Liberal Arts: Arithmetic

Detail from a photo by user Dnalor_01, From Wikimedia Commons, License CC-BY-SA 3.0

I’ve fallen down a bit of a rabbit hole, the last few weeks, as I’ve tried to reconstruct or at least think about what the liberal arts might have looked like in a medieval university.  Grammar was sort of easy, although that post is mostly about the earliest words. This more about mathematics, and specifically arithmetic.

It turns out that the curriculum of the liberal arts distinguished significantly between logistic, which was accounting and bookkeeping and the management of numerable things like bolts of cloth and kegs of wine and pounds of coffee — and arithmetic, which was more like abstract mathematics.  Logistic was for merchants and merchants’ sons; Arithmetic was for masters of the sciences.  It’s odd to think about, but most of medieval mathematics is mostly encapsulated within a rigorous primary through high school mathematics education — and the high school math education is likely to be more complete still, between calculus, statistics, game theory, and probability.

The Shift in Operations

It took more than two hundred years, from about AD 1000 until AD 1220, for Europe to completely retrain its population to use Arabic-Hindu numerals (1, 2, 3, 4, 5, 6, 7, 8, 9), and place-counting (introduced possibly by Gerbert of Aurillac, later Pope Sylvester II), and to accept the idea of the cipher or zero (0).  There wasn’t much conflict between the abacists, who used the Roman abacus shown below (and which apparently predates the Chinese and Japanese abacus by a good 200 years); and the algorists, who believed in both a kind of complex finger-counting, and doing mathematics by procedures we would recognize today as algorithms.Screen Shot 2018-04-29 at 11.17.19 AM The Roman abacus remained in use in some medieval universities into the early 1600s, but was gradually supplanted even as Roman numerals were supplanted.

It’s hard to understand the abacist point of view without having a sense of the device they used, which was about the size of a modern shirt-pocket calculator (tools always wind up fitting the hands that use them, huh?)

The Roman abacus has place values — one million in the extreme left column (marked by |X|), then 100,000s column indicated by |M|, then a 10,000 column marked by |C|, then a thousands column marked by M, then a hundreds column marked by C, a tens column marked by X, a ones column marked by 1. (I’ve greatly simplified the markers for the large numbers of 10,000 and higher).

Counting was done by sliding beads upwards in the lower groove, 1, 2, 3, 4, and then sliding the bead in the upper groove in each column for 5.  The beads in the lower column would then be slid downward for 6, 7, 8, and 9. The upper bead would then be pushed down in one column, and the first bead in the next column would be pushed up to indicate a change of magnitude.  Tada! Counting by tens.

Final results would be recorded in Roman Numbers — I for 1s, V for 5s, X for 10s, C for 100s, D for 500s, M for 1000s, and then these more complicated symbols that I’ve already simplified for higher numbers.  Three abaci or counting-tables would be put side by side in order to do complex and higher-order mathematics: one to hold the answer and two to hold the original numbers and to gradually cancel out parts of the operation.

What about fractions?

Turns out that the Romans mostly counted in twelfths, called an uncia: twelfths of a mile, twelfths of a As (the principle Roman gold coin), twelfths of a pound (ounces or uncia‘s), twelfths of an amphora, twelfths of a food (pes = foot, uncia in the context of distance = inch!). The symbol for an uncia, regardless of the type of measure, looked like the Greek letter theta, or θ.  What about fractions of an uncia? I can’t even replicate their symbols, so I’ve simply marked them with 1/2, 1/4 and 1/3 — but one-quarter of one-twelfth, or 1/48th of an uncia, appears to be as far down as the Roman abacus could go.

COMPUTOR was a person who owned and knew how to use an abacus.  The abacus and the person together made the computor, rather than a separate machine.

The Algorists, on the other hand, first with a short-hand system of using modified Roman symbols and a rough system of place-holding values, began to develop systematic ideas about orders of operation.  They came into contact with Arabic ideas about mathematics, and began to teach a combination of finger-counting (using a binary system that allowed them to count to 1023 on both hands, or a surviving Hellenistic system that allowed them to count to 9,999 on both hands).  It was a slow transition from abacism to algorism, and the abacists really weren’t completely replaced in all European universities until the early 1700s.  (which seems crazy).

