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Review by Augustus De Morgan
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\noindent\hangindent=1em
ART.\ IV.---\textit{Th\'eorie Analytique des Probabilit\'es}. Par M.\ le
Marquis de Laplace, \&c.\ \&c. 3\`eme edition. Paris 1820.
\medskip
\noindent
MONTUCLA remarked, that if any subject might be expected to baffle the
mathematicians, it would be \textit{chance}. The same might have been
said of the motions of the heavenly bodies; not at the time when the
first rude theories sufficiently well represented the results of still
ruder observations, but while successive improvements in the latter
department were overwhelming the successive attempts at improvement of
the former. In truth, the notion of chance, probability, likelihood, or
by whatever name it may be called, is as much of its own nature the
object of mathematical reasoning, as force or colour: it contains in
itself a distinct application of the notion of relative magnitude: it is
\textit{more} or \textit{less}, and the only difficulty (as in many
other cases) lies in the assignment of the test of quantity, \textit{how
much} more or less.
Worse understood than any of the applications of mathematics, a science
has been growing for a century and a half, which must end by playing
an even a more important part in the adjustment of social relations, than
astronomy in international communication. We make this assertion most
deliberately and most positively, to be controverted by some who are at
least as well able to judge as ourselves, to be looked upon with
derision by others, and with doubt by most educated men. If the public
mind has not yet been made to feel that the preceding prophecy is
actually in process of fulfilment, it is because one primary agent has
not yet been awakened to a sense of the importance of his share of the
work. The mathematician has done his part, and a more difficult task he
never had; the statesman is only just awakened to so much as a
disposition to accumulate \textit{some} of the data which are necessary.
We speak especially of England, and by the English statesman we now
begin to understand all the monied and educated part of the English
public. Among the liberties in which we pride ourselves, is that of
refusing to the executive all the information which is necessary to
\textit{know the country}, or at least a very considerable portion of
the statistics necessary for large legislation. And yet we expect
ministers to be accurately informed upon the bearings of every measure
they propose, at a period when it is demonstrable that the greater
portion of the community neither knows, nor has the means of being
told, within twelve per cent., what its stake in the national property
amounts to. This is a curious assertion, and we proceed to make it good.
All who hold life incomes, whether salaries of professional emoluments,
and all who expect reversions, have a tenure which depends for its value
upon two things---the average duration of life at \textit{every} age
(the mathematician will understand that we do not fall into the common
error), and the rate of interest which money will obtain. As to the
second, who will undertake to say what rate of interest actually
\textit{is} made, not by large companies, always ready with means of
investment, or by clever men of business, who live in the metropolis,
but by the average transactions of all who use money throughout the
country. Let us only suppose it to be a question of one per cent.: that
is, that it lies somewhere, say between 3 and 4 per cent.\ (if
$3\frac{1}{2}$ and $4\frac{1}{2}$ be taken, it will hardly affect the
final result). Now, with regard to the first point, all are agreed that
the Northampton Tables are below the general average at present
existing, and the Government Tables absolve it. The latter are so near
the Carlisle Tables, that, for our present rough purpose, the two can
hardly be distinguished. What stress we are to lay on the following
circumstances we hardly know, but if we take the results of a
neighbouring country, Belgium, where statistical enquiries are in a
state of rapid prosecution, we find the general average of the whole
country to be extremely near the mean between the Northampton and
Carlisle Tables. As follows:
\begin{center}
\begin{tabular}{cccc}
&\multicolumn{3}{c}{Mean duration of life in years} \\
Age. \\
& On the average of & M.\ Quetelet's \\
& the Carlisle and & Belgian Tables. & Difference \\
\phantom{0}0 & 31.95 & 32.15 & $+.20$ \\
\phantom{0}5 & 46.04 & 45.72 & $-.32$ \\
10 & 44.30 & 43.86 & $-.44$ \\
15 & 40.26 & 40.50 & $+.24$ \\
20 & 37.45 & 37.34 & $-.11$ \\
25 & 34.35 & 34.72 & $+.37$ \\
30 & 31.31 & 31.96 & $+.65$ \\
35 & 28.34 & 28.93 & $+.59$ \\
40 & 25.35 & 25.84 & $+.49$ \\
45 & 22.49 & 22.68 & $+.19$ \\
50 & 19.55 & 19.48 & $-.07$ \\
55 & 16.58 & 16.44 & $-.34$ \\
60 & 13.78 & 13.44 & $-.34$ \\
65 & 11.34 & 10.76 & $-.49$ \\
70 & \phantom{0}8.89 & \phantom{0}8.40 & $-.49$ \\
75 & \phantom{0}6.78 & \phantom{0}6.39 & $-.39$ \\
80 & \phantom{0}5.13 & \phantom{0}5.04 & $-.09$ \\
85 & \phantom{0}3.75 & \phantom{0}3.83 & $+.08$ \\
90 & \phantom{0}2.85 & \phantom{0}3.12 & $+.27$ \\
95 & \phantom{0}2.14 & \phantom{0}2.13 & $-.01$
\end{tabular}
\end{center}
This agreement is remarkably close, but it is useless. We have not the
means of forming an opinion as to whether life in Belgium much exceeds
of falls short of that of England. By a rough calculation for the age
of 40, made from M.\ Ansell's Table\footnote{This table refers entirely
to the labouring classes, members of Friendly Societies. It is from
work published by the Society for the Diffusion of Useful Knowledge.},
we find $24\frac{1}{2}$ for the mean duration of life at that age: but
we are not prepared to go any farther into the subject. Incidentally,
we may press upon those who are actually engaged in such matters, the
propriety of taking steps to ascertain what is the proper mean between
the Carlisle and Northampton tables which represents the grand average.
Resuming our subject, we conceive the great extremes of the question to
be represented by the Carlisle table at 3 per cent.: and the Northampton
table at 4 per cent. That is to say, 13 years and 17 years' purchase
are the limits of the remaining value of a life income in the hands of a
person aged 40, supposed to have just received a year's income. Taking
15 years as the mean, we think he must be a bold man who will undertake
to pronounce, for the whole country, where between 14 and 16 years the
truth would lie. Or, one man with another throughout England, the value
of existing interests cannot be pronounced upon within 12 per cent.
That the preceding will be denied from all quarters, only favours our
assertion. For some will lean to one table, some to another. We remain
uncontradicted, so long as authorities differ to the amount which we
have stated. The insurance offices, which deal in select lives, now
stand upon a proper basis of knowledge, the permanence of which,
however, rests upon their demanding what would be called, if means of
information were more extensive, enormous premiums. The reason why they
are not to be so styled, shows the consequences which result to the
country from insufficient statistics. Let us take the instance of the
Equitable Society, an insurance office which has accumulated enormous
wealth. How was this accumulation made? By demanding the ignorance of
those who came to insure. We deny the correctness of this view, but it
was an ignorance which was common both to the office and to its
customers; and known to the former, who were therefore obliged to make
such charges as would cover, not only the risk which their tables
showed with regard to the individual, but also the danger of attempting
insurance at all with such limited knowledge. Having demanded premiums
of which in the first instance, it was only known that they were
\textit{safe}, the result has been that they were \textit{much more than
safe:} the profit really belongs to those who now possess it, and it
has been bought and paid for. The consequence to the public is, that
the want of foresight in an existing government, whether blameable or
not we will not undertake to pronounce, has caused a large body of
subjects of the realm to make a provision for their families at the
expense of some millions sterling more than was necessary for the purpose.
