A History of Mathematical Notations/Volume 1/Greeks

by Florian Cajori

GREEKS

32. On the island of Crete, near Greece, there developed, under Egyptian influence, a remarkable civilization. Hieroglyphic writing on clay, of perhaps about 1500 B.C., discloses number symbols as follows: 𐅀 or 𐄇 for 1, 𐅀𐅀𐅀𐅀𐅀 or 𐄇𐄇𐄇𐄇𐄇 or 𐄋 for 5, (symbol characters) for 10, (symbol characters) or (symbol characters) for 100, (symbol characters) for 1,000, (symbol characters) for ¼ (probably), (symbol characters) for 483.^[1] In this combination of symbols only the additive principle is employed. Somewhat later,^[2] 10 is represented also by a horizontal dash; the sloping line indicative of 100 and the lozenge-shaped figure used for 1,000 were replaced by the forms O for 100, and X for 1,000.

XX  OO     OO      ——— ——— ———  ||  ||  ||   stood for 2,496.

33. The oldest strictly Greek numeral symbols were the so-called Herodianic signs, named after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them. These signs occur frequently in Athenian inscriptions and are, on that account, now generally called Attic. They were the initial letters of numeral adjectives.^[3] They were used as early as the time of Solon, about 600 B.C., and continued in use for several centuries, traces of them being found as late as the time of Cicero. From about 470 to 350 B.C. this system existed in competition with a newer one to be described presently. The Herodianic signs were

𐅂	Iota for 1
Π or 𐅃 or 𐅃	Pi for 5
Δ	Delta for 10
Η	Eta for 100
Χ	Chi for 1,000
Μ	My for 10,000

34. Combinations of the symbols for 5 with the symbols for 10, 100, 1,000 yielded symbols for 50, 500, 5,000. These signs appear on an abacus found in 1847, represented upon a Greek marble monument on the island of Salamis.^[4] This computing table is represented in Figure 12.

The four right-hand signs Ι Ϲ Τ Χ, appearing on the horizontal line below, stand for the fractions 1/4, 1/12, 1/24, 1/48, respectively. Proceeding next from right to left, we have the symbols for 1, 5, 10, 50, 100, 500, 1,000, 5,000, and finally the sign Τ for 6,000. The group of symbols drawn on the left margin, and that drawn above, do not contain the two symbols for 5,000 and 6,000. The pebbles in the columns represent the number 9,823. The four columns represented by the five vertical lines on the right were used for the representation of the fractional values 1/4, 1/12, 1/24, 1/48, respectively.

35. Figure 13 shows the old Herodianic numerals in an Athenian state record of the fifth century B.C. The last two lines are: Κεφάλαιον ὰνα[λώατοϛτ] οῦ ἐπὶ τ[ης] ἀπχῆς 𐅋𐅋𐅋𐅄ΤΤΤ . . . .; i.e., “Total of expenditures during our office three hundred and fifty-three talents. . . . .”

Fig. 12.—The computing table of Salamis

Fig. 12.—The computing table of Salamis

36. The exact reason for the displacement of the Herodianic symbols by others is not known. It has been suggested that the commercial intercourse of Greeks with the Phoenicians, Syrians, and Hebrews brought about the change. The Phoenicians made one important contribution to civilization by their invention of the alphabet. The Babylonians and Egyptians had used their symbols to represent whole syllables or words. The Phoenicians borrowed hieratic signs from Egypt and assigned them a more primitive function as letters. But the Phoenicians did not use their alphabet for numerical purposes. As previously seen, they represented numbers by vertical and horizontal bars. The earliest use of an entire alphabet for designating numbers has been attributed to the Hebrews. As previously noted, the Syrians had an alphabet representing numbers. The Greeks are supposed by some to have copied the idea from the Hebrews. But Moritz Cantor^[5] argues that the Greek use is the older and that the invention of alphabetic numerals must be ascribed to the Greeks. They used the twenty-four letters of their alphabet, together with three strange and antique letters, ϛ (old van), ϙ (koppa), ϡ (sampi), and the symbol Μ. This change was decidedly for the worse, for the old Attic numerals were less burdensome on the memory

