Part 20

There must be a singular charm about insoluble problems, since there are never wanting persons who are willing to attack them. I doubt not that at this moment there are persons who are devoting their energies to Squaring the Circle, in the full belief that important advantages would accrue to science—and possibly a considerable pecuniary profit to themselves—if they could succeed in solving it. Quite recently, applications have been made to the Paris Academy of Sciences, to ascertain what was the amount which that body was authorised to pay over to anyone who should square the circle. So seriously, indeed, was the secretary annoyed by applications of this sort, that it was found necessary to announce in the daily journals that not only was the Academy not authorised to pay any sum at all, but that it had determined never to give the least attention to those who fancied they had mastered the famous problem.

It is a singular circumstance that people have even attacked the problem without knowing exactly what its nature is. One ingenious workman, to whom the difficulty had been propounded, actually set to work to invent an arrangement for measuring the circumference of the circle; and was perfectly satisfied that he had thus solved a problem which had mastered all the mathematicians of ancient and modern times. That we may not fall into a similar error, let us clearly understand what it is that is required for the solution of the problem of ‘squaring the circle.’

To begin with, we must note that the term ‘squaring the circle’ is rather a misnomer; because the true problem to be solved is the determination of the length of a circle’s circumference when the diameter is known. Of course, the solution of this problem, or, as it is termed, the _rectification_ of the circle, involves the solution of the other, or the _quadrature_, of the circle. But it is well to keep the simpler issue before us.

Many have supposed that there exists some exact relation between the circumference and the diameter of the circle, and that the problem to be solved is the determination of this relation. Suppose, for example, that the approximate relation discovered by Archimedes (who found, that if a circle’s diameter is represented by _seven_, the circumference may be almost exactly represented by _twenty-two_) were strictly correct, and that Archimedes had proved it to be so; then, according to this view, he would have solved the great problem; and it is to determine a relation of some such sort that many persons have set themselves. Now, undoubtedly, if any relation of this sort could be established, the problem would be solved; but as a matter of fact no such relation exists, and the solution of the problem does not require that there should be any relation of the sort. For example, we do not look on the determination of the diagonal of a square (whose side is known) as an insoluble, or as otherwise than a very simple problem. Yet in this case no exact relation exists. We cannot possibly express both the side and the diagonal of a square in whole numbers, no matter what unit of measurement we adopt: or, to put the matter in another way, we cannot possibly divide both the side and the diagonal into equal parts (which shall be the same along each), no matter how small we take the parts. If we divide the side into 1,000 parts, there will be 1,414 such parts, _and a piece over_ in the diagonal; if we divide the side into 10,000 parts, there will be 14,142, and still a little piece over, in the diagonal; and so on for ever. Similarly, the mere fact that no exact relation exists between the diameter and the circumference of a circle is no bar whatever to the solution of the great problem.

Before leaving this part of the subject, however, I may mention a relation which is very easily remembered, and is very nearly exact—much more so, at any rate, than that of Archimedes. Write down the numbers 113,355, that is, the first three odd numbers each repeated twice over. Then separate the six numbers into two sets of three, thus,—113) 355, and proceed with the division thus indicated. The result, 3·1415929‍‍..., expresses the circumference of a circle whose diameter is 1, correctly to the sixth decimal place, the true relation being 3·14159265.

Again, many people imagine that mathematicians are still in a state of uncertainty as to the relation which exists between the circumference and the diameter of the circle. If this were so, scientific societies might well hold out a reward to anyone who could enlighten them; for the determination of this relation (with satisfactory exactitude) may be held to lie at the foundation of the whole of our modern system of mathematics. I need hardly say that no doubt whatever rests on the matter. A hundred different methods are known to mathematicians by which the circumference may be calculated from the diameter with any required degree of exactness. Here is a simple one, for example:—Take any number of the fractions formed by putting _one_ as a numerator over the successive odd numbers. Add together the alternate ones beginning with the first, which, of course, is unity. Add together the remainder. Subtract the second sum from the first. The remainder will express the circumference (the diameter being taken as unity) to any required degree of exactness. We have merely to take enough fractions. The process would, of course, be a very laborious one, if great exactness were required, and as a matter of fact mathematicians have made use of much more convenient methods for determining the required relation: but the method is strictly exact.

