If we allow ourselves to mix for a moment what is with what might be, we can see how unlimited is the field of possible growth of numerals by mere adoption of the names of familiar things. Following the example of the Sleswigers we might make shilling a numeral for 12, and go on to express 4 by groat; week would provide us with a name for 7, and clover for 3. But this simple method of description is not the only available one for the purpose of making numerals. The moment any series of names is arranged in regular order in our minds, it becomes a counting-machine. I have read of a little girl who was set to count cards, and she counted them accordingly, January, February, March, April. She might, of course, have reckoned them as Monday, Tuesday, Wednesday. It is interesting to find a case coming under the same class in the language of grown people. We know that the numerical value of the Hebrew letters is given with reference to their place in the alphabet, which was arranged for reasons that can hardly have had anything to do with arithmetic. The Greek alphabet is modified from a Semitic one, but instead of letting the numeral value of their letters follow throughout their newly-arranged alphabet, they reckon α, β, γ, δ, ε, properly, as 1, 2, 3, 4, 5, then put in σ for 6, and so manage to let ι stand for 10, as י does in Hebrew, where it is really the 10th letter. Now, having this conventional arrangement of letters made, it is evident that a Greek who had to give up the regular 1, 2, 3,—εἷς, δύο, τρεῖς, could supply their places at once by adopting the names of the letters which had been settled to stand for them, thus calling 1 alpha, 2 bēta, 3 gamma, and so onward. The thing has actually happened; a remarkable slang dialect of Albania, which is Greek in structure, though full of borrowed and mystified words and metaphors and epithets understood only by the initiated, has, as its equivalent for ‘four’ and ‘ten,’ the words δέλτα and ἰῶτα.[[327]]
While insisting on the value of such evidence as this in making out the general principles of the formation of numerals, I have not found it profitable to undertake the task of etymologizing the actual numerals of the languages of the world, outside the safe limits of the systems of digit-numerals among the lower races, already discussed. There may be in the languages of the lower races other relics of the etymology of numerals, giving the clue to the ideas according to which they were selected for an arithmetical purpose, but such relics seem scanty and indistinct.[[328]] There may even exist vestiges of a growth of numerals from descriptive words in our Indo-European languages, in Hebrew and Arabic, in Chinese. Such etymologies have been brought forward,[[329]] and they are consistent with what is known of the principles on which numerals or quasi-numerals 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 digit-numerals, 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 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 tune in honore fuit.
Seu quia tot digiti, per quos numerare solemus:
Seu quia bis quino femina mense parit:
Seu quod adusque decem numero crescente venitur,
Principium spatiis sumitur inde novis.’[[330]]
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. For 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, 5·1 ... 10, 10·1 ...10·5, 10·5·1 ... 20, ... 30, ... 40, &c. Pure decimal may be instanced from Hebrew 1, 2 ... 10, 10·1 ... 20, 20·1 ... &c. Pure vigesimal is not usual, for the obvious 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, 10·1 ... 10·5, 10·5·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40, &c.; or from decimal, as in Basque, 1 ... 10, 10·1 ... 20, 20·1 ... 20·10, 20·10·1 ... 40 &c.[[331]] 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,[[332]] 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.