Conceptions of Number not always definite. 429 symbol, for that idea which it denotes, is itself symbolical it is a conventional representation of a certain vast number of units, far too great to be individually contemplated and apprehended. As we rise higher still from millions, say for example, into the class of billions, the vagueness increases. The million is now become a sort of new unit, and the relation of two millions to one million, is thus pretty clearly apprehended as being double; but this too becomes obscured as we mount, and even (for example) the relation of quantity between ten billions of wheat-corns, and an hundred billions of the same, is far less determinately conveyed to the mind, than the relation between ten wheat-corns and one. At this high level, the nouns of number approximate to the indefinite character of the class of algebraic symbols called known quantities. In proportion as our conception of numbers is definite, the idea of them, instead of being suited for an address to the imagination, remains unsuited for poetic handling, and thrives within the sphere of the understanding only. But when we pass beyond the scale of determinate into that of practically indeterminate amounts, then the use of numbers becomes highly poetical. I would quote, as a very noble example of this use of number, a verse in the Revelations of St. John. And I beheld, and I heard the voice of many angels round about the throne, and the beasts and the elders: and the number of them was ten thousand times ten thousand, and thousands of thousands. As a proof of the power of this fine passage, I would observe, that the descent from ten thousand times ten thousand to thousands of thousands, though it is in fact numerically very great, has none of the chilling effect of anticlimax, because these numbers a Rev. v. II. are not arithmetically conceived, and the last member of the sentence is simply, so to speak, the trail of light which the former draws behind it. Now we must keep clearly before our minds the idea, that this poetical and figurative use of number among the Greeks at least preceded what I may call its calculative use. We shall find in Homer nothing that can strictly be called calculation. He repeatedly gives us what may be termed the factors of a sum in multiplication; but he never even partially combines them, even as they are combined for example in Cowper's ballad, John Gilpin's spouse said to her dear, Though wedded we have been These twice ten tedious years, yet we Reference has been made to the convenience which we find in using number as a measure of quantity, and as a means of comparing things of every species in their own kind. But we never meet with this use of it in Homer. He has not even the words necessary to enable him to say, 'This house is five times as large as that.' If he had the idea to express, he would say, Five houses, each as large as that, would hardly be equal to this. The word 7pis may be called an adverb of multiplication; but it is never used for these comparisons. Indeed, Damm observes, that in a large majority of instances it signifies an indefinite number, not a precise one. Terpakis is found only once, and in a sense wholly indeterminate : the passage isb τρισμάκαρες Δαναοὶ καὶ τετράκις. Πεντάκις does not even exist. Ajax lifts a stone, not twice as large as a mortal of to-day could raise', but so large that it would require two such mortals to raise it. All Homer's numerical expressions are Greek estimate of the discovery of Number. 431 in the most elementary forms; such forms, as are without composition, and refuse all further analysis. His use of number appears to have been confined to simple addition: and it is probable that all the higher numbers which we find in the poems, were figurative and most vaguely conceived. If we are able to make good the proof of these propositions from the Homeric text, we shall then be well able to understand the manner in which Numeration, or the science of number, is spoken of by the Greeks of the historic age as a marvellous invention. It appears in Eschylus, as among the very greatest of the discoveries of Prometheus": καὶ μὴν ἀριθμὸν, ἔξοχον σοφισμάτων, ἐξεῦρον αὐτοῖς· he goes on to add, γραμμάτων τε συνθέσεις. So that the use of numbers by rule was to the Greek mind as much a discovery as the letters of the alphabet, and is even described here as a greater one: much as in later times men have viewed the use of logarithms, or of the method of fluxions or the calculus. In full conformity with this are the superlative terms, in which Plato speaks of number. Number, in fact, seems to be exhibited in great part of the Greek philosophy, as if it had actually been the guide of the human mind in its progress towards realizing all the great and cardinal ideas of order, measure, proportion, and relation. Up to what point human intelligence, in the time of Homer, was able to push the process of simple addition, we do not precisely know. It is not, however, hastily to be assumed that, in any one of his faculties, Homer was behind his age; and it is safer to believe c Esch. Prom. V. 468. see also Soph. Naupl. Fragm. v. that the poems, even in these points, represent it advantageously. Now, in one place at least, we have a primitive account of a process of addition. The passage is in the Fourth Odyssey, where Menelaus relates, how Proteus counted upon his fingers the number of his seals". That it was a certain particular number is obvious, because when four of them had been killed by Eidothee, their skins were put upon Menelaus and his three comrades, and the four Greeks were then counted into the herd, so that the word apieμòs here evidently means a definite total. This addition by Proteus, however, was not addition in the proper arithmetical sense, and would be more properly called enumeration it was probably effected simply by adding each unit singly, in succession, to the others, with the aid of the fingers, (proved through the word Teμπάσσerai,) but not by the aid of any scale or combination of units, either decimal or quinal. In the word dexas we have, indeed, the first step towards a decimal scale; but we have not even that in the case of the number five, there being no Tevtàs or πεμπτάς. The meaning of πεμπάσσεται evidently is, not that he arranged the numeration in fives, but that, by means of the fingers of one hand, employed upon those of the other, he assisted the process of simple enumeration. Homer's highest numeral is uúpio. He describes the Myrmidons as being uúpio, though, if we assume a mean strength of about eighty-five for their crews, the force would but little have exceeded four thousand : and at the maximum of one hundred and twenty for each ship, it would only come to six thousand. Again, Homer uses the expression uúpia n, to denote a person of instructed and accomplished minds. e Od. iv. 412, 451. f Il. xxiii. 29. g Od. ii. 16. Highest numerals of the poems. 433 Next to the uúpia, the highest numerals employed in the poems are those contained in the passage where the Poet says that the howl of Mars, on being wounded by Diomed, was as loud as the shout of an army of nine thousand or ten thousand menh: ὅσσον τ' ἐννεάχιλοι ἐπίαχον ἢ δεκάχιλοι ἀνέρες ἐν πολέμῳ. But it is clear that the expressions are purely poetical and figurative. For he never comes near the use of such high numbers elsewhere; and yet it obviously lay in his path to use these, and higher numbers still, when he was describing the strength of the Greek and Trojan armies. The highest Homeric number, after those which have been named, is found in the three thousand horses of Erichthonius. This we must also consider poetical, because it is so far beyond the ordinary range of the poems, and in some degree likewise because of the obvious unlikelihood of his having possessed that particular number of maresi. Only thrice, besides the instances already quoted, does Homer use the fourth power of numbers; it is in the case of the single thousand. A thousand measures of wine were sent by Euneos as a present to Agamemnon and Menelaus. A thousand watch-fires were kindled by the Trojans on the plain. Iphidamas, having given an hundred oxen in order to obtain his wife, then promised a thousand goats and sheep out of his countless herds. In all these three cases, it is more than doubtful whether the word thousand is not roughly and loosely used as a round number. The combination of the thousand sheep and goats with the hundred i Il. xxi. 251. h Il. v. 860. k Il. vii. 571. viii. 562. xi. 244. F f |