TRANSACTIONS OF THE SECTIONS. 7 



studyino' in Eg-y^rt, and 12 in Babylon, opened school when 5G or 57 years old in 

 Magna Grfecia, married a young- wife when past GO, and died, carrying on his work 

 with energy unspent to the last, at the age of 99. The niatheniatician lives long 

 and lives young ; the wings of his soid do not early drop off, nor do its pores 

 become clogged with the earthy particles blown from the dusty highways of 

 vulgar life. 



Some people have been found to regard all mathematics, after the 47th proposi- 

 tion of Euclid, as a sort of morbid secretion, to be compared only with the pearl 

 said to be generated in the diseased oyster, or, as I have heard it described, "una 

 excroissance maladive de I'esprit humain." Others find its justification, its " raison 

 d'etre," in its being either the torch-bearer leading the way, or the handmaiden 

 holding up the train of Physical Science ; and a very clever ■ni'iter in a recent maga- 

 zine article, expresses his doubts whether it is, in itself, a more serious pursuit, or 

 more N\-orthy of interesting an intellectual human being, than the study of chess pro- 

 blems or Chinese puzzles. What is it to us, they say, if the three angles of a tri- 

 angle ai'e equal to two right angles, or if every even number is, or may be, the sum 

 of two primes, or if every equation of an odd degree must have a real root. How 

 dull, stale, fiat, and unprofitable are such and such like annoimcements ! Much 

 more interesting to read an account of a marriage in high life, or the details of an 

 international boiit-race. But this is like judging of architecture from being shown 

 some of the bride and mortar, or even a quarried stone of a public building, or of 

 painting from the colours mixed on the palette, or of music by listening to the thin 

 and screechy sounds produced by a bow passed haphazard over the strings of a 

 A'ioliu. The world of ideas wliich it discloses or illuuiinates, the contemplation of 

 divine beauty and order which it induces, the harmonious connexion of its parts, 

 the infinite hierarchy and absolute CA-idence of the truths with which it is con- 

 cerned, these, and such like, are the surest grounds of the title of mathematics to 

 human regard, and would remain rmimpeached and unimpaired were the plan of 

 the universe unrolled like a map at our feet, and the mind of man qualified to 

 take in the whole scheme of creation at a glance. 



In conformity with general usage, I have used the word mathematics in the 

 phu'al; but I think it would be desirable that this form of word shordd be re- 

 served for the applications of the science, aud that we should use mathematic in 

 the singular number to denote the science itself, in the same way as we speak of 

 logic, rhetoric, or (own sister to algebra *) music. Time was when all the parts of 

 the subject were dissevered, when algebra, geometry, and aritlimetic either lived 

 apart or kept up cold relations of acquaintance confined to occasional calls upon 

 one another ; but that is now at an end ; they are drav^m together and are constantly 

 becoming more and more intimately related and connected by a thousand fresh ties, 

 and we may confidently look forward to a time when they shall form but one body 

 with one soul. Geometry formerly was the chief borrower from arithmetic and 

 algebra, but it has since repaid its obligations -with abundant usury ; and if I were 

 asked to name, in one Avord, the pole-star round which the mathematical firmanent 

 revolves, the central idea which pervades as a hidden spirit the whole corpus of 

 mathematical doctrine, I should point to Continuity as contained in our notions of 

 space, and say, it is this, it is this ! Space is the Grand Cmitiiuimn from which, as 

 from an inexhaustible reservoir, all the fertilizing ideas of modem analysis are de- 

 rived ; and as Ih'indley, the engineer, once allowed before a parliamentary com- 

 mittee that, in his opinion, rivers were made to feed naA'igable canals, I feel almost 

 tempted to say that one principal reason for the existence of space, or at least one 

 principal function which it discharges, is that of feeding mathematical invention. 

 Everybody knows what a wonderful influence geometrj' has exercised in the hands 

 of Cauchy, Puiseux, Riemann, and his followers Clebsch, Gordan, and others, over 

 the very form and presentment of the modem calculus, and how it has come to 

 pass that the tracing of curves, which was once to be regarded as a puerile amuse- 



* I have elsewhere (in my Trilogy published in the 'Philosophical Transactions') re- 

 ferred to the close connexion between these two cultures, not merely as having Arithmetic 

 for their common ])arent, but as similar in their habits and affections. I have called 

 " Music the Algebra of sense, Algebra the Music of the reason ; Music the dream. Algebra 

 the waking life, — the soul of each the same !'' 



