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. 3 is 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.
A PAPER ON THE PRINCIPLES WHICH GOVERN THE FORMATION AND
APPLICATION OF ACUTE EDGES, WITH SPECIAL REFERENCE TO FIXED
TURNING-TOOLS, CONTRIBUTED BY MR. DODSWORTH HAYDON.
"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 when _applied_ as well as _constructed_ on 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 extreme edge 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. Thus
_cutting_ and _scraping_ remain 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 in Fig. 2, or by the material
giving way on one side, as in Figs. 1 and 3. These last two figures
illustrate the action of all _paring_ tools, to which class cutting
lathe tools belong. The dotted lines are added in Fig. 2, to show that
the action of the edge is the same, whether it be formed by one or two
bevels.
[Illustration: ILLUS. No. 1]
[Illustration: ILLUS. No. 2]
[Illustration: ILLUS. No. 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 place the 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.
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 the _upper surface_ of
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 such an 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. 4
and 5 will illustrate this. Both edges are supposed to be of the same
acuteness, viz., 60°, and in Fig. 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 in
Fig. 5 increases 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° }
ditto edge 60° } 90°
ditto 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 of Fig. 5 might 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 in Fig. 5; for
the lower face here points too much _into_ the work, creating the
tendency to dig explained above. The same arguments and illustrations
apply with equal force to drills and boring tools, and Fig. 5 may
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
in Fig. 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 5 shows the direction in which the strain of the cut will fall on
the edges respectively. It has been shown that the position of Fig.
5 increases 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 on Fig. 5 _across_ the 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 in Fig. 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 in Fig. 6, bisects the
edge, making each face equidistant from the surface which moves across
it. Thus, Fig. 6 represents the _scraping_ position; and it is obvious
that all bow-drills or other tools, which are _said 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 require _cutting_, 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
edge and 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 metal _below_ the 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 a _little 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 edge _still more above_
the 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 if Fig. 4 were 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
that Fig. 7 is 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. 7 will explain the reasons for this rule
and the consequences of neglecting it when there is much strain put on
the edge.
Let E, 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 is important
to recognise this principle, because while it shows that every tool
in which the top of the edge stands _above_ the 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. 9 shows 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 in Fig. 8. Referring again
to Fig. 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.
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. The
modification 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:--
Wrought iron and steel 60°
Cast do. do. 70°
Roughing brass 80°
Finishing do. 90°
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 in Fig. 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 be
_torn_ from 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 in Fig. 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: ILLUS. No. 4]
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 the _section_ and _plan_ angles of the point
_conjointly_ (Fig. 8 is a _section_ view; Figs. 10, 11 and 12 are
_plan_ views). 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.
[Illustration:
+-------------------------------------+
|PLAN ANGLE WITH SECTION ANGLES |
| +-------+-------+-------+
| 140° | 79.5° | 69° | 58° |
| 120° | 78.5 | 67 | 55 |
| 100° | 77 | 63.5 | 49.5 |
| 90° | 76 | 61 | 45 |
| 70° | 72.5 | 53.5 | 29 |
| +-------+-------+-------+
|WILL GIVE ------- CUTTING EDGES |
| -------+-------+-------+
| 80° | 70° | 60° |
+---------------------+-------+-------+
]
The graver will again serve to illustrate the use of this table;
for although only one edge is employed at the same time in hand
turning, it belongs properly to the double edged class. This will be
very apparent if a graver is held point upwards side by side with a
point tool, and the dotted lines are added in Fig. 8, to make the
similarity of form more evident. Now Holtzapffel has said of the
graver when employed for its original purpose of engraving, that "no
instrument works more perfectly," pointing out that, while both the
edges are engaged in cutting the same shaving, both the lower faces
of the edges are respectively inclined at the smallest possible
angle from the sides of the V-shaped groove. Fig. 10 has already
been used to illustrate the best position of the slide-rest tool,
and if the illustration be turned round until the letters Q, R, read
horizontally, and this line be taken to represent a flat surface
with the graver acting upon it, it will be seen that the shaving is
removed in exactly the same manner in both cases; the only difference
being that the section of the shaving is triangular in one case and
rectangular in the other. But so far as the tool is concerned the
action is identical in each case, thus proving that every point
tool may be so made and placed as to merit Holtzapffel's eulogium
on the graver, and that this simple tool is in fact the type of all
double-edged tools.
