# The Book of Squares

The Book of Squares, (Liber Quadratorum in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225.[1] It was dedicated to Frederick II, Holy Roman Emperor.[2]

After being brought to Pisa by Master Dominick to the feet of your celestial majesty, most glorious prince, Lord F.,

Leonardo Fibonacci, Liber quadratorum, epistle (translated by L. E. Sigler[3])

The Liber quadratorum has been passed down by a single XV century-manuscript, the so-called ms. E 75 Sup. of the Biblioteca Ambrosiana (Milan, Italy), ff. 19r-39v[4]. During the XIX century, the work has been published for the first time in a printed edition by Baldassarre Boncompagni Ludovisi, prince of Piombino[5].

Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication, appears in it. The book anticipated the works of later mathematicians like Fermat and Euler.[6] The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the first ${\displaystyle n}$ odd integers is ${\displaystyle n^{2}}$, and the solution to the congruum problem.[7]

## Notes

1. Fibonacci, Leonardo Pisano, The Book of Squares (Liber Quadratorum). An annotated translation into modern English by L. E. Sigler (1987) Orlando, FL: Academic Press. ISBN 978-0-12-643130-8
2. "Full text of "Leonardo of Pisa and his Liber Quadratorum"". archive.org. Retrieved 2016-10-04.
3. The Book of Squares (Liber Quadratorum). An annotated translation into modern English by L. E. Sigler (1987) Orlando, FL: Academic Press.
4. E. Picutti, Il «Libro dei Quadrati» di Leonardo Pisano, in «Physis. Rivista Internazionale di Storia della Scienza» XXI, 1979, pp. 195-339
5. Baldassarre 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, Tipografia delle scienze matematiche e fisiche, 1862, pp. 253-283.
6. Berlinghoff, William P. and Fernando Q. Gouvêa (2004). Math through the ages: a gentle history for teachers and others. MAA, p. 34. ISBN 0-88385-736-7
7. McClenon, R. B., "Leonardo of Pisa and his Liber Quadratorum", American Mathematical Monthly, Vol. 26, No. 1, January 1919, pp. 1–8.
• 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.