Was it for the math? Or the money?

Gabrielle Birchak/ May 3, 2022/ Early Modern History, Modern History

In my last pod­cast and show, I note that in 1494 Fra Luca Paci­oli pub­lished his book Sum­ma de Arith­meti­ca Geome­tria Pro­por­tione, com­mon­ly referred to as the Sum­ma. In the con­clu­sion of his book, Paci­oli stat­ed that the solu­tion of 

x^3+2x^2+10x=20

was as impos­si­ble as squar­ing the cir­cle. Thus, he was say­ing that this is an impos­si­ble prob­lem.[1]

It might have been impos­si­ble for Paci­oli but not for Sci­pio del Fer­ro, the bril­liant math­e­mati­cian who emerged on the math scene as Paci­oli was enter­ing his gold­en years.

Del Fer­ro was born on Feb­ru­ary 6, 1465, about twen­ty years after Paci­oli. In 1496, the Uni­ver­si­ty of Bologna appoint­ed del Fer­ro as a lec­tur­er of arith­metic and geom­e­try. It is unfor­tu­nate that we can­not see his works because he kept them hid­den. Dur­ing the fif­teenth cen­tu­ry, the math­e­mat­ics com­mu­ni­ty was extreme­ly com­pet­i­tive because of the aca­d­e­m­ic com­pe­ti­tions that offered mon­ey and a posi­tion in the uni­ver­si­ty. Thus, aca­d­e­mics kept their solu­tions secret and rarely shared them with others.

The only rea­son we know about del Ferro’s math­e­mat­i­cal works is that he kept his work in a secret note­book. On his deathbed in 1526, he gave this note­book to his son-in-law Han­ni­bal Nave, also a math­e­mati­cian. Con­ve­nient­ly, his son-in-law was able to take over del Ferro’s posi­tion at the uni­ver­si­ty. In an inter­est­ing turn of events, he also took the name del Ferro.

How­ev­er, Nave was not the only math­e­mati­cian in del Ferro’s cir­cle to wit­ness and see his math­e­mat­ics. Del Fer­ro also taught a stu­dent named Anto­nio Maria Fiore, whom he con­sid­ered a trust­ed stu­dent. As a result, on his deathbed, del Fer­ro shared his math­e­mat­ics with Fiore.[2]

 Del Ferro’s work remained hid­den from the pub­lic for many years. Pos­si­bly Nave and Fiore hid their knowl­edge of del Ferro’s work from each oth­er. And right­ly so, the rea­son is that del Fer­ro had a valu­able secret. It so hap­pened that when del Fer­ro worked at the Uni­ver­si­ty of Bologna in 1501, he met Luca Paci­oli when Paci­oli arrived in Bologna to teach at the uni­ver­si­ty. They became friend­ly asso­ciates and often dis­cussed their work with each oth­er. When Paci­oli dis­cussed the infa­mous cubic equa­tion that he pub­lished in 1497 and the lack of an alge­bra­ic solu­tion, del Fer­ro was inspired to prove this equa­tion algebraically.

Dur­ing del Ferro’s time, it was already known that cer­tain gen­er­al cubic equa­tions could be reduced. These forms were among a list of proven equa­tions deter­mined by the eleventh-cen­tu­ry Per­sian math­e­mati­cian Omar Khayyam.[3] Khayyam pro­vid­ed one of the first sys­tem­at­ic clas­si­fi­ca­tions of equa­tions based on their degree. In this clas­si­fi­ca­tion, he list­ed eigh­teen cubic equa­tions.[4]

How­ev­er, del Fer­ro had deter­mined a solu­tion for the cubic equa­tion[5]

x^3+ax=b

How­ev­er, at this time, these solu­tions only involved pos­i­tive num­bers because, geo­met­ri­cal­ly, neg­a­tive num­bers didn’t make sense. Also, zero was not applied to the solu­tion, and it was not known that the qua­drat­ic could have two roots through the qua­drat­ic equa­tion[6]

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Fur­ther­more, at this time, nobody knew that del Fer­ro had solved it because del Fer­ro kept the solu­tion a secret.

