Fibonacci's Liber Abaci (Book of Calculation)

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Liber Abaci (Book of Calculation)

In 1202, Leonardo Pisano (Leonardo of Pisa), also called Leonardo Fibonacci (Filius Bonaccii, son of Bonaccio), published one of the most influential books ever published in mathematics.

His book, Liber Abaci (Book of Calculation), introduced the Hindu numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to Europe, along with the Latinized Arabic word, zephirum, which became zefiro in Italian, and in the Venetian dialect, zero, the name by which we know it in English, and several other European languages, today.

Fibonacci's book also introduced to Europe the notion of an algorithm, which derives from the name of the Persian scholar al-Khwarizmi (ca. 780--850), and the subject of algebra, which comes from the title of al-Khwarizmi's book, Hisab Al-Jabr wal Mugabalah (Book of Calculations, Restoration and Reduction).

The cover page of the 1857 reprint edited by Baldassarre Boncompagni reads


but page 1 begins Incipit liber Abaci Compositus a leonardo filio Bonacij Pisano in Anno. Mccij. Most English-language references to the book call it Liber Abaci, but since both that spelling and Liber Abbaci are used in the book, either is probably acceptable. A Web search in late 2005 showed that the single-b form, Abaci, is about 40 times more common than the double-b spelling, Abbaci, in Web documents.

In his famous book series, The Art of Computer Programming , Donald Knuth refers several times to Fibonacci's work, citing the 1857 Latin reprint.

Liber Abaci in English translation

In 2002, the 800th anniversary of Liber Abaci, the book was republished for the first time in a modern language:

@String{pub-SV                  = "Spring{\-}er-Ver{\-}lag"}
@String{pub-SV:adr              = "Berlin, Germany~/ Heidelberg, Germany~/
                                  London, UK~/ etc."}

  author =       "L. E. (Laurence E.) Sigler",
  title =        "{Fibonacci}'s Liber Abaci: {A} Translation into Modern
                 {English} of {Leonardo Pisano}'s {Book of Calculation}",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "viii + 636",
  year =         "2002",
  ISBN =         "0-387-95419-8",
  LCCN =         "QA32 .F4713 2002",
  bibdate =      "Tue Jun 10 11:44:26 2003",
  price =        "US\$99.00",
  series =       "Sources and studies in the history of mathematics and
                 physical sciences",
  acknowledgement = ack-nhfb,
  annote =       "First English of the original Latin, 800 years later.
                 This book introduced to Europe the Hindu numerals 0
                 through 9, the word zero, the notion of an algorithm
                 (named after the Persian scholar Abu 'Abd Allah
                 Muhammad ibn Musa al-Khwarizmi (ca. 780--850)), and the
                 subject of algebra, which comes from the title of
                 al-Khwarizmi's book, {\em Hisab Al-Jabr wal Mugabalah}
                 ({\em Book of Calculations, Restoration and

Unfortunately, Prof. Sigler did not live to see this book in print.

Large pictures are available for the front cover in GIF, PNG, and TIFF formats, and similarly for the back cover: GIF, PNG, and TIFF, There are also small front and back views in PNG format.

Fibonacci's rapidly-reproducing rabbits

Pages 404--405 of the English edition discusses the famous Fibonacci rabbit population problem, which can be summarized like this:

How many pairs of rabbits can be produced in a year from a single pair if each pair produces a new pair every month, each new pair reproduces starting at the age of one month, and rabbits never die?

The general mathematical solution for the rabbit population is a series with terms 1, 1, 2, 3, 5, 8, 11, ..., expressed by the initial conditions F(0) = 0 and F(1) = 1, with the recursion F(n) = F(n-1) + F(n-2). The total number of rabbit pairs after the births at the start of month n is F(n+1). The answer to Fibonacci's question is the count at the start of year 2, F(14) = 377. At the start of year 3, F(26) = 121,393. At the start of year 4, F(38) = 39,088,169. At the start of year 5, F(50) = 12,586,269,025.

  n                                          F(n)
  0                                             0
 10                                            55
 20                                         6,765
 30                                       832,040
 40                                   102,334,155
 50                                12,586,269,025
 60                             1,548,008,755,920
 70                           190,392,490,709,135
 80                        23,416,728,348,467,685
 90                     2,880,067,194,370,816,120
100                   354,224,848,179,261,915,075
110                43,566,776,258,854,844,738,105
120             5,358,359,254,990,966,640,871,840
130           659,034,621,587,630,041,982,498,215
140        81,055,900,096,023,504,197,206,408,605
150     9,969,216,677,189,303,386,214,405,760,200

Problem solving in the Middle Ages

Life was harder in 1202, especially for students solving word problems. Here is Sigler's translation of Fibonacci's original statement; imagine doing this problem in Latin:

How Many Pairs of Rabbits Are Created by One Pair in One Year. [26]

A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Because the abovewritten pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs; of these in one month two are pregnant, and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; in this month 3 pairs are pregnant, and in the fourth month there are 8 pairs, of which 5 pairs bear another 5 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the sixth month 21 pairs; [p284] to these are added the 13 pairs that are born in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will be 55 pairs in this month; to these are added the 34 pairs that are born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; there will be 144 pairs in this month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the abovewritten pair in the mentioned place at the end of the one year.

