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SECT. III.
Homer’s perceptions and use of Number

While the faculties of Homer were in many respects both intense and refined in their action, beyond all ordinary, perhaps we might say beyond all modern, examples, there were other points in which they bear the marks of having been less developed than is now common even among the mass of many civilized nations. In the power of abstraction and distinct introspective contemplation, it is not improbable that he was inferior to the generality of educated men in the present day. In some other lower faculties, he is probably excelled by the majority of the population of this country, nay even by many of the children in its schools. I venture to specify, as examples of the last-named proposition, the faculties of number, and of colour. It may be true of one or both of these, that a certain indistinctness in the perception of them is incidental everywhere to the early stages of society. But yet it is surprising to find it where, as with Homer, it accompanies a remarkable quickness and maturity not only of great mental powers, but of certain other perceptions more akin to number and colour, such as those of motion, of sound, and of form. But let us proceed to examine, in the first place, the former of these two subjects.

It may be observed at the outset, that probably none of us are aware to how great an extent our aptitudes with respect to these matters are traditionary, and dependent therefore not upon ourselves, but upon the acquisitions made by the human race before our birth, and upon the degree in which those acquisitions have circulated, and have been as it were filtered through and through the community, so as to take their place among the elementary ideas, impressions, and habits of the population. For such parts of human knowledge, as have attained to this position, are usually gained by each successive generation through the medium of that insensible training, which begins from the very earliest infancy, and which precedes by a great interval all the systematic, and even all the conscious, processes of education. Nor am I for one prepared by any means to deny that there may be an actual ‘traducianism’ in the case: on the contrary, in full consistency with the teaching of experience, we may believe that the acquired aptitudes of one generation may become, in a greater or a less degree, the inherited and inborn aptitudes of another.

We must, therefore, reckon upon finding a set of marked differences in the relative degrees of advancement among different human faculties in different stages of society, which shall be simply referable to the source now pointed out, and distinct altogether from such variations as are referable to other causes. It is not difficult to admit this to be true in general: but the question, whether in the case before us it applies to number and colour, can of course only be decided by an examination of the Homeric text.

Yet, before we enter upon this examination, let us endeavour to throw some further light upon the general aspect of the proposition, which has just been laid down.

Of all visible things, colour is to our English eye the most striking. Of all ideas, as conceived by the English mind, number appears to be the most rigidly definite, so that we adopt it as a standard for reducing all other things to definiteness; as when we say that this field or this house is five, ten, or twenty times as large as that. Our merchants, and even our schoolchildren, are good calculators. So that there is a sense of something strikingly paradoxical, to us in particular, when we speak of Homer as having had only indeterminate ideas of these subjects.

Conceptions of Number not always definite.

There are however two practical instances, which may be cited to illustrate the position, that number is not a thing to be as matter of course definitely conceived in the mind. One of these is the case of very young children. To them the very lowest numbers are soon intelligible, but all beyond the lowest are not so, and only present a vague sense of multitude, that cannot be severed into its component parts. The distinctive mark of a clear arithmetical conception is, that the mind at one and the same time embraces the two ideas, first of the aggregate, secondly of each one of the units which make it up. This double operation of the brain becomes more arduous, as we ascend higher in the scale. I have heard a child, put to count beads or something of the sort, reckon them thus: ‘One, two, three, four, a hundred.’ The first words express his ideas, the last one his despair. Up to four, his mind could contain the joint ideas of unity and of severalty, but not beyond; so he then passed to an expression wholly general, and meant to express a sense like that of the word multitude.

But though the transition from number definitely conceived to number without bounds is like launching into a sea, yet the conception of multitude itself is in one sense susceptible of degree. We may have the idea of a limited, or of an unbounded, multitude. The essential distinction of the first is, that it might possibly be counted; the notion of the second is, that it is wholly beyond the power of numeration to overtake. Probably even the child, to whom the word ‘hundred’ expressed an indefinite idea, would have been faintly sensible of a difference in degree between ‘hundred’ and ‘million,’ and would have known that the latter expressed something larger than the former. The circumscribing outline of the idea apprehended is loose, but still there is such an outline. The clearness of the double conception is indeed effaced; the whole only, and not the whole together with each part, is contemplated by the mind; but still there is a certain clouded sense of a real difference in magnitude, as between one such whole and another.

