return to Vignettes of Ancient Mathematics
(the sand escapes numbering)--Pindar, Olympian Ode 2, line 98
Chapter 1, §§1-22: introduction, hypotheses, description of the measurement of the arc of the sun and proof that the diameter of the sun is larger than the side of a chiliagon inscribed in the same circle about the earth as the sun.
Chapter 2, §§1-4: dimensions of the world on the basis of the hypotheses
Chapter 3, §§1-8: the system of numbers from his book to Zeuxippus, including a theorem on how to multiply Archimedes numbers.
Chapter 4, §§1-20: the grains of sand filling up a sphere equal to the world and conclusion
How to translate numbers
Brief survey of the Hellenistic Greek Number System
The text used is I. Heiberg's as revised by Stamatis (v. 2, Stuttgart: Teubner, 1972). Ch. Mugler's text and French translation was also consulted, cf. vol 3, (Paris: Société d'Édition «Les Belles Lettres», 1971).
Archimedes, The Arenarius of Archimedes, ed. by E.J. Dijksterhuis (Leiden: Brill, 1956). This is a student text of the Greek with a short glossary in Dutch and English.
Archimedes, Les Oeuvres completes d'Archimede (French trans. by Paul ver Eecke; Paris: Descelée de Brouwer, 1921).
Archimedes, Apanta (ed. with Mod. Greek trans. by Evangelos S. Stamatis; Technikou Epimelitirou tis Ellados, 1970). Archimedes, .Über schwimmende Körper und die Sandzahl (trans. Arthur Czwalina-Allenstein; Leipzig: Akademische Verlagsgesellschaft, 1922). German translation.
E.J. Dijksterhuis, Archimedes (trans. by C. Dikshoorn, forword (this
edition) by Wilbur Knorr; Princeton: Princeton University Press, 1986) . A translation
of the introduction of Ch. 1 and the conclusion (4.20) with a detailed analysis
of the rest.
T.L. Heath, The Works of Archimedes (Cambridge: Cambridge U. P., 1897). A translation of much of Ch. 1 and the conclusion (4.20) with an analysis of the rest.
T.L. Heath, Aristarchus of Samos, the Ancient Copernicus (Oxford: Oxford
O. Neugebaurer, A History of Ancient Mathematical Astronomy (Berlin: Springer, 1975)
On Greek number systems, cf.
T.L. Heath, Greek Mathematics, vol. 1 (Oxford: Oxford University Press, 1921)
M.N. Todd, Ancient Greek Numerical Systems (Chicago: Ares, 1979)
Since Archimedes counts in groups of 10000 (myriad), higher numbers are separated by a comma after the fourth and not the third unit, as is our practice. E.g. 10,0000 instead of 100,000.
In the notes a number in his system is given with its rank as a subscript. 13247 is 1324 of the seventh rank, i.e., 1324 * 1049-1 or more in our notation 1.324 *1051.
|unit||not a number||one||one something|
|octet||eight units grouped together||eight||eight somethings|
|decad||ten units grouped together||ten||ten somethings|
|hekatontad||one hundred units grouped together||hundred||one-hundred somethings|
|chiliad||one thousand units grouped together||thousand||one thousand somethings|
|myriad||10,000 units grouped together||ten-thousand||ten-thousand somethings|
Where we commonly use only three words, although we have many alternative words for some numbers, Archimedes uses five different words for numbers, as is standard in Ancient Greek:
1. cardinal adjective, which can be used as a noun with a plural neuter article,
2. cardinal noun, which is a feminine singular
3. adverb indicating multiplication,
4. ordinal adjective.
5. numerical symbols.
How to translate three of these is straight-forward.
1. Cardinal adjectives may be translated by the standard English words for number, 'one', 'two', 'three', ..., except where the words are used as nouns, where English misses the plural. Nonetheless, I translate the noun form by these.
4. Ordinals may be translated as 'first', 'second', etc.
5. Numerical symbols may be translated with Arabic numerals, but with a loss, since the difficulty that Archimedes is attempting to deal with is not obvious. See the survey below of the Greek number system based on the alphabet. Keep in mind that numerical symbols could be used as ordinals as well (just as we write '5th'). Also the older, and possibly more common system, even in the time of Archimedes, was based on iterations of symbols for 1, 5, 10, etc. In the case of symbols for 5 and up, this was based on abbreviations of the name of the number, e.g. for 20 or 2 tens or 2 (deka).
The other two are not so obvious. How are we to translate the other two without artificiality?
The adverb indicating multiplication is translated by the number with the affix 'times', e.g. 'thirty-six-times', rather than attempting to form general forms from 'triple', etc.
The cardinal noun is not so obvious. Here are some nouns for numbers in English:
monad, dyad, ennead, octad, decad (or decade). The Greek occurs for some numbers in English.
duet, trio, quartet, quintet, ..., octet, ...
doublet, triplet, quadruplet, ..., octuplet, ...
Any of these, except the last, creates an artificiality. Hence, it is tempting to translate as 'ten-thousand-units'. This proves even more artificial and prevents our speaking of more than one myriad without further linguistic contortion. For this reason I chose 'unit' and the Anglicizations of Greek nouns for numbers.
Note that it is wrong to think of an octad as a set of eight units. It is eight units grouped together as a single unit.
In the fifth century, a system of numbers was developed based on the Ionian alphabet supplemented. Although the system was not commonly adopted for a long time, it becomes the preferred system for Hellenistic mathematicians.
The system has a distinct digit for the units, 1, ..., 9, for the tens, for the hundreds, etc. To indicate that a sequence of symbols is a number, it is common to place a horizontal line above the number, although there were other conventions. Fractional parts (one nth), incidentally, will normally have a line drawn almost vertically above the same numerical symbol.
There are three sets of 9 numerals from the Ionian alphabet (24 letters) with an additional 3 letters (6, 90, 900). In this way one can write numbers up to 999 or .
|1 A||2 B||3||4||5 E||6||7 Z||8 H||9|
|10 I||20 K||30||40 M||50 N||60||70 O||80||90|
|100||200||300 T||400||500||600||700||800||900 or|
A slash on the left side of a numeral up to multiplies the digit by 1000, i.e., 1000 = . In this way, one can write numbers up to 9999 or . This is the highest number in this series, e.g. does not occur as a symbol for 1,0000. Part of the reason for this must be that people conceived of 1,0000 as a unit of large counting, as reflected in Archimedes' treatise.
There are several ways of extending the system by taking M , for 'myriad' (keep in mind that M is the digit for 40). Placed beneath a number multiplies it by 1,0000. For example, 1,0000 is . In this way, one can write a number up to 9999,9999 or . However, this is as high as one can go. Hence, Archimedes presents his proposal for larger numbers. Another method is to place the abbreviation before the number of myriads.
Later systems are based on iterating myriads with a separator.