METHODS OF INDUCTION DEPENDING ON RESEMBLANCE.
APHORISM XLIX.
The Law of Continuity _is this:--that a quantity cannot pass from
one amount to another by any change of conditions, without passing
through all intermediate magnitudes according to the intermediate
conditions. This Law may often be employed to disprove distinctions
which have no real foundation._
APHORISM L.
The Method of Gradation _consists in taking a number of stages of a
property in question, intermediate between two extreme cases which
appear to be different. This Method is employed to determine whether
the extreme cases are really distinct or not._
APHORISM LI.
_The Method of Gradation, applied to decide the question, whether the
existing_ geological _phenomena arise from existing causes, leads to
this result:--That the phenomena do appear to arise from Existing
Causes, but that the action of existing causes may, in past times,
have transgressed, to any extent, their_ recorded _limits of
intensity._
APHORISM LII.
The Method of Natural Classification _consists in classing cases,
not according to any_ assumed _Definition, but according to the
connexion of the facts themselves, so as to make them the means of
asserting general truths._ {221}
SECT. I.--_The Law of Continuity._
1. THE Law of Continuity is applicable to quantity primarily, and
therefore might be associated with the methods treated of in the
last chapter: but inasmuch as its inferences are made by a
transition from one degree to another among contiguous cases, it
will be found to belong more properly to the Methods of Induction of
which we have now to speak.
The _Law of Continuity_ consists in this proposition,--That a
quantity cannot pass from one amount to another by any change of
conditions, without passing through all intermediate degrees of
magnitude according to the intermediate conditions. And this law may
often be employed to correct inaccurate inductions, and to reject
distinctions which have no real foundation in nature. For example,
the Aristotelians made a distinction between motions according to
nature, (as that of a body falling vertically downwards,) and
motions contrary to nature, (as that of a body moving along a
horizontal plane:) the former, they held, became naturally quicker
and quicker, the latter naturally slower and slower. But to this it
might be replied, that a horizontal line may pass, by gradual
motion, through various inclined positions, to a vertical position:
and thus the retarded motion may pass into the accelerated; and
hence there must be some inclined plane on which the motion
downwards is naturally uniform: which is false, and therefore the
distinction of such kinds of motion is unfounded. Again, the proof
of the First Law of Motion depends upon the Law of Continuity: for
since, by diminishing the resistance to a body moving on a
horizontal plane, we diminish the retardation, and this without
limit, the law of continuity will bring us at the same time to the
case of no resistance and to the case of no retardation.
2. The Law of Continuity is asserted by Galileo in a particular
application; and the assertion which it {222} suggests is by him
referred to Plato;--namely[36\3] that a moveable body cannot pass
from rest to a determinate degree of velocity without passing
through all smaller degrees of velocity. This law, however, was
first asserted in a more general and abstract form by
Leibnitz[37\3]: and was employed by him to show that the laws of
motion propounded by Descartes must be false. The Third Cartesian
Law of Motion was this[38\3]: that when one moving body meets
another, if the first body have a less momentum than the second, it
will be reflected with its whole motion: but if the first have a
greater momentum than the second, it will lose a part of its motion,
which it will transfer to the second. Now each of these cases leads,
by the Law of Continuity, to the case in which the two bodies have
_equal_ momentums: but in this case, by the first part of the law the
body would _retain all_ its motion; and by the second part of the law
it would _lose_ a portion of it: hence the Cartesian Law is false.
[Note 36\3: _Dialog._ iii. 150. iv. 32.]
[Note 37\3: _Opera_, i. 366.]
[Note 38\3: Cartes, _Prin._ p. 35.]
3. I shall take another example of the application of this Law from
Professor Playfair's Dissertation on the History of Mathematical and
Physical Science[39\3]. 'The Academy of Sciences at Paris having (in
1724) proposed, as a Prize Question, the Investigation of the Laws
of the Communication of Motion, John Bernoulli presented an Essay on
the subject very ingenious and profound; in which, however, he
denied the existence of hard bodies, because in the collision of
such bodies, a finite change of motion must take place in an
instant: an event which, on the principle just explained, he
maintained to be impossible.' And this reasoning was justifiable:
for we can form a _continuous_ transition from cases in which the
impact manifestly occupies a finite time, (as when we strike a large
soft body) to cases in which it is apparently instantaneous.
