Mechanical Drawing Self-Taught
by Joshua Rose · English · Gutenberg #23319 · 418,185 chars · public domain (US)
The Project Gutenberg eBook of Mechanical Drawing Self-Taught
This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online
at www.gutenberg.org. If you are not located in the United States,
you will have to check the laws of the country where you are located
before using this eBook.
Title: Mechanical Drawing Self-Taught
Author: Joshua Rose
Release date: November 4, 2007 [eBook #23319]
Language: English
Other information and formats: www.gutenberg.org/ebooks/23319
Credits: Produced by Joseph R. Hauser, Ross Wilburn and the Online
Distributed Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK MECHANICAL DRAWING SELF-TAUGHT ***
Produced by Joseph R. Hauser, Ross Wilburn and the Online
Distributed Proofreading Team at http://www.pgdp.net
MECHANICAL DRAWING SELF-TAUGHT:
COMPRISING
INSTRUCTIONS IN THE SELECTION AND
PREPARATION OF DRAWING
INSTRUMENTS,
_ELEMENTARY INSTRUCTION IN PRACTICAL
MECHANICAL DRAWING_;
TOGETHER WITH
EXAMPLES IN SIMPLE GEOMETRY AND ELEMENTARY MECHANISM,
INCLUDING SCREW THREADS, GEAR WHEELS,
MECHANICAL MOTIONS, ENGINES AND BOILERS.
BY JOSHUA ROSE, M.E.,
AUTHOR OF "THE COMPLETE PRACTICAL MACHINIST,"
"THE PATTERN MAKER'S ASSISTANT,"
"THE SLIDE VALVE"
ILLUSTRATED BY THREE HUNDRED AND THIRTY ENGRAVINGS.
PHILADELPHIA:
HENRY CAREY BAIRD & CO.,
INDUSTRIAL PUBLISHERS, BOOKSELLERS AND IMPORTERS,
810 WALNUT STREET.
LONDON:
SAMPSON LOW, MARSTON, SEARLE & RIVINGTON,
CROWN BUILDINGS, 188 FLEET STREET.
1887.
Copyright by
JOSHUA ROSE.
1883.
PHILADELPHIA.
COLLINS, PRINTER
PREFACE.
The object of this book is to enable the beginner to learn to make
simple mechanical drawings without the aid of an instructor, and to
create an interest in the subject by giving examples such as the
machinist meets with in his every-day workshop practice. The plan of
representing in many examples the pencil lines, and numbering the order
in which they are marked, the author believes to possess great
advantages for the learner, since it is the producing of the pencil
lines that really proves the study, the inking in being merely a
curtailed repetition of the pencilling. Similarly when the drawing of a
piece, such, for example, as a fully developed screw thread, is shown
fully developed from end to end, even though the pencil lines were all
shown, yet the process of construction will be less clear than if the
process of development be shown gradually along the drawing. Thus
beginning at an end of the example the first pencil lines only may be
shown, and as the pencilling progresses to the right-hand, the
development may progress so that at the other or left-hand end, the
finished inked in and shaded thread may be shown, and between these two
ends will be found a part showing each stage of development of the
thread, all the lines being numbered in the order in which they were
marked. This prevents a confusion of lines, and makes it more easy to
follow or to copy the drawing.
It is the numerous inquiries from working machinists for a book of this
kind that have led the author to its production, which he hopes and
believes will meet the want thus indicated, giving to the learner a
sufficiently practical knowledge of mechanical drawing to enable him to
proceed further by copying such drawings as he may be able to obtain, or
by the aid of some of the more expensive and elaborate books already
published on the subject.
He believes that in learning mechanical drawing without the aid of an
instructor the chief difficulty is overcome when the learner has become
sufficiently familiar with the instruments to be enabled to use them
without hesitation or difficulty, and it is to attain this end that the
chapter on plotting mechanical motions and the succeeding examples have
been introduced; these forming studies that are easily followed by the
beginner; while sufficiently interesting to afford to the student
pleasure as well as profit.
NEW YORK, _February, 1883_.
CONTENTS.
CHAPTER I.
THE DRAWING BOARD.
The T square 18
The triangles 19
Curves 21
Selecting and testing drawing instruments 22
Lead pencils 23
Mixing India ink 25
The drawing paper 26
Tracing paper 29
The ink 30
Testing and selecting India ink 30
Draftsmen's measuring rules 33
CHAPTER II.
THE PREPARATION AND USE OF THE INSTRUMENTS.
Preparing the lining pen for use 34
The shapes of the lining pen points 35
Oil stoning pen points 36
Preparing the circle pen for use 38
The shape for circle pen points 38
Shaping circle pens for very small circles 39
A form of pen point recently introduced; forming the pen point 39
The method of oil-stoning circle pen points 40
The needle point and pen point 42
How to use the circle pen 43
German instrument to avoid slipping of a needle point 44
How to use the lining pen 45
Applying the ink to the bow-pen 46
Using a straight line or lining pen with a T square 47
CHAPTER III.
LINES AND CURVES.
Explanation of simple geometrical terms; radius; explanation of
conventional dotted lines 48
A line at a right angle to another; a point; parallel lines 49
A line produced; a line bisected; a line bounding a circle; an arc of a
circle; segments of a circle; the chord of an arc; a quadrant of a
circle 50
A sector of a circle; a line tangent to a circle; a semicircle; centre of
a circle; axis of a cylinder; to draw a circle that shall pass through
three given points 51
To find the centre from which an arc of a circle has been struck; the
degrees of a circle 52
The protractor 53
To find the angle of one line to another 54
To find the angles of three lines one to the other 55
Acute angles and obtuse angles 57
Triangles; right angle triangle; obtuse angle triangle; equilateral
triangle; isosceles triangle 58
Scalene triangle; a quadrangle; quadrilateral or tetragon 59
Rhomboid; trapezoid; trapezium 60
The construction of polygons 61
The names of regular polygons 62
The angles of regular polygons; the ellipse 63
Form of a true ellipse 69
The use of a trammel for drawing an ellipse 72
To draw a parabola mechanically 73
To draw a parabola by lines 74
To draw a heart cam 75
CHAPTER IV.
SHADOW LINES AND LINE-SHADING.
Section lining or cross-hatching 77
To represent cylindrical pieces one within the other; to represent a
number of pieces one within the other 78
To represent pieces put together and having slots or keyways through
them. 79
Effects of shading or cross-hatching 80
Lines in sectional shading or cross-hatching made to denote the material
of which the piece is composed--lead, wood, steel, brass, wrought
iron, cast iron 81
Line-shading 82
The shade line to indicate the shape of piece; representation of a
washer 83
A key drawn with a shade line; shade line applied to a nut; a German
pen regulated to draw lines of various breadths 84
Example of line-shading in perspective drawing, shown in a pipe threading
stock and die 85
A cylindrical pin line-shaded; two cylindrical pieces that join each
other; a lathe centre; a piece having a curved outline 86
Line-shading applied to a ball or sphere; applied to a pin in a socket
shown in section 87
A piece of tube, where the thickness of the tube is shown; where the
hollow or hole is seen, the piece shown in section; where the body is
bell-mouthed and the hollow curve shown by shading 88
Example of line-shading to denote the relative distances of various
surfaces from the eye 89
Line-shading to denote that the piece represented is of wood; shade-lines
being regular or irregular 90
CHAPTER V.
MARKING DIMENSIONS.
Examples in marking dimensions 91
CHAPTER VI.
THE ARRANGEMENT OF DIFFERENT VIEWS.
The different views of a mechanical drawing; elevation; plan; general
view; a figure to represent a solid cylinder 94
To represent the different sides of a cube; the use of a cross to denote a
square 95
A triangular piece requires two or three views 96
To represent a ring having hexagon cross section; examples; a rectangular
piece in two views 98
The position of the piece when in its place determines the name of the
view in the drawing 103
View of a lever 105
Best method of projecting one view from another; the two systems of
different views of a piece 106
CHAPTER VII.
EXAMPLES IN BOLTS, NUTS AND POLYGONS.
To represent the thread of a small screw 112
A bolt with a hexagon head 113
United States standard sizes for forged or unfinished bolts and nuts 116
The basis of the Franklin Institute or United States standard for bolts
and nuts; hexagonal or hexagon heads of bolts 118
Comparison of hexagon and square heads of bolts; chamfers 120
Without chamfer; best plan for view of both square and hexagon heads 123
Drawing different views of hexagon heads 125
To draw a square-headed bolt; to draw the end view of a hexagon head 125
Use of the triangle to divide circles 129
Scales giving the length of the sides of polygons 135
To find what a square body which measures one inch on each side measures
across the corners; to find what diameter a cylindrical piece of
wood must be turned to which is to be squared, and each side of which
square must measure an inch 136
To find a radius across corners of a hexagon or a six sided figure, the
length of a side being an inch 138
To draw a stud 142
To pencil in a cap nut; pencilling for a link having the hubs on one side
only 145
Link with hubs on both sides; pencil lines for a double eye or a knuckle
joint 146
Double eye or knuckle joint with an offset; a connecting rod end 147
A rod end with a round stem 148
A bolt with a square under the head 149
Example in which the corner where the round stem meets the square
under the head is sharp; a centre punch giving an example in
which the flat sides gradually run out upon a circle, the edges
forming curves 150
CHAPTER VIII.
SCREW THREADS AND SPIRALS.
Screw threads for small bolts with the angles of the thread drawn in, and
the method of doing this 152
A double thread; a round top and bottom thread such as the Whitworth
thread; a left hand thread; to draw screw threads of a large
diameter 156
Drawing the curves for screw threads 157
To draw the United States standard thread 160
To draw a square thread 162
Form of template for drawing the curves of threads 165
To show the thread depth in a top or end view of a nut; to draw a spiral
spring 166
To obtain an accurate division of the lines that divide the pitch 167
CHAPTER IX.
EXAMPLES FOR PRACTICE.
A locomotive spring; a stuffing box and gland; working drawings of a
coupling rod; dimensions and directions marked; a connecting rod
drawn and put together as it would be for the lathe, vise, or erecting
shop 169
Drawings for the blacksmith 172
A locomotive frame 174
Reducing scales 175
Making a drawing to scale 177
CHAPTER X.
PROJECTIONS.
A spiral wound around a cylinder whose end is cut off at an angle 178
A cylindrical body joining another at a right-angle; a Tee for
example 180
Other examples of Tees 181
Example of a cylinder intersecting a cone 186
A cylindrical body whose top face if viewed from one point would appear as
a straight line, or from another a circle 188
CHAPTER XI.
DRAWING GEAR WHEELS.
Names of the curves and lines of gear teeth 193
How to draw spur wheel teeth 194
Professor Willis' scale of tooth proportions 195
The application of the scale 197
How to find the curve for the tooth face 198
To trace hypocycloides for the flanks of teeth 200
Sectional view of a section of a wheel for showing the dimensions through
the arms and hub 202
To draw an edge view of a wheel; rules for drawing the teeth of wheels;
bevel gear wheels 203
The construction to find the curves 204
To draw the arcs for the teeth 205
To draw the pitch circle of the inner and small end of the pinion
teeth 206
One-half of a bevel gear and an edge view projected from the same 207
A pair of bevel wheels shown in section; drawing of a part of an Ames
lathe feed motion; small bevel gears 208
Example in which part of the gear is shown with teeth in, and the
remainder illustrated by circles; drawings of part of the feed
motion of a Niles horizontal tool work boring mill 209
Three bevel gears, one of which is line-shaded; the construction of oval
gearing; Professor Rankine's process for rectifying and subdividing
circular arcs 210
Various examples of laying out gear wheels 214
CHAPTER XII.
PLOTTING MECHANICAL MOTIONS.
To find how much motion an eccentric will give to its rod 223
To find how much a given amount of motion of a long arm will move the
short arm of a lever 224
Example of the end of a lever acting directly on a shoe; a short arm
having a roller acting upon a larger roller 225
A link introduced in the place of the roller to find the amount of motion
of the rod; a lever actuating a plunger in a vertical line, to find
how much a given amount of motion of the long arm will actuate the
plunger 226
Two levers upon their axles or shafts, the arms connected by a link and
one arm connected to a rod 227
A lever arm and cam in one piece on a shaft, a shoe sliding on the line,
and held against the cam face by the rod, to find the position of the
face of the shoe against the cam 228
To find the amount of motion imparted in a straight line to a rod,
attached to an eccentric strap 229
Examples in drawing the cut off cams employed instead of eccentrics on
river steamboats in the Western and Southern States. Different views
of a pair of cams 232
The object of using a cam instead of an eccentric 234
Method of drawing or marking out a full stroke cam 237
Illustration of the lines embracing cut off cams of varying limits of
cut-off 240
Part played by the stroke of the engine in determining the conformation
of cut-off cams; manner of finding essential points of drawings of
cutoff cams 241
A cam designed to cut off the steam at five-eighths of the piston
stroke 244
Three-fourths and seven-eighths cams 246
Necessary imperfections in the operations of cut-off cams 247
Drawing representing the motion which a crank imparts to a connecting
rod 249
Plotting out the motion of a shaper link quick return 250
Plotting out the Whitworth quick return motion employed in machines 253
Finding the curves for moulding cutters 257
CHAPTER XIII.
EXAMPLES IN LINE-SHADING AND DRAWING FOR LINE-SHADED
ENGRAVINGS.
Arrangement of idle pulleys to guide bolts from one pulley to another;
representation of a cutting tool for a planing machine 264
Drawings for photo-engraving 267
Drawing for an engraver in wood; drawings for engravings by the wax
process 268
Engraving made by the wax process from a print from a wood engraving;
engravings of a boiler drilling machine 269
CHAPTER XIV.
SHADING AND COLORING DRAWINGS.
Coloring the journals of shafts; simple shading; drawing cast-iron,
wrought iron, steel and copper 277
Points to be observed in coloring and shading; colored drawings to be
glued around their edges to the drawing board; to maintain an even
shade of color; mixing colors 278
To graduate the depth of tint for a cylindrical surface 279
The size and use of brushes; light in shading; example for shading a
Medart pulley 280
Brush shading 281
To show by the shading that the surfaces are highly polished;
representation of an oil cup; representation of an iron planing
machine 282
Example in shading of Blake's patent direct acting steam pump 284
Example of shading an independent condenser 288
CHAPTER XV.
EXAMPLES OF ENGINE WORK.
Drawings of an automatic high speed engine; side and end views of the
engine; vertical section of the cylinder through the valve face 289
Valve motion; governor 292
Pillow box, block crank-pin, wheel and main journal 294
Side and edge view of the connecting rod 295
A two hundred horse power horizontal steam boiler for a stationary engine;
cross sectional view of the boiler shell 296
Side elevation, end view of the boiler, and setting 297
Working drawings of a one hundred horse power engine; plan and side
view of the bed plate, with the main bearing and guide bars; cross
sections of the bed plate; side elevation of the cylinder, with end
view of the same 299
Steam chest side and horizontal cross section of the cylinder; steam chest
and the valves; cam wrist plate and cut-off mechanism; shaft for the
cam plate; cross head; side view and section through the centre of
the eccentric and strap 301
Construction of the connecting rod 303
INDEX 305
+-----------------------------------------------------------------------+
|Transcriber's note: In this text $T$ indicates a larger capital letter.|
+-----------------------------------------------------------------------+
MECHANICAL DRAWING
SELF-TAUGHT.
CHAPTER I.
_THE DRAWING BOARD._
A Drawing Board should be of soft pine and free from knots, so that it
will easily receive the pins or tacks used to fasten down the paper. Its
surface should be flat and level, or a little rounding, so that the
paper shall lie close to its surface, which is one of the first
requisites in making a good drawing. Its edges should be straight and at
a right angle one to the other, and the ends of the battens B B in
Figure 1 should fall a little short of the edge A of the board, so that
if the latter shrinks they will not protrude. The size of the board of
course depends upon the size of the paper, hence it is best to obtain a
board as small as will answer for the size of paper it is intended to
use. The student will find it most convenient as well as cheapest to
learn on small drawings rather than large ones, since they take less
time to make, and cost less for paper; and although they require more
skill to make, yet are preferable for the beginner, because he does not
require to reach so far over the board, and furthermore, they teach him
more quickly and effectively. He who can make a fair drawing having
short lines and small curves can make a better one if it has large
curves, etc., because it is easier to draw a large than a very small
circle or curve. It is unnecessary to enter into a description of the
various kinds of drawing boards in use, because if the student purchases
one he will be duly informed of the kinds and their special features,
while if he intends to make one the sketch in Figure 1 will give him all
the information he requires, save that, as before noted, the wood must
be soft pine, well seasoned and free from knots, while the battens B
should be dovetailed in and the face of the board trued after they are
glued and driven in. To true the edges square, it is best to make the
two longest edges parallel and straight, and then the ends may be
squared from those long edges.
[Illustration: Fig. 1.]
THE $T$ SQUARE.
Drawing squares or T squares, as they are termed, are made of wood, of
hard rubber and of steel.
There are several kinds of T squares; in one the blade is solid, as it
is shown in Figure 5 on page 20; in another the back of the square is
pivoted, so that the blade can be set to draw lines at an angle as well
as across the board, which is often very convenient, although this
double back prevents the triangles, when used in some positions, from
coming close enough to the left hand side of the board. In an improved
form of steel square, with pivoted blade, shown in Figure 2, the back is
provided with a half circle divided into the degrees of a circle, so
that the blade can be set to any required degree of angle at once.
[Illustration: Fig. 2.]
[Illustration: Fig. 3.]
[Illustration: Fig. 4.]
THE TRIANGLES.
[Illustration: Fig. 5.]
Two triangles are all that are absolutely necessary for a beginner. The
first is that shown in Figure 3, which is called a triangle of 45
degrees, because its edge A is at that angle to edges B and C. That in
Figure 4 is called a triangle of 60 degrees, its edge A being at 60
degrees to B, and at 30 degrees to C. The edges P and C are at a right
angle or an angle of 90 degrees in both figures; hence they are in this
respect alike. By means of these triangles alone, a great many straight
line drawings may be made with ease without the use of a drawing square;
but it is better for the beginner to use the square at first. The manner
of using these triangles with the square is shown in Figure 5, in which
the triangle, Figure 3, is shown in three positions marked D E F, and
that shown in Figure 4 is shown in three positions, marked respectively
G H and I. It is obvious, however, that by turning I over, end for end,
another position is attained. The usefulness in these particular
triangles is because in the various positions shown they are capable of
use for drawing a very large proportion of the lines that occur in
mechanical drawing. The principal requirement in their use is to hold
them firmly to the square-blade without moving it, and without
permitting them to move upon it. The learner will find that this is best
attained by so regulating the height of the square-blade that the line
to be drawn does not come down too near the bottom of the triangle or
edge of the square-blade, nor too high on the triangle; that is to say,
too near its uppermost point. It is the left-hand edge of the triangle
that is used, whenever it can be done, to produce the required line.
[Illustration: Fig. 6.]
CURVES.
To draw curves that are not formed of arcs or parts of circles,
templates called curves are provided, examples of these forms being
given in Figure 6. They are made in wood and in hard rubber, the latter
being most durable; their uses are so obvious as to require no
explanation. It may be remarked, however, that the use of curves gives
excellent practice, because they must be adjusted very accurately to
produce good results, and the drawing pen must be held in the same
vertical plane, or the curve drawn will not be true in its outline.
DRAWING INSTRUMENTS.
It is not intended or necessary to enter into an elaborate discussion of
the various kinds of drawing instruments, since the purchaser can obtain
a good set of drawing instruments from a reputable dealer by paying a
proportionate price, and must _per force_ learn to use such as his means
enable him to purchase. It is recommended that the beginner purchase as
good a set of instruments as his means will permit, and that if his
means are limited he purchase less than a full set of instruments,
having the same of good quality.
All the instruments that need be used in the examples of this book are
as follows:
A small spring bow-pen for circles, a lining pen or pen for straight
lines, a small spring bow-pencil for circles, a large bow-pen with a
removable leg to replace by a divider leg or a pencil leg, and having an
extension piece to increase its capacity.
The spring bow-pen should have a stiff spring, and should be opened out
to its full capacity to see that the spring acts well when so opened
out, keeping the legs stiff when opened for the larger diameters. The
purchaser should see that the joint for opening and closing the legs is
an easy but not a loose fit on the screw, and that the legs will not
move sideways. To test this latter, which is of great importance in the
spring bow-pencil as well as in the pen, it is well to close the legs
nearly together and taking one leg in one hand and the other leg in the
other hand (between the forefinger and thumb), pushing and pulling them
sideways, any motion in that direction being sufficient to condemn the
instrument. It is safest and best to have the two legs of the bow-pen
and pencil made from one piece of metal, and not of two separate pieces
screwed together at the top, as the screw will rarely hold them firmly
together. The points should be long and fine, and as round as possible.
In very small instruments separate points that are fastened with a screw
are objectionable, because, in very small circles, they hide the point
and make it difficult to apply the instrument to the exact proper point
or spot on the drawing.
