Part 8 (2/2)

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 diminis.h.i.+ng 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.

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