The same indifference to statistical information on the part of the
government, or fear of the disinclination of the people to inform it,
still exists and produces its effects. We see it in every large
financial measure which is proposed to Parliament. The roughness of the
guesses on which such plans are built, is only exceeded by the boldness
of the mathematical steps by which the results are to be deduced. It is
hard to say where the wedge is to be introduced into the massy obstacles
which are to be cleft asunder. Shall we point to the good effects which
have resulted, and do result, form the application of sound principles
to actual measurements of facts? The inertness of the legislative power
never attempts to originate utility, unless acted on by pressure from
without. Shall we address ourselves to the individual inhabitant of the
country, and endeavour to show him that the power which governs his
interests is not knowledge? We shall find him so busily employed in
watching the intentions of his rulers, that he has no time to think
about their fitness to carry their intentions into useful effect. All
the countries of Europe, which are in a state of commercial progress,
are searching for information; while with us it is much that the
government should compile for the use of the legislature, just that
information which the collection of the revenue brings with itself, or
which the demand of individual members of the House of Commons has
caused to be furnished. The tables now produced by the Board of Trade,
useful as they are compared with any thing which had previously existed,
are but a poor provision for the growing wants of the community. We
hail them, nevertheless, as harbingers of better things.
Whenever a demand arises for the creation of some new method of meeting
the uncertainty of individual prospects, a process takes place which we
may briefly describe as follows. To meet the chance of fundamental
error, arising from ignorance of the subject, a large capital must be
subscribed as an insurance fund in case the whole speculation should
fail. The subscribers of this capital are of course indemnified for the
risk, by receiving a return in the success of the undertaking. Those
who receive this benefit must pay for it; and hence the ant of
statistical knowledge is an immediate and direct tax upon the people, in
favour of those who may be called the \textit{ignorance-insurers}. But
some extreme and enormous case of failure must be anticipated and
provided for. And through the principle of \textit{mutual} insurance,
so little understood, yet there remains the evil of requiring those who
would stipulate for a fixed sum, and have perhaps better means of
employing their money, to buy, not what they want, but a something
between that and half as much again, without any very definite means of
saying what it shall be.
Among the various projects of this kind, we observe one recently
established, which promises to be of great utility. It is a society for
providing fixed sums to be paid on account of each child of a marriage,
on his attaining a given age, in consideration of a life premium to be
paid by the father, to date from the time of the marriage. We see here,
firstly, the apparatus of an insuring capital; secondly, the
construction of tables. superintended, it is said, by a great
mathematician. Now this mathematician, be he great or small, could not
make bricks without straw, or tables without data. Doubtless he
enquired, what is the average age of marriage? What are the relative
numbers of such contracts made by parties at different ages? What is
the number of children produced by each, on an average, and what is the
average interval between their births? He need be no conjurer to see,
that all this and more, was necessary for his purpose; but we must
confess, we should think him one, if he found the answers to all these
questions. No doubt his province was to investigate the premiums which
should be paid on some supposition which the most cautious theoriser
would admit to be above the mark, and require the office to adopt them.
Both the office and the insurers may thus be made safe; but neither
party can undertake to say what the other buys and the other sells. The
nominal \pounds100 must be something between \pounds100 and \pounds150,
a part of the surplus being deducted in favour of the owners of the
subscribed capital.
Such is the state of our commercial relations in regard to the
employment of life interests for the creation of certainties. And yet
we see daily variations of such interests, to which even the courts of
law are continually obliged to appeal. The office of actuary has
received a legal character, though what constitutes an actuary is not
defined. Without the diploma of a college, or the initiation of an
apprenticeship, a class of professional arithmeticians has arisen, whose
verdicts are, in fact, as binding upon our courts as those of a jury
where they agree, without any distinct rule as to what is to be done
when they disagree. The statute relating Friendly Societies requires
that their rules should be certified by an ``actuary or person skilled
in calculation.'' Is the second necessarily in the first, or the first
necessarily the second? We do not at all quarrel with the legal
uncertainty, because the consequence is, that an actuary is in fact he
who is shown to be one, by proof that men will pay him money to have his
opinions. No class of men, taken as a whole, has acted with more
judgment in the multifarious and important questions which have been
submitted to them. They seem to have been fully aware, that in the
absence of perfect information, it was at least desirable to throw the
difficulties of the subject entire upon the data, and to make everything
sure from that point. Apply to one actuary for the value of a
contingent reversion, and he answers boldly, say \pounds2539.
14\textit{s}. $7\frac{3}{4}$\textit{d}. Apply to another and his answer
is as ready, say \pounds2092. 16\textit{s}. $\frac{1}{2}$\textit{d}.
Whence arises this difference between two mean, each of whom might
almost be supposed to contend for that last farthing? \textit{There is
a theory in dispute between them}, about which the public knows
nothing, and each avoids distressing the mercantile man by using round
numbers. For the latter, in common with the rest of the world, has got
a notion that the mathematical process must give exact results, whatever
the nature of the \textit{data} may be. But both, the trader and
actuary employ a course of proceeding which, as far as it goes, is one
of safety. The first is generally no mathematician, and the second very
often not more so than is absolutely required for his purpose. Now, to
know what to throw away without thereby rendering the result more
imperfect than the data, is the more difficult and delicate part of the
province of the mathematician. It is, therefore, most desirable that
such abbreviation should not be handled by any one who is not fully
competent, as well by experience in this as in other branches of
practical applications.
We have said that it may frequently happen, that two actuaries differ in
their results, by differing on a point of theory, and it may be useful
to the general reader, to know the general characteristics of this
difference. The table know by the name of \textit{Northampton Tables},
were published by Dr.\ Price in 1771, by means of registers kept in the
town of Northampton, from the year 1741. This table (with some
theoretical alterations, for the sake of introducing equality of
decrements,) is formed form 4,700 deaths at various ages, of which,
however, only 2,700 occurred above the age of 20. It was formed with
that degree of caution, in such matters, for which Dr.\ Price was
distinguished; and to which, we have no doubt whatever, the community is
indebted for this, that no insurance office has ever failed, nor so far
as we know, ever been generally believed to be close to failure. A
bolder theorist might very easily, and upon sufficiently plausible
grounds, have hazarded tables. which would have retarded this important
social improvement for fifty years at least. The Northampton Tables
were made the basis of the transactions of all the insurance offices;
and, considered as a whole, must be looked upon as a great commercial
benefit to the country. But it was soon supposed that they contained
defects which made them unfit to adjust the relative interests of
parties at different ages, and it was frequently affirmed, that while
the younger lives were represented as too low in value, the older lives
were made too high.
The \textit{Carlisle Tables} were published in 1815, by Mr.\ Milne, then
and now actuary to the Sun Life Assurance Society. They exhibit (with
theoretical alterations as before,) the results of 1,840 deaths, which
took place at Carlisle between 1779 and 1787: and 861 of these were
above the age of twenty. With reference, therefore, to numbers of
deaths, they are inferior in authority to the Northampton Tables, but
not so much as would be generally supposed. For it is a principle
\textit{perfectly} demonstrable, but not \textit{easily}, that when
chance selections are used for the purpose of constructing a probable
general law, the degree of confidence which is to be placed in the
superior numbers of one selection, does not increase with the numbers,
\textit{but with their square roots}. Thus, to construct a table which
should be \textit{twice} as good as another, \textit{ceteris paribus},
\textit{four} times as many deaths must be recorded; for \textit{thrice},
\textit{nine} times as many, and so on. Exclusive, therefore of every
circumstance except mere numbers, the goodness of the Carlisle and
Northampton tables is not (for above 20 years of age) as 861 to 2400,
but as 29 to 49, or thereabouts. In every circumstance except mere
numbers, Mr.\ Milne had the advantage of Dr.\ Price: and he used it with
an energy which deserved distinguished success, and, as it turned out,
obtained it. For there can now be no question, that the Carlisle
tables, represent the state of life among the better classes (in wealth)
of this country with an approach towards precision which is remarkable,
considering the scanty character of the materials.
