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"ex veto" on page 173.
A PHILOSOPHICAL ESSAY
ON
PROBABILITIES.
BY
PIERRE SIMON, MARQUIS DE LAPLACE.
_TRANSLATED FROM THE SIXTH FRENCH EDITION_
BY
FREDERICK WILSON TRUSCOTT, PH.D. (HARV.),
_Professor of Germanic Languages in the West Virginia University_,
AND
FREDERICK LINCOLN EMORY, M.E. (WOR. POLY. INST.),
_Professor of Mechanics and Applied Mathematics in the West Virginia
University; Mem. Amer. Soc. Mech. Eng._
_FIRST EDITION._
FIRST THOUSAND.
NEW YORK:
JOHN WILEY & SONS.
LONDON: CHAPMAN & HALL, LIMITED.
1902.
Copyright, 1902,
BY
F. W. TRUSCOTT
AND
F. L. EMORY.
ROBERT DRUMMOND PRINTER, NEW YORK
TABLE OF CONTENTS.
PAGE
PART I.
_A PHILOSOPHICAL ESSAY ON PROBABILITIES._
CHAPTER I.
Introduction 1
CHAPTER II.
Concerning Probability 3
CHAPTER III.
General Principles of the Calculus of Probabilities 11
CHAPTER IV.
Concerning Hope 20
CHAPTER V.
Analytical Methods of the Calculus of Probabilities 26
PART II.
_APPLICATION OF THE CALCULUS OF PROBABILITIES._
CHAPTER VI.
Games of Chance 53
CHAPTER VII.
Concerning the Unknown Inequalities which may Exist among
Chances Supposed to be Equal 56
CHAPTER VIII.
Concerning the Laws of Probability which result from the
Indefinite Multiplication of Events 60
CHAPTER IX.
Application of the Calculus of Probabilities to Natural
Philosophy 73
CHAPTER X.
Application of the Calculus of Probabilities to the Moral
Sciences 107
CHAPTER XI.
Concerning the Probability of Testimonies 109
CHAPTER XII.
Concerning the Selections and Decisions of Assemblies 126
CHAPTER XIII.
Concerning the Probability of the Judgments of Tribunals 132
CHAPTER XIV.
Concerning Tables of Mortality, and the Mean Durations
of Life, Marriage, and Some Associations 140
CHAPTER XV.
Concerning the Benefits of Institutions which Depend upon
the Probability of Events 149
CHAPTER XVI.
Concerning Illusions in the Estimation of Probabilities 160
CHAPTER XVII.
Concerning the Various Means of Approaching Certainty 176
CHAPTER XVIII.
Historical Notice of the Calculus of Probabilities to 1816 185
ERRATA.
Page 89, line 22, _for_ Pline _read_ Pliny
" 102, lines 14, 16, " minutes " days
" 143, line 25, " sun " soil
" 177, lines 15, 17, 18, 21, 22, 24, _for_ primary _read_ prime
" 182, line 5, _for_ conjunctions _read_ being binary
A PHILOSOPHICAL ESSAY ON PROBABILITIES.
CHAPTER I.
_INTRODUCTION._
This philosophical essay is the development of a lecture on
probabilities which I delivered in 1795 to the normal schools whither I
had been called, by a decree of the national convention, as professor of
mathematics with Lagrange. I have recently published upon the same
subject a work entitled _The Analytical Theory of Probabilities_. I
present here without the aid of analysis the principles and general
results of this theory, applying them to the most important questions of
life, which are indeed for the most part only problems of probability.
Strictly speaking it may even be said that nearly all our knowledge is
problematical; and in the small number of things which we are able to
know with certainty, even in the mathematical sciences themselves, the
principal means for ascertaining truth—induction and analogy—are based
on probabilities; so that the entire system of human knowledge is
connected with the theory set forth in this essay. Doubtless it will be
seen here with interest that in considering, even in the eternal
principles of reason, justice, and humanity, only the favorable chances
which are constantly attached to them, there is a great advantage in
following these principles and serious inconvenience in departing from
them: their chances, like those favorable to lotteries, always end by
prevailing in the midst of the vacillations of hazard. I hope that the
reflections given in this essay may merit the attention of philosophers
and direct it to a subject so worthy of engaging their minds.
CHAPTER II.
_CONCERNING PROBABILITY._
All events, even those which on account of their insignificance do not
seem to follow the great laws of nature, are a result of it just as
necessarily as the revolutions of the sun. In ignorance of the ties
which unite such events to the entire system of the universe, they have
been made to depend upon final causes or upon hazard, according as they
occur and are repeated with regularity, or appear without regard to
order; but these imaginary causes have gradually receded with the
widening bounds of knowledge and disappear entirely before sound
philosophy, which sees in them only the expression of our ignorance of
the true causes.
Present events are connected with preceding ones by a tie based upon the
evident principle that a thing cannot occur without a cause which
produces it. This axiom, known by the name of _the principle of
sufficient reason_, extends even to actions which are considered
indifferent; the freest will is unable without a determinative motive to
give them birth; if we assume two positions with exactly similar
circumstances and find that the will is active in the one and inactive
in the other, we say that its choice is an effect without a cause. It is
then, says Leibnitz, the blind chance of the Epicureans. The contrary
opinion is an illusion of the mind, which, losing sight of the evasive
reasons of the choice of the will in indifferent things, believes that
choice is determined of itself and without motives.
We ought then to regard the present state of the universe as the effect
of its anterior state and as the cause of the one which is to follow.
Given for one instant an intelligence which could comprehend all the
forces by which nature is animated and the respective situation of the
beings who compose it—an intelligence sufficiently vast to submit these
data to analysis—it would embrace in the same formula the movements of
the greatest bodies of the universe and those of the lightest atom; for
it, nothing would be uncertain and the future, as the past, would be
present to its eyes. The human mind offers, in the perfection which it
has been able to give to astronomy, a feeble idea of this intelligence.
Its discoveries in mechanics and geometry, added to that of universal
gravity, have enabled it to comprehend in the same analytical
expressions the past and future states of the system of the world.
Applying the same method to some other objects of its knowledge, it has
succeeded in referring to general laws observed phenomena and in
foreseeing those which given circumstances ought to produce. All these
efforts in the search for truth tend to lead it back continually to the
vast intelligence which we have just mentioned, but from which it will
always remain infinitely removed. This tendency, peculiar to the human
race, is that which renders it superior to animals; and their progress
in this respect distinguishes nations and ages and constitutes their
true glory.
Let us recall that formerly, and at no remote epoch, an unusual rain or
an extreme drought, a comet having in train a very long tail, the
eclipses, the aurora borealis, and in general all the unusual phenomena
were regarded as so many signs of celestial wrath. Heaven was invoked in
order to avert their baneful influence. No one prayed to have the
planets and the sun arrested in their courses: observation had soon made
apparent the futility of such prayers. But as these phenomena, occurring
and disappearing at long intervals, seemed to oppose the order of
nature, it was supposed that Heaven, irritated by the crimes of the
earth, had created them to announce its vengeance. Thus the long tail of
the comet of 1456 spread terror through Europe, already thrown into
consternation by the rapid successes of the Turks, who had just
overthrown the Lower Empire. This star after four revolutions has
excited among us a very different interest. The knowledge of the laws of
the system of the world acquired in the interval had dissipated the
fears begotten by the ignorance of the true relationship of man to the
universe; and Halley, having recognized the identity of this comet with
those of the years 1531, 1607, and 1682, announced its next return for
the end of the year 1758 or the beginning of the year 1759. The learned
world awaited with impatience this return which was to confirm one of
the greatest discoveries that have been made in the sciences, and fulfil
the prediction of Seneca when he said, in speaking of the revolutions of
those stars which fall from an enormous height: "The day will come when,
by study pursued through several ages, the things now concealed will
appear with evidence; and posterity will be astonished that truths so
clear had escaped us." Clairaut then undertook to submit to analysis the
perturbations which the comet had experienced by the action of the two
great planets, Jupiter and Saturn; after immense calculations he fixed
its next passage at the perihelion toward the beginning of April, 1759,
which was actually verified by observation. The regularity which
astronomy shows us in the movements of the comets doubtless exists also
in all phenomena.
