Next Steps in Algebra, 11/18 We have seen evidence of the way medieval mathematicians such as Al-Khwarizmi and Thabit ibn-Qurra helped to preserve and transmit older Greek works (through translation, dissemination) extended that tradition with new contributions and critical re-examinations, and introduced new ideas into western mathematics like the Hindu-Arabic numerals and systematic ideas on algebra

“Fibonacci” and the Liber Abaci Now, we want to examine a few of the ways these ideas were assimilated and combined in Europe starting around a century before the start of the Renaissance Start with a topic that has the most connection with “practical mathematics” taught in schools to this day (though by different methods!) The Hindu-Arabic numerals were first introduced in Europe through the Liber Abaci by Leonardo Pisano (“Fibonacci”, ca – 1250)

“Fibonacci,” author of the Liber Abaci

“Fibonacci” and the Liber Abaci We know that Fibonacci traveled in North Africa and the Arabic world, certainly “picked up” new mathematics there Including, no doubt, works by al-Khwarizmi and others Also created the “Fibonacci numbers” 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … (each number is the sum of the two before)

Hindu-Arabic numerals Older systems of calculation persisted until roughly the 1500's, though Fibonacci's work was in Latin, aimed at small educated class For the computations needed in commerce, crafts, etc. other methods (abacus, even Roman numerals) were often still used Tide was finally decisively turned by works like that of Adam Ries (1522, Rechnung auff der linien und federn) – note: written in German, not Latin – intended for those without extensive formal educations.

Luca Pacioli, Probably best known for his Summa de arithmetica, geometria, proportioni et proportionalità (Venice 1494, in Italian) A textbook and summary of most mathematical knowledge of the time (practical and “pure”) Shows that Europeans at this point were still pretty much at the stage of assimilating and learning the mathematics of the Greek and Islamic traditions – not very much new or original Also did a Latin translation of the Elements

Luca Pacioli,

Algebra in the Renaissance In the early-to-mid 1500's we start to see dramatic progress in several mathematical areas For the rest of today, we will focus on algebra, and in particular on the solution of polynomial equations Probably not coincidentally, we also start to see the beginnings of our symbolic algebraic notation about this time But people were definitely “making it up as they went along” and nothing was standardized until much later

Algebraic notation The + and – symbols for addition and subtraction appear first in the late 1400's The = sign for equality was invented by Robert Recorde ( , at University of Cambridge in England) inspired by parallel lines(!) The √ symbol for roots is first found in Descartes ( ) Curious mixtures of words, symbols, numbers, … are very common at this time for algebraic expressions

Algebraic notation, some examples François Viète (1540 – 1603), an important figure in this story, wrote the binomial expansion of (a + b)³ as: a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo aequalia a+b cubo One of Viète's influential works: In Artem Analyticien Isagoge (Introduction to the art of analysis) – tried to do for algebra what Euclid had done for geometry

Algebraic notation, some examples Curiously, one of the last parts of our modern algebraic notation to get standardized was the use of numerical superscripts (exponents) to mean powers of numbers and symbolic expressions. For instance, can still find mathematics books and papers from the 1700's where the authors write xx instead of x² for the second power Wasn't completely standard until roughly 1800 to do this(!)

Algebraic notation, another example Girolamo Cardano (1501 – 1576) wrote expressions like √ 7 +√5 like this: R. V. 7. p: R 5. One of Cardano's influential works: Ars Magna (The Great Art) – we'll see one of his original mathematical contributions next!

An episode in Renaissance algebra First some background: On the basis of problem texts like YBC 6967, it is often said that methods for solving quadratic equations go back to the Old Babylonian period We know, from the work of Jens Hoyrup, Eleanor Robson (and others) that the “real story” about what the Babylonians knew and how they thought about it is more complicated and interesting! By al-Khwarizmi (~ 800) at least, general methods for solving quadratic equations essentially equivalent to the quadratic formula (but still not written down in that form) were well-understood

An episode in Renaissance algebra Even so, there was still some controversy, confusion about issues like negative roots, complex (non-real) roots and similar things. Mathematicians like Omar Khayyam (and many others) had also worked on solving various types of higher degree equations – cubics, quartics, etc. “Big question” in algebra, though, was: Were there general methods for higher degree equations, and what was needed to express the roots of cubics, quartics, higher degree equations?

An episode in Renaissance algebra About 1500 – Scipione dal Ferro (Bologna, ) finds a general method for solving cubic equations of the form x³ + px = q (with p, q > 0). He doesn't publish this in any form, though he does communicate it to a friend – Antonio Maria del Fior It was the practice of the time for Italian mathematicians to keep their methods secret so they could publicly challenge others to solve problems and then assert their own superiority if the rivals failed(!)

An episode in Renaissance algebra In 1535, Fior in fact challenged another mathematician, Nicolo Fontana (Brescia, 1499 – 1557), known as “Tartaglia” – “the stammerer” -- to just such a contest Tartaglia came from a very poor underprivileged background but taught himself math, Latin, and Greek. He earned a living teaching in different cities. The contest was to solve 35 cubic equations(!) including both x³ + px = q and x³ + px² = q forms Tartaglia already know how to solve the second type, figured out the first too – won the challenge!