Within the last few years, the two insurance offices which possessed the
largest amount of experience, the Amicable and the Equitable, have
published their results. The first of these dates back for more than a
century, the second for more than fifty years. The selection of its
lives, in the first, was, for a long time, anything but rigorous, as we
are informed: the latter has always been distinguished by more than
usual care in this respect. Taking the mean durations of life at
different ages, a test which have several reasons for preferring to the
one in more common use, we subjoin the following table:--
\begin{center}
\begin{tabular}{ccccc}
&\multicolumn{4}{c}{Mean Duration according to the} \\
Age \\
& Northampton & Carlisle & Amicable & Equitable \\ \hline
20 & 33.4 & 41.5 & 36.1 & 41.7 \\
30 & 28.3 & 31.3 & 31.1 & 34.5 \\
40 & 23.1 & 27.6 & 24.4 & 27.4 \\
50 & 18.0 & 21.1 & 17.9 & 20.4 \\
60 & 13.2 & 14.3 & 12.5 & 13.9 \\
70 & \phantom{0}8.6 &\phantom{0}9.2 & \phantom{0}7.8 & \phantom{0}8.7 \\
80 & \phantom{0}4.8 &\phantom{0}5.5 & \phantom{0}5.0 & \phantom{0}4.8 \\
\hline
\end{tabular}
\end{center}
\noindent
The Amicable Table contains 2800 deaths above the age of 20 and the
Equitable 5100. On looking at these tables, we see not only a
remarkable connexion between the Northampton and the Amicable, and
between the Carlisle and Equitable, but also some similarity between the
circumstances under which each pair was made. The Northampton table is
older than the Carlisle; the Amicable is on the whole older than the
Equitable. The town of Northampton is shown, by the documents of Dr.\
Price, to be less healthy than Carlisle, by those of Mr.\ Milne; the
election of the Amicable, on the whole term of its existence, was
believed, before their tables appeared, to be inferior to that of the
Equitable. And in both there is the same anomaly with regard to the
older lives; the difference between the Amicable and the Equitable,
which is very great at 20 years of age, is materially lessened as we
approach the older ages. But the particular point on which the
Northampton Tables were long suspected, appears even from the comparison
with its own companion; for whereas at 20 years of age, the Northampton
gives considerably less than the Amicable, at 60 years and upwards the
case is reversed. We do not speak of various other tables, as we only
wish to convey to the reader who is entirely new to the question, some
slight notion of the state in which we stand with respect to the results
of tables.
Now the question among actuaries is this: which are the tables to be
actually used in the computation of money results, those of long or
short life, the Carlisle of the Northampton. There are great
authorities, so far as authorities go, on both sides of the question: and
we apprehend that some would use one table in one set of circumstances,
and another in another. Discretion must decide; but in the meanwhile it
is of importance that the public in general, and the courts of law in
particular, should distinctly know, that the actuary does not merely
deduce a result of pure arithmetic: for he has not only to use the
tables, but to settle which of the conflicting tables he shall use. And
this alone is frequently a question of two or three years' purchase in
the value of contingency. It has happened more than once, that
litigation has been rendered more complicated, by the opposing values
producing very different opinions upon the estimated value of life
interests. On what principles the judges settle the matter in such a
case, we are not aware; but it most unquestionably belongs to them to
enquire \textit{what tables have been used, and why?} For the question,
whether a given individual shall be considered a good or a bad life, is
one which admits of being determined by the evidence, and it would be
much better that the court, acting upon information, should decide
whether one of the other table should be used, or whether any and what
mean between them should be taken, than permit such a matter to be
settled by the actuaries consulted,---the point in dispute having
considerable authorities on both sides. It is also to be remembered,
that even the professional men consulted are not always in possession of
the information necessary to decide: a case may begin, ``\textit{A
person}, aged fifty,'' \&c.\ without the least information as to what
the class and habits of this person may be; and parties, interested in
the result, may wilfully put such a case, with \textit{algebraical}
description only, for the purpose of taking into court such an opinion
as may suit their purpose. We are convinced, that, in the process of
time, and as the eyes of the public become open to the very extensive
character of \textit{life interests} in this country, an officer will be
appointed, a new species of \textit{Master in Chancery}, whose duty it
will be to decide those points which are now settled by reading the
opinions given upon \textit{cases before caused by parties}.
Among all the confusion which unfortunately exists in the ramifications
of an extensive branch of the subject we are considering, there seems
to us but one point which is very clear; namely, that though such
progress has been made as secures safety as to those who are interested
\textit{en masse}, the equitable apportionment of the relative claims of
the different parts of the whole, is by no means in the same state of forwardness.
The state of probability in general, as applied to the preceding
questions, may be divided into two parts; of which the knowledge of the
first is easily attainable, in comparison with that of the second. The
latter of the two is the guide of the formed, and often the method of
checking too hasty conclusions drawn from it. The mathematical analysis
of the former is easy, while that of the latter is almost as complicated
as the planetary theory, perhaps even more so length for length. We need
hardly add, that we refer to those extensions of the subject which
were first struck out by De Moivre, and which have been raised to a high
degree of development by La Place. Of all the masterpieces of analysis,
this is perhaps the best known; if does not address its powers to the
consideration of a vast and prominent subject, such as astronomy or
optics, but confines itself to a branch of enquiry of which the first
principles are so easily mastered (in appearance), that the student who
attempts the higher parts feels almost deprived of this rights when he
begins to encounter the steepness of the subsequent ascent. The
\textit{Th\'eorie des Probabilit\'es} is the Mont Blanc of mathematical
analysis; but the mountain has this advantage over the book, that there
are guides always ready near the former, whereas the student has been
left to his own method of encountering the latter.
The genius of Laplace was a perfect sledge hammer in bursting purely
mathematical obstacles; but, like that useful instrument, it gave
neither finish nor beauty to the results. In truth, in truism if the
reader please, Laplace was neither Lagrange nor Euler, as every student
is made to feel. The second is power and symmetry, the third power and
simplicity. But, nevertheless, Laplace never attempted the
investigation of a subject without leaving upon it the marks of
difficulties conquered; sometimes clumsily, sometimes indirectly, always
without minuteness of design or arrangement of detail; but still his end
is obtained, and the difficulty is conquered. There are several
circumstances connected with the writings of this great mathematician,
which indicate vices peculiar to himself, and others which are common to
his countrymen in general, we shall begin with one of the latter.