The curve described by a simple molecule of air or vapor is regulated in
a manner just as certain as the planetary orbits; the only difference
between them is that which comes from our ignorance.
Probability is relative, in part to this ignorance, in part to our
knowledge. We know that of three or a greater number of events a single
one ought to occur; but nothing induces us to believe that one of them
will occur rather than the others. In this state of indecision it is
impossible for us to announce their occurrence with certainty. It is,
however, probable that one of these events, chosen at will, will not
occur because we see several cases equally possible which exclude its
occurrence, while only a single one favors it.
The theory of chance consists in reducing all the events of the same
kind to a certain number of cases equally possible, that is to say, to
such as we may be equally undecided about in regard to their existence,
and in determining the number of cases favorable to the event whose
probability is sought. The ratio of this number to that of all the cases
possible is the measure of this probability, which is thus simply a
fraction whose numerator is the number of favorable cases and whose
denominator is the number of all the cases possible.
The preceding notion of probability supposes that, in increasing in the
same ratio the number of favorable cases and that of all the cases
possible, the probability remains the same. In order to convince
ourselves let us take two urns, A and B, the first containing four white
and two black balls, and the second containing only two white balls and
one black one. We may imagine the two black balls of the first urn
attached by a thread which breaks at the moment when one of them is
seized in order to be drawn out, and the four white balls thus forming
two similar systems. All the chances which will favor the seizure of one
of the balls of the black system will lead to a black ball. If we
conceive now that the threads which unite the balls do not break at all,
it is clear that the number of possible chances will not change any more
than that of the chances favorable to the extraction of the black balls;
but two balls will be drawn from the urn at the same time; the
probability of drawing a black ball from the urn A will then be the same
as at first. But then we have obviously the case of urn B with the
single difference that the three balls of this last urn would be
replaced by three systems of two balls invariably connected.
When all the cases are favorable to an event the probability changes to
certainty and its expression becomes equal to unity. Upon this
condition, certainty and probability are comparable, although there may
be an essential difference between the two states of the mind when a
truth is rigorously demonstrated to it, or when it still perceives a
small source of error.
In things which are only probable the difference of the data, which each
man has in regard to them, is one of the principal causes of the
diversity of opinions which prevail in regard to the same objects. Let
us suppose, for example, that we have three urns, A, B, C, one of which
contains only black balls while the two others contain only white balls;
a ball is to be drawn from the urn C and the probability is demanded
that this ball will be black. If we do not know which of the three urns
contains black balls only, so that there is no reason to believe that it
is C rather than B or A, these three hypotheses will appear equally
possible, and since a black ball can be drawn only in the first
hypothesis, the probability of drawing it is equal to one third. If it
is known that the urn A contains white balls only, the indecision then
extends only to the urns B and C, and the probability that the ball
drawn from the urn C will be black is one half. Finally this probability
changes to certainty if we are assured that the urns A and B contain
white balls only.
It is thus that an incident related to a numerous assembly finds various
degrees of credence, according to the extent of knowledge of the
auditors. If the man who reports it is fully convinced of it and if, by
his position and character, he inspires great confidence, his statement,
however extraordinary it may be, will have for the auditors who lack
information the same degree of probability as an ordinary statement made
by the same man, and they will have entire faith in it. But if some one
of them knows that the same incident is rejected by other equally
trustworthy men, he will be in doubt and the incident will be
discredited by the enlightened auditors, who will reject it whether it
be in regard to facts well averred or the immutable laws of nature.
It is to the influence of the opinion of those whom the multitude judges
best informed and to whom it has been accustomed to give its confidence
in regard to the most important matters of life that the propagation of
those errors is due which in times of ignorance have covered the face of
the earth. Magic and astrology offer us two great examples. These errors
inculcated in infancy, adopted without examination, and having for a
basis only universal credence, have maintained themselves during a very
long time; but at last the progress of science has destroyed them in the
minds of enlightened men, whose opinion consequently has caused them to
disappear even among the common people, through the power of imitation
and habit which had so generally spread them abroad. This power, the
richest resource of the moral world, establishes and conserves in a
whole nation ideas entirely contrary to those which it upholds elsewhere
with the same authority. What indulgence ought we not then to have for
opinions different from ours, when this difference often depends only
upon the various points of view where circumstances have placed us! Let
us enlighten those whom we judge insufficiently instructed; but first
let us examine critically our own opinions and weigh with impartiality
their respective probabilities.
The difference of opinions depends, however, upon the manner in which
the influence of known data is determined. The theory of probabilities
holds to considerations so delicate that it is not surprising that with
the same data two persons arrive at different results, especially in
very complicated questions. Let us examine now the general principles of
this theory.
CHAPTER III.
_THE GENERAL PRINCIPLES OF THE CALCULUS OF PROBABILITIES._
_First Principle._—The first of these principles is the definition
itself of probability, which, as has been seen, is the ratio of the
number of favorable cases to that of all the cases possible.
_Second Principle._—But that supposes the various cases equally
possible. If they are not so, we will determine first their respective
possibilities, whose exact appreciation is one of the most delicate
points of the theory of chance. Then the probability will be the sum of
the possibilities of each favorable case. Let us illustrate this
principle by an example.
Let us suppose that we throw into the air a large and very thin coin
whose two large opposite faces, which we will call heads and tails, are
perfectly similar. Let us find the probability of throwing heads at
least one time in two throws. It is clear that four equally possible
cases may arise, namely, heads at the first and at the second throw;
heads at the first throw and tails at the second; tails at the first
throw and heads at the second; finally, tails at both throws. The first
three cases are favorable to the event whose probability is sought;
consequently this probability is equal to ¾; so that it is a bet of
three to one that heads will be thrown at least once in two throws.
We can count at this game only three different cases, namely, heads at
the first throw, which dispenses with throwing a second time; tails at
the first throw and heads at the second; finally, tails at the first and
at the second throw. This would reduce the probability to ⅔ if we should
consider with d'Alembert these three cases as equally possible. But it
is apparent that the probability of throwing heads at the first throw is
½, while that of the two other cases is ¼, the first case being a simple
event which corresponds to two events combined: heads at the first and
at the second throw, and heads at the first throw, tails at the second.
If we then, conforming to the second principle, add the possibility ½ of
heads at the first throw to the possibility ¼ of tails at the first
throw and heads at the second, we shall have ¾ for the probability
sought, which agrees with what is found in the supposition when we play
the two throws. This supposition does not change at all the chance of
that one who bets on this event; it simply serves to reduce the various
cases to the cases equally possible.
_Third Principle._—One of the most important points of the theory of
probabilities and that which lends the most to illusions is the manner
in which these probabilities increase or diminish by their mutual
combination. If the events are independent of one another, the
probability of their combined existence is the product of their
respective probabilities. Thus the probability of throwing one ace with
a single die is ⅙; that of throwing two aces in throwing two dice at the
same time is 1/36. Each face of the one being able to combine with the
six faces of the other, there are in fact thirty-six equally possible
cases, among which one single case gives two aces. Generally the
probability that a simple event in the same circumstances will occur
consecutively a given number of times is equal to the probability of
this simple event raised to the power indicated by this number. Having
thus the successive powers of a fraction less than unity diminishing
without ceasing, an event which depends upon a series of very great
probabilities may become extremely improbable. Suppose then an incident
be transmitted to us by twenty witnesses in such manner that the first
has transmitted it to the second, the second to the third, and so on.
