acquainted with parallel lines as a system of certain figure, before we can recognise the fact, that lines in such a position may be prolonged for ever without meeting. In like manner, the form of a plane surface must previously be known as a substantive object of thought, in order to supply us with the system of straight lines by coincidence with which the planeness of the same surface is to be established under this definition.

The analysis, then, of what is fundamentally meant by the attributes of planeness, or of parallelism, as well as of straightness, is to be sought for in other quarters. In this research it must be borne in mind, that figure is considered in geometry as extended in empty space, and therefore, as marked exclusively by position, the only character by which the parts of space are distinguished. A certain point will be a point occupying a definite position in space. A figure will be conceived as a line or a surface, extending through a succession of points arranged in a certain scheme. of relative position, the enunciation of which will be the object to be aimed at in definition, and the analysis will be pushed to its utmost limits, when the definition expresses the fundamental relation of each individual point in the figure to its immediate neighbours, and consequently, to the remainder of the system.

What the fundamental relations of position

are, I have elsewhere endeavoured to shew, from a careful examination of the active process by which the knowledge of space and all the relations which it involves are originally acquired. But without at present entering upon the metaphysical inquiry, it will be seen, that the position of an object is given in the knowledge of a track by which it may be reached from a known station. When the organ employed in reaching the object is conceived as an individual point, the station from whence the motion commences, as well as the position attained at any moment, will both be single points, while the track of motion will be a mathematical line. Thus the position of a point is determined by the nature of the line by which it is united to a point antecedently known, and the identity of points is accordingly proved in geometry, by shewing the coincidence or entire identity of the lines by which the points in question are united with the same given point.

Now motion, in as far as the relations of space are concerned, admits of variation in two ways, each giving rise to the conception of an elementary attribute, or one, which can only be explained by reference to the various phases exhibited in actual existence, in the same way that colour can only be explained, as the attribute, of which white, blue, red, etc., are particular phases.

The elementary attributes of motion (and there

fore of a line, as the track of a point in motion) are: first, the longitudinal extent of the track, which keeps continually increasing from the commencement of the motion; and secondly, the direction of the motion at any given instant, a character admitting of variation in each infinitesimal-element into which the track may be divided. The nature of the entire track or figure of the line will be conceived by combining in a single act of thought the continuous succession of infinitesimal elements, each with their distinctive character of direction and distance from the origin. Thus, the position of a point will be determined by the character, in respect of distance and direction, of a line by which it is united with a point already known. But the same fixed point may be attained from a given station by tracks of a wholly different description; the same two points may be united by lines which have nothing in common except the beginning and the end. The very conception of a triangle ABC, implies the possibility of recognising the identity of the point C attained by

otion from a given station A either through the track A B, BC, or straight through AC. That is to say, the aggregate character of the broken line AB, BC, in respect of distance and direction, must be recognised as equivalent, in the determination of position, to that of the straight line A C. There must then be some fundamental

connection between distance and direction, some means of reducing distance in different directions to a common standard, in order to render possible the equivalence, in the determination of position, of different combinations of those elements.

The discrimination of the infinite variety of directions in which motion is possible from any point in space, is based on the two fundamental relations of opposition and transverseness. If, after moving through a certain distance in any one direction, we stop to contemplate, from the station so attained, the position of the point from whence we started, it will be conceived as lying at the distance traversed in attaining our present station, and in a direction, the relation of which to that of the original motion is designated by the term opposition. In other words, an object which has moved through a certain distance in a given direction will be brought back, by the same extent of motion in the opposite direction, to the position originally occupied : and the fundamental characteristic of the relation will be, that motion in a given direction is exactly nullified, in the determination of position, by the same extent of motion in the opposite direction. Thus a direction and the one opposed to it may be considered as the positive and negative modifications of a common direction.

The origin of the notion of transverseness was

argument for the possibility of attaining the end which reformers have had in view, and nothing but his predecessors' want of success to discourage the efforts of any new competitor in the same arena. The figures which form the subject of geometrical reasoning being wholly the creation of the understanding, it would seem that they can be endowed with no essential qualities except such as are derived from the plan on which the elements of figure are put together in the conception of the geometrical species. If, therefore, we were able to analyse the first step taken by the understanding in the conception of figure, and to indicate the immediate relation by which the ultimate elements of surface and of line are combined in the conception of the simplest species of figure, the propositions enouncing this primitive synthesis, together with those laying down, in like manner, the composition of the more complicated species, should constitute premises, from whence might directly be deduced every possible relation of the geometrical system. The question then arises, Have any of the proposed amendments been based on the ultimate analysis of all the species of geometrical figure?—and specially, Has the true analysis of a plane as yet been propounded?

In the complete conception of every kind of surface, each infinitesimal element of the surface must be brought successively before the mind and arranged in proper relation to the rest of the system, and whatever can be distinctly conceived