The first duty of a mathematical investigator, in the manner of sating
his results, is the most distinct recognition of the rights of others;
and this is a duty which he owes as much to himself as to others. He
owes it to himself, because the value of every work diminishes with
time, so far as it is a statement of principles or development of
methods; others will in time present all such information in a shape
better suited to the habits of a succeeding age. But the
\textit{historical} value of a work never diminished, but rather
increases, with time; theory may be overthrown, processes may be
simplified, but historical information remains, and becomes an authority
which renders it necessary to preserve and refer to any work in which
it exists. No one now thinks of consulting the work of the erudite
Longomontanus; while that of his contemporary Riccoli is esteemed and
sought after. The reason is, that the first contains little or nothing
of history, while the second is full of it. That such attention to the
rights of others, is due to those others, need hardly be here insisted
on. Now, what we assert is, that there runs throughout most of the
writings of the French national school, a thorough and culpable
indifference to the necessity of clearly stating how much has been done
by the writer himself, and how much by his predecessors. We do not by
any means charge them with nationality; on the contrary, they are most
impartially unfair both to their own countrymen and to foreigners; we
may even say, that, to a certain extent, they behave properly to the
latter, while of each other they are almost uniformly neglectful.
Laplace himself set the most striking example of this disingenuous
practice. For instance, Lagrange, proceeding on a route suggested by a
theorem of Lambert, discovered the celebrated method of expansion, which
all foreigners call \textit{Lagrange's theorem}. Other and subordinate
methods (in generality, only, not in utility) had been given by Taylor
and Maclaurin, and are sufficiently well known by their names. Now,
Laplace has occasion to demonstrate these theorems in the
\textit{M\'echanique C\'eleste}, and how does he proceed? ``Nous
donnerons sur la r\'eduction des fonctions en series, quelques
th\'eor\`emes g\'en\'eraux qui nous seront utilis\'e dans la suite.''
(Book II, No.\ 20.) Would not any one imagine that these were some
theorems which Laplace was producing for the first time? In the sequel,
the theorem which is know to the mere beginner, as Taylor's Theorem, is
described as ``la formule (i) du num\'ero 21.'' Let us even grant that
it is natural to refer back throughout any one work to any fixed part of
it, and we have not done with this strange determination not to mention
the writings of any other mathematician. For in \textit{Th\'eorie des
Probabilit\'es}, a work totally unconnected with the one just mentioned.
Lagrange's theorem has no other designation than ``la formule (p) du
num\'ero 21 du second Livre de la \textit{M\'echanique C\'eleste}.''
And the exception only of professedly historical summaries upon points
which have for the most part no connexion with his own researches, a
studied suppression of the names of his predecessors and contemporaries,
insomuch that had he had occasion to cite a proposition of Euclid, we
have little doubt that it would have appeared as ``le th\'eoreme que
j'ai demonstr\'e dans un tel num\'ero.'' The consequence is, that the
student of the \textit{M\'echanique C\'eleste} begins by forming an
estimate of the author, which is too high, even for Laplace; and ends by
discovering that the author has frequently, even where he appears most
original, been only using the materials, and working nupon the track, of
Lagrange, or some other. If the reaction be greater than it should be,
and if the estimate formed of Laplace should be lower than it really
ought to be, it would be no more than a proper lesson for living
analysts of the same country, who, as we could easily show, if we were
concerned with their writings, have closely copied the not very
creditable example of Laplace.
The preceding remarks have a particular bearing upon the
\textit{Th\'eorie des Probabilit\'es}, for it is in this work that the
author has furnished the most decided proof of grand originality and
power. It is not that the preceding fault is avoided; for to whatever
extent De Moivre, Euler, or any other, had furnished either isolated
results, of hints as to method of proceeding, to precisely the same
extent have their names been suppressed. Nevertheless, since less had
been done to master the difficulties of this subject than in the case of
the theory of gravitation, it is here that Laplace most shines as a
creator of resources. It is not for us to say that, failing such
predecessors as he had (Newton only excepted), he would not by his won
genius have opened a route for himself. Certainly, if the power of any
one man would have sufficed for the purpose, that man might have been
Laplace. As it is, we can only, looking at the \textit{Th\'eorie des
Probabilit\'es}, in which he is most \textit{himself}, congratulate the
student upon the fact of more symmetrical heads having preceded him in
his \textit{M\'echanique C\'eleste}. Sharing, as does the latter work,
in the defects of the former, what would its five volumes have presented
if Laplace had no forerunner?
It might appear to be our intention to decry the work which we have
placed at the head of this article. We cannot but demur to such a
charge, because to \textit{decry} is, we presume, to try to alter the
tone of a cry already existing. Now, even meaning by the world the
mathematical world, there is not a sufficient proportion of that little
public which has read the work in question, to raid any such collective
sound as a cry either on one side or the other. The subject of the work
is, in its highest parts, comparatively isolated and detached, though
admitted to be of great importance in the sciences of observation. The
pure theorist has no immediate occasion for the results, as results, and
therefore contents himself in many instances with a glance at the
processes, sufficient for admiration, though hardly so for use. The
practical observer and experimenter obtains a knowledge of results and
nothing more, will knowing in most cases, that the analysis is above his
reach. We could number upon the finders of one hand, all the men we
know \textit{in Europe} who have \textit{used} the results in their
\textit{published} writings in a manner which makes it clear that they
could both \textit{use} and \textit{demonstrate}.
In pointing out, therefore, the defects of the work in question---in
detaching them form the subject and laying them upon the author---taking
care at the same time to distinguish between the high praise which is
due to the originality and invention of the latter, and the expression
of regret that he should, like Newton, have retarded the progress of his
most original views by faults of style and manner---we conceive that we
are doing good service, not only to the subject itself, but even to the
fame of its investigator. If, at the same time, we can render it
somewhat more accessible to the student, and help to create a larger
class of readers, we are forwarding the creation of the opinion that the
results of this theory, in its more abstruse parts, may and should be
made both practical and useful, even in the restricted and commercial
sense of the former term. Such must be the impression of all who have
examined the evidence for this theory.
It is not our intention to conclude the subject in the present number:
the length of this article (for such articles should not be very long)
warns us to conclude for the present by finishing our account of the
difficulties which have been placed in the way of the student previously
entering upon the consideration of the subject matter of the treatise.
The \textit{Th\'eorie des Probabilit\'es} consists of three great
divisions. 1. An introductory essay, explanatory of general principles
and results, without any appearance of mathematical symbols. 2. A
purely mathematical introduction, developing the analytical methods
which are finally to be employed. 3. The application of the second
part to the details of the solution of questions connected with
probabilities. The first of these has also been published in a separate
form, under the title of \textit{Essai Philosophique}, \&c., and is
comparatively well known. Our business here is mostly with the second
and third. The arrangement will seem simple and natural, but there is a
secret which does not appear immediately, and refers to a point which
distinguishes this and several other works from most of the same
magnitude. The work is not an independent treatment of the subject, but
a collection of memoirs taken \textit{verbatim} from those which the
author had previously inserted in the Transactions of the Academy of
Sciences. Thus in the volume for 1782, appears a paper on the valuation
of functions of very high numbers, with an historical and explanatory
introduction. Now this introduction being omitted, the rest of the
memoir is, substantially, and the most part word for word, inserted in
the work we are now describing. And the same may be said of other
memoirs published at a later period: so that the \textit{Th\'eorie des
Probabilit\'es}, first published in 1812, may be considered as a
collection of the various papers which had appeared in the Transactions
cited from 1778 up to 1812.
This materially alters the view which must be taken of the treatise,
considered as intended for the mathematical student. It also makes a
change in the idea which must be formed of the real difficulty of the
subject, as distinguished from that which is actually found in reading
Laplace. The course taken has both its advantages and disadvantages: on
which it may be worth while to say a few words.
