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April 2, 1997


Algebra in the Renaissance

The general cultural movement of the renaissance in Europe had a profound impact also on the mathematics of the time. Italy was especially impacted.

tex2html_wrap_inline348 The Italian merchants of the time travelled widely throughout the East, bringing goods back in hopes of making a profit. They needed little by way of mathematics. Only the elementary needs of finance were required.

tex2html_wrap_inline348 After the crusades, the commercial revolution changed this system. New technologies in ship building and saftey on the seas allows the single merchant to become a shipping magnate. These sedentary merchants could remain at home and hire others to make the journeys.

tex2html_wrap_inline348 This allowed and required them to make deals, and finance capital, arrange letters of credit, create bills of exchange, and make interest calculations.

tex2html_wrap_inline348 Double-entry bookkeeping began as a way of tracking the continuous flow of goods and money. The economy of barter was slowly replaced by the economy of money we have today.

tex2html_wrap_inline348 Needing more mathematics, they inspired the emergence of a new class of mathematician called abacist, who wrote the texts from which they taught the necessary mathematics to the sons of merchants in schools created for this purpose. There are hundreds of different ones still in existance. (Compare quadrivium (arithmetic, geometry, music, astronomy. Compare trivium: (grammar, rhetoric, and dialectics)).

The Italian Abacists

tex2html_wrap_inline348 The Italian abacists of the 14th century were instrumental in teaching the merchants the ``new" Hindu-Arabic decimal place-value system and the algorithms for using it. There was formidable resistance to this system, in Italy and most of Europe.

tex2html_wrap_inline348 These abacists had thoroughly studied arabic mathematics, which emphasized algebraic methods.

tex2html_wrap_inline348 In fact, for many years Roman numerals were used to keep account ledgers. The old system of counting boards required the board plus a bag of counters. The new system required only pen and paper. By and by, as with new technologies in general, the superior Hindu-Arabic system won out.

Note. ``Believe it or not" ....The decreasing costs and availability of paper was a factor in this.

Mathematical Texts

tex2html_wrap_inline364 Mathematical texts were mostly practical, teaching only those problems young merchants would need in carrying out daily transactions. Problems and their solutions were described in detail, with all steps fully described.

tex2html_wrap_inline364 Besides the business problems required for their profession there were also recreational problems. There were problems in geometry, elementary number theory, the calendar, and astrology.

tex2html_wrap_inline364 The texts did not dwell on problems without a solution. Therefore, some student-teacher interaction would accompany the learning.

tex2html_wrap_inline364 During the 14th and 15th centuries the abacist extended the Islamic methods by introducing abbreviations and symbolisms, developing new methods for dealing with complex algebraic problems.

tex2html_wrap_inline364 Perhaps most important were the lessons learned in the use of algebra to solve practical problems.

Partly because of this practical need
for mathematics the new direction of
mathematics was toward
algebraic methods.

New Algebraic Techniques

tex2html_wrap_inline374 Unlike Islamic algebra, which was entirely rhetorical, the abacists allowed the use of symbols for unknowns. Standard words were:



tex2html_wrap_inline374 Text, p.316 example.

tex2html_wrap_inline374 From Antonio de' Mazzinhi (1353-1383), known for his cleverness in solving algebraic problems, we have the example. `` Find two numbers such that multiplying one by the other makes 8 and the sum of their squares is 27."

The solution begins by supposing that the first number is un cosa meno la radice d'alchuna quantità ( a thing minus the root of some quantity) while the second number equals una cosa più la radice d'alchuna quantità (another thing plus the root of some quantity. We have


Answer: tex2html_wrap_inline384 . Solve the problem.

Higher Degree Equations

tex2html_wrap_inline364 Another innovation of the abacists was their extention of the Islamic quadratic solving techniques to higher order equations.

tex2html_wrap_inline364 Of course, each text began with the standard six type of quadratics as described by . But many went further.

tex2html_wrap_inline364 Maestro Dardi of Pisa in a 1344 work extended this list to 198 types of equations of degree up to four, some involving radicals. He gave an example of how to solve a particular cubic equation, but the methods would not generalize. tex2html_wrap_inline364 Another mathematician of this age was Luca Pacioli (1445-1517), who was reknown for his teaching.

