Who Murdered Ferrari?! And Why?!

Gabrielle Birchak/ May 17, 2022/ Modern History, Post Classical

When I last signed off, it was 1535, and Tartaglia had won the Tartaglia-Fiore math­e­mat­i­cal duel. Tartaglia was a rock star on the math­e­mat­i­cal scene. He had solved all Fiore’s prob­lems and was also able to find a solu­tion for the cubic equa­tion x³=ax+b.

How­ev­er, before I begin my sto­ry, I want to back­track to the late fif­teenth cen­tu­ry, around 1482, when a man named Bar­tolomeo Fer­rari had two sons, Vin­cent and Alexan­der. Alexan­der had two chil­dren: a daugh­ter named Mad­dale­na and a son named Lodovi­co, who was born in 1522. Sad­ly, like Tartaglia, Lodovi­co and Mad­dale­na became orphaned when their father was killed. As a result, Lodovi­co was raised by his uncle Vin­cent and grew up with his cousin Luca. Luca was a trou­ble­mak­er and was writ­ten into his­to­ry as a dif­fi­cult young man. Luca like­ly felt equal­ly chal­lenged by his father. So, he ran away from home. Luca thus found him­self in Milan, look­ing for work. Even­tu­al­ly, he land­ed a posi­tion work­ing as a ser­vant for the 35-year-old math­e­mati­cian Giro­lamo Cardano.

Car­dano was an intel­li­gent math­e­mati­cian with a hot tem­per. He was born an ille­git­i­mate child in 1501 in Pavia. At that time, Pavia was in the Duchy of Milan, which means it was French ter­ri­to­ry. I men­tioned this in my last arti­cle about Tartaglia, where I talked about the war in the north­ern part of Italy when France’s King Louis XII was the Duke of Milan.

Girolamo’s father was Fazio Car­dano, a lawyer in Milan. He was also a bril­liant math­e­mati­cian. Fazio taught geom­e­try at the Uni­ver­si­ty of Pavia and the Piat­ti Foun­da­tion in Milan. Fazio was so knowl­edge­able about geom­e­try that Leonar­do da Vin­ci had reached out to him for expla­na­tions on pro­por­tions and geom­e­try. Pos­si­bly, this was when he and Paci­oli were work­ing on Pacioli’s book De Div­ina Pro­por­tione. So clear­ly, the math­e­mat­i­cal com­mu­ni­ty in Italy was small, and every­body knew every­body because now we have come full cir­cle with thir­ty-six years of association.

When Fazio was fifty years old, he met Chiara Micheria, a sin­gle par­ent rais­ing three chil­dren. Chiara had Giro­lamo out of wed­lock, even though even­tu­al­ly Fazio and Chiara mar­ried. Sad­ly, Chiara’s three oth­er chil­dren died when the plague took over Milan. Giro­lamo learned math­e­mat­ics from his father and found inter­est in acad­e­mia, specif­i­cal­ly med­i­cine. He attend­ed Pavia Uni­ver­si­ty, where he did well. How­ev­er, he also often argued with his peers and supe­ri­ors. Giro­lamo wrote, “this I rec­og­nize as unique and out­stand­ing amongst my faults — the habit, which I per­sist in, of pre­fer­ring to say above all things what I know to be dis­pleas­ing to the ears of my hear­ers. I’m aware of this, yet I keep it up will­ful­ly, in no way igno­rant of how many ene­mies it makes for me.”[1]

Giro­lamo spent his father’s mon­ey and need­ed to earn more mon­ey. As a result, he began to gam­ble by play­ing card games, dice, and chess. Even though he earned his doc­tor­ate in med­i­cine in 1525, his gam­bling addic­tion taint­ed his reputation.