The Curriculum

One article I read suggested that the curriculum was a sort of Christianized Pythagoreanism; others suggested that there was genuine instruction in abstract mathematics.  In general, there seem to be four overlapping periods of medieval mathematical instruction (these are general and vague, not academically reliable):

  • Phase I: AD 500 to AD 1000 — The computus, that is, being able to calculate a calendar and find the date of Easter and the moveable feasts; ratios and proportions, and geometric methods of calculating squares and square-roots.  A lot of number-mysticism lightly Christianized from Pythagoreanism. The phasing-out at the end of the period of Roman numerals.
  • Phase II: AD 900 to AD 1250 — abacism and the shift to Arabic numerals and place-value systems, the introduction of the number zero. The computus remains important. Still a lot of number-mysticism.
  • Phase III: AD 1200 to AD 1500 — the development of algorism, the introduction of algebraic methods to the medieval arithmetic curriculum (though they didn’t look like our algebra at all), the computus.  A declining amount of number-mysticism.
  • Phase IV: AD 1400 to AD 1600 — The triumph of the algorists (it was never much of a fight, anyway). Arabic numerals fully integrated into the curriculum, finger-counting, mental calculation, algebraic methods fully integrated, the computus, very little number-mysticism, very little abacism left.
Square Numbers
The gnomon is the upside-down L shape that forms the increase of each square.

What did this mathematics look like? What sorts of number theory was involved? Some of it involved the calculation of polygonal numbers. Four, for example, is a square number, since its four unities can be arranged into a 2×2 grid. Square numbers are still useful today, still a common part of mathematics. The other polygonal numbers, though, seem less useful to us because we’re not used to thinking in terms of these kinds of geometric progressions. But they were clearly important to the medieval mind, which was used to thinking in both geometric and arithmetic ways about ratios and proportions.  Consider the pentagonal numbers, which represented pentagons of increasing size. The first pentagonal number is 5, of course, because that’s the first number where each vertex of a pentagon or each side can be represented by a single unit.  But then things get odd.  The second number is twelve, because you have to add seven units to five units in order to create a pentagon with a side of three.  And the third pentagonal number is 22, where one has to add ten units to twelve, in order to image a pentagon with a side of 4.  We begin to see why proportions and ratios were so important to the medieval mathematical curriculum — these underlying patterns formed the structure of the number-mysticism that underlay the Christianized Pythagorean thinking at the heart of the training of the Quadrivium.

Screen Shot 2018-04-29 at 12.29.29 PM
Pentagonal numbers are a little weirder, in that they include the parts of the previous pentagon.

Prime numbers were sort of the next thing in the curriculum.  What is a number divisible by?  Just itself?  How many fractions can it be sorted into?

It was related to the idea of proportion and ratio, which forms part of Plato’s Timaeus and the Demiurge’s construction of the world, by dividing and compounding it by units, halves, thirds, sixths, ninths, eighths, twenty-sevenths, thirty-sixths, forty-eighths…. by units, tens, nines, and eights, by twenty-sevens, by fifty-fours, and by one-hundred-eights…

I’m beginning to sense a theme and pattern here.  A great deal of the arithmetic curriculum of the medieval university was focused on getting the smartest people in Europe to think about number in a lot of complicated ways — not to find solutions, exactly, or to know how to calculate, but how to think about parts, wholes, multiples and divisors.   Does this number resolve into a plane shape — a hexagon, a pentagon, an octagon (this is not the same thing as divisible by six, five or eight, mind you)?  Does this number form a solid (i.e., does it break into parts that can be used to define a volume such as a triangular prism, a pyramid, or a cube)? Is it superabundant, perfect, or deficient (that is, what is its relationship to its divisors including itself?):

  • superabundant: 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 ≥ 24.  Here, 24 is a superabundant number because the sum of its divisors is greater than the number.
  • perfect: 1+2+3 = 6 ≥ 6.  Here, 6 is a perfect number because the sum of its divisors is equal to the number itself.
  • deficient: 1 + 2 + 4 + = 7 ≥ 8. Here, 8 is a deficient number, because the sum of its divisors is less than the number itself.

So… Yes, the algorists won.  The idea that number could be reduced to a series of procedures, of formulae, won out over this kind of wild experimentalism.  Certain kinds of numbers were seen as particularly useful and structurally-valuable: squares and cubes, prime numbers, order of operations, exponents.

But these were interesting things going on, nonetheless.  This is an arithmetic rooted not in counting things, but in counting and in numeracy, and in trying to suss out patterns inherent in the universe.  Yes, it’s also a kind of number-mysticism, but it’s a number-mysticism trying to uncover some methodologies for understanding a complex and mystifying cosmos… and to understand number is to understand the underpinnings of reality.  Maybe.

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