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.
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.
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.
This allowed and required them to make deals, and finance capital, arrange letters of credit, create bills of exchange, and make interest calculations.
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.
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
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.
These abacists had thoroughly studied arabic mathematics, which emphasized algebraic methods.
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
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.
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.
The texts did not dwell on problems without a solution. Therefore, some student-teacher interaction would accompany the learning.
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.
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
Unlike Islamic algebra, which was entirely rhetorical,
the abacists allowed the use of symbols for unknowns. Standard words were:
ß
Text, p.316 example.
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: 
. Solve the problem.
Higher Degree Equations
Another innovation of the abacists was their extention of the
Islamic quadratic solving techniques to higher order equations.
Of course, each text began with the standard six type of quadratics as described by . But many went further.
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.
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
Nicholas of Cusa was ordained in 1440 and was a cardinal in Brixon (now
Bressanone) and in 1450 became bishop there.
He was interested in geometry
and logic. He contributed to the study of infinity, studying the infinitely
large and the infinitely small.
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.
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
Regiomontanus was born Johann üller of Königsberg but he used the Latin version of his name (Königsberg = "King's mountain").
Regiomontanus
Regiomontanus (1436-1478)
. Actual: 
.
Nicolas Chuquet, (c. 1445 -c. 1500)
French physician
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.
The mathematics in Triparty was known to the Islamic algebraists, but Triparty is
the first detailed algebra in French.
Triparty - Part I
- He uses the Hindu-Arabic place-value system
- Use the result for fractions between: Given a,b,c,d>0.
Then 
is between 
and 
.
e.g. 
is between 
and 
.
No proof is given. He uses this to approximate roots of quadratics.
Triparty - Part II
- uses the fraction rule to extract square roots: Find 
.
Solution. (1) 2<r<3. (2) 
. (3) 
.
- introduces some notation
-plus
-minus,
-square root,
-cube root.
- allows negative coefficients
Triparty - Part III
- introduces exponential notation
Therefore,
party par 
egualx 
- novelty
egualx 
4x=-2
- solves 
. 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
In 1509 he published a Latin translation of Euclid's Elements , the first printed edition.
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.
Pacioli was a teacher of some reknown. He also assembled a vast collection of mathematical materials over some 20 years.
Pacioli
wrote Summa de arithmetica, geometrica proportioni et
proportionalita (1494) It gives a summary of the mathematics known at that time.
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.
It overshadowed Chuquet's Triparty but was not mathematically significant. Subsequently, Chuquet's
work was scarcely mentioned.
The German School
Johann Widman (b. 
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:\
Robert Recorde
Born: 1510 in Tenby, Wales
Died: 1558 in London, England
Recorde virtually established the English school of mathematics and first introduced algebra into England.
Recorde is best known for inventing the 'equals' symbol 
which appears in his book The Whetstone of Witte (1557).
Recorde was educated at Oxford and Cambridge. He became physician to King Edward VI and Queen Mary. He served for a time in Ireland as 'Comptroller of Mines and Monies'.
He wrote many textbooks, for example The Grounde of Artes in 1540 was a very successful commercial arithmetic book teaching the ``perfect work and practice of Arithmeticke etc", in Recorde's own words. The book discusses operations with arabic numerals, computation with
counters, proportion, the `rule of three', and fractions.
In 1551 Recorde wrote Pathwaie to Knowledge which some consider an abridged version of Euclid's Elements. It is the only one of his books not written in the form of a dialogue between a master and scholar.
The 'equals' symbol appears in Recorde's book The Whetstone of Witte published in 1557. He justifies using two parallel line segments ``bicause noe 2 thynges can be moare equalle".
The symbol was not immediately popular. The symbol || was used by some and 'ae' ( or 'oe' ), from the word 'aequalis' meaning equal, was widely used into the 1700's.
Recorde died in King's Bench prison in Southwark, where he was committed for debt. Although no official record remains of other crimes, some historians think he was guilty of much more serious offences.
Large Summary of notation
The solution of the cubic
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)
Also starring:
Antonio Maria Fiore ( 
half 16 
century), student,
ß
Annibale della Nave (1500-1558), student
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)
Cardano is famed for his work Ars Magna which was the
first Latin treatise devoted solely to algebra.
Girolamo Cardano's name was Cardan in Latin and in English he is sometimes known as Jerome Cardan.
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.
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.
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.
In Ars Magna appears the first computation with complex numbers although Cardano did not properly understand it.
Cardano's Liber de ludo aleae in 1563 was the first study of the theory of probability.
De vita propria liber in 1575 is Cardano's autobiography. It is one of the first modern psychological autobiographies.
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.
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.
Tartaglia was famed for his algebraic solution of cubic equations which was published in Cardan's Ars Magna.
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'.
Tartaglia was self taught in mathematics but, having an extraordinary ability, was able to earn his living teaching at Verona and Venice.
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.
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.
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
Ferrari'was orphaned at the age of fourteen.
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.
Years later, in 1564, he returned to Bologna where he earned a doctorate in philosophy.
Scientific Disciplines -- Primary: Mathematics. Subordinate: Geography, Astronomy
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.
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.
Ferrari was then (c. 1548-56) in the service of Ercole Gonzaga, Cardinal of Mantua, for some eight years.
From 1564 until his death in 1565, he was lecturer in mathematics at the University of Bologna.
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
Scipione dal Ferro lectured at Bologna where he was a colleague of Pacioli.
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).
He kept his discovery secret and only told his student Fior shortly before his death.
Ferrari reports seeing a notebook in del Ferro's handwriting where the solution is clearly written down.
The solution of the cubic
Transform 
into 
, using the
transformations 
and then y=x.
Next define u and v by
This gives
Using v =(p/3)/u, substitute to get:
Solve for u:
and then for v; finally for x.
Cardano gives the solutions:
However, his solution is .
Example. Solve 
.
Solution.
Example. In Ars Magna we find 
. 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 
then it is a simple calculation to show that
b=1. Thus
Hence, the root of the cubic is
Cardano also posed the problem with complex roots
Connection with angle trisection.
So
The cosine of an angle and its trisection are related through the solution of a cubic.
Summary: Solution of cubic and quartic
greatest feat of algebra since ancient times
not practical - purely theoretical
renewed interest in the trisection problem
inspired work to solve general quintics
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
Bombelli was an engineer whose projects included reclaiming land.
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.
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.
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