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, 
FREDERICK LINCOLN EMORY, M.E. (WOR. POLY. INST.), 
Professor of Mechanics and Applied Mathematics in the West Virginia 
University ; Mem. Amer. Soc. Mtch. Eng. 


NEW YORK: JOHN WILEY & SONS. 
LONDON : CHAPMAN & HALL, LIMITED. 
Copyright, 1902, by F. W. TRUSCOTT and F. L. EMORY. 


TABLE OF CONTENTS. 

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 6 

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 


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; 


2 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

so that the entire system of human knowledge is con- 
nected with the theory set forth in this essay. Doubt- 
less 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 con- 
stantly attached to them, there is a great advantage in 
following these principles and serious inconvenience in 
departing from them: their chances, like those favor- 
able to lotteries, always end by prevailing in the midst 
of the vacillations of hazard. I hope that the reflec- 
tions given in this essay may merit the attention of 
philosophers and direct it to a subject so worthy of 
engaging their minds. 

ToC

3 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 revolu- 
tions 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 determi- 
native 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 
 

4 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 in- 
telligence. 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 

CONCERNING PROBABILITY. 5 

in this respect distinguishes nations and ages and con- 
stitutes 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 over- 
thrown the Lower Empire. This star after four revolu- 
tions 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 

6 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 cal- 
culations 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 occur- 
rence, 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 


CONCERNING PROBABILITY. 7 

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 differ- 
ence that the three balls of this last urn would be 
replaced by three systems of two balls invariably con- 
nected. 

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 



8 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 


CONCERNING PROBABILITY. 9 

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 differ- 
ence 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. 



10 A PHILOSOPHICAL ESSAY ON 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 con- 
siderations 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. 

ToC

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 



12 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

three cases are favorable to the event whose probability 
is sought; consequently this probability is equal to 3/4; 
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 
2/3 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 1/2, 
while that of the two other cases is J, 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 f of heads at the first throw to the possi- 
bility J of tails at the first throw and heads at the 
second, we shall have f 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 prob- 
abilities increase or diminish by their mutual combina- 
tion. If the events are independent of one another, the 
probability of their combined existence is the product 
of their respective probabilities. Thus the probability 



CALCULUS OF PROBABILITIES. 13 

of throwing one ace with a single die is ^; that of 
throwing two aces in throwing two dice at the same 
time is --$. Each face of the one being able to com- 
bine 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 T 9 ; that of the 
incident resulting from the testimonies will be less 
than . 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 princi- 



14 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

pie 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 pre- 
cedes 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 f , 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 ; the product of \ by , or , 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 prob- 
ability of drawing a white ball from the urn B, which 
primarily is f, becomes \ 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 follow- 



CALCULUS OF PROBABILITIES. 15 

ing principle, which is a corollary of the preceding 
one. 

Fifth Principle. If we calculate a priori the prob- 
ability 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 con- 
stantly 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 prob- 
ability 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, con- 
sidered a priori, are unequally probable, it is necessary, 



16 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 funda- 
mental 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, 
Constantinople, and we judge that this arrange- 
ment 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 



CALCULUS OF PROBABILITIES. 17 

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 com- 
binations 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 prob- 
ability that, this cause existing, the future event will 



18 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 prob- 
ability 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 J; it is unity or certainty in the 
second. Thus in regarding these hypotheses as so 
many causes, we shall have for the sixth principle 
% and | for their respective probabilities. But if the 
first hypothesis occurs, the probability of drawing a 
white ball at the third draw is ^ ; 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 T 9 ^, 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 prin- 
ciple, 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 



CALCULUS OF PROBABILITIES. 19 

we multiply each fraction by the probability of the 
future event, determined in the corresponding hypothe- 
sis, 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 182623 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, recogniz- 
ing 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. 

ToC

20 CHAPTER IV. 
CONCERNING HOPE. 

THE probability of events serves to determine the 
hope or the fear of persons interested in their exist- 
ence. 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 



CONCERNING HOPE. 21 

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 of the first 
case, and five francs by the probability 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 prob- 
ability of throwing heads at the second throw being , 
multiplying two francs and five francs by 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 



22 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 happi- 
ness, with which the majority of people feed their self- 
love. 

The application of the preceding principles to the 
following question has greatly exercised the geometri- 
cians. Paul plays at heads and tails with the condition 
of receiving two francs if he throws heads at the first 
thro\v, 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, accord- 
ing 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 differ- 
ence 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 pro- 
cures 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. 



CONCERNING HOPE. 23 

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 expecta- 
tions 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 coin- 
cides 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 



24 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

the two hopes may differ very materially among them- 
selves. 

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 



CONCERNING HOPE. 25 

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. 

ToC

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 sub- 
tract 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 pre- 
ceding product will become the nth power of the 
binomial one plus the first letter; thus the number of 
combinations of n letters taken s by s times will be the 
coefficient of the sth power of the first letter in the 



THE CALCULUS OF PROBABILITIES. 27 

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 combina- 
tions. 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 combina- 
tions. From this it is easy to conclude that the 
number of arrangements of which s letters are suscepti- 
ble 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 possi- 
ble 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 



28 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 -&> or T V ; 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 3-^-5-, 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 advan- 
tages. 

