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SHADING EFFECTS.--Now, to examine it properly so as to observe what the draughtsman wishes to express, look at Fig. 138, in which the three diverging lines (A, B, C) are increased in thickness, and the cube appears plainly. On the other hand, in Fig. 139, the thickening of the lines (D, E, F) shows an entirely different structure.

[Ill.u.s.tration: _Fig. 137._]

[Ill.u.s.tration: _Fig. 138._]

[Ill.u.s.tration: _Fig. 139._]

It must be remembered, therefore, that to show raised surfaces the general direction is to shade heavily the lower horizontal and the right vertical lines. (See Fig. 133.)

HEAVY LINES.--But there is an exception to this rule. See two examples (Fig. 140). Here two parallel lines appear close together to form the edge nearest the eye. In such cases the second, or upper, line is heaviest. On vertical lines, as in Fig. 141, the second line from the right is heaviest. These examples show plain geometrical lines, and those from Figs. 138 to 141, inclusive, are in perspective.

[Ill.u.s.tration: _Fig. 140._]

[Ill.u.s.tration: _Fig. 141._]

PERSPECTIVE.--A perspective is a most deceptive figure, and a cube, for instance, may be drawn so that the various lines will differ in length, and also be equidistant from each other. Or all the lines may be of the same length and have the distances between them vary. Supposing we have two cubes, one located above the other, separated, say, two feet or more from each other. It is obvious that the lines of the two cubes will not be the same to a camera, because, if they were photographed, they would appear exactly as they are, so far as their positions are concerned, and not as they appear. But the cubes do appear to the eye as having six equal sides. The camera shows that they do not have six equal sides so far as measurement is concerned. You will see, therefore, that the position of the eye, relative to the cube, is what determines the angle, or $the relative$ angles of all the lines.

[Ill.u.s.tration: _Fig. 142._]

[Ill.u.s.tration: _Fig. 143._]

A TRUE PERSPECTIVE OF A CUBE.--Fig. 142 shows a true perspective--that is, it is true from the measurement standpoint. It is what is called an _isometrical_ view, or a figure in which all the lines not only are of equal length, but the parallel lines are all s.p.a.ced apart the same distances from each other.

ISOMETRIC CUBE.--I enclose this cube within a circle, as in Fig. 143. To form this cube the circle (A) is drawn and bisected with a vertical line (B). This forms the starting point for stepping off the six points (C) in the circle, using the dividers without resetting, after you have made the circle. Then connect each of the points (C) by straight lines (D).

These lines are called chords. From the center draw two lines (E) at an angle and one line (F) vertically. These are the radial lines. You will see from the foregoing that the chords (D) form the outline of the cube--or the lines farthest from the eye, and the radial lines (E, F) are the nearest to the eye. In this position we are looking at the block at a true diagonal--that is, from a corner at one side to the extreme corner on the opposite side.

[Ill.u.s.tration: _Fig. 144._]

Let us contrast this, and particularly Fig. 142, with the cube which is placed higher up, viewed from the same standpoint.

FLATTENED PERSPECTIVE.--Fig. 144 shows the new perspective, in which the three vertical lines (A, A, A) are of equal length, and the six angularly disposed lines (B, C) are of equal length, but shorter than the lines A. The only change which has been made is to shorten the distance across the corner from D to D, but the vertical lines (A) are the same in length as the corresponding lines in Fig. 143.

Notwithstanding this change the cubes in both figures appear to be of the same size, as, in fact, they really are.

[Ill.u.s.tration: _Fig. 145._]

In forming a perspective, therefore, it would be a good idea for the boy to have a cube of wood always at hand, which, if laid down on a horizontal support, alongside, or within range of the object to be drawn, will serve as a guide to the perspective.

TECHNICAL DESIGNATIONS.--As all geometrical lines have designations, I have incorporated such figures as will be most serviceable to the boy, each figure being accompanied by its proper definition.

[Ill.u.s.tration: _Fig. 146._]

[Ill.u.s.tration: _Fig. 147._]

Before pa.s.sing to that subject I can better show some of the simple forms by means of suitable diagrams.

Referring to Fig. 145, let us direct our attention to the body (G), formed by the line (D) across the circle. This body is called a segment.

A chord (D) and a curve comprise a segment.

SECTOR AND SEGMENT.--Now examine the shape of the body formed by two of the radial lines (E, E) and that part of the circle which extends from one radial line to the other. The body thus formed is a sector, and it is made by two radiating lines and a curved line. Learn to distinguish readily, in your mind, the difference between the two figures.

TERMS OF ANGLES.--The relation of the lines to each other, the manner in which they are joined together, and their comparative angles, all have special terms and meanings. Thus, referring to the isometric cube, in Fig. 145, the angle formed at the center by the lines (B, E) is different from the angle formed at the margin by the lines (E, F). The angle formed by B, E is called an exterior angle; and that formed by E, F is an interior angle. If you will draw a line (G) from the center to the circle line, so it intersects it at C, the lines B, D, G form an equilateral or isosceles triangle; if you draw a chord (A) from C to C, the lines H, E, F will form an obtuse triangle, and B, F, H a right-angled triangle.

CIRCLES AND CURVES.--Circles, and, in fact, all forms of curved work, are the most difficult for beginners. The simplest figure is the circle, which, if it represents a raised surface, is provided with a heavy line on the lower right-hand side, as in Fig. 146; but the proper artistic expression is shown in Fig. 147, in which the lower right-hand side is shaded in rings running only a part of the way around, gradually diminishing in length, and s.p.a.ced farther and farther apart as you approach the center, thus giving the appearance of a sphere.