Suppose again that the probability of each testimony be equal to the
fraction 9/10; that of the incident resulting from the testimonies will
be less than ⅛. We cannot better compare this diminution of the
probability than with the extinction of the light of objects by the
interposition of several pieces of glass. A relatively small number of
pieces suffices to take away the view of an object that a single piece
allows us to perceive in a distinct manner. The historians do not appear
to have paid sufficient attention to this degradation of the probability
of events when seen across a great number of successive generations;
many historical events reputed as certain would be at least doubtful if
they were submitted to this test.
In the purely mathematical sciences the most distant consequences
participate in the certainty of the principle from which they are
derived. In the applications of analysis to physics the results have all
the certainty of facts or experiences. But in the moral sciences, where
each inference is deduced from that which precedes it only in a probable
manner, however probable these deductions may be, the chance of error
increases with their number and ultimately surpasses the chance of truth
in the consequences very remote from the principle.
_Fourth Principle._—When two events depend upon each other, the
probability of the compound event is the product of the probability of
the first event and the probability that, this event having occurred,
the second will occur. Thus in the preceding case of the three urns A,
B, C, of which two contain only white balls and one contains only black
balls, the probability of drawing a white ball from the urn C is ⅔,
since of the three urns only two contain balls of that color. But when a
white ball has been drawn from the urn C, the indecision relative to
that one of the urns which contain only black balls extends only to the
urns A and B; the probability of drawing a white ball from the urn B is
½; the product of ⅔ by ½, or ⅓, is then the probability of drawing two
white balls at one time from the urns B and C.
We see by this example the influence of past events upon the probability
of future events. For the probability of drawing a white ball from the
urn B, which primarily is ⅔, becomes ½ when a white ball has been drawn
from the urn C; it would change to certainty if a black ball had been
drawn from the same urn. We will determine this influence by means of
the following principle, which is a corollary of the preceding one.
_Fifth Principle._—If we calculate _à priori_ the probability of the
occurred event and the probability of an event composed of that one and
a second one which is expected, the second probability divided by the
first will be the probability of the event expected, drawn from the
observed event.
Here is presented the question raised by some philosophers touching the
influence of the past upon the probability of the future. Let us suppose
at the play of heads and tails that heads has occurred oftener than
tails. By this alone we shall be led to believe that in the constitution
of the coin there is a secret cause which favors it. Thus in the conduct
of life constant happiness is a proof of competency which should induce
us to employ preferably happy persons. But if by the unreliability of
circumstances we are constantly brought back to a state of absolute
indecision, if, for example, we change the coin at each throw at the
play of heads and tails, the past can shed no light upon the future and
it would be absurd to take account of it.
_Sixth Principle._—Each of the causes to which an observed event may be
attributed is indicated with just as much likelihood as there is
probability that the event will take place, supposing the event to be
constant. The probability of the existence of any one of these causes is
then a fraction whose numerator is the probability of the event
resulting from this cause and whose denominator is the sum of the
similar probabilities relative to all the causes; if these various
causes, considered _à priori_, are unequally probable, it is necessary,
in place of the probability of the event resulting from each cause, to
employ the product of this probability by the possibility of the cause
itself. This is the fundamental principle of this branch of the analysis
of chances which consists in passing from events to causes.
This principle gives the reason why we attribute regular events to a
particular cause. Some philosophers have thought that these events are
less possible than others and that at the play of heads and tails, for
example, the combination in which heads occurs twenty successive times
is less easy in its nature than those where heads and tails are mixed in
an irregular manner. But this opinion supposes that past events have an
influence on the possibility of future events, which is not at all
admissible. The regular combinations occur more rarely only because they
are less numerous. If we seek a cause wherever we perceive symmetry, it
is not that we regard a symmetrical event as less possible than the
others, but, since this event ought to be the effect of a regular cause
or that of chance, the first of these suppositions is more probable than
the second. On a table we see letters arranged in this order,
_C o n s t a n t i n o p l e_,
and we judge that this arrangement is not the result of chance, not
because it is less possible than the others, for if this word were not
employed in any language we should not suspect it came from any
particular cause, but this word being in use among us, it is
incomparably more probable that some person has thus arranged the
aforesaid letters than that this arrangement is due to chance.
This is the place to define the word _extraordinary_. We arrange in our
thought all possible events in various classes; and we regard as
_extraordinary_ those classes which include a very small number. Thus at
the play of heads and tails the occurrence of heads a hundred successive
times appears to us extraordinary because of the almost infinite number
of combinations which may occur in a hundred throws; and if we divide
the combinations into regular series containing an order easy to
comprehend, and into irregular series, the latter are incomparably more
numerous. The drawing of a white ball from an urn which among a million
balls contains only one of this color, the others being black, would
appear to us likewise extraordinary, because we form only two classes of
events relative to the two colors. But the drawing of the number 475813,
for example, from an urn that contains a million numbers seems to us an
ordinary event; because, comparing individually the numbers with one
another without dividing them into classes, we have no reason to believe
that one of them will appear sooner than the others.
From what precedes, we ought generally to conclude that the more
extraordinary the event, the greater the need of its being supported by
strong proofs. For those who attest it, being able to deceive or to have
been deceived, these two causes are as much more probable as the reality
of the event is less. We shall see this particularly when we come to
speak of the probability of testimony.
_Seventh Principle._—The probability of a future event is the sum of the
products of the probability of each cause, drawn from the event
observed, by the probability that, this cause existing, the future event
will occur. The following example will illustrate this principle.
Let us imagine an urn which contains only two balls, each of which may
be either white or black. One of these balls is drawn and is put back
into the urn before proceeding to a new draw. Suppose that in the first
two draws white balls have been drawn; the probability of again drawing
a white ball at the third draw is required.
Only two hypotheses can be made here: either one of the balls is white
and the other black, or both are white. In the first hypothesis the
probability of the event observed is ¼; it is unity or certainty in the
second. Thus in regarding these hypotheses as so many causes, we shall
have for the sixth principle ⅕ and ⅘ for their respective probabilities.
But if the first hypothesis occurs, the probability of drawing a white
ball at the third draw is ½; it is equal to certainty in the second
hypothesis; multiplying then the last probabilities by those of the
corresponding hypotheses, the sum of the products, or 9/10, will be the
probability of drawing a white ball at the third draw.
When the probability of a single event is unknown we may suppose it
equal to any value from zero to unity. The probability of each of these
hypotheses, drawn from the event observed, is, by the sixth principle, a
fraction whose numerator is the probability of the event in this
hypothesis and whose denominator is the sum of the similar probabilities
relative to all the hypotheses. Thus the probability that the
possibility of the event is comprised within given limits is the sum of
the fractions comprised within these limits. Now if we multiply each
fraction by the probability of the future event, determined in the
corresponding hypothesis, the sum of the products relative to all the
hypotheses will be, by the seventh principle, the probability of the
future event drawn from the event observed. Thus we find that an event
having occurred successively any number of times, the probability that
it will happen again the next time is equal to this number increased by
unity divided by the same number, increased by two units. Placing the
most ancient epoch of history at five thousand years ago, or at 1826213
days, and the sun having risen constantly in the interval at each
revolution of twenty-four hours, it is a bet of 1826214 to one that it
will rise again to-morrow. But this number is incomparably greater for
him who, recognizing in the totality of phenomena the principal
regulator of days and seasons, sees that nothing at the present moment
can arrest the course of it.
Buffon in his _Political Arithmetic_ calculates differently the
preceding probability. He supposes that it differs from unity only by a
fraction whose numerator is unity and whose denominator is the number 2
raised to a power equal to the number of days which have elapsed since
the epoch. But the true manner of relating past events with the
probability of causes and of future events was unknown to this
illustrious writer.
CHAPTER IV.
_CONCERNING HOPE._
The probability of events serves to determine the hope or the fear of
persons interested in their existence. The word _hope_ has various
acceptations; it expresses generally the advantage of that one who
expects a certain benefit in suppositions which are only probable. This
advantage in the theory of chance is a product of the sum hoped for by
the probability of obtaining it; it is the partial sum which ought to
result when we do not wish to run the risks of the event in supposing
that the division is made proportional to the probabilities. This
division is the only equitable one when all strange circumstances are
eliminated; because an equal degree of probability gives an equal right
to the sum hoped for. We will call this advantage _mathematical hope_.
