The phenomena of spiritual photography were first observed some years since, and a set of carte photographs were sent from America to Dr. Walker, of Edinburgh, in which photographic phantoms were very obviously, however indistinctly, discernible. More recently an English photographer noticed a yet stranger circumstance, though he was too sensible to seek for a supernatural interpretation of it. When he took a photograph with a particular lens, there could be seen not only the usual portrait of the sitter, but at some little distance a faint ‘double,’ exactly resembling the principal image. Superstitious minds might find this result even more distressing than the phantom photographic friend. To be visited by the departed through the medium of a lens, is at least not more unpleasing than to hold converse with spirits through an ordinary ‘rapping’ medium. But the appearance of a ‘double,’ or ‘fetch,’ has ever been held by the learned in ghostly lore to signify approaching death.

Fortunately both one and the other appearance can be very easily accounted for without calling in the aid of the supernatural. At a recent meeting of the Photographical Society it was shown that an image may often be so deeply impressed on the glass that the subsequent cleaning of the plate, even with strong acids, will not completely remove the picture. When the plate is used for receiving another picture, the original image makes its reappearance, and as it is too faint to be recognisable, a highly susceptible imagination may readily transform it into the image of a departed friend. The ‘double’ is generated by the well-known property of double refraction, obtained by a lens under certain circumstances of unequal pressure, or sometimes by inequalities in the process of annealing. So vanish two ghosts which might have been more or less troublesome to those who are ready to see the supernatural in commonplace phenomena. Will the time ever come when no more such phantoms will remain to be exorcised?

(From the Daily News, March 2, 1869.)

THE OXFORD AND CAMBRIDGE ROWING STYLES.

Whatever opinion we may have of the result of the approaching contest (1869), there can be no doubt that this year, as in former years, there is a striking dissimilarity between the rowing styles of the dark blue and the light blue oarsmen. This dissimilarity makes itself obvious whether we compare the two boats as seen from the side, or when the line of sight is directed along the length of either. Perhaps it is in the latter aspect that an unpractised eye will most readily detect the difference I am speaking of. Watch the Cambridge boat approaching you from some distance, or receding, and you will notice in the rise and fall of the oars, as so seen, the following peculiarities—a long stay of the oar in the water, a quick rise from and return to the water, the oars remaining out of the water for the briefest possible interval of time. In the case of the Oxford boat quite a different appearance is presented—there is a short stay in the water, a sharp rise from and return to it, and between these the oars appear to hang over the water for a perceptible interval. It is, however, when the boats are seen from the side that the meaning of these peculiarities is detected, and also that the fundamental distinction between the two styles is made apparent to the experienced eye. In the Cambridge boat we recognise the long stroke and ‘lightning feather’ inculcated in the old treatises on rowing: in the Oxford boat we see these conditions reversed, and in their place the ‘waiting feather’ and lightning stroke. By the ‘waiting feather’ I do not refer to what is commonly understood by slow feathering, but to a momentary pause (scarcely to be detected when the crew is rowing hard) before the simultaneous dash of the oars upon the first grip of the stroke.15 And observing more closely—which, by the way, is no easy matter—as either boat dashes swiftly past, we detect the distinctive peculiarities of ‘work’ by which the two styles are severally arrived at. In the Cambridge crew we see the first part of the stroke done with the shoulders—precisely according to the old-fashioned models—the arms straight until the body has fallen back to an almost upright position; then comes the sharp drop back of the shoulders beyond the perpendicular, the arms simultaneously doing their work, so that as the swing back is finished, the backs of the hands just touch the ribs in feathering. All these things are quite in accordance with what used to be considered the perfection of rowing; and, indeed, this style of rowing has some important good qualities and a very handsome appearance. The lightning feather, also, which follows the long sweeping stroke, is theoretically perfect. Now, in the case of the Oxford crew, we observe a style which at first sight seems less excellent. As soon as the oars are dashed down and catch their first hold of the water, the arms as well as the shoulders of each oarsman are at work.16 The result is, that when the back has reached an upright position, the arms have already reached the chest, and the stroke is finished. Thus the Oxford stroke takes a perceptibly shorter time than the Cambridge stroke; it is also, necessarily, somewhat shorter in the water. One would, therefore, say it must be less effective. Especially would an unpractised observer form this opinion, because the Oxford stroke seems to be much shorter in range than it is in reality. There we have the secret of its efficiency. It is actually as long as the Cambridge stroke, but is taken in a perceptibly shorter time. What does this mean but that the oar is taken more sharply, and, therefore, much more effectively, through the water?