The largest circle we have much to do with in scientific questions is the earth’s equator. As a matter of curiosity, we may inquire what the circumference of the earth’s orbit is; but as we are far from being sure of the exact length of the radius of that orbit (that is, of the earth’s distance from the sun), it is clear that we do not need a very exact relation between the circumference and the diameter in dealing with that enormous circle. Confining ourselves, therefore, to the circle of the earth’s equator, let us see what exactness we seem to require. We will suppose for a moment that it is possible to measure round the earth’s equator without losing count of a single yard, and that we want to gather from our estimate what the diameter of this great circle may be. This seems, indeed, the only use to which, in this case, we can put our knowledge of the relation we are dealing with. We have then a circle some twenty-five thousand miles round, and each mile contains one thousand seven hundred and sixty yards: or in all there are some forty-four million yards in the circumference, and therefore (roughly) some fourteen million yards in the diameter of this great circle. Hence, if our relation is correct within a fourteen-millionth part of the diameter, or a forty-four millionth part of the circumference, we are safe from any error exceeding a yard. All we want, then, is that the number expressing the circumference (the diameter being unity) should be true to the eighth decimal place, as quoted above (p. 291, l. 5).

But as I have said, mathematicians have not been content with a computation of this sort. They have calculated the number not to the _eighth_, but to the _six hundred and twentieth_ decimal place. Now, if we remember that each new decimal makes the result ten times more exact, we shall begin to see what a waste of time there has been in this tremendous calculation. We all remember the story of the horse which had twenty-four nails in its shoes, and was valued at the sum obtained by adding together a farthing for the first nail, a halfpenny for the next, a penny for the next, and so on, doubling twenty-four times. The result was counted by thousands of pounds. The old miser who paid at a similar rate for a grave eighteen feet deep (doubling for each foot), killed himself when he heard the total. But now consider the effect of multiplying by ten, six hundred and twenty times. A fraction, with that enormous number for denominator, and unity for numerator, expresses the minuteness of the error which would result if the ‘long value’ of the circumference were made use of. Let an illustration show the force of this:—

It has been estimated that light, which could eight times circle the earth in a second, takes 50,000 years in reaching us from the faintest stars seen in Lord Rosse’s giant reflector. Suppose we knew the exact length of the tremendous line which extends from the earth to such a star, and wanted, for some inconceivable purpose, to know the length of the circumference of a circle, of which that line was the radius. The value deduced from the above-mentioned calculation of the relation between the circumference and the diameter would differ from the truth by a length which would be imperceptible under the most powerful microscope ever yet constructed. Nay, the radius we have conceived, enormous as it is, might be increased a million-fold, or a million times a million-fold, with the same result. And the area of the circle formed with this increased radius would be determinable with so much accuracy, that the error, if presented in the form of a minute square, would be utterly imperceptible under a microscope a million times more powerful than the best ever yet constructed by man.

Not only has the length of the circumference been calculated once in this unnecessarily exact manner, but a second calculator has gone over the work independently. The two results are of course identical figure for figure.

It will be asked then, what _is_ the problem about which so great a work has been made? The problem is, in fact, utterly insignificant; its only interest lies in the fact that it is insoluble—a property which it shares along with many other problems, as the trisection of an angle, the duplication of a cube, and so on.

The problem is simply this: _Having given the diameter of a circle, to determine, by a geometrical construction, in which only straight lines and circles shall be made use of, the side of a square, equal in area to the circle_. As I have said, the problem is solved, if, by a construction of the kind described, we can determine the length of the circumference; because then the rectangle under half this length and the radius is equal in area to the circle, and it is a simple problem to describe a square equal to a given rectangle.

To illustrate the kind of construction required, I give an approximate solution which is remarkably simple, and, so far as I am aware, not generally known. Describe a square about the given circle, touching it at the ends of two diameters, AOB, COB, at right angles to each other, and join CA; let COAE be one of the quarters of the circumscribing square, and from E draw EG, cutting off from AO a fourth part AG of its length, and from AC the portion AH. Then three sides of the circumscribing square together with AH are very nearly equal to the circumference of the circle. The difference is so small, that in a circle two feet in diameter, it would be less than the two-hundredth part of an inch. If this construction were exact, the great problem would have been solved.

One point, however, must be noted; the circle is of all curved lines the easiest to draw by mechanical means. But there are others which can be so drawn. And if such curves as these be admitted as available, the problem of the quadrature of the circle can be readily solved. There is a curve, for instance, invented by Dinostratus, which can readily be described mechanically, and has been called the quadratrix of Dinostratus, because it has the property of thus solving the problem we are dealing with.

As such curves can be described with quite as much accuracy as the circle—for, be it remembered, an absolutely perfect circle has never yet been drawn—we see that it is only the limitations which geometers have themselves invented that give this problem its difficulty. Its solution has, as I have said, no value; and no mathematician would ever think of wasting a moment over the problem—for this reason, simply, that it has long since been demonstrated to be insoluble by simple geometrical methods. So that, when a man says he has squared the circle (and many will say so, if one will only give them a hearing), he shows that either he wholly misunderstands the nature of the problem, or that his ignorance of mathematics has led him to mistake a faulty for a true solution.

(From _Chambers’s Journal_, January 16, 1869.)

