_THE APPLICATION OF THE CALCULUS OF PROBABILITIES TO NATURAL PHILOSOPHY._
The phenomena of nature are most often enveloped by so many strange
circumstances, and so great a number of disturbing causes mix their
influence, that it is very difficult to recognize them. We may arrive at
them only by multiplying the observations or the experiences, so that
the strange effects finally destroy reciprocally each other, the mean
results putting in evidence those phenomena and their divers elements.
The more numerous the number of observations and the less they vary
among themselves the more their results approach the truth. We fulfil
this last condition by the choice of the methods of observations, by the
precision of the instruments, and by the care which we take to observe
closely; then we determine by the theory of probabilities the most
advantageous mean results or those which give the least value of the
error. But that is not sufficient; it is further necessary to appreciate
the probability that the errors of these results are comprised in the
given limits; and without this we have only an imperfect knowledge of
the degree of exactitude obtained. Formulæ suitable to these matters are
then true improvements of the method of sciences, and it is indeed
important to add them to this method. The analysis which they require is
the most delicate and the most difficult of the theory of probabilities;
it is one of the principal objects of the work which I have published
upon this theory, and in which I have arrived at formulæ of this kind
which have the remarkable advantage of being independent of the law of
the probability of errors and of including only the quantities given by
the observations themselves and their expressions.
Each observation has for an analytic expression a function of the
elements which we wish to determine; and if these elements are nearly
known, this function becomes a linear function of their corrections. In
equating it to the observation itself there is formed _an equation of
condition._ If we have a great number of similar equations, we combine
them in such a manner as to obtain as many final equations as there are
elements whose corrections we determine then by resolving these
equations. But what is the most advantageous manner of combining
equations of condition in order to obtain final equations? What is the
law of the probabilities of errors of which the elements are still
susceptible that we draw from them? This is made clear to us by the
theory of probabilities. The formation of a final equation by means of
the equation of condition amounts to multiplying each one of these by an
indeterminate factor and by uniting the products; it is necessary to
choose the system of factors which gives the smallest opportunity for
error. But it is apparent that if we multiply the possible errors of an
element by their respective probabilities, the most advantageous system
will be that in which the sum of these products all, taken, positively
is a _minimum_; for a positive or a negative error ought to be
considered as a loss. Forming, then, this sum of products, the condition
of the _minimum_ will determine the system of factors which it is
expedient to adopt, or the most advantageous system. We find thus that
this system is that of the coefficients of the elements in each equation
of condition; so that we form a first final equation by multiplying
respectively each equation of condition by its coefficient of the first
element and by uniting all these equations thus multiplied. We form a
second final equation by employing in the same manner the coefficients
of the second element, and so on. In this manner the elements and the
laws of the phenomena obtained in the collection of a great number of
observations are developed with the most evidence.
The probability of the errors which each element still leaves to be
feared is proportional to the number whose hyperbolic logarithm is unity
raised to a power equal to the square of the error taken as a minus
quantity and multiplied by a constant coefficient which may be
considered as the modulus of the probability of the errors; because, the
error remaining the same, its probability decreases with rapidity when
the former increases; so that the element obtained weighs, if I may thus
speak toward the truth, as much more as this modulus is greater. I would
call for this reason this modulus the _weight_ of the element or of the
result. This weight is the greatest possible in the system of
factors—the most advantageous; it is this which gives to this system
superiority over others. By a remarkable analogy of this weight with
those of bodies compared at their common centre of gravity it results
that if the same element is given by divers systems, composed each of a
great number of observations, the most advantageous, the mean result of
their totality is the sum of the products of each partial result by its
weight. Moreover, the total weight of the results of the divers systems
is the sum of their partial weights; so that the probability of the
errors of the mean result of their totality is proportional to the
number which has unity for an hyperbolic logarithm raised to a power
equal to the square of the error taken as minus and multiplied by the
sum of the weights. Each weight depends in truth upon the law of the
probability of error of each system, and almost always this law is
unknown; but happily I have been able to eliminate the factor which
contains it by means of the sum of the squares of the variations of the
observations in this system from their mean result. It would then be
desirable in order to complete our knowledge of the results obtained by
the totality of a great number of observations that we write by the side
of each result the weight which corresponds to it; analysis furnishes
for this object both general and simple methods. When we have thus
obtained the exponential which represents the law of the probability of
errors, we shall have the probability that the error of the result is
included within given limits by taking within the limits the integral of
the product of this exponential by the differential of the error and
multiplying it by the square root of the weight of the result divided by
the circumference whose diameter is unity. Hence it follows that for the
same probability the errors of the results are reciprocal to the square
roots of their weights, which serves to compare their respective
precision.
In order to apply this method with success it is necessary to vary the
circumstances of the observations or the experiences in such a manner as
to avoid the constant causes of error. It is necessary that the
observations should be numerous, and that they should be so much the
more so as there are more elements to determine; for the weight of the
mean result increases as the number of observations divided by the
number of the elements. It is still necessary that the elements follow
in these observations a different course; for if the course of the two
elements were exactly the same, which would render their coefficients
proportional in equation of conditions, these elements would form only a
single unknown quantity and it would be impossible to distinguish them
by these observations. Finally it is necessary that the observations
should be precise; this condition, the first of all, increases greatly
the weight of the result the expression of which has for a divisor the
sum of the squares of the deviations of the observations from this
result. With these precautions we shall be able to make use of the
preceding method and measure the degree of confidence which the results
deduced from a great number of observations merit.
