_HISTORICAL NOTICE CONCERNING THE CALCULUS OF PROBABILITIES._
Long ago were determined, in the simplest games, the ratios of the
chances which are favorable or unfavorable to the players; the stakes
and the bets were regulated according to these ratios. But no one before
Pascal and Fermat had given the principles and the methods for
submitting this subject to calculus, and no one had solved the rather
complicated questions of this kind. It is, then, to these two great
geometricians that we must refer the first elements of the science of
probabilities, the discovery of which can be ranked among the remarkable
things which have rendered illustrious the seventeenth century—the
century which has done the greatest honor to the human mind. The
principal problem which they solved by different methods, consists, as
we have seen, in distributing equitably the stake among the players, who
are supposed to be equally skilful and who agree to stop the game before
it is finished, the condition of play being that, in order to win the
game, one must gain a given number of points different for each of the
players. It is clear that the distribution should be made proportionally
to the respective probabilities of the players of winning this game, the
probabilities depending upon the numbers of points which are still
lacking. The method of Pascal is very ingenious, and is at bottom only
the equation of partial differences of this problem applied in
determining the successive probabilities of the players, by going from
the smallest numbers to the following ones. This method is limited to
the case of two players; that of Fermat, based upon combinations,
applies to any number of players. Pascal believed at first that it was,
like his own, restricted to two players; this brought about between them
a discussion, at the conclusion of which Pascal recognized the
generality of the method of Fermat.
Huygens united the divers problems which had already been solved and
added new ones in a little treatise, the first that has appeared on this
subject and which has the title _De Ratiociniis in ludo aleæ_. Several
geometricians have occupied themselves with the subject since: Hudde,
the great pensionary, Witt in Holland, and Halley in England, applied
calculus to the probabilities of human life, and Halley published in
this field the first table of mortality. About the same time Jacques
Bernoulli proposed to geometricians various problems of probability, of
which he afterwards gave solutions. Finally he composed his beautiful
work entitled _Ars conjectandi_, which appeared seven years after his
death, which occurred in 1706. The science of probabilities is more
profoundly investigated in this work than in that of Huygens. The author
gives a general theory of combinations and series, and applies it to
several difficult questions concerning hazards. This work is still
remarkable on account of the justice and the cleverness of view, the
employment of the formula of the binomial in this kind of questions, and
by the demonstration of this theorem, namely, that in multiplying
indefinitely the observations and the experiences, the ratio of the
events of different natures approaches that of their respective
probabilities in the limits whose interval becomes more and more narrow
in proportion as they are multiplied, and become less than any
assignable quantity. This theorem is very useful for obtaining by
observations the laws and the causes of phenomena. Bernoulli attaches,
with reason, a great importance to his demonstration, upon which he has
said to have meditated for twenty years.
In the interval, from the death of Jacques Bernoulli to the publication
of his work, Montmort and Moivre produced two treatises upon the
calculus of probabilities. That of Montmort has the title _Essai sur les
Jeux de hasard_; it contains numerous applications of this calculus to
various games. The author has added in the second edition some letters
in which Nicolas Bernoulli gives the ingenious solutions of several
difficult problems. The treatise of Moivre, later than that of Montmort,
appeared at first in the _Transactions philosophiques_ of the year 1711.
Then the author published it separately, and he has improved it
successively in three editions. This work is principally based upon the
formula of the binomial and the problems which it contains have, like
their solutions, a grand generality. But its distinguishing feature is
the theory of recurrent series and their use in this subject. This
theory is the integration of linear equations of finite differences with
constant coefficients, which Moivre made in a very happy manner.
In his work, Moivre has taken up again the theory of Jacques Bernoulli
in regard to the probability of results determined by a great number of
observations. He does not content himself with showing, as Bernoulli
does, that the ratio of the events which ought to occur approaches
without ceasing that of their respective probabilities; but he gives
besides an elegant and simple expression of the probability that the
difference of these two ratios is contained within the given limits. For
this purpose he determines the ratio of the greatest term of the
development of a very high power of the binomial to the sum of all its
terms, and the hyperbolic logarithm of the excess of this term above the
terms adjacent to it.
The greatest term being then the product of a considerable number of
factors, his numerical calculus becomes impracticable. In order to
obtain it by a convergent approximation, Moivre makes use of a theorem
of Stirling in regard to the mean term of the binomial raised to a high
power, a remarkable theorem, especially in this, that it introduces the
square root of the ratio of the circumference to the radius in an
expression which seemingly ought to be irrelevant to this transcendent.
Moreover, Moivre was greatly struck by this result, which Stirling had
deduced from the expression of the circumference in infinite products;
Wallis had arrived at this expression by a singular analysis which
contains the germ of the very curious and useful theory of definite
integrals.
