By a line, Euclid does not mean something long, but the abstraction LENGTH; and although our experience has never brought us acquainted with length, but in connexion with something long; yet the geometry of lines contemplates length only, the "something" in which it inheres not being recognized in this science. So surface, or superficial extension, which can never be actually presented to us in nature or in art but as the boundary of body, is considered by the geometer in the abstract, or apart from all material connexion. Solidity even, in the geometrical sense, is equally abstract. Every finite portion of space-whether actually occupied by a body or not-is called a solid; for it is with bulk or volume, only, that the geometer has to do; and he takes no cognizance of what may be comprised under, or may occupy, this bulk or volume. Every geometrical solid, like every material substance, being limited in all directions, has length, breadth, and thickness; using these terms in their ordinary acceptation. The boundary, or surface of the solid, being no part of it, can of course have but two dimensions-length and breadth. Further, the bounds of a surface, being no part of that surface, can have but one dimension-length; in other words, these bounds are lines. And finally, the limits or terminations of these, marking position only, are points. We thus see how, from the idea of a solid, we may, by successive abstractions, acquire accurate conceptions of a surface, a line, and a point; and these conceptions we shall find to be embodied in the definitions which Euclid has formally given, and which, (such is their precision,) will serve to remove whatever vagueness there may happen to be in the notions to be tested by them. The first six books of Euclid are occupied with the properties of lines and surfaces-not lines and surfaces of every variety of form-but only with those fundamental and simpler classes, of which the properties furnish the basis, and enter into the superstructure, of the doctrine of figure in general; and hence the propriety of the term "Elements of Geometry." In order to assist our minds in these enquiries about figure in the abstract, it is desirable to give a sensible representation to our geometrical conceptions; and hence arise the figures and diagrams of Geometry. But, from what we have already said the student will not confound these visible symbols with the abstractions for which they stand: he will not, for example, imagine that the properties he proves of lines, belong to the slender black marks which he employs as the sensible symbols of lengths. They do not fulfil the conditions of the definition, because they are not the things defined; but only convenient representations of them. A There is as much difference between the conceptions of the Geometer, and the diagrams by which he represents them, as there is between thought and language. From this the student will at once see that what is called graphical accuracy in his diagrams is a matter of no moment in investigating the truths of pure geometry. We cannot, by the aid of rule and compasses, draw a geometrical line, nor describe a geometrical circle; and the things which we call by these names are always supposed to be stripped of every physical circumstance, and instrumental imperfection; and to be thus brought within the restrictions of the definitions. Whatever be the extent of our graphical inaccuracy, if we only make these suppositions,necessary in all cases,-every purpose of a representation will be answered. There is, indeed, some advantage in disregarding all such unavailing attempts at graphical accuracy; inasmuch as the physical lines which the student roughly sketches, cannot but proclaim their own real character and office, in connexion with the train of reasoning he may be pursuing; they serve merely the purpose of recording, by sensible and permanent marks, the several stages of his intellectual progress:-of representing, by visible symbols, those mental abstractions which are the real objects of contemplation. THE ELEMENTS OF EUCLID. BOOK I. DEFINITIONS. I. A point has no magnitude; but position only. II. A line is length without breadth. Therefore the extremities of a line are points; and the intersection of one line with another is also a point. III. A straight line is that which lies evenly between its extreme points. The idea of a geometrical straight line may be gained from that of a physical stretched line or thread. It is distinguished from other lines by uniformity of direction. IV. A superficies or surface is that which has only length and breadth. V. Therefore the boundaries of a surface are lines. VI. A plane surface, or simply a plane, is that in which any two points being taken, the straight line between them lies wholly in that surface. The meaning of this is that a plane surface is such that whatever two points in it be taken for the extremities of a straight line, not only these extremities, but all the intermediate points, must also be posited in the surface. VII. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. It is better to say that a plane rectilineal angle, or simply a rectilineal angle, is the opening between two straight lines which meet one another. N. B. 'When several angles are at one point B, any one 'of them is expressed by three letters, of which the letter 'that is at the vertex of the angle, that is, at the point in 'which the straight lines that contain the angle meet one 'another, is put between the other two letters; and one of these two is somewhere upon one of those straight lines, 'and the other upon the other line: Thus the angle which 'is contained by the straight lines, AB, CB, is named the 'angle ABC, or CBA; that which is contained by AB, DB, 'is named the angle ABD, or DBA; and that which is con'tained by DB, CB, is called the angle DBC, or CBD; but, "if there be only one angle at a point, it may be expressed 'by a letter placed at that point; as the angle at E.' VIII. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight lines are said to be perpendicular to each other. IX. An obtuse angle is that which is greater than a right angle. 1 X. An acute angle is that which is less than a right angle. |