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 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 -j- 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 th power of the binomial a + b 



THE CALCULUS OF PROBABILITIES. 29 

expresses the number of all the cases possible in n 
draws; and that in the development of this power the 
term multiplied by the mth power of a expresses the 
number of cases in which m white balls and n in 
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/ and g, the probability of drawing m white 
balls in n draws will be the term multiplied by the mth 
power of/ in the development of the 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 prob- 
ability when we increase the variables by their respect- 
ive 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 con- 
sider 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 



30 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

victor plays against the second until one of the players 
has defeated consecutively the two others, which ends 
the game. The probability is demanded that the game 
will be finished in a certain number n of plays. Let 
us find the probability that it will end precisely at the 
nth play. For that the player who wins ought to enter 
the game at the play n I and win it thus at the fol- 
lowing play. But if in place of winning the play n i 
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 I and will win it is 
equal to the probability that the game will end pre- 
cisely with this play; and as this player ought to win 
the following play in order that the game may be 
finished at the nth play, the probability of this last case 
will be only one half of the preceding one. This 
probability is evidently a function of the number ; 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 . 
Hence by virtue of the preceding equation we conclude 
that the successive probabilities of the end of the game 
are for the third play, \ for the fourth play, and so 



THE CALCULUS OF PROBABILITIES. 31 

on ; and in general raised to the power n I for the 
nth play. The sum of all these powers of 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 propor- 
tional to the respective probabilities of winning the 
game. The question is reduced then to the determina- 
tion of these probabilities. They depend evidently 
upon the number of points which each player lacks of 
having attained the given number. Hence the prob- 
ability 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 \ ; the function sought is then equal to one half 
of this function in which we diminish by unity the first 



32 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 observ- 
ing 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. Sup- 
posing thus that the player A lacks only one point, we 
find that his probability is f, f, |, etc., according as B 
lacks one point, two, three, etc. Generally it is then 
unity less the power of , 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 J, , \\, 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 accord- 
ing 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 



THE CALCULUS OF PROBABILITIES. 33 

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 con- 
nected by a general equation among several consecutive 
terms, taken as much in the horizontal as in the ver- 
tical 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 differ- 
ences by three indices. 

Finally, considering the matter in an abstract way 



34 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 differ- 
ences to these indices and the indices themselves, as 
many equations as there are magnitudes ; these equa- 
tions will be partial finite differences by a certain num- 
ber 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 
known 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 elimina- 
tions 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. 



THE CALCULUS OF PROBABILITIES. 35 

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 conse- 
quently 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 coeffi- 
cient. 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 equa- 
tion of this new progression. In this condition he 
multiplies it by the ratio of the terms of the first pro- 
gression, 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 



36 A PHILOSOPHICAL ESSAY CW PROBABILITIES. 

similar to that which he has found for the first progres- 
sion. 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 progres- 
sions an equation of the second degree, whose roots are 
these ratios. Thus Moivre decomposes the recurrent 
series into two geometrical progressions, each multi- 
plied 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 equa- 
tions of infinitely small differences with constant coeffi- 
cients, 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 / 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, 


THE CALCULUS OF PROBABILITIES. 37 

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 func- 
tion of the index x in V plus twice the same function 
in which the index is diminished by unity. This func- 
tion 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 d 
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 /. If we multiply anew by T the product of 
V by T, which is equivalent to multiplying V by T 2 , 
we shall form a third discriminant function, in which 
the coefficient of the jrth power of t will be a derivative 
similar to the corresponding coefficient of the preceding 
product ; it may be expressed by the same character 8 
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 th power of T, we shall have the coefficient 
of the ;rth power of t in the product of V by the nth 
power of T by placing before the primitive function the 
character 6 with n for an exponent. 

Let us suppose, for example, that T be unity divided 



38 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

by /; then in the product of Fby T the coefficient of 
the .rth power of / will be the coefficient of the power 
greater by unity in V\ this coefficient in the product 
of V by the nth power of T will then be the primitive 
function in which x is augmented by n units. 

Let us consider now a new function Z of /, developed 
like V and T according to the powers of /; let us 
designate by the character A placed before the primi- 
tive function the coefficient of the .rth power of / in the 
product of V by Z; this coefficient in the product of V 
by the nth power of Z will be expressed by the char- 
acter A affected by the exponent n and placed before 
the primitive function of x. 

If, for example, Z is equal to unity divided by t less 
one, the coefficient of the xth power of / in the product 
of V by Z will be the coefficient of the x -\- I power 
of t in V less the coefficient of the xth power. It will 
be then the finite difference of the primitive function of 
the index x. Then the character A indicates a finite 
difference of the primitive function in the case where 
the index varies by unity; and the nth power of this 
character placed before the primitive function will indi- 
cate the finite nth difference of this function. If we 
suppose that T be unity divided by /, we shall have 7 
equal to the binomial Z -j- I . The product of V by 
the nth power of T will then be equal to the product 
of V by the nth power of the binomial Z-\-\. Develop- 
ing 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 



THE CALCULUS OF PROBABILITIES. 39 

corresponding finite differences of the primitive function 
of the index. 

Now the product of V by the nth power of T is the 
primitive function in which the index x is augmented 
by n units; repassing from the discriminant functions 
to their coefficients, we shall have this primitive function 
thus augmented equal to the development of the nth 
power of the binomial Z-\- \, 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 i ; 
the product of V by the #th power of Z will then be 
equal to the product of V by the development of the 
#th power of the binomial T I . Repassing from the 
discriminant functions to their coefficients as has just 
been done, we shall have the nth difference of the 
primitive function expressed by the development of the 
?zth power of the binomial T I , in which we substi- 
tute 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 S before the primitive function expressing the 
derivative of this function, which multiplies the x power 
of / in the product of V by T, and A expressing the 
same derivative in the product of V by Z, we are led 



40 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

by that which precedes to this general result : whatever 
may be the function of the variable / 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 d and // 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 trans- 
form 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 / i , and that Z be always unity divided by / I ; 
then the coefficient of the x power of / in the pro- 
duct of V by T will be the coefficient of the x -\- i 
power of / 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-f- I 
and unity. The wth power of T is equal to the th 
power of this difference. If in this equality we substi- 
tute in place of T and Z the characters 6 and J, and 
after the development we place at the end of each term 
the primitive function of the index x> we shall have the 
wth difference of this function in which x varies by * 
units expressed by a series of differences of the same 
function in which x varies by unity. This series is 



