MEDIEVAL MATHEMATICS

MATHEMATICS

Medieval abacus, based on the Roman/Greek model

During the centuries in which the Chinese, Indian and Islamic mathematicians had been in the ascendancy, Europe had fallen into the Dark Ages, in which science, mathematics and almost all intellectual endeavour stagnated. Scholastic scholars only valued studies in the humanities, such as philosophy and literature, and spent much of their energies quarrelling over subtle subjects in metaphysics and theology, such as "How many angels can stand on the point of a needle?"
From the 4th to 12th Centuries, European knowledge and study of arithmetic, geometry, astronomy and music was limited mainly to Boethius’ translations of some of the works of ancient Greek masters such as Nicomachus and Euclid.
All trade and calculation was made using the clumsy and inefficient Roman numeral system, and with an abacus based on Greek and Roman models.
By the 12th Century, though, Europe, and particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester translated Al-Khwarizmi's important book on algebra into Latin in the 12th Century, and the complete text of Euclid's “Elements” was translated in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited to the academic realm.
The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more important position in education.
Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century, which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

Oresme was one of the first to use graphical analysis

An important (but largely unknown and underrated) mathematician and scholar of the 14th Century was the Frenchman Nicole Oresme. He used a system of rectangular coordinates centuries before his countryman René Descartes popularized the idea, as well as perhaps the first time-speed-distance graph. Also, leading from his research into musicology, he was the first to use fractional exponents, and also worked on infinite series, being the first to prove that the harmonic series ¹⁄₁ + ¹⁄₂ + ¹⁄₃ + ¹⁄₄ + ¹⁄₅... is a divergent infinite series (i.e. not tending to a limit, other than infinity).
The German scholar Regiomontatus was perhaps the most capable mathematician of the 15th Century, his main contribution to mathematics being in the area of trigonometry. He helped separate trigonometry from astronomy, and it was largely through his efforts that trigonometry came to be considered an independent branch of mathematics. His book "De Triangulis", in which he described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to appear in print.
Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried Leibniz and Georg Cantor. He also held some distinctly non-standard intuitive ideas about the universe and the Earth's position in it, and about the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler

FIBONACCI

Leonardo of Pisa (Fibonacci) (c.1170-1250)

The 13th Century Italian Leonardo of Pisa, better known by his nickname Fibonacci, was perhaps the most talented Western mathematician of the Middle Ages. Little is known of his life except that he was the son of a customs offical and, as a child, he travelled around North Africa with his father, where he learned about Arabic mathematics. On his return to Italy, he helped to disseminate this knowledge throughout Europe, thus setting in motion a rejuvenation in European mathematics, which had lain largely dormant for centuries during the Dark Ages.
In particular, in 1202, he wrote a hugely influential book called “Liber Abaci” ("Book of Calculation"), in which he promoted the use of the Hindu-Arabic numeral system, describing its many benefits for merchants and mathematicians alike over the clumsy system of Roman numerals then in use in Europe. Despite its obvious advantages, uptake of the system in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify than Roman numerals. However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century, making the Roman system obsolete. The horizontal bar notation for fractions was also first used in this work (although following the Arabic practice of placing the fraction to the left of the integer).

The discovery of the famous Fibonacci sequence

Fibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence. He discovered the sequence - the first recursive number sequence known in Europe - while considering a practical problem in the “Liber Abaci” involving the growth of a hypothetical population of rabbits based on idealized assumptions. He noted that, after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc, and identified how the sequence progressed by adding the previous two terms (in mathematical terms, F_n = F_n-1 + F_n-2), a sequence which could in theory extend indefinitely.
The sequence, which had actually been known to Indian mathematicians since the 6th Century, has many interesting mathematical properties, and many of the implications and relationships of the sequence were not discovered until several centuries after Fibonacci's death. For instance, the sequence regenerates itself in some surprising ways: every third F-number is divisible by 2 (F₃ = 2), every fourth F-number is divisible by 3 (F₄ = 3), every fifth F-number is divisible by 5 (F₅ = 5), every sixth F-number is divisible by 8 (F₆ = 8), every seventh F-number is divisible by 13 (F₇ = 13), etc. The numbers of the sequence has also been found to be ubiquitous in nature: among other things, many species of flowering plants have numbers of petals in the Fibonacci Sequence; the spiral arrangements of pineapples occur in 5s and 8s, those of pinecones in 8s and 13s, and the seeds of sunflower heads in 21s, 34s, 55s or even higher terms in the sequence; etc.

