s had ample time
to germinate and bear fruit in the fertile brain upon which it was cast.
It is almost certain that the treatise as a whole--leaving out of account
the special question of the solution of cubic equations--must have gained
enormously in completeness and lucidity from the fresh knowledge revealed
to the writer thereof by Tartaglia's reluctant disclosure, and, over and
beyond this, it must be borne in mind that Cardan had been working for
several years at Giovanni Colla's questions in conjunction with Ferrari,
an algebraist as famous as Tartaglia or himself. The opening chapters of
the book show that Cardan was well acquainted with the chief properties of
the roots of equations of all sorts. He lays it down that all square
numbers have two different kinds of root, one positive and one
negative,[110] _vera_ and _ficta_: thus the root of 9 is either 3. or -3.
He shows that when a case has all its roots, or when none are impossible,
the number of its positive roots is the same as the number of changes in
the signs of the terms when they are all brought to one side. In the case
of _x^3 + 3bx = 2c_, he demonstrates his first resolution of a cubic
equation, and gives his own version of his dealings with Tartaglia. His
chief obligation to the Brescian was the information how to solve the
three cases which follow, _i.e. x^3 + bx = c. x^3 = bx + c._ and
_x^3 + c = bx_, and this he freely acknowledges, and furthermore admits
the great service of the system of geometrical demonstration which
Tartaglia had first suggested to him, and which he always employed
hereafter. He claims originality for all processes in the book not
ascribed to others, asserting that all the demonstrations of existing
rules were his own except three which had been left by Mahommed ben Musa,
and two invented by Ludovico Ferrari.

With this vantage ground beneath his feet Cardan raised the study of
Algebra to a point it had never reached before, and climbed himself to a
height of fame to which Medicine had not yet brought him. His name as a
mathematician was known throughout Europe, and the success of his book
was remarkable. In the _De Libris Propriis_ there is a passage which
indicates that he himself was not unconscious of the renown he had won, or
disposed to underrate the value of his contribution to mathematical
science. "And even if I were to claim this art (Algebra) as my own
invention, I should perhaps be speaking only the truth, though N