The graver being made from a square bar has of course a plan angle of
90°, and using it to illustrate the table, we will suppose that it is
desired to give it two cutting edges of 60° each. Referring to 90°
under the heading of "plan angle," 45° will be found on this line in
the column over "cutting edge" 60°; denoting that the section angle of
the point, _i.e._, the slope at which the graver is ground, must be
45° to give the desired edges. In the same way, if the section angle
were 61° the cutting edges would be 70°. But taking the plan angle of
120°, the table shows that this would produce the same cutting edges
of 60° with the larger section of 55°; and from this we have the
important rule that, _in obtaining cutting edges of any given degree
of acuteness the larger the plan angle is made, the larger also may
be the section angle_. Thus pointed tools though constructed on the
principle of the graver are an improvement on it in its simple form;
for by making both plan and section angles as wide as possible, it is
obvious that the strength and durability of the point will be much
increased. It is, therefore, always better to give slide-rest tools
a large plan angle, as in Fig. 12; and plan 120°, with section 55°,
will be found a very useful and durable tool for surfacing purposes
with wrought iron. When there are rectangular corners to cut in and
out of, of course the plan angle cannot be more than 90°, and then
it is well to sacrifice a little of the acuteness, as the section of
45° makes the point rather too weak. It is also worthy of remark as
a mathematical fact, that unless the plan angle exceed 60°, it is
impossible to obtain two cutting edges of that degree of acuteness;
and in any case, such a plan angle would be radically bad, because it
could not be used on the double-edged principle without _undercutting_
the shaving. It is somewhat remarkable, in connection with this point,
that while Holtzapffel, vol. ii. p. 536, recommends prismatic cutters,
he should add a footnote which indirectly but most conclusively
corroborates Prof. Willis' condemnation of that particular form, by
admitting that the proper degree of acuteness cannot be given to both
edges.
As the practical result of a circular edge is to cut in two opposite
directions,--the edge passing gradually from one to the other,--so
round-nosed tools belong properly to the double-edged class, and are
open to great objections unless carefully formed on this principle.
[Illustration: ILLUS. No. 5]
This is illustrated in Fig. 13, representing an oblique section of a
round bar; and supposing the section to be made at an angle of 45°,
it is obvious that the highest part of the edge at S will be exactly
of this angle, while the lower point at T will be 135°, and the side
points at U and V will be 90° each. Thus, between the point S and the
points U and V on each side respectively, the edge will gradually pass
through a range of 45°, consequently no two adjacent points on the
same side will be of the same angle, and the highest point S may be
too acute to stand while the lowest, U or V, is too blunt to cut.
Whenever, therefore, it is intended to round the nose of the tool, it
should be first formed as a double-edged point tool, with a section
angle agreeing as nearly as practicable with the intended degree of
acuteness in the edge, so as to secure the highest points from being
too weak, and the table given above will show what plan angle must be
used, in combination with this section, to secure any part from being
too blunt.
Thus, supposing a circular edge of about 60° is desired, the section
of 58° approaches this most nearly, and if the plan of 140°, which,
with this section, gives two straight cutting edges of 60°, be
adopted, there can only be a variation of 2° in different parts of the
edge. But it will be observed that this combination admits of very
little rounding; and although less acute edges, being obtainable with
a smaller plan-angle, admit of rather more rounding, it may be taken
as a general rule that when any tool is much rounded on the nose, so
as to present a large segment of a circle, different parts of the edge
must vary considerably in acuteness. Although Professor Willis objects
to rounding the nose of a tool at all on account of the necessary
variation in the character of the edge and some other reasons, I
am disposed to think that when the tool is carefully formed on the
principles given above, it is very advantageous to round off the point
slightly for taking heavy cuts; and I have found this form a favourite
one in such workshops as Woolwich Arsenal and Portsmouth Dockyard. But
care must be taken to place the nose of the tool towards the width
of the shaving (presuming that this is taken from the matter to be
removed, as it usually is,) for unless the edge is straight, and
almost parallel with the face of the work as it leaves it, the face
would be marked with a series of concave grooves of greater or less
width, according to the feed given to the tool; and even when this is
very slow, if the curved part of the edge were placed towards the face
of the work it would present the appearance of corrugated iron, when
examined under a magnifying lens. Willis further advances against the
round nose that, as the shaving removed by it must be of a curvilinear
section, it will oppose more force in rolling itself off the edge,
than a flat shaving. This would be quite true if a flat shaving could
be rolled up on itself like a piece of tape or ribbon; but I think the
professor has overlooked the fact that when two cutting edges have one
common upper face the shaving must be bent laterally as well as in its
length; and I am disposed to think, from practical experiment, that
there is very little difference on the point of the force required,
and that when a point of large plan angle is just rounded off it
stands better and cuts sweeter than when the point is not so rounded.
