Boethius Blunders
The bumpy transition from the abacus to modern arithmetic
06 January 2025
In 1503, a German monk by the name of Gregor Reisch wrote an educational encyclopaedia called Margarita Philosophica - which roughly translates to "Pearl of Wisdom". The book has twelve chapters on subjects ranging from Grammar to Astronomy.
Chapter IV is on Arithmetic and at the start of the chapter is a famous woodcut image.
On the right is a despondent-looking Pythagoras using a counting board, which was the traditional method for doing arithmetic. On the left is a happy-looking Boethius, a scholar from ancient Rome, using our modern 'Indo Arabic' numerals. In the centre Dame Arithmetic appears to be giving her approval to Boethius. The two men are competing to see who can calculate faster, and from the expression on his face, it's clear that Boethius thinks he has won.
But not so fast, Boethius! I don't suppose the illustrator intended people to check his working but I thought I'd take a closer look at Boethius' calculations.
The numerals are a bit hard to read, but at the top is a 2, then 365, 1234 and 970.
I can't be certain, but I'm pretty sure Boethius is doing a subtraction, because 1234 take away 365 is...very close to 970. Boethius has got the 1234 and 365 the wrong way around (understandable, because the sum was drawn upside down), and there's no subtraction symbol because this hasn't been invented yet.
If we do the subtraction, we can see where the solitary 2 at the top might have come from. Starting at the right, 5 from 4 can't be done so take 10 from the next column, 14 - 5 = 9, and now cross out 3(0) and make it 2(0).
Finishing off the subtraction, do we end up with 970? No! Instead it comes to 869 - each of the digits is one away from what it should be. The illustrator has made at least a couple of mistakes! *
That was the problem with this new method of doing arithmetic: it required a lot of practice, and it was easy to get it wrong. This is why, for a couple of centuries, many accountants were more confident and more accurate using traditional counting tables than using the new-fangled written arithmetic.
In Shakespeare's The Winter's Tale, a shepherd is attempting to work out the value of the wool that has been shorn from his sheep. "Every eleven sheep yields a pound and odd shilling. Fifteen hundred have been shorn. What comes the wool to?"
The arithmetic is too hard for him. "I cannot do it without counters", he sighs. Like most of the general public in 1600, the shepherd needed counters and a counting table** to do his sums. It would be another hundred years before the written arithmetic we do today was standard practice in England.
* Another possible explanation is that the illustrator started calculating from the left. 1200 - 300 = 900 (so far, so good). Then 30 - 60 = 70 (as long as you 'borrow' 100 from the left column, which might explain the crossed out 2?) But you need to 'pay back' the 1 from the borrowed 100, and maybe the illustrator paid it back to completely the wrong place, by adding it to the 4 in the units column to make 5, and 5 - 5 = 0. That leads to his incorrect answer of 970. I know this sounds convoluted, but modern children make mistakes like this all the time when doing arithmetic.
** In modern classrooms counters are known by the ungainly term "manipulatives"
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