The lost mathematicians: Numbers in the (not so) dark ages
"Europe had fallen into the dark ages, in which science, mathematics and almost all intellectual endeavor stagnated." From The story of mathematics.
A commonly held belief about early medieval Europe is that academic pursuits, particularly those appertaining to mathematics and science, had fallen into a dark age, lost between the insights of antiquity and the achievements of the renaissance. The majority of learned scholars were churchmen, and the subject of their enquiry usually related to some principle of church practice. In the modern era, where religion and science are so often at loggerheads, the concept of a monkish mathematician may seem unusual; could our appreciation of medieval mathematicians be compromised by our lack of engagement with the issues of their time? Much of the priorities and practices of this era may seem obscure to us, but is there a value to respecting the tenacity of historic mathematicians?
Figure 1: Venerable Bede, the man himself (in a later representation).
At the beginning of the eighth century the Venerable Bede (in those days not so much Venerable as highly prolific, see figure 1) was a monk with a problem. Each year Easter had to be predicted with accuracy; all other moveable feasts in the Christian annual cycle depended on its date. The problem was that opinion was divided on when exactly that date might be. So critical was the issue that an entire (now lost) branch of mathematics was devoted to the subject: computus.
Computus needed to respect the rules of the Church, which were by no means straightforward. The crucifixion and resurrection of Christ celebrated over Easter were fundamentally linked to the Jewish festival of Passover. Passover was calculated to occur after dusk on the fourteenth day after the first full moon of the first month of the Hebrew lunar calendar (after dusk on Nisan 14), a date derived from astronomical knowledge of lunar cycles. The Church had chosen to diverge from the Judaic system and had decided that Easter should always fall on a Sunday on or after the first full Moon following the spring equinox. The Julian calendar of the age, a solar calendar, had a fixed date for the equinox, which was at Bede's time set at 21st March (though, just to be awkward, some communities used 25th March).
To make matters complicated, the lunar and solar cycles didn't (and still don't) match very well. A lunar month is 29.5306 days (approximated by the Julian calendar as 29 or 30 days); a solar year is 356.2422 days, which does not equate to 12 lunar months — the lunar calendar is eleven days shorter, meaning that, without intervention, on any calendar date the lunar date would be eleven days older the next year.
To predict Easter computists needed to develop a cyclical table based around a common multiple m of solar years and lunar months: that is, one in which a whole number of solar years equated to a whole number of lunar months. The general idea was that m years after some reference year, Easter will be on the same date as in the reference year itself because a whole number of lunar months will have passed. The number of lunar months in those m solar years wouldn't be an exact multiple of 12 (since there are more lunar months than solar ones), so the period wouldn't equate to a whole number of lunar years. For this reason, an extra month, called an embolismic month, was added to some years in the lunar calendar to make sure that years counted in lunar months would not gradually creep more and more ahead of years counted in solar months.
Figure 2: An abacus as shown on page 819 of Opera historica et philologica, 1682, by Marcus Welser.
The closest approximate cycles (the number of embolismic lunar months/solar years) were 3/8; 4/11; 7/19 and 31/84. But these cycles, summarised on tables without much supporting description, were unlikely to be universally accepted throughout a diverse Christian world; Bede's De temporum ratione (725) (On the reckoning of time) attempted a practical and universal solution by explaining as well as promoting his preferred avenue of calculation.
Bede started by reviewing, assessing and evaluating the tables of the age, preferring the 7/19 cycle, that is, 19 solar years, equivalent to 19x365 = 6935 days. The number of lunar months in these 19 years is 6935/29.5 = 235.08, which we'll take as 235. This is compared to 19x12 = 228 solar months, so seven of the lunar months are embolismic (the actual calculations were slightly more complicated, taking account of extra days creeping in here and there, but this gives the general drift). The key advocate of this system was Dionysius Exiguus, a man whose tables were considered admirably robust but whose name, unfortunately, loses some gravitas in modern translation: Denis the Titchy. Bede, never one to shrink from a challenge, focused his energies not only onto calculating Easter but also onto describing why the maths mattered as much as the result. In this, his elevated rhetoric is balanced by a very human enthusiasm — it's hard not to love a writer who signposts his core hypotheses with phrases such as 'now to gut the bowels of this question!'
To the modern mind, the arithmetic multiplications Bede had to tackle were straightforward, but a further challenge for a computist of his era was the fact that Arabic numerals were not yet around. Bede had a (literally) manual method for designating numbers for even basic arithmetic calculations, a form of sign language equivalent to doing the Macarena while juggling eggs (see figure 3). With only the Roman numeral system to help him, even the arithmetic aids of the day were unlikely to make his task that much easier (see figure 2). It is therefore to his credit that Bede had the mental dexterity to formulate his dates whilst explicating his processes to others in a way that found near-universal acceptance.
