Scipione dal Ferro

Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2. 
It is known that dal Ferro succeeded in solving the cubic ax³ + bx = c sometime between 1500 and 1515, and possibly in 1504. In keeping up with the customs of the time, dal Ferro kept his discovery a closely guarded secret, revealing it only to a very privileged few. Among the privileged were his son-in-law, the mathematician Annibale della Nave (c. 1500-58) and his student, Antonio Maria Fiore. The solution was not published; it was by no means disseminated; it was private and precious property. dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated. The problem was to find the roots by adding, subtracting, multiplying, dividing and taking roots of expressions in the coefficients.

It is believed that dal Ferro could only solve cubic equations of the form
x³ + mx = n.
In fact this is all that is required. For, given the general cubic:
y³ - by² + cy - d = 0,      put   y = x + b/3      to get
x³+ mx = n     where m = c - b/3,       and n = d - bc/3 + 2b/27.

In modern notation his solution of x³ + px = q will look like this:

Say x = (y-b) and take the cubic of both sides: (y-b) ³ = y³ - 3y²b +3yb² - b³
rearrange: (y-b)³ + 3yb(y-b) = y³- b³ , identify and let 3yb = p
which gives b = p/3y and y³- b³ = q , then x=(y-b) will be a solution to x³ +px =q.
If now b = p/_3y is inserted in to y³- b³ = q the result will be y³- p³/_{27 y}3 = q,
which gives us y⁶ -qy³ - p³/27 =0, where we can get y³ from this
equation of second degree and we can get b as b = p/_3y

and finally: x = ³[(p³/27 + q²/4) + q/2] - - ³[(p³/27 + q²/4) - q/2]

However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations. Remarkably, dal Ferro solved this cubic equation most probably before 1515, but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.

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