The Foundations of Chance [pp. 471-510]

THE FOUNDA TIONS OF CHANCE.
vice. The fact is, that we are so used to taking the average in
simple cases that we look upon it as self-evident. If two men,
after having measured the length of a wall, give us respectively
fifty-one and fifty-four feet, we should settle the discordance by
taking the mean, namely, fifty-two and one half. And many
would clinch this by an a priori argument that it must be so,
for, having no possible reason to prefer one result to the other,
it would be irrational to do any thing else than put them on
precisely equal terms by just halving their difference. But, as
regards this argument, compare the following case: Two sailors
from a ship are sent to steal into a fort and take the calibre of a
cannon. We may suppose that the fort is going to be taken
and used against the enemy, and that it is therefore of importance to provide shot of the right calibre. One sailor says he
made the calibre eight inches, the other made it nine. What
should we say of the wisdom of the captain who should go
there provided with shot of eight and one half inches, there being, let us say, no such calibre in the service? It would be as
if a man who was uncertain whether or not he hail been asked
to stop to dinner somewhere were to meet the case half-way by
going in dress trousers and waistcoat, but in a flannel shirt and
check coat. In the case of the cannon, it would be better far
to toss up than to try to "combine" the observations.'
1 A rather curious case of this kind has recently fallen under our notice. It
is an attempt to determine the length of certain ancient measures, such as the
cubit, simply from the examination and measurement of ancient buildings (we
ought to premise that we only know the work in question at second-hand, not
having had an opportunity of consulting it). At first sight, the attempt may
seem absurd; for why, it may be asked, should any part of a building be of
one particular length rather than of another? The only assumption with which
we start is that fractions of cubits are less likely to present themselves than
whole ones, and irregular fractions less likely than such simple ones as halves
and quarters. In modern buildings, of course, the exigencies of space in a
crowded city are paramount; but in more roomy times it seems plausible
enough that builders would have preferred unbroken units to work with, when
possible. Starting with this assumption, the problem becomes feasible. We
collect a large quantity of measures of lengths and heights, etc., of these ancient
buildings; we examine their differences and ratios, and ascertain whether these
show a tendency to yield frequent multiples of some particular magnitude. If
they do, this magnitude is presumably the lost ancient measure, or some fraction of it. Of course, all this may be the mere result of coincidence; whether
it is so or not must be decided on principles of probability, by seeing how
many independent magnitudes have been examined for the purpose.
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