_Eighth Principle._—When the advantage depends on several events it is
obtained by taking the sum of the products of the probability of each
event by the benefit attached to its occurrence.
Let us apply this principle to some examples. Let us suppose that at the
play of heads and tails Paul receives two francs if he throws heads at
the first throw and five francs if he throws it only at the second.
Multiplying two francs by the probability ½ of the first case, and five
francs by the probability ¼ of the second case, the sum of the products,
or two and a quarter francs, will be Paul's advantage. It is the sum
which he ought to give in advance to that one who has given him this
advantage; for, in order to maintain the equality of the play, the throw
ought to be equal to the advantage which it procures.
If Paul receives two francs by throwing heads at the first and five
francs by throwing it at the second throw, whether he has thrown it or
not at the first, the probability of throwing heads at the second throw
being ½, multiplying two francs and five francs by ½ the sum of these
products will give three and one half francs for Paul's advantage and
consequently for his stake at the game.
_Ninth Principle._—In a series of probable events of which the ones
produce a benefit and the others a loss, we shall have the advantage
which results from it by making a sum of the products of the probability
of each favorable event by the benefit which it procures, and
subtracting from this sum that of the products of the probability of
each unfavorable event by the loss which is attached to it. If the
second sum is greater than the first, the benefit becomes a loss and
hope is changed to fear.
Consequently we ought always in the conduct of life to make the product
of the benefit hoped for, by its probability, at least equal to the
similar product relative to the loss. But it is necessary, in order to
attain this, to appreciate exactly the advantages, the losses, and their
respective probabilities. For this a great accuracy of mind, a delicate
judgment, and a great experience in affairs is necessary; it is
necessary to know how to guard one's self against prejudices, illusions
of fear or hope, and erroneous ideas, ideas of fortune and happiness,
with which the majority of people feed their self-love.
The application of the preceding principles to the following question
has greatly exercised the geometricians. Paul plays at heads and tails
with the condition of receiving two francs if he throws heads at the
first throw, four francs if he throws it only at the second throw, eight
francs if he throws it only at the third, and so on. His stake at the
play ought to be, according to the eighth principle, equal to the number
of throws, so that if the game continues to infinity the stake ought to
be infinite. However, no reasonable man would wish to risk at this game
even a small sum, for example five francs. Whence comes this difference
between the result of calculation and the indication of common sense? We
soon recognize that it amounts to this: that the moral advantage which a
benefit procures for us is not proportional to this benefit and that it
depends upon a thousand circumstances, often very difficult to define,
but of which the most general and most important is that of fortune.
Indeed it is apparent that one franc has much greater value for him who
possesses only a hundred than for a millionaire. We ought then to
distinguish in the hoped-for benefit its absolute from its relative
value. But the latter is regulated by the motives which make it
desirable, whereas the first is independent of them. The general
principle for appreciating this relative value cannot be given, but here
is one proposed by Daniel Bernoulli which will serve in many cases.
_Tenth Principle._—The relative value of an infinitely small sum is
equal to its absolute value divided by the total benefit of the person
interested. This supposes that every one has a certain benefit whose
value can never be estimated as zero. Indeed even that one who possesses
nothing always gives to the product of his labor and to his hopes a
value at least equal to that which is absolutely necessary to sustain
him.
If we apply analysis to the principle just propounded, we obtain the
following rule: Let us designate by unity the part of the fortune of an
individual, independent of his expectations. If we determine the
different values that this fortune may have by virtue of these
expectations and their probabilities, the product of these values raised
respectively to the powers indicated by their probabilities will be the
physical fortune which would procure for the individual the same moral
advantage which he receives from the part of his fortune taken as unity
and from his expectations; by subtracting unity from the product, the
difference will be the increase of the physical fortune due to
expectations: we will call this increase _moral hope_. It is easy to see
that it coincides with mathematical hope when the fortune taken as unity
becomes infinite in reference to the variations which it receives from
the expectations. But when these variations are an appreciable part of
this unity the two hopes may differ very materially among themselves.
This rule conduces to results conformable to the indications of common
sense which can by this means be appreciated with some exactitude. Thus
in the preceding question it is found that if the fortune of Paul is two
hundred francs, he ought not reasonably to stake more than nine francs.
The same rule leads us again to distribute the danger over several parts
of a benefit expected rather than to expose the entire benefit to this
danger. It results similarly that at the fairest game the loss is always
greater than the gain. Let us suppose, for example, that a player having
a fortune of one hundred francs risks fifty at the play of heads and
tails; his fortune after his stake at the play will be reduced to
eighty-seven francs, that is to say, this last sum would procure for the
player the same moral advantage as the state of his fortune after the
stake. The play is then disadvantageous even in the case where the stake
is equal to the product of the sum hoped for, by its probability. We can
judge by this of the immorality of games in which the sum hoped for is
below this product. They subsist only by false reasonings and by the
cupidity which they excite and which, leading the people to sacrifice
their necessaries to chimerical hopes whose improbability they are not
in condition to appreciate, are the source of an infinity of evils.
The disadvantage of games of chance, the advantage of not exposing to
the same danger the whole benefit that is expected, and all the similar
results indicated by common sense, subsist, whatever may be the function
of the physical fortune which for each individual expresses his moral
fortune. It is enough that the proportion of the increase of this
function to the increase of the physical fortune diminishes in the
measure that the latter increases.
CHAPTER V.
_CONCERNING THE ANALYTICAL METHODS OF THE CALCULUS OF PROBABILITIES._
The application of the principle which we have just expounded to the
various questions of probability requires methods whose investigation
has given birth to several methods of analysis and especially to the
theory of combinations and to the calculus of finite differences.
If we form the product of the binomials, unity plus the first letter,
unity plus the second letter, unity plus the third letter, and so on up
to _n_ letters, and subtract unity from this developed product, the
result will be the sum of the combination of all these letters taken one
by one, two by two, three by three, etc., each combination having unity
for a coefficient. In order to have the number of combinations of these
_n_ letters taken _s_ by _s_ times, we shall observe that if we suppose
these letters equal among themselves, the preceding product will become
the _n_th power of the binomial one plus the first letter; thus the
number of combinations of _n_ letters taken _s_ by _s_ times will be the
coefficient of the _s_th power of the first letter in the development in
this binomial; and this number is obtained by means of the known
binomial formula.
Attention must be paid to the respective situations of the letters in
each combination, observing that if a second letter is joined to the
first it may be placed in the first or second position which gives two
combinations. If we join to these combinations a third letter, we can
give it in each combination the first, the second, and the third rank
which forms three combinations relative to each of the two others, in
all six combinations. From this it is easy to conclude that the number
of arrangements of which _s_ letters are susceptible is the product of
the numbers from unity to _s_. In order to pay regard to the respective
positions of the letters it is necessary then to multiply by this
product the number of combinations of _n_ letters _s_ by _s_ times,
which is tantamount to taking away the denominator of the coefficient of
the binomial which expresses this number.
Let us imagine a lottery composed of _n_ numbers, of which _r_ are drawn
at each draw. The probability is demanded of the drawing of _s_ given
numbers in one draw. To arrive at this let us form a fraction whose
denominator will be the number of all the cases possible or of the
combinations of _n_ letters taken _r_ by _r_ times, and whose numerator
will be the number of all the combinations which contain the given _s_
numbers. This last number is evidently that of the combinations of the
other numbers taken _n_ less _s_ by _n_ less _s_ times. This fraction
will be the required probability, and we shall easily find that it can
be reduced to a fraction whose numerator is the number of combinations
of _r_ numbers taken _s_ by _s_ times, and whose denominator is the
number of combinations of _n_ numbers taken similarly _s_ by _s_ times.