_A NEW THEORY OF ACHILLES’ SHIELD._

A distinguished classical authority has remarked that the description of Achilles’ shield occupies an anomalous position in Homer’s ‘Iliad.’ On the one hand, it is easy to show that the poem—for the description may be looked on as a complete poem—is out of place in the ‘Iliad;’ on the other, it is no less easy to show that Homer has carefully led up to the description of the shield by a series of introductory events.

I propose to examine, briefly, the evidence on each of these points, and then to exhibit a theory respecting the shield which may appear _bizarre_ enough on a first view, but which seems to me to be supported by satisfactory evidence.

An argument commonly urged against the genuineness of the ‘Shield of Achilles’ is founded on the length and laboured character of the description. Even Grote, whose theory is that Homer’s original poem was not an _Iliad_, but an _Achilleis_, has admitted the force of this argument. He finds clear evidence that from Book II. to Book XX. Homer has been husbanding his resources for the more effective description of the final conflict. He therefore concedes the possibility that the ‘Shield of Achilles’ may be an interpolation—perhaps the work of another hand.

It appears to me, however, that the mere length of the description is no argument against the genuineness of the passage. Events have, indeed, been hastening to a crisis up to the end of Book XVII., and the action is checked in a marked manner by the ‘Oplopœia’ in Book XVIII. Yet it is quite in Homer’s manner to introduce, between two series of important events, an interval of comparative inaction, or at least of events wholly different in character from those of either series. We have a marked instance of this in Books IX. and X. Here the appeal to Achilles and the night-adventure of Diomed and Ulysses are interposed between the first victory of the Trojans and the great struggle in which Patroclus is slain, and Agamemnon, Ulysses, Diomed, Machaon, and Eurypylus wounded.[19] In fact, one cannot doubt that in such an arrangement Homer exhibits admirable taste and judgment. The contrast between action and inaction, or between the confused tumult of a heady conflict and the subtle advance of the two Greek heroes, is conceived in the true poetic spirit. The dignity and importance of the action, and the interest of the interposed events, are alike enhanced. Indeed, there is scarcely a noted author whose works do not afford instances of corresponding contrasts. How skilfully, for example, has Shakespeare interposed the ‘bald, disjointed chat’ of the sleepy porter between the conscience-wrought horror of Duncan’s murderers and the ‘horror, horror, horror’ which ‘tongue nor heart could not conceive nor name’ of his faithful followers. Nor will the reader need to be reminded of the frequent and effective use of the contrast between the humorous and the pathetic by others.

The laboured character of the description of the shield is an argument—though not, perhaps, a very striking one—for the independent origin of the poem.

But the arguments on which I am disposed to lay most stress lie nearer the surface.

Scarcely anyone, I think, can have read the description of the shield without a feeling of wonder that Homer should describe the shield of a mortal hero as adorned with so many and such important objects. We find the sun and moon, the constellations, the waves of ocean, and a variety of other objects, better suited to adorn the temple of a great deity than the shield of a warrior, however noble and heroic. The objects depicted even on the Ægis of Zeus are much less important. There is certainly no trace in the ‘Iliad’ of a wish on Homer’s part to raise the dignity of mortal heroes at the expense of Zeus, yet the Ægis is thus succinctly described:—

Fring’d round with ever-fighting snakes, though it was drawn to life, The miseries and deaths of fight; in it frown’d bloody Strife, In it shone sacred Fortitude, in it fell Pursuit flew, In it the monster Gorgon’s head, in which held out to view Were all the dire ostents of Jove.—_Chapman’s_ Translation.

Five lines here, as in the original, suffice for the description of Jove’s Ægis, while one hundred and thirty lines are employed in the description of the celestial and terrestrial objects depicted on the shield of Achilles.

Another circumstance attracts notice in the description of Achilles’ armour—the disproportionate importance attached to the shield. Undoubtedly, the shield was that portion of a hero’s armour which admitted of the freest application of artistic skill. Yet this consideration is not sufficient to account for the fact, that while so many lines are given to the shield, the helmet, corselet, and greaves are disposed of in four.

