ANALYSIS OF THE PROCESS OF INDUCTION.
APHORISM XXXIV.
_The Process of Induction may be resolved into three steps; the_
Selection of the Idea, _the_ Construction of the Conception, _and
the_ Determination of the Magnitudes.
APHORISM XXXV.
_These three steps correspond to the determination of the_
Independent Variable, _the_ Formula, _and the_ Coefficients, _in
mathematical investigations; or to the_ Argument, _the_ Law, _and
the_ Numerical Data, _in a Table of an astronomical or other_
Inequality.
APHORISM XXXVI.
_The Selection of the Idea depends mainly upon inventive sagacity:
which operates by suggesting and trying various hypotheses. Some
inquirers try erroneous hypotheses; and thus, exhausting the forms
of errour, form the Prelude to Discovery._
APHORISM XXXVII.
_The following Rules may be given, in order to the selection of the
Idea for purposes of Induction:--the Idea and the Facts must be_
homogeneous; _and the Rule must be_ tested by the Facts.
SECT. I.--_The Three Steps of Induction._
1. WHEN facts have been decomposed and phenomena measured, the
philosopher endeavours to combine them into general laws, by the aid
of {187} Ideas and Conceptions; these being illustrated and
regulated by such means as we have spoken of in the last two
chapters. In this task, of gathering laws of nature from observed
facts, as we have already said[24\3], the natural sagacity of gifted
minds is the power by which the greater part of the successful
results have been obtained; and this power will probably always be
more efficacious than any Method can be. Still there are certain
methods of procedure which may, in such investigations, give us no
inconsiderable aid, and these I shall endeavour to expound.
[Note 24\3: B. ii. c. vi.]
2. For this purpose, I remark that the Colligation of ascertained
Facts into general Propositions may be considered as containing
three steps, which I shall term the _Selection of the Idea_, _the
Construction of the Conception_, and _the Determination of the
Magnitudes_. It will be recollected that by the word _Idea_, (or
Fundamental Idea,) used in a peculiar sense, I mean certain wide and
general fields of intelligible relation, such as Space, Number,
Cause, Likeness; while by _Conception_ I denote more special
modifications of these ideas, as a _circle_, a _square number_, a
_uniform force_, a _like form_ of flower. Now in order to establish
any law by reference to facts, we must select the _true Idea_ and the
_true Conception_. For example; when Hipparchus found[25\3] that the
distance of the bright star Spica Virginis from the equinoxial point
had increased by two degrees in about two hundred years, and desired
to reduce this change to a law, he had first to assign, if possible,
the _idea_ on which it depended;--whether it was regulated for
instance, by _space_, or by _time_; whether it was determined by the
positions of other stars at each moment, or went on progressively
with the lapse of ages. And when there was found reason to select
_time_ as the regulative _idea_ of this change, it was then to be
determined how the change went on with the time;--whether uniformly,
or in some other manner: the _conception_, or the rule of the
progression, was to be {188} rightly constructed. Finally, it being
ascertained that the change did go on uniformly, the question then
occurred what was its _amount_:--whether exactly a degree in a
century, or more, or less, and how much: and thus the determination
of the _magnitude_ completed the discovery of the law of phenomena
respecting this star.
[Note 25\3: _Hist. Ind. Sc._ b. iii. c. iv. sect. 3.]
3. Steps similar to these three may be discerned in all other
discoveries of laws of nature. Thus, in investigating the laws of
the motions of the sun, moon or planets, we find that these motions
may be resolved, besides a uniform motion, into a series of partial
motions, or Inequalities; and for each of these Inequalities, we
have to learn upon what it directly depends, whether upon the
progress of time only, or upon some configuration of the heavenly
bodies in space; then, we have to ascertain its law; and finally, we
have to determine what is its amount. In the case of such
Inequalities, the fundamental element on which the Inequality
depends, is called by mathematicians the _Argument_. And when the
Inequality has been fully reduced to known rules, and expressed in
the form of a Table, the Argument is the fundamental Series of
Numbers which stands in the margin of the Table, and by means of
which we refer to the other Numbers which express the Inequality.
Thus, in order to obtain from a Solar Table the Inequality of the
sun's annual motion, the Argument is the Number which expresses the
day of the year; the Inequalities for each day being (in the Table)
ranged in a line corresponding to the days. Moreover, the Argument
of an Inequality being assumed to be known, we must, in order to
calculate the Table, that is, in order to exhibit the law of nature,
know also the _Law_ of the Inequality, and its _Amount_. And the
investigation of these three things, the Argument, the Law, and the
Amount of the Inequality, represents the three steps above
described, the Selection of the Idea, the Construction of the
Conception, and the Determination of the Magnitude.
