Page:The New International Encyclopædia 1st ed. v. 14.djvu/761

* NOTATION. 649 NOTATION. as Heroilianic iuiiiiIris. Shortly after B.C. 500 two new systems a|)peared. One used the 24 letters of the Ionic alphabet in their natural order for the nunihers from 1 to 24. The oilier arranged these letters apparently at random, but actually in an order fixed arbitrarily; thus, o=l,/3=:'2, . =10, K = -20,, p — 100, ff = 200, etc. For 6, 90, and !)00 ex- ception was made, and the symbols ^ 9 ^ were used respectively. Here, too, there is no special .symbol for the zero. The Roman numerals were probably inherited from the Etruscans. The noteworthy peculiari- ties are the lack of the zero; the subtractive principle whereby the value of a .symbol was diminished bv placing before it one of a lower order ( I V = 4. IX = 9, XL = 40, XC = 90 ) , even in cases where the languaoe itself did not signify a subtraction; and finally the multipli- cative effect of a bar over the numerals (X3CX = .30.000, C = 100,000). Also for certain fractions there were special symbols and names. According to ilommsen, the Roman number-sym- bols I, V, X represent the finger, the hand, and the double hand, but they are more probably from old Etruscan letter forms. The use of the bar (vinculum or titulus) was very uncertain. Thus in the tenth century we find it used over the I and X (as among later Romans) to in- crease the value 1000-fold, but over the JI it had no significance. The symbols, too, were carelessly used. Thus, X ll TlM meant 10. 1000' and C M meant 100. 1000. T!ie subtractive prin- cijile, although known to the later Romans, was little used until very recently, as witness IIII for IV, continued from the early clock faces to those of to-day. 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 now in use. Their oldest symbols for the num- bers from 4 to 9 seem to have been merely the initials of their number-words, and the use of letters as figures seems to have been quite prevalent from the third century n.c. The zero is of later origin, and its introduction is not proved with certainty until after 400 a.d. The writing of numbers was carried on. chiefly ac- conling to the position system, in various ways. One plan, which Aryabhatta records, represented the numbers from 1 to 2.5 by the twenty-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 consisting of pow- ers of ten. ga meaning .3, f/i 300, f/ii 30.000, fjau 3. 10'°. In this there is no application of the position system, but it apjjears in two other nielhiids of writing numbers in use among the arithmeticians of Southern India. Both of these l)lans 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 1 to 9 be denoted by the consonants h. r,, c, so that after two countings one finally has 1 = 2, and zero be denoted by every vowel or combination of vowels, the number 00502 might be indicated by xircn or heron, and might be introduced by some other words in the text. The second method employed type-words. Thus uhdhi (one of the 4 seas) ^4, suri/u (the sun with its 12 houses) := 12, a(rin (the two sons of the sun):=2. The combination abdlii- suryin'iinu.i denoted the number 2124. In the eighth century (c.772) the Arabs, whose numerals consisted of abbreviated number- words of an inferior type, the Divuni, became acquainted with the Hindu system. From these figures there arose, among the Western Arabs, the (Jubar numerals (dust numerals). These (iubar 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- nence to Gerbert, afterwards Pope Sylvester 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 Ronuin symbols were called German numerals, in distinction from the symbols of .Vrab-IIindu origin, which included the zero (Arabic, as-sifr; Sanskrit, suni/a, the void). The latter were called ciphers (Zifferii). From the fifteenth century on these Arab-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 Petrarch with page-numbers in Hindu figures at the top. In 1478 the first printed arithmetic appeared at Treviso. and in 14S2 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 Hindu 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 Jf T T %-J LJ by Gerbert (latter part of v^ "-J ' 1 ^ the tenth centurv). the ^^, . >» f-. Eastern Arab, the West- (j ^ ^^ Q em Arab Gubar numer- q als, the numerals of the 1.-' _p V/ y eleventh, thirteenth, and ' /? sixteenth centuries: '"^ In the sixteenth cen- ^ '-^ y P) tury the Hindu position f f fjf arithmetic and its nota- "• I ^ l A tion first found complete O ^-^^ V / introduction among all ^^ the civilized peoples of 'T^ g ^ ^ O the West. By this means ' ^^ ' » O was ful tilled one of the ,^ •_ ^^ ^ indispensable conditions j[^ ^ ^g for the development of common arithmetic in the schools and in the service of trade and commerce.