Working Numbers

In the thirteenth-century Mediterranean, commerce transformed as merchants shifted from Roman to Indo-Arabic numerals-an alternative that better facilitated complex calculations. It has long been known that this transition stemmed from Europe's increasing exchanges with India, Persia, and the Arabic world. Yet much remains to be understood about how Indo-Arabic numerals-and the practical arithmetic they enabled-actually spread across Europe. As Raffaele Danna shows, it was hundreds of ordinary merchants, schoolmasters, and artisans who nurtured these changes, thereby driving key advances in both commerce and mathematics.

Drawing on an original catalogue of more than 1,200 practical arithmetic manuals, Danna charts the incremental spread of the new figures with unprecedented precision. While Italian merchants were the early adopters, it took nearly three centuries for Indo-Arabic numerals to become established in northern Europe. As Danna shows, adoption did not follow the routes of maritime trade. Rather, Indo-Arabic numerals moved gradually across the continent through inland networks of practitioners. Everywhere they went, the ten figures enhanced commercial practices and facilitated the emergence of a coherent language of mathematical craft. The growing social circulation of this knowledge, in turn, had a lasting impact on the economic trajectory of Western Europe. By the late sixteenth century, even academics were absorbing lessons from the vernacular tradition-a development that led to the first major breakthroughs in European mathematical theory since antiquity.

Combining economic history with the social history of mathematics, Working Numbers illuminates the integral role of practical arithmetic in both intellectual and commercial transformations across Western Europe.