380 



NATURE 



[December 5, 1912 



theorem there stated about resolving N into sets 

 of sequences. The theorem actually follows from 

 the most elementary considerations; the only 

 reason that can be suggested for Sylvester miss- 

 ing the proof is that he failed to note a one-one 

 correspondence between two arithmetical expres- 

 sions for sets of numbers. 



The lectures on reciprocants aroused great in- 

 terest at the time of their delivery, and the subject 

 was taken up by English mathematicians with 

 considerable vigour. Here, as in other cases, 

 Sylvester had to suffer for his neglect of current 

 mathematical literature. It was pointed out with 

 truth, but unnecessary acrimony, that the whole 

 theory of reciprocants was a mere particular case 

 of Lie's transformation-theory ; with this, and 

 even with Halphen's papers on differential in- 

 variants, Sylvester seems to have been practically 

 unacquainted until his attention was directed to 

 them. Still more remarkable is the fact that he 

 makes no allusion to Cockle's papers on criticoids. 

 However, his work on reciprocants is permanently 

 valuable, as giving a calculus of differential in- 

 variants analogous to that employed in the case 

 of algebraic forms. Dr. Baker points out 

 (p. xxxiii) that Sylvester was familiar with the 

 idea of infinitesimal transformations ; this, of 

 course, is quite true, but he does not explicitly 

 construct a theory of groups of such transforma- 

 tions, although his results constantly depend on 

 the existence of such groups. 



There are many brief notes of a stimulating 

 character, such as those on matrices, on the dis- 

 tribution of primes, and on Goldbach's theorem. 

 The theorem of Goldbach (or Euler) is to the 

 effect that every even number, zn, can be ex- 

 pressed in at least one way as the sum of two odd 

 primes. Sylvester adds, on the strength of ex- 

 periment, that in at least one such representation, 

 supposing n is composite, each of the primes will 

 lie between n/2 and 3«/2. On this assumption he 

 makes a remarkable first step towards the proof 

 of the theorem; for if p, q, r, &c. , are the odd 

 primes between 11/2 and 3)2/2, it easily follows 

 that the power-series equivalent to 



.-j^+ ' +...r 

 I -.f'' I -.1-' I 



must give, as the coefficient of a-", the number of 

 compositions 211 = p + q, if we reckon (p + q), 

 (q + p) as different compositions when p, q are 

 unequal. All that has to be done now is to show 

 that the coefficient of .v"" does not vanish. 



This volume, like its predecessors, illustrates 

 Sylvester's power of attracting willing and 

 devoted disciples. Thus Franklin, and more parti- 

 cularly Hammond, toiled ungrudgingly in the 

 NO. 2249, VOL. 901 



service of the master; and their services deserve 

 cordial recognition and gratitude. 



A single remark will show how much we are 

 indebted to the editor, Dr. Baker, for his care in 

 revision. Mr. Morgan Jenkins supplied no fewer 

 than twenty-seven corrections of errors and mis- 

 prints in the " construction theory of partitions " 

 — that is, about one for every three pages. Prob- 

 ably Dr. Baker found others, and it is not at all 

 likely that this memoir was exceptionally inaccu- 

 rate. So the corrections involved in the four 

 volumes run into many hundreds. We have only 

 noticed one place where a note seems required 

 and has not been added. On p. 358 it is said 

 that "the extent is not altered by the operation 

 of V " ; this is only true if the operand contains 

 the highest letter to the second or higher degree. 

 Oddly enough, Sylvester immediately takes an 

 example where V diminishes the extent ! 



The biographical notice gives an interesting 

 account of Sylvester's life and character, and a 

 judicious estimate of his mathematical work. In 

 particular, it is pointed out how strictly he con- 

 fined himself to what may be called combinatorial 

 analysis ; writing nothing on groups, for example, 

 and never dealing with function-theory properly 

 so called. In addition to the notice, we have a 

 photogravure reproduction of the portrait which 

 hangs in the hall of St. John's College, Cam- 

 bridge, and an engraving of the obverse of the 

 Royal Society Sylvester medal, which represents 

 his face in profile. G. B. M. 



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MODERN science supplies an abundance of 

 material, such as the philosophers of the 

 past could not have dreamed of, for the psycho- 

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 man of science has worked through it. This 

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 new lines of philosophic thought, as the work of 

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 has shown. It is significant, ho\ve\er, that the 