Mean­while, about 180 kilo­me­ters north of Bologna, there lived a young math­e­mati­cian by the name of Nic­colò Fontana in the city of Bres­cia. He was born around 1499 to Michele Fontana. His father was also known as “Michelet­to the Rid­er” because he deliv­ered mail to oth­er towns around Bres­cia. Sad­ly, and for rea­sons unknown, his father was mur­dered while deliv­er­ing mail when Nic­colò was only six years old.[7]

In 1511, the north­ern part of Italy was in a pre­car­i­ous posi­tion. France’s King Louis XII was the Duchy of Milan. Pope Julius II was aware that France’s wealth over Milan could take over the Ital­ian penin­su­la, which could over­throw the papa­cy. And so, Pope Julius began to form anti-French alliances. He aligned the papa­cy with Eng­land, Rome, Spain, Venice, and the Swiss Con­fed­er­a­tion. The pope’s goal was to dri­ve out the French. On Octo­ber 5, 1511, the papa­cy formed the Holy League, a large group of mil­i­tary forces. By Feb­ru­ary 1512, the Bat­tle of Raven­na was under­way. France and the papa­cy were at war. Unfor­tu­nate­ly, Italy and the cit­i­zens of Bres­cia would be the ones to suf­fer the most. On April 11, 1512, French Mil­i­tary Com­man­der Gas­ton de Foix instruct­ed his forces to enter Italy. The Holy League forces in Bres­cia man­aged to push back the French mil­i­tary ini­tial­ly. How­ev­er, embar­rassed by this defeat, the French army decid­ed to car­ry out ret­ri­bu­tion. As a result, they entered Bres­cia and mur­dered tens of thou­sands of res­i­dents of this great Ital­ian city. The inno­cent cit­i­zens of Bres­cia had been dec­i­mat­ed by French troops, and bod­ies were every­where.[8]

It was a tumul­tuous time as peo­ple were pulled from their homes and mur­dered. And so Nic­colò, his lit­tle sis­ter, and his moth­er ran and hid in a church. Unfor­tu­nate­ly, the French sol­diers found them. One sol­dier who was deter­mined to kill Nic­colò and his fam­i­ly sliced the 12-year-old’s jaw and palate with a saber. After the sol­diers left the church, his moth­er looked for him and found him bleed­ing pro­fuse­ly. He would have bled to death if she had not found him when she did. Sad­ly, Niccolò’s face had been muti­lat­ed so much that he had a per­ma­nent scar across his face and strug­gled with stam­mer­ing, which in Ital­ian is known as tartagliare. Thus, Nic­colò had tak­en on the nick­name Tartaglia, which is why we now refer to him as Nic­colò Tartaglia. As he grew old­er, he grew a beard to cov­er his scars.

He didn’t receive much edu­ca­tion because the fam­i­ly income sole­ly relied on his moth­er. She had saved up just a lit­tle bit of mon­ey to afford a writ­ing tutor for Tartaglia. How­ev­er, accord­ing to Tartaglia, his moth­er ran out of mon­ey by the time he learned about the let­ter K. How­ev­er, this did not stop him. He pur­loined his master’s notes and taught him­self, com­plet­ing the course through his own efforts. He wrote upon tomb­stones when he need­ed writ­ing slates to learn his math­e­mat­ics.[9]

The only area where his edu­ca­tion fell short was learn­ing Latin, which at the time would have seemed to be a hin­drance to his math­e­mat­i­cal career. How­ev­er, writ­ing his trea­tis­es in Ital­ian served the Ital­ian math­e­mat­i­cal com­mu­ni­ty and his career quite well. As a result, the Ital­ian math­e­mat­i­cal com­mu­ni­ty would even­tu­al­ly ben­e­fit from his trans­la­tions of Archimedes and Euclid.[10]

 As a young man, he was a very skilled math­e­mati­cian. His moth­er rec­og­nized this and found Tartaglia a patron named Ludovi­co Bal­biso­nio. Tartaglia and Bal­biso­nio trav­eled to Pad­ua, where he would learn math­e­mat­ics. He returned to Bres­cia with his patron, proud and filled with ego­tis­ti­cal hubris. How­ev­er, his inflat­ed ego did not do well for his pop­u­lar­i­ty in Brescia.

And so he moved to Verona, where he taught between 1516 and 1518 at Palaz­zo Maz­zan­ti. Even though he had a sus­tain­able income as an arith­metic teacher, his earn­ings were mea­ger. At that time, he was referred to as an “aba­cus mas­ter.”[11] And, much like Luca Paci­oli, he trav­eled as a teacher.

While Tartaglia was liv­ing in Verona, he had been con­tact­ed by Zuanne de Toni­ni da Coi, an Ital­ian pro­fes­sor from his home­town. Da Coi pre­sent­ed Tartaglia with two alge­bra­ic, cubic equa­tions. Tartaglia was intrigued and set out to solve them.[12] These equa­tions were of the form

x^3+3x^2=5y

and

x^3+6x^2+8x=1000.

These equa­tions began Tartaglia’s ongo­ing work on cubic equations.

