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SCIENTIFIC THOUGHT. 



Before Weierstrass, Cauchy and Eiemann had at- 

 tempted to define the vague term " function " or 

 mathematical dependence. Both clung to the graphical 

 representation so common and so helpful in analysis 

 since Descartes invented it. We have, of course, in 

 abstract science, a right to begin with any definition 

 we choose. Only the definition must be such that it 



remarkable tract on "Oscillating 

 functions," in which he drew 

 attention to the existence of func- 

 tions which admit of an integral, 

 but where the existence of a differ- 

 ential coefficient remains doubtful. 

 In fact, it appears that the question 

 as to the latter had never been 

 raised ; the only attempt in this 

 direction being that of Ampere in 

 1806, which failed (Hankel, p. 7). 

 Hankel in his original investigation 

 showed that a continuous curve 

 might be supposed to be generated 

 by the motion of a point which 

 oscillated to and fro, these oscilla- 

 tions at the limit becoming in- 

 finitely numerous and infinitely 

 small : a curve thus generated 

 would present what he called "a 

 condensation of singularities " at 

 every point, but would possess no 

 definite direction, hence also no 

 differential coefficient. The argu- 

 ments and illustrations of Hankel 

 have been criticised and found fault 

 with. He nevertheless deserves the 

 credit of having among the first 

 attempted " to gain a firm footing 

 on a slippery road which had only 

 been rarely trodden " (p. 8). In 

 this tract (which is reprinted in 

 'Math. Ann.,' vol. xx.), as well as 

 in his valuable article on " Limit " 

 (Ersch und Gruber, 'Encyk.,' vol. 

 xc. p. 185, art. "Grenze"), Hankel 

 did much to establish clearly the 

 essential point on which depends 

 the entire modern revolution in 

 our ideas regarding the foundations 



of the so-called infinitesimal cal- 

 culus ; reverting to the idea of a 

 "limit," both in the definition of 

 the derived function (limit of a 

 ratio) and of the integral (limit of 

 a sum) as contained in the writings 

 both of Newton and Leibniz, 

 but obscured by the method of 

 " Fluxions " of the former and the 

 method of " Infinitesimals " of the 

 latter. Lagrange and Cauchy had 

 begun this revolution, but it was 

 not consistently and generally 

 carried through till the researches 

 of Riemann, Hankel, Weierstrass, 

 and others made rigorous defini- 

 tions necessary and generally ac- 

 cepted. It is, however, well to 

 note that in this country A. de 

 Morgan very early expressed clear 

 views on this subject. Prof. Voss, 

 in his excellent chapter on the 

 Differential and Integral Calculus 

 ('Encyk. Math. Wiss.,' vol. ii. i. p. 

 54, &c.), calls the later period the 

 period of the purely arithmetical 

 examination of infinitesimal con- 

 ceptions, and says (p. 60), " The 

 purely arithmetical definition of 

 the infinitesimal operations which 

 is characteristic of the present 

 critical period of mathematics has 

 shown that most of the theorems 

 established by older researches, 

 which aimed at a formal extension 

 of method, only possess a validity 

 limited by very definite assump- 

 tions." Such assumptions were 

 tacitly made by earlier writers, but 

 not explicitly stated. 



