The Dark Ages of Arithmetic

I

AN excellent way to see one’s own country, or one’s own century, is to visit another.

While I was making just such a mental pilgrimage to the eighth century, a recent interest in unconventional mathematics1 led me to examine with unusual care the arithmetic of the Franks in the days of Charlemagne and his great schoolmaster, Alcuin.

These were the Dark Ages of arithmetic, as of most other forms of learning. The Hindu-Arabic numerals we use to-day had not been imported by way of Spain, and even among the Arabs there is no evidence that the magic zero — symbol for nothing on which most mathematical science rests — was yet invented. The cumbersome Roman numerals were those used, with knowledge among the learned of the still worse Greek ones. Even geometry, which the Greeks under Euclid and Apollonius had developed to a fine science capable of performing operations awkward or impossible with their numerals, was being partly forgotten.

One might suppose that nothing useful or interesting could be learned from examining the arithmetic texts of that Dark Age. I have found the opposite true. The very difficulty men had in performing simple multiplications which a child of ten now does with ease resulted in ingenious devices. Because men had no good set of simple formulas on which to glide to automatic answers, they had to think hard about quantity, and what it did and why. Their strange and often crude ideas of number have in several instances guided modern mathematicians to important discoveries.

II

There are two chief sources for our knowledge of the arithmetic of the Middle Ages, both textbooks. The first is the Introduction to Arithmetic of Nicomachus of Gerasa, concerning whom almost nothing is known except that he lived about 100 A.D., probably in a town not far from Jerusalem, and wrote in Greek what appears to be the world’s first true arithmetic. His book contributed little that was new, but it summarized most of the arithmetical knowledge and beliefs of his time. Boethius, prolific writer and Christian martyr of the sixth century, popularized Nicomachus by rewriting him in Latin, the common language of the scholars of the day. The textbooks of these two men, together with minor contributions of other writers, constituted the arithmetic of the Dark Ages.

There were dreary wastes and blind stumblings in this arithmetic, but there were also some facts which most of the world has since forgotten. How many people know to-day that any number can be squared without multiplication, simply by adding a series of odd numbers equal to the number to be squared? For instance, the square of 3 (which is 9) is the sum of the first three odd numbers, l+3+5 = 9. And the square of 8 (or 64) is the sum of the first eight odd numbers, 1 + 3+5 + 7+9+11 + 13 +15 = 64. And so on infinitely, through all the possible integral squares.

We no longer need this method since multiplication has been reduced to an efficient formula. But imagine the difficulty the powerful emperor Charlemagne had in trying to multiply, say, XXXVII by XXIX. Just these practical difficulties, in either the Roman or the Greek numerals, led to some concepts of number that sound strange in our ears, and will bear examining.

Nicomachus thought the creation could be divided into two varieties — magnitudes and multitudes. Magnitudes were things like the earth itself, or a tree, and were infinitely divisible. Multitudes were like a heap of stones or a flock of sheep, and these were infinitely increasable. It followed that ‘sciences are always sciences of limited things,’ or things that could be numbered. Arithmetic itself occupied a special position, for without it no science could exist, but it needed no science for its own existence.

Boethius used number for his division of the famous quadrivium of the Middle Ages. ‘Numbers absolute’ constituted arithmetic. ‘Numbers in mutual relationship’ were the foundation of music. ‘Quantity at rest’ was the subject matter of geometry; ‘quantity in motion,’ of astronomy.

III

The ideas of number and actual physical quantity never got wholly dissociated in the Middle Ages. For centuries it was denied that 1 was a number at all. The idea of number seemed necessarily plural, and 1 was not plural. (We still frequently use the word ‘number’ with a plural verb, as ‘A number of men were killed.’) Moreover, any number multiplied by itself should produce a different number, but 1 so multiplied reproduced itself. Even 2 was regarded as rather a representation of the duality of the universe than a number, and for many scholars the number system really began with 3. That is why the Latin poets use three (terque quaterque beatus) as the first symbol for many.

