Even the simple axiom that extension is boundless (boundlessness, since Riemann and the theory of curved space, is to be distinguished from endlessness) at once contradicts the essential character of all immediate perception, in that the latter depends upon the existence of light-resistances and _ipso facto_ has material bounds. But abstract principles of boundary can be imagined which transcend, in an entirely new sense, the possibilities of optical definition. For the deep thinker, there exists even in the Cartesian geometry the tendency to get beyond the three dimensions of _experiential_ space, regarded as an unnecessary restriction on the symbolism of number. And although it was not till about 1800 that the notion of _multi-dimensional space_ (it is a pity that no better word was found) provided analysis with broader foundations, the real first step was taken at the moment when powers— that is, really, logarithms—were released from their original relation with sensually realizable surfaces and solids and, through the employment of irrational and complex exponents, brought within the realm of function as perfectly general relation-values. It will be admitted by everyone who understands anything of mathematical reasoning that directly we passed from the notion of a³ as a natural maximum to that of a^{_n_}, the unconditional necessity of three-dimensional space was done away with. Once the space-element or point had lost its last persistent relic of visualness and, instead of being represented to the eye as a cut in co- ordinate lines, was defined as a group of three independent numbers, there was no longer any inherent objection to replacing the number 3 by the general number _n_. The notion of dimension was radically changed. It was no longer a matter of treating the properties of a point metrically with reference to its position in a visible system, but of representing the entirely abstract properties of a number-group by means of any dimensions that we please. The number-group—consisting of _n_ independent ordered elements—is an _image_ of the point and it is _called_ a point. Similarly, an equation logically arrived therefrom is _called_ a plane and is the image of a plane. And the aggregate of all points of _n_ dimensions is _called_ an _n_-dimensional space.[76] In these transcendent space-worlds, which are remote from every sort of sensualism, lie the relations which it is the business of analysis to investigate and which are found to be consistently in agreement with the data of experimental physics. This space of higher degree is a symbol which is through-and-through the peculiar property of the Western mind. That mind alone has attempted, and successfully too, to capture the “become” and the extended in _these_ forms, to conjure and bind—to “know”—the alien by _this_ kind of appropriation or taboo. Not until such spheres of number-thought are reached, and not for any men but the