January 20, 1898] 



NATURE 



279 



a satisfactory endowment for a complete professorial body in 

 connection with the University. In addition to the chairs which 

 were already endowed they wanted endowments varying from 

 400/. to 600/. a year, which he did not think would be con- 

 sidered by any one as excessive, for fifteen chairs. The cost of 

 such an endowment was, therefore, from 12,000/. to 20,000/. 

 If this work was to be carried on they must not expect too 

 much from individuals. They might expect something from 

 the great trade of Birmingham, a little from each member of the 

 trade, if they would organise themselves for the purpose. The 

 great industries when appealed to would see that they had a 

 great duty in that matter, and would be glad to be represented 

 in the work. They had already made some progress, and from 

 two or three whom they had approached they might certainly 

 expect the endowment of such a professorship. His duty that 

 day was to put the question before them and ask them to accept 

 the principle and give their good will and assistance in carrying 

 it out. 



FRANCESCO BRIOSCHI 



BIOGRAPHICAL notices of this veteran Italian mathe- 

 matician, who died on December 13 last, have 

 been given in the Comptes rendus by M. Hermite, and 

 by Cremona and Beltrami in the Amiali di Matematica, 

 the journal of which Brioschi was chief editor, and one 

 of the founders in 1858 ; from these notices many of the 

 following details of Brioschi's scientific work have been 

 extracted. 



This work covers about half a century in time of pro- 

 duction, and ranges over the subjects of Analysis, 

 Geometry, Higher Algebra, Differential Equations, 

 Elliptic and Abelian Functions, Mechanics and Mathe- 

 matical Physics. 



Brioschi occupied himself at first with various dy- 

 namical questions, in continuation of the researches of 

 Lagrange and Dirichlet. He next turned to the develop- 

 ment of Gauss's analytic theory of surfaces, which had 

 hitherto not attracted the attention it deserved. He also 

 discussed the theory of the Correspondence of Poncelet's 

 Polygons ; as subsequently developed by Moutard and 

 Halphen, this theory appears likely to provide the sim- 

 plest analytical view of the Multiplication and Division 

 of Elliptic Functions, at present engaging the attention 

 of Prof. H. Weber. 



But the true bent of Brioschi's genius was found when 

 he attacked the general theory of algebraical equations, 

 being inspired by Hermite's discovery of the transcen- 

 dental solution of the quintic equation, which has recently 

 been summarised with developments in Klein's Lectures 

 on the " Ikosahedron." 



The history of the quintic equation throws a curious 

 light upon the ways of a certain typical British school of 

 mathematicians, who are accustomed to jog along in in- 

 tellectual isolation, knowing and caring nothing for the 

 advances made by others ; like " Rip van Winkles," as 

 Clifford called them. 



Working in this hermit-like way, Mr. Jerrard made the 

 important discovery that it was possible by the solution 

 of algebraical equations of an order not higher than the 

 third, to reduce the general quintic equation to the tri- 

 nomial form ; but Klein has pointed out, in the 

 " Ikosahedron," that all this had been done a hundred 

 years ago, by the Swedish mathematician Bring, in 

 1786. 



Arguing by analogy, Mr. Jerrard, it is related, was 

 firmly convinced that by the solution of a quartic, it 

 would be possible to reduce the quintic to the binomial 

 form, when the algebraical solution would be complete ; 

 and he died in happiness before having discovered his 

 error, which a slight acquaintance with the work of Abel 

 and Galois would have revealed. 



The reduction by Hermite of the general quintic 

 equation to the form of the Modular Equation of the 

 Transformation of the Fifth Order of Elliptic Functions 



NO. 1473. VOL. 57] 



suggested to Brioschi the examination of the correspond- 

 ing equations of higher order ; and Brioschi's last com- 

 munication was one to the Mathematical Congress at 

 Zurich, 1897, on the particular case of the Transformation 

 of the Eleventh Order. 



Galois's statements (" Les iddes prdcipitamment emises," 

 Halphen, " Fonctions elliptiques," iii. p. 124), which he did 

 not live to demonstrate, that the Modular Equations of 

 the Fifth, Seventh, and Eleventh Order have Resolvents 

 of the same order, had long baffled analysts, and they 

 did not receive universal acceptance till the appearance 

 of the article in "Tortolini," 1853, by Betti, Brioschi's 

 co-editor, who succeeded in retracing Galois's line of 

 argument ; Betti's article being followed up by Hermite, 

 in the Comptes rendus, 1859. 



In his biographical memoir M. Hermite points out 

 that Brioschi was the pioneer in another line of generalisa- 

 tion in the theory of algebraical equations, in his dis- 

 covery of the solution of the general sextic equation, em- 

 ploying for that purpose the six even 9 functions, of two 

 variables. The details of the development of this theory 

 will afford plenty of employment to young mathematicians 

 for some time to come. 



In addition to his scientific labours, Brioschi found 

 time to devote to public duties ; he acted as an Under- 

 Secretary of State, and was a Senator of the Upper 

 House of the Italian Parliament ; he was an organiser of 

 the railway system of Italy, and he served on the In- 

 ternational Committee of the Metric System. 



He was a member of most of the Academies and 

 Scientific Societies of Europe and America, and Pre- 

 sident of the Royal Academy of Lincei. The biograph- 

 ical notices by those who were personally acquainted with 

 him speak highly of the respect and esteem which he 

 inspired. G. 



REV. C. L. DODGSON. 



A FORMIDABLE champion of Euclidean methods 

 in the elementary teaching of geometry has just 

 passed away after a short illness. The Rev. Charles 

 Lutwidge Dodgson was born in 1832 at Daresbury in 

 Cheshire ; and, after passing five years at Rugby School, 

 matriculated in 1850 at Christ Church, Oxford, where he 

 was appointed a student in 1852, and graduated in 1854 

 with honours in both classics and mathematics. He 

 was appointed Mathematical Lecturer in the College in 

 1855, and retained that office till 1881 ; he further served 

 the University as Mathematical Examiner in 1863, and 

 Moderator in 1868. 



The mathematical subject in which he was most in- 

 terested was the elementary teaching of geometry ; of 

 this he had a personal experience of twenty-six years. 

 Without stint of labour he submitted to rigid logical 

 analysis every text-book on the subject that came to his 

 notice, undismayed by their surprising number, the re- 

 sult being the amusing and, at the same time, deep 

 "Euclid and his Modern Rivals," published in 1879, in 

 which he demonstrated the logical superiority of Euclid's 

 method over all the others examined. The Appendices 

 of this book are very valuable. A " Supplement " to it 

 appeared in 1885. In 1882 he edited Euclid, Books I. 

 and II., with an introduction ; and in 1888 he published 

 " A New Theory of Parallels," in the third edition of 

 which (1890) he simplified his fundamental axiom. 



His other mathematical work comprises "A Syllabus 

 of Plane Analytical Geometry" (i860), "Formula; of 

 Plain Trigonometry" (1861), "An Elementary Treatise 

 on Determinants" (1867), "Euclid, Book V., proved 

 Algebraically" (1874), and "Pillow Problems" (1893). 

 He invented a new method of evaluating determinants, 

 which is published in the Proceedings of the Royal 

 Society for 1866, and also a method (which was pub- 

 lished in Nature) of easily determining the day of the 