An episode in Renaissance algebra But the story gets even better! Tartaglia was urged to reveal his solution by Girolamo Cardano, whom we met before. Tartaglia agreed, but did so in an obscure poetical form, and he also made Cardano agree to keep it secret (probably for the same reason dal Ferro had done) Cardano then figured out the method and published it in his book called Ars Magna Ever since, the method for solving the general cubic (by complicated expressions with square and cube roots) is called “Cardano's formulas”

An episode in Renaissance algebra Tartaglia was enraged by this, perhaps understandably and got into a very public dispute over what he understood as a theft of his intellectual property, mostly with a student of Cardano named Ludovico Ferrari ( ) Ferrari, in the mean time, also generalized what Tartaglia/Cardano had done and came up with a similar (but even more complicated) way to solve quartic (fourth-degree) equations. Probably didn't see any of this in high school, though.

The next steps in geometry, 11/20 Around the same period we discussed last time in which Tartaglia, Cardano, Ferrari, et. al.were extending the range of algebra, two somewhat parallel developments happened in geometry. The first has some interesting roots in the visual arts during this era; discuss that today – we'll see the other next time. In some Roman decorative painting such as the following fresco from a house in the city of Pompeii (buried by the eruption of Vesuvius in 79 CE), we see tentative steps toward representation of physical space in a 2D plane

A Roman painting from Pompeii

The next steps in geometry The artist here was pretty clearly trying to show something of what our eyes would see if we stood in the scene and looked through the arches into the distance. We see more distant columns as shorter, closer ones as larger. But what would the actual colonnades do? Presumably all the same height! Geometrically, the cornices are parts of two parallel lines in the same plane, so they would not meet. However, our eyes see parallel lines in a plane appear to converge “at infinity” – “one point perspective”

Medieval European Art But European medieval art typically had much different goals and agendas Much (most?) of it was religious in origin and in purpose (that is, not decorative for a home as in the Pompeii fresco, but intended either as part of the religious symbols contained in a church, or as religious icons for private contemplation in a (almost always a wealthy person's) home. Can you see differences between the Pompeii fresco and the following painting “The Temptation of Christ on the Mount” by Duccio di Buoninsegna (Siena, ca )

Duccio, The Temptation

Work originally created as part of a group of scenes of the life of Christ for the high altar in the Cathedral of Siena, (Duccio's home city in Italy) Now in the Frick Collection in New York City According to the on-line catalog of the Frick, “a majestically towering Christ is shown rejecting the devil, who offers Him `all the kingdoms of the world' if Christ will worship him (Matthew 4:8–11)” What is the “organizing principle” here? If not how a human being's eyes would see the scene, then what?

Humanism and Renaissance art Within one hundred years or so of Duccio's time, the Renaissance was well under way Renewed interest in and engagement with the learning and world-views of the Greek and Roman tradition in all of their forms Not just the parts that had always been more or less accepted by the Church during the medieval period (e.g. Aristotle, Euclid, … ) In art, a renewed interest in the portrayal of scenes naturalistically – as the human eye would see them Meant renewed interest in perspective

Albrecht Dürer (1471 – 1528), Draughtsman Drawing a Lute

Raphael (Raffaelo Sanzio da Urbino, 1483 – 1520) The School of Athens

Geometry and perspective in Renaissance art What does it take to create accurate perspective representations of 3D scenes on a 2D canvas or piece of paper? Good geometric understanding (at least intuitively) of how light rays from a scene reach the picture plane (or the retina of the eye)! It's actually very mathematical as you might be able to tell from the Dürer print And in fact the Italian artists who did the most to develop the theory did so in mathematical, geometric terms

Geometry and perspective in Renaissance art Filipo Brunelleschi ( ), architect of the dome of the Duomo in Florence, studied geometry and took up painting to practice perspective(!) Leon Battista Alberti ( ), laid out the theory geometrically in his text Della Pittura Piero della Francesca (1415 – 1492), known as a mathematician and an artist (work on perspective in painting: De Prospectiva Pingendi) also created an illustrated manuscript of works of Archimedes(!) Taken up by Leonardo da Vinci, many others

Brunelleschi's dome

Alberti's perspective method

From art back into mathematics The next step illustrates a frequent pattern When existing mathematics is used to study something in the real world, in visual arts, etc. it often happens that there is a sort of “feedback loop” by which the ways the mathematics is used inspires mathematicians to go back and develop new mathematical ideas Here, the ideas of “vanishing points on the horizon” led to the development of a new type of geometry called projective geometry.

From art, back to mathematics Idea: To the usual Euclidean plane, adjoin an ideal line at infinity consisting of the vanishing points for all families of parallel lines. The resulting projective plane has a beautifully symmetrical structure: as in Euclid's Postulate 1, Every pair of distinct points is contained in a unique line, but also Every pair of distinct lines meet in a unique point (possibly at infinity). The properties of 2D and 3D versions of this geometry were extensively studied first by Girard Desargues (1591 – 1661) and then by many others, to the present time(!)

Desargues' Theorem Theorem. If triangles ΔABC and ΔA'B'C' are “in point perspective” with respect to the point O (that is, if the lines AA', BB', CC' meet in O), then they are also “in axial perspective” (that is, the three pairs of lines: AB and A'B', AC and A'C', BC and B'C' meet at three points that are collinear). Note: The two triangles can either lie in one plane, or in two different planes in 3D space Appeared in a work of his friend and pupil Abraham Bosse (1602 – 1676)

Desargues' Theorem of Projective Geometry