The joints of the large bow or circle-pen should also be somewhat stiff,
and quite free from side motion, and the extension piece should be
rigidly secured when held by the screw. It is a good plan in purchasing
to put in the extension piece, open the joint and the pen to their
fullest, and draw a circle, moving the pen in one direction, and then
redraw it, moving it in the other direction, and if one line only
appears and that not thickened by the second drawing, the pen is a good
one.
The lead pencil should be of hard lead, and it is recommended that they
be of the H, H, H, H, H, H, in the English grades, which corresponds to
the V, V, H, of the Dixon grade. The pencil lines should be made as
lightly as possible; first, because the presence of the lead on the
paper tends to prevent the ink from passing to the paper; and, secondly,
because in rubbing out the pencil lines the ink lines are reduced in
blackness and the surface of the paper becomes roughened, so that it
will soil easier and be harder to clean. In order to produce fine pencil
lines without requiring a very frequent sharpening of the pencil it is
best to sharpen the pencil as in Figures 7 and 8, so that the edge shall
be long in the direction in which it is moved, which is denoted by the
arrow in Figure 7. But when very fine work is to be done, as in the case
of Patent Office drawings, a long, round point is preferable, because
the eye can see plainer just where the pencil will begin to mark and
leave off; hence the pencil lines will not be so liable to overrun.
[Illustration: Fig. 7.]
[Illustration: Fig. 8.]
In place of the ordinary wood-covered lead pencils there may be obtained
at the drawing material stores pencil holders for holding the fine,
round sticks of lead, and these are by far the best for a learner. They
are easier to sharpen, and will slip in the holder, giving warning when
the draftsman is pressing them too hard on the paper, as he is apt to
do. The best method of trimming these leads, as also lead pencils after
they have been roughly shaped, is with a small fine file, holding the
file still and moving the pencil; or a good piece of emery paper or sand
paper is good, moving the pencil as before.
All lines in pencilling as in inking in should begin at the left hand
and be drawn towards the right, or when triangles are used the lines are
begun at the bottom and drawn towards the top or away from the
operator. The rubber used should not be of a harsh grade, since that
will roughen the face of the paper and probably cause the ink to run.
The less rubbing out the better the learner will progress, and the more
satisfaction he will receive from the results. If it becomes necessary
to scratch out it is best done with a penknife well sharpened, and not
applied too forcibly to the paper but somewhat lightly, and moved in
different and not all in one direction. After the penknife the rubber
may sometimes be used to advantage, since it will, if of a smooth grade,
leave the paper smoother than the knife. Finally, before inking in, the
surface that has been scraped should be condensed again by rubbing some
clean, hard substance over it which will prevent the ink from spreading.
The end of a paper-cutter or the end of a rounded ivory handled drawing
instrument is excellent for this purpose.
[Illustration: Fig. 9.]
[Illustration: Fig. 10.]
It is well to use the rubber for general purposes in such a way as to
fit it for special purposes; thus, in cleaning the sheet of paper, the
rubber may be applied first, as in Figure 9, as at A, and then as at B,
and if it be moved sideways at the same time it will wear to the form
shown in Figure 10, which will enable it to be applied along a line that
may require to be rubbed out without removing other and neighboring
lines. If the rubber is in the form of a square stick one end may be
bevelled, as in Figure 11, which is an excellent form, or it may be made
to have a point, as in Figure 12. The object is in each case to enable
the rubber action to be confined to the desired location on the paper,
so as to destroy its smooth surface as little as possible.
[Illustration: Fig. 11.]
[Illustration: Fig 12.]
For simple cleaning purposes, or to efface the pencil lines when they
are drawn very lightly, squares of sponge-rubber answer admirably, these
being furnished by the dealers in drawing materials.
A piece of bread will answer a similar purpose, but it is less
convenient.
For glazed surface paper, as Bristol-board, the smoothest rubber must be
used, the grade termed velvet rubber answering well.
THE DRAWING PAPER.
Whatever kind of drawing paper be used it should be kept dry, or the
ink, however good it may be, will be apt to run and make a thick line
that will not have the sharp, clean edges necessary to make lines look
well.
Drawing paper is made in various qualities, kinds, and forms, as
follows: The sizes and names of paper made in sheets are:
Cap, 13 x 16 inches.
Demy, 20 x 15 "
Medium, 22 x 17 "
Royal, 24 x 19 "
Super Royal, 27 x 19 "
Imperial, 30 x 21 "
Elephant, 28 x 22 "
Columbier, 34 x 23 "
Atlas, 33 x 26 "
Theorem, 34 x 28 "
Double Elephant, 40 x 26 "
Antiquarian, 52 x 31 "
Emperor, 40 x 60 "
Uncle Sam, 48 x 120 "
the thickness of the sheets increasing with their size. Some sheets of
paper are hot pressed, to give a smoother surface, and thus enable
cleaner-edged lines to be drawn.
[Illustration: Fig. 13.]
For large drawings paper is made in rolls of various widths, but as
rolled paper is troublesome to lay flat upon the drawing board, it is
recommended to the learner to obtain the sheets, which may be laid
sufficiently flat by means of broad headed pins, such as shown in Figure
13, which are called thumb tacks. These are forced through the paper
into the board at each corner, as in Figure 14 at _f_. On account of the
large diameter of the stems of these thumb tacks, which unduly pierce
and damage the board, and on account also of their heads, by reason of
their thickness, coming in the way of the square blade, it will be
found preferable to use the smallest sizes of ordinary iron tacks, with
flat heads, whose stems are much finer and heads much thinner than thumb
tacks. The objection to ordinary tacks is that they are more difficult
to remove, but they are, as stated, more desirable for use.
[Illustration: Fig. 14.]
[Illustration: Fig. 15.]
If the paper is nearly the full size of the board, it does not much
matter as to its precise location on the board, but otherwise it is best
to place it as near the left-hand edge of the board as convenient, as is
shown in Figure 14.
The lower edge, D, Figure 15, of the paper, however, should not be
placed too near the edge, A, of the board, because if the end P of the
square back comes down below the edge of the board, it is more difficult
to keep the square back true against the end of the board.
The paper must lie flat upon and close to the surface of the board, and
a sufficient number of tacks must be used to effect this purpose.
Drawings that are to be intricate, or to contain a great many lines, as
a drawing of an engine or of a machine, are best pasted or glued all
around the edges of the paper, which should first be dampened; but as
the learner will scarcely require to make such drawings until he is
somewhat familiar with and well practised in the use of the instruments,
this part of the subject need not be treated here.
TRACING PAPER.
For taking tracings from drawings tracing paper or tracing cloth is
used. They require to be stretched tightly and without wrinkles upon the
drawing. To effect this object the mucilage should be thick, and the
tracing paper should be dampened with a sponge after it is pasted. It
must be thoroughly dry before use, or the ink will run.
Tracing cloth must be fastened by pins or thumb tacks, and not dampened.
The drawing should be made on the polished side of the cloth, and any
coloring to be done should be on the other side, and done after the
tracing is removed from the drawing.
THE INK.
India ink should always be used for mechanical drawing: First, because
it lies upon and does not sink into the paper, and is, therefore, easily
erased; and, secondly, because it does not corrode or injure the drawing
instruments.
India ink is prepared in two forms--in the stick and in a liquid form.
The stick ink is mixed in what are termed saucers, or cabinet saucers,
one being placed above the other, so as to exclude the dust from
settling in it, and also to prevent the rapid evaporation to which it is
subject.
The surface of the saucer should be smooth, as any roughness grinds the
ink too coarsely, whereas the finer it is ground or mixed the easier it
will flow, the less liability to clog the instruments, and the smoother
and more flat it will lie upon the paper. In mixing the ink only a small
quantity of water should be used, the stick of ink being pressed
_lightly_ upon the saucer and moved quickly, the grinding being
continued until the ink is mixed quite thickly. This will grind the ink
fine as it is mixed, and more water may be added to thin it. It is best,
however, to let the ink be somewhat thick for use, and to keep it
covered when not in use; and though water may be added if it gets too
thick, yet ink that has once dried should not be mixed up again, as it
will not work so well after having once dried.
Of liquid inks the Higgins ink is by far the best, being quite equal to
and much more convenient for use than the best stick ink.
The difference between a good and an inferior India ink lies chiefly in
the extent to which the lamp-black, which is the coloring matter, forms
with the water a chemical solution rather than a mechanical mixture. In
inferior ink the lamp-black is more or less held in suspension, and by
prolonged exposure to the air will separate, so that on being spread the
solid particles will aggregate by themselves and the water by itself.
This explains why draughtsmen will, after the ink has been exposed to
the air for an hour or two, add a drop of mucilage to it; the mucilage
thickening the solution, adding weight to the water, and deferring the
separation of the lamp-black.
A good India ink is jet black, flows easily, lies close to, does not
stand upon or sink into the paper, and has an even lustre, the latter
being an indication of fineness. The more perfect the incorporation of
the lamp-black with the water the easier the ink will flow, the less
liable it is to clog the instruments, the more even and sharp the edges
of the lines, and the finer the lines that may be drawn.
Usually India ink can only be tested by actual trial; but since it is
desirable to test before purchasing it, it may be mentioned that one
method is to mix a little on the finger nail, and if it has a "bronzy"
gloss it is a good indication. It should also spread out and dry without
any tendency to separate.
The best method of testing is to mix a very little, and drop a single
drop in a tumbler of clear water. The best ink will diffuse itself over
the surface, and if the water is disturbed will diffuse itself through
the water, leaving it translucent and black, with a slight tinge of
bronze color. A coarser ink will act in a similar manner, but make the
water somewhat opaque, with a blue-black, or dull, ashy color. A still
coarser ink will, when diffused over the surface of the water, show fine
specks, like black dust, on the surface. This is readily apparent,
showing that the mixture of the ink is not homogeneous.
When it is an object to have the lines of a drawing show as black as
possible, as for drawings that are to be photo-engraved, the ink should
be mixed so thickly as to have a tendency to lift when a body, such as a
lead pencil, is lifted out of it. For Patent Office drawings some will
mix it so thickly that under the above test it appears a little stringy.
The thicker the ink can be used the better, because the tendency of the
carbon to separate is less; and it is for this reason that the test
mentioned with a tumbler of water is so accurate. When ink is to be used
on parchment, or glossy tracing-paper, it will flow perfectly if a few
drops of ox-gall be mixed with it; but on soft paper, or on bristol
board, this will cause the ink to spread.
For purposes of measurement, there are special rules or scales of steel
and of paper manufactured. The steel rules are finely and accurately
divided, and some are of triangular form, so that when laid upon the
paper the lines divided will lie close to the paper, and the light will
fall directly on the ruled surface. Triangular rules or scales are
therefore much superior to flat ones. The object of having a paper rule
or scale is, that the paper will expand and contract under varying
degrees of atmospheric moisture, the same as the drawing paper does.
Figure 16 represents a triangular scale, having upon it six different
divisions of the inch. These are made in different patterns, having
either decimal divisions or the vulgar fractions. Being made of steel,
and nickel-plated, they are proof against the moisture of the fingers,
and are not subject to the variation of the wooden scale.
[Illustration: Fig. 16.]
CHAPTER II.
_THE PREPARATION AND USE OF THE INSTRUMENTS._
The points of drawing instruments require to be very accurately prepared
and shaped, to enable them to make clean, clear lines. The object is to
have the points as sharp as they can be made without cutting the paper,
and the curves as even and regular as possible.
[Illustration: Fig. 17.]
[Illustration: Fig. 18.]
The lining pen should be formed as in Figure 17, which presents an edge
and a front view of the points. The inside faces should be flat across,
and slightly curved in their lengths, as shown. If this curve is too
great, as shown exaggerated in Figure 18, the body of the ink lies too
near the point and is apt to flow too freely, running over the pen-point
and making a thick, ragged line. On the other hand, if the inside faces,
between which the ink lies, are too parallel and narrow near the points,
the ink dries in the pen, and renders a too frequent cleaning necessary.
Looking at the face of the pen as at A in Figure 17, its point should
have an even curve, as shown, the edge being as sharp as it can be made
without cutting the drawing paper. Upon this quality depends the
fineness and cleanness of the lines it will make. This thin edge should
extend around the curve as far as the dotted line, so that it will be
practicable to slant the pen in either of the directions shown in Figure
19; and it is obvious that its thickness must be equal around the arc,
so that the same thickness of line will be drawn whether the pen be held
vertical or slanted in either direction.
[Illustration: Fig. 19.]
[Illustration: Fig. 20.]
The outside faces of the pen should be slightly curved, so that when
held vertically, as in Figure 20 (the dotted line representing the
centre of the length of the instrument), and against the square blade S,
the point will meet the paper a short distance from the lower edge of S
as shown. By this means it is not necessary to adjust the square edge
exactly coincident with the line, but a little way from it. This is an
advantage for two reasons: first, the trouble of setting the square-edge
exactly coincident is avoided, and, secondly, the liability of the ink
to adhere to the edge of the square-blade and flow on to the paper and
make a thick, ragged line, is prevented.
The square being set as near to the line as desired, the handle may be
held at such an angle that the pen-point will just meet the line when
sloped either as in Figure 21 or 22. If, however, the slope be too much
in the direction shown in Figure 21, practice is necessary to enable the
drawing of straight lines if they be long ones, because any variation in
the angle of the instrument to the paper obviously vitiates the
straightness of the line. If, on the other hand, the square be too close
to the line, and the pen therefore requires to be sloped as in Figure
22, the ink flowing from the pen-point is apt to adhere to the
square-edge, and the result will be a ragged, thick line, as shown in
Figure 23.
[Illustration: Fig. 21.]
[Illustration: Fig. 22.]
[Illustration: Fig. 23.]
[Illustration: Fig. 24.]
[Illustration: Fig. 25.]
[Illustration: Fig. 26.]
Each of the legs should be of equal thickness at the pen-point edge, so
that when closed together the point will be in the middle of the edge.
The width and curve of each individual point should be quite equal, and
the easiest method of attaining this end is as follows:
Take a small slip of Arkansas oil-stone, and with the pen-points closed
firmly by the screw trim the pen-edges to the required curve as shown at
A, Figure 17, making the curve as even as possible. Then stone the faces
until this curve is brought up to a sharp edge at the point between the
two pen-legs forming the point.
Next take a piece of 000 French emery paper, lay it upon some flat body
like the blade of a square, and smooth the curve of the edge enough to
take off the fine, sharp edge left by the oil-stone; then apply the
outside flat faces of the pen to the emery paper again, bringing the
pen-edge up sharp.
The emery paper will simply have smoothed and polished the surfaces,
still leaving them too sharp, so sharp as to cut the paper, and to take
off this sharp edge (which must first be done on the inside faces) open
the pen-points as wide as the screw will permit. Then wrap one thickness
of the emery paper upon a thin blade, as upon a drawing-triangle, and
pass the open pen-points over it, and move the instrument endwise,
taking care to keep the inside face level with the surface of the emery
paper, so that the pen-points shall not cut through. Next close the
pen-points with the screw until they nearly, but not quite, touch, and
sweep the edge of the pen-point along the emery paper under a slight
pressure, so moving the handle that at each stroke the whole length
around the curved end of the pen will meet the emery surface. During
this motion the inside faces of the pen-point must be held as nearly
vertical as possible, so as to keep the two halves of the pen-point
equal.
The pen is now ready for use, and will draw a fine and clean line.
It is not usual to employ emery paper for the purpose indicated, but it
will be found very desirable, since it leaves a smoother surface and
edge than the oil-stone alone.
Circle-pens are more difficult to put in order than the straight-line
pen, especially those for drawing the smallest circles, which cannot be
well drawn unless the pen is of the precise right shape and in the best
condition.
A circle-pen is shown in Figure 24, in which A represents the point-leg
and B the pen-leg. The point-leg must be the longest because it requires
to enter the drawing paper before the pen meets the surface. The point
should be sharp and round, for any edges or angles on it will cause it
to widen the hole in the paper when it is rotated. To shape the points
to prevent the enlargement of the centre in the paper is one of the most
important considerations in the use of this instrument, especially when
several circles require to be drawn from the same centre. To accomplish
this end the inside of the point-leg should be, as near as possible,
parallel to the length of the instrument (which is denoted in Figure 24
by the dotted line) when the legs are closed, as in the figure. If the
point is at an angle, as shown in Figure 25, it is obvious that rotating
it will enlarge the top of the centre in the drawing paper. The point
should be sharp and smooth on its circumferential surface, and so much
longer than the pen-point that it will have sufficient hold in the paper
when the instrument stands vertical and the pen-point meets the surface
of it, which amount is about 1/64th of an inch.
We may now consider the shape of the pen-point. Its inside surfaces
should be flat across and to the curve shown in Figure 24, not as shown
exaggerated in Figure 25, because in the latter the body of the ink will
be too near the pen-point, and but little can be placed in it without
causing it sometimes to flow over the edges and down the outside of the
pen.
A form of pen-point recently introduced is shaped as in Figure 26, the
object being to have a thin stream of ink near the marking pen-point and
the main body of the ink near at hand, instead of extending up the pen,
as would be the case with Figure 24. The advantage thus gained is that
the ink lies in a more solid body, and having less area of surface
exposed to the air will not dry so quickly in the pen; but this is more
than offset by the liability of the ink to flow over the crook at A, and
cause the pen to draw a thick ragged line. The pen-point must be
slightly inclined toward the needle-point, to the end that they may
approach each other close enough for drawing very small circles, but it
should also stand as nearly vertical as will permit that end to be
attained. As this pen is for drawing small circles only, it does not
require much ink, and hence may be somewhat close together, as in Figure
24; this has the advantage that the point is not hidden from
observation.
In forming the pen-point the greatest refinement is necessary to enable
the drawing of very small true circles, say 1/16th of an inch, or less,
in diameter. The requirements are that the pen-point shall meet the
surface of the paper when the needle-point has entered it sufficiently
to give the necessary support, and that the instrument shall stand
vertical, as shown by the dotted line in Figure 24. Also, that the pen
shall then touch the paper at a point only, this point being the apex of
a fine curve; that this curve be equal on each side of the point of
contact with the paper; that both halves forming the pen be of equal
thickness and width at the pointed curve; and that the point be as sharp
as possible without cutting the paper.
The best method of attaining these ends is as follows: On each side of
the pen make, with an oil-stone, a flat place, as C D, Figure 27 (where
the pen-point is shown magnified), thus bringing both halves to an edge
of exactly equal length, and leaving the point flat at D. These flat
places must be parallel to one another and to the joint between the two
halves of the pen. As the oil-stone may leave a slightly ragged edge, it
is a good plan to take a piece of 00 French emery paper, lay it on a
flat surface, and holding the instrument vertically remove the fine edge
D until it will not cut. Then with the oil-stone shape the curved edge
as in Figure 28, taking care that the curve no more than brings the flat
place D up to a true curve and leaves the edge sharp, with only the very
point touching the paper, which is represented in the cut by the
horizontal line.
[Illustration: Figures 27. 28. 29. 30. 31. 32.]
The point must have a sharp edge all around the curve, and the two
halves must be exactly equal in width, for if one half is wider than the
other, as in Figure 29 at a, or as in Figure 30 at b, it will be
impossible to draw a very small circle true. So, likewise, the two
halves of the pen must be of exactly equal length, and not one half
longer than the other, as in Figures 31 or 32, which would tend to cut
the paper, and also render the drawing of true small circles
impracticable.
When the pen is closed to draw a very small circle the two halves of the
pen-leg should have an equal degree of contact with the surface of the
paper, and then as the legs are opened out to draw larger circles the
contact of the outside half of the pen will have less contact with the
paper. The smaller the circle, the more difficult it is to keep the
point-leg from slipping out of the centre, and the more difficult it is
to draw a clear line and true circle; hence the points should be shaped
to the best advantage for drawing these small circles, by oil-stoning
the pen, as already described, and then finishing it as follows:
After the oil-stoning, open the two valves of the pen-leg wide enough to
admit a piece of 000 French emery paper wrapped once around a very thin
blade, and move the pen endwise as described for the straight-line pen.
This will smooth the inner surfaces and remove any fine wire-edge that
the oil-stone may leave. Close the two halves of the pen again, and
lightly emery-paper the outside faces, which will leave the edge sharp
enough to cut the paper. The removal of the sharp edge still left, to
the exact degree, requires great care. It may best be done by closing
the pen until its two halves very nearly, but not quite, touch, then
adjust it to mark a circle of about 3/16 inch diameter, and strike a
number of circles in different locations upon the surface of a piece of
0000 French emery paper.
In marking these circles, however, let the instrument stand out of the
perpendicular, and do very little while standing vertically. Indeed, it
is well to strike a number of half-circles, first from right to left and
then from left to right, and finally draw a full circle, sloping the pen
on one side, gradually raising it vertically, and finally sloping it to
the other side. This will insure that the pen has contact at its extreme
point, and leave that point fine and keen, but not enough so to cut the
paper. To test the pen, draw small circles with the pen rotated first in
one direction and then in the other, closing its points so as to mark a
fine line, which, if the pen is properly shaped, will be clear and fine,
while if improperly formed the circle drawn with the pen rotated in one
direction will not coincide with that drawn while rotating it in the
other. The same circle may be drawn over several times to make a
thorough test. If a drawing instrument will draw a fine line correctly,
it will be found to answer for thick lines which are more easily made.
