604 



SCIENCE. 



[N. S. Vol. V. No. 120. 



result from Gordan's researches, wliicli are 

 indubitably correct. This supposed conse- 

 quence must have arisen from a misappre- 

 hension, on the part of M. de Bruno, of the 

 nature of Professor Cayley's rectification 

 of the error of reasoning contained in his 

 second memoir on Quantics, which had led 

 to results discordant with Gordan's. Thus 

 error breeds error, unless and until the per- 

 nicious brood is stamped out for good and 

 all under the iron heel of rigid demonstra- 

 tion. In the early part of this year Mr. 

 Halsted, a ifellow of John's Hopkins Uni- 

 versity, called my attention to this passage 

 in M. de Bruno's book ; and all I could say 

 in reply was that ' the extrinsic evidence 

 in support of the independence of the equa- 

 tions which had been impugned rendered 

 it in my mind as certain as any fact in na- 

 ture could be, but that to reduce it to an 

 exact demonstration transcended,! thought, 

 the powers of the human understanding.' " 



In 1883 Sylvester was made Savilian pro- 

 fessor of geometry at Oxford, the first Cam- 

 bridge man so honored since the appoint- 

 ment of Wallis in 1649. 



To greet the new environment, he created 

 a new subject for his researches — Recipro- 

 cants, which has inspired, among others, 

 J. Hammond, of Oxford; McMahon, of 

 "Woolwich ; A. E. Forsyth, of Cambridge ; 

 Leudesdorf, Elliott and Halphen. 



Sylvester never solved exercise problems 

 such as are proposed in the Educational Times, 

 though he made them all his life long down 

 to his latest years. For example, unsolved 

 problems by him will be found even in Vol. 

 LXII. and Vol. LXIII. of the Educational 

 Times reprints (1895). If at the time of 

 meeting his own problem he met also a 

 neat solution he would communicate them 

 together, but he never solved any. In the 

 meagre notices that have been given of 

 Sylvester the strangest errors abound. Thus 

 C. S. Pierce, in the Post, March 16th, speaks 

 of his accepting, ' with much diffidence,' a 



word whose meaning he never knew; and 

 gives 1862 as the date of his retirement from 

 Woolwich, which is eight years wrong, as 

 this forced retirement was July 31, 1870, 

 after his 55th birthday. Cajori, in his in- 

 adequate account (History of Mathematics, 

 p. 326), puts the studying of law before the 

 professorship at University College and the 

 professorship at the University of Virginia, 

 both of which it followed. Effect must fol- 

 low cause. And strange, that of the few 

 things he ascribes to Sylvester, he should 

 have hit upon something not his, " the dis- 

 covery of the partial differential equations 

 satisfied by the invariants and covariants 

 of binary quantics." But Sylvester has 

 explicitly said in Section VI. of his ' Cal- 

 culus of Forms :' "I alluded to the partial 

 differential equations by which every in- 

 variant may be defined. M. Aronhold, as 

 I collect from private information, was the 

 first to think of the application of this 

 method to the subject; but it was Mr. 

 Cayley who communicated to me the equa- 

 tions which define the invariants of func- 

 tions of two variables." 



Surely he needs nothing but his very own, 

 this marvellous man who gave so lavishly 

 to every one devoted to mathematics, or, 

 indeed, to the highest advance of human 

 thought in any form. 



Geoege Bextoe Halsted. 



Univeesity of Texas. 



THE GREAT FAULT AND ACCOMPANYING 

 SANDSTONE DIKES OF UTE PASS, COLO- 

 RADO* 

 Theeb year years ago Whitman Cross 

 first directed the attention of geologists to 

 the fact that dike-like masses of sandstone 

 occur in the granite of the Pike's Peak 

 massif, forming a belt about one mile wide 

 extending north-northwest from the vicinity 

 of Green Mountain Falls, in Ute Pass, 



*Abstract of a paper read before the Boston Society 

 of Na al History, January 20, 1897. 



