MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Biographical Vignettes 153
a diplomatic post. He traveled widely with his father and became thoroughly
familiar with the Hindu-Arabic numerals and their arithmetic. He returned to
Pisa and published his Liber Abaci (Book of Calculation) in 1202. This most
famous of his works is based upon the arithmetic and algebra that he had
accumulated in his travels and served to introduce the Hindu-Arabic placevalued
decimal system, as well as their numerals, into Europe. An example
from Liber Abaci involving the breeding of rabbits gave rise to the so-called
Fibonacci numbers which he did not discover. (See Property 73 in Chapter 2.)
Simultaneous linear equations are also studied in this work. It also contains
problems involving perfect numbers, the Chinese Remainder Theorem, and the
summation of arithmetic and geometric series. In 1220, he published Practica
geometriae which contains a large collection of geometry problems arranged
into eight chapters together with theorems and proofs based upon the work of
Euclid. This book also includes practical information for surveyors. The final
chapter contains “geometrical subtleties” such as inscribing a rectangle and a
square in an equilateral triangle! In 1225 he published his mathematically most
sophisticated work, Liber quadratorum. This is a book on number theory and
includes a treatment of Pythagorean triples as well as such gems as: “There
are no x, y such that x 2 +y 2 and x 2 −y 2 are both squares” and “x 4 −y 4 cannot
be a square”. This book clearly establishes Fibonacci as the major contributor
to number theory between Diophantus and Fermat. He died in Pisa, aged 80.
Source material for Fibonacci is available in [42, 297].
Vignette 8 (Leonardo da Vinci: 1452-1519).
Leonardo da Vinci was born the illegitimate son of a wealthy Florentine
notary in the Tuscan town of Vinci [44]. He grew up to become one of the
greatest painters of all time and perhaps the most diversely talented person
ever to have lived. A bona fide polymath, he was painter, sculptor, architect,
musician, inventor, anatomist, geologist, cartographer, botanist, writer, engineer,
scientist and Mathematician. In what follows, attention is focused on the
strictly mathematical contributions dispersed amongst his legacy of more than
7,000 surviving manuscript pages. In Chapter 1, I have already described the
circumstances surrounding his illustrations of the Platonic solids for Pacioli’s
De Divina Proportione. The knowledge of the golden section which he gained
through this collaboration is reflected through his paintings. His masterpiece
on perspective, Trattato della Pittura, opens with the injunction “Let no one
who is not a Mathematician read my works.”. This admonition seems more
natural when considered together with the observations of Morris Kline: “It
is no exaggeration to state that the Renaissance artist was the best practicing
Mathematician and that in the fifteenth century he was also the most learned
and accomplished theoretical Mathematician.” [197, p. 127]. Leonardo was
engaged in rusty compass constructions [97, p. 174] and also gave an innovative
congruency-by-subtraction proof of the Pythagorean Theorem [98, p. 29].