Thus in the lottery of France, formed as is known of 90 numbers of which
five are drawn at each draw, the probability of drawing a given
combination is 5/90, or 1/18; the lottery ought then for the equality of
the play to give eighteen times the stake. The total number of
combinations two by two of the 90 numbers is 4005, and that of the
combinations two by two of 5 numbers is 10. The probability of the
drawing of a given pair is then 1/4005, and the lottery ought to give
four hundred and a half times the stake; it ought to give 11748 times
for a given tray, 511038 times for a quaternary, and 43949268 times for
a quint. The lottery is far from giving the player these advantages.
Suppose in an urn _a_ white balls, _b_ black balls, and after having
drawn a ball it is put back into the urn; the probability is asked that
in _n_ number of draws _m_ white balls and _n_ - _m_ black balls will be
drawn. It is clear that the number of cases that may occur at each
drawing is _a_ + _b_. Each case of the second drawing being able to
combine with all the cases of the first, the number of possible cases in
two drawings is the square of the binomial _a_ + _b_. In the development
of this square, the square of a expresses the number of cases in which a
white ball is twice drawn, the double product of _a_ by _b_ expresses
the number of cases in which a white ball and a black ball are drawn.
Finally, the square of _b_ expresses the number of cases in which two
black balls are drawn. Continuing thus, we see generally that the _n_th
power of the binomial _a_ + _b_ expresses the number of all the cases
possible in _n_ draws; and that in the development of this power the
term multiplied by the _m_th power of _a_ expresses the number of cases
in which _m_ white balls and _n_ - _m_ black balls may be drawn.
Dividing then this term by the entire power of the binomial, we shall
have the probability of drawing _m_ white balls and _n_ - _m_ black
balls. The ratio of the numbers _a_ and _a_ + _b_ being the probability
of drawing one white ball at one draw; and the ratio of the numbers _b_
and _a_ + _b_ being the probability of drawing one black ball; if we
call these probabilities _p_ and _q_, the probability of drawing _m_
white balls in _n_ draws will be the term multiplied by the _m_th power
of _p_ in the development of the _n_th power of the binomial _p_ + _q_;
we may see that the sum _p_ + _q_ is unity. This remarkable property of
the binomial is very useful in the theory of probabilities. But the most
general and direct method of resolving questions of probability consists
in making them depend upon equations of differences. Comparing the
successive conditions of the function which expresses the probability
when we increase the variables by their respective differences, the
proposed question often furnishes a very simple proportion between the
conditions. This proportion is what is called _equation of ordinary or
partial differentials_; _ordinary_ when there is only one variable,
_partial_ when there are several. Let us consider some examples of this.
Three players of supposed equal ability play together on the following
conditions: that one of the first two players who beats his adversary
plays the third, and if he beats him the game is finished. If he is
beaten, the victor plays against the second until one of the players has
defeated consecutively the two others, which ends the game. The
probability is demanded that the game will be finished in a certain
number _n_ of plays. Let us find the probability that it will end
precisely at the _n_th play. For that the player who wins ought to enter
the game at the play _n_ - 1 and win it thus at the following play. But
if in place of winning the play _n_ - 1 he should be beaten by his
adversary who had just beaten the other player, the game would end at
this play. Thus the probability that one of the players will enter the
game at the play _n_ - 1 and will win it is equal to the probability
that the game will end precisely with this play; and as this player
ought to win the following play in order that the game may be finished
at the _n_th play, the probability of this last case will be only one
half of the preceding one. This probability is evidently a function of
the number _n_; this function is then equal to the half of the same
function when _n_ is diminished by unity. This equality forms one of
those equations called _ordinary finite differential equations_.
We may easily determine by its use the probability that the game will
end precisely at a certain play. It is evident that the play cannot end
sooner than at the second play; and for this it is necessary that that
one of the first two players who has beaten his adversary should beat at
the second play the third player; the probability that the game will end
at this play is ½. Hence by virtue of the preceding equation we conclude
that the successive probabilities of the end of the game are ¼ for the
third play, ⅛ for the fourth play, and so on; and in general ½ raised to
the power _n_ - 1 for the _n_th play. The sum of all these powers of ½
is unity less the last of these powers; it is the probability that the
game will end at the latest in _n_ plays.
Let us consider again the first problem more difficult which may be
solved by probabilities and which Pascal proposed to Fermat to solve.
Two players, A and B, of equal skill play together on the conditions
that the one who first shall beat the other a given number of times
shall win the game and shall take the sum of the stakes at the game;
after some throws the players agree to quit without having finished the
game: we ask in what manner the sum ought to be divided between them. It
is evident that the parts ought to be proportional to the respective
probabilities of winning the game. The question is reduced then to the
determination of these probabilities. They depend evidently upon the
number of points which each player lacks of having attained the given
number. Hence the probability of A is a function of the two numbers
which we will call _indices_. If the two players should agree to play
one throw more (an agreement which does not change their condition,
provided that after this new throw the division is always made
proportionally to the new probabilities of winning the game), then
either A would win this throw and in that case the number of points
which he lacks would be diminished by unity, or the player B would win
it and in that case the number of points lacking to this last player
would be less by unity. But the probability of each of these cases is ½;
the function sought is then equal to one half of this function in which
we diminish by unity the first index plus the half of the same function
in which the second variable is diminished by unity. This equality is
one of those equations called _equations of partial differentials_.
We are able to determine by its use the probabilities of A by dividing
the smallest numbers, and by observing that the probability or the
function which expresses it is equal to unity when the player A does not
lack a single point, or when the first index is zero, and that this
function becomes zero with the second index. Supposing thus that the
player A lacks only one point, we find that his probability is ½, ¾, ⅞,
etc., according as B lacks one point, two, three, etc. Generally it is
then unity less the power of ½, equal to the number of points which B
lacks. We will suppose then that the player A lacks two points and his
probability will be found equal to ¼, ½, 11/16, etc., according as B
lacks one point, two points, three points, etc. We will suppose again
that the player A lacks three points, and so on.
This manner of obtaining the successive values of a quantity by means of
its equation of differences is long and laborious. The geometricians
have sought methods to obtain the general function of indices that
satisfies this equation, so that for any particular case we need only to
substitute in this function the corresponding values of the indices. Let
us consider this subject in a general way. For this purpose let us
conceive a series of terms arranged along a horizontal line so that each
of them is derived from the preceding one according to a given law. Let
us suppose this law expressed by an equation among several consecutive
terms and their index, or the number which indicates the rank that they
occupy in the series. This equation I call the _equation of finite
differences by a single index_. The order or the degree of this equation
is the difference of rank of its two extreme terms. We are able by its
use to determine successively the terms of the series and to continue it
indefinitely; but for that it is necessary to know a number of terms of
the series equal to the degree of the equation. These terms are the
arbitrary constants of the expression of the general term of the series
or of the integral of the equation of differences.
Let us imagine now below the terms of the preceding series a second
series of terms arranged horizontally; let us imagine again below the
terms of the second series a third horizontal series, and so on to
infinity; and let us suppose the terms of all these series connected by
a general equation among several consecutive terms, taken as much in the
horizontal as in the vertical sense, and the numbers which indicate
their rank in the two senses. This equation is called the _equation of
partial finite differences by two indices_.
Let us imagine in the same way below the plan of the preceding series a
second plan of similar series, whose terms should be placed respectively
below those of the first plan; let us imagine again below this second
plan a third plan of similar series, and so on to infinity; let us
suppose all the terms of these series connected by an equation among
several consecutive terms taken in the sense of length, width, and
depth, and the three numbers which indicate their rank in these three
senses. This equation I call the _equation of partial finite differences
by three indices_.
Finally, considering the matter in an abstract way and independently of
the dimensions of space, let us imagine generally a system of
magnitudes, which should be functions of a certain number of indices,
and let us suppose among these magnitudes, their relative differences to
these indices and the indices themselves, as many equations as there are
magnitudes; these equations will be partial finite differences by a
certain number of indices.