Renaissance Mathematicians

Nicholas Cusa
Born: 1401 in Kues, Trier
Died: 11 Aug 1464 in Todi, Papal States

tex2html_wrap_inline364 Nicholas of Cusa was ordained in 1440 and was a cardinal in Brixon (now Bressanone) and in 1450 became bishop there.

tex2html_wrap_inline364 He was interested in geometry and logic. He contributed to the study of infinity, studying the infinitely large and the infinitely small.

tex2html_wrap_inline364 He looked at the circle as the limit of regular polygons and used it in his religious teaching to show how one can approach truth but never reach it completely.

tex2html_wrap_inline364 Cusa is best known as a philosopher who argued the incomplete nature of man's knowledge of the universe. He claimed that the search for truth was equal to the task of squaring the circle.

Johann Müller Regiomontanus
Born: 6 June 1436 in Königsberg,
(now Kaliningrad, Russia)
Died: 8 July 1476 in Rome, Italy

tex2html_wrap_inline364 Regiomontanus was born Johann üller of Königsberg but he used the Latin version of his name (Königsberg = "King's mountain").

tex2html_wrap_inline364 Regiomontanus

Regiomontanus (1436-1478)

Nicolas Chuquet, (c. 1445 -c. 1500)
French physician

tex2html_wrap_inline374   Chuquet wrote Triparty en la science des nombres (1484), a work on algebra and arithmetic in three parts. However, it was not printed until 1880.

tex2html_wrap_inline374   The mathematics in Triparty was known to the Islamic algebraists, but Triparty is the first detailed algebra in French.

tex2html_wrap_inline374   Triparty - Part I

- He uses the Hindu-Arabic place-value system

- Use the result for fractions between: Given a,b,c,d>0. Then tex2html_wrap_inline422 is between tex2html_wrap_inline424 and tex2html_wrap_inline426 . e.g. tex2html_wrap_inline428 is between tex2html_wrap_inline430 and tex2html_wrap_inline432 . No proof is given. He uses this to approximate roots of quadratics.

tex2html_wrap_inline374   Triparty - Part II

- uses the fraction rule to extract square roots: Find tex2html_wrap_inline436 . Solution. (1) 2<r<3. (2)  tex2html_wrap_inline440 . (3)  tex2html_wrap_inline442 .

- introduces some notation



- allows negative coefficients

tex2html_wrap_inline374   Triparty - Part III

- introduces exponential notation


Therefore, tex2html_wrap_inline478 party par tex2html_wrap_inline480 egualx tex2html_wrap_inline482

- novelty

tex2html_wrap_inline484 egualx tex2html_wrap_inline486 4x=-2

- solves tex2html_wrap_inline490 . This generalizes al Kashi.

- notes that there are multiple solutions of some system of two equations in three unknowns.

Luca Pacioli
Born: 1445 in Sansepolcro, Italy
Died: 1517 in Sansepolcro, Italy

tex2html_wrap_inline364 In 1509 he published a Latin translation of Euclid's Elements , the first printed edition.

tex2html_wrap_inline364 He left unpublished a work on recreational problems, geometrical problems and proverbs. It makes frequent reference to Leonardo da Vinci who worked with him on the project.

tex2html_wrap_inline364 Pacioli was a teacher of some reknown. He also assembled a vast collection of mathematical materials over some 20 years.

tex2html_wrap_inline364 Pacioli wrote Summa de arithmetica, geometrica proportioni et proportionalita (1494) It gives a summary of the mathematics known at that time.

tex2html_wrap_inline364 Summa studies arithmetic, algebra, geometry and trigonometry and provided a basis for the major progress in mathematics which took place in Europe shortly after this time.

tex2html_wrap_inline364 It overshadowed Chuquet's Triparty but was not mathematically significant. Subsequently, Chuquet's work was scarcely mentioned.

The German School

Johann Widman (b. tex2html_wrap_inline504 1460), German, wrote Rechnung uff allen Kauffmanschafften used + and - first time in print.

Adam Riese (1492-1559), German, wrote Die Coss (1524), an influential arithmetic and algebra text. In Germany, the term ``nach Adam Riese" still survives.

Christoph Rudolff, (ca. 1500-ca. 1545), German, wrote Coss (1525). In it he uses decimal fractions and modern notation for roots.

Peter Apian, (1495-1552), German, wrote Rechnung (1527). In it we see Pascal's triangle (!!) appears on title page -- fully one century before Pascal.