THE CALCULUS OF PROBABILITIES. 41 

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 tf and A indicate the integrals. Indeed the 
nth difference of the primitive function having for a 
discriminant function the product of V by the nth power 
of the binomial one divided by t less unity, the primi- 
tive function which is the nth integral of this difference 
has for a discriminant function that of the same differ- 
ence multiplied by the nth power taken less than the 
binomial one divided by / minus one, a power to which 
the same power of the character A corresponds ; this 
power indicates then an integral of the same order, the 
index x varying by unity; and the negative powers of 
6 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 $ placed before the 
primitive function is zero, we shall have an equation of 
finite differences, and Fwill 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 / ought to disappear except the 
powers inferior to the order of the equation of differ- 
ences; 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 



42 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

equation of differences. The arbitrary coefficients of 
the various powers of / 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 / in place of the primitive function in 
which this index is diminished by unity; the second 
power of / for the primitive function where this index 
is diminished by two units, and so on. The coefficient 
of the x\\\ power of / in the development of the preced- 
ing 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 / and /' developed according to the powers 
and products of these variables ; the coefficient of any 
product of the powers x and x' of / and /' 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 /' 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 



THE CALCULUS OF PROBABILITIES. 43 

variable / 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 d placed before the primitive 
function, this derivative, the product of V by the wth 
power of T, will be the discriminant function of the 
derivative of the primitive function before which one 
places the ;/th power of the character 8. Hence result 
the theorems analogous to those which are relative to 
functions of a single variable. 

Suppose the function indicated by the character $ be 
zero; one will have an equation of partial differences. 
If, for example, we make as before T equal to the 
variable / phis 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 / by t' ; but Fmay include separately 
the powers of t and those of t' , that is to say, an arbi- 
trary function of t and an arbitrary function of /'; 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 ;rth power of t by the 
x' power of /' 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 



44 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 geometri- 
cians. It is thus that I have demonstrated the possi- 
bility 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 understand- 
ing, general signs whose true meaning may be ascer- 
tained only by repassing by metaphysical analysis to 
the elementary ideas which have led to them ; this 
often presents great difficulties, for the human mind 
tries still less to transport itself into the future than to 
retire within itself. The comparison of infinitely small 
differences with finite differences is able similarly to 
shed great light upon the metaphysics of infinitesimal 
calculus. 

It is easily proven that the finite nth difference of a 
function in which the increase of the variable is E 
being divided by the nth power of E, the quotient 
reduced in series by ratio to the powers of the increase 
E is formed by a first term independent of E. In the 
measure that E diminishes, the series approaches more 
and more this first term from which it can differ only 
by quantities less than any assignable magnitude. 



THE CALCULUS OF PROBABILITIES. 45 

This term is then the limit of the series and expresses 
in differential calculus the infinitely small nth difference 
of the function divided by the nth power of the infinitely 
small increase. 

Considering from this point of view the infinitely 
small differences, we see that the various operations of 
differential calculus amount to comparing separately in 
the development of identical expressions the finite 
terms or those independent of the increments of the 
variables which are regarded as infinitely small ; this 
is rigorously exact, these increments being indetermi- 
nant. 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 sub- 
secant; 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 differ- 
ence 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 sub- 
secants, a limit which is evidently the subtangent. 

This singularly happy method of obtaining the sub- 
tangent is due to Fermat, who has extended it to 



46 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

transcendent curves. This great geometrician ex- 
presses 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 differen- 
tial 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 nota- 
tion which, by indicating the passage from the finite to 
the infinitely small, adds to the advantage of express- 
ing 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 formulae in order to know with what probability 
the results are indicated, which the events develop by 
multiplication. It is necessary especially to have the 



THE CALCULUS OF PROBABILITIES. 47 

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 differen- 
tials multiplied by the factors raised to great powers 
would give by integration the formulae 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 remark- 
able 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 "nore 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 circum- 
ference to the diameter; sometimes it depends upon 
other transcendents whose number is infinite. 

An important remark which pertains to great gen- 
erality of analysis, and which permits us to extend this 
method to formulae 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 



48 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 nega- 
tive 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 accord- 
ingly 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 com- 
pared 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 



THE CALCULUS OF PROBABILITIES. 49 

germs of new calculi. This is rendered appreciable by 
this example. 

Wallis, who in his work entitled Arithmetica Infini- 
torum, one of those which have most contributed to the 
progress of analysis, has interested himself especially 
in following the thread of induction and analogy, con- 
sidered 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 frac- 
tion namely, of the quantity raised to a power indi- 
cated 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 



50 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

number. The observation then of the law of the num- 
bers which express these integrals, a series of inter- 
polations 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 frac- 
tional 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 deter- 
mined the law of these coefficients and extended k 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 con- 
stitutes exponential calculus, one of the most fruitful 



THE CALCULUS OF PROBABILITIES. 51 

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 tile variable. It is thus 
that a variable has for a logarithm the exponent of the 
power which is equal to it hi 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 differ- 
entiations 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 
formula which are as curious as useful on the trans- 
formations of differences and of integrals the ones into 
the others when the variables have divers finite incre- 
ments 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 



52 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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. 


53  PART II. 

APPLICATIONS OF THE CALCULUS OF 
PROBABILITIES. 

ToC

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 compli- 
cated, 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 


54 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

numbers depends upon the 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 / which renders this probability equal to J 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 approxima- 
tions 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 \ 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, 

GAMES OF CHANCE. 55 

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. 

ToC

56  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 and that of throwing it twice in succes- 
sion is J. 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 be ; 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 



UNKNOWN INEQUALITIES AMONG CHANCES. 57 

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 prob- 
ability of the simple event which it favors. If this 
event is heads, its probability will be plus -fo, or \, 
and the probability of throwing it twice in succession 
will be the square of -j^-, or |f . If the favored event is 
tails, the probability of heads, will be | minus ^ > or *V 
and the probability of throwing it twice in succession 
will be T Vo- 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 ^| for this 
probability, which exceeds by ^J-g- or by the square of 
the favor -fa that the inequality adds to the possibilities 
of the event which it favors. The probability of throw- 
ing tails tails is similarly f^, but the probability of 
throwing heads tails or tails heads is each jV T ; for 
the sum of these four probabilities ought to equal cer- 
tainty 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 



58 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

both to happen either an even number of times or odd 
number of times. The probability of each of these 
cases is 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 gam* 
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 or the same as the 
probability of the other player in the case where the 
number of their counters should become infinite, pre- 
serving 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 prob- 
ability of throwing heads twice in succession with the 
first coin will approach much nearer \ 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 



UNKNOWN INEQUALITIES AMONG CHANCES. 59 

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 i 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. 