Fig. 13.—Account of disbursements of the Athenian state, 418–415 B.C., British Museum, Greek Inscription No. 23. (Taken from R. Brown, A History of Accounting and Accountants [Edinburgh, 1905], p. 26.)

much as they contained fewer symbols. The following are the Greek alphabetic numerals and their respective values:

α
1 β
2 γ
3 δ
4 ε
5 ϛ
6 ζ
7 η
8 θ
9 ι
10 κ
20 λ
30 μ
40 ν
50 ξ
60 ο
70 π
80 ϙ
90 ρ
100 σ
200 τ
300 υ
400 φ
500 χ
600 ψ
700 ω
800 ϡ
900 ͵α
1,000 ͵β
2,000 ͵γ
3,000 Μ
10,000 ${\stackrel {\beta }{\mathrm {M} }}$
20,000 ${\stackrel {\gamma }{\mathrm {M} }}$
30,000 etc.

37. A horizontal line drawn over a number served to distinguish it more readily from words. The coefficient for Μ was sometimes placed before or behind instead of over the Μ. Thus 43,678 was written δΜ͵γχοη. The horizontal line over the Greek numerals can hardly be considered an essential part of the notation; it does not seem to have been used except in manuscripts of the Byzantine period.^[6] For 10,000 or myriad one finds frequently the symbol Μ or Μυ, sometimes simply the dot ·, as in β·οδ for 20,074. Often^[7] the coefficient of the myriad is found written above the symbol μ^υ.

38. The paradox recurs, Why did the Greeks change from the Herodianic to the alphabet number system? Such a change would not be made if the new did not seem to offer some advantages over the old. And, indeed, in the new system numbers could be written in a more compact form. The Herodianic representation of 1,739 was Χ𐅅ΗΗΔΔΔΠΙΙΙΙ; the alphabetic was ͵αψλθ. A scribe might consider the latter a great innovation. The computer derived little aid from either. Some advantage lay, however, on the side of the Herodianic, as Cantor pointed out. Consider ΗΗΗΗ+ΗΗ=𐅅Η, ΔΔΔΔ+ΔΔ=𐅄Δ; there is an analogy here in the addition of hundred's and of ten's. But no such analogy presents itself in the alphabetic numerals, where the corresponding steps are υ+σ=χ and μ+κ=ξ; adding the hundred's expressed in the newer notation affords no clew as to the sum of the corresponding ten's. But there was another still more important consideration which placed the Herodianic far above the alphabetical numerals. The former had only six symbols, yet they afforded an easy representation of numbers below 100,000; the latter demanded twenty-seven symbols for numbers below 1,000! The mental effort of remembering such an array of signs was comparatively great. We are reminded of the centipede having so many legs that it could hardly advance.

39. We have here an instructive illustration of the fact that a mathematical topic may have an amount of symbolism that is a hindrance rather than a help, that becomes burdensome, that obstructs progress. We have here an early exhibition of the truth that the movements of science are not always in a forward direction. Had the Greeks not possessed an abacus and a finger symbolism, by the aid of which computations could be carried out independently of the numeral notation in vogue, their accomplishment in arithmetic and algebra might have been less than it actually was.

40, Notwithstanding the defects of the Greek system of numeral notation, its use is occasionally encountered long after far better systems were generally known. A Calabrian monk by the name of Barlaam,^[8] of the early part of the fourteenth century, wrote several mathematical books in Greek, including arithmetical proofs of the second book of Euclid’s Elements, and six books of Logistic, printed in 1564 at Strassburg and in several later editions. In the Logistic he develops the computation with integers, ordinary fractions, and sexagesimal fractions; numbers are expressed by Greek letters. The appearance of an arithmetical book using the Greek numerals at as late a period as the close of the sixteenth century in the cities of Strass burg and Paris is indeed surprising.

