Leonardo Pisano

Leonardo Pisano was the first great mathematician of medieval
Christian Europe. He played an important role in reviving
ancient mathematics and made great contributions of his own.
After his death in 1240, Leonardo Pisano became known as Leonardo
Fibonacci.

Leonardo Fibonacci was born in Pisa in about 1180, the son
of a member of the government of the Republic of Pisa. When he
was 12 years old, his father was made administer of Pisa\'s
trading colony in Algeria. It was in Algeria that he was taught
the art of calculating. His teacher, who remains completely
unknown seemed to have imparted to him not only an excellently
practical and well-rounded foundation in mathematics, but also a
true scientific curiosity.

In 1202, two years after finally settling in Pisa, Fibonacci
produced his most famous book, Liber abaci (the book of the
Calculator). The book consisted of four parts, and was revised by
him a quarter of a century later (in 1228). It was a thorough
treatise on algebraic methods and problems which strongly
emphasized and advocated the introduction of the Indo-Arabic
numeral system, comprising the figures one to nine, and the
innovation of the "zephirum" the figure zero. Dealing with
operations in whole numbers systematically, he also proposed the
idea of the bar (solidus) for fractions, and went on to develop
rules for converting fraction factors into the sum of unit
factors.

At the end of the first part of the book, he presented
tables for multiplication, prime numbers and factor numbers. In
the second part he demonstrated mathematical applications to
commercial transactions.

In part three he gave many examples of recreational
mathematical problems, much like the type which are enjoyed
today. Next he prepared a thesis on series from which was derived
what is now called the "Fibonnaci series." The "Fibonacci
Sequence" is also named after Fibonacci. The Fibonacci sequence
is a sequence in which each term is the sum of two terms
immediately preceding it. The Fibonacci Sequence that has one as
its first term is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. . . . The
numbers may also be referred to as Fibonacci numbers. Fibonacci
sequences have proven useful in number theory, geometry, the
theory of continued fractions, and genetics. They also arise in
many unrelated phenomena, for example, the Golden Section, (whose
value is 1.6180) a shape valued in art and architecture because
of its pleasing proportions, and spiral arrangement of petals and
branches on certain types of flowers and plants.

In the final part of the book Fibonnaci, a student of
Euclid, applied the algebraic method. Fibonacci\'s book, the Liber
abaci remained a standard text for the next two centuries.

In 1220 he published Practica geometriae, a book on geometry
that was very significant to future studies of the subject. In it
he uses algebraic methods to solve many arithmetical and
geometrical problems. He also published Flos (flowers) in 1224.
In this work he combined Euclidean methodology with techniques of
Chinese and Arabic origin in solving determinate problems.

Liber quadratorum was published in 1225("Book of Square
Numbers") was dedicated to the Holy Roman emperor, Frederick II.
This book was devoted entirely to Diophantine equations of the
second degree (i.e., containing squares). The Liber quadratorum
may be considered Fibonacci\'s masterpiece. It is a systematically
arranged collection of theorems, many invented by the author, who
used his own proofs to work out general solutions. Probably his
most creative work was in congruent numbers- numbers that give
the same remainder when divided by a given number. He worked out
an original solution for finding a number that, when added to or
subtracted from a square number, leaves a square number.
Leonardo\'s statement that X + Y and X - Y could not both be
squares was of great importance to the detemination of the area
of rational right triangles. Although the Liber abaci was more
influential and broader in scope, the Liber quadratorum alone
ranks its author as the major contributor to number theory
between Diophantus and Pierre de Fermat, the 17th-century French
mathematician.

Except for his roll of spreading the use of the Hindu-Arabic
numerals, Fibonacci\'s contribution to mathematics has been
largely overlooked. His name is known to modern mathematicians
mainly because of the Fibonacci Sequence dervived from a problem
in the Liber abaci:

A certain man puts a pair of rabbits in a place surrounded on
all sides by a wall. How many pairs of rabbits can be produced
from that pair in a year, if it is supposed that every month
each pair begets a new pair which from the second