The rule which we have just given to conclude equations of condition,
final equations, amount to rendering a minimum the sum of the squares of
the errors of observations; for each equation of condition becomes exact
by substituting in it the observation plus its error; and if we draw
from it the expression of this error, it is easy to see that the
condition of the _minimum_ of the sum of the squares of these
expressions gives the rule in question. This rule is the more precise as
the observations are more numerous; but even in the case where their
number is small it appears natural to employ the same rule which in all
cases offers a simple means of obtaining without groping the corrections
which we seek to determine. It serves further to compare the precision
of the divers astronomical tables of the same star. These tables may
always be supposed as reduced to the same form, and then they differ
only by the epochs, the mean movements and the coefficients of the
arguments; for if one of them contains a coefficient which is not found
in the others, it is clear that this amounts to supposing zero in them
as the coefficient of this argument. If now we rectify these tables by
the totality of the good observations, they would satisfy the condition
that the sum of the squares of the errors should be a minimum; the
tables which, compared to a considerable number of observations,
approach nearest this condition merit then the preference.
It is principally in astronomy that the method explained above may be
employed with advantage. The astronomical tables owe the truly
astonishing exactitude which they have attained to the precision of
observations and of theories, and to the use of equations of conditions
which cause to concur a great number of excellent observations in the
correction of the same element. But it remains to determine the
probability of the errors that this correction leaves still to be
feared; and the method which I have just explained enables us to
recognize the probability of these errors. In order to give some
interesting applications of it I have profited by the immense work which
M. Bouvard has just finished on the movements of Jupiter and Saturn, of
which he has formed very precise tables. He has discussed with the
greatest care the oppositions and quadratures of these two planets
observed by Bradley and by the astronomers who have followed him down to
the last years; he has concluded the corrections of the elements of
their movement and their masses compared to that of the sun taken as
unity. His calculations give him the mass of Saturn equal to the 3512th
part of that of the sun. Applying to them my formulæ of probability, I
find that it is a bet of 11,000 against one that the error of this
result is not 1/100 of its value, or that which amounts to almost the
same—that after a century of new observations added to the preceding
ones, and examined in the same manner, the new result will not differ by
1/100 from that of M. Bouvard. This wise astronomer finds again the mass
of Jupiter equal to the 1071th part of the sun; and my method of
probability gives a bet of 1,000,000 to one that this result is not
1/100 in error.
This method may be employed again with success in geodetic operations.
We determine the length of the great arc on the surface of the earth by
triangulation, which depends upon a base measured with exactitude. But
whatever precision may be brought to the measure of the angles, the
inevitable errors can, by accumulating, cause the value of the arc
concluded from a great number of triangles to deviate appreciably from
the truth. We recognize this value, then, only imperfectly unless the
probability that its error is comprised within given limits can be
assigned. The error of a geodetic result is a function of the errors of
the angles of each triangle. I have given in the work cited general
formulæ in order to obtain the probability of the values of one or of
several linear functions of a great number of partial errors of which we
know the law of probability; we may then by means of these formulæ
determine the probability that the error of a geodetic result is
contained within the assigned limits, whatever may be the law of the
probability of partial errors. It is moreover more necessary to render
ourselves independent of the law, since the most simple laws themselves
are always infinitely less probable, seeing the infinite number of those
which may exist in nature. But the unknown law of partial errors
introduces into the formulæ an indeterminant which does not permit of
reducing them to numbers unless we are able to eliminate it. We have
seen that in astronomical questions, where each observation furnishes an
equation of condition for obtaining the elements, we eliminate this
determinant by means of the sum of the squares of the remainders when
the most probable values of the elements have been substituted in each
equation. Geodetic questions not offering similar equations, it is
necessary to seek another means of elimination. The quantity by which
the sum of the angles of each observed triangle surpasses two right
angles plus the spherical excess furnishes this means. Thus we replace
by the sum of the squares of these quantities the sum of the squares of
the remainders of the equations of condition; and we may assign in
numbers the probability that the error of the final result of a series
of geodetic operations will not exceed a given quantity. But what is the
most advantageous manner of dividing among the three angles of each
triangle the observed sum of their errors? The analysis of probabilities
renders it apparent that each angle ought to be diminished by a third of
this sum, provided that the weight of a geodetic result be the greatest
possible, which renders the same error less probable. There is then a
great advantage in observing the three angles of each triangle and of
correcting them as we have just said. Simple common sense indicates this
advantage; but the calculation of probabilities alone is able to
appreciate it and to render apparent that by this correction it becomes
the greatest possible.
