Fibonacci

Fibonacci, also known as Leonardo of Pisa, was an Italian mathematician born around 1170 and living into the early 1240s. His writings helped disseminate the Hindu–Arabic numeral system and its methods of calculation in Latin Europe. Although best known today for the Fibonacci sequence, his work ranged across commercial arithmetic, algebra, geometry, and number theory. His principal book, Liber abaci, was completed in 1202 and revised in 1228. (library.ethz.ch)

Life and name

Much of the surviving information about Leonardo’s early life comes from the autobiographical preface to Liber abaci. His father, Guglielmo, represented Pisan merchants at Bugia, now Béjaïa in Algeria. Leonardo received instruction there in calculation with Indian numerals. He subsequently encountered mathematical practices in Egypt, Syria, Greece, Sicily, and Provence, and returned to Pisa around 1200. His education thus developed within the commercial and intellectual networks of the Mediterranean. (mathshistory.st-andrews.ac.uk)

Leonardo attracted the attention of scholars associated with Emperor Frederick II. Around 1225, Johannes of Palermo presented mathematical challenges to him; Leonardo recorded solutions in Flos. A surviving municipal document also records payment by Pisa for his services. Accounts date this document to 1240 or 1241; his exact death date is unknown, and the frequently given date of approximately 1250 is conjectural. (mathshistory.st-andrews.ac.uk)

“Fibonacci” is a posthumous designation, not a securely documented contemporary name. It became established through later historical writing. ETH Library’s historical account associates it with membership in the Bonacci family and notes that the familiar explanation “son of Bonacci” is not the only interpretation of its formation. Leonardo himself used the name Bigollo, whose meaning is uncertain. (library.ethz.ch)

Liber abaci and written calculation

*Liber abaci*, usually translated as The Book of Calculation, systematically explained arithmetic using nine numerical digits and zero. Its central technique was decimal positional notation: a digit’s value depends on its position within a number. The book connected numerical procedures with practical problems rather than treating notation as an isolated subject. It also drew on earlier mathematical traditions, including those of al-Khwarizmi, Abu Kamil, and Euclid. (mathshistory.st-andrews.ac.uk)

Its commercial examples concerned prices, profits, currency conversions, and other calculations arising in Mediterranean commerce. More advanced sections addressed equations, numerical approximations, and recreational problems. This combination made the work both a mathematical treatise and a resource for practical calculation. (mathshistory.st-andrews.ac.uk)

Fibonacci did not invent the numeral system. His contribution was to explain and promote an existing body of arithmetic in a substantial Latin work. Its adoption was gradual: written calculation with the new digits coexisted and competed with established methods using Roman numerals and the abacus. (library.ethz.ch)

The rabbit problem

One problem in Liber abaci asks how a population of rabbit pairs increases when each mature pair produces another pair monthly and newly born pairs require time before reproducing. The calculation yields the numbers

with each number obtained by adding the preceding two. In modern notation, the sequence is commonly defined by the recurrence relation

The recurrence expresses the addition of new pairs to those already present under the problem’s stipulated reproduction rules. (mathshistory.st-andrews.ac.uk)

The sequence was known before Fibonacci: Indian scholars, including Gopāla and Hemachandra, had described the same numbers while counting rhythmic patterns involving short and long units in Sanskrit verse. Its modern association with Leonardo derives from his rabbit problem; Édouard Lucas named the sequence after him in the nineteenth century. (fibonacci-numbers.surrey.ac.uk)

Geometry and the theory of squares

Practica geometriae, written in 1220, combined geometrical propositions with practical measurement. It treated plane and solid figures and included methods useful to surveyors, such as determining otherwise inaccessible heights through similar triangles. Its theoretical foundations drew substantially on Euclid’s *Elements*. (library.ethz.ch)

Liber quadratorum, or The Book of Squares, written in 1225, investigated relationships among square numbers and rational squares. Among its problems was finding a rational number whose square remains a square after either adding or subtracting five:

The work also examined constructions related to the Pythagorean theorem. Unlike modern presentations, its arguments used verbal descriptions rather than contemporary symbolic algebra. (mathshistory.st-andrews.ac.uk)

Transmission and later recognition

Leonardo’s works circulated as handwritten manuscripts. Their practical arithmetic and geometry influenced later teaching, but much of his more advanced work became less widely known. Renaissance authors preserved awareness of him, notably through Luca Pacioli’s references to his writings. Historical scholarship in the eighteenth and nineteenth centuries renewed interest in his achievements and recovered a broader picture of his contribution than the rabbit sequence alone conveys. (library.ethz.ch)

References

1. Leonardo Pisano Fibonacci — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk
2. Biography of Leonardo of Pisa — ETH Librarylibrary.ethz.ch
3. Rediscovery of Fibonacci in the eighteenth and nineteenth centuries — ETH Librarylibrary.ethz.ch
4. Fibonacci as mathematician — ETH Librarylibrary.ethz.ch
5. Fibonacci — Dictionary of Scientific Biographymathshistory.st-andrews.ac.uk
6. Who was Fibonacci?fibonacci-numbers.surrey.ac.uk
7. A000045 — OEISoeis.org
8. Fibonacci's legacy — ETH Librarylibrary.ethz.ch