310 



Functions ; on the Calculus of Probabilities ; on the Calculus of Princi- 

 pal Relations ; on the Argument of Abel to prove the Insolubility of 

 Equations of the Fifth Degree ; on Differences of Zero ; on Geometrical 

 ISTets in Space. Any one of these memoirs would have been sufficient to 

 make the reputation of a mathematician. 



Hamilton was gifted with a rare combination of those qualities 

 which are essential instruments of discovery. He had that fine percep- 

 tion of analogy by which the investigator is guided in his passage from 

 the known to the unknown. This is an instrument by which many 

 important mathematical discoveries have been effected. Sometimes the 

 mathematician devises some happy modification in the statement of a 

 theorem or a method, by which its application may be extended. 

 Sometimes, by analyzing different demonstrations, he even sees that a 

 particular proposition may be made the starting point from which he 

 ascends to more than one generalization. In the investigations of Ha- 

 milton - we find abundant instances of the skilful use of all the ordinary 

 expedients and instruments of inventive sagacity. But he seems, also, 

 to have possessed a higher power of divination — an intuitive perception 

 that new truths lay in a particular direction, and that patient and sys- 

 tematic search, carried on within definite limits, must certainly be 

 rewarded by the discovery of a path leading into regions hitherto 

 unexplored. Something like this was the unshaken assurance which 

 led Coluinbus to turn his back upon Europe, to launch upon the broad 

 Atlantic, and seek a ISTew World in the far-off West. 



And our illustrious countryman's diligence in research was not less 

 admirable than his prescient sagacity. No amount of labour to be in- 

 curred could deter him from entering upon the calculations by which 

 the correctness of his conjectures was to be tested. The confident 

 expectation of obtaining results instructive in one way or another re- 

 conciled him to the irksomeness of the most tedious and complicated 

 calculations. He felt that the great object to be sought, in the first 

 instance, was the discovery of the result itself ; and he trusted that, 

 once it was reached, he would be able to strike out some more direct 

 and more elegant method of investigation. His MSS., even his pub- 

 lished researches, furnish many examples of this. Once he had reached 

 the conclusion at which he had been aiming, he resumed the considera- 

 tion of the principal steps in his argument ; he interpreted them Avith 

 care ; he traced their connexion, and seldom failed to arrive at simpli- 

 fications and generalizations, which amply compensated for the labour 

 spent upon his first essays. By this habit of grappling courageously 

 with the difficulties of calculation he was distinguished from some 

 other eminent mathematicians. Averse to plunge into depths of cal- 

 culation from which they see no certain hope of emerging in the end, 

 they are tempted to expend an undue amount of intellectual energy in 

 the endeavour to force their way by a direct method to the desired 

 result. 



Whilst touching on this point, I cannot help reverting to ano- 

 ther mathematician of whom Ireland is justly proud — the late Pro- 



