Part 3

The first effort should, therefore, be to produce a line of points exactly parallel with the upper edge of the paper, and at equal distances from each other. Commence with point _A_ and place the point _B_ carefully level with it, now place a slip of paper against these points in the original, mark their distance apart, and see if the same proportion has been given in your copy; if not, make the necessary correction. Proceed with the next point, examine it, and so on to the end of the line. When this is complete, examine each point in succession, to try if it is at the same distance from the top of the paper; when this is correct, proceed to draw the first level line. Hold the pencil as in the drawing, fig. 57, keeping the elbow near the side; join _A_ to _B_ by one light, steady stroke, produced by a movement of the wrist, and add stroke upon stroke until the line is of the required depth. Continue this process to the end of the line of points. Now place the point _D_ at the right distance below the _A_, proceed with the points for another line as before, and continue the lines until the paper is covered. In producing the stroke the pencil should not be jerked, or any stop be made between the points, but the movement should be even throughout, and it is much better to produce each line by several soft strokes, as _the repetition of delicate lines induces lightness of touch and freedom of hand_; and it is also no small advantage that lines thus produced are more easily removed by the India rubber, should they require correction.

TO DRAW THE FIRST OBLIQUE LINE.

Prepare three rows of points down the side of the paper, on the left hand; examine them to see that they are at equal distances from the side and from each other; hold the pencil as in the drawing, fig. 58, move the elbow a little from the side, and join the points _A_ and _B_ with one light line, produced by a movement of the fingers and thumb, repeating the strokes until the line is of the requisite depth; proceed to join _B_ to _C_, taking care previously to bring the hand a little down the paper, as the line from _A_ to _C_ is too long to be produced from one position. When the three rows of points are filled, make another set, examine them and proceed as before. By these means the paper will be covered with oblique lines, and if the points have been placed exactly, the sheet will have a neat and regular appearance.

NOTE.--The drawings of hands are introduced to show the positions for holding the pencil, and are not intended for copying.

It is a common, and at the same time highly injurious habit, to draw this line by a movement of the wrist, the fingers remaining rigid. This may be detected by watching the action of the thumb; if it bends as the line is produced, all is right; but if it does not the wrist is at work.

TO DRAW THE UPRIGHT OR PERPENDICULAR LINE.

This line demands the greatest attention, and any care bestowed upon it will be amply repaid in the after studies.

Commence by placing a line of points down the side of the paper, examine them very carefully to see that they are all the same distance from its edge, hold the pencil as in the drawing, fig. 59, move the elbow well out from the side, and join the points by a movement of the fingers and thumb. When one line is complete, place the points for the next, and examine them from the edge of the paper, not from the line just drawn. Proceed in this manner until the paper is covered.

There is in most cases a tendency to place the points for this line in a slightly inclined direction, as in writing, though in some instances the tendency is the opposite, a thoroughly correct eye in this respect being a rare gift: and it may be useful to suggest that the paper be so placed that the line of points to be produced may be exactly in front of the eye.

TO DRAW THE SECOND OBLIQUE LINE.

Prepare three rows of points down the side of the paper, examine them for correctness of position, hold the pencil as in the drawing, fig. 60, remove the elbow as far as possible from the side, and join the points by a movement of the fingers and thumb, and continue the exercise until the paper is covered.

It will be noticed that each change in direction of the line to be drawn, has been accompanied with a corresponding change in the position of the elbow and wrist. The following simple rule will assist the memory when placing the hand for any given line; the pencil should be held so that it may form a T with the line to be drawn:

For the horizontal line, elbow near the side.

For the first oblique, elbow a little removed.

For the perpendicular, elbow more removed.

For the second oblique, elbow most removed.

It may also be interesting to notice, with regard to the movements by which lines are produced, that they are divided into two systems; the first is that of the wrist, which includes the horizontal, and lines in nearly the same direction; the second is that of the fingers and thumb, by which all other lines are formed. The following diagram exhibits the two systems and their various lines grouped, and it will be observed that there is a space marked (a) between the two sets, which may be considered neutral ground. Lines in this direction may be produced by either movement, as may be most convenient, but it will always be found that these lines are the most trying to the hand.

ON FIGURES FORMED OF STRAIGHT LINES.

Before commencing this subject, let it be clearly understood that future success will, in a great measure, depend upon the amount of care bestowed upon it. The aim should be to obtain absolute accuracy, and for this end the copies should be tested by the most careful measurements, and corrected until they are true with the originals, but it should be distinctly understood that these measurements are only to be made after the eye and hand have done their best.

NOTE.--To some it may appear that too much time and care has been bestowed on mere lines, but let it be understood that a good system of line drawing is the basis of all education--the slightest outline by a hand thus trained has a bold, free and masterly character; and with regard to shading, which is simply an aggregation of good lines, it is only by such a practiced hand its most charming effects can be produced.

Fig. 66: Place the points _A_, _B_. Examine them to see that they are the same distance apart as in the original, and that they are level; place the point _C_ exactly under _A_, and make _A C_ equal in distance to _A B_; now place the point _D_ opposite _C_ and under _B_; try the distances between each point to see that they are the same; divide each side by a point half way, and then draw the lines.

Fig. 67: Repeat the last figure and add the lines _A_ and _B_, taking great care that the points for them are correctly placed.

Fig. 68: Commence with the square as before; then join the half-way points.

Fig. 69: After the square is drawn, place the points _A_ and _B_ at the right height above the half-way points, and _C_, _D_ at the proper distance from the corners, then draw the figure.

Fig. 70: The greatest care should be taken with the squares for this and the following figure, as the slightest error in them will destroy the symmetry of the drawing within; when the square is completed, join the opposite corners, and place on the crossed lines the points _B_, _C_, _D_, _E_; examine these to see that they are each at the same distance from the centre _A_, and that this distance is equal to the space from _A_ to the sides of the square; when all are proved to be correct, complete the figure.

Fig. 71: Repeat the last drawing with, if possible, greater exactness, and outside the octagon place the points _A_, _B_, _C_, _D_, etc.; examine each of these points to see that they are all at the same distance from the centre, and then complete the figure.

ON CURVED LINES.

The right position of the hand for drawing any curved line is that required for a straight line which would touch the extremities of the curve. The straight lines given in the exercises are valuable, not only as a guide to the position of the hand, but as an assistance to the eye when forming the curves or examining them after they are produced.

The direction given for drawing a straight line was to form it by one steady movement from point to point, without any jerk or stop by the way. This instruction requires to be changed for the curve, _which is better produced by several short strokes_, thus:

or by overlapping lines, any outside bits being cleared away with India rubber.

These exercises will test the drawing power and try the patience of the pupil, but they are worthy of all the care which can be bestowed, which in future efforts will meet with its full reward.

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_.