478 



SCIENCE. 



[Vol. II., Ko. 35. 



dent and trustee of the association ; one wlio has done 

 for it so much, and has so often attended the meet- 

 ings; whose presence among us at this meeting we 

 might have hoped for, — the president of the Royal 

 society, William Spottiswoode. It is unnecessary to 

 say any thing of his various merits. The place of his 

 burial, the crowd of sorrowing friends who were pres- 

 ent in the Abbey, bear witness to the esteem in which 

 he was held. 



I take the opportunity of mentioning the comple- 

 tion of a work promoted by the association, — the 

 determination, by Mr. James Glaisher, of the least 

 factors of the missing three out of the first nine 

 million numbers. The volume containing the sixth 

 million is now published. 



I wish to speak to you to-night upon mathematics. 

 I am quite aware of the difficulty arising from the 

 abstract nature of my subject; and if, as I fear, 

 many or some of you, recalling the presidential ad- 

 dresses at former meetings, — for instance, the resume 

 and survey which we had at York of the progress, 

 during the half-century of the lifetime of the associa- 

 tion, of a whole circle of sciences (biology, paleontol- 

 ogy, geology, astronomy, chemistry) so much more 

 familiar to you, and in which there was so much to 

 tell of the fairy-tales of science; or, at Southampton, 

 the discourse of my friend, who has in such kind terms 

 introduced me to you, on the wondrous practical appli- 

 cations of science to electric lighting, telegraphy, the 

 St. Gothard Tunnel and the Suez Canal, gun-cotton, 

 and a host of other purposes, and with the grand 

 concluding speculation on the conservation of solar 

 energy: — if, I say, recalling these or any earlier ad- 

 dresses, you should wish that you were now about to 

 have, from a difierent president, a discourse on a dif- 

 ferent subject, I can very well sympathize with you 

 in the feeling. 



But, be this as it may, I think it is more respectful 

 to you that I should speak to you upon, and do my 

 best to interest you in, the subject which has occu- 

 pied me, and in which I am myself most interested. 

 And, in another point of view, I think it is right 

 that the address of a president should be on his own 

 subject, and that different subjects should be thus 

 brought in turn before the meetings. So much the 

 worse, it maybe, for a particular meeting; but the 

 meeting is the individual, which, on evolution princi- 

 ples, must he sacrificed for the development of the 

 race. 



Mathematics connect themselves, on the one side, 

 with common life and the physical sciences; on the 

 other side, with philosophy in regard to our notions 

 of space and time, and in the questions which have 

 arisen as to the universality and necessity of the 

 truths of mathematics, and the foundation of our 

 knowledge of them. I would remark here, that the 

 connection (if it exists) of arithmetic and algebra 

 with the notion of time is far less obvious than that 

 of geometry with the notion of space. 



As to the former side: 1 am not making before you 

 a defence of mathematics; but, if I were, I should 

 desire to do it in such manner as in the ' Kepublic ' 

 Socrates was required to defend justice, — quite irre- 



spectively of the worldly advantages which may ac- 

 company a life of virtue and justice, — and to show, 

 that, independently of all these, justice was a thing 

 desirable in itself and for its own sake, not by 

 speaking to you of the utility of mathematics in any 

 of the questions of common life or of jihysical sci- 

 ence. Still less would I speak of this utility before, 

 I trust, a friendly audience, interested or willing to 

 appreciate an interest in mathematics in itself and 

 for its own sake. I would, on the contrary, rather 

 consider the obligations of mathematics to these dif- 

 ferent subjects as the sources of mathematical theo- 

 ries, now as remote from them, and in as different a 

 region of thought, — for instance, geometry from the 

 measurement of land, or the theory of numbers 

 from arithmetic, — as a river at its mouth is from its 

 mountain source. 



On the other side: the general opinion has been, 

 and is, that it is indeed by experience that we arrive 

 at the truths of mathematics, but tbat experience is 

 not their proper foundation. The mind itself contrib- 

 utes something. This is involved in the Platonic 

 theory of reminiscence. Looking at two things — 

 trees or stones or any thing else — which seem to us 

 more or less equal, we arrive at the idea of equality; 

 hut we must have had this idea of equality before the 

 time when, first seeingt he two things, we were led 

 to regard them as coming up more or less perfectly to 

 this idea of equality; and the lilie as regards our idea 

 of the beautiful, and in other cases. 



The same view is expressed in the answer of Leib- 

 nitz, the 'nisi intellectus ipse,' to the scholastic dic- 

 tum, 'Nihil in intellectu quod non prius in sensu' 

 ('There is nothing in the intellect which was not 

 first in sensation' — ' except [said Leibnitz] the intel- 

 lect itself). And so again, in the ' Critick of pure 

 reason,' Kant's view is, that while there is no doubt 

 but that 'all our cognition begins with experience, 

 we are nevertheless in possession of cognitions a 

 priori, independent, not of this or that experience, 

 but absolutely so of all experience, and in particular 

 that the axioms of mathematics furnish an example 

 of such cognitions a priori. Kant holds, further, 

 that space is no empirical conception which has 

 been derived from external experiences, but that, 

 in order that sensations may be referred to some- 

 thing external, the representation of space must al- 

 ready lie at the foundation, and that the external 

 experience is itself first only possible by this repre- 

 sentation of space. And, in lilve manner, time is no 

 empirical conception which can be deduced from an 

 experience, but it is a necessary representation lying 

 at the foundation of all intuitions. 



And so in regard to mathematics. Sir W. R. Hamil- 

 ton, in an introductory lecture on astronomy (1S36), 

 observes, " These purely mathematical sciences of 

 algebra and geometry are sciences of the pure reason, 

 deriving no weight and no assistance from experi- 

 ment, and isolated, or at least isolable, from all out- 

 ward and accidental phenomena. The idea of order, 

 with its subordinate ideas of number and figure, we 

 must not, indeed, call innate ideas, if that phrase be 

 defined to imply that all men must possess them with 



