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Title: A Philosophical Essay on Probabilities

Author: marquis de Pierre Simon Laplace

Translator: Frederick Lincoln Emory
Frederick Wilson Truscott

Release date: February 13, 2019 [eBook #58881]
Most recently updated: May 3, 2019

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK A
PHILOSOPHICAL ESSAY ON PROBABILITIES ***

Transcriber's Note:

Apparent typographical errors
have been corrected.
The corrections noted in the Errata
section have been incorporated in
the text. Three further corrections
have also been made: 9/10 in place
of 1/10, and 9/10 in place of 6/10,
on page 110; and "ex voto" in
place of "ex veto" on page 173.

A PHILOSOPHICAL ESSAY
ON

PROBABILITIES.
BY

PIERRE SIMON, Marquis de LAPLACE.

TRANSLATED FROM THE SIXTH FRENCH EDITION
BY

FREDERICK WILSON TRUSCOTT, Ph.D. (Harv.),
Professor of Germanic Languages in the West Virginia University,

AND

FREDERICK LINCOLN EMORY, M.E. (Wor. Poly. Inst.),
Professor of Mechanics and Applied Mathematics in the West Virginia
University; Mem. Amer. Soc. Mech. Eng.

FIRST EDITION.
FIRST THOUSAND.

NEW YORK:
JOHN WILEY & SONS.
London: CHAPMAN & HALL, Limited.
1902.

Copyright, 1902,
BY
F. W. TRUSCOTT
AND
F. L. EMORY.
ROBERT DRUMMOND PRINTER, NEW YORK

TABLE OF CONTENTS.

PAGE

PART I.
A PHILOSOPHICAL ESSAY ON PROBABILITIES.

CHAPTER I.
Introduction 1

CHAPTER II.
Concerning Probability 3

CHAPTER III.
General Principles of the Calculus of Probabilities 11

CHAPTER IV.
Concerning Hope 20

CHAPTER V.
Analytical Methods of the Calculus of Probabilities 26

PART II.
APPLICATION OF THE CALCULUS OF
PROBABILITIES.

CHAPTER VI.
Games of Chance 53

CHAPTER VII.
Concerning the Unknown Inequalities which may
Exist among Chances Supposed to be Equal 56

CHAPTER VIII.
Concerning the Laws of Probability which result
from the Indefinite Multiplication of Events 60

CHAPTER IX.
Application of the Calculus of Probabilities to
Natural Philosophy 73

CHAPTER X.
Application of the Calculus of Probabilities to the
Moral Sciences 107

CHAPTER XI.
Concerning the Probability of Testimonies 109

CHAPTER XII.
Concerning the Selections and Decisions of
Assemblies 126

CHAPTER XIII.
Concerning the Probability of Testimonies 132

CHAPTER XIV.
Concerning Tables of Mortality, and the Mean
Durations of Life, Marriage, and Some
Associations 140

CHAPTER XV.
Concerning the Benefits of Institutions which
Depend upon the Probability of Events 149

CHAPTER XVI.
Concerning Illusions in the Estimation of
Probabilities 160

CHAPTER XVII.
Concerning the Various Means of Approaching
Certainty 176

CHAPTER XVIII.
Historical Notice of the Calculus of Probabilities to
1816 185

ERRATA.

Page 89, line 22, for Pline read Pliny
" 102, lines 14, 16, " minutes " days
" 143, line 25, " sun " soil
lines 15, 17, 18, 21, 22, 24, for
" 177,
primary read prime
" 182, line 5, for conjunctions read being binary

A PHILOSOPHICAL ESSAY ON
PROBABILITIES.

CHAPTER I.
INTRODUCTION.
This philosophical essay is the development of a lecture on probabilities
which I delivered in 1795 to the normal schools whither I had been called,
by a decree of the national convention, as professor of mathematics with
Lagrange. I have recently published upon the same subject a work entitled
The Analytical Theory of Probabilities. I present here without the aid of
analysis the principles and general results of this theory, applying them to
the most important questions of life, which are indeed for the most part only
problems of probability. Strictly speaking it may even be said that nearly all
our knowledge is problematical; and in the small number of things which
we are able to know with certainty, even in the mathematical sciences
themselves, the principal means for ascertaining truth—induction and
analogy—are based on probabilities; so that the entire system of human
knowledge is connected with the theory set forth in this essay. Doubtless it
will be seen here with interest that in considering, even in the eternal
principles of reason, justice, and humanity, only the favorable chances
which are constantly attached to them, there is a great advantage in
following these principles and serious inconvenience in departing from
them: their chances, like those favorable to lotteries, always end by
prevailing in the midst of the vacillations of hazard. I hope that the

reflections given in this essay may merit the attention of philosophers and
direct it to a subject so worthy of engaging their minds.

CHAPTER II.
CONCERNING PROBABILITY.
All events, even those which on account of their insignificance do not
seem to follow the great laws of nature, are a result of it just as necessarily
as the revolutions of the sun. In ignorance of the ties which unite such
events to the entire system of the universe, they have been made to depend
upon final causes or upon hazard, according as they occur and are repeated
with regularity, or appear without regard to order; but these imaginary
causes have gradually receded with the widening bounds of knowledge and
disappear entirely before sound philosophy, which sees in them only the
expression of our ignorance of the true causes.
Present events are connected with preceding ones by a tie based upon the
evident principle that a thing cannot occur without a cause which produces
it. This axiom, known by the name of the principle of sufficient reason,
extends even to actions which are considered indifferent; the freest will is
unable without a determinative motive to give them birth; if we assume two
positions with exactly similar circumstances and find that the will is active
in the one and inactive in the other, we say that its choice is an effect
without a cause. It is then, says Leibnitz, the blind chance of the
Epicureans. The contrary opinion is an illusion of the mind, which, losing
sight of the evasive reasons of the choice of the will in indifferent things,
believes that choice is determined of itself and without motives.
We ought then to regard the present state of the universe as the effect of
its anterior state and as the cause of the one which is to follow. Given for
one instant an intelligence which could comprehend all the forces by which
nature is animated and the respective situation of the beings who compose it
—an intelligence sufficiently vast to submit these data to analysis—it would
embrace in the same formula the movements of the greatest bodies of the
universe and those of the lightest atom; for it, nothing would be uncertain
and the future, as the past, would be present to its eyes. The human mind

offers, in the perfection which it has been able to give to astronomy, a
feeble idea of this intelligence. Its discoveries in mechanics and geometry,
added to that of universal gravity, have enabled it to comprehend in the
same analytical expressions the past and future states of the system of the
world. Applying the same method to some other objects of its knowledge, it
has succeeded in referring to general laws observed phenomena and in
foreseeing those which given circumstances ought to produce. All these
efforts in the search for truth tend to lead it back continually to the vast
intelligence which we have just mentioned, but from which it will always
remain infinitely removed. This tendency, peculiar to the human race, is that
which renders it superior to animals; and their progress in this respect
distinguishes nations and ages and constitutes their true glory.
Let us recall that formerly, and at no remote epoch, an unusual rain or an
extreme drought, a comet having in train a very long tail, the eclipses, the
aurora borealis, and in general all the unusual phenomena were regarded as
so many signs of celestial wrath. Heaven was invoked in order to avert their
baneful influence. No one prayed to have the planets and the sun arrested in
their courses: observation had soon made apparent the futility of such
prayers. But as these phenomena, occurring and disappearing at long
intervals, seemed to oppose the order of nature, it was supposed that
Heaven, irritated by the crimes of the earth, had created them to announce
its vengeance. Thus the long tail of the comet of 1456 spread terror through
Europe, already thrown into consternation by the rapid successes of the
Turks, who had just overthrown the Lower Empire. This star after four
revolutions has excited among us a very different interest. The knowledge
of the laws of the system of the world acquired in the interval had
dissipated the fears begotten by the ignorance of the true relationship of
man to the universe; and Halley, having recognized the identity of this
comet with those of the years 1531, 1607, and 1682, announced its next
return for the end of the year 1758 or the beginning of the year 1759. The
learned world awaited with impatience this return which was to confirm one
of the greatest discoveries that have been made in the sciences, and fulfil
the prediction of Seneca when he said, in speaking of the revolutions of
those stars which fall from an enormous height: "The day will come when,
by study pursued through several ages, the things now concealed will

appear with evidence; and posterity will be astonished that truths so clear
had escaped us." Clairaut then undertook to submit to analysis the
perturbations which the comet had experienced by the action of the two
great planets, Jupiter and Saturn; after immense calculations he fixed its
next passage at the perihelion toward the beginning of April, 1759, which
was actually verified by observation. The regularity which astronomy
shows us in the movements of the comets doubtless exists also in all
phenomena.
The curve described by a simple molecule of air or vapor is regulated in a
manner just as certain as the planetary orbits; the only difference between
them is that which comes from our ignorance.
Probability is relative, in part to this ignorance, in part to our knowledge.
We know that of three or a greater number of events a single one ought to
occur; but nothing induces us to believe that one of them will occur rather
than the others. In this state of indecision it is impossible for us to announce
their occurrence with certainty. It is, however, probable that one of these
events, chosen at will, will not occur because we see several cases equally
possible which exclude its occurrence, while only a single one favors it.
The theory of chance consists in reducing all the events of the same kind
to a certain number of cases equally possible, that is to say, to such as we
may be equally undecided about in regard to their existence, and in
determining the number of cases favorable to the event whose probability is
sought. The ratio of this number to that of all the cases possible is the
measure of this probability, which is thus simply a fraction whose
numerator is the number of favorable cases and whose denominator is the
number of all the cases possible.
The preceding notion of probability supposes that, in increasing in the
same ratio the number of favorable cases and that of all the cases possible,
the probability remains the same. In order to convince ourselves let us take
two urns, A and B, the first containing four white and two black balls, and
the second containing only two white balls and one black one. We may
imagine the two black balls of the first urn attached by a thread which
breaks at the moment when one of them is seized in order to be drawn out,
and the four white balls thus forming two similar systems. All the chances

which will favor the seizure of one of the balls of the black system will lead
to a black ball. If we conceive now that the threads which unite the balls do
not break at all, it is clear that the number of possible chances will not
change any more than that of the chances favorable to the extraction of the
black balls; but two balls will be drawn from the urn at the same time; the
probability of drawing a black ball from the urn A will then be the same as
at first. But then we have obviously the case of urn B with the single
difference that the three balls of this last urn would be replaced by three
systems of two balls invariably connected.
When all the cases are favorable to an event the probability changes to
certainty and its expression becomes equal to unity. Upon this condition,
certainty and probability are comparable, although there may be an essential
difference between the two states of the mind when a truth is rigorously
demonstrated to it, or when it still perceives a small source of error.
In things which are only probable the difference of the data, which each
man has in regard to them, is one of the principal causes of the diversity of
opinions which prevail in regard to the same objects. Let us suppose, for
example, that we have three urns, A, B, C, one of which contains only black
balls while the two others contain only white balls; a ball is to be drawn
from the urn C and the probability is demanded that this ball will be black.
If we do not know which of the three urns contains black balls only, so that
there is no reason to believe that it is C rather than B or A, these three
hypotheses will appear equally possible, and since a black ball can be
drawn only in the first hypothesis, the probability of drawing it is equal to
one third. If it is known that the urn A contains white balls only, the
indecision then extends only to the urns B and C, and the probability that
the ball drawn from the urn C will be black is one half. Finally this
probability changes to certainty if we are assured that the urns A and B
contain white balls only.
It is thus that an incident related to a numerous assembly finds various
degrees of credence, according to the extent of knowledge of the auditors. If
the man who reports it is fully convinced of it and if, by his position and
character, he inspires great confidence, his statement, however
extraordinary it may be, will have for the auditors who lack information the

same degree of probability as an ordinary statement made by the same man,
and they will have entire faith in it. But if some one of them knows that the
same incident is rejected by other equally trustworthy men, he will be in
doubt and the incident will be discredited by the enlightened auditors, who
will reject it whether it be in regard to facts well averred or the immutable
laws of nature.
It is to the influence of the opinion of those whom the multitude judges
best informed and to whom it has been accustomed to give its confidence in
regard to the most important matters of life that the propagation of those
errors is due which in times of ignorance have covered the face of the earth.
Magic and astrology offer us two great examples. These errors inculcated in
infancy, adopted without examination, and having for a basis only universal
credence, have maintained themselves during a very long time; but at last
the progress of science has destroyed them in the minds of enlightened men,
whose opinion consequently has caused them to disappear even among the
common people, through the power of imitation and habit which had so
generally spread them abroad. This power, the richest resource of the moral
world, establishes and conserves in a whole nation ideas entirely contrary to
those which it upholds elsewhere with the same authority. What indulgence
ought we not then to have for opinions different from ours, when this
difference often depends only upon the various points of view where
circumstances have placed us! Let us enlighten those whom we judge
insufficiently instructed; but first let us examine critically our own opinions
and weigh with impartiality their respective probabilities.
The difference of opinions depends, however, upon the manner in which
the influence of known data is determined. The theory of probabilities holds
to considerations so delicate that it is not surprising that with the same data
two persons arrive at different results, especially in very complicated
questions. Let us examine now the general principles of this theory.

CHAPTER III.
THE GENERAL PRINCIPLES OF THE CALCULUS
OF PROBABILITIES.
First Principle.—The first of these principles is the definition itself of
probability, which, as has been seen, is the ratio of the number of favorable
cases to that of all the cases possible.
Second Principle.—But that supposes the various cases equally possible.
If they are not so, we will determine first their respective possibilities,
whose exact appreciation is one of the most delicate points of the theory of
chance. Then the probability will be the sum of the possibilities of each
favorable case. Let us illustrate this principle by an example.
Let us suppose that we throw into the air a large and very thin coin whose
two large opposite faces, which we will call heads and tails, are perfectly
similar. Let us find the probability of throwing heads at least one time in
two throws. It is clear that four equally possible cases may arise, namely,
heads at the first and at the second throw; heads at the first throw and tails
at the second; tails at the first throw and heads at the second; finally, tails at
both throws. The first three cases are favorable to the event whose
probability is sought; consequently this probability is equal to ¾; so that it
is a bet of three to one that heads will be thrown at least once in two throws.
We can count at this game only three different cases, namely, heads at the
first throw, which dispenses with throwing a second time; tails at the first
throw and heads at the second; finally, tails at the first and at the second
throw. This would reduce the probability to ⅔ if we should consider with
d'Alembert these three cases as equally possible. But it is apparent that the
probability of throwing heads at the first throw is ½, while that of the two
other cases is ¼, the first case being a simple event which corresponds to
two events combined: heads at the first and at the second throw, and heads

at the first throw, tails at the second. If we then, conforming to the second
principle, add the possibility ½ of heads at the first throw to the possibility
¼ of tails at the first throw and heads at the second, we shall have ¾ for the
probability sought, which agrees with what is found in the supposition
when we play the two throws. This supposition does not change at all the
chance of that one who bets on this event; it simply serves to reduce the
various cases to the cases equally possible.
Third Principle.—One of the most important points of the theory of
probabilities and that which lends the most to illusions is the manner in
which these probabilities increase or diminish by their mutual combination.
If the events are independent of one another, the probability of their
combined existence is the product of their respective probabilities. Thus the
probability of throwing one ace with a single die is ⅙; that of throwing two
aces in throwing two dice at the same time is 1⁄36. Each face of the one being
able to combine with the six faces of the other, there are in fact thirty-six
equally possible cases, among which one single case gives two aces.
Generally the probability that a simple event in the same circumstances will
occur consecutively a given number of times is equal to the probability of
this simple event raised to the power indicated by this number. Having thus
the successive powers of a fraction less than unity diminishing without
ceasing, an event which depends upon a series of very great probabilities
may become extremely improbable. Suppose then an incident be
transmitted to us by twenty witnesses in such manner that the first has
transmitted it to the second, the second to the third, and so on. Suppose
again that the probability of each testimony be equal to the fraction 9⁄10; that
of the incident resulting from the testimonies will be less than ⅛. We cannot
better compare this diminution of the probability than with the extinction of
the light of objects by the interposition of several pieces of glass. A
relatively small number of pieces suffices to take away the view of an
object that a single piece allows us to perceive in a distinct manner. The
historians do not appear to have paid sufficient attention to this degradation
of the probability of events when seen across a great number of successive
generations; many historical events reputed as certain would be at least
doubtful if they were submitted to this test.

In the purely mathematical sciences the most distant consequences
participate in the certainty of the principle from which they are derived. In
the applications of analysis to physics the results have all the certainty of
facts or experiences. But in the moral sciences, where each inference is
deduced from that which precedes it only in a probable manner, however
probable these deductions may be, the chance of error increases with their
number and ultimately surpasses the chance of truth in the consequences
very remote from the principle. Fourth Principle.—When two events
depend upon each other, the probability of the compound event is the
product of the probability of the first event and the probability that, this
event having occurred, the second will occur. Thus in the preceding case of
the three urns A, B, C, of which two contain only white balls and one
contains only black balls, the probability of drawing a white ball from the
urn C is ⅔, since of the three urns only two contain balls of that color. But
when a white ball has been drawn from the urn C, the indecision relative to
that one of the urns which contain only black balls extends only to the urns
A and B; the probability of drawing a white ball from the urn B is ½; the
product of ⅔ by ½, or ⅓, is then the probability of drawing two white balls
at one time from the urns B and C.
We see by this example the influence of past events upon the probability
of future events. For the probability of drawing a white ball from the urn B,
which primarily is ⅔, becomes ½ when a white ball has been drawn from
the urn C; it would change to certainty if a black ball had been drawn from
the same urn. We will determine this influence by means of the following
principle, which is a corollary of the preceding one.
Fifth Principle.—If we calculate à priori the probability of the occurred
event and the probability of an event composed of that one and a second
one which is expected, the second probability divided by the first will be
the probability of the event expected, drawn from the observed event.
Here is presented the question raised by some philosophers touching the
influence of the past upon the probability of the future. Let us suppose at
the play of heads and tails that heads has occurred oftener than tails. By this
alone we shall be led to believe that in the constitution of the coin there is a
secret cause which favors it. Thus in the conduct of life constant happiness

is a proof of competency which should induce us to employ preferably
happy persons. But if by the unreliability of circumstances we are
constantly brought back to a state of absolute indecision, if, for example, we
change the coin at each throw at the play of heads and tails, the past can
shed no light upon the future and it would be absurd to take account of it.
Sixth Principle.—Each of the causes to which an observed event may be
attributed is indicated with just as much likelihood as there is probability
that the event will take place, supposing the event to be constant. The
probability of the existence of any one of these causes is then a fraction
whose numerator is the probability of the event resulting from this cause
and whose denominator is the sum of the similar probabilities relative to all
the causes; if these various causes, considered à priori, are unequally
probable, it is necessary, in place of the probability of the event resulting
from each cause, to employ the product of this probability by the possibility
of the cause itself. This is the fundamental principle of this branch of the
analysis of chances which consists in passing from events to causes.
This principle gives the reason why we attribute regular events to a
particular cause. Some philosophers have thought that these events are less
possible than others and that at the play of heads and tails, for example, the
combination in which heads occurs twenty successive times is less easy in
its nature than those where heads and tails are mixed in an irregular manner.
But this opinion supposes that past events have an influence on the
possibility of future events, which is not at all admissible. The regular
combinations occur more rarely only because they are less numerous. If we
seek a cause wherever we perceive symmetry, it is not that we regard a
symmetrical event as less possible than the others, but, since this event
ought to be the effect of a regular cause or that of chance, the first of these
suppositions is more probable than the second. On a table we see letters
arranged in this order, C o n s t a n t i n o p l e , and we judge that
this arrangement is not the result of chance, not because it is less possible
than the others, for if this word were not employed in any language we
should not suspect it came from any particular cause, but this word being in
use among us, it is incomparably more probable that some person has thus
arranged the aforesaid letters than that this arrangement is due to chance.

This is the place to define the word extraordinary. We arrange in our
thought all possible events in various classes; and we regard as
extraordinary those classes which include a very small number. Thus at the
play of heads and tails the occurrence of heads a hundred successive times
appears to us extraordinary because of the almost infinite number of
combinations which may occur in a hundred throws; and if we divide the
combinations into regular series containing an order easy to comprehend,
and into irregular series, the latter are incomparably more numerous. The
drawing of a white ball from an urn which among a million balls contains
only one of this color, the others being black, would appear to us likewise
extraordinary, because we form only two classes of events relative to the
two colors. But the drawing of the number 475813, for example, from an
urn that contains a million numbers seems to us an ordinary event; because,
comparing individually the numbers with one another without dividing
them into classes, we have no reason to believe that one of them will appear
sooner than the others.
From what precedes, we ought generally to conclude that the more
extraordinary the event, the greater the need of its being supported by
strong proofs. For those who attest it, being able to deceive or to have been
deceived, these two causes are as much more probable as the reality of the
event is less. We shall see this particularly when we come to speak of the
probability of testimony.
Seventh Principle.—The probability of a future event is the sum of the
products of the probability of each cause, drawn from the event observed,
by the probability that, this cause existing, the future event will occur. The
following example will illustrate this principle.
Let us imagine an urn which contains only two balls, each of which may
be either white or black. One of these balls is drawn and is put back into the
urn before proceeding to a new draw. Suppose that in the first two draws
white balls have been drawn; the probability of again drawing a white ball
at the third draw is required.
Only two hypotheses can be made here: either one of the balls is white
and the other black, or both are white. In the first hypothesis the probability
of the event observed is ¼; it is unity or certainty in the second. Thus in

regarding these hypotheses as so many causes, we shall have for the sixth
principle ⅕ and ⅘ for their respective probabilities. But if the first
hypothesis occurs, the probability of drawing a white ball at the third draw
is ½; it is equal to certainty in the second hypothesis; multiplying then the
last probabilities by those of the corresponding hypotheses, the sum of the
products, or 9⁄10, will be the probability of drawing a white ball at the third
draw.
When the probability of a single event is unknown we may suppose it
equal to any value from zero to unity. The probability of each of these
hypotheses, drawn from the event observed, is, by the sixth principle, a
fraction whose numerator is the probability of the event in this hypothesis
and whose denominator is the sum of the similar probabilities relative to all
the hypotheses. Thus the probability that the possibility of the event is
comprised within given limits is the sum of the fractions comprised within
these limits. Now if we multiply each fraction by the probability of the
future event, determined in the corresponding hypothesis, the sum of the
products relative to all the hypotheses will be, by the seventh principle, the
probability of the future event drawn from the event observed. Thus we find
that an event having occurred successively any number of times, the
probability that it will happen again the next time is equal to this number
increased by unity divided by the same number, increased by two units.
Placing the most ancient epoch of history at five thousand years ago, or at
1826213 days, and the sun having risen constantly in the interval at each
revolution of twenty-four hours, it is a bet of 1826214 to one that it will rise
again to-morrow. But this number is incomparably greater for him who,
recognizing in the totality of phenomena the principal regulator of days and
seasons, sees that nothing at the present moment can arrest the course of it.
Buffon in his Political Arithmetic calculates differently the preceding
probability. He supposes that it differs from unity only by a fraction whose
numerator is unity and whose denominator is the number 2 raised to a
power equal to the number of days which have elapsed since the epoch. But
the true manner of relating past events with the probability of causes and of
future events was unknown to this illustrious writer.

CHAPTER IV.
CONCERNING HOPE.
The probability of events serves to determine the hope or the fear of
persons interested in their existence. The word hope has various
acceptations; it expresses generally the advantage of that one who expects a
certain benefit in suppositions which are only probable. This advantage in
the theory of chance is a product of the sum hoped for by the probability of
obtaining it; it is the partial sum which ought to result when we do not wish
to run the risks of the event in supposing that the division is made
proportional to the probabilities. This division is the only equitable one
when all strange circumstances are eliminated; because an equal degree of
probability gives an equal right to the sum hoped for. We will call this
advantage mathematical hope.
Eighth Principle.—When the advantage depends on several events it is
obtained by taking the sum of the products of the probability of each event
by the benefit attached to its occurrence.
Let us apply this principle to some examples. Let us suppose that at the
play of heads and tails Paul receives two francs if he throws heads at the
first throw and five francs if he throws it only at the second. Multiplying
two francs by the probability ½ of the first case, and five francs by the
probability ¼ of the second case, the sum of the products, or two and a
quarter francs, will be Paul's advantage. It is the sum which he ought to give
in advance to that one who has given him this advantage; for, in order to
maintain the equality of the play, the throw ought to be equal to the
advantage which it procures.
If Paul receives two francs by throwing heads at the first and five francs
by throwing it at the second throw, whether he has thrown it or not at the
first, the probability of throwing heads at the second throw being ½,
multiplying two francs and five francs by ½ the sum of these products will

give three and one half francs for Paul's advantage and consequently for his
stake at the game.
Ninth Principle.—In a series of probable events of which the ones
produce a benefit and the others a loss, we shall have the advantage which
results from it by making a sum of the products of the probability of each
favorable event by the benefit which it procures, and subtracting from this
sum that of the products of the probability of each unfavorable event by the
loss which is attached to it. If the second sum is greater than the first, the
benefit becomes a loss and hope is changed to fear.
Consequently we ought always in the conduct of life to make the product
of the benefit hoped for, by its probability, at least equal to the similar
product relative to the loss. But it is necessary, in order to attain this, to
appreciate exactly the advantages, the losses, and their respective
probabilities. For this a great accuracy of mind, a delicate judgment, and a
great experience in affairs is necessary; it is necessary to know how to
guard one's self against prejudices, illusions of fear or hope, and erroneous
ideas, ideas of fortune and happiness, with which the majority of people
feed their self-love.
The application of the preceding principles to the following question has
greatly exercised the geometricians. Paul plays at heads and tails with the
condition of receiving two francs if he throws heads at the first throw, four
francs if he throws it only at the second throw, eight francs if he throws it
only at the third, and so on. His stake at the play ought to be, according to
the eighth principle, equal to the number of throws, so that if the game
continues to infinity the stake ought to be infinite. However, no reasonable
man would wish to risk at this game even a small sum, for example five
francs. Whence comes this difference between the result of calculation and
the indication of common sense? We soon recognize that it amounts to this:
that the moral advantage which a benefit procures for us is not proportional
to this benefit and that it depends upon a thousand circumstances, often
very difficult to define, but of which the most general and most important is
that of fortune.
Indeed it is apparent that one franc has much greater value for him who
possesses only a hundred than for a millionaire. We ought then to

distinguish in the hoped-for benefit its absolute from its relative value. But
the latter is regulated by the motives which make it desirable, whereas the
first is independent of them. The general principle for appreciating this
relative value cannot be given, but here is one proposed by Daniel Bernoulli
which will serve in many cases.
Tenth Principle.—The relative value of an infinitely small sum is equal to
its absolute value divided by the total benefit of the person interested. This
supposes that every one has a certain benefit whose value can never be
estimated as zero. Indeed even that one who possesses nothing always gives
to the product of his labor and to his hopes a value at least equal to that
which is absolutely necessary to sustain him.
If we apply analysis to the principle just propounded, we obtain the
following rule: Let us designate by unity the part of the fortune of an
individual, independent of his expectations. If we determine the different
values that this fortune may have by virtue of these expectations and their
probabilities, the product of these values raised respectively to the powers
indicated by their probabilities will be the physical fortune which would
procure for the individual the same moral advantage which he receives from
the part of his fortune taken as unity and from his expectations; by
subtracting unity from the product, the difference will be the increase of the
physical fortune due to expectations: we will call this increase moral hope.
It is easy to see that it coincides with mathematical hope when the fortune
taken as unity becomes infinite in reference to the variations which it
receives from the expectations. But when these variations are an
appreciable part of this unity the two hopes may differ very materially
among themselves.
This rule conduces to results conformable to the indications of common
sense which can by this means be appreciated with some exactitude. Thus
in the preceding question it is found that if the fortune of Paul is two
hundred francs, he ought not reasonably to stake more than nine francs. The
same rule leads us again to distribute the danger over several parts of a
benefit expected rather than to expose the entire benefit to this danger. It
results similarly that at the fairest game the loss is always greater than the
gain. Let us suppose, for example, that a player having a fortune of one

hundred francs risks fifty at the play of heads and tails; his fortune after his
stake at the play will be reduced to eighty-seven francs, that is to say, this
last sum would procure for the player the same moral advantage as the state
of his fortune after the stake. The play is then disadvantageous even in the
case where the stake is equal to the product of the sum hoped for, by its
probability. We can judge by this of the immorality of games in which the
sum hoped for is below this product. They subsist only by false reasonings
and by the cupidity which they excite and which, leading the people to
sacrifice their necessaries to chimerical hopes whose improbability they are
not in condition to appreciate, are the source of an infinity of evils.
The disadvantage of games of chance, the advantage of not exposing to
the same danger the whole benefit that is expected, and all the similar
results indicated by common sense, subsist, whatever may be the function
of the physical fortune which for each individual expresses his moral
fortune. It is enough that the proportion of the increase of this function to
the increase of the physical fortune diminishes in the measure that the latter
increases.