But this only applies to heavy cuts, and for ordinary surfacing work
nothing can act more perfectly than a point tool of wide plan-angle
placed as described above. If a double-edged tool, with edges of 60°,
be thus used in turning wrought iron or steel, and be well lubricated
with clean water during the progress of the work, its face may be left
with a burnished brilliancy that a touch of the finest emery would
spoil. But, as the scheme of this paper is confined to the principles
which determine the action of edges, and the rules by which those
edges may be formed with certainty, it will be well to conclude these
remarks with a few hints as to the construction of ordinary slide-rest
tools. Bearing in mind that all double-edged tools belong to the
graver class, it is well to form the side faces carefully in the first
instance, and then never to alter these, but to keep the tool in order
by grinding in the upper surface only, just as the graver is treated.
Nasmyth's cone gauge, illustrated in Holtzapffel, vol. ii., p. 534,
and also in "Baker's Mechanism," p. 236, affords a ready means of
forming the side faces with accuracy. But the range and convenience
of this gauge is much increased by dispensing with every part of the
arrangement, except the cone itself. This can be made of any piece
of stout metal bar turned truly, with the slide-rest set at an angle
of 3°, and the base should be broad enough to stand steady by itself
when squared off truly in the lathe, as in Fig. 14. Two marks can
then be made upon it: one as at W, showing the exact height of the
lathe-centre when the cone stands on the bed of the lathe, and another
as at X, showing the height of the centre when the cone is placed on
any given part of the slide-rest. Thus, in whatever direction the tool
is to be used, its adjustment can be accurately made in the first
instance on the slide-rest itself, and again tested after the tool is
clamped down.
It is too common a practice in setting slide-rest tools to wedge up
one end or the other, with regard only to the application of the edge
on the central line. But this generally sacrifices the position of the
lower faces, which is essential, to a consideration which has been
shown to be only secondary. The best plan is to keep several strips,
varying from 1/32 to 1/2 inch in thickness, but all about as long and
wide as the shaft of the tool. These can be made of bar iron for the
thicker strips, and sheet iron or tin for the thinner ones; and by
using any two or three of these together, the tool can be packed up
parallel to the bed of the slide-rest. The adjustment of the edge can
thus be made with the greatest ease and certainty without altering the
relative position of the lower faces.
It may be well to remark that in using the cone gauge, it is the lower
faces of each edge which are to be tried against it, and not the front
line of the point, as the inclination of this rule will vary slightly
with variations in the plan-angle of the tool, although the slope of
the faces remains the same. But the section angle is always to be
estimated from the front line, whatever its slope may be.
[Illustration: No. 6]
When the principles which this paper has endeavoured to embody are
once thoroughly understood, no handy workmen need ever be at a loss
to form and apply his edges with the best effect under any shape the
circumstances may require. The first point to be observed is the
manner in which the work should be attacked--that is to say, whether
the removal of the shaving or scraping requires the use of a single or
double-edged tool. The next point is the position of the lower face
or faces of the edges, so that they may be applied in the required
direction, and in the position explained above. This involves the
nature of the treatment best suited to the material, both as regards
the kind of edge employed and the principle on which it should be
applied--viz., cutting or scraping. In double-edged tools the position
of the two lower faces determines that of the point, which is simply
an accident resulting from the meeting of the cutting edges; but
which, when so determined, affords a guide for the slope of the upper
face. This must be so ground that it gives each edge the same degree
of acuteness. Thus, in Fig. 15, the point of the tool being at A, the
slope must be made in the direction A, B; while, in Fig. 16, the point
being at C, the slope of the upper face must be in the direction C, D.
The writer is fully aware that those who expect to find "a rule of
thumb" in this paper, will be miserably disappointed. But while he
is conscious that the principles of which he has treated admit of
a much fuller and yet more concise definition, he would remind the
novice that there is "no royal road to learning," and that where
practice of hand is wanting it can only be supplied by greater
knowledge of principle. His object will therefore be fulfilled if
this supplementary paper can supply any explanation or illustration
of principle that may add to the practical utility of a work so
exhaustive of its subject as "the Lathe and its Uses."