Much more effectively so far as the actual conditions of the contest are concerned. The modern racing outrigger requires a sharp impulse, because it will take almost any speed we can apply to it. It will also retain that speed between the strokes, a consideration of great importance. The old-fashioned racing-eights required to be continually under propulsion. The lightning-feather was a necessity in their case, for between every stroke the boat would lag terribly with a slow-feathering crew. I do not say, of course, that the speed of a light outrigged craft does not diminish between the strokes. Anyone who has watched a closely contested bumping-race, and noticed the way in which the sharply cut bow of the pursuing boat draws up to the rudder of the other as by a succession of impulses, although either boat seen alone would seem to sweep on with almost uniform speed, will know that the motion of the lightest boat is not strictly uniform. But there is an immense difference between the almost imperceptible loss of way of a modern eight and the dead ‘lag’ in the old-fashioned craft. And hence we get the following important consideration. Whereas with the old boats it was useless for a crew to attempt to give a very quick motion to their boat by a sharp, sudden ‘lift,’ this plan is calculated to be, of all others, the most effective with the modern racing-eight.

It may seem, at first sight, that, after all, the result of the Cambridge style should be as effective as that of the other. If arms and shoulders do their work in both crews with equal energy—which we may assume to be the case—and if the number of strokes per minute is equal, the actual propulsive energy ought to be equal likewise. A little consideration will show that this is a fallacy. If two men pull at a weight together they will move it farther with a given expenditure of energy than if first one and then the other apply his strength to the work. And what is more to the purpose, they will be able to move it faster. So shoulders and arms working simultaneously will give a greater propulsive power than when working separately, even though in the latter case each works with its fullest energy. And not only so, but by the simultaneous use of arms and shoulders, that sharpness of motion can alone be given which is essential to the propulsion of a modern racing-boat.

I have said that the two crews are severally rowing in the style which has lately been peculiar to their respective Universities. But the Cambridge crew is rowing in that form of the Cambridge style which brings it nearest to the requirements of modern racing. The faults of the style are subdued, so to speak, and its best qualities brought out effectively. In one or two of the long series of defeats lately sustained by Cambridge the reverse has been the case. At present, too, there is a certain roughness about the Oxford crew which encourages the hopes of the light blue supporters. But it must be admitted that this roughness is rather apparent than real, great as it seems, and it will doubtless disappear before the day of encounter. I venture to predict that the ‘time’ of the approaching race, taken in conjunction with the state of the tide, will show the present crews to be at least equal to the average.17

(From the Daily News, April 1869.)

BETTING ON HORSE RACES: OR, THE STATE OF THE ODDS.

There appears every day in the newspapers an account of the betting on the principal forthcoming races. The betting on such races as the Two Thousand Guineas, the Derby, and the Oaks, often begins more than a year before the races are run; and during the interval, the odds laid against the different horses engaged in them vary repeatedly, in accordance with the reported progress of the animals in their training, or with what is learned respecting the intentions of their owners. Many who do not bet themselves, find an interest in watching the varying fortunes of the horses which are held by the initiated to be leading favourites, or to fall into the second rank, or merely to have an outside chance of success. It is amusing to notice, too, how frequently the final state of the odds is falsified by the event; how some ‘rank outsider’ will run into the first place, while the leading favourites are not even ‘placed.’

