During the Dark Ages in Europe from the 4th to 12th centuries, mathematical study stagnated while Chinese, Indian, and Islamic mathematicians advanced the field. European knowledge was limited until the 12th century when trade with the East began spreading foreign ideas and practical needs drove more study of arithmetic. The 15th century printer furthered mathematics education, and figures like Fibonacci, Oresme, and Regiomontatus made important contributions that advanced European mathematics.

1. 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

2. 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.
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

3. 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.
LEONARDO FIBONACCI – ITALIAN
MATHEMATICIAN (WROTE LEBER ABACI)
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

4. 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).
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, Fn = Fn-1 + Fn-2), a
sequence which could in theory extend indefinitely.

5. 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.

6. The Golden Ratio φ
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

7. 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.
Lattice Multiplication
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
.

8. 16TH CENTURY MATHEMATICS
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.
The Supermagic Square
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 “super magic 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.

9. The supermagic square shown in Albrecht Dürer’s engraving “Melencolia I”
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

10. 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

11. 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.