In thus preparing the instruments, the operator will find that if he
occasionally holds the points in the right position with regard to the
light, he will be able to see plainly if the work is proceeding evenly
and equally, for if one-half of the pen is thicker at the point or edge
than the other, it will show a brighter line. This is especially the
case with instruments that have become dull by use, for in that case the
edges will be found quite bright, and any inequality of thickness shows
plainly.
[Illustration: Fig 33.]
[Illustration: Fig. 34.]
It follows, from what has been said, that the needle-point and pen-point
should stand vertical when in use, and to effect this the instruments,
except in the smallest sizes, are provided with joints, such as shown at
A and B in the bow-pencil or circle-pencil, in Figure 33. These joints
should be sufficiently stiff that they will not move too easily, and yet
will move rather than that the legs should sensibly spring without
moving at the joint. The needle-point leg should be adjusted by means of
the joint, to stand vertical, and the same remarks apply equally to the
pen-leg; but in the case of the pencil-leg it is the pencil itself and
not the leg that requires attention, the joint B being so adjusted that
the pencil either stands vertical, or, what is perhaps preferable, so
that it stands inclined slightly towards the needle-point. In
sharpening the pencil the inner face C may be made concave or at least
vertical and flat, and the outer convex or else bevelled and flat,
producing a fine and long edge rounded in its length of edge. In using
the circle-pencil and circle-pen it will be found more convenient to
rotate it in the direction of the arrow in Figure 34. It should be held
lightly to the paper, and the learner will find that he has a natural
tendency to hold it too firmly and press it too heavily, which is
_especially to be avoided_.
If in drawing a small circle the needle-point slips out of the paper, it
is because the pencil-point is too long; or, what is the same thing, the
needle-point does not protrude far enough out from the leg. Or if the
instrument requires to be leaned over too much to make the pencil or pen
mark, it is because the pen or pencil is not far enough out, and this
again may cause the needle-point to slip out of the paper.
[Illustration: Fig. 35.]
[Illustration: Fig. 36.]
In Figure 35 is shown a German instrument especially designed to avoid
this slipping. The peculiarity of this instrument consists in the
arrangement of the centre point, which remains stationary whilst the pen
or pencil, resting by its own weight on the paper, is guided round by
gently turning, without pressure, the small knob at the upper end of the
tube. By this means the misplacing or sliding of the centre-point and
the cutting of the paper by the pen are avoided. By means of this fixed
centre-point any number of concentric circles may be drawn, without
making a hole of very distinguishable size on the paper.
[Illustration: Fig 37.]
In applying the ink to the bow-pen as to all other instruments, care
must be taken that the ink lies between the points only and not on the
outside, for in the latter case the ink will flow down too freely and
make a broad, ragged line, perhaps getting on the edge of the square
blade or triangle, and causing a blot of ink on the drawing.
In using a straight line or lining pen with a T square it may be used as
in Figure 36, being nearly vertical, as shown, and moved from left to
right as denoted by the arrow, S representing the square blade. But in
using it, or a pencil, with a straight edge or a triangle unsupported by
the square blade, the latter should be steadied by letting the fingers
rest upon it while using the instrument, the operation being shown in
Figure 37. The position, Figure 36, is suitable for long lines, and that
in Figure 37 for small drawings, where the pen requires close adjustment
to the lines.
CHAPTER III.
_LINES AND CURVES._
Although the beginner will find that a study of geometry is not
essential to the production of such elementary examples of mechanical
drawing as are given in this book, yet as more difficult examples are
essayed he will find such a study to be of great advantage and
assistance. Meantime the following explanation of simple geometrical
terms is all that is necessary to an understanding of the examples
given.
The shortest distance between two points is termed the radius; and, in
the case of a circle, means the distance from the centre to the
perimeter measured in a straight line.
[Illustration: Fig. 38.]
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
Dotted lines, thus, <----->, mean the direction and the points at which a
dimension is taken or marked. Dotted lines, thus,-----, simply connect
the same parts or lines in different views of the object. Thus in
Figure 38 are a side and an end view of a rivet, and the dotted lines
show that the circles on the end view correspond to the circle of the
diameters of the head and of the stem, and therefore represent their
diameters while showing that both are round. A straight line is in
geometry termed a right line.
A line at a right angle to another is said to be perpendicular to it;
thus, in Figures 39, 40, and 41, lines A are in each case perpendicular
to line B, or line B is in each case perpendicular to line A.
A point is a position or location supposed to have no size, and in cases
where necessary is indicated by a dot.
[Illustration: Fig. 41.]
[Illustration: Fig. 42.]
[Illustration: Fig. 43.]
Parallel lines are those equidistant one from the other throughout their
length, as in Figure 42. Lines maybe parallel though not straight; thus,
in Figure 43, the lines are parallel.
[Illustration: Fig. 44.]
[Illustration: Fig. 45.]
[Illustration: Fig. 46.]
A line is said to be _produced_ when it is extended beyond its natural
limits: thus, in Figure 44, lines A and B are _produced_ in the point C.
A line is bisected when the centre of its length is marked: thus, line A
in Figure 45 is bisected, at or in, as it is termed, _e_.
The line bounding a circle is termed its circumference or periphery and
sometimes the perimeter.
A part of this circumference is termed an arc of a circle or an arc;
thus Figure 46 represents an arc. When this arc has breadth it is termed
a segment; thus Figures 47 and 48 are segments of a circle. A straight
line cutting off an arc is termed the chord of the arc; thus, in Figure
48, line A is the chord of the arc.
[Illustration: Fig. 47.]
[Illustration: Fig. 48.]
[Illustration: Fig. 49.]
[Illustration: Fig. 50.]
[Illustration: Fig. 51.]
A quadrant of a circle is one quarter of the same, being bounded on two
of its sides by two radial lines, as in Figure 49.
When the area of a circle that is enclosed within two radial lines is
either less or more than one quarter of the whole area of the circle the
figure is termed a sector; thus, in Figure 50, A and B are both sectors
of a circle.
A straight line touching the perimeter of a circle is said to be tangent
to that circle, and the point at which it touches is that to which it is
tangent; thus, in Figure 51, line A is tangent to the circle at point B.
The half of a circle is termed a semicircle; thus, in Figure 52, A B and
C are each a semicircle.
[Illustration: Fig. 52.]
[Illustration: Fig. 53.]
The point from which a circle or arc of a circle is drawn is termed its
centre. The line representing the centre of a cylinder is termed its
axis; thus, in Figure 53, dot _d_ represents the centre of the circle,
and line _b b_ the axial line of the cylinder.
To draw a circle that shall pass through any three given points: Let A B
and C in Figure 54 be the points through which the circumference of a
circle is to pass. Draw line D connecting A to C, and line E connecting
B to C. Bisect D in F and E in G. From F as a centre draw the semicircle
O, and from G as a centre draw the semicircle P; these two semicircles
meeting the two ends of the respective lines D E. From B as a centre
draw arc H, and from C the arc I, bisecting P in J. From A as a centre
draw arc K, and from C the arc L, bisecting the semicircle O in M. Draw
a line passing through M and F, and a line passing through J and Q, and
where these two lines intersect, as at Q, is the centre of a circle R
that will pass through all three of the points A B and C.
[Illustration: Fig. 54.]
[Illustration: Fig. 55.]
To find the centre from which an arc of a circle has been struck: Let A
A in Figure 55 be the arc whose centre is to be found. From the extreme
ends of the arc bisect it in B. From end A draw the arc C, and from B
the arc D. Then from the end A draw arc G, and from B the arc F. Draw
line H passing through the two points of intersections of arcs C D, and
line I passing through the two points of intersection of F G, and where
H and I meet, as at J, is the centre from which the arc was drawn.
A degree of a circle is the 1/360 part of its circumference. The whole
circumference is supposed to be divided into 360 equal divisions, which
are called the degrees of a circle; but, as one-half of the circle is
simply a repetition of the other half, it is not necessary for
mechanical purposes to deal with more than one-half, as is done in
Figure 56. As the whole circle contains 360 degrees, half of it will
contain one-half of that number, or 180; a quarter will contain 90, and
an eighth will contain 45 degrees. In the protractors (as the
instruments having the degrees of a circle marked on them are termed)
made for sale the edges of the half-circle are marked off into degrees
and half-degrees; but it is sufficient for the purpose of this
explanation to divide off one quarter by lines 10 degrees apart, and the
other by lines 5 degrees apart. The diameter of the circle obviously
makes no difference in the number of decrees contained in any portion of
it. Thus, in the quarter from 0 to 90, there are 90 degrees, as marked;
but suppose the diameter of the circle were that of inner circle _d_,
and one-quarter of it would still contain 90 degrees.
[Illustration: Fig. 56.]
So, likewise, the degrees of one line to another are not always taken
from one point, as from the point O, but from any one line to another.
Thus the line marked 120 is 60 degrees from line 180, or line 90 is 60
degrees from line 150. Similarly in the other quarter of the circle 60
degrees are marked. This may be explained further by stating that the
point O or zero may be situated at the point from which the degrees of
angle are to be taken. Here it may be remarked that, to save writing the
word "degrees," it is usual to place on the right and above the figures
a small deg., as is done in Figure 56, the 60 deg. meaning sixty degrees, the
deg., of course, standing for degrees.
[Illustration: Fig. 57.]
Suppose, then, we are given two lines, as _a_ and _b_ in Figure 57, and
are required to find their angle one to the other. Then, if we have a
protractor, we may apply it to the lines and see how many degrees of
angle they contain. This word "contain" means how many degrees of angle
there are between the lines, which, in the absence of a protractor, we
may find by prolonging the lines until they meet in a point as at _c_.
From this point as a centre we draw a circle D, passing through both
lines _a_, _b_. All we now have to do is to find what part, or how much
of the circumference, of the circle is enclosed within the two lines. In
the example we find it is the one-twelfth part; hence the lines are 30
degrees apart, for, as the whole circle contains 360, then one-twelfth
must contain 30, because 360/12 = 30.
[Illustration: Fig. 58.]
If we have three lines, as lines A B and C in Figure 58, we may find
their angles one to the other by projecting or prolonging the lines
until they meet as at points D, E, and F, and use these points as the
centres wherefrom to mark circles as G, H, and I. Then, from circle H,
we may, by dividing it, obtain the angle of A to B or of B to A. By
dividing circle I we may obtain the angle of A to C or of C to A, and by
dividing circle G we may obtain the angle of B to C or of C to B.
[Illustration: Fig. 59.]
It may happen, and, indeed, generally will do so, that the first attempt
will not succeed, because the distance between the lines measured, or
the arc of the circle, will not divide the circle without having the
last division either too long or too short, in which case the circle may
be divided as follows: The compasses set to its radius, or half its
diameter, will divide the circle into 6 equal divisions, and each of
these divisions will contain 60 degrees of angle, because 360 (the
number of degrees in the whole circle) /6 (the number of divisions) =
60, the number of degrees in each division. We may, therefore, subdivide
as many of the divisions as are necessary for the two lines whose
degrees of angle are to be found. Thus, in Figure 59, are two lines, C,
D, and it is required to find their angle one to the other. The circle
is divided into six divisions, marked respectively from 1 to 6, the
division being made from the intersection of line C with the circle. As
both lines fall within less than a division, we subdivide that division
as by arcs _a_, _b_, which divide it into three equal divisions, of
which the lines occupy one division. Hence, it is clear that they are at
an angle of 20 degrees, because twenty is one-third of sixty. When the
number of degrees of angle between two lines is less than 90, the lines
are said to form an acute angle one to the other, but when they are at
more than 90 degrees of angle they are said to form an obtuse angle.
Thus, in Figure 60, A and C are at an acute angle, while B and C are at
an obtuse angle. F and G form an acute angle one to the other, as also
do G and B, while H and A are at an obtuse angle. Between I and J there
are 90 degrees of angle; hence they form neither an acute nor an obtuse
angle, but what is termed a right-angle, or an angle of 90 degrees. E
and B are at an obtuse angle. Thus it will be perceived that it is the
amount of inclination of one line to another that determines its angle,
irrespective of the positions of the lines, with respect to the circle.
[Illustration: Fig. 60.]
TRIANGLES.
A right-angled triangle is one in which two of the sides are at a right
angle one to the other. Figure 61 represents a right-angled triangle, A
and B forming a right angle. The side opposite, as C, is called the
hypothenuse. The other sides, A and B, are called respectively the base
and the perpendicular.
[Illustration: Fig. 61.]
[Illustration: Fig. 62.]
[Illustration: Fig. 63.]
[Illustration: Fig. 64.]
An acute-angled triangle has all its angles acute, as in Figure 63.
An obtuse-angled triangle has one obtuse angle, as A, Figure 62.
When all the sides of a triangle are equal in length and the angles are
all equal, as in Figure 63, it is termed an equilateral triangle, and
either of its sides may be called the base. When two only of the sides
and two only of the angles are equal, as in Figure 64, it is termed an
isosceles triangle, and the side that is unequal, as A in the figure, is
termed the base.
[Illustration: Fig. 65.]
[Illustration: Fig. 66.]
When all the sides and angles are unequal, as in Figure 65, it is termed
a scalene triangle, and either of its sides may be called the base.
The angle opposite the base of a triangle is called the vertex.
[Illustration: Fig. 67.]
[Illustration: Fig. 68.]
A figure that is bounded by four straight lines is termed a quadrangle,
quadrilateral or tetragon. When opposite sides of the figure are
parallel to each other it is termed a parallelogram, no matter what the
angle of the adjoining lines in the figure may be. When all the angles
are right angles, as in Figure 66, the figure is called a rectangle. If
the sides of a rectangle are of equal length, as in Figure 67, the
figure is called a square. If two of the parallel sides of a rectangle
are longer than the other two sides, as in Figure 66, it is called an
oblong. If the length of the sides of a parallelogram are all equal and
the angles are not right angles, as in Figure 68, it is called a rhomb,
rhombus or diamond. If two of the parallel sides of a parallelogram are
longer than the other two, and the angles are not right angles, as in
Figure 69, it is called a rhomboid. If two of the parallel sides of a
quadrilateral are of unequal lengths and the angles of the other two
sides are not equal, as in Figure 70, it is termed a trapezoid.
[Illustration: Fig. 69.]
[Illustration: Fig. 70.]
[Illustration: Fig. 71.]
If none of the sides of a quadrangle are parallel, as in Figure 71, it
is termed a trapezium.
THE CONSTRUCTION OF POLYGONS.
[Illustration: Fig. 71_a_.]
[Illustration: Fig. 72.]
The term polygon is applied to figures having flat sides equidistant
from a common centre. From this centre a circle may be struck that will
touch all the corners of the sides of the polygon, or the point of each
side that is central in the length of the side. In drawing a polygon,
one of these circles is used upon which to divide the figure into the
requisite number of divisions for the sides. When the dimension of the
polygon across its corners is given, the circle drawn to that dimension
circumscribes the polygon, because the circle is without or outside of
the polygon and touches it at its corners only. When the dimension
across the flats of the polygon is given, or when the dimension given is
that of a circle that can be inscribed or marked within the polygon,
touching its sides but not passing through them, then the polygon
circumscribes or envelops the circle, and the circle is inscribed or
marked within the polygon. Thus, in Figure 71 _a_, the circle is
inscribed within the polygon, while in Figure 72 the polygon is
circumscribed by the circle; the first is therefore a circumscribed and
the second an inscribed polygon. A regular polygon is one the sides of
which are all of an equal length.
NAMES OF REGULAR POLYGONS.
A figure of 3 sides is called a Trigon.
" 4 " " Tetragon.
polygon 5 " " Pentagon.
" 6 " " Hexagon.
" 7 " " Heptagon.
" 8 " " Octagon.
" 9 " " Enneagon or Nonagon.
[Illustration: Fig. 73.]
[Illustration: Fig. 74.]
The angles of regular polygons are designated by their degrees of angle,
"at the centre" and "at the circumference." By the angle at the centre
is meant the angle of a side to a radial line; thus in Figure 73 is a
hexagon, and at C is a radial line; thus the angle of the side D to C is
60 degrees. Or if at the two ends of a side, as A, two radial lines be
drawn, as B, C, then the angles of these two lines, one to the other,
will be the "angle at the centre." The angle at the circumference is the
angle of one side to its next neighbor; thus the angle at the
circumference in a hexagon is 120 degrees, as shown in the figure for
the sides E, F. It is obvious that as all the sides are of equal length,
they are all at the same angle both to the centre and to one another. In
Figure 74 is a trigon, the angles at its centre being 120, and the angle
at the circumference being 60, as marked.
The angles of regular polygons:
Trigon, at the centre, 120 deg., at the circumference, 60 deg..
Tetragon, " 90 deg., " " 90 deg..
Pentagon, " 72 deg., " " 108 deg..
Hexagon, " 60 deg., " " 120 deg..
Octagon, " 45 deg., " " 135 deg..
Enneagon, " 40 deg., " " 140 deg..
Decagon, " 36 deg., " " 144 deg..
Dodecagon, " 30 deg., " " 150 deg..
THE ELLIPSE.
An ellipse is a figure bounded by a continuous curve, whose nature will
be shown presently.
The dimensions of an ellipse are taken at its extreme length and
narrowest width, and they are designated in three ways, as by the length
and breadth, by the major and minor axis (the major axis meaning the
length, and the minor the breadth of the figure), and the conjugate and
transverse diameters, the transverse meaning the shortest, and the
conjugate the longest diameter of the figure.
In this book the terms major and minor axis will be used to designate
the dimensions.
The minor and major axes are at a right angle one to the other, and
their point of intersection is termed the axis of the ellipse.
In an ellipse there are two points situated upon the line representing
the major axis, and which are termed the foci when both are spoken of,
and a focus when one only is referred to, foci simply being the plural
of focus. These foci are equidistant from the centre of the ellipse,
which is formed as follows: Two pins are driven in on the major axis to
represent the foci A and B, Figure 75, and around these pins a loop of
fine twine is passed; a pencil point, C, is then placed in the loop and
pulled outwards, to take up the slack of the twine. The pencil is held
vertical and moved around, tracing an ellipse as shown.
[Illustration: Fig. 75.]
Now it is obvious, from this method of construction, that there will be
at every point in the pencil's path a length of twine from the final
point to each of the foci, and a length from one foci to the other, and
the length of twine in the loop remaining constant, it is demonstrated
that if in a true ellipse we take any number of points in its curve, and
for each point add together its distance to each focus, and to this add
the distance apart of the foci, the total sum obtained will be the same
for each point taken.
[Illustration: Fig. 76.]
[Illustration: Fig. 77.]
In Figures 76 and 77 are a series of ellipses marked with pins and a
piece of twine, as already described. The corresponding ellipses, as A
in both figures, were marked with the same loop, the difference in the
two forms being due to the difference in distance apart of the foci.
Again, the same loop was used for ellipses B in both figures, as also
for C and D. From these figures we perceive that--
1st. With a given width or distance apart of foci, the larger the
dimensions are the nearer the form of the figure will approach to that
of a circle.
2d. The nearer the foci are together in an ellipse, having any given
dimensions, the nearer the form of the figure will approach that of a
circle.
3d. That the proportion of length to width in an ellipse is determined
by the distance apart of the foci.
4th. That the area enclosed within an ellipse of a given circumference
is greater in proportion as the distance apart of the foci is
diminished; and,
5th. That an ellipse may be given any required proportion of width to
length by locating the foci at the requisite distance apart.
The form of a true ellipse may be very nearly approached by means of the
arcs of circles, if the centres from which those arcs are struck are
located in the most desirable positions for the form of ellipse to be
drawn.
[Illustration: Fig. 78.]
Thus in Figure 78 are three ellipses whose forms were pencilled in by
means of pins and a loop of twine, as already described, but which were
inked in by finding four arcs of circles of a radius that would most
closely approach the pencilled line; _a b_ are the foci of all three
ellipses A, B, and C; the centre for the end curves of _a_ are at _c_
and _d_, and those for its side arcs are at _e_ and _f_. For B the end
centres are at _g_ and _h_, and the side centres at _i_ and _j_. For C
the end centres are at _k_, _l_, and the side centres at _m_ and _n_.
