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SCIENTIFIC THOUGHT. 



the seventeenth century afforded two grand occasions 

 for such progress, and the creation through it of novel 

 mathematical ideas. The translation of geometrical con- 



secoud class is the following : 

 having defined the symbols 

 dy d?y d n y 



dx dx 2 ' ' dx 11 ' 

 an operation suggests itself in the 

 inverse order, the indices or their 

 reciprocals (inversions) being taken 

 negatively. Can any meaning be 

 attached to these latter symbols ? 

 Further, if the operation denoted 

 by going on from one of these 

 symbols to the next is known and 

 feasible, how can the inverse oper- 

 ation be carried out ? In the first 

 class of problems we proceed from 

 an intuitively given order to a 

 purely logical order, and have in 

 the sequel to go back from the 

 purely logical order to an intuitive 

 order of ideas. In the second 

 case, having followed a certain 

 logical order, we desire to know 

 what the inversion of this order will 

 produce and how it can be carried 

 put. The view that the direct and 

 indirect processes of thought form 

 the basis of all mathematical 

 reasoning, and an alternation of 

 the two the principle of progress, 

 has been for the first time con- 

 sistently expounded by Hermann 

 Hankel in his 'Theorie der Com- 

 plexen Zahlen - Systeme, ' Leipzig, 

 1867. But it had already been 

 insisted on by George Peacock in 

 his "Report," &c., contained in 

 the 3rd vol. of the ' Reports of 

 the Brit. Assoc.,' 1833, where he 

 says (p. 223): "There are two 

 distinct processes in Algebra, the 

 direct and the inverse, presenting 

 generally very different degrees of 

 difficulty. In the first case, we 

 proceed from defined operations, 

 and by various processes of de- 

 monstrative reasoning we arrive 

 at results which are general in 



form though particular in value, 

 and which are subsequently gen- 

 eralised in value likewise ; in the 

 second, we commence from the 

 general result, and we are either 

 required to discover from its form 

 and composition some equivalent 

 result, or, if defined operations 

 have produced it, to discover the 

 primitive quantity from which those 

 operations have commenced. Of 

 all these processes we have already 

 given examples, and nearly the 

 whole business of analysis will 

 consist in their discussion and 

 development, under the infinitely 

 varied forms in which they will 

 present themselves." 



It is extraordinary how little in- 

 fluence this very interesting, com- 

 prehensive, and up - to - date re- 

 port on Continental mathematics, 

 including the works of Gauss, 

 Cauchy, and Abel, seems to have 

 had on the development of English 

 mathematics. But the latter have 

 through an independent movement 

 viz., the invention of the 

 Calculus of Operations led on 

 to the radical change which has 

 taken place in recent mathematical 

 thought. This change, which can 

 be explained by saying that the 

 science of Magnitude must be 

 preceded by the doctrine of Forms 

 or Relations, and that the science 

 of Magnitude is only a special 

 application of the science of Forms, 

 was independently prepared by 

 Hermann Grassman, of whom 

 Hankel says (loc. cit., p. 16) : "The 

 idea of a doctrine of Forms which 

 should precede a doctrine of Mag- 

 nitude, and of considering the 

 latter from the point of view 

 of the former, . . . remained of 

 little value for the development 



