Accounting History - Greek Numeration Basics

History of Accounting > Greek Numeration Basics

The cardinal numbers in Greek are—efr, one; Svo, two;

s, three; re(rcrapa, four; Wi/re, five; e£, six; e-n-ra, seven; W, eight; ewea, nine ; SfKa, ten.

The names for the numbers 11-19 are formed by conjoining the units with ten, with in general an intervening and (koi). A few of the higher numbers are etKoa-t, twenty; TpioKovra, thirty; e/caroV, a hundred; x’^’°’, & thousand ; nvptot, ten thousand. The last number, often termed a myriad (/u^idy), was the highest denomination used by the Greeks, though they could without difficulty express numbers higher than 10,000. Thus 100,000 would be Ska pvpuiSes, and a billion ;.’ » •**•••* 7 (that is, amillion)millions) would be nvpiaKis nvpiai nvpidSes.

In the statement of large and also of comparatively small numbers it was common to begin with the units and to ascend to the tens, hundreds, but the reverse order was frequently adopted. Thus they could say five and

twenty as well as twenty and Jive; when and was omitted they put the tens first, thus twenty five (e’/coo-t TreVre).

Instead of eighteen or nineteen the Greeks (and the Romans also, as we shall see) often said twenty wanting two or twenty •wanting one. A similar mode of expression was employed for 28, 29; and Thucydides uses a myriad wanting three hundred for nine thousand seven hundred.

Some peculiarities, which are not easy to account for, occur in the names of the cardinal numbers. Thus the words for 1, 2, 3, 4 are declinable, those for 5 to 199 inclusive are indeclinable, while those for 200, 300-1000, 2000-10,000 are declinable.

The names for the ordinal numbers are mostly derived from those of the cardinal by a process which is in general uniform throughout. The first of the ordinals (irpaiTo? from irpo) has the ending of the Greek superlative—compare the form -n-poaros and the comparative Trpo-repos—and the second (Sevrepot) has the ending of the comparative. It may here be noted, however, that some tribes exist which have names for many cardinal numbers, but the only names they have for ordinals are first and last.

The Greeks expressed fractions whose numerator is unity much as we do; thus -j = To ii/j-tav, J = To rpirov, ^ = To Tgraprov. Sometimes they conjoined the name for the ordinal with a word for part; thus J = Tprninopiov, £ = Trenirrrinopiov. Fractions with other numerators than unity they expressed by stating the denominator first and then the numerator:

thus | = T irevre Tu Tpia fJ.eptj

(of the five the three parts);

fy = riav e-TTTu at Svo paipai

(of the seven the two portions).

Fractions whose numerator was one less than the denominator they expressed by stating the numerator only with the word for parts; thus 9 = ra Svo /me’pri, the two parts (out of three).

The way in which they expressed certain mixed numbers may be seen from such a phrase as efiSonov -fifjuTaXavrov. It means the seventh a half talent, and is equivalent to six whole talents and the seventh a half talent, that is, to 6i talents.

The cardinal numbers in Latin are unus, one; duo, two; tres, three; quatuor, four; quinque, five; sex, six; septem, seven; octo, eight; novem, nine; decem, ten.

The names for the numbers 11-19 are formed by prefixing the units to ten, the word deccm being modified to decim. A few of the higher numbers are viginti, twenty; triginta, thirty; centum, a hundred; mile, a thousand. The last number, mille, is the highest denomination of the Romans.

The numbers between 20 and 100 are expressed either by the larger number first and the smaller number after it without a connecting et (and), or by the smaller number first and the larger number after it with a connecting ct. Thus 21 is viginti unus, or unus et viginti. Numbers above 100 always have the larger number first.

The numbers 18, 19, 28, 29, &c., are often expressed by two from twenty, one from twenty, &c. The smaller number is put first, and de is inserted for from; thus 18 = duodeviginti, 19 = undeviginti, 28 = duodetriginta, 29 = undetriginta, and so on to undecentum for 99. 98 however is either nonaginta octo or octo et nonaginta.

As in Greek, the names of the ordinal numbers are derived from those of the cardinal, with the exception of primus and secundus. Primus is the superlative of prae or pro (before), and secundus the present participle of the verb sequor (follow).

Fractions are expressed by the ordinal numbers, with part or parts (pars or partes) expressed or understood; thus J = tertia pars, f = tres septimae. Other forms are employed when the numerator is one less than the denominator; thus § = duae partes (two parts out of three), f = tres partes (three parts out of four). Sometimes a fraction is expressed as the sum of two fractions ; thus pars quarto, et septima = { +f = H: sometimes as the product of two fractions ; thus quarta septima = ixl = *V

The Roman unit of weight, length, area was called as (our ace), and it was divided into twelve equal parts, called unciae, whence came our ounce and inch. The names of the parts are—

Uncia = -fa of the unit. Septunx = ^ of the unit.

Sextans = T or J „ „ Bes = & or § „

Quadrans = & or J „ „ Dodrans = ^ or f „ „

Triens = T or J „ „ Dextans = |£ or £ „ „

Quincunx = ^ „ „ Deunx = f£ » »

Semis = A or £ „ „

Here we have the first occurrence of duodecimal fractions.

In the expression of mixed numbers the fractional part is followed by that ordinal number which is one more than the given integer; thus 3$ = quadrans quartus, the fourth a quarter (three wholes being understood), 2£ = semis tertius, the third a half (two wholes being understood). Semis tertius, contracted to sestertius, was written with the symbol for two (II), and S, the initial of semis, after it (IIS). The horizontal stroke that was drawn through the whole symbol, as in our ft, £ for pounds, is represented by the printer’s HS.

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