In order to assure oneself of the exactitude of the value of a great arc
which rests upon a base measured at one of its extremities one measures
a second base toward the other extremity; and one concludes from one of
these bases the length of the other. If this length varies very little
from the observation, there is all reason to believe that the chain of
triangles which unites these bases is very nearly exact and likewise the
value of the large arc which results from it. One corrects, then, this
value by modifying the angles of the triangles in such a manner that the
base is calculated according to the bases measured. But this may be done
in an infinity of ways, among which is preferred that of which the
geodetic result has the greatest weight, inasmuch as the same error
becomes less probable. The analysis of probabilities gives formulæ for
obtaining directly the most advantageous correction which results from
the measurements of the several bases and the laws of probability which
the multiplicity of the bases makes—laws which become very rapidly
decreasing by this multiplicity.
Generally the errors of the results deduced from a great number of
observations are the linear functions of the partial errors of each
observation. The coefficients of these functions depend upon the nature
of the problem and upon the process followed in order to obtain the
results. The most advantageous process is evidently that in which the
same error in the results is less probable than according to any other
process. The application of the calculus of probabilities to natural
philosophy consists, then, in determining analytically the probability
of the values of these functions and in choosing their indeterminant
coefficients in such a manner that the law of this probability should be
most rapidly descending. Eliminating, then, from the formulæ by the data
of the question the factor which is introduced by the almost always
unknown law of the probability of partial errors, we may be able to
evaluate numerically the probability that the errors of the results do
not exceed a given quantity. We shall thus have all that may be desired
touching the results deduced from a great number of observations.
Very approximate results may be obtained by other considerations.
Suppose, for example, that one has a thousand and one observations of
the same quantity; the arithmetical mean of all these observations is
the result given by the most advantageous method. But one would be able
to choose the result according to the condition that the sum of the
variations from each partial value all taken positively should be a
_minimum_. It appears indeed natural to regard as very approximate the
result which satisfies this condition. It is easy to see that if one
disposes the values given by the observations according to the order of
magnitude, the value which will occupy the mean will fulfil the
preceding condition, and calculus renders it apparent that in the case
of an infinite number of observations it would coincide with the truth;
but the result given by the most advantageous method is still preferable.
We see by that which precedes that the theory of probabilities leaves
nothing arbitrary in the manner of distributing the errors of the
observations; it gives for this distribution the most advantageous
formulæ which diminishes as much as possible the errors to be feared in
the results.
The consideration of probabilities can serve to distinguish the small
irregularities of the celestial movements enveloped in the errors of
observations, and to repass to the cause of the anomalies observed in
these movements.
In comparing all the observations it was Ticho-Brahé who recognized the
necessity of applying to the moon an equation of time different from
that which had been applied to the sun and to the planets. It was
similarly the totality of a great number of observations which made
Mayer recognize that the coefficient of the inequality of the precession
ought to be diminished a little for the moon. But since this diminution,
although confirmed and even augmented by Mason, did not appear to result
from universal gravitation, the majority of astronomers neglect it in
their calculations. Having submitted to the calculation of probabilities
a considerable number of lunar observations chosen for this purpose and
which M. Bouvard consented to examine at my request, it appeared to me
to be indicated with so strong a probability that I believed the cause
of it ought to be investigated. I soon saw that it would be only the
ellipticity of the terrestrial spheroid, neglected up to that time in
the theory of the lunar movement as being able to produce only
imperceptible terms. I concluded that these terms became perceptible by
the successive integrations of differential equations. I determined then
those terms by a particular analysis, and I discovered first the
inequality of the lunar movement in latitude which is proportional to
the sine of the longitude of the moon, which no astronomer before had
suspected. I recognized then by means of this inequality that another
exists in the lunar movement in longitude which produces the diminution
observed by Mayer in the equation of the precession applicable to the
moon. The quantity of this diminution and the coefficient of the
preceding inequality in latitude are very appropriate to fix the
oblateness of the earth. Having communicated my researches to M. Burg,
who was occupied at that time in perfecting the tables of the moon by
the comparison of all the good observations, I requested him to
determine with a particular care these two quantities. By a very
remarkable agreement the values which he has found give to the earth the
same oblateness, 1/305, which differs little from the mean derived from
the measurements of the degrees of the meridian and the pendulum; but
those regarded from the point of view of the influence of the errors of
the observations and of the perturbing causes in these measurements, did
not appear to me exactly determined by these lunar inequalities.
It was again by the consideration of probabilities that I recognized the
cause of the secular equation of the moon. The modern observations of
this star compared to the ancient eclipses had indicated to astronomers
an acceleration in the lunar movement; but the geometricians, and
particularly Lagrange, having vainly sought in the perturbations which
this movement experienced the terms upon which this acceleration
depends, reject it. An attentive examination of the ancient and modern
observations and of the intermediary eclipses observed by the Arabians
convinced me that it was indicated with a great probability. I took up
again then from this point of view the lunar theory, and I recognized
that the secular equation of the moon is due to the action of the sun
upon this satellite, combined with the secular variation of the
eccentricity of the terrestrial orb; this brought me to the discovery of
the secular equations of the movements of the nodes and of the perigees
of the lunar orbit, which equations had not been even suspected by
astronomers. The very remarkable agreement of this theory with all the
ancient and modern observations has brought it to a very high degree of
evidence.
