearly parallel with it as possible, because it is only that face which leaves any trace of the tool's action on the face of the work—the action of the other edge being lost with the shaving.

Illustration No. 4.(10, 11, 12)

Thus tools may be broadly divided into two classes—viz., single-edged and double-edged—remembering always that this distinction refers to the manner in which they should act, and not to the number of edges which it may be convenient to form on the same tool. In single-edged tools, whether there be one or many edges, each edge acts independently in removing its own shaving, and may therefore be formed separately. In this case a longitudinal section, showing the angle of the point, will give a true idea of that of the cutting edge. But, in the case of double-edged tools, as the two edges should co-operate in the removal of the same shaving, they must also be formed so that, while each lower face can occupy its proper position with regard to that surface of the work opposed to it, both edges shall possess the same degree of acuteness. In this case the two edges are formed by three planes—viz., two side faces and one upper surface common to both; and the angle of the point is now not only not that of the cutting edges, but has not even any fixed relation to them, for the cutting edges may vary some 25° or more on the very same longitudinal section of the point.

Prof. Willis has pointed out that in these tools the angles of the cutting edges depend on thesectionandplanangles of the pointconjointly(Fig. 8is asectionview;Figs. 10, 11 and 12areplanviews). From this it follows that cutting edges of exactly the same angle may be obtained by a great variety of combinations in the plan and section angles; and in note A, U, of Holtzapffel's work, vol. ii. p. 994, Prof. Willis has given a table, showing some of the different combinations by which cutting edges of certain angles may be produced with accuracy and simplicity. The following short table is arranged from this source and though much abbreviated will be found sufficient for all ordinary purposes.

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