any Knowledge of Self. That is the last word of philosophy. The impossibility of any such existence is clearly proved by the greatest thinker of our age.

But we have not even yet exhausted everything; there are still certain so-called abstract truths left-truths independent of material forms -which must surely always be, and always have been. A rather ingenious little book was published some years ago by an anonymous author, called The Two Kinds of Truth.' There is the Empirical or Particular Truth, and the Universal or Necessary Truth, and it is very important carefully to distinguish between them, never to mistake one for the other. For example, the law of gravitation comes under the former category. It is true, as far as our observations go, that all material bodies attract with a force varying directly as their mass, and indirectly as the squares of their distances. But there appears no intrinsic necessity why they should always do so. Surely it is legitimate to imagine a Universe where equilibrium was maintained by some other process. Gravitation is, after all, only

an arbitrary mechanical contrivance to bring about certain ends which might conceivably be brought about by some other contrivance, equally arbitrary and equally mechanical. But there are other truths which are absolute, universal, eternal, necessary; and of these are the truths of geometry. There never has been, and there never can be, a circle the radii of which are unequal, or a right-angled triangle in which the square of the hypotenuse is not equal to the two squares on the subtending sides. We need not bother ourselves with Mr. J. S. Mill's contention that there are no objective realities corresponding to our subjective mathematical conceptions. It is possible, of course, as he says, that in the material world (so called) 'there exist no points without magnitude, no lines without breadth or perfectly straight, no circles with all their radii exactly equal, or squares with all their angles perfectly right.' This is no more than saying that we possess no instruments sufficiently delicate to form mathematical figures which, under the most transcendently powerful microscope, should exhibit no irregularity of line;

and it is, at any rate, as possible that accident should produce a perfect right-angle or an equalradii'd circle as an imperfect and irregular one. But, of course, the main answer is obvious. Show me a figure, and say, 'There is a circle, and its radii are not exactly equal,' and I reply at once, 'Then it is not a circle.' A circle with unequal radii, a square whose angles are not true right angles, is simply a contradiction in terms. Two parallel lines in the same plane, even if indefinitely prolonged, can never meet; were two lines under such conditions eventually to intersect each other, they would not be parallel. One might as well talk about a round square, or a present past, or a footless stocking without a leg. Here, then, one would think, surely we cannot escape from a Belief in Something. The abstract truths of mathematics are absolute and eternal truths. Whatever else we are able to explain away, the axioms and postulates of Euclid are irrefutable.

And yet even they may be challenged. There is a school of mathematicians who traverse the very foundations of the Euclidean system, and

call in question the actuality of Euclidean space. Now, space is popularly supposed to be flat, as far as such an attribute can be predicated of mere extension-flat in the sense that there is nothing in it to prevent anybody from drawing a perfectly straight line between any two given points, a line which shall lie flat or evenly between such points. But cases have been

known in which astronomers have observed displacements in the line of vision; in which, on attempting to ascertain the parallax of a certain star, the angle of intersection between the lines of vision was found to be different from that required by the known facts and laws of optics and astronomy. How is this to be accounted for? Well, there are those who tell us that it is because space is not flat, but curved. This curvature of space is a favourite hypothesis of Transcendental Geometers; and if it is a true. hypothesis, there can be no such thing, even in geometry, as a straight line. Every line we call straight must be curved, and therefore two so-called straight lines, being really curved, may very easily enclose a portion of space, and two

apparently parallel lines may actually meet. According to Lobatchewsky, it is conceivable that there may be two non-parallel lines in the same plane which on being produced indefinitely would yet never meet; and it has even been suggested that the curvature of space itself is not regular, but that space may be compared to a piece of paper which has been lightly crumpled up, leaving some parts flat-or nearly so-others slightly curved, and others, again, almost angular. As, then, in the rush of the Solar System through this irregularly constituted space we may find ourselves now in flat space and now in curved, it follows that our astronomical observations and mathematical measurements may vary; that we have no constant basis for either; and that whereas in flat space an object, say, a million miles off would appear a million miles off, in curved space there would be an appreciable difference between its real and apparent distance—a difference equal to that between the base or third side of a triangle and the sum of its subtending sides. In this case Euclid must be laid on the shelf. If space is