41. Greek Writers often express fractional values in words. Thus Archimedes says that the length of a circle amounts to three diameters and a part of one, the size of which lies between one-seventh and ten-seventy-firsts.^[9] Eratosthenes expresses ⅛⅓ of a unit arc of the earth’s meridian by stating that the distance in question “amounts to eleven parts of which the meridian has eighty-three.”^[10] When expressed in symbols, fractions were often denoted by first writing the numerator marked with an accent, then the denominator marked with two accents and written twice. Thus,^[11] ιζʹ καʹʹ καʹʹ=17/21. Archimedes, Eutocius, and Diophantus place the denominator in the position of the modern exponent; thus^[12] Archimedes and Eutocius use the notation (symbol characters) or (symbol characters) for 17/21, and Diophantus (§§ 101–6), in expressing large numbers, writes (Arithmetica, Vol. IV, p. 17), (symbol characters) for 36,621/2,704. Here the sign (symbol characters) takes the place of the accent. Greek writers, even as late as the Middle Ages, display a preference for unit fractions, which played a dominating rôle in old Egyptian arithmetic.^[13] In expressing such fractions, the Greeks omitted the αʹ for the numerator and wrote the denominator only once. Thus μδʺ=¹⁄₄₄. Unit fractions in juxtaposition were added,^[14] as in ζʺ κηʺ ριβʺ σκδʺ=⅐+¹⁄₂₈+¹⁄₁₁₂+¹⁄₂₂₄. One finds also a single accent,^[15] as in δʹ=¼. Frequent use of unit fractions is found in Geminus (first century B.C.), Diophantus (third century A.D.), Eutocius and Proclus (fifth century A.D.). The fraction ½ had a mark of its own,^[16] namely, 𐅵 or 𐅵ʹ, but this designation was no more adopted generally among the Greeks than were the other notations of fractions. Ptolemy^[17] wrote 38°50′ (i.e., 38°½ ⅓) thus, ληʹ 𐅵ʺγʺ. Hultsch has found in manuscripts other symbols for ½, namely, the semicircles (symbol characters), 𐅁, and the sign (symbol characters); the origin of the latter is uncertain. He found also a symbol for ⅔, resembling somewhat the small omega (𐅷).^[18] Whether these symbols represent late practice, but not early usage, it is difficult to determine with certainty.

42. A table for reducing certain ordinary fractions to the sum of unit fractions is found in a Greek papyrus from Egypt, described by L. C. Karpinski,^[19] and supposed to be intermediate between the Ahmes papyrus and the Akhmim papyrus. Karpinski (p. 22) says: “In the table no distinction is made between integers and the corresponding unit fractions; thus γʹ may represent either 3 or ⅓, and actually γʹγʹ in the table represents 3⅓. Commonly the letters used as numerals were distinguished in early Greek manuscripts by a bar placed above the letters but not in this manuscript nor in the Akhmim papyrus.” In a third document dealing with unit fractions, a Byzantine table of fractions, described by Herbert Thompson,^[20] ⅔ is written (symbol characters); ½, (symbol characters); ⅓, (symbol characters) (from (symbol characters)); ¼, (symbol characters) (from Δʹ); ⅕, (symbol characters) (from (symbol characters)); ⅛, (symbol characters) (from Ηʹ). As late as the fourteenth century, Nicolas Rhabdas of Smyrna wrote two letters in the Greek language, on arithmetic, containing tables for unit fractions.^[21] Here letters of the Greek alphabet used as integral numbers have bars placed above them.

43. About the second century before Christ the Babylonian sexagesimal numbers were in use in Greek astronomy; the letter omicron, which closely resembles in form our modern zero, was used to designate a vacant space in the writing of numbers. The Byzantines wrote it usually ο̄, the bar indicating a numeral significance as it has when placed over the ordinary Greek letters used as numerals.^[22]

44. The division of the circle into 360 equal parts is found in Hypsicles.^[23] Hipparchus employed sexagesimal fractions regularly, as did also C. Ptolemy^[24] who, in his Almagest, took the approximate value of π to be 3+8/60+30/60×60. In the Heiberg edition this value is written γ̄ η̄ λ̄, purely a notation of position. In the tables, as printed by Heiberg, the dash over the letters expressing numbers is omitted. In the edition of N. Halma^[25] is given the notation γ̄ ηʹ λʹʹ, which is probably the older form. Sexagesimal fractions were used during the whole of the Middle Ages in India, and in Arabic and Christian countries. One encounters them again in the sixteenth and seventeenth centuries. Not only sexagesimal fractions, but also the sexagesimal notation of integers, are explained by John Wallis in his Mathesis universalis (Oxford, 1657), page 68, and by V. Wing in his Astronomia Britannica (London, 1652, 1669), Book I.