You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth, and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the abovewritten sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

Liber Abaci is full of word problems like this, going from simple arithmetic through fractions, then on to commercial calculations, unit conversions, square roots, cube roots, binomials, root finding, and Euclidean geometry.

Latin and rabbits, or, Lingua latina et coniculorii

There isn't yet much text from the year 1202 on the World-Wide Web, but here is the initial part of the text of the original Latin, from pp. 283--284 of the 1857 Scritti di Leonardo Pisano: mathematico del secolo decimoterzo, edited by Baldassarre Boncompagni, Roma, as posted on the Math Forum's Epigone discussion archives by Michel Ballieu <michel.ballieu at>:, and then completed with data obtained by optical character recognition of a photocopy of those two pages. Imagine reading it in a cold, dank, and dimly-light room, with a flickering candle, trying to translate the hand-written Latin into your native tongue ...

Quot paria coniculorum in uno anno ex uno pario germinentur.

Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique pariete circundatus, ut sciret, quot ex eo paria germinarentur in uno anno: cum natura eorum sit per singulum mensem aliud par germinare; et in secundo mense ab eorum natiuitate germinant. Quia suprascriptum par in primo mense germinat, duplicabis ipsum, erunt paria duo in uno mense. Ex quibus unum, scilicet primum, in secundo mense geminat; et sic sunt in secundo mense paria 3 ; ex quibus in uno mense duo pregnantur; et geminantur in tercio mense paria 2 coniculorum ; et sic sunt paria 5 in ipso mense; ex quibus in ipso pregnantur paria 3; et sunt in quarto mense paria 8; ex quibus paria 5 geminant alia paria 5: quibus additis cum parijs 8, faciunt paria 13 in quinto mense; ex quibus paria 5, que geminata fuerunt in ipso mense, non concipiunt in ipso mense, sed alia 8 paria pregnantur; et sic sunt in sexto mense paria 21; cum quibus additis parijs 13, que geminantur in septimo , erunt in ipso paria 34 ; cum quibus additis parijs 21, que geminantur in octauo mense, erunt in ipso paria 55; cum quibus additis parjis [sic] 34, que geminantur in nono mense, erunt in ipso paria 89; cum quibus additis rursum parijs 55, que geminantur in decimo mense 144; cum quibus additis rursum parijs 89, que geminantur in undecimo mense, erunt in ipso paria 233. Cum quibus etiam additis parijs 144 , que geminantur in ultimo mense, erunt paria 377; et tot paria peperit suprascriptum par in prefato loco in capite unius anni. Potes enim uidere in hac margine, qualiter hoc operati fuimus, scilicet quod iunximus primum numerum cum secundo, uidelicet 1 cum 2; et secundum cum tercio; et tercium cum quarto; et quartum cum quinto, et sic deinceps, donec iunximus decimum cum undecimo, uidelicet 144 cum 233; et habuimus suprascriptorum cuniculorum summam, uidelicet 377 ; et sic posses facere per ordinem de infinitis numeris mensibus.

This table appears in the left margin of page 284, beside the text above:


Sources of Fibonacci's work in Latin

The University of Utah Marriott Library has the 1857 Scritti di Leonardo Pisano book on a microform (many pages of reduced black text on a solid white background) issued by the New York Public Library. Unfortunately, while the Marriott Library has a machine that can magnify the page images, they are still quite hard to read, and the companion machine that can print them is broken, unrepairable, and no longer manufactured. ... Sic transit technologicum [pardon my idiomatic Latin/Greek]. I managed to get a photocopy of the relevant pages from Smith College via Interlibrary Loan that I scanned and converted with OCR (optical character recognition) to the full original text, and used to correct errors in Michel Ballieu's posting. The word marked [sic] in the Latin text is a typographical error of transposition in the book.

In December 2005, a Web search turned up an electronic copy in PDF of Edouard Lucas' 1877 French-language book Recherches sur plusieuers ouvrages de Léonard de Pise et sur diverses questions d'arithmétique supérieuer (Research on several works of Leonardo of Pisa and on various questions of higher arithmetic). For safety, a local copy is preserved here. Page 7 of that book reproduces the typographical display of the rabbit problem from Liber Abaci.