fingers were erect, the palms facing outwards. The 3rd, 4th and 5th fingers (to use the German description) might be exтELνόμενοι or straight, συστελλόμενοι bent or half-closed, κλινόμevol or 'closed.' The subsequent gestures may be thus described: μενοι (a) On the left hand: for 1, half-close the 5th finger only: 2, 3, 4, 5, 6, the 4th and 5th fingers only: the 4th finger only: 7, close the 5th finger only: 8, the 4th and 5th fingers only: (b) The same operations on the right hand gave the thousands, from 1000 to 9000. (c) On the left hand: for 10, apply the tip of the forefinger to the bottom of the thumb, so that the resulting figure resembles 8: 20, the forefinger is straight and is separated by the thumb from the remaining fingers, which are slightly bent: 30, join the tips of the forefinger and thumb: 40, place the thumb behind (on the knuckle of) the forefinger: 50, place the thumb in front (on the ball) of the fore finger: 60, place the thumb as for 50 and bend the forefinger over it, so as to touch the ball of the thumb: 70, rest the forefinger on the tip of the thumb: ,, 80, lay the thumb on the palm, bend the forefinger close over the first joint of the thumb and slightly bend the remaining fingers: 90, close the forefinger only as completely as possible. (d) The same operations on the right hand gave the hundreds, from 100 to 900. Nicolaus himself does not give signs for numbers above 9000, but Martianus Capella, a writer of the 5th century, says (De Nuptiis, Lib. VII. p. 244 of Grotius edn. 1599) nonnulli Graeci etiam pupía adjecisse videntur' by means, apparently, of 'quaedam brachiorum contorta saltatio' of which he does not approve. The motions were probably the same as those described by Bede in his tract 'De loquela per gestum digitorum'.' Different positions of the left hand on the left breast and hip gave the numbers from 10,000 to 90,000: the same motions with the right hand gave the hundred thousands and the hands folded together represented a million. 20. The finger-symbolism here described was in use, in practically the same form, in Greece and Italy and throughout the East certainly from the beginning of our era2, but there is unfortunately no evidence as to where or when it was invented. By far the oldest passage in which any reference to it may be supposed to occur is Aristophanes, Vespae, 11. 656—664, where Bdelycleon tells his father to do an easy sum, où pois årλ' aπò xeɩρós. "The income of the state," says he, "is nearly 2000 talents: the yearly payment to the 6000 dicasts is only 150 talents." "Why," answers the old man, "we don't get a tenth of the revenue." It is clear, from this reply, that the 'easy sum' in question amounted only to dividing 2000 by 10 or multiplying 150 by 10, an operation which does not require the more elaborate finger-signs. Failing this passage, there is 1 Opera, Basileae, col. 171-173. The material part is given by Roediger. The finger positions described by Bede differ slightly, in one or two cases, from those of Nicolaus Smyrnaeus, and both again vary slightly from those used in the East, where the units and tens were represented (not always, v. the Arabic poem in Bulletino Boncompagni, 1863, 1. pp. 236, 237) on the right hand and not on the left. The reader is referred to Roediger's article above mentioned. Plates will be found in Journal of Philology, Vol. 11. p. 247, in Stoy's pamphlet Zur Gesch. des Rechenunterrichts (Jena, 1876), in Neue Jahrb. für Phil. u. Päd. 15th supplbd. p. 511, and in many other places. A large collection of references is given by Prof. Mayor in his note to Juvenal, Sat. x. 248. More, esp. to late Jewish and Arabic writers, in Steinschneider's Bibliogr. Hebr. Vol. XXI. pp. 39, 40. 2 The same or something like it is still used by Persian merchants. See De Sacy in Journ. Asiatique, Vol. 2, and Tylor, Primitive Culture, I. p. 246, n. another possible reference, equally doubtful, to this system of finger-symbolism in Plautus, Miles Gloriosus, II. 3, but the first clear references to it occur in Plutarch and authors of his time'. Pliny, indeed, says that there was, in his time, a statue of Janus, erected by Numa, of which the fingers indicated 365 or 355 (the reading is doubtful, cf. also Macrobius, Conv. Sat. I. 9), the number of days in the year, but no importance can be attached to such a statement. All that we can allege of the system is that it is mentioned only in later classical literature, that it then appears to be of universal diffusion and that it was far more persistent in the East than in the West. If we consider that such a system can have been of no use in calculation, save as a memoria technica for some number with which the mind of the reckoner was not immediately engaged-if, in other words, we consider that such a system was useful to represent numbers but not to calculate with them, then it becomes probable that it was invented in the first instance as a secret means of communication between merchants or as a numerical gesture-language between persons who were ignorant of one another's tongues. Phoenician and Greek commerce would make it widely known: the later diffusion of Latin and Greek and the larger use of writing would ensure its gradual extinction in the West, but it would still preserve its original utility in the motley and ignorant crowds of the Eastern bazaars. 21. In reckoning with pebbles, no doubt at first each pebble represented one of the objects to be counted, the advantage of course being that space was saved and the memory relieved by a good coup d'oeil, for it will be conceded that it is easier to count 100 pebbles than 100 cows or to find 10 times 1 Plut. Apophth. 