[Ill.u.s.tration: Fig. 41.]

[Ill.u.s.tration: Fig. 42.]

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 44.]

[Ill.u.s.tration: Fig. 45.]

[Ill.u.s.tration: 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 circ.u.mference or periphery and sometimes the perimeter.

A part of this circ.u.mference 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.

[Ill.u.s.tration: Fig. 47.]

[Ill.u.s.tration: Fig. 48.]

[Ill.u.s.tration: Fig. 49.]

[Ill.u.s.tration: Fig. 50.]

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 52.]

[Ill.u.s.tration: 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 pa.s.s through any three given points: Let A B and C in Figure 54 be the points through which the circ.u.mference of a circle is to pa.s.s. 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 pa.s.sing through M and F, and a line pa.s.sing through J and Q, and where these two lines intersect, as at Q, is the centre of a circle R that will pa.s.s through all three of the points A B and C.

[Ill.u.s.tration: Fig. 54.]

[Ill.u.s.tration: 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 pa.s.sing through the two points of intersections of arcs C D, and line I pa.s.sing 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 circ.u.mference. The whole circ.u.mference 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 repet.i.tion 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.

[Ill.u.s.tration: 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 , as is done in Figure 56, the 60 meaning sixty degrees, the , of course, standing for degrees.

[Ill.u.s.tration: 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, pa.s.sing through both lines _a_, _b_. All we now have to do is to find what part, or how much of the circ.u.mference, 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 36012 = 30.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: 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 compa.s.ses 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 61.]

[Ill.u.s.tration: Fig. 62.]

[Ill.u.s.tration: Fig. 63.]

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 65.]

[Ill.u.s.tration: 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.