TRANSACTIONS OF THE SECTIONS. 



luniinaiy, the Darwin of the English school of mathematicians, started and elabo- 

 rated at an early age, and witli happy consequences, tlie same bold hypothesis. 



Most, if not all, of the great ideas of modern mathematics have had their origin 

 in observation. Take, for instance, the arithmetical theory of forms, of which the 

 foundation was laid in the diophantine theorems of Fermat, left without proof by 

 their author, which resisted all the eflbrts of the myriad-minded Euler to reduce to 

 demonstration, and only yielded up their cause of being when turned over in the 

 blowpipe flame of Gauss's transcendent genius; or Ihe doctrine of double periodi- 

 city, which resulted from the observation by Jacobi of a purely analytical fact of 

 transformation ; or Legendre's law of reciprocity; or Sturm's theorem about the roots 

 of equations, which, as lie informed me with his own lips, stared him in the face in 

 the midst of some mechanical investigations connected with the motion of compound 

 pendulums ; or Iluyghens' method of continued fractions, characterized by Lagrange 

 as one of the principal discoveries of " that great mathematician, and to which he 

 appears to have been led by the construction of his Planetary Automaton ; " or tlie 

 now algebra, speaking of which one of my predecessors ( Mr. Spottiswoode) has 

 said,_not without just reason and authority, from this Chair, " that it reaches out 

 and indissolubly connects itself each year with fresh branches of mathematics, that 

 the theory of equations has almost become new through it, algebraic geometry 

 transfigured in its light, that tlie calculus of variations, molecular physics, and me- 

 chanics" (he might, if speaking at the present moment, go on to add the theory of 

 elasticity and the highest developments of the integral calculus) " have all felt 

 its influence." 



Now this gigantic outcome of modern analytical thought, itself, too, only the pre- 

 cursor_ and progenitor of a future still more heaven-reaching tlieoiy, w-hich will 

 comprise a complete study of the interoperation, the actions and reactions, of 

 algebraic forms (Analytical Morphology in its absolute sense), how did this origi- 

 nate ? In the accidental observation liy Eisensteiu, some score or more years ago, 

 of a single invariant (the Quadriuvariant of a Binary Quartic) which he niet with in 

 the course of certain researches just as accidentally and unexpectedly as M. Du 

 Chaillu might meet a Gorilla in the country of the Fantees, or any one of us in London 

 a White Polar Bear escaped from the Zoological Gardens. Fortunately he pounced 

 down upon his prey and preserved it for the contemplation and study of future 

 mathematicians. It occupies only part of a page in his collected posthumous works. 

 This single result of observation (as well entitled to be so called as the discovery of 

 Globigerime in chalk or of the Coufoco-ellipsoidal structure of the shells of the 

 Foraminifera), which remained unproducti^-e in the hands of its distinguished 

 author, has served to set in motion a train of thought and to propagate an impulse 

 which have led to a complete revolution in the whole aspect of inodern analysis, 

 and whose consequences will .continue to be felt until Mathematics are forgotten 

 and British Associations meet no more. 



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

 mount importance of the faculty of observation to the process of mathematical 

 diseoveiy*. Were it not unbecoming to dilate on one's personal experience, 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 



bookworm in an unrumpled page : but what if the page should be undergoing a process 

 of gradual bending into a curved form ? Mr. W. K. Clifford lias indulged in some re- 

 markable speculations as to tlie possibility of our being able to infer, from certain unex- 

 plained phenomena of light and magnetism, the fact of our level space of three dimensions 

 being in the act of imdergoing in space of four dimensions (space as inconceivable to us 

 as our space to the supposititious bookworm) a distortion analogous to tlie rumpUng of 

 tlie page to which that creature's powers of direct perception have been postulated to be 

 limited. 



* Newton's Eule was to all appearance, and according to the more received opinion, 

 obtained inductively by its author. My own reduction of Eulcr's problem of tlie Virgins 

 (or rather one slightly more general than this) to tlie form of a question (or, to speak more 

 exactly, a set of questions) in simple partitions was, strange to say, first obtained by myself 

 inductively, the result communicated to Prof. Cayley, and proved subsequently by each of 

 ns independently, and by perfectly distinct methods. 