ToC

60 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 spon 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  



INDEFINITE MULTIPLICATION OF EVENTS. 61 

mutually in the totality of a great number of draws, 
allowing us to perceive more and more the ratio cf 
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 follow- 
ing 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 diffi- 
cult to demonstrate by analysis. Accordingly the 
illustrious geometrician Jacques Bernouilli, 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 num- 
ber 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 



62 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 con- 
forming 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 



INDEFINITE MULTIPLICATION OF EVENTS. 63 

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 happi- 
ness 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 1hat of offering to man 
most useful lessons. Sometimes we attribute the 
inevitable results of these causes to the accidental cir- 
cumstances 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 



64 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

strong to give to a subjugated people its natural inde- 
pendence 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 prob- 
ability 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 num- 
bers of repetitions should become infinite. 

There are two kinds of approximations: the one is 
relative to the limits taken on all sides of the possibili- 
ties which give to the past the greatest probability; the 
other approximation is related to the probability that 
these possibilities fall within these limits. The repeti- 
tion 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 



INDEFINITE MULTIPLICATION OF EVENTS. 65 

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 Naptes 
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 AritJimctic several communi- 



66 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

ties of Bourgogne where the births of girls have sur- 
passed those of boys. Among these communities that 
of Carcelle-le-Grignon presents in 20x39 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 prob- 
ability of -fa, and this probability, smaller than that cf 
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 prob- 
ability, would disappear if one should follow during a 
century ihe 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 dis- 
tricts; 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 



INDEFINITE MULTIPLICATION OF EVENTS. 67 

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, 
1 80 1, and 1802 have given: 

Births. Marriages. Deaths. 

1 103 1 2 boys 46037 103659 men 
105287 girls 99443 women 

The ratio of the population to annual births is 
then 28 T 3 I7 5 oWb7r> ^ 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 pop- 
ulation 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 I 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 



68 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

epoch in which one has commenced to distinguish the 
sexes upon the birth-registers, up to the end of 1/84, 
there have been baptized in this capital 393386 boys 
and 377555 girls. The ratio of the two numbers is 
almost that of 2 5 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 i in favor of the existence of this cause, which is 
sufficient to authorize the investigation. Upon reflec- 
tion 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 -$$ 
the second, which it ought to have surpassed at least 
by ?V- 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 



INDEFINITE MULTIPLICATION Of-' EVENTS. 69 

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 T 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 ff ; the greatest is of the year 
1817, when it was ff. These ratios vary appreciably 
from the ratio of |f 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 ff , smaller than the mean ratio of ff . 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 T | 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 



70 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 investi- 
gation of the probability that this constancy will main- 
tain 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 I 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 



INDEFINITE MULTIPLICATION OF EVENTS. 71

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 recom- 
menced 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 indefi- 
nitely 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 



72 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 prob- 
ability 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 prob- 
ability, 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 varia- 
tion 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. 

ToC

73 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 



74 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

of exactitude obtained. Formulas 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 prob-^ 
abilities; it is one of the principal objects of the work 
which I have published upon this theory, and in which 
I have arrived at formulas 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 ele- 
ments whose corrections we determine then by resolv- 
ing 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 forma- 
tion 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 75 

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. Form- 
ing, 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 respect- 
ively 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 em- 
ploying in the same manner the coefficients of tl.e 
second element, and so on. In this manner the ele- 
ments 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 weigJit of the element or of the result. 
This weight is the greatest possible in the system of 



76 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 proba- 
bility 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 77 

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 equa- 
tions 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 



78 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 com- 
pare 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 coeffi- 
cient 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 79 

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 cor- 
rections 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 formulas of probability, I find that it is a bet of 
n,ooo against one that the error of this result is not 
T ^ 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 T L ff from that of 
M. Bouvard. This wise astronomer finds again the 
mass of Jupiter equal to the ro/ith part of the sun; 
and my method of probability gives a bet of 1,000,000 
to one that this result is not T ^o- 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 accumulat- 
ing, cause the value of the arc concluded from a great 
number of triangles to deviate appreciably from the 



80 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 
formulae 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 prob- 
ability; we may then by means of these formulae deter- 
mine 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 more- 
over 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 
formula:: an indeterminant which does not permit of 
reducing them to numbers unless we are able to elimi- 
nate it. We have seen that in astronomical questions, 
where each observation furnishes an equation of condi- 
tion 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 ele- 
ments 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 tri- 
angle 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 ; 



PROBABILITIES AND NATURAL PHILOSOPHY. 81 

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 cor- 
rects, 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 prob- 
able. The analysis of probabilities gives formulae for 



82 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 coeffi- 
cients 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 analyti- 
cally 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 
formulae 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 83 

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 obser- 
vations 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 formulae which 
diminishes as much as possible the errors to be feared 
in the results. 

The consideration of probabilities can serve to dis- 
tinguish the small irregularities of the celestial move- 
ments 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-Brahe 
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 



84 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

of astronomers neglect it in their calculations. Having 
submitted to the calculation of probabilities a consider- 
able 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 move- 
ment 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 1o 
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 observa- 
tions, 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, 7 J T , which differs little from 
the mean derived from the measurements of the degrees 
of the meridian and the pendulum ; but those regarded 



PROBABILITIES AND NATURAL PHILOSOPHY. 85 

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 geometri- 
cians, 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 ter- 
restrial 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 



86 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

to conciliate the observations he reduced the move- 
ments 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 intro- 
duced 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 87 

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 integra- 
tions 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-Brahe, 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 com- 
parison. 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, accord- 
ing 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 



88 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 estab- 
lished 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 move- 
ments about Jupiter. 