Mean­while, he need­ed to make more mon­ey for his fam­i­ly, so at the age of 34, in 1534, he set­tled in Venice, Italy. At the time, he was an unknown. How­ev­er, he began to earn him­self a cred­i­ble rep­u­ta­tion by tak­ing part in debates and math­e­mat­i­cal duels.[13]

Tartaglia became an expert in Archimedean physics and pub­lished a few trea­tis­es that boost­ed his rep­u­ta­tion as an expert in mil­i­tary appli­ca­tions. Pupils trav­eled to him from around Italy and from Spain and Eng­land.[14]  Fur­ther­more, he worked and con­sult­ed with the engi­neers and sol­diers of the Venet­ian arse­nal. While in Venice, he pub­lished two books that were sequels. The first one, La Nuo­va Sci­en­tia, which means The New Sci­ence of Artillery, was pub­lished in 1537. The sec­ond book Que­si­ti et Inven­tioni Diverse, which means Prob­lems and Var­i­ous Inven­tions, was pub­lished in 1546. And as a ref­er­ence to his empir­i­cal obser­va­tions, he pref­aced his sec­ond book with a promise that the math­e­mat­ics is based on “new inven­tions, not stolen from Pla­to, from Plot­i­nus, or from any oth­er Greek or Roman, but obtained only by art, mea­sure­ment, and rea­son­ing.”[15] What is inter­est­ing about these two books is that he wrote them in the form of a dia­logue. Galileo uti­lized this unique form of writ­ing 100 years later.

He also pub­lished Travagli­a­ta Inven­tione, which applied his empir­i­cal obser­va­tions in physics and explained his inven­tion to retrieve a sunken ship. He sub­mit­ted this inven­tion for copy­right in 1551.[16]

Tartaglia’s books and his teach­ings had begun to earn him mon­ey. How­ev­er, a por­tion of his income also relied on math­e­mat­i­cal duels. Often, the win­ner of these math­e­mat­i­cal duels would give him cred­i­bil­i­ty and an aca­d­e­m­ic posi­tion with­in the university.

Mean­while, there were rum­blings that there was one per­son from Bologna who could solve cubic equa­tions, which intrigued the math­e­mat­i­cal com­mu­ni­ty. This one per­son was Anto­nio del Fiore, the indi­vid­ual who obtained the secret equa­tions from his teacher Sci­pio del Fier­ro. Tartaglia, who worked on these same cubic equa­tions, was intrigued. So, when Fiore chal­lenged Tartaglia to a math­e­mat­i­cal duel in 1535, Tartaglia accept­ed. The dif­fer­ence between the two was that Fiore was con­fi­dent that he could win because he had obtained the secret solu­tions. Tartaglia, how­ev­er, was con­fi­dent that he could win because he had been work­ing on these equa­tions for the last sev­er­al years.

The com­pe­ti­tions were arranged as fol­lows; each com­pet­ing mem­ber would present the oth­er per­son with thir­ty ques­tions for the oth­er per­son to solve. Fiore sub­mit­ted his thir­ty ques­tions to Tartaglia. All these equa­tions resolved to

x^3+ax=b.

Tartaglia had been work­ing on cubic equa­tions for years. As a result, he was already famil­iar with reduc­ing cubic to this form. Addi­tion­al­ly, Tartaglia was famil­iar with reduc­ing cubic equa­tions to the form

x^3+ax^2=b

As a result, he sub­mit­ted his thir­ty ques­tions to del Fiore that were reduced to this form.[17]

Tartaglia’s equa­tions caught Fiore by sur­prise because he expect­ed to receive equa­tions sim­i­lar to the ones that del Fer­ro had shared with him. How­ev­er, Tartaglia, who had been work­ing on these equa­tions for years, sub­mit­ted many equa­tions that were unfa­mil­iar to Fiore. After receiv­ing their equa­tions, they each went their sep­a­rate way. Tartaglia was con­fi­dent that Fiore would lose based on Fra Luca Pacioli’s state­ment about the dif­fi­cul­ty of his cubic equa­tions. Tartaglia wrote that he “Thought that not a sin­gle one could be solved because Fra Luca Paci­oli assures us of their dif­fi­cul­ty that such an equa­tion can­not be solved by a gen­er­al for­mu­la.”[18]

They each had fifty days to com­plete the equa­tions and sub­mit them to a notary for ver­i­fi­ca­tion. How­ev­er, after forty-one days passed, Tartaglia heard gos­sip that Fiore had a secret that was aid­ing him in solv­ing the cubic equation

x^3+ax=b

This rumor roused pan­ic with­in Tartaglia, who real­ized that Paci­oli might be wrong. And so, ear­ly the fol­low­ing day, eight days before the dead­line, Tartaglia sat down and, with­in two hours, solved all of Fiore’s prob­lems, includ­ing the cubic equa­tion[19]

x^3=ax+b

Mean­while, Fiore was still con­fused by Tartaglia’s equa­tions and could not solve any of them using del Ferro’s meth­ods. Tartaglia had won the math­e­mat­i­cal duel.

