“Bread crumbs”(perhaps how you got here)

January: Otto III and Sylvester II

The splendid illustration from the Gospels of Otto III was executed sometime in his reign, at least when Pope Sylvester II (Gerbert of Aurillac) was still alive. The artist is usually designated simply as the “Master of the Reichenau School”. The man nearest the Emperor’s right hand is the famous Pope.

 Technically, the illustration is not especially remarkable. In common with many earlier (and even late) Medieval works, there is no proper sense of perspective; the treatment of the feet is awkward and unrealistic (at least to modern eyes). Even the faces are depicted in a remarkably stereotyped, expressionless way.

On the other hand, the use of color is strong and the colors are probably still today more or less what the artist intended. (Sometimes I wonder what the works of Hans Hoffmann, for example, will look like in a thousand years - perhaps they will be mistaken for mere daubing of paint on a canvas, something to throw away and forget.)

The symbolism in the picture is important and quite clear. The Emperor himself holds the orb and sacred lance, emblems of his imperial authority. The Pope, to his right, holds what is probably a Bible or perhaps part of it; the general at Otto’s left hand has a sword. I am unaware of this man’s identity; perhaps it is not known today. But the point is clear: the Emperor is supported by both the church and the army.

It is usually thought that much of Otto’s agenda as Holy Roman Emperor was inculcated by Gerbert of Aurillac; that agenda seems still to be clear today - the establishment of firm, universal Empire, safe and indestructible. The agenda was short-lived, however; Otto became involved in local Italian politics and passions, and died in somewhat mysterious circumstances in 1002. No subsequent Holy Roman Emperor had such an ambitious program.

In some ways, Gerbert of Aurillac is a much more interesting man. Most people with at least a casual acquaintance with the Tenth Century CE tend to consider him the greatest scholar of his time. He is often credited with “reintroducing” or “rediscovering” the abacus, and perhaps with “introducing” the “Arabic” numerals to the West. This is rubbish. Gerbert was, first and foremost, a rhetorician. Most of his scholarly works are concerned with rhetoric and various theological topics. It is not at all impossible that he knew of the “Arabic” numerals, although there is no direct proof of this.

Gerbert’s mathematics was primarily concerned with the dominant mathematical issue of the Early Middle Ages in the West - the calculation of the date of Easter. Traditionally, this task was accomplished by the use of tables and (if necessary) the abacus. There is no doubt that Gerbert worked with the abacus and he seems to have made at least one extremely large abacus. This device was supposedly capable of calculating with numbers in the billions (the details are not important); it may have been physically very large. We do not, however, actually know what this immense academic showpiece was used for. It may have been used to establish (in a rhetorical way) that there is no such thing as a largest integer.

The traditional abacus of ancient times was certainly passed down through the Middle Ages and was still in use in the Renaissance. There were, at any given time, probably thousands of these devices all over Western Europe (and the entire Ancient World, probably even in the Orient). The abacus that is sometimes pictured today has beads on metal rods; this is a comparatively modern device, dating from something like the Sixteenth Century CE (and probably a Chinese invention).

The common commercial abacus that was used in ancient and medieval times looked (in its deluxe version) like a modern checkerboard. Counters placed in the righthand-most column represent integers from 0 to 9; the next column over (to the left) represented the tens, and the third column represented the hundreds. Obviously, by the time one reached the lefthand-most column, counters in that column represented the ten millions. At some time (perhaps around Gerbert’s time, perhaps earlier) there evolved the practice of putting a counter on the line between adjacent columns. The counter, when present, represented 5 ( or 50, 500, etc). This permitted the common 8 x 8 checkerboard (actually, in those days almost certainly a checkered cloth) to have two registers, much like a modern calculator that maintains subtotals and totals. In fact, given that few Medieval commercial activities involved numbers much larger than 10,000, if one wished an 8 x 8 checkerboard could be divided vertically in the middle, giving a cheap, portable, and easily understood calculator that had four registers. (Numbers larger than 10,000 are known from a few examples in ancient commerce, but I am not aware of examples in the Middle Ages.)

By a curious turn of linguistic evolution, the term calculus, which (in a mathematical context) originally referred to the “stones” or “pebbles” used in the abacus, came to be used to refer to an entirely different mathematical topic, the study of variations, rates of variations, and cumulative variations.

Gerbert wrote at least one treatise on the use of the abacus and discussed techniques for multiplication and division - but his discussion seems to be oriented around the standard calculating device, the abacus. There is no clear evidence that Gerbert knew anything like modern decimal notation, especially as to fractions.

It may well be that one of the reasons for Gerbert’s election (other than that Otto III really admired him) as Pope Sylvester II in 999 was his apparent demonstration that there was nothing very special about the year “1000.” There is no direct evidence of this, but perhaps the cardinals that voted on him liked the idea of electing a man with good mathematical credentials.

Unfortunately, as noted above, both Sylvester II and Otto III died in 1002. The papacy returned to a kind of ecclesiastical hum-drum existence, and the Holy Roman Empire gradually abandoned the plan of a universal reign of peace.

 All text on this page is the work of J W Durham and is licensed only under terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License. Other licensing terms may be available. E-mail me