Chapter II: Part 2

Fig. 76: Draw first the square as directed in the previous lesson, join the points _A_, _B_, _C_ and add the short lines at _E_ and _F_, proceed with the curve _A B_, drawing it with faint lines at first, and adding stroke upon stroke until the required depth is obtained; the curve _A C_ is more difficult to produce, in consequence of the formation of the hand; it should, therefore, be drawn in shorter pieces, joining them together afterwards by over strokes.

Fig. 77: Draw the square and straight lines first, then add the curves, taking care to give the greatest fullness at the right place.

Fig. 78: Draw the square and straight lines, proceed with the curves, taking care to make each of the same proportion.

Figs. 79 and 80: The ovals contained in these figures are simply foreshortened circles, and as such forms are of frequent occurrence in sketching from objects, in bridges, wheels, ends of timber, etc., they should be carefully studied; the greatest difficulty is to turn the narrow ends, and prevent their looking like corners. For this purpose it is better to draw the short curves first, thus:

and then join the longer sides to them.

Fig. 81: If this figure can be drawn correctly, a great success has been achieved; the circle is a most difficult form to delineate, and without system could not be accomplished. Draw the square and straight lines within it with great care, examine each point of the octagon to see that it is at the same distance from the centre, and then draw the circle.

EXAMPLES FOR PRACTICE.

Several figures, 83 to 96, representing more or less familiar parts of machines, utilities, etc., are introduced for practice in free-hand, but--

It must be noted that even in free-hand the wise student will occasionally use the straight edge and compasses, so as to make his first attempts fairly creditable. Many good draughtsmen have begun by simply copying such figures and illustrations as are used throughout this volume and other similar sources; perhaps there is nothing better for practice or training than the copying and reproducing of samples of good mechanical drawings, yet it must always be remembered that advancement in free-hand must be made in the line of less to greater efforts, and that the why and wherefore will be constantly asked by the aspiring student; that good and correct drawings are to be aimed for at all times in every line and dimension--never forgetting the law of proportion in the smallest outlines of objects to be represented.

Fig. 83 is a section, or end view of a bar of angle iron; the student will find helpful practice in attempting this figure; he may be allowed to use a straight-edge in drawing the lines, but no measurements; the work should be tested on completion by a rule, or better by penciling from the original on tracing paper, and comparing the free-hand with the copy, when the defective proportions, if any, will be clearly exhibited.

Fig. 84 is a section of tee iron, and fig. 85 is a section of channel iron. These three figures on page 75 should be practiced alternately, although seeming similar in shape.

Fig. 86 is a side and end view of an angle plate shaded. Fig. 87 is a wrench shaded.

Examples of bolt ends are shown in the two next numbers; fig. 88 exhibits the common square-head bolt, and fig. 89 the hexagon or six-sided bolt-head; these are also examples of _straight-line shading_. Fig. 90 is a lathe-dog, and shows an example of _curved shading_; fig. 92 is an engine crank, and an example of _straight and curved shading_; fig. 91 is a screw clamp.

Fig. 93 is a section of boiler plates riveted together; a caulking tool is also shown.

In the example, fig. 94--a hand-wheel--the principal difficulty, even for the most advanced student in free-hand, will be in drawing the circles; a coin, if convenient, can be used to scribe about, in drawing these; the other parts can afterwards be filled in around the circle. Fig. 96 is introduced for practice in penciling and shading; the figure represents a water-wheel on a stone pier.

The familiar oil can is shown in fig. 95. These all are excellent objects for practice.

Geometrical Drawing.

Geometry is the science of measurement; it has been known for more than three thousand years; many lives have been devoted to its development, and it exists to-day as the foundation of all mathematics.

Geometrical drawing is the art of representing, to the eye, the problems “worked out” by geometricians, and the importance of a knowledge of geometrical drawing is paramount. The student will find that the figures delineated and explained in the next few pages constantly occur in mechanical drawing. Says Walter Smith, State Director of Art Education in Massachusetts, “I have never known a case where a student did not progress more satisfactorily in his studies after a course of practical geometry.”

The elementary conceptions of geometry are few:

1.--A point.
2.--A line.
3.--A surface.
4.--A solid, and
5.--An angle.

All of which elements are used in mechanical drawings.

From these, as data, a vast number of mathematical problems have been deduced; of which a few of the most elementary will be illustrated in this work; but these few will repay the attention of the student.

In “freehand” drawing the crayon and pencil are used; in geometrical drawings the dividers, as shown in illustration, fig. 97, together with a rule, are all that is necessary to accomplish the work.

A problem is something to be done, and geometry has been defined as the science of measurement; the relation between geometry and mechanical drawing is very close, hence the term “geometrical problem.”

Before proceeding with the examples, a few elementary statements belonging to the science of geometry are presented; these will be useful to the student, not only while “doing” the problems, but in many cases of every-day--future--experience.

Geometry is one of the oldest and simplest of sciences; it may be defined as _the science of measurement_; geometry is _the root_ from which all regular mathematical calculations issue. It has claimed the best thought of practical men from the times of the Greeks and Romans two thousand years ago; they derived their knowledge of the science from the Egyptians, who in turn were indebted to the Chaldeans and Hindoos in times beyond any authentic history; hence it was under the operations of the laws explained in geometry, that the pyramids of Egypt and the temples of Greece were constructed, as well as the engines of war and appliances of peace of ancient times.

