Archimedes (287–212 BC) invented a method of describing large numbers for purely theoretical purposes, to show that the number of grains of sand contained in the universe was not infinite, but just very large. This is how the Arénaire begins:
There are people, O King Gelon, who think that the number of grains of sand is infinite. I am not talking about the sand which is around Syracuse [but] of a volume of sand which was equal to that of the Earth.
To do this, Archimedes begins by evaluating the perimeter of the Earth, wanting to be sure that the actual measurement is less than the one he gives, so he multiplies the known measurements by ten:
That said, let the Earth’s outline be about three hundred myriad stages but no larger. Because you are aware that others have wanted to demonstrate that the contour of the Earth is about thirty myriad stages.
In the Greek number system, the myriad was the unit directly following the thousand. It was therefore worth ten thousand. The stadium is a measure that we all have in mind because it has given the length of our stadiums. He was therefore a little less than 200 meters, but that does not matter here. From this data it is possible to calculate the volume of the Earth. Archimedes then estimates that, in a volume equivalent to a poppy seed, there are no more than a myriad of grains of sand, before finding that 40 seeds had to be aligned to obtain the width of a finger. Archimedes then has all the elements to make his calculation. It just lacks a numbering system.
Archimedes’ number system
Archimedes begins by describing the system in use in Greece in his time:
Numbers have been given names up to a myriad and beyond a myriad, the names given to the numbers are fairly well known, since one only repeats a myriad to ten thousand myriads.
He makes it the basis of his system:
That the numbers just discussed which go up to a myriad of myriads be called prime numbers [not in the present sense], and that a myriad of myriads of prime numbers be called the unit of seconds; let’s count by those units, and by the tens, hundreds, thousands, myriads of those same units, up to a myriad of myriads.
These prime and second numbers can go up to the thousands of billions of Nicolas Chuquet, that is to say the billiards! (see the table of equivalents of prime and second numbers in Nicolas Chuquet’s system).
Archimedes continues in the same way to define the third numbers and so on. It reaches the limits of Nicolas Chuquet’s system, that is, the nonillion, with the hundreds of myriads of first-order seventh numbers! It continues to eighth numbers:
Let a myriad of myriads of second numbers be called the unit of third numbers; let’s count by those units, and by the tens, hundreds, thousands, myriads of those same units, to a myriad of myriads; that a myriad of myriads of third numbers be called the unit of fourth numbers; let a myriad of myriads of fourth numbers be called the unit of fifths, and let’s continue to give names to the following numbers …
Archimedes calls “first period” the numbers he defined up to eighth numbers and begins a second period:
Although this large quantity of known numbers is certainly more than sufficient, we can still go further. Indeed, let the numbers we just discussed be called the numbers of the first period, and the last number of the first period be called the unit of prime numbers of the second period. Moreover, a myriad of myriads of prime numbers of the second period is called the unit of seconds of the second period …
By making calculations of the order of magnitude, for the universe, as it was seen in his time, Archimedes found:
it follows that the number of grains of sand contained in a sphere as large as that of fixed stars assumed by Aristarchus, is less than a thousand myriads of eighth numbers.
This is much more than can be counted in the original Nicolas Chuquet system, since this number is equal to 1 followed by 63 zeros! If we extend it by decillions each worth one million nonillions, this number is equal to 1000 decillions. We can compare to the estimated number of electrons in the universe, which is equal to 1 followed by 81 zeros, which we note 1081. In the Archimedes system, this number is equal to ten of the third numbers of the second period.