And this leads me to the second of the two illustrations, to which reference has been made. That loss of definiteness in the conception of number, which the child in our day suffers before he has counted over his fingers, the grown man suffers also, though at a point commonly much higher in the scale. What point that may be, depends very much upon the particular habits and aptitudes of the individual. A student in a library of a thousand volumes, an officer before his regiment of a thousand men upon parade, may have a pretty clear idea of the units as well as of the totals; but when we come to a thousand times a thousand, or a thousand times a million, all view of the units, for most men, probably for every man, is lost: the million for the grown man is in a great degree like the hundred for the child. The numerical term has now become essentially a symbol; not only as every word is by its essence a symbol in reference to the idea it immediately denotes; but, in a further sense, it is a symbol of a 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 thousands⁷⁹⁰.’ 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 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
No holiday have seen.
 

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 τρὶς 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. Τετράκις is found only once, and in a sense wholly indeterminate: the passage is⁷⁹¹ 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 in the most elementary forms; such forms, as are without composition, and refuse all further analysis.

Greek estimate of the discovery of Number.

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 Æschylus, as among the very greatest of the discoveries of Prometheus⁷⁹²:

 
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 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 seals⁷⁹³. 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 ἀριθμὸς 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 πεμπάσσεται,) but not by the aid of any scale or combination of units, either decimal or quinal. In the word δεκὰς 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 πεντὰς 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.

Highest numerals of the poems.

Homer’s highest numeral is μύριοι. He describes the Myrmidons as being μύριοι⁷⁹⁴, 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 μύρια ᾔδη, to denote a person of instructed and accomplished mind⁷⁹⁵.

Next to the μύρια, 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 men⁷⁹⁶:

 
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 mares⁷⁹⁷.

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 herds⁷⁹⁸. 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 oxen, immediately awakens the recollection that even the Homeric hecatomb, though meaning etymologically an hundred oxen, practically meant nothing of the kind, but only what we should call a lot or batch of oxen. Again, it is so obviously improbable that the Trojans should in an hurried bivouac have lighted just a thousand fires, and placed just fifty men by each, that we may take this passage as plainly figurative, and as conveying no more than a very rude approximation, of such a kind as would be inadmissible where the practice of calculation is familiar. It is then most likely, that in the remaining one of the three passages, the Poet means only to convey that a large and liberal present of wine was sent by Euneus, as the consideration for his being allowed to trade with the army. There is certainly more of approximation to a definite use of the single thousand, than of the three, the nine, or the ten: but this difference in definiteness is in reality a main point in the evidence. Most of all does this become palpable, when we consider how strange is in itself the omission to state the numbers of the combatants on either side of this great struggle: an omission so strange, of what would be to ourselves a fact of such elementary and primary interest, that we can hardly account for it otherwise than by the admission, that to the Greeks of the Homeric age the totals of the armies, even if the Poet himself could have reckoned them, would have been unintelligible.

Among all the numbers found in Homer, the highest which he appears to use with a clearly determinate meaning, is that of the three hundred and sixty fat hogs under the care of Eumæus in Ithaca⁷⁹⁹;

οἱ δὲ τριηκόσιοί τε καὶ ἑξήκοντα πέλοντο.

The reason for considering this number as having a pretty definite sense in the Poet’s mind (quite a different matter, let it be borne in mind, from the question whether the circumstance is meant to be taken as historical) is, that it stands in evident association with the number of days, as it was probably then reckoned, in the year. It seems plain that he meant to describe the whole circle of the year, where he says, that for each of the days and nights which Jupiter has given, or, in his own words⁸⁰⁰,

ὅσσαι γὰρ νύκτες τε καὶ ἡμέραι ἐκ Διός εἰσιν,

the greedy Suitors are not contented with the slaughter of one animal, or even of two. Eumæus then gives an account of the wealth of Ulysses in live stock, both within the isle and on the mainland, from whence the animals were supplied: and adds, that from the Ithacan store a goatherd took down daily a fat goat, while he himself as often sent down a fat hog. I have dwelt thus particularly on the detail of this case, because it may fairly be inferred from the correspondence between the number of the hogs and the days of the year, that for once, at all events, the Poet intended to speak, though somewhat at random, yet in a degree arithmetically, and that of so high a number as 360.

There are other cases of lower numbers in different parts of the poems, where it may be argued, with varying measures of probability, that Homer had a similar intention.

The ἑκατομβὴ and numerals of value.

The word ἑκατομβὴ, without doubt, affords a striking proof of vagueness in the ideas of the heroic age with respect to number: and this vagueness extends, yet apparently in varying degrees, to the adjective ἑκατομβοῖος. I have elsewhere⁸⁰¹ referred to adjectives of this formation as indicative of the fact, that for those generations of mankind oxen may be said to have constituted a measure of value; and this fact certainly involves an aim at numerical exactitude. It seems, indeed, on general grounds far from improbable, that the business of exchange may have been the original guide of our race into the art, and thus into the science, of arithmetic.