_A point_ is mere position, and has no magnitude.

_A line_ is that which has extension in length only. The extremities of lines are points.

_A surface_ is that which has extension in length and breadth only.

_A solid_ is that which has extension in length, breadth and thickness.

_An angle_ is the difference in the direction of two lines proceeding from the same point.

Lines, Surfaces, Angles and Solids constitute the different kinds of quantity called _geometrical_ magnitudes.

_Parallel lines_ are lines which have the same direction; hence parallel lines can never meet, however far they may be produced; for two lines taking the same direction cannot approach or recede from each other.

An _Axiom_ is a self-evident truth, not only too simple to require, _but too simple to admit of demonstration_.

A _Proposition_ is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

A _Problem_ is something proposed to be done.

A _Theorem_ is something proposed to be demonstrated.

A _Hypothesis_ is a supposition made with a view to draw from it some consequence which establishes the truth or falsehood of a proposition, or solves a problem.

A _Lemma_ is something which is premised, or demonstrated, in order to render what follows more easy.

A _Corollary_ is a consequent truth derived immediately from some preceding truth or demonstration.

A _Scholium_ is a remark or observation made upon something going before it.

A _Postulate_ is a problem, the solution of which is self-evident.

Let it be granted--

I. That a straight line can be drawn from any one point to any other point;

II. That a straight line can be produced to any distance, or terminated at any point;

III. That the circumference of a circle can be described about any center, at any distance from that center.

The common algebraic signs are used in Geometry, and it is necessary that the student in geometry should understand some of the more simple operations of algebra. As the terms circle, angle, triangle, hypothesis, axiom, theorem, corollary and definition are constantly occurring in a course of geometry, they are abbreviated as shown in the following list:

Addition is expressed by +
Subtraction is expressed by -
Multiplication is expressed by ×
Equality and Equivalency are expressed by =
Greater than, is expressed by >
Less than, is expressed by <
Thus _B_ is greater than _A_, is written _B_ > _A_
_B_ is less than _A_, is written _B_ < _A_
A circle is expressed by O
An angle is expressed by L
A right angle is expressed by R. L
Degrees, minutes and seconds are expressed by ° ′ ″
A triangle is expressed by △
The term Hypothesis is expressed by (Hy.)
The term Axiom is expressed by (Ax.)
The term Theorem is expressed by (Th.)
The term Corollary is expressed by (Cor.)
The term Definition is expressed by (Def.)
The term Perpendicular is expressed by ⊥
The difference of two quantities, when it is not
known which is the greater, is expressed by
the symbol ~
Thus, the difference between _A_ and _B_ is written _A_ ~ _B_

GEOMETRICAL AXIOMS.

1. _Things which are equal to the same thing are equal to each other._

2. _When equals are added to equals the wholes are equal._

3. _When equals are taken from equals the remainders are equal._

4. _When equals are added to unequals the wholes are unequal._

5. _When equals are taken from unequals the remainders are unequal._

6. _Things which are double of the same thing, or equal things, are equal to each other._

7. _Things which are halves of the same thing, or of equal things, are equal to each other._

8. _The whole is greater than any of its parts._

9. _Every whole is equal to all its parts taken together._

10. _Things which coincide, or fill the same space, are identical, or mutually equal in all their parts._

11. _All right angles are equal to one another._

12. _A straight line is the shortest distance between two points._

13. _Two straight lines cannot enclose a space._

Problems in Geometrical Drawing.

EXAMPLE 1.--_To bisect_ (cut in two) _a straight line or an arc of a circle_, Fig. 99. From the ends of _A B_ as centers, describe arcs cutting each other at _C_ and _D_, and draw _C D_, which cuts the line at _E_ or the arc at _F_.

EX. 2.--_To draw a perpendicular to a straight line, or a radial line to a circular arc_, Fig. 99. Operate as in the foregoing problem. The line _C D_ is perpendicular to _A B_; the line _C D_ is also radial to the arc _A B_.

EX. 3.--_To draw a perpendicular to a straight line, from a given point in that line_, Fig. 100. With any radius from any given point _A_ in the line _B C_, cut the line at _B_ and _C_. Next, with a longer radius, describe arcs from _B_ and _C_, cutting each other at _D_, and draw the perpendicular _D A_.

_Second Method_, Fig. 101. From any center _F_ above _B C_, describe a circle passing through the given point _A_, and cutting the given line at _D_; draw _D F_, and produce it to cut the circle at _E_; and draw the perpendicular _A E_.

_Third Method_, Fig. 102. From _A_ describe an arc _E C_, and from _E_, with the same radius, the arc _A C_ cutting the other at _C_; through _C_ draw a line _E C D_ and set off _C D_ equal to _C E_, and through _D_ draw the perpendicular _A D_.

EX. 4.--_To draw a perpendicular to a straight line from any point without it_, Fig. 103. From the point _A_ with a sufficient radius cut the given line at _F_ and _G_; and from these points describe arcs cutting at _E_. Draw the perpendicular _A E_.

If there be no room below the line, the intersection may be taken above the line; that is to say, between the line and the given point.

_Second Method_, Fig. 104. From any two points _B C_ at some distance apart, in the given line, and with the radii _B A_, _C A_, respectively, describe arcs cutting at _A D_. Draw the perpendicular _A D_.

EX. 5.--_To draw a parallel line through a given point_, Fig. 105. With a radius equal to the given point _C_ from the given line _A B_, describe the arc _D_ from _B_, taken considerably distant from _C_. Draw the parallel through _C_ to touch the arc _D_.