Accordingly it is seen how necessary it is to be at- 
tentive 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 appreci- 
able 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 deter- 
mined 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 89 

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 circum- 
stances 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 observa- 
tions 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 phenom- 
ena of nature, the flow and the ebb of the sea. 

Pline 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 



90 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 exam- 
ple, 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 investi- 
gation, 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 91 

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 equa- 
tions of partial differences which are relative has com- 
pelled 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 cricumstances of each port and are 
reduced to the smallest number possible ; it is only 
necessary to compare it to a great number of observa- 
tions. 

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 



92 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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. More- 
over, the tides being considerable at Brest, the acciden- 
tal 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 , 1 806 ; 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 calcula- 
tions which the comparison of my analysis with the 
observations has demanded. There have been used 
about six thousand observations, made during the year 
1 807 and the fifteen years following. It results from 
this comparison that my formulae represent with a 
remarkable precision all the varieties of the tides rela- 
tive 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 93 

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 observa- 
tions arranged in the order most suitable to indicate 
them. This calculation again determines their prob- 
ability 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 conse- 
quently to feel its influence and to be subjected to 
movements similar to those of the sea. 

These movements produce in the barometer periodic 



94 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 meteoro- 
logical 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 I, 1815, to October I, 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 milli- 
meter 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 mete- 
orology. 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 suffi- 
ciently 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 95 

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 mani- 
fested 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 arrang- 
ing 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 pres- 
sure 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 influ- 
ence 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 maxivntin 
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 



96 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

appreciable than in the tropics. M. Ramond has 
recognized and determined it at Clermont, the chief 
place of the district of Puy- de-Dome, 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 influ- 
ence of attractions of the sun and the moon upon the 
barometric heights at Paris have served me in deter- 
mining 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 I, 1817, to 
January I, 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 obser- 
vations of this phenomenon, I found that it was a bet 01 
more than 300,00x3 against one that a regular cause 
produced it. I do not seek to determine this cause; I 



PROBABILITIES AND NATURAL PHILOSOPHY. 97 

content myself with stating its existence. The period 
of the diurnal variation regulated by the solar day indi- 
cates 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 observa- 
tions 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 movemens 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 move- 
ments. 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 com- 
posed 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 



98 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 analy- 
sis 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 recog- 
nize 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. 



PROBABILITIES AND NATURAL PHILOSOPHY. 99 

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 sepa- 
rate 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 nebulae 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 pre- 
ceded 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 exist- 
ence. 

Such is indeed the first state of the nebulae 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 t6 us except after 



100 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 con- 
densed about one or several faintly luminous nebulae. 
In the other nebulae 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 
nebulas 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 nebulae which are called planetary. 
Finally a greater degree of condensation transforms all 
these nebulae into stars. The nebulae classed accord- 
ing 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 tis are the results of a primitive cause 



PROBABILITIES AND NATURAL PHILOSOPHY. 1O1 

or of a general law of nature. These groups are a 
necessary result of the condensation of the nebulae 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 nebulae 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 6ist of the Swan and its following one 
in which Bessel has just recognized particular move- 
ments 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 con- 
sideration 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 prob- 
ability 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 



102 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

clivers 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 con- 
sidering 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. Obser- 
vation 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 nebulae they may be regarded as small nebulae 
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 
nebulae that they are often confounded with them ; and 
it is only by their movement, or by the knowledge of 
all the nebulae 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 



PROBABILITIES AND NATURAL PHILOSOPHY. 103 

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 nebulse 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 conform- 
able 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 hyper- 
bolic ; and if the axes of these hyperbolae 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 ele- 
ments, 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 inconsider- 
able. 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 



104 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 determin- 
ing 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 par- 



PROBABILITIES AND NATURAL PHILOSOPHY. 105 

ticular 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 physi- 
cists 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 recogniz- 
ing 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 



106 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 condi- 
tions 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. 

ToC

107 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 compli- 
cated 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 a 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 reestablished 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  



108 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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. 

ToC

109 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 wit- 
nesses 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 approxi- 
mations to guide and to defend us againt 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 obtain- 
ing it. 

A single number has been drawn from an urn which 
contains a thousand of them. A witness to this draw- 
ing 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



110 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

deceives one time in ten, so that the probability of his 
testimony is T V Here the event observed is the wit- 
ness 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 a priori the probabil- 
ity 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, TTTOTT- It is necessary to multiply it by 
the probability j 6 ff of the veracity of the witness ; one 
will have then T |hn5 f r 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 $$$$. 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 -577; multiplying, then, this probability by the pre- 
ceding one, we shall have y^Vo f r the probability that 
the witness will announce number 79 in the second 
hypothesis. It is necessary again to multiply this 
probability by T V of the hypothesis itself, which gives 
uriinr f r t^ e probability of the event relative to this 
hypothesis. Now if we form a fraction whose numera- 
tor is the probability relative to the first hypothesis, and 
whose denominator is the sum of the probabilities rela- 
tive to the two hypotheses, we shall have, by the sixth 
principle, the probability of the first hypothesis, and 



CONCERNING THE PROBABILITIES OF TESTIMONIES, 111 

this probability will be T 9 ff ; 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 fa. 

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, -jfg, it will then be , , etc., according to the 
interest that he will have in announcing its drawing. 
Supposing it to be |, it will be necessary to multiply 
by this fraction the probability T VVo m order to get in 
the hypothesis of the falsehood the probability of the 
event observed, which it is necessary still to multiply 
by y^, which gives TihhjT f r 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 yfg-. It is then 
very much decreased by the consideration of the in- 
terest which the witness may have in announcing the 
drawing of number 79. In truth this same interest 
increases the probability -^ that the witness will speak 
the truth if number 79 is drawn. But this probability 
cannot exceed unity or | ; thus the probability of the 
drawing of number 79 will not surpass T y>T- Common 
sense tells us that this interest ought to inspire distrust, 
but calculus appreciates the influence of it. 

