brought forward,1 and they are consistent with what is known of the principles on which numerals or quasinumerals are really formed. But so far as I have been able to examine the evidence, the cases all seem so philologically doubtful, that I cannot bring them forward in aid of the theory before us, and, indeed, think that if they succeed in establishing themselves, it will be by the theory supporting them, rather than by their supporting the theory. This state of things, indeed, fits perfectly with the view here adopted, that when a word has once been taken up to serve as a numeral, and is thenceforth wanted as a mere symbol, it becomes the interest of language to allow it to break down into an apparent nonsense-word, from which all traces of original etymology have disappeared. Etymological research into the derivation of numeral words thus hardly goes with safety beyond showing in the languages of the lower culture frequent instances of digitnumerals, words taken from direct description of the gestures of counting on fingers and toes. Beyond this, another strong argument is available, which indeed covers almost the whole range of the problem. The numerical systems of the world, by the actual schemes of their arrangement, extend and confirm the opinion that counting on fingers and toes was man's original method of reckoning, taken up and represented in language. To count the fingers on one hand up to 5, and then go on with a second 1 See Farrar, 'Chapters on Language,' p. 223. Benloew, 'Recherches sur l'Origine des Noms de Nombre;' Pictet, 'Origines Indo-Europ.' part ii. ch. ii.; Pott, Zählmethode,' p. 128, &c.; A. v. Humboldt's plausible comparison between Skr. pancha, 5, and Pers. penjeh, the palm of the hand with the fingers spread out; the outspread foot of a bird,' as though 5 were called pancha from being like a hand, is erroneous. The Persian penjeh is itself derived from the numeral 5, as in Skr. the hand is called panchaçâkha, 'the five-branched.' The same formation is found in English; slang describes a man's hand as his 'fives,' or 'bunch of fives,' thence the name of the game of fives, played by striking the ball with the open hand, a term which has made its way out of slang into accepted language. Burton describes the polite Arab at a meal, calling his companion's attention to a grain of rice fallen into his beard. 'The gazelle is in the garden,' he says, with a smile. We will hunt her with the five,' is the reply. five, is a notation by fives, or as it is called, a quinary notation. To count by the use of both hands to 10, and thence to reckon by tens, is a decimal notation. To go on by hands and feet to 20, and thence to reckon by twenties, is a vigesimal notation. Now though in the larger proportion of known languages, no distinct mention of fingers and toes, hands and feet, is observable in the numerals themselves, yet the very schemes of quinary, decimal, and vigesimal notation remain to vouch for such hand-and-foot-counting having been the original method on which they were founded. There seems no doubt that the number of the fingers led to the adoption of the not especially suitable number 10 as a period in reckoning, so that decimal arithmetic is based on human anatomy. This is so obvious, that it is curious to see Ovid in his well-known lines putting the two facts close together, without seeing that the second was the consequence of the first. 'Annus erat, decimum cum luna receperat orbem. Hic numerus magno tunc in honore fuit. Seu quia tot digiti, per quos numerare solemus: Seu quod adusque decem numero crescente venitur, For In surveying the languages of the world at large, it is found that among tribes or nations far enough advanced in arithmetic to count up to five in words, there prevails, with scarcely an exception, a method founded on hand-counting, quinary, decimal, vigesimal, or combined of these. perfect examples of the quinary method, we may take a Polynesian series which runs 1, 2, 3, 4, 5, 5·1, 5·2, &c.; or a Melanesian series which may be rendered as 1, 2, 3, 4, 5, 2nd 1, 2nd 2, &c. Quinary leading into decimal is well shown in the Fellata series 1 . . . 5, 51 . . . 10, 10.1 . . . 105, 10:51. . . 20, . . . 30, 40, &c. Pure decimal may be instanced from Hebrew 1, 2 . . . 10, 10.1 ... 20, 201... &c. Pure vigesimal is not usual, for the obvious 1 Ovid, Fast. iii. 121. ... reason that a set of independent numerals to 20 would be inconvenient, but it takes on from quinary, as in Aztec, which may be analyzed as 1, 2 . . . 5, 5·1 . . . 10, 101 . . . 105, 10·5·1 . . . 20, 201 . . . 2010, 20·101 . . . 40, &c.; or from decimal, as in Basque, 1 . . . 10, 101 . . . 20, 20·1 . . . 20·10, 20·101 . . . 