The idea of numbers as composed of individual physical particles gave rise to a geometrical conception of arithmetic, with description and handling of numbers in geometrical patterns. While special symbols for large numbers were available, it seemed more useful to regard a number as just a collection of like particles (the Greeks used α, since that was their 1) which could be moved about in geometrical patterns. One (α), being a mere point, was non-dimensional. Two (α α) was a straight line, or onedimensional. Then followed the plane numbers, which were two-dimensional. These began with the triangles, 3, 6, 10 (any bowler will recognize 10 as a triangle), 15,21 . . . Thus: —

After the triangular numbers came the square numbers (our term ‘square’ is, of course, just a relic of this geometrical treatment) such as 4, 9, 16, 25, 36 . . .

Nicomachus pointed out that any two successive triangular numbers could be arranged into a square number (3 + 6 = 9; 6+10=16), and that any square, as one can see in the last example on page 65, is the sum of two unequal triangles.

He also noted, as we have already done, that all the squares could be derived in their order from the sum of the series of odd numbers, a fact earlier known to Pythagoras and Aristotle.

The plane numbers were continued through pentagons, hexagons, and so on. Nicomachus formed a table of these plane numbers which demonstrated some rather curious number relationships, which will be evident on careful examination of this section of his table: —

Triangular numbers 1 3 6 10 15 21 28
Square numbers 1 4 9 16 25 36 49
Pentagonal numbers 1 5 12 22 35 51 70
Hexagonal numbers 1 6 15 28 45 06 91

After two-dimensional numbers Nicomachus went on to three-dimensional or solid numbers. The simplest solid number was the three-sided pyramid, represented by 4, 10, 20, 35, 56 . . . To see how Nicomachus derived his pyramids it is necessary to consider his units piled up like cannon balls in a Civil War memorial, and count the successive layers. Here are the successive layers of 20, treated as a solid three-sided pyramid: —

It will be observed that the solid pyramids of three sides can be derived mathematically by adding the successive triangular numbers in the table just given. Similarly, solid pyramids of four sides are derived by adding the successive square numbers. The cubes, which we retain, Nicomachus derived by adding as many layers of the same square number as there were numbers on one side of the square.

It is unnecessary to go further into the representation of single numbers as geometrical figures. We may even feel relieved that Nicomachus did not think of a theoretical fourth dimension, and get out an additional series. Nevertheless an examination of what we may call the physical composition of various of these numbers led to certain ideas of their interrelation, and even to a singularly clear concept of multiplication.

IV

Most numbers, when expressed as particles, can be arranged in the form of rectangles; a few numbers cannot. Let us take for example 12, which makes two rectangles, and 5, which makes none: —

Numbers like 5, which cannot be arranged in a rectangle, are prime; that is to say, they have no factors but 1 and themselves. Numbers which can be arranged in rectangular form possess length, longitudo, and breadth, latitudo; and they have as many pairs of factors (in addition to 1 and themselves) as there are different rectangles into which they can be formed. For example, 12 makes two rectangles, and from them its factors can be counted off to be 2 and 6, 3 and 4. If the rectangle is also a square, then there is a doubled factor, as, for instance, 5 and 5 for 25.

From this conception of rectangular numbers, multiplication springs as full-bodied as Minerva. To multiply is simply to give a number a second dimension. To multiply 4 by 3, we express 4 as a linear number (α α α α), give it a breadth of 3, and fill in the rest of the rectangle. The result is twelve α’s.

Similarly, division can be done by arranging the number to be divided into a rectangle with one side equal to the proposed divisor. The other dimension will then be the answer; and if the number does not make a perfect rectangle, then it cannot be evenly divided by the selected divisor.