The probability a priori of the number announced 
by the witness is unity divided by the number of the 



112 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 for the 
probability a priori of the drawing of number I , and if 
the veracity of a witness who announces it is T %, 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 . 
But if there are three numbers in the urn the probability 
a priori of the drawing of number I is increased by 
the affirmation of a witness whose veracity surpasses . 
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 a priori in the first hypothesis, will be here, 
as in the preceding question, equal to -foooir- But m 
the hypothesis where the witness deceives, the white 
ball is not drawn and the probability of this case 
is T V(TV It ls necessary to multiply it by the prob- 
ability T V of the falsehood, which gives T |||^ for the 
probability of the event observed relative to the second 
hypothesis. This probability was only T ol7nr m tne 
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 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 113 

to each hypothesis, and whose common denominator is 
the sum of these probabilities, one will have i^^ for 
the probability of the first hypothesis and of the drawing 
of a white ball, and T Vir 9 ff f r 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 
TVoVoVs 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. Deter- 
mining a priori in each of these hypotheses the prob- 
ability 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 mis- 
taken, 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 



114 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 draw- 
ing 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 com- 
bination is extraordinary. 

But if the conclusion which results from the com- 
bination of the testimonies was impossible one of them 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 115 

would be necessarily false; but an impossible conclu- 
sion 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 prob- 
abilities. 

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 yc-UFor- ^ n order to 
determine the diminution which results in the prob- 
ability 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 T 9 T the 
probability that he announces the truth, which can 



116 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 Ihe probability of this event is |, since the ball 
drawn al 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 
the probability of the fact announced is then 
Multiplying it by the product of the probabilities -fa 
and y 9 ^ that the witnesses speak the truth one will 
have ^nnrVoinr f r the probability of the event ob- 
served 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 . Then this ball 
having been placed in the urn B a black ball has been 
drawn from it: the probability of such drawing is 
|_o_o 0.0.0 . one has then #{$!$ for the probability of 
the compound event. Multiplying it by the product 
of the two probabilities T 9 and T V that the first witness 
speaks the truth and that the second does not, one 
will have y^^fjfo. 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 



CONCERNING THE PROBABILITIES OF TESTIMONIES, 117 

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 and that of the second is T #$S$T ; the 
probability of the compound event is then i^^-^f. 
Multiplying it by the product of the probabilities 1 3 j r 
and yV that the first witness does not speak the truth 
and that the second announces it, one will have 
siiHfTmHta for tne 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 prob- 
ability of this compound event is aooooo?- Multiply- 
ing it by the product of the probabilities -fa and y 1 ^- that 
each witness does not speak the truth one will have 
200000200 f r tne 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 hy- 
pothesis by the sum of the probabilities relative to 
the four hypotheses ; and then one has for this prob- 
ability y-g-ooooas' 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 y\V; it will then 



118 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 de- 
creases 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 his- 
torical 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. 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 119 

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 Mile. 
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 perse- 
cuted 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 



120 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 per- 
formed 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 Under- 
standing by the philosopher Locke, who says, in 
speaking of the degree of assent: " Though the com- 
mon 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 lact 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 
maybe 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 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 121 

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 ot 
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 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 num- 
bers by the probability of falsehood considered a 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 



122 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 prob- 
ability 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 -j-J-j-. It is necessary to multiply it by 
the product of the veracities of the witnesses, veracities 
which we will suppose to be T 9 7 and T \: one will have 
then T^VTFIT 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 y 9 ^. 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 5 \ by itself; it becomes 
necessary then to multiply these two probabilities 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 123 

together, and by the product of the probabilities y 1 ^ and 
Y 3 ^ that the witnesses deceive; one will have thus 
sygVuir f r 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 f$|-|, and the 
probability of the failure to draw this number and of 
the falsehood of the witnesses will be ^ViF' 

If the urn should contain only the numbers I and 2 
one would find in the same manner f for the prob- 
ability of the drawing of number 2, and consequently 
^ 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 prob- 
ability 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 diffi- 
cult, at least when they do not have a secret agree- 
ment, which we do not suppose here at all. 

In the preceding case where the urn contained only 
two numbers the a priori probability of the fact attested 
is ^, 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 



124 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 testi- 
monies 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 or" the 
probability of his falsehood to the number of the num- 
bers in the urn less one; so that the limit of the prob- 
ability of the fact is that of this fact considered a 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 



CONCERNING THE PROBABILITIES OF TESTIMONIES. 125 

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 ; sup- 
posing 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. 

ToC

126 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 prob- 
ability 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 informa- 
tion be generally diffused and that good works founded 
upon reason and experience should enlighten those  



SELECTIONS AND DECISIONS OF ASSEMBLIES. 127 

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 candi- 
dates. 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 



128 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 num- 
ber 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 I, 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 indi- 
cate 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 



SELECTIONS AND DECISIONS OF ASSEMBLIES. 129 

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 candi- 
dates. 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 neces- 
sary 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 dis- 
tributes 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 



130 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 quanti- 
ties : 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 



SELECTIONS AND DECISIONS OF ASSEMBLIES. 131 

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. 