It will be noted that, first, all the centres for the end curves fall on
the line of the length or major axis, while all those for the sides fall
on the line of width or the minor axis; and, second, that as the
dimensions of the ellipses increase, the centres for the arcs fall
nearer to the axis of the ellipse. Now in proportion as a greater number
of arcs of circles are employed to form the figure, the nearer it will
approach the form of a true ellipse; but in practice it is not usual to
employ more than eight, while it is obvious that not less than four can
be used. When four are used they will always fall somewhere on the lines
on the major and minor axis; but if eight are used, two will fall on the
line of the major axis, two on the line of the minor axis, and the
remaining four elsewhere.
[Illustration: Fig. 79.]
In Figure 79 is a construction wherein four arcs are used. Draw the line
_a b_, the major axis, and at a right angle to it the line _c d_, the
minor axis of the figure. Now find the difference between the length of
half the two axes as shown below the figure, the length of line _f_
(from _g_ to _i_) representing half the length of the figure (as from
_a_ to _e_), and the length or radius from _g_ to _h_ equalling that
from _e_ to _d_; hence from _h_ to _i_ is the difference between half
the major and half the minor axis. With the radius (_h i_), mark from
_e_ as a centre the arcs _j k_, and join _j k_ by line _l_. Take half
the length of line _l_ and from _j_ as a centre mark a line on _a_ to
the arc _m_. Now the radius of _m_ from _e_ will be the radius of all
the centres from which to draw the figure; hence we may draw in the
circle _m_ and draw line _s_, cutting the circle. Then draw line _o_,
passing through _m_, and giving the centre _p_. From _p_ we draw the
line _q_, cutting the intersection of the circle with line _a_ and
giving the centre _r_. From _r_ we draw line _s_, meeting the circle and
the line _c, d_, giving us the centre _t_. From _t_ we draw line _u_,
passing through the centre _m_. These four lines _o_, _q_, _s_, _u_ are
prolonged past the centres, because they define what part of the curve
is to be drawn from each centre: thus from centre _m_ the curve from _v_
to _w_ is drawn, from centre _t_ the curve from _w_ to _x_ is drawn.
From centre _r_ the curve from _x_ to _y_ is drawn, and from centre _p_
the curve from _y_ to _v_ is drawn. It is to be noted, however, that
after the point _m_ is found, the remaining lines may be drawn very
quickly, because the line _o_ from _m_ to _p_ may be drawn with the
triangle of 45 degrees resting on the square blade. The triangle may be
turned over, set to point _p_ and line _q_ drawn, and by turning the
triangle again the line _s_ may be drawn from point _r_; finally the
triangle may be again turned over and line _u_ drawn, which renders the
drawing of the circle _m_ unnecessary.
To draw an elliptical figure whose proportion of width to breadth shall
remain the same, whatever the length of the major axis may be: Take any
square figure and bisect it by the line A in Figure 80. Draw, in each
half of the square, the diagonals E F, G H. From P as a centre with the
radius P R draw the arc S E R. With the same radius draw from O as a
centre the arc T D V. With radius L C draw arc R C V, and from K as a
centre draw arc S B T.
[Illustration: Fig. 80.]
[Illustration: Fig. 81.]
A very near approach to the true form of a true ellipse may be drawn by
the construction given in Figure 81, in which A A and B B are centre
lines passing through the major and minor axis of the ellipse, of which
_a_ is the axis or centre, _b c_ is the major axis, and _a e_ half the
minor axis. Draw the rectangle _b f g c_, and then the diagonal line _b
e_; at a right angle to _b e_ draw line _f h_, cutting B B at _i_. With
radius _a e_ and from _a_ as a centre draw the dotted arc _e j_, giving
the point _j_ on line B B. From centre _k_, which is on the line B B and
central between _b_ and _j_, draw the semicircle _b m j_, cutting A A at
_l_. Draw the radius of the semicircle _b m j_, cutting it at _m_, and
cutting _f g_ at _n_. With the radius _m n_ mark on A A at and from _a_
as a centre the point _o_. With radius _h o_ and from centre _h_ draw
the arc _p o q_. With radius _a l_ and from _b_ and _c_ as centres, draw
arcs cutting _p o q_ at the points _p q_. Draw the lines _h p r_ and _h
q s_ and also the lines _p i t_ and _q v w_. From _h_ as a centre draw
that part of the ellipse lying between _r_ and _s_, with radius _p r_;
from _p_ as a centre draw that part of the ellipse lying between _r_ and
_t_, with radius _q s_, and from _q_ as a centre draw the ellipse from
_s_ to _w_, with radius _i t_; and from _i_ as a centre draw the ellipse
from _t_ to _b_ and with radius _v w_, and from _v_ as a centre draw the
ellipse from _w_ to _c_, and one-half of the ellipse will be drawn. It
will be seen that the whole construction has been performed to find the
centres _h_, _p_, _q_, _i_ and _v_, and that while _v_ and _i_ may be
used to carry the curve around on the other side of the ellipse, new
centres must be provided for _h_ _p_ and _q_, these new centres
corresponding in position to _h_ _p_ _q_. Divesting the drawing of all
the lines except those determining its dimensions and the centres from
which the ellipse is struck, we have in Figure 82 the same ellipse drawn
half as large. The centres _v_, _p_, _q_, _h_ correspond to the same
centres in Figure 81, while _v'_, _p'_, _q'_, _h'_ are in corresponding
positions to draw in the other half of the ellipse. The length of curve
drawn from each centre is denoted by the dotted lines radiating from
that centre; thus, from _h_ the part from _r_ to _s_ is drawn; from _h'_
that part from _r'_ to _s'_. At the ends the respective centres _v_ are
used for the parts from _w_ to _w'_ and from _t_ to _t'_ respectively.
[Illustration: Fig. 82.]
[Illustration: Fig. 83.]
The most correct method of drawing an ellipse is by means of an
instrument termed a trammel, which is shown in Figure 83. It consists of
a cross frame in which are two grooves, represented by the broad black
lines, one of which is at a right angle to the other. In these grooves
are closely fitted two sliding blocks, carrying pivots E F, which may be
fastened to the sliding blocks, while leaving them free to slide in the
grooves at any adjusted distance apart. These blocks carry an arm or rod
having a tracing point (as pen or pencil) at G. When this arm is swept
around by the operator, the blocks slide in the grooves and the
pen-point describes an ellipse whose proportion of width to length is
determined by the distance apart of the sliding blocks, and whose
dimensions are determined by the distance of the pen-point from the
sliding block. To set the instrument, draw lines representing the major
and minor axes of the required ellipse, and set off on these lines
(equidistant from their intersection), to mark the required length and
width of ellipse. Place the trammel so that the centre of its slots is
directly over the point or centre from which the axes are marked (which
may be done by setting the centres of the slots true to the lines
passing through the axis) and set the pivots as follows: Place the
pencil-point G so that it coincides with one of the points as C, and
place the pivot E so that it comes directly at the point of intersection
of the two slots, and fasten it there. Then turn the arm so that the
pencil-point G coincides with one of the points of the minor axis as D,
the arm lying parallel to B D, and place the pivot F over the centre of
the trammel and fasten it there, and the setting is complete.
[Illustration: Fig. 84.]
To draw a parabola mechanically: In Figure 84 C D is the width and H J
the height of the curve. Bisect H D in K. Draw the diagonal line J K
and draw K E, cutting K at a right angle to J K, and produce it in E.
With the radius H E, and from J as a centre, mark point F, which will be
the focus of the curve. At any convenient distance above J fasten a
straight-edge A B, setting it parallel to the base C D of the parabola.
Place a square S with its back against the straight-edge, setting the
edge O N coincident with the line J H. Place a pin in the focus F, and
tie to it one end of a piece of twine. Place a tracing-point at J, pass
the twine around the tracing-point, bringing down along the square-blade
and fasten it at N, with the tracing-point kept against the edge of the
square and the twine kept taut; slide the square along the
straight-edge, and the tracing-point will mark the half J C of the
parabola. Turn the square over and repeat the operation to trace the
other half J D. This method corresponds to the method of drawing an
ellipse by the twine and pins, as already described.
[Illustration: Fig. 85.]
To draw a parabola by lines: Bisect the width A B in Figure 85, and
divide each half into any convenient number of equal divisions; and
through these points of division draw vertical lines, as 1, 2, 3, etc.
(in each half). Divide the height A D at one end and B E at the other
into as many equal divisions as the half of A B is divided into. From
the points of divisions 1, 2, 3, etc., on lines A D and B E, draw lines
pointing to C, and where these lines intersect the corresponding
vertical lines are points through which the curve may be drawn. Thus on
the side A D of the curve, the intersection of the two lines marked 1 is
a point in the curve; the intersection of the two lines marked 2 is
another point in the curve, and so on.
TO DRAW A HEART CAM.
[Illustration: Fig. 86.]
Draw the line A B, Figure 86, equal to the length of stroke required.
Divide it into any number of equal parts, and from C as a centre draw
circles through the points of division. Draw the outer circle and divide
its circumference into twice as many equal divisions as the line A B was
divided into. Draw radial lines from each point of division on the
circle, and the points of intersection of the radial lines with the
circles are points for the outline of the cam, and through these points
a curved line may be drawn giving the shape of the cam. It is obvious
that the greater the number of divisions on A B, the more points and the
more perfect the curve may be drawn.
CHAPTER IV.
_SHADOW LINES AND LINE SHADING._
SECTION LINING OR CROSS-HATCHING.
When the interior of a piece is to be shown as a piece cut in half, or
when a piece is broken away, as is done to make more of the parts show,
or show more clearly, the surface so broken away or cut off is
section-lined or cross-hatched; that is to say, diagonal lines are drawn
across it, and to distinguish one piece from another these lines are
drawn at varying angles and of varying widths apart. In Figure 87 is
given a view of three cylindrical pieces. It may be known to be a
sectional view by the cross-hatching or section lines. It would be a
difficult matter to represent the three pieces put together without
showing them in section, because, in an outline view, the collars and
recesses would not appear. Each piece could of course be drawn
separately, but this would not show how they were placed when put
together. They could be shown in one view if they were shaded by lines
and a piece shown broken out where the collars and, recesses are, but
line shading is too tedious for detail drawings, beside involving too
much labor in their production.
[Illustration: Fig. 87.]
Figure 88 represents a case in which there are three cylindrical pieces
one within the other, the two inner ones being fastened together by a
screw which is shown dotted in in the end view, and whose position along
the pieces is shown in the side view. The edges of the fracture in the
outer piece are in this case cross-hatched, to show the line of
fracture.
[Illustration: Fig. 88.]
[Illustration: Fig. 89.]
In cross-hatching it is better that the diagonal lines do not quite meet
the edges of the piece, than that they should in the least overrun, as
is shown in Figure 89, where in the top half the diagonals slightly
overrun, while in the lower half they do not quite meet the outlines of
the piece.
In Figure 90 are shown in section a number of pieces one within the
other, the central bore being filled with short plugs. All the
cross-hatching was done with the triangle of 60 degrees and that of 90
degrees. It is here shown that with these two triangles only, and a
judicious arrangement of the diagonals, an almost infinite number of
pieces may be shown in cross section without any liability of mistaking
one for the other, or any doubt as to the form and arrangement of the
pieces; for, beside the difference in spacing in the cross-hatching,
there are no two adjoining pieces with the diagonals running in the same
direction. It will be seen that the narrow pieces are most clearly
defined by a close spacing of the cross-hatching.
[Illustration: Fig. 90.]
In Figure 91 are shown three pieces put together and having slots or
keyways through them. The outer shell is shown to be in one piece from
end to end, because the cross-hatching is not only equally spaced, but
the diagonals are in the same direction; hence it would be known that D,
F, H, and E were slots or recesses through the piece. The same remarks
apply to piece B, wherein G, J, K are recesses or slots. Piece C is
shown to have in its bore a recess at L. In the case of B, as of A,
there would be no question as to the piece being all one from end to
end, notwithstanding that the two ends are completely severed where the
slots G, I, come, because the spacing and direction of the
cross-hatching are equal on each side of the slots, which they would not
be if they were separate pieces.
[Illustration: Fig. 91.]
[Illustration: Fig. 92.]
Section shading or cross-hatching may sometimes cause the lines of the
drawing to appear crooked to the eye. Thus, in Figure 92, the key edge
on the right appears curved inwards, while on the left the key edge
appears curved outwards, although such is not actually the case. The
same effect is produced in Figure 93 on the right-hand edge of the key,
but not on the left-hand edge.
[Illustration: Fig. 93]
[Illustration: Fig. 94.]
A remarkable instance of this kind is shown in Figure 94, when the
vertical lines appear to the eye to be at a considerable angle one to
the other, although they are parallel.
The lines in sectional shading or cross-hatching may be made to denote
the material of which the piece is to be composed. Thus Professor Unwin
has proposed the system shown in the Figures 95 and 96. This may be of
service in some cases, but it would involve very much more labor than it
is worth in ordinary machine shop drawings, except in the case of cast
iron and wood, these two being shown in the simplest and the usual
manner. It is much better to write the name of the material beneath the
piece in a detail drawing.
[Illustration: Fig. 95.]
[Illustration: Fig. 96.]
LINE SHADING.
Mechanical drawings are made to look better and to show more distinctly
by being line shaded or shaded by lines. The simplest form of line
shading is by the use of the shade or shadow line.
In a mechanical drawing the light is supposed, for the purposes of line
shading or of coloring, to come in from the upper left-hand corner of
the drawing paper; hence it falls directly upon the upper and left-hand
lines of each piece, which are therefore represented by fine lines,
while the right hand and lower edges of the piece being on the shadow
side may therefore, with propriety, be represented by broader lines,
which are called shadow or shade lines. These lines will often serve to
indicate the shape of some part of the piece represented, as will be
seen from the following examples. In Figure 97 is a piece that contains
a hole, the fact being shown by the circle being thickened at A. If the
circle were thickened on the other side as at B, in Figure 98, it would
show that it represented a cylindrical stem instead of a hole.
[Illustration: Fig. 97.]
[Illustration: Fig. 98.]
[Illustration: Fig. 99.]
In Figure 99 is represented a washer, the surfaces that are in the
shadow side being shown in a shade line or shadow line, as it is often
called.
In Figure 100 is a key drawn with a shade line, while in Figure 101 the
shade line is shown applied to a nut. The shade line may be produced in
straight lines by drawing the line twice over, and slightly inclining
the pen, or by opening the pen points a little. For circles, however, it
may be produced either by slightly moving the centre from which the
circle is drawn, or by going over the shade part twice, and slightly
pressing the instrument as it moves, so as to gradually spring the legs
farther apart, the latter plan being generally preferable.
[Illustration: Fig. 100.]
[Illustration: Fig. 101.]
[Illustration: Fig. 102.]
Figure 102 shows a German pen, that can be regulated to draw lines of
various breadths. The head of the adjusting screw is made rather larger
than usual, and is divided at the under side into twenty divisional
notches, each alternate notch being marked by a figure on the face. By
this arrangement a uniform thickness of line may be maintained after
filling or clearing the pen, and any desired thickness may be repeated,
without any loss of time in trial of thickness on the paper. A small
spring automatically holds the divided screw-head in any place. With
very little practice the click of the spring in the notches becomes a
sufficient guide for adjustment, without reference to the figures on the
screw-head. Another meritorious feature of this pen is that it is armed
with sapphire points, which retain their sharpness very long, and thus
save the time and labor required to keep ordinary instruments in order
for the performance of fine work.
An example of line shading in perspective drawing is shown in the
drawing of a pipe threading stock and die in Figure 103.
[Illustration: Fig. 103.]
Shading by means of lines may be used with excellent effect in
mechanical drawing, not only to distinguish round from flat surfaces,
but also to denote to the eye the relative distances of surfaces. Figure
104 represents a cylindrical pin line shaded. As the light is supposed
to come in from the upper left-hand corner, it will evidently fall more
upon the left-hand half of the stem, and of the collar or bead, hence
those parts are shaded with lighter or finer lines than the right-hand
sides are.
[Illustration: Fig. 104.]
[Illustration: Fig. 105.]
Two cylindrical pieces that join each other may be line shaded at
whatever angle they may join. Figure 105 represents two such pieces, one
at a right angle to the other, both being of equal diameter.
[Illustration: Fig. 106.]
Figure 106 represents a drawing of a lathe centre shaded by lines, the
lines on the taper parts meeting those on the parallel part A, and
becoming more nearly parallel to the axis of the piece as the centre of
the piece is approached. The same is the case where a piece having a
curved outline is drawn, which is shown in Figure 107, where the set of
the bow-pen is gradually increased for drawing the shade lines of the
curves. The centres of the shade curves fall in each case upon a line at
a right angle to the axis of the piece, as upon the lines A, B, C, the
dotted lines showing the radius for each curve.
[Illustration: Fig. 107.]
The lines are made finer by closing the pen points by means of the screw
provided for that purpose. The pen requires for this purpose to be
cleaned of the ink that is apt to dry in it.
In Figure 108 line shading is shown applied to a ball or sphere, while
in Figure 109 it is shown applied to a pin in a socket which is shown in
section. By showing the hollow in connection with the round piece, the
difference between the two is quite clearly seen, the light falling
most upon the upper half of the pin and the lower half of the hole. This
perhaps is more clearly shown in the piece of tube in Figure 110, where
the thickness of the tube showing is a great aid to the eye. So,
likewise, the hollow or hole is more clearly seen where the piece is
shown in section, as in Figure 111, which is the case even though the
piece be taper as in the figure. If the body be bell-mouthed, as in
Figure 112, the hollow curve is readily shown by the shading; but to
line shade a hollow curve without any of these aids to the eye, as say,
to show a half of a tin tube, is a very difficult matter if the piece
is to look natural; and all that can be done is to shade the top darkly
and let the light fall mostly at and near the bottom. An example of line
shading to denote the relative distances from the eye of various
surfaces is given in Figure 113, where the surfaces most distant are the
most shaded. The flat surfaces are lined with lines of equal breadth,
the degrees of shading being governed by the width apart of the lines.
[Illustration: Fig. 108.]
[Illustration: Fig. 109.]
[Illustration: Fig. 110.]
[Illustration: Fig. 111.]
[Illustration: Fig. 112.]
[Illustration: Fig. 113.]
Line shading is often used to denote that the piece represented is to be
of wood, the shade lines being in some cases regular in combination with
regular ones, or entirely irregular, as in Figure 114.
[Illustration: Fig. 114.]
CHAPTER V.
_MARKING DIMENSIONS._
The dimensions of mechanical drawings are best marked in red ink so that
they will show plainly, and that the lines denoting the points at which
the dimension is given shall not be confounded with the lines of the
drawing.
The dimension figures should be as large as the drawing will
conveniently admit; and should be marked at every point at which a
shoulder or change of form or dimension occurs, except in the case of
straight tapers which have their dimensions marked at each end of the
taper.
In the case of a single piece standing by itself the dimension figures
may be marked all standing one way, so as to be read without changing
the position of the operator or requiring to turn the drawing around.
This is done in Figure 115, which represents the drawing of a key. The
figures are here placed outside the drawing in all cases where it can be
done, which, in the case of a small drawing, leaves the same clearer.
[Illustration: Fig. 115.]
In Figure 116 the dimensions are marked, running parallel to the
dimension for which they are given, so that all measures of length stand
lengthwise, and those of breadth across the drawing.
[Illustration: Fig. 116.]
Figure 117 represents a key with a sharp-cornered step in it. Here the
two dimensions forming the steps cannot both be coincident with it;
hence they are marked as near to it as convenient, it being understood
that they apply to the step, and not to one side of it. When the step
has a round instead of a sharp corner, the radius of the arc of the
corner may be marked, as shown in Figure 118.
[Illustration: Fig. 117.]
Figure 119 represents a key drawn in perspective, so that all the
dimensions may be marked on one view. Perspective sketches may be used
for single pieces, as they denote the shape of the piece more clearly to
the eye. On account of the skill required in their production, they are
not, however, used in mechanical drawing, except as in the case of
Patent-Office or similar drawings, where the form and construction
rather than the dimension is the information sought to be conveyed.
[Illustration: Fig. 118.]
[Illustration: Fig. 119.]
CHAPTER VI.
_THE ARRANGEMENT OF DIFFERENT VIEWS._
THE DIFFERENT VIEWS OF A MECHANICAL DRAWING.
The word _elevation_, as applied to mechanical drawing, means simply a
view; hence a side elevation is a side view, or an end elevation is an
end view.
The word _plan_ is employed in place of the word top; hence a plan view
is a top view, or a view looking down upon the top of the piece.
A _general_ view means a view showing the machine put together or
assembled, while a detail drawing is one containing a detail, as a part
of the machine or a single piece disconnected from the other parts of
the whole machine.