CHAPTER V.
CONCERNING THE ANALYTICAL METHODS OF
THE CALCULUS OF PROBABILITIES.
The application of the principle which we have just expounded to the
various questions of probability requires methods whose investigation has
given birth to several methods of analysis and especially to the theory of
combinations and to the calculus of finite differences.
If we form the product of the binomials, unity plus the first letter, unity
plus the second letter, unity plus the third letter, and so on up to n letters,
and subtract unity from this developed product, the result will be the sum of
the combination of all these letters taken one by one, two by two, three by
three, etc., each combination having unity for a coefficient. In order to have
the number of combinations of these n letters taken s by s times, we shall
observe that if we suppose these letters equal among themselves, the
preceding product will become the nth power of the binomial one plus the
first letter; thus the number of combinations of n letters taken s by s times
will be the coefficient of the sth power of the first letter in the development
in this binomial; and this number is obtained by means of the known
binomial formula.
Attention must be paid to the respective situations of the letters in each
combination, observing that if a second letter is joined to the first it may be
placed in the first or second position which gives two combinations. If we
join to these combinations a third letter, we can give it in each combination
the first, the second, and the third rank which forms three combinations
relative to each of the two others, in all six combinations. From this it is
easy to conclude that the number of arrangements of which s letters are
susceptible is the product of the numbers from unity to s. In order to pay
regard to the respective positions of the letters it is necessary then to
multiply by this product the number of combinations of n letters s by s

times, which is tantamount to taking away the denominator of the
coefficient of the binomial which expresses this number.
Let us imagine a lottery composed of n numbers, of which r are drawn at
each draw. The probability is demanded of the drawing of s given numbers
in one draw. To arrive at this let us form a fraction whose denominator will
be the number of all the cases possible or of the combinations of n letters
taken r by r times, and whose numerator will be the number of all the
combinations which contain the given s numbers. This last number is
evidently that of the combinations of the other numbers taken n less s by n
less s times. This fraction will be the required probability, and we shall
easily find that it can be reduced to a fraction whose numerator is the
number of combinations of r numbers taken s by s times, and whose
denominator is the number of combinations of n numbers taken similarly s
by s times. Thus in the lottery of France, formed as is known of 90 numbers
of which five are drawn at each draw, the probability of drawing a given
combination is 5⁄90, or 1⁄18; the lottery ought then for the equality of the play
to give eighteen times the stake. The total number of combinations two by
two of the 90 numbers is 4005, and that of the combinations two by two of
5 numbers is 10. The probability of the drawing of a given pair is then 1⁄4005,
and the lottery ought to give four hundred and a half times the stake; it
ought to give 11748 times for a given tray, 511038 times for a quaternary,
and 43949268 times for a quint. The lottery is far from giving the player
these advantages.
Suppose in an urn a white balls, b black balls, and after having drawn a
ball it is put back into the urn; the probability is asked that in n number of
draws m white balls and n - m black balls will be drawn. It is clear that the
number of cases that may occur at each drawing is a + b. Each case of the
second drawing being able to combine with all the cases of the first, the
number of possible cases in two drawings is the square of the binomial a +
b. In the development of this square, the square of a expresses the number
of cases in which a white ball is twice drawn, the double product of a by b
expresses the number of cases in which a white ball and a black ball are
drawn. Finally, the square of b expresses the number of cases in which two
black balls are drawn. Continuing thus, we see generally that the nth power

of the binomial a + b expresses the number of all the cases possible in n
draws; and that in the development of this power the term multiplied by the
mth power of a expresses the number of cases in which m white balls and n
- m black balls may be drawn. Dividing then this term by the entire power
of the binomial, we shall have the probability of drawing m white balls and
n - m black balls. The ratio of the numbers a and a + b being the probability
of drawing one white ball at one draw; and the ratio of the numbers b and a
+ b being the probability of drawing one black ball; if we call these
probabilities p and q, the probability of drawing m white balls in n draws
will be the term multiplied by the mth power of p in the development of the
nth power of the binomial p + q; we may see that the sum p + q is unity.
This remarkable property of the binomial is very useful in the theory of
probabilities. But the most general and direct method of resolving questions
of probability consists in making them depend upon equations of
differences. Comparing the successive conditions of the function which
expresses the probability when we increase the variables by their respective
differences, the proposed question often furnishes a very simple proportion
between the conditions. This proportion is what is called equation of
ordinary or partial differentials; ordinary when there is only one variable,
partial when there are several. Let us consider some examples of this.
Three players of supposed equal ability play together on the following
conditions: that one of the first two players who beats his adversary plays
the third, and if he beats him the game is finished. If he is beaten, the victor
plays against the second until one of the players has defeated consecutively
the two others, which ends the game. The probability is demanded that the
game will be finished in a certain number n of plays. Let us find the
probability that it will end precisely at the nth play. For that the player who
wins ought to enter the game at the play n - 1 and win it thus at the
following play. But if in place of winning the play n - 1 he should be beaten
by his adversary who had just beaten the other player, the game would end
at this play. Thus the probability that one of the players will enter the game
at the play n - 1 and will win it is equal to the probability that the game will
end precisely with this play; and as this player ought to win the following
play in order that the game may be finished at the nth play, the probability
of this last case will be only one half of the preceding one. This probability

is evidently a function of the number n; this function is then equal to the
half of the same function when n is diminished by unity. This equality
forms one of those equations called ordinary finite differential equations.
We may easily determine by its use the probability that the game will end
precisely at a certain play. It is evident that the play cannot end sooner than
at the second play; and for this it is necessary that that one of the first two
players who has beaten his adversary should beat at the second play the
third player; the probability that the game will end at this play is ½. Hence
by virtue of the preceding equation we conclude that the successive
probabilities of the end of the game are ¼ for the third play, ⅛ for the fourth
play, and so on; and in general ½ raised to the power n - 1 for the nth play.
The sum of all these powers of ½ is unity less the last of these powers; it is
the probability that the game will end at the latest in n plays.
Let us consider again the first problem more difficult which may be
solved by probabilities and which Pascal proposed to Fermat to solve. Two
players, A and B, of equal skill play together on the conditions that the one
who first shall beat the other a given number of times shall win the game
and shall take the sum of the stakes at the game; after some throws the
players agree to quit without having finished the game: we ask in what
manner the sum ought to be divided between them. It is evident that the
parts ought to be proportional to the respective probabilities of winning the
game. The question is reduced then to the determination of these
probabilities. They depend evidently upon the number of points which each
player lacks of having attained the given number. Hence the probability of
A is a function of the two numbers which we will call indices. If the two
players should agree to play one throw more (an agreement which does not
change their condition, provided that after this new throw the division is
always made proportionally to the new probabilities of winning the game),
then either A would win this throw and in that case the number of points
which he lacks would be diminished by unity, or the player B would win it
and in that case the number of points lacking to this last player would be
less by unity. But the probability of each of these cases is ½; the function
sought is then equal to one half of this function in which we diminish by
unity the first index plus the half of the same function in which the second

variable is diminished by unity. This equality is one of those equations
called equations of partial differentials.
We are able to determine by its use the probabilities of A by dividing the
smallest numbers, and by observing that the probability or the function
which expresses it is equal to unity when the player A does not lack a single
point, or when the first index is zero, and that this function becomes zero
with the second index. Supposing thus that the player A lacks only one
point, we find that his probability is ½, ¾, 7⁄8, etc., according as B lacks one
point, two, three, etc. Generally it is then unity less the power of ½, equal to
the number of points which B lacks. We will suppose then that the player A
lacks two points and his probability will be found equal to ¼, ½, 11⁄16, etc.,
according as B lacks one point, two points, three points, etc. We will
suppose again that the player A lacks three points, and so on.
This manner of obtaining the successive values of a quantity by means of
its equation of differences is long and laborious. The geometricians have
sought methods to obtain the general function of indices that satisfies this
equation, so that for any particular case we need only to substitute in this
function the corresponding values of the indices. Let us consider this
subject in a general way. For this purpose let us conceive a series of terms
arranged along a horizontal line so that each of them is derived from the
preceding one according to a given law. Let us suppose this law expressed
by an equation among several consecutive terms and their index, or the
number which indicates the rank that they occupy in the series. This
equation I call the equation of finite differences by a single index. The order
or the degree of this equation is the difference of rank of its two extreme
terms. We are able by its use to determine successively the terms of the
series and to continue it indefinitely; but for that it is necessary to know a
number of terms of the series equal to the degree of the equation. These
terms are the arbitrary constants of the expression of the general term of the
series or of the integral of the equation of differences.
Let us imagine now below the terms of the preceding series a second
series of terms arranged horizontally; let us imagine again below the terms
of the second series a third horizontal series, and so on to infinity; and let us
suppose the terms of all these series connected by a general equation among

several consecutive terms, taken as much in the horizontal as in the vertical
sense, and the numbers which indicate their rank in the two senses. This
equation is called the equation of partial finite differences by two indices.
Let us imagine in the same way below the plan of the preceding series a
second plan of similar series, whose terms should be placed respectively
below those of the first plan; let us imagine again below this second plan a
third plan of similar series, and so on to infinity; let us suppose all the terms
of these series connected by an equation among several consecutive terms
taken in the sense of length, width, and depth, and the three numbers which
indicate their rank in these three senses. This equation I call the equation of
partial finite differences by three indices.
Finally, considering the matter in an abstract way and independently of
the dimensions of space, let us imagine generally a system of magnitudes,
which should be functions of a certain number of indices, and let us
suppose among these magnitudes, their relative differences to these indices
and the indices themselves, as many equations as there are magnitudes;
these equations will be partial finite differences by a certain number of
indices.
We are able by their use to determine successively these magnitudes. But
in the same manner as the equation by a single index requires for it that we
know a certain number of terms of the series, so the equation by two indices
requires that we know one or several lines of series whose general terms
should be expressed each by an arbitrary function of one of the indices.
Similarly the equation by three indices requires that we know one or several
plans of series, the general terms of which should be expressed each by an
arbitrary function of two indices, and so on. In all these cases we shall be
able by successive eliminations to determine a certain term of the series.
But all the equations among which we eliminate being comprised in the
same system of equations, all the expressions of the successive terms which
we obtain by these eliminations ought to be comprised in one general
expression, a function of the indices which determine the rank of the term.
This expression is the integral of the proposed equation of differences, and
the search for it is the object of integral calculus.

Taylor is the first who in his work entitled Metodus incrementorum has
considered linear equations of finite differences. He gives the manner of
integrating those of the first order with a coefficient and a last term,
functions of the index. In truth the relations of the terms of the arithmetical
and geometrical progressions which have always been taken into
consideration are the simplest cases of linear equations of differences; but
they had not been considered from this point of view. It was one of those
which, attaching themselves to general theories, lead to these theories and
are consequently veritable discoveries.
About the same time Moivre was considering under the name of
recurring series the equations of finite differences of a certain order having
a constant coefficient. He succeeded in integrating them in a very ingenious
manner. As it is always interesting to follow the progress of inventors, I
shall expound the method of Moivre by applying it to a recurring series
whose relation among three consecutive terms is given. First he considers
the relation among the consecutive terms of a geometrical progression or
the equation of two terms which expresses it. Referring it to terms less than
unity, he multiplies it in this state by a constant factor and subtracts the
product from the first equation. Thus he obtains an equation among three
consecutive terms of the geometrical progression. Moivre considers next a
second progression whose ratio of terms is the same factor which he has
just used. He diminishes similarly by unity the index of the terms of the
equation of this new progression. In this condition he multiplies it by the
ratio of the terms of the first progression, and he subtracts the product from
the equation of the second progression, which gives him among three
consecutive terms of this progression a relation entirely similar to that
which he has found for the first progression. Then he observes that if one
adds term by term the two progressions, the same ratio exists among any
three of these consecutive terms. He compares the coefficients of this ratio
to those of the relation of the terms of the proposed recurrent series, and he
finds for determining the ratios of the two geometrical progressions an
equation of the second degree, whose roots are these ratios. Thus Moivre
decomposes the recurrent series into two geometrical progressions, each
multiplied by an arbitrary constant which he determines by means of the
first two terms of the recurrent series. This ingenious process is in fact the

one that d'Alembert has since employed for the integration of linear
equations of infinitely small differences with constant coefficients, and
Lagrange has transformed into similar equations of finite differences.
Finally, I have considered the linear equations of partial finite
differences, first under the name of recurro-recurrent series and afterwards
under their own name. The most general and simplest manner of integrating
all these equations appears to me that which I have based upon the
consideration of discriminant functions, the idea of which is here given.
If we conceive a function V of a variable t developed according to the
powers of this variable, the coefficient of any one of these powers will be a
function of the exponent or index of this power, which index I shall call x. V
is what I call the discriminant function of this coefficient or of the function
of the index.
Now if we multiply the series of the development of V by a function of
the same variable, such, for example, as unity plus two times this variable,
the product will be a new discriminant function in which the coefficient of
the power x of the variable t will be equal to the coefficient of the same
power in V plus twice the coefficient of the power less unity. Thus the
function of the index x in the product will be equal to the function of the
index x in V plus twice the same function in which the index is diminished
by unity. This function of the index x is thus a derivative of the function of
the same index in the development of V, a function which I shall call the
primitive function of the index. Let us designate the derivative function by
the letter Alembert placed before the primitive function. The derivation
indicated by this letter will depend upon the multiplier of V, which we will
call T and which we will suppose developed like V by the ratio to the
powers of the variable t. If we multiply anew by T the product of V by T,
which is equivalent to multiplying V by T², we shall form a third
discriminant function, in which the coefficient of the xth power of t will be
a derivative similar to the corresponding coefficient of the preceding
product; it may be expressed by the same character δ placed before the
preceding derivative, and then this character will be written twice before the
primitive function of x. But in place of writing it thus twice we give it 2 for
an exponent.

Continuing thus, we see generally that if we multiply V by the nth power
of T, we shall have the coefficient of the xth power of t in the product of V
by the nth power of T by placing before the primitive function the character
δ with n for an exponent.
Let us suppose, for example, that T be unity divided by t; then in the
product of V by T the coefficient of the xth power of t will be the coefficient
of the power greater by unity in V; this coefficient in the product of V by the
nth power of T will then be the primitive function in which x is augmented
by n units.
Let us consider now a new function Z of t, developed like V and T
according to the powers of t; let us designate by the character Δ placed
before the primitive function the coefficient of the xth power of t in the
product of V by Z; this coefficient in the product of V by the nth power of Z
will be expressed by the character Δ affected by the exponent n and placed
before the primitive function of x.
If, for example, Z is equal to unity divided by t less one, the coefficient of
the xth power of t in the product of V by Z will be the coefficient of the x +
1 power of t in V less the coefficient of the xth power. It will be then the
finite difference of the primitive function of the index x. Then the character
Δ indicates a finite difference of the primitive function in the case where the
index varies by unity; and the nth power of this character placed before the
primitive function will indicate the finite nth difference of this function. If
we suppose that T be unity divided by t, we shall have T equal to the
binomial Z + 1. The product of V by the nth power of T will then be equal to
the product of V by the nth power of the binomial Z + 1. Developing this
power in the ratio of the powers of Z, the product of V by the various terms
of this development will be the discriminant functions of these same terms
in which we substitute in place of the powers of Z the corresponding finite
differences of the primitive function of the index.
Now the product of V by the nth power of T is the primitive function in
which the index x is augmented by n units; repassing from the discriminant
functions to their coefficients, we shall have this primitive function thus
augmented equal to the development of the nth power of the binomial Z + 1,
provided that in this development we substitute in place of the powers of Z

the corresponding differences of the primitive function and that we multiply
the independent term of these powers by the primitive function. We shall
thus obtain the primitive function whose index is augmented by any number
n by means of its differences.
Supposing that T and Z always have the preceding values, we shall have
Z equal to the binomial T - 1; the product of V by the nth power of Z will
then be equal to the product of V by the development of the nth power of
the binomial T - 1. Repassing from the discriminant functions to their
coefficients as has just been done, we shall have the nth difference of the
primitive function expressed by the development of the nth power of the
binomial T - 1, in which we substitute for the powers of T this same
function whose index is augmented by the exponent of the power, and for
the independent term of t, which is unity, the primitive function, which
gives this difference by means of the consecutive terms of this function.
Placing δ before the primitive function expressing the derivative of this
function, which multiplies the x power of t in the product of V by T, and Δ
expressing the same derivative in the product of V by Z, we are led by that
which precedes to this general result: whatever may be the function of the
variable t represented by T and Z, we may, in the development of all the
identical equations susceptible of being formed among these functions,
substitute the characters δ and Δ in place of T and Z, provided that we write
the primitive function of the index in series with the powers and with the
products of the powers of the characters, and that we multiply by this
function the independent terms of these characters.
We are able by means of this general result to transform any certain
power of a difference of the primitive function of the index x, in which x
varies by unity, into a series of differences of the same function in which x
varies by a certain number of units and reciprocally. Let us suppose that T
be the i power of unity divided by t - 1, and that Z be always unity divided
by t - 1; then the coefficient of the x power of t in the product of V by T will
be the coefficient of the x + i power of t in V less the coefficient of the x
power of t; it will then be the finite difference of the primitive function of
the index x in which we vary this index by the number i. It is easy to see
that T is equal to the difference between the i power of the binomial Z + 1

and unity. The nth power of T is equal to the nth power of this difference. If
in this equality we substitute in place of T and Z the characters δ and Δ, and
after the development we place at the end of each term the primitive
function of the index x, we shall have the nth difference of this function in
which x varies by i units expressed by a series of differences of the same
function in which x varies by unity. This series is only a transformation of
the difference which it expresses and which is identical with it; but it is in
similar transformations that the power of analysis resides.
The generality of analysis permits us to suppose in this expression that n
is negative. Then the negative powers of δ and Δ indicate the integrals.
Indeed the nth difference of the primitive function having for a discriminant
function the product of V by the nth power of the binomial one divided by t
less unity, the primitive function which is the nth integral of this difference
has for a discriminant function that of the same difference multiplied by the
nth power taken less than the binomial one divided by t minus one, a power
to which the same power of the character Δ corresponds; this power
indicates then an integral of the same order, the index x varying by unity;
and the negative powers of δ indicate equally the integrals x varying by i
units. We see, thus, in the clearest and simplest manner the rationality of the
analysis observed among the positive powers and differences, and among
the negative powers and the integrals.
If the function indicated by δ placed before the primitive function is zero,
we shall have an equation of finite differences, and V will be the
discriminant function of its integral. In order to obtain this discriminant
function we shall observe that in the product of V by T all the powers of t
ought to disappear except the powers inferior to the order of the equation of
differences; V is then equal to a fraction whose denominator is T and whose
numerator is a polynomial in which the highest power of t is less by unity
than the order of the equation of differences. The arbitrary coefficients of
the various powers of t in this polynomial, including the power zero, will be
determined by as many values of the primitive function of the index when
we make successively x equal to zero, to one, to two, etc. When the
equation of differences is given we determine T by putting all its terms in
the first member and zero in the second; by substituting in the first member
unity in place of the function which has the largest index; the first power of

t in place of the primitive function in which this index is diminished by
unity; the second power of t for the primitive function where this index is
diminished by two units, and so on. The coefficient of the xth power of t in
the development of the preceding expression of V will be the primitive
function of x or the integral of the equation of finite differences. Analysis
furnishes for this development various means, among which we may choose
that one which is most suitable for the question proposed; this is an
advantage of this method of integration.
Let us conceive now that V be a function of the two variables t and t´
developed according to the powers and products of these variables; the
coefficient of any product of the powers x and x´ of t and t´ will be a
function of the exponents or indices x and x´ of these powers; this function I
shall call the primitive function of which V is the discriminant function.
Let us multiply V by a function T of the two variables t and t´ developed
like V in ratio of the powers and the products of these variables; the product
will be the discriminant function of a derivative of the primitive function; if
T, for example, is equal to the variable t plus the variable t´ minus two, this
derivative will be the primitive function of which we diminish by unity the
index x plus this same primitive function of which we diminish by unity the
index x´ less two times the primitive function. Designating whatever T may
be by the character δ placed before the primitive function, this derivative,
the product of V by the nth power of T, will be the discriminant function of
the derivative of the primitive function before which one places the nth
power of the character δ. Hence result the theorems analogous to those
which are relative to functions of a single variable.
Suppose the function indicated by the character δ be zero; one will have
an equation of partial differences. If, for example, we make as before T
equal to the variable t plus the variable t´ - 2, we have zero equal to the
primitive function of which we diminish by unity the index x plus the same
function of which we diminish by unity the index x´ minus two times the
primitive function. The discriminant function V of the primitive function or
of the integral of this equation ought then to be such that its product by T
does not include at all the products of t by t´; but V may include separately
the powers of t and those of t´, that is to say, an arbitrary function of t and

an arbitrary function of t´; V is then a fraction whose numerator is the sum
of these two arbitrary functions and whose denominator is T. The
coefficient of the product of the xth power of t by the x´ power of t´ in the
development of this fraction will then be the integral of the preceding
equation of partial differences. This method of integrating this kind of
equations seems to me the simplest and the easiest by the employment of
the various analytical processes for the development of rational fractions.
More ample details in this matter would be scarcely understood without
the aid of calculus.
Considering equations of infinitely small partial differences as equations
of finite partial differences in which nothing is neglected, we are able to
throw light upon the obscure points of their calculus, which have been the
subject of great discussions among geometricians. It is thus that I have
demonstrated the possibility of introducing discontinued functions in their
integrals, provided that the discontinuity takes place only for the
differentials of the order of these equations or of a superior order. The
transcendent results of calculus are, like all the abstractions of the
understanding, general signs whose true meaning may be ascertained only
by repassing by metaphysical analysis to the elementary ideas which have
led to them; this often presents great difficulties, for the human mind tries
still less to transport itself into the future than to retire within itself. The
comparison of infinitely small differences with finite differences is able
similarly to shed great light upon the metaphysics of infinitesimal calculus.
It is easily proven that the finite nth difference of a function in which the
increase of the variable is E being divided by the nth power of E, the
quotient reduced in series by ratio to the powers of the increase E is formed
by a first term independent of E. In the measure that E diminishes, the
series approaches more and more this first term from which it can differ
only by quantities less than any assignable magnitude. This term is then the
limit of the series and expresses in differential calculus the infinitely small
nth difference of the function divided by the nth power of the infinitely
small increase.
Considering from this point of view the infinitely small differences, we
see that the various operations of differential calculus amount to comparing

separately in the development of identical expressions the finite terms or
those independent of the increments of the variables which are regarded as
infinitely small; this is rigorously exact, these increments being
indeterminant. Thus differential calculus has all the exactitude of other
algebraic operations.
The same exactitude is found in the applications of differential calculus
to geometry and mechanics. If we imagine a curve cut by a secant at two
adjacent points, naming E the interval of the ordinates of these two points,
E will be the increment of the abscissa from the first to the second ordinate.
It is easy to see that the corresponding increment of the ordinate will be the
product of E by the first ordinate divided by its subsecant; augmenting then
in this equation of the curve the first ordinate by this increment, we shall
have the equation relative to the second ordinate. The difference of these
two equations will be a third equation which, developed by the ratio of the
powers of E and divided by E, will have its first term independent of E,
which will be the limit of this development. This term, equal to zero, will
give then the limit of the subsecants, a limit which is evidently the
subtangent.
This singularly happy method of obtaining the subtangent is due to
Fermat, who has extended it to transcendent curves. This great geometrician
expresses by the character E the increment of the abscissa; and considering
only the first power of this increment, he determines exactly as we do by
differential calculus the subtangents of the curves, their points of inflection,
the maxima and minima of their ordinates, and in general those of rational
functions. We see likewise by his beautiful solution of the problem of the
refraction of light inserted in the Collection of the Letters of Descartes that
he knows how to extend his methods to irrational functions in freeing them
from irrationalities by the elevation of the roots to powers. Fermat should
be regarded, then, as the true discoverer of Differential Calculus. Newton
has since rendered this calculus more analytical in his Method of Fluxions,
and simplified and generalized the processes by his beautiful theorem of the
binomial. Finally, about the same time Leibnitz has enriched differential
calculus by a notation which, by indicating the passage from the finite to
the infinitely small, adds to the advantage of expressing the general results
of calculus that of giving the first approximate values of the differences and

