The number system of the Hindus is of special interest. because it is to these Aryans or to the Arabs that we owe the valuable position system DOW in use. 'Their oldest symbols for the num bers from 4 to 9 seem to have been merely the initials of their nnmber-words, and the use of letters as figures seems to have been quite prevalent from the third century B.C. The zero is of later origin, and its introduction is not proved with certainty after 400 A.D. The writing of numbers was carried on. chiefly ac cording to the position system, in various ways. One plan. which Aryahhatta records, represented the numbers from 1 to 25 by the hventy-five consonants of the Sanskrit alphabet, and the succeeding tens (30, 40....100) by the semi vowels and sibilants. A series of vowels and diphthongs formed multipliers eonsisting of pow ers of ten, go meaning 3, gi 300, go 30,000, gun 3. In this there is no application of the position system, hut it appears in two other methods of writing numbers in use among the arithmeticians of Southern India. Both of these plans are distinguished by the fact that the same number can be made up in various ways. The first method consisted in allowing the alphabet, in groups of nine symbols. to denote the numbers from I to 9 repeatedly, while certain vowels represented the zeros. If in the English al phabet, according to this method, the numbers from I to 9 he denoted by the consonants 1), e, , z, so that after two countings one finally has z = 2, and zero be denoted IT every vowel or combination of vowels, the number 60502 might be indicated by sir, n or heron, and might be introduced by some other words in the text. The second method employed type-words. Thus alalhi (one of the 4 seas) =-I, surya (the sun with its 12 houses)= 12. acrin (the two sons of the sun) = 2. The combination abdhi suryorrinas denoted the number 2124.
In the eighth century (c.772) the Arabs. whose numerals consisted of abbreviated number-words of an inferior type, the Dirani, heeame acquainted with the Ilindu system. From these figures there arose, among the Western Arabs. the unbar numerals (dust numerals). These Glubar numer als. almost entirely forgotten to-day among the Arabs themselves, are the ancestors of our mod ern numerals. These primitive Western forms,
used in the abacus-calculation, are found in the West European manuscripts of the eleventh and twelfth centuries, and owe much of their promi nenee to Gerbert, afterwards Pope :-.4ylvester II. (q.v.), and to Leonardo Fibonacci (q.v.).
The arithmetic of the Western nations, cul tivated to a considerable extent in the cloister schools from the ninth century on. employed, besides the abacus, the Roman numerals, and consequently did not use a symbol for zero. In Germany. up to the year 1500, the Boman symbols were called German numerals, in distinction from the symbols of Arabilindu origin, which included the zero (Arabic, as-sifr; Sanskrit. sunya, the void). The latter were called ciphers (Ziffern). From the fifteenth century on these Arah-Hindu numerals appear more frequently in Germany on monuments and in churches, but at that time they had not become common property. A fre quent and free use of the zero in the thirteenth century is shown in tables for the calculation of the tides at London, and of the duration of moonlight. In the year 1471 there appeared in Cologne a work of Tetrarch with page-numbers in Hindu figures at the top. In 1478 the first printed arithmetic appeared at Treviso. and in 1482 the first German arithmetic at Bamberg, and these explained the system. Besides the ordinary forms of numerals everywhere used to-day. the following forms for 4, 5, 7 were used in Germany at the time of the struggle between the Roman and Ilindn notations: The derivation of the modern numerals is illus trated by the examples below. which are taken in succession from the Sanskrit, the figures used by Gerbert (latter part of the tenth century), the Eastern Arah, the West ern Arab Gubar numer als, the numerals of the eleventh, thirteenth. and sixteenth eenturies: In the sixteenth cen tury the Hindu position arithmetic and its nota tion first found eomplete introduction the civilized peoples of the West. By this means was fulfilled one of the indispensable conditions for the development of common arithmetie in the schools and in the service of trade and commerce.