4. In a great body of cases, _mathematical_ language and calculation
are used to express the connexion {189} between the general law and
the special facts. And when this is done, the three steps above
described may be spoken of as the Selection of the _Independent
Variable_, the Construction of the _Formula_, and the Determination
of the _Coefficients_. It may be worth our while to attend to an
exemplification of this. Suppose then, that, in such observations as
we have just spoken of, namely, the shifting of a star from its
place in the heavens by an unknown law, astronomers had, at the end
of three successive years, found that the star had removed by 3, by
8, and by 15 minutes from its original place. Suppose it to be
ascertained also, by methods of which we shall hereafter treat, that
this change depends upon the time; we must then take the _time_,
(which we may denote by the symbol _t_,) for the _independent
variable_. But though the star changes its place _with_ the time,
the change is not _proportional_ to the time; for its motion which
is only 3 minutes in the first year, is 5 minutes in the second
year, and 7 in the third. But it is not difficult for a person a
little versed in mathematics to perceive that the series 3, 8, 15,
may be obtained by means of two terms, one of which is proportional
to the time, and the other to the square of the time; that is, it is
expressed by the _formula at + btt_. The question then occurs, what
are the values of the _coefficients_ _a_ and _b_; and a little
examination of the case shows us that _a_ must be 2, and _b_, 1: so
that the formula is 2_t_ + _tt_. Indeed if we add together the series
2, 4, 6, which expresses a change proportional to the time, and 1,
4, 9, which is proportional to the square of the time, we obtain the
series 3, 8, 15, which is the series of numbers given by
observation. And thus the three steps which give us the Idea, the
Conception, and the Magnitudes; or the Argument, the Law, and the
Amount, of the change; give us the Independent Variable, the
Formula, and the Coefficients, respectively.
We now proceed to offer some suggestions of methods by which each of
these steps may be in some degree promoted. {190}
SECT. II.--_Of the Selection of the Fundamental Idea._
5. When we turn our thoughts upon any assemblage of facts, with a
view of collecting from them some connexion or law, the most
important step, and at the same time that in which rules can least
aid us, is the Selection of the Idea by which they are to be
collected. So long as this idea has not been detected, all seems to
be hopeless confusion or insulated facts; when the connecting idea
has been caught sight of, we constantly regard the facts with
reference to their connexion, and wonder that it should be possible
for any one to consider them in any other point of view.
Thus the different seasons, and the various aspects of the heavenly
bodies, might at first appear to be direct manifestations from some
superior power, which man could not even understand: but it was soon
found that the ideas of time and space, of motion and recurrence,
would give coherency to many of the phenomena. Yet this took place
by successive steps. Eclipses, for a long period, seemed to follow
no law; and being very remarkable events, continued to be deemed the
indications of a supernatural will, after the common motions of the
heavens were seen to be governed by relations of time and space. At
length, however, the Chaldeans discovered that, after a period of
eighteen years, similar sets of eclipses recur; and, thus selecting
the idea of _time_, simply, as that to which these events were to be
referred, they were able to reduce them to rule; and from that time,
eclipses were recognized as parts of a regular order of things. We
may, in the same manner, consider any other course of events, and
may enquire by what idea they are bound together. For example, if we
take the weather, years peculiarly wet or dry, hot and cold,
productive and unproductive, follow each other in a manner which, at
first sight at least, seems utterly lawless and irregular. Now can
we in any way discover some rule and order in these occurrences? Is
there, for example, in these events, as in eclipses, a certain cycle
of years, after which like {191} seasons come round again? or does
the weather depend upon the force of some extraneous body--for
instance, the moon--and follow in some way her aspects? or would the
most proper way of investigating this subject be to consider the
effect of the moisture and heat of various tracts of the earth's
surface upon the ambient air? It is at our choice to _try_ these and
other modes of obtaining a science of the weather: that is, we may
refer the phenomena to the idea of _time_, introducing the
conception of a cycle;--or to the idea of external _force_, by the
conception of the moon's action;--or to the idea of _mutual action_,
introducing the conceptions of thermotical and atmological agencies,
operating between different regions of earth, water, and air.
6. It may be asked, How are we to decide in such alternatives? How
are we to select the one right idea out of several conceivable ones?
To which we can only reply, that this must be done by _trying_ which
will succeed. If there really exist a cycle of the weather, as well
as of eclipses, this must be established by comparing the asserted
cycle with a good register of the seasons, of sufficient extent. Or
if the moon really influence the meteorological conditions of the
air, the asserted influence must be compared with the observed
facts, and so accepted or rejected. When Hipparchus had observed the
increase of longitude of the stars, the idea of a motion of the
celestial sphere suggested itself as the explanation of the change;
but this thought was _verified_ only by observing several stars. It
was conceivable that each star should have an independent motion,
governed by time only, or by other circumstances, instead of being
regulated by its place in the sphere; and this possibility could be
rejected by trial alone. In like manner, the original opinion of the
composition of bodies supposed the compounds to derive their
properties from the elements according to the law of _likeness_; but
this opinion was overturned by a thousand facts; and thus the really
applicable Idea of Chemical Composition was introduced in modern
times. In what has already been said on the History of Ideas, we
have seen how each science was in a state {192} of confusion and
darkness till the right idea was introduced.
