37° THE POPULAR SCIENCE MONTHLY 



obvious that they are called self-evident, and the rest of his work consists of 

 subtle deductions from them. The teaching of languages, at any rate as ordi- 

 narily practised, is of the same general nature — authority and tradition furnish 

 the data, and the mental operations are deductive." It would seem that from 

 the above somewhat singularly juxtaposed paragraphs that, according to Pro- 

 fessor Huxley, the business of a mathematical student is from a limited number 

 of propositions (bottled up and labelled ready for future use) to deduce any 

 required result by a process of the same general nature as a student of language 

 employs in declining and conjugating his nouns and verbs — that to make out a 

 mathematical proposition and to construe or parse a sentence are equivalent or 

 identical mental operations. Such an oijinion scarcely seems to need serious 

 refutation. 



Further on Sylvester says: 



We are told that "mathematics is that study which knows nothing of 

 observation, nothing of experiment, nothing of induction, nothing of causation." 

 I think no statement could have been more opposite to the undoubted facts of 

 the case; that mathematical analysis is constantly invoking the aid of new prin- 

 ciples, new ideas and new methods, not capable of being defined by any form of 

 words, but springing direct from the inherent powers and activity of the human 

 mind, and from continually renewed introspection of that inner world of thought 

 of which the phenomena are as varied and require as close attention to discern 

 as those of the outer physical world, . . . that it is unceasingly calling forth the 

 faculties of observation and comparison, that one of its principal weapons is 

 induction, that it has frequent recourse to experimental trial and verification, and 

 that it affords a boundless scope for the exercise of the highest efforts of 

 imagination and invention. 



Lagrange . . . has expressed emphatically his belief in the importance to 

 the mathematician of the faculty of observation ; Gauss has called mathematics 

 a science of the eye . . . ; the ever to be lamented Eiemann has written a thesis 

 to show that the basis of our conception of space is purely empirical, and our 

 knowledge of its laws the result of observation, that other kinds of space might 

 be conceived to exist subject to laws different from those which govern the 

 actual space in which we are immersed. . . . Most, if not all, of the great ideas 

 of modern mathematics have had their origin in observation. Take, for instance, 

 . . . Sturm's theorem about the roots of equations, which, as he informed me 

 with his own lips, stared him in the face in the midst of some mechanical investi- 

 gations connected with the motion of compound pendulums. 



After citing many other instances, Sylvester says: 



I might go on, were it necessary, piling instance upon instance, to prove the 

 paramount importance of the faculty of observation to the progress of mathe- 

 matical discovery. Were it not unbecoming to dilate on one's personal experi- 

 ence, I could tell a story of almost romantic interest about my own latest 

 researches in a field where Geometry, Algebra, and the Theory of Numbers melt 

 in a surprising manner into one another, . . . which would very strikingly illus- 

 trate how much observation, divination, induction, experimental trial, and veri- 

 fication, cavisation, too (if that means, as I suppose it must, mounting from 

 phenomena to their reasons or causes of being), have to do with the work of the 

 mathematician. In the face of these facts, which every analyst in this room 

 or out of it can vouch for out of his own knowledge and personal experience, 

 how can it be maintained, in the words of Professor Huxley, who, in this 

 instance, is speaking of the sciences as they are in themselves and without any 

 reference to scholastic discipline, that Mathematics "is that study which knows 