of the sums of the quantities; this notation is adapted of itself to the calculus
of partial differentials.
We are often led to expressions which contain so many terms and factors
that the numerical substitutions are impracticable. This takes place in
questions of probability when we consider a great number of events.
Meanwhile it is necessary to have the numerical value of the formulæ in
order to know with what probability the results are indicated, which the
events develop by multiplication. It is necessary especially to have the law
according to which this probability continually approaches certainty, which
it will finally attain if the number of events were infinite. In order to obtain
this law I considered that the definite integrals of differentials multiplied by
the factors raised to great powers would give by integration the formulæ
composed of a great number of terms and factors. This remark brought me
to the idea of transforming into similar integrals the complicated
expressions of analysis and the integrals of the equation of differences. I
fulfilled this condition by a method which gives at the same time the
function comprised under the integral sign and the limits of the integration.
It offers this remarkable thing, that the function is the same discriminant
function of the expressions and the proposed equations; this attaches this
method to the theory of discriminant functions of which it is thus the
complement. Further, it would only be a question of reducing the definite
integral to a converging series. This I have obtained by a process which
makes the series converge with as much more rapidity as the formula which
it represents is more complicated, so that it is more exact as it becomes
more necessary. Frequently the series has for a factor the square root of the
ratio of the circumference to the diameter; sometimes it depends upon other
transcendents whose number is infinite.
An important remark which pertains to great generality of analysis, and
which permits us to extend this method to formulæ and to equations of
difference which the theory of probability presents most frequently, is that
the series to which one comes by supposing the limits of the definite
integrals to be real and positive take place equally in the case where the
equation which determines these limits has only negative or imaginary
roots. These passages from the positive to the negative and from the real to
the imaginary, of which I first have made use, have led me further to the

values of many singular definite integrals, which I have accordingly
demonstrated directly. We may then consider these passages as a means of
discovery parallel to induction and analogy long employed by
geometricians, at first with an extreme reserve, afterwards with entire
confidence, since a great number of examples has justified its use. In the
mean time it is always necessary to confirm by direct demonstrations the
results obtained by these divers means.
I have named the ensemble of the preceding methods the Calculus of
Discriminant Functions; this calculus serves as a basis for the work which I
have published under the title of the Analytical Theory of Probabilities. It is
connected with the simple idea of indicating the repeated multiplications of
a quantity by itself or its entire and positive powers by writing toward the
top of the letter which expresses it the numbers which mark the degrees of
these powers.
This notation, employed by Descartes in his Geometry and generally
adopted since the publication of this important work, is a little thing,
especially when compared with the theory of curves and variable functions
by which this great geometrician has established the foundations of modern
calculus. But the language of analysis, most perfect of all, being in itself a
powerful instrument of discoveries, its notations, especially when they are
necessary and happily conceived, are so many germs of new calculi. This is
rendered appreciable by this example.
Wallis, who in his work entitled Arithmetica Infinitorum, one of those
which have most contributed to the progress of analysis, has interested
himself especially in following the thread of induction and analogy,
considered that if one divides the exponent of a letter by two, three, etc., the
quotient will be accordingly the Cartesian notation, and when division is
possible the exponent of the square, cube, etc., root of the quantity which
represents the letter raised to the dividend exponent. Extending by analogy
this result to the case where division is impossible, he considered a quantity
raised to a fractional exponent as the root of the degree indicated by the
denominator of this fraction—namely, of the quantity raised to a power
indicated by the numerator. He observed then that, according to the
Cartesian notation, the multiplication of two powers of the same letter

amounts to adding their exponents, and that their division amounts to
subtracting the exponents of the power of the divisor from that of the power
of the dividend, when the second of these exponents is greater than the first.
Wallis extended this result to the case where the first exponent is equal to or
greater than the second, which makes the difference zero or negative. He
supposed then that a negative exponent indicates unity divided by the
quantity raised to the same exponent taken positively. These remarks led
him to integrate generally the monomial differentials, whence he inferred
the definite integrals of a particular kind of binomial differentials whose
exponent is a positive integral number. The observation then of the law of
the numbers which express these integrals, a series of interpolations and
happy inductions where one perceives the germ of the calculus of definite
integrals which has so much exercised geometricians and which is one of
the fundaments of my new Theory of Probabilities, gave him the ratio of the
area of the circle to the square of its diameter expressed by an infinite
product, which, when one stops it, confines this ratio to limits more and
more converging; this is one of the most singular results in analysis. But it
is remarkable that Wallis, who had so well considered the fractional
exponents of radical powers, should have continued to note these powers as
had been done before him. Newton in his Letters to Oldembourg, if I am not
mistaken, was the first to employ the notation of these powers by fractional
exponents. Comparing by the way of induction, of which Wallis had made
such a beautiful use, the exponents of the powers of the binomial with the
coefficients of the terms of its development in the case where this exponent
is integral and positive, he determined the law of these coefficients and
extended it by analogy to fractional and negative powers. These various
results, based upon the notation of Descartes, show his influence on the
progress of analysis. It has still the advantage of giving the simplest and
fairest idea of logarithms, which are indeed only the exponents of a
magnitude whose successive powers, increasing by infinitely small degrees,
can represent all numbers.
But the most important extension that this notation has received is that of
variable exponents, which constitutes exponential calculus, one of the most
fruitful branches of modern analysis. Leibnitz was the first to indicate the
transcendents by variable exponents, and thereby he has completed the

system of elements of which a finite function can be composed; for every
finite explicit function of a variable may be reduced in the last analysis to
simple magnitudes, combined by the method of addition, subtraction,
multiplication, and division and raised to constant or variable powers. The
roots of the equations formed from these elements are the implicit functions
of the variable. It is thus that a variable has for a logarithm the exponent of
the power which is equal to it in the series of the powers of the number
whose hyperbolic logarithm is unity, and the logarithm of a variable of it is
an implicit function.
Leibnitz thought to give to his differential character the same exponents
as to magnitudes; but then in place of indicating the repeated
multiplications of the same magnitude these exponents indicate the repeated
differentiations of the same function. This new extension of the Cartesian
notation led Leibnitz to the analogy of positive powers with the
differentials, and the negative powers with the integrals. Lagrange has
followed this singular analogy in all its developments; and by series of
inductions which may be regarded as one of the most beautiful applications
which have ever been made of the method of induction he has arrived at
general formulæ which are as curious as useful on the transformations of
differences and of integrals the ones into the others when the variables have
divers finite increments and when these increments are infinitely small. But
he has not given the demonstrations of it which appear to him difficult. The
theory of discriminant functions extends the Cartesian notations to some of
its characters; it shows with proof the analogy of the powers and operations
indicated by these characters; so that it may still be regarded as the
exponential calculus of characters. All that concerns the series and the
integration of equations of differences springs from it with an extreme
facility.

PART II.
APPLICATIONS OF THE CALCULUS OF
PROBABILITIES.

CHAPTER VI.
GAMES OF CHANCE.
The combinations which games present were the object of the first
investigations of probabilities. In an infinite variety of these combinations
many of them lend themselves readily to calculus; others require more
difficult calculi; and the difficulties increasing in the measure that the
combinations become more complicated, the desire to surmount them and
curiosity have excited geometricians to perfect more and more this kind of
analysis. It has been seen already that the benefits of a lottery are easily
determined by the theory of combinations. But it is more difficult to know
in how many draws one can bet one against one, for example that all the
numbers will be drawn, n being the number of numbers, r that of the
numbers drawn at each draw, and i the unknown number of draws. The
expression of the probability of drawing all the numbers depends upon the
nth finite difference of the i power of a product of r consecutive numbers.
When the number n is considerable the search for the value of i which
renders this probability equal to ½ becomes impossible at least unless this
difference is converted into a very converging series. This is easily done by
the method here below indicated by the approximations of functions of very
large numbers. It is found thus since the lottery is composed of ten thousand
numbers, one of which is drawn at each draw, that there is a disadvantage in
betting one against one that all the numbers will be drawn in 95767 draws

and an advantage in making the same bet for 95768 draws. In the lottery of
France this bet is disadvantageous for 85 draws and advantageous for 86
draws.
Let us consider again two players, A and B, playing together at heads and
tails in such a manner that at each throw if heads turns up A gives one
counter to B, who gives him one if tails turns up; the number of counters of
B is limited, while that of A is unlimited, and the game is to end only when
B shall have no more counters. We ask in how many throws one should bet
one to one that the game will end. The expression of the probability that the
game will end in an i number of throws is given by a series which
comprises a great number of terms and factors if the number of counters of
B is considerable; the search for the value of the unknown i which renders
this series ½ would then be impossible if we did not reduce the same to a
very convergent series. In applying to it the method of which we have just
spoken, we find a very simple expression for the unknown from which it
results that if, for example, B has a hundred counters, it is a bet of a little
less than one against one that the game will end in 23780 throws, and a bet
of a little more than one against one that it will end in 23781 throws.
These two examples added to those we have already given are sufficient
to shows how the problems of games have contributed to the perfection of
analysis.

CHAPTER VII.
CONCERNING THE UNKNOWN INEQUALITIES
WHICH MAY EXIST AMONG CHANCES WHICH
ARE SUPPOSED EQUAL.
Inequalities of this kind have upon the results of the calculation of
probabilities a sensible influence which deserves particular attention. Let us
take the game of heads and tails, and let us suppose that it is equally easy to
throw the one or the other side of the coin. Then the probability of throwing
heads at the first throw is ½ and that of throwing it twice in succession is ¼.
But if there exist in the coin an inequality which causes one of the faces to
appear rather than the other without knowing which side is favored by this
inequality, the probability of throwing heads at the first throw will always
½; because of our ignorance of which face is favored by the inequality the
probability of the simple event is increased if this inequality is favorable to
it, just so much is it diminished if the inequality is contrary to it. But in this
same ignorance the probability of throwing heads twice in succession is
increased. Indeed this probability is that of throwing heads at the first throw
multiplied by the probability that having thrown it at the first throw it will
be thrown at the second; but its happening at the first throw is a reason for
belief that the inequality of the coin favors it; the unknown inequality
increases, then, the probability of throwing heads at the second throw; it
consequently increases the product of these two probabilities. In order to
submit this matter to calculus let us suppose that this inequality increases by
a twentieth the probability of the simple event which it favors. If this event
is heads, its probability will be ½ plus 1⁄20, or 11⁄20, and the probability of
throwing it twice in succession will be the square of 11⁄20, or 121⁄400. If the
favored event is tails, the probability of heads, will be ½ minus 1⁄20, or 9⁄20,
and the probability of throwing it twice in succession will be 81⁄400. Since we

have at first no reason for believing that the inequality favors one of these
events rather than the other, it is clear that in order to have the probability of
the compound event heads heads it is necessary to add the two preceding
probabilities and take the half of their sum, which gives 101⁄400 for this
probability, which exceeds ¼ by 1⁄400 or by the square of the favor 1⁄20 that
the inequality adds to the possibilities of the event which it favors. The
probability of throwing tails tails is similarly 101⁄400, but the probability of
throwing heads tails or tails heads is each 99⁄400; for the sum of these four
probabilities ought to equal certainty or unity. We find thus generally that
the constant and unknown causes which favor simple events which are
judged equally possible always increase the probability of the repetition of
the same simple event.
In an even number of throws heads and tails ought both to happen either
an even number of times or odd number of times. The probability of each of
these cases is ½ if the possibilities of the two faces are equal; but if there is
between them an unknown inequality, this inequality is always favorable to
the first case.
Two players whose skill is supposed to be equal play on the conditions
that at each throw that one who loses gives a counter to his adversary, and
that the game continues until one of the players has no more counters. The
calculation of the probabilities shows us that for the equality of the play the
throws of the players ought to be an inverse ratio to their counters. But if
there is between the players a small unknown inequality, it favors that one
of the players who has the smallest number of counters. His probability of
winning the game increases if the players agree to double or triple their
counters; and it will be ½ or the same as the probability of the other player
in the case where the number of their counters should become infinite,
preserving always the same ratio.
One may correct the influence of these unknown inequalities by
submitting them themselves to the chances of hazard. Thus at the play of
heads and tails, if one has a second coin which is thrown each time with the
first and one agrees to name constantly heads the face turned up by the
second coin, the probability of throwing heads twice in succession with the

first coin will approach much nearer ¼ than in the case of a single coin. In
this last case the difference is the square of the small increment of
possibility that the unknown inequality gives to the face of the first coin
which it favors; in the other case this difference is the quadruple product of
this square by the corresponding square relative to the second coin.
Let there be thrown into an urn a hundred numbers from 1 to 100 in the
order of numeration, and after having shaken the urn in order to mix the
numbers one is drawn; it is clear that if the mixing has been well done the
probabilities of the drawing of the numbers will be the same. But if we fear
that there is among them small differences dependent upon the order
according to which the numbers have been thrown into the urn, we shall
diminish considerably these differences by throwing into a second urn the
numbers according to the order of their drawing from the first urn, and by
shaking then this second urn in order to mix the numbers. A third urn, a
fourth urn, etc., would diminish more and more these differences already
inappreciable in the second urn.

CHAPTER VIII.
CONCERNING THE LAWS OF PROBABILITY
WHICH RESULT FROM THE INDEFINITE
MULTIPLICATION OF EVENTS.
Amid the variable and unknown causes which we comprehend under the
name of chance, and which render uncertain and irregular the march of
events, we see appearing, in the measure that they multiply, a striking
regularity which seems to hold to a design and which has been considered
as a proof of Providence. But in reflecting upon this we soon recognize that
this regularity is only the development of the respective possibilities of
simple events which ought to present themselves more often when they are
more probable. Let us imagine, for example, an urn which contains white
balls and black balls; and let us suppose that each time a ball is drawn it is
put back into the urn before proceeding to a new draw. The ratio of the
number of the white balls drawn to the number of black balls drawn will be
most often very irregular in the first drawings; but the variable causes of
this irregularity produce effects alternately favorable and unfavorable to the
regular march of events which destroy each other mutually in the totality of
a great number of draws, allowing us to perceive more and more the ratio of
white balls to the black balls contained in the urn, or the respective
possibilities of drawing a white ball or black ball at each draw. From this
results the following theorem.
The probability that the ratio of the number of white balls drawn to the
total number of balls drawn does not deviate beyond a given interval from
the ratio of the number of white balls to the total number of balls contained
in the urn, approaches indefinitely to certainty by the indefinite
multiplication of events, however small this interval.

This theorem indicated by common sense was difficult to demonstrate by
analysis. Accordingly the illustrious geometrician Jacques Bernoulli, who
first has occupied himself with it, attaches great importance to the
demonstrations he has given. The calculus of discriminant functions applied
to this matter not only demonstrates with facility this theorem, but still more
it gives the probability that the ratio of the events observed deviates only in
certain limits from the true ratio of their respective possibilities.
One may draw from the preceding theorem this consequence which
ought to be regarded as a general law, namely, that the ratios of the acts of
nature are very nearly constant when these acts are considered in great
number. Thus in spite of the variety of years the sum of the productions
during a considerable number of years is sensibly the same; so that man by
useful foresight is able to provide against the irregularity of the seasons by
spreading out equally over all the seasons the goods which nature
distributes in an unequal manner. I do not except from the above law results
due to moral causes. The ratio of annual births to the population, and that of
marriages to births, show only small variations; at Paris the number of
annual births is almost the same, and I have heard it said at the post-office
in ordinary seasons the number of letters thrown aside on account of
defective addresses changes little each year; this has likewise been observed
at London.
It follows again from this theorem that in a series of events indefinitely
prolonged the action of regular and constant causes ought to prevail in the
long run over that of irregular causes. It is this which renders the gains of
the lotteries just as certain as the products of agriculture; the chances which
they reserve assure them a benefit in the totality of a great number of
throws. Thus favorable and numerous chances being constantly attached to
the observation of the eternal principles of reason, of justice, and of
humanity which establish and maintain societies, there is a great advantage
in conforming to these principles and of grave inconvenience in departing
from them. If one consult histories and his own experience, one will see all
the facts come to the aid of this result of calculus. Consider the happy
effects of institutions founded upon reason and the natural rights of man
among the peoples who have known how to establish and preserve them.
Consider again the advantages which good faith has procured for the

governments who have made it the basis of their conduct and how they
have been indemnified for the sacrifices which a scrupulous exactitude in
keeping their engagements has cost them. What immense credit at home!
What preponderance abroad! On the contrary, look into what an abyss of
misfortunes nations have often been precipitated by the ambition and the
perfidy of their chiefs. Every time that a great power intoxicated by the love
of conquest aspires to universal domination the sentiment of independence
produces among the menaced nations a coalition of which it becomes
almost always the victim. Similarly in the midst of the variable causes
which extend or restrain the divers states, the natural limits acting as
constant causes ought to end by prevailing. It is important then to the
stability as well as to the happiness of empires not to extend them beyond
those limits into which they are led again without cessation by the action of
the causes; just as the waters of the seas raised by violent tempests fall
again into their basins by the force of gravity. It is again a result of the
calculus of probabilities confirmed by numerous and melancholy
experiences. History treated from the point of view of the influence of
constant causes would unite to the interest of curiosity that of offering to
man most useful lessons. Sometimes we attribute the inevitable results of
these causes to the accidental circumstances which have produced their
action. It is, for example, against the nature of things that one people should
ever be governed by another when a vast sea or a great distance separates
them. It may be affirmed that in the long run this constant cause, joining
itself without ceasing to the variable causes which act in the same way and
which the course of time develops, will end by finding them sufficiently
strong to give to a subjugated people its natural independence or to unite it
to a powerful state which may be contiguous.
In a great number of cases, and these are the most important of the
analysis of hazards, the possibilities of simple events are unknown and we
are forced to search in past events for the indices which can guide us in our
conjectures about the causes upon which they depend. In applying the
analysis of discriminant functions to the principle elucidated above on the
probability of the causes drawn from the events observed, we are led to the
following theorem.

When a simple event or one composed of several simple events, as, for
instance, in a game, has been repeated a great number of times the
possibilities of the simple events which render most probable that which has
been observed are those that observation indicates with the greatest
probability; in the measure that the observed event is repeated this
probability increases and would end by amounting to certainty if the
numbers of repetitions should become infinite.
There are two kinds of approximations: the one is relative to the limits
taken on all sides of the possibilities which give to the past the greatest
probability; the other approximation is related to the probability that these
possibilities fall within these limits. The repetition of the compound event
increases more and more this probability, the limits remaining the same; it
reduces more and more the interval of these limits, the probability
remaining the same; in infinity this interval becomes zero and the
probability changes to certainty.
If we apply this theorem to the ratio of the births of boys to that of girls
observed in the different countries of Europe, we find that this ratio, which
is everywhere about equal to that of 22 to 21, indicates with an extreme
probability a greater facility in the birth of boys. Considering further that it
is the same at Naples and at St. Petersburg, we shall see that in this regard
the influence of climate is without effect. We might then suspect, contrary
to the common belief, that this predominance of masculine births exists
even in the Orient. I have consequently invited the French scholars sent to
Egypt to occupy themselves with this interesting question; but the difficulty
in obtaining exact information about the births has not permitted them to
solve it. Happily, M. de Humboldt has not neglected this matter among the
innumerable new things which he has observed and collected in America
with so much sagacity, constancy, and courage. He has found in the tropics
the same ratio of the births as we observe in Paris; this ought to make us
regard the greater number of masculine births as a general law of the human
race. The laws which the different kinds of animals follow in this regard
seem to me worthy of the attention of naturalists.
The fact that the ratio of births of boys to that of girls differs very little
from unity even in the great number of the births observed in a place would

offer in this regard a result contrary to the general law, without which we
should be right in concluding that this law did not exist. In order to arrive at
this result it is necessary to employ great numbers and to be sure that it is
indicated by great probability. Buffon cites, for example, in his Political
Arithmetic several communities of Bourgogne where the births of girls have
surpassed those of boys. Among these communities that of Carcelle-le-
Grignon presents in 2009 births during five years 1026 girls and 983 boys.
Although these numbers are considerable, they indicate, however, only a
greater possibility in the births of girls with a probability of 9⁄10, and this
probability, smaller than that of not throwing heads four times in succession
in the game of heads and tails, is not sufficient to investigate the cause for
this anomaly, which, according to all probability, would disappear if one
should follow during a century the births in this community.
The registers of births, which are kept with care in order to assure the
condition of the citizens, may serve in determining the population of a great
empire without recurring to the enumeration of its inhabitants—a laborious
operation and one difficult to make with exactitude. But for this it is
necessary to know the ratio of the population to the annual births. The most
precise means of obtaining it consists, first, in choosing in the empire
districts distributed in an almost equal manner over its whole surface, so as
to render the general result independent of local circumstances; second, in
enumerating with care for a given epoch the inhabitants of several
communities in each of these districts; third, by determining from the
statement of the births during several years which precede and follow this
epoch the mean number corresponding to the annual births. This number,
divided by that of the inhabitants, will give the ratio of the annual births to
the population in a manner more and more accurate as the enumeration
becomes more considerable. The government, convinced of the utility of a
similar enumeration, has decided at my request to order its execution. In
thirty districts spread out equally over the whole of France, communities
have been chosen which would be able to furnish the most exact
information. Their enumerations have given 2037615 individuals as the
total number of their inhabitants on the 23d of September, 1802. The
statement of the births in these communities during the years 1800, 1801,
and 1802 have given:

Births Marriages Deaths
110312 boys 46037 103659 men
105287 girls 99443 women

The ratio of the population to annual births is then 28352845⁄1000000; it is
greater than had been estimated up to this time. Multiplying the number of
annual births in France by this ratio, we shall have the population of this
kingdom. But what is the probability that the population thus determined
will not deviate from the true population beyond a given limit? Resolving
this problem and applying to its solution the preceding data, I have found
that, the number of annual births in France being supposed to be 1000000,
which brings the population to 28352845 inhabitants, it is a bet of almost
300000 against 1 that the error of this result is not half a million.
The ratio of the births of boys to that of girls which the preceding
statement offers is that of 22 to 21; and the marriages are to the births as 3
is to 4.
At Paris the baptisms of children of both sexes vary a little from the ratio
of 22 to 21. Since 1745, the epoch in which one has commenced to
distinguish the sexes upon the birth-registers, up to the end of 1784, there
have been baptized in this capital 393386 boys and 377555 girls. The ratio
of the two numbers is almost that of 25 to 24; it appears then at Paris that a
particular cause approximates an equality of baptisms of the two sexes. If
we apply to this matter the calculus of probabilities, we find that it is a bet
of 238 to 1 in favor of the existence of this cause, which is sufficient to
authorize the investigation. Upon reflection it has appeared to me that the
difference observed holds to this, that the parents in the country and the
provinces, finding some advantage in keeping the boys at home, have sent
to the Hospital for Foundlings in Paris fewer of them relative to the number
of girls according to the ratio of births of the two sexes. This is proved by
the statement of the registers of this hospital. From the beginning of 1745 to
the end of 1809 there were entered 163499 boys and 159405 girls. The first
of these numbers exceeds only by 1⁄38 the second, which it ought to have
surpassed at least by 1⁄24. This confirms the existence of the assigned cause,

namely, that the ratio of births of boys to those of girls is at Paris that of 22
to 21, no attention having been paid to foundlings.
The preceding results suppose that we may compare the births to the
drawings of balls from an urn which contains an infinite number of white
balls and black balls so mixed that at each draw the chances of drawing
ought to be the same for each ball; but it is possible that the variations of
the same seasons in different years may have some influence upon the
annual ratio of the births of boys to those of girls. The Bureau of
Longitudes of France publishes each year in its annual the tables of the
annual movement of the population of the kingdom. The tables already
published commence in 1817; in that year and in the five following years
there were born 2962361 boys and 2781997 girls, which gives about 16⁄15 for
the ratio of the births of boys to that of girls. The ratios of each year vary
little from this mean result; the smallest ratio is that of 1822, where it was
only 17⁄16; the greatest is of the year 1817, when it was 15⁄14. These ratios vary
appreciably from the ratio of 22⁄21 found above. Applying to this deviation
the analysis of probabilities in the hypothesis of the comparison of births to
the drawings of balls from an urn, we find that it would be scarcely
probable. It appears, then, to indicate that this hypothesis, although closely
approximated, is not rigorously exact. In the number of births which we
have just stated there are of natural children 200494 boys and 190698 girls.
The ratio of masculine and feminine births was then in this regard 20⁄19,
smaller than the mean ratio of 16⁄15. This result is in the same sense as that of
the births of foundlings; and it seems to prove that in the class of natural
children the births of the two sexes approach more nearly equality than in
the class of legitimate children. The difference of the climates from the
north to the south of France does not appear to influence appreciably the
ratio of the births of boys and girls. The thirty most southern districts have
given 16⁄15 for this ratio, the same as that of entire France.
The constancy of the superiority of the births of boys over girls at Paris
and at London since they have been observed has appeared to some
scholars to be a proof of Providence, without which they have thought that
the irregular causes which disturb without ceasing the course of events

ought several times to have rendered the annual births of girls superior to
those of boys.
But this proof is a new example of the abuse which has been so often
made of final causes which always disappear on a searching examination of
the questions when we have the necessary data to solve them. The
constancy in question is a result of regular causes which give the superiority
to the births of boys and which extend it to the anomalies due to hazard
when the number of annual births is considerable. The investigation of the
probability that this constancy will maintain itself for a long time belongs to
that branch of the analysis of hazards which passes from past events to the
probability of future events; and taking as a basis the births observed from
1745 to 1784, it is a bet of almost 4 against 1 that at Paris the annual births
of boys will constantly surpass for a century the births of girls; there is then
no reason to be astonished that this has taken place for a half-century.
Let us take another example of the development of constant ratios which
events present in the measure that they are multiplied. Let us imagine a
series of urns arranged circularly, and each containing a very great number
of white balls and black balls; the ratio of white balls to the black in the
urns being originally very different and such, for example, that one of these
urns contains only white balls, while another contains only black balls. If
one draws a ball from the first urn in order to put it into the second, and,
after having shaken the second urn in order to mix well the new ball with
the others, one draws a ball to put it into the third urn, and so on to the last
urn, from which is drawn a ball to put into the first, and if this series is
recommenced continually, the analysis of probability shows us that the
ratios of the white balls to the black in these urns will end by being the
same and equal to the ratio of the sum of all the white balls to the sum of all
the black balls contained in the urns. Thus by this regular mode of change
the primitive irregularity of these ratios disappears eventually in order to
make room for the most simple order. Now if among these urns one
intercalate new ones in which the ratio of the sum of the white balls to the
sum of the black balls which they contain differs from the preceding,
continuing indefinitely in the totality of the urns the drawings which we
have just indicated, the simple order established in the old urns will be at
first disturbed, and the ratios of the white balls to the black balls will

become irregular; but little by little this irregularity will disappear in order
to make room for a new order, which will finally be that of the equality of
the ratios of the white balls to the black balls contained in the urns. We may
apply these results to all the combinations of nature in which the constant
forces by which their elements are animated establish regular modes of
action, suited to bring about in the very heart of chaos systems governed by
admirable laws.
The phenomena which seem the most dependent upon hazard present,
then, when multiplied a tendency to approach without ceasing fixed ratios,
in such a manner that if we conceive on all sides of each of these ratios an
interval as small as desired, the probability that the mean result of the
observations falls within this interval will end by differing from certainty
only by a quantity greater than an assignable magnitude. Thus by the
calculations of probabilities applied to a great number of observations we
may recognize the existence of these ratios. But before seeking the causes it
is necessary, in order not to be led into vain speculations, to assure
ourselves that they are indicated by a probability which does not permit us
to regard them as anomalies due to hazard. The theory of discriminant
functions gives a very simple expression for this probability, which is
obtained by integrating the product of the differential of the quantity of
which the result deduced from a great number of observations varies from
the truth by a constant less than unity, dependent upon the nature of the
problem, and raised to a power whose exponent is the ratio of the square of
this variation to the number of observations. The integral taken between the
limits given and divided by the same integral, applied to a positive and
negative infinity, will express the probability that the variation from the
truth is comprised between these limits. Such is the general law of the
probability of results indicated by a great number of observations.

