This woodcut illustration from the early 1500s shows what appears to be a competition between an algorist and an abacist. What may be the muse of arithmetic looks over two men; one on the left engaged with Hindu-Arabic algorithmic computation and one on the right using counters on a calculation “banc” to perform an abacus type of computation. The expressions on the faces of the two men suggest that the algorist has the advantage; the algorist looking somewhat smugly at the abacist’s table, the abacist looking forlornly into space contemplating perhaps his limited future. The muse favours the algorist with her gaze and half-smile.
The banners with the names Boetius and Pythagoras are, to my mind, enigmatic. Although Boetius seems to be associated with the algorist, Hindu-Arabic notation was not known in Europe at his time (c. 500 CE). And it’s unlikely Pythagoras was interested in computation itself. The interests of both men centred on number theory and the relationship of arithmetic to music rather than the use of numbers for everyday computation.
The woodcut is symbolic of the protracted period of transition from Roman numeration and the use of the abacus for calculation to the Hindu-Arabic numeration system and the use of symbolic computation.
In Mathematics Education Across Time and Place
The following extract from the memoirs of Olinto Bernardini of Pisa (Chapter 3: The Italian Renaissance) gives us some understanding of how students of that time were prepared for their future in the world of trade and commerce.
Naturally, it was assumed that my two brothers and I would each assume a place in the family Company. It was also assumed that we would become competent in the mathematical skills used to run businesses in our parts. Therefore, at the age of ten, as was the custom, my father sent me to begin the study of abaco. This course, under Maestro Pietro Cataneo of Pisa, lasted two years, but well before it was over, it had changed my life forever.
Abaco schooling normally began with an introduction to the Hindu numbers, and with an explanation of the place value that gave them meaning. But already familiar with this numeration, and well-versed in multiplication and addition facts, I quickly advanced to the next four mute [stages] where students were taught division and fractions. All this I devoured eagerly and before long had even caught up to students who were one year older. From there, the maestro introduced me to the core of abaco learning: the mathematics of business that even my great-grandfather had used in his quest to make a fortune.
At this level, that is in the sixth and seventh mute, we were taught to work with prices, barter, partnerships, alligation, proportions, monetary systems, measurements, interest and discount. (I will discuss these aspects of abaco in detail during a description of my own career as a maestro in Pisa and in Lucca) Our teacher took great care in imparting this knowledge to us, for he appreciated that he had but two years to prepare us for important positions in business.