Gerardis attempt

For the first time in Western mathematical literature, a mathematician, Gerardi gave general, albeit incorrect, solutions for the irreducible cubics:

ax³ = bx + N,

ax³ = bx² + N, and

ax³ = bx² + cx + N.

His solutions were merely naïve applications of the quadratic formula to cubic equations.

Thus, for ax³ = bx + N,
he claimed that:

which is the solution of the quadratic ax² = bx + N. Since he did not check his answers by reapplying them to the original problem or testing the answersby insering them in the equations, he did not recognize that his solution techniques yielded erroneous results.

Nevertheless, Gerardi's treatment of irreducible cubics categorically proved that the quest for solutions to such equations did not begin in the sixteenth century with the celebrated controversy involving Cardan and Niccolò Tartaglia (c. 1499-1557).

Gerardi wrote mathematics rethorically. These were not the first cubic equations to appear in the Western mathematical literature. Borrowing either directly or indirectly from al-Khayyami (Omar Khayyam), Fibonacci included the cubic equation
x³ + 2x² + 10x = 20 in his text entitled, Flos (c. 1225)

In fact, "... Gerardi's rules, his problems, and even his erroneous formulations are repeated in similar abacus manuscripts dating from about 1340 to the time of Paciolo .... Thus Gerardi's treatise was only the beginning of a long tradition in the study of higher order equations that did not bear fruit until the sixteenth century.

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