_Second Method_, Fig. 106. From _A_, the given point, describe the arc _F D_, cutting the given line at _F_; from _F_, with the same radius, describe the arc _E A_, and set off _F D_, equal to _E A_. Draw the parallel through the points _A D_.

When a series of parallels are required perpendicular to a base line _A B_, they may be drawn as in fig. 107 through points in the base line set off at the required distances apart. This method is convenient also where a succession of parallels are required to a given line _C D_, for the perpendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular.

EX. 6.--_To divide a line into a number of equal parts_, Fig. 108.

To divide the line _A B_ into, say, five parts. From _A_ and _B_ draw parallels _A C_, _B D_ on opposite sides; set off any convenient distance four times (one less than the given number), from _A_ on _A C_, and on _B_ on _B D_; join the first on _A C_ to the fourth on _B D_, and so on. The lines so drawn divide _A B_ as required.

_Second Method_, Fig. 109. Draw the line at _A C_, at an angle from _A_, set off, say, five equal parts; draw _B_ 5, and draw parallels to it from the other points of division in _A C_. These parallels divide _A B_ as required.

EX. 7.--_Upon a straight line to draw an angle equal to a given angle_, Fig. 110. Let _A_ be the given angle and _F G_ the line. With any radius from the points _A_ and _F_, describe arcs _D E_, _I H_, cutting the sides of the angle _A_ and the line _F G_.

Set off the arc _I H_, equal to _D E_ and draw _F H_. The angle _F_ is equal to _A_ as required.

EX. 8.--_To bisect an angle_, Fig. 111. Let _A C B_ be the angle; on the center _C_ cut the sides at _A B_. On _A_ and _B_ as centers describe arcs cutting at _D_ dividing the angle into two equal parts.

EX. 9.--_To find the center of a circle or of an arc of a circle._ Fig. 112. Draw the chord _A B_, bisect it by the perpendicular _C D_, bounded both ways by the circle; and bisect _C D_ for the center _G_.

EX. 10.--_Through two given points to describe an arc of a circle with a given radius_, Fig. 113. On the points _A_ and _B_ as centers, with the given radius, describe arcs cutting at _C_; and from _C_, with the same radius, describe an arc _A B_ as required.

Second, for a circle or an arc, Fig. 114. Select three points _A_, _B_, _C_ in the circumference, well apart; with the same radius describe arcs from these three points cutting each other, and draw two lines _D E_, _F G_, through their intersections according to Fig. 107. The point where they cut is the center of the circle or arc.

EX. 11.--_To describe a circle passing through three given points_, Fig. 114. Let _A_, _B_, _C_ be the given points and proceed as in last problem to find the center _O_, from which the circle may be described.

This problem is variously useful; in finding the diameter of a large fly-wheel, or any other object of large diameter when only a part of the circumference is accessible; in striking out arches when the span and rise are given, etc.

EX. 12.--_To draw a tangent to a circle from a given point in the circumference_, Fig. 115. From _A_ set off equal segments _A B_, _A D_, join _B D_ and draw _A E_, parallel to it, for the tangent.

EX. 13.--_To draw tangents to a circle from points without it_, Fig. 116. From _A_ with the radius _A C_ describe an arc _B C D_, and from _C_ with a radius equal to the diameter of the circle, cut the arc at _B D_, join _B C_, _C D_, cutting the circle at _E F_, and draw _A E_, _A F_, the tangents.

EX. 14.--_Between two inclined lines to draw a series of circles touching these lines and touching each other_, Fig. 117. Bisect the inclination of the given lines _A B_, _C D_ by the line _N O_. From a point _P_ in this line draw the perpendicular _P B_ to the line _A B_, and on _P_ describe the circle _B D_, touching the lines and cutting the center lines at _E_. From _E_ draw _E F_ perpendicular to the center line, cutting _A B_ at _F_, and from _F_ describe an arc _E G_, cutting _A B_ at _G_. Draw _G H_ parallel to _B P_, giving _H_, the center of the next circle, to be described with the radius _H E_, and so on for the next circle, _I N_.

EX. 15.--_To construct a triangle on a given base, the sides being given._

First. An equilateral triangle, Fig. 118. On the ends of a given base _A B_, with _A B_ as a radius describe arcs cutting at _C_, and draw _A C_, _C B_.

Second. Triangle of unequal sides, Fig. 119. On either end of the base _A D_, with the side _B_ as a radius describe an arc; and with the side _C_ as a radius, on the other end of the base as a center, describe arcs cutting the arc at _E_; join _A E_, _D E_.

This construction may be used for finding the position of a point _C_ or _E_ at given distances from the ends of a base, not necessarily to form a triangle.

EX. 16.--_To construct a square rectangle on a given straight line._

First. A square, Fig. 120. On the ends _B A_ as centers, with the line _A B_ as radius, describe arcs cutting at _C_; on _C_ describe arcs cutting the others at _D E_; and on _D_ and _E_ cut these at _F G_. Draw _A F_, _B G_ and join the intersections _H I_.

Second. A rectangle, Fig. 121. On the base _E F_ draw the perpendiculars _E H_, _F G_, equal to the height of the rectangle, and join _G H_.

