With the old Semitic alphabet of 22 letters this system broke
down at ת＝400, and the higher hundreds had to be got by
juxtaposition; but when the Hebrew square character got the
distinct final forms ץ ,ף ,ן ,ם ,ך these served for the hundreds from
500 to 900. The Greeks with their larger alphabet required but
three supplemental signs, which they got by keeping for this
purpose two old Phoenician letters which were not used in
writing (Ϝ or ϛ＝ו＝6, and Ϙ＝ק＝90), and by adding sampi 
for 900.[1]
Among the Greeks the first certain use of this system seems to be on coins of Ptolemy II. The first trace of it on Semitic ground is on Jewish coins of the Ḥasmoneans. It is the foundation of gematria as we find it in Jewish book and in the apocalyptic number of the beast (נרון קסר＝666). But we do not know how old gematria is; the name is borrowed from the Greek.
The most familiar case of the use of letters as numerals is
the Roman system. Here C is the initial of centum and M of
mille; but instead of these signs we find older forms, consisting
of a circle divided vertically for 1000 and horizontally, ⊖, or in
the cognate Etruscan system divided into quadrants, ⊕, for
100. From the sign for 1000, still sometimes roughly shown
in print as cIↄ, comes D, the half of the symbol for half the number;
and the older forms of L, viz. ⊥ or 
, suggest that this also was
once half of the hundred symbol. So V (Etruscan Λ) is half of X,
which itself is not a true Roman letter. The system, therefore,
is hardly alphabetic in origin, though the idea has been thrown
out that the signs for 10, 50, and 100 were originally the Greek
Χ, Ψ, Φ, which were not used in writing Latin.[2]
When high numbers had to be expressed systems such as we have described became very cumbrous, and in alphabetic systems it became inevitable to introduce a principle of periodicity by which, for example, the signs for 1, 2, 3, &c., might be used with a difference to express the same number of thousands. Language itself suggested this principle, and so we find in Hebrew א̄ or in Greek ,α＝1000. So further βΜυ, βΜ., or simply β.＝20,000 (2 myriads). If now the larger were always written to the left of the smaller elements of a number the diacritic mark could be dispensed with in such a case as βωλα (instead of ,βωλα＝2831, for here it was plain that β＝2000, not＝2, since otherwise it would not have preceded ω＝800. We have here the germ of the very important notion that the value of a symbol may be periodic and defined by its position. The same idea had appeared much earlier among the Babylonians, who reckoned by powers of 60, calling 60 a soss and 60 sixties a sar. On the tablets of Senkerah a list of squares and cubes is given on this principle, and here the square of 59 is written 58·1—that is, 58✕60+1; and the cube of 30 is 7·30—that is, 7 sar+30 soss＝7✕602+30✕60. Here again we have value by position; but, as there is no zero, it is left to the judgment of the reader to know which power of 60 is meant in each case. The sexagesimal system, long specially associated with astronomy, has left a trace in our division of the hour and of the circle, but as language goes by powers of 10 it is practically very inconvenient for most purposes of reckoning. The Greek mathematicians used a sort of decimal system; thus Archimedes was able to solve his problem of stating a number greater than that of the grains of sand which would fill the sphere of the fixed stars by dividing numbers into octades, the unit of the second octade being 108 and of the third 1016. So too Apollonius of Perga teaches multiplication by regarding 7 as the πυθμήν or 70, 700 and so forth. One must then find successively the product of the several pythmens of the multiplier and the multiplicand, noticing in each case what are tens, what hundreds, and so on, and adding the results. The want of a sign for zero made it impossible mechanically to distinguish the tens, hundreds, &c., as we now do.
Very early, however, a mechanical contrivance, the abacus, had been introduced for keeping numbers of different denominations apart. This was a table with compartments or columns for counters, each column representing a different value to be given to a counter placed on it. This might be used either for concrete arithmetic—say with columns for pence, shillings and pounds; or for abstract reckoning—say with the Babylonian sexagesimal system. An old Greek abacus found at Salamis has columns which, taken from right to left, give a counter the value of 1, 10, 100, 1000 drachms, and finally of 1 talent (6000 drachms) respectively. An abacus on the decimal system might be ruled on paper or on a board strewed with fine sand, and was then a first step to the decimal system. Two important steps, however, were still lacking: the first was to use instead of counters distinctive marks (ciphers) for the digits from one to nine; the second and more important was to get a sign for zero, so that the columns might be dispensed with, and the denomination of each cipher seen at once by counting the number of digits following it. These two steps taken, we have at once the modern so-called Arabic numerals and the possibility of modern arithmetic; but the invention of the ciphers and zero came but slowly, and their history is a most obscure problem. What is quite certain is that our present decimal system, in its complete form, with the zero which enables us to do without the ruled columns of the abacus, is of Indian origin. From the Indians it passed to the Arabians, probably along with the astronomical tables brought to Bagdad by an Indian ambassador in 773 A.D. At all events the system was explained in Arabic in the early part of the 9th century by the famous Abū Jaʽfar Mohammed b. Musa al-Khwārizmī (Hovarezmi), and from that time continued to spread, though at first slowly, through the Arabian world.
In Europe the complete system with the zero was derived from the Arabs in the 12th century, and the arithmetic based on this system was known by the name of algoritmus, algorithm, algorism. This barbarous word is nothing more than a transcription of Al-Khwārizmī, as was conjectured by Reinaud, and has become plain since the publication of a unique Cambridge MS. containing a Latin translation—perhaps by Adelhard of Bath—of the lost arithmetical treatise of the Arabian mathematician.[3] The arithmetical methods of Khwārizmī were simplified by later Eastern Writers, and these simpler methods were introduced to Europe by Leonardo of Pisa in the West and Maximus Planudes in the East. The term zero appears to come from the Arabic ṣifr through the form zephyro used by Leonardo.
Thus far modern inquirers are agreed. The disputed points are—(1) the origin and age of the Indian system, and (2) whether or not a less developed Indian system, without the zero but with the nine other ciphers used on an abacus, entered Europe before the rise of Islam, and prepared the way for a complete decimal notation.
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1. The use of numerals in India can be followed back to the Nana Ghat inscriptions, supposed to date from the early part of the
3rd century B.C. These are signs for units, tens and hundreds, as- ↑ The Arabs, who quite changed the order of the alphabet and extended it to twenty-eight letters, kept the original values of the old letters (putting س for ם and ش for ש), while the hundreds from 500 to 1000 were expressed by the new letters in order from ث to ع. In the time of Caliph Walīd (A.D. 705–715) the Arabs had as yet no signs of numeration.
- ↑ See further Fabretti, Paläographische Studien.
- ↑ Published by Boncompagni in Trattati d’ aritmetica (Rome, 1857).