Many scholars, among whom one ought to name Deparcieux, Kersseboom,
Wargentin, Dupré de Saint-Maure, Simpson, Sussmilch, Messène, Moheau,
Price, Bailey, and Duvillard, have collected a great amount of precise
data in regard to population, births, marriages, and mortality. They
have given formulæ and tables relative to life annuities, tontines,
assurances, etc. But in this short notice I can only indicate these
useful works in order to adhere to original ideas. Of this number
special mention is due to the mathematical and moral hopes and to the
ingenious principle which Daniel Bernoulli has given for submitting the
latter to analysis. Such is again the happy application which he has
made of the calculus of probabilities to inoculation. One ought
especially to include, in the number of these original ideas, direct
consideration of the possibility of events drawn from events observed.
Jacques Bernoulli and Moivre supposed these possibilities known, and
they sought the probability that the result of future experiences will
more and more nearly represent them. Bayes, in the _Transactions
philosophiques_ of the year 1763, sought directly the probability that
the possibilities indicated by past experiences are comprised within
given limits; and he has arrived at this in a refined and very ingenious
manner, although a little perplexing. This subject is connected with the
theory of the probability of causes and future events, concluded from
events observed. Some years later I expounded the principles of this
theory with a remark as to the influence of the inequalities which may
exist among the chances which are supposed to be equal. Although it is
not known which of the simple events these inequalities favor,
nevertheless this ignorance itself often increases the probability of
compound events.
In generalizing analysis and the problems concerning probabilities, I
was led to the calculus of partial finite differences, which Lagrange
has since treated by a very simple method, elegant applications of which
he has used in this kind of problems. The theory of generative functions
which I published about the same time includes these subjects among
those it embraces, and is adapted of itself and with the greatest
generality to the most difficult questions of probability. It determines
again, by very convergent approximations, the values of the functions
composed of a great number of terms and factors; and in showing that the
square root of the ratio of the circumference to the radius enters most
frequently into these values, it shows that an infinity of other
transcendents may be introduced.
Testimonies, votes, and the decisions of electoral and deliberative
assemblies, and the judgments of tribunals, have been submitted likewise
to the calculus of probabilities. So many passions, divers interests,
and circumstances complicate the questions relative to the subjects,
that they are almost always insoluble. But the solution of very simple
problems which have a great analogy with them, may often shed upon
difficult and important questions great light, which the surety of
calculus renders always preferable to the most specious reasonings.
One of the most interesting applications of the calculus of
probabilities concerns the mean values which must be chosen among the
results of observations. Many geometricians have studied the subject,
and Lagrange has published in the _Mémoires de Turin_ a beautiful method
for determining these mean values when the law of the errors of the
observations is known. I have given for the same purpose a method based
upon a singular contrivance which may be employed with advantage in
other questions of analysis; and this, by permitting indefinite
extension in the whole course of a long calculation of the functions
which ought to be limited by the nature of the problem, indicates the
modifications which each term of the final result ought to receive by
virtue of these limitations. It has already been seen that each
observation furnishes an equation of condition of the first degree,
which may always be disposed of in such a manner that all its terms be
in the first member, the second being zero. The use of these equations
is one of the principal causes of the great precision of our
astronomical tables, because an immense number of excellent observations
has thus been made to concur in determining their elements. When there
is only one element to be determined Côtes prescribed that the equations
of condition should be prepared in such a manner that the coefficient of
the unknown element be positive in each of them; and that all these
equations should be added in order to form a final equation, whence is
derived the value of this element. The rule of Côtes was followed by all
calculators, but since he failed to determine several elements, there
was no fixed rule for combining the equations of condition in such a
manner as to obtain the necessary final equations; but one chose for
each element the observations most suitable to determine it. It was in
order to obviate these gropings that Legendre and Gauss concluded to add
the squares of the first members of the equations of condition, and to
render the sum a minimum, by varying each unknown element; by this means
is obtained directly as many final equations as there are elements. But
do the values determined by these equations merit the preference over
all those which may be obtained by other means? This question, the
calculus of probabilities alone was able to answer. I applied it, then,
to this subject, and obtained by a delicate analysis a rule which
includes the preceding method, and which adds to the advantage of
giving, by a regular process, the desired elements that of obtaining
them with the greatest show of evidence from the totality of
observations, and of determining the values which leave only the
smallest possible errors to be feared.
However, we have only an imperfect knowledge of the results obtained, as
long as the law of the errors of which they are susceptible is unknown;
we must be able to assign the probability that these errors are
contained within given limits, which amounts to determining that which I
have called the _weight_ of a result. Analysis leads to general and
simple formulæ for this purpose. I have applied this analysis to the
results of geodetic observations. The general problem consists in
determining the probabilities that the values of one or of several
linear functions of the errors of a very great number of observations
are contained within any limits.