We are able by their use to determine successively these magnitudes. But
in the same manner as the equation by a single index requires for it
that we know a certain number of terms of the series, so the equation by
two indices requires that we know one or several lines of series whose
general terms should be expressed each by an arbitrary function of one
of the indices. Similarly the equation by three indices requires that we
know one or several plans of series, the general terms of which should
be expressed each by an arbitrary function of two indices, and so on. In
all these cases we shall be able by successive eliminations to determine
a certain term of the series. But all the equations among which we
eliminate being comprised in the same system of equations, all the
expressions of the successive terms which we obtain by these
eliminations ought to be comprised in one general expression, a function
of the indices which determine the rank of the term. This expression is
the integral of the proposed equation of differences, and the search for
it is the object of integral calculus.
Taylor is the first who in his work entitled _Metodus incrementorum_ has
considered linear equations of finite differences. He gives the manner
of integrating those of the first order with a coefficient and a last
term, functions of the index. In truth the relations of the terms of the
arithmetical and geometrical progressions which have always been taken
into consideration are the simplest cases of linear equations of
differences; but they had not been considered from this point of view.
It was one of those which, attaching themselves to general theories,
lead to these theories and are consequently veritable discoveries.
About the same time Moivre was considering under the name of recurring
series the equations of finite differences of a certain order having a
constant coefficient. He succeeded in integrating them in a very
ingenious manner. As it is always interesting to follow the progress of
inventors, I shall expound the method of Moivre by applying it to a
recurring series whose relation among three consecutive terms is given.
First he considers the relation among the consecutive terms of a
geometrical progression or the equation of two terms which expresses it.
Referring it to terms less than unity, he multiplies it in this state by
a constant factor and subtracts the product from the first equation.
Thus he obtains an equation among three consecutive terms of the
geometrical progression. Moivre considers next a second progression
whose ratio of terms is the same factor which he has just used. He
diminishes similarly by unity the index of the terms of the equation of
this new progression. In this condition he multiplies it by the ratio of
the terms of the first progression, and he subtracts the product from
the equation of the second progression, which gives him among three
consecutive terms of this progression a relation entirely similar to
that which he has found for the first progression. Then he observes that
if one adds term by term the two progressions, the same ratio exists
among any three of these consecutive terms. He compares the coefficients
of this ratio to those of the relation of the terms of the proposed
recurrent series, and he finds for determining the ratios of the two
geometrical progressions an equation of the second degree, whose roots
are these ratios. Thus Moivre decomposes the recurrent series into two
geometrical progressions, each multiplied by an arbitrary constant which
he determines by means of the first two terms of the recurrent series.
This ingenious process is in fact the one that d'Alembert has since
employed for the integration of linear equations of infinitely small
differences with constant coefficients, and Lagrange has transformed
into similar equations of finite differences.
Finally, I have considered the linear equations of partial finite
differences, first under the name of _recurro-recurrent_ series and
afterwards under their own name. The most general and simplest manner of
integrating all these equations appears to me that which I have based
upon the consideration of discriminant functions, the idea of which is
here given.
If we conceive a function _V_ of a variable _t_ developed according to
the powers of this variable, the coefficient of any one of these powers
will be a function of the exponent or index of this power, which index I
shall call _x_. _V_ is what I call the discriminant function of this
coefficient or of the function of the index.
Now if we multiply the series of the development of _V_ by a function of
the same variable, such, for example, as unity plus two times this
variable, the product will be a new discriminant function in which the
coefficient of the power _x_ of the variable _t_ will be equal to the
coefficient of the same power in _V_ plus twice the coefficient of the
power less unity. Thus the function of the index _x_ in the product will
be equal to the function of the index _x_ in _V_ plus twice the same
function in which the index is diminished by unity. This function of the
index _x_ is thus a derivative of the function of the same index in the
development of _V_, a function which I shall call the _primitive
function_ of the index. Let us designate the derivative function by the
letter Alembert placed before the primitive function. The derivation
indicated by this letter will depend upon the multiplier of _V_, which
we will call _T_ and which we will suppose developed like _V_ by the
ratio to the powers of the variable _t_. If we multiply anew by _T_ the
product of _V_ by _T_, which is equivalent to multiplying _V_ by _T²_,
we shall form a third discriminant function, in which the coefficient of
the _x_th power of _t_ will be a derivative similar to the corresponding
coefficient of the preceding product; it may be expressed by the same
character _δ_ placed before the preceding derivative, and then this
character will be written twice before the primitive function of _x_.
But in place of writing it thus twice we give it 2 for an exponent.
Continuing thus, we see generally that if we multiply _V_ by the _n_th
power of _T_, we shall have the coefficient of the _x_th power of _t_ in
the product of _V_ by the _n_th power of _T_ by placing before the
primitive function the character _δ_ with _n_ for an exponent.
Let us suppose, for example, that _T_ be unity divided by _t_; then in
the product of _V_ by _T_ the coefficient of the _x_th power of _t_ will
be the coefficient of the power greater by unity in _V_; this
coefficient in the product of _V_ by the _n_th power of _T_ will then be
the primitive function in which _x_ is augmented by _n_ units.
Let us consider now a new function _Z_ of _t_, developed like _V_ and
_T_ according to the powers of _t_; let us designate by the character
_Δ_ placed before the primitive function the coefficient of the _x_th
power of _t_ in the product of _V_ by _Z_; this coefficient in the
product of _V_ by the _n_th power of _Z_ will be expressed by the
character _Δ_ affected by the exponent _n_ and placed before the
primitive function of _x_.
If, for example, _Z_ is equal to unity divided by _t_ less one, the
coefficient of the _x_th power of _t_ in the product of _V_ by _Z_ will
be the coefficient of the _x_ + 1 power of _t_ in _V_ less the
coefficient of the _x_th power. It will be then the finite difference of
the primitive function of the index _x_. Then the character _Δ_
indicates a finite difference of the primitive function in the case
where the index varies by unity; and the _n_th power of this character
placed before the primitive function will indicate the finite _n_th
difference of this function. If we suppose that _T_ be unity divided by
_t_, we shall have _T_ equal to the binomial _Z_ + 1. The product of _V_
by the _n_th power of _T_ will then be equal to the product of _V_ by
the _n_th power of the binomial _Z_ + 1. Developing this power in the
ratio of the powers of _Z_, the product of _V_ by the various terms of
this development will be the discriminant functions of these same terms
in which we substitute in place of the powers of _Z_ the corresponding
finite differences of the primitive function of the index.
Now the product of _V_ by the _n_th power of _T_ is the primitive
function in which the index _x_ is augmented by _n_ units; repassing
from the discriminant functions to their coefficients, we shall have
this primitive function thus augmented equal to the development of the
_n_th power of the binomial _Z_ + 1, provided that in this development
we substitute in place of the powers of _Z_ the corresponding
differences of the primitive function and that we multiply the
independent term of these powers by the primitive function. We shall
thus obtain the primitive function whose index is augmented by any
number _n_ by means of its differences.
Supposing that _T_ and _Z_ always have the preceding values, we shall
have _Z_ equal to the binomial _T_ - 1; the product of _V_ by the _n_th
power of _Z_ will then be equal to the product of _V_ by the development
of the _n_th power of the binomial _T_ - 1. Repassing from the
discriminant functions to their coefficients as has just been done, we
shall have the _n_th difference of the primitive function expressed by
the development of the _n_th power of the binomial _T_ - 1, in which we
substitute for the powers of _T_ this same function whose index is
augmented by the exponent of the power, and for the independent term of
_t_, which is unity, the primitive function, which gives this difference
by means of the consecutive terms of this function.
Placing _δ_ before the primitive function expressing the derivative of
this function, which multiplies the _x_ power of _t_ in the product of
_V_ by _T_, and _Δ_ expressing the same derivative in the product of _V_
by _Z_, we are led by that which precedes to this general result:
whatever may be the function of the variable _t_ represented by _T_ and
_Z_, we may, in the development of all the identical equations
susceptible of being formed among these functions, substitute the
characters _δ_ and _Δ_ in place of _T_ and _Z_, provided that we write
the primitive function of the index in series with the powers and with
the products of the powers of the characters, and that we multiply by
this function the independent terms of these characters.
