the normal acuity of perception. In infancy and early childhood the individual's mental development is largely bound up in learning to use his organs of sense, in learning to perceive as well as to act. Similarities and dissimilarities which are not perceptible to him at one stage of development become perceptible at a later stage. Admitting some such process of exploration and perception of linear shape as we briefly suggested above, it then becomes perfectly intelligible, as we shall presently see, that a judgement such as Mill's and Cayley's, as to the nonexistence of straight lines in nature, is compatible with an experiential origin of the conception of straightness.

It must be borne in mind that we carry with us no recollection of the long laborious process of learning to perceive. Thus it is not until we begin to philosophize that the question presents itself whether the entities of geometry exist in nature. The individual who is either too practical or too busy to philosophize accepts without further reflection, or has accepted at some time without reflection, that the edge of a mathematical ruler is perfectly straight. He is then invited to look at the edge of such a ruler through a powerful microscope, and he perceives it to be not perfectly straight. Is it necessary to assume that he has an a priori conception of straightness which enables him to conclude that even the edge of a mathematical ruler is not perfectly straight? Evidently not; he perceives the edge as not straight through the microscope by contrast with his perception of it as straight by the naked eye. The microscope does here for the mature individual, on a very large scale and in a brief moment, what, on a very much smaller scale and in a much longer period of time, was effected for him in his immaturity through the training of his visual apparatus. An untrained thinker, for the first time making this experience with the ruler and microscope, would not improbably assert that the edge of the ruler is quite straight, though it is 'rough'; and the conception of uniformity of direction which we all possess would at once enable us to understand what he meant by the assertion. The judgement would of course be a rash one, for we could feel no assurance whatever that the edge of the ruler might not be after all slightly curved, and that the microscope may be as ineffective to make us sensible of the curvature as the naked eye is to make us aware of the roughness.

From the moment we realize, be it even only in a perfunctory manner, how the conception of straightness can be elaborated out of the material of experience, we are no longer intellectually justified in assigning to this conception an a priori origin as explicative of our possessing it. To do so would be very much like persisting in the explanation of the existence of natural species by the hypothesis of special acts of creation, after Darwin and his followers have made it plain that these species may perfectly well have originated not supernaturally, but naturally. But whoever admits the straight line and the other 'purely imaginary objects' of geometry to originate in experience, rejects the theory that geometry is an a priori science; like all other 'pure' sciences, it is a process of reasoning about abstractions from experience, about abstract ways of regarding the concrete. Admit that in nature there exist no straight lines, planes, spheres, &c. ; you admit also that in nature there exist no rigid levers, inextensible strings, frictionless pulleys, &c. These latter notions also originate in experience and by processes very similar to those which yield the geometrical entities. Elementary theoretical mechanics is a process of reasoning about these and similar abstractions, just as geometry is a process of reasoning about the others.

The consciousness of that break, in the continuity of linear shape, which arrests attention in the course of the exploration of bodily edge or outline, develops into the conception of the angle, rectilinear and curvilinear; and, in accordance with the view here advocated of the origin of the fundamental geometrical conceptions, those of the rectilinear angle and of difference of direction complete and establish one another so soon as the percipient has correlated the straight line with uniformity of direction. Hence also there is, properly speaking, neither rectilinear nor curvilinear angle: the 'break' is rectilinear, or curvilinear, or partly one and partly the other, while the angle is the difference of initial direction, or, if the lines are straight, the inclination of the one to the other. But, the straight line and uniformity of direction having been correlated and in a loose sense identified, we find the so-called curvilinear angle defined as formed by the meeting of the tangents to two curved lines at their point of intersection (cf. The Century Dictionary, under 'Angle ').

Clifford, in his Common Sense of the Exact Sciences, tells us that 'shape is a matter of angles', and illustrates this proposition in various ways. The statement is manifestly inapplicable to linear and surface shape. But we might say, employing the term figure in a restricted sense, that figure is a matter of angles. In this sense the solid has no shape, other than that or those of its surfaces and edges, but it may have figure. The point, however, is of little importance; and it would certainly be inconvenient thus to restrict the meaning of figure. Euclid's definitions of the line, surface, and solid are definitions of dimension rather than of figure.

