FINGER NOTATION, a method of rep resenting numbers by the position of the fingers or hands, analogous to the digital language of deaf-mutes. The system is an ancient one and has had three distinct ramifications: (1) The obsolete one of mere numerical representation, use among the common people in ancient times. In the early centuries of the Christian era it was well known, for it is mentioned by Tertul lian (c. 200 A.D.), Saint •Ambrose (c. 375 A.D.), and Saint Augustine (c. 400 A.D.). To the Venerable Bede, however, we are indebted for our most definite knowledge of the mediaeval system in the West, and to his treatise, loquela per gestum digitorum,) in his omnia,) (Paris 1862, tomus I, p. 689), the stu dent should refer for the best description ex tant. The same notation, not so fully ex plained, appears in a work written about 800 A.D., although assigned to Cyril of Alexandria de computo S. Cyrilli Alexandrini,' published by L. A. Muratori in his (Anecdota,) Napoli 1776), and in one by Rudulph of Laon, c. 1100 A.D. (Consult 'Abhandlungen zur Geschichte der Mathematik,' Bd. V, p. 91). The international character of the system is further evident from two poems, one in Arabic (Marne, (Maniere de compter des anciens avec les doigts des mains, d' apres un petit poente, arabe de Chems-Eddin el Mossouli,' in the Bon compagni t. I, p. 309), and the other in Hebrew, by Samuel ibn Abbas (Steinschneider, 'Die Mathematik bei den the Mathematica,' Ser. II, Vol. X, p. 81), and from a description by Nicholas Rhabdas of Smyrna. In his on Arithmetic,' written about 1341, Rhabdas gives the digital notation as it had probably been used in Asia Minor for many centuries, and, aside from Bede's description, this is our best source of information. For a brief digest consult Gow, 'History of Greek Mathematics' (Cambridge 1884, p. 24). For the Greek text, with French translation by Tannery, consult 'Notices et extraits des manuscrits de la Bibliotheque nationale,' (t. 32, p. 147, Paris 1886). The system had not yet died out in Europe when printing i was invented and Paci nolo sets it forth n a page of illustra tions in the first great modern treatise on mathematics ('Soma de Arithmetica Geometria Proportioni & Proportionalita) . . . vinegia .. . M.cccc.lxliiij, fol. 36, v.). Several arithmeti cians of the 16th century gave descriptions of the method, including Recorde (q.v.), the author of the first arithmetic printed in English. The accompanying illustration is from the most elab orate treatise on the subject, that of Aventinus ('Abacvs at qve vetvssissima vetervm lati forum per digitos manusq numerandi (quip etaim loquendi) cosuetudo, Ex beda cu picturis et imaginabus . Ratispone . . . 1532), a work
enough esteemed to warrant an edition as late as 1710.
The most common form of finger reckoning was that involving multiplication, the factors being between 5 and 10, thus rendering the table beyond 5 X 10 superfluous. This form is very old and is not yet obsolete. For example, to multiply 9 by 8, think: 5 and 4 are 9 (raising 4 fingers on one hand) ; 5 and 3 are 8 (raising 3 on the other hand) ; 4 + 3 (raised fingers)' 7 tens; 1 X 2 (bent fingers)=2 units; 70 + 72.
In addition to the sources above cited, con sult Stoy, (Zur Geschichte des Rechenunter dissertation (Jena 1876); Treutlein, in the (Abhandlungen zur Geschichte der Mathe matik> (Bd. I, p. 21); Bombelli, antica nu merazione Italica' (Roma 1876, p. 102) ; Gunther, 'Geschichte des mathematischen Un terrichts im deutschen Mittelalter> (Berlin 1887, p. 12), and 'Untersuchungen zur Ge schichte der mathematischen Wissenschaften) (Leipzig 1876).
awn) EUGENE SMITH, Professor of Mathematics, Teachers College, Columbia University, New York.
impressions from the terminal phalange of the fingers or thumbs, valuable in the Identification of individuals be cause of the fact that the lines vary in each individual from all others. This method of identification of persons has been known and practised in the East since a very early period. In later days it was used with success in legal documents, to prevent false personation, on re ceipts paid to laborers for wages, etc. More recently it has obtained greater prominence on account of the certainty of its results in identi fying criminals, particularly when used in con nection with Bertillon's Signal ment' and his 'Signalment by peculiar marks.' It is claimed that the use of the finger-print makes forgery impracticable and is of value on documents where illiteracy is prevalent. Some elaborate plans have been perfected, after much experiment, of classifying finger-prints. Some experts have distinguished four principal types, known as arches, loops, composites and whorls, with seven subclasses. Others have distin guished nine chief classes with many subordi nate classes. Investigators, however, find real difficulty in agreeing on a clear, comprehensible method of classification which would be of universal application and this perhaps is the chief handicap to the system at present.