EX. 17.--_To construct a parallelogram of which the sides and one of the angles are given_, Fig. 122. Draw the side _D E_ equal to the given length _A_, and set off the other side _D F_ equal to the other length _B_, forming the given angle _C_. From _E_ with _D F_ as radius, describe an arc, and from _F_, with the radius _D E_ cut the arc at _G_. Draw _F G_, _E G_. Or, the remaining sides may be drawn as parallels to _D E_, _D F_.

EX. 18.--_To describe a circle about a triangle_, Fig. 123. Bisect two sides _A B_, _A C_ of the triangle at _E F_, and from these points draw perpendiculars cutting at _K_. On the center _K_, with the radius _K A_ draw the circle _A B C_.

EX. 19.--_To describe a circle about a square, and to inscribe a square in a circle_, Fig. 124.

First. To describe the circle. Draw the diagonals _A B_, _C D_ of the square, cutting at _E_; on the center _E_ with the radius _E A_ describe the circle.

Second. To inscribe the square. Draw the two diameters _A B_, _C D_ at right angles and join the points _A B_, _C D_ to form the square.

_In the same way a circle may be described about a triangle._

EX. 20.--_To inscribe a circle on a square, and to describe a square about a circle_, Fig. 125.

First. To inscribe the circle. Draw the diagonals _A B_, _C D_ of the square, cutting at _E_; draw the perpendicular _E F_ to one side, and with the radius _E F_ describe the circle.

Second. To describe the square. Draw two diameters _A B_, _C D_ at right angles, and produce them; bisect the angle _D E B_ at the center by the diameter _F G_, and through _F_ and _G_ draw perpendiculars _A C_, _B D_, and join the points _A D_ and _B C_ where they cut the diagonals to complete the square.

EX. 21.--_To inscribe a circle in a triangle_, Fig. 126. Bisect two of the angles _A C_ of the triangle by lines cutting at _D_; from _D_ draw a perpendicular _D E_ to any side, and with _D E_ as radius describe a circle.

EX. 22.--_To inscribe a pentagon in a circle_, Fig. 127. Draw two diameters _A C_, _B D_ at right angles cutting at _O_; bisect _A O_ at _E_, and from _B_ with radius _B E_ cut the circumference at _G H_ and with the same radius step round the circle to _I_ and _K_; join the points to form the pentagon.

EX. 23.--_To construct a hexagon upon a given straight line_, Fig. 128. From _A_ and _B_, the ends of the given line, describe arcs cutting at _G_; from _G_ with the radius _G A_ describe a circle. With the same radius set off the arcs _A C_, _C F_ and _B D_, _D E_; join the points so found to form the hexagon.

EX. 24.--_To inscribe a hexagon in a circle_, Fig. 129. Draw a diameter _A C B_; from _A_ and _B_ as centers, with the radius of the circle _A C_ cut the circumference at _D_, _E_, _F_, _G_, and draw _A D_, _D E_, etc., to form the hexagon. The points _D E_, etc., may be found by stepping the radius (with the dividers) six times round the circle.

EX. 25.--_To describe an octagon on a given straight line_, Fig. 130. Produce the given line _A B_ both ways and draw perpendiculars _A E_, _B F_; bisect the external angles _A_ and _B_ by the lines _A H_, _B C_, which make equal to _A B_. Draw _C D_ and _H G_ parallel to _A E_ and equal to _A B_; from the center _G D_, with the radius _A B_, cut the perpendiculars at _E F_, and draw _E F_ to complete the hexagon.

EX. 26.--_To convert a square into an octagon_, Fig. 131.--Draw the diagonals of the square cutting at _E_; from the corners _A_, _B_, _C_, _D_, with _A E_ as radius, describe arcs cutting the sides at _G_, _H_, etc., and join the points so found to complete the octagon.

EX. 27.--_To inscribe an octagon in a circle_, Fig. 132. Draw two diameters _A C_, _B D_, at right angles; bisect the arcs _A B_, _B C_, at _E_, _F_, etc., to form the octagon.

EX. 28.--_To describe an octagon about a circle_, Fig. 133. Describe a square about the given circle _A B_, draw perpendiculars _H_ and _K_, to the diagonals, touching the circle to form the octagon. Or, the points _H_, _K_, etc., may be found by cutting the sides from the corners, by lines parallel to the diagonals.

EX. 29.--_To describe an ellipse when the length and breadth are given_, Fig. 134. On the center _C_, with _A E_ as radius, cut the axis _A B_ at _F_ and _G_, the foci, fix a couple of pins into the axis at _F_ and _G_, and loop on a thread or cord upon them equal in length to the axis _A B_, so as when stretched to reach the extremity _C_ of the conjugate axis, as shown in dot-lining. Place a pencil or drawpoint inside the cord, as at _H_, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins _F_, _G_, and so describe the ellipse.

NOTE.--The ellipse is an oval figure, like a circle in perspective.
The line that divides it equally in the direction of its great length
is the _transverse axis_, and the line which divides the opposite way
is the _conjugate axis_.

_Second Method._ Along the straight edge of a piece of stiff paper mark off a distance _a c_ equal to _A C_, half the transverse axis; and from the same point a distance _a b_ equal to _C D_, half the conjugate axis. Place the slip so as to bring the point _b_ on the line _A B_ of the transverse axis, and the point _c_ on the line _D E_; and set off on the drawing the position of the point _a_. Shifting the slip, so that the point travels on the transverse axis, and the point _c_ on the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. See fig. 135.

Trigonometry.

