in space may be defined by the proportion in which motion in that direction is composed of motion in the directions up and down, right and left, and fore and aft, respectively.

We have it now in our power to explain the fact formerly observed, that the same position may be determined by tracks of a wholly different description, from the same starting point. If we suppose the motion in each elementary portion of one of two different tracks from one point to another, to be resolved in the direction of the three transverse axes, the motion in the entire track will be equivalent to the aggregate motion in the direction of each of the three axes. In like manner, the motion in the other track may be resolved into a certain amount of motion in the direction of the same three axes; and, in this condition, will admit of direct comparison with the aggregate motion in the former track. When the motion in the direction of each of the three axes is of like extent in either track, the entire spaces traversed will be the same in respect of distance and direction, and the same position will be attained in both cases.

The position of a point may now be defined as the relation depending on the character of the space by which it is separated from a given point in respect of distance and direction; whence it follows, that points identical in position lie at a like distance from a point antecedently known, in

whatever direction their respective distances may

be compared.

Having thus supplied the first great want in the premises of the ordinary system of geometry, by a thorough investigation of the relation of position, we shall proceed to construct definitions of the elementary species of geometrical figure on the principles established in the foregoing inquiry.

We have seen that the shape of a line or nature of the track pursued by a point in motion, depends upon the direction of the motion, or of the line at the points successively brought under notice in the apprehension or imagination of the entire line. The very conception then of linear figure, supposes the capacity of comparing the direction from one instant to another, in the track of motion. The moment we lose count of our direction, as in wandering in a wood or in the streets of a crowded city, we lose all knowledge of the track we are pursuing, as completely as if we were carried along in the cabin of a ship or in a railway carriage.

We may then suppose a point to move for any extent in the same direction; or, after moving for a certain extent in a given direction, it may be supposed to diverge for a while in a track of any other description and again to return to the original direction. In the former case, the point will move in a straight line; in the latter (neglecting

that part of the path traversed in the intermediate period), it will move first in one straight line and afterwards in a second one parallel to the first.

A straight line may, accordingly, be defined as a line lying throughout in the same direction, or a line passing through each successive point in space situate in a certain given direction from a given point. In like manner, the definition of parallel straight lines will be, straight lines lying in the same direction in a system and not forming parts of the same straight line.

Here it will be observed, that the characters of straightness and parallelism, each of them attributes of the entire line, are reduced to the single relation of identity of direction, a character of each infinitesimal element of the line, and a real advance in analysis is embodied in the proposed definitions.

If the fundamental analysis of a plane had been equally obvious, it is probable that little difficulty would have arisen respecting the validity of the former two, but as long as the necessity of resorting to premises of a description other than definitions remained, it would be open to doubt with which of the actual premises the blame of failure ought to lie. Thus the question began to occur, What is direction? Is it not simply the position of a certain straight line, and is not relative direction fundamentally measured by the angle inter

cepted between straight lines in the directions compared? Does not, therefore, the idea of direction rest upon that of a straight line rather than vice versâ? Nor was there any escape from the dilemma until the notion of relative direction was placed, as in the foregoing inquiry, on a basis independent of angular magnitude.

In our system, the objection meets with a ready answer. Direction is a relation incapable of logical analysis, designating the mode in which motion admits of variation, and of which it exhibits a definite phase, at every point in the track pursued; and the relation between two directions is fundamentally measured by the proportion in which distance in the second admits of resolution into distance in the first, and in a transverse or wholly different direction respectively.

The notion of a plane is doubtless originally derived from the experience of a solid surface of uniform inclination, that is to say, a surface whose absolute resistance to motion is everywhere in the same direction. In every direction transverse to this fundamental direction, the surface may be freely traversed, while all motion in the direction of the resistance is opposed by the solid substance of the body. Thus the motion of a point along a solid plane will be limited by the sole condition of a total absence of motion in a certain given direction, and conversely, if a point be supposed

to move in a track the direction of which is everywhere transverse to a certain constant direction, it will pass through a series of positions related to each other as those successively traversed by a point moving on a solid plane. Thus a plane may be defined as a surface passing through all the the points which can be reached from a given point by motion transverse to a given direction, called the normal to the plane. From such a definition it is obvious, that the series of straight lines diverging from a point in directions transverse to a given direction will be included throughout their entire length in the same plane; and as such a series includes directions in every possible relation which one direction can bear to another, it follows, that any two straight lines meeting in a point may be included in a single plane.

Let CA be a straight line pointing to the left, CB transverse to C A, and let a moveable straight line, CP, be supposed to revolve round C in the plane of CA and C B from left to right, passing successively through every direction intermediate between CA and C B. Then the portion of plane surface intercepted between CA and CP will continually increase in magnitude in the part abutting upon the point C as the arm CP sweeps over a fresh segment of the plane in its progress from left to right. In other words, the angular distance, as it is called, between CA and C P, or the