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 dia

meter 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 we 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, we give an approximate solution which is remarkably simple, and, so far as we are aware, not generally known. Describe a square about the given circle, touching it at the ends of two diameters, AOB, COD, 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 constructtion 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 But there are others which can be so drawn. And, if such curves as these be admitted as available,

means.

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 we 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.)

299

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 interpolationperhaps 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 actionis checked in a marked manner by the Oplopoeia' 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.* 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

* Another well-known instance, where Patroclus sent in hot haste for news by a man of the most fiery impatience, is button-held by Nestor, and though he has no time to sit down, yet is obliged to endure a speech of 152 lines,' is accounted for by Gladstone in a different manner.