[Ill.u.s.tration: _Fig. 148._]

IRREGULAR CURVES.--But the irregular curves require the most care to form properly. Let us try first the elliptical curve (Fig. 148). The proper thing is, first, to draw a line (A), which is called the "major axis." On this axis we mark for our guidance two points (B, B). With the dividers find a point (C) exactly midway, and draw a cross line (D).

This is called the "minor axis." If we choose to do so we may indicate two points (E, E) on the minor axis, which, in this case, for convenience, are so s.p.a.ced that the distance along the major axis, between B, B, is twice the length across the minor axis (D), along E, E.

Now find one-quarter of the distance from B to C, as at F, and with a compa.s.s pencil make a half circle (G). If, now, you will set the compa.s.s point on the center mark (C), and the pencil point of the compa.s.s on B, and measure along the minor axis (D) on both sides of the major axis, you will make two points, as at H. These points are your centers for scribing the long sides of the ellipse. Before proceeding to strike the curved lines (J), draw a diagonal line (K) from H to each marking point (F). Do this on both sides of the major axis, and produce these lines so they cross the curved lines (G). When you ink in your ellipse do not allow the circle pen to cross the lines (K), and you will have a mechanical ellipse.

ELLIPSES AND OVALS.--It is not necessary to measure the centering points (F) at certain specified distances from the intersection of the horizontal and vertical lines. We may take any point along the major axis, as shown, for instance, in Fig. 149. Let B be this point, taken at random. Then describe the half circle (C). We may, also, arbitrarily, take any point, as, for instance, D on the minor axis E, and by drawing the diagonal lines (F) we find marks on the circle (C), which are the meeting lines for the large curve (H), with the small curve (C). In this case we have formed an ovate or an oval form. Experience will soon make perfect in following out these directions.

FOCAL POINTS.--The focal point of a circle is its center, and is called the _focus_. But an ellipse has two focal points, called _foci_, represented by F, F in Fig. 148, and by B, B in Fig. 149.

A _produced line_ is one which extends out beyond the marking point.

Thus in Fig. 148 that part of the line K between F and G represents the produced portion of line K.

[Ill.u.s.tration: _Fig. 149._]

SPIRALS.--There is no more difficult figure to make with a bow or a circle pen than a spiral. In Fig. 150 a horizontal and a vertical line (A, B), respectively, are drawn, and at their intersection a small circle (C) is formed. This now provides for four centering points for the circle pen, on the two lines (A, B). Intermediate these points indicate a second set of marks halfway between the marks on the lines.

If you will now set the point of the compa.s.s at, say, the mark 3, and the pencil point of the compa.s.s at D, and make a curved mark one-eighth of the way around, say, to the radial line (E), then put the point of the compa.s.s to 4, and extend the pencil point of the compa.s.s so it coincides with the curved line just drawn, and then again make another curve, one-eighth of a complete circle, and so on around the entire circle of marking points, successively, you will produce a spiral, which, although not absolutely accurate, is the nearest approach with a circle pen. To make this neatly requires care and patience.

[Ill.u.s.tration: _Fig. 150._]

PERPENDICULAR AND VERTICAL.--A few words now as to terms. The boy is often confused in determining the difference between _perpendicular_ and _vertical_. There is a p.r.o.nounced difference. Vertical means up and down. It is on a line in the direction a ball takes when it falls straight toward the center of the earth. The word _perpendicular_, as usually employed in astronomy, means the same thing, but in geometry, or in drafting, or in its use in the arts it means that a perpendicular line is at right angles to some other line. Suppose you put a square upon a roof so that one leg of the square extends up and down on the roof, and the other leg projects outwardly from the roof. In this case the projecting leg is _perpendicular_ to the roof. Never use the word _vertical_ in this connection.

SIGNS TO INDICATE MEASUREMENTS.--The small circle () is always used to designate _degree_. Thus 10 means ten degrees.

Feet are indicated by the single mark '; and two closely allied marks "

are for inches. Thus five feet ten inches should be written 5' 10". A large cross () indicates the word "by," and in expressing the term six feet by three feet two inches, it should be written 6' 3'2".

The foregoing figures give some of the fundamentals necessary to be acquired, and it may be said that if the boy will learn the principles involved in the drawings he will have no difficulty in producing intelligible work; but as this is not a treatise on drawing we cannot go into the more refined phases of the subject.

DEFINITIONS.--The following figures show the various geometrical forms and their definitions:

[Ill.u.s.tration: _Fig. 151.-Fig. 165._]

151. _Abscissa._--The point in a curve, A, which is referred to by certain lines, such as B, which extend out from an axis, X, or the ordinate line Z.

152. _Angle._--The inclosed s.p.a.ce near the point where two lines meet.

153. _Apothegm._--The perpendicular line A from the center to one side of a regular polygon. It represents the radial line of a polygon the same as the radius represents half the diameter of a circle.

154. _Apsides_ or _Apsis_.--One of two points, A, A, of an orbit, oval or ellipse farthest from the axis, or the two small dots.

155. _Chord._--A right line, as A, uniting the extremities of the arc of a circle or a curve.

156. _Convolute_ (see also _Involute_).--Usually employed to designate a wave or folds in opposite directions. A double involute.