Of the highest and most vigorous class of mathematical students, it may
be easily guessed that they are most benefitted by the works which are
least intended for them. Complete digestion and arrangement, so far
from being essential to aid them in the formation power, are rather
injurious. The best writer is he who shows most clearly by his process
where the difficulty lies,. and who meets it in the most direct manner.
All the artifice by which the road is smoothed and levelled, all the
contrivance by which difficulty is actually overcome without perception
of its existence, though a desirable study for the proficient, and most
useful with reference to the application of science, is a loss of
advantageous prospect to the student who wishes to become an original
investigator. An officer who has never seen any but well-drilled
soldiers, may \textit{command} an army of them; but he who would
\textit{raise} an army must have been used to the machine he wishes to
create in every stage of its process of creation, from a disorderly
assembly of clowns up to a completely organised force. It is on such a
ground as this that we take our stand, when we say that Euler, from the
almost infinite simplicity with which he presents the most difficult
subjects, and Lagrange, from the unattainable combination of power and
generality which he uses \textit{for} (more than \textit{through}) the student,
are not the best guides for one who would practise investigation. It is
Laplace whose writings we should recommend for this purpose, for those
very reasons which induce us to point him out as one of the most rough
and clumsy of mathematical writers. A student is more likely
\textit{pro ingenio suo}, to be able to imitate Laplace by reading
Laplace, than Lagrange by Lagrange, or Euler by Euler.
In the next place, of all the works which any one has produced, the most
effective for the formation of original power are those which lie nearest
to his own source of invention. All the difference between analysis and
synthesis will exist, for the most part, between the memoir in which the
discoverer first opened his views, and the ultimate method which he
considered as most favourable for their deduction from his first
principles. Hence we should recommend to the student to leave the
elementary works and the arranged treatises as soon as possible, and
betake himself to the original memoirs. He will find them not only
absolutely more clear than compilations from them, but what is of much
more importance, they state with distinctness what has been done on each
particular point, and what is attempted to be done. If there should
arise confusion from the student not perceiving that he is employed upon
an isolated part of the whole which is not yet complete, there are
safeguards in the \textit{Memoir} which do not exist in the
\textit{Treatise}. Take any work on the differential calculus, from the
time of Leibnitz downwards, and the formality of chapters, distinctions
of subjects, and treatment of nothing but what is complete, or appears
so, will leave the impression that the whole is exhausted, and that all
apparent difficulty arises from the student not being able to see all
that is presented to him. Now the fact is that in many cases the
obstacle is of another kind, namely, that the reader is not made aware
that there is more to be looked for than is presented. The assertion,
\textit{je n'en sais rien}, by which Lagrange frequently astonished
those who imagined that a grand mathematician knew every thing, is
frequently embodied in the spirit, or enspirited into the body, of a
memoir, but seldom into that of a formal treatise. It happened to us
not long ago to be very much puzzled with the account of a process given
in the great work of Lacroix, one of the best of methodical writers.
Chance threw in our way the original memoir of Legendre, from which the
process was taken, and we found that, word for word nearly, the former
writer agreed with the latter, so far as he went. But a few sentences
of omission in which the original writer had limited himself were, it
should seem, inconsistent with the vastness of the general design
indicated in the excellent compiler's chapter. The difficulty vanished
at once, since it merely arose form venturing to hint to ourselves, in
the way of doubt, precisely what the original writer had proposed as a
limitation.
So far then, as the great work before us preserves the actual contents
of the original memoirs, it must be looked upon as very wholesome
exercise for the student. But there are still some defects, arising
from not completing the plan. The short historical notice and general
explanation is omitted, in consequence, we suppose, of the humiliation
which the writer of a treatise would feel, were he compelled to name
another man. The extravagance of an original memoir lights the candle
at both ends: not merely is an author permitted to say clearly where he
ends, but also where he began. Did Stirling give a result which might
have afforded a hint as to the direction in which more was to be looked
for? Laplace may and does confess it in the Transactions of the
Academy. But the economy of a finished work will not permit such
freedoms; and while on the one hand the student has no direct reason
for supposing that there ever \textit{will} be any body but Laplace,
he has, on the other, no means of knowing that there ever \textit{was}
any body but Laplace.
In the next place, the difficulty of the subject is materially increased
by the practice of placing general descriptions at the beginning,
instead of the end. Our present work begins with a tremendous account
of the theory of generating functions, which we doubt not has deterred
many a reader, who has imagined that it was necessary to master this
first part of the work before there was an old memoir ready to reprint
from. And where in the subsequent part of the work is it used? In some
isolated problems connected with gambling, which in the first place
might be omitted without rendering the material part of the work more
difficult; and in the second place there are applications of the theory
of generating functions of so simple a character, that the preliminaries
connected with it might be discussed in two pages. And in what future
part of the work do the very tedious (though skilful) methods of
development become useful which are formally treated in the introductory
chapter? Nowhere.
Here the reader may begin to suspect that the difficulty of this work
does not lie entirely in the subject, but is to be attributed in great
part to the author's method. That such difficulty is in part wholesome,
may be very true; but it is also discouraging. Believing as we do that
in spite of all we have said, the \textit{Th\'eorie des Probabilit\'es}
is one of the points to which the attention of the future analyst
should be directed, as the subject is any way within his power, we shall
here finish what we have to say on the character of the work, and
proceed in a future article with that of its results.
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\noindent\hangindent=1em
ART.\ IV.---\textit{Th\'eorie Analytique des Probabilit\'es}. Par M.\ le
Marquis de Laplace, \&c.\ \&c. 3\`eme edition. Paris 1820.
\medskip
\noindent
IN continuing our remarks upon the work of which the title is now before
the reader's eye, we may remind him that we have not room to enter at
length upon the subject. We have already discussed considerations of a
practical character, tending to shew that upon several questions, in
which recourse is actually had to to the theory of probabilities,
insufficiency of information produces effects prejudicial to the
pecuniary interests of those concerned. This is indeed a strong point:
we might urge in any plan or prospective utility upon the English
public, till we were tired, and without awakening the least attention.
Nor would there be any reason to complain of such a result; for the
present is an age of suggestions, and every person who can read and
write has some scheme in hand, by which the community is to be
advantage: no wonder, then, that so few of the speculations in question
have more than one investigator. But when we speak of the theory of
probabilities, we bring forward a something upon which, right or wrong,
many tens of millions of pounds sterling depend. The insurance
offices, the friendly societies, all annuitants, and all who hold life
interests of any species---again, all who insure their goods from fire,
or their ships from wreck---are visibly and immediately interested in
the dissemination of correct principles upon probability in general. So
much for that which actually is invested: now with regard to that which
might be, let it be remembered, that whenever money is hazarded in
commerce or manufactures, by those who would resign the possibility of
more than average profit, if they might thereby be secured from the risk
of disastrous loss, the desired arrangement is rendered impossible, by
want of knowledge how to apply the theory of probabilities, combined
with the defect of methodized information upon the contingencies in
question.
The name of the \textit{theory of probabilities} is odious in the eyes
of many, for, as all the world knows, it is the new phrase for the
computation of chances, the instrument of gamblers, and, for a long
time, of gamblers only; meaning, by that word, not the people who play
with stocks and markets, but with cards, dice, and horses. Such an
impression was the inevitable consequence of the course pursued by
earlier writers on the subject, who filled their books entirely with
problems related to games of chance. This was not so much a consequence
of the nature of the subject, as of the state of mathematical knowledge
at the time: games of chance, involving a given and comparatively small
number of cases, are of easy calculation, and require only the
application of simple methods; while questions of natural philosophy, or
concerning the common affairs of life, involve very large numbers of
cases, and require a more powerful analysis. Consequently, the older
works abound with questions upon games of chance, while later writings
begin to display the power of applying the very same principles to wider
as well as more useful inquiries.
