ARITHMETIC 525 which computations can be effected, is in most cases alpha betical. In the Hebrew and Greek notations, for instance, the letters were taken in the main in their alphabetical order, being divided into three groups, of which the first represented units, the second tens, and the third hundreds; and very similar systems have been in use among other nations. As the Hebrew alphabet contained no more than twenty-two letters, the numbers from 500 to 900 were represented by five final forms, and sometimes by additive combinations with n, 400. The following is the usual Hebrew notation, numbers exceeding ten being made up by placing the larger numeral characters first : 1 2 G 3 7 D J D 10 20 30 40 50 GO p 1 C> n 1 D 100 200 300 400 500 GOO r n B 789 y B * 70 80 90 t n r roo 800 900 The ordinary Greek notation has not merely a general resemblance of structure to the Hebrew, but corresponds to it closely, character for character, up to 80. The Greek alphabet consisting of twenty-four letters, three additional characters were introduced. The first, for 6, occupying the place of 1, is 5- (named crrav), which was afterwards used as a contraction for or; the others, for 90 and 900, were named KOTTTTO. and aa.ij.-7rl respectively, and written ? or ^ and ft). The notation is thus as follows : a (3 7 123 i K A 10 20 30 40 50 GO o 70 80 90 pa- 7 v < x "A w /?) 100 200 300 400 500 GOO 700 800 900 To distinguish the numeral letters an accent was written after the last, thus, 38 was A^, while thousands were indi cated by writing the accent below, thus 8 was 4000. The letter M (for pvpLoi) increased the numeral 10,000 times. Fractions were separated by a space from the integers they were affixed to, and the denominator was written like an index or power in our notation, thus, fj.y 6 ia was 43^-. In addition to this another entirely different notation is found in Greek inscriptions. It more nearly resembles the Roman system, the numeral characters (except the first) being the initials of the numeral names, and being repeated till the number to be signified was expressed. I represents, 1, IT (TTCVTC) 5, A (Se /ca) 10, II (eVaro v, the aspirate being written as an H) 100, X (xi Atot) 1000, M (jjivpioC) 10,000. Characters enclosed within three lines (forming a II) are thereby multiplied by 5. To give an example, in this notation I jj HH LA] AIIIII represents 768. The Roman system, with which we are still familiar nearly in its completeness, employs the letters of the alphabet, but -is not based upon alphabetical order. Many attempts have been made to account for the symbols, and the com plete solution of the problem of their origination is perhaps unattainable. Sir John Leslie, following some writers of the 1 Gth and 1 7th centuries (see in particular the Cursus Matlie- maticus of Dechales, vol. i. 1G74, 2d. ed. 1690), advocated the opinion that, one line or stroke being taken to repre sent the unit, when ten of these were set down, a stroke would be drawn across them in a slanting direction to cancel them, and the unit stroke and cancelling stroke would thus give the form X for 10; that a repetition of this proceeding would give a third stroke when 100 was reached, and the three might take the form Q or C; and that, similarly, the combination of four strokes would give M for 1000. This explanation is perhaps too ingenious. It has the merit of accounting for the X (the crux of the method), which may have been introduced in some such way. But one does not readily see how the Q could be formed from the X, or the M from the LT ; and it appears far more likely that the signs for 100 and 1000 are merely the initial letters of Centum and Mille, all the more that the very ancient notation noticed above as found in Greek inscriptions has evidently an origin of this kind. In any case V, L, and D appear to be respectively the halves of X, the angular C (C), and the rounded M (co). The ancient forms of D and M, viz., IO and CIO, have ceased to be familiar. By an extension of this style of characters, IOO denoted 5000; CCIOO, 100,000; CCCIOOO, 1,000,000, <fcc. To represent two, three, &c., millions, the CCCIOOO was repeated the required number of times. The Roman notation employs fewer characters than the Greek, and makes greater use of combinations. One is repeated up to four; to the new character for five, ones are repeated up to nine; ones are added to the ten character; at fifteen the five enters, and so at twenty-five, &c. ; for the tens up to forty the ten is repeated, and so on, the symbol* I, V, X, L, C, D expressing all numbers by regular com binations up to M, a thousand. The subtractive colloca tions, IV, IX, XL, XC, now the ordinary, were originally alternative forms, as were also the rarer combinations IIX for 8, XIIX for 18, XXC for 0, fec. To the extent of these subtractive forms, the values of the characters depended on their position, a smaller number being added to a greater when it followed, and subtracted from it when it preceded it. This element of position is, however, an irregular and exceptional one; and, instead of being of such advantage as the local value of the Arabic notation, is rather a hindrance in calculation. Far superior to all the ancient systems, and indeed to every other system that exists, is the arithmetical nota tion that is in common use. The Arabic numerals, as they are called, are ten in number, nine of them repre senting the first nine numbers, and the tenth, the cipher or zero, indicating the absence or negation of numerical value. The significance of these figures or digits depends on their relative position, and the great merit of the system is due to this element of local value. Standing singly, the figures denote simply one, two, three, &c. ; but in combina tions of them every removal towards the left increases the value of the figure ten times. In 5673, for instance, the 3 denotes three, the 7 seven tens (70), the 6 six times ten tens (GOO), the 5 five times ten hundreds (5000). Should any of the series of tens be wanting, as in nine thousand and forty-eight, where hundreds do not occur, the place is supplied by a cipher, which throws back the digit that expresses thousands into its proper place, thus 9048. It is by this use of the cipher in supplying blanks, and so- regulating the places of the significant digits, that the principle of local value is carried out. See further the sec tions on notation and numeration below, p. 527. The ordinary numerals are called Arabic, and it appears to have been through the Arabians that they were introduced into Europe; but they are now generally acknowledged to be of Indian origin. As may be imagined, they have passed through a great variety of forms, one of the earliest types of them being the Devanagari, a species of Sanskrit numerals. In the early Indian treatise of Bhascara and Brahmegupta, translated by Mr Colebrooke (see article ALGEBRA, p. 517 of vol. i.), as well as in the still earlier writings of Arya-Bhatta, there are both indications and
illustrations of the use of the nine digits and the cipher;