As a result, Tartaglia was a super­star on the math­e­mat­i­cal scene. He wrote that he knew and under­stood how to solve these cubic equa­tions that includ­ed “squares and cubes equal to num­bers.”[20]

Tartaglia had estab­lished him­self as a cred­i­ble math­e­mati­cian in Venice and the win­ner of the infa­mous Tartaglia-Fiore duel. Many want­ed to know Tartaglia’s secret, but like del Fer­ro, he held the answer close to his chest. He intend­ed that nobody would ever learn this secret. But some­one would, and it would be in the most mali­cious, dia­bol­i­cal way pos­si­ble. My next arti­cle will address those whose love for mon­ey trumped their love for math.

[1] Gindikin, Semy­on. Tales of Math­e­mati­cians and Physi­cists. Trans­lat­ed by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988.

[2] O’Connor, John J., and Edmund F. Robert­son. “Nico­lo Tartaglia.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, Sep­tem­ber 2005. https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia.

[3] Eves, Howard. “Omar Khayyam’s Solu­tion of Cubic Equa­tions.” The Math­e­mat­ics Teacher 51, no. 4 (1958): 285–86.

[4] Smith, David Eugene. His­to­ry of Math­e­mat­ics. Vol. II. New York: Dover Pub­li­ca­tions, 1958, 442–443.

[5] Nor­gaard, Mar­tin. “Side­lights on the Car­dan-Tartaglia Con­tro­ver­sy.” Nation­al Math­e­mat­ics Mag­a­zine 12, no. 7 (April 1938): 327–46.

[6] O’Connor, John J., and Edmund F. Robert­son. “Sci­p­i­one Del Fer­ro.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, July 1999. https://mathshistory.st-andrews.ac.uk/Biographies/Ferro.

[7] O’Connor, John J., and Edmund F. Robert­son. “Sci­p­i­one Del Fer­ro.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, July 1999. https://mathshistory.st-andrews.ac.uk/Biographies/Ferro/

[8] Don­vi­to, Fil­ip­po. “Storm of Steel.” Medieval War­fare 2, no. 5 (2012): 17–21.

[9] Feld­mann, Richard. “The Car­dano-Tartaglia Dis­pute.” The Math­e­mat­ics Teacher 54, no. 3 (March 1961): 160–163.

[10] Gindikin, Semy­on. Tales of Math­e­mati­cians and Physi­cists. Trans­lat­ed by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988, 5.

[11] Gindikin, Semy­on. Tales of Math­e­mati­cians and Physi­cists. Trans­lat­ed by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988, 3.

[12] O’Connor, John J., and Edmund F. Robert­son. “Nico­lo Tartaglia.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, Sep­tem­ber 2005. https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia.

[13] O’Connor, John J., and Edmund F. Robert­son. “Nico­lo Tartaglia.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, Sep­tem­ber 2005. https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia.

[14] Keller, Alex. “Archimedean Hydro­sta­t­ic The­o­rems and Sal­vage Oper­a­tions in 16th-Cen­tu­ry Venice.” Tech­nol­o­gy and Cul­ture 12, no. 4 (1971): 602–17. https://doi.org/10.2307/3102573.

[15] Gindikin, Semy­on. Tales of Math­e­mati­cians and Physi­cists. Trans­lat­ed by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988, 4.

[16] Keller, Alex. “Archimedean Hydro­sta­t­ic The­o­rems and Sal­vage Oper­a­tions in 16th-Cen­tu­ry Venice.” Tech­nol­o­gy and Cul­ture 12, no. 4 (1971): 604. https://doi.org/10.2307/3102573.

[17] Feld­mann, Richard. “The Car­dano-Tartaglia Dis­pute.” The Math­e­mat­ics Teacher 54, no. 3 (March 1961): 160–163.

[18] Gindikin, Semy­on. Tales of Math­e­mati­cians and Physi­cists. Trans­lat­ed by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988, 4.

[19] Smith, David Eugene. His­to­ry of Math­e­mat­ics. Vol. II. New York: Dover Pub­li­ca­tions, 1958, 428.

[20] O’Connor, John J., and Edmund F. Robert­son. “Nico­lo Tartaglia.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, Sep­tem­ber 2005. https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia.