About 1830 was published, in the _Library of Useful Knowledge_, the tract
on _Probability_, the joint work of the late Sir John Lubbock[611] and Mr.
Drinkwater (Bethune).[612] It is one of the best elementary openings of the
subject. A binder put my name on the outside (the work was anonymous) and
the consequence was that nothing could drive out of people's heads that it
was written by me. I do not know how many denials I have made, from a
passage in one of my own works to a letter in the _Times_: and I am not
sure that I have succeeded in establishing the truth, even now. I
accordingly note the fact once more. But as a book has no right here unless
it contain a paradox--or thing counter to general opinion or practice--I
will produce two small ones. Sir John Lubbock, with whom lay the executive
arrangement, had a strong objection to the last word in "Theory of
Probabilities," he maintained that the singular _probability_, should be
used; and I hold him quite right.
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The second case was this: My friend Sir J. L., with a large cluster of
intellectual qualities, and another of social qualities, had one point of
character which I will not call bad and cannot call good; he never used a
slang expression. To such a length did he carry his dislike, that he could
not bear _head_ and _tail_, even in a work on games of chance: so he used
_obverse_ and _reverse_. I stared when I first saw this: but, to my
delight, I found that the force of circumstances beat him at last. He was
obliged to take an example from the race-course, and the name of one of the
horses was _Bessy Bedlam_! And he did not put her down as _Elizabeth
Bethlehem_, but forced himself to follow the jockeys.
[Almanach Romain sur la Loterie Royale de France, ou les Etrennes
necessaires aux Actionnaires et Receveurs de la dite Loterie. Par M.
Menut de St.-Mesmin. Paris, 1830. 12mo.
This book contains all the drawings of the French lottery (two or three,
each month) from 1758 to 1830. It is intended for those who thought they
could predict the future drawings from the past: and various sets of
_sympathetic_ numbers are given to help them. The principle is, that
anything which has not happened for a long time must be soon to come. At
_rouge et noir_, for example, when the red has won five times running,
sagacious gamblers stake on the black, for they think the turn which must
come at last is nearer than it was. So it is: but observation would have
shown that if a large number of those cases had been registered which show
a run of five for the red, the next game would just as often have made the
run into six as have turned in favor of the black. But the gambling
reasoner is incorrigible: if he would but take to squaring the circle, what
a load of misery would be saved. A writer of 1823, who appeared to be
thoroughly acquainted with the gambling of Paris and London, says that the
gamesters by {281} profession are haunted by a secret foreboding of their
future destruction, and seem as if they said to the banker at the table, as
the gladiators said to the emperor, _Morituri te salutant_.[613]
In the French lottery, five numbers out of ninety were drawn at a time. Any
person, in any part of the country, might stake any sum upon any event he
pleased, as that 27 should be drawn; that 42 and 81 should be drawn; that
42 and 81 should be drawn, and 42 first; and so on up to a _quine
determine_, if he chose, which is betting on five given numbers in a given
order. Thus, in July, 1821, one of the drawings was
8 46 16 64 13.
A gambler had actually predicted the five numbers (but not their order),
and won 131,350 francs on a trifling stake. M. Menut seems to insinuate
that the hint what numbers to choose was given at his own office. Another
won 20,852 francs on the quaterne, 8, 16, 46, 64, in this very drawing.
These gains, of course, were widely advertised: of the multitudes who lost
nothing was said. The enormous number of those who played is proved to all
who have studied chances arithmetically by the numbers of simple quaternes
which were gained: in 1822, fourteen; in 1823, six; in 1824, sixteen; in
1825, nine, &c.
The paradoxes of what is called chance, or hazard, might themselves make a
small volume. All the world understands that there is a long run, a general
average; but great part of the world is surprised that this general average
should be computed and predicted. There are many remarkable cases of
verification; and one of them relates to the quadrature of the circle. I
give some account of this and another. Throw a penny time after time until
_head_ arrives, which it will do before long: let this be called a _set_.
Accordingly, H is the smallest set, TH the next smallest, then TTH, &c. For
abbreviation, let a set in which seven _tails_ {282} occur before _head_
turns up be T^{7}H. In an immense number of trials of sets, about half will
be H; about a quarter TH; about an eighth, T^{2}H. Buffon[614] tried 2,048
sets; and several have followed him. It will tend to illustrate the
principle if I give all the results; namely, that many trials will with
moral certainty show an approach--and the greater the greater the number of
trials--to that average which sober reasoning predicts. In the first column
is the most likely number of the theory: the next column gives Buffon's
result; the three next are results obtained from trial by correspondents of
mine. In each case the number of trials is 2,048.
H 1,024 1,061 1,048 1,017 1,039
TH 512 494 507 547 480
T^{2}H 256 232 248 235 267
T^{3}H 128 137 99 118 126
T^{4}H 64 56 71 72 67
T^{5}H 32 29 38 32 33
T^{6}H 16 25 17 10 19
T^{7}H 8 8 9 9 10
T^{8}H 4 6 5 3 3
T^{9}H 2 3 2 4
T^{10}H 1 1 1
T^{11}H 0 1
T^{12}H 0 0
T^{13}H 1 1 0
T^{14}H 0 0
T^{15}H 1 1
&c. 0 0
----- ----- ----- ----- -----
2,048 2,048 2,048 2,048 2,048
{283}
In very many trials, then, we may depend upon something like the predicted
average. Conversely, from many trials we may form a guess at what the
average will be. Thus, in Buffon's experiment the 2,048 first throws of the
sets gave _head_ in 1,061 cases: we have a right to infer that in the long
run something like 1,061 out of 2,048 is the proportion of heads, even
before we know the reasons for the equality of chance, which tell us that
1,024 out of 2,048 is the real truth. I now come to the way in which such
considerations have led to a mode in which mere pitch-and-toss has given a
more accurate approach to the quadrature of the circle than has been
reached by some of my paradoxers. What would my friend[615] in No. 14 have
said to this? The method is as follows: Suppose a planked floor of the
usual kind, with thin visible seams between the planks. Let there be a thin
straight rod, or wire, not so long as the breadth of the plank. This rod,
being tossed up at hazard, will either fall quite clear of the seams, or
will lay across one seam. Now Buffon, and after him Laplace, proved the
following: That in the long run the fraction of the whole number of trials
in which a seam is intersected will be the fraction which twice the length
of the rod is of the circumference of the circle having the breadth of a
plank for its diameter. In 1855 Mr. _Ambrose_ Smith, of Aberdeen, made
3,204 trials with a rod three-fifths of the distance between the planks:
there were 1,213 clear intersections, and 11 contacts on which it was
difficult to decide. Divide these contacts equally, and we have 1,2181/2 to
3,204 for the ratio of 6 to 5[pi], presuming that the greatness of the
number of trials gives something near to the final average, or result in
the long run: this gives [pi] = 3.1553. If all the 11 contacts had been
treated as intersections, the result would have been {284} [pi] = 3.1412,
exceedingly near. A pupil of mine made 600 trials with a rod of the length
between the seams, and got [pi] = 3.137.
This method will hardly be believed until it has been repeated so often
that "there never could have been any doubt about it."
The first experiment strongly illustrates a truth of the theory, well
confirmed by practice: whatever can happen will happen if we make trials
enough. Who would undertake to throw tail eight times running?
Nevertheless, in the 8,192 sets tail 8 times running occurred 17 times; 9
times running, 9 times; 10 times running, twice; 11 times and 13 times,
each once; and 15 times twice.]
ON CURIOSITIES OF [pi].
1830. The celebrated interminable fraction 3.14159..., which the
mathematician calls [pi], is the ratio of the circumference to the
diameter. But it is thousands of things besides. It is constantly turning
up in mathematics: and if arithmetic and algebra had been studied without
geometry, [pi] must have come in somehow, though at what stage or under
what name must have depended upon the casualties of algebraical invention.
This will readily be seen when it is stated that [pi] is nothing but four
times the series
1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
_ad infinitum_.[616] It would be wonderful if so simple a series {285} had
but one kind of occurrence. As it is, our trigonometry being founded on the
circle, [pi] first appears as the ratio stated. If, for instance, a deep
study of probable fluctuation from average had preceded, [pi] might have
emerged as a number perfectly indispensable in such problems as: What is
the chance of the number of aces lying between a million + x and a million
- x, when six million of throws are made with a die? I have not gone into
any detail of all those cases in which the paradoxer finds out, by his
unassisted acumen, that results of mathematical investigation _cannot be_:
in fact, this discovery is only an accompaniment, though a necessary one,
of his paradoxical statement of that which _must be_. Logicians are
beginning to see that the notion of _horse_ is inseparably connected with
that of _non-horse_: that the first without the second would be no notion
at all. And it is clear that the positive affirmation of that which
contradicts mathematical demonstration cannot but be accompanied by a
declaration, mostly overtly made, that demonstration is false. If the
mathematician were interested in punishing this indiscretion, he could make
his denier ridiculous by inventing asserted results which would completely
take him in.
More than thirty years ago I had a friend, now long gone, who was a
mathematician, but not of the higher branches: he was, _inter alia_,
thoroughly up in all that relates to mortality, life assurance, &c. One
day, explaining to him how it should be ascertained what the chance is of
the survivors of a large number of persons now alive lying between given
limits of number at the end of a certain time, I came, of course upon the
introduction of [pi], which I could only describe as the ratio of the
circumference of a circle to its diameter. "Oh, my dear friend! that must
be a delusion; what can the circle have to do with the numbers alive at the
end of a given time?"--"I cannot demonstrate it to you; but it is
demonstrated."--"Oh! stuff! I think you can prove anything with your
differential calculus: figment, {286} depend upon it." I said no more; but,
a few days afterwards, I went to him and very gravely told him that I had
discovered the law of human mortality in the Carlisle Table, of which he
thought very highly. I told him that the law was involved in this
circumstance. Take the table of expectation of life, choose any age, take
its expectation and make the nearest integer a new age, do the same with
that, and so on; begin at what age you like, you are sure to end at the
place where the age past is equal, or most nearly equal, to the expectation
to come. "You don't mean that this always happens?"--"Try it." He did try,
again and again; and found it as I said. "This is, indeed, a curious thing;
this _is_ a discovery." I might have sent him about trumpeting the law of
life: but I contented myself with informing him that the same thing would
happen with any table whatsoever in which the first column goes up and the
second goes down; and that if a proficient in the higher mathematics chose
to palm a figment upon him, he could do without the circle: _a corsaire,
corsaire et demi_,[617] the French proverb says. "Oh!" it was remarked, "I
see, this was Milne!"[618] It was _not_ Milne: I remember well showing the
formula to him some time afterwards. He raised no difficulty about [pi]; he
knew the forms of Laplace's results, and he was much interested. Besides,
Milne never said stuff! and figment! And he would not have been taken in:
he would have quietly tried it with the Northampton and all the other
tables, and would have got at the truth.
{287}