It is obviously desirable in a mechanical drawing to present the piece
of work in as few views as possible, but in all cases there must be a
sufficient number to permit of the dimensions in every necessary
direction to be marked on the drawing. Suppose, then, that in Figure 120
we have to represent a solid cylinder, whose length equals its diameter,
and it is obvious that both the diameter and length may be marked in the
one view given; hence, a second view, such as shown by the circle in
Figure 121, is unnecessary, except it be to distinguish the body from a
cube, in which the one view would also be sufficient whereon to mark
all the dimensions necessary to enable the piece to be made. It happens,
however, that a cube and a cylinder are the only two figures upon which
all the dimensions can be marked on one view of the piece, and as
cylindrical pieces are much more common in machine work than cubes are,
it is taken for granted that, where the pieces are cylindrical, but one
view shall be used, and that where they are cubes either two views shall
be given, or where they are square a cross shall be marked upon the
parts that are square; thus, in Figure 122, is shown a cross formed by
the lines A B across the face of the drawing, which saves making a
second view.
[Illustration: Fig. 120.]
[Illustration: Fig. 121.]
[Illustration: Fig. 122.]
[Illustration: Fig. 123.]
It would appear that under some conditions this might lead to error; as,
for example, take the piece in Figure 123, and there is nothing to
denote which is the length and which is the diameter of the piece, but
there is a certain amount of custom in such cases than will usually
determine this point; thus, the piece will be given a name, as pin or
disk, the one denoting that its diameter is less than its length, and
the other that its diameter is greater than its length. In the absence
of any such name, it would be in practice assumed that it was a pin and
not a disk; because, if it were a disk, it would either be named or
shaded, or a second view given to show its unusual form, the disk being
a more unusual form than the pin-form in mechanical structures. As an
example of the use of the cross to denote a square, we have Figure 124,
which represents a piece having a hexagon head, section _a_, _a'_, that
is rectangular, a collar _b_, a square part _c_, and a round stem _d_.
Here it will be noted that it is the rectangular part _a_, _a'_, that
renders necessary two views, and that in the absence of the cross, yet
another view would be necessary to show that part _c_ is square.
[Illustration: Fig. 124.]
[Illustration: Fig. 125.]
A rectangular piece always requires two views and sometimes three. In
Figure 125, for example, is a piece that would require a side view to
show the length and breadth, and an edge view to show the thickness.
Suppose the piece to be wedge-shaped in any direction; then another view
will be necessary, as is shown in Figs. 126 and 127. In the former the
wedge or taper is in the direction of its length, while in the latter it
is in the direction of its thickness. Outline views, however, will not
in some cases show the form of the figure, however many views be
presented. An example of this is given in Figure 128, which represents a
ring having a hexagon cross section. A sectional edge view is here
necessary in order to show the hexagonal form. Another example of this
kind, which occurs more frequently in practice, is a cupped ring such as
shown in Figure 129.
[Illustration: Fig. 126.]
[Illustration: Fig. 127.]
[Illustration: Fig. 128.]
[Illustration: Fig. 129.]
EXAMPLES.
Let it be required to draw a rectangular piece such as is shown in two
views in Figure 130, and the process for the pencil lines is as follows:
[Illustration: Fig. 130.]
With the bow-pencil set to half the required length and breadth of the
square the arcs 1, 2, 3 and 4, in Figure 131, are marked, and then the
lines 5 and 6, letting them run past the width of the arcs 3 and 4.
There is no need to pencil in lines 7 and 8, since they can be inked in
without pencilling, because it is known that they must meet the arcs 3
and 4 and terminate at the lines 5 and 6. The top and bottom lines of
the edge view are merely prolongations of lines 5 and 6; hence the lines
9 and 10 are drawn the requisite distance apart for the thickness and to
meet the top and bottom lines. The lines are then inked in, the pencil
lines rubbed out, and the drawing will appear as in Figure 130.
[Illustration: Fig. 131.]
[Illustration: Fig. 132.]
Suppose, however, that the piece has a step in it, as in Figure 132, and
the pencilling will be as in Figure 133. From the centre, the arcs 1, 2,
3 and 4 for the outer, and arcs 5, 6, 7 and 8 for the inner square are
marked; lines 9 and 10, and their prolongations, 11 and 12, for the
edge view, are then pencilled; lines 13 and 14, and their prolongations,
15 and 16, are then pencilled, and dots to show the locations for lines
21 and 22 maybe marked and the pencilling is complete. Lines 17, 18, 19,
20, 21, 22, and 23 may then be inked in, in the order named, and then
lines 9, 10, 11, 12, 13, 14, 15 and 16, when the inking in will be
complete.
[Illustration: Fig. 133.]
In inking in horizontal lines begin at the top and mark in each line as
the square comes to it; and in inking the vertical ones begin always at
the left hand line and mark the lines as they are come to, moving the
square or the triangle to the right, and great care should be taken not
to let the lines cross where they meet, as at the corners, since this
would greatly impair the appearance of the drawing.
These figures have been drawn without the aid of a centre line, because
from their shapes it was easy to dispense with it, but in most cases a
centre line is necessary; thus in Figure 134 we have a body having a
number of steps. The diameters of these steps are marked by arcs, as in
the previous examples, and their lengths may be marked by applying the
measuring rule direct to the drawing paper and making the necessary
pencil mark.
But it would be tedious to mark the successive steps true one with the
other by measuring each step, because one step would require to be
pencilled in before the next could be marked. To avoid this the centre
line 1, Figure 134, is first marked, and the arcs for the steps are then
marked as shown. Centre lines are also necessary to show the alignment
of one part to another; thus in Figure 135 is a cube with a hole passing
through it. The dotted lines in the side view show that the hole passes
clear through the piece and is a parallel one, while the centre line,
being central to the outline throughout the piece, shows that the hole
is equidistant, all through, from the walls of the piece.
[Illustration: Fig. 134.]
[Illustration: Fig. 135.]
The pencil lines for this piece would be marked as in Figure 136, line 1
representing the centre line from which all the arcs are marked. It will
be noted that the length of the piece is marked by arcs which occur,
because being a cube the set of the compasses for arcs 2, 3, 4 and 5
will answer without altering to mark arcs 6 and 7.
[Illustration: Fig. 136.]
If the hole in the piece were a taper or conical one, it would be
denoted by the dotted lines, as in Figure 137, and that the taper is
central to the body is shown by these dotted lines being equidistant
from the centre line.
[Illustration: Fig. 137.]
Suppose one of the sides to be tapered, as is the side A, in Figure 138,
and that the hole is not central, and both facts will be shown by the
centre lines 1 and 2 in the figure. The measurement of face A would be
marked from A to line B at each end, but the distance the hole was out
of the centre would be marked by the distance between the centre line 2
and the edge C of the piece.
[Illustration: Fig. 138.]
If the hole did not pass entirely through the piece, the dotted lines
would show it, as in Figure 139.
[Illustration: Fig. 139.]
[Illustration: Fig. 140.]
The designations of the views of a piece of work depend upon the
position in which the piece stands, when in place upon the machine of
which it forms a part. Thus in Figure 140 is a lever, and if its shaft
stood horizontal when the piece is in place in the machine, the view
given is an end one, but suppose that the shaft stood vertical, and the
same view becomes a plan or top view.
[Illustration: Fig. 141.]
[Illustration: Fig. 142.]
In Figure 142 is a view of a lever which is a side view if the lever
stands horizontal, and lever B hangs down, or a plan view if the shaft
stands horizontal, but lever B stands also horizontal. We may take the
same drawing and turn it around on the paper as in Figure 143, and it
becomes a side view if the shaft stands vertical, and a plan view if the
shaft stands horizontal and arm D vertical above it.
In a side or an end view, the piece that projects highest in the drawing
is highest when upon the machine; also in a side elevation the piece
that is at the highest point in the drawing extends farthest upward when
the piece is on the machine. But in a plan or top view the height of
vertical pieces is not shown, as appears in the case of arm D in Figure
143.
[Illustration: Fig. 143.]
In either of the levers, Figures 142 or 143, all the dimensions could be
marked if an additional view were given, but this will not be the case
if an eye have a slot in it, as at E, in Figure 144, or a jaw have a
tongue in it, as at F: hence, end views of the eye and the jaw must be
given, which may be most conveniently done by showing them projected
from the ends of those parts as in the figure.
This naturally brings us to a consideration as to the best method of
projecting one view from another. As a general rule, the side elevation
or side view is the most important, because it shows more of the parts
and details of the work; hence it should be drawn first, because it
affords more assistance in drawing the other views.
[Illustration: Fig. 144.]
There are two systems of placing the different views of a piece. In the
first the views are presented as the piece would present itself if it
were laid upon the paper for the side view, and then turned or rolled
upon the paper for the other views, as shown in Figure 145, in which the
piece consists of five sections or members, marked respectively A, B, C,
D, and E. Now if the piece were turned or rolled so that the end face of
B were uppermost, and the member E was beneath, it will, by the
operation of turning it, have assumed the position in the lower view
marked position 2; while if it were turned over upon the paper in the
opposite direction it would assume the position marked 3. This gives to
the mind a clear idea of the various views and positions; but it
possesses some disadvantages: thus, if position 1 is a side elevation or
view of the piece, as it stands when in place of the machine, then E is
naturally the bottom member; but it is shown in the top view of the
drawing, hence what is actually the bottom view of the piece (position
3) becomes the top view in the drawing. A second disadvantage is that if
we desire to put in dotted lines, to show how one view is derived from
the other, and denote corresponding parts, then these dotted lines must
be drawn across the face of the drawing, making it less distinct; thus
the dotted lines connecting stem E in position 1 to E in position 3,
pass across the faces of both A and B of position 1.
[Illustration: Fig. 145.]
[Illustration: Fig. 146.]
In a large drawing, or one composed of many members or parts, it would,
therefore, be out of the question to mark in the dotted lines. A
further disadvantage in a large drawing is that it is necessary to go
from one side of the drawing to the other to see the construction of the
same part.
[Illustration: Fig. 147.]
To obviate these difficulties, a modern method is to suppose the piece,
instead of rolling upon the paper, to be lifted from it, turned around
to present the required view, and then moved upwards on the paper for a
top view, sideways for a side view, and below for a bottom view. Thus
the three views of the piece in Figure 145 would be as in Figure 146,
where position 2 is obtained by supposing the piece to be lifted from
position 1, the bottom face turned uppermost, and the piece moved down
the paper to position 2, which is a bottom view of the piece, and the
bottom view in the drawing. Similarly, if the piece be lifted from
position 1, and the top face in that figure is turned uppermost, and the
piece is then slid upwards on the paper, view 3 is obtained, being a top
view of the piece as it lies in position 1, and the top view in the
drawing. Now suppose we require to find the shape of member B, then in
Figure 145 we require to look at the top of position 1, and then down
below to position 2.
[Illustration: Fig. 148.]
But in Figure 146 we have the side view and end view both together,
while the dotted lines do not require to cross the face of the side
view. Now suppose we take a similar piece, and suppose its end faces,
as F, G, to have holes in them, which require to be shown in both views,
and under the one system the drawing would, if the dotted lines were
drawn across, appear as in Figure 147, whereas under the other system
the drawing would appear as in Figure 148. And it follows that in cases
where it is necessary to draw dotted lines from one view to the other,
it is best to adopt the new system.
CHAPTER VII.
_EXAMPLES IN BOLTS, NUTS, AND POLYGONS._
[Illustration: Fig. 149.]
[Illustration: Fig. 150.]
[Illustration: Fig. 151.]
Let it be required to draw a machine screw, and it is not necessary, and
therefore not usual in small screws to draw the full outline of the
thread, but to represent it by thick and thin lines running diagonally
across the bolt, as in Figure 149, the thick ones representing the
bottom, and the thin ones the top of the thread. The pencil lines would
be drawn in the order shown in Figure 150. Line 1 is the centre line,
and line 2 a line to represent the lower side of the head; from the
intersection of these two lines as a centre (as at A) short arcs 3 and
6, showing the diameter of the thread, are marked, and the arcs 5 and 6,
representing the depth of the thread, are marked. The arc 7,
representing the head, is then marked. The vertical lines 8, 9, 10, and
11 are then marked, and the outline of the screw is complete. The thick
lines representing the bottom of the thread are next marked in, as in
Figure 151, extending from line 9 to line 10. Midway between these lines
fine ones are made for the tops of the thread. All the lines being
pencilled in, they may be inked in with the drawing instruments, taking
care that they do not overrun one another. When the pencil lines are
rubbed out, the sketch will appear as in Figure 149.
[Illustration: Fig. 152.]
For a bolt with a hexagon head the lines would be drawn in the order
shown in Figure 152. At a right-angle to centre line 1, line two is
drawn. The pencil-compasses are then set to half the diameter of the
bolt, and from point A arcs 3 and 6 are pencilled, thus showing the
width of the front flat of the head, as well as the diameter of the
stem. From the point where these arcs meet line 2, and with the same
radius, arcs 5 and 6 are marked, showing the widths of the other two
flats of the head. The thickness of the head and the length of the bolt
head may then be marked either by placing a rule on line 1 and marking
the short lines (such as line 7) a cross line 1, or the pencil-compasses
may be set to the rule and the lengths marked from point A. In the
United States standard for bolt heads and nuts the thickness of the head
is made equal to the diameter of the bolt. With the compasses set for
the arcs 3 and 4, we may in two steps, from A along the centre line,
mark off the thickness of the head without using the rule. But as the
rule has to be applied along line 1 to mark line 7 for the length of the
bolt, it is just as easy to mark the head thickness at the same time.
The line 8 showing the length of the thread may be marked at the same
time as the other lengths are marked, and the outlines 9, 10, 11, 12, 13
may be drawn in the order named. We have now to mark the arcs at the top
of the flats of the head to show the chamfer, and to explain how these
arcs are obtained we have in Figure 153 an enlarged view of the head. It
is evident that the smallest diameter of the chamfer is represented by
the circle A, and therefore the length of the line B must equal A. It is
also evident that the outer edge of the chamfer will meet the corners at
an equal depth (from the face of the nut), as represented by the line C
C, and it is obvious that the curves that represent the outline of the
chamfer on each side of the head or nut will approach the face of the
head or nut at an equal distance, as denoted by the line D D. It follows
that the curve must in each case be such as will, at each of its ends,
meet the line C, and at its centre meet the line D D, the centres of the
respective curves being marked in the figure by X.
[Illustration: Fig. 153.]
It is sufficiently accurate, therefore, for all practical purposes to
set the pencil on the centre-line at the point A in Figure 152 and mark
the curve 14, and to then set the compasses by trial to mark the other
two curves of the chamfer, so that they shall be an equal distance with
arc 14 from line 9, and join lines 10 and 13 at the same distance from
line 9 that 14 joins lines 3 and 4, so that as in Figure 153 all three
of the arcs would touch a line as C, and another line as D.
[Illustration: Fig. 154.]
The United States standard sizes for forged or unfinished bolts and nuts
are given in the following table, Figure 154 showing the dimensions
referred to in the table.
UNITED STATES STANDARD DIMENSIONS OF BOLTS AND NUTS.
KEY:
A: Nominal. D.
B: Effective.[*]
C: Standard Number of threads per inch.
------------------------+----------------------------+---------------------
BOLT. | BOLT HEAD AND NUT. |
------------------------+-----------------+----------+----------+----------
Diameter. | | Long diameter, | Short | |
-------+------| | I, or diameter | diameter | |
| | | across corners. | of | |
| | |--------+--------| hexagon | |
A | B | C | Hexa- | Square.| and | |
| | | gon. | | square, | Depth of | Depth of
| | | | | or width | nut, | bolt
| | | | | across J.| H. | head, K.
-------+------+---------+--------+--------+----------+----------+----------
1/4 | .185 | 20 | 9/16| 23/32| 1/2 | 1/4 | 1/4
5/16 | .240 | 18 | 11/16| 27/32| 19/32 | 5/16 | 19/64
3/8 | .294 | 16 | 25/32| 31/32| 11/16 | 3/8 | 11/32
7/16 | .345 | 14 | 29/32| 1-3/32 | 25/32 | 7/16 | 25/64
1/2 | .400 | 13 | 1 | 1-1/4 | 7/8 | 1/2 | 7/16
9/16 | .454 | 12 | 1-1/8 | 1-3/8 | 31/32 | 9/16 | 31/64
5/8 | .507 | 11 | 1-7/32 | 1-1/2 | 1-1/16 | 5/8 | 17/32
3/4 | .620 | 10 | 1-7/16 | 1-3/4 | 1-1/4 | 3/4 | 5/8
7/8 | .731 | 9 | 1-21/32| 2-1/32 | 1-7/16 | 7/8 | 23/32
1 | .837 | 8 | 1-7/8 | 2-5/16 | 1-5/8 | 1 | 13/16
1-1/8 | .940 | 7 | 2-3/32 | 2-9/16 | 1-13/16 | 1-1/8 | 29/32
1-1/4 |1.065 | 7 | 2-5/16 | 2-27/32| 2 | 1-1/4 | 1
1-3/8 |1.160 | 6 | 2-17/32| 3-3/32 | 2-3/16 | 1-3/8 | 1-3/32
1-1/2 |1.284 | 6 | 2-3/4 | 3-11/32| 2-3/8 | 1-1/2 | 1-3/16
1-5/8 |1.389 | 5-1/2 | 2-31/32| 3-5/8 | 2-9/16 | 1-5/8 | 1-9/32
1-3/4 |1.491 | 5 | 3-3/16 | 3-7/8 | 2-3/4 | 1-3/4 | 1-3/8
1-7/8 |1.616 | 5 | 3-13/32| 4-5/32 | 2-15/16 | 1-7/8 | 1-15/32
2 |1.712 | 4-1/2 | 3-19/32| 4-13/32| 3-1/8 | 2 | 1-9/16
2-1/4 |1.962 | 4-1/2 | 4-1/32 | 4-15/16| 3-1/2 | 2-1/4 | 1-3/4
2-1/2 |2.176 | 4 | 4-15/32| 5-15/32| 3-7/8 | 2-1/2 | 1-15/16
2-3/4 |2.426 | 4 | 4-29/32| 6 | 4-1/4 | 2-3/4 | 2-1/8
3 |2.629 | 3-1/2 | 5-11/32| 6-17/32| 4-5/8 | 3 | 2-5/16
3-1/4 |2.879 | 3-1/2 | 5-25/32| 7-1/16 | 5 | 3-1/4 | 2-1/2
3-1/2 |3.100 | 3-1/4 | 6-7/32 | 7-19/32| 5-3/8 | 3-1/2 | 2-11/16
3-3/4 |3.317 | 3 | 6-5/8 | 8-1/8 | 5-3/4 | 3-3/4 | 2-7/8
... |3.567 | 3 | 7-1/16 | 8-21/32| 6-1/8 | 4 | 3-1/16
4-1/4 |3.798 | 2-7/8 | 7-1/2 | 9-3/16 | 6-1/2 | 4-1/4 | 3-1/4
4-1/2 |4.028 | 2-3/4 | 7-15/16| 9-23/32| 6-7/8 | 4-1/2 | 3-7/16
4-3/4 |4.256 | 2-5/8 | 8-3/8 |10-1/4 | 7-1/4 | 4-3/4 | 3-5/8
5 |4.480 | 2-1/2 | 8-13/16|10-25/32| 7-5/8 | 5 | 3-13/16
5-1/4 |4.730 | 2-1/2 | 9-1/4 |11-5/16 | 8 | 5-1/4 | 4
5-1/2 |4.953 | 2-3/8 | 9-11/16|11-27/32| 8-3/8 | 5-1/2 | 4-3/16
5-3/4 |5.203 | 2-3/8 |10-3/32 |12-3/8 | 8-3/4 | 5-3/4 | 4-3/8
6 |5.423 | 2-1/4 |10-17/32|12-29/32| 9-1/8 | 6 | 4-9/16
-------+------+---------+--------+--------+----------+----------+----------
* Diameter at the root of the thread.
The basis of the Franklin Institute or United States standard for the
heads of bolts and for nuts is as follows:
The short diameter or width across the flats is equal to one and
one-half times the diameter plus 1/8 inch for rough or unfinished bolts
and nuts, and one and one-half times the bolt diameter plus, 1/16 inch
for finished heads and nuts. The thickness is, for rough heads and nuts,
equal to the diameter of the bolt, and for finished heads and nuts 1/16
inch less.
[Illustration: Fig. 155.]
[Illustration: Fig. 156.]
The hexagonal or hexagon (as they are termed in the shop) heads of bolts
may be presented in two ways, as is shown in Figures 155 and 156.
The latter is preferable, inasmuch as it shows the width across the
flats, which is the dimension that is worked to, because it is where the
wrench fits, and therefore of most importance; whereas the latter gives
the length of a flat, which is not worked to, except incidentally, as it
were. There is the objection to the view of the head, given in Figure
156, however, that unless it is accompanied by an end view it somewhat
resembles a similar view of a square head for a bolt. It may be
distinguished therefrom, however, in the following points:
If the amount of chamfer is such as to leave the chamfer circle (as
circle A, in Figure 153) of smaller diameter than the width across the
flats of the bolt-head, the outline of the sides of the head will pass
above the arcs at the top of the flats, and there will be two small flat
places, as A and B, in Figure 156 (representing the angle of the
chamfer), which will not meet the arcs at the top of the flats, but will
join the sides above those arcs, as in the figure; which is also the
case in a similar view of a square-headed bolt. It may be distinguished
therefrom, however, in the following points:
If the amount of chamfer is such as to leave the chamfer circle (A,
Figure 153) of smaller diameter than the width across the flats of the
bolt-head, the outline of the sides will pass above the arc on the
flats, as is shown in Figure 157, in which the chamfer A meets the side
of the head at B, and does not, therefore, meet the arc C. The length of
side lying between B and D in the side view corresponds with the part
lying between E and F in the end view.