The calculus of probabilities has led me similarly to the cause of the
great irregularities of Jupiter and Saturn. Comparing modern
observations with ancient, Halley found an acceleration in the movement
of Jupiter and a retardation in that of Saturn. In order to conciliate
the observations he reduced the movements to two secular equations of
contrary signs and increasing as the squares of the times passed since
1700. Euler and Lagrange submitted to analysis the alterations which the
mutual attraction of these two planets ought to produce in these
movements. They found in doing this the secular equations; but their
results were so different that one of the two at least ought to be
erroneous. I determined then to take up again this important problem of
_celestial mechanics_, and I recognized the invariability of the mean
planetary movements, which nullified the secular equations introduced by
Halley in the tables of Jupiter and Saturn. Thus there remain, in order
to explain the great irregularity of these planets, only the attractions
of the comets to which many astronomers had effective recourse, or the
existence of an irregularity over a long period produced in the
movements of the two planets by their reciprocal action and affected by
contrary signs for each of them. A theorem which I found in regard to
the inequalities of this kind rendered this inequality very probable.
According to this theorem, if the movement of Jupiter is accelerated,
that of Saturn is retarded, which has already conformed to what Halley
had noticed; moreover, the acceleration of Jupiter resulting from the
same theorem is to the retardation of Saturn very nearly in the ratio of
the secular equations proposed by Halley. Considering the mean movements
of Jupiter and Saturn I was enabled easily to recognize that two times
that of Jupiter differed only by a very small quantity from five times
that of Saturn. The period of an irregularity which would have for an
argument this difference would be about nine centuries. Indeed its
coefficient would be of the order of the cubes of the eccentricities of
the orbits; but I knew that by virtue of successive integrations it
acquired for divisor the square of the very small multiplier of the time
in the argument of this inequality which is able to give it a great
value; the existence of this inequality appeared to me then very
probable. The following observation increased then its probability.
Supposing its argument zero toward the epoch of the observations of
Ticho-Brahé, I saw that Halley ought to have found by the comparison of
modern with ancient observations the alterations which he had indicated;
while the comparison of the modern observations among themselves ought
to offer contrary alterations similar to those which Lambert had
concluded from this comparison. I did not then hesitate at all to
undertake this long and tedious calculation necessary to assure myself
of this inequality. It was entirely confirmed by the result of this
calculation, which moreover made me recognize a great number of other
inequalities of which the totality has inclined the tables of Jupiter
and Saturn to the precision of the same observations.
It was again by means of the calculus of probabilities that I recognized
the remarkable law of the mean movements of the three first satellites
of Jupiter, according to which the mean longitude of the first minus
three times that of the second plus two times that of the third is
rigorously equal to the half-circumference. The approximation with which
the mean movements of these stars satisfy this law since their discovery
indicates its existence with an extreme probability. I sought then the
cause of it in their mutual action. The searching examination of this
action convinced me that it was sufficient if in the beginning the
ratios of their mean movements had approached this law within certain
limits, because their mutual action had established and maintained it
rigorously. Thus these three bodies will balance one another eternally
in space according to the preceding law unless strange causes, such as
comets, should change suddenly their movements about Jupiter.
Accordingly it is seen how necessary it is to be attentive to the
indications of nature when they are the result of a great number of
observations, although in other respects they may be inexplicable by
known means. The extreme difficulty of problems relative to the system
of the world has forced geometricians to recur to the approximation
which always leaves room for the fear that the quantities neglected may
have an appreciable influence. When they have been warned of this
influence by the observations, they have recurred to their analysis; in
rectifying it they have always found the cause of the anomalies
observed; they have determined the laws and often they have anticipated
the observations in discovering the inequalities which it had not yet
indicated. Thus one may say that nature itself has concurred in the
analytical perfection of the theories based upon the principle of
universal gravity; and this is to my mind one of the strongest proofs of
the truth of this admirable principle.
In the cases which I have just considered the analytical solution of the
question has changed the probability of the causes into certainty. But
most often this solution is impossible and it remains only to augment
more and more this probability. In the midst of numerous and
incalculable modifications which the action of the causes receives then
from strange circumstances these causes conserve always with the effects
observed the proper ratios to make them recognizable and to verify their
existence. Determining these ratios and comparing them with a great
number of observations if one finds that they constantly satisfy it, the
probability of the causes may increase to the point of equalling that of
facts in regard to which there is no doubt. The investigation of these
ratios of causes to their effects is not less useful in natural
philosophy than the direct solution of problems whether it be to verify
the reality of these causes or to determine the laws from their effects;
since it may be employed in a great number of questions whose direct
solution is not possible, it replaces it in the most advantageous
manner. I shall discuss here the application which I have made of it to
one of the most interesting phenomena of nature, the flow and the ebb of
the sea.
Pliny has given of this phenomenon a description remarkable for its
exactitude, and in it one sees that the ancients had observed that the
tides of each month are greatest toward the syzygies and smallest toward
the quadratures; that they are higher in the perigees than in the
apogees of the moon, and higher in the equinoxes than in the solstices.