174 b. Pliny, Hist. Nat. XXXIV. s. 33. For other reff. see Prof. Mayor's note on Juv. x. 249, above referred to, or Dean Peacock's article Arithmetic in Encycl. Metropolitana. 2 Erasmus, in his ed. of Jerome (III. 25 B c) published in 1516, confessed his ignorance of the finger-symbolism referred to by the saint. He under stood it afterwards (see ibid. p. 313). 3 The Persian system mentioned by De Sacy and Tylor (see note above) is used only in secret, when for instance a dragoman wishes to have one price with the seller and the other with his master. See also the opening words of Roediger's article. Another suggestion as to the origin of this symbolism will be made below, § 25. 10 in pebbles than in sacks or such other articles of commerce. So soon as the heap contained one pebble for each object, the calculator would begin afresh and by arranging the pebbles in groups of 10, arrive at the total and the name of the total, without having his attention embarrassed by petty circumstances'. This use of pebbles in mere counting, where each represents a real object, would naturally precede their use in calculations where some pebbles would represent imaginary objects. A great number of pebbles could be dispensed with if the operator, on completing a group of 10, laid aside a large pebble or a white one and then began again with the pebbles of the original group. He would soon find that there would be no need for a variety of pebbles, if he always laid pebbles representing 10 in a separate place from those representing units. In this way, he would arrive at a neat visible symbolism for a high number, which would greatly facilitate operations in the four rules of arithmetic. Such an advanced pebble-symbolism the Egyptians and the Chinese had from a time 'whereof the memory of man runneth not to the contrary.' It can hardly be doubted that they invented it independently and imparted it to the nations around. Wherever and whenever invented or borrowed, the Greeks and Italians had it also and used it by preference for all ordinary calculations down to the 15th century of our era. The evidence for its use, however, is singularly late. Homer and Pindar do not allude to it, but it is plain that it was in regular use by the 5th century B. C., though the authorities even of that time do not state explicitly how the calculation with pebbles was conducted2. It cannot be doubted, 1 In a London night-school I have often seen a boy, in order to multiply say 12 by 10, make 120 dots on his slate and then count these. What he wanted was the name of the total and he did not always get this right. With primitive man, I imagine, the use of pebbles would not arise till numeral names had partly superseded fingercounting. If, for instance, a savage sold something for 50 cows, he would indicate his price by naming it, and would then, with the aid of pebbles, ascertain whether he had got the price he bargained for. Thus the Mexicans acquired a set of numerals, used in counting animals and things, which runs centetl, ontetl, etc. or 'one-stone,' 'two-stone,' etc. Other similar examples are cited in Tylor, Early Hist. p. 163. 2 Diogenes Laertius (1. 59) ascribes to Solon a saying that courtiers were like the pebbles on a reckoning-board, for they sometimes stood for more, however, that the pebbles were arranged in lines, either horizontal or perpendicular, and that the pebbles on the first line represented units, those on the second tens, those on the third. hundreds and so on. How many lines there were and how many pebbles might be placed on each there is no evidence to show. It may be added that fractions in the form of 'submultiples' would not present any difficulty when the system of local values for the pebbles was once introduced. If for instance a line were appropriated to pebbles of the value of 1, it would be as easy to discern that 12 pebbles on that line are equal in value to 1 on the units line, as to perceive that 10 pebbles on the unit line may be replaced by 1 on the tens line. But since a great many lines devoted to fractions would have been inconvenient, probably a few lines only were devoted to certain selected fractions, and all other fractions were reduced as nearly as possible to terms of these. 22. The surface on which such lines were drawn, or the frame on which strings or wires were stretched, for the purpose of pebble-reckoning, was called by the Greeks ἄβαξ or ἀβάκιον. This name seems to point to the common Semitic word abaq meaning 'sand,' and it is said that a board strewn with sand, on which lines might be drawn with a stick, was and still is a common instrument for calculation in the East. It is the more desirable also that some Oriental origin for the aßag should be found because, in late Greek writers, we find a general tradition that Pythagoras, who certainly studied out of Greece, was the inventor or introducer of the instrument. It cannot, however, be considered that the Semitic origin of ǎßağ is rendered at all probable by such considerations. The sometimes for less. This, if genuine (but cf. Polyb. v. 26, 13), is the first and also one of the most explicit references to the pebble-symbolism. If this be doubted, then the earliest authentic reference is probably a fragment of Epicharmus (ed. Ahrens, 94, 8): then Aeschylus (Agam. 570), then perhaps Herodotus (11. 36), who says that, in pebble-reckoning, the Egyptians count ed as they wrote from right to left, the Greeks from left to right. It may be that the abacus with the Greeks was not so old as writing, for the Greeks did not originally write from left to right, but either from right to left or βουστροφηδόν. They may have counted from right to left, but can hardly have counted βουστροφηδόν. |