It is in reality a simple matter to understand the betting on races (or contests of any kind), yet it is astonishing how seldom those who do not actually bet upon races have any inkling of the meaning of those mysterious columns which indicate the opinion of the betting world respecting the probable results of approaching contests, equine or otherwise.

Let us take a few simple cases of ‘odds,’ to begin with; and, having mastered the elements of our subject, proceed to see how cases of greater complexity are to be dealt with.

Suppose the newspapers inform us that the betting is 2 to 1 against a certain horse for such and such a race, what inference are we to deduce? To learn this let us conceive a case in which the true odds against a certain event are as 2 to 1. Suppose there are three balls in a bag, one being white, the others black. Then, if we draw a ball at random, it is clear that we are twice as likely to draw a black as to draw a white ball. This is technically expressed by saying that the odds are 2 to 1 against drawing a white ball; or 2 to 1 on (that is, in favour of) drawing a black ball. This being understood, it follows that, when the odds are said to be 2 to 1 against a certain horse, we are to infer that, in the opinion of those who have studied the performance of the horse, and compared it with that of the other horses engaged in the race, his chance of winning is equivalent to the chance of drawing one particular ball out of a bag of three balls.

Observe how this result is obtained: the odds are 2 to 1, and the chance of the horse is as that of drawing one ball out of a bag of three—three being the sum of the two numbers 2 and 1. This is the method followed in all such cases. Thus, if the odds against a horse are 7 to 1, we infer that the cognoscenti consider his chance equal to that of drawing one particular ball out of a bag of eight.

A similar treatment applies when the odds are not given as so many to one. Thus, if the odds against a horse are as 5 to 2, we infer that the horse’s chance is equal to that of drawing a white ball out of a bag containing five black and two white balls—or seven in all.

We must notice also that the number of balls may be increased to any extent, provided the proportion between the total number and the number of a specified colour remains unchanged. Thus, if the odds are 5 to 1 against a horse, his chance is assumed to be equivalent to that of drawing one white ball out of a bag containing six balls, only one of which is white; or to that of drawing a white ball out of a bag containing sixty balls, of which ten are white-and so on. This is a very important principle, as we shall now see.

Suppose there are two horses (amongst others) engaged in a race, and that the odds are 2 to 1 against one, and 4 to 1 against the other-what are the odds that one of the two horses will win the race? This case will doubtless remind my readers of an amusing sketch by Leech, called—if I remember rightly—‘Signs of the Commission.’ Three or four undergraduates are at a ‘wine,’ discussing matters equine. One propounds to his neighbour the following question: I say, Charley, if the odds are 2 to 1 against Rataplan, and 4 to 1 against Quick March, what’s the betting about the pair?’—‘Don’t know, I’m sure,’ replies Charley; ‘but I’ll give you 6 to 1 against them.’ The absurdity of the reply is, of course, very obvious; we see at once that the odds cannot be heavier against a pair of horses than against either singly. Still, there are many who would not find it easy to give a correct reply to the question. What has been said above, however, will enable us at once to determine the just odds in this or any similar case. Thus-the odds against one horse being 2 to 1, his chance of winning is equal to that of drawing one white ball out of a bag of three, one only of which is white. In like manner, the chance of the second horse is equal to that of drawing one white ball out of a bag of five, one only of which is white. Now we have to find a number which is a multiple of both the numbers three and five. Fifteen is such a number. The chance of the first horse, modified according to the principle explained above, is equal to that of drawing a white ball out of a bag of fifteen of which five are white. In like manner, the chance of the second is equal to that of drawing a white ball out of a bag of fifteen of which three are white. Therefore the chance that one of the two will win is equal to that of drawing a white ball out of a bag of fifteen balls of which eight (five added to three) are white. There remain seven black balls, and therefore the odds are 8 to 7 on the pair.