They concluded from this that this phenomenon is due to the action of
the sun and moon upon the sea. In the preface of his work _De Stella
Martis_ Kepler admits a tendency of the waters of the sea toward the
moon; but, ignorant of the law of this tendency, he was able to give on
this subject only a probable idea. Newton converted into certainty the
probability of this idea by attaching it to his great principle of
universal gravity. He gave the exact expression of the attractive forces
which produced the flood and the ebb of the sea; and in order to
determine the effects he supposed that the sea takes at each instant the
position of equilibrium which is agreeable to these forces. He explained
in this manner the principal phenomena of the tides; but it followed
from this theory that in our ports the two tides of the same day would
be very unequal if the sun and the moon should have a great declination.
At Brest, for example, the evening tide would be in the syzygies of the
solstices about eight times greater than the morning tide, which is
certainly contrary to the observations which prove that these two tides
are very nearly equal. This result from the Newtonian theory might hold
to the supposition that the sea is agreeable at each instant to a
position of equilibrium, a supposition which is not at all admissible.
But the investigation of the true figure of the sea presents great
difficulties. Aided by the discoveries which the geometricians had just
made in the theory of the movement of fluids and in the calculus of
partial differences, I undertook this investigation, and I gave the
differential equations of the movement of the sea by supposing that it
covers the entire earth. In drawing thus near to nature I had the
satisfaction of seeing that my results approached the observations,
especially in regard to the little difference which exists in our ports
between the two tides of the solstitial syzygies of the same day. I
found that they would be equal if the sea had everywhere the same depth;
I found further that in giving to this depth convenient values one was
able to augment the height of the tides in a port conformably to the
observations. But these investigations, in spite of their generality,
did not satisfy at all the great differences which even adjacent ports
present in this regard and which prove the influence of local
circumstances. The impossibility of knowing these circumstances and the
irregularity of the basin of the seas and that of integrating the
equations of partial differences which are relative has compelled me to
make up the deficiency by the method I have indicated above. I then
endeavored to determine the greatest ratios possible among the forces
which affect all the molecules of the sea, and their effects observable
in our ports. For this I made use of the following principle, which may
be applied to many other phenomena.
"The state of the system of a body in which the primitive conditions of
the movement have disappeared by the resistances which this movement
meets is periodic as the forces which animate it."
Combining this principle with that of the coexistence of very small
oscillations, I have found an expression of the height of the tides
whose arbitraries contain the effect of local circumstances of each port
and are reduced to the smallest number possible; it is only necessary to
compare it to a great number of observations.
Upon the invitation of the Academy of Sciences, observations were made
at the beginning of the last century at Brest upon the tides, which were
continued during six consecutive years. The situation of this port is
very favorable to this sort of observations; it communicates with the
sea by a canal which empties into a vast roadstead at the far end of
which the port has been constructed. The irregularities of the sea
extend thus only to a small degree into the port, just as the
oscillations which the irregular movement of a vessel produces in a
barometer are diminished by a throttling made in the tube of this
instrument. Moreover, the tides being considerable at Brest, the
accidental variations caused by the winds are only feeble; likewise we
notice in the observations of these tides, however little we multiply
them, a great regularity which induced me to propose to the government
to order in this port a new series of observations of the tides,
continued during a period of the movement of the nodes of the lunar
orbit. This has been done. The observations began June 1, 1806; and
since this time they have been made every day without interruption. I am
indebted to the indefatigable zeal of M. Bouvard, for all that interests
astronomy, the immense calculations which the comparison of my analysis
with the observations has demanded. There have been used about six
thousand observations, made during the year 1807 and the fifteen years
following. It results from this comparison that my formulæ represent
with a remarkable precision all the varieties of the tides relative to
the digression of the moon, from the sun, to the declination of these
stars, to their distances from the earth, and to the laws of variation
at the _maximum_ and _minimum_ of each of these elements. There results
from this accord a probability that the flow and the ebb of the sea is
due to the attraction of the sun and moon, so approaching certainty that
it ought to leave room for no reasonable doubt. It changes into
certainty when we consider that this attraction is derived from the law
of universal gravity demonstrated by all the celestial phenomena.
The action of the moon upon the sea is more than double that of the sun.
Newton and his successors in the development of this action have paid
attention only to the terms divided by the cube of the distance from the
moon to the earth, judging that the effects due to the following terms
ought to be inappreciable. But the calculation of probabilities makes it
clear to us that the smallest effects of regular causes may manifest
themselves in the results of a great number of observations arranged in
the order most suitable to indicate them. This calculation again
determines their probability and up to what point it is necessary to
multiply the observations to make it very great. Applying it to the
numerous observations discussed by M. Bouvard I recognized that at Brest
the action of the moon upon the sea is greater in the full moons than in
the new moons, and greater when the moon is austral than when it is
boreal—phenomena which can result only from the terms of the lunar
action divided by the fourth power of the distance from the moon to the
earth.
To arrive at the ocean the action of the sun and the moon traverses the
atmosphere, which ought consequently to feel its influence and to be
subjected to movements similar to those of the sea.
These movements produce in the barometer periodic oscillations. Analysis
has made it clear to me that they are inappreciable in our climates. But
as local circumstances increase considerably the tides in our ports, I
have inquired again if similar circumstances have made appreciable these
oscillations of the barometer. For this I have made use of the
meteorological observations which have been made every day for many
years at the royal observatory. The heights of the barometer and of the
thermometer are observed there at nine o'clock in the morning, at noon,
at three o'clock in the afternoon, and at eleven o'clock in the evening.
