postulates until we have answered all these questions. Even in pure mathematics one has, in the first instance, to proceed tentatively, to venture on the work of drawing inferences from what seem to be plausible postulates before one can pass a verdict on the merits of the postulates themselves. The consequence of this tentative character of our inquiries is that, so far as there is a difference between Philosophy and Science at all, it is a difference in _thoroughness_. The more philosophic a man's mind is, the less ready will he be to let an assertion pass without examination as obviously true. Thus Euclid makes a famous assumption--the 'parallel-postulate'--which amounts to the assertion that if three of the angles of a rectilinear quadrilateral are right angles, the fourth will be a right angle. The mathematicians of the eighteenth and early nineteenth centuries, again, generally assumed that if a function is continuous it can always be differentiated. A comparatively unphilosophical mind may let such plausible assertions pass unexamined, but a more philosophical mind will say to itself, when it comes across them, 'You great duffer, aren't you going to ask _Why_?' Suppose that, by way of experiment, I assume that the fourth angle of my quadrilateral will be acute, or again obtuse, will the body of conclusions I can now deduce from my set of postulates be free from contradictions or not? If I really give my mind to the task, cannot I define a continuous function which is _not_ differentiable? The raising of the first question led in fact to the discovery of what is called 'non-Euclidean' geometry, the raising of the second has banished from the text-books of the Calculus the masses of bad reasoning which long made that branch of mathematics a scandal to logic and led distinguished philosophers--Kant among them--to suspect that there are hopeless contradictions in the very foundations of mathematical science. Now, the effect of such careful scrutiny of first principles is not, of course, to upset any conclusions which have been correctly drawn from a set of premisses. All that happens is that the conclusion is no longer asserted by itself as a truth; what is asserted is that the conclusion is true _if_ the premisses are true. Thus we no longer assert the 'theorem of Pythagoras' as a categorical proposition; what we assert is that the theorem follows as a consequence from the assertion of some half-dozen ultimate postulates which will be found on analysis to be the premisses of Euclid's proof of his forty-seventh proposition.