Figure 3: Old school counting. Top row, right to left: 1, 2, 3, 4, 5 and 6. Middle row, right to left: 7,8,9, 10, 20 and 30. Bottom row, right to left: 40, 50, 60, 70, 80 and 90.
Despite the high acclaim and wide acceptance of Bede's work, centuries later, medieval computists were still working on the same problem. Given the (relatively) confined scope of the mathematics involved, how did computus manage to remain such a core intellectual pursuit? In part, computists refined the basic model Bede had popularised. The introduction of a golden number in circa 1150 enabled computists to reference a given year's position in the 19-year cycle. It was calculated as (year number, divided modulo 19) + 1. By this reckoning, 2013/ 19, remainder 18; 18+1 = 19. Thus, 2013 is the 19th year in the 19-year cycle.
Key works from Islamic scholars also stimulated areas of debate: Al-Khwarizmi's Algebra of 825 was a core text, with dialogue between the Islamic and Christian mathematicians persisting for centuries through the work of Omar Khayyam, Al-Samaw'a' and Al-Kashi. Islamic algebraic systems enabled more sophisticated calculations, clarifying the areas where computists had needed to make approximations. The Islamic knowledge of astronomy further exacerbated awareness that artificially setting an equinox inevitably led to discrepancies: without adjustment, the date of the true astronomical equinox creeps backwards relative to the calendar, a divergence that was becoming difficult to ignore. By the thirteenth century, Roger Bacon, a leading thinker, commented in Opus Tertium LXX: "Any computist knows that the prime [of the moon] is off by three or four days in our time; and any rustic can see this error in the sky". The final (and somewhat ironic) solution was to accept that the equinox is governed by the heavens not humans, and to invent the Gregorian calendar we use today, complete with equinoxes determined by the stars.
Figure 4: A computus table from the later medieval period.
Yet while computists continued with their equations from a desire to map the Christian calendar, their quest for a divinely perfect solution amidst a series of numbers that just didn't seem to add up often prompted deeper reflection. Medieval church mathematicians had inherited a canon of principles from pagan authors, and needed to translate the meta-physics of antiquity into a contemplative format acceptable to Christian philosophy. In doing so, they tried to build a congruence between the Pythagorean-Platonic concept of number as the structure through which the world is made manifest, and evidence of the divine organisation as revealed in Wisdom 11:20: But thou hast arranged all things by measure and number and might. (The book of wisdom is from the vulgate bible, it is 11:21 in modern Catholic bibles and omitted in Protestant bibles.)
Numbers, therefore, meant something; but what, and how, and why was a matter of some debate. Byrhtferth, a computist working around 1000, was an advocate of allegorical numerology, an extinct discipline of mathematics in which numbers derive meaning through biblical interpretation. His manual of computus included a section extending the reckoning of time to the meaning of numbers. Byrhtferth makes the case that 7 is a special number as it derives from the addition of the number of the gospels to the number of the holy trinity. Abbo of Fleury, Byrhtferth's teacher, also considered 7 a sacred number, but for a different reason: it is the only number among the first ten which is not a factor or multiple of any other number among the first ten, a "virgin number" that represented the soul. To a modern mathematician, any attribution of mystical powers to a number feels less like maths and more like magic, but the difference in the reasoning behind the symbolism is significant. For Byrhtferth, what was important was the story that the number related to in biblical exegesis; for Abbo, the number's symbolic meaning was derived from its numerical properties.
Deeper consideration of numerical properties brought more evidence of a systematised world: divine proportionality, popularised by the sequence Fibonacci devised in the thirteenth century, had been a preoccupation of a much earlier era. Back in the sixth century, amid the ruins of the Roman empire, the Christian philosopher Boethius found beauty in triples of whole numbers satisfying and For example, 182 + 294 = 476, 294 ÷ 182 ≈ 476 ÷ 294. The geometric mean of the extremes and in this case is nearly a whole number, , which is at the same time their difference. If this happens then the ratios and are approximately equal to the golden ratio, linked to Fibonacci's sequence and considered the most harmonious proportion (see this article). This concept, familiar to modern mathematicians, has its roots in the contemplation of divinely mediated beauty.
Mathematical enquiry provided the foundation of divine order; it made manifest the dynamism of the spiritual world. Numbers were the means by which the spiritual world became incarnate, creating form and order yet remaining unchanged. The concerns of the medieval period, so entwined with religious philosophy, often seem alien, yet there were then, as in every age, individuals whose mathematical abilities were profoundly impressive. The tenacity with which our monastic predecessors balanced their convictions of faith with their observations of mathematics deserves some credit: perhaps the dark ages were not, after all, quite so dark.
About the author
Charlotte Mulcare read biological anthropology at Cambridge and completed a PhD at University College London in human genetics. She now works as a freelance science and technical writer and lives in Chester.