CHAPTER IX.
THE APPLICATION OF THE CALCULUS OF
PROBABILITIES TO NATURAL PHILOSOPHY.
The phenomena of nature are most often enveloped by so many strange
circumstances, and so great a number of disturbing causes mix their
influence, that it is very difficult to recognize them. We may arrive at them
only by multiplying the observations or the experiences, so that the strange
effects finally destroy reciprocally each other, the mean results putting in
evidence those phenomena and their divers elements. The more numerous
the number of observations and the less they vary among themselves the
more their results approach the truth. We fulfil this last condition by the
choice of the methods of observations, by the precision of the instruments,
and by the care which we take to observe closely; then we determine by the
theory of probabilities the most advantageous mean results or those which
give the least value of the error. But that is not sufficient; it is further
necessary to appreciate the probability that the errors of these results are
comprised in the given limits; and without this we have only an imperfect
knowledge of the degree of exactitude obtained. Formulæ suitable to these
matters are then true improvements of the method of sciences, and it is
indeed important to add them to this method. The analysis which they
require is the most delicate and the most difficult of the theory of
probabilities; it is one of the principal objects of the work which I have
published upon this theory, and in which I have arrived at formulæ of this
kind which have the remarkable advantage of being independent of the law
of the probability of errors and of including only the quantities given by the
observations themselves and their expressions.
Each observation has for an analytic expression a function of the
elements which we wish to determine; and if these elements are nearly
known, this function becomes a linear function of their corrections. In

equating it to the observation itself there is formed an equation of condition.
If we have a great number of similar equations, we combine them in such a
manner as to obtain as many final equations as there are elements whose
corrections we determine then by resolving these equations. But what is the
most advantageous manner of combining equations of condition in order to
obtain final equations? What is the law of the probabilities of errors of
which the elements are still susceptible that we draw from them? This is
made clear to us by the theory of probabilities. The formation of a final
equation by means of the equation of condition amounts to multiplying each
one of these by an indeterminate factor and by uniting the products; it is
necessary to choose the system of factors which gives the smallest
opportunity for error. But it is apparent that if we multiply the possible
errors of an element by their respective probabilities, the most advantageous
system will be that in which the sum of these products all, taken, positively
is a minimum; for a positive or a negative error ought to be considered as a
loss. Forming, then, this sum of products, the condition of the minimum will
determine the system of factors which it is expedient to adopt, or the most
advantageous system. We find thus that this system is that of the
coefficients of the elements in each equation of condition; so that we form a
first final equation by multiplying respectively each equation of condition
by its coefficient of the first element and by uniting all these equations thus
multiplied. We form a second final equation by employing in the same
manner the coefficients of the second element, and so on. In this manner the
elements and the laws of the phenomena obtained in the collection of a
great number of observations are developed with the most evidence.
The probability of the errors which each element still leaves to be feared
is proportional to the number whose hyperbolic logarithm is unity raised to
a power equal to the square of the error taken as a minus quantity and
multiplied by a constant coefficient which may be considered as the
modulus of the probability of the errors; because, the error remaining the
same, its probability decreases with rapidity when the former increases; so
that the element obtained weighs, if I may thus speak toward the truth, as
much more as this modulus is greater. I would call for this reason this
modulus the weight of the element or of the result. This weight is the
greatest possible in the system of factors—the most advantageous; it is this

which gives to this system superiority over others. By a remarkable analogy
of this weight with those of bodies compared at their common centre of
gravity it results that if the same element is given by divers systems,
composed each of a great number of observations, the most advantageous,
the mean result of their totality is the sum of the products of each partial
result by its weight. Moreover, the total weight of the results of the divers
systems is the sum of their partial weights; so that the probability of the
errors of the mean result of their totality is proportional to the number
which has unity for an hyperbolic logarithm raised to a power equal to the
square of the error taken as minus and multiplied by the sum of the weights.
Each weight depends in truth upon the law of the probability of error of
each system, and almost always this law is unknown; but happily I have
been able to eliminate the factor which contains it by means of the sum of
the squares of the variations of the observations in this system from their
mean result. It would then be desirable in order to complete our knowledge
of the results obtained by the totality of a great number of observations that
we write by the side of each result the weight which corresponds to it;
analysis furnishes for this object both general and simple methods. When
we have thus obtained the exponential which represents the law of the
probability of errors, we shall have the probability that the error of the
result is included within given limits by taking within the limits the integral
of the product of this exponential by the differential of the error and
multiplying it by the square root of the weight of the result divided by the
circumference whose diameter is unity. Hence it follows that for the same
probability the errors of the results are reciprocal to the square roots of their
weights, which serves to compare their respective precision.
In order to apply this method with success it is necessary to vary the
circumstances of the observations or the experiences in such a manner as to
avoid the constant causes of error. It is necessary that the observations
should be numerous, and that they should be so much the more so as there
are more elements to determine; for the weight of the mean result increases
as the number of observations divided by the number of the elements. It is
still necessary that the elements follow in these observations a different
course; for if the course of the two elements were exactly the same, which
would render their coefficients proportional in equation of conditions, these

elements would form only a single unknown quantity and it would be
impossible to distinguish them by these observations. Finally it is necessary
that the observations should be precise; this condition, the first of all,
increases greatly the weight of the result the expression of which has for a
divisor the sum of the squares of the deviations of the observations from
this result. With these precautions we shall be able to make use of the
preceding method and measure the degree of confidence which the results
deduced from a great number of observations merit.
The rule which we have just given to conclude equations of condition,
final equations, amount to rendering a minimum the sum of the squares of
the errors of observations; for each equation of condition becomes exact by
substituting in it the observation plus its error; and if we draw from it the
expression of this error, it is easy to see that the condition of the minimum
of the sum of the squares of these expressions gives the rule in question.
This rule is the more precise as the observations are more numerous; but
even in the case where their number is small it appears natural to employ
the same rule which in all cases offers a simple means of obtaining without
groping the corrections which we seek to determine. It serves further to
compare the precision of the divers astronomical tables of the same star.
These tables may always be supposed as reduced to the same form, and
then they differ only by the epochs, the mean movements and the
coefficients of the arguments; for if one of them contains a coefficient
which is not found in the others, it is clear that this amounts to supposing
zero in them as the coefficient of this argument. If now we rectify these
tables by the totality of the good observations, they would satisfy the
condition that the sum of the squares of the errors should be a minimum;
the tables which, compared to a considerable number of observations,
approach nearest this condition merit then the preference.
It is principally in astronomy that the method explained above may be
employed with advantage. The astronomical tables owe the truly
astonishing exactitude which they have attained to the precision of
observations and of theories, and to the use of equations of conditions
which cause to concur a great number of excellent observations in the
correction of the same element. But it remains to determine the probability
of the errors that this correction leaves still to be feared; and the method

which I have just explained enables us to recognize the probability of these
errors. In order to give some interesting applications of it I have profited by
the immense work which M. Bouvard has just finished on the movements
of Jupiter and Saturn, of which he has formed very precise tables. He has
discussed with the greatest care the oppositions and quadratures of these
two planets observed by Bradley and by the astronomers who have
followed him down to the last years; he has concluded the corrections of the
elements of their movement and their masses compared to that of the sun
taken as unity. His calculations give him the mass of Saturn equal to the
3512th part of that of the sun. Applying to them my formulæ of probability,
I find that it is a bet of 11,000 against one that the error of this result is not
1
⁄100 of its value, or that which amounts to almost the same—that after a
century of new observations added to the preceding ones, and examined in
the same manner, the new result will not differ by 1⁄100 from that of M.
Bouvard. This wise astronomer finds again the mass of Jupiter equal to the
1071th part of the sun; and my method of probability gives a bet of
1,000,000 to one that this result is not 1⁄100 in error.
This method may be employed again with success in geodetic operations.
We determine the length of the great arc on the surface of the earth by
triangulation, which depends upon a base measured with exactitude. But
whatever precision may be brought to the measure of the angles, the
inevitable errors can, by accumulating, cause the value of the arc concluded
from a great number of triangles to deviate appreciably from the truth. We
recognize this value, then, only imperfectly unless the probability that its
error is comprised within given limits can be assigned. The error of a
geodetic result is a function of the errors of the angles of each triangle. I
have given in the work cited general formulæ in order to obtain the
probability of the values of one or of several linear functions of a great
number of partial errors of which we know the law of probability; we may
then by means of these formulæ determine the probability that the error of a
geodetic result is contained within the assigned limits, whatever may be the
law of the probability of partial errors. It is moreover more necessary to
render ourselves independent of the law, since the most simple laws
themselves are always infinitely less probable, seeing the infinite number of

those which may exist in nature. But the unknown law of partial errors
introduces into the formulæ an indeterminant which does not permit of
reducing them to numbers unless we are able to eliminate it. We have seen
that in astronomical questions, where each observation furnishes an
equation of condition for obtaining the elements, we eliminate this
determinant by means of the sum of the squares of the remainders when the
most probable values of the elements have been substituted in each
equation. Geodetic questions not offering similar equations, it is necessary
to seek another means of elimination. The quantity by which the sum of the
angles of each observed triangle surpasses two right angles plus the
spherical excess furnishes this means. Thus we replace by the sum of the
squares of these quantities the sum of the squares of the remainders of the
equations of condition; and we may assign in numbers the probability that
the error of the final result of a series of geodetic operations will not exceed
a given quantity. But what is the most advantageous manner of dividing
among the three angles of each triangle the observed sum of their errors?
The analysis of probabilities renders it apparent that each angle ought to be
diminished by a third of this sum, provided that the weight of a geodetic
result be the greatest possible, which renders the same error less probable.
There is then a great advantage in observing the three angles of each
triangle and of correcting them as we have just said. Simple common sense
indicates this advantage; but the calculation of probabilities alone is able to
appreciate it and to render apparent that by this correction it becomes the
greatest possible.
In order to assure oneself of the exactitude of the value of a great arc
which rests upon a base measured at one of its extremities one measures a
second base toward the other extremity; and one concludes from one of
these bases the length of the other. If this length varies very little from the
observation, there is all reason to believe that the chain of triangles which
unites these bases is very nearly exact and likewise the value of the large
arc which results from it. One corrects, then, this value by modifying the
angles of the triangles in such a manner that the base is calculated according
to the bases measured. But this may be done in an infinity of ways, among
which is preferred that of which the geodetic result has the greatest weight,
inasmuch as the same error becomes less probable. The analysis of

probabilities gives formulæ for obtaining directly the most advantageous
correction which results from the measurements of the several bases and the
laws of probability which the multiplicity of the bases makes—laws which
become very rapidly decreasing by this multiplicity.
Generally the errors of the results deduced from a great number of
observations are the linear functions of the partial errors of each
observation. The coefficients of these functions depend upon the nature of
the problem and upon the process followed in order to obtain the results.
The most advantageous process is evidently that in which the same error in
the results is less probable than according to any other process. The
application of the calculus of probabilities to natural philosophy consists,
then, in determining analytically the probability of the values of these
functions and in choosing their indeterminant coefficients in such a manner
that the law of this probability should be most rapidly descending.
Eliminating, then, from the formulæ by the data of the question the factor
which is introduced by the almost always unknown law of the probability of
partial errors, we may be able to evaluate numerically the probability that
the errors of the results do not exceed a given quantity. We shall thus have
all that may be desired touching the results deduced from a great number of
observations.
Very approximate results may be obtained by other considerations.
Suppose, for example, that one has a thousand and one observations of the
same quantity; the arithmetical mean of all these observations is the result
given by the most advantageous method. But one would be able to choose
the result according to the condition that the sum of the variations from
each partial value all taken positively should be a minimum. It appears
indeed natural to regard as very approximate the result which satisfies this
condition. It is easy to see that if one disposes the values given by the
observations according to the order of magnitude, the value which will
occupy the mean will fulfil the preceding condition, and calculus renders it
apparent that in the case of an infinite number of observations it would
coincide with the truth; but the result given by the most advantageous
method is still preferable.

We see by that which precedes that the theory of probabilities leaves
nothing arbitrary in the manner of distributing the errors of the
observations; it gives for this distribution the most advantageous formulæ
which diminishes as much as possible the errors to be feared in the results.
The consideration of probabilities can serve to distinguish the small
irregularities of the celestial movements enveloped in the errors of
observations, and to repass to the cause of the anomalies observed in these
movements.
In comparing all the observations it was Ticho-Brahé who recognized the
necessity of applying to the moon an equation of time different from that
which had been applied to the sun and to the planets. It was similarly the
totality of a great number of observations which made Mayer recognize that
the coefficient of the inequality of the precession ought to be diminished a
little for the moon. But since this diminution, although confirmed and even
augmented by Mason, did not appear to result from universal gravitation,
the majority of astronomers neglect it in their calculations. Having
submitted to the calculation of probabilities a considerable number of lunar
observations chosen for this purpose and which M. Bouvard consented to
examine at my request, it appeared to me to be indicated with so strong a
probability that I believed the cause of it ought to be investigated. I soon
saw that it would be only the ellipticity of the terrestrial spheroid, neglected
up to that time in the theory of the lunar movement as being able to produce
only imperceptible terms. I concluded that these terms became perceptible
by the successive integrations of differential equations. I determined then
those terms by a particular analysis, and I discovered first the inequality of
the lunar movement in latitude which is proportional to the sine of the
longitude of the moon, which no astronomer before had suspected. I
recognized then by means of this inequality that another exists in the lunar
movement in longitude which produces the diminution observed by Mayer
in the equation of the precession applicable to the moon. The quantity of
this diminution and the coefficient of the preceding inequality in latitude are
very appropriate to fix the oblateness of the earth. Having communicated
my researches to M. Burg, who was occupied at that time in perfecting the
tables of the moon by the comparison of all the good observations, I
requested him to determine with a particular care these two quantities. By a

very remarkable agreement the values which he has found give to the earth
the same oblateness, 1⁄305, which differs little from the mean derived from
the measurements of the degrees of the meridian and the pendulum; but
those regarded from the point of view of the influence of the errors of the
observations and of the perturbing causes in these measurements, did not
appear to me exactly determined by these lunar inequalities.
It was again by the consideration of probabilities that I recognized the
cause of the secular equation of the moon. The modern observations of this
star compared to the ancient eclipses had indicated to astronomers an
acceleration in the lunar movement; but the geometricians, and particularly
Lagrange, having vainly sought in the perturbations which this movement
experienced the terms upon which this acceleration depends, reject it. An
attentive examination of the ancient and modern observations and of the
intermediary eclipses observed by the Arabians convinced me that it was
indicated with a great probability. I took up again then from this point of
view the lunar theory, and I recognized that the secular equation of the
moon is due to the action of the sun upon this satellite, combined with the
secular variation of the eccentricity of the terrestrial orb; this brought me to
the discovery of the secular equations of the movements of the nodes and of
the perigees of the lunar orbit, which equations had not been even suspected
by astronomers. The very remarkable agreement of this theory with all the
ancient and modern observations has brought it to a very high degree of
evidence.
The calculus of probabilities has led me similarly to the cause of the great
irregularities of Jupiter and Saturn. Comparing modern observations with
ancient, Halley found an acceleration in the movement of Jupiter and a
retardation in that of Saturn. In order to conciliate the observations he
reduced the movements to two secular equations of contrary signs and
increasing as the squares of the times passed since 1700. Euler and
Lagrange submitted to analysis the alterations which the mutual attraction
of these two planets ought to produce in these movements. They found in
doing this the secular equations; but their results were so different that one
of the two at least ought to be erroneous. I determined then to take up again
this important problem of celestial mechanics, and I recognized the

invariability of the mean planetary movements, which nullified the secular
equations introduced by Halley in the tables of Jupiter and Saturn. Thus
there remain, in order to explain the great irregularity of these planets, only
the attractions of the comets to which many astronomers had effective
recourse, or the existence of an irregularity over a long period produced in
the movements of the two planets by their reciprocal action and affected by
contrary signs for each of them. A theorem which I found in regard to the
inequalities of this kind rendered this inequality very probable. According
to this theorem, if the movement of Jupiter is accelerated, that of Saturn is
retarded, which has already conformed to what Halley had noticed;
moreover, the acceleration of Jupiter resulting from the same theorem is to
the retardation of Saturn very nearly in the ratio of the secular equations
proposed by Halley. Considering the mean movements of Jupiter and Saturn
I was enabled easily to recognize that two times that of Jupiter differed only
by a very small quantity from five times that of Saturn. The period of an
irregularity which would have for an argument this difference would be
about nine centuries. Indeed its coefficient would be of the order of the
cubes of the eccentricities of the orbits; but I knew that by virtue of
successive integrations it acquired for divisor the square of the very small
multiplier of the time in the argument of this inequality which is able to
give it a great value; the existence of this inequality appeared to me then
very probable. The following observation increased then its probability.
Supposing its argument zero toward the epoch of the observations of Ticho-
Brahé, I saw that Halley ought to have found by the comparison of modern
with ancient observations the alterations which he had indicated; while the
comparison of the modern observations among themselves ought to offer
contrary alterations similar to those which Lambert had concluded from this
comparison. I did not then hesitate at all to undertake this long and tedious
calculation necessary to assure myself of this inequality. It was entirely
confirmed by the result of this calculation, which moreover made me
recognize a great number of other inequalities of which the totality has
inclined the tables of Jupiter and Saturn to the precision of the same
observations.
It was again by means of the calculus of probabilities that I recognized
the remarkable law of the mean movements of the three first satellites of

Jupiter, according to which the mean longitude of the first minus three
times that of the second plus two times that of the third is rigorously equal
to the half-circumference. The approximation with which the mean
movements of these stars satisfy this law since their discovery indicates its
existence with an extreme probability. I sought then the cause of it in their
mutual action. The searching examination of this action convinced me that
it was sufficient if in the beginning the ratios of their mean movements had
approached this law within certain limits, because their mutual action had
established and maintained it rigorously. Thus these three bodies will
balance one another eternally in space according to the preceding law
unless strange causes, such as comets, should change suddenly their
movements about Jupiter.
Accordingly it is seen how necessary it is to be attentive to the
indications of nature when they are the result of a great number of
observations, although in other respects they may be inexplicable by known
means. The extreme difficulty of problems relative to the system of the
world has forced geometricians to recur to the approximation which always
leaves room for the fear that the quantities neglected may have an
appreciable influence. When they have been warned of this influence by the
observations, they have recurred to their analysis; in rectifying it they have
always found the cause of the anomalies observed; they have determined
the laws and often they have anticipated the observations in discovering the
inequalities which it had not yet indicated. Thus one may say that nature
itself has concurred in the analytical perfection of the theories based upon
the principle of universal gravity; and this is to my mind one of the
strongest proofs of the truth of this admirable principle.
In the cases which I have just considered the analytical solution of the
question has changed the probability of the causes into certainty. But most
often this solution is impossible and it remains only to augment more and
more this probability. In the midst of numerous and incalculable
modifications which the action of the causes receives then from strange
circumstances these causes conserve always with the effects observed the
proper ratios to make them recognizable and to verify their existence.
Determining these ratios and comparing them with a great number of
observations if one finds that they constantly satisfy it, the probability of the

causes may increase to the point of equalling that of facts in regard to which
there is no doubt. The investigation of these ratios of causes to their effects
is not less useful in natural philosophy than the direct solution of problems
whether it be to verify the reality of these causes or to determine the laws
from their effects; since it may be employed in a great number of questions
whose direct solution is not possible, it replaces it in the most advantageous
manner. I shall discuss here the application which I have made of it to one
of the most interesting phenomena of nature, the flow and the ebb of the
sea.
Pliny has given of this phenomenon a description remarkable for its
exactitude, and in it one sees that the ancients had observed that the tides of
each month are greatest toward the syzygies and smallest toward the
quadratures; that they are higher in the perigees than in the apogees of the
moon, and higher in the equinoxes than in the solstices. They concluded
from this that this phenomenon is due to the action of the sun and moon
upon the sea. In the preface of his work De Stella Martis Kepler admits a
tendency of the waters of the sea toward the moon; but, ignorant of the law
of this tendency, he was able to give on this subject only a probable idea.
Newton converted into certainty the probability of this idea by attaching it
to his great principle of universal gravity. He gave the exact expression of
the attractive forces which produced the flood and the ebb of the sea; and in
order to determine the effects he supposed that the sea takes at each instant
the position of equilibrium which is agreeable to these forces. He explained
in this manner the principal phenomena of the tides; but it followed from
this theory that in our ports the two tides of the same day would be very
unequal if the sun and the moon should have a great declination. At Brest,
for example, the evening tide would be in the syzygies of the solstices about
eight times greater than the morning tide, which is certainly contrary to the
observations which prove that these two tides are very nearly equal. This
result from the Newtonian theory might hold to the supposition that the sea
is agreeable at each instant to a position of equilibrium, a supposition which
is not at all admissible. But the investigation of the true figure of the sea
presents great difficulties. Aided by the discoveries which the geometricians
had just made in the theory of the movement of fluids and in the calculus of
partial differences, I undertook this investigation, and I gave the differential

equations of the movement of the sea by supposing that it covers the entire
earth. In drawing thus near to nature I had the satisfaction of seeing that my
results approached the observations, especially in regard to the little
difference which exists in our ports between the two tides of the solstitial
syzygies of the same day. I found that they would be equal if the sea had
everywhere the same depth; I found further that in giving to this depth
convenient values one was able to augment the height of the tides in a port
conformably to the observations. But these investigations, in spite of their
generality, did not satisfy at all the great differences which even adjacent
ports present in this regard and which prove the influence of local
circumstances. The impossibility of knowing these circumstances and the
irregularity of the basin of the seas and that of integrating the equations of
partial differences which are relative has compelled me to make up the
deficiency by the method I have indicated above. I then endeavored to
determine the greatest ratios possible among the forces which affect all the
molecules of the sea, and their effects observable in our ports. For this I
made use of the following principle, which may be applied to many other
phenomena.
"The state of the system of a body in which the primitive conditions of
the movement have disappeared by the resistances which this movement
meets is periodic as the forces which animate it."
Combining this principle with that of the coexistence of very small
oscillations, I have found an expression of the height of the tides whose
arbitraries contain the effect of local circumstances of each port and are
reduced to the smallest number possible; it is only necessary to compare it
to a great number of observations.
Upon the invitation of the Academy of Sciences, observations were made
at the beginning of the last century at Brest upon the tides, which were
continued during six consecutive years. The situation of this port is very
favorable to this sort of observations; it communicates with the sea by a
canal which empties into a vast roadstead at the far end of which the port
has been constructed. The irregularities of the sea extend thus only to a
small degree into the port, just as the oscillations which the irregular
movement of a vessel produces in a barometer are diminished by a

throttling made in the tube of this instrument. Moreover, the tides being
considerable at Brest, the accidental variations caused by the winds are only
feeble; likewise we notice in the observations of these tides, however little
we multiply them, a great regularity which induced me to propose to the
government to order in this port a new series of observations of the tides,
continued during a period of the movement of the nodes of the lunar orbit.
This has been done. The observations began June 1, 1806; and since this
time they have been made every day without interruption. I am indebted to
the indefatigable zeal of M. Bouvard, for all that interests astronomy, the
immense calculations which the comparison of my analysis with the
observations has demanded. There have been used about six thousand
observations, made during the year 1807 and the fifteen years following. It
results from this comparison that my formulæ represent with a remarkable
precision all the varieties of the tides relative to the digression of the moon,
from the sun, to the declination of these stars, to their distances from the
earth, and to the laws of variation at the maximum and minimum of each of
these elements. There results from this accord a probability that the flow
and the ebb of the sea is due to the attraction of the sun and moon, so
approaching certainty that it ought to leave room for no reasonable doubt. It
changes into certainty when we consider that this attraction is derived from
the law of universal gravity demonstrated by all the celestial phenomena.
The action of the moon upon the sea is more than double that of the sun.
Newton and his successors in the development of this action have paid
attention only to the terms divided by the cube of the distance from the
moon to the earth, judging that the effects due to the following terms ought
to be inappreciable. But the calculation of probabilities makes it clear to us
that the smallest effects of regular causes may manifest themselves in the
results of a great number of observations arranged in the order most suitable
to indicate them. This calculation again determines their probability and up
to what point it is necessary to multiply the observations to make it very
great. Applying it to the numerous observations discussed by M. Bouvard I
recognized that at Brest the action of the moon upon the sea is greater in the
full moons than in the new moons, and greater when the moon is austral
than when it is boreal—phenomena which can result only from the terms of

