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Lecture 21. The Solution of Third andFourth Degree Equations
In the early sixteenth, as an enormous and exhilarating
surprise, Italian mathematicians,summarized in the names of de
Ferro, Tartaglia, Cardan, Ferrari and Bombelli, showed thatit was
possible to develop a new mathematical theory which the ancient and
Arabs hadmissed.
Cubic equations As early as the 5th century B.C., ancient
Babylonians, ancient Greeksand ancient Indians were able to solve
certain cubic equations.
Omar Khayyam (1048 - 1123), who was a Persian mathematician,
philosopher, as-tronomer and poet, made significant contributions
to the theory of cubic equations. Hediscovered that a cubic
equation can have more than one solution, that it cannot be
solvedusing earlier compass and straight-edge constructions, and
found a geometric solution whichcould be used to get a numerical
answer.
A Persian mathematician, Sharaf al-Din al-Tusi (1135 - 1213),
wrote the Al-Muadalat(Treatise on Equations), in which he dealt
with eight types of cubic equations with positivesolutions and five
types of cubic equations which may not have positive solutions. In
histime, many mathematicians were working on such problems.
Figure 21.1 Bologna University where Scipione del Ferro
(1465-1526) was working.
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Ferros discovery Sometime between 1500 and 1515, Scipione del
Ferro (1465-1526), aprofessor at the University of Bologna,
discovered an algebraic method of solving the cubicequation x3+cx =
d with c, d > 0. By the way, today we know that a general cubic
equationcan be reduced to this form if we allow negative numbers.
However at that time negativenumbers were not known.
After Ferro found the breakthrough, he did not publish it; he
did not even announce itpublicly but kept it a secret. Why ? In
modern universities, professors announce and publishnew results as
quickly as possible. But academic life in the 16th century Italy
was completelydifferent. University appointments were mostly
temporary and subject to periodic renewalby the university senate.
One of the ways a professor convinced the senate to renew orappoint
to a position was by winning public challenges. Usually, two
candidates for onegiven position would present each other with a
list of problems. Before a public forum sometime later, each would
present his solutions to the others problems. Often,
considerableamounts of money, in addition to the university
positions, were dependent on the outcomeof such a challenge.
Consequently, if a professor discovered a new method for solving
certainproblems, it was to his advantage to keep it secret. He
could then pose these problems tohis opponents to win the
contest.
Just before he died, del Ferro disclosed his solution to his
pupil, Antonio Maria Fioreand to his successor (his son-in-law) at
Bologna, Annibale delia Nave (1500 - 1558).
Figure 21.2 Niccolo Tartaglia and the title page of his
book.
Tartaglias solution Another mathematician, Niccolo Tartaglia
(1499-1557) also discov-
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ered a solution to a form of the cubic, x3 + bx2 = d. In 1535,
Fiore challenged Tartaglia toa public contest, as described above.
By the rules of the contest, each should submit thirtyquestions for
the other to solve. Since Fiore had a solution from his teacher del
Ferro, hesubmitted all of his problems on solutions of a class of
cubic equations. For example, one ofthe problems was as
follows,
A man sells a sapphire for 500 ducats, making a profit of the
cube root of hiscapital. How much is the profit? (x3 + x = 500)
On the other hand, Tartaglia submitted a variety of different
questions, reflecting a broadrange of mathematics. Tartaglia was
able to solve two special types of cubic equations fiveyears
previously, and before the contest, Tartaglia worked very hard on
the cubic problems.As he later wrote, on the night of February 12,
1535, he discovered all the solutions. At thecontest, Tartaglia was
able to solve all thirty of Fiores problems in less than two hours.
Onthe other hand, Fiore did poorly on Tartaglias questions. As an
obvious winner. Tartagliadeclined the prize and accepted only the
honor of the victory.
Figure 2.3 Gerolamo Cardan (1501-1576)
Gardans book Gerolamo Cardano (1501 - 1576), better known as
Cardan, was a trueItalian Renaissance man: mathematician,
physician, astrologer and gambler. While he wasat the Piatti
Foundation, in Milan, to give public lectures on mathematics,
Cardan heardof the news about the contest and the breakthough about
solving the cubic equations.Before this news, Cardan always
believed what Pacioli 1 said: solutions of cubic equationswere
impossible. Stimulated by this news, Cardan immediately decided to
try to discoverTartaglias method by himself, but he did not
succeed.
1Fra Luca Bartolomeo de Pacioli(1446/7?1517) was an Italian
mathematician.
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In 1539 Cardan approached Tartaglia, and tried to get him to
divulge the method sothat it could be included in his arithmetic
book with full credit.