In the description of the Shield of Minerva, which had an hundred golden drops or tassels, we are told that each of them was ἑκατομβοῖος, or worth an hundred oxen. This use of the word must be regarded as strongly charged with figure. Minerva was arming to mingle among men upon the plain of Troy⁸⁰², and it is not likely, therefore, that the Poet would represent her in dimensions utterly inordinate. He judiciously reserves this license of exaggeration without bounds for scenes where he is beyond the sphere of relations properly human, as for example, the Theomachy and the Under-world. Now we may venture to take the Homeric value of an ox before Troy at half an ounce of gold. In the prizes of the wrestling match, where a tripod was worth twelve oxen, a highly skilled woman (πολλὰ δ’ ἐπίστατο ἔργα) was worth four⁸⁰³. Two ounces of gold would be a low price for such a person in almost any age. According to this computation, each drop on the Ægis of Minerva would weigh fifty ounces: the whole would weigh above 300 lbs. avoirdupois, and if we were to assume the purely ornamental fringe in a work of this kind to weigh one tenth part of the whole, the Ægis itself would weigh nearly a ton and a half. Primâ facie, this is susceptible of explanation in either of two ways: the one, that the numbers are used poetically and not arithmetically; the other, that of sheer intentional exaggeration in bulk. The rules of the Poet, as they are elsewhere applied, oblige us to reject the latter solution, and consequently throw us back upon the former.

The numerals of value.

Again, we are told that, when Diomed obtained the exchange of arms from Glaucus, he gave a suit of copper, and obtained in return a suit of gilt⁸⁰⁴;

χρύσεα χαλκείων, ἑκατόμβοι’ ἐννεαβοίων.

Here there seems to be a mixture of the metaphorical and the arithmetical use. For, on the one hand, it is singular that he should have chosen numbers which require the aid of a fraction to express their relation to one another. He could certainly not have meant to say that the values of the two suits were precisely as 100:9, or as 11 :1. And yet, on the one hand, he could scarcely use the term ἐννεαβοῖα, except with reference to the known and usual value of a suit of armour, while the ἑκατομβοῖα, from its use in other places, must be suspected of having no more than a merely indeterminate force.

With this fractional relation of 100:9, may be compared the arrangement at the feast in Pylos, where each division of five hundred persons was supplied with nine oxen. These numbers, however, are probably less vague than in some other cases: for the provision stated, though large, is not beyond what a rude plenty might suggest on a great public occasion.

Again, Lycaon, when captured for the second time by Achilles, reminds that hero of what he had fetched or been worth to him on the former occasion⁸⁰⁵: ἑκατόμβοιον δέ τοι ἦλφον. Here we have a decisive proof of the figurative use of number. Had the young prince been ransomed by Priam, a great price, no doubt, would have been given. But Achilles sold him into Lemnos, ἄνευθεν ἄγων πατρός τε φίλων τε: and to the Lemnians he could hardly have value but as a labourer, although indeed it chanced that he was afterwards redeemed, by a ξεῖνος of Priam⁸⁰⁶, at a high price. We cannot, then, suppose that he had brought any such return as would be represented by a full hundred of oxen.

The evidence thus far, I think, tends powerfully to support the hypothesis, that there is an amount of vagueness in Homer’s general use of numbers, unless indeed as to very low ones, which cannot be explained otherwise than as metaphorical or purely poetical: and that his mind never had before it any of those processes, simple as they are to all who are familiar with them, of multiplication, subtraction, or division.

I admit it to be possible, that his manner of treating number may have been owing to his determination to be intelligible, and to the state of the faculties of his hearers, as much as, or even more than, his own. But to me the supposition of the infant condition even of his faculties with respect to number, though at first sight startling, approves itself on reflection as one thoroughly in conformity with analogy and nature. Indeed the experience of life may convince us that to this hour we should be mistaken, if we supposed arithmetical conceptions to be uniform in different minds; that the relations of number are faintly and imperfectly apprehended, except by either practised or else peculiarly gifted persons; and that, in short, there is nothing more mysterious than arithmetic to those who do not understand it. As one illustration of this opinion, I will cite the difficulty which most educated persons, when studying history, certainly feel in mastering its chronology; while to those who are apt at figures it is not only acquired with ease, but it even serves as the nexus and support of the whole chain of events.

There were several occasions, upon which it would have been most natural and appropriate for Homer to use the faculty of multiplication; yet on no one of these has he used it. He constantly supplies us with the materials of a sum, but never once performs the process.

Silence as to the numbers of the armies.