40, &c.1 It seems unnecessary to bring forward here the mass of linguistic details required for any general demonstration of these principles of numeration among the races of the world. Prof. Pott, of Halle, has treated the subject on elaborate philological evidence, in a special monograph,2 which is incidentally the most extensive collection of details relating to numerals, indispensable to students occupied with such enquiries. For the present purpose the following rough generalization may suffice, that the quinary system is frequent among the lower races, among whom also the vigesimal system is considerably developed, but the tendency of the higher nations has been to avoid the one as too scanty, and the other as too cumbrous, and to use the intermediate decimal system. These differences in the usage of various tribes and nations do not interfere with, but rather confirm, the general principle which is their common cause, that man originally learnt to reckon from his fingers and toes, and in various ways stereotyped in language the result of this primitive method. Some curious points as to the relation of these systems may be noticed in Europe. It was observed of a certain deaf-and-dumb boy, Oliver Caswell, that he learnt to count as high as 50 on his fingers, but always 'fived,' reckoning, for instance, 18 objects as 'both hands, one hand, three fingers.' The suggestion has been made that the Greek use 1 The actual word-numerals of the two quinary series are given as examples. Triton's Bay, 1, samosi; 2, roëeti; 3, touwroe; 4, faat; 5, rimi; 6, rim-samos; 7, rim-roëeti; 8, rim-touwroe; 9, rim-faat; 10, woetsja. Lifu, 1, pacha; 2, lo; 3, kun ; 4, thack; 5, thabumb; 6, lo-acha; 7, lo-a-lo; 8, lo-kunn; 9, lo-thack; 10, te-bennete. 2 A. F. Pott, Die Quinäre und Vigesimale Zahlmethode bei Völkern aller Welttheile,' Halle, 1847; supplemented in 'Festgabe zur XXV. Versammlung Deutscher Philologen, &c., in Halle' (1867). 3 Account of Laura Bridgman,' London, 1845, p. 159. of Teμπáčew, 'to five,' as an expression for counting, is a trace of rude old quinary numeration (compare Finnish lokket ‘to count,' from lokke ten'). Certainly, the Roman numerals I, II, . . . V, VI . . . X, XI. . . XV, XVI, &c., form a remarkably well-defined written quinary system. Remains of vigesimal counting are still more instructive. Counting by twenties is a strongly marked Keltic characteristic. The cumbrous vigesimal notation could hardly be brought more strongly into view in any savage race than in such examples as Gaelic aon deug is da fhichead 'one, ten, and two twenties,' i.e., 51; or Welsh unarbymtheg ar ugain ‘one and fifteen over twenty,' i.e., 36; or Breton unnek ha triugent eleven and three twenties,' i.e., 71. Now French, being a Romance language, has a regular system of Latin tens up to 100; cinquante, soixante, septante, huitante, nonante, which are to be found still in use in districts within the limits of the French language, as in Belgium. Nevertheless, the clumsy system of reckoning by twenties has broken out through the decimal system in France. The septante is to a great extent suppressed, soixantequatorze, for instance, standing for 74; quatre-vingts has fairly established itself for 80, and its use continues into the nineties, quatre-vingt-treize for 93; in numbers above 100 we find six-vingts, sept-vingts, huit-vingts, for 120, 140, 160, and a certain hospital has its name of Les Quinzevingts from its 300 inmates. It is, perhaps, the most reasonable explanation of this curious phenomenon, to suppose the earlier Keltic system of France to have held its ground, modelling the later French into its own ruder shape. In England, the Anglo-Saxon numeration is decimal, hund-seofontig, 70; hund-eahtatig, 80; hund-nigontig, 90; hund-teontig, 100; hund-enlufontig, 110; hundtwelftig, 120. It may be here also by Keltic survival that the vigesimal reckoning by the score,' threescore and ten, fourscore and thirteen, &c., gained a position in English which it has not yet totally lost.1 1 Compare the Rajmahali tribes adopting Hindi numerals, yet reckoning From some minor details in numeration, ethnological hints may be gained. Among rude tribes with scanty series of numerals, combination to make out new numbers is very soon resorted to. Among Australian tribes addition makes two-one,' 'two-two,' express 3 and 4; in Guachi 'two-two' is 4; in San Antonio four and two-one' is 7. The plan of making numerals by subtraction is known in North America, and is well shown in the Aino language of Yesso, where the words for 8 and 9 obviously mean 'two from ten,' 'one from ten.' Multiplication appears, as in San Antonio, 'two-and-one-two,' and in a Tupi dialect 'two-three,' to express 6. Division seems not known for such purposes among the lower races, and quite exceptional among the higher. Facts of this class show variety in the inventive devices of mankind, and independence in their formation of language. They are consistent at the same time with the general principles of hand-counting. The traces of what might be called binary, ternary, quaternary, senary reckoning, which turn on 2, 3, 4, 6, are mere varieties, leading up to, or lapsing into, quinary and decimal methods. The contrast is a striking one between the educated European, with his easy use of his boundless numeral series, and the Tasmanian, who reckons 3, or anything beyond 2, as 'many,' and makes shift by his whole hand to reach the limit of man,' that is to say, 5. This contrast is due to arrest of development in the savage, whose mind remains in the childish state which the beginning of one of our nursery number-rhymes illustrates curiously. It runs by twenties. Shaw, 1.c. 'One's none, Two's some, Four's a penny, Five's a little hundred.' The use of a 'score' as an indefinite number in England, and similarly of 20 in France, of 40 in the Hebrew of the Old Testament and the Arabic of the Thousand and One Nights, may be among other traces of vigesimal reckoning. |