To impress the method of treating such cases on the mind of the reader, let us take the betting about three horses—say 3 to 1, 7 to 2, and 9 to 1 against the three horses respectively. Then their respective chances are equal to the chance of drawing (1) one white ball out of four, one only of which is white; (2) a white ball out of nine, of which two only are white; and (3) one white ball out of ten, one only of which is white. The least number which contains four, nine, and ten is 180; and the above chances, modified according to the principle explained above, become equal to the chance of drawing a white ball out of a bag containing 180 balls, when 45, 40, and 18 (respectively) are white. Therefore, the chance that one of the three will win is equal to that of drawing a white ball out of a bag containing 180 balls, of which 103 (the sum of 45, 40, and 18) are white. Therefore, the odds are 103 to 77 on the three.

One does not hear in practice of such odds as 103 to 77. But betting-men (whether or not they apply just principles of computation to such questions, is unknown to me) manage to run very near the truth. For instance, in such a case as the above, the odds on the three would probably be given as 4 to 3—that is, instead of 103 to 77 (or 412 to 308), the published odds would be equivalent to 412 to 309.

And here a certain nicety in betting has to be mentioned. In running the eye down the list of odds, one will often meet such expressions as 10 to 1 against such a horse offered, or 10 to 1 wanted. Now, the odds of 10 to 1 taken may be understood to imply that the horse’s chance is equivalent to that of drawing a certain ball out of a bag of eleven. But if the odds are offered and not taken, we cannot infer this. The offering of the odds implies that the horse’s chance is not better than that above mentioned, but the fact that they are not taken implies that the horse’s chance is not so good. If no higher odds are offered against the horse, we may infer that his chance is very little worse than that mentioned above. Similarly, if the odds of 10 to 1 are asked for, we infer that the horse’s chance is not worse than that of drawing one ball out of eleven; if the odds are not obtained, we infer that his chance is better; and if no lower odds are asked for, we infer that his chance is very little better.

Thus, there might be three horses (A, B, and C) against whom the nominal odds were 10 to 1, and yet these horses might not be equally good favourites, because the odds might not be taken, or might be asked for in vain. We might accordingly find three such horses arranged thus:—

Or these different stages might mark the upward or downward progress of the same horse in the betting. In fact, there are yet more delicate gradations, marked by such expressions respecting certain odds, as—offered freely, offered, offered and taken (meaning that some offers only have been accepted), taken, taken and wanted, wanted, and so on.

As an illustration of some of the principles I have been considering, let us take from the day’s paper,18 the state of the odds respecting the ‘Two Thousand Guineas.’ It is presented in the following form:—

TWO THOUSAND GUINEAS.

This table is interpreted thus: bettors are willing to lay the same odds against Rosicrucian as would be the true mathematical odds against drawing a white ball out of a bag containing two white and seven black balls; but no one is willing to back the horse at this rate; on the other hand, higher odds are not offered against him. Hence it is presumable that his chance is somewhat less than that above indicated. Again, bettors are willing to lay the same odds against Pace as might fairly be laid against drawing one white ball out of a bag of seven, one only of which is white; but backers of the horse consider that they ought to get the same odds as might be fairly laid against drawing the white ball when an additional black ball had been put into the bag. As respects Green Sleeve and Blue Gown, bettors are willing to lay the odds which there would be, respectively, against drawing a white ball out of a bag containing—(1) eleven balls, one only of which is white, and (2) one hundred and seven balls, seven only of which are white. Now, the three horses, Rosicrucian, Green Sleeve, and Blue Gown, all belong to Sir Joseph Hawley, so that the odds about the three are referred to in the last statement of the list just given. And since none of the offers against the three horses have been taken, we may expect the odds actually taken about ‘Sir Joseph Hawley’s lot’ to be more favourable than those obtained by summing up the three former in the manner we have already examined. It will be found that the resulting odds (offered) against Sir J. Hawley’s lot—estimated in this way—should be, as nearly as possible, 132 to 80. We find, however, that the odds taken are 180 to 80. Hence, we learn that the offers against some or all of the three horses are considerably short of what backers require; or else that some person has been induced to offer far heavier odds against Sir J. Hawley’s lot than are justified by the fair odds against his horses, severally.