ToC

132 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  



PROBABILITY OF THE JUDGMENTS OF TRIBUNALS. 133 

probabi litres to demand a majority of at least two votes 
in a tribunal of appeal in order to invalidate the sen- 
tence 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 crimi- 
nal 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 sus- 
ceptible. 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 con- 
demned, 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 ascer- 
tain. Such is the eminence of danger which would 



134 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 insolu- 
ble 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 



PROBABILITY OF THE JUDGMENTS OF TRIBUNALS. 135 

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 compar- 
ing among themselves the different tribunals. The 
majority of a single vote in a numerous tribunal indi- 
cates 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 neces- 
sary, 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 tri- 
bunal 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 



136 A PHILOSOPHICAL ESSAY OV PROBABILITIES. 

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 ; 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 deter- 
mine this probability. I suppose thus that it can vary 
from to unity, but that it cannot be below . 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 ^. 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 ^. 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 



PROBABILITY OF THE JUDGMENTS OF TRIBUNALS. 137 

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 numer- 
ous. 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 attain- 
ing 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 appre- 
ciate 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 prob- 
ability of error in the first case, and only -^-^ for this 
probability in the second case, a probability which is 
not one thousandth of the first. It is a confirmation 



138 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 % 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 ^ 6 / ff , or more than ^. 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 reluc- 
tance a culprit in the accused submitted to his judg- 
ment. 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 



PROBABILITY OF THE JUDGMENTS OF TRIBUNALS. 139 

demanded for the condemnation is eight of twelve 
votes, the probability of the error to be feared |fff , or 
a little more than one eighth, it is almost -fa if this 
plurality consists of nine votes. In the case of una- 
nimity the probability of the error to be feared is -g- T 1 7 j, 
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 circum- 
stances 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. 

ToC

140 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 num- 
ber 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 I 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  



CONCERNING TABLES OF MORTALITY, ETC. 141

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 differ- 
ence of the numbers of individuals inscribed at the side 
of the years o and I. 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 
formulae for appreciating the probability that the num- 



142 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

bers indicated in these tables will vary from the truth 
only within narrow limits. We see by these formulae 
that the interval of the limits diminishes and that the 
probability increases in proportion as we take into con- 
sideration more births; so that the tables would repre- 
sent 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 com- 
prised 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 



CONCERNING TABLES OF MORTALITY, ETC. 143 

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 I . There is subtracted from this first re- 
mainder 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 accord- 
ing 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 indi- 
cated. Thus the ratio of the population to annual 
births, which one has seen raised in France to twenty- 



144 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 advan- 
tage of inoculation, joined to those of not altering the 
appearance and of preserving from the grievous conse- 
quences 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 



CONCERNING TABLES OF MORTALITY, ETC. 145 

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 Ber- 
noulli: 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 dis- 
appears when one considers a great number of indi- 
viduals, 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 



146 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 con- 
sists in determining by observation the number of indi- 
viduals 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 sub- 
tracting 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 prob- 
abilities 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 dura- 
tion 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 



CONCERNING TABLES OF MORTALITY, ETC. 147 

all kinds of animals and vegetables, nature* tends with- 
out 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 genera- 
tions, 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 simul- 
taneously in geometric progression. But when clear- 
ings become more difficult and more rare then the 
increase of population diminishes; it approaches con- 
tinually the variable state of subsistences, making 
oscillations about it just as a pendulum whose periodicity 
is retarded by changing the point of suspension, oscil- 
lates 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 favor- 
able 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 popula- 
tion, births, marriages, mortality, all increase accord- 
ing to the same geometric progression of which we have 
the constant ratio of consecutive terms by the observa- 
tion 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 



148 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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. 

ToC

149 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 repe-  



150 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

titions; 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 



INSTITUTIONS BASED UPON PROBABILITIES. 151 

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 exam- 
ple, the rate of interest is --$ 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. Favor- 
able 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 legisla- 
tive power is divided among several independent 
powers. The confidence which the necessary coopera- 
tion 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. 



152 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 indi- 
viduals 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. Form- 
ing, 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 numera- 
tors 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 



INSTITUTIONS BASED UPON PROBABILITIES. 153 

life annuity to assure to his heirs an amount payable 
at the end of the year of his death. In order to deter- 
mine 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 assur- 
ance 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 pro- 
vide 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 disadvan- 



154 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

tage 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 



INSTITUTIONS BASED UPON PROBABILITIES. 155 

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 advan- 
tage 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 pro- 
portion to the benefit that they give, give a moral 
advantage always inferior to the mathematical advan- 
tage. 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 asso- 
ciation whose members secure mutually their proper- 
ties 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 com- 
mon to all and without doubt~the system of weights, 
measures, and moneys proposed by the French sci- 
entists would be adopted in this congress as one of 
the things most useful to commerical 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 



156 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 exist- 
ence. 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 progres- 
sion 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 pro- 
gression 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 



INSTITUTIONS BASED UPON PROBABILITIES. 157 

acquired would be formed by these funds and from 
this annuity the State contributes annually to the sink- 
ing 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 pro- 
portion 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 con- 
cerning 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 
g, q ', q" , etc., the corrections of the following results. 
Let us name e, e' , e" , etc., the errors of the observa- 
tions 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 observa- 
tion w r ill be equal to the product of x by a determined 
coefficient. Likewise the error e' of the second obser- 
vation 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 pre- 
ceding 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 simul- 



158 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

taneous existence of the errors e, f', c" , etc., will be 
then proportional to the product of these divers func- 
tions, 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 com- 
mitted 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. Multi- 
plying 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 cor- 
rection ; the integral of the product of x by the corre- 
sponding 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 



INSTITUTIONS BASED UPON PROBABILITIES. 159 

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. 

ToC

160 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 exag- 
gerated 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  



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 161 

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 suffi- 
ciently guard himself. 

It is principally at games of chance that a multitude 
of illusions support hope and sustain it against unfavor- 
able 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 imagi- 
nation 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 com- 
binations 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 excite- 
ment 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 



162 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 163 

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 mul- 
tiplied, 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 combina- 
tions 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 calcula- 



164 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

tion 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 con- 
vince 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 con- 
siderable 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 prob- 
abilities is to teach us to distrust first opinions. As we 
recognize that they often deceive when they may be 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 165 

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 
of extracting a black ball at each drawing ; it would 
be then T ^. 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 |; it is if one supposes two 
white balls and two black; finally it is reduced to J if 
one supposes three black balls and one white. Follow- 
ing the principle of the probability of causes drawn 
from events the probabilities of these three suppositions 
are among themselves as the quantities , f, ; they 
are consequently equal to |, f, . It is thus a bet of 
5 against i 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 



166 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

of , of |, and of | 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 f , f , and , the sum of the products 
will be the probability of extracting successively four 
black balls. One has thus for this probability ^ 2 9 , a 
fraction greater than -fa. 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 f or ^ ; but this 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 167 

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 -|, a 
fraction greater than f . 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 Mere, 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 num- 
bers 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 under- 
taking 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 Mere, 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 



168 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 suppos- 
ing 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 draw- 
ings 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 draw- 
ing 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 fol- 
lowing 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 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 169 

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 cal- 
culus 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 , 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 , 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 methematics in 
China, in the hope that this emblem of creation would 
convert to Christianity the emperor there who particu- 
larly loved the sciences. I report this incident only 
to show to what extent the prejudices of infancy can 
mislead the greatest men. 



