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SCIENCE. 



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



form simultaneous invariants of a system 

 of two functions. The paper appeared in 

 November, 1841, and shortly after, in 

 February, 1842, Boole showed that the 

 polars of a Form lead to a broad class of 

 covariants. Here he extended the results 

 of the first article to more than two Forms. 

 Boole's papers led Cayley, nearly three 

 years later (1846), to propose to himself the 

 problem to determine a priori what functions 

 of the coefficients of an equation possess 

 this property of invariance, and he dis- 

 covered its possession by other functions 

 besides discriminants, for example the 

 quadrinvariants of binary quantics, and in 

 particular the invariant S of a quartic. 



Boole next discovered the other invariant 

 T of a quartic and the expression of the 

 discriminant in terms of S and T. Cayley 

 next (1846) published a symbolic method 

 of finding invariants. Early in 1851 Boole 

 reproduced, with additions, his paper on 

 Linear Transformations ; then at last be- 

 gan Sylvester. He always mourned what 

 he called 'the years he lost fighting the 

 world '; but, after all, it was he who made 

 the Theory of Invariants. 



Says Meyer : " sehen wir in dem Cyklus 

 Sylvester^ Boh-QV Publicationen (1851-1854) 

 bereits die Grundziige einer allgemeinen 

 Theorie erstehen, welche die Elemente von 

 den verschiedenartigsten Zweigen der 

 spateren Disciplin umfasst." "Sylvester 

 beginnt damit, die Ergebnisse seiner 

 Vorganger unter einem einzigen Gesichts- 

 punkte zuvereinigen." 



"With deepest foresight Sylvester intro- 

 duced, together with the original variables, 

 those dual to them, and created the theory 

 of contravariants and intermediate forms. 

 He introduced, with many other processes 

 for producing invariantive forms, the prin- 

 ciple of mutual differentiation. 



Hilbert attributes the sudden growth of 

 the theory to these processes for producing 

 and handling invariantive creatures. " Die 



Theorie dieser Gebilde erhob sich, von 

 speciellen Aufgaben ausgehend, rasch zu 

 grosser Allgemeinheit — dank vor AUem 

 dem Umstande, dass es gelang, eine Reihe 

 von besonderen der Invariantentheorie 

 eigenthiimlichen Prozessen zu entdecken, 

 deren Anwendung die Aufstellung und 

 Behandlung invarianter Bildungen be- 

 trachtlich erleichterte." 



" Was die Theorie der algebraischen In- 

 varianten anbetrifft so sind die ersten Be- 

 griinder derselben, Cayley und Sylvester, zu- 

 gleich auch als die Vertreter der naiven 

 Periode anzusehen : an der Aufstellung der 

 einfachsten Invariantenbildungen und an 

 den eleganten Anwendungen auf die Auf- 

 losung der Gleichungen der ersten 4 Grade 

 hatten sie die unmittelbare Freude der 

 ersten Entdeckung." It was Sylvester alone 

 who created the theory of canonic forms 

 and proceeded to apply it with astonishing 

 power. What marvelous mass of brand 

 new being he now brought forth ! 



Moreover he trumpeted abroad the erup- 

 tion. He called for communications to him- 

 self in English, French, Italian, Latin or 

 German, so only the ' Latin character ' 

 were used. 



From 1851 to 1854 he produces forty-six 

 different memoirs. Then comes a dead 

 silence of a whole year, broken in 1856 by 

 a feeble chirp called ' A Trifle on Projec- 

 tiles." 



What has happened ? Some more ' fight- 

 ing the world. ' Sylvester declared himself a 

 candidate for the vacant professorship of 

 geometry in Gresham College, delivered a 

 probationary lecture on the 4th of Decem- 

 ber, 1854, and was ignominiously ' turned 

 down.' Let us save a couple of sentences 

 from this lecture : 



" He who would know what geometry is 

 must venture boldly into its depths and 

 learn to think and feel as a geometer. I 

 believe that it is impossible to do this, to 

 study geometry as it admits of being stud- 



