4 REPORT — 18G9. 



juxtaposed paragrraplis that, according to Prof. Huxley, the business of the mathe- 

 matical 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 a? 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 ec|uivalent or identical mental operations. Such an opinion 

 scarcely seems to need serious refutation. The passage is taken from an article 

 in * Macmillau's Magazine ' for June last, entitled " Scientific Education — Xotes of 

 an After-dinner Speech," and I cannot but think would have been couched in more 

 guarded terms bj^ my distinguished friend had his speech been made before dinner 

 instead of after. 



The notion that mathematical truth rests on the nan'ow basis of a limited 

 number of elementary propositions from which all others are to be derived by a 

 process of logical inference and verbal deduction, has been stated still more strongly 

 and explicitly by the same eminent writer in an article of even date with the pre- 

 ceding in the ' Fortnightly Review,' where we are told that " Mathematics is that 

 study which knows nothing of observation, nothing of experiment, nothing of in- 

 duction, nothing of causation." I think no statement could have been made more 

 opposite to the undoubted facts of the case, that mathematical analysis is constantly 

 invoking the aid of new principles, new ideas, and new methods, not capable of being 

 defined by any form of words, but springing direct from the inherent powers and 

 acti-^ity 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 atten- 

 tion to discern as those of the outer physical world (to which the inner one in each 

 individual man may, I think, be conceived to stand in somewhat the same general 

 relation of con-espondence as a shadow to the object from which it is projected, or 

 as the hollow palm of one hand to the closed fist which it grasps of the otlier), 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 A'erification, and that it aftbrds a boundless scope for the exercise of 

 the highest efforts of imagination and invention. 



Lagi-ange, than whom no greater authority coidd be quoted, has expressed em- 

 phatically his belief in the importance to the mathematician of the faculty of ob- 

 servation ; Gauss has called mathematics a science of the eye, and in conformity 

 with this view always paid the most punctilious attention to preserve his text free 

 from typographical errors ; the ever to be lamented Riemann has written a thesis to 

 show that the basis of om- conception of space is purely empirical, and our know- 

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

 conceived to exist subject to laws difierent from those which govern the actual 

 space in which we are immersed, and that there is no evidence of these laws ex- 

 tending to the ultimate infinitesimal elements of which space is composed. Like 

 his master Gauss, Riemann refuses to accept Kant's doctrine of space and time 

 being forms of intuition, and regards them as possessed of physical and objective 

 reality. I may mention that Baron Sartorius von Waltershause'n (a member of this 

 Association) in his biogTaphy of Gauss ("Gauss zu gedachtniss "), published 

 shortly after his death, relates that this great man was used to say that he had laid 

 aside several questions which he had treated analytically, and hoped to apply to them 

 geometrical methods in a future state of existence, when his conceptions of space 

 .should have become amplified and extended ; for as we can conceive beings (like 

 infinitely attenuated book-wonns* in an infinite!}' thin sheet of paper) which possess 

 only the notion of space of two dimensions, so we may imagine beings capable 

 of realizing space of four or a gi-eater number of dimensionsf. Our Cayley, the centi-al 



* I have read or been told that eye of observer has never lighted on these depredators, living 

 or dead. Nature has gifted nie with eyes of exceptional microscopic power, and I can speak 

 witli some assurance of having repeatedly seen the creature \vriggling on the learned page. 

 On approaching it with breath or finger-nail it stiffens out into tlic semblance of a streak 

 of dirt, and so eludes detection. 



t It is well known to those who have gone into these views that tlie laws of motion 

 accepted as a fact suffice to prove in a general way that the space we live in is a flat or 

 level space (a 'liomaloid"), our existence therein being assimilable to the life of the 



