SECTION X.

GEOMETRICAL QUANTITIES, METHODS OF EXPRESSION, AND

DEFINITIONS.

In the general account of the elementary notions of geometry and geometrical quantities given in the last section, we studiously avoided all allusion to the methods of expressing quantities geometrically, to the short elementary definitions, self-evident principles (or axioms), and also to the objects of geometrical operations or inquiries, as usually given in books on the elements. When the general notion is explained by allusion to a particular symbol, there is some danger that the symbol will lay hold of the student's conception, and particularise it; and this is especially the case when the quantity under explanation is a relation, and as such, not expressible by any separate symbol, but merely by the position of those other quantities of which it is a relation. Thus, for instance, a line, a surface, or a solid may be represented by a picture, or diagram, as it is usually called ; but no diagram can represent simply and singly that which we mean by an angle. This magnitude (an angle) can be represented in a diagram only by the two lines of whose inclination it is the measure; and as these lines must, in any diagram which can be drawn, have some visible length, and also include between them, as far as they extend from the point of meeting, some visible portion of surface, it is very difficult for a beginner to avoid mixing up the notion of the lengths of the lines, and also that of the space between them, with the proper notion of the angle. It will be found that, in consequence of this confusion, those who have made but little progress in geometry,

even though they have made some, have very vague and confused notions of what is really meant by an angle, independently of the lengths of the lines of whose inclination it is the measure, and of the quantity of surface which may be contained within or between those lines. If the explanation which we gave in the last section has been read and studied with due attention, the reader will not find much difficulty in forming a correct notion of what an angle is, without mixing it up with any notion of the length of particular lines, or the extent of particular surfaces; and if we have succeeded in doing this, the reader will have gained more than he is aware of.

We may now, therefore, proceed to point out the modes by which quantities are geometrically represented, and the names which are given to a few of the simpler modifications of them; and in the mean time we shall confine ourselves to the elements of PLANE GEOMETRY, that is, to lines, plane rectilineal angles, and surfaces; only, because the knowledge of the only curve which enters into the elements of plane geometry is necessary, in order rightly to understand the distinctions of those leading varieties of angles which we require to define at the outset, we shall include that curve among lines, though it cannot appear as a line without appearing at the same time as the boundary of a surface or figure.

1. OF LINES.

There are only two kinds of line in elementary geometry, the straight line and the circle.

Straight lines have been already defined. They, when we consider them as single lines, have length only; but a straight line may be of any length, known or unknown; and no straight line can be of a known length, unless we can measure it, and

have measured it, or can deduce its length by correct reasoning from that of some line which we have measured.

A straight line is represented geometrically by a line drawn as straight as possible; and it is named by two letters, placed either one at each end of the line, or at any point in it; and the line is named by those two letters, it being of no consequence which of them is mentioned first.

In geometry it is customary and advisable to use capital letters, as a distinction from the letters used in algebra, just as in algebra we use italic letters to distinguish them from the Roman letters generally used in printing common language. But when a geometrical magnitude is affected by a number stated generally by means of a letter, it is customary to use small or lowercase letters, and generally to use Roman ones, to distinguish them from the algebraical representations of quantities, just in the same manner as it is desirable to use Roman letters for exponents in algebra. We give an instance of the representation and naming of a straight line geometrically; thus, the following line is the line a B, if viewed from left to right, or the line BA, when viewed from right to left, but it is exactly the same line both ways:→

B

A circle is a plane figure, or portion of surface, bounded by one line, which is called the circumference, and the property of the circle by which it is defined, and from which all its other properties are derived, is that the circumference is everywhere equally distant from a point within the figure, which is called the centre of the circle.

The circumference, which means the measure round, or, literally, the "carrying round," and sometimes the periphery, which has the same meaning, is often called a circle, as well as

the surface which it incloses. The circumference is named by any number of letters more than two, marked either without it or within; the centre by a letter marked as near to it as possible; and the surface of the circle either by the letters which mark the circumference, or by any letter within the figure. In the meantime we are considering the circumference only, and the relation which it has to the centre, namely, that above stated, that of being everywhere equally distant from it. Thus the following is any circle, A B D, of which c is the centre :—

The property of a circle, upon which its definition is founded, follows immediately from the way in which the circle is drawn or described. Thus, suppose a bit of thread, a bit of stick, or anything else of a constant length, as the line a c, has one end made fast at the centre c, and being kept perfectly and equally stretched so as to represent a straight line, and has its other extremity a carried round, either by в and D, or by D and B, till it comes back to the position a, the circle will be described; and if a pen, a pencil, or anything else that will leave a mark, is carried round at the point or extremity a, and made to mark á plane surface, a circle will be drawn upon that surface.

The line c A, extending from the centre to the circumference, is called the radius, or ray of the circle; and it is evident, from the manner in which the circle is described, that the magnitude or size of the circle depends upon the length of the radius.

DESCRIBING A CIRCLE.

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It follows from this, that, as the radius is simply a straight line, which has no property but length, and the length of which is the same at every part of the circle, the circumferences of different circles can vary only in the same proportion as the lengths of the radii; that is to say, if the radius is double, the circumference must be double; if the radius is three times, the circumference must be three times, and so on in all other proportions.

In practice, circles of small dimensions are usually drawn with an instrument called a pair of compasses, the two points of the compasses being set at exactly the same distance from each other as the radius of the intended circle; and this is an instance in which the representation of a line by its extreme points, answers the same purpose as the line itself; from which we may conclude generally, that, if the two points which are the extremities of a line are determined, the line itself is determined.

In describing a circle with compasses, it is necessary that the distance between the points should remain exactly the same, otherwise the fundamental property of the circle is departed from, and there can, in fact, be no circle.

The postulate, or operation assumed as being self-evidently possible in the case of circles, is, that "a circle may be drawn from any point as a centre, and at any distance from that centre." The word 66 any," in both parts of this postulate, includes all points and all distances which we can by possibility imagine; and it is not confined to circles which we can actually draw or describe in practice, and show them after they are drawn. Hence we have a distinction between geometrical possibility and practical or mechanical possibility. Geometrically, it is possible to take the sun as a centre, and imagine a circle to be drawn passing through the most distant star which we