Maclaurin and others are disposed, in order to avoid the conclusion
of Bernoulli, to reject the Law of {223} Continuity. This, however,
would not only be, as Playfair says, to deprive ourselves of an
auxiliary, commonly useful though sometimes deceptive; but what is
much worse, to acquiesce in false propositions, from the want of
clear and patient thinking. For the Law of Continuity, when rightly
interpreted, is _never_ violated in actual fact. There are not
really any such bodies as have been termed _perfectly hard_: and if
we approach towards such cases, we must learn the laws of motion
which rule them by attending to the Law of Continuity, not by
rejecting it.
[Note 39\3: In the _Encyc. Brit._ p. 537.]
4. Newton used the Law of Continuity to suggest, but not to prove,
the doctrine of universal gravitation. Let, he said, a terrestrial
body be carried as high as the moon: will it not still fall to the
earth? and does not the moon fall by the same force[40\3]? Again: if
any one says that there is a material ether which does not
gravitate[41\3], this kind of matter, by condensation, may be
gradually transmuted to the density of the most intensely
gravitating bodies: and these gravitating bodies, by taking the
internal texture of the condensed ether, may cease to gravitate; and
thus the weight of bodies depends, not on their quantity of matter,
but on their texture; which doctrine Newton conceived he had
disproved by experiment.
[Note 40\3: _Principia_, lib. iii. prop. 6.]
[Note 41\3: _Ib._ cor. 2.]
5. The evidence of the Law of Continuity resides in the universality
of those Ideas, which enter into our apprehension of Laws of Nature.
When, of two quantities, one depends upon the other, the Law of
Continuity necessarily governs this dependence. Every philosopher
has the power of applying this law, in proportion as he has the
faculty of apprehending the Ideas which he employs in his induction,
with the same clearness and steadiness which belong to the
fundamental ideas of Quantity, Space and Number. To those who
possess this faculty, the Law is a Rule of very wide and decisive
application. Its use, as has appeared in the above examples, is seen
rather in the disproof of erroneous views, and in the correction of
false propositions, {224} than in the invention of new truths. It is
a test of truth, rather than an instrument of discovery.
Methods, however, approaching very near to the Law of Continuity may
be employed as positive means of obtaining new truths; and these I
shall now describe.
SECT. II.--_The Method of Gradation._
6. To gather together the cases which resemble each other, and to
separate those which are essentially distinct, has often been
described as the main business of science; and may, in a certain
loose and vague manner of speaking, pass for a description of some
of the leading procedures in the acquirement of knowledge. The
selection of instances which agree, and of instances which differ,
in some prominent point or property, are important steps in the
formation of science. But when classes of things and properties have
been established in virtue of such comparisons, it may still be
doubtful whether these classes are separated by distinctions of
opposites, or by differences of degree. And to settle such
questions, the _Method of Gradation_ is employed; which consists in
taking intermediate stages of the properties in question, so as to
ascertain by experiment whether, in the transition from one class to
another, we have to leap over a manifest gap, or to follow a
continuous road.
7. Thus for instance, one of the early _Divisions_ established by
electrical philosophers was that of _Electrics_ and _Conductors_.
But this division Dr. Faraday has overturned as an essential
opposition. He takes[42\3] a _Gradation_ which carries him from
Conductors to Non-conductors. Sulphur, or Lac, he says, are held to
be non-conductors, but are not rigorously so. Spermaceti is a bad
conductor: ice or water better than spermaceti: metals so much
better that they are put in a different class. But even in metals
the transit of the electricity is not instantaneous: we have in them
proof of a retardation of the electric current: 'and what {225}
reason," Mr. Faraday asks, "why this retardation should not be of
the same kind as that in spermaceti, or in lac, or sulphur? But as,
in them, retardation is insulation, [and insulation is
induction[43\3]] why should we refuse the same relation to the same
exhibitions of force in the metals?"
[Note 42\3: _Researches_, 12th series, art. 1328.]
[Note 43\3: These words refer to another proposition, also
established by the Method of Gradation.]