M. Bouvard has indeed wished to take up the consideration of
observations of the eight years elapsed from October 1, 1815, to October
1, 1823, on the registers. In disposing the observations in the manner
most suitable to indicate the lunar atmospheric flood at Paris, I find
only one eighteenth of a millimeter for the extent of the corresponding
oscillation of the barometer. It is this especially which has made us
feel the necessity of a method for determining the probability of a
result, and without this method one is forced to present as the laws of
nature the results of irregular causes which has often happened in
meteorology. This method applied to the preceding result shows the
uncertainty of it in spite of the great number of observations employed,
which it would be necessary to increase tenfold in order to obtain a
result sufficiently probable.
The principle which serves as a basis for my theory of the tides may be
extended to all the effects of hazard to which variable causes are
joined according to regular laws. The action of these causes produces in
the mean results of a great number of effects varieties which follow the
same laws and which one may recognize by the analysis of probabilities.
In the measure which these effects are multiplied those varieties are
manifested with an ever-increasing probability, which would approach
certainty if the number of the effects of the results should become
infinite. This theorem is analogous to that which I have already
developed upon the action of constant causes. Every time, then, that a
cause whose progress is regular can have influence upon a kind of
events, we may seek to discover its influence by multiplying the
observations and arranging them in the most suitable order to indicate
it. When this influence appears to manifest itself the analysis of
probabilities determines the probability of its existence and that of
its intensity; thus the variation of the temperature from day to night
modifying the pressure of the atmosphere and consequently the height of
the barometer, it is natural to think that the multiplied observations
of these heights ought to show the influence of the solar heat. Indeed
there has long been recognized at the equator, where this influence
appears to be greatest, a small diurnal variation in the height of the
barometer of which the _maximum_ occurs about nine o'clock in the
morning and the _minimum_ about three o'clock in the afternoon. A second
_maximum_ occurs about eleven o'clock in the evening and a second
_minimum_ about four o'clock in the morning. The oscillations of the
night are less than those of the day, the extent of which is about two
millimeters. The inconstancy of our climate has not taken this variation
from our observers, although it may be less appreciable than in the
tropics. M. Ramond has recognized and determined it at Clermont, the
chief place of the district of Puy-de-Dôme, by a series of precise
observations made during several years; he has even found that it is
smaller in the months of winter than in other months. The numerous
observations which I have discussed in order to estimate the influence
of attractions of the sun and the moon upon the barometric heights at
Paris have served me in determining their diurnal variation. Comparing
the heights at nine o'clock in the morning with those of the same days
at three o'clock in the afternoon, this variation is manifested with so
much evidence that its mean value each month has been constantly
positive for each of the seventy-two months from January 1, 1817, to
January 1, 1823; its mean value in these seventy-two months has been
almost .8 of a millimeter, a little less than at Clermont and much less
than at the equator. I have recognized that the mean result of the
diurnal variations of the barometer from 9 o'clock A.M. to 3 P.M. has
been only .5428 millimeter in the three months of November, December,
January, and that it has risen to 1.0563 millimeters in the three
following months, which coincides with the observations of M. Ramond.
The other months offer nothing similar.
In order to apply to these phenomena the calculation of these
probabilities, I commenced by determining the law of the probability of
the anomalies of the diurnal variation due to hazard. Applying it then
to the observations of this phenomenon, I found that it was a bet of
more than 300,000 against one that a regular cause produced it. I do not
seek to determine this cause; I content myself with stating its
existence. The period of the diurnal variation regulated by the solar
day indicates evidently that this variation is due to the action of the
sun. The extreme smallness of the attractive action of the sun upon the
atmosphere is proved by the smallness of the effects due to the united
attractions of the sun and the moon. It is then by the action of its
heat that the sun produces the diurnal variation of the barometer; but
it is impossible to subject to calculus the effects of its action on the
height of the barometer and upon the winds. The diurnal variation of the
magnetic needle is certainly a result of the action of the sun. But does
this star act here as in the diurnal variation of the barometer by its
heat or by its influence upon electricity and upon magnetism, or finally
by the union of these influences? A long series of observations made in
different countries will enable us to apprehend this.
One of the most remarkable phenomena of the system of the world is that
of all the movements of rotation and of revolution of the planets and
the satellites in the sense of the rotation of the sun and about in the
same plane of its equator. A phenomenon so remarkable is not the effect
of hazard: it indicates a general cause which has determined all its
movements. In order to obtain the probability with which this cause is
indicated we shall observe that the planetary system, such as we know it
to-day, is composed of eleven planets and of eighteen satellites at
least, if we attribute with Herschel six satellites to the planet
Uranus. The movements of the rotation of the sun, of six planets, of the
moon, of the satellites of Jupiter, of the ring of Saturn, and of one of
its satellites have been recognized. These movements form with those of
revolution a totality of forty-three movements directed in the same
sense; but one finds by the analysis of probabilities that it is a bet
of more than 4000000000000 against one that this disposition is not the
result of hazard; this forms a probability indeed superior to that of
historical events in regard to which no doubt exists. We ought then to
believe at least with equal confidence that a primitive cause has
directed the planetary movements, especially if we consider that the
inclination of the greatest number of these movements at the solar
equator is very small.
