1

alphabetic numerals were first employed in Alexandria early in the 3rd century B. C. It remains to be added that two of the foremost Greek mathematicians were during this century very much interested in the further abbreviation of Greek numerals. Archimedes (B. C. 287-212) and Apollonius of Perga (flor. temp. Ptol. Euergetes B. C. 247-222) both suggested new modes of stating extremely high numbers, the former in his ψαμμίτης, the latter probably in his ωκυτόκιον. These will be described later on but are mentioned here to show that probably arithmetical symbolism was one of the Alexandrian subjects of inquiry at precisely the time when the new symbolism first appears on Alexandrian records.

34. But it is time to return to the alphabetic numerals as used in calculation. Fractions (AETTá) do not appear on inscriptions but are represented in MSS. in various ways. The most common methods are either to write the denominator over the numerator or to write the numerator with one accent and the denominator

twice with two accents each time (e.g. S or i oг iç′ kа" κa”). Submultiples, or fractions of which the numerator is unity, are the most common. With these, the numerator is omitted, and the denominator is written above the line or is written once with two accents, (e. g. λβ' or ẞ" =)1. Some special signs are found, viz. signs similar to L, C' and S for and w' for . Brugsch gives, on the authority of Greek papyri?, the signs for addition, for subtraction, and for a total. Another common compendium is the form X for exárTwv and its inflexions 3.

1 For some more minute details see Nesselmann, Alg. der Gr. pp. 112-115. Hultsch, Metrol. Scriptt. 1. pp. 172— 175. Friedlein Zahlz. pp. 13–14.

It is to be remembered that though fractions with high numerators occur in Greek writers, yet they represented only the ratio between the numerator and denominator. In calculation, they were reduced, as among the Egyptians, to a series with unity for numerator and these two conceptions of a fraction,

as a ratio and as a portion of the unit,

were alone permissible in Greek arithmetic. See Cantor Vorl. pp. 107, 174, 405. Hankel, p. 62, and Hultsch, loc. cit.

2 Numerorum Demoticorum Doctrina, 1849, p. 31. See plate 1. appended to Friedlein Zahlz. and reff. there given.

3 In Heron's Dioptra (ed. Vincent p. 173) and the scholia to the Vatican Pappus (ed. Hultsch, Vol. 1. p. 128). Nesselmann, Alg. Gr. p. 305 and n. 17.

It remains to be mentioned only that the Greeks had no cipher. The ō which Delambre found in the Almagest is a contraction of ovdév, and occurs only in the measurements of angles, which happen to contain no degrees or no minutes1. It stands therefore always alone and is not used as a digit of a high number. The stroke which Ottfried Müller found on an Athenian inscription, and which Böckh thought to be a cipher, is clearly explained by Cantor as the iota, the alphabetical symbol for 10.

2

ἄβαξ.

35. Of calculation with these alphabetic numerals very little mention is made in any Greek literature. It would seem from the technical names for addition and subtraction (viz. συντιθέναι and ἀφαιρεῖν, ὑπεξαιρεῖν) and from some passages of classical authors that these operations were generally performed on the aßağ3. Multiplication, also, was, if possible, performed by addition, but it cannot be doubted that an expert reckoner would master a multiplication-table and have the alphabetic signs at his finger-ends. For such a person, for a mathematician, that is, who was competent to read Archimedes, Eutocius, a commentator of the 6th century after Christ, performs a great number of multiplications with alphabetical numerals 5. The date of the writer and the work to which they are appended alike show that these are masterpieces of Greek arithmetic. A specimen or two, with modern signs added for more convenient explanation, may be here inserted :

1 Astronomie Ancienne, I. p. 547, II. pp. 14 and 15. Theon in his commentary says nothing of this ō which indeed may be only the introduction of late transcribers who knew the Arabic signs (v. Nesselmann, Alg. Gr. p. 133, and note 25. Friedlein, Zahlz. p. 82).

2 See Math. Beitr. p. 121 sqq. and plate 28. Hultsch, Scriptt. Metrol. Graeci, Praef., pp. v. vI., Friedlein, Zahlz. p. 74.

3 Cf. Theophrastus Char. (ed. Jebb) Iy. n. 10 and XIII, n. 2.

4 Lucian, 'Epuóтiμos, 48. Friedlein, Zahlz. p. 75.

5 Torelli's ed. of Archimedes (Oxford, 1792), Circuli Dimensio, p. 208 sqq. The forms as they stand in MSS. are given p. 216. See also Nesselmann, Alg. Gr. pp. 116-118. Hankel, p. 56. Friedlein, Zahlz. p. 76, where many misprints in Nesselmann are corrected.

