### The Sand Reckoner

The Sand Reckoner is a remarkable work in which Archimedes of Syracuse (287-212 BC)  developed, on the background of reckoning the number of sand grains that could be contained in a sphere of the size of our "universe", a system of forming  a system which is capable of expressing huge numbers. At that time used  Greek numerical notation  was very cumbersome to work with. 1

The Sand Reckoner is the standard English translation of the original Greek name Psammites  ψσαμμιτησ. A better translation is Having to do with sand  (psammos=Gr. sand), the Latin translation of the name is Arenarius.  It is addressed to King Gelo(n), King of Syracuse, and can be considered as the first research-expository paper.

Archimedes starts with the fact that the Greek numeral system allowed without problems to name numbers up to a myriad.  This is the biggest number for which the Greeks had a separate name and denoted 10,000.  In Greek numeral system this number was represented by M from the Greek word murious μυριος (uncountable, pl. murioi; also a word for infinity in Ancient Greek).  The Romans converted it to myriad. The largest number named in Ancient Greek was a myriad myriad and this because Archimedes starts with this quantity as the basis for a numeration system.

Archimedes took myriad myriad as the basis for his numeration system of large powers of ten.  Since myriad myriad equals 100,000,000 Archimedes created a positional system of notation with base 100,000,000. Since it is hard to believe that he knew about contemporary Babylonian positional system with base 60,  this was apparently his completely original idea.

To be able to work with greater numbers Archimedes invented the following system for very large numbers. He calls the numbers up to , that is numbers up to myriad myriads as the numbers of the first order. The numbers greater than   to he called the numbers of the second order. The numbers from here to he called the numbers of the third order, and so on up to the order of a myriad, that is up to the number myriad myriad to the myriad myriad power .

The numbers in the segment he called the numbers of the first period. The numbers between and he called the numbers of the first order of the second period up to the order of the second period.  Then there follows the second order of the first period, and he continues up to the myriad myriads order of the myriad myriads period. The latest number ends with zeros. In Greek he wrote: αι  μυριακιδμυριοστας  περιοδου  μυριακισμυριοστϖν  αριϑμϖν  μυριαι  μυριαδες (a myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period) .

Archimedes’ estimate is the grains of sand to fill our universe.

Archimedes developed a system with base 10,000 and therefore it is not surprising that he discovered and proved the law of exponents necessary to manipulate powers of 10.  The ancient users of abacus when performed the multiplication of two numbers multiplied from the left to the right that is they started with highest power of the multiplier. In abacus application this means that it was necessary estimate the order of the product in order to be able to place the first counter of the product. The rule is clear from our point of view: if is the number of columns occupied by the multiplicand, the number occupied by the multiplier then is the number of columns occupied by the product. This rule can be found in the treatise :

Psammites, Chap.3(6) : This too is usefully ascertained. If when numbers are proportional from the unit, some of the numbers from the same proportion multiply one another, the number which arises will be from the same proportion and will be distant from the larger of the numbers which multiplied one another as much as the smaller of the numbers which multiplied one another is distant proportionally from the unit, but it will be distant from the unit by one less than the number of the sum [of the distances] which the numbers which multiplied one another are distant from the unit.

### Notes

 1 The work is also of interest from other related reasons. Archimedes gives here the most detailed surviving description of the heliocentric system of Aristarchus of Samos. The treatise also contains an account of an ingenious procedure that Archimedes used to determine the Sun’s apparent diameter by observation with an instrument.

Cite this web-page as:

Štefan Porubský: The Sand Reckoner.

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