the lunar action divided by the fourth power of the distance from the moon
to the earth.
To arrive at the ocean the action of the sun and the moon traverses the
atmosphere, which ought consequently to feel its influence and to be
subjected to movements similar to those of the sea.
These movements produce in the barometer periodic oscillations.
Analysis has made it clear to me that they are inappreciable in our climates.
But as local circumstances increase considerably the tides in our ports, I
have inquired again if similar circumstances have made appreciable these
oscillations of the barometer. For this I have made use of the meteorological
observations which have been made every day for many years at the royal
observatory. The heights of the barometer and of the thermometer are
observed there at nine o'clock in the morning, at noon, at three o'clock in
the afternoon, and at eleven o'clock in the evening. M. Bouvard has indeed
wished to take up the consideration of observations of the eight years
elapsed from October 1, 1815, to October 1, 1823, on the registers. In
disposing the observations in the manner most suitable to indicate the lunar
atmospheric flood at Paris, I find only one eighteenth of a millimeter for the
extent of the corresponding oscillation of the barometer. It is this especially
which has made us feel the necessity of a method for determining the
probability of a result, and without this method one is forced to present as
the laws of nature the results of irregular causes which has often happened
in meteorology. This method applied to the preceding result shows the
uncertainty of it in spite of the great number of observations employed,
which it would be necessary to increase tenfold in order to obtain a result
sufficiently probable.
The principle which serves as a basis for my theory of the tides may be
extended to all the effects of hazard to which variable causes are joined
according to regular laws. The action of these causes produces in the mean
results of a great number of effects varieties which follow the same laws
and which one may recognize by the analysis of probabilities. In the
measure which these effects are multiplied those varieties are manifested
with an ever-increasing probability, which would approach certainty if the
number of the effects of the results should become infinite. This theorem is

analogous to that which I have already developed upon the action of
constant causes. Every time, then, that a cause whose progress is regular
can have influence upon a kind of events, we may seek to discover its
influence by multiplying the observations and arranging them in the most
suitable order to indicate it. When this influence appears to manifest itself
the analysis of probabilities determines the probability of its existence and
that of its intensity; thus the variation of the temperature from day to night
modifying the pressure of the atmosphere and consequently the height of
the barometer, it is natural to think that the multiplied observations of these
heights ought to show the influence of the solar heat. Indeed there has long
been recognized at the equator, where this influence appears to be greatest,
a small diurnal variation in the height of the barometer of which the
maximum occurs about nine o'clock in the morning and the minimum about
three o'clock in the afternoon. A second maximum occurs about eleven
o'clock in the evening and a second minimum about four o'clock in the
morning. The oscillations of the night are less than those of the day, the
extent of which is about two millimeters. The inconstancy of our climate
has not taken this variation from our observers, although it may be less
appreciable than in the tropics. M. Ramond has recognized and determined
it at Clermont, the chief place of the district of Puy-de-Dôme, by a series of
precise observations made during several years; he has even found that it is
smaller in the months of winter than in other months. The numerous
observations which I have discussed in order to estimate the influence of
attractions of the sun and the moon upon the barometric heights at Paris
have served me in determining their diurnal variation. Comparing the
heights at nine o'clock in the morning with those of the same days at three
o'clock in the afternoon, this variation is manifested with so much evidence
that its mean value each month has been constantly positive for each of the
seventy-two months from January 1, 1817, to January 1, 1823; its mean
value in these seventy-two months has been almost .8 of a millimeter, a
little less than at Clermont and much less than at the equator. I have
recognized that the mean result of the diurnal variations of the barometer
from 9 o'clock A.M. to 3 P.M. has been only .5428 millimeter in the three
months of November, December, January, and that it has risen to 1.0563

millimeters in the three following months, which coincides with the
observations of M. Ramond. The other months offer nothing similar.
In order to apply to these phenomena the calculation of these
probabilities, I commenced by determining the law of the probability of the
anomalies of the diurnal variation due to hazard. Applying it then to the
observations of this phenomenon, I found that it was a bet of more than
300,000 against one that a regular cause produced it. I do not seek to
determine this cause; I content myself with stating its existence. The period
of the diurnal variation regulated by the solar day indicates evidently that
this variation is due to the action of the sun. The extreme smallness of the
attractive action of the sun upon the atmosphere is proved by the smallness
of the effects due to the united attractions of the sun and the moon. It is then
by the action of its heat that the sun produces the diurnal variation of the
barometer; but it is impossible to subject to calculus the effects of its action
on the height of the barometer and upon the winds. The diurnal variation of
the magnetic needle is certainly a result of the action of the sun. But does
this star act here as in the diurnal variation of the barometer by its heat or
by its influence upon electricity and upon magnetism, or finally by the
union of these influences? A long series of observations made in different
countries will enable us to apprehend this.
One of the most remarkable phenomena of the system of the world is that
of all the movements of rotation and of revolution of the planets and the
satellites in the sense of the rotation of the sun and about in the same plane
of its equator. A phenomenon so remarkable is not the effect of hazard: it
indicates a general cause which has determined all its movements. In order
to obtain the probability with which this cause is indicated we shall observe
that the planetary system, such as we know it to-day, is composed of eleven
planets and of eighteen satellites at least, if we attribute with Herschel six
satellites to the planet Uranus. The movements of the rotation of the sun, of
six planets, of the moon, of the satellites of Jupiter, of the ring of Saturn,
and of one of its satellites have been recognized. These movements form
with those of revolution a totality of forty-three movements directed in the
same sense; but one finds by the analysis of probabilities that it is a bet of
more than 4000000000000 against one that this disposition is not the result
of hazard; this forms a probability indeed superior to that of historical

events in regard to which no doubt exists. We ought then to believe at least
with equal confidence that a primitive cause has directed the planetary
movements, especially if we consider that the inclination of the greatest
number of these movements at the solar equator is very small.
Another equally remarkable phenomenon of the solar system is the small
degree of the eccentricity of the orbs of the planets and the satellites, while
those of the comets are very elongated, the orbs of the system not offering
any intermediate shades between a great and a small eccentricity. We are
again forced to recognize here the effect of a regular cause; chance has
certainly not given an almost circular form to the orbits of all the planets
and their satellites; it is then that the cause which has determined the
movements of these bodies has rendered them almost circular. It is
necessary, again, that the great eccentricities of the orbits of the comets
should result from the existence of this cause without its having influenced
the direction of their movements; for it is found that there are almost as
many retrograde comets as direct comets, and that the mean inclination of
all their orbits to the ecliptic approaches very nearly half a right angle, as it
ought to be if the bodies had been thrown at hazard.
Whatever may be the nature of the cause in question, since it has
produced or directed the movement of the planets, it is necessary that it
should have embraced all the bodies and considered all the distances which
separate them, it can have been only a fluid of an immense extension.
Therefore in order to have given them in the same sense an almost circular
movement about the sun it is necessary that this fluid should have
surrounded this star as an atmosphere. The consideration of the planetary
movements leads us then to think that by virtue of an excessive heat the
atmosphere of the sun was originally extended beyond the orbits of all the
planets, and that it has contracted gradually to its present limits.
In the primitive state where we imagine the sun it resembled the nebulæ
that the telescope shows us composed of a nucleus more or less brilliant
surrounded by a nebula which, condensing at the surface, ought to
transform it some day into a star. If one conceives by analogy all the stars
formed in this manner, one can imagine their anterior state of nebulosity
itself preceded by other stars in which the nebulous matter was more and

more diffuse, the nucleus being less and less luminous and dense. Going
back, then, as far as possible, one would arrive at a nebulosity so diffuse
that one would be able scarcely to suspect its existence.
Such is indeed the first state of the nebulæ which Herschel observed with
particular care by means of his powerful telescopes, and in which he has
followed the progress of condensation, not in a single one, these stages not
becoming appreciable to us except after centuries, but in their totality, just
about as one can in a vast forest follow the increase of the trees by the
individuals of the divers ages which the forest contains. He has observed
from the beginning nebulous matter spread out in divers masses in the
different parts of the heavens, of which it occupies a great extent. He has
seen in some of these masses this matter slightly condensed about one or
several faintly luminous nebulæ. In the other nebulæ these nuclei shine,
moreover, in proportion to the nebulosity which surrounds them. The
atmospheres of each nucleus becoming separated by an ulterior
condensation, there result the multifold nebulæ formed of brilliant nuclei
very adjacent and surrounded each by an atmosphere; sometimes the
nebulous matter, by condensing in a uniform manner, has produced the
nebulæ which are called planetary. Finally a greater degree of condensation
transforms all these nebulæ into stars. The nebulæ classed according to this
philosophic view indicate with an extreme probability their future
transformation into stars and the anterior state of nebulosity of existing
stars. The following considerations come to the aid of proofs drawn from
these analogies.
For a long time the particular disposition of certain stars visible to the
naked eye has struck the attention of philosophical observers. Mitchel has
already remarked how improbable it is that the stars of the Pleiades, for
example, should have been confined in the narrow space which contain
them by the chances of hazard alone, and he has concluded from this that
this group of stars and the similar groups that the heaven presents us are the
results of a primitive cause or of a general law of nature. These groups are a
necessary result of the condensation of the nebulæ at several nuclei; it is
apparent that the nebulous matter being attracted continuously by the divers
nuclei, they ought to form in time a group of stars equal to that of the
Pleiades. The condensation of the nebulæ at two nuclei forms similarly very

adjacent stars, revolving the one about the other, equal to those whose
respective movements Herschel has already considered. Such are, further,
the 61st of the Swan and its following one in which Bessel has just
recognized particular movements so considerable and so little different that
the proximity of these stars to one another and their movement about the
common centre of gravity ought to leave no doubt. Thus one descends by
degrees from the condensation of nebulous matter to the consideration of
the sun surrounded formerly by a vast atmosphere, a consideration to which
one repasses, as has been seen, by the examination of the phenomena of the
solar system. A case so remarkable gives to the existence of this anterior
state of the sun a probability strongly approaching certainty.
But how has the solar atmosphere determined the movements of rotation
and revolution of the planets and the satellites? If these bodies had
penetrated deeply the atmosphere its resistance would have caused them to
fall upon the sun; one is then led to believe with much probability that the
planets have been formed at the successive limits of the solar atmosphere
which, contracting by the cold, ought to have abandoned in the plane of its
equator zones of vapors which the mutual attraction of their molecules has
changed into divers spheroids. The satellites have been similarly formed by
the atmospheres of their respective planets.
I have developed at length in my Exposition of the System of the World
this hypothesis, which appears to me to satisfy all the phenomena which
this system presents us. I shall content myself here with considering that the
angular velocity of rotation of the sun and the planets being accelerated by
the successive condensation of their atmospheres at their surfaces, it ought
to surpass the angular velocity of revolution of the nearest bodies which
revolve about them. Observation has indeed confirmed this with regard to
the planets and satellites, and even in ratio to the ring of Saturn, the
duration of whose revolution is .438 days, while the duration of the rotation
of Saturn is .427 days.
In this hypothesis the comets are strangers to the planetary system. In
attaching their formation to that of the nebulæ they may be regarded as
small nebulæ at the nuclei, wandering from systems to solar systems, and
formed by the condensation of the nebulous matter spread out in such great

profusion in the universe. The comets would be thus, in relation to our
system, as the aerolites are relatively to the Earth, to which they would
appear strangers. When these stars become visible to us they offer so
perfect resemblance to the nebulæ that they are often confounded with
them; and it is only by their movement, or by the knowledge of all the
nebulæ confined to that part of the heavens where they appear, that we
succeed in distinguishing them. This supposition explains in a happy
manner the great extension which the heads and tails of comets take in the
measure that they approach the sun, and the extreme rarity of these tails
which, in spite of their immense depth, do not weaken at all appreciably the
light of the stars which we look across.
When the little nebulæ come into that part of space where the attraction
of the sun is predominant, and which we shall call the sphere of activity of
this star, it forces them to describe elliptic or hyperbolic orbits. But their
speed being equally possible in all directions they ought to move
indifferently in all the senses and under all inclinations of the elliptic, which
is conformable to that which has been observed.
The great eccentricity of the cometary orbits results again from the
preceding hypothesis. Indeed if these orbits are elliptical they are very
elongated, since their great axes are at least equal to the radius of the sphere
of activity of the sun. But these orbits may be hyperbolic; and if the axes of
these hyperbolæ are not very large in proportion to the mean distance from
the sun to the earth, the movement of the comets which describe them will
appear sensibly hyperbolic. However, of the hundred comets of which we
already have the elements, not one has appeared certainly to move in an
hyperbola; it is necessary, then, that the chances which give an appreciable
hyperbola should be extremely rare in proportion to the contrary chances.
The comets are so small that, in order to become visible, their perihelion
distance ought to be inconsiderable. Up to the present this distance has
surpassed only twice the diameter of the terrestrial orbit, and most often it
has been below the radius of this orbit. It is conceived that, in order to
approach so near the sun, their speed at the moment of their entrance into its
sphere of activity ought to have a magnitude and a direction confined
within narrow limits. In determining by the analysis of probabilities the

ratio of the chances which, in these limits, give an appreciable hyperbola, to
the chances which give an orbit which may be confounded with a parabola,
I have found that it is a bet of at least 6000 against one that a nebula which
penetrates into the activity of the sun in such a manner as to be observed
will describe either a very elongated ellipse or an hyperbola. By the
magnitude of its axis, the latter will be appreciably confounded with a
parabola in the part which is observed; it is then not surprising that, up to
this time, hyperbolic movements have not been recognized.
The attraction of the planets, and, perhaps further, the resistance of the
ethereal centres, ought to have changed many cometary orbits in the ellipses
whose great axis is less than the radius of the sphere of activity of the sun,
which augments the chances of the elliptical orbits. We may believe that
this change has taken place with the comet of 1759, and with the comet
whose duration is only twelve hundred days, and which will reappear
without ceasing in this short interval, unless the evaporation which it meets
at each of its returns to the perihelion ends by rendering it invisible.
We are able further, by the analysis of probabilities, to verify the
existence or the influence of certain causes whose action is believed to exist
upon organized beings. Of all the instruments that we are able to employ in
order to recognize the imperceptible agents of nature the most sensitive are
the nerves, especially when particular causes increase their sensibility. It is
by their aid that the feeble electricity which the contact of two
heterogeneous metals develops has been discovered; this has opened a vast
field to the researches of physicists and chemists. The singular phenomena
which results from extreme sensibility of the nerves in some individuals
have given birth to divers opinions about the existence of a new agent
which has been named animal magnetism, about the action on ordinary
magnetism, and about the influence of the sun and moon in some nervous
affections, and finally, about the impressions which the proximity of metals
or of running water makes felt. It is natural to think that the action of these
causes is very feeble, and that it may be easily disturbed by accidental
circumstances; thus because in some cases it is not manifested at all its
existence ought not to be denied. We are so far from recognizing all the
agents of nature and their divers modes of action that it would be
unphilosophical to deny the phenomena solely because they are

inexplicable in the present state of our knowledge. But we ought to examine
them with an attention as much the more scrupulous as it appears the more
difficult to admit them; and it is here that the calculation of probabilities
becomes indispensable in determining to just what point it is necessary to
multiply the observations or the experiences in order to obtain in favor of
the agents which they indicate, a probability superior to the reasons which
can be obtained elsewhere for not admitting them.
The calculation of probabilities can make appreciable the advantages and
the inconveniences of the methods employed in the speculative sciences.
Thus in order to recognize the best of the treatments in use in the healing of
a malady, it is sufficient to test each of them on an equal number of patients,
making all the conditions exactly similar; the superiority of the most
advantageous treatment will manifest itself more and more in the measure
that the number is increased; and the calculation will make apparent the
corresponding probability of its advantage and the ratio according to which
it is superior to the others.

CHAPTER X.
APPLICATION OF THE CALCULUS OF
PROBABILITIES TO THE MORAL SCIENCES.
We have just seen the advantages of the analysis of probabilities in the
investigation of the laws of natural phenomena whose causes are unknown
or so complicated that their results cannot be submitted to calculus. This is
the case of nearly all subjects of the moral sciences. So many unforeseen
causes, either hidden or inappreciable, influence human institutions that it is
impossible to judge à priori the results. The series of events which time
brings about develops these results and indicates the means of remedying
those that are harmful. Wise laws have often been made in this regard; but
because we had neglected to conserve the motives many have been
abrogated as useless, and the fact that vexatious experiences have made the
need felt anew ought to have reëstablished them.
It is very important to keep in each branch of the public administration an
exact register of the results which the various means used have produced,
and which are so many experiences made on a large scale by governments.
Let us apply to the political and moral sciences the method founded upon
observation and upon calculus, the method which has served us so well in
the natural sciences. Let us not offer in the least a useless and often
dangerous resistance to the inevitable effects of the progress of knowledge;
but let us change only with an extreme circumspection our institutions and
the usages to which we have already so long conformed. We should know
well by the experience of the past the difficulties which they present; but we
are ignorant of the extent of the evils which their change can produce. In
this ignorance the theory of probability directs us to avoid all change;
especially is it necessary to avoid the sudden changes which in the moral
world as well as in the physical world never operate without a great loss of
vital force.

Already the calculus of probabilities has been applied with success to
several subjects of the moral sciences. I shall present here the principal
results.

CHAPTER XI.
CONCERNING THE PROBABILITIES OF
TESTIMONIES.
The majority of our opinions being founded on the probability of proofs
it is indeed important to submit it to calculus. Things it is true often become
impossible by the difficulty of appreciating the veracity of witnesses and by
the great number of circumstances which accompany the deeds they attest;
but one is able in several cases to resolve the problems which have much
analogy with the questions which are proposed and whose solutions may be
regarded as suitable approximations to guide and to defend us against the
errors and the dangers of false reasoning to which we are exposed. An
approximation of this kind, when it is well made, is always preferable to the
most specious reasonings. Let us try then to give some general rules for
obtaining it.
A single number has been drawn from an urn which contains a thousand
of them. A witness to this drawing announces that number 79 is drawn; one
asks the probability of drawing this number. Let us suppose that experience
has made known that this witness deceives one time in ten, so that the
probability of his testimony is 9⁄10. Here the event observed is the witness
attesting that number 79 is drawn. This event may result from the two
following hypotheses, namely: that the witness utters the truth or that he
deceives. Following the principle that has been expounded on the
probability of causes drawn from events observed it is necessary first to
determine à priori the probability of the event in each hypothesis. In the
first, the probability that the witness will announce number 79 is the
probability itself of the drawing of this number, that is to say, 1⁄1000. It is
necessary to multiply it by the probability 9⁄10 of the veracity of the witness;
one will have then 9⁄10000 for the probability of the event observed in this

hypothesis. If the witness deceives, number 79 is not drawn, and the
probability of this case is 999⁄1000. But to announce the drawing of this
number the witness has to choose it among the 999 numbers not drawn; and
as he is supposed to have no motive of preference for the ones rather than
the others, the probability that he will choose number 79 is 1⁄999;
multiplying, then, this probability by the preceding one, we shall have 1⁄1000
for the probability that the witness will announce number 79 in the second
hypothesis. It is necessary again to multiply this probability by 1⁄10 of the
hypothesis itself, which gives 1⁄10000 for the probability of the event relative
to this hypothesis. Now if we form a fraction whose numerator is the
probability relative to the first hypothesis, and whose denominator is the
sum of the probabilities relative to the two hypotheses, we shall have, by
the sixth principle, the probability of the first hypothesis, and this
probability will be 9⁄10; that is to say, the veracity itself of the witness. This
is likewise the probability of the drawing of number 79. The probability of
the falsehood of the witness and of the failure of drawing this number is 1⁄10.
If the witness, wishing to deceive, has some interest in choosing number
79 among the numbers not drawn,—if he judges, for example, that having
placed upon this number a considerable stake, the announcement of its
drawing will increase his credit, the probability that he will choose this
number will no longer be as at first, 1⁄999, it will then be ½, ⅓, etc., according
to the interest that he will have in announcing its drawing. Supposing it to
be 1⁄9, it will be necessary to multiply by this fraction the probability 999⁄1000
in order to get in the hypothesis of the falsehood the probability of the event
observed, which it is necessary still to multiply by 1⁄10, which gives 111⁄10000
for the probability of the event in the second hypothesis. Then the
probability of the first hypothesis, or of the drawing of number 79, is
reduced by the preceding rule to 9⁄120. It is then very much decreased by the
consideration of the interest which the witness may have in announcing the
drawing of number 79. In truth this same interest increases the probability
9
⁄10 that the witness will speak the truth if number 79 is drawn. But this
probability cannot exceed unity or 10⁄10; thus the probability of the drawing

of number 79 will not surpass 10⁄121. Common sense tells us that this interest
ought to inspire distrust, but calculus appreciates the influence of it.
The probability à priori of the number announced by the witness is unity
divided by the number of the numbers in the urn; it is changed by virtue of
the proof into the veracity itself of the witness; it may then be decreased by
the proof. If, for example, the urn contains only two numbers, which gives
½ for the probability à priori of the drawing of number 1, and if the
veracity of a witness who announces it is 4⁄10, this drawing becomes less
probable. Indeed it is apparent, since the witness has then more inclination
towards a falsehood than towards the truth, that his testimony ought to
decrease the probability of the fact attested every time that this probability
equals or surpasses ½. But if there are three numbers in the urn the
probability à priori of the drawing of number 1 is increased by the
affirmation of a witness whose veracity surpasses ⅓.
Suppose now that the urn contains 999 black balls and one white ball,
and that one ball having been drawn a witness of the drawing announces
that this ball is white. The probability of the event observed, determined à
priori in the first hypothesis, will be here, as in the preceding question,
equal to 9⁄10000. But in the hypothesis where the witness deceives, the white
ball is not drawn and the probability of this case is 999⁄1000. It is necessary to
multiply it by the probability 1⁄10 of the falsehood, which gives 999⁄10000 for the
probability of the event observed relative to the second hypothesis. This
probability was only 1⁄10000 in the preceding question; this great difference
results from this—that a black ball having been drawn the witness who
wishes to deceive has no choice at all to make among the 999 balls not
drawn in order to announce the drawing of a white ball. Now if one forms
two fractions whose numerators are the probabilities relative to each
hypothesis, and whose common denominator is the sum of these
probabilities, one will have 9⁄1008 for the probability of the first hypothesis
and of the drawing of a white ball, and 999⁄1008 for the probability of the
second hypothesis and of the drawing of a black ball. This last probability
strongly approaches certainty; it would approach it much nearer and would
become 999999⁄1000008 if the urn contained a million balls of which one was

white, the drawing of a white ball becoming then much more extraordinary.
We see thus how the probability of the falsehood increases in the measure
that the deed becomes more extraordinary.
We have supposed up to this time that the witness was not mistaken at
all; but if one admits, however, the chance of his error the extraordinary
incident becomes more improbable. Then in place of the two hypotheses
one will have the four following ones, namely: that of the witness not
deceiving and not being mistaken at all; that of the witness not deceiving at
all and being mistaken; the hypothesis of the witness deceiving and not
being mistaken at all; finally, that of the witness deceiving and being
mistaken. Determining à priori in each of these hypotheses the probability
of the event observed, we find by the sixth principle the probability that the
fact attested is false equal to a fraction whose numerator is the number of
black balls in the urn multiplied by the sum of the probabilities that the
witness does not deceive at all and is mistaken, or that he deceives and is
not mistaken, and whose denominator is this numerator augmented by the
sum of the probabilities that the witness does not deceive at all and is not
mistaken at all, or that he deceives and is mistaken at the same time. We see
by this that if the number of black balls in the urn is very great, which
renders the drawing of the white ball extraordinary, the probability that the
fact attested is not true approaches most nearly to certainty.
Applying this conclusion to all extraordinary deeds it results from it that
the probability of the error or of the falsehood of the witness becomes as
much greater as the fact attested is more extraordinary. Some authors have
advanced the contrary on this basis that the view of an extraordinary fact
being perfectly similar to that of an ordinary fact the same motives ought to
lead us to give the witness the same credence when he affirms the one or
the other of these facts. Simple common sense rejects such a strange
assertion; but the calculus of probabilities, while confirming the findings of
common sense, appreciates the greatest improbability of testimonies in
regard to extraordinary facts.
These authors insist and suppose two witnesses equally worthy of belief,
of whom the first attests that he saw an individual dead fifteen days ago
whom the second witness affirms to have seen yesterday full of life. The

one or the other of these facts offers no improbability. The reservation of
the individual is a result of their combination; but the testimonies do not
bring us at all directly to this result, although the credence which is due
these testimonies ought not to be decreased by the fact that the result of
their combination is extraordinary.
But if the conclusion which results from the combination of the
testimonies was impossible one of them would be necessarily false; but an
impossible conclusion is the limit of extraordinary conclusions, as error is
the limit of improbable conclusions; the value of the testimonies which
becomes zero in the case of an impossible conclusion ought then to be very
much decreased in that of an extraordinary conclusion. This is indeed
confirmed by the calculus of probabilities.
In order to make it plain let us consider two urns, A and B, of which the
first contains a million white balls and the second a million black balls. One
draws from one of these urns a ball, which he puts back into the other urn,
from which one then draws a ball. Two witnesses, the one of the first
drawing, the other of the second, attest that the ball which they have seen
drawn is white without indicating the urn from which it has been drawn.
Each testimony taken alone is not improbable; and it is easy to see that the
probability of the fact attested is the veracity itself of the witness. But it
follows from the combination of the testimonies that a white ball has been
extracted from the urn A at the first draw, and that then placed in the urn B
it has reappeared at the second draw, which is very extraordinary; for this
second urn, containing then one white ball among a million black balls, the
probability of drawing the white ball is 1⁄1000001. In order to determine the
diminution which results in the probability of the thing announced by the
two witnesses we shall notice that the event observed is here the affirmation
by each of them that the ball which he has seen extracted is white. Let us
represent by 9⁄10 the probability that he announces the truth, which can occur
in the present case when the witness does not deceive and is not mistaken at
all, and when he deceives and is mistaken at the same time. One may form
the four following hypotheses:
1st. The first and second witness speak the truth. Then a white ball has at
first been drawn from the urn A, and the probability of this event is ½, since

the ball drawn at the first draw may have been drawn either from the one or
the other urn. Consequently the ball drawn, placed in the urn B, has
reappeared at the second draw; the probability of this event is 1⁄1000001, the
probability of the fact announced is then 1⁄2000002. Multiplying it by the
product of the probabilities 9⁄10 and 9⁄10 that the witnesses speak the truth one
will have 81⁄200000200 for the probability of the event observed in this first
hypothesis.
2d. The first witness speaks the truth and the second does not, whether he
deceives and is not mistaken or he does not deceive and is mistaken. Then a
white ball has been drawn from the urn A at the first draw, and the
probability of this event is ½. Then this ball having been placed in the urn B
a black ball has been drawn from it: the probability of such drawing is
1000000
⁄1000001; one has then 1000000⁄2000002 for the probability of the compound
event. Multiplying it by the product of the two probabilities 9⁄10 and 1⁄10 that
the first witness speaks the truth and that the second does not, one will have
9000000
⁄200000200 for the probability for the event observed in the second
hypothesis.
3d. The first witness does not speak the truth and the second announces
it. Then a black ball has been drawn from the urn B at the first drawing, and
after having been placed in the urn A a white ball has been drawn from this
urn. The probability of the first of these events is ½ and that of the second is
1000000
⁄1000001; the probability of the compound event is then 1000000⁄2000002.
Multiplying it by the product of the probabilities 1⁄10 and 9⁄10 that the first
witness does not speak the truth and that the second announces it, one will
have 9000000⁄200000200 for the probability of the event observed relative to this
hypothesis.
4th. Finally, neither of the witnesses speaks the truth. Then a black ball
has been drawn from the urn B at the first draw; then having been placed in
the urn A it has reappeared at the second drawing: the probability of this
compound event is 1⁄2000002. Multiplying it by the product of the probabilities

