TRANSACTIONS OF SECTION A, 525 



nence by the imaginations of great intellects, were fraught with difficulty. There 

 seemed to be an atjsence, partial or complete, of the law and order that investi- 

 gators had been accustomed to find in the wide realm of continuous quantity. 

 The country as explored was found to be full of pitfalls for the unwary. JIany 

 a lesson concerning the danger of hasty generalisation had to be learnt and 

 taken to heart. Many a false step had to be retraced. Many a road which a 

 first reconnaissance had shown to be straight for a short distance, was found on 

 further exploration, to suddenly change its direction and to break up into a 

 number of paths which wandered in a fitful manner in country of increasing 

 natural difficulty. There were few vanishing points in the perspective. Few, 

 also, and insignificant were the peaks from which a general view could be 

 gathered of any considerable portion of the country. The surveying instruments 

 were inadequate to cope with the physical characters of the land. The province 

 of the Theory of Numbers was forbidding. Many a man returned empty-handed 

 and baffled from the pursuit, or else was drawn into the vortex of a kind of 

 Maelstrom and had his heart crushed out of him. But early in the last century 

 the dawn of a brighter day was breaking. A combination of great intellects — 

 Legendre, Gauss, Eisenstein, Stephen Smith, &c. — succeeded in adapting some 

 of the existing instruments of research in continuous quantity to effective use 

 in discontinuous quantity. These adaptations are of so difficult and ingenious 

 a nature that they are to-day, at the commencement of a new century, the 

 wonder and, I may add, the delight of beholders. True it is that the beholders 

 are few. To attain to the point of vantage is an arduous task demanding alike 

 devotion and courage. I am reminded, to take a geographical analogy, of the 

 Hamilton Falls, near Hamilton Inlet, in Labrador. I have been informed that 

 to obtain a view of this wonderful natural feature demands so much time and 

 intrepidity, and necessitates so many collateral arrangements, that a few years 

 ago only nine white men had feasted their eyes on falls which are liner than 

 those of Niagara. The labours of the mathematicians named have resulted 

 in the formation of a large body of doctrine in the Theory of Numbers. Much 

 that, to the superficial observer, appears to lie on the threshold of the subject 

 is found to be deeply set in it and to be only capable of attack after problems 

 at first sight much more complicated have been solved. The miras'e that 

 distorted the scenery and obscured the perspective has been to some" extent 

 dissipated ; certain vanishing points have been ascertained ; certain elevated 

 spots giving extensive views have been either found or constructed. The point 

 I wish to urge is, that these specialists in the Theory of Numbers were successful 

 for the reason that they were not specialists at all in any nan-ow meaning of the 

 word. Success was only possible because of the wide learning of the investigator ; 

 because of his accurate knowledge of the instruments that had been made elective in 

 other branches ; and because he had grasped the underlying principles which caused 

 those instruments to be effective in particular cases. I am confident that many a 

 worker who, from the supposed extremely special character of his researches 

 has been the mark of sneer and of sarcasm, would be found to have devoted the 

 larger portion of his time to the study of methods which had been available in 

 other branches, perhaps remote from the one which was particularly attracting 

 his attention. He would be found to have realised that analogy is often the 

 finger-post that points the way to useful .advance ; that his mind had been trained, 

 and his work assisted, by studying exhaustively the succe.sses and failures of his 

 fellow- workers. But it is not only existing methods that may be available in a 

 special research. 



Furthermore, a special study frequently creates new methods which may be 

 subsequently found applicable to other branches. Of this the Theory of Numbers 

 furnishes several beautiful illustrations. Generall)', the method is more important 

 than the immediate result. Though the result is the offspring of the method, the 

 method is the offspring of the search after the result. The Law of Qaudratic 

 Reciprocity, a corner-stone of the edifice, stands out not only for the influence it hag 

 exerted in many branches, but also for the number of new methods to which it 

 has given birth, which are now a portion of the stock-in-trade of a mathematician. 



