418 On a particular Differential Equation. 



must be retained, those which aie introduced by the other in- 

 verse factors niayindifferently be retained or rejected. There are 

 two or perhaps three distinct fallacies in Mr. Bronwin's reason- 

 ings on these points ; but the remarks alreatly offered, — offered, 

 I trust, in no captious spirit, — may be deemed sufficient. 



It is indeed with sincere pleasure that I quit a topic, upon 

 which, had my reputation alone been concerned, I would 

 willingly have been spared the necessity of writing, and ac- 

 knowledge that upon some other points Mr. Bronwin has 

 added to the completeness of the investigation. To each of 

 the two cases already determined, he has added anotlier form 

 of solution ; and the value which lie has assigned to tt,,,, though 

 not quite the most general*, is much more general than the one 

 which I had contented myself with employing. 



It is further but just to Mr. Bronwin to remark, that in 

 questions like that which he has undertaken to examine, far more 

 than the ordinary caution which the mathematician is bound 

 to exercise appears to be demanded. A settled conviction 

 that there is no anomaly which may not be explained, no ex- 

 ception but which may and which ought to be referred to some 

 governing principle, with the corresponding habit of patient 

 research and cautious inference, is essential to the success of 

 this class of speculations. Dismissing, however, the more 

 immediate subject of these comments, I cannot but observe 

 upon the fact as somewhat remarkable, that in pure mathe- 

 matics, controversy and misunderstanding should be so rife, 

 as lor some years they appear to have been. It would, I con- 

 ceive, be interesting to inquire into the causes of a state of 

 things which, judging it priori, we should so little expect. One 

 reason is undoubtedly to be found in the unmeasured capabi- 

 lities of the modern analysis for the expression of general 

 theorems, and in the practically frequent employment of ana- 

 logy and induction, especially as suggestive aids, in contra- 

 distinction to the purely deductive processes and more limited 

 conclusions of the ancient geometry. But it may be doubted 

 whether this is a sufficient explanation of the fact in question. 

 A far more inffuential cause is, I believe, to be found in 

 the almost entire absence of any direct study of the laws of 

 correct reasoning in connexion with the practical discipline of 

 modern science. But this is a topic which I do not venture 

 upon the present occasion more fully to discuss. 

 I remain. Gentlemen, 



Your obedient faithful Servant, 



Lincoln, April 8, 1848. George BoolE. 



* Even of itie kind; there also exists, as I discovered shortly after pub- 

 lishing the original paper, another value of a quite distinct kind. 