7. No general method of evolving such ideas can be given. Such
events appear to result from a peculiar sagacity and felicity of
mind;--never without labour, never without preparation;--yet with no
constant dependence upon preparation, or upon labour, or even
entirely upon personal endowments. Newton explained the colours
which refraction produces, by referring each colour to a peculiar
_angle of refraction_, thus introducing the right idea. But when the
same philosopher tried to explain the colours produced by
diffraction, he erred, by attempting to apply the same idea, (_the
course of a single ray_,) instead of applying the truer idea, of the
_interference of two rays_. Newton gave a wrong rule for the double
refraction of Iceland spar, by making the refraction depend on the
_edges_ of the rhombohedron: Huyghens, more happy, introduced the
idea of the _axis of symmetry_ of the solid, and thus was able to
give the true law of the phenomena.
8. Although the selected idea is proved to be the right one, only
when the true law of nature is established by means of it, yet it
often happens that there prevails a settled conviction respecting
the relation which must afford the key to the phenomena, before the
selection has been confirmed by the laws to which it leads. Even
before the empirical laws of the tides were made out, it was not
doubtful that these laws depended upon the places and motions of the
sun and moon. We know that the crystalline form of a body must
depend upon its chemical composition, though we are as yet unable to
assign the law of this dependence.
Indeed in most cases of great discoveries, the right idea to which
the facts were to be referred, was selected by many philosophers,
before the decisive demonstration that it was the right idea, was
given by the discoverer. Thus Newton showed that the motions of the
planets might be explained by means of a central force in the sun:
but though he established, he did not first select the idea involved
in the conception of a {193} central force. The idea had already
been sufficiently pointed out, dimly by Kepler, more clearly by
Borelli, Huyghens, Wren, and Hooke. Indeed this anticipation of the
true idea is always a principal part of that which, in the _History
of the Sciences_, we have termed the _Prelude_ of a Discovery. The
two steps of _proposing_ a philosophical problem, and of _solving_
it, are, as we have elsewhere said, both important, and are often
performed by different persons. The former step is, in fact, the
Selection of the Idea. In explaining any change, we have to discover
first the _Argument_, and then the _Law_ of the change. The
selection of the Argument is the step of which we here speak; and is
that in which inventiveness of mind and justness of thought are
mainly shown.
9. Although, as we have said, we can give few precise directions for
this cardinal process, the Selection of the Idea, in speculating on
phenomena, yet there is one Rule which may have its use: it is
this:--_The idea and the facts must be homogeneous_: the elementary
Conceptions, into which the facts have been decomposed, must be of
the same nature as the Idea by which we attempt to collect them into
laws. Thus, if facts have been observed and measured by reference to
space, they must be bound together by the idea of space: if we would
obtain a knowledge of mechanical forces in the solar system, we must
observe mechanical phenomena. Kepler erred against this rule in his
attempts at obtaining physical laws of the system; for the facts
which he took were the _velocities_, not the _changes of velocity_,
which are really the mechanical facts. Again, there has been a
transgression of this Rule committed by all chemical philosophers
who have attempted to assign the relative position of the elementary
particles of bodies in their component molecules. For their purpose
has been to discover the _relations_ of the particles in _space_;
and yet they have neglected the only facts in the constitution of
bodies which have a reference to space--namely, _crystalline form_,
and _optical properties_. No progress can be made in the theory of
the elementary structure of bodies, {194} without making these
classes of facts the main basis of our speculations.
10. The only other Rule which I have to offer on this subject, is
that which I have already given:--_the Idea must be tested by the
facts_. It must be tried by applying to the facts the conceptions
which are derived from the idea, and not accepted till some of these
succeed in giving the law of the phenomena. The justice of the
suggestion cannot be known otherwise than by making the trial. If we
can discover a _true law_ by employing any conceptions, the idea
from which these conceptions are derived is the _right_ one; nor can
there be any proof of its rightness so complete and satisfactory, as
that we are by it led to a solid and permanent truth.
This, however, can hardly be termed a Rule; for when we would know,
to conjecture and to try the truth of our conjecture by a comparison
with the facts, is the natural and obvious dictate of common sense.
Supposing the Idea which we adopt, or which we would try, to be now
fixed upon, we still have before us the range of many Conceptions
derived from it; many Formulæ may be devised depending on the same
Independent Variable, and we must now consider how our selection
among these is to be made.
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