But the argument on which I am inclined to lay most stress is the occurrence _elsewhere_ of a description which is undoubtedly only another version of the ‘Shield of Achilles.’ The ‘Shield of Hercules’ occurs in a poem ascribed to Hesiod. But whatever opinion may be formed respecting the authorship of the description, there can be no doubt that it is not Hesiod’s work. It exhibits no trace of his dry, didactic, somewhat heavy style. Elton ascribes the ‘Shield of Hercules’ to an imitator of Homer, and in support of this view points out those respects in which the poem resembles, and those in which it is inferior to, the ‘Shield of Achilles.’ The two descriptions are, however, absolutely identical in many places; and this would certainly not have happened if one had been an honest imitation of the other. And those parts of the ‘Shield of Hercules,’ which have no counterparts in the ‘Shield of Achilles,’ are too well conceived and expressed to be ascribed to a very inferior poet—a poet so inferior as to be reduced to the necessity of simply reproducing Homer’s words in other parts of the poem. Those parts which admit of comparison—where, for instance, the same objects are described, but in different terms—are certainly inferior in the ‘Shield of Hercules.’ The description is injured by the addition of unnecessary or inharmonious details. Elton speaks, accordingly, of these portions as if they were expansions of the corresponding parts of the ‘Shield of Achilles.’ This appears to me a mistake. It seems far more likely that both descriptions are by the same poet. It is not necessary for the support of my theory that this poet should be Homer, but I think both descriptions show undoubted traces of his handiwork. Indeed, all known imitations of Homer are so easily recognisable as the work of inferior poets, that I should have thought no doubt could exist on this point, but for the attention which the German theory respecting the ‘Iliad’ has received. Assigning both poems to Homer, the ‘Shield of Hercules’ may be regarded, not as an expansion (in parts) of the ‘Shield of Achilles,’ but as an earlier work of Homer’s, improved and pruned by his maturer judgment, when he desired to fit it into the plan of the ‘Iliad.’ Or rather, each poem may be looked on as an abridgment (the ‘Shield of Hercules’ the earlier) of an independent work on a subject presently to be mentioned.

It is next to be shown that in the events preceding the ‘Oplopœia,’ there is a preparation for the introduction of a separate poem.

In the first place, every reader of Homer is familiar with the fact that the poet constantly makes use, when occasion serves, of expressions, sentences, often even of complete passages, which have been already applied in a corresponding, or occasionally even in a wholly different relation. The same epithets are repeatedly applied to the same deity or hero. A long message is delivered in the very words which have been already used by the sender of the message. In one well-known instance (in Book II.), not only is a message delivered thus, but the person who has received it repeats it to others in precisely the same terms. In the combat between Hector and Ajax (Book VI.), the flight of Ajax’s spear and the movement by which Hector avoids the missile, are described in six lines, differing only as to proper names from those which had been already used in describing the encounter between Paris and Menelaus (Book III.).

This peculiarity would be a decided blemish in a written poem. Tennyson, indeed, occasionally copies Homer’s manner—for instance, in ‘Enid,’ he twice repeats the line—

As careful robins eye the delver’s toil;—

but with a good taste which prevents the repetition from becoming offensive. The fact is, that the peculiarity marks Homer as the _singer_, not the _writer_, of poetry. I would not be understood as accepting the theory, according to which the ‘Iliad’ is a mere string of ballads. I imagine that no one who justly appreciates that noble poem would be willing to countenance such a theory. But that the whole poem was sung by Homer at those prolonged festivals which formed a characteristic peculiarity of Achaian manners seems shown, not only by what we learn respecting the later ‘rhapsodists,’ but by the internal evidence of the poem itself.[20]

Homer, reciting a long and elaborate poem of his own composition, occasionally varying the order of events, or adding new episodes, extemporized as the song proceeded, would exhibit the peculiarity invariably observed in the ‘improvisatore,’ of using, more than once, expressions, sentences, or passages which happened to be conveniently applicable. The art of extemporizing depends on the capacity for composing fresh matter while the tongue is engaged in the recital of matter already composed. Anyone who has watched a clever improvisatore cannot fail to have noticed that, though gesture is aptly wedded to words, the thoughts are elsewhere. In the case, therefore, of an improvisatore, or even of a rhapsodist reciting from memory, the occasional recurrence of a well-worn form of words serves as a relief to the strained invention or memory.

We have reason then for supposing that if Homer had, in his earlier days, composed a poem which was applicable, with slight alterations, to the story of the ‘Iliad,’ he would endeavour, by a suitable arrangement of the plan of his narrative, to introduce the lines whose recital had long since become familiar to him.

Evidence of design in the introduction of the ‘Shield of Achilles’ certainly does not seem wanting.

It is by no means necessary to the plot of the ‘Iliad’ that Achilles should lose the celestial armour given to Peleus as a dowry with Thetis. On the contrary, Homer has gone out of his way to render the labours of Vulcan necessary. Patroclus has to be so ingeniously disposed of, that while the armour he had worn is seized by Hector, his body is rescued, as are also the horses and chariot of Achilles.

We have the additional improbability that the armour of the great Achilles should fit the inferior warriors Patroclus and Hector. Indeed, that the armour should fit Hector, or rather that Hector should fit the armour, the aid of Zeus and Ares has to be called in—

To this Jove’s sable brows did bow; and he made fit his limbs To those great arms, to fill which up the war-god enter’d him Austere and terrible, his joints and every part extends With strength and fortitude.—_Chapman’s_ Translation.