The process employed by the same sagacious philosopher to show the
_identity_ of Voltaic and Franklinic electricity, is another example
of the same kind[44\3]. Machine [Franklinic] electricity was made to
exhibit the same phenomena as Voltaic electricity, by causing the
discharge to pass through a bad conductor, into a very extensive
discharging train: and thus it was clearly shown that Franklinic
electricity, not so conducted, differs from the other kinds, only in
being in a state of successive tension and explosion instead of a
state of continued current.
[Note 44\3: _Hist. Ind. Sc._ b. xiv. c. ix. sect. 2.]
Again; to show that the decomposition of bodies in the Voltaic
circuit was not due to the _Attraction_ of the Poles[45\3], Mr.
Faraday devised a beautiful series of experiments, in which these
supposed _Poles_ were made to assume all possible electrical
conditions:--in some cases the decomposition took place against air,
which according to common language is not a conductor, nor is
decomposed;--in others, against the metallic poles, which are
excellent conductors but undecomposable;--and so on: and hence he
infers that the decomposition cannot justly be considered as due to
the Attraction, or Attractive Powers, of the Poles.
[Note 45\3: _Ibid. Researches_, art. 497.]
8. The reader of the _Novum Organon_ may perhaps, in looking at such
examples of the Rule, be reminded of some of Bacon's Classes of
Instances, as his _instantiæ absentiæ in proximo_, and his
_instantiæ migrantes_. But we may remark that Instances classed and
treated as Bacon recommends in those parts of his work, could hardly
lead to scientific truth. His {226} processes are vitiated by his
proposing to himself the _form_ or _cause_ of the property before
him, as the object of his inquiry; instead of being content to
obtain, in the first place, the _law of phenomena_. Thus his
example[46\3] of a Migrating Instance is thus given. "Let the
_Nature inquired into_ be that of Whiteness; an Instance Migrating
to the production of this property is glass, first whole, and then
pulverized; or plain water, and water agitated into a foam; for
glass and water are transparent, and not white; but glass powder and
foam are white, and not transparent. Hence we must inquire what has
happened to the glass or water in that Migration. For it is plain
that the _Form of Whiteness_ is conveyed and induced by the crushing
of the glass and shaking of the water." No real knowledge has
resulted from this line of reasoning:--from taking the Natures and
Forms of things and of their qualities for the primary subject of
our researches.
[Note 46\3: _Nov. Org._ lib. ii. Aph. 28.]
9. We may easily give examples from other subjects in which the
Method of Gradation has been used to establish, or to endeavour to
establish, very extensive propositions. Thus Laplace's Nebular
Hypothesis,--that systems like our solar system are formed by
gradual condensation from diffused masses, such as the nebulæ among
the stars,--is founded by him upon an application of this Method of
Gradation. We see, he conceives, among these nebulæ, instances of
all degrees of condensation, from the most loosely diffused fluid,
to that separation and solidification of parts by which suns, and
satellites, and planets are formed: and thus we have before us
instances of systems in all their stages; as in a forest we see
trees in every period of growth. How far the examples in this case
satisfy the demands of the Method of Gradation, it remains for
astronomers and philosophers to examine.
Again; this method was used with great success by Macculloch and
others to refute the opinion, put in currency by the Wernerian
school of geologists, that {227} the rocks called _trap rocks_ must
be classed with those to which a _sedimentary_ origin is ascribed.
For it was shown that a gradual _transition_ might be traced from
those examples in which trap rocks most resembled stratified rocks,
to the lavas which have been recently ejected from volcanoes: and
that it was impossible to assign a different origin to one portion,
and to the other, of this kind of mineral masses; and as the
volcanic rocks were certainly not sedimentary, it followed, that the
trap rocks were not of that nature.
Again; we have an attempt of a still larger kind made by Sir C.
Lyell, to apply this Method of Gradation so as to disprove all
distinction between the causes by which geological phenomena have
been produced, and the causes which are now acting at the earth's
surface. He has collected a very remarkable series of changes which
have taken place, and are still taking place, by the action of
water, volcanoes, earthquakes, and other terrestrial operations; and
he conceives he has shown in these a _gradation_ which leads, with
no wide chasm or violent leap, to the state of things of which
geological researches have supplied the evidence.