1
⁄10 and 1⁄10 that each witness does not speak the truth one will have 1⁄200000200
for the probability of the event observed in this hypothesis.
Now in order to obtain the probability of the thing announced by the two
witnesses, namely, that a white ball has been drawn at each draw, it is
necessary to divide the probability corresponding to the first hypothesis by
the sum of the probabilities relative to the four hypotheses; and then one has
for this probability 81⁄18000082, an extremely small fraction.
If the two witnesses affirm the first, that a white ball has been drawn
from one of the two urns A and B; the second that a white ball has been
likewise drawn from one of the two urns A´ and B´, quite similar to the first
ones, the probability of the thing announced by the two witnesses will be
the product of the probabilities of their testimonies, or 81⁄100; it will then be at
least a hundred and eighty thousand times greater than the preceding one.
One sees by this how much, in the first case, the reappearance at the second
draw of the white ball drawn at the first draw, the extraordinary conclusion
of the two testimonies decreases the value of it.
We would give no credence to the testimony of a man who should attest
to us that in throwing a hundred dice into the air they had all fallen on the
same face. If we had ourselves been spectators of this event we should
believe our own eyes only after having carefully examined all the
circumstances, and after having brought in the testimonies of other eyes in
order to be quite sure that there had been neither hallucination nor
deception. But after this examination we should not hesitate to admit it in
spite of its extreme improbability; and no one would be tempted, in order to
explain it, to recur to a denial of the laws of vision. We ought to conclude
from it that the probability of the constancy of the laws of nature is for us
greater than this, that the event in question has not taken place at all—a
probability greater than that of the majority of historical facts which we
regard as incontestable. One may judge by this the immense weight of
testimonies necessary to admit a suspension of natural laws, and how
improper it would be to apply to this case the ordinary rules of criticism. All
those who without offering this immensity of testimonies support this when
making recitals of events contrary to those laws, decrease rather than
augment the belief which they wish to inspire; for then those recitals render

very probable the error or the falsehood of their authors. But that which
diminishes the belief of educated men increases often that of the
uneducated, always greedy for the wonderful.
There are things so extraordinary that nothing can balance their
improbability. But this, by the effect of a dominant opinion, can be
weakened to the point of appearing inferior to the probability of the
testimonies; and when this opinion changes an absurd statement admitted
unanimously in the century which has given it birth offers to the following
centuries only a new proof of the extreme influence of the general opinion
upon the more enlightened minds. Two great men of the century of Louis
XIV.—Racine and Pascal—are striking examples of this. It is painful to see
with what complaisance Racine, this admirable painter of the human heart
and the most perfect poet that has ever lived, reports as miraculous the
recovery of Mlle. Perrier, a niece of Pascal and a day pupil at the monastery
of Port-Royal; it is painful to read the reasons by which Pascal seeks to
prove that this miracle should be necessary to religion in order to justify the
doctrine of the monks of this abbey, at that time persecuted by the Jesuits.
The young Perrier had been afflicted for three years and a half by a
lachrymal fistula; she touched her afflicted eye with a relic which was
pretended to be one of the thorns of the crown of the Saviour and she had
faith in instant recovery. Some days afterward the physicians and the
surgeons attest the recovery, and they declare that nature and the remedies
have had no part in it. This event, which took place in 1656, made a great
sensation, and "all Paris rushed," says Racine, "to Port-Royal. The crowd
increased from day to day, and God himself seemed to take pleasure in
authorizing the devotion of the people by the number of miracles which
were performed in this church." At this time miracles and sorcery did not
yet appear improbable, and one did not hesitate at all to attribute to them the
singularities of nature which could not be explained otherwise.
This manner of viewing extraordinary results is found in the most
remarkable works of the century of Louis XIV.; even in the Essay on the
Human Understanding by the philosopher Locke, who says, in speaking of
the degree of assent: "Though the common experience and the ordinary
course of things have justly a mighty influence on the minds of men, to
make them give or refuse credit to anything proposed to their belief; yet

there is one case, wherein the strangeness of the fact lessens not the assent
to a fair testimony of it. For where such supernatural events are suitable to
ends aimed at by him who has the power to change the course of nature,
there, under such circumstances, they may be the fitter to procure belief, by
how much the more they are beyond or contrary to ordinary observation."
The true principles of the probability of testimonies having been thus
misunderstood by philosophers to whom reason is principally indebted for
its progress, I have thought it necessary to present at length the results of
calculus upon this important subject.
There comes up naturally at this point the discussion of a famous
argument of Pascal, that Craig, an English mathematician, has produced
under a geometric form. Witnesses declare that they have it from Divinity
that in conforming to a certain thing one will enjoy not one or two but an
infinity of happy lives. However feeble the probability of the proofs may
be, provided that it be not infinitely small, it is clear that the advantage of
those who conform to the prescribed thing is infinite since it is the product
of this probability and an infinite good; one ought not to hesitate then to
procure for oneself this advantage.
This argument is based upon the infinite number of happy lives promised
in the name of the Divinity by the witnesses; it is necessary then to
prescribe them, precisely because they exaggerate their promises beyond all
limits, a consequence which is repugnant to good sense. Also calculus
teaches us that this exaggeration itself enfeebles the probability of their
testimony to the point of rendering it infinitely small or zero. Indeed this
case is similar to that of a witness who should announce the drawing of the
highest number from an urn filled with a great number of numbers, one of
which has been drawn and who would have a great interest in announcing
the drawing of this number. One has already seen how much this interest
enfeebles his testimony. In evaluating only at ½ the probability that if the
witness deceives he will choose the largest number, calculus gives the
probability of his announcement as smaller than a fraction whose numerator
is unity and whose denominator is unity plus the half of the product of the
number of the numbers by the probability of falsehood considered à priori
or independently of the announcement. In order to compare this case to that
of the argument of Pascal it is sufficient to represent by the numbers in the

urn all the possible numbers of happy lives which the number of these
numbers renders infinite; and to observe that if the witnesses deceive they
have the greatest interest, in order to accredit their falsehood, in promising
an eternity of happiness. The expression of the probability of their
testimony becomes then infinitely small. Multiplying it by the infinite
number of happy lives promised, infinity would disappear from the product
which expresses the advantage resultant from this promise which destroys
the argument of Pascal.
Let us consider now the probability of the totality of several testimonies
upon an established fact. In order to fix our ideas let us suppose that the fact
be the drawing of a number from an urn which contains a hundred of them,
and of which one single number has been drawn. Two witnesses of this
drawing announce that number 2 has been drawn, and one asks for the
resultant probability of the totality of these testimonies. One may form
these two hypotheses: the witnesses speak the truth; the witnesses deceive.
In the first hypothesis the number 2 is drawn and the probability of this
event is 1⁄100. It is necessary to multiply it by the product of the veracities of
the witnesses, veracities which we will suppose to be 9⁄10 and 7⁄10: one will
have then 63⁄10000 for the probability of the event observed in this hypothesis.
In the second, the number 2 is not drawn and the probability of this event is
99
⁄100. But the agreement of the witnesses requires then that in seeking to
deceive they both choose the number 2 from the 99 numbers not drawn: the
probability of this choice if the witnesses do not have a secret agreement is
the product of the fraction 1⁄99 by itself; it becomes necessary then to
multiply these two probabilities together, and by the product of the
probabilities 1⁄10 and 3⁄10 that the witnesses deceive; one will have thus 1⁄330000
for the probability of the event observed in the second hypothesis. Now one
will have the probability of the fact attested or of the drawing of number 2
in dividing the probability relative to the first hypothesis by the sum of the
probabilities relative to the two hypotheses; this probability will be then
2079
⁄2080, and the probability of the failure to draw this number and of the
falsehood of the witnesses will be 1⁄2080.

If the urn should contain only the numbers 1 and 2 one would find in the
same manner 21⁄22 for the probability of the drawing of number 2, and
consequently 1⁄22 for the probability of the falsehood of the witnesses, a
probability at least ninety-four times larger than the preceding one. One
sees by this how much the probability of the falsehood of the witnesses
diminishes when the fact which they attest is less probable in itself. Indeed
one conceives that then the accord of the witnesses, when they deceive,
becomes more difficult, at least when they do not have a secret agreement,
which we do not suppose here at all.
In the preceding case where the urn contained only two numbers the à
priori probability of the fact attested is ½, the resultant probability of the
testimonies is the product of the veracities of the witnesses divided by this
product added to that of the respective probabilities of their falsehood.
It now remains for us to consider the influence of time upon the
probability of facts transmitted by a traditional chain of witnesses. It is clear
that this probability ought to diminish in proportion as the chain is
prolonged. If the fact has no probability itself, such as the drawing of a
number from an urn which contains an infinity of them, that which it
acquires by the testimonies decreases according to the continued product of
the veracity of the witnesses. If the fact has a probability in itself; if, for
example, this fact is the drawing of the number 2 from an urn which
contains an infinity of them, and of which it is certain that one has drawn a
single number; that which the traditional chain adds to this probability
decreases, following a continued product of which the first factor is the
ratio of the number of numbers in the urn less one to the same number, and
of which each other factor is the veracity of each witness diminished by the
ratio of the probability of his falsehood to the number of the numbers in the
urn less one; so that the limit of the probability of the fact is that of this fact
considered à priori, or independently of the testimonies, a probability equal
to unity divided by the number of the numbers in the urn.
The action of time enfeebles then, without ceasing, the probability of
historical facts just as it changes the most durable monuments. One can
indeed diminish it by multiplying and conserving the testimonies and the
monuments which support them. Printing offers for this purpose a great

means, unfortunately unknown to the ancients. In spite of the infinite
advantages which it procures the physical and moral revolutions by which
the surface of this globe will always be agitated will end, in conjunction
with the inevitable effect of time, by rendering doubtful after thousands of
years the historical facts regarded to-day as the most certain.
Craig has tried to submit to calculus the gradual enfeebling of the proofs
of the Christian religion; supposing that the world ought to end at the epoch
when it will cease to be probable, he finds that this ought to take place 1454
years after the time when he writes. But his analysis is as faulty as his
hypothesis upon the duration of the moon is bizarre.

CHAPTER XII.
CONCERNING THE SELECTIONS AND THE
DECISIONS OF ASSEMBLIES.
The probability of the decisions of an assembly depends upon the
plurality of votes, the intelligence and the impartiality of the members who
compose it. So many passions and particular interests so often add their
influence that it is impossible to submit this probability to calculus. There
are, however, some general results dictated by simple common sense and
confirmed by calculus. If, for example, the assembly is poorly informed
about the subject submitted to its decision, if this subject requires delicate
considerations, or if the truth on this point is contrary to established
prejudices, so that it would be a bet of more than one against one that each
voter will err; then the decision of the majority will be probably wrong, and
the fear of it will be the better based as the assembly is more numerous. It is
important then, in public affairs, that assemblies should have to pass upon
subjects within reach of the greatest number; it is important for them that
information be generally diffused and that good works founded upon reason
and experience should enlighten those who are called to decide the lot of
their fellows or to govern them, and should forewarn them against false
ideas and the prejudices of ignorance. Scholars have had frequent occasion
to remark that first conceptions often deceive and that the truth is not
always probable.
It is difficult to understand and to define the desire of an assembly in the
midst of a variety of opinions of its members. Let us attempt to give some
rules in regard to this matter by considering the two most ordinary cases:
the election among several candidates, and that among several propositions
relative to the same subject.
When an assembly has to choose among several candidates who present
themselves for one or for several places of the same kind, that which

appears simplest is to have each voter write upon a ticket the names of all
the candidates according to the order of merit that he attributes to them.
Supposing that he classifies them in good faith, the inspection of these
tickets will give the results of the elections in such a manner that the
candidates may be compared among themselves; so that new elections can
give nothing more in this regard. It is a question now to conclude the order
of preference which the tickets establish among the candidates. Let us
imagine that one gives to each voter an urn which contains an infinity of
balls by means of which he is able to shade all the degrees of merit of the
candidates; let us conceive again that he draws from his urn a number of
balls proportional to the merit of each candidate, and let us suppose this
number written upon a ticket at the side of the name of the candidate. It is
clear that by making a sum of all the numbers relative to each candidate
upon each ticket, that one of all the candidates who shall have the largest
sum will be the candidate whom the assembly prefers; and that in general
the order of preference of the candidates will be that of the sums relative to
each of them. But the tickets do not mark at all the number of balls which
each voter gives to the candidates; they indicate solely that the first has
more of them than the second, the second more than the third, and so on. In
supposing then at first upon a given ticket a certain number of balls all the
combinations of the inferior numbers which fulfil the preceding conditions
are equally admissible; and one will have the number of balls relative to
each candidate by making a sum of all the numbers which each
combination gives him and dividing it by the entire number of
combinations. A very simple analysis shows that the numbers which must
be written upon each ticket at the side of the last name, of the one before the
last, etc., are proportional to the terms of the arithmetical progression 1, 2,
3, etc. Writing then thus upon each ticket the terms of this progression, and
adding the terms relative to each candidate upon these tickets, the divers
sums will indicate by their magnitude the order of their preference which
ought to be established among the candidates. Such is the mode of election
which The Theory of Probabilities indicates. Without doubt it would be
better if each voter should write upon his ticket the names of the candidates
in the order of merit which he attributes to them. But particular interests and
many strange considerations of merit would affect this order and place
sometimes in the last rank the candidate most formidable to that one whom

one prefers, which gives too great an advantage to the candidates of
mediocre merit. Likewise experience has caused the abandonment of this
mode of election in the societies which had adopted it.
The election by the absolute majority of the suffrages unites to the
certainty of not admitting any one of the candidates whom this majority
rejects, the advantage of expressing most often the desire of the assembly. It
always coincides with the preceding mode when there are only two
candidates. Indeed it exposes an assembly to the inconvenience of rendering
elections interminable. But experience has shown that this inconvenience is
nil, and that the general desire to put an end to elections soon unites the
majority of the suffrages upon one of the candidates.
The choice among several propositions relative to the same object ought
to be subjected, seemingly, to the same rules as the election among several
candidates. But there exists between the two cases this difference, namely,
that the merit of a candidate does not exclude that of his competitors; but if
it is necessary to choose among propositions which are contrary, the truth of
the one excludes the truth of the others. Let us see how one ought then to
view this question.
Let us give to each voter an urn which contains an infinite number of
balls, and let us suppose that he distributes them upon the divers
propositions according to the respective probabilities which he attributes to
them. It is clear that the total number of balls expressing certainty, and the
voter being by the hypothesis assured that one of the propositions ought to
be true, he will distribute this number at length upon the propositions. The
problem is reduced then to this, namely, to determine the combinations in
which the balls will be distributed in such a manner that there may be more
of them upon the first proposition of the ticket than upon the second, more
upon the second than upon the third, etc.; to make the sums of all the
numbers of balls relative to each proposition in the divers combinations,
and to divide this sum by the number of combinations; the quotients will be
the numbers of balls that one ought to attribute to the propositions upon a
certain ticket. One finds by analysis that in going from the last proposition
these quotients are among themselves as the following quantities: first,
unity divided by the number of propositions; second, the preceding

quantity, augmented by unity, divided by the number of propositions less
one; third, this second quantity, augmented by unity, divided by the number
of propositions less two, and so on for the others. One will write then upon
each ticket these quantities at the side of the corresponding propositions,
and adding the relative quantities to each proposition upon the divers tickets
the sums will indicate by their magnitude the order of preference which the
assembly gives to these propositions.
Let us speak a word about the manner of renewing assemblies which
should change in totality in a definite number of years. Ought the renewal
to be made at one time, or is it advantageous to divide it among these years?
According to the last method the assembly would be formed under the
influence of the divers opinions dominant during the time of its renewal; the
opinion which obtained then would be probably the mean of all these
opinions. The assembly would receive thus at the time the same advantage
that is given to it by the extension of the elections of its members to all
parts of the territory which it represents. Now if one considers what
experience has only too clearly taught, namely, that elections are always
directed in the greatest degree by dominant opinions, one will feel how
useful it is to temper these opinions, the ones by the others, by means of a
partial renewal.

CHAPTER XIII.
CONCERNING THE PROBABILITY OF THE
JUDGMENTS OF TRIBUNALS.
Analysis confirms what simple common sense teaches us, namely, the
correctness of judgments is as much more probable as the judges are more
numerous and more enlightened. It is important then that tribunals of appeal
should fulfil these two conditions. The tribunals of the first instance
standing in closer relation to those amenable offer to the higher tribunal the
advantage of a first judgment already probable, and with which the latter
often agree, be it in compromising or in desisting from their claims. But if
the uncertainty of the matter in litigation and its importance determine a
litigant to have recourse to the tribunal of appeals, he ought to find in a
greater probability of obtaining an equitable judgment greater security for
his fortune and the compensation for the trouble and expense which a new
procedure entails. It is this which had no place in the institution of the
reciprocal appeal of the tribunals of the district, an institution thereby very
prejudicial to the interest of the citizens. It would be perhaps proper and
conformable to the calculus of probabilities to demand a majority of at least
two votes in a tribunal of appeal in order to invalidate the sentence of the
lower tribunal. One would obtain this result if the tribunal of appeal being
composed of an even number of judges the sentence should stand in the
case of the equality of votes.
I shall consider particularly the judgments in criminal matters.
In order to condemn an accused it is necessary without doubt that the
judges should have the strongest proofs of his offence. But a moral proof is
never more than a probability; and experience has only too clearly shown
the errors of which criminal judgments, even those which appear to be the
most just, are still susceptible. The impossibility of amending these errors is
the strongest argument of the philosophers who have wished to proscribe

the penalty of death. We should then be obliged to abstain from judging if it
were necessary for us to await mathematical evidence. But the judgment is
required by the danger which would result from the impunity of the crime.
This judgment reduces itself, if I am not mistaken, to the solution of the
following question: Has the proof of the offence of the accused the high
degree of probability necessary so that the citizens would have less reason
to doubt the errors of the tribunals, if he is innocent and condemned, than
they would have to fear his new crimes and those of the unfortunate ones
who would be emboldened by the example of his impunity if he were guilty
and acquitted? The solution of this question depends upon several elements
very difficult to ascertain. Such is the eminence of danger which would
threaten society if the criminal accused should remain unpunished.
Sometimes this danger is so great that the magistrate sees himself
constrained to waive forms wisely established for the protection of
innocence. But that which renders almost always this question insoluble is
the impossibility of appreciating exactly the probability of the offence and
of fixing that which is necessary for the condemnation of the accused. Each
judge in this respect is forced to rely upon his own judgment. He forms his
opinion by comparing the divers testimonies and the circumstances by
which the offence is accompanied, to the results of his reflections and his
experiences, and in this respect a long habitude of interrogating and judging
accused persons gives great advantage in ascertaining the truth in the midst
of indices often contradictory.
The preceding question depends again upon the care taken in the
investigation of the offence; for one demands naturally much stronger
proofs for imposing the death penalty than for inflicting a detention of some
months. It is a reason for proportioning the care to the offence, great care
taken with an unimportant case inevitably clearing many guilty ones. A law
which gives to the judges power of moderating the care in the case of
attenuating circumstances is then conformable at the same time to
principles of humanity towards the culprit, and to the interest of society.
The product of the probability of the offence by its gravity being the
measure of the danger to which the acquittal of the accused can expose
society, one would think that the care taken ought to depend upon this
probability. This is done indirectly in the tribunals where one retains for

some time the accused against whom there are very strong proofs, but
insufficient to condemn him; in the hope of acquiring new light one does
not place him immediately in the midst of his fellow citizens, who would
not see him again without great alarm. But the arbitrariness of this measure
and the abuse which one can make of it have caused its rejection in the
countries where one attaches the greatest price to individual liberty.
Now what is the probability that the decision of a tribunal which can
condemn only by a given majority will be just, that is to say, conform to the
true solution of the question proposed above? This important problem well
solved will give the means of comparing among themselves the different
tribunals. The majority of a single vote in a numerous tribunal indicates that
the affair in question is very doubtful; the condemnation of the accused
would be then contrary to the principles of humanity, protectors of
innocence. The unanimity of the judges would give very strong probability
of a just decision; but in abstaining from it too many guilty ones would be
acquitted. It is necessary, then, either to limit the number of judges, if one
wishes that they should be unanimous, or increase the majority necessary
for a condemnation, when the tribunal becomes more numerous. I shall
attempt to apply calculus to this subject, being persuaded that it is always
the best guide when one bases it upon the data which common sense
suggests to us.
The probability that the opinion of each judge is just enters as the
principal element into this calculation. If in a tribunal of a thousand and one
judges, five hundred and one are of one opinion, and five hundred are of the
contrary opinion, it is apparent that the probability of the opinion of each
judge surpasses very little ½; for supposing it obviously very large a single
vote of difference would be an improbable event. But if the judges are
unanimous, this indicates in the proofs that degree of strength which entails
conviction; the probability of the opinion of each judge is then very near
unity or certainty, provided that the passions or the ordinary prejudices do
not affect at the same time all the judges. Outside of these cases the ratio of
the votes for or against the accused ought alone to determine this
probability. I suppose thus that it can vary from ½ to unity, but that it cannot
be below ½. If that were not the case the decision of the tribunal would be
as insignificant as chance; it has value only in so far as the opinion of the

judge has a greater tendency to truth than to error. It is thus by the ratio of
the numbers of votes favorable, and contrary to the accused, that I
determine the probability of this opinion.
These data suffice to ascertain the general expression of the probability
that the decision of a tribunal judging by a known majority is just. In the
tribunals where of eight judges five votes would be necessary for the
condemnation of an accused, the probability of the error to be feared in the
justice of the decision would surpass ¼. If the tribunal should be reduced to
six members who are able to condemn only by a plurality of four votes, the
probability of the error to be feared would be below ¼. There would be then
for the accused an advantage in this reduction of the tribunal. In both cases
the majority required is the same and is equal to two. Thus the majority
remaining constant, the probability of error increases with the number of
judges; this is general whatever may be the majority required, provided that
it remains the same. Taking, then, for the rule the arithmetical ratio, the
accused finds himself in a position less and less advantageous in the
measure that the tribunal becomes more numerous. One might believe that
in a tribunal where one might demand a majority of twelve votes, whatever
the number of the judges was, the votes of the minority, neutralizing an
equal number of votes of the majority, the twelve remaining votes would
represent the unanimity of a jury of twelve members, required in England
for the condemnation of an accused; but one would be greatly mistaken.
Common sense shows that there is a difference between the decision of a
tribunal of two hundred and twelve judges, of which one hundred and
twelve condemn the accused, while one hundred acquit him, and that of a
tribunal of twelve judges unanimous for condemnation. In the first case the
hundred votes favorable to the accused warrant in thinking that the proofs
are far from attaining the degree of strength which entails conviction; in the
second case, the unanimity of the judges leads to the belief that they have
attained this degree. But simple common sense does not suffice at all to
appreciate the extreme difference of the probability of error in the two
cases. It is necessary then to recur to calculus, and one finds nearly one fifth
for the probability of error in the first case, and only 1⁄8192 for this probability
in the second case, a probability which is not one thousandth of the first. It
is a confirmation of the principle that the arithmetical ratio is unfavorable to

the accused when the number of judges increases. On the contrary, if one
takes for a rule the geometrical ratio, the probability of the error of the
decision diminishes when the number of judges increases. For example, in
the tribunals which can condemn only by a plurality of two thirds of the
votes, the probability of the error to be feared is nearly one fourth if the
number of the judges is six; it is below 1⁄7 if this number is increased to
twelve. Thus one ought to be governed neither by the arithmetical ratio nor
by the geometrical ratio if one wishes that the probability of error should
never be above nor below a given fraction.
But what fraction ought to be determined upon? It is here that the
arbitrariness begins and the tribunals offer in this regard the greatest variety.
In the special tribunals where five of the eight votes suffice for the
condemnation of the accused, the probability of the error to be feared in
regard to justice of the judgment is 65⁄256, or more than ¼. The magnitude of
this fraction is dreadful; but that which ought to reassure us a little is the
consideration that most frequently the judge who acquits an accused does
not regard him as innocent; he pronounces solely that it is not attained by
proofs sufficient for condemnation. One is especially reassured by the pity
which nature has placed in the heart of man and which disposes the mind to
see only with reluctance a culprit in the accused submitted to his judgment.
This sentiment, more active in those who have not the habitude of criminal
judgments, compensates for the inconveniences attached to the inexperience
of the jurors. In a jury of twelve members, if the plurality demanded for the
condemnation is eight of twelve votes, the probability of the error to be
feared 1093⁄8192, or a little more than one eighth, it is almost 1⁄22 if this plurality
consists of nine votes. In the case of unanimity the probability of the error
to be feared is 1⁄8192, that is to say, more than a thousand times less than in
our juries. This supposes that the unanimity results only from proofs
favorable or contrary to the accused; but motives that are entirely strange,
ought oftentimes to concur in producing it, when it is imposed upon the jury
as a necessary condition of its judgment. Then its decisions depending upon
the temperament, the character, the habits of the jurors, and the
circumstances in which they are placed, they are sometimes contrary to the
decisions which the majority of the jury would have made if they had

listened only to the proofs; this seems to me to be a great fault of this
manner of judging.
The probability of the decision is too feeble in our juries, and I think that
in order to give a sufficient guarantee to innocence, one ought to demand at
least a plurality of nine votes in twelve.