Initially Tartaglia refused Cardans suggestion. They began to
exchange insulting andthreatening letters. Cardan then invited
Tartaglia for a visit, lavishing him with gifts andcompliments.
Tartaglia eventually agreed after getting Cardan to swear an oath
that hewould not publish the method until Tartaglia had himself
published it. Here is Cardansoath:
I swear to you, by Gods holy Gospels, and as a true man of
honour, not onlynever to publish your discoveries, if you teach me
them, but I also promise you,and I pledge my faith as a true
Christian, to note them down in code, so thatafter my death no one
will be able to understand them.
Tartaglia divulged the secrets of the three different forms of
the cubic equation to Cardanin the form of a poem. Meanwhile,
Tartaglia planned to publish them himself at some laterdate.
Cardan kept his promise not to publish Tartaglias result in his
arithmetic book and hesent Tartaglia a copy off the press to show
his good faith.
Cardan began to work on the problem himself, probably assisted
by his servant and stu-dent, Lodovico Ferrari (1522-1565). Over the
next several years he worked out the solutions.They then heard that
Scipiore del Ferro had solved the problem, and that before he
diedshared the secret with his student Antonio Fiore.
Cardan and Ferrari visited Annibale delia Nave at Bologna. Nave
graciously gave thempermission to inspect del Ferros papers, and
they were able to verify that del Ferro haddiscovered the solution
first. Since someone else had solved the same equation, Cardan
felthis promise to Tartaglia was void. He quickly published his
most important mathematicalwork, the Ars Magna, sive de Regulis
Algebricis (The Great Art, or on the Rules of Algebra),chiefly
devoted to the solution of cubic and quadratic equations. The
discovery of quadraticequations was by his servant and student
Lodovico Ferrari.
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Figure 21.4 Title page of Cardans Ars Magna.
Cardans book was published, and Cardan did mention that Tartglia
was one of theoriginal discovers of the method. Feeling cheated,
Tartaglia was very furious, but his protestsproved to be useless.
In an attempt to recoup his prestige, he had another public
contest,and this time it was with Ferrari.
Lodovico Ferrari (1522-1565) was an Italian mathematician, born
in Milan, Italy. He set-tled in Bologna, Italy and he was the
servant of Gerolamo Cardan. Ferrari had mathematicaltalent and he
learned mathematics from Cardan. Besides assisting Cardan on work
of thesolutions for cubic equations, Ferrari was credited with the
solution of quartic equationsthat Cardan published.
In the contest, Targlia was defeated by the bright young
mathematician Ferrari. Tartaglialeft Milan that night at the end of
the first day and left the contest unresolved.
Figure 21.5 Milan, Italy.
Cardans formula for cubic equations We first divide the standard
equation by the
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leading coefficient to arrive at an equation of the form
x3 + ax2 + bx + c = 0.
The substitution x = y a3
eliminates the quadratic term, and we obtain
y3 = py + q, where p = b a2
3and q = c 2a
3 9ab27
, (1)
which is called the depressed cubic. By setting y = u + v, the
left hand side of (1) becomes
(u3 + v3) + 3uv(u + v) = 3uvy + (u3 + v3), (2)
which will equal the right hand side of (1) if
3uv = p, and u3 + v3 = q.
Eliminating v gives a quadratic in u3,
u3 +( p
3u
)3= q
with roots
u3 =q
2(q
2
)2 (p3
)3. (3)
By symmetry, we obtain the same value for v3. And since u3 + v3
= q, if one of the roots istaken to be u3, the other is v3. Without
loss of generality we can take
u3 =q
2+
(q2
)2 (p3
)3, v3 =
q
2(q
2
)2 (p3
)3,
and hence we find one solution
y = u + v =3
q
2+
(q2
)2 (p3
)3+
3
q
2(q
2
)2 (p3
)3.
Now we know that a cubic equation always has three roots. It was
Leonhard Euler who,in 1732, gave the first complete discussion of
Cardans solution of the cubic, in which heemphasized that there are
always three roots and pointed out how these are obtained.
Since
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u in (3) has three complex solutions: the principal root and the
ones multiplied by 12 i
32
,in addition to the root y obtained above, we get other two
roots:(
12
+ i
3
2
)3
q
2+
(q2
)2 (p3
)3+
( 1
2+ i
3
2
)3
q
2(q
2
)2 (p3
)3,
and ( 1
2 i
3
2
)3
q
2+
(q2
)2 (p3
)3+
( 1
2 i
3
2
)3
q
2(q
2
)2 (p3
)3.
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