The Golden Ratio φ can be derived from the Fibonacci Sequence

In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1 : 1.6180339887 (it is actually an irrational number equal to ^{(1 + √5)}⁄₂ which has since been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ). Essentially, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The Golden Ratio itself has many unique properties, such as ¹⁄_φ = φ - 1 (0.618...) and φ² = φ + 1 (2.618...), and there are countless examples of it to be found both in nature and in the human world.
A rectangle with sides in the ratio of 1 : φ is known as a Golden Rectangle, and many artists and architects throughout history (dating back to ancient Egypt and Greece, but particularly popular in the Renaissance art of Leonardo da Vinci and his contemporaries) have proportioned their works approximately using the Golden Ratio and Golden Rectangles, which are widely considered to be innately aesthetically pleasing. An arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral. The Golden Ratio and Golden Spiral can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human bodies to storm systems to complete galaxies.
It should be remembered, though, that the Fibonacci Sequence was actually only a very minor element in “Liber Abaci” - indeed, the sequence only received Fibonacci's name in 1877 when Eduouard Lucas decided to pay tribute to him by naming the series after him - and that Fibonacci himself was not responsible for identifying any of the interesting mathematical properties of the sequence, its relationship to the Golden Mean and Golden Rectangles and Spirals, etc.

Fibonacci introduced lattice multiplication to Europe

However, the book's influence on medieval mathematics is undeniable, and it does also include discussions of a number of other mathematical problems such as the Chinese Remainder Theorem, perfect numbers and prime numbers, formulas for arithmetic series and for square pyramidal numbers, Euclidean geometric proofs, and a study of simultaneous linear equations along the lines of Diophantus and Al-Karaji. He also described the lattice (or sieve) multiplication method of multiplying large numbers, a method - originally pioneered by Islamic mathematicians like Al-Khwarizmi - algorithmically equivalent to long multiplication.
Neither was “Liber Abaci” Fibonacci’s only book, although it was his most important one. His “Liber Quadratorum” (“The Book of Squares”), for example, is a book on algebra, published in 1225 in which appears a statement of what is now called Fibonacci's identity - sometimes also known as Brahmagupta’s identity after the much earlier Indian mathematician who also came to the same conclusions - that the product of two sums of two squares is itself a sum of two squares e.g. (1² + 4²)(2² + 7²) = 26² + 15² = 30² + 1²

16TH CENTURY MATHEMATICS

The supermagic square shown in Albrecht Dürer's engraving "Melencolia I"

The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon taking place.
It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed German artist Albrecht Dürer included an order-4 magic square in his engraving "Melencolia I". In fact, it is a so-called "supermagic square" with many more lines of addition symmetry than a regular 4 x 4 magic square (see image at right). The year of the work, 1514, is shown in the two bottom central squares.
An important figure in the late 15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who published a book on arithmetic, geometry and book-keeping at the end of the 15th Century which became quite popular for the mathematical puzzles it contained. It also introduced symbols for plus and minus for the first time in a printed book (although this is also sometimes attributed to Giel Vander Hoecke, Johannes Widmann and others), symbols that were to become standard notation. Pacioli also investigated the Golden Ratio of 1 : 1.618... (see the section on Fibonacci) in his 1509 book "The Divine Proportion", concluding that the number was a message from God and a source of secret knowledge about the inner beauty of things.

Basic mathematical notation, with dates of first use

During the 16th and early 17th Century, the equals, multiplication, division, radical (root), decimal and inequality symbols were gradually introduced and standardized. The use of decimal fractions and decimal arithmetic is usually attributed to the Flemish mathematician Simon Stevin the late 16th Century, although the decimal point notation was not popularized until early in the 17th Century. Stevin was ahead of his time in enjoining that all types of numbers, whether fractions, negatives, real numbers or surds (such as √2) should be treated equally as numbers in their own right.
In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competion that the unlikely figure of the young, self-taught Niccolò Fontana Tartaglia revealed to the world the formula for solving first one type, and later all types, of cubic equations (equations with terms including x³), an achievement hitherto considered impossible and which had stumped the best mathematicians of China, India and the Islamic world.
Building on Tartaglia’s work, another young Italian, Lodovico Ferrari, soon devised a similar method to solve quartic equations (equations with terms including x⁴) and both solutions were published by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers (although it fell to another Bologna resident, Rafael Bombelli, to explain what imaginary numbers really were and how they could be used). Tartaglia went on to produce other important (although largely ignored) formulas and methods, and Cardano published perhaps the first systematic treatment of probability.
With Hindu-Arabic numerals, standardized notation and the new language of algebra at their disposal, the stage was set for the European mathematical revolution of the 17th Century.
TARTAGLIA, CARDANO & FERRARI

Niccolò Fontana Tartaglia (1499-1557)

In the Renaissance Italy of the early 16th Century, Bologna University in particular was famed for its intense public mathematics competitions. It was in just such a competition, in 1535, that the unlikely figure of the young Venetian Tartaglia first revealed a mathematical finding hitherto considered impossible, and which had stumped the best mathematicians of China, India and the Islamic world.
Niccolò Fontana became known as Tartaglia (meaning “the stammerer”) for a speech defect he suffered due to an injury he received in a battle against the invading French army. He was a poor engineer known for designing fortifications, a surveyor of topography (seeking the best means of defence or offence in battles) and a bookkeeper in the Republic of Venice.
But he was also a self-taught, but wildly ambitious, mathematician. He distinguised himself by producing, among other things, the first Italian translations of works by Archimedes and Euclid from uncorrupted Greek texts (for two centuries, Euclid's "Elements" had been taught from two Latin translations taken from an Arabic source, parts of which contained errors making them all but unusable), as well as an acclaimed compilation of mathematics of his own.

