144 A History ofMathematics
- A realization that the Greek writings were—depending on the author’s particular take—too
difficult, or too slow, or even mistaken, and that better methods could and should be found to
solve the pressing new problems. The ‘misunderstandings’ of Euclid which dogged themedieval
writers now change into something more creative: the invention of a method (a symbolic
algebra, a primitive calculus) which masquerades as a true understanding, but is in fact
something quite new.
The general solution of cubic equations (first half of the sixteenth century) is a good starting
off point, because with it we leave the limitations of both the university and the abbacus school
traditions. Although it was only one of several important developments around 1500, it illustrates
a number of points about this period in mathematics. Briefly, the problem was to solve equations
involving cubes of the unknown in the same way that, since al-Khw ̄arizm ̄i, quadratics had been
solved—that is, by some sort of recipe. Omar Khayyam had hoped that such a solution could be
found, but had to settle for his geometric constructions (Chapter 5).
The first point to note is that the history of the solution bridges the gap between university and
non-university study. The first case was found by Scipione dal Ferro, a professor at Bologna; he did
not publish it, but passed it on to his student Antonio Fiore. The general case, also not published,
was found by Niccolò Tartaglia, a prolific mathematician working outside the university. He taught
in Venetian schools, translated Greek texts—or sometimes passed off others’ translations as his
own—and wrote original works on algebra, the art of warfare and much else. Hieronimo Cardano,
to whom Tartaglia revealed his ‘secret’ was again a university man, but a very unusual one, whose
most celebrated work was in medicine and astrology; having allegedly promised not to publish
before Tartaglia did, he ‘broke’ his promise and published in hisArs Magna.
This context of secrecy was very different from what we think of as research,^8 and was connected,
at least in Tartaglia’s case, with the chance of winning a reputation, and sometimes money, by
competitions in which mathematicians set each other problems and tried to defeat each other.
Clearly public knowledge of the method would ruin the contest.
The second point is that the mathematics itself is complicated and non-obvious, if all you have
at your disposal is 1500-style algebraic methods. Tartaglia could probably justify his method in
any particular case by calculation, but did he have the language for a general proof? Cardano gave
a proof derived from Euclid, as an Islamic algebraist like ab ̄uK ̄amil would have done. Tartaglia’s
well-known rhyme—in his version, a mnemonic to help him remember how to get the solution—
goes as follows, for the case ‘cube and things equal to numbers’. (Resisting the temptation to
translate the sixteenth century mathematical rap song into verse, I will quote Fauvel and Gray’s
literal translation with its modern equivalents.)When the cube and the things together
Are equal to some discrete number
[To solvex^3 +cx=d,]
Find two other numbers differing in this one.
Then you will keep this as a habit
That their product should always be equal
Exactly to the cube of the third of the things.
[Findu,vsuch thatu−v=danduv=(c/ 3 )^3 .]
- Although cases have occurred in more recent times—one could even mention Andrew Wiles’s actions on the proof of Fermat’s
Last Theorem (see Chapter 10).