( 222 ) 
ON MALET'S PEOOF THAT EVEEY EQUATION HAS BOOTS, 
EEAL OE IMAGINAEY, EQUAL IN NUMBEE TO ITS 
DEGEEE. 
By Professor W. N. Eoseveare, M.A. 
(Communicated by Professor Lawrence Crawford.) 
(Eead June 17, 1914.) 
The writer has not found it easy, with the books of reference at his 
command, to trace the final result of the mathematical search for a proof 
that " every equation has a root real or imaginary." Cauchy's proof by 
vectors is of course well known, but the text-books give no purely 
analytical proof. Burnside and Panton have an instructive note in an 
Appendix ; and the German " Encyklopadie," on pp. 233 et seq., gives 
summaries of various attempts. In a " Note " of 1870 Clilford suggests 
a form of proof. As will be shown below this proof is unsound ; but 
Clifford's idea was taken up in 1882 by Malet, who published a sound 
proof, of which the following paper gives the substance. It seems to be 
generally accepted that this is the earliest straightforward proof of the 
" Fundamental Theorem of Algebra." Elliott in Proc. Lond. Math. Soc, 
XXV., March and April, 1894, discusses Clifford's and Malet's proofs, 
dismisses both as unsound, and suggests another proof which he confesses 
that he cannot complete owing to the intractability of a determinant. (To 
the present writer Elliott's reasons for rejecting the earlier proofs are not 
convincing.) Clifford's proof ends thus : Eg (an equation of degree 8) has 
a quadratic factor if E,_,8 has a real root ; E28 has a quadratic factor if 
Ei4^27 bas a real root; E^^.ay bas a quadratic factor if E^^^ has a real 
root. This last condition is true: therefore ... E28 h.di>s real root; . * . Eg 
has a quadratic factor. Thus stated the unsoundness is obvious. On the 
other hand, Malet's proof seems to be sound. The following paper gives 
the substance of the proof, and develops some of the detailed work by 
means of the treatment of certain determinants which the writer has 
suggested in a previous paper. 
