Leonardo of Pisa’s Liber Abaci (Book of Calculation) is one of the most important books in the history of European mathematics. Leonardo learned his mathematics from the Islamic mathematical tradition (some have even argued that he should be considered part of that tradition), but he wrote his books in Latin. As a result, he became one of the most important mediators of that tradition to European readers.
The Liber Abaci focused on “calculation.” From our point of view, most of the contents are either arithmetic or algebra. It starts off with an extensive and detailed discussion of the use of “the nine Indian figures” to represent numbers and to perform arithmetical operations with them. The “nine figures” are, of course, 1, 2, 3, 4, 5, 6, 7, 8, and 9, with 0 being treated as a separate sign. The book then goes on to deal with a large number of problems that we would describe as algebraic. Most of these reduce to linear equations (this includes systems of linear equations and also linear diophantine equations). The final chapter discusses quadratic equations. All of this is done, of course, without any sort of algebraic symbolism.
As Heinz Lueneberg has pointed out in a recent issue of FOCUS, this is an amazing book, and it is somewhat frustrating that we had to wait for the 800th anniversary of its original publication to see an English translation. But here it is at last, the Liber Abaci in English. It is now possible for mathematicians who are interested in history to read it, and for students to read portions of it in their history courses.
One final comment: attentive readers will note that the name “Fibonacci” does not occur in this brief review. That is because Leonardo never used that name: it is a nickname given to him by a 19th century historian of mathematics. I’m enough of a purist to want to refer to Leonardo as he referred to himself: Leonardo of Pisa, of the Bonacci family. He was a great mathematician, and here is an accessible and readable edition of his most famous book. Don’t miss the opportunity to get a copy.
Fernando Q. Gouvêa is the author of several books, including, most recently, Math through the Ages, written in collaboration with William Berlinghoff.
Liber Abaci: Here Begins the First Chapter
On the Multiplication of Whole Numbers
On the Addition of Whole Numbers
On the Subtraction of Lesser Numbers from Greater Numbers
On the Divisions of Integral Numbers
On the Multiplication of Integral Numbers with Fractions
On the Addition and Subtraction and Division of Numbers with Fractions and the Reduction of Several Parts to a Single Part
On Finding the Value of Merchandise by the Principal Method
On the Barter of Merchandise and Similar Things
On Companies and Their Members
On the Alloying of Monies
Here Begins Chapter Twelve
On the Method Elchataym and How with It Nearly All Problems of Mathematics Are Solved
On Finding Square and Cubic Roots, and on the Multiplication, Division, and Subtraction of Them, and On the Treatment of Binomials and Apotomes and their Roots
On Pertinent Geometric Rules And on Problems of Algebra and Almuchabala
Notes for Liber Abaci