We may remark, however, with reference to Euclid's definition of the line, that it appears to involve a conflict of opinion between him and some modern geometers. The line, says Euclid, is length without breadth, and-so we must read it-without thickness. Whatever we may think of this as a definition of the line (it shows us, of course, that he is defining an abstraction '1), we can at least gather from it what was Euclid's conception of length. Length, for Euclid, is a feature or attribute of all lines. According to this view the notion of length is not what it is, e.g. for Mr. Russell, 'originally derived from the straight line, and extended to other curves by dividing them into infinitesimal straight lines.'2 Euclid's definition is quite general and precedes that of the straight line. What Mr. Russell describes as extending the notion of length from the straight line to other curves, is the notion of comparing a straight line and a curve in length by the intermediation of a broken line, the comparison becoming more and more precise as the number of breaks increases. But it is obviously a pre-condition of this notion, that the straight line, the broken line, and the curve are conceived under this common attribute, viz. length; we could not otherwise think of thus comparing them. The notion is, at bottom, the breaking of the straight line so that it becomes more and more nearly congruent with the curve. In the limit' of this process the length of the broken line is the length of the curve. No doubt,

1 I do not by any means assert, however, that for Euclid the geometrical entities which he defined were nothing but abstractions. Euclid was by philosophical profession a Platonist, and, for aught I know, the geometrical entities thus defined may have been for him, as they were for Cayley, among the 'only realities'.

2 The Foundations of Geometry, p. 17.

but 'in the limit' of this process the broken line is just the straight line bent into congruence with the curve. It is obviously because we conceive all lines, whether perceived or imagined, under this common attribute which we call length, that we have been led to the invention of the tape as well as of the wand. That the straight line, being the simplest shape of line and unique among linear shapes in the possession of a proper name, should have become in geometry the standard to which all other linear shapes are referred for length was inevitable, and is perfectly compatible with the fact that the notion of length is necessarily involved in that of the line, while the derivation of the notion of length from that of the straight line in particular, and its extension to other lines, is incompatible with that fact.

Length being a common, but simple and unanalysable attribute of lines, it follows that linear shape, with the exception of the straight shape, which involves uniformity of direction, must be analysable into relations of length and change of direction. In other words, non-straightness or curvature of line is, in analysis, a function of length and change of direction. But change of direction in the motion of a point describing a line is itself a complex notion, unless we confine the moving point to a plane. In this simple case differences of linear shape, whether we compare distinct lines, or different parts of the same line, are definable as differences in the rate of change of direction, length being the independent variable. The analysis of plane linear shape thus closely resembles that of velocity. Just as we have velocity, uniform, variable, and 'at an instant', so we have linear shape or curvature, uniform, variable, and at a point'. The usual geometrical way of analysing curvature of line springs from the customary definition of the circle, and is no doubt simpler in expression, but it hides from us rather too easily that the conception of change of direction is involved in that of curvature of line, and where the curvature is variable does not any more than the former avoid the notion of limit, which is contained in that of the circle of curvature 'at a point'.

The conclusion which, in terminating this chapter, I wish particularly to emphasize, is that the conceptions of direction and of length are fundamental in geometry, even where, as in projective geometry, it is not our intention to investigate either general or particular relations of direction or of length.

CHAPTER XII

DEFINITIONS AND AXIOMS IN GEOMETRY

Self-evidence and the object of Demonstration.-Discordant views regarding the distinction between Definitions and Axioms.-The Assumptions alleged to be hidden in Euclid's Definitions.-The Sense in which Geometrical Entities may be said to exist.—Irrelevance of Assumption to this sense of existence.-Mr. Poincaré and Professor Klein on the nature of Geometrical Axioms.—Similarity of Klein's views and those of Cayley.— Euclid's geometrical Axioms and Postulates.-Alleged distinction between the infinitude and the unboundedness of Space.-Some of the propositions usually classed as geometrical axioms are definitions of geometrical abstractions, in regard to which neither assumption nor convention has any rele

vance.

'EVERY conclusion rests upon premisses. These premisses are either self-evident and require no demonstration, or else they can be established only by derivation from other propositions. Since we cannot thus proceed ad infinitum, every deductive science, and geometry in particular, must be founded upon a certain number of indemonstrable axioms.'1

Robert Simson in the Notes to his edition of Euclid's Elements, tells us that

Proclus, in his commentary, relates that the Epicureans derided Prop. 20' (any two sides of a triangle are greater than the third side)' as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third; but the right answer to this objection against this and the 21st, and some other plain propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated.'

I do not know that Simson's answer is any better than that of Proclus. Some modern geometers reject, or believe that they reject, all appeal to intuition in geometry. 'L'intuition ne doit avoir aucune part réelle dans les raisonnements géométriques ... ceux-ci, pour être rigoureux, doivent être purement logiques,' 1 Poincaré, La Science et l'Hypothèse, p. 49.