CHAPTER XIV.
CONCERNING TABLES OF MORTALITY, AND OF
MEAN DURATIONS OF LIFE, OF MARRIAGES,
AND OF ASSOCIATIONS.
The manner of preparing tables of mortality is very simple. One takes in
the civil registers a great number of individuals whose birth and death are
indicated. One determines how many of these individuals have died in the
first year of their age, how many in the second year, and so on. It is
concluded from these the number of individuals living at the
commencement of each year, and this number is written in the table at the
side of that which indicates the year. Thus one writes at the side of zero the
number of births; at the side of the year 1 the number of infants who have
attained one year; at the side of the year 2 the number of infants who have
attained two years, and so on for the rest. But since in the first two years of
life the mortality is very great, it is necessary for the sake of greater
exactitude to indicate in this first age the number of survivors at the end of
each half year.
If we divide the sum of the years of the life of all the individuals
inscribed in a table of mortality by the number of these individuals we shall
have the mean duration of life which corresponds to this table. For this, we
will multiply by a half year the number of deaths in the first year, a number
equal to the difference of the numbers of individuals inscribed at the side of
the years 0 and 1. Their mortality being distributed over the entire year the
mean duration of their life is only a half year. We will multiply by a year
and a half the number of deaths in the second year; by two years and a half
the number of deaths in the third year; and so on. The sum of these products
divided by the number of births will be the mean duration of life. It is easy
to conclude from this that we will obtain this duration, by making the sum

of the numbers inscribed in the table at the side of each year, dividing it by
the number of births and subtracting one half from the quotient, the year
being taken as unity. The mean duration of life that remains, starting from
any age, is determined in the same manner, working upon the number of
individuals who have arrived at this age, as has just been done with the
number of births. But it is not at the moment of birth that the mean duration
of life is the greatest; it is when one has escaped the dangers of infancy and
it is then about forty-three years. The probability of arriving at a certain age,
starting from a given age is equal to the ratio of the two numbers of
individuals indicated in the table at these two ages.
The precision of these results demands that for the formation of tables we
should employ a very great number of births. Analysis gives then very
simple formulæ for appreciating the probability that the numbers indicated
in these tables will vary from the truth only within narrow limits. We see by
these formulæ that the interval of the limits diminishes and that the
probability increases in proportion as we take into consideration more
births; so that the tables would represent exactly the true law of mortality if
the number of births employed were infinite.
A table of mortality is then a table of the probability of human life. The
ratio of the individuals inscribed at the side of each year to the number of
births is the probability that a new birth will attain this year. As we estimate
the value of hope by making a sum of the products of each benefit hoped
for, by the probability of obtaining it, so we can equally evaluate the mean
duration of life by adding the products of each year by half the sum of the
probabilities of attaining the commencement and the end of it, which leads
to the result found above. But this manner of viewing the mean duration of
life has the advantage of showing that in a stationary population, that is to
say, such that the number of births equals that of deaths, the mean duration
of life is the ratio itself of the population to the annual births; for the
population being supposed stationary, the number of individuals of an age
comprised between two consecutive years of the table is equal to the
number of annual births, multiplied by half the sum of the probabilities of
attaining these years; the sum of all these products will be then the entire
population. Now it is easy to see that this sum, divided by the number of

annual births, coincides with the mean duration of life as we have just
defined it.
It is easy by means of a table of mortality to form the corresponding table
of the population supposed to be stationary. For this we take the
arithmetical means of the numbers of the table of mortality corresponding
to the ages zero and one year, one and two years, two and three years, etc.
The sum of all these means is the entire population; it is written at the side
of the age zero. There is subtracted from this sum the first mean and the
remainder is the number of individuals of one year and upwards; it is
written at the side of the year 1. There is subtracted from this first
remainder the second mean; this second remainder is the number of
individuals of two years and upwards; it is written at the side of the year 2,
and so on.
So many variable causes influence mortality that the tables which
represent it ought to be changed according to place and time. The divers
states of life offer in this regard appreciable differences relative to the
fatigues and the dangers inseparable from each state and of which it is
indispensable to keep account in the calculations founded upon the duration
of life. But these differences have not been sufficiently observed. Some day
they will be and then will be known what sacrifice of life each profession
demands and one will profit by this knowledge to diminish the dangers.
The greater or less salubrity of the soil, its elevation, its temperature, the
customs of the inhabitants, and the operations of governments have a
considerable influence upon mortality. But it is always necessary to precede
the investigation of the cause of the differences observed by that of the
probability with which this cause is indicated. Thus the ratio of the
population to annual births, which one has seen raised in France to twenty-
eight and one third, is not equal to twenty-five in the ancient duchy of
Milan. These ratios, both established upon a great number of births, do not
permit of calling into question the existence among the Milanese of a
special cause of mortality, which it is of moment for the government of our
country to investigate and remove.
The ratio of the population to the births would increase again if we could
diminish and remove certain dangerous and widely spread maladies. This

has happily been done for the smallpox, at first by the inoculation of this
disease, then in a manner much more advantageous, by the inoculation of
vaccine, the inestimable discovery of Jenner, who has thereby become one
of the greatest benefactors of humanity.
The smallpox has this in particular, namely, that the same individual is
not twice affected by it, or at least such cases are so rare that they may be
abstracted from the calculation. This malady, from which few escaped
before the discovery of vaccine, is often fatal and causes the death of one
seventh of those whom it attacks. Sometimes it is mild, and experience has
taught that it can be given this latter character by inoculating it upon healthy
persons, prepared for it by a proper diet and in a favorable season. Then the
ratio of the individuals who die to the inoculated ones is not one three
hundredth. This great advantage of inoculation, joined to those of not
altering the appearance and of preserving from the grievous consequences
which the natural smallpox often brings, caused it to be adopted by a great
number of persons. The practice was strongly recommended, but it was
strongly combated, as is nearly always the case in things subject to
inconvenience. In the midst of this dispute Daniel Bernoulli proposed to
submit to the calculus of probabilities the influence of inoculation upon the
mean duration of life. Since precise data of the mortality produced by the
smallpox at the various ages of life were lacking, he supposed that the
danger of having this malady and that of dying of it are the same at every
age. By means of these suppositions he succeeded by a delicate analysis in
converting an ordinary table of mortality into that which would be used if
smallpox did not exist, or if it caused the death of only a very small number
of those affected, and he concludes from it that inoculation would augment
by three years at least the mean duration of life, which appeared to him
beyond doubt the advantage of this operation. D'Alembert attacked the
analysis of Bernoulli: at first in regard to the uncertainty of his two
hypotheses, then in regard to its insufficiency in this, that no comparison
was made of the immediate danger, although very small, of dying of
inoculation, to the very great but very remote danger of succumbing to
natural smallpox. This consideration, which disappears when one considers
a great number of individuals, is for this reason immaterial for governments
and the advantages of inoculation for them still remain; but it is of great

weight for the father of a family who must fear, in having his children
inoculated, to see that one perish whom he holds most dear and to be the
cause of it. Many parents were restrained by this fear, which the discovery
of vaccine has happily dissipated. By one of those mysteries which nature
offers to us so frequently, vaccine is a preventive of smallpox just as certain
as variolar virus, and there is no danger at all; it does not expose to any
malady and demands only very little care. Therefore the practice of it has
spread quickly; and to render it universal it remains only to overcome the
natural inertia of the people, against which it is necessary to strive
continually, even when it is a question of their dearest interests.
The simplest means of calculating the advantage which the extinction of
a malady would produce consists in determining by observation the number
of individuals of a given age who die of it each year and subtracting this
number from the number of deaths at the same age. The ratio of the
difference to the total number of individuals of the given age would be the
probability of dying in the year at this age if the malady did not exist.
Making, then, a sum of these probabilities from birth up to any given age,
and subtracting this sum from unity, the remainder will be the probability of
living to that age corresponding to the extinction of the malady. The series
of these probabilities will be the table of mortality relative to this
hypothesis, and we may conclude from it, by what precedes, the mean
duration of life. It is thus that Duvilard has found that the increase of the
mean duration of life, due to inoculation with vaccine, is three years at the
least. An increase so considerable would produce a very great increase in
the population if the latter, for other reasons, were not restrained by the
relative diminution of subsistences.
It is principally by the lack of subsistences that the progressive march of
the population is arrested. In all kinds of animals and vegetables, nature
tends without ceasing to augment the number of individuals until they are
on a level of the means of subsistence. In the human race moral causes have
a great influence upon the population. If easy clearings of the forest can
furnish an abundant nourishment for new generations, the certainty of being
able to support a numerous family encourages marriages and renders them
more productive. Upon the same soil the population and the births ought to
increase at the same time simultaneously in geometric progression. But

when clearings become more difficult and more rare then the increase of
population diminishes; it approaches continually the variable state of
subsistences, making oscillations about it just as a pendulum whose
periodicity is retarded by changing the point of suspension, oscillates about
this point by virtue of its own weight. It is difficult to evaluate the maximum
increase of the population; it appears after observations that in favorable
circumstances the population of the human race would be doubled every
fifteen years. We estimate that in North America the period of this doubling
is twenty-two years. In this state of things, the population, births, marriages,
mortality, all increase according to the same geometric progression of
which we have the constant ratio of consecutive terms by the observation of
annual births at two epochs.
By means of a table of mortality representing the probabilities of human
life, we may determine the duration of marriages. Supposing in order to
simplify the matter that the mortality is the same for the two sexes, we shall
obtain the probability that the marriage will subsist one year, or two, or
three, etc., by forming a series of fractions whose common denominator is
the product of the two numbers of the table corresponding to the ages of the
consorts, and whose numerators are the successive products of the numbers
corresponding to these ages augmented by one, by two, by three, etc., years.
The sum of these fractions augmented by one half will be the mean duration
of marriage, the year being taken as unity. It is easy to extend the same rule
to the mean duration of an association formed of three or of a greater
number of individuals.

CHAPTER XV.
CONCERNING THE BENEFITS OF INSTITUTIONS
WHICH DEPEND UPON THE PROBABILITY OF
EVENTS.
Let us recall here what has been said in speaking of hope. It has been
seen that in order to obtain the advantage which results from several simple
events, of which the ones produce a benefit and the others a loss, it is
necessary to add the products of the probability of each favorable event by
the benefit which it procures, and subtract from their sum that of the
products of the probability of each unfavorable event by the loss which is
attached to it. But whatever may be the advantage expressed by the
difference of these sums, a single event composed of these simple events
does not guarantee against the fear of experiencing a loss. One imagines
that this fear ought to decrease when one multiplies the compound event.
The analysis of probabilities leads to this general theorem.
By the repetition of an advantageous event, simple or compound, the real
benefit becomes more and more probable and increases without ceasing; it
becomes certain in the hypothesis of an infinite number of repetitions; and
dividing it by this number the quotient or the mean benefit of each event is
the mathematical hope itself or the advantage relative to the event. It is the
same with a loss which becomes certain in the long run, however small the
disadvantage of the event may be.
This theorem upon benefits and losses is analogous to those which we
have already given upon the ratios which are indicated by the indefinite
repetition of events simple or compound; and, like them, it proves that
regularity ends by establishing itself even in the things which are most
subordinated to that which we name hazard.

When the events are in great number, analysis gives another very simple
expression of the probability that the benefit will be comprised within
determined limits. This is the expression which enters again into the general
law of probability given above in speaking of the probabilities which result
from the indefinite multiplication of events.
The stability of institutions which are based upon probabilities depends
upon the truth of the preceding theorem. But in order that it may be applied
to them it is necessary that those institutions should multiply these
advantageous events for the sake of numerous things.
There have been based upon the probabilities of human life divers
institutions, such as life annuities and tontines. The most general and the
most simple method of calculating the benefits and the expenses of these
institutions consists in reducing these to actual amounts. The annual interest
of unity is that which is called the rate of interest. At the end of each year
an amount acquires for a factor unity plus the rate of interest; it increases
then according to a geometrical progression of which this factor is the ratio.
Thus in the course of time it becomes immense. If, for example, the rate of
interest is 1⁄20 or five per cent, the capital doubles very nearly in fourteen
years, quadruples in twenty-nine years, and in less than three centuries it
becomes two million times larger.
An increase so prodigious has given birth to the idea of making use of it
in order to pay off the public debt. One forms for this purpose a sinking
fund to which is devoted an annual fund employed for the redemption of
public bills and without ceasing increased by the interest of the bills
redeemed. It is clear that in the long run this fund will absorb a great part of
the national debt. If, when the needs of the State make a loan necessary, a
part of this loan is devoted to the increasing of the annual sinking fund, the
variation of public bills will be less; the confidence of the lenders and the
probability of retiring without loss of capital loaned when one desires will
be augmented and will render the conditions of the loan less onerous.
Favorable experiences have fully confirmed these advantages. But the
fidelity in engagements and the stability, so necessary to the success of such
institutions, can be guaranteed only by a government in which the
legislative power is divided among several independent powers. The

confidence which the necessary coöperation of these powers inspires,
doubles the strength of the State, and the sovereign himself gains then in
legal power more than he loses in arbitrary power.
It results from that which precedes that the actual capital equivalent to a
sum which is to be paid only after a certain number of years is equal to this
sum multiplied by the probability that it will be paid at that time and
divided by unity augmented by the rate of interest and raised to a power
expressed by the number of these years.
It is easy to apply this principle to life annuities upon one or several
persons, and to savings banks, and to assurance societies of any nature.
Suppose that one proposes to form a table of life annuities according to a
given table of mortality. A life annuity payable at the end of five years, for
example, and reduced to an actual amount is, by this principle, equal to the
product of the two following quantities, namely, the annuity divided by the
fifth power of unity augmented by the rate of interest and the probability of
paying it. This probability is the inverse ratio of the number of individuals
inscribed in the table opposite to the age of that one who settles the annuity
to the number inscribed opposite to this age augmented by five years.
Forming, then, a series of fractions whose denominators are the products of
the number of persons indicated in the table of mortality as living at the age
of that one who settles the annuity, by the successive powers of unity
augmented by the rate of interest, and whose numerators are the products of
the annuity by the number of persons living at the same age augmented
successively by one year, by two years, etc., the sum of these fractions will
be the amount required for the life annuity at that age.
Let us suppose that a person wishes by means of a life annuity to assure
to his heirs an amount payable at the end of the year of his death. In order to
determine the value of this annuity, one may imagine that the person
borrows in life at a bank this capital and that he places it at perpetual
interest in the same bank. It is clear that this same capital will be due by the
bank to his heirs at the end of the year of his death; but he will have paid
each year only the excess of the life interest over the perpetual interest. The
table of life annuities will then show that which the person ought to pay
annually to the bank in order to assure this capital after his death.

Maritime assurance, that against fire and storms, and generally all the
institutions of this kind, are computed on the same principles. A merchant
having vessels at sea wishes to assure their value and that of their cargoes
against the dangers that they may run; in order to do this, he gives a sum to
a company which becomes responsible to him for the estimated value of his
cargoes and his vessels. The ratio of this value to the sum which ought to be
given for the price of the assurance depends upon the dangers to which the
vessels are exposed and can be appreciated only by numerous observations
upon the fate of vessels which have sailed from port for the same
destination.
If the persons assured should give to the assurance company only the
sum indicated by the calculus of probabilities, this company would not be
able to provide for the expenses of its institution; it is necessary then that
they should pay a sum much greater than the cost of such insurance. What
then is their advantage? It is here that the consideration of the moral
disadvantage attached to an uncertainty becomes necessary. One conceives
that the fairest game becomes, as has already been seen, disadvantageous,
because the player exchanges a certain stake for an uncertain benefit;
assurance by which one exchanges the uncertain for the certain ought to be
advantageous. It is indeed this which results from the rule which we have
given above for determining moral hope and by which one sees moreover
how far the sacrifice may extend which ought to be made to the assurance
company by reserving always a moral advantage. This company can then in
procuring this advantage itself make a great benefit, if the number of the
assured persons is very large, a condition necessary to its continued
existence. Then its benefits become certain and the mathematical and moral
hopes coincide; for analysis leads to this general theorem, namely, that if
the expectations are very numerous the two hopes approach each other
without ceasing and end by coinciding in the case of an infinite number.
We have said in speaking of mathematical and moral hopes that there is a
moral advantage in distributing the risks of a benefit which one expects
over several of its parts. Thus in order to send a sum of money to a distant
part it is much better to send it on several vessels than to expose it on one.
This one does by means of mutual assurances. If two persons, each having
the same sum upon two different vessels which have sailed from the same

port to the same destination, agree to divide equally all the money which
may arrive, it is clear that by this agreement each of them divides equally
between the two vessels the sum which he expects. Indeed this kind of
assurance always leaves uncertainty as to the loss which one may fear. But
this uncertainty diminishes in proportion as the number of policy-holders
increases; the moral advantage increases more and more and ends by
coinciding with the mathematical advantage, its natural limit. This renders
the association of mutual assurances when it is very numerous more
advantageous to the assured ones than the companies of assurance which, in
proportion to the benefit that they give, give a moral advantage always
inferior to the mathematical advantage. But the surveillance of their
administration can balance the advantage of the mutual assurances. All
these results are, as has already been seen, independent of the law which
expresses the moral advantage.
One may look upon a free people as a great association whose members
secure mutually their properties by supporting proportionally the charges of
this guaranty. The confederation of several peoples would give to them
advantages analogous to those which each individual enjoys in the society.
A congress of their representatives would discuss objects of a utility
common to all and without doubt the system of weights, measures, and
moneys proposed by the French scientists would be adopted in this congress
as one of the things most useful to commercial relations.
Among the institutions founded upon the probabilities of human life the
better ones are those in which, by means of a light sacrifice of his revenue,
one assures his existence and that of his family for a time when one ought
to fear to be unable to satisfy their needs. As far as games are immoral, so
far these institutions are advantageous to customs by favoring the strongest
bents of our nature. The government ought then to encourage them and
respect them in the vicissitudes of public fortune; since the hopes which
they present look toward a distant future, they are able to prosper only when
sheltered from all inquietude during their existence. It is an advantage that
the institution of a representative government assures them.
Let us say a word about loans. It is clear that in order to borrow
perpetually it is necessary to pay each year the product of the capital by the

rate of interest. But one may wish to discharge this principal in equal
payments made during a definite number of years, payments which are
called annuities and whose value is obtained in this manner. Each annuity in
order to be reduced at the actual moment ought to be divided by a power of
unity augmented by the rate of interest equal to the number of years after
which this annuity ought to be paid. Forming then a geometric progression
whose first term is the annuity divided by unity augmented by the rate of
interest, and whose last term is this annuity divided by the same quantity
raised to a power equal to the number of years during which the payment
should have been made, the sum of this progression will be equivalent to
the capital borrowed, which will determine the value of the annuity. A
sinking fund is at bottom only a means of converting into annuities a
perpetual rent with the sole difference that in the case of a loan by annuities
the interest is supposed constant, while the interest of funds acquired by the
sinking fund is variable. If it were the same in both cases, the annuity
corresponding to the funds acquired would be formed by these funds and
from this annuity the State contributes annually to the sinking fund.
If one wishes to make a life loan it will be observed that the tables of life
annuities give the capital required to constitute a life annuity at any age, a
simple proportion will give the rent which one ought to pay to the
individual from whom the capital is borrowed. From these principles all the
possible kinds of loans may be calculated.
The principles which we have just expounded concerning the benefits
and the losses of institutions may serve to determine the mean result of any
number of observations already made, when one wishes to regard the
deviations of the results corresponding to divers observations. Let us
designate by x the correction of the least result and by x augmented
successively by q, q´, q´´, etc., the corrections of the following results. Let
us name e, e´, e´´, etc., the errors of the observations whose law of
probability we will suppose known. Each observation being a function of
the result, it is easy to see that by supposing the correction x of this result to
be very small, the error e of the first observation will be equal to the
product of x by a determined coefficient. Likewise the error e´ of the second
observation will be the product of the sum q plus x, by a determined
coefficient, and so on. The probability of the error e being given by a

known function, it will be expressed by the same function of the first of the
preceding products. The probability of e´ will be expressed by the same
function of the second of these products, and so on of the others. The
probability of the simultaneous existence of the errors e, e´, e´´, etc., will be
then proportional to the product of these divers functions, a product which
will be a function of x. This being granted, if one conceives a curve whose
abscissa is x, and whose corresponding ordinate is this product, this curve
will represent the probability of the divers values of x, whose limits will be
determined by the limits of the errors e, e´, e´´, etc. Now let us designate by
X the abscissa which it is necessary to choose; X diminished by x will be the
error which would be committed if the abscissa x were the true correction.
This error, multiplied by the probability of x or by the corresponding
ordinate of the curve, will be the product of the loss by its probability,
regarding, as one should, this error as a loss attached to the choice X.
Multiplying this product by the differential of x the integral taken from the
first extremity of the curve to X will be the disadvantage of X resulting from
the values of x inferior to X. For the values of x superior to X, x less X
would be the error of X if x were the true correction; the integral of the
product of x by the corresponding ordinate of the curve and by the
differential of x will be then the disadvantage of X resulting from the values
x superior to x, this integral being taken from x equal to X up to the last
extremity of the curve. Adding this disadvantage to the preceding one, the
sum will be the disadvantage attached to the choice of X. This choice ought
to be determined by the condition that this disadvantage be a minimum; and
a very simple calculation shows that for this, X ought to be the abscissa
whose ordinate divides the curve into two equal parts, so that it is thus
probable that the true value of x falls on neither the one side nor the other of
X.
Celebrated geometricians have chosen for X the most probable value of x
and consequently that which corresponds to the largest ordinate of the
curve; but the preceding value appears to me evidently that which the
theory of probability indicates.

