Moreover, a proof of the theorem had been accepted. It appeared in text-books, and was often to be heard in lecture rooms; nor was a shadow of a doubt suggested. For it was not merely a _demonstratio ad oculos_, but it appeared directly to our sense of intuition. And we may safely say that up to the present day no one has ever been able to _imagine_ a continuously curved line which has no tangent; no one has been able to picture even one point of such a curve at which no tangent could be drawn. Nevertheless, scientists appeared who began to entertain doubts. In the case of Riemann and Schwarz these doubts assumed a concrete form, in that they proved that certain functions are refractory at certain points. But Weierstrass was the first to make a real breach in the old belief that was so firmly rooted. He set up a function that is continuous at every point, but differentiable at no point. The graphical picture would thus have to be a continuous curve having no tangent at all. What is the appearance of such a configuration? We do not know, nor shall we presumably ever get to know. During a conversation in which this problem of Weierstrass arose, Einstein said that such a curve lay beyond the power of imagination. It must be remarked that, although the mathematical expression of the Weierstrass function is not exactly simple, it is not inordinately complex. Moreover, seeing that one such function (or curve) exists, others will soon be added to it (Poincaré mentions that Darboux actually gave other examples even in the same year that the first was discovered); there will, indeed, be found an infinite number of them. We may go still further, and say that, corresponding to each curve that has tangents, there are an infinite number that have no tangents, so that the former form the exception and not the rule. This is an overwhelming confession that shakes the foundations of our mathematical convictions, yet there is no escape. How may we apply the principle of "approximation" to these considerations? May we say that the theorem that was believed earlier is an approximation to a mathematical truth?