Page:Mind (New Series) Volume 15.djvu/524

 510 HUGH MACCOLL : point of the curve between A and B the radius of curvature is infinite, and the curvature consequently infinitesimal. Even an infinite section AB may be called virtually straight A ~R when is infinitesimal, E being any of the radii of cur- vature ; x for in this case the infinity AB (infinite in regard to any finite unit) is infinitesimal compared with the infinity E. The symbolic and linguistic conventions here proposed would, I think, greatly increase the logical accuracy of modern geometry without in the least impairing its great power as a practical instrument of discovery and research. 9. Metaphysicians sometimes ask whether space the actual space of our perceptive experience the space filled with the entity called matter, or with the entity called ether, or with both is really infinite. Considering we can give no satisfactory definition either of 'matter' or of 'ether,' the question hardly admits of a clear and intelligible answer, or, at any rate, of any answer that logicians, metaphysicians, and physicists would be all likely to accept. The purely ideal space of the mathematician is far easier to understand, and this space must, I think, be pronounced infinite that is to say, infinite in the sense of the word explained in 3 for the simple reason that the opposite supposition plunges us at once into logical contradictions. We can all, without any conflict of opposing concepts, imagine a sphere whose radius is too large to be expressible (whatever be our conventional notation) in terms of finite ratios alone, and that sphere is, by our very definition, infinite. And we cannot stop there. By our definitions and conventions also, it follows that Hj + H., = H 3 ; that the sum of the real infinities Hj and H., make a third real infinity H 3 greater that either. Similarly we get H! + H 2 + H 3 = H 4, and so on for ever. But we cannot, without further data, assert H : - H 2 = H 3 ; for Hj - H may = F 17 or may = 0, since neither the supposition H T = H 2 + F, nor the supposition B^ = H. 2, involves any formal inconsistency. 10. It may be objected that the definitions which I here propose of the finite, the infinite, and the infinitesimal are quite arbitrary. To this I reply, firstly, that all definitions are more or less arbitrary ; secondly, that, however arbitrary my definitions may be, they are mutually consistent, and in 1 By this is meant that -, -, -, etc., are respectively infinitesimal R! Ra Rg for the separate points Pj, P 2, P 3 , etc., between A and B. The infinities HI, RQ, RS, etc., are generally, though not necessarily, unequal.