 

1. Arthur J. Evans, Scripta Minoa, Vol. I (1909), p. 258, 256.
2. Arthur J. Evans, The Palace of Minos (London, 1921), Vol. I, p. 646; see also p. 279.
3. See, for instance, G. Friedlein, Die Zahlzeichen und das elementare Rechnen der Griechen und Römer (Erlangen, 1869), p. 8; M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 120; H. Hankel, Zur Geschichte der Mathematik im Alterthum und Mittelalter (Leipzig, 1874), p. 37.
4. Kubitschek, “Die Salaminische Rechentafel,” Numismatische Zeitschrift (Vienna, 1900), Vol. XXXI, p. 393–98; A. Nagl, ibid., Vol. XXXV (1903), p. 131–43; M. Cantor, Kulturleben der Völker (Halle, 1863), p. 132, 136; M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed), p. 133.
5. Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed., 1907), p. 25.
6. Encyc. des scien. math., Tome I, Vol. I (1904), p. 12.
7. Ibid.
8. All our information on Barlaam is drawn from M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 509, 510; A. G. Kästner, Geschichte der Mathematik (Göttingen, 1796), Vol. I, p. 45; J. C. Heilbronner, Historia matheseos universae (Lipsiae, 1742), p. 488, 489.
9. Archimedis opera omnia (ed. Heiberg; Leipzig, 1880), Vol. I, p. 262.
10. Ptolemäus, Μεγάλη σύνταξις (ed. Heiberg), Pars I, Lib. 1, Cap. 12, p. 68.
11. Heron, Stereometrica (ed. Hultsch; Berlin, 1864), Pars I, Par. 8, p. 155.
12. G. H. F. Nesselmann, Algebra der Griechen (Berlin, 1842), p. 114.
13. J. Baillet describes a papyrus, “Le papyrus mathématique d’Akhmîm,” in Mémoire publiés par les membres de la Mission archéologique française au Caire (Paris, 1892), Vol. IX, p. 1–89 (8 plates). This papyrus, found at Akhmîm, in Egypt, is written in Greek, and is supposed to belong to the period between 500 and 800 A.D. It contains a table for the conversion of ordinary fractions into unit fractions.
14. Fr. Hultsch, Metrologicorum scriptorum reliquiae (1864—66), p. 173–75; M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 129.
15. Nesselmann, op. cit., p. 112.
16. Ibid., James Gow, Short History of Greek Mathematics (Cambridge, 1884), p. 48, 50.
17. Geographia (ed. Carolus Müllerus; Paris, 1883), Vol. I, Part I, p. 151.
18. Metrologicorum scriptorum reliquiae (Leipzig, 1864), Vol. I. p. 173, 174. On p. 175 and 176 Hultsch collects the numeral symbols found in three Parisian manuscripts, written in Greek, which exhibit minute variations in the symbolism. For instance, 700 is found to be ψ, ψ^π, ψʹ.
19. “The Michigan Mathematical Papyrus No. 621,” Isis, Vol. V (1922), p. 20—25.
20. “A Byzantine Table of Fractions,” Ancient Egypt, Vol. I (1914), p. 52–54.
21. The letters were edited by Paul Tannery in Notices et extraits des manuscrits de la Bibliothèque Nationale, Vol. XXXII, Part I (1886), p. 121–252.
22. C. Ptolemy, Almagest (ed. N. Halma; Paris, 1813), Book I, chap. ix, p. 38 and later; J. L. Heiberg, in his edition of the Almagest (Syntaxis mathematica) (Leipzig, 1898; 2d ed., Leipzig, 1903), Book I, does not write the bar over the ο but places it over all the significant Greek numerals. This procedure has the advantage of distinguishing between the ο which stands for 70 and the ο which stands for zero. See Encyc. des scien. math., Tome I, Vol. I (1904), p. 17, n. 89.
23. Αναφορικός (ed. K. Manitius), p. xxvi.
24. Syntaxis mathematica (ed. Heiberg), Vol. I, Part I, p. 513.
25. Composition math. de Ptolémée (Paris, 1813), Vol. I, p. 421; see also Encyc. des scien. math., Tome I, Vol. I (1904), p. 53, n. 181.