_CONCERNING THE VARIOUS MEANS OF APPROACHING CERTAINTY._
Induction, analogy, hypotheses founded upon facts and rectified
continually by new observations, a happy tact given by nature and
strengthened by numerous comparisons of its indications with experience,
such are the principal means for arriving at truth.
If one considers a series of objects of the same nature one perceives
among them and in their changes ratios which manifest themselves more
and more in proportion as the series is prolonged, and which, extending
and generalizing continually, lead finally to the principle from which
they were derived. But these ratios are enveloped by so many strange
circumstances that it requires great sagacity to disentangle them and to
recur to this principle: it is in this that the true genius of sciences
consists. Analysis and natural philosophy owe their most important
discoveries to this fruitful means, which is called _induction_. Newton
was indebted to it for his theorem of the binomial and the principle of
universal gravity. It is difficult to appreciate the probability of the
results of induction, which is based upon this that the simplest ratios
are the most common; this is verified in the formulæ of analysis and is
found again in natural phenomena, in crystallization, and in chemical
combinations. This simplicity of ratios will not appear astonishing if
we consider that all the effects of nature are only mathematical results
of a small number of immutable laws.
Yet induction, in leading to the discovery of the general principles of
the sciences, does not suffice to establish them absolutely. It is
always necessary to confirm them by demonstrations or by decisive
experiences; for the history of the sciences shows us that induction has
sometimes led to inexact results. I shall cite, for example, a theorem
of Fermat in regard to prime numbers. This great geometrician, who had
meditated profoundly upon this theorem, sought a formula which,
containing only prime numbers, gave directly a prime number greater than
any other number assignable. Induction led him to think that two, raised
to a power which was itself a power of two, formed with unity a prime
number. Thus, two raised to the square plus one, forms the prime number
five; two raised to the second power of two, or sixteen, forms with one
the prime number seventeen. He found that this was still true for the
eighth and the sixteenth power of two augmented by unity; and this
induction, based upon several arithmetical considerations, caused him to
regard this result as general. However, he avowed that he had not
demonstrated it. Indeed, Euler recognized that this does not hold for
the thirty-second power of two, which, augmented by unity, gives
4,294,967,297, a number divisible by 641.
We judge by induction that if various events, movements, for example,
appear constantly and have been long connected by a simple ratio, they
will continue to be subjected to it; and we conclude from this, by the
theory of probabilities, that this ratio is due, not to hazard, but to a
regular cause. Thus the equality of the movements of the rotation and
the revolution of the moon; that of the movements of the nodes of the
orbit and of the lunar equator, and the coincidence of these nodes; the
singular ratio of the movements of the first three satellites of
Jupiter, according to which the mean longitude of the first satellite,
less three times that of the second, plus two times that of the third,
is equal to two right angles; the equality of the interval of the tides
to that of the passage of the moon to the meridian; the return of the
greatest tides with the syzygies, and of the smallest with the
quadratures; all these things, which have been maintained since they
were first observed, indicate with an extreme probability, the existence
of constant causes which geometricians have happily succeeded in
attaching to the law of universal gravity, and the knowledge of which
renders certain the perpetuity of these ratios.
The chancellor Bacon, the eloquent promoter of the true philosophical
method, has made a very strange misuse of induction in order to prove
the immobility of the earth. He reasons thus in the _Novum Organum_, his
finest work: "The movement of the stars from the orient to the occident
increases in swiftness, in proportion to their distance from the earth.
This movement is swiftest with the stars; it slackens a little with
Saturn, a little more with Jupiter, and so on to the moon and the
highest comets. It is still perceptible in the atmosphere, especially
between the tropics, on account of the great circles which the molecules
of the air describe there; finally, it is almost inappreciable with the
ocean; it is then nil for the earth." But this induction proves only
that Saturn, and the stars which are inferior to it, have their own
movements, contrary to the real or apparent movement which sweeps the
whole celestial sphere from the orient to the occident, and that these
movements appear slower with the more remote stars, which is conformable
to the laws of optics. Bacon ought to have been struck by the
inconceivable swiftness which the stars require in order to accomplish
their diurnal revolution, if the earth is immovable, and by the extreme
simplicity with which its rotation explains how bodies so distant, the
ones from the others, as the stars, the sun, the planets, and the moon,
all seem subjected to this revolution. As to the ocean and to the
atmosphere, he ought not to compare their movement with that of the
stars which are detached from the earth; but since the air and the sea
make part of the terrestrial globe, they ought to participate in its
movement or in its repose. It is singular that Bacon, carried to great
prospects by his genius, was not won over by the majestic idea which the
Copernican system of the universe offers. He was able, however, to find
in favor of that system, strong analogies in the discoveries of Galileo,
which were continued by him. He has given for the search after truth the
precept, but not the example. But by insisting, with all the force of
reason and of eloquence, upon the necessity of abandoning the
insignificant subtleties of the school, in order to apply oneself to
observations and to experiences, and by indicating the true method of
ascending to the general causes of phenomena, this great philosopher
contributed to the immense strides which the human mind made in the
grand century in which he terminated his career.