This objection to the tendency of the theory of probability, or the
doctrine of chances, is as old as the time of De Moivre; who was not,
however, able to meet it, by extending the subject matter of his
celebrated treatise. In the second edition, published in 1738. he
writes thus, in his dedication to a Lord Carpenter: ``There are many
people in the world who are preposessed with an opinion, that the
doctrine of chances has a tendency to promote play; but they soon will
be undeceived, if they think fit to look into the general design of this
book. In the mean while, it will not be improper to inform them, that
your lordship is pleased to espouse the patronage of this second
edition,'' \&c.\ \&c. The general design of De Moivre's work appears to
be, the analysis of every game of chance which prevailed in his time;
and the author seems to have imagined that he could not attract
attention to any other species of problem.
In reviewing the \textit{general design} of the work of Laplace, we
desire to make the description of a book mark the present state of a
science. In any other point of view, it would be superfluous to give an
account of a standard treatise, which is actually in the hands of a
larger number of persons than are able to read it.
In considering the simple questions of chances, we place ourselves, at
the outset, in hypothetical possession of a set of circumstances, and
attribute to ourselves exact and rigorous knowledge. We assume that we
positively know every case that can arrive, and also that we can
estimate the relative probabilities of the several causes. This of
itself has a tendency to mislead the beginner, because the known
circumstances are generally expressed by means of some simple gambling
hypothesis. A set of balls which have been drawn, 83 white and 4 black,
places ourselves in the same position with regard to our disposition to
expect black or white for the future, as that in which we should stand
if we had observed 83 successful and 43 unsuccessful speculations in a
matter of business: it matters nothing as to the amount of chances for
the future, whether the observed even be called the drawing of a white
ball, or the acquirement of a profit. Nevertheless, the abstraction of
the idea of probability from the circumstances under which it is
presented, sometimes throws a difficulty in the way.
The science of probability has also this in common with others, that the
problems which most naturally present themselves are of an inverse
character, as compared with those which an elementary and deductive
course first enables the student to solve. If we know that out of 1000
infants born, 900 live a year, it is sufficiently easy to understand why
we say it is nine to one any specified individual of them will live a
year. But seeing that we can only arrive at such knowledge by
observation, and also that such observation must be limited, there
arises this very obvious preliminary question---Having registered a
certain thousand infants, and found that, \textit{of that thousand},
nine hundred were alive at the end of a year, what presumption would
arise from thence that something like the same proportion would obtain
if a second thousand were registered? For instance, would it be wise to
lay an even bet that the the results of the second trial would exhibit
something between 850 and 950, in place of 900? Or, to generalize the
form of the question, let us imagine a thousand balls to have been drawn
from a lottery containing an infinite number; of which it is found that
there are 721 white, 116 red, and 163 black. We may then ask, what
degree of presumption ought to be considered as established---1. That
the contents of the lottery all white, red, and black, and not other
colour? 2. That the white and red balls are distributed throughout the
whole mass, nearly in the proportion of 721 white to 116 red? This is a
question which must present itself previously to the definition of any
inference upon the probable results of future drawings: but at the same
time, it is not of the most and direct easy class, requiring, in fact,
the previous discussion of many methods which are subsequent in order of
application.
It is common to assume that any considerable number of observations will
give a result nearly coinciding with the average of the whole. The
constructors of the Northampton and Carlisle tables (see the last
number, p.344) did not think it necessary to ask whether 2,400 and 861
cases of mortality would themselves furnish a near approximation to the
law which actually prevails in England. It had long been admitted, or
supposed, that a considerable number of deaths (no definite number being
specified) would present a table of mortality. such as might be depended
on for pecuniary transactions. It is true that such is the case; but
the proposition is one requiring that sort of examination and
demonstration which Laplace has given. We shall not stop to rebut any
conclusion which might be drawn against the utility of the theory, from
the circumstances of common sense having felt for and attained some of
its most elaborate results: but we \textit{shall} stop to remark, that
in the case of a speculation, so very delicate, so very liable to be
misunderstood, and, above all, accessible to so small a part of the
educated world, it is a great advantage that there exist such
landmarks, as propositions which, though distant results of theory, yet
coincide with notions of the world at large, and are supposed to have
evidence of their own.
When we have learnt that the result of analysis agrees with general
opinion, in admitting the safety of relying upon a comparatively small
number of cases to determine a general average, we then become disposed
to rely on the same analysis for correctly determining the probable
limits of accidental fluctuation.
The two-fold object of the theory is, then, firstly, to determine the
mean, or average state of things; secondly, to ascertain what degree of
fluctuation may be reasonably expected. Let it be remarked, that the
common theory of chances applies itself almost entirely to the
first-mentioned problem; when we say that we determined the probability
of an event to be two-sevenths, we mean, that, taking every possible
case in which the said event can happen, we shall find it happen twice
out of seven times. Such is the general average; but, supposing that
we select 700 possible cases out of the whole, it does not therefore
become probable, or more likely than not, that the event shall happen
precisely 200 times, and fail precisely 500 times. All that becomes
very likely is, that the number of arrivals shall be nearly 200, and of
non-arrivals nearly 500; and if is one of the most important objects of
the theory, to ascertain within what limits there is a given amount of
probability that the departure from the general average shall be
contained.
The question thus enunciated is of no small practical importance, and to
the neglect of it we must attribute the supposed necessity for the large
capitals with which many undertakings are commenced. (See last Number,
p.342.) Let us imagine an insurance office to be founded, and, for the
sake of simplicity, let it take no life except at the age of 30. Let
the materials for its management consist in the examination of a register
of 1,000 lives, which have been found to drop in the manner pointed out,
say by the Carlisle table. The premium which should be demanded is then
easily ascertained; but its security depends upon two circumstances---1.
That the 1,000 lives so recorded, shall represent the general
mortality. 2. That the amount of business obtained by the office,
shall be so large as to render their actual experience another
representation of the same general average. Neither of these
conditions can be precisely attained; some small allowance must be
made for both; and the question is, what amount of additional premium
is necessary to cover the risk of fluctuation?---what number of insured
lives will be sufficient to begin with?---or, supposing that all risks
are to be taken, what is the smallest capital upon which a commencement
can prudently be made, without any security for a large amount of business?
Perhaps we could not in fewer words convey an idea of the different
states of the science in the times of De Moivre and Laplace, than by
stating, that the former could have ascertained the requisite premium,
and the latter could have made the necessary additions for the
fluctuation, \&c.