[Illustration: Fig. 157.]
If we compare this head with similar views of a square head G, both
being of equal widths, and having their chamfer circles at an equal
distance from the sides of the flats, and at the same angle, we perceive
at once that the amount of chamfer necessary to give the same distance
between the chamfer circle and the side of the bolt (that is, the
distance from J to K, being equal to that from L to M), the length of
the chamfer N for the square head so greatly exceeds the length A for
the hexagon head that the eye detects the difference at once, and is
instinctively informed that G must be square, independently of the fact
that in the case of the square head, N meets the arc O, while in the
hexagon head, A, which corresponds to N, does not meet the arc C, which
corresponds to O.
When, however, the chamfer is drawn, but just sufficient to meet the
flats, as in the case of the hexagon H, and the square I, in Figure 157,
the chamfer line passes from the chamfer circle to the side of the head,
and the distinction is greater, as will be seen by comparing head H with
head I, both being of equal width, having the same angle of chamfer, and
an amount just sufficient to meet the sides of the flats. Here it will
be seen that in the hexagon H, each side of the head, as P, meets the
chamfer circle A. Whereas, in the square head these two lines are joined
by the chamfer line Q, the figures being quite dissimilar.
[Illustration: Fig. 158.]
It is obvious that whatever the degree or angle of the chamfer may be,
the diameter of the chamfer circle will be the same in any view in
which the head may be presented. Thus, in Figure 158, the line G in the
side view is in length equal to the diameter of circle G, in the end
view, and so long as the angle of the chamfer is forty-five degrees, as
in all the views hitherto given, the width of the chamfer will be equal
at corresponding points in the different views; thus in the figure the
widths A and B in the two views are equal.
[Illustration: Fig. 159.]
If the other view showing a corner of the head in front of the head be
given, the same fact holds good, as is shown in Figure 159. That the two
outside flats should appear in the drawing to be half the width of the
middle flat is also shown in Figure 158, where D and E are each half the
width of C. Let us now suppose, that the chamfer be given some other
angle than that of 45 degrees, and we shall find that the effect is to
alter the curves of the chamfer arcs on the flats, as is shown in Figure
160, where these arcs E, C, D are shown less curved, because the chamfer
B has more angle to the flats. As a result, the width or distance
between the arcs and line G is different in the two views. On this
account it is better to draw the chamfer at 45 degrees, as correct
results may be obtained with the least trouble.
If no chamfer at all is to be given, a hexagon head may still be
distinguished from a square one, providing that the view giving three
sides of the head, as in Figure 158, is shown, because the two sides D
and E being half the width of the middle one C, imparts the information
that it is a hexagon head. If, however, the view showing but two of the
sides and a corner in front is given, and no chamfer is used, it could
not be known whether the head was to be hexagon or square, unless an end
view be given, as in Figure 161.
[Illustration: Fig. 160.]
If the view showing a full side of the head of a square-headed bolt is
given, then either an end view must be given, as in Figure 162, or else
a single view with a cross on its head, as in Figure 163, may be given.
It is the better plan, both in square and hexagon heads, to give the
view in which the full face of a flat is presented, that is, as in
Figures 155 and 163; because, in the case of the square, the length of a
side and the width across the head are both given in that view; whereas
if two sides are shown, as in Figure 161, the width across flats is not
given, and this is the dimension that is wanted to work to, and not the
width across corners. In the case of a hexagon the middle of the three
flats is equal in width to the diameter of the bolt, and the other two
are one-half its width; all three, therefore, being marked with the same
set of compasses as gives the diameter of the body of the bolt, were as
shown in Figure 152. For the width across flats there is an accepted
standard; hence there is no need to mark it upon the drawing, unless in
cases where the standard is to be departed from, in which event an end
view may be added, or the view showing two sides may be given.
[Illustration: Fig 161.]
[Illustration: Fig. 162.]
[Illustration: Fig. 163.]
[Illustration: Fig. 164.]
To draw a square-headed bolt, the pencil lines are marked in the order
shown by figures in Figure 164. The inking in is done in the order of
the letters _a_, _b_, _c_, etc. It will be observed that pencil lines 2,
9, and 10 are not drawn to cross, but only to meet the lines at their
ends, a point that, as before stated, should always be carefully
attended to.
[Illustration: Fig. 165]
To draw the end view of a hexagon head, first draw a circle of the
diameter across the flats, and then rest the triangle of 60 degrees on
the blade _s_ of the square, as at T 1, in Figure 165, and mark the
lines _a_ and _b_. Reverse the triangle, as at T 2, and draw lines _c_
and _d_. Then place the triangle as in Figure 166, and draw the lines
_e_ and _f_.
[Illustration: Fig. 166.]
If the other view of the head is to be drawn, then first draw the lines
_a_ and _b_ in Figure 167 with the square, then with the 60 degree
triangle, placed on the square S, as at T 1, draw the lines _c_, _d_,
and turning the square over, as at T 2, mark lines _e_ and _f_.
[Illustration: Fig. 167.]
If the diameter across corners of a square head is given, and it be
required to draw the head, the process is as follows: For a view showing
one corner in front, as in Figure 168, a circle of the given diameter
across corners is pencilled, and the horizontal centre-line _a_ is
marked, and the triangle of 45 degrees is rested against the square
blade S, as in position T 1, and lines _b_ and _c_ marked, _b_ being
marked first; and the triangle is then slid along the square blade to
position T 1, when line _c_ is marked, these two lines just meeting the
horizontal line _a_, where it meets the circle. The triangle is then
moved to the left, and line _d_, joining the ends of _b_ and _c_, is
marked, and by moving it still farther to the left to position T 2, line
_e_ is marked. Lines _b_, _c_, _d_, and _e_ are, of course, the only
ones inked in.
[Illustration: Fig. 168.]
[Illustration: Fig. 169.]
If the flats are to lie in the other direction, the pencilling will be
done as in Figure 169. The circle is marked as before, and with the
triangle placed as shown at T 1, line _a_, passing through the centre of
the circle, is drawn. By moving the triangle to the right its edge B
will be brought into position to mark line _b_, also passing through
the centre of the circle. All that remains is to join the ends of these
two lines, using the square blade for lines _c_, _d_, and the triangle
for _e_ and _f_, its position on the square blade being denoted at T 3;
lines _c_, _d_, _e_, _f_, are the ones inked in.
[Illustration: Fig. 170.]
For a hexagon head we have the processes, Figures 170 and 171. The
circle is struck, and across it line _a_, Figure 170, passing through
its centre, the triangle of sixty degrees will mark the sides _b_, _c_,
and _d_, _e_, as shown, and the square blade is used for _f_, _g_.
[Illustration: Fig. 171.]
The chamfer circles are left out of these figures to reduce the number
of lines and so keep the engraving clear. Figure 171 shows the method of
drawing a hexagon head when the diameter across corners is given, the
lines being drawn in the alphabetical order marked, and the triangle
used as will now be understood.
[Illustration: Fig. 172.]
[Illustration: Fig. 173.]
It may now be pointed out that the triangle may be used to divide
circles much more quickly than they could be divided by stepping around
them with compasses. Suppose, for example, that we require to divide a
circle into eight equal parts, and we may do so as in Figure 172, line
_a_ being marked from the square, and lines _b_, _c_ and _d_ from the
triangle of forty-five degrees; the lines to be inked in to form an
octagon need not be pencilled, as their location is clearly defined,
being lines joining the ends of the lines crossing the circle, as for
example, lines _e_, _f_.
Let it be required to draw a polygon having twelve equal sides, and the
triangle of sixty is used, marking all the lines within the circle in
Figure 173, except _a_, for which the square blade is used; the only
lines to be inked in are such as _b_, _c_. In this example there is a
corner at the top and bottom, but suppose it were required that a flat
should fall there instead of a corner; then all we have to do is to set
the square blade S at the required angle, as in Figure 174, and then
proceed as before, bearing in mind that the point of the circle nearest
to the square blade, straight-edge, or whatever the triangle is rested
on, is always a corner of a polygon having twelve sides.
[Illustration: Fig. 174.]
[Illustration: Fig. 175.]
In both of these examples we have assumed that the diameter across
corners of the polygon was given, but suppose the diameter across the
flats were given, and the construction is a little more complicated.
Circle _a_, _a_, in Figure 175, is drawn of the required diameter across
the flats, and the lines of division are drawn across with the triangle
of 60 as before; the triangle of 45 is then used to draw the four lines,
_b_, _c_, _d_, _e_, joining the ends of lines _i_, _j_, _k_, _l_, and
touching the inner circle, _a_, _a_. The outer circle is then pencilled
in, touching the lines of division where they meet the lines _b_, _c_,
_d_, _e_, and the rest of the lines for the sides of the polygon may
then be drawn within the outer circle, as at _g_, _h_.
[Illustration: Fig. 176.]
It is obvious, also, that the triangle may be used to draw slots
radiating from a centre, as in Figure 176, where it is desired to draw a
chuck-plate having 6 slots. The triangle of 60 is used to draw the
centre lines, _a_, _b_, _c_, etc., for the slots. From the centre, the
arcs _e_, _f_, _g_, _h_, etc., are marked, showing where the centres
will fall for describing the half circles forming the ends of the slots.
Then half circles, _i_, _j_, _k_, _l_, etc., being drawn, the sides of
the slots may be drawn in with the triangle, and the outer circle and
the slots inked in.
If the slots are not to radiate from the centre of the circle the
process is as follows:
The outer circle _a_, Figure 177, being drawn, an inner one _b_ is
drawn, its radius equalling the amount; the centres of the slots are to
point to one side of the centre of circle _a_. The triangle is then used
to divide the circle into the requisite number of divisions _c_ for the
slots, and arcs _i, j_, are then drawn for the lengths of the slots. The
centre lines _e_ are then drawn, passing through the lines _c_, and the
arcs _i, j_, etc., and touching the perimeter of the inner circle _b_;
arcs _f, g_, are then marked in, and their sides joined with the
triangle adjusted by hand. All that would be inked in black are the
outer circle and the slots, but the inner circle _b_ and a centre line
of one of the slots should be marked in red ink to show how the
inclination of the slot was obtained, and therefore its amount.
[Illustration: Fig. 177.]
For a five-sided figure it is best to step around the circumference of
the circle with the compasses, but for a three-sided one, or trigon, the
construction is as follows: It will be found that the compasses set to
the radius of a circle will accurately divide it into six equal
divisions, as is shown in Figure 178; hence every other one of these
divisions will be the location for a corner of a trigon.
The circle being drawn, a line A, 179, is drawn through its centre, and
from its intersection with the circle as at _b_, here a step on each
side is marked as _c_, _d_, then lines _c_ to _d_, and _c_ and _d_ to
_e_, where A meets, the circle will describe a trigon. If the figure is
to stand vertical, all that is necessary is to draw the line _a_
vertical, as in Figure 180. A ready method of getting the dimension
across corners, across the flats, or the length of a side of a given
polygon, is by means of diagrams, such as shown in the following
figures, which form excellent examples for practice.
[Illustration: Fig. 178.]
[Illustration: Fig. 179.]
[Illustration: Fig. 180.]
Draw the line O P, Figure 181, and at a right angle to it the line O B;
divide these two lines into parts of one inch, as shown in the cut,
which is divided into inches and quarter inches, and from these points
of division draw lines crossing each other as shown.
[Illustration: Fig. 181.]
From the point O, draw diagonal lines, at suitable angles to the line O
P. As shown in the cut, these diagonal lines are marked:
40 degrees for 5 sided figures.
45 " " 6 " "
49 " " 7 " "
52-1/2 " " 8 " "
55-1/2 " " 9 " "
But still others could be added for figures having a greater number of
sides.
1. Now it will be found as follows: Half the diameter, or the radius of
a piece of cylindrical work being given, and the number of sides it is
to have being stated, the length of one side will be the distance
measured horizontally from the line O B to the diagonal line for that
particular number of sides.
EXAMPLE.--A piece of work is 2-1/2 inches in diameter, and is required
to have 9 sides: what will be the length of the sides or flats?
Now the half diameter or radius of 2-1/2 inches is 1-1/4 inches. Then
look along the line O B for 1-1/4, which is denoted in the cut by
figures and the arrow A; set one point of the compasses at A, and the
other at the point of crossing of the diagonal line with the 1-1/4
horizontal line, as shown in the figure at _a_, and from A to _a_ is the
length of one side.
Again: A piece of work, 4 inches in diameter, is to have 9 sides: how
long will each side be?
Now half of 4 is 2, hence from B to _b_ is the length of each side.
But suppose that from the length of each side, and the number of sides,
it is required to find the diameter to which to turn the piece; that is,
its diameter across corners, and we simply reverse the process thus: A
body has 9 sides, each side measures 27/32: what is its diameter across
corners?
Take a rule, apply it horizontally on the figure, and pass it along till
the distance from the line O B to the diagonal line marked 9 sides
measures 27/32, which is from 1-1/4 on O B to _a_, and the 1-1/4 is the
radius, which, multiplied by 2, gives 2-1/2 inches, which is the
required diameter across corners.
For any other number of sides the process is just the same. Thus: A
body is 3-1/2 inches in diameter, and is to have 5 sides: what will be
the length of each side? Now half of 3-1/2 is 1-3/4; hence from 1-3/4 on
the line O B to the point C, where the diagonal line crosses the 1-3/4
line, is the length of each of the sides.
2. It will be found that the length of a side of a square being given,
the size of the square, measured across corners, will be the length of
the diagonal line marked 45 degrees, from the point O to the figures
indicating, on the line O B or on the line O P, the length of one side.
EXAMPLE.--A square body measures 1 inch on each side: what does it
measure across the corners? Answer: From the point O, along diagonal
line marked 45 degrees, to the point where it crosses the lines 1 (as
denoted in the figure by a dot).
Again: A cylindrical piece of wood requires to be squared, and each side
of the square must measure an inch: what diameter must the piece be
turned to?
Now the diagonal line marked 45 degrees passes through the 1-inch line
on O B, and the inch line on O P, at the point where these lines meet;
hence all we have to do is to run the eye along either of the lines
marked inch, and from its point of meeting the 45 degrees line, to the
point O, is the diameter to turn the piece to.
There is another way, however, of getting this same measurement, which
is to set a pair of compasses from the line 1 on O B, to line 1 on O P,
as shown by the line D, which is the full diameter across corners. This
is apparent, because from point O, along line O B, to 1, thence to the
dot, thence down to line 1 on O P, and along that to O, encloses a
square, of which either from O to the dot, or the length of the line D,
is the measurement across corners, while the length of each side, or
diameter across the flats, is from point O to either of the points 1, or
from either of the points 1 to the dot.
[Illustration: Fig. 182.]
After graphically demonstrating the correctness of the scale we may
simplify it considerably. In Figure 182, therefore, we have applications
shown. A is a hexagon, and if one of its sides be measured, it will be
found that it measures the same as along line 1 from O B to the diagonal
line 45 degrees, which distance is shown by a thickened line.
At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and
radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened
horizontal line), to the diagonal marked 49 degrees, be measured, it
will be found exactly equal to the length of a side on the polygon.
At C is shown part of a nine-sided polygon, of 2-inch radius, and the
length of one of its sides will be found to equal the distance from the
diagonal line marked 52-1/2 degrees, and the line O B at 2.
Let it now be noted that if from the point O, as a centre, we describe
arcs of circles from the points of division on O B to O P, one end of
each arc will meet the same figure on O P as it started from at O B, as
is shown in Figure 181, and it becomes apparent that in the length of
diagonal line between O and the required arc we have the radius of the
polygon.
EXAMPLE.--What is the radius across corners of a hexagon or six-sided
figure, the length of a side being an inch?
Turning to our scale we find that the place where there is a horizontal
distance of an inch between the diagonal 45 degrees, answering to
six-sided figures, is along line 1 (Figure 182), and the radius of the
circle enclosing the six-sided body is, therefore, an inch, as will be
seen on referring to circle A. But it will be noted that the length of
diagonal line 45 degrees, enclosed between the point O and the arc of
circle from 1 on O B to one on O P, measures also an inch. Hence we may
measure the radius along the diagonal lines if we choose. This, however,
simply serves to demonstrate the correctness of the scale, which, being
understood, we may dispense with most of the lines, arriving at a scale
such as shown in Figure 183, in which the length of the side of the
polygon is the distance from the line O B, measured horizontally to the
diagonal, corresponding to the number of sides of the polygon. The
radius across corners of the polygon is that of the distance from O
along O B to the horizontal line, giving the length of the side of the
polygon, and the width across corners for a given length of one side of
the square, is measured by the length of the lines A, B, C, etc. Thus,
dotted line 2 shows the length of the side of a nine-sided figure, of
2-inch radius, the radius across corners of the figure being 2 inches.
[Illustration: Fig. 183.]
The dotted line 2-1/2 shows the length of the side of a nine-sided
polygon, having a radius across corners of 2-1/2 inches. The dotted line
1 shows the diameter, across corners, of a square whose sides measure an
inch, and so on.
[Illustration: Fig. 184.]
This scale lacks, however, one element, in that the diameter across the
flats of a regular polygon being given, it will not give the diameter
across the corners. This, however, we may obtain by a somewhat similar
construction. Thus, in Figure 184, draw the line O B, and divide it into
inches and parts of an inch. From these points of division draw
horizontal lines; from the point O draw the following lines and at the
following angles from the horizontal line O P.
[Illustration: Fig. 185.]
A line at 75 deg. for polygons having 12 sides.
" 72 deg. " " 10 "
" 67-1/2 deg. " " 8 "
" 60 deg. " " 6 "
From the point O to the numerals denoting the radius of the polygon is
the radius across the flats, while from point O to the horizontal line
drawn from those numerals is the radius across corners of the polygon.
[Illustration: Fig. 186.]
A hexagon measures two inches across the flats: what is its diameter
measured across the corners? Now from point O to the horizontal line
marked 1 inch, measured along the line of 60 degrees, is 1 5-32nds
inches: hence the hexagon measures twice that, or 2 5-16ths inches
across corners. The proof of the construction is shown in the figure,
the hexagon and other polygons being marked simply for clearness of
illustration.
[Illustration: Fig. 187.]
[Illustration: Fig. 188.]
Let it be required to draw the stud shown in Figure 185, and the
construction would be, for the pencil lines, as shown in Figure 186; line
1 is the centre line, arcs, 2 and 3 give the large, and arcs 4 and 5
the small diameter, to touch which lines 6, 7, 8, and 9 may be drawn.
Lines 10, 11, and 12 are then drawn for the lengths, and it remains to
draw the curves in. In drawing these curves great exactitude is required
to properly find their centres; nothing looks worse in a drawing than an
unfair or uneven junction between curves and straight lines. To find the
location for these centres, set the compasses to the required radius for
the curve, and from the point or corner A draw the arcs _b_ and _c_,
from _c_ mark the arc _e_, and from _b_ the arc _d_, and where _d_ and
_e_ cross is the centre for the curve _f_.
[Illustration: Fig. 189.]
Similarly for the curve _h_, set the compasses on _i_ and mark the arc
_g_, and from the point where it crosses line 6, draw the curve _h_. In
inking in it is best to draw in all curves or arcs of circles first, and
the straight lines that join them afterward, because, if the straight
lines are drawn first, it is a difficult matter to alter the centres of
the curves to make them fall true, whereas, after the curves are drawn
it is an easy matter, if it should be necessary, to vary the line a
trifle, so as to make it join the curves correctly and fair. In inking
in these curves also, care must be taken not to draw them too short or
too long, as this would impair the appearance very much, as is shown in
Figure 187.
[Illustration: Fig. 190.]
[Illustration: Fig. 191.]
To draw the piece shown in Figure 188, the lines are drawn in the order
indicated by the letters in Figure 189, the example being given for
practice. It is well for the beginner to draw examples of common
objects, such as the hand hammer in Figure 190, or the chuck plate in
Figure 191, which afford good examples in the drawing of arcs and
circles.
In Figure 191 _a_ is a cap nut, and the order in which the same would be
pencilled in is indicated by the respective numerals. The circles 3 and
4 represent the thread.
[Illustration: Fig. 191 _a_.]
In Figure 192 is shown the pencilling for a link having the hubs on one
side only, so that a centre line is unnecessary on the edge view, as
all the lengths are derived from the top view, while the thickness of
the stem and height of the hubs may be measured from the line A. In
Figure 193 there are hubs (on both sides of the link) of unequal height,
hence a centre line is necessary in both views, and from this line all
measurements should be marked.
