# 1911 Encyclopædia Britannica/Numeral

**NUMERAL** (from Lat. *numerus*, a number), a figure used to represent a number. The use of visible signs to represent numbers and aid reckoning is not only older than writing, but older than the development of numerical language on the denary system; we count by tens because our ancestors counted on their fingers and named numbers accordingly. So used, the fingers are really numerals, that is, visible numerical signs; and in antiquity the practice of counting by these natural signs prevailed in all classes of society. In the later times of antiquity the finger symbols were developed into a system capable of expressing all numbers below io,000. The left hand was held up flat with the fingers together. The units from I to 9 were expressed by various positions of the third, fourth, and fifth fingers alone, one or more of these being either closed on the palm or simply bent at the middle joint, according to the number meant. The thumb and index were thus left free to express the tens by a variety of relative positions, e.g. for 30 their points were brought together and stretched forward; for 50 the thumb was bent like the Greek F and brought against the ball of the index. The same set of signs if executed with the thumb and index of the right hand meant hundreds instead of tens, and the unit signs if performed on the right hand meant thousands.^{[1]}

The fingers serve to express numbers, but to make a permanent note of numbers some kind of mark or tally is needed. A single stroke is the obvious representation of unity; higher numbers are indicated by groups of strokes. But when the strokes become many they are confusing, and so a new sign must be introduced, perhaps for 5, at any rate for 10, Ioo, 1000, and so forth. Intermediate numbers are expressed by the addition of symbols, as in the Roman system ccxxxvi= 236. This simplest way of writing numbers is well seen in the Babylonian inscriptions, where all numbers from I to 99 are got by repetition of the vertical arrowhead T =z, and a barbed sign- (= 10. But the most interesting case is the Egyptian, because from its hieratic form sprang the Phoenician numerals, and from them in turn those of Palmyra and the Syrians, as illustrated in table I. Two things are to be noted in this table - first, the way in which groups of units come to be joined by a cross line, and then run together into a single symbol, and, further, the substitution in the hundreds of a principle of multiplication for the mere addition of symbols. The same thing appears in Babylonia, where a smaller number put to the right of the sign for ioo {T ?) is to be added to it, but put to the left gives the number of hundreds. Thus (T =1000, but T. ' < = iio. The Egyptians had hieroglyphics for a thousand, a myriad, ioo,000 (a frog), a million (a man with arms stretched out in admiration), and even for ten millions.

Alphabetic writing did not do away with the use of numerical symbols, which were more perspicuous, and compendious than words written at length. But the letters of the alphabet themselves came to be used as numerals. One way of doing this was to use the initial letter of the name of a number as its sign. This was the old Greek notation, said to go back to the time of Solon, and usually named after the grammarian Herodian, who described it about A.D. 200. I stood for I, H for 5, A for io, H for 100, X for 1000, and M for 10,000; II with A in its bosom was 50, with H in its bosom it was Soo. Another way common to the Greeks, Hebrews, and Syrians, and which in Greece gradually displaced the Herodian numbers, was to make the first nine letters stand for the units and the rest for the tens and hundreds.

Syriac. Palmyrene. Phoenician. Hieratic. Hieroglyphic.

With the old Semitic alphabet of 22 letters this system broke down at n= 400, and the higher hundreds had to be got by juxtaposition; but when the Hebrew square character got the distinct final forms 1, o, 1, + l, these served for the hundreds from Soo to coo. 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 (F or g= i = 6, and y = p = 90), and by adding sampi-m for 900.^{[2]}

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 Hasmoneans. It is the foundation of gematria as we find it in Jewish book and in the apocalyptic number of the beast ("DP 1113= 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 z000 and horizontally, 0, or in the cognate Etruscan system divided into quadrants, G, for too. From the sign for loco, still sometimes roughly shown in print as dIo, comes D, the half of the symbol for half the number; and the older forms of L, viz. or .L, suggest that this also was once half of the hundred symbol. So V (Etruscan A) 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 io, 50, and zoo were originally the Greek X, 'Y, 4), which were not used in writing Latin.^{[3]}

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 I, 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 N or in Greek ,a = z000. So further 1 3111v, or simply 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 f wXa (instead of ,/3coXa) = 2831, for here it was plain that (3 = 2000, not = 2, since otherwise it would not have preceded w = Soo. 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 r - that is, 58 X 60+1; and the cube of 30 is 7.30 - that is, 7 sar+30 soss = 7 X 60 2 -30 X 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 10 8 and of the third io' 6. So too Apollonius of Perga teaches multiplication by regarding 7 as the 7rvOµnv 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 a, 10, zoo, 1000 drachms, and finally of i 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 Abu Ja`far Mohammed b. Musa al-Khwarizmi (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-Khwarizmi, 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.^{[4]} The arithmetical methods of Khwarizmi 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 sifr through the form zephyro used by Leonardo.

Nana Ghat (Indian)^{[5]}

Cave Inscriptions (Indian)^{[6]}

Devanagari^{[7]}

Eastern Arabic.^{[8]}

Ghobar 8. ^{[9]}

Boetius 9^{[10]}

- ↑ The system is described by Nicolaus Rhabda of Smyrna (8th century A.D.), ap. N. Caussinus, De eloquentia sacra et humana (Paris, 1636). The Venerable Bede gives essentially the same system, and it long survived in the East; see especially Rodiger, "Ober die im Orient gebrauchliche Fingersprache, &c.," D.M.G. (1845), and Palmer in Journ. of Philology, ii. 247 sqq.
- ↑ 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 for ? and w for vi), while the hundreds from 500 to woo were expressed by the new letters in order from v to t. In the time of Caliph Walid (A.D. 705-715) the Arabs had as yet no signs of numeration.
- ↑ See further Fabretti, Palaographische Studien.
- ↑ Published by Boncompagni in Trattati d'aritmetica (Rome, 1857).
- ↑ From Sir E. C. Bayley's paper in J.R.A.S. (1882).
- ↑ From Burnell's South Indian Palaeography (1874).
- ↑ Of the 10th century. (From Burnell, op. cit.)
- ↑ Of the 10th century; from a MS. written at Shiraz. (From Woepcke, Memoire sur la propagation des chiffres indiens.)
- ↑ From a MS. at Paris. (From Woepcke, op. cit.)
- ↑ Erlangen (Altdorf) MS. (From Woepcke, op. cit.)