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221 



THURSDAY, MAY 17, 1917. 



ELEVEN BRITISH MATHEMATICIANS. 

 Mathematical Monographs. No. 17. Lectures 

 on Ten British Mathematicians of the Nine- 

 teenth Century. By Alexander MacFarlane. 

 Pp. 148. (New York : John Wiley and Sons, 

 Inc.; London: Chapman and Hall, Ltd., igi6.) 

 Price 55. 6d. net. 

 T^^HE lives of mathematicians, as a rule, are 

 -^ free from sensational episodes, and provide 

 no material for spicy biographies. But these un- 

 obtrusive being-s form a quaint and varied set ; and 

 many people would be surprised to hear how many 

 g-ood stories -are on record about their oddities, 

 .their accomplishments — nay, even .their displays 

 of wit. 



The author of these lectures was himself a 

 noteworthy man. A Highlander, and a pupil of 

 Tait, he became an ardent quaternionist, and 

 helped to found the International Association for 

 Promoting the Study of Quaternions. Tait was a 

 prejudiced and pugnacious champion of quater- 

 nions as against all other vector algebras, and he 

 infected with his enthusiasm quite a considerable 

 number of people. MacFarlane, in the larnd of 

 Willard Gibbs, shook off most of this obsession, 

 but there are traces of it even in these addresses, 

 as on p. 45, where he says "most analysts are 

 still crawling in Flatland," after a reference to 

 Hamilton's application of quaternions to three- 

 dimensional space. One might have thought this 

 a place for some reference to Grassmann, whose 

 calculus applies to space of any number of dimen- 

 sions ; not so, and although the index gives 

 twenty-three references to Hamilton and fifteen to 

 Tait, Grassmann is not so much as mentioned. 



Apart from this blemish, the lectures deserve the 

 warmest praise. Ii^ the first place, the list is truly 

 representative, consisting of Peacock, De Mor- 

 gan, Hamilton, Boole, Cayley, Clifford, H. J. S. 

 Smith, Sylvester, Kirkman, and Todhunter. A 

 few remarks may be offered on these worthies, 

 each in his turn, and as each lecture suggests a 

 reflection. 



George Peacock will always be. associated with 

 the "principle of equivalent forms." As a prin- 

 ciple it is as dead as a doornail, at any rate as 

 it used to be appealed to; but, all the same, Pea- 

 cock was one of the first to realise the general 

 character of pure algebra as an abstract sym- 

 bolism with more or less arbitrary fundamental 

 rules. More than this, he was able, by his posi- 

 tion at Cambridge, to make a vast improvement 

 in the study of mathematics there. It was in his 

 time that Leibniz's notation in the differential 

 calculus obtained oflficial recognition in England, 

 partly owing to the efforts of the Analytical 

 Society, "the object of which was stated to be 

 to advocate the d'ism of the Continent versus 

 the dot-age "of the university." Peacock's 

 " Examples " and " Symbolical Algebra " are still 

 worth consulting, even apart from their historical 

 interest. 



NO. 2481, VOL. 99] 



The lecture on De Morgan is very good, but 

 after our recent review of the reprint of the 

 "Budget," we content ourselves with quoting his 

 description of himself as homo paucaruju litera- 

 rum, apparently as a man who declined to be either 

 F.R.S. or LL.D.' 



As might be expected, the lecture on Rowan 

 Hamilton is well-informed and appreciative, 

 though the reference to his work in dynamics is 

 meagre indeed as compared with the account of 

 his calculus ef quaternions. It may be noted 

 that both Salmon and Cayley attended Hamilton's 

 first course of lectures on quaternions. 



George Boole, in some ways, is typically 

 English. He may fairly be called the inventor of 

 symbolic logic, and his work on the so-called 

 symbolical method of solving differential equations 

 is that of a pioneer. His predecessor is Peacock, 

 and we may be proud to think that among his 

 successors are Russell and Whitehead, even if 

 the former, in a fit of just indignation, should be- 

 come an American citizen. MacFarlane 's com- 

 ments on Boole's logical calculus are, technically, 

 the weakest things in his course ; on pp. 57—58 he 

 shows that he has not appreciated the modern 

 meaning of "class," and pp. 59—62 are simply 

 "obfuscation," except to a man who cannot 

 improve on the old Euler diagrams. 



Cayley comes next, and the gem of the lecture 

 is Clerk Maxwell's poem on the occasion of 

 Cayley's portrait being presented; we cannot help 

 quoting a stanza, if only to show that some 

 mathematicians appreciate the witchery of 

 words : — 



First, ye Determinants, in ordered row 

 And massive columns ranged, before 'him go, 

 To form a phalanx for his safe protection. 

 Ye powers of the nth root of — i ! 

 Around his head in endless cycles run, 

 As unembodied spirits of direction. 



• Cayley's presidential address to the British 

 Association receives a proper amount of attention ; 

 oddly enough, Cayley either never read v. Staudt's 

 "Geometric der Lage " or did not appreciate it, 

 for he seems to have died without realising an 

 "imaginary point "as an actual geometric entity, 

 although he did so much to found the projective 

 theory of metrics. Cayley's kindness and 

 courtesy and help to young mathematicians are 

 duly recorded. 



Clifford, owing to his early death, is an un- 

 solved problem. All his mathematical work is 

 brilliant, and, considering his years, original; but 

 his philosophy was as bad as Herbert Spencer's, 

 if not worse, and his cocksureness was irritating, 

 even to his friends. But the author of " Common- 

 sense of the Exact Sciences " deserves immor- 

 tality, even thodgh (or because) we agree with 

 MacFarlane where he says: "The ' Phaedo ' of 

 Plato is more satisfying' to the mind than the 

 ' Unseen L'niverse ' of Tait and Stewart." 



If we were asked to name the Admirable 

 Crichton among British mathematicians,, we 



5i-(i. we may add, who was accustomed to «pell liticra wilh one t. 



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