[Illustration: Fig. 192.]
[Illustration: Fig. 193.]
In Figure 194 are represented the pencil lines for a double eye or
knuckle joint, as it is sometimes termed, an example that it is
desirable for the student to draw in various sizes, as it is
representative of a large class of work.
These eyes often have an offset, and an example of this is given in
Figure 195, in which A is the centre line for the stem distant from the
centre line B of the eyes to the amount of offset required.
[Illustration: Fig. 194.]
[Illustration: Fig. 195.]
[Illustration: Fig. 196.]
[Illustration: Fig. 197.]
In Figure 196 is an example of a connecting rod end. From a point, as A,
we draw arcs, as B C for the width, and E D for the length of the block,
and through A we draw the centre line. It is obvious, however, that we
may draw the centre line first, and apply the measuring rule direct to
the paper, and mark lines in place of the arcs B, C, D, E, and F, G,
which are for the stem. As the block joins the stem in a straight line,
the latter is evidently rectangular, as will be seen by referring to
Figure 197, which represents a rod end with a round stem, the fact that
the stem is round being clearly shown by the curves A B. The radius of
these curves is obtained as follows: It is obvious that they will join
the rod stem at the same point as the shoulder curves do, as denoted by
the dotted vertical line. So likewise they join the curves E F at the
same point in the rod length as the shoulder curves, both curves in fact
being formed by the same round corner or shoulder. The centre of the
radius of A or B must therefore be the same distance from the centre of
the rod as is the centre from which the shoulder curve is struck, and at
the same time at such a distance from the corner (as E or F) that the
curve will meet the centre line of the rod at the same point in its
length as the shoulder curves do.
[Illustration: Fig. 198.]
Figure 198 gives an example, in which the similar curved lines show that
a part is square. The figure represents a bolt with a square under the
head. As but one view is given, that fact alone tells us that it must be
round or square. Now we might mark a cross on the square part, to denote
that it is square; but this is unnecessary, because the curves F G show
such to be the case. These curves are marked as follows: With the
compasses set to the radius E, one point is rested at A, and arc B is
drawn; then one point of the compass is rested at C, and arc D is drawn;
giving the centre for the curve F by a similar process on the other
side of the figure, curve G is drawn. Point C is obtained by drawing the
dotted line across where the outline curve meets the stem. Suppose that
the corner where the round stem meets the square under the head was a
sharp one instead of a curve, then the traditional cross would require
to be put on the square, as in Figure 199; or the cross will be
necessary if the corner be a round one, if the stem is reduced in
diameter, as in Figure 200.
[Illustration: Fig. 199.]
[Illustration: Fig. 200.]
[Illustration: Fig. 201.]
Figure 201 represents a centre punch, giving an example, in which the
flat sides gradually run out upon a circle, the edges forming curves, as
at A, B, etc. The length of these curves is determined as follows: They
must begin where the taper of the punch joins the parallel, or at C, C,
and they must end on that part of the taper stem where the diameter is
equal to the diameter across the flats of the octagon. All that is to be
done then is to find the diameter across the flats on the end view, and
mark it on the taper stem, as at D, D, which will show where the flats
terminate on the taper stem. And the curved lines, as A, B, may be drawn
in by a curve that must meet at the line C, and also in a rounded point
at line D.
CHAPTER VIII.
_SCREW THREADS AND SPIRALS._
[Illustration: Fig. 202.]
[Illustration: Fig. 203.]
The screw thread for small bolts is represented by thick and thin lines,
such as was shown in Figure 152, but in larger sizes; the angles of the
thread also are drawn in, as in Figure 202, and the method of doing this
is shown in Figure 203. The centre line 1 and lines 2 and 3 for the full
diameter of the thread being drawn, set the compasses to the required
pitch of the thread, and stepping along line 2, mark the arcs 4, 5, 6,
etc., for the full length the thread is to be marked. With the triangle
resting against the $T$-square, the lines 7, 8, 9, etc. (for the full
length of the thread), are drawn from the points 4, 5, 6, on line 2.
These give one side of the thread. Reversing the drawing triangle,
angles 10, 11, etc., are then drawn, which will complete the outline of
the thread at the top of the bolt. We may now mark the depth of the
thread by drawing line 12, and with the compasses set on the centre line
transfer this depth to the other side of the bolt, as denoted by the
arcs 13 and 14. Touching arc 14 we mark line 15 for the thread depth on
that side. We have now to get the slant of the thread across the bolt.
It is obvious that in passing once around the bolt the thread advances
to the amount of the pitch as from _a_ to _b_; hence, in passing half
way around, it will advance from _a_ to _c_; we therefore draw line 16
at a right-angle to the centre line, and a line that touches the top of
the threads at _a_, where it meets line 2, and also meets line 16, where
it touches line 3, is the angle or slope for the tops of the threads,
which may be drawn across by lines, as 18, 19, 20, etc. From these lines
the sides of the thread may be drawn at the bottom of the bolt, marking
first the angle on one side, as by lines 21, 22, 23, etc., and then the
angles on the other, as by lines 24, 25, etc.
[Illustration: Fig. 204.]
There now remain the bottoms of the thread to draw, and this is done by
drawing lines from the bottom of the thread on one side of the bolt to
the bottom on the other, as shown in the cut by a dotted line; hence, we
may set a square blade to that angle, and mark in these lines, as 26,
27, 28, etc., and the thread is pencilled in complete.
If the student will follow out this example upon paper, it will appear
to him that after the thread had been marked out on one side of the
bolt, the angle of the thread might be obtained, as shown by lines 16
and 17, and that the bottoms of the thread as well as the tops might be
carried across the bolt to the other side. Figure 204 represents a case
in which this has been done, and it will be observed that the lines
denoting the bottom of the thread do not meet the bottoms of the thread,
which occurs for the reason that the angle for the bottom is not the
same as that for the top of the thread.
[Illustration: Fig. 205.]
[Illustration: Fig. 206.]
In inking in the thread, it enhances the appearance to give the bottom
of the thread and the right-hand side of the same, heavy shade lines,
as in Figure 202, a plan that is usually adopted for threads of large
diameter and coarse pitch.
A double thread, such as in Figure 205, is drawn in the same way, except
that the slant of the thread is doubled, and the square is to be set for
the thread-pitch A, A, both for the tops and bottoms of the thread.
[Illustration: Fig 207.]
A round top and bottom thread, as the Whitworth thread, is drawn by
single lines, as in Figure 206. A left-hand thread, Figure 207, is
obviously drawn by the same process as a right-hand one, except that the
slant of the thread is given in the opposite direction.
For screw threads of a large diameter it is not uncommon to draw in the
thread curves as they appear to the eye, and the method of doing this is
shown in Figure 208. The thread is first marked on both sides of the
bolt, as explained, and instead of drawing, straight across the bolt,
lines to represent the tops and bottoms of the thread, a template to
draw the curves by is required. To get these curves, two half-circles,
one equal in diameter to the top, and one equal to the bottom of the
thread, are drawn, as in Figure 208.
[Illustration: Fig. 208.]
These half-circles are divided into any convenient number of equal
divisions: thus in Figure 208, each has eight divisions, as _a_, _b_,
_c_, etc., for the outer, and _i_, _j_, _k_, etc., for the inner one.
The pitch of the thread is then divided off by vertical lines into as
many equal divisions as the half-circles are divided into, as by the
lines _a_, _b_, _c_, etc., to _o_. Of these, the seven from _a_, to _h_,
correspond to the seven from _a'_ to _g'_, and are for the top of the
thread, and the seven from _i_ to _o_ correspond to the seven on the
inner half-circle, as _i_, _j_, _k_, etc. Horizontal lines are then
drawn from the points of the division to meet the vertical lines of
division; thus the horizontal dotted line from _a'_ meets the vertical
line _a_, and where they meet, as at A, a dot is made. Where the dotted
line from _b'_ meets vertical line _b_, another dot is made, as at B,
and so on until the point G is found. A curve drawn to pass from the top
of the thread on one side of the bolt to the top of the other side, and
passing through these points, as from A to G, will be the curve for the
top of the thread, and from this curve a template may be made to mark
all the other thread-tops from, because manifestly all the tops of the
thread on the bolt will be alike.
For the bottoms of the thread, lines are similarly drawn, as from _i'_
to meet _i_, where dot I is marked. J is got from _j'_ and _j_, while K
is got from the intersection of _k'_ with _k_, and so on, the dots from
I to O being those through which a curve is drawn for the bottom of the
thread, and from this curve a template also may be made to mark all the
thread bottoms. We have in our example used eight points of division in
each half-circle, but either more or less points maybe used, the only
requisite being that the pitch of the thread must be divided into as
many divisions as the two half-circles are. But it is not absolutely
necessary that both half-circles be divided into the same number of
equal divisions. Thus, suppose the large half-circle were divided into
ten divisions, then instead of the first half of the pitch being divided
into eight (as from _a_ to _h_) it would require to have ten lines. But
the inner half-circle may have eight only, as in our example. It is more
convenient, however, to use the same number of divisions for both
circles, so that they may both be divided together by lines radiating
from the centre. The more the points of division, the greater number of
points to draw the curves through; hence it is desirable to have as many
as possible, which is governed by the pitch of the thread, it being
obvious that the finer the pitch the less the number of distinct and
clear divisions it is practicable to divide it into. In our example the
angles of the thread are spread out to cause these lines to be thrown
further apart than they would be in a bolt of that diameter; hence it
will be seen that in threads of but two or three inches in diameter the
lines would fall very close together, and would require to be drawn
finely and with care to keep them distinct.
[Illustration: Fig. 208 _a_.]
[Illustration: Fig. 209.]
The curves for a United States standard form of thread are obtained in
the same manner as from the $V$ thread in Figure 208, but the thread
itself is more difficult to draw. The construction of this thread is
shown in Figure 208, it having a flat place at the top and at the bottom
of the thread. A common $V$ thread has its sides at an angle of 60
degrees, one to the other, the top and bottom meeting in a point. The
United States standard is obtained from drawing a common $V$ thread and
dividing its depth into eight equal divisions, as at _x_, in Figure 208
_a_, and cutting off one of these divisions at the top and filling in
one at the bottom to form flat places, as shown in the figure. But the
thread cannot be sketched on a bolt by this means unless temporary lines
are used to get the thread from, these temporary lines being drawn to
represent a bolt one-fourth the depth of the thread too large in
diameter. Thus, in Figure 208 _a_, it is seen that cutting off
one-eighth the depth of the thread reduces the diameter of the thread.
It is necessary, then, to draw the flat place on top of the thread
first, the order of procedure being shown in Figure 209. The lines for
the full diameter of the thread being drawn, the pitch is stepped off by
arcs, as 1, 2, 3, etc.; and from these, arcs, as 4, 5, 6, etc., are
marked for the width of the flat places at the tops of the threads.
Then one side of the thread is marked off by lines, as 7, which meet the
arcs 1, 2, 3, etc., as at _a_, _c_, etc. Similar lines, as 8 and 9, are
marked for the other side of the thread, these lines, 7, 8 and 9,
projecting until they cross each other. Line 10 is then drawn, making a
flat place at the bottom of the thread equal in width to that at the
top. Line 12 is then drawn square across the bolt, starting from the
bottom of the thread, and line 13 is drawn starting from the corner _f_
on one side of the thread and meeting line 12 on the other side of the
thread, which gives the angle for the tops of the thread. The depth of
the thread may then be marked on the other side of the bolt by the arcs
_d_ and _e_, and the line 14. The tops of all the threads may then be
drawn in, as by lines 15, 16, 17 and 18, and by lines, as 19, etc., the
thread sides may be drawn on the other side of the bolt. All that
remains is to join the bottoms of the threads by lines across the bolt,
and the pencil lines will be complete, ready to ink in. If the thread is
to be shown curved instead of drawn straight across, the curve may be
obtained by the construction in Figure 208, which is similar to that in
Figure 207, except that while the pitch is divided off into 16
divisions, the whole of these 16 divisions are not used to get the
curves, some of them being used twice over; thus for the bottom the
eight divisions from _b_ to _i_ are used, while for the tops the eight
from _g_ to _o_ are used. Hence _g_, _h_ and _i_ are used for getting
both curves, the divisions from _a_ to _b_ and from _o_ to _p_ being
taken up by the flat top and bottom of the thread. It will be noted that
in Figure 207, the top of the thread is drawn first, while in Figure
208 the bottom is drawn first, and that in the latter (for the U.S.
standard) the pitch is marked from centre to centre of the flats of the
thread.
[Illustration: Fig. 210.]
To draw a square thread the pencil lines are marked in the order shown
in Figure 210, in which 1 represents the centre line and 2, 3, 4 and 5,
the diameter and depth of the thread. The pitch of the thread is marked
off by arcs, as 6, 7, etc., or by laying a rule directly on the centre
line and marking with a lead pencil. To obtain the slant of the thread,
lines 8 and 9 are drawn, and from these line 10, touching 8 and 9 where
they meet lines 2 and 5; the threads may then be drawn in from the arcs
as 6, 7, etc. The side of the thread will show at the top and the bottom
as at A B, because of the coarse pitch and the thread on the other or
unseen side of the bolt slants, as denoted by the lines 12, 13; and
hence to draw the sides A B, the triangle must be set from one thread to
the next on the opposite side of the bolt, as denoted by the dotted
lines 12 and 13.
[Illustration: Fig. 211.]
If the curves of the thread are to be drawn in, they may be obtained as
in Figure 211, which is substantially the same as described for a V
thread. The curves _f_, representing the sides of the thread, terminate
at the centre line _g_, and the curves _e_ are equidistant with the
curves _c_ from the vertical lines _d_. As the curves _f_ above the
line are the same as _f_ below the line, the template for _f_ need not
be made to extend the whole distance across, but one-half only; as is
shown by the dotted curve _g_, in the construction for finding the curve
for square-threaded nuts in Figure 212.
[Illustration: Fig. 212.]
[Illustration: Fig. 213.]
A specimen of the form of template for drawing these curves is shown in
Figure 213; _g_ _g_, is the centre line parallel to the edges R, S;
lines _m_, _n_, represent the diameter of the thread at the top, and
_o_, _p_, that at the bottom or root; edge _a_ is formed to the points
(found by the constructions in the figures as already explained) for the
tops of the thread, and edge _f_ is so formed for the curve at the
thread bottoms. The edge, as S or R, is laid against the square-blade to
steady it while drawing in the curves. It may be noted, however, that
since the curve is the same below the centre line as it is above, the
template may be made to serve for one-half the thread diameter, as at
_f_, where it is made from _o_ to _g_, only being turned upside down to
draw the other half of the curve; the notches cut out at _x_, _x_, are
merely to let the pencil-lines in the drawing show plainly when setting
the template.
When the thread of a nut is shown in section, it slants in the opposite
direction to that which appears on the bolt-thread, because it shows the
thread that fits to the opposite side of the bolt, which, therefore,
slants in the opposite direction, as shown by the lines 12 and 13 in
Figure 210.
In a top or end view of a nut the thread depth is usually shown by a
simple circle, as in Figure 214.
[Illustration: Fig. 214.]
To draw a spiral spring, draw the centre line A, and lines B, C, Figure
215, distant apart the diameter the spring is to be less the diameter of
the wire of which it is to be made. On the centre line A mark two lines
_a b_, _c d_, representing the pitch of the spring. Divide the distance
between _a_ and _b_ into four equal divisions, as by lines 1, 2, 3,
letting line 3 meet line B. Line _e_ meeting the centre line at line
_a_, and the line B at its intersection with line 3, is the angle of the
coil on one side of the spring; hence it may be marked in at all the
locations, as at _e f_, etc. These lines give at their intersections
with the lines C and B the centres for the half circles _g_, which being
drawn, the sides _h_, _i_, _j_, _k_, etc., of the spring, may all be
marked in. By the lines _m_, _n_, _o_, _p_, the other sides of the
spring may be marked in.
[Illustration: Fig. 215.]
The end of the spring is usually marked straight across, as at L. If it
is required to draw the coils curved instead of straight across, a
template must be made, the curve being obtained as already described for
threads. It may be pointed out, however, that to obtain as accurate a
division as possible of the lines that divide the pitch, the pitch may
be divided upon a diagonal line, as F, Figure 216, which will greatly
facilitate the operation.
[Illustration: Fig. 216.]
Before going into projections it may be as well to give some examples
for practice.
[Illustration: Fig. 219. (Page 169.)]
CHAPTER IX.
_EXAMPLES FOR PRACTICE._
Figure 217 represents a simple example for practice, which the student
may draw the size of the engraving, or he may draw it twice the size. It
is a locomotive spring, composed of leaves or plates, held together by a
central band.
[Illustration: Fig. 217.]
Figure 218 is an example of a stuffing box and gland, supposed to stand
vertical, hence the gland has an oil cup or receptacle.
In Figure 219 are working drawings of a coupling rod, with the
dimensions and directions marked in.
It may be remarked, however, that the drawings of a workshop are, where
large quantities of the same kind of work is done, varied in character
to suit some special departments--that is to say, special extra drawings
are made for these departments. In Figures 220 and 221 is a drawing of a
connecting rod drawn, put together as it would be for the lathe, vise or
erecting shop.
[Illustration: Fig. 218.]
[Illustration: Fig. 219.]
[Illustration: Fig. 220.]
[Illustration: Fig. 221.]
To the two views shown there would be necessary detail sketches of the
set screws, gibbs, and keys, all the rest being shown; the necessary
dimensions being, of course, marked on the general drawing and on the
details.
In so simple a thing as a connecting rod, however, there would be no
question as to how the parts go together; hence detail drawings of each
separate piece would answer for the lathe or vise bands.
But in many cases this would not be the case, and the drawing would
require to show the parts put together, and be accompanied with such
detail sketches as might be necessary to show parts that could not be
clearly defined in the general views.
The blacksmith, for example, is only concerned with the making of the
separate pieces, and has no concern as to how the parts go together.
Furthermore, there are parts and dimensions in the general drawing with
which the blacksmith has nothing to do.
Thus the location and dimensions of the keyways, the dimensions of the
brasses, and the location of the bolt holes, are matters that have no
reference to the blacksmith's work, because the keyways, bolt holes, and
set-screw holes would be cut out of the solid in the machine shop. It is
customary, therefore, to send to the blacksmith shop drawings containing
separate views of each piece drawn to the shape it is to be forged; and
drawn full size, or else on a scale sufficiently large to make each part
show clearly without close inspection, marking thereon the full sizes,
and stating beneath the number of pieces of each detail. (As in Figure
222, which represents the iron work of the connecting rod in Figure
220). In some cases the finished sizes are marked, and it is left to the
blacksmith's judgment how much to leave for the finishing. This is
undesirable, because either the blacksmith is left to judge what parts
are to be finished, or else there must be on the drawing instructions on
this point, or else signs or symbols that are understood to convey the
information. It is better, therefore, to make for the blacksmith a
special sketch, and mark thereon the full-forged sizes, stating on the
drawing that such is the case.
[Illustration: Fig. 222]
As to the material of which the pieces are to be made, the greater part
of blacksmith work is made of wrought iron, and it is, therefore,
unnecessary to write "wrought iron" beneath each piece. When the pieces
are to be of steel, however, it should be marked on the drawing and
beneath the piece. In special cases, as where the greater part of the
work of the shop is of steel, the rule may, of course, be reversed, and
the parts made of iron may be the ones marked, whereas when parts are
sometimes of iron, and at others of steel, each piece should be marked.
As a general rule the blacksmith knows, from the custom of the shop or
the nature of the work, what the quality or kind of iron is to be, and
it is, therefore, only in exceptional cases that they need to be
mentioned on the drawing. Thus in a carriage manufactory, Norway or
Swede iron will be found, as well as the better grades of refined iron,
but the blacksmith will know what iron to use, for certain parts, or the
shop may be so regulated that the selection of the iron is not left to
him. In marking the number of pieces required, it is better to use the
word "thus" than the words "of this," or "off this," because it is
shorter and more correct, for the forging is not taken off the drawing,
nor is it of the same; the drawing gives the shape and the size, and the
word "thus" conveys that idea better than "of," "off," or "like this."
In shops where there are many of the same pieces forged, the blacksmith
is furnished with sheet-iron templates that he can lay directly upon the
forging and test its dimensions at once, which is an excellent plan in
large work. Such templates are, of course, made from the drawings, and
it becomes a question as to whether their dimensions should be the
forged or the finished ones. If they are the forged, they may cause
trouble, because a forging may have a scant place that it is difficult
for the blacksmith to bring up to the size of the template, and he is in
doubt whether there is enough metal in the scant place to allow the job
to clean up. It is better, therefore, to make them to finished sizes, so
that he can see at once if the work will clean up, notwithstanding the
scant place. This will lead to no errors in large work, because such
work is marked out by lines, and the scant part will therefore be
discovered by the machinist, who will line out the piece accordingly.