We are able by means of this general result to transform any certain
power of a difference of the primitive function of the index _x_, in
which _x_ varies by unity, into a series of differences of the same
function in which _x_ varies by a certain number of units and
reciprocally. Let us suppose that _T_ be the _i_ power of unity divided
by _t_ - 1, and that _Z_ be always unity divided by _t_ - 1; then the
coefficient of the _x_ power of _t_ in the product of _V_ by _T_ will be
the coefficient of the _x_ + _i_ power of _t_ in _V_ less the
coefficient of the _x_ power of _t_; it will then be the finite
difference of the primitive function of the index _x_ in which we vary
this index by the number _i_. It is easy to see that _T_ is equal to the
difference between the _i_ power of the binomial _Z_ + 1 and unity. The
_n_th power of _T_ is equal to the _n_th power of this difference. If in
this equality we substitute in place of _T_ and _Z_ the characters _δ_
and _Δ_, and after the development we place at the end of each term the
primitive function of the index _x_, we shall have the _n_th difference
of this function in which _x_ varies by _i_ units expressed by a series
of differences of the same function in which _x_ varies by unity. This
series is only a transformation of the difference which it expresses and
which is identical with it; but it is in similar transformations that
the power of analysis resides.
The generality of analysis permits us to suppose in this expression that
_n_ is negative. Then the negative powers of _δ_ and _Δ_ indicate the
integrals. Indeed the _n_th difference of the primitive function having
for a discriminant function the product of _V_ by the _n_th power of the
binomial one divided by _t_ less unity, the primitive function which is
the _n_th integral of this difference has for a discriminant function
that of the same difference multiplied by the _n_th power taken less
than the binomial one divided by _t_ minus one, a power to which the
same power of the character _Δ_ corresponds; this power indicates then
an integral of the same order, the index _x_ varying by unity; and the
negative powers of _δ_ indicate equally the integrals _x_ varying by _i_
units. We see, thus, in the clearest and simplest manner the rationality
of the analysis observed among the positive powers and differences, and
among the negative powers and the integrals.
If the function indicated by _δ_ placed before the primitive function is
zero, we shall have an equation of finite differences, and _V_ will be
the discriminant function of its integral. In order to obtain this
discriminant function we shall observe that in the product of _V_ by _T_
all the powers of _t_ ought to disappear except the powers inferior to
the order of the equation of differences; _V_ is then equal to a
fraction whose denominator is _T_ and whose numerator is a polynomial in
which the highest power of _t_ is less by unity than the order of the
equation of differences. The arbitrary coefficients of the various
powers of _t_ in this polynomial, including the power zero, will be
determined by as many values of the primitive function of the index when
we make successively _x_ equal to zero, to one, to two, etc. When the
equation of differences is given we determine _T_ by putting all its
terms in the first member and zero in the second; by substituting in the
first member unity in place of the function which has the largest index;
the first power of _t_ in place of the primitive function in which this
index is diminished by unity; the second power of _t_ for the primitive
function where this index is diminished by two units, and so on. The
coefficient of the _x_th power of _t_ in the development of the
preceding expression of _V_ will be the primitive function of _x_ or the
integral of the equation of finite differences. Analysis furnishes for
this development various means, among which we may choose that one which
is most suitable for the question proposed; this is an advantage of this
method of integration.
Let us conceive now that _V_ be a function of the two variables _t_ and
_t´_ developed according to the powers and products of these variables;
the coefficient of any product of the powers _x_ and _x´_ of _t_ and
_t´_ will be a function of the exponents or indices _x_ and _x´_ of
these powers; this function I shall call the _primitive function_ of
which _V_ is the discriminant function.
Let us multiply _V_ by a function _T_ of the two variables _t_ and _t´_
developed like _V_ in ratio of the powers and the products of these
variables; the product will be the discriminant function of a derivative
of the primitive function; if _T_, for example, is equal to the variable
_t_ plus the variable _t´_ minus two, this derivative will be the
primitive function of which we diminish by unity the index _x_ plus this
same primitive function of which we diminish by unity the index _x´_
less two times the primitive function. Designating whatever _T_ may be
by the character _δ_ placed before the primitive function, this
derivative, the product of _V_ by the _n_th power of _T_, will be the
discriminant function of the derivative of the primitive function before
which one places the _n_th power of the character _δ_. Hence result the
theorems analogous to those which are relative to functions of a single
variable.
Suppose the function indicated by the character _δ_ be zero; one will
have an equation of partial differences. If, for example, we make as
before _T_ equal to the variable _t_ plus the variable _t´_ - 2, we have
zero equal to the primitive function of which we diminish by unity the
index _x_ plus the same function of which we diminish by unity the index
_x´_ minus two times the primitive function. The discriminant function
_V_ of the primitive function or of the integral of this equation ought
then to be such that its product by _T_ does not include at all the
products of _t_ by _t´_; but _V_ may include separately the powers of
_t_ and those of _t´_, that is to say, an arbitrary function of _t_ and
an arbitrary function of _t´_; _V_ is then a fraction whose numerator is
the sum of these two arbitrary functions and whose denominator is _T_.
The coefficient of the product of the _x_th power of _t_ by the _x´_
power of _t´_ in the development of this fraction will then be the
integral of the preceding equation of partial differences. This method
of integrating this kind of equations seems to me the simplest and the
easiest by the employment of the various analytical processes for the
development of rational fractions.
More ample details in this matter would be scarcely understood without
the aid of calculus.
Considering equations of infinitely small partial differences as
equations of finite partial differences in which nothing is neglected,
we are able to throw light upon the obscure points of their calculus,
which have been the subject of great discussions among geometricians. It
is thus that I have demonstrated the possibility of introducing
discontinued functions in their integrals, provided that the
discontinuity takes place only for the differentials of the order of
these equations or of a superior order. The transcendent results of
calculus are, like all the abstractions of the understanding, general
signs whose true meaning may be ascertained only by repassing by
metaphysical analysis to the elementary ideas which have led to them;
this often presents great difficulties, for the human mind tries still
less to transport itself into the future than to retire within itself.
The comparison of infinitely small differences with finite differences
is able similarly to shed great light upon the metaphysics of
infinitesimal calculus.
It is easily proven that the finite _n_th difference of a function in
which the increase of the variable is _E_ being divided by the _n_th
power of _E_, the quotient reduced in series by ratio to the powers of
the increase _E_ is formed by a first term independent of _E._ In the
measure that _E_ diminishes, the series approaches more and more this
first term from which it can differ only by quantities less than any
assignable magnitude. This term is then the limit of the series and
expresses in differential calculus the infinitely small _n_th difference
of the function divided by the _n_th power of the infinitely small
increase.
Considering from this point of view the infinitely small differences, we
see that the various operations of differential calculus amount to
comparing separately in the development of identical expressions the
finite terms or those independent of the increments of the variables
which are regarded as infinitely small; this is rigorously exact, these
increments being indeterminant. Thus differential calculus has all the
exactitude of other algebraic operations.
The same exactitude is found in the applications of differential
calculus to geometry and mechanics. If we imagine a curve cut by a
secant at two adjacent points, naming _E_ the interval of the ordinates
of these two points, _E_ will be the increment of the abscissa from the
first to the second ordinate. It is easy to see that the corresponding
increment of the ordinate will be the product of _E_ by the first
ordinate divided by its subsecant; augmenting then in this equation of
the curve the first ordinate by this increment, we shall have the
equation relative to the second ordinate. The difference of these two
equations will be a third equation which, developed by the ratio of the
powers of _E_ and divided by _E_, will have its first term independent
of _E_, which will be the limit of this development. This term, equal to
zero, will give then the limit of the subsecants, a limit which is
evidently the subtangent.
