Little Harvard on the River Sauer

Who is still familiar nowadays with the septem artes liberales, the “seven liberal arts”, viz. the trivium and the quadrivium, which formed the educational canon of Late Antiquity and whose mastery was a prerequisite in the Early Middle Ages for pursuing higher studies at one of the three faculties (theology, law and medicine)?

The trivium (“threefold way”) taught the arts of language: grammar (reading and writing), rhetoric (eloquent expression) and dialectic (logically sound argumentation). The quadrivium (“fourfold way”), meanwhile, was devoted to the sciences of number: arithmetic, geometry, music and astronomy.

Preserved among the collections of the BnL is a single parchment leaf, written on animal skin and measuring approximately 600 x 420 mm, that proves the above-said canon was taught in Luxembourg, specifically at Echternach, around the year 1000. The Luxembourgish state acquired it in 1951 together with the so-called Giant Bible of Echternach (BnL Ms. 264), where it had been used as a paste-down in order to reinforce the lower plate. Today, it has been separated from the plate and is kept separately under the shelfmark Ms 770.

On the verso page (not illustrated here), more than 80 diagrams can be found, all of which relate to the fields of propositional calculus and syllogistics, i.e. the final part of the trivium, whose mastery, in the ancient understanding, necessarily presupposed proficiency in the propedeutic sciences of grammar and rhetoric. When looking for the source of this outline of classical logic, one comes across the name of the last great Roman polymath of Late Antiquity, Boethius, who was beheaded around 525 as the victim of a defamatory campaign mounted against him, accused as he was of high treason under the reign of the Ostrogoth king Theodoric. Boethius had pursued an ambitious educational project: he had intended to translate the complete works of Plato and of Aristotle into Latin, and to provide them with full-fledged commentaries, a plan cut short by his premature death. What does survive, however, is his translation of the so-called Isagoge by the Neo-Platonist philosopher Porphyry (3^rd century AD). This introduction to philosophy in general and to logic in particular, known under the title Quinque voces (“Five terms”, referring to the so-called predicables: genus, species, difference, proprium and accident), was to have a decisive influence on medieval thought until well into the 12^th century. The Echternach fragment bears witness to an intensive engagement with this very text.

More interesting, however, is the front side shown here. The lower half contains elements from Boethius’ De institutione musicae, mainly concerned with the theory of intervals. The fact that questions of music theory, i.e. of the third part of the quadrivium, were discussed at a high level at the Abbey of Echternach during the Early and High Middle Ages and undoubtedly also found their way into musical practice, is confirmed by a number of other manuscripts that are still held at the BnL today.

The upper half, however, deserves particular attention. What is depicted there is a so-called “abacus”, a kind of computer avant la lettre, which made it possible to calculate powers of ten using counting stones placed in vertically arranged columns. In our fragment (which is to be read from right to left), the first three columns are missing. A complete abacus, when used correctly, allowed calculations up to the value of 10²⁶ (i.e. one hundred million billion billion) to be carried out with relative ease. The important point to notice here is that, for the first time, the numerical value was determined by position, and zero was used as a neutral number, in a sense as a placeholder, which was not possible when calculating with Roman numerals. This is precisely what makes the Echternach fragment so remarkable: in the Roman numeral system, the letters with numerical values (I, V, X, L, C, D, M) have an intrinsic value, but no positional value. Seven times fourteen is 98, but the result of VII times XIV is not immediately apparent. (The result can, incidentally, be written both as XCVIII and as IIC.)

Among historians of mathematics, it is generally agreed that the Benedictine monk Gerbert of Aurillac (946–1003) was responsible for introducing both the “Arabic” numerals (based on Indian models) and the number “zero” to Europe. On our abacus, one of the earliest surviving forms in the West of the numerals 2–9 can be seen in the smallest of the three semicircles. Gerbert taught at the cathedral school of Reims from 972 to 983; among his pupils was the young German emperor Otto III, who in 999 had him crowned pope under the name Sylvester II. He also maintained close relations with the Benedictine abbey of Mettlach, which under Abbot Ruotwic (941–975) was a centre of learning in Lotharingia. It is possible that the English monk Leofsin, who fled from Mettlach to Echternach in 993, was the person who introduced the abacus, the “Arabic” numerals and zero there. But regardless of how this transfer of knowledge may have taken place in detail, Luxembourg had a little Harvard on the River Sauer – a thousand years before the foundation of the University of Luxembourg.

This article was originally published in German. The English version was translated using AI and reviewed by the author.