Figure 223 is a drawing of a locomotive frame, which the student may as
well draw three or four times as large as the engraving, which brings us
to the subject of enlarging or reducing scales.
REDUCING SCALES.
[Illustration: Fig. 223.]
[Illustration: Fig. 224.]
[Illustration: Fig. 225.]
It is sometimes necessary to reduce a drawing to a smaller scale, or to
find a minute fraction of a given dimension, such fraction not being
marked on the lineal measuring rules at hand. Figure 224 represents a
scale for finding minute fractions. Draw seven lines parallel to each
other, and equidistant draw vertical lines dividing the scale into
half-inches, as at _a_, _b_, _c_, etc. Divide the first space _e d_ into
equal halves, draw diagonal lines, and number them as in the figure. The
distance of point 1, which is at the intersection of diagonal with the
second horizontal line, will be 1/24 inch from vertical line _e_. Point
2 will be 2/24 inch from line _e_, and so on. For tenths of inches there
would require to be but six horizontal lines, the diagonals being drawn
as before. A similar scale is shown in Figure 225. Draw the lines A B, B
D, D C, C A, enclosing a square inch. Divide each of these lines into
ten equal divisions, and number and letter them as shown. Draw also the
diagonal lines A 1, _a_ 2, B 3, and so on; then the distances from the
line A C to the points of intersection of the diagonals with the
horizontal lines represent hundredths of an inch.
Suppose, for example, we trace one diagonal line in its path across the
figure, taking that which starts from A and ends at 1 on the top
horizontal line; then where the diagonal intersects _horizontal_ line 1,
is 99/100 from the line B D, and 1/100 from the line A C, while where it
intersects _horizontal_ line 2, is 98/100 from line B D, and 2/100 from
line A C, and so on. If we require to set the compasses to 67/100 inch,
we set them to the radius of _n_, and the figure 3 on line B D, because
from that 3 to the vertical line _d_ 4 is 6/10 or 60/100 inch, and from
that vertical line to the diagonal at _n_ is seven divisions from the
line C D of the figure.
In making a drawing to scale, however, it is an excellent plan to draw a
line and divide it off to suit the required scale. Suppose, for example,
that the given scale is one-quarter size, or three inches per foot; then
a line three inches long may be divided into twelve equal divisions,
representing twelve inches, and these may be subdivided into half or
quarter inches and so on. It is recommended to the beginner, however, to
spend all his time making simple drawings, without making them to scale,
in order to become so familiar with the use of the instruments as to
feel at home with them, avoiding the complication of early studies that
would accompany drawing to scale.
CHAPTER X.
_PROJECTIONS._
In projecting, the lines in one view are used to mark those in other
views, and to find their shapes or curvature as they will appear in
other views. Thus, in Figure 225_a_ we have a spiral, wound around a
cylinder whose end is cut off at an angle. The pitch of the spiral is
the distance A B, and we may delineate the curve of the spiral looking
at the cylinder from two positions (one at a right-angle to the other,
as is shown in the figure), by means of a circle having a circumference
equal to that of the cylinder.
The circumference of this circle we divide into any number of
equidistant divisions, as from 1 to 24. The pitch A B of the spiral or
thread is then divided off also into 24 equidistant divisions, as marked
on the left hand of the figure; vertical lines are then drawn from the
points of division on the circle to the points correspondingly numbered
on the lines dividing the pitch; and where line 1 on the circle
intersects line 1 on the pitch is one point in the curve. Similarly,
where point 2 on the circle intersects line 2 on the pitch is another
point in the curve, and so on for the whole 24 divisions on the circle
and on the pitch. In this view, however, the path of the spiral from
line 7 to line 19 lies on the other side of the cylinder, and is marked
in dotted lines, because it is hidden by the cylinder. In the
right-hand view, however, a different portion of the spiral or thread is
hidden, namely from lines 1 to 13 inclusive, being an equal proportion
to that hidden in the left-hand view.
[Illustration: Fig. 225 _a_.]
The top of the cylinder is shown in the left-hand view to be cut off at
an angle to the axis, and will therefore appear elliptical; in the
right-hand view, to delineate this oval, the same vertical lines from
the circle may be carried up as shown on the right hand, and horizontal
lines may be drawn from the inclined face in one view across the end of
the other view, as at P; the divisions on the circle may be carried up
on the right-hand view by means of straight lines, as Q, and arcs of
circle, as at R, and vertical lines drawn from these arcs, as line S,
and where these vertical lines S intersect the horizontal lines as P,
are points in the ellipse.
Let it be required to draw a cylindrical body joining another at a
right-angle; as for example, a Tee, such as in Figure 226, and the
outline can all be shown in one view, but it is required to find the
line of junction of one piece, A, with the other, B; that is, find or
mark the lines of junction C. Now when the diameters of A and B are
equal, the line of junction C is a straight line, but it becomes a
curved one when the diameter of A is less than that of B, or _vice
versa_; hence it may be as well to project it in both cases. For this
purpose the three views are necessary. One-quarter of the circle of B,
in the end view, is divided off into any number of equal divisions; thus
we have chosen the divisions marked _a_, _b_, _c_, _d_, _e_, etc.; a
quarter of the top view is similarly divided off, as at _f_, _g_, _h_,
_i_, _j_; from these points of division lines are projected on to the
side view, as shown by the dotted lines _k_, _l_, _m_, _n_, _o_, _p_,
etc., and where these lines meet, as denoted by the dots, is in each
case a point in the line of junction of the two cylinders A, B.
[Illustration: Fig. 226.]
[Illustration: Fig. 227.]
Figure 227 represents a Tee, in which B is less in diameter than A;
hence the two join in a curve, which is found in a similar manner, as is
shown in Figure 227. Suppose that the end and top views are drawn, and
that the side view is drawn in outline, but that the curve of junction
or intersection is to be found. Now it is evident that since the centre
line 1 passes through the side and end views, that the face _a_, in the
end view, will be even with the face _a'_ in the side view, both being
the same face, and as the full length of the side of B in the end view
is marked by line _b_, therefore line _c_ projected down from _b_ will
at its junction with line _b'_, which corresponds to line _b_, give the
extreme depth to which _b'_ extends into the body A, and therefore, the
apex of the curve of intersection of B with A. To obtain other points,
we divide one-quarter of the circumference of the circle B in the top
view into four equal divisions, as by lines _d_, _e_, _f_, and from the
points of division we draw lines _j_, _i_, _g_, to the centre line
marked 2, these lines being thickened in the cut for clearness of
illustration. The compasses are then set to the length of thickened line
_g_, and from point _h_, in the end view, as a centre, the arc _g'_ is
marked. With the compasses set to the length of thickened line _i_, and
from _h_ as a centre, arc _i'_ is marked, and with the length of
thickened line _j_ as a radius and from _h_ as a centre arc _j'_ is
marked; from these arcs lines _k_, _l_, _m_ are drawn, and from the
intersection of _k_, _l_, _m_, with the circle of A, lines _n_, _o_, _p_
are let fall. From the lines of division, _d_, _e_, _f_, the lines _q_,
_r_, _s_ are drawn, and where lines _n_, _o_, _p_ join lines _q_, _r_,
_s_, are points in the curve, as shown by the dots, and by drawing a
line to intersect these dots the curve is obtained on one-half of B.
Since the axis of B is in the same plane as that of A, the lower half of
the curve is of the same curvature as the upper, as is shown by the
dotted curve.
[Illustration: Fig. 228.]
In Figure 228 the axis of piece B is not in the same plane as that of D,
but to one side of it to the distance between the centre lines C, D,
which is most clearly seen in the top view. In this case the process is
the same except in the following points: In the side view the line _w_,
corresponding to the line _w_ in the end view, passes within the line
_x_ before the curve of intersection begins, and in transferring the
lengths of the full lines _b_, _c_, _d_, _e_, _f_ to the end view, and
marking the arcs _b'_, _c'_, _d'_, _e'_, _f'_, they are marked from the
point _w_ (the point where the centre line of B intersects the outline
of A), instead of from the point _x_. In all other respects the
construction is the same as that in Figure 227.
[Illustration: Fig. 229.]
In these examples the axis of B stands at a right-angle to that of A.
But in Figure 229 is shown the construction where the axis of B is not
at a right-angle to A. In this case there is projected from B, in the
side view, an end view of B as at B', and across this end at a
right-angle to the centre line of B is marked a centre line C C of B',
which is divided as before by lines _d_, _e_, _f_, _g_, _h_, their
respective lengths being transferred from W as a centre, and marked by
the arcs _d'_, _e'_, _f'_, which are marked on a vertical line and
carried by horizontal lines, to the arc of A as at _i_, _j_, _k_. From
these points, _i_, _j_, _k_, the perpendicular lines _l_, _m_, _n_, _o_,
are dropped, and where these lines meet lines _p_, _q_, _r_, _s_, _t_,
are points in the curve of intersection of B with A. It will be observed
that each of the lines _m_, _n_, _o_, serves for two of the points in
the curve; thus, _m_ meets _q_ and _s_, while _n_ meets _p_ and _t_, and
_o_ meets the outline on each side of B, in the side view, and as _i_,
_j_, _k_ are obtained from _d_ and _e_, the lines _g_ and _h_ might have
been omitted, being inserted merely for the sake of illustration.
In Figure 230 is an example in which a cylinder intersects a cone, the
axes being parallel. To obtain the curve of intersection in this case,
the side view is divided by any convenient number of lines, as _a_, _b_,
_c_, etc., drawn at a right-angle to its axis A A, and from one end of
these lines are let fall the perpendiculars _f_, _g_, _h_, _i_, _j_;
from the ends of these (where they meet the centre line of A in the top
view), half-circles _k_, _l_, _m_, _n_, _o_, are drawn to meet the
circle of B in the top view, and from their points of intersection with
B, lines _p_, _q_, _r_, _s_, _t_, are drawn, and where these meet lines
_a_, _b_, _c_, _d_ and _e_, which is at _u_, _v_, _w_, _x_, _y_, are
points in the curve.
[Illustration: Fig. 230.]
[Illustration: Fig. 231.]
It will be observed, on referring again to Figure 229, that the branch
or cylinder B appears to be of elliptical section on its end face,
which occurs because it is seen at an angle to its end surface; now the
method of finding the ellipse for any given degree of angle is as in
Figure 231, in which B represents a cylindrical body whose top face
would, if viewed from point I, appear as a straight line, while if
viewed from point J it would appear in outline a circle. Now if viewed
from point E its apparent dimension in one direction will obviously be
defined by the lines S, Z. So that if on a line G G at a right angle to
the line of vision E, we mark points touching lines S, Z, we get points
1 and 2, representing the apparent dimension in that direction which is
the width of the ellipse. The length of the ellipse will obviously be
the full diameter of the cylinder B; hence from E as a centre we mark
points 3 and 4, and of the remaining points we will speak presently.
Suppose now the angle the top face of B is viewed from is denoted by the
line L, and lines S', Z, parallel to L, will be the width for the
ellipse whose length is marked by dots, equidistant on each side of
centre line G' G', which equal in their widths one from the other the
full diameter of B. In this construction the ellipse will be drawn away
from the cylinder B, and the ellipse, after being found, would have to
be transferred to the end of B. But since centre line G G is obviously
at the same angle to A A that A A is to G G, we may start from the
centre line of the body whose elliptical appearance is to be drawn, and
draw a centre line A A at the same angle to G G as the end of B is
supposed to be viewed from. This is done in Figure 231 _a_, in which the
end face of B is to be drawn viewed from a point on the line G G, but at
an angle of 45 degrees; hence line A A is drawn at an angle of 45
degrees to centre line G G, and centre line E is drawn from the centre
of the end of B at a right angle to G G, and from where it cuts A A, as
at F, a side view of B is drawn, or a single line of a length equal to
the diameter of B may be drawn at a right angle to A A and equidistant
on each side of F. A line, D D, at a right angle to A A, and at any
convenient distance above F, is then drawn, and from its intersection
with A A as a centre, a circle C equal to the diameter of B is drawn;
one-half of the circumference of C is divided off into any number of
equal divisions as by arcs _a_, _b_, _c_, _d_, _e_, _f_. From these
points of division, lines _g_, _h_, _i_, _j_, _k_, _l_ are drawn, and
also lines _m_, _n_, _o_, _p_, _q_, _r_. From the intersection of these
last lines with the face in the side view, lines _s_, _t_, _u_, _t_,
_w_, _x_, _y_, _z_ are drawn, and from point F line E is drawn. Now it
is clear that the width of the end face of the cylinder will appear the
same from any point of view it may be looked at, hence the sides H H are
made to equal the diameter of the cylinder B and marked up to centre
line E.
[Illustration: Fig. 231 _a_.]
[Illustration: Fig. 232.]
It is obvious also that the lines _s_, _z_, drawn from the extremes of
the face to be projected will define the width of the ellipse, hence we
have four of the points (marked respectively 1, 2, 3, 4) in the ellipse.
To obtain the remaining points, lines _t_, _u_, _v_, _w_, _x_, _y_
(which start from the point on the face F where the lines _m_, _n_, _o_,
_p_, _q_, _r_, respectively meet it) are drawn across the face of B as
shown. The compasses are then set to the radius _g_; that is, from
centre line D to division _a_ on the circle, and this radius is
transferred to the face to be projected the compass-point being rested
at the intersection of centre line G and line _t_, and two arcs as 5 and
6 drawn, giving two more points in the curve of the ellipse. The
compasses are then set to the length of line _h_ (that is, from centre
line D to point of division _b_), and this distance is transferred,
setting the compasses on centre line G where it is intersected by line
_u_, and arcs 7, 8 are marked, giving two more points in the ellipse. In
like manner points 9 and 10 are obtained from the length of line _i_, 11
and 12 from that of _j_; points 13 and 14 from the length of _k_, and 15
and 16 from _l_, and the ellipse may be drawn in from these points.
It may be pointed out, however, that since points 5 and 6 are the same
distance from G that points 15 and 16 are, and since points 7 and 8 are
the same distance from G that points 13 and 14 are, while points 9 and
10 are the same distance from G that 11 and 12 are, the lines, _j_, _k_,
_l_ are unnecessary, since _l_ and _g_ are of equal length, as are also
_h_ and _k_ and _i_ and _j_. In Figure 232 the cylinders are line shaded
to make them show plainer to the eye, and but three lines (_a_, _b_,
_c_) are used to get the radius wherefrom to mark the arcs where the
points in the ellipse shall fall; thus, radius _a_ gives points 1, 2, 3
and 4; radius _b_ gives points 5, 6, 7 and 8, and radius _c_ gives 9,
10, 11 and 12, the extreme diameter being obtained from lines S, Z, and
H, H.
CHAPTER XI.
_DRAWING GEAR WHEELS._
The names given to the various lines of a tooth on a gear-wheel are as
follows:
In Figure 233, A is the face and B the flank of a tooth, while C is the
point, and D the root of the tooth; E is the height or depth, and F the
breadth. P P is the pitch circle, and the space between the two teeth,
as H, is termed a space.
[Illustration: Fig. 233.]
[Illustration: Fig. 234.]
It is obvious that the points of the teeth and the bottoms of the
spaces, as well as the pitch circle, are concentric to the axis of the
wheel bore. And to pencil in the teeth these circles must be fully
drawn, as in Figure 234, in which P P is the pitch circle. This circle
is divided into as many equal divisions as the wheel is to have teeth,
these divisions being denoted by the radial lines, A, B, C, etc. Where
these divisions intersect the pitch circle are the centres from which
all the teeth curves may be drawn. The compasses are set to a radius
equal to the pitch, less one-half the thickness of the tooth, and from a
centre, as R, two face curves, as F G, may be marked; from the next
centre, as at S, the curves D E may be marked, and so on for all the
faces; that is, the tooth curves lying between the outer circle X and
the pitch circle P. For the flank curves, that is, the curve from P to
Y, the compasses are set to a radius equal to the pitch; and from the
sides of the teeth the flank curves are drawn. Thus from J, as a centre
flank, K is drawn; from V, as a centre flank, H is drawn, and so on.
The proportions of the teeth for cast gears generally accepted in this
country are those given by Professor Willis, as average practice, and
are as follows:
Depth to pitch line, 3/10 of the pitch.
Working depth, 6/10 " "
Whole depth, 7/10 " "
Thickness of tooth, 5/11 " "
Breadth of space, 6/11 " "
Instead, however, of calculating the dimensions these proportions give
for any particular pitch, a diagram or scale may be made from which they
may be taken for any pitch by a direct application of the compasses. A
scale of this kind is given in Figure 235, in which the line A B is
divided into inches and parts to represent the pitches; its total length
representing the coarsest pitch within the capacity of the scale; and,
the line B C (at a right-angle to A B) the whole depth of the tooth for
the coarsest pitch, being 7/10 of the length of A B.
[Illustration: Fig. 235.]
The other diagonal lines are for the proportion of the dimensions marked
on the figure. Thus the depth of face, or distance from the pitch line
to the extremity or tooth point for a 4 inch pitch, would be measured
along the line B C, from the vertical line B to the first diagonal. The
thickness of the tooth would be for a 4 inch pitch along line B C from B
to the second diagonal, and so on. For a 3 inch pitch the measurement
would be taken along the horizontal line, starting from the 3 on the
line A B, and so on. On the left of the diagram or scale is marked the
lbs. strain each pitch will safely transmit per inch width of wheel
face, according to Professor Marks.
[Illustration: Fig. 236.]
The application of the scale as follows: The pitch circles P P and P'
P', Figure 236, for the respective wheels, are drawn, and the height of
the teeth is obtained from the scale and marked beyond the pitch
circles, when circles Q and Q' may be drawn. Similarly, the depths of
the teeth within the pitch circles are obtained from the scale or
diagram and marked within the respective pitch circles, and circles R
and R' are marked in. The pitch circles are divided off into as many
points of equal division, as at _a_, _b_, _c_, _d_, _e_, etc., as the
respective wheels are to have teeth, and the thickness of tooth having
been obtained from the scale, this thickness is marked from the points
of division on the pitch circles, as at _f_ in the figure, and the tooth
curves may then be drawn in. It may be observed, however, that the tooth
thicknesses will not be strictly correct, because the scale gives the
same chord pitch for the teeth on both wheels which will give different
arc pitches to the teeth on the two wheels; whereas, it is the arc
pitches, and not the chord pitches, that should be correct. This error
obviously increases as there is a greater amount of difference between
the two wheels.
The curves given to the teeth in Figure 234 are not the proper ones to
transmit uniform motion, but are curves merely used by draughtsmen to
save the trouble of finding the true curves, which if it be required,
may be drawn with a very near approach to accuracy, as follows, which is
a construction given by Rankine:
Draw the rolling circle D, Figure 237, and draw A D, the line of
centres. From the point of contact at C, mark on D, a point distant from
C one-half the amount of the pitch, as at P, and draw the line P C of
indefinite length beyond C. Draw the line P E passing through the line
of centres at E, which is equidistant between C and A. Then increase the
length of line P F to the right of C by an amount equal to the radius A
C, and then diminish it to an amount equal to the radius E D, thus
obtaining the point F and the latter will be the location of centre for
compasses to strike the face curve.
[Illustration: Fig. 237.]
[Illustration: Fig. 238.]
Another method of finding the face curve, with compasses, is as follows:
In Figure 238 let P P represent the pitch circle of the wheel to be
marked, and B C the path of the centre of the generating or describing
circle as it rolls outside of P P. Let the point B represent the centre
of the generating circle when it is in contact with the pitch circle at
A. Then from B mark off, on B C, any number of equidistant points, as
D, E, F, G, H, and from A mark on the pitch circle, with the same
radius, an equal number of points of division, as 1, 2, 3, 4, 5. With
the compasses set to the radius of the generating circle, that is, A B,
from B, as a centre, mark the arc I, from D, the arc J, from E, the arc
K, from F, and so on, marking as many arcs as there are points of
division on B C. With the compasses set to the radius of divisions 1, 2,
etc., step off on arc M the five divisions, N, O, S, T, V, and at V will
be a point on the epicycloidal curve. From point of division 4, step off
on L four points of division, as _a_, _b_, _c_, _d_; and _d_ will be
another point on the epicycloidal curve. From point 3, set off three
divisions, and so on, and through the points so obtained draw by hand,
or with a scroll, the curve.
[Illustration: Fig. 239.]
Hypocycloids for the flanks of the teeth maybe traced in a similar
manner. Thus in Figure 239, P P is the pitch circle, and B C the line of
motion of the centre of the generating circle to be rolled within P P.
From 1 to 6 are points of equal division on the pitch circle, and D to I
are arc locations for the centre of the generating circle. Starting
from A, which represents the location for the centre of the generating
circle, the point of contact between the generating and base circles
will be at B. Then from 1 to 6 are points of equal division on the pitch
circle, and from D to I are the correspo
Showing first 200,000 characters. Full text: gutenberg.org/ebooks/23319