Leonardo Fibonacci

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Leonardo Fibonacci
Liber abbaci , MS Biblioteca Nazionale di Firenze, Codice Magliabechiano cs cI 2616, fol. 124r: Calculation of the "rabbit problem" with Fibonacci sequence

Leonardo da Pisa , also called Fibonacci (Italian: [fiboˈnattʃi]) (* around 1170 in Pisa ; † after 1240 there), was an arithmetic master in Pisa and is considered one of the most important mathematicians of the Middle Ages .

On his travels to Africa, Byzantium and Syria, he familiarized himself with Arabic mathematics and, using the knowledge gained, wrote the arithmetic book Liber ab (b) aci in 1202 (revision 1228). The Fibonacci sequence named after him , which is related to the golden ratio, is best known today.

Origin of name

Leonardo is referred to in the manuscripts as Leonardus Pisanus , Leonardus filius Bonacij , Leonardus Pisanus de filiis Bonaccij and Leonardus Bigollus . Bonaccio (from Latin bonatius "good, cheap, pleasant") was the grandfather name that Leonardo's father Guglielmo and his brothers Alberto and Matteo had as patronyms and which had already become a family name in Leonardo's own generation. From filius Bonacii or figlio di Bonaccio ( "Son of Bonaccio") was in Italian then contraction not yet testified to the treasury in Leonardo's time Zunamensform Fibonacci , under the Leonardo mainly because of since Edouard Lucas after him named Fibonacci sequence today is best known. The epithet Bigollus , only used in the genitive form Leonardi Bigolli and therefore sometimes erroneously given in the literature as the patronymic Leonardo Bigolli , is not certain in its interpretation, but is mostly interpreted in the sense of "the well-traveled".

Life and writings

Little is known about Leonardo's biography; most of the information goes back to the dedication prologue of his arithmetic book Liber abbaci and to a document from the Commune of Siena.

Leonardo was born in Pisa in the second half of the 12th century as one of at least two sons of Guglielmo Bonacci, where the family can be traced back to Leonardo's great-grandfather, a Bonito who died at the beginning of the 12th century. When the father was sent by the city as a notary to the branch of the Pisan merchants in Bougie, today's Bejaia , in Algeria - which is assumed to be around 1192 - he also had Leonardo come to him to have him instructed in arithmetic there . There Leonardo learned to calculate with the novem figurae indorum (“nine digits of the Indians”), our current (Indo-Arabic) digits , which the Arabic mathematicians in Baghdad had known from India since the second half of the 8th century Century from Spain ( Toledo ) through Latin translations of the Arabic scripts of Al-Chwarizmi were gradually spread in the West as well.

In the nineties of the 12th century Leonardo was consequently not the first Latin to learn to calculate with the new digits, but he evidently acquired a mathematical foundation in Bougie, which he valued more than anything he found in further studies in trading places “in Egypt, Syria, Greece, Sicily and the South of France “still learned. He especially called the "algorism", which was understood as the elementary numerical calculation according to Al-Chwarizmi , from which Leonardo's own mathematics actually differed only through the more demanding application of the procedures, as well as one method , which he assessed as comparatively minor, "as it were as an error" He described the method as "Arches of Pythagoras": what is meant is the abazistic arithmetic on Gerbert's abacus , which was in use in the 10th to 12th centuries and which was largely out of use in Leonardo's time , which was considered to be the invention of Pythagoras and on the other hand to the later medieval abacus with numbered calculating stones (numbered with Arabic Ghubar digits 1–9).

His travels seem to have also taken him to Constantinople towards the end of the 12th century , as he states that one of the tasks in his Liber abbaci was given to him in Constantinople by a highly learned master named "Muscus" ( a peritissimo magistro musco constantinopolitano , ed.Boncompagni, vol. I, p. 249). A mathematician by that name, presumably Μόσκος, is not otherwise known.

After Leonardo had deepened his knowledge, as he explains in the dedication prologue, partly through his own observations and partly through studying the geometry of Euclid , he finally laid down the "summa" of his mathematical knowledge in his main work, the Liber abbaci . The title can best be translated as “book of arithmetic”, as the original ab (b) acus meaning, which was tied to the abacus , had expanded in Italy and assumed the general meaning “arithmetic” in Leonardo's time. The first version of this work, no longer preserved today, is said to have been written as early as 1202 (or 1201?), But this date is only known from the colophon of a manuscript in the second, only surviving version. In addition, with the explicit dating of Leonardo's writings, there is generally the difficulty that the year after mos pisanus began on March 25 of the previous year - from the point of view of ordinary annual counting - so that a year must be deducted from such dates if the date is not in last quarter of the year (from January to March 24th).