Short­ly after his father’s death, Car­dano need­ed to make mon­ey. Thus, he moved about ten miles away to a tiny vil­lage area now referred to as Piove di Sac­co.[2] He trav­eled back to Milan to apply to the Col­lege of Physi­cians. How­ev­er, the col­lege reject­ed him because of his gam­bling and abra­sive rep­u­ta­tion. How­ev­er, the col­lege need­ed a rea­son to deny him entrance into the pro­gram, so they used his ille­git­i­ma­cy as a child. Mov­ing back to Piove di Sac­co, he ran a small med­ical prac­tice and, in 1532, mar­ried Lucia Ban­dari­ni. He and Lucia had three chil­dren. Unfor­tu­nate­ly, his small prac­tice did not earn him enough mon­ey. So, he and his wife moved 290 kilo­me­ters west to Gal­larate, intend­ing to make more mon­ey. How­ev­er, he fell into deep­er debt. He writes, “I ceased to be poor because I had noth­ing left.”[3]  Luck­i­ly, through patrons and pri­vate fund­ing, he could teach math­e­mat­ics publicly.

In 1535 he began to author his book Prac­ti­ca arith­meti­ca Gen­er­alis, also known as The Prac­tice of Arith­metic and Sim­ple Men­su­ra­tion. As a side note, this was the same year that Tartaglia beat Fiore in the math com­pe­ti­tion that I wrote about in my last arti­cle. While work­ing on his book, Car­dano encoun­tered an equa­tion that he couldn’t solve. It was the same cubic equa­tion that Paci­oli stat­ed could have no gen­er­al solu­tion. He was unsuccessful.

Also, in 1535, he hired the young Luca Fer­rari to be his ser­vant. As not­ed before, Luca was a prob­lem­at­ic kid. Luca didn’t like work­ing and pos­si­bly real­ized that he could live for free at home if he just tried to get along with his father.[4]  Thus, short­ly after he arrived in Milan, Luca ran away from Car­dano and returned to Bologna.

Car­dano was angry because Luca left Car­dano before com­plet­ing his employ­ment. Feel­ing slight­ed, Car­dano reached out to Vin­cent Fer­rari and demand­ed that he send Luca back to fin­ish his employ­ment. How­ev­er, Vin­cent decid­ed that instead of just send­ing his son back, he would also send his nephew Lodovico.

Car­dano wrote that on Novem­ber 30, 1536, a mag­pie in his court­yard “kept up such an end­less and alto­geth­er unwont­ed chat­ter­ing that we were look­ing for some­one to arrive.”[5] This day was when 14-year-old Lodovi­co Fer­rari and his cousin Luca arrived on Cardano’s doorstep. Luca was to com­plete his employ­ment as a ser­vant, and Lodovi­co began his work as a ser­vant. How­ev­er, Car­dano learned that Lodovi­co was edu­cat­ed and could read and write. As a result, Car­dano appoint­ed Lodovi­co as his sec­re­tary. While work­ing as his sec­re­tary, Car­dano real­ized that Lodovi­co was bril­liant. And so, he began to teach him math­e­mat­ics. Lodovi­co even­tu­al­ly became one of his three most out­stand­ing stu­dents, there­by enter­tain­ing Car­dano to ques­tion if the magpie’s chirring was a sign.

In 1537, Car­dano applied to the Col­lege of Physi­cians in Milan for a sec­ond time. The col­lege reject­ed him again. Mean­while, he con­tin­ued to work on his book and attempt­ed to solve sev­er­al cubic equa­tions. He still was unsuccessful.

By 1539, many indi­vid­u­als had with­drawn their objec­tions toward Car­dano at the col­lege of Physi­cians in Milan. Fur­ther­more, two of his friends pro­vid­ed glow­ing refer­rals for Car­dano to teach med­i­cine at the same col­lege. As a result, the col­lege final­ly hired him to teach.

Though Car­dano still had a rep­u­ta­tion for being abra­sive, he found suc­cess teach­ing med­i­cine and writ­ing. Addi­tion­al­ly, his lessons with Fer­rari proved to be ben­e­fi­cial. Fer­rari helped Car­dano with his man­u­scripts and became quite a math­e­mati­cian in his own right. They had a suc­cess­ful work­ing rela­tion­ship, although both had short tem­pers. Fur­ther­more, Fer­rari was known for hav­ing a “stormy tem­per,” so much that even Car­dano would avoid him at times. At one point, three years into their work­ing rela­tion­ship, Fer­rari had come back from a fight miss­ing some fin­gers on his right hand.[6]

While in Milan, Car­dano had an inter­est­ing encounter with a char­ac­ter from my last pod­cast, Zuanne da Coi. Da Coi was the indi­vid­ual who gave Tartaglia two cubic equa­tions and sparked Tartaglia’s inter­est in solv­ing these equa­tions. Da Coi had moved to Milan and met Car­dano, who was still strug­gling to solve cer­tain cubic equa­tions. Car­dano had pub­lished Prac­ti­ca and began work­ing on his next book Ars Magna. Upon telling da Coi about his chal­lenges with his cubic equa­tions, da Coi told Car­dano about Tartaglia’s solutions.

