NOTATION. 650 NOTHNAGEL.
Scaij:s of Xotatiox. The explanation of the fact that 10 is almost everywhere found as the base of the system of countinjr is seen in the common use of the finders in elementary ealeula- tions. In all ancient civilizations finder reckon- ing was kiio«n, and even to-day it is cairied on to a 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 being the same as the number of units in the base. .Some languages contain words l)eloiiging fundamentally to the scales of ;"> and 20. with- out these systems having been completely elabo- rated. In the Roman and Babylonian systems 12 and GO appear as bases. The Xew Zealanders have a scale of 11, their language possessing words for the first few powers of 11, and con- seiiuently 12 is represented as II and 1, 13 as 11 and 2, 22 as two U's, and so on. (See Xljiebation.) What has been said concerning the delevopmcnt of the number symbols illus- trates the ])Ower of a well-arranged numlier sys- tem and its necessity for progress in mathe- matic-il science. For reasons already stated, the world has generally adopted the decimal notation. In this system each place has a value tenfold that of the place at its right, the general form of the integers being ID"-)/! + ]0"-'-7 + IQ'd + 10»-c + 10/) + a. and that of the fractions being 10-'-«' + IQ-^- b'+ IQ-'-c' + 10-°-m'. The decimal fraction was a relatively late Ueveloimicnt of the system. During the Middle Ages the sexagesimal fractions (see Fkaction.S) , inherited from the Uabylonians through the later Greeks, had been generally used by physicists and astronomers, and had therefore received the names 'jihysical' and 'astronomical fractions.' We have the remains of the system in our degrees, minutes, an<l seconds. The inediieval fractions were not limited to seconds, however, but ex- tended to 'thirds.' 'fourths,' and so on. For example, 12° 5' 3" 16'" 18 means in modern symbols 12 + ^+ g5^ + Q^y^ + ^,. To distin- guish the fractions of trade from the 'fractiones astronomicie,' the former were called 'fractiones vulgares,' from which come the English 'vulgar fractions' and the .merican 'common fractions.' The constant advance of science, calling for larger numbers and more elaborate fraction-*, finally demanded some improvement on the sexagesimal system. As early as the latter part of the fifteenth century some indications of the approach of the (h'cimal fraction are seen. Dur- ing the sixteenth century several elforls were made in the same direction, notably that by iStevin (q.v.). But it was the advent of logar- ithms at the opening of the seventeenth century that made the necessity apparent and gave to decimal fractions a general recognition in the sclent i(ic world. It was, however, fully a cen- tury later that they began to be rccognizeil in business: the establishment of the metric system (q.v.) and the decimal coinages of the various countries finally compelled their general use. . comparison of the three sy.stems is seen in the following representation of one-tenth:^ (com- mon), 0.1 (<lecinial) , 0' (sexagesimal). Index notation may also be mentioned as a recent example of the power of symlxilism. .s- tronomers and physicists, having to employ both exceedingly large and small numlfcrs in calcula- tion, find it advantageous in approximations to introduce powers of ten. Thus 2S4.(lii{).()0(),000 may be expressed by 284 • 1(1". and 0.01100000003.) by 3..5-10"'". If these numbers an- to be multi- plied, the process is simply 284:3-5- lO"-'" or
- ii)4-10-' = 00.4.
Consult: Cantor. Torlcsungen iiber Gcschichte tlrr ilathemiitik (Leipzig. IS.SO: 2d ed. 1804); I'nger, Die Mclhodik dcr pnihlischeit Arithntctik (ih., 1888) ; GUnther, Gcxrhichtc dix ninlhc- matisclien Untcrrichts im dfulschcn Mitti lulter bis ::um Jahr lo2'> (Berlin, 1887); Woepcke, Kitr Vintrodiiction de rurilluiitiitjuc indiruiic vn Occident (Rome, 1850) ; ilenioirs sur la i)ropa- gation des chilTres indiens." in Journal Asinlii/iic (Vl&rae seric, part i., Paris, 18(53) ; Friedlein, Die Zahlzeichen und das eleincnlaic I'rchnrn dcr Gricchen und RUmer (Erlangcu, 18(iO) ; Pihan, Expose des signes de nuinerutioii usites ehr: les peuples orientanx (Paris, 18(J0). For the history and bibliography, consult: Treutlein, Gcsehiehte iinserrr Zalil:cielien (Karlsruhe, 1875) ; and Can- tor, Gcschichte der Muthrmntik (Leipzig. 2 edi- tions, 1880-08), both of which give extensive tables showing the development of the forms of the nmnerals. NOTATION, Musical. See JIusical Xota- T10-. NOTE (OF., Fr. note, from Lat. nota. mark). In music, a character which l)y the degree it occupies on the staff re])resents a sound, and by its form the period of time or duration of that sound. The notes commonly in use in modern music are the semibreve,^; minim, f^; crotchet,, ; quaver, |* ; semiquaver, • ; demisemiquaver, > ; Taking the and senii-demisemiquaver, , . semibreve as a unit, the minim is its duration, the crotchet ',, the quaver ;, the scmi<|uaver t^ji, the demisemiquaver ',, and the semi-clemisemi- quaver f,'^. Xotes of greater length than the semibreve were formerly in use — viz. the breve, twice the duration of the .semibreve; the long, four times; and the large, eight times the semi- breve. Of these, the breve, ZH or H-sl, is still sometimes met with in ecclesiastical music. Ac- cording to their numerical ]>ro]>ortions, it is cus- tomary to speak of notes as whole, half, quarter, eighth, sixteenth, etc. See LicADlMi To.nej McsicAi, Notation. NOTE (in Law), See PROMIS.SORY XoTE. NOTHING TO WEAR. A humorous poem by William Allen Butler (q.v.). which appeared anonymously in IJnrpcr's Wccklif in 1857. The heroine is Miss Flora McFlim.sey. Many edi- (ions of this popular satire on dress have been issued. NOTHNAGEL, not'nil'grl. Hermann (1841 — ). A (Jerman ])hysician. horn at .Mt-Lietze- giiricke. He studied medicine at Berlin, taught at Kiinigsberg. Berlin, and Breshui. and was ap- pointed professor at Freiburg (1872). at .Tena (1874). and at Vienna (1882). ]?esides his Topisehe Diiignostik dcr Ochirnkrnnkhciirn (1879). and lieitriige zur Phi/sinltiiiic und I'ntho- loflie des Danns. Xothnagel wrote Ifundhuch dcr Arzneimittellchre (1870: 7th ed. 1«04). and contributions to Speziclle Pntholoflie und Thera- pie (1894 et seq.), of which he is an editor. J
