NUMERALS 625 numerical signs ; and in antiquity the practice of counting by these natural signs prevailed in all classes of society. Even at this day if a Wallachian peasant wants to multiply 8 by 9 he effects this by making the fingers of each hand, counting from the thumb, represent the numbers from 6 to 10 consecutively. He therefore sets his question by stretching out the ring finger of the right hand and the middle finger of the left. He then counts that in the direction from the thumb there is one more finger on the right hand and two on the left. Multiplying 1 by 2 he gets 2 as the units of the product sought. Again the out stretched fingers are respectively the third and fourth from the thumb ; adding 3 and 4 he gets 7 for the tens of the product. By this rule he does not need to know the multiplication table above 4 times 4. In the later times of antiquity the finger symbols were developed into a system capable of expressing all numbers below 10,000. The left hand was held up flat with the fingers together. The units from 1 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. Thus the Romans kept count of years by yearly driving a nail into the temple of Minerva. The nail in this case is a sort of hieroglyphic, and in all systems of hieroglyphics signs for numbers naturally occur. 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, 100, 1000, and so forth. Intermediate numbers are ex pressed 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 1 to 99 are got by repetition of the vertical arrow head T= 1, and a barbed sign { = 10. But the most inter esting 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 1. 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 multipli cation for the mere addition of symbols. The same thing appears in Babylonia, where a smaller number put to the right of the sign for 100 (1) is to be added to it, but put to the left gives the number of hundreds. Thus {T~- = 1 000, but Y- < = 1 1 0. The Egyptians had hieroglyphics for a thousand, a myriad, 100,000 (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 com pendious than words written at length. But the letters 1 The system is described by Nicolaus Ehabda of Smyrna (8th century A.D.), ap. N. Canssiuus, De eloqmntia sacra et humana, Paris, 1 636. Bede gives essentially the same system, and it has survived in the East to the present day ; see especially Rodiger, " Ueber die im Orient gebriiuchliche Fingersprache, &c.," D. M. G., 1845, and Palmer in Journ. of Philology, ii. 247 sqq. 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 200 A.D. I stood for 1, II for 5, A for 10, 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 500. Another way of using the alphabet depended on the fixed order of its letters. ^ The simplest application of this principle is to use consecutive letters for consecutive numbers. Thus the 24 letters of the Ionic alphabet stood for the numbers 1 to 24, as we still see in the letters attached to the books of the Iliad. Another way common to the Greeks, Hebrews, and Syrians, and which in Greece gradually displaced the Phoenician. Hipi-atir. Hieroglyphic. Mil Syriac. Palmyrene. 1 / t II Yl III W Illl ^ IV //H
///H 493 7 ~7 7
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o 3 7 r4 O 33 joe -^9 ooo f f ^t ~^-x"O "O JOOO -^333 oooo 3333 JOGQO -^3333 I ~^ 1? -S! ># WJ 1 1 qj| II 2 tO M 111 3 i tea au Illl 4 M} II III 5 tt nun 6 S* in mi 1 ^> *=) Illl Illl 8
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111 111 III 9 ^ /< n 10 IX in 11 ^x immiin 19
nn 20 ^ inn 21 X nnn 30 -L- nnnn 40 < nnnnn 50 VI nnnnnn 60 ^ nnn nnnn 70 >H nnnn nnnn 80 J% nnnnnnnnn 90 ^ 9 100 ^J (j) G) 200 J* ff)C)O 300 HI Table 1. Herodian numbers, was to make the first nine letters stand for the units and the rest for the tens and hundreds. 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, D, f, *, Y 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 (F or = = 1 = 6, and ? = p = 90), and by adding sampi TA for 900. 2 2 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 o- for D and (A for >), while the hundreds from 500 to 1000 were expressed by the new letters in order from vi> to i. In the time of Caliph Walid (705-715 A.D.) the Arabs had as yet no signs of numeration. XVII. 79
