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ιβ', δώδεκα, δυώδεκα, and δυοκαίδεκα, twelve.

ιγ', τρισκαίδεκα and δεκατρεῖς, thirteen.

ιδ', τεσσαρεσκαίδεκα and τεσσαρακαίδεκα, fourteen.
ιέ, πεντεκαίδεκα, fifteen.

ις', εκκαίδεκα, sixteen.

ιζ', ἑπτακαίδεκα, seventeen.

ιή, ὀκτωκαίδεκα, eighteen.

ιθ', ἐννεακαίδεκα, nineteen.

κ', εἴκοσι(ν), twenty.

κα', εἷς καὶ εἴκοσι", twenty-one.

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1 Instead of " eighteen” or “ nineteen,” the Greeks frequently said " twenty wanting two" or one:" e. g. νῆες δυείν or μιᾶς δέουσαι εἴκοσι, twenty ships wanting tuo or one, i. e. 18 or 19 ships; and so for 28, 29, 38; and this applies to detailed numbers, as in Thucyd. VII. 31: τετρακόσιοι καὶ δυοῖν δέοντες πεντήκοντα ἄνδρες, “four hundred and forty-eight men ;" and to hundreds as well as to tens; thus we have in Thucyd. II. 31: τριακοσίων ἀποδέοντα μύρια, “ nine thousand seven hundred.”

2 In the composition of two numbers, if the smaller precedes, the two are joined by καί ; if the greater, the conjunction is omitted: as πέντε καὶ εἴκοσι, Herod. ; εἴκοσι πέντε, Demosth. In Pindar, Οl. I. 128, we have the two numbers combined by τε καί.

κβ', δύο καὶ εἴκοσι, twenty-two.
κγ', τρεῖς καὶ εἴκοσι, twenty-three.
κδ', τέσσαρες καὶ εἴκοσι, twenty-four.
κέ, πέντε καὶ εἴκοσι, twenty-five.
κ5', ἓξ καὶ εἴκοσι, twenty-six.
κζ', ἑπτὰ καὶ εἴκοσι, twenty-seven.
κή, ὀκτὼ καὶ εἴκοσι, twenty-eight.
κθ', ἐννέα καὶ εἴκοσι, twenty-nine.

λ', τριάκοντα, thirty, Ionie τριήκοντα.

μ', τεσσαράκοντα, forty, Ionic τεσσαρήκοντα.
ν, πεντήκοντα, fifty.

ξ', ἑξήκοντα, sixty.

ό, ἑβδομήκοντα, seventy.

π', ὀγδοήκοντα, eighty, Ionic ὀγδώκοντα.

2, 4, or 4, ἐνενήκοντα, ninety.

ρ', ἑκατόν, a hundred.

σ', διακόσιοι, two hundred, Ionic διηκόσιοι, declined thus:

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τ', τριακόσιοι, three hundred.
υ, τετρακόσιοι, four hundred.
φ', πεντακόσιοι, five hundred.
χ', ἑξακόσιοι, six hundred.
ψ', ἑπτακόσιοι, seven hundred.
ω', ὀκτακόσιοι, eight hundred.

7, ἐνακόσιοι (ἐννακόσιοι), nine hundred.

α, χίλιοι, a thousand lit. " as in a heap of fodder").

α

β, δισχίλιοι, two thousand.

γ, τρισχίλιοι, three thousand.

In the Odyss. XVI. 249, we ought perhaps to read πίσυρες καὶ Γείκοσι for πίσυρές τε καὶ είκοσι. The cardinal numbers compounded with σύν signify:

Together," σύνδυο, two together.

2 "At a time," or the distribution of a whole number into equal aliquot parts, σύντρεις, three at a time.

The same meaning may be expressed by ȧvá and els, the latter being used of the array of soldiers, as εἰς ὀκτώ, " eight deep."

δ, τεσσαρακισχίλιοι, four thousand.

P, pupio, ten thousand (lit. "as in drops of water").

Obs. 1 The following are the alterations observable in the cardinal numbers:

(a) In Homer we find ἴα, τῆς for μία, μιής, and even iῷ once for ἑνί (Il. vi. 422). The forms μín, ovdeμíny are hyper-Ionic, and are found only in Hippocrates.