CHAPTER XVI.
CONCERNING ILLUSIONS IN THE ESTIMATION
OF PROBABILITIES.
The mind has its illusions as the sense of sight; and in the same manner
that the sense of feeling corrects the latter, reflection and calculation correct
the former. Probability based upon a daily experience, or exaggerated by
fear and by hope, strikes us more than a superior probability but it is only a
simple result of calculus. Thus we do not fear in return for small advantages
to expose our life to dangers much less improbable than the drawing of a
quint in the lottery of France; and yet no one would wish to procure for
himself the same advantages with the certainty of losing his life if this quint
should be drawn.
Our passions, our prejudices, and dominating opinions, by exaggerating
the probabilities which are favorable to them and by attenuating the
contrary probabilities, are the abundant sources of dangerous illusions.
Present evils and the cause which produced them effect us much more
than the remembrance of evils produced by the contrary cause; they prevent
us from appreciating with justice the inconveniences of the ones and the
others, and the probability of the proper means to guard ourselves against
them. It is this which leads alternately to despotism and to anarchy the
people who are driven from the state of repose to which they never return
except after long and cruel agitations.
This vivid impression which we receive from the presence of events, and
which allows us scarcely to remark the contrary events observed by others,
is a principal cause of error against which one cannot sufficiently guard
himself.
It is principally at games of chance that a multitude of illusions support
hope and sustain it against unfavorable chances. The majority of those who

play at lotteries do not know how many chances are to their advantage, how
many are contrary to them. They see only the possibility by a small stake of
gaining a considerable sum, and the projects which their imagination brings
forth, exaggerate to their eyes the probability of obtaining it; the poor man
especially, excited by the desire of a better fate, risks at play his necessities
by clinging to the most unfavorable combinations which promise him a
great benefit. All would be without doubt surprised by the immense number
of stakes lost if they could know of them; but one takes care on the contrary
to give to the winnings a great publicity, which becomes a new cause of
excitement for this funereal play.
When a number in the lottery of France has not been drawn for a long
time the crowd is eager to cover it with stakes. They judge since the number
has not been drawn for a long time that it ought at the next drawing to be
drawn in preference to others. So common an error appears to me to rest
upon an illusion by which one is carried back involuntarily to the origin of
events. It is, for example, very improbable that at the play of heads and tails
one will throw heads ten times in succession. This improbability which
strikes us indeed when it has happened nine times, leads us to believe that at
the tenth throw tails will be thrown. But the past indicating in the coin a
greater propensity for heads than for tails renders the first of the events
more probable than the second; it increases as one has seen the probability
of throwing heads at the following throw. A similar illusion persuades many
people that one can certainly win in a lottery by placing each time upon the
same number, until it is drawn, a stake whose product surpasses the sum of
all the stakes. But even when similar speculations would not often be
stopped by the impossibility of sustaining them they would not diminish the
mathematical disadvantage of speculators and they would increase their
moral disadvantage, since at each drawing they would risk a very large part
of their fortune.
I have seen men, ardently desirous of having a son, who could learn only
with anxiety of the births of boys in the month when they expected to
become fathers. Imagining that the ratio of these births to those of girls
ought to be the same at the end of each month, they judged that the boys
already born would render more probable the births next of girls. Thus the
extraction of a white ball from an urn which contains a limited number of

white balls and of black balls increases the probability of extracting a black
ball at the following drawing. But this ceases to take place when the
number of balls in the urn is unlimited, as one must suppose in order to
compare this case with that of births. If, in the course of a month, there
were born many more boys than girls, one might suspect that toward the
time of their conception a general cause had favored masculine conception,
which would render more probable the birth next of a boy. The irregular
events of nature are not exactly comparable to the drawing of the numbers
of a lottery in which all the numbers are mixed at each drawing in such a
manner as to render the chances of their drawing perfectly equal. The
frequency of one of these events seems to indicate a cause slightly favoring
it, which increases the probability of its next return, and its repetition
prolonged for a long time, such as a long series of rainy days, may develop
unknown causes for its change; so that at each expected event we are not, as
at each drawing of a lottery, led back to the same state of indecision in
regard to what ought to happen. But in proportion as the observation of
these events is multiplied, the comparison of their results with those of
lotteries becomes more exact.
By an illusion contrary to the preceding ones one seeks in the past
drawings of the lottery of France the numbers most often drawn, in order to
form combinations upon which one thinks to place the stake to advantage.
But when the manner in which the mixing of the numbers in this lottery is
considered, the past ought to have no influence upon the future. The very
frequent drawings of a number are only the anomalies of chance; I have
submitted several of them to calculation and have constantly found that they
are included within the limits which the supposition of an equal possibility
of the drawing of all the numbers allows us to admit without improbability.
In a long series of events of the same kind the single chances of hazard
ought sometimes to offer the singular veins of good luck or bad luck which
the majority of players do not fail to attribute to a kind of fatality. It
happens often in games which depend at the same time upon hazard and
upon the competency of the players, that that one who loses, troubled by his
loss, seeks to repair it by hazardous throws which he would shun in another
situation; thus he aggravates his own ill luck and prolongs its duration. It is
then that prudence becomes necessary and that it is of importance to

convince oneself that the moral disadvantage attached to unfavorable
chances is increased by the ill luck itself.
The opinion that man has long been placed in the centre of the universe,
considering himself the special object of the cares of nature, leads each
individual to make himself the centre of a more or less extended sphere and
to believe that hazard has preference for him. Sustained by this belief,
players often risk considerable sums at games when they know that the
chances are unfavorable. In the conduct of life a similar opinion may
sometimes have advantages; but most often it leads to disastrous
enterprises. Here as everywhere illusions are dangerous and truth alone is
generally useful.
One of the great advantages of the calculus of probabilities is to teach us
to distrust first opinions. As we recognize that they often deceive when they
may be submitted to calculus, we ought to conclude that in other matters
confidence should be given only after extreme circumspection. Let us prove
this by example.
An urn contains four balls, black and white, but which are not all of the
same color. One of these balls has been drawn whose color is white and
which has been put back in the urn in order to proceed again to similar
drawings. One demands the probability of extracting only black balls in the
four following drawings.
If the white and black were in equal number this probability would be the
fourth power of the probability ½ of extracting a black ball at each drawing;
it would be then 1⁄16. But the extraction of a white ball at the first drawing
indicates a superiority in the number of white balls in the urn; for if one
supposes in the urn three white balls and one black the probability of
extracting a white ball is ¾; it is 2⁄4 if one supposes two white balls and two
black; finally it is reduced to ¼ if one supposes three black balls and one
white. Following the principle of the probability of causes drawn from
events the probabilities of these three suppositions are among themselves as
the quantities ¾, 2⁄4, ¼; they are consequently equal to 3⁄6, 2⁄6, ⅙. It is thus a
bet of 5 against 1 that the number of black balls is inferior, or at the most
equal, to that of the white. It seems then that after the extraction of a white

ball at the first drawing, the probability of extracting successively four
black balls ought to be less than in the case of the equality of the colors or
smaller than one sixteenth. However, it is not, and it is found by a very
simple calculation that this probability is greater than one fourteenth.
Indeed it would be the fourth power of ¼, of 2⁄4, and of ¾ in the first, the
second, and the third of the preceding suppositions concerning the colors of
the balls in the urn. Multiplying respectively each power by the probability
of the corresponding supposition, or by 3⁄6, 2⁄6, and ⅙ , the sum of the
products will be the probability of extracting successively four black balls.
One has thus for this probability 29⁄384, a fraction greater than 1⁄14. This
paradox is explained by considering that the indication of the superiority of
white balls over the black ones at the first drawing does not exclude at all
the superiority of the black balls over the white ones, a superiority which
excludes the supposition of the equality of the colors. But this superiority,
though but slightly probable, ought to render the probability of drawing
successively a given number of black balls greater than in this supposition
if the number is considerable; and one has just seen that this commences
when the given number is equal to four. Let us consider again an urn which
contains several white and black balls. Let us suppose at first that there is
only one white ball and one black. It is then an even bet that a white ball
will be extracted in one drawing. But it seems for the equality of the bet that
one who bets on extracting the white ball ought to have two drawings if the
urn contains two black and one white, three drawings if it contains three
black and one white, and so on; it is supposed that after each drawing the
extracted ball is placed again in the urn.
We are convinced easily that this first idea is erroneous. Indeed in the
case of two black and one white ball, the probability of extracting two black
in two drawings is the second power of ⅔ or 4⁄9; but this probability added
to that of drawing a white ball in two drawings is certainty or unity, since it
is certain that two black balls or at least one white ball ought to be drawn;
the probability in this last case is then 5⁄9, a fraction greater than ½. There
would still be a greater advantage in the bet of drawing one white ball in
five draws when the urn contains five black and one white ball; this bet is

even advantageous in four drawings; it returns then to that of throwing six
in four throws with a single die.
The Chevalier de Meré, who caused the invention of the calculus of
probabilities by encouraging his friend Pascal, the great geometrician, to
occupy himself with it, said to him "that he had found error in the numbers
by this ratio. If we undertake to make six with one die there is an advantage
in undertaking it in four throws, as 671 to 625. If we undertake to make two
sixes with two dice, there is a disadvantage in undertaking in 24 throws. At
least 24 is to 36, the number of the faces of the two dice, as 4 is to 6, the
number of faces of one die." "This was," wrote Pascal to Fermat, "his great
scandal which caused him to say boldly that the propositions were not
constant and that arithmetic was demented.... He has a very good mind, but
he is not a geometrician, which is, as you know, a great fault." The
Chevalier de Meré, deceived by a false analogy, thought that in the case of
the equality of bets the number of throws ought to increase in proportion to
the number of all the chances possible, which is not exact, but which
approaches exactness as this number becomes larger.
One has endeavored to explain the superiority of the births of boys over
those of girls by the general desire of fathers to have a son who would
perpetuate the name. Thus by imagining an urn filled with an infinity of
white and black balls in equal number, and supposing a great number of
persons each of whom draws a ball from this urn and continues with the
intention of stopping when he shall have extracted a white ball, one has
believed that this intention ought to render the number of white balls
extracted superior to that of the black ones. Indeed this intention gives
necessarily after all the drawings a number of white balls equal to that of
persons, and it is possible that these drawings would never lead a black ball.
But it is easy to see that this first notion is only an illusion; for if one
conceives that in the first drawing all the persons draw at once a ball from
the urn, it is evident that their intention can have no influence upon the
color of the balls which ought to appear at this drawing. Its unique effect
will be to exclude from the second drawing the persons who shall have
drawn a white one at the first. It is likewise apparent that the intention of
the persons who shall take part in the new drawing will have no influence
upon the color of the balls which shall be drawn, and that it will be the same

at the following drawings. This intention will have no influence then upon
the color of the balls extracted in the totality of drawings; it will, however,
cause more or fewer to participate at each drawing. The ratio of the white
balls extracted to the black ones will differ thus very little from unity. It
follows that the number of persons being supposed very large, if
observation gives between the colors extracted a ratio which differs sensibly
from unity, it is very probable that the same difference is found between
unity and the ratio of the white balls to the black contained in the urn.
I count again among illusions the application which Liebnitz and Daniel
Bernoulli have made of the calculus of probabilities to the summation of
series. If one reduces the fraction whose numerator is unity and whose
denominator is unity plus a variable, in a series prescribed by the ratio to
the powers of this variable, it is easy to see that in supposing the variable
equal to unity the fraction becomes ½, and the series becomes plus one,
minus one, plus one, minus one, etc. In adding the first two terms, the
second two, and so on, the series is transformed into another of which each
term is zero. Grandi, an Italian Jesuit, concluded from this the possibility of
the creation; because the series being always ½, he saw this fraction spring
from an infinity of zeros or from nothing. It was thus that Liebnitz believed
he saw the image of creation in his binary arithmetic where he employed
only the two characters, unity and zero. He imagined, since God can be
represented by unity and nothing by zero, that the Supreme Being had
drawn from nothing all beings, as unity with zero expresses all the numbers
in this system of arithmetic. This idea was so pleasing to Liebnitz that he
communicated it to the Jesuit Grimaldi, president of the tribunal of
mathematics in China, in the hope that this emblem of creation would
convert to Christianity the emperor there who particularly loved the
sciences. I report this incident only to show to what extent the prejudices of
infancy can mislead the greatest men.
Liebnitz, always led by a singular and very loose metaphysics,
considered that the series plus one, minus one, plus one, etc., becomes unity
or zero according as one stops at a number of terms odd or even; and as in
infinity there is no reason to prefer the even number to the odd, one ought
following the rules of probability, to take the half of the results relative to
these two kinds of numbers, and which are zero and unity, which gives ½

for the value of the series. Daniel Bernoulli has since extended this
reasoning to the summation of series formed from periodic terms. But all
these series have no values properly speaking; they get them only in the
case where their terms are multiplied by the successive powers of a variable
less than unity. Then these series are always convergent, however small one
supposes the difference of the variable from unity; and it is easy to
demonstrate that the values assigned by Bernoulli, by virtue of the rule of
probabilities, are the same values of the generative fraction of the series,
when one supposes in these fractions the variable equal to unity. These
values are again the limits which the series approach more and more, in
proportion as the variable approaches unity. But when the variable is
exactly equal to unity the series cease to be convergent; they have values
only as far as one arrests them. The remarkable ratio of this application of
the calculus of probabilities with the limits of the values of periodic series
supposes that the terms of these series are multiplied by all the consecutive
powers of the variable. But this series may result from the development of
an infinity of different fractions in which this did not occur. Thus the series
plus one, minus one, plus one, etc., may spring from the development of a
fraction whose numerator is unity plus the variable, and whose denominator
is this numerator augmented by the square of the variable. Supposing the
variable equal to unity, this development changes, in the series proposed,
and the generative fraction becomes equal to ⅔; the rules of probabilities
would give then a false result, which proves how dangerous it would be to
employ similar reasoning, especially in the mathematical sciences, which
ought to be especially distinguished by the rigor of their operations.
We are led naturally to believe that the order according to which we see
things renewed upon the earth has existed from all times and will continue
always. Indeed if the present state of the universe were exactly similar to
the anterior state which has produced it, it would give birth in its turn to a
similar state; the succession of these states would then be eternal. I have
found by the application of analysis to the law of universal gravity that the
movement of rotation and of revolution of the planets and satellites, and the
position of the orbits and of their equators are subjected only to periodic
inequalities. In comparing with ancient eclipses the theory of the secular
equation of the moon I have found that since Hipparchus the duration of the

day has not varied by the hundredth of a second, and that the mean
temperature of the earth has not diminished the one-hundredth of a degree.
Thus the stability of actual order appears established at the same time by
theory and by observations. But this order is effected by divers causes
which an attentive examination reveals, and which it is impossible to
submit to calculus.
The actions of the ocean, of the atmosphere, and of meteors, of
earthquakes, and the eruptions of volcanoes, agitate continually the surface
of the earth and ought to effect in the long run great changes. The
temperature of climates, the volume of the atmosphere, and the proportion
of the gases which constitute it, may vary in an inappreciable manner. The
instruments and the means suitable to determine these variations being new,
observation has been unable up to this time to teach us anything in this
regard. But it is hardly probable that the causes which absorb and renew the
gases constituting the air maintain exactly their respective proportions. A
long series of centuries will show the alterations which are experienced by
all these elements so essential to the conservation of organized beings.
Although historical monuments do not go back to a very great antiquity
they offer us nevertheless sufficiently great changes which have come about
by the slow and continued action of natural agents. Searching in the bowels
of the earth one discovers numerous débris of former nature, entirely
different from the present. Moreover, if the entire earth was in the
beginning fluid, as everything appears to indicate, one imagines that in
passing from that state to the one which it has now, its surface ought to have
experienced prodigious changes. The heavens itself in spite of the order of
its movements, is not unchangeable. The resistance of light and of other
ethereal fluids, and the attraction of the stars ought, after a great number of
centuries, to alter considerably the planetary movements. The variations
already observed in the stars and in the form of the nebulæ give us a
presentiment of those which time will develop in the system of these great
bodies. One may represent the successive states of the universe by a curve,
of which time would be the abscissa and of which the ordinates are the
divers states. Scarcely knowing an element of this curve we are far from
being able to go back to its origin; and if in order to satisfy the imagination,
always restless from our ignorance of the cause of the phenomena which

interest it, one ventures some conjectures it is wise to present them only
with extreme reserve.
There exists in the estimation of probabilities a kind of illusions, which
depending especially upon the laws of the intellectual organization
demands, in order to secure oneself against them, a profound examination
of these laws. The desire to penetrate into the future and the ratios of some
remarkable events, to the predictions of astrologers, of diviners and
soothsayers, to presentiments and dreams, to the numbers and the days
reputed lucky or unlucky, have given birth to a multitude of prejudices still
very widespread. One does not reflect upon the great number of non-
coincidences which have made no impression or which are unknown.
However, it is necessary to be acquainted with them in order to appreciate
the probability of the causes to which the coincidences are attributed. This
knowledge would confirm without doubt that which reason tells us in
regard to these prejudices. Thus the philosopher of antiquity to whom is
shown in a temple, in order to exalt the power of the god who is adored
there, the ex voto of all those who after having invoked it were saved from
shipwreck, presents an incident consonant with the calculus of probabilities,
observing that he does not see inscribed the names of those who, in spite of
this invocation, have perished. Cicero has refuted all these prejudices with
much reason and eloquence in his Treatise on Divination, which he ends by
a passage which I shall cite; for one loves to find again among the ancients
the thunderbolts of reason, which, after having dissipated all the prejudices
by its light, shall become the sole foundation of human institutions.
"It is necessary," says the Roman orator, "to reject divination by dreams
and all similar prejudices. Widespread superstition has subjugated the
majority of minds and has taken possession of the feebleness of men. It is
this we have expounded in our books upon the nature of the gods and
especially in this work, persuaded that we shall render a service to others
and to ourselves if we succeed in destroying superstition. However (and I
desire especially in this regard my thought be well comprehended), in
destroying superstition I am far from wishing to disturb religion. Wisdom
enjoins us to maintain the institutions and the ceremonies of our ancestors,
touching the cult of the gods. Moreover, the beauty of the universe and the
order of celestial things force us to recognize some superior nature which

ought to be remarked and admired by the human race. But as far as it is
proper to propagate religion, which is joined to the knowledge of nature, so
far it is necessary to work toward the extirpation of superstition, for it
torments one, importunes one, and pursues one continually and in all
places. If one consult a diviner or a soothsayer, if one immolates a victim, if
one regards the flight of a bird, if one encounters a Chaldean or an aruspex,
if it lightens, if it thunders, if the thunderbolt strikes, finally, if there is born
or is manifested a kind of prodigy, things one of which ought often to
happen, then superstition dominates and leaves no repose. Sleep itself, this
refuge of mortals in their troubles and their labors, becomes by it a new
source of inquietude and fear."
All these prejudices and the terrors which they inspire are connected with
physiological causes which continue sometimes to operate strongly after
reason has disabused us of them. But the repetition of acts contrary to these
prejudices can always destroy them.

CHAPTER XVII.
CONCERNING THE VARIOUS MEANS OF
APPROACHING CERTAINTY.
Induction, analogy, hypotheses founded upon facts and rectified
continually by new observations, a happy tact given by nature and
strengthened by numerous comparisons of its indications with experience,
such are the principal means for arriving at truth.
If one considers a series of objects of the same nature one perceives
among them and in their changes ratios which manifest themselves more
and more in proportion as the series is prolonged, and which, extending and
generalizing continually, lead finally to the principle from which they were
derived. But these ratios are enveloped by so many strange circumstances
that it requires great sagacity to disentangle them and to recur to this
principle: it is in this that the true genius of sciences consists. Analysis and
natural philosophy owe their most important discoveries to this fruitful
means, which is called induction. Newton was indebted to it for his theorem
of the binomial and the principle of universal gravity. It is difficult to
appreciate the probability of the results of induction, which is based upon
this that the simplest ratios are the most common; this is verified in the
formulæ of analysis and is found again in natural phenomena, in
crystallization, and in chemical combinations. This simplicity of ratios will
not appear astonishing if we consider that all the effects of nature are only
mathematical results of a small number of immutable laws.
Yet induction, in leading to the discovery of the general principles of the
sciences, does not suffice to establish them absolutely. It is always
necessary to confirm them by demonstrations or by decisive experiences;
for the history of the sciences shows us that induction has sometimes led to
inexact results. I shall cite, for example, a theorem of Fermat in regard to
prime numbers. This great geometrician, who had meditated, profoundly

upon this theorem, sought a formula which, containing only prime numbers,
gave directly a prime number greater than any other number assignable.
Induction led him to think that two, raised to a power which was itself a
power of two, formed with unity a prime number. Thus, two raised to the
square plus one, forms the prime number five; two raised to the second
power of two, or sixteen, forms with one the prime number seventeen. He
found that this was still true for the eighth and the sixteenth power of two
augmented by unity; and this induction, based upon several arithmetical
considerations, caused him to regard this result as general. However, he
avowed that he had not demonstrated it. Indeed, Euler recognized that this
does not hold for the thirty-second power of two, which, augmented by
unity, gives 4,294,967,297, a number divisible by 641.
We judge by induction that if various events, movements, for example,
appear constantly and have been long connected by a simple ratio, they will
continue to be subjected to it; and we conclude from this, by the theory of
probabilities, that this ratio is due, not to hazard, but to a regular cause.
Thus the equality of the movements of the rotation and the revolution of the
moon; that of the movements of the nodes of the orbit and of the lunar
equator, and the coincidence of these nodes; the singular ratio of the
movements of the first three satellites of Jupiter, according to which the
mean longitude of the first satellite, less three times that of the second, plus
two times that of the third, is equal to two right angles; the equality of the
interval of the tides to that of the passage of the moon to the meridian; the
return of the greatest tides with the syzygies, and of the smallest with the
quadratures; all these things, which have been maintained since they were
first observed, indicate with an extreme probability, the existence of
constant causes which geometricians have happily succeeded in attaching to
the law of universal gravity, and the knowledge of which renders certain the
perpetuity of these ratios.
The chancellor Bacon, the eloquent promoter of the true philosophical
method, has made a very strange misuse of induction in order to prove the
immobility of the earth. He reasons thus in the Novum Organum, his finest
work: "The movement of the stars from the orient to the occident increases
in swiftness, in proportion to their distance from the earth. This movement
is swiftest with the stars; it slackens a little with Saturn, a little more with

Jupiter, and so on to the moon and the highest comets. It is still perceptible
in the atmosphere, especially between the tropics, on account of the great
circles which the molecules of the air describe there; finally, it is almost
inappreciable with the ocean; it is then nil for the earth." But this induction
proves only that Saturn, and the stars which are inferior to it, have their own
movements, contrary to the real or apparent movement which sweeps the
whole celestial sphere from the orient to the occident, and that these
movements appear slower with the more remote stars, which is conformable
to the laws of optics. Bacon ought to have been struck by the inconceivable
swiftness which the stars require in order to accomplish their diurnal
revolution, if the earth is immovable, and by the extreme simplicity with
which its rotation explains how bodies so distant, the ones from the others,
as the stars, the sun, the planets, and the moon, all seem subjected to this
revolution. As to the ocean and to the atmosphere, he ought not to compare
their movement with that of the stars which are detached from the earth; but
since the air and the sea make part of the terrestrial globe, they ought to
participate in its movement or in its repose. It is singular that Bacon, carried
to great prospects by his genius, was not won over by the majestic idea
which the Copernican system of the universe offers. He was able, however,
to find in favor of that system, strong analogies in the discoveries of
Galileo, which were continued by him. He has given for the search after
truth the precept, but not the example. But by insisting, with all the force of
reason and of eloquence, upon the necessity of abandoning the insignificant
subtleties of the school, in order to apply oneself to observations and to
experiences, and by indicating the true method of ascending to the general
causes of phenomena, this great philosopher contributed to the immense
strides which the human mind made in the grand century in which he
terminated his career.
Analogy is based upon the probability, that similar things have causes of
the same kind and produce the same effects. This probability increase as the
similitude becomes more perfect. Thus we judge without doubt that beings
provided with the same organs, doing the same things, experience the same
sensations, and are moved by the same desires. The probability that the
animals which resemble us have sensations analogous to ours, although a
little inferior to that which is relative to individuals of our species, is still

exceedingly great; and it has required all the influence of religious
prejudices to make us think with some philosophers that animals are mere
automatons. The probability of the existence of feeling decreases in the
same proportion as the similitude of the organs with ours diminishes, but it
is always very great, even with insects. In seeing those of the same species
execute very complicated things exactly in the same manner from
generation to generation, and without having learned them, one is led to
believe that they act by a kind of affinity analogous to that which brings
together the molecules of crystals, but which, together with the sensation
attached to all animal organization, produces, with the regularity of
chemical combinations, combinations that are much more singular; one
might, perhaps, name this mingling of elective affinities and sensations
animal affinity. Although there exists a great analogy between the
organization of plants and that of animals, it does not seem to me sufficient
to extend to vegetables the sense of feeling; but nothing authorizes us in
denying it to them.
Since the sun brings forth, by the beneficent action of its light and of its
heat, the animals and plants which cover the earth, we judge by analogy that
it produces similar effects upon the other planets; for it is not natural to
think that the cause whose activity we see developed in so many ways
should be sterile upon so great a planet as Jupiter, which, like the terrestrial
globe, has its days, its nights, and its years, and upon which observations
indicate changes which suppose very active forces. Yet this would be giving
too great an extension to analogy to conclude from it the similitude of the
inhabitants of the planets and of the earth. Man, made for the temperature
which he enjoys, and for the element which he breathes, would not be able,
according to all appearance, to live upon the other planets. But ought there
not to be an infinity of organization relative to the various constitutions of
the globes of this universe? If the single difference of the elements and of
the climates make so much variety in terrestrial productions, how much
greater the difference ought to be among those of the various planets and of
their satellites! The most active imagination can form no idea of it; but their
existence is very probable.
We are led by a strong analogy to regard the stars as so many suns
endowed, like ours, with an attractive power proportional to the mass and

reciprocal to the square of the distances; for this power being demonstrated
for all the bodies of the solar system, and for their smallest molecules, it
appears to appertain to all matter. Already the movements of the small stars,
which have been called double, on account of their being binary, appear to
indicate it; a century at most of precise observations, by verifying their
movements of revolution, the ones about the others, will place beyond
doubt their reciprocal attractions.
The analogy which leads us to make each star the centre of a planetary
system is far less strong than the preceding one; but it acquires probability
by the hypothesis which has been proposed in regard to the formation of the
stars and of the sun; for in this hypothesis each star, having been like the
sun, primitively environed by a vast atmosphere, it is natural to attribute to
this atmosphere the same effects as to the solar atmosphere, and to suppose
that it has produced, in condensing, planets and satellites.
A great number of discoveries in the sciences is due to analogy. I shall
cite as one of the most remarkable, the discovery of atmospheric electricity,
to which one has been led by the analogy of electric phenomena with the
effects of thunder.
The surest method which can guide us in the search for truth, consists in
rising by induction from phenomena to laws and from laws to forces. Laws
are the ratios which connect particular phenomena together: when they have
shown the general principle of the forces from which they are derived, one
verifies it either by direct experiences, when this is possible, or by
examination if it agrees with known phenomena; and if by a rigorous
analysis we see them proceed from this principle, even in their small
details, and if, moreover, they are quite varied and very numerous, then
science acquires the highest degree of certainty and of perfection that it is
able to attain. Such, astronomy has become by the discovery of universal
gravity. The history of the sciences shows that the slow and laborious path
of induction has not always been that of inventors. The imagination,
impatient to arrive at the causes, takes pleasure in creating hypotheses, and
often it changes the facts in order to adapt them to its work; then the
hypotheses are dangerous. But when one regards them only as the means of
connecting the phenomena in order to discover the laws; when, by refusing

to attribute them to a reality, one rectifies them continually by new
observations, they are able to lead to the veritable causes, or at least put us
in a position to conclude from the phenomena observed those which given
circumstances ought to produce.
If we should try all the hypotheses which can be formed in regard to the
cause of phenomena we should arrive, by a process of exclusion, at the true
one. This means has been employed with success; sometimes we have
arrived at several hypotheses which explain equally well all the facts
known, and among which scholars are divided, until decisive observations
have made known the true one. Then it is interesting, for the history of the
human mind, to return to these hypotheses, to see how they succeed in
explaining a great number of facts, and to investigate the changes which
they ought to undergo in order to agree with the history of nature. It is thus
that the system of Ptolemy, which is only the realization of celestial
appearances, is transformed into the hypothesis of the movement of the
planets about the sun, by rendering equal and parallel to the solar orbit the
circles and the epicycles which he causes to be described annually, and the
magnitude of which he leaves undetermined. It suffices, then, in order to
change this hypothesis into the true system of the world, to transport the
apparent movement of the sun in a sense contrary to the earth.
It is almost always impossible to submit to calculus the probability of the
results obtained by these various means; this is true likewise for historical
facts. But the totality of the phenomena explained, or of the testimonies, is
sometimes such that without being able to appreciate the probability we
cannot reasonably permit ourselves any doubt in regard to them. In the
other cases it is prudent to admit them only with great reserve.

CHAPTER XVIII.
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.

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