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enough, must meet the length required for this purpose, also varying with the inclination; secondly, lines which have no inclination to each other, but are in the same direction, and which, consequently, have not the property which depends upon inclination, namely, that of meeting each other when produced. Lines which have no inclination are called PARALLEL LINES, which means that they are "spoken of as away or apart from each other," which name comes very nearly to the definition, as it involves the notion that, produce them ever so far, there is no point at which we can speak of them as being together: and this differs but little from Euclid's definition, which, though it has not been adopted by all geometers, is at once the simplest and the best :-" Straight lines which are in the same plane, and being produced ever so far both ways do not meet, are called parallel lines."

It will be perceived that the words "being produced ever so far both ways" in this definition take for granted that very principle of motion which the more rigid geometers profess to reject; and not only motion, but motion "both ways,” that is, from both extremities of the lines, and continued indefinitely

both ways 66 ever so far." Now if motion is assumed as a

postulate of action, the possibility of performing which is selfevident; and farther, if the same notion is involved in the three postulates of, joining two points, producing a straight line, and drawing a circle, it is not easy to see upon what principle it can be excluded in the definition of straight lines in the same plane which are inclined to each other; and that this should be the case is the more wonderful, and the more to be regretted, that its admission there removes much of the labour and more of the difficulty of elementary geometry.

It may be said that the words "inclined to and from each other" stand in need of explanation; but the word "inclined"

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is as generally understood as any word in language; and if we were to give an elaborate definition of every word we make use of in our attempts to explain the principles of science, we should never be able to let anybody see one of those principles through such a misty multitude of words as would in that case envelope them all.

We may now alter the expression a little, and say, "all straight lines in the same plane which are not parallel to each other, must, if produced far enough both ways, meet each other either the one way or the other, but not both.”

They must meet when produced at those extremities where they are inclined to each other; and they must become farther apart the farther they are produced at the opposite extremities, or at those at which they are inclined from each other; but throughout the whole of their length, whatever it may be, and whether they are produced till they meet or not, their inclination to and from each other is exactly the same, and cannot vary unless one or both change in direction, and thus lose the character of straight lines. We have used the words "from" and "to" together in the course of these explanations, because they are inseparable by the very nature of the case; an inclination to the one way being as necessarily an inclination from the other way, as a road which when taken at the one end leads directly to London, leads as directly from London if we take it at the other end.

The inclination of lines to each other furnishes us with another set or kind of quantities in elementary geometry besides lines, surfaces, and solids. The measure of this inclina'tion is called an ANGLE, which means a corner; and as we have spoken of straight lines, which are also called right lines, in the same plane, the simplest angle which occurs in elementary geometry is

A PLANE RECTILINEAL ANGLE, which may be defined as

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"the corner, or opening made at the point where they meet by two straight lines which are inclined to each other."

As the inclination and the angle which measures it are quantities, they must, like other quantities, admit of variation, that is, there may be an endless variety of angles answering to and measuring an endless variety of inclinations, just as there may be an endless variety of lengths of lines, extent of surfaces, and capacity or content of solids; but as the angle depends altogether upon the inclination of the lines, and not upon the length of any one of them or of them both, or even on the direction in which any one of them extends, an angle is not a quantity of the same kind with a line, neither is it a quantity which can be accurately expressed by any product of straight lines, as a surface is by the product of length and breadth, and a solid by the product of length, breadth, and thickness.

The four different kinds of quantities, lines, angles, surfaces, and solids (for that is the proper order in which to take them), of which we have attempted to give as clear and simple an explanation as possible, are the only quantities which are purely geometrical; and with the exception of one curved line (to be afterwards described), and the curved surfaces and solids which are founded on the line, the relations of straight lines, rectilineal angles, plane surfaces, and plane solids, include all which belongs to elementary geometry.

We have endeavoured to explain the nature of the quantities at greater length than is usually done, and in different terms, though not upon different principles; for we trust it has been shown that motion, which we have taken as an element, has been tacitly assumed by every geometer; and we feel convinced that it has been this latent principle, which works powerfully, but unconfessedly, which has made the science of geometry—a science beautifully clear in itself-so perplexing to the majority of students.

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,

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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

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