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 84_l._ (because 84 contains the numbers 3,
2, 4, and 14). The layer of the odds wagers 84_l._ to 56_l._ against
the leading favourite, 84_l._ to 42_l._ against the second horse,
84_l._ to 21_l._ against the third, and 84_l._ to 6_l._ against the
fourth. Whichever horse wins, the layer has to pay 84_l._; but if
the favourite wins, he receives only 42_l._ on one horse, 21_l._ on
another, and 6_l._ on the third—that is 69_l._ in all, so that he loses
15_l._; if the second horse wins, he has to receive 56_l._, 21_l._,
and 6_l._—or 83_l._ in all, so that he loses 1_l._; if the third horse
wins, he receives 56_l._, 42_l._, and 6_l._—or 104_l._ in all, and thus
gains 20_l._; and lastly, if the fourth horse wins, he has to receive
56_l._, 42_l._, and 2l_l._—or 119_l._ in all, so that he gains 35_l._
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 140_l._
as respects the favourite, 126_l._ as respects the second, 105_l._
as respects the third, and 90_l._ 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 84_l._ to 56_l._, 84_l._ to 42_l._, 84_l._
to 21_l._, and 84_l._ to 6_l._, a book-maker had laid 8400_l._ to
5600_l._, 840_l._ to 420_l._, 84_l._ to 21_l._, and 14_l._ to 1_l._,
respectively—then it will easily be seen that he would lose 7958_l._
by the success of the favourite; whereas he would gain 4782_l._ by
the success of the second horse, 5937_l._ by that of the third, and
6027_l._ 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
9_l._ to 6_l._, 10_l._ to 5_l._, 12_l._ to 3_l._, and 14_l._ to 1_l._,
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 8_l._ to 6_l._, 9_l._ to 5_l._, 11_l._ to
3_l._ and 13_l._ to 1_l._, against the four horses respectively; it
will be found that he _must_ win 1_l._ Or if he _takes_ the odds 18_l._
to 11_l._, 20_l._ to 9_l._, 24_l._ to 5_l._, and 28_l._ to 1_l._ (the
just odds being 18_l._ to 12_l._, 20_l._ to 10_l._, 24_l._ to 6_l._,
and 28_l._ to 2_l._ respectively), he will win 1_l._ 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.)
FOOTNOTES:
[17] The race (that of 1869) was one of the best ever rowed, and the
time of the winners (Oxford) better than in any former race.
[18] This article was written early in March 1868.
_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:—
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.