In 1204 the Fourth Crusade, reaching the peak of its mission to undermine Christianity in the eastern Mediterranean, sacked Constantinople and established a Latin ruler in the remains of the Roman Empire, which we dub today the Byzantine Empire. This I mention because I’m reading John Julius Norwich’s A History of Venice, and it discusses one of the consequences. Venice had supported the expedition, in no small part to divert the Fourth Crusaders from attacking its trading partners in Egypt, and also to reduce Constantinople as a threat to Venice’s power. Venice got direct material rewards too, and Norwich mentions one of them:
When, on 5 August 1205, Sebastiano Ziani’s son Pietro was unanimously elected Doge of Venice, the first question that confronted him was one of identity. To the long list of sonorous but mostly empty titles which had gradually become attached to the ducal throne, there had now been added a new one which meant exactly what it said: Lord of a Quarter and Half a Quarter of the Roman Empire.
This I mention because the reward of three-eighths of the Byzantine Empire (the Byzantines considered themselves the Roman Empire, quite reasonably, and called themselves that) is phrased here in a way that just wouldn’t be said today. Why go to the circumlocution of “a quarter and half a quarter” instead of “three-eights”?
My guess is that “three-eighths” was too exotic a phrasing for the early 13th century. In ancient times while the reciprocals of numbers — a half, a third, a quarter — were tolerably familiar, fractions which weren’t reciprocals tended to be made by adding together these reciprocals. Otto Neugebauer, from whom much of our understanding of the ancient world’s computations and astronomy is derived, demonstrated the Ancient Egyptian system of numbers as working in just that fashion. I wouldn’t be surprised if the habit continued for a long, long while.
To speak of three-eights as “a quarter and half a quarter” seems longwinded these days. But it’s also got an immediately obvious, visualizable interpretation, and I wouldn’t be surprised if someone who asked for three-eights of a pie got it in two slices, one a quarter and one half that.
I do find, in historic references, points where a tax or fee would be charged as, for example, a tenth and a fifth of the amount in question. (At least, I would find them if I remembered just which book it was mentioned them, and where it was in the book, and where the book was; it’s possible the one I’m thinking of there was in Asia last I checked.) When decimal arithmetic is comfortable and easy — and when calculating devices are so abundant it’s almost hard to not have one on hand — the labor- and time-saving values of setting a fee as “a tenth and a fifth” instead of “thirty percent” isn’t obvious. If you try doing any amount of computation in your head — or if you’re trying to work out a 15 percent tip at the restaurant — the practicality of this ancient scheme returns.
That said it does make a fun little exercise to work out a good added-reciprocals form for fractions such as two-sevenths or four-fifths or 99/100ths (you can add 1/100 to itself 99 times over but that is so painfully long) or such. Not surprisingly, the people who actually had to work with this system, rather than use it for recreation, worked up tables of the sorts of numbers they’d have to deal with so that their work wouldn’t be slowed down by figuring out a good way to represent 13/89ths.