Cubic equations were first solved algebraically by del Ferro and Tartaglia

Tartaglia's greates legacy to mathematical history, though, occurred when he won the 1535 Bologna University mathematics competition by demonstrating a general algebraic formula for solving cubic equations (equations with terms including x³), something which had come to be seen by this time as an impossibility, requiring as it does an understanding of the square roots of negative numbers. In the competition, he beat Scipione del Ferro (or at least del Ferro's assistant, Fior), who had coincidentally produced his own partial solution to the cubic equation problem not long before. Although del Ferro's solution perhaps predated Tartaglia’s, it was much more limited, and Tartaglia is usually credited with the first general solution. In the highly competitive and cut-throat environment of 16th Century Italy, Tartaglia even encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it.
Tartaglia’s definitive method was, however, leaked to Gerolamo Cardano (or Cardan), a rather eccentric and confrontational mathematician, doctor and Renaissance man, and author throughout his lifetime of some 131 books. Cardano published it himself in his 1545 book "Ars Magna" (despite having promised Tartaglia that he would not), along with the work of his own brilliant student Lodovico Ferrari. Ferrari, on seeing Tartaglia's cubic solution, had realized that he could use a similar method to solve quartic equations (equations with terms including x⁴).
In this work, Tartaglia, Cardano and Ferrari between them demonstrated the first uses of what are now known as complex numbers, combinations of real and imaginary numbers of the type a + bi, where i is the imaginary unit √-1. It fell to another Bologna resident, Rafael Bombelli, to explain, at the end of the 1560's, exactly what imaginary numbers really were and how they could be used.

Gerolamo Cardano (1501-1576)

Although both of the younger men were acknowledged in the foreword of Cardano's book, as well as in several places within its body, Tartgalia engaged Cardano in a decade-long fight over the publication. Cardano argued that, when he happened to see (some years after the 1535 competition) Scipione del Ferro's unpublished independent cubic equation solution, which was dated before Tartaglia's, he decided that his promise to Tartaglia could legitimately be broken, and he included Tartaglia's solution in his next publication, along with Ferrari's quartic solution.
Ferrari eventually came to understand cubic and quartic equations much better than Tartaglia. When Ferrari challenged Tartaglia to another public debate, Tartaglia initially accepted, but then (perhaps wisely) decided not to show up, and Ferrari won by default. Tartaglia was thoroughly discredited and became effectively unemployable.
Poor Tartaglia died penniless and unknown, despite having produced (in addition to his cubic equation solution) the first translation of Euclid’s “Elements” in a modern European language, formulated Tartaglia's Formula for the volume of a tetrahedron, devised a method to obtain binomial coefficients called Tartaglia's Triangle (an earlier version of Pascal's Triangle), and become the first to apply mathematics to the investigation of the paths of cannonballs (work which was later validated by Galileo's studies on falling bodies). Even today, the solution to cubic equations is usually known as Cardano’s Formula and not Tartgalia’s.
Ferrari, on the other hand, obtained a prestigious teaching post while still in his teens after Cardano resigned from it and recommended him, and was eventually able to retired young and quite rich, despite having started out as Cardano’s servant.
Cardano himself, an accomplished gambler and chess player, wrote a book called "Liber de ludo aleae" ("Book on Games of Chance") when he was just 25 years old, which contains perhaps the first systematic treatment of probability (as well as a section on effective cheating methods). The ancient Greeks, Romans and Indians had all been inveterate gamblers, but none of them had ever attempted to understand randomness as being governed by mathematical laws.

The circles used to generate hypocycloids are known as Cardano Circles

The book described the - now obvious, but then revolutionary - insight that, if a random event has several equally likely outcomes, the chance of any individual outcome is equal to the proportion of that outcome to all possible outcomes. The book was far ahead of its time, though, and it remained unpublished until 1663, nearly a century after his death. It was the only serious work on probability until Pascal's work in the 17th Century.
Cardano was also the first to describe hypocycloids, the pointed plane curves generated by the trace of a fixed point on a small circle that rolls within a larger circle, and the generating circles were later named Cardano (or Cardanic) circles.
The colourful Cardano remained notoriously short of money thoughout his life, largely due to his gambling habits, and was accused of heresy in 1570 after publishing a horoscope of Jesus (apparently, his own son contributed to the prosecution, bribed by Tartaglia).