G. G. M.

4

The mode of proceeding is apparent on the face of this example. Each digit of the multiplier, beginning with the highest, is applied successively to each digit of the multiplicand beginning with the highest. Examples of multiplication, where fractions are involved, are also given by Eutocius. One of them is as follows1:

Multiplications are given also by Heron of Alexandria (flor. B.C. 100) and are conducted in precisely the same way as those of Eutocius. In other words, for 700 years after the introduction

1 In this specimen, the letter L represents the Greek sign for . See above, p. 48.

2 It will be observed here that Eutocius treats 13 as a single digit. He knew the multiplication table for 13.

3 Geometria, ed. Hultsch, 36 and 83, pp. 81 and 110. They are printed also by Friedlein, Zahlz. pp. 76, 77. The second of them begins: μονάδες

ιδ' καὶ λεπτὰ τριακοστότριτα κγ'· ὧν ὁ πολυπλασιασμὸς γίνεται οὕτως ιδ' ιδ' ρος· καὶ ιδ' τὰ κγ' λγ' λγ' τκβ' λγ' λγ' K.T.λ. In modern figures, the problem is 143 × 1438.

It is worked out as follows: 14 × 14=196: 14×33=322: 33×14= and (=33·33)=1++33.33. The sum (oμoû) is 1960+33 · 33= 216 + 3's • 3's •

of the alphabetic numerals, no improvement was made in the style of Greek calculation. And if such were the performances of professional calculators, it may be conceived that those of the unlearned were yet more clumsy. Thus Hankel1 quotes from a work written as late as 944 A. D., some multiplications in which the writer finds by addition that 5 times 400 is 2000 and that 5 times 9 are 45! It can hardly be doubted that some Greek compiled a multiplication table and that children at school were practised in the use of it, as Roman children were, but no trace of such a table survives nor is any clear mention of it made in any Greek writer.

36. No example of simple division nor any rules for division are found in Greek arithmetical literature. The operation must have been performed by subtracting the divisor or some easily ascertained multiple of the divisor from the dividend and repeating this process with the successive remainders. The several quotients were then added together2. But the Greeks had no name for a quotient and did not conceive the result of a division as we do. To a Greek 5 was not the quotient of . The operation did not discover the fact that 5 times 7 is 35 but that a seventh part of 35 contains 5, and so generally in Greek a division sum is not stated in the form "Divide a by b," but in the form "Find the bth part of a." This is the sort of nomenclature which would naturally be expected among a people who were constantly compelled to resort to the aẞağ with its concrete symbols.

But though there is no instance of a simple division, there is more than one of what, in our schools, is called 'compound' division, where the dividend and the divisor both consist of a

1

p. 55, citing De argumentis lunae, wrongly attributed to Bede. (Patrologia, ed. Migne, Vol. 90, p. 702.) On p. 56 Hankel gives a division from the same book. To divide 6152 by 15, multiples of 15 are first tried in order up to 6000. The remainder is 152. Then 15, 30, 60, 90, 120, 150. Remainder 2. The answer is 400+10 and 2 over.

2 The process with whole numbers may be inferred from that with fractions. Heron (Geometria, ed. Hultsch, 12. 4, p. 56) divides 25 by 13, finds a quotient 1++++ and adds these terms together to 113. Obviously the intermediate stages were 13= = {+}}=}+; etc. See Friedlein, Zahlz. p. 79.

whole number with fractions. These occur in Theon's commentary on Ptolemy's μeyáλn ouvragis (the Almagest).

Here

for astronomical purposes it is frequently necessary to conduct operations with degrees and the sexagesimal fractions, minutes, seconds etc. (πρῶτα ἑξηκοστά, δεύτερα ἑξηκοστά, etc.). The rules for such operations are easy to perceive, if it be remembered that degrees are the units, minutes ths and seconds 360ths of the unit. Hence Theon rightly premises that where a dividend consists of degrees, minutes, seconds, etc., division by degrees produces a quotient of the same denomination as the dividend division by minutes produces a quotient of the next higher denomination to the dividend: division by seconds a quotient of two denominations higher than the dividend etc. And in multiplication, of course, the denominations are similarly lowered. There is no occasion here to give a specimen of Theon's multiplication, for it follows precisely the same lines as that of Eutocius, exhibited above, p. 50. But it is desirable to show his method of division, since no other specimen of the process is procurable. He divides apie к'' te'" (i.e. 1515° 20′ 15′′) by ké iẞ'''" (i.e. 25° 12′ 10′′) in the following manner ":

1 The Latin for these was partes minutae, partes minutae secundae. The sexagesimal system is beyond question of Babylonian origin. In Greek mathematical literature, the circle is divided into 360 parts (Tμμата .or μοῖραι) first in the 'Αναφορικός of Hypsicles (cir. B. c. 180). The division of the diameter into 120 parts with sexagesimal fractions appears first in Ptolemy (cir. A.D. 140), but was probably introduced by Hipparchus (cir. B.c. 130). This trigonometrical reckoning was never used save by astronomers. See Cantor, Vorles. pp. 70, 76, 274, 311,

336, 351. Hankel, p. 65. Friedlein, Zahlz. pp. 81-82. Nesselmann, Alg. der Griech. pp. 139–147. Theon's Commentary (ed. Halma) pp. 110-119. 185-6. A summary of Hypsicles' book is given by Delambre, Astron. Anc. 1. See also post §§ 55, 140.

2 Theon does not himself give a scheme of a division, as he does of a multiplication. He merely describes the process. The scheme in the text with modern figures is from Delambre, Astron. Anc. II. p. 25. A translation of Theon's words is given by Nesselmann, p. 142,