Several other works by Leonardo have survived: a Practica geometriae from 1220 (1219?), Dedicated to a friend and teacher Dominicus, which was also translated into Italian by Cristoforo Gherardo di Dino in the 15th century; a Liber quadratorum from 1225 (1224?), which is dedicated to Frederick II and mentions that the latter had already read a book by Leonardus, which is commonly referred to as the Liber abbaci ; Furthermore, an undated text Flos super solutionibus quarumdam questionum ad numerum et ad geometriam uel ad utrumque pertinentium , which is dedicated to Cardinal Raniero Capocci of Viterbo and deals with questions that are said to have been presented to Leonardo in the presence of Frederick II by a Magister Johannes from Palermo ; and finally a letter to a Magister Theodorus. From Leonardo's writings it emerges that he also wrote two other writings that are no longer extant today, a shorter arithmetic book and a commentary on the tenth book of the elements of Euclid.

According to the dedication prologue , the second version of Liber abbaci was written for Michael Scotus († around 1236) after the latter asked Leonardo for a copy of the work and Leonardo had made some additions and cuts on the occasion. Since Michael Scotus is attested from the autumn of 1227 at the court of Frederick II, 1227 has also been assumed as the date of origin for the second version of Liber abbaci that has survived , but it can actually have been written earlier or later, but not before 1220 (1219?), since it already refers to the Practica geometriae .

The last mention of Leonardos can be found in a decree of the Commune of Pisa, which honors him as the respected Magister Leonardus Bigollus for his services as tax appraiser and arithmetic master of the city and an annual salary of twenty pounds pfennigs plus the usual for such officials for future services of this kind Granted in kind. The editor Bonaini had dated the document, which was not dated in the text, to 1241, but without giving reasons. If the dating is correct, Leonardo died not before 1241 (because of the divergence of the mos pisanus in the research sometimes also stated “not before 1240”), so that if the year of birth is set to 1180, he is an age of not insignificant for the time would have reached at least sixty years and even at this age would have been earmarked for further services by the municipality.

The content of the Liber abbaci

The Liber abbaci explicitly focuses more on theory than practice ( magis ad theoricam spectat quam ad practicam ) and actually goes far beyond everything that was known to the Latin Middle Ages up to then or up to the 16th century got known. The specialty lies not so much in the difficulty of the tasks, but in the mathematical intelligence of the author, his penetration of matter and the special value that he places on not only demonstrating solutions and rules, but also proving them mathematically. The Liber abbaci is divided into 15 capitula:

  1. De cognitione nouem figurarum yndorum, et qualiter cum eis omnis numerus scribatur; et qui numeri, et qualiter retineri debeant in manibus, et de introductionibus abbaci : About the knowledge of the nine numerals of the Indians, and how every number is written with them; and how the numbers are to be noted with the hands, and of the introduction of arithmetic.
  2. De multiplicatione integrorum numerorum : From the multiplication of natural numbers.
  3. De additione ipsorum ad inuicem : From the addition of these together.
  4. De extractione minorum numerorum ex maioribus : From the subtraction of smaller numbers from larger ones.
  5. De diuisione integrarum (sic) numerorum per integros : From the division of natural numbers by natural numbers.
  6. De multiplicatione integrarum (sic) numerorum cum ruptis atque ruptorum sine sanis : From the multiplication of natural numbers with fractions and the multiplication of fractions without a whole.
  7. De additione ac extractione et diuisione numerorum integrarum cum ruptis atque partium numerorum in singulis partibus reductione : Of the addition, subtraction and division of natural numbers with fractions and the decomposition of fractions into parent fractions.
  8. De emptione et venditione rerum uenalium et similium : On buying and selling goods and similar things. - Covers the rule of three, conversion of currencies, cloth and other dimensions and weights.
  9. De baractis rerum uenalium et de emptione bolsonalie, et quibusdam regulis similibus : About bartering with goods and buying Bolsonalien (coins whose value depends on their silver content), and some similar rules.
  10. De societatibus factis inter consocios : From the companies among shareholders. - First arithmetic problems on forage, tree felling and food are dealt with, then the distribution of profits among shareholders according to their title according to their share of the capital employed.
  11. De consolamine monetarum atque eorum regulis, que ad consolamen pertinent : On the alloy of money and the rules that affect the alloy. - It is specifically a matter of producing a new alloy with a specified silver content from copper-silver alloys with a known silver content.
  12. De solutionibus multarum positarum questionum quas erraticas appellamus : Of the solutions to many questions that we call erratic. - The most extensive chapter, which takes up about a third of the entire work, is in turn divided into nine sub-chapters:
    1. De collectionibus numerorum, et quarundam aliarum similium questionum : From the collections of numbers and some similar questions. - The summation of arithmetic series is dealt with.
    2. De proportionibus numerorum : Of number proportions. - Covers systems of linear equations.
    3. De questionibus arborum, atque aliarum similium, quarum solutiones fiunt : Of problems with trees, and other similar problems whose solutions they (i.e. the proportions) offer. - Application of the rules discussed in the previous subsection.
    4. De inuentione bursarum : Finding purses. - Continuation of the topic with arithmetic problems that revolve around found purses.
    5. De emptione equorum inter consocios, secundum datam proportionem : From the purchase of horses among shareholders, according to a given proportion.
    6. De uiagiis, atque equorum questionum, que habent similitudinem uiagiorum questionibus : Of traveling and tasks with horses that are similar to tasks with journeys. - Treated u. a. Interest duties.
    7. De reliquis erraticis, que ad inuicem in eorum regulis uariantur : Of the other erratic tasks, which differ from one another in their solution paths . - Contains u. a. the famous rabbit problem, which Leonardo deals with rather briefly and casually, and which is apparently the only application of the Fibonacci sequence in his writings .
    8. De quibusdam diuinationibus : Of some advice tasks. - Tasks on residual problems, where z. B. A number can be guessed by the remainder of its division by several other numbers.
    9. De Duplicatione scacherii, et quibusdam aliis questionibus : Of the doubling on the chessboard, and some other tasks. - Exercises around the number (2 ^ 64) -1
  13. De regula elcataym qualiter per ipsam fere omnes erratice questiones soluantur : From the rule “al-hata'ain”, how it can solve almost all wrong tasks. - Treats the rule of the double wrong approach ( regula duarum falsarum posicionum ), now also called regula falsi or "linear input", which calculates the correct solution from two incorrect solutions for linear problems.
  14. De reperiendis radicibus quadratis et cubitis ex multiplicatione et diuisione seu extractione earum inter se, et de tractatu binomiorum et recisorum et eorum radicum : From finding square and cube roots by multiplying and dividing or subtracting them from one another, and from binomials and differences and their roots .
  15. De regulis proportionibus geometrie pertinentibus: de questionibus aliebre almuchabale : Of the rules that concern the proportions of geometry: of the tasks of algebra and almuchabala. - About quadratic equations.

Biographical evidence

From the dedicatory prologue of Liber abbaci , ed. B. Boncompagni, vol. I, Rome 1857, p. 1:

"Cvm genitor meus a patria publicus scriba in duana bugee pro pisanis mercatoribus ad eam confluentibus constitutus preesset, me in pueritia mea ad se uenire faciens, inspecta utilitate et commoditate futura, ibi me studio abbaci per aliquot dies stare uoluit et doceri. Vbi ex mirabili magisterio in arte [m] per nouem figuras indorum introductus, scientia artis in tantum mihi pre ceteris placuit, et intellexi ad illam, quod quicquid studebatur ex ea apud egyptum, syriam, graeciam, siciliam et prouinciam modis suis uariis que loca negotiationis tam postea peragraui per multum studium et disputationis didici conflictum. Sed hoc totum etiam et algorismum atque arcus pictagore quasi errorem computaui respectu modi indorum. "

“When my producer was seconded from his hometown to the Bougie trading post for the sake of the Pisan merchants who met there, he had me come to when I was a boy. In view of the future utility and advantage, he wanted me to stay there for a few days and receive instruction at the school of arithmetic. When, out of admirable mastery, I was introduced to the art of the nine numerals of the Indians, and so much I liked the science of this art more than any other, and I tried to gain insight into it, that I could do whatever of it with its various kinds to be learned in Egypt, Syria, Greece, Sicily and southern France, I acquired on later trips to these trading locations with a great deal of study and disputations. But I thought all this and also the algorism and the arcs of Pythagoras to be a mistake in comparison to the calculation method of the Indians. "

From the Constitutum usus pisanae civitatis , quotation and translation from H. Lüneburg: Leonardo Pisanos Liber abbaci . In: Der Mathematik- Lehr 42,3 (1996), p. 31–42, p. 31:

“Considerantes nostre civitatis et civium honorem atque profectum, qui eis tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium, et aliis quoties expedit, conferunter; ut eidem Leonardo, merito, dilectionis et gratie, atque scientie sue prerogativa, in recompensatione laboris sui, quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a communi et camerariis publicis de communi et pro communi mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant; ipseque Pisano communi et eius officialibus in abbacatione de cetero, more solito, servat; presenti constitutione firmamus (...). "

“In view of our city and the citizens, honor and advantage, which, as is often the case when necessary, comes to them both through the erudition and the diligent service of the excellent and clever man and teacher Leonardo Bigollo, who is responsible for calculating (tax) estimates and There are invoices for the city and its officials and others, we stipulate by this constitution that this same Leonardo out of appreciation and favor, due to the merit and due to the priority of his knowledge to compensate for his work, which he carries out by examination and determination of the above Estimates and bills, by the community and its treasurers - appointed by the community and acting on behalf of the community - must be given as wages or his salary XX pennies annually and the usual natural benefits and that he will henceforth like the community of Pisa and its officials used by executing invoices. "

To the statue of Leonardos

Leonardo's statue, Camposanto di Pisa , 1863

In Pisa, in the cloister of the historic Camposanto cemetery, there is a statue of Leonardos with the inscription: A Leonardo Fibonacci Insigne Matematico Pisano del Secolo XII . As a portrait , the representation is a product of artistic imagination, as there are no images from Leonardo's own time and no tradition about his appearance.

The statue goes back to the initiative of two members of the provisional government of the former Grand Duchy of Tuscany , Bettino Ricasoli and Cosimo Ridolfi, who passed a decree on September 23, 1859 to finance the statue. The Florentine sculptor Giovanni Paganucci was commissioned and completed the work in 1863. The statue was placed in Pisa on the Campo Santo, where funerary monuments of Pisan citizens, together with ancient sarcophagi and newly added works of art, form a unique tomb and memorial ensemble since the Middle Ages.

At the time of fascism , the authorities in Pisa decided to move the statue of Leonardos in 1926 as well as two statues of other well-known citizens of Pisa out of the sacred seclusion of Campo Santo to more publicly visible locations. The statue of Leonardos was placed at the southern end of the Ponte di Mezzo. During the Second World War in 1944, the bridge was destroyed in the fighting for Pisa and the statue was also damaged, which initially remained at its location, then was kept in a warehouse and was temporarily forgotten. In the 1950s it was rediscovered, poorly restored and placed in the Giardino Scotto Park at the eastern entrance to the old town. It was not until the 1990s that the Pisan city administration decided to restore the statue and put it back in its original location in Campo Santo.


  • Baldassare Boncompagni , Tre scritti inediti di Leonardo Pisano pubblicati da Baldassare Boncompagni secondo la lezione di un codice della Biblioteca Ambrosiana di Milano , Florence: Tipografia Galileiana di M. Cellini e C., 1854 ( digitized from Google Books ), 2nd edition: Opuscoli di Leonardo Pisano pubblicati da Baldassare Boncompagni secondo la lezione di un codice della Biblioteca Ambrosiana di Milano, Seconda edizione , Florence: Tipografia Galileiana di M. Cellini e C., 1856 ( digitized at Google Books ; digitized at the Göttingen digitization center)
  • Baldassare Boncompagni, Scritti di Leonardo Pisano matematico del secolo decimoterzo , Roma: Tipografia delle scienze matematiche e fisiche; vol. I: Il liber abbaci pubblicato secondo la lezione del codice Magliabechiano C. I, 2616, Badia Fiorentina, no. 73 (1857); vol. II: Practica Geometriae et Opuscoli (1862) ( digitized copies of both volumes in the Göttingen digitization center; digitized copies of volume 1 and volume 2 in the Munich digitization center)
  • Paul ver Eecke , Léonard de Pise, Le livre des nombres carrées. Traduit pour la première fois du latin médiéval en français, with an introduction et des notes . Bruges: Desclée, De Brouwer, 1952
  • Gino Arrighi, La pratica di geometria volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano, dal codice 2186 della Biblioteca Riccardiana di Firenze . Pisa: Domus Galilaeana, 1966 (= Testimonianze di storia della scienza, 3)
  • Lucia Salomone, È chasi della terza parte del XV capitolo del Liber Abaci nella trascelta a cura di maestro Benedetto: secondo la lezione del codice L.IV.21 (sec. XV) dell Biblioteca Comunale di Siena . Siena: Servizio Editoriale dell'Università, 1984 (= Quaderni del Centro Studi della Matematica Medioevale, 10)
  • Laurence E. Sigler, Leonardo Pisano Fibonacci, The book of squares: an annotated translation into modern English , Boston / London: Academic Press, 1987, ISBN 0-12-643130-2
  • Jean-Pierre Levet, Léonard de Pise, Des chiffres hindous aux racines cubiques: extraits du Liber abaci, introduction, traduction et brefs commentaires mathématiques et philologiques , Poitiers: IREM, 1997 (= Cahiers d'histoire des mathématiques et d'épistémologie)
  • Jean-Pierre Levet, Léonard de Pise, Divisions et portions, perles et animaux , Poitiers: IREM, 1997 (= Cahiers d'histoire des mathématiques et d'épistémologie)
  • Laurence E. Sigler, Fibonacci's Liber Abaci. A Translation into Modern English of Leonardo Pisano's Book of Calculation , New York: Springer, 2002, ISBN 0-387-95419-8 , on this critical Heinz Lüneburg, review ( Memento of March 21, 2009 in the Internet Archive )
  • Barnabas Hughes, Fibonacci's De Practica Geometrie , New York: Springer, 2008, ISBN 978-0-387-72930-5 (English translation with commentary, without reproduction of the Latin text)