Tak­ing this tip from da Coi, he sent the pub­lish­er Zuan da Bas­sano to Tartaglia with sev­en cubic equa­tions, ask­ing for the solu­tions and the for­mu­la to solve them. It was Cardano’s goal to pub­lish them in his book Ars Magna. Thus, da Bas­sano arrived with the sev­en equa­tions and stat­ed that if he didn’t want to impart his solu­tions to Car­dano, at least give Car­dano the thir­ty equa­tions that he solved in the Tartaglia-Fiore duel along with all the solu­tions.[7]

Offend­ed at this request, Tartaglia refused his offer and told Zuan to “tell his Excel­len­cy that he must par­don me, that when I pub­lish my inven­tion, it will be in my own work and not in that of oth­ers.”[8] Tartaglia informed da Bas­sano that Car­dano could obtain the ques­tions from the court and that he would not share the solution.

Then, after observ­ing the sev­en equa­tions that da Bas­sano brought from Car­dano, Tartaglia had a rev­e­la­tion. He replied to da Bas­sano again, stat­ing, “these ques­tions are from Mess­er Zuanne da Coi and from no one else, for I rec­og­nized the last two. Two years ago, he pro­posed to me a ques­tion like the sixth and I made him own up that he nei­ther under­stood the prob­lem nor knew the solu­tion. He also pro­posed one sim­i­lar to the last one, which involves work­ing in cen­sus and cubes equal to a num­ber and out of my kind­ness I gave him the solu­tion less than a year ago. For such solu­tions I found a par­tic­u­lar rule applic­a­ble to sim­i­lar prob­lems.”[9] In this let­ter, Tartaglia point­ed out Cardano’s lack of orig­i­nal­i­ty and use of Tartaglia’s ini­tial for­mu­la he shared with da Coi two years prior.

Car­dano was furi­ous. He replied on Feb­ru­ary 12, 1539, and accused Tartaglia of being a greedy and unchar­i­ta­ble fraud.[10] Car­dano then attempt­ed to coax Tartaglia to solve two of the equa­tions he sent.[11]

The equa­tions were:

1. Make me of ten four quan­ti­ties in con­tin­ued pro­por­tion whose squares added shall make sixty.

2. Two per­sons were in com­pa­ny, and pos­sessed I know not how many ducats. They gained the cube of the tenth part of their cap­i­tal, and if they had gained three less than they did gain, they would have gained an amount equal to their cap­i­tal. How many ducats had they?[12]

Cardano’s ulte­ri­or motives were evi­dent, and Tartaglia was aware of this. Regard­less, on Feb­ru­ary 18, Tartaglia replied. To the first equa­tion, he gave an ele­gant solu­tion in poet­ry.[13] How­ev­er, the sec­ond, he did not pro­vide a solution.

Now that Car­dano had a clue about how to solve these cubic equa­tions, Car­dano put Fer­rari to work to try and solve the cubic equa­tions to dis­cov­er how Tartaglia solved them.

Then, on March 13, 1539, Car­dano sent a let­ter to Tartaglia, only this time, his let­ter was filled with flat­tery and praise. He begins his let­ter stat­ing, “Mess­er Nico­lo, mio caris­si­mo.” In Ital­ian, Mio Caris­si­mo trans­lates to My Dear­est. He tells Tartaglia not to be offend­ed by his for­mer words. He blames his insults on da Coi, who had left him a wrong impres­sion of Tartaglia. He tells him that da Coi left the uni­ver­si­ty “uncer­e­mo­ni­ous­ly” and left behind 60 pupils that need­ed a pro­fes­sor.[14] He con­tin­ues his flat­tery by invit­ing Tartaglia to vis­it him telling him that the March­ese del Vas­to, the Span­ish Gov­er­nor of Lom­bardy, want­ed to meet him.