I have heard it asked why a horse is said to be a favourite, though the odds may be against him. This is very easily explained. Let us take as an illustration the case of a race in which four horses are engaged to run. If all these horses had an equal chance of winning, it is very clear that the case would correspond to that of a bag containing four balls of different colours; since, in this case, we should have an equal chance of drawing a ball of any assigned colour. Now, the odds against drawing a particular ball would clearly be 3 to 1. This, then, should be the betting against each of the three horses. If any one of the horses has less odds offered against him, he is a favourite. There may be more than one of the four horses thus distinguished; and, in that case, the horse against which the least odds are offered is the first favourite. Let us suppose there are two favourites, and that the odds against the leading favourite are 3 to 2, those against the other 2 to 1, and those against the best non-favourite 4 to 1; and let us compare the chances of the four horses. I have not named any odds against the fourth, because, if the odds against all the horses but one are given, the just odds against that one are determinable, as we shall see immediately. The chance of the leading favourite corresponds to the chance of drawing a ball out of a bag in which are three black and two white balls, five in all; that of the next to the chance of drawing a ball out of a bag in which are two black and one white ball, three in all; that of the third, to the chance of drawing a ball out of a bag in which are four black balls and one white one, five in all. We take, then, the least number containing both five and three—that is, fifteen; and then the number of white balls, corresponding to the chances of the three horses, are respectively six, five, and three, or fourteen in all; leaving only one to represent the chance of the fourth horse (against which the odds are therefore 14 to 1). Hence the chances of the four horses are respectively as the numbers six, five, three and one.

I have spoken above of the published odds. The statements made in the daily papers commonly refer to wagers actually made, and therefore the uninitiated might suppose that everyone who tried would be able to obtain the same odds. This is not the case. The wagers which are laid between practised betting-men afford very little indication of the prices which would be forced (so to speak) upon an inexperienced bettor. Book-makers—that is, men who make a series of bets upon several or all of the horses engaged in a race—naturally seek to give less favourable terms than the known chances of the different horses engaged would suffice to warrant. As they cannot offer such terms to the initiated, they offer them-and in general success—fully—to the inexperienced.

It is often said that a man may so lay his wagers about a race as to make sure of gaining money whichever horse wins the race. This is not strictly the case. It is of course possible to make sure of winning if the bettor can only get persons to lay or take the odds he requires to the amount he requires. But this is precisely the problem which would remain insoluble if all bettors were equally experienced.

Suppose, for instance, that there are three horses engaged in a race with equal chances of success. It is readily shown that the odds are 2 to 1 against each. But if a bettor can get a person to take even betting against the first horse (A), a second person to do the like about the second horse (B), and a third to do the like about the third horse (C), and if all these bets are made to the same amount—say 1000l.—then, inasmuch as only one horse can win, the bettor loses 1000l. on that horse (say A), and gains the same sum on each of the two horses B and C. Thus, on the whole, he gains 1000l., the sum laid out against each horse.

If the layer of the odds had laid the true odds to the same amount on each horse, he would neither have gained nor lost. Suppose, for instance, that he laid 1000l. to 500l. against each horse, and A won; then he would have to pay 1000l. to the backer of A, and to receive 500l. from each of the backers of B and C. In like manner, a person who had backed each horse to the same extent would neither lose nor gain by the event. Nor would a backer or layer who had wagered different sums necessarily gain or lose by the race; he would gain or lose according to the event. This will at once be seen, on trial.

