SCA. LES or NoT.yrnix. The explanation of the fact that 10 is almost everywhere found a. the base of the system of counting is seen in the common u.r of the in elementary calcula tions. In all ancient civilizations linger reckon ing was known, and even to-day it is can on to ;t remarkable extent among savage peoples. It is evident that any integer may be made the base of a scale of notation, the number of symbols the sane as the number of units in the base. Some languages contain word: I whinging fundamentally to the scales of 5 and 20. with out these system- having been completely elabo rated. In the Roman and Babylonian systems 12 and GO appear as bases. The New Zealanders have a scale Of 11, their language possessing words for the first few powers of I I, and con sequently 12 is represented as 11 and 1, 13 as II and 2, 22 as two 11's, and so on. (See NUNIER.vrtoN.) NVIiat has been said couit'erning the delevopment of the number symbols illus trates the power of a well-arranged number sys teln and its necessity for progress in Inathe matieal science. For reasons already stated, the world has generally adopted the decimal notation. In this system eault place has a value tin-fold that of the place at its right, the general form of the integers being 10". m +.........
1 + + 101, g, and that of t he fract ions being in + TLr (1(Tittial fraction was a relatively late development of the system. During the Aliddle the sexagesinial fractions (see FnAcrioNs) inherited from the Babylonians through the later (;reeks, had been generally used by physicists and astronomers, and had therefore reecived the nalnes and 'ast ninon] lea 1 fractions,' NVe have the remains of the system in our degrees, mintit(s. and seconds. The medineval fractions were not. I limited to seconds, lloW4.ver, hut ex tended to 'third.,' fourths; and so on. For example, 12° 5' 3" Ili"' DV' means in modern 7) :3 1 (1 1 „symbols 12 + 60 , I o (list in guish the fractions of trade front the 'fraetiones astromanie;e: the former Were called 'fi.aetiones vul,aares,' from which come the English 'vulgar fractions' :1111.1 t hi' 111PriVall fraction..' The constant advance of scieno . calling for numbers and more elaborate fractions, finally .log11;111(1(41 Olt the
sexagesinial system. .\ s early as the hitter part I ti ft Vent h sonar 111.111'0601N of of t111` Ill'el1na1 fraction an' seen. ing the sixteenth centnry several efforts were made in the same Ilircct ion. notably that by ti1galui (IV.). But it was the advent of ithms at the opening of the seventeenth century Illat made the rweessity apparent and pi) to (leen' ;d fractions a general recognition in the s. die world. It Na., howi.ver, fully a I Ory liter that they began to be reeognized in business; the establishment of the nictrie system (q.v.) :Ind the deeinial coinages of the various countries thiallv compelled their general A 41.11111:11rj.1111 1)f the three systems is in the following representatiim of one-tenth: (com mon). 0.1 Isexagesinial).
Index notation may also be mentioned as a recent example of the power of symbolism. .\ I ronianers and physicists, having to employ both exceedingly large and small 'lumbers iu ealeula lion, find it advantageous in approximations to introduce powers of till. Tim, 284,000.000,000 may be expressed by 281 • and 0.00000000035 by 3.5 •11) it if these numbers are to be multi plied, the process is simply or 991 = 99.4.
Consult: Cantor, Forlcsungen iiber Gcsehieltle (ler 11iillienoilik (Leipzig, I8811; 2(1 ed. 18114); 1111' I/ethodik der pnikliselien ( lb., 1888) ; Gil/tither, Gesehichte des writhe unitise/lc» Utilerrielas but ilculselien his zit»; dithr 152.; (Berlin, 1887) NVocpeke, Sur Ph/too/action de Inrillitio'ligue inilienne iii Deciticul (Nome. 1859) "Winoirs sue la propa gation des chitIres in Journal/ Asia/bine ( Vli(tic s6ric, part i., 1'a cis, 1863) ; Friedlein, Die Zultlzeicheit um/ flits eleineitlirre Prehnen der t;rieelien und Romer (Erlangen. 1S119) ; Pilian, Exioa.' des signes de num ("ration asiti's chez les peup/cs orb /aux (Paris, 18(10). For the history and bibliography. consult : Treutlein, GcsuItic/ne unserer Zahl:eichen (Karlsruhe. 1875) : and Can tor, Dese/uic/iIc der (Leipzig, 2 tions, 18811-981, both of which give extensive tables showing the development of the forms of the numerals.