Produced by Turgut Dincer, Chris Pinfield, and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive)









Transcriber's Note:

Apparent typographical errors have been corrected.

Italics are indicated by _underscores_. Small capitals have been
replaced by full capitals.

The corrections noted in the Errata section have been incorporated in
the text. Two 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 the Judgments of Tribunals            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

   "  177, lines 15, 17, 18, 21, 22, 24, _for_ 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 _n_th 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 _s_th 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 _n_th
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 _m_th 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 _m_th power
of _p_ in the development of the _n_th 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 _n_th 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 _n_th 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 _n_th 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, ⅞,
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 _x_th 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 _n_th
power of _T_, we shall have the coefficient of the _x_th power of _t_ in
the product of _V_ by the _n_th 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 _x_th power of _t_ will
be the coefficient of the power greater by unity in _V_; this
coefficient in the product of _V_ by the _n_th 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 _x_th
power of _t_ in the product of _V_ by _Z_; this coefficient in the
product of _V_ by the _n_th 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 _x_th 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 _x_th 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 _n_th power of this character
placed before the primitive function will indicate the finite _n_th
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 _n_th power of _T_ will then be equal to the product of _V_ by
the _n_th 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 _n_th 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
_n_th 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 _n_th
power of _Z_ will then be equal to the product of _V_ by the development
of the _n_th power of the binomial _T_ - 1. Repassing from the
discriminant functions to their coefficients as has just been done, we
shall have the _n_th difference of the primitive function expressed by
the development of the _n_th 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
_n_th power of _T_ is equal to the _n_th 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 _n_th 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 _n_th difference of the primitive function having
for a discriminant function the product of _V_ by the _n_th power of the
binomial one divided by _t_ less unity, the primitive function which is
the _n_th integral of this difference has for a discriminant function
that of the same difference multiplied by the _n_th 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 _x_th 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 _n_th power of _T_, will be the
discriminant function of the derivative of the primitive function before
which one places the _n_th 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 _x_th 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 _n_th difference of a function in
which the increase of the variable is _E_ being divided by the _n_th
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 _n_th difference
of the function divided by the _n_th 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 _n_th 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 28 + (352845/1000000);
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 1/10. 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 6/10 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 veto_ 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.