Tartaglia ques­tioned the valid­i­ty of this invi­ta­tion and wrote to a friend that if he did not go to Milan, the Mar­quis might take offense. As a result, his absence might not look good for him. Tartaglia went unwill­ing­ly. Thus, on March 25, 1539, Tartaglia arrived at Cardano’s house.

How­ev­er, Tartaglia arrived only to dis­cov­er that March­ese del Vas­to was not there. Tartaglia not­ed the con­ver­sa­tion and pub­lished the dia­logue in his book Que­si­ti et Inven­tioni Diverse. He writes that Car­dano began by say­ing, “it is con­ve­nient for us that the Mar­quis has just left for Vigevano so we can talk about our affairs until he returns. You sure­ly have not been any too oblig­ing and not show­ing me your solu­tions of the cube and cose equal to a num­ber that I have so earnest­ly asked you to do.”[15]

Is it me, or is Car­dano a snake?

Tartaglia told him that if he gave away the solu­tion, Car­dano would prob­a­bly pub­lish them as his own and com­plete­ly spoil Tartaglia’s upcom­ing book. He then goes on to accuse Car­dano of poten­tial plagiarism.

Car­dano pressed on. He promised he would keep the solu­tions a secret, telling Tartaglia, “I swear you by the holy Evan­gels of God and as a true man of hon­or that I will not only nev­er pub­lish it, but I will write it for myself in code so that no one find­ing them after my death can under­stand. If you will now believe me, believe; if not, let it pass.”[16]

Tartaglia caved in and gave him the solu­tion for the two equa­tions x³+ax=b and x³+b=ax.

How­ev­er, he gave him the solu­tion in twen­ty-five lines of poetry.

Though Tartaglia made the solu­tion dif­fi­cult for Car­dano, Car­dano put Fer­rari to work to try and solve the cubic equa­tions for his book Ars Magna.

On some lev­el, Tartaglia’s poem thwart­ed Car­dano. Even though Car­dano had Fer­rari help him solve these equa­tions, he still wrote to Tartaglia on April 9 because he had trou­ble with his math­e­mat­i­cal poet­ry. Tartaglia replied, cor­rect­ing his mis­un­der­stand­ings and giv­ing him anoth­er hint about the solu­tion. Nev­er­the­less, Tartaglia was still uncom­fort­able giv­ing him this information.

Over the fol­low­ing months, Car­dano wrote sev­er­al let­ters to Tartaglia. Tartaglia nev­er respond­ed because he was busy work­ing on his trans­la­tions of Euclid.[17]

Final­ly, still work­ing on his book Ars Magna, Car­dano wrote to Tartaglia on August 4, 1539, “I have sent to enquire after the solu­tion to var­i­ous prob­lems for which you have giv­en me no answer, one of which con­cerns the cube equal to an unknown plus a num­ber. I have cer­tain­ly grasped this rule, but when the cube of one-third of the coef­fi­cient of the unknown is greater in the val­ue than the square of one-half of the num­ber, then, it appears, I can­not make it fit into the equa­tion.”[18]

Tartaglia real­ized that Car­dano was close to fig­ur­ing it out. Tartaglia was filled with regret. Also, he had received a rumor that Car­dano pre­pared to pub­lish Tartaglia’s solu­tions, claim­ing “these new rules in Alge­bra” as his own.[19] In a let­ter to Car­dano, Tartaglia cor­rect­ed him with false infor­ma­tion, there­by hop­ing to con­fuse him and thwart his project, Ars Magna. Addi­tion­al­ly, he called Car­dano out on his pla­gia­rism. Tartaglia told him he knew of his plans to pub­lish his secret solu­tions to the cubic equations.