10. Of the value of this Method in geological speculations, no doubt
can be entertained. Yet it must still require a grave and profound
consideration, in so vast an application of the Method as that
attempted by Sir C. Lyell, to determine what extent we may allow to
the steps of our _gradation_; and to decide how far the changes
which have taken place in distant parts of the series may exceed
those of which we have historical knowledge, without ceasing to be
of the _same kind_. Those who, dwelling in a city, see, from time to
time, one house built and another pulled down, may say that such
_existing causes_, operating through past time, sufficiently explain
the existing condition of the city. Yet we arrive at important
political and historical truths, by considering the _origin_ of a
city as an event of a _different order_ from those daily changes.
The causes which are now working to produce geological results, may
be supposed to have been, at some former epoch, so far exaggerated
in their operation, that the changes {228} should be paroxysms, not
degrees;--that they should violate, not continue, the gradual
series. And we have no kind of evidence whether the duration of our
historical times is sufficient to give us a just measure of the
limits of such degrees;--whether the terms which we have under our
notice enable us to ascertain the average rate of progression.
11. The result of such considerations seems to be this:--that we may
apply the Method of Gradation in the investigation of geological
causes, provided we leave the Limits of the Gradation undefined.
But, then, this is equivalent to the admission of the opposite
hypothesis: for a continuity of which the successive intervals are
not limited, is not distinguishable from discontinuity. The
geological sects of recent times have been distinguished as
_uniformitarians_ and _catastrophists_: the Method of Gradation
seems to prove the doctrine of the uniformitarians; but then, at the
same time that it does this, it breaks down the distinction between
them and the catastrophists.
There are other exemplifications of the use of gradations in Science
which well deserve notice: but some of them are of a kind somewhat
different, and may be considered under a separate head.
SECT. III. _The Method of Natural Classification._
12. The Method of Natural Classification consists, as we have seen,
in grouping together objects, not according to any selected
properties, but according to their most important resemblances; and
in combining such grouping with the assignation of certain marks of
the classes thus formed. The examples of the successful application
of this method are to be found in the Classificatory Sciences
through their whole extent; as, for example, in framing the Genera
of plants and animals. The same method, however, may often be
extended to other sciences. Thus the classification of Crystalline
Forms, according to their Degree of Symmetry, (which is really an
important distinction,) as introduced by Mohs and Weiss, was a great
improvement {229} upon Haüy's arbitrary division according to
certain assumed primary forms. Sir David Brewster was led to the
same distinction of crystals by the study of their optical
properties; and the scientific value of the classification was thus
strongly exhibited. Mr. Howard's classification of Clouds appears to
be founded in their real nature, since it enables him to express the
laws of their changes and successions. As we have elsewhere said,
the criterion of a true classification is, that it makes general
propositions possible. One of the most prominent examples of the
beneficial influence of a right classification, is to be seen in the
impulse given to geology by the distinction of strata according to
the organic fossils which they contain[47\3]: which, ever since its
general adoption, has been a leading principle in the speculations
of geologists.
[Note 47\3: _Hist. Ind. Sc._ b. xviii. c. ii. sect. 3.]
13. The mode in which, in this and in other cases, the Method of
Natural Classification directs the researches of the philosopher, is
this:--his arrangement being adopted, at least as an instrument of
inquiry and trial, he follows the course of the different members of
the classification, according to the guidance which Nature herself
offers; not prescribing beforehand the marks of each part, but
distributing the facts according to the total resemblances, or
according to those resemblances which he finds to be most important.
Thus, in tracing the course of a series of strata from place to
place, we identify each stratum, not by any single character, but by
all taken together;--texture, colour, fossils, position, and any
other circumstances which offer themselves. And if, by this means,
we come to ambiguous cases, where different indications appear to
point different ways, we decide so as best to preserve undamaged
those general relations and truths which constitute the value of our
system. Thus although we consider the organic fossils in each
stratum as its most important characteristic, we are not prevented,
by the disappearance of some fossils, or the addition of others, or
by the total absence of fossils, {230} from identifying strata in
distant countries, if the position and other circumstances authorize
us to do so. And by this Method of Classification, the doctrine of
_Geological Equivalents_[48\3] has been applied to a great part of
Europe.
[Note 48\3: _Hist. Ind. Sc._ b. xviii. c. iii. sect. 4.]