Analogy is based upon the probability, that similar things have causes
of the same kind and produce the same effects. This probability increase
as the similitude becomes more perfect. Thus we judge without doubt that
beings provided with the same organs, doing the same things, experience
the same sensations, and are moved by the same desires. The probability
that the animals which resemble us have sensations analogous to ours,
although a little inferior to that which is relative to individuals of
our species, is still exceedingly great; and it has required all the
influence of religious prejudices to make us think with some
philosophers that animals are mere automatons. The probability of the
existence of feeling decreases in the same proportion as the similitude
of the organs with ours diminishes, but it is always very great, even
with insects. In seeing those of the same species execute very
complicated things exactly in the same manner from generation to
generation, and without having learned them, one is led to believe that
they act by a kind of affinity analogous to that which brings together
the molecules of crystals, but which, together with the sensation
attached to all animal organization, produces, with the regularity of
chemical combinations, combinations that are much more singular; one
might, perhaps, name this mingling of elective affinities and sensations
_animal affinity_. Although there exists a great analogy between the
organization of plants and that of animals, it does not seem to me
sufficient to extend to vegetables the sense of feeling; but nothing
authorizes us in denying it to them.
Since the sun brings forth, by the beneficent action of its light and of
its heat, the animals and plants which cover the earth, we judge by
analogy that it produces similar effects upon the other planets; for it
is not natural to think that the cause whose activity we see developed
in so many ways should be sterile upon so great a planet as Jupiter,
which, like the terrestrial globe, has its days, its nights, and its
years, and upon which observations indicate changes which suppose very
active forces. Yet this would be giving too great an extension to
analogy to conclude from it the similitude of the inhabitants of the
planets and of the earth. Man, made for the temperature which he enjoys,
and for the element which he breathes, would not be able, according to
all appearance, to live upon the other planets. But ought there not to
be an infinity of organization relative to the various constitutions of
the globes of this universe? If the single difference of the elements
and of the climates make so much variety in terrestrial productions, how
much greater the difference ought to be among those of the various
planets and of their satellites! The most active imagination can form no
idea of it; but their existence is very probable.
We are led by a strong analogy to regard the stars as so many suns
endowed, like ours, with an attractive power proportional to the mass
and reciprocal to the square of the distances; for this power being
demonstrated for all the bodies of the solar system, and for their
smallest molecules, it appears to appertain to all matter. Already the
movements of the small stars, which have been called _double_, on
account of their being binary, appear to indicate it; a century at most
of precise observations, by verifying their movements of revolution, the
ones about the others, will place beyond doubt their reciprocal
attractions.
The analogy which leads us to make each star the centre of a planetary
system is far less strong than the preceding one; but it acquires
probability by the hypothesis which has been proposed in regard to the
formation of the stars and of the sun; for in this hypothesis each star,
having been like the sun, primitively environed by a vast atmosphere, it
is natural to attribute to this atmosphere the same effects as to the
solar atmosphere, and to suppose that it has produced, in condensing,
planets and satellites.
A great number of discoveries in the sciences is due to analogy. I shall
cite as one of the most remarkable, the discovery of atmospheric
electricity, to which one has been led by the analogy of electric
phenomena with the effects of thunder.
The surest method which can guide us in the search for truth, consists
in rising by induction from phenomena to laws and from laws to forces.
Laws are the ratios which connect particular phenomena together: when
they have shown the general principle of the forces from which they are
derived, one verifies it either by direct experiences, when this is
possible, or by examination if it agrees with known phenomena; and if by
a rigorous analysis we see them proceed from this principle, even in
their small details, and if, moreover, they are quite varied and very
numerous, then science acquires the highest degree of certainty and of
perfection that it is able to attain. Such, astronomy has become by the
discovery of universal gravity. The history of the sciences shows that
the slow and laborious path of induction has not always been that of
inventors. The imagination, impatient to arrive at the causes, takes
pleasure in creating hypotheses, and often it changes the facts in order
to adapt them to its work; then the hypotheses are dangerous. But when
one regards them only as the means of connecting the phenomena in order
to discover the laws; when, by refusing to attribute them to a reality,
one rectifies them continually by new observations, they are able to
lead to the veritable causes, or at least put us in a position to
conclude from the phenomena observed those which given circumstances
ought to produce.
If we should try all the hypotheses which can be formed in regard to the
cause of phenomena we should arrive, by a process of exclusion, at the
true one. This means has been employed with success; sometimes we have
arrived at several hypotheses which explain equally well all the facts
known, and among which scholars are divided, until decisive observations
have made known the true one. Then it is interesting, for the history of
the human mind, to return to these hypotheses, to see how they succeed
in explaining a great number of facts, and to investigate the changes
which they ought to undergo in order to agree with the history of
nature. It is thus that the system of Ptolemy, which is only the
realization of celestial appearances, is transformed into the hypothesis
of the movement of the planets about the sun, by rendering equal and
parallel to the solar orbit the circles and the epicycles which he
causes to be described annually, and the magnitude of which he leaves
undetermined. It suffices, then, in order to change this hypothesis into
the true system of the world, to transport the apparent movement of the
sun in a sense contrary to the earth.
It is almost always impossible to submit to calculus the probability of
the results obtained by these various means; this is true likewise for
historical facts. But the totality of the phenomena explained, or of the
testimonies, is sometimes such that without being able to appreciate the
probability we cannot reasonably permit ourselves any doubt in regard to
them. In the other cases it is prudent to admit them only with great
reserve.