Michael Stifel (1487-1567), German monk turned itinerant Lutheran preacher; for a time a professor at Jena. We wrote Arithmetica integra (1544). It was the most important of the 16th centrury algebras: Features:\

Large Summary of notation


The solution of the cubic

tex2html_wrap_inline374   The Cast:

Geronimo Cardano (1501-1576) Ars magna (1545)ß

Niccolo Tartaglia (ca. 1500-1577)ß

Ludovico Ferrari (1522-1565)ß

Scipio del Ferro (ca. 1465-1526)

tex2html_wrap_inline374   Also starring:

Antonio Maria Fiore ( tex2html_wrap_inline558 half 16 tex2html_wrap_inline560 century), student, ß

Annibale della Nave (1500-1558), student

tex2html_wrap_inline374   You will witness:  mathematical competitions, intrigue, murder, poison, broken promises, blasphemy, rascals and scoundrels.

Girolamo Cardano
Born: 24 Sept 1501 in Pavia, Duchy of Milan (now Italy)
Died: 21 Sept 1576 in (now Italy)

tex2html_wrap_inline364 Cardano is famed for his work Ars Magna which was the first Latin treatise devoted solely to algebra.

tex2html_wrap_inline364 Girolamo Cardano's name was Cardan in Latin and in English he is sometimes known as Jerome Cardan.

tex2html_wrap_inline364 Cardano studied at Pavia and Padua receiving a doctorate in medicine in 1525. He was professor of mathematics at Milan, Pavia and Bologna leaving each after some scandal.

tex2html_wrap_inline364 Cardano lectured and wrote on mathematics, medicine, astronomy, astrology, alchemy, and physics. In fact his fame as a doctor was such that the Archbishop of St Andrews, on suffering as he thought from consumption, sent for Cardan. Cardano is reported to have visited Scotland to treat the Archbishop who was not suffering from consumption and made a complete recovery.

tex2html_wrap_inline364 Cardano is famed for his work Ars Magna which was the first Latin treatise devoted solely to algebra and is one of the important steps in the rapid development in mathematics which began around this time (and still continues today). Ars Magna made known the solution of the cubic by radicals and the solution of the quartic by radicals. These were proved by Tartaglia and Ferrari respectively. Ferrari was in fact a pupil of Cardan's.

tex2html_wrap_inline364 In Ars Magna appears the first computation with complex numbers although Cardano did not properly understand it.

tex2html_wrap_inline364 Cardano's Liber de ludo aleae in 1563 was the first study of the theory of probability.

tex2html_wrap_inline364 De vita propria liber in 1575 is Cardano's autobiography. It is one of the first modern psychological autobiographies.

tex2html_wrap_inline364 Cardano was eventually forbidden to lecture or publish books. In 1570 he was imprisioned on a charge of having cast the horoscope of Christ. In 1571 Pope Pius V granted him an annuity for life and he settled in Rome and became astrologer to the papal court.

tex2html_wrap_inline364 Cardano is reported to have correctly predicted the exact date of his own death. He achieved this by committing suicide.

Nicolo Fontana Tartaglia
Born: 1500, Brescia;
Died: 13 Dec 1557 in Venice.

tex2html_wrap_inline364 Tartaglia was famed for his algebraic solution of cubic equations which was published in Cardan's Ars Magna.

tex2html_wrap_inline364 Tartaglia's proper name was Niccolo Fontana although he is always known by his nickname. When the French sacked Brescia in 1512 the soldiers killed Tartaglia's father and left Tartaglia for dead with a sabre wound that cut his jaw and palate. The nickname Tartaglia means the 'stammerer'.

tex2html_wrap_inline364 Tartaglia was self taught in mathematics but, having an extraordinary ability, was able to earn his living teaching at Verona and Venice.

tex2html_wrap_inline364 The first person known to have solved cubic equations algebraically was dal Ferro. On his deathbed dal Ferro passed on the secret to his (rather poor) student Fior. A competition to solve cubic equation was arranged between Fior and Tartaglia. Tartaglia, by winning the competition in 1535, is famed as the discoverer of a formula to solve cubic equations. Because negative numbers were not used there was more than one type of cubic equation and Tartaglia could solve all types, Fior only one type. Tartaglia confided his solution to Cardan on the condition that he would keep it secret. The method was, however, published by Cardan in Ars Magna in 1545.

tex2html_wrap_inline364 Tartaglia wrote Nova Scientia (1537) on the application of mathematics to artillery fire. He described new ballistic methods and instruments, including the first firing tables.

tex2html_wrap_inline364 Tartaglia also wrote a popular arithmetic text and was the first Italian translator and publisher of Euclid's Elements in 1543. He also published Latin editions of Archimedes's works.

