Mark Kac’s First Publication: A Translation of "O nowym sposobie rozwiązywania równań stopnia trzeciego" – Epilogue

Author(s): 
David Derbes (University of Chicago Laboratory School, retired)

In our correspondence, I asked Prof. Roy why he had not published his paper on Kac’s derivation of Cardano’s formula. He replied that he had subsequently decided that Kac had independently rediscovered a derivation given nearly a century earlier by George Boole (1815–1864) in [Boole 1842], and he suggested that I consult his book [Roy 2011, 732–733]. Boole’s derivation of the cubic is also discussed by Wolfson [2008]. I have not had a chance to consult Prof. Roy’s book at length, but thanks to his friend and Beloit colleague Prof. Mehmet Dik, I was able to read a scan of the relevant pages; therein Prof. Roy suggested that it was fortunate that Rusiecki did not know Boole’s work. In my opinion the methods, though similar, are not the same, and Boole’s derivation in any case should not have prevented the publication of Kac’s work.

Interestingly, as a postscript to his prologue, Kac described being at a talk given by Gian-Carlo Rota (1932–1999) which involved a theorem of J. J. Sylvester (1814–1897). Sylvester’s work followed Boole’s in invariant theory. Kac concluded his postscript:

Almost en passant [Rota] said, “I’ll now show you how one can use Sylvester’s theorem to solve cubic equations.” After only a few words a feeling of dèjà vu came over me; it was the method I had discovered in the summer of 1930 [Kac 1985, 5].

Rota’s derivation, like Boole’s, was based on invariant theory. Mark Kac did not need any of that.