170 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 
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 multi- 
plied 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., 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 171 

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 f ; 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 rota- 
tion and of revolution of the planets and satellites, and 
the position of the orbits and of their equators are sub- 
jected 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 atten- 



172 A PHILOSOPHICAL ESSAY ON PROBABILITIES, 

tive 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 tempera- 
ture 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 respec- 
tive 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 debris 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 move- 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 173 

ments. The variations already observed in the stars 
and in the form of the nebulae 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 predic- 
tions 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-coinci- 
dences 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 



174 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 on.e 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 insti- 
tutions. 

"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 super- 
stition 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, impor- 
tunes one, and pursues one continually and in all places. 



ILLUSIONS IN THE ESTIMATION OF PROBABILITIES. 175 

If one consult a diviner or a soothsayer, if one immo- 
lates 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. 

ToC

176 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 inditction. Newton was 
indebted to it for his theorem of the binomial and the 
principle of universal gravity. It is difficult to appre- 
ciate the probability of the results of induction, which is  



VARIOUS MEANS OF APPROACHING CERTAINTY. 177 

based upon this that the simplest ratios are the most 
common ; this is verified in the formulae 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 experi- 
ences; 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 num- 
ber 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 considera- 
tions, 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. 



178 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

We judge by induction that if various events, move- 
ments, 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 prob- 
ability, the existence of constant causes which geome- 
tricians 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 



VARIOUS MEANS OF APPROACHING CERTAINTY. 1?9 

the moon and the highest comets. It is still percepti- 
ble 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 



180 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

subtilities 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 simili- 
tude 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, combina- 
tions that are much more singular; one might, perhaps, 
name this mingling of elective affinities and sensations 



VARIOUS MEANS OF APPROACHING CERTAINTY. 181 

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 deny- 
ing it to them. 

Since the sun brings forth, bythe beneficent action 
of its light and of its heat, the animals and plants 
Avhich 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 simili- 
tude 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 demon- 



182 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

strated 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, primi- 
tively 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 exami- 
nation if it agrees with known phenomena; and if by 
a rigorous analysis we see them proceed from this 



YAR1OUS MEJNS OF APPROACHING CERTAINTY. 183 

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 perfec- 
tion 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 ; some- 
times 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 01 
Ptolemy, which is only the realization of celestial 



184 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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. 

ToC

185  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 sup- 
posed 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



186 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

is clear that the distribution should be made propor- 
tionally 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 alece. 
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 conjcctandi, 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 



THE CALCULUS OF PROBABILITIES. 187 

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 demon- 
stration, 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 probabili- 
ties. That of Montmort has the title Ess at 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 diffi- 
cult problems. The treatise of Moivre, later than that 
of Montmort, appeared at first in the Transactions 
pliilosopliiqucs of the year 1711. Then the author 
published it separately, and he has improved it succes- 
sively 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 



188 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 con- 
siderable 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 singlar 



THE CALCULUS OF PROBABILITIES. 189 

analysis which contains the germ of the very curious 
and useful theory of definite intergrals. 

Many scholars, among whom one ought to name 
Deparcieux, Kersseboom, Wargentin, Dupre de Saint- 
Maure, Simpson, Sussmilch, Messene, Moheau, Price, 
Bailey, and Duvillard, have collected a great amount 
of precise data in regard to population, births, mar- 
riages, and mortality. They have given formulae 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 inocula- 
tion. 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 possibili- 
ties known, and they sought the probability that the 
result of future experiences will more and more nearly 
represent them. Bayes, in the Transactions pliiloso- 
phiqncs 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 



190 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 concern- 
ing 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 deter- 
mines 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 cal- 



THE CALCULUS OF PROBABILITIES. 191 

culus 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 Memoircs 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 Cotes 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 equa- 
tions should be added in order to form a final equation, 
whence is derived the value of this element. The rule 
of Cotes was followed by all calculators, but since he 
failed to determine several elements, there \vas no fixed 
rule for combining the equations of condition in such a 



192 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

manner as to obtain the necessary final equations; but 
one chose for each element the observations most suit- 
able 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 vary- 
ing 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 prob- 
abilities 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 observa- 
tions, 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 deter- 
mining that which I have called the weight of a result. 
Analysis leads to general and simple formulae 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 CALCULUS OF PROBABILITIES. 193 

The law of the possibility of the errors of observa- 
tions introduces into the expressions of these prob- 
abilities 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 opera- 
tions 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 deter- 
mined by means of the meridian telescope and of the 
circle, both susceptible of errors whose law of prob- 
ability 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 



194 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

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 formulae 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 observa- 
tions. 

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 



THE CALCULUS OF PROBABILITIES. 195 

regard to the correctness of the results. The analytic 
formulae of probabilities satisfy perfectly this require- 
ment; 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 cal- 
culus; 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 observa- 
tions 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 con- 
stant 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 knowlegdge. 



196 A PHILOSOPHICAL ESSAY ON PROBABILITIES. 

I have collected all these methods in my TJieorie 
analytique des Probabilite's, in which I have proposed 
to expound in the most general manner the principles 
and the analysis of the calculus of probabilities, like- 
wise 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 arbitrari- 
ness 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 impor- 
tant 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. 

ToC