OF MATHEMATICS

sixteen, and while a student at the college of Louis-le-Grand . . .
Galois occupied himself with this difficult subject.' Liouville
then states that the referees at the Academy had rejected
Galois' memoirs on account of their obscurity. He continues:
kAn exaggerated desire for conciseness was the cause of this
defect which one should strive above all else to avoid when
treating the abstract and mysterious matters of pure Algebra.
Clarity is, indeed, all the more necessary when one essays to
lead the reader farther from the beaten path and into wilder
territory. As Descartes said, "\Yhen transcendental questions
are under discussion be transcendentally clear." Too often
Galois neglected this precept; and we can understand how
illustrious mathematicians may have judged it proper to try,
by the harshness of their sage advice, to turn a beginner, full
of genius but inexperienced, back on the right road. The author
they censured was before them, ardent, active; he could profit
by their advice.
'But now everything is changed. Galois is no more! Let us not
indulge in useless criticisms; let us leave the defects there and
look at the merits.' Continuing, Liouville tells how he studied
the manuscripts, and singles out one perfect gem for special
mention.
'My zeal was well rewarded, and I experienced an intense
pleasure at the moment when, having filled in some slight gaps,
I saw the complete correctness of the method by which Galois
proves, in particular, this beautiful theorem: In order that an
irreducible equation of prime degree be solvable by radicals it is
necessary and sufficient that aU its roots be rational functions of
any two of them.9*
Galois addressed his wifl to his faithful friend Auguste
Chevalier, to whom the world owes its preservation. *My dear
friend% he began, *I have made some new discoveries in analy-
sis/ He then proceeds to outline such as he has time for. They
were epoch-making. He" concludes: 'Ask Jacobi or Gauss
publicly to give their opinion, not as to the truth, but as to the
importance of these theorems. Later there will be, I hope, some
* The significance of this theorem will he clear if the reader will
glance thiongh the extracts from Abel in Chapter 17*
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