Trigonometry is that portion of geometry which has for its object the measurement of triangles. When it treats of plane triangles, it is called _Plane Trigonometry_; and as the engineer will continually meet in his studies of higher mathematics _the terms_ used in plane trigonometry, it is advantageous for him to become familiar with some of the principles and definitions relating to this branch of mathematics.

The circumferences of all circles contain the same number of degrees, but the greater the radius the greater is the absolute measures of a degree. The circumference of a fly wheel or the circumference of the earth have the same number of degrees; yet the same number of degrees in each and every circumference is the measure of precisely the same angle.

The circumference of a circle is supposed to be divided into 360 degrees or divisions, and as the total angularity about the center is equal to four right angles, each right angle contains 90 degrees, or 90°, and half a right angle contains 45°. Each degree is divided into 60 minutes, or 60′; and for the sake of still further minuteness of measurement, each minute is divided into 60″. In a whole circle there are, therefore, 360 × 60 × 60 = 1,296,000 seconds. The annexed diagram, fig. 136, exemplifies the relative positions of the

Sine,
Co-sine,
Versed Sine,

Tangent,
Co-Tangent,
Secant and
Co-secant

of an angle.

These may be defined thus:

DEFINITIONS.

1. The _Complement_ of an arc is 90° minus the arc.

2. The _Supplement_ of an arc is 180° minus the arc.

3. The _Sine_ of an angle, or of an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end.

4. The _Cosine_ of an arc is the perpendicular distance from the center of the circle to the sine of the arc; or, it is the same in magnitude as the sine of the complement of the arc.

5. The _Tangent_ of an arc is a line touching the circle in one extremity of the arc, and continued from thence, to meet a line drawn through the center and the other extremity.

6. The _Cotangent_ of an arc is the tangent of the complement of the arc. The _Co_ is but a contraction of the word complement.

7. The _Secant_ of an arc is a line drawn from the center of the circle to the extremity of the tangent.

8. The _Cosecant_ of an arc is the secant of the complement.

9. The _Versed Sine_ of an arc is the distance from the extremity of the arc to the foot of the sine.

For the sake of brevity, these technical terms are contracted thus: for sine _A B_, we write _sin. A B_; for cosine _A B_, we write _cos. A B_; for tangent _A B_, we write _tan. A B_, etc.

Drawing Materials and Instruments.

Drawing tools or instruments are contrived solely for mechanical drawing; aside from this use they are perfectly worthless, hence the quality of these special utensils is a matter of the first consideration to the earnest student.

There are several degrees of excellence to be found in the make-up of drawing instruments and materials; it may be remarked with truth that “any kind are good enough, and the best none too good,” i. e., a learner in this delightful art should not stop at the lack of goodness or the low grade existing in his “tools,” but rather do the best work possible with the means at hand.

However, in order that acceptable work may be accomplished, fairly good instruments should be procured. The advice of some one experienced in the use and care of draughting tools should be sought before purchasing. A drawing board, a single sheet of paper and a pencil is the simplest “outfit” to be thought of; to this small beginning may be added, soon afterwards, an inexpensive pair of compasses, a T-square and a couple of triangles; a vast range of work can be executed with these few tools.

Nothing else will be needed to do fine work except, perhaps, one or two pairs of better compasses and a few sweeps or means of drawing irregular curves; all these had best be purchased separately; for in buying a “box of instruments,” it may contain some articles which are not desired, or that are of a wrong size, or even duplicates of those already possessed.

An outfit recommended by the author of “Reed’s Hand Book” is as follows.

Large compasses with movable leg.

A pair of dividers.

Bow pencil.

Bow pen.

Pencil leg for large compasses.

Pen leg for large compasses.

Drawing pen.

Louis Rouillon, B. S., Instructor of Drawing in Pratt Institute, New York, recommends the following:

Compasses, 5¹⁄₂ inches, with needle point; pen pencil and lengthening bar.

Drawing pen, 4¹⁄₂ inches.

T-square, 24-inch blade.

45-degree triangle, 9 inches.

30 and 60 degree triangle, 9 inches.

1 Scroll.

Dixon’s V. H. pencil.

12-inch boxwood scale, flat, graduated ¹⁄₁₆ inch the entire length.

Bottle of liquid India ink, four thumb-tacks, pen and ink eraser.

20 sheets drawing paper, 11 × 15 inches, and a drawing-board about 16 × 23 inches will also be necessary; students can usually make the board themselves for less money than it can be bought.

NOTE.--The purchasing of drawing tools is one of the most difficult
points to settle that can present itself to a person about to buy a
drawing outfit for the first time. Nothing can be so productive of
distress to a person drawing as to have his tools getting out of
order, joints one day too tight, next day too slack, points getting
blunt or perhaps turning up altogether; if needle points, then the
needles slip up, and drawing spoiled; in fact, the purchaser can be
annoyed in numberless different ways.--W. H. THORN.

THE DRAWING BOARD.

A drawing-board should be made of well seasoned pine of a convenient size, say 23 × 16, which will take half a sheet of imperial paper, leaving ¹⁄₂-inch margin all around.

The working surface of the board--or its front side--should be perfectly smooth, but instead of being flat it should have a very slight camber, or rounding, breadthways, this latter feature in its construction being to prevent the possibility of a sheet of paper when stretched on its surface having any vacuity beneath it.