We now pass from matters of business,---as to which we can only say what
might be done,---to questions connected with the sciences of observation
and inquiry which involves the actual use of our physical senses, the
repetition of a process will always afford a series of discordances,
varying in amount with the method used, the skill of the observed, and
the nature of the observation. If the observed discordances present
anything like uniformity of character, we are naturally led to conclude,
that they are not, properly speaking, the results of errors of
observation, but of some unknown law, by which the predicted or expected
result than might have been expected, and sometimes smaller. Now,
having noticed a set of observations which do not agree, it is one of
the first objects of the theory to settle what presumption should exist
that the variations are accidental (that is, totally unregulated by
apparent or discoverable law), or that they follow a law which then
becomes the object on investigation. The case taken by Laplace, as an
illustration, will do for the same purpose here. It was suspected that,
independently of local fluctuations, the barometer was always a little
higher in the morning than in the afternoon. To settle this point, four
hundred days were chosen, in which the barometer was remarkably steady,
not varying four millimetres in any one day. This was done to avoid the
large fluctuations, which would have rendered the changes in question,
if such there were, imperceptible. It was found that, the sum of the
heights at four in the afternoon, by four hundred millimetres, or, one
day with another, by a millimetre a day. But what can we infer from
such a circumstance, is the first suggestion? A millimetre, or about
one twenty-fifth part of an inch, is so very small a variation, that
considering the nature of observation, and the imperfections of the
instrument, that mere instrumental error might have occasioned such a
discrepancy. The theory of probabilities gives an entirely different
notion: it appears that it is many millions to one against such a
phenomenon presenting itself, upon the supposition that it was produced
by nothing but the casual imperfections of the instrument. A very great
probability was therefore given to the supposition, that there really
exists a diurnal variation of the barometer, in virtue of which,
\textit{ceteris paribus}, it is a little higher at one particular part
of the day than at another.
In this way, Laplace actually used the theory of probabilities as a
method of discovery. He expressly affirms (p.355), that the
irregularity of the lunar motion, which he afterwards showed to depend
on the figure of the earth, was pointed out to him as not being of a
merely casual character, by having ``soumis son existence au calcul des
probabilit\'es.'' Of another of his most brilliant results, he says as
distinctly (p.356), ``L'Analyse des probabilit\'es m'a conduit
pareillement a la cause des grands irregularit\'es de Jupiter et de
Saturne.'' There is much in these assertions which will appear not a
little singular, even to those versed in the subject. But there are
two circumstances which afford presumption, not only of the good faith
of Laplace, but of his freedom from a mistaken bias for a favourite
subject. In the first place, it somewhat lowers the opinion which the
world at large entertains of a philosopher, when he is found using
means, instead of penetrating mysteries by pure thought. The Newton of
the world at large sat down under a tree, saw an apple fall, and after
an immense reveries, the length of which is not stated, got up, with the
theory of gravitation will planned, if not fit to print. It is painful
to be obliged to add, that the Newton of Trinity College Cambridge, of
whom there is no manner of doubt that he was the hero of the preceding
myth, not only was to a large extent indebted to the perusal of what his
predecessors had written, but went through years of deduction and
comparison,---abandoned his theory, on account of its non-agreement with
some existing observations,---took it up again upon trial when new sets
of observations had been made,---and, in point of fact, went through a
detail which was a great deal more like a book-keeping operation, than
the poetical process of the fable. Partial as Laplace might be wedded
to it, as to wish it should appear that he had used a method, instead of
unassisted sagacity. The fault of discoverers generally lies in the
opposite extreme; they conceal the simple suggestions which led them on
the road, and by presenting a finished and elaborate results, as well as
establish them. One of the most difficult and original inquiries in
which he engaged, was the question of \textit{tides in the atmosphere},
answering to those in the ocean, and produced by the same causes. That
such tides must exist, to some degree or other, cannot be questioned by
any one who admits the theory of gravitation: the point was to ascertain
whether corresponding appearances could be detected to any
\textit{sensible} extent. Laplace investigated the deduction of the law
in a brilliant manner,---and carefully examined barometrical
observations, which of course exhibited a mixed amount of error and
actually prevailing law. But upon submitting the result to the test of
the theory of probabilities, there was not found to be strong
presumption that any part of the diurnal variation arose from such a law
as was shewn by theory to be a consequence of the luni-solar action: and
the theory, beautiful as it is, was honestly abandoned. We assume then,
that Laplace did not deceive himself, when he attributed a part of this
success in the explanation of the phenomena, to his use of the theory of
probabilities; and we pass to another division of the subject.
All observations are liable to error; if we were to take, for instance,
all the altitudes which had ever been measured by a given theodolite and
a given observer,---and if we could ascertain what the correct truth was
in each instance, we should find many observations wrong by
half-a-minute or less; but much fewer in number wrong by something more
than half a minute. The \textit{law of facility} of error, is a term we
use to express the chance of an error under a given amount; to speak
mathematically, let $\phi x$ express the chance, that the error of a
single observation is not so great as $x$ is called the law of facility.
Nothing can be more obvious than that the law of facility may vary with
the phenomenon to be observed, the general character of the observer,
his state of mind for the time, \&c.\ \&c.
At the same time, there is one conclusion on which all the scientific
world was agreed, on every subject, for every instrument, \&c.; namely,
that when a number of observations disagreed with each other, the way of
determining their most probable result, was to take the \textit{average}
of all the observations. But it must be obviously proper to ask, can
this method be true, whatever might have been the qualities of the
observer, the instrument, \&c.? Is it likely that the same rule for
deducing the probable truth would apply to the bungler and the practised
observer, the near and the far-sighted,---to Hipparchus without a
telescope, missing whole degrees and Bradley, with his zenith sector,
missing seconds? There never was perhaps a case, in which the
applications of strict investigation was more likely to play havoc with
the prevailing opinion of preceding ages. Such was not, however, the
case; and we have here a striking instance of the manner in which
existing notions have been confirmed by the march of science.
The theory of probabilities draws a remarkable distinction between
observations which have been made, and those which are to be made.
Suppose it required of an experimenter, that he should choose his method
of treating results previously to obtaining them, and then, whatever his
tendency to err may be, provided only that he is not more likely to
measure too much than too little,---or in technical language, that
positive and negative errors are equally likely,---the method of
averaging is the best which he can take. But let him be allowed to
defer his choice of a process until the observations are finished, and
the process of averaging is not then the best which can be chosen,
unless it can be shown that one particular law of facility, pointed out
by the theory, is the one to which he is really subject. Some little
account of the reason of this paradox may be easily given. The
probability of any event is not a quality of the event itself, but an
impression of the mind, depending upon our state of knowledge with
regard to the causes of the event. If A feel certain that an urn
contains nothing but white balls, and B that half its contents are
black, the two are really in different circumstances, and the
probability of a drawing being white is not the same to both. Now
\textit{before} the observations are made, there is no presumption to
guide the observer in suspecting any law of facility; but afterwards,
the observations themselves furnish an imperfect knowledge of the law of
facility. For instance, this much at least will be seen, that if the
results of observation be near to each other, the tendency to error is
small, and if they differ very much, the same tendency is considerable.
Now since it is always competent to the observer to choose his method of
proceeding when he pleases; it follows, that the common notion cannot be
strictly applicable to the results of any case.
But at the same time it appeared, singularly enough, that whatever the
law of facility may be, the more numerous the observations, the more
nearly does their average present the most probable result. And more
than this, the approximation implied in the preceding sentence takes
place so rapidly, that a moderate number of observations is sufficient
to allow of its application. There is another consideration, which
cannot be explained to any but the mathematician; namely, that the law
of facility, under which the average is strictly the most probable
result, contains an arbitrary constant, by means of which a particular
case of it may be made a sufficient approximation to any law of facility
which can be believed to exist. Practically then, the method of
averaging, as universally used, has that tendency to promote
correctness, as compared with other methods, which it has always been
thought to have.