The law of the possibility of the errors of observations introduces into
the expressions of these probabilities a constant, whose value seems to
require the knowledge of this law, which is almost always unknown.
Happily this constant can be determined from the observations.
In the investigation of astronomical elements it is given by the sum of
the squares of the differences between each observation and the
calculated one. The errors equally probable being proportional to the
square root of this sum, one can, by the comparison of these squares,
appreciate the relative exactitude of the different tables of the same
star. In geodetic operations these squares are replaced by the squares
of the errors of the sums observed of the three angles of each triangle.
The comparison of the squares of these errors will enable us to judge of
the relative precision of the instruments with which the angles have
been measured. By this comparison is seen the advantage of the repeating
circle over the instruments which it has replaced in geodesy.
There often exists in the observations many sources of errors: thus the
positions of the stars being determined by means of the meridian
telescope and of the circle, both susceptible of errors whose law of
probability ought not to be supposed the same, the elements that are
deduced from these positions are affected by these errors. The equations
of condition, which are made to obtain these elements, contain the
errors of each instrument and they have various coefficients. The most
advantageous system of factors by which these equations ought to be
multiplied respectively, in order to obtain, by the union of the
products, as many final equations as there are elements to be
determined, is no longer that of the coefficients of the elements in
each equation of condition. The analysis which I have used leads easily,
whatever the number of the sources of error may be, to the system of
factors which gives the most advantageous results, or those in which the
same error is less probable than in any other system. The same analysis
determines the laws of probability of the errors of these results. These
formulæ contain as many unknown constants as there are sources of error,
and they depend upon the laws of probability of these errors. It has
been seen that, in the case of a single source, this constant can be
determined by forming the sum of the squares of the residuals of each
equation of condition, when the values found for these elements have
been substituted. A similar process generally gives values of these
constants, whatever their number may be, which completes the application
of the calculus of probabilities to the results of observations.
I ought to make here an important remark. The small uncertainty that the
observations, when they are not numerous, leave in regard to the values
of the constants of which I have just spoken, renders a little uncertain
the probabilities determined by analysis. But it almost always suffices
to know if the probability, that the errors of the results obtained are
comprised within narrow limits, approaches closely to unity; and when it
is not, it suffices to know up to what point the observations should be
multiplied, in order to obtain a probability such that no reasonable
doubt remains in regard to the correctness of the results. The analytic
formulæ of probabilities satisfy perfectly this requirement; and in this
connection they may be viewed as the necessary complement of the
sciences, based upon a totality of observations susceptible of error.
They are likewise indispensable in solving a great number of problems in
the natural and moral sciences. The regular causes of phenomena are most
frequently either unknown, or too complicated to be submitted to
calculus; again, their action is often disturbed by accidental and
irregular causes; but its impression always remains in the events
produced by all these causes, and it leads to modifications which only a
long series of observations can determine. The analysis of probabilities
develops these modifications; it assigns the probability of their causes
and it indicates the means of continually increasing this probability.
Thus in the midst of the irregular causes which disturb the atmosphere,
the periodic changes of solar heat, from day to night, and from winter
to summer, produce in the pressure of this great fluid mass and in the
corresponding height of the barometer, the diurnal and annual
oscillations; and numerous barometric observations have revealed the
former with a probability at least equal to that of the facts which we
regard as certain. Thus it is again that the series of historical events
shows us the constant action of the great principles of ethics in the
midst of the passions and the various interests which disturb societies
in every way. It is remarkable that a science, which commenced with the
consideration of games of chance, should be elevated to the rank of the
most important subjects of human knowledge.
I have collected all these methods in my _Théorie analytique des
Probabilités_, in which I have proposed to expound in the most general
manner the principles and the analysis of the calculus of probabilities,
likewise the solutions of the most interesting and most difficult
problems which calculus presents.
It is seen in this essay that the theory of probabilities is at bottom
only common sense reduced to calculus; it makes us appreciate with
exactitude that which exact minds feel by a sort of instinct without
being able ofttimes to give a reason for it. It leaves no arbitrariness
in the choice of opinions and sides to be taken; and by its use can
always be determined the most advantageous choice. Thereby it
supplements most happily the ignorance and the weakness of the human
mind. If we consider the analytical methods to which this theory has
given birth; the truth of the principles which serve as a basis; the
fine and delicate logic which their employment in the solution of
problems requires; the establishments of public utility which rest upon
it; the extension which it has received and which it can still receive
by its application to the most important questions of natural philosophy
and the moral science; if we consider again that, even in the things
which cannot be submitted to calculus, it gives the surest hints which
can guide us in our judgments, and that it teaches us to avoid the
illusions which ofttimes confuse us, then we shall see that there is no
science more worthy of our meditations, and that no more useful one
could be incorporated in the system of public instruction.