''The Book of Squares'', ''(Liber Quadratorum'' in the original
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
) is a book on
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
by
Leonardo Fibonacci
Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
, published in 1225. It was dedicated to
Frederick II, Holy Roman Emperor
Frederick II (German language, German: ''Friedrich''; Italian language, Italian: ''Federico''; Latin: ''Federicus''; 26 December 1194 – 13 December 1250) was King of Sicily from 1198, King of Germany from 1212, King of Italy and Holy Roman Em ...
.
The ''Liber quadratorum'' has been passed down by a single 15th-century
manuscript
A manuscript (abbreviated MS for singular and MSS for plural) was, traditionally, any document written by hand – or, once practical typewriters became available, typewritten – as opposed to mechanically printing, printed or repr ...
, the so-called ms. ''E 75 Sup.'' of the
Biblioteca Ambrosiana
The Biblioteca Ambrosiana is a historic library in Milan, Italy, also housing the Pinacoteca Ambrosiana, the Ambrosian art gallery. Named after Ambrose, the patron saint of Milan, it was founded in 1609 by Cardinal Federico Borromeo, whose agen ...
(Milan, Italy), ff. 19r-39v. During the 19th century, the work has been published for the first time in a printed edition by
Baldassarre Boncompagni Ludovisi, prince of Piombino.
Appearing in the book is
Fibonacci's identity, establishing that the set of all sums of two
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
is closed under multiplication. The book anticipated the works of later mathematicians such as
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
and
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. The book examines several topics in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, among them an
inductive method for finding
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s based on the sequence of odd
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, the fact that the sum of the first
odd integers is
, and the solution to the
congruum problem.
[McClenon, R. B., "Leonardo of Pisa and his Liber Quadratorum", ''American Mathematical Monthly'', Vol. 26, No. 1, January 1919, pp. 1–8.]
Notes
Further reading
* B. Boncompagni Ludovisi, ''Opuscoli di Leonardo Pisano secondo un codice della Biblioteca Ambrosiana di Milano contrassegnato E.75. Parte Superiore'', in Id., ''Scritti di Leonardo Pisano matematico del secolo decimoterzo'', vol. II, Roma 1862, pp. 253–283
* P. Ver Eecke, ''Léonard de Pise. Le livre des nombres carrés.'' Traduit pour la première fois du Latin Médiéval en Français, Paris, Blanchard-Desclée - Bruges 1952.
* G. Arrighi, ''La fortuna di Leonardo Pisano alla corte di Federico II'', in ''Dante e la cultura sveva. Atti del Convegno di Studi, Melfi, 2-5 novembre 1969'', Firenze 1970, pp. 17–31.
* E. Picutti, ''Il Libro dei quadrati di Leonardo Pisano e i problemi di analisi indeterminata nel Codice Palatino 557 della Biblioteca Nazionale di Firenze'', in «Physis. Rivista Internazionale di Storia della Scienza» XXI, 1979, pp. 195–339.
* L.E. Sigler, ''Leonardo Pisano Fibonacci, the book of squares.'' An annotated translation into modern English, Boston 1987.
* M. Moyon, ''Algèbre & Practica geometriæ en Occident médiéval latin: Abū Bakr, Fibonacci et Jean de Murs'', in ''Pluralité de l’algèbre à la Renaissance'', a cura di S. Rommevaux, M. Spiesser, M.R. Massa Esteve, Paris 2012, pp. 33–65.
External links
Fibonacci and Square Numbersa
{{DEFAULTSORT:Book of squaress
1225 books
13th-century Latin books
Mathematics books
Squares in number theory