As it is rather our object to shew the bearings of the science on the
notions of mankind, than to make a digest of results, we shall here take
notice of another theorem, in which propositions, generally admitted,
but apparently wholly unconnected, are shewn to be dependent, so that
one of them cannot be true without the other. It has not been noticed
by Laplace, but has been deduced by ourselves form the principles
employed by him and others.
Firstly,---the value of any sum of money is always considered as
dependent upon the whole of which it forms a part. A guinea is
\textit{nothing} to a rich man, but \textit{a great deal} to a poor man
and, on the same principle, no trader contemplates the gain or loss of a
given sum, otherwise than with reference to the whole capital which is
invested to produce it. Among the various ways in which a part may be
compared with the hole, the simple proportion, per centage, or whatever
it may be called, is that which is universally adopted; we shall say,
then, that the value of any piece of money is to be measured by its
proportion to the whole sum of which it is considered to be a part.
Secondly,---the effect of life assurance is considered, in a point of
view imported by its name: it is not called the insurance of a certain
sum of money \textit{at death}, but the insurance \textit{of life}. It
is then taken as placing every person who avails himself of it, in the
position of being sure to live a certain time. But, if we consider that
those who live long must pay more than they receive, in order that those
who die before their time may receive more than they pay, it is clear,
that life insurance amounts to an equalization of life, or the assigning
to each person the average share of life. Thus the effect of
guaranteeing sums of money at death, for premiums properly calculated,
is equivalent to insuring the average term of life.
These two propositions, both, to all appearances, highly reasonable to
themselves, are not visibly connected with each other: either might be
true, it should seem, without the other. But this is not the fact; for
it can be shown, that if either of them be false, the other falls with
it. If, for instance, a person should affirm, that a guinea to a man
who is insured for a hundred, is to be considered as precisely the same
thing as the same sum is to another person insured for a thousand, then
it can be proved that he contradicts himself, if he imagines that the
effect of life insurance is equivalent to the equalization of life in
all persons who begin at the same age. There is great analogy between
the dependence just explained, and that which prevails between the
method of averaging, and the existence of one particular law of
facility; and many common notions, examined by the test of the theory of
probability, will either confute or confirm each other.
The crowning proposition in the application of the theory to natural
philosophy, is undoubtedly that known as the \textit{method of least
squares}, to which astronomy, in particular, lies under very great
obligations. In fact, we may safely say, that the time must have
arrived, when, but for this aid, additional observation would have
ceased to carry additional accuracy into our knowledge of the celestial
motions. It will somewhat diminish the effect of the technical term
``method of least squares,'' if we state, that the method of averaging is
a particular case of it, so that a farmer, who calculates his probable
crop by taking an average bushel from several soils, proceeds by the
method of least squares, as much as an astronomer, who uses it to
determine the elements of a comet's orbit. We remember having heard the
following problem proposed, which is an ingenious illustration of the
cases to which the method applies. A large target is erected, with a
small chalk mark, (not necessarily in the middle) and a number of
persons, all of whom are tolerably certain of hitting the target, and
all of whom are equally likely to miss the chalk in any direction from
it, fire in succession, say with sharp-pointed arrows. The chalk is
then rubbed out, and the target, with all the arrows sticking in it, is
presented to a mathematician, who is required to say what point, judging
from the position of the arrows, is the one which was fired at. His
investigation will lead them to the following result; he must ascertain
that point in the target, from which, if lines were drawn to all the
points of the arrows, the sum of the squares of those lines would be the
least possible. From the answer to such questions always requiring the
sum of certain squares to be made the least possible, the method derives
its name. It is not of course asserted, that the process described
would infallibly discover the place where the chalk mark existed; but if
the same person were to try the method upon a hundred such targets,
losing at the rate of a given sum for every inch by which he was wrong,
he would certainly lose less by acting in the manner described than by
any other process.
Singularly enough, it was not as a result of the theory of
probabilities, but as a convenient and easily practicable process, that
the method of least squares first appeared. Legendre and Gauss,
independently of each other (though the former first published it) saw
the utility of such an addition to astronomical computation. It is to
Laplace that we owe its introduction as the best theoretical mode of
ascertaining the \textit{most probable} result of discordant
observations. His investigation wants clearness and elegance; but is in
other respects one of his most brilliant labours. The beauty,
generality, and simplicity of the result secured for it an immediate
admission into every process, though the demonstration is of a kind
which there are not many to understand; the process is one which has the
air of being highly probable, and seems in itself to be free from
objections which might be proposed against any other method. But at the
same time it appears to us, that many have used it without a thorough
comprehension of its meaning; and just as we now say that astronomy must have
stopped its career of increasing accuracy, if the method of least
squares had not been introduced, so we will venture to hope that the
time must come when the same remark shall be made upon an improved and
extended way of using it.
The difficulty of admitting several points connected with the theory of
probabilities arises from the neglect to make an important distinction;
namely, between the correctness or incorrectness of the hypothesis
assumed, and that of the inferences which are drawn from it. Let it be
proposed to apply mathematical reasoning to the valuation of the
credibility of evidence, and the answer appears to be simple---namely,
that such a proposition must be the result of an overheated imagination.
That would be a fair answer if it were required to apply calculation to
the character and actions of a given man, with a view of ascertaining
whether he was likely or not to tell the truth in a particular case.
Mathematics will not tell us whether A and B are credible witnesses, not
whether, supposing them credible, their evidence will be as much as
should in prudence be considered sufficient for the establishment of any
particular point. Nor will mathematics enable us to measure a length in
feet, or to reason upon it, unless we first know by other than
mathematical means, what is that length which it is agreed to call a
foot. But let a foot be known, and we can then assign lines, areas, and
solids, by means of numbers; and, in like manner, let the credibility of
one witness be given, and we can then determine that which results from
the evidence of any number, contradicted by any other number. By the
credibility of a witness, we are supposed to mean the probability that
an assertion advanced by him will be correct, the moment before the
assertion is made.
For instance, suppose it admitted that a jury of twelve men, all equally
likely to be correct in any particular verdict, decide wrongly once out
of fifty times. It is a matter of pure algebra to find out how often
each of them, using his own unassisted judgement, would come to
erroneous decisions. It is also the province of algebra to determine
how often a jury would err, if, upon the preceding hypothesis as to the
correctness of twelve men, the number were reduced or increased.
Laplace, and others before him, have made extensive applications of
analysis to such questions; but their labours in this respect have been
misunderstood, and always must be, until the province of mathematical
reasoning is better understood by the world at large.
We have now, we believe, briefly touched upon the principle subjects
which are to be found in the Th\'eorie des Probabilit\'es. The subject
is one which must make its way slowly, having to extricate itself from
its old connexion with games of chance, before it can take its proper
place as an agent in statistical and political enquiry. One of our
principal objects in writing the present articles has been to show that
the nature of probability may be treated, and its results applied,
without mention of dice or cards. Laplace himself has introduced a few
problems connected with common gambling, in some instances on account of
their historical notoriety, in others because they afforded easy and
striking examples of the applications of generating functions, the
theory of which was introduced in his work. But the greater part of the
treatise is full of such questions as those which have been alluded to
in the preceding pages, bearing in the most direct manner on the way to
draw correct inferences from physical and statistical facts.
If we can make a few reflecting individuals understand, that, be the
theory of probabilities true or false, valuable or useless, its merits
must be settled by reference to something more than the consideration of
a few games at cards, we shall have done all which we ventured to
propose to ourselves.
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\textit{Dublin Review} \textbf{2} (1837), 338--354 \& \textbf{3}
(1838), 237--248.
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