Greek and Islamic mathematics were basically “academic” enterprises, having little interaction with day-to-day matters involving building, transportation, and commerce. This situation first began to change in Italy in the 13th and 14th centuries. In particular, the rise of Italian mercantile companies and their use of modern financial instruments for trade with the East, such as letters of credit, bills of exchange, promissory notes, and interest calculations, led to a need for improved methods of bookkeeping.
Leonardo Pisano, known to history as Fibonacci, studied the works of Kāmil and other Arabic mathematicians as a boy while accompanying his father’s trade mission to North Africa on behalf of the merchants of Pisa. In 1202, soon after his return to Italy, Fibonacci wrote Liber Abbaci (“Book of the Abacus”). Although it contained no specific innovations, and although it strictly followed the Islamic tradition of formulating and solving problems in purely rhetorical fashion, it was instrumental in communicating the Hindu-Arabic numerals to a wider audience in the Latin world. Early adopters of the “new” numerals became known as abacists, regardless of whether they used the numerals for calculating and recording transactions or employed an abacus for doing the actual calculations. Soon numerous abacist schools sprang up to teach the sons of Italian merchants the “new math.”
The abacists first began to introduce abbreviations for unknowns in the 14th century—another important milestone toward the full-fledged manipulation of abstract symbols. For instance, c stood for cossa (“thing”), ce for censo (“square”), cu for cubo (“cube”), and R for Radice (“root”). Even combinations of these symbols were introduced for obtaining higher powers. This trend eventually led to works such as the first French algebra text, Nicolas Chuquet’s Triparty en la science des nombres (1484; “The Science of Numbers in Three Parts”). As part of a discussion on how to use the Hindu-Arabic numerals, Triparty contained relatively complicated symbolic expressions, such as
R214pR2180
(meaning:Chuquet solution ).
Chuquet also introduced a more flexible way of denoting powers of the unknown—i.e., 122 (for 12 squares) and even m12m (to indicate −12x−2). This was, in fact, the first time that negative numbers were explicitly used in European mathematics. Chuquet could now write an equation as follows:
.3.2p.12 egaulx a .9.1
(meaning: 3x2 + 12 = 9x).
Following the ancient tradition, coefficients were always positive, and thus the above was only one of several possible equations involving an unknown and squares of it. Indeed, Chuquet would say that the above was an impossible equation, since its solution would involve the square root of −63. This illustrates the difficulties involved in reaching a more general and flexible concept of number: the same mathematician would allow negative numbers in a certain context and even introduce a useful notation for dealing with them, but he would completely avoid their use in a different, albeit closely connected, context.
In the 15th century, the German-speaking countries developed their own version of the abacist tradition: the Cossists, including mathematicians such as Michal Stiffel, Johannes Scheubel, and Christoff Rudolff. There one finds the first use of specific symbols for the arithmetic operations, equality, roots, and so forth. The subsequent process of standardizing symbols was, nevertheless, lengthy and involved.