The dis­pute became so heat­ed that Fer­rari decid­ed to inject him­self into the argu­ment. The rea­son is that he had been solv­ing these equa­tions for Car­dano. Thus, Fer­rari wrote back to Tartaglia defend­ing Car­dano, stat­ing, “You have the infamy to say that Car­dano is igno­rant in math­e­mat­ics, and you call him uncul­tured and sim­ple-mind­ed, a man of low stand­ing and core stock and oth­er sim­i­lar offend­ing words too tedious to repeat. …This mat­ter con­cerns me per­son­al­ly since I am his crea­ture, I have tak­en it upon myself to make known pub­licly your deceit and mal­ice.”[20]

Thus, tak­ing the solu­tion that Tartaglia used to solve x³+ax=b Car­dano and Fer­rari found solu­tions for

x³+ax²=b

x³=ax²+b

and

x³+b=ax²

These solu­tions cre­at­ed a dis­tinct gap in any oppor­tu­ni­ty for cor­dial dis­cus­sion about these cubic equa­tions. Tartaglia was furious.

How­ev­er, Tartaglia was not Cardano’s only vic­tim. In 1541, Car­dano resigned from the Piat­ti Foun­da­tion. He intend­ed to have Fer­rari take over his posi­tion. How­ev­er, da Coi want­ed the posi­tion, so a duel was arranged between Fer­rari and da Coi. Hav­ing stud­ied all da Coi’s ear­li­er prob­lems, Fer­rari under­stood how much da Coi knew about cubic equa­tions. As a result, Fer­rari won the duel and the pub­lic lec­tur­er posi­tion at the Piat­ti Foun­da­tion. Fer­rari was only 20 years old.

Car­dano and Fer­rari con­tin­ued to work togeth­er on the cubic equa­tions. In 1545, Car­dano pub­lished his book, Ars Magna, and all the answers to the cubic equa­tions con­tained Tartaglia’s solu­tions. Tartaglia was furi­ous and made it offi­cial­ly known in his book Que­si­ti et Inven­tioni Diverse, which he pub­lished in 1546. These pub­lished con­ver­sa­tions infu­ri­at­ed Fer­rari. So, on Feb­ru­ary 10, 1547, Fer­rari chal­lenged Tartaglia to a duel on geom­e­try, arith­metic, astrol­o­gy, music, cos­mol­o­gy, per­spec­tive, and archi­tec­ture.[21]

Tartaglia ignored the request for a duel. How­ev­er, in March 1548, Tartaglia was invit­ed to Bres­cia to con­duct pub­lic lec­tures and pri­vate lessons. How­ev­er, his patrons would not give him this posi­tion unless he fol­lowed up on the duel with Ferrari.

Thus, on August 10, 1548, in Milan, Tartaglia and Fer­rari con­duct­ed an intel­lec­tu­al duel. How­ev­er, this com­pe­ti­tion was noth­ing like the Tartaglia-Fiore duel. It was a pub­lic pre­sen­ta­tion where both par­ties would debate against each oth­er. Tartaglia knew more about cubic equa­tions. How­ev­er, Fer­rari pro­vid­ed stel­lar argu­ments on the oth­er top­ics. As a result, Fer­rari won, and Tartaglia nev­er obtained the posi­tion that he want­ed in his home­town of Bres­cia. Instead, he moved to Venice, where he would con­tin­ue to seek work and teach pub­licly through the fund­ing of patrons. He died nine years lat­er in Venice.

Fer­rari was the new rock star of math in Italy. He took over Cardano’s posi­tion at the Piat­ti Foun­da­tion. After work­ing at the foun­da­tion, Fer­rari obtained employ­ment work­ing as a tax asses­sor. He then accept­ed the same role as a tax asses­sor for the church, which served him well. As a result, Fer­rari became excep­tion­al­ly rich and was able to retire as a young man. In his wealthy retire­ment, he moved back to his home­town of Bologna to take care of his wid­owed sis­ter Mad­dale­na. While back home, Fer­rari began a posi­tion as a pro­fes­sor of math­e­mat­ics at the Uni­ver­si­ty of Bologna in 1565. Sad­ly, he died a year lat­er. The rea­son? Well, he was poi­soned. Why was he poi­soned? Was it for math­e­mat­i­cal solu­tions? Did Tartaglia poi­son him because of his loss to Fer­rari? Was it the unem­ployed da Coi who lost the posi­tion at the foun­da­tion because of him? Or was it Car­dano, who need­ed more mon­ey because he was los­ing a court bat­tle because his own son mur­dered his daugh­ter-in-law? Who mur­dered Ferrari?

