TheLiber Abaci orLiber Abbaci[1] (Latin for "The Book of Calculation") was a 1202 Latin work onarithmetic by Leonardo of Pisa, posthumously known asFibonacci. It is primarily famous for introducing bothbase-10 positional notation and the symbols known asArabic numerals in Europe.
Liber Abaci was among the first Western books to describe theHindu–Arabic numeral system and to use symbols resembling modern "Arabic numerals". By addressing the applications of both commercial tradesmen and mathematicians, it promoted the superiority of the system and the use of these glyphs.[2]
Although the book's title is sometimes translated as "The Book of the Abacus",Sigler (2002) notes that it is an error to read this as referring to theabacus as a calculating device. Rather, the word "abacus" was used at the time to refer to calculation in any form; the spelling "abbacus" with two "b"s was, and still is in Italy, used to refer to calculation using Hindu-Arabic numerals, which can avoid confusion.[3] The book describes methods of doing calculations without aid of an abacus, and asOre (1948) confirms, for centuries after its publication thealgorismists (followers of the style of calculation demonstrated inLiber Abaci) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals). The historian of mathematicsCarl Boyer emphasizes in hisHistory of Mathematics that although "Liber abaci...isnot on the abacus"per se, nevertheless "...it is a very thorough treatise on algebraic methods and problems in which the use of the Hindu-Arabic numerals is strongly advocated."[4]
The first section introduces the Hindu–Arabic numeral system, including its arithmetic and methods for converting between different representation systems.[5] This section also includes the first known description oftrial division for testing whether a number iscomposite and, if so,factoring it.[6]
The second section presents examples from commerce, such as conversions ofcurrency and measurements, and calculations ofprofit andinterest.[7]
The third section discusses a number of mathematical problems; for instance, it includes theChinese remainder theorem,perfect numbers andMersenne primes as well as formulas forarithmetic series and forsquare pyramidal numbers. Another example in this chapter involves the growth of a population of rabbits, where the solution requires generating a numerical sequence.[8] Although the resultingFibonacci sequence dates back long before Leonardo,[9] its inclusion in his book is why the sequence is named after him today.
The fourth section derives approximations, both numerical and geometrical, ofirrational numbers such as square roots.[10]
The book also includes proofs inEuclidean geometry.[11] Fibonacci's method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematicianAbū Kāmil Shujāʿ ibn Aslam.[12]
In readingLiber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between theEgyptian fractions commonly used until that time and thevulgar fractions still in use today.[13]
Fibonacci's notation differs from modern fraction notation in three key ways:
The complexity of this notation allows numbers to be written in many different ways, and Fibonacci described several methods for converting from one style of representation to another. In particular, chapter II.7 contains a list of methods for converting an improper fraction to an Egyptian fraction, including thegreedy algorithm for Egyptian fractions, also known as the Fibonacci–Sylvester expansion.
In theLiber Abaci, Fibonacci says the following introducing the affirmativeModus Indorum (the method of the Indians), today known asHindu–Arabic numeral system or base-10 positional notation. It also introduced digits that greatly resembled the modernArabic numerals.
As my father was a public official away from our homeland in theBugia customshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle Euclidean geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.[15]
The nine Indian figures are:
9 8 7 6 5 4 3 2 1
With these nine figures, and with the sign 0 which the Arabs call zephir any number whatsoever is written...[16]
In other words, in his book he advocated the use of the digits 0–9, and ofplace value. Until this time Europe used Roman numerals, making modern mathematics almost impossible. The book thus made an important contribution to the spread of decimal numerals. The spread of the Hindu-Arabic system, however, as Ore writes, was "long-drawn-out", takingmany more centuries to spread widely, and did not become complete until the later part of the 16th century, accelerating dramatically only in the 1500s with the advent of printing.[17]
The first appearance of the manuscript was in 1202. No copies of this version are known. A revised version ofLiber Abaci, dedicated toMichael Scot, appeared in 1228.[18][19] There are at least nineteen manuscripts extant containing parts of this text.[20] There are three complete versions of this manuscript from the thirteenth and fourteenth centuries.[21] There are a further nine incomplete copies known between the thirteenth and fifteenth centuries, and there may be more not yet identified.[20][21]
There were no known printed versions ofLiber Abaci until Boncompagni's edition of 1857. The first complete English translation was Sigler's text of 2002.[20]