(b) Homer and the later epic poets write Sotós, doin, Solov, in addition to δύω and δύο. In Herodotus we find δυῶν, δυοῖσι; in Dorie δυῶν, and in olic, according to Eustathius, δύεσιν (δύεσσιν).

(e) In old Doric inscriptions we have τρῖς for τρεῖς ; and τριοῖσι for rpuri is quoted from Hipponax.

(d) For Téσσapes, which is the form adopted by the old Ionic, by Thucydides, and the tragedians, we have TérTapes in newer Attic, réorepes in Herodotus, rérepes in Doric and in Hesiod, Op. et D. 698, and πίσυρες οι πέσσυρες in Holic.

(e) The Æolic form of Tévre was Téμne, from which come the Homeric and Attic derivatives πεμπάς, πεμπάζω, &c., and the common ordinal πέμπτος. Κ. Ο. Müller has proposed to write πέμπων for the gen. pl. in an admirable emendation of Alcæus, ap. Strab. p. 617.

(f) Homer writes Fείκοσι and εείκοσι. The Doric forms are εἴκατι, FIKATI, Feikari, Beíkari (Ahrens, Dor. Dial. p. 279), and the initial letter is retained in the Latin viginti.

(g) It will be observed that the formative a in Tplákovтa, &c. is always long after a vowel, but otherwise short or changed into ŋ. In epic and Ionic Greek we have τριήκοντα and even τριηκόντων (Hesiod, Op. et D. 694). For Teoσapákovтa the Doric dialect gives TETρÓKоVTа (Ahrens, Dor. Dial. p. 280).

(h) As far as can be determined from the variations of the MSS., it is best to write ἐνενήκοντα and ἐνακόσιοι with a single v.

(1) For χίλιοι the Lesbians wrote έλλιοι, the Boeotians χείλιοι, and the Doric xλo (Ahrens, Dor. Dial. p. 281).

(j) It was customary to distinguish between μúpio, "ten thousand," and pvpíos, "infinitely numerous." Homer has always pupíos both in the singular and in the plural, and always uses the word in the indefinite signification. With the definite value, as an indication of number, the singular is used with collective nouns in the historical writers; thus we have pupía donis in Xenophon for "ten thousand heavy-armed men;" similarly χιλίη ἵππος, “ a thousand cavalry,” in Herodotus; and διακοσία TOS, "two hundred horsemen," in Thucydides.

Obs. 2 In statements of considerable and detailed numbers, it is common to begin with the digits and so ascend to the tens, hundreds, &c. Thus the number 63,974 will be expressed Téttapa kai éßdoμÝKOVтa καὶ ἐνακόσια καὶ τρισχίλια καὶ ἑξακισμύρια. But the reversed order is often adopted, especially when the sum consists of only two numbers;

thus we have in Xenophon, Anab. v. 5, § 4: σταθμοὶ ἑκατὸν εἴκοσι δύο, παρασάγγαι ἑξακόσιοι καὶ εἴκοσι, στάδιοι μύριοι καὶ ὀκτακισχίλιοι καὶ ἑξαKOOLOL Very large numbers are frequently expressed by multiples of the substantive μvpiádes, to which the smaller amounts are added; thus we have δέκα μυριάδες, 100,000; τρισχίλια καὶ πέντε μυριάδες, 53,000; μυριάκις μύριαι μυριάδες, “ a billion.” We have also colloquial exaggerations, such as vaμμakóσio in the comedians and Athenæus, p. 671 a.

66

Obs. 3 In combining the numerical signs, which are indicated before each number, only the last of the same series bears the appropriate accent; thus we write x, 27; og', 260; awve', 1859; YTB, 53,682.