Well, actu­al­ly, it was none of those peo­ple. It was Mad­dale­na, his sis­ter, who poi­soned Fer­rari. She took his entire for­tune upon his death and remar­ried two weeks lat­er. Upon trans­fer­ring all her pos­ses­sions over to her new hus­band, he left her. Mad­dale­na died in poverty.

Such was the life of a math­e­mati­cian in Renais­sance Italy. Or, as they say in Italy: “Fidar­si é bene ma non fidar­si é meglio,” which means, “trust­ing is good but not trust­ing is better.”

---

[1] Car­dano, Giro­lamo. The Book of My Life. Trans­lat­ed by Jean Ston­er. New York, NY: E.P. Dut­ton and Co., 1930, 52.

[2] Car­dano, Giro­lamo. The Book of My Life. Trans­lat­ed by Jean Ston­er. New York, NY: E.P. Dut­ton and Co., 1930, 13.

[3] Car­dano, Giro­lamo. The Book of My Life. Trans­lat­ed by Jean Ston­er. New York, NY: E.P. Dut­ton and Co., 1930, 15.

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

[5] Car­dano, Giro­lamo. The Book of My Life. Trans­lat­ed by Jean Ston­er. New York, NY: E.P. Dut­ton and Co., 1930, 195.

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

[7] Nordgaard, Mar­tin A. “Side­lights on the Car­dan-Tartaglia Con­tro­ver­sy.” Nation­al Math­e­mat­ics Mag­a­zine 12, no. 7 (1938): 327–46. https://doi.org/10.2307/3028578.

[8] Ore, Oys­tein. Car­dano, the Gam­bling Schol­ar. Prince­ton, NJ: Prince­ton Uni­ver­si­ty Press, 1953, 66

[9] Nordgaard, Mar­tin A. “Side­lights on the Car­dan-Tartaglia Con­tro­ver­sy.” Nation­al Math­e­mat­ics Mag­a­zine 12, no. 7 (1938): 327–46. https://doi.org/10.2307/3028578.

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

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

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

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

[14] Nordgaard, Mar­tin A. “Side­lights on the Car­dan-Tartaglia Con­tro­ver­sy.” Nation­al Math­e­mat­ics Mag­a­zine 12, no. 7 (1938): 327–46. https://doi.org/10.2307/3028578

[15] Tartaglia, Nic­co­lo. Que­si­ti et inu­en­tioni diuerse. appres­so de l’auttore, 1554. http://archive.org/details/bub_gb_1Z5k28O8RBcC, 120.

[16] Tartaglia, Nic­co­lo. Que­si­ti et inu­en­tioni diuerse. appres­so de l’auttore, 1554. http://archive.org/details/bub_gb_1Z5k28O8RBcC, 120.

[17] Nordgaard, Mar­tin A. “Side­lights on the Car­dan-Tartaglia Con­tro­ver­sy.” Nation­al Math­e­mat­ics Mag­a­zine 12, no. 7 (1938): 327–46. https://doi.org/10.2307/3028578.

[18] O’Connor, John J., and Edmund F. Robert­son. “Giro­lamo Car­dano.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, June 1998. https://mathshistory.st-andrews.ac.uk/Biographies/Cardan/.

[19] O’Connor, John J., and Edmund F. Robert­son. “Tartaglia ver­sus Car­dan.” Mac­Tu­tor His­to­ry of Math­e­mat­ics, Sep­tem­ber 2005. https://mathshistory.st-andrews.ac.uk/HistTopics/Tartaglia_v_Cardan/.

[20] Tartaglia, Nic­co­lo. Que­si­ti et inu­en­tioni diuerse. appres­so de l’auttore, 1554. http://archive.org/details/bub_gb_1Z5k28O8RBcC, 120.

[21] Ore, Oys­tein. Car­dano, the Gam­bling Schol­ar. Prince­ton, NJ: Prince­ton Uni­ver­si­ty Press, 1953, 88.