Another equally remarkable phenomenon of the solar system is the small
degree of the eccentricity of the orbs of the planets and the
satellites, while those of the comets are very elongated, the orbs of
the system not offering any intermediate shades between a great and a
small eccentricity. We are again forced to recognize here the effect of
a regular cause; chance has certainly not given an almost circular form
to the orbits of all the planets and their satellites; it is then that
the cause which has determined the movements of these bodies has
rendered them almost circular. It is necessary, again, that the great
eccentricities of the orbits of the comets should result from the
existence of this cause without its having influenced the direction of
their movements; for it is found that there are almost as many
retrograde comets as direct comets, and that the mean inclination of all
their orbits to the ecliptic approaches very nearly half a right angle,
as it ought to be if the bodies had been thrown at hazard.
Whatever may be the nature of the cause in question, since it has
produced or directed the movement of the planets, it is necessary that
it should have embraced all the bodies and considered all the distances
which separate them, it can have been only a fluid of an immense
extension. Therefore in order to have given them in the same sense an
almost circular movement about the sun it is necessary that this fluid
should have surrounded this star as an atmosphere. The consideration of
the planetary movements leads us then to think that by virtue of an
excessive heat the atmosphere of the sun was originally extended beyond
the orbits of all the planets, and that it has contracted gradually to
its present limits.
In the primitive state where we imagine the sun it resembled the nebulæ
that the telescope shows us composed of a nucleus more or less brilliant
surrounded by a nebula which, condensing at the surface, ought to
transform it some day into a star. If one conceives by analogy all the
stars formed in this manner, one can imagine their anterior state of
nebulosity itself preceded by other stars in which the nebulous matter
was more and more diffuse, the nucleus being less and less luminous and
dense. Going back, then, as far as possible, one would arrive at a
nebulosity so diffuse that one would be able scarcely to suspect its
existence.
Such is indeed the first state of the nebulæ which Herschel observed
with particular care by means of his powerful telescopes, and in which
he has followed the progress of condensation, not in a single one, these
stages not becoming appreciable to us except after centuries, but in
their totality, just about as one can in a vast forest follow the
increase of the trees by the individuals of the divers ages which the
forest contains. He has observed from the beginning nebulous matter
spread out in divers masses in the different parts of the heavens, of
which it occupies a great extent. He has seen in some of these masses
this matter slightly condensed about one or several faintly luminous
nebulæ. In the other nebulæ these nuclei shine, moreover, in proportion
to the nebulosity which surrounds them. The atmospheres of each nucleus
becoming separated by an ulterior condensation, there result the
multifold nebulæ formed of brilliant nuclei very adjacent and surrounded
each by an atmosphere; sometimes the nebulous matter, by condensing in a
uniform manner, has produced the nebulæ which are called _planetary_.
Finally a greater degree of condensation transforms all these nebulæ
into stars. The nebulæ classed according to this philosophic view
indicate with an extreme probability their future transformation into
stars and the anterior state of nebulosity of existing stars. The
following considerations come to the aid of proofs drawn from these
analogies.
For a long time the particular disposition of certain stars visible to
the naked eye has struck the attention of philosophical observers.
Mitchel has already remarked how improbable it is that the stars of the
Pleiades, for example, should have been confined in the narrow space
which contain them by the chances of hazard alone, and he has concluded
from this that this group of stars and the similar groups that the
heaven presents us are the results of a primitive cause or of a general
law of nature. These groups are a necessary result of the condensation
of the nebulæ at several nuclei; it is apparent that the nebulous matter
being attracted continuously by the divers nuclei, they ought to form in
time a group of stars equal to that of the Pleiades. The condensation of
the nebulæ at two nuclei forms similarly very adjacent stars, revolving
the one about the other, equal to those whose respective movements
Herschel has already considered. Such are, further, the 61st of the Swan
and its following one in which Bessel has just recognized particular
movements so considerable and so little different that the proximity of
these stars to one another and their movement about the common centre of
gravity ought to leave no doubt. Thus one descends by degrees from the
condensation of nebulous matter to the consideration of the sun
surrounded formerly by a vast atmosphere, a consideration to which one
repasses, as has been seen, by the examination of the phenomena of the
solar system. A case so remarkable gives to the existence of this
anterior state of the sun a probability strongly approaching certainty.
But how has the solar atmosphere determined the movements of rotation
and revolution of the planets and the satellites? If these bodies had
penetrated deeply the atmosphere its resistance would have caused them
to fall upon the sun; one is then led to believe with much probability
that the planets have been formed at the successive limits of the solar
atmosphere which, contracting by the cold, ought to have abandoned in
the plane of its equator zones of vapors which the mutual attraction of
their molecules has changed into divers spheroids. The satellites have
been similarly formed by the atmospheres of their respective planets.
I have developed at length in my _Exposition of the System of the World_
this hypothesis, which appears to me to satisfy all the phenomena which
this system presents us. I shall content myself here with considering
that the angular velocity of rotation of the sun and the planets being
accelerated by the successive condensation of their atmospheres at their
surfaces, it ought to surpass the angular velocity of revolution of the
nearest bodies which revolve about them. Observation has indeed
confirmed this with regard to the planets and satellites, and even in
ratio to the ring of Saturn, the duration of whose revolution is .438
days, while the duration of the rotation of Saturn is .427 days.