The _four edges_ of the board need not form an exact rectangle, as much valuable time is often wasted in the attempts to produce such a board; but it will answer every purpose of the draughtsman so long as the adjacent edges at the lower left-hand corner of it are at right angles, or square to each other.

An English authority recommends the use of two drawing-boards, 42 inches long and 30 inches wide, made of plain stuff, without cleets, 1¹⁄₄ inches thick--seasoned--with edges perfectly straight and at right angles to each other. _With two boards, one may be used for sketching and drawing details and the other for the finished drawing._

The board should be ³⁄₄ inch in thickness, and fitted at the back, at right angles to its longest side, with a couple of hardwood battens, about 2 inches wide and ³⁄₄ inch thick; the use of these battens being to keep the board from casting or winding and to allow of its expansion or contraction through changes of temperature. This latter purpose, however, is only effected by attaching the battens to the back of the board in the following manner: ... At the middle of the length of each batten--which should be one inch less than the width of the board--a stout, well-fitted wood screw is firmly inserted into it, and made to penetrate the board for about ¹⁄₂ inch, the head of the screw being made flush with the surface of the batten; on either side of the central screw, two others, about 3¹⁄₂ inches apart, are passed through oblong holes in the battens, and screwed into the body of the board until their heads are flush with the central one; fitted in this way the board itself can expand or contract lengthwise or crosswise, while its surface is prevented from warping or bending.

A further improvement in such a drawing board as above shown is made by cutting lengthwise along its ends a narrow groove and inserting an ebony or hardwood strip; this is cut or sawn apart at about every inch to admit of contraction; this strip serves as a guide to the stock of the drawing square, allowing an easy sliding movement.

To produce really good work in the shape of a mechanical drawing, one perfect straight edge only is required on a drawing board, and that the left one, which is always known as the _working-edge_; but for the convenience of being able to draw a long line across the board at right angles to its lower edge, this edge is made truly square with that on the left side of the board.

The details for building these drawing boards are given, because they are easy to be made by one who understands the use of a few wood-working tools; while the boards themselves are difficult of transportation--in case of the change of residence of their owners--quite unlike the instruments which are to accompany them.

Fig. 141 represents the board which has been described in the text, with provisions for the contraction and expansion; the very dark lines are intended to represent the ebony insertions--as described. Fig. 140 represents a plain pine board with dovetailed battens.

Fig. 142 represents the common means used to attach or secure slightly or temporarily the drawing paper to the drawing board; these are called thumb-tacks, and are usually forced through the paper into the wood by the hand, whence they are easily detached. These are made to have as slight a projection as may be, so as not to interfere with the free movement of the tee-square.

For mechanical drawing the invariable practice is to secure the paper on which the drawing is to be made to the drawing board by pinning it; this is effected by various kinds of _drawing pins_ or _thumb-tacks_.

The best kind for this purpose have a head as thin as possible without cutting at its edges, slightly concave on the under side next the paper, and only so much convex on its upper side as will give it sufficient thickness to enable the pin to be secured to it; better use four or more small pins along the edge of a sheet of paper, than use one clumsy, badly made pin at each end.

Fig. 143 and fig. 144 represent a pair of plain trestles or horses in common use for supporting large size drawing boards. This pattern is found frequently in the laying-out shop. Fig. 145 and fig. 146 represent _adjustable_ horses or trestles--these are designed, primarily, for office use. As will be seen by viewing the illustration, the upper part is supported by two hard-wood sliding pieces; these are provided with strong pins and numerous holes, and pass through the frame of the trestle, as shown, so that the upper portions can be arranged at any angle convenient to the draughtsman, as he lies over his work or stands by it.

Fig. 137 is introduced to exhibit the paper attached to the drawing board with the thumb-tacks, and with the T-square and set-squares arranged to commence work; the paper should not extend to the edges of the board; three, four or more tacks may be used on each edge of the sheet of paper, instead of two, as shown in illustration.

A most convenient--and except for its extreme lightness, which is not good in a drawing stand--a most admirable device is shown in fig. 138. The drawing table is simply a drawing board with folding legs; these are made from hard-wood, while the top is made of soft, seasoned pine, with square corners; while the device is strong and well braced, it can be folded and easily carried about--all as shown in the illustration.

THE TEE-SQUARE.

This is an instrument in the form of a letter T, as shown in the figures 149 and 150; the two parts are known as _stock_ and _blade_; the horizontal part of the letter (T) is the stock, and the vertical part the blade--hence the name, T-square; to form the square, the two parts are joined together in such a way as to make them exactly at right angles to each other; the stock, which is applied to the working edge of the drawing board, being about one-third the length of the blade, and about three times its thickness.

To be perfect in construction, a tee-square should be as light as is consistent with its necessary strength and stiffness of parts; it should be made of suitable material easily manufactured, put together, and repaired, and withal as truly correct as is possible to be made. Such a square is represented in fig. 148; it has a taper blade, which is generally about double the width where secured to the stock as it is at the end.

The manner in which the stock is united to the blade determines its adaptability or otherwise to the use made of it; in some the stock is rectangular in section, and the blade mortised into it; in others the blade is dovetailed and let into the stock for the whole of its thickness.

ADJUSTABLE BLADED SQUARE.