Since this shifting about of number particles is something of a bother, one soon begins to plot out the length and breadth of the lower numbers. A section of Boethius’s chart — which he continued to X in both directions — is given below. He set clown on the first horizontal line the units of length (longitudo); he arranged in the first vertical column the units of breadth (latitudo). Then, just as one might chart cities by their longitude and latitude, he filled in the centre with the rectangular numbers which had the length and breadth their columns called for. There emerges — the multiplication table!

Longitudo
I II III IIII V VI etc.
II IIII VI VIII X XII
Longitudo III VI VIIII XII XV XVIII
IIII VIII XII XVI XX XVIIII
V X XV XX XXV XXX
VI XII XVIII XXIIII XXX XXXVI
etc.

This form of the multiplication table is so much more compact than our common 2-line, 3-line, 4-line arrangement that it is still sometimes used; it has also certain further advantages in showing number relations. The godly Boethius, normally a moderate and reserved writer, issima d all over the page with Latin superlatives in praise of it. In addition to showing instantly both the products and the factors of all low numbers, it presented the successive squares in the diagonal proceeding from the upper left-hana corner of the table down to the lower right-hand corner.

With the aid of such a table, or the still earlier Greek one, the arithmetical Dark Ages could perform simple multiplications. But since their numbers lacked ‘place value,’ they ran into severe complications with larger figures. Apparently a few mathematical geniuses could remember what happened when one multiplied LI by XV, but most people could not. For larger figures they frequently resorted to the abacus, where beads on separate strings gave them precisely the place value they had not the wit to invent in number symbols, and a vacant string represented the zero they still did not think of.

Sometimes they performed multiplications by successive additions of doubles, technically called ‘duplation.’ Here is how it worked, using our own numerals for simplicity: —

Problem: Multiply 19 by 25
1 nineteen = 19
Doubled, 2 nineteens = 38
4 = 76
8 = 152
16 = 304
Then, by selection:
1 nineteen = 19
8 nineteens = 152
16 = 304
25 475

A little experimenting will prove that this method of duplation will suffice for all whole numbers, and reaches the higher ones rather speedily; it was therefore not necessary to know more than how to add successive numbers to themselves to achieve, a trifle laboriously, large multiplications.

A more mysterious method, based upon the same principle, is cited by David Eugene Smith in his History of Mathematics as being still in use among Russian peasants; I have found no demonstration of it in medieval textbooks, but it is so similar it may very well have been known. In this method we halve the one number (neglecting fractions) until unity is reached, and beneath these figures place the other number and its successive doubles. The final answer is the sum of those lower numbers which are beneath an odd number. Why this is so a demonstration of the same problem may help make clear. To multiply 25 and 19: —

25 12 6 3 1
19 38 76 152 304

Now, adding figures beneath odd numbers:

19+152+301 = 475 — Answer

Nicomaehus also recognized ten varieties of proportions, but, the world still being without the aid of algebra, his treatment of them is scarcely enlightening. One item may be of interest to musicians. The so-called harmonic proportions were often not derived by mathematical means at all, but solved by striking chords and measuring the respective lengths of strings. Musician-readers are invited to solve on their own guitars 6: 8:: 9: x

Nicomaehus called 6,8,9,12 the most perfect proportion . . . most useful for all progress in music, and in the theory of the nature of the universe.’2

V

There are further curiosities of the mediaeval arithmetic, such as the quest for ‘perfect numbers,’ which the reader may follow out for himself. And if perfect numbers and other similar interests of these experimenters in arithmetic seem to us mystical and quite useless, it may be well to point out, as L. E. Dickson does in his recent History of the Theory of Numbers, that modern number theory stems largely from Fermat’s experimentations with these very perfect numbers of the Dark Ages. If number and the nature of the universe are as intimately related as many modern physicists believe, it may be that the laborious concepts of the Dark Ages of arithmetic hold still further revelations for the understanding modern mind.

  1. See the Atlantic for October 1934. — EDITOR
  2. Translated by Martin Luther D’Ooge, University of Michigan Studies, Humanistic Series. — AUTHOR