Obs. 4 Besides the cyphers given above, there is an older method of expressing numbers by the initial letters of "Ios for eis, IIévre, Aéka, Ἡεκατόν, Χίλιοι and Μύριοι. According to this method I is I, II 2, III 3, IIII 4; II is 5, III 6, ПIII 9; A is 10, AI 11, AП 15, AПIII 18, and so on; AA is 20, ▲▲▲▲ 40, and so on. In the same way H is 100, HH 200, X 1,000, XX 2,000, M 10,000. There were also abbreviated combinations of II and other letters; thus 50 was written F, F, i.e. mevtákis déka; 500 was written F or H, i.e. TeνTÁKIS ikaróv; 5,000 was written H, F, i.e. Tevтákis xíλioi, &c. Similarly they

β

expressed multiples of 10,000; thus 20,000 was M, 100,000 was M, 1,000,000 was M. It was also possible to express powers of 10,000 by repeating the letter M; thus MM was 100,000,000. In writing fractions eithery, & alone meant,, or the denominator was placed above é 26 the numerator; thus was written In the y' 49 works of the Greek mathematicians there are abundant examples of these numerical signs.

3 5

was written

με
K5

The rhapsodies of Homer, the symbols of the Heliastæ, and other conventional numbers, were indicated by the letters of the alphabet, counted a-w=1-24. For recollecting the place of a letter in the alphabet the combination ŋ, indicating the first letters of the last three hexads, furnishes a convenient memoria technica.

254

Ordinal Numbers (answering to the question "Which of the number?")

πρOTоs, first.

δεύτερος, second.

TρíTOs, third.

τέταρτος and τέτρατος, fourth.

πέμπτος, fifth.

ἕκτος, sixth.

ἕβδομος, seventh.

oydoos, eighth.

ἔνατος, ninth.

SéκaTOS, tenth.

évdékaтos, eleventh.

δωδέκατος and δυοκαιδέκατος, twelfth.

τρισκαιδέκατος and τρίτος καὶ δέκατος, thirteenth. τεσσαρακαιδέκατος and τέταρτος καὶ δέκατος, fourteenth.

EiKOσTós, twentieth.

EiKOσTÓS πρŴTOS, twenty-first.

Tρiaкоστós, thirtieth.

τεσσαρακοστός, fortieth.

πεντηκοστός, fiftieth.

ἑξηκοστός, sixtieth.

ἑβδομηκοστός, seventieth.

bydoŋkooτós, eightieth.

ἐνενηκοστός, ninetieth.

EKATоOTÓS, hundredth.

Siakoσioσтós, two-hundredth.

трiaкоσioσтós, three-hundredth.
Tεσσаракоσιοστós, four-hundredth.
Xiλoorós, thousandth.

μvρioσтós, ten-thousandth.

And so on.

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Obs. 1 The ordinals are 'used idiomatically to denote broken numbers. (a) When the fraction only is expressed, we have a compound of the ordinals TpíTOS, TéTapTos, &c. with the word uóptov, "a part;" thus τρίτος, τέταρτος, μόριον, TрITημópiov, "one-third;" Terаprηuópiov, "one-fourth," &c. The adjective ἥμισυς expresses "one-half;" and for definite magnitudes we have compounds with ἡμι-, e. g. ἡμιτάλαντον, “ half a talent,” ἡμιώβολον οι ἡμιwfóliov, "half an obol." We may also express a fraction by saying Tŵv πέντε αἱ δύο μοῖραι οι τῶν πέντε μερῶν τὰ δύο, i. e. “ "two-fifths (Thucyd. I. 10'; Arist. Pol. II. 6), or тav eέ ai тpeîs μoîpai, i. e. "three-sixths." Or if the denominator exceeds the numerator by one only, we may omit the former, as ra dúo μépn, "two-thirds." (b) When a whole number is expressed as well as its fractional part, we may either use cardinal numbers, as πévτe yμutáλavтa, “two talents and a half;" or ordinals, as πέντε ἡμιτάλαντα, ßdoμov μтáλavrov, "six talents and the seventh a half talent," i. e. 64 talents; and as σðaμý is half a ŷxus, we have in Herodotus, II. 106, teμntîs oñibaμîs, "four cubits and a half." Compare the Latin sestertius, &c.

1 There is no reference in this passage to the usual and territorial divisions of the Peloponnesus, which were six and not five, but it is merely a computation of the relative extent of territory belonging to the Lacedæmonians.

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