As in all such matters, the origin of these numerals is obscure, although the changes in their forms since the 3rd century B.C. are well known. Of the various theories that of Mommsen (185o) has had the widest acceptance. This was that the use of V for 5 is due to the fact that it is a kind of hieroglyphic representing the open hand with its five fingers. Two of these gave the X for ten. Three of the other symbols, he asserted, were modifications of Greek letters not needed in the Etruscan and early Latin alphabet. These were chi, which appears in inscriptions not only as X but also in such forms as t, and which later became the L that was arbitrarily chosen for so; theta, 0, which was selected for ioo, being finally changed to C under the influence of the word centum (hundred) ; and phi, 1, to which was assigned the value 1,000, and which finally took the forms (I), and M, the last being chosen because of the word mille (thousand). There is consider able epigraphical evidence in support of these contentions made by Mommsen. (See the bibliography for other theories.) The oldest noteworthy inscription containing numerals rep resenting very large numbers is on the columna rostrata, a monu ment erected in the Roman forum to commemorate the victory of 26o B.C. over the Carthaginians. In this a symbol for ioo,000, which was an early form of ((OM, was repeated 23 times, making 2,300,000. This illustrates not only the early Roman use of re peated symbols, but also a custom which extended to modern times—that of using (I) for 1,000, ((I)) for io,000, (((I))) for 100,000, and ((((I)))) for ,000,000. The symbol (I) for i,000 frequently appears in various other forms, including the cursive 00 . All these symbols persisted until long after printing became common. In the middle ages a bar (vinculum, titulus) was placed over a number to multiply it by 1,000, but this use is not found in the Roman inscriptions. When the bar appears in early manuscripts it was merely for the purpose of distinguishing numerals from nouns, as in the case IIVIR for duumviri. We also find in the middle ages such forms as rx1 or I xI for ten hundred thousand and jrni for one thousand hundred thousand.
Of the later use of the numerals, a few of the special types are as follows: The first represents the use of the vinculum; (2) represents the place value as it occasionally appears in Roman numerals; (3) illustrates the not infrequent use of a (like D, originally half of (I), the symbol for I,o00); (4) illustrates the persistence of the old Roman form for i,000 and Soo, and the subtractive principle so rarely used by the Romans for a number like 99; (5) shows the use of quatre vingt for 8o, commonly found in French manu scripts until the i 7th century, and occasionally later, the numbers often being written like iiijxx, vijxx, and so on; (6) represents the coefficient method, "four C" meaning 40o, a method often lead ing to forms like ijM or I IM for 2,000, as shown in (7).
The Maya Use of Place Value.—Evidence of some apprecia tion of place value is found in various systems of notation. For example, the cuneiform inscriptions contain frequent examples like Ittl (using I to represent the vertical wedge and t to represent the tens symbol), in which the first t stands for 6o and the last I for one, the number being 8i. Similarly the later Roman forms occasionally contain cases like II • CX for 2,110, although these are not examples of the systematic use of place value as we under stand it. In Yucatan, however, the highly developed Maya civilization used, for calendar purposes, a system of numerals based upon a scale which has elements of both 5 and 20, the horizontal bar (—) representing 5 and the dot (.) representing unity. The following are examples: Our Common Numerals.—Several different claims, each hav ing a certain amount of justification, have been made with respect to the origin of our present numerals, commonly spoken of as Arabic but preferably as Hindu-Arabic. These include the as sertion that the origin is to be found among the Arabs, the Per sians, the Egyptians, and the Hindus. It is not improbable that the intercourse between traders served to carry such symbols from country to country, so that our numerals may be a con glomeration from different sources. The country, however, which first used, so far as we know, the largest number of our numeral forms is India. The I, 4, and 6 are found in the Agoka inscrip tions (3rd century B.c.); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 3, 4, 5, 7, and 9, in the Nasik caves of the ist or 2nd century of our era— all in forms that have considerable resemblance to our own, our 2 and 3 being well-recognized cursive derivations from the an cient = and = . None of these early Indian inscriptions gives any evidence of place value, or of a zero that would make our place value possible. Hindu literature gives some evidence that the zero may have been known before our era, but we have no actual inscription containing such a symbol before the 9th century.
The first definite external reference to the Hindu numerals is in a note by Severus Sebokht, a bishop who lived in Mesopotamia c. 65o. Since he speaks of "nine signs," the zero seems not to have been known to him. By the close of the 8th century, how ever, some astronomical tables of India are said to have been translated into Arabic at Baghdad, and in any case the numerals became known to Arabic scholars about this time. About 825 al-Khowarizmi (q.v.) wrote a small book upon the subject, and this was translated into Latin by Adelard of Bath (c. 112o) under the title of Liber Algorismi de numero Indorum. There is some reason for believing that the numerals found their way into Eu rope even earlier than into Baghdad, but the earliest European manuscript that is known to contain them was written in Spain in 976. The table on preceding page shows their subsequent development until the time of printing.