The first example in the Iliad is supplied by that passage of the unhappy speech of Agamemnon to the Assembly in the Second Book, which causes the fever-fit of home-sickness. He compares the strength of the Greek army with that of the Trojans; and he only effects the purpose by this feeble but elaborate contrivance. ‘Should the Greeks and Trojans agree to be numbered respectively, and should the Trojans properly so called be placed one by one, but the Greeks in tens, and every Trojan made cupbearer to a Greek ten, many of our tens would be without a cupbearer⁸⁰⁷.’ In the first place, the fact that he calls this ascertaining of comparative force numbering ἀριθμηθημέναι is remarkable; for it would not have shown the numbers of either army; nor even the difference, by which the Greeks exceeded a tenfold ratio to the Trojans; but simply, by leaving an unexhausted residue, the fact that they were more, whether by much or by little, than ten times as many as the besieged. Secondly, it seems plain that, if Homer had known what was meant by multiplication, he would have used the process in this instance, in lieu of the elaborate (yet poetical) circumlocution which he has adopted; and would have said the Greeks were ten times, or fifteen times, or twenty times, as many as the inhabitants of Troy.

After this, Ulysses reminds the Assembly of the apparition of the dragon they had seen at Aulis. The phrase χθιζά τε καὶ πρώιζα, which he employs, may grammatically either belong to the epoch of the gathering at Aulis, or to the time of the plague, which had carried off a part of the force a fortnight or three weeks before. In whichever connection of the two we place it, it affords an instance of extreme indefiniteness in the use of two adverbs which are at once expressive of time and of number; for on one supposition he must use them to express whole years, and on the other they must mean near a fortnight, and therefore a certain number of days.

The next case is remarkable. It is that of the Catalogue.

The resolution, which introduces it, was not a resolution to number the host; but simply to make a careful division and distribution of the men under their leaders, with a view to a more effective responsibility, both of officers and men⁸⁰⁸. But when the Poet comes to enumerate the divisions, it is evidently a great object with him to make known the relative forces, and thus the relative prominence and power, of the different States of Greece. Yet nothing can be more imperfect than the manner in which the enumerating portion of his task is executed. In the first place, we trace again the old habit of the loose and figurative use of numbers. For Homer could hardly mean us to take literally all the numbers of ships, which he has stated in the Catalogue: since, in every case where they come up to or exceed twenty, they run in complete decades without odd numbers; subject to the single exception of the twenty-two ships of Gouneus. Podalirius and Machaon have thirty, the Phocians forty, Achilles fifty, Menelaus sixty, Diomed eighty, Nestor ninety, Agamemnon an hundred: the only full multiple of ten omitted being the utterly intractable ἑβδομήκοντα. But again, he gives us no effectual clue to the numbers of the crews. Each of the fifty ships of the Bœotians had one hundred and twenty men, and each of the seven ships of Philoctetes had fifty⁸⁰⁹. Thus he supplies us with the two factors of the sum, which would find the number of men, in each of these two cases; but in neither case does he perform the sum; and such is the uniform practice throughout the poems. For the Greek force generally, he has not even given us the factors. It has indeed been conjectured, that fifty may have been the smallest ship’s company, and one hundred and twenty the largest: but this is mere conjecture; and even if it be well founded, still we do not know whether the generality of the ships were about the mean, or nearer one or the other of the extremes. Again, it would appear probable from the Odyssey, that these numbers, of fifty and one hundred and twenty, are exclusive at least of pilots and commanders, if not also of the stewards⁸¹⁰ and the minor officers⁸¹¹; for the number mentioned by Alcinous⁸¹² is fifty-two; and although he says that all were to sit down to row, the texts when compared cannot but suggest, that the number fifty was an usual complement of oars, and that the two were the captain and pilot respectively⁸¹³.

Plainly, there must have been very great inequalities in the crews of the Greek armament; or Homer could not have said, after giving Agamemnon an hundred ships, that he had by far the largest force of all the chiefs⁸¹⁴;

 
ἅμα τῷγε πολὺ πλεῖστοι καὶ ἄριστοι
λαοὶ ἕποντ’.
 

For Diomed and Idomeneus have each eighty ships, and Nestor has ninety, so that their numbers would come very near Agamemnon’s, unless their ships were smaller. But to sum up this discussion. It is evident that, if only we suppose the Greeks of Homer’s time to have had a definite and well developed sense of number, the mention by Homer of the amount of force in the Trojan expedition would have been a fact of the highest national interest and importance. Yet he has left us nothing, which can be said even definitely to approximate to a record of it, though the enumeration of the Catalogue appears almost to force the subject upon him. The fair inferences seem to be, that he did not understand the calculative use of numbers at all, or beyond some very limited range; and that, even within that range, he for the most part employed them poetically and ornamentally; they were decorative and effective, like epithets to his song, but they were not statistical; as expressions of force they were no more than (as it were) tentative, and that but very rudely.