Let us next take the case of horses with unequal prospects of success—for instance, take the case of the four horses considered above, against which the odds were respectively 3 to 2, 2 to 1, 4 to 1, and 14 to 1. Here, suppose the same sum laid against each, and for convenience let this sum be 84l. (because 84 contains the numbers 3, 2, 4, and 14). The layer of the odds wagers 84l. to 56l. against the leading favourite, 84l. to 42l. against the second horse, 84l. to 21l. against the third, and 84l. to 6l. against the fourth. Whichever horse wins, the layer has to pay 84l.; but if the favourite wins, he receives only 42l. on one horse, 21l. on another, and 6l. on the third—that is 69l. in all, so that he loses 15l.; if the second horse wins, he has to receive 56l., 21l., and 6l.—or 83l. in all, so that he loses 1l.; if the third horse wins, he receives 56l., 42l., and 6l.—or 104l. in all, and thus gains 20l.; and lastly, if the fourth horse wins, he has to receive 56l., 42l., and 2ll.—or 119l. in all, so that he gains 35l. He clearly risks much less than he has a chance (however small) of gaining. It is also clear that in all such cases the worst event for the layer of the odds is, that the favourite should win. Accordingly, as professional book-makers are nearly always layers of odds, one often finds the success of a favourite spoken of in the papers as a ‘great blow for the book-makers,’ while the success of a rank outsider will be described as ‘a misfortune to backers.’

But there is another circumstance which tends to make the success of a favourite a blow to layers of the odds and vice versâ. In the case we have supposed, the money actually pending about the four horses (that is, the sum of the amount laid for and against them) was 140l. as respects the favourite, 126l. as respects the second, 105l. as respects the third, and 90l. as respects the fourth. But as a matter of fact the amounts pending about the favourites bear always a much greater proportion than the above to the amounts pending about outsiders. It is easy to see the effect of this. Suppose, for instance, that instead of the sums 84l. to 56l., 84l. to 42l., 84l. to 21l., and 84l. to 6l., a book-maker had laid 8400l. to 5600l., 840l. to 420l., 84l. to 21l., and 14l. to 1l., respectively—then it will easily be seen that he would lose 7958l. by the success of the favourite; whereas he would gain 4782l. by the success of the second horse, 5937l. by that of the third, and 6027l. by that of the fourth. I have taken this as an extreme case; as a general rule, there is not so great a disparity as has been here assumed between the sums pending on favourites and outsiders.

Finally, it may be asked whether, in the case of horses having unequal chances, it is possible that wagers can be so proportioned (just odds being given and taken), that, as in the former case, a person backing or laying against all the four shall neither gain nor lose. It is so. All that is necessary is, that the sum actually pending about each horse shall be the same. Thus, in the preceding case, if the wagers 9l. to 6l., 10l. to 5l., 12l. to 3l., and 14l. to 1l., are either laid or taken by the same person, he will neither gain nor lose by the event, whatever it may be. And therefore, if unfair odds are laid or taken about all the horses, in such a manner that the amounts pending on the several horses are equal (or nearly so), the unfair bettor must win by the result. Say, for instance, that instead of the above odds, he lays 8l. to 6l., 9l. to 5l., 11l. to 3l. and 13l. to 1l., against the four horses respectively; it will be found that he must win 1l. Or if he takes the odds 18l. to 11l., 20l. to 9l., 24l. to 5l., and 28l. to 1l. (the just odds being 18l. to 12l., 20l. to 10l., 24l. to 6l., and 28l. to 2l. respectively), he will win 1l. by the race. So that, by giving or taking such odds to a sufficiently great amount, a bettor would be certain of pocketing a large sum, whatever the event of a given race might be.

In every instance, a man who bets on a race must risk his money, unless he can succeed in taking unfair advantages over those with whom he bets. My readers will conceive how small must be the chance that an unpractised bettor will gain anything but dearly-bought experience by speculating on horse-races. I would recommend those who are tempted to hold another opinion to follow the plan suggested by Thackeray in a similar case—to take a good look at professional and practised betting-men, and to decide ‘which of those men they are most likely to get the better of’ in turf transactions.

(From Chambers’s Journal, July 1869.)

SQUARING THE CIRCLE.

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:—