  • Leonardo Fibonacci: matematica e società nel Mediterraneo nel secolo XIII , Pisa: Istituti editoriali e poligrafici internazionali, 2005, ISBN 88-8147-374-7 , special issues of the Bollettino di storia delle scienze matematiche, anno 23, num. 2 (Dec. 2003), anno 24, num. 1 (June 2004)
  • Heinz Lüneburg : Liber Abbaci or reading pleasure of a mathematician . 2nd, revised and expanded edition, Mannheim [et al.]: BI Wissenschaftsverlag, 1999, ISBN 3-411-15462-4
  • Heinz Lüneburg: Leonardo Pisanos Liber abbaci. In: Der Mathematik-Lehr 42,3 (1996), pp. 31–42
  • Marcello Morelli / Marco Tangheroni (eds.): Leonardo Fibonacci: il tempo, le opere, l'eredita scientifica . Pisa: Pacini, 1994
  • Maria Muccillo:  Fibonacci, Leonardo. In: Massimiliano Pavan (ed.): Dizionario Biografico degli Italiani (DBI). Volume 40:  DiFausto – Donadoni. Istituto della Enciclopedia Italiana, Rome 1991.
  • Helmuth Gericke : Mathematics in the West: From the Roman surveyors to Descartes . Berlin [et al.]: Springer, 1990, pp. 96-104, ISBN 3-540-51206-3
  • Moritz Cantor : Lectures on the history of mathematics , II: From the year 1200 to the year 1668. 2nd edition 1900, repr. New York / Stuttgart 1965 (= Bibliotheca mathematica Teubneriana, 7)
  • Édouard Lucas : Recherches sur plusieurs ouvrages de Léonard de Pise et sur diverse questions d'arithmétique supérieure . In: Bulletino di bibliografia e di storia delle scienze matematiche e fisiche 10 (1877), pp. 129-193, pp. 239-293
  • Francesco Bonaini : Memoria unica sincrona di Leonardo Fibonacci, nuovamente trovata. Pisa: Nistri, 1858
  • Baldassare Boncompagni: Intorno ad alcune opere di Leonardo Pisano, matematico del secolo decimoterzo. Rome: Tipografia delle Belle Arti, 1854 ( digitized at Google Books ; digitized in the Munich digitization center)
  • Baldassare Boncompagni: Della vita e delle opere di Leonardo Pisano matematico del secolo decimoterzo. In: Atti dell'Accademia Pontifica dei Nuovi Lincei 5 (1852), pp. 5-91, pp. 208-246
  • Keith Devlin : Finding Fibonacci - The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World . Princeton: Princeton University Press, 2017.
  • Kurt Vogel: Fibonacci, Leonardo . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 4 : Richard Dedekind - Firmicus Maternus . Charles Scribner's Sons, New York 1971, p. 604-613 .
  • R. Flood, R. Wilson: Fibonacci. In: The Great Mathematicians , Arcturus, London 2012, ISBN 978-1-84858-843-1 , pp. 89-92

Web links

Wikisource: Leonardo Fibonacci  - Sources and full texts (Latin)
Commons : Fibonacci  - collection of images, videos and audio files