In this hypothesis the comets are strangers to the planetary system. In
attaching their formation to that of the nebulæ they may be regarded as
small nebulæ at the nuclei, wandering from systems to solar systems, and
formed by the condensation of the nebulous matter spread out in such
great profusion in the universe. The comets would be thus, in relation
to our system, as the aerolites are relatively to the Earth, to which
they would appear strangers. When these stars become visible to us they
offer so perfect resemblance to the nebulæ that they are often
confounded with them; and it is only by their movement, or by the
knowledge of all the nebulæ confined to that part of the heavens where
they appear, that we succeed in distinguishing them. This supposition
explains in a happy manner the great extension which the heads and tails
of comets take in the measure that they approach the sun, and the
extreme rarity of these tails which, in spite of their immense depth, do
not weaken at all appreciably the light of the stars which we look
across.
When the little nebulæ come into that part of space where the attraction
of the sun is predominant, and which we shall call the _sphere of
activity_ of this star, it forces them to describe elliptic or
hyperbolic orbits. But their speed being equally possible in all
directions they ought to move indifferently in all the senses and under
all inclinations of the elliptic, which is conformable to that which has
been observed.
The great eccentricity of the cometary orbits results again from the
preceding hypothesis. Indeed if these orbits are elliptical they are
very elongated, since their great axes are at least equal to the radius
of the sphere of activity of the sun. But these orbits may be
hyperbolic; and if the axes of these hyperbolæ are not very large in
proportion to the mean distance from the sun to the earth, the movement
of the comets which describe them will appear sensibly hyperbolic.
However, of the hundred comets of which we already have the elements,
not one has appeared certainly to move in an hyperbola; it is necessary,
then, that the chances which give an appreciable hyperbola should be
extremely rare in proportion to the contrary chances.
The comets are so small that, in order to become visible, their
perihelion distance ought to be inconsiderable. Up to the present this
distance has surpassed only twice the diameter of the terrestrial orbit,
and most often it has been below the radius of this orbit. It is
conceived that, in order to approach so near the sun, their speed at the
moment of their entrance into its sphere of activity ought to have a
magnitude and a direction confined within narrow limits. In determining
by the analysis of probabilities the ratio of the chances which, in
these limits, give an appreciable hyperbola, to the chances which give
an orbit which may be confounded with a parabola, I have found that it
is a bet of at least 6000 against one that a nebula which penetrates
into the activity of the sun in such a manner as to be observed will
describe either a very elongated ellipse or an hyperbola. By the
magnitude of its axis, the latter will be appreciably confounded with a
parabola in the part which is observed; it is then not surprising that,
up to this time, hyperbolic movements have not been recognized.
The attraction of the planets, and, perhaps further, the resistance of
the ethereal centres, ought to have changed many cometary orbits in the
ellipses whose great axis is less than the radius of the sphere of
activity of the sun, which augments the chances of the elliptical
orbits. We may believe that this change has taken place with the comet
of 1759, and with the comet whose duration is only twelve hundred days,
and which will reappear without ceasing in this short interval, unless
the evaporation which it meets at each of its returns to the perihelion
ends by rendering it invisible.
We are able further, by the analysis of probabilities, to verify the
existence or the influence of certain causes whose action is believed to
exist upon organized beings. Of all the instruments that we are able to
employ in order to recognize the imperceptible agents of nature the most
sensitive are the nerves, especially when particular causes increase
their sensibility. It is by their aid that the feeble electricity which
the contact of two heterogeneous metals develops has been discovered;
this has opened a vast field to the researches of physicists and
chemists. The singular phenomena which results from extreme sensibility
of the nerves in some individuals have given birth to divers opinions
about the existence of a new agent which has been named _animal
magnetism_, about the action on ordinary magnetism, and about the
influence of the sun and moon in some nervous affections, and finally,
about the impressions which the proximity of metals or of running water
makes felt. It is natural to think that the action of these causes is
very feeble, and that it may be easily disturbed by accidental
circumstances; thus because in some cases it is not manifested at all
its existence ought not to be denied. We are so far from recognizing all
the agents of nature and their divers modes of action that it would be
unphilosophical to deny the phenomena solely because they are
inexplicable in the present state of our knowledge. But we ought to
examine them with an attention as much the more scrupulous as it appears
the more difficult to admit them; and it is here that the calculation of
probabilities becomes indispensable in determining to just what point it
is necessary to multiply the observations or the experiences in order to
obtain in favor of the agents which they indicate, a probability
superior to the reasons which can be obtained elsewhere for not
admitting them.
The calculation of probabilities can make appreciable the advantages and
the inconveniences of the methods employed in the speculative sciences.
Thus in order to recognize the best of the treatments in use in the
healing of a malady, it is sufficient to test each of them on an equal
number of patients, making all the conditions exactly similar; the
superiority of the most advantageous treatment will manifest itself more
and more in the measure that the number is increased; and the
calculation will make apparent the corresponding probability of its
advantage and the ratio according to which it is superior to the others.