DEVELOPMENT OF MATHEMATICAL THOUGHT. 705 



mathematical thought. Up to that time " one would 

 have said that a continuous function is essentially cap- 

 able of being represented by a curve, and that a curve 

 has always a tangent. Such reasoning has no mathe- 

 matical value whatever; it is founded on intuition, or 

 rather on a visible representation. But such representa- 

 tion is crude and misleading. We think we can figure 

 to ourselves a curve without thickness; but we only 

 figure a stroke of small thickness. In like manner we 

 see the tangent as a straight band of small thickness, 

 and when we say that it touches the curve, we wish 

 merely to say that these two bands coincide without 

 crossing. If that is what we call a curve and a tangent, 

 it is clear that every curve has a tangent ; but this has 

 nothing to do with the theory of functions. We see to 

 what error we are led by a foolish confidence in what 

 we take to be visual evidence. By the discovery of this 

 striking example Weierstrass has accordingly given us a 

 useful reminder, and has taught us better to appreciate 

 the faultless and purely arithmetical methods with which 

 he more than any one has enriched our science." l 



" metaphysics and theory of the tion by Weierstrass (in the year 



fundamental conceptions in mathe- 1872, ' Trans. Berlin Academy,' re- 



matics : quantity, limit, argument, i printed in Weierstrass's ' Math, 



and function " (Tubingen). This Werke,' vol. ii. p. 71) of the proof 



work touches the borderland of 

 mathematics and philosophy, as 

 does the same author's posthumous 

 work, ' Uber die Grundlagen der 

 Erkenntniss in den exacten Wissen- 

 schaf ten ' (Tubingen, 1890), and will 

 occupy us in another place. 



1 M. Poincare in the ' Acta 

 Mathematical vol. xxii., " L'couvre 



of the existence of a continuous 

 function which nowhere possessed 

 a definite (finite or infinite) differ- 

 ential coefficient. This example 

 cleared up a point brought into 

 prominence by Riemann in his 

 posthumously (1867) published 

 Inaugural Dissertation of 1854 

 ('Werke,' p. 213). The question 



mathe"matique de Weierstrass," p. had already, following on Rie- 



6. The " test-case " referred to in mann's suggestions, been dis- 



the text consisted in the publica- cussed by Hermann Hankel in a 



VOL. II. 2 Y 