Ferrari, Ludovico

Born: Bologna, 2 Feb. 1522
Died: Italy, Oct. 1565

tex2html_wrap_inline364 Ferrari'was orphaned at the age of fourteen.

tex2html_wrap_inline364 Having no formal education, he was sent as a refugee to Milan where he joined the household of Girolamo Cardano in 1536. Cardano introduced him to Latin, Greek, and Mathematics. He became a member of Cardano's household as his amanuensis, disciple, and ultimately collaborator.

tex2html_wrap_inline364 Years later, in 1564, he returned to Bologna where he earned a doctorate in philosophy.

tex2html_wrap_inline364 Scientific Disciplines -- Primary: Mathematics. Subordinate: Geography, Astronomy

tex2html_wrap_inline364 He collaborated with Cardano in researches on the cubic and quartic equations, the results of which were published in the Ars magna (1545). He found a method of solving the quartic equation. Feud with Niccolo Tartaglia, which was caused by the publication of the Ars magna in 1545.


tex2html_wrap_inline364 In 1540, he was appointed by Ferrante Gonzaga, the governor or Milan, public lecturer in mathematics in Milan. In this capacity he gave lessons on the Geography of Ptolemy.

tex2html_wrap_inline364 Ferrari was then (c. 1548-56) in the service of Ercole Gonzaga, Cardinal of Mantua, for some eight years.

tex2html_wrap_inline364 From 1564 until his death in 1565, he was lecturer in mathematics at the University of Bologna.

tex2html_wrap_inline364 He received an offer from Emperor Charles V, who wanted a tutor for his son.

Scipione dal Ferro
Born: 6 Feb 1465 in Bologna, Italy
Died: 5 Nov 1526 in Bologna, Italy

tex2html_wrap_inline364 Scipione dal Ferro lectured at Bologna where he was a colleague of Pacioli.

tex2html_wrap_inline364 Dal Ferro is the first to solve the cubic equation by radicals. He only solved one of the two cases (the fact that 0 and negative numbers were not in use made many distinct cases).

tex2html_wrap_inline364 He kept his discovery secret and only told his student Fior shortly before his death.

tex2html_wrap_inline364 Ferrari reports seeing a notebook in del Ferro's handwriting where the solution is clearly written down.

The solution of the cubic

tex2html_wrap_inline374   Transform tex2html_wrap_inline626 into tex2html_wrap_inline628 , using the transformations tex2html_wrap_inline630 and then y=x.

tex2html_wrap_inline374   Next define u and v by


This gives


tex2html_wrap_inline374   Using v =(p/3)/u, substitute to get:


tex2html_wrap_inline374   Solve for u:


and then for v; finally for x.

tex2html_wrap_inline374   Cardano gives the solutions:


However, his solution is .

Example. Solve tex2html_wrap_inline660 .



Example. In Ars Magna we find tex2html_wrap_inline662 . The solution here, by the formula, is


Cardano knew there was no square root of -121 and he knew x=4 was a solution. He did not know to proceed. He referred to such square roots as ``sophists'', and thought taking square roots of negative numbers was ``as subtle as it is useless''.

However, from this example Rafael Bombelli's (ca. 1526-1573) made the first step toward complex numbers. Bombelli's idea was that the radicals themselves might be related in much the way that the radicans are related. That is, in modern notation: They are conjugate so each has the form 2+ib. If tex2html_wrap_inline672 then it is a simple calculation to show that b=1. Thus


Hence, the root of the cubic is


tex2html_wrap_inline374   Cardano also posed the problem with complex roots


Connection with angle trisection.




The cosine of an angle and its trisection are related through the solution of a cubic.

Summary: Solution of cubic and quartic

tex2html_wrap_inline374   greatest feat of algebra since ancient times

tex2html_wrap_inline374   not practical - purely theoretical

tex2html_wrap_inline374   renewed interest in the trisection problem

tex2html_wrap_inline374   inspired work to solve general quintics


tex2html_wrap_inline374   demanded awareness of negative and imaginary numbers - through the Tartaglia-Cardano formula.

Rafael Bombelli

Born: Jan 1526 in Bologna, Italy
Died: 1573 in (probably) Rome, Italy

tex2html_wrap_inline364 Bombelli was an engineer whose projects included reclaiming land.

tex2html_wrap_inline364 Bombelli was the first person to write down the rules for addition and multiplication of complex numbers. He showed that, using his methods, correct real solutions could be obtained from the Cardano-Tartaglia formula for the solution to a cubic. His work was strictly formal, though.

tex2html_wrap_inline364 He wrote Algebra in 1850, his only publication, a few years after the publication of Ars Magna. It was not published until 1872. It was nonetheless a very influential work and Leibnitz cited Bombelli as an outstanding master of the analytical art. Features:

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Don Allen
Wed Apr 2 09:40:09 CST 1997