193

PARTS OF A CIRCLE.

ference and surface, into two equal parts, each of which is called a semicircle, which is only another name for a half circle. That a diameter divides both the circumference and the surface equally, is evident without any proof, because there is nothing affects the portion of either, on the one side of the diameter, which does not equally affect the portion on the other.

If a line, not passing through the centre, is drawn till it meet the circumference both ways, as, for instance, the line DE in the above circle, it is called a chord, and the portion of the circumference which is cut or marked off by a chord, is called an arch, or arc. Thus, the chord DE cuts the circumference of the above circle into two arcs, a greater one surrounding the portion of surface in which the centre is situated, and a smaller one, in which the centre is not situated.

The portions into which the surface of a circle is divided by a chord, are called segments; and when a circle is divided into two segments, one is always greater than a semicircle, and the other less.

It may not be improper to mention here that the word area is often used for surface, and that the two words have exactly the same meaning.

A circle may be unequally cut by two radii, as well as by a chord, and in this case the parts into which it is cut are called

B

D

sectors. Thus, in the circle ABD, either of the portions divided off by the radii, a c and BC, opposite to D, and towards D, is a sector.

SEGMENT AND SECTOR.

199

The circular part of the boundary of a sector is called an arc, as well as that of a segment; but there is this difference between them, that a segment has only one straight boundary, while a sector has two; that the straight boundary of a segment may be any line less than the diameter, but that the two straight boundaries of a sector, being each equal to the radius, are always, both together, equal to the diameter: and that, while a sector always extends to the centre of the circle of which it is a sector, a segment never does. A segment, too, is never more than a two-sided figure, the one side a chord, and the other an arc; a sector is always a three-sided figure, one of the sides being an arc, and two being radii.

Rectilineal figures are named from the number of their sides or angles, the number of sides and of angles in every rectilineal figure being equal. It is easy to understand why this must be the case every side has two extremities, and every angle is formed by the meeting of one extremity of each of two sides. Of course no rectilineal figure has fewer than three sides and three angles, because three is the smallest number of straight lines that can inclose a space or surface.

Figures with three sides are called triangles; those with four sides, quadrilateral figures, or quadrilaterals; and those with more than four sides, multilateral, or many-sided; the last are also sometimes called polygons, or many-angled figures; but that name is, perhaps, better restricted to one particular form of figure, whatever may be the number of sides.

There are thus three particulars in all rectilineal figures having the same number of sides and angles, in which one figure may agree with or differ from another. There are, first, the magnitude of the angles taken in the same order; and when the figures have all their angles equally taken in this way, and

all their sides taken in the same order, in the same proportion, the first to the first, the second to the second, the third to the third, and so on, the figures are said to be similar, which means that they are all of the same shape. Thus, the following figures, ▲ and B, are similar; for all their angles, taken in the same order, are equal, and the sides of the figure a are, to those of the figure B, in the proportion of 1 to 2; that is, the sides of B, taken in the same order as the sides of a, are each twice as long.

A

B

The other relations which similar figures bear to each other can be better explained afterwards.

Rectilineal figures, which have all their sides and all their angles equal, are called polygons, or regular polygons; and if the name polygon is applied to a figure which has not all its sides and angles equal, the name polygon is qualified by the epithet irregular.

A three-sided regular polygon is called a trigon, or, more generally, an equilateral triangle; one with four sides is called a tetragon, or, more frequently, a square; one with five sides is a pentagon; one with six sides, an hexagon; one with seven sides, a heptagon; one with eight sides, an octagon; and so on, the name being compounded of the Greek term for the number of sides or angles, and the Greek name for angles.

Plane triangles are also distinguished into three kinds:

equilateral, having all the sides equal; isosceles, having two equal sides; and scalene, having all the sides unequal. Isosceles means having “equal legs," the third, or unequal side being called the base; and scalene means having unequal sides.

Four-sided figures are also distinguished into several species; but the nature of them can be better understood afterwards.

3. OF ANGLES.

The general nature of a plane rectilineal angle has been explained in the preceding section, and the comparison of angles with each other, together with their measurement, and the standard by which they are measured, will be explained afterwards; so that all that requires to be done in this place is to mention how an angle is represented and named geometrically. Now an angle is represented by two lines which meet at a point, the point where they meet being called the angular point, or the apex, or the vertex of the angle.

If there is only one angle at a point, it may be named by a single letter at the point, as, in the following figure, we would "the angle A," or 66 the angle at ▲.”

say,

A

But if there are several lines which meet at a point, then there are more angles than one, and it becomes necessary to place a letter on each line, at some distance from the point; and when we name any of the angles, we name, first, the letter

202

REPRESENTATION OF ANGLES.

on one of the lines, next the letter at the point, and thirdly the letter on the other line.

In the above figure there are five lines which meet at the point a, and thus there are four distinct and separate angles, all angles at A. Not only this, but there are as many more as can be formed of combining those in juxta-position, or taking those which are beside each other.

Taking the single ones from left to right, they are the angles BAC, CAD, DAE, and EAF, four angles.

Next, taking them two and two, there are BAD BAC + CAD, CAE CAD + DAE, and DAF DAE + EAF, three angles.

Again, taking them three and three, there are, BAE=BAC+ CAD +DAE, and CAF CAD +DAE+EAF, two angles.

Lastly, there is the whole angle, BAF=BAC+CAD+DAE+ EAF, one angle.

So that these five lines meeting at the point a, form ten distinct angles. Cases in which there are more than one angle at the same point, require some attention from beginners, in order that they may avoid confounding the one with the other; and no small part of the difficulty which is felt in the case of complicated diagrams, arises from not having sufficiently studied the simple parts of which they are made up. "Take time, and get on fast," is no bad maxim in most matters, and there is none