In cases where many parallel lines have to be drawn, of lengths beyond the capabilities of ordinary set-squares, and in directions other than square with or parallel to the working edge of the drawing board it is convenient to have for use an _adjustable_ bladed tee-square, or one whose blade can be set at any desired angle. The blade of such a square should be tapered as in illustration, but shaped at its wide end as shown, and having a stock wide enough to allow for the surface required in the washers of the fittings necessary to make the blade adjustable. These fittings, though requiring to be well made and neatly finished, are not expensive or difficult to make, as they consist merely of two washers, a square-necked bolt, and a fly nut.

The tee-square, as shown, has four parts: 1, blade; 2, fixed head; 3, shifting head; 4, swivel. The head is held firmly by the left hand to the left edge of the drawing board, and the blade serves as a straight-edge for horizontal lines that may extend the whole length of the paper. It can be used for either horizontal, or, by reversing to the bottom of the board, for vertical lines; and, by turning it over, so that the shifting head is against the edge of the drawing board instead of the fixed head, lines at different angles may be drawn. The length of the blade should be the length of the drawing board; if it is shorter, inconvenience will be experienced when lines the whole length of the board are wanted.

TRIANGLES, OR SET-SQUARES.

Set-squares are invariably used in connection with the tee-square, as shown in fig. 148. The illustrations below show several patterns of the device; by these, vertical lines, triangles, squares and hexagonal, octagonal and twelve-sided figures, diagonal section lines, etc., can be easily drawn. For ordinary purposes, a triangle or set-square with angles of 45° may be 4 inches long and the other 8 inches in length, but a six-inch set-square having angles 90°, 45° and 45°, and an eight-inch one having angles of 90°, 60° and 30°, will be found sufficient for all purposes; there are other triangles used specially for making letters.

In practice the triangles or set-squares are slid along the edge of the blade, and need not be any thicker than it.

PARALLEL RULE.

This instrument is used to mark lines which are neither horizontal nor vertical (usually these are drawn by the square and set-square), and which are parallel to one another; by adjusting the edge of the parallel ruler to a line, it can be extended or opened out (or vice-versa closed), and the line or lines drawn will be parallel or equally distant from the base or first line it was set by. See fig. 157.

Fig. 158 is a parallel ruler, constructed with two rollers fixed on a rod so that they move the same distance, carrying the ruler parallel to the starting line.

NOTE.--It has been said that “a workman may be known by his tools,”
but the statement must be taken with a good deal of allowance. Some
workmen may possess a very fine set of tools and never use them,
because they have not the ability or inclination to learn how;
especially is this the case with drawing instruments.

If all the fine sets of drawing instruments that are owned by workmen
were put to frequent use the owners would find a marked improvement in
their abilities in other lines as well as drawing; for it is a
noticeable fact that when a person’s mind has been trained in a
business that requires close calculation and a knowledge of materials,
he is capable of showing good qualifications in other lines, and the
more skilled he is in one the easier can he acquire skillfulness in
another, if he applies the same amount of energy, thought and interest
as he did to acquire skill in the first.

SECTION LINER.

Fig. 159 shows an improved section liner which can be adjusted to any angle, and will space the parallel lines at any desired regular distance.

IRREGULAR CURVE OR SCROLL.

Irregular curves, or, as they are sometimes termed, sweeps, represented by figs. 160-166, are used for curves that cannot be put in by the other instruments. They are very useful when elliptical or parabolic curves are desired, in preference to circles or arcs of a circle. They are much used in design and architectural drawing. They are made of thin hardwood or rubber, and sometimes of horn.

Curves are irregular lines; a circle is a regular line. If a curve is to be passed through a number of predetermined points it should be first sketched in lightly, free-hand; a section of the scroll is then applied to the curve so as to embrace as many points as possible; only the central points of those thus embraced should be inked in; this process is continued until the desired curve is completed.

Curves are made of various material, pearwood, cardboard, xylonite, hard rubber, and a strip of soft lead is sometimes used, which may be easily adjusted to the curve required.

The curves generally used in mechanical drawing are shown on previous page.

Fig. 208, page 134, is a logarithmic _spiral_ curve. It is mathematically constructed and contains every curve within the limit of its size.

ELLIPSES.

An ellipse is a geometrical figure, and can be drawn as described in geometrical problem 29, page 96; many drawing offices keep sets of hard rubber ellipses, to economize time constructing them.

DRAWING PENCILS.

These are instruments for marking, drawing or writing, formed of graphite, colored chalk or materials of similar properties, and having a tapering end, inclosed, generally, in a cylinder of softwood. Fig. 167 represents a ruling pencil; its point is a parallelogram or of a wedge shape. In ruling, the length view rests against the square; its shape gives considerable strength to the lead and allows the making of a very fine line. Fig. 168 differs in the point of the pencil shown, as may be observed in the illustration.

A pencil that is hard is best for mechanical drawing; one that will retain a good point for some considerable time. Pencil lines should be made as light as possible; the presence of lead on the surface of the paper tends to prevent the ink passing to the paper, and in rubbing out pencil lines the ink is reduced in blackness, and the surface of paper is roughened, which is a disadvantage. As little erasing or rubbing out as possible should be done.

DIVIDERS AND COMPASSES.

These instruments, while they appear alike, have a separate use: the dividers are used to space off distances and dimensions; especially are they necessary in reading drawings made to scale. _Compasses_ are used for describing circles, curves, etc., _dividers_ are used for marking out spaces.

