suggests that universality was not the system’s main design objective, and that it was not greatly valued when it was achieved.
Perhaps an insight into this recurring oddity is provided by a remarkable episode in the third century BCE involving the ancient Greek scientist and mathematician Archimedes. His research in astronomy and pure mathematics led him to a need to do arithmetic with some rather large numbers, so he had to invent his own system of numerals. His starting point was a Greek system with which he was familiar, similar to the Roman one but with a highest-valued symbol M for 10,000 (one myriad). The range of the system had already been extended with the rule that digits written above an M would be multiplied by a myriad. For instance, the symbol for twenty was κ and the symbol for four was δ, so they could write twenty-four myriad (240,000) as .
If only they had allowed that rule to generate multi-tier numerals, so that would mean twenty-four myriad myriad, the system would have been universal. But apparently they never did. Even more surprisingly, nor did Archimedes. His system used a different idea, similar to modern ‘scientific notation’ (in which, say, two million is written 2×106), except that instead of powers of ten it used powers of a myriad myriad. But, again, he then required the exponent (the power to which the myriad myriad was raised) to be an existing Greek numeral – that is to say, it could not easily exceed a myriad myriad or so. Hence this construction petered out after the number that we call 10800,000,000. If only he had not imposed that additional rule, he would have had a universal system, albeit an unnecessarily awkward one.
Even today, only mathematicians ever need numbers above 10800,000,000, and only rarely at that. But that cannot be why Archimedes imposed the restriction, for he did not stop there. Exploring the concept of numbers further, he set up yet another extension, this time amounting to an even more unwieldy system with base 10800,000,000. Yet, once again, he allowed this number to be raised only to powers not exceeding 800,000,000, thus imposing an arbitrary limit somewhere in excess of 106.4×1017.
Why? Today it seems very perverse of Archimedes to have placed limits on which symbols could be used at which positions in his numerals. There is no mathematical justification for them. But, if Archimedes had been willing to allow his rules to be applied without arbitrary limits, he could have invented a much better universal system just by removing the arbitrary limits from the existing Greek system. A few years later the mathematician Apollonius invented yet another system of numerals which fell short of universality for the same reason. It is as though everyone in the ancient world was avoiding universality on purpose.
The mathematician Pierre Simon Laplace (1749–1827) wrote, of the Indian system, ‘We shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.’ But was this really something that escaped them, or something that they chose to steer clear of? Archimedes must have been aware that his method of extending a number system – which he used twice in succession – could be continued indefinitely. But perhaps he doubted that the resulting numerals would refer to anything about which one could validly reason. Indeed, one motivation for that whole project was to contradict the idea – which was a truism at the time – that the grains of sand on a beach could literally not be numbered. So he used his system to calculate the number of grains of sand that would be needed to fill the entire celestial sphere. This suggests that he, and ancient Greek culture in general, may not have had the concept of an abstract number at all, so that, for them, numerals could refer only to objects – if only objects of the imagination. In that case universality would have been a difficult property to grasp, let alone to aspire to. Or maybe he merely felt that he had to avoid aspiring to infinite reach in order to make a convincing case. At any rate, although from our perspective Archimedes’ system repeatedly ‘tried’ to jump to universality, he apparently did not want it to.
Here is an even more speculative possibility. The largest benefits of any universality, beyond whatever parochial problem it is intended to solve, come from its being useful for further