Posted on Tuesday 21 February 2006

He showed marked respect to Livia Augusta [wife of Caesar Augustus], to whose favour he owed great influence during her lifetime and by whose last will he almost became a rich man; for he had the largest bequest among her legatees, one of fifty million sesterces. But because the sum was designated in figures and not written out in words, Tiberius, who was her heir, reduced the bequest to five hundred thousand, and Galba never received even that amount.

— Suetonius,The Lives of the Caesars : Galba

Of course Tiberius never had any intention of giving Galba a penny but the technicality that he used to slash Galba's entitlement to a thousandth of the original amount relied upon the exploitation of a simple fact, one that was obvious even at the height of Roman power in the first century AD, that

Roman numerals totally suck!

I've noted previously the problems that many ancient number systems encountered when they came to trying to represent very large numbers. In that article I referred to the weaknesses of the Greek system which used letters to represent numbers 1, 2 ... 9 but then used a different set of letters to represent 10, 20 ... 90 and yet another to represent the hundreds. Naturally the Greeks found it easy to count up to 1000 (and be able to play also sorts of fun mathematical word games) but for higher numbers they then had to fall back on to various scaling rules. The system was inelegant but at least it was systematic, the Roman system was even less elegant and certainly less systematic.

To denote a number greater than 1,000 using the Roman system, the standard way was to rehash the old symbols but marking them with a bar over the top (a "vinculum") to indicate multiplication by one thousand. So V (i.e 5) with a bar over the top indicated 5,000, L (i.e. 50) with a bar indicated 50,000. Larger amounts could be shown by surrounding the number in a three cornered frame (open at the bottom) which indicated multiplication by one million. Livia's bequest to Galba was denoted as CCCCC enclosed in a three cornered frame (i.e 50 million) which Tiberius deliberately took to mean CCCCC with a bar over the top (i.e. 50 thousand).

Though cheated by Tiberius, Galba eventually got the last laugh by ending the Julio-Claudian dynasty when he became Empereror. Within only a few months, however, he was murdered in the Forum by his praetorian guard.

It's interesting to contrast this clumsy incremental approach to big numbers with what was already established practice in the old Mesopotamian city-states which were already extremely ancient even in the time of the Romans. Nearly two thousand years before Livia's last will and testament was read out, the scribes of ancient Sumer and Akkad routinely dealt with with large numbers which they wrote down by impressing tablets of soft clay with reed styluses. Their system starts out familiarly enough with the number of reed marks directly corresponding to the numeric value to be represented. By 10 however this becomes unwieldy so a different symbol is placed to the left which counts the tens.

This sequence could have theoretically carried on until 99 was reached but instead the Mesopotamians chose to limit it to 59. Just as with our modern decimal system where the unique symbols stop at 9, they used the place of the numerals as a way to multiply them by the base (in this case 60) raised by a power.

For example, a "1" placed to the left of a "57" would represent 1 x 60 + 57 i.e 117.

The sequence 1 57 46 40 would be 1 x (60 x 60 x 60) + 57 x (60 x 60) + 46 x 60 + 40

The base of 60 added a complicating wrinkle to the whole thing (one that we still see today reflected in the number of minutes we count in an hour and the number of degrees in a circle) but despite this it can be seen that the Mesopotamians used a place notation system which was remarkably similar to our own decimal system in many respects.

The system had some shortcomings, it was still possible to be ambiguous, unused columns had to contain blanks so that numbers like 3601 didn't get mixed up with 61. They did eventually come up with a symbol for zero which they could use to pad out numbers but they never fully systematised it as a true zero. They never used it at the end, for example, so the order of magnitude of a round number still needed to be guessed at by looking at its context.

Below is a famous example known simply as Plimpton 322. It was excavated from the ruins of the city of Larsa near Ur and dates from around 1800 BC during its last flourish of independence before all the old city-states of Sumer and Akkad were conquered by Hammurabi and unified under the control of the Babylon. The tablet was ruled with 15 lines and divided into four columns.


Here it is transcribed into Indian numerals. First as base 60

	width	diagonal	name
59 0 15	1 59	2 49	1
56 56 58 14 50 6 15	56 07	1 20 25	2
55 7 41 15 33 45	1 16 41	1 50 49	3
53 10 29 32 52 16	3 31 49	5 9 01	4
48 54 1 40	1 05	1 37	5
47 6 41 40	5 19	8 01	6
43 11 56 28 26 40	38 11	59 01	7
41 33 59 3 45	13 19	20 49	8
38 33 36 36	8 01	12 49	9
35 10 2 28 27 24 26 40	1 22 41	2 16 01	10
33 45	45	1 15	11
29 21 54 2 15	27 59	48 49	12
27 0 3 45	2 41	4 49	13
25 48 51 35 6 40	29 31	53 49	14
23 13 46 40	28	53	15

and then as decimal.

	width	diagonal	name
0.9834	119	169	1
0.9492	3367	4825	2
0.9188	4601	6649	3
0.8862	12709	18541	4
0.8150	65	97	5
0.7852	319	481	6
0.7200	2291	3541	7
0.6928	799	1249	8
0.6427	481	769	9
0.5861	4961	8161	10
0.5625	45	75	11
0.4894	1679	2929	12
0.4500	161	289	13
0.4302	1771	3229	14
0.3872	28	53	15

Each of the columns has a heading. The fourth (rightmost) column heading is "its name" and its values number from 1 to 15. The second and third column headings are the "square of the short side" and "square of the diagonal" respectively. Both contain whole numbers which taken together form the short side and diagonal of a right angled triangle. The third side of this triangle which is not represented on this tablet would also be a whole number. Neat correspondences such as these do not just happen by chance, they need to be found. At the very least these values demonstrate that the ancient Mesopotamians understood Pythagoras' theorem and they knew it a good 1,300 years before Pythagoras was even born.

The first (leftmost) column heading is more cryptic "The takiltum-square of the diagonal from which 1 is torn out, so that the short side comes up", this implies some kind of of geometrical operation but the numbers speak clearly enough for themselves. They are perfect squares but ones that are fractional where we have to imagine a decimal point to their left. Another remarkable thing about them is that if you add 1 to them the numbers that result are also perfect squares.

Relating the first column to the second and third has been the subject of much mathematical debate over the past sixty years. Two theories predominate, one argues that their values are derived by trigonometric functions from the angle of the the triangle, in other words this tablet represents the first ever found trigonometric table (a quite advanced one in fact showing the squares of cosecants). However, this theory may be projecting greater mathematical prowess onto the Mesopotamians than they may have actually possessed, in the history of mathematics this would place them somewhere equivalent to the early European Renaissance.

A recent and perhaps more historically more satisfying theory argues that they are based on a list of reciprocals pairs. A complementary article provides a working out of the formulas involved.