Two forms of the dividers are shown in figs. 169 and 170; the simplest, plainest form is shown in fig. 169; these are used for rough spacings; fig. 170 represents a pair of dividers fitted with an adjustable screw controlled by a steel spring in one leg; by this a very exact measurement can be made. Fig. 170 is intended to exhibit what is called a “hair-spring divider.”

PROPORTIONAL DIVIDERS.

These dividers differ from the ordinary ones shown in figs. 169 and 170 in that they are provided with four steel points, one pair of which being set to the full dimension will be reproduced by the other pair, but in a smaller, or reduced size.

Fig. 171 are “bisecting” dividers, being proportional dividers, which, when open, one end measures double the distance of the other.

Fig. 172 are proportional dividers; the points at one end are capable of being changed, to measure practically any desired proportion at the other end, by altering the position of the pivot where the legs cross one another. The lower connecting link is a micrometer adjustment, for minute measurements.

Fig. 173 are proportional dividers which are marked for the proportions of lines and radii of circles, being provided with a rack movement for adjustment.

Fig. 174 represents three-leg dividers, used for taking the position of three points; this instrument is very useful in finding the position of a point in a figure.

COMPASSES.

Compasses consist of two pointed legs; they are instruments for describing circles or for--sometimes--measuring figures, in absence of dividers. Fig. 175 represents compasses fitted as dividers.

Compasses should have jointed legs, which will allow the points to be placed at right angles to the paper, whatever the size of the circle to be drawn. Compasses should not be used for circles which are too large to allow the points to be thus placed; a lengthening bar is generally provided, which greatly increases the diameters of circles which may be drawn by this attachment; it is shown in fig. 176.

One leg of the compasses is usually provided with a socket to which are fitted three points: a divider point, fig. 179; a pencil point, fig. 177; and a point, fig. 178, carrying a special pen for the inking of circles. Each of these points is generally provided with a joint, so that it may be placed at right angles to the paper.

The other leg should be jointed; it is often provided with a socket which receives two points, one a divider point, and the other carrying a needle point. Such an instrument may be used as dividers for spacing, or as compasses for penciling or inking circles.

The joint at the head of the compasses (see fig. 175) is the most important feature. It should hold the legs firmly in any position, so that in going over a circle several times only one line will result. It should allow the legs to move smoothly and evenly, and should be capable of adjustment.

As shown in fig. 174, one leg has a hinge or joint, and a needle point, which can be regulated by a thumb screw; the other leg has a socket or recess into which interchangeable parts can be inserted. The four figures to the right of the compasses show the parts which are provided with shanks or insertion pieces. Fig. 180 and fig. 181 represent compasses specially used for making small circles, and work too minute for the larger instruments described above.

To do work of this nature easily a pair of spring dividers are frequently used. This instrument has one point attached to a spring, which is regulated by a screw, so that very slight changes in the space may be made with ease.

Compasses specially used for putting in fine circles and dimensions are called “bows.” When a pen point it is a “bow pen,” with a pencil point a “bow pencil,” and if with needle point a “bow dividers.” Fig. 180 is a “bow dividers”, this fitted with screw for fine adjustment in one leg, fig. 181, is called a “hair-spring bow dividers”; for small details, bows with steel spring legs without any joint are used; these are called “steel-spring bows.”

SPRING BOWS.

These were originally developed from the common form of compasses, with a single spring leg; later, the demand for smaller sizes made changes necessary, and spring bows are now made symmetrical, both sides of the bow being made to “spring.”

Fig. 182 are spring dividers.

Fig. 183 is a spring pencil.

Fig. 184 is a spring pen.

In these figures it will be seen that the two threads, a right and a left, are moved with one central thumbscrew; in the figs. 185 to 187 a single screw is used.

In choosing spring bows, care must be exercised to select a sufficiently strong, stiff spring, as the relation between spring pressure and thumbscrew is important.

BEAM COMPASSES AND TRAMMELS.

In fig. 188 is shown a set of beam compasses, together with a portion of the wooden rod or beam on which they are used.

The latter, as will be seen by the section drawn to one side, _A_, is in the shape of a T. This form has considerable strength and rigidity. Beam compasses are provided with extra points for pencil or ink work, as shown.

While the general adjustment is effected by means of the clamp against the wood, minute variations are made by the screw, _B_, shifting one of the points, as shown in the figure.

This instrument is quite delicate, and, when in good order, is very accurate. It should be used only for fine work on paper, and never for scribing on metal.

A coarser instrument, and one especially designed for use upon metal, is shown in fig. 189, and is called a trammel. There are various forms of this instrument, all being the same in principle. The engraving shows a form in common use. A heavier stick is used with it than with the beam compasses, and no other adjustment is provided than that which is afforded by clamping against the stick.

In the illustration, a carrier at the side is shown, in which a pencil may be placed. Some trammels are arranged in such a manner, that either of the points may be detached and a pencil substituted.

A trammel, by careful arrangement, can be made to describe very accurate curves, and hence can be used in place of the beam compasses in many instances. For all coarse work it is to be preferred to beam compasses. It is useful for all short sweeps upon sheets of metal, but for curves of a very long radius a strip of sheet iron or a piece of wire will be found of a more practical service than even this tool.

The length of rods for both beam compasses and tramels, up to certain limits, is determined by the nature of the work to be done. The extreme length is determined by the strength and rigidity of the rod itself. It is usually convenient to have two rods for each instrument, one about 1¹⁄₂ or 2 feet in length and the other considerably longer--as long as the strength of the material will admit.

DRAWING SCALES.