This singularly happy method of obtaining the subtangent is due to
Fermat, who has extended it to transcendent curves. This great
geometrician expresses by the character _E_ the increment of the
abscissa; and considering only the first power of this increment, he
determines exactly as we do by differential calculus the subtangents of
the curves, their points of inflection, the _maxima_ and _minima_ of
their ordinates, and in general those of rational functions. We see
likewise by his beautiful solution of the problem of the refraction of
light inserted in the _Collection of the Letters of Descartes_ that he
knows how to extend his methods to irrational functions in freeing them
from irrationalities by the elevation of the roots to powers. Fermat
should be regarded, then, as the true discoverer of Differential
Calculus. Newton has since rendered this calculus more analytical in his
_Method of Fluxions_, and simplified and generalized the processes by
his beautiful theorem of the binomial. Finally, about the same time
Leibnitz has enriched differential calculus by a notation which, by
indicating the passage from the finite to the infinitely small, adds to
the advantage of expressing the general results of calculus that of
giving the first approximate values of the differences and of the sums
of the quantities; this notation is adapted of itself to the calculus of
partial differentials.
We are often led to expressions which contain so many terms and factors
that the numerical substitutions are impracticable. This takes place in
questions of probability when we consider a great number of events.
Meanwhile it is necessary to have the numerical value of the formulæ in
order to know with what probability the results are indicated, which the
events develop by multiplication. It is necessary especially to have the
law according to which this probability continually approaches
certainty, which it will finally attain if the number of events were
infinite. In order to obtain this law I considered that the definite
integrals of differentials multiplied by the factors raised to great
powers would give by integration the formulæ composed of a great number
of terms and factors. This remark brought me to the idea of transforming
into similar integrals the complicated expressions of analysis and the
integrals of the equation of differences. I fulfilled this condition by
a method which gives at the same time the function comprised under the
integral sign and the limits of the integration. It offers this
remarkable thing, that the function is the same discriminant function of
the expressions and the proposed equations; this attaches this method to
the theory of discriminant functions of which it is thus the complement.
Further, it would only be a question of reducing the definite integral
to a converging series. This I have obtained by a process which makes
the series converge with as much more rapidity as the formula which it
represents is more complicated, so that it is more exact as it becomes
more necessary. Frequently the series has for a factor the square root
of the ratio of the circumference to the diameter; sometimes it depends
upon other transcendents whose number is infinite.
An important remark which pertains to great generality of analysis, and
which permits us to extend this method to formulæ and to equations of
difference which the theory of probability presents most frequently, is
that the series to which one comes by supposing the limits of the
definite integrals to be real and positive take place equally in the
case where the equation which determines these limits has only negative
or imaginary roots. These passages from the positive to the negative and
from the real to the imaginary, of which I first have made use, have led
me further to the values of many singular definite integrals, which I
have accordingly demonstrated directly. We may then consider these
passages as a means of discovery parallel to induction and analogy long
employed by geometricians, at first with an extreme reserve, afterwards
with entire confidence, since a great number of examples has justified
its use. In the mean time it is always necessary to confirm by direct
demonstrations the results obtained by these divers means.
I have named the ensemble of the preceding methods the _Calculus of
Discriminant Functions_; this calculus serves as a basis for the work
which I have published under the title of the _Analytical Theory of
Probabilities_. It is connected with the simple idea of indicating the
repeated multiplications of a quantity by itself or its entire and
positive powers by writing toward the top of the letter which expresses
it the numbers which mark the degrees of these powers.
This notation, employed by Descartes in his _Geometry_ and generally
adopted since the publication of this important work, is a little thing,
especially when compared with the theory of curves and variable
functions by which this great geometrician has established the
foundations of modern calculus. But the language of analysis, most
perfect of all, being in itself a powerful instrument of discoveries,
its notations, especially when they are necessary and happily conceived,
are so many germs of new calculi. This is rendered appreciable by this
example.
Wallis, who in his work entitled _Arithmetica Infinitorum_, one of those
which have most contributed to the progress of analysis, has interested
himself especially in following the thread of induction and analogy,
considered that if one divides the exponent of a letter by two, three,
etc., the quotient will be accordingly the Cartesian notation, and when
division is possible the exponent of the square, cube, etc., root of the
quantity which represents the letter raised to the dividend exponent.
Extending by analogy this result to the case where division is
impossible, he considered a quantity raised to a fractional exponent as
the root of the degree indicated by the denominator of this
fraction—namely, of the quantity raised to a power indicated by the
numerator. He observed then that, according to the Cartesian notation,
the multiplication of two powers of the same letter amounts to adding
their exponents, and that their division amounts to subtracting the
exponents of the power of the divisor from that of the power of the
dividend, when the second of these exponents is greater than the first.
Wallis extended this result to the case where the first exponent is
equal to or greater than the second, which makes the difference zero or
negative. He supposed then that a negative exponent indicates unity
divided by the quantity raised to the same exponent taken positively.
These remarks led him to integrate generally the monomial differentials,
whence he inferred the definite integrals of a particular kind of
binomial differentials whose exponent is a positive integral number. The
observation then of the law of the numbers which express these
integrals, a series of interpolations and happy inductions where one
perceives the germ of the calculus of definite integrals which has so
much exercised geometricians and which is one of the fundaments of my
new _Theory of Probabilities_, gave him the ratio of the area of the
circle to the square of its diameter expressed by an infinite product,
which, when one stops it, confines this ratio to limits more and more
converging; this is one of the most singular results in analysis. But it
is remarkable that Wallis, who had so well considered the fractional
exponents of radical powers, should have continued to note these powers
as had been done before him. Newton in his _Letters to Oldembourg_, if I
am not mistaken, was the first to employ the notation of these powers by
fractional exponents. Comparing by the way of induction, of which Wallis
had made such a beautiful use, the exponents of the powers of the
binomial with the coefficients of the terms of its development in the
case where this exponent is integral and positive, he determined the law
of these coefficients and extended it by analogy to fractional and
negative powers. These various results, based upon the notation of
Descartes, show his influence on the progress of analysis. It has still
the advantage of giving the simplest and fairest idea of logarithms,
which are indeed only the exponents of a magnitude whose successive
powers, increasing by infinitely small degrees, can represent all
numbers.
But the most important extension that this notation has received is that
of variable exponents, which constitutes exponential calculus, one of
the most fruitful branches of modern analysis. Leibnitz was the first to
indicate the transcendents by variable exponents, and thereby he has
completed the system of elements of which a finite function can be
composed; for every finite explicit function of a variable may be
reduced in the last analysis to simple magnitudes, combined by the
method of addition, subtraction, multiplication, and division and raised
to constant or variable powers. The roots of the equations formed from
these elements are the implicit functions of the variable. It is thus
that a variable has for a logarithm the exponent of the power which is
equal to it in the series of the powers of the number whose hyperbolic
logarithm is unity, and the logarithm of a variable of it is an implicit
function.
Leibnitz thought to give to his differential character the same
exponents as to magnitudes; but then in place of indicating the repeated
multiplications of the same magnitude these exponents indicate the
repeated differentiations of the same function. This new extension of
the Cartesian notation led Leibnitz to the analogy of positive powers
with the differentials, and the negative powers with the integrals.
Lagrange has followed this singular analogy in all its developments; and
by series of inductions which may be regarded as one of the most
beautiful applications which have ever been made of the method of
induction he has arrived at general formulæ which are as curious as
useful on the transformations of differences and of integrals the ones
into the others when the variables have divers finite increments and
when these increments are infinitely small. But he has not given the
demonstrations of it which appear to him difficult. The theory of
discriminant functions extends the Cartesian notations to some of its
characters; it shows with proof the analogy of the powers and operations
indicated by these characters; so that it may still be regarded as the
exponential calculus of characters. All that concerns the series and the
integration of equations of differences springs from it with an extreme
facility.
PART II.
APPLICATIONS OF THE CALCULUS OF PROBABILITIES.