14. We may further observe, that the same method of natural
classification which thus enables us to identify strata in remote
situations, notwithstanding that there may be great differences in
their material and contents, also forbids us to assume the identity
of the series of rocks which occur in different countries, when this
identity has not been verified by such a continuous exploration of
the component members of the series. It would be in the highest
degree unphilosophical to apply the special names of the English or
German strata to the rocks of India, or America, or even of southern
Europe, till it has appeared that in those countries the geological
series of northern Europe really exists. In each separate country,
the divisions of the formations which compose the crust of the earth
must be made out, by applying the Method of Natural Arrangement _to
that particular case_, and not by arbitrarily extending to it the
nomenclature belonging to another case. It is only by such
precautions, that we can ever succeed in obtaining geological
propositions, at the same time true and comprehensive; or can obtain
any sound general views respecting the physical history of the
earth.
15. The Method of Natural Classification, which we thus recommend,
falls in with those mental habits which we formerly described as
resulting from the study of Natural History. The method was then
termed the _Method of Type_, and was put in opposition to the
_Method of Definition_.
The Method of Natural Classification is directly opposed to the
process in which we assume and apply _arbitrary_ definitions; for in
the former Method, we find our classes in nature, and do not make
them by marks of our own imposition. Nor can any advantage {231} to
the progress of knowledge be procured, by laying down our characters
when our arrangements are as yet quite loose and unformed. Nothing
was gained by the attempts to _define_ Metals by their weight, their
hardness, their ductility, their colour; for to all these marks, as
fast as they were proposed, exceptions were found, among bodies
which still could not be excluded from the list of Metals. It was
only when elementary substances were divided into _Natural Classes_,
of which classes Metals were one, that a true view of their
distinctive characters was obtained. Definitions in the outset of
our examination of nature are almost always, not only useless, but
prejudicial.
16. When we obtain a Law of Nature by induction from phenomena, it
commonly happens, as we have already seen, that we introduce, at the
same time, a Proposition and a Definition. In this case, the two are
correlative, each giving a real value to the other. In such cases,
also, the Definition, as well as the Proposition, may become the
basis of rigorous reasoning, and may lead to a series of deductive
truths. We have examples of such Definitions and Propositions in the
Laws of Motion, and in many other cases.
17. When we have established Natural Classes of objects, we seek for
Characters of our classes; and these Characters may, to a certain
extent, be called the _Definitions_ of our classes. This is to be
understood, however, only in a limited sense: for these Definitions
are not absolute and permanent. They are liable to be modified and
superseded. If we find a case which manifestly belongs to our
Natural Class, though violating our Definition, we do not shut out
the case, but alter our definition. Thus, when we have made it part
of our Definition of the _Rose_ family, that they have _alternate
stipulate leaves_, we do not, therefore, exclude from the family the
genus _Lowæa_, which has _no stipulæ_. In Natural Classifications,
our Definitions are to be considered as temporary and provisional
only. When Sir C. Lyell established the distinctions of the tertiary
strata, which he termed _Eocene_, _Miocene_, and _Pliocene_, he took
a numerical criterion {232} (the proportion of recent species of
shells contained in those strata) as the basis of his division. But
now that those kinds of strata have become, by their application to
a great variety of cases, a series of Natural Classes, we must, in
our researches, keep in view the natural connexion of the formations
themselves in different places; and must by no means allow ourselves
to be governed by the numerical proportions which were originally
contemplated; or even by any amended numerical criterion equally
arbitrary; for however amended, Definitions in natural history are
never immortal. The etymologies of _Pliocene_ and _Miocene_ may,
hereafter, come to have merely an historical interest; and such a
state of things will be no more inconvenient, provided the natural
connexions of each class are retained, than it is to call a rock
_oolite_ or _porphyry_, when it has no roelike structure and no
fiery spots.
The Methods of Induction which are treated of in this and the
preceding chapter, and which are specially applicable to causes
governed by relations of Quantity or of Resemblance, commonly lead
us to _Laws of Phenomena_ only. Inductions founded upon other ideas,
those of Substance and Cause for example, appear to conduct us
somewhat further into a knowledge of the essential nature and real
connexions of things. But before we speak of these, we shall say a
few words respecting the way in which inductive propositions, once
obtained, may be verified and carried into effect by their
application.
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