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we shall render an important service to reason if we succeed in simply
indicating the path along which it must travel； in order to arrive
at any results… even if it should be found necessary to abandon many
of those aims which； without reflection； have been proposed for its
attainment。
That logic has advanced in this sure course； even from the
earliest times； is apparent from the fact that； since Aristotle； it
has been unable to advance a step and； thus； to all appearance has
reached its pletion。 For； if some of the moderns have thought to
enlarge its domain by introducing psychological discussions on the
mental faculties； such as imagination and wit； metaphysical；
discussions on the origin of knowledge and the different kinds of
certitude； according to the difference of the objects （idealism；
scepticism； and so on）； or anthropological discussions on
prejudices； their causes and remedies： this attempt； on the part of
these authors； only shows their ignorance of the peculiar nature of
logical science。 We do not enlarge but disfigure the sciences when
we lose sight of their respective limits and allow them to run into
one another。 Now logic is enclosed within limits which admit of
perfectly clear definition； it is a science which has for its object
nothing but the exposition and proof of the formal laws of all
thought； whether it be a priori or empirical； whatever be its origin
or its object； and whatever the difficulties… natural or accidental…
which it encounters in the human mind。
The early success of logic must be attributed exclusively to the
narrowness of its field； in which abstraction may； or rather must；
be made of all the objects of cognition with their characteristic
distinctions； and in which the understanding has only to deal with
itself and with its own forms。 It is； obviously； a much more difficult
task for reason to strike into the sure path of science； where it
has to deal not simply with itself； but with objects external to
itself。 Hence； logic is properly only a propaedeutic… forms； as it
were； the vestibule of the sciences； and while it is necessary to
enable us to form a correct judgement with regard to the various
branches of knowledge； still the acquisition of real； substantive
knowledge is to be sought only in the sciences properly so called；
that is； in the objective sciences。
Now these sciences； if they can be termed rational at all； must
contain elements of a priori cognition； and this cognition may stand
in a twofold relation to its object。 Either it may have to determine
the conception of the object… which must be supplied extraneously；
or it may have to establish its reality。 The former is theoretical；
the latter practical； rational cognition。 In both； the pure or a
priori element must be treated first； and must be carefully
distinguished from that which is supplied from other sources。 Any
other method can only lead to irremediable confusion。
Mathematics and physics are the two theoretical sciences which
have to determine their objects a priori。 The former is purely a
priori； the latter is partially so； but is also dependent on other
sources of cognition。
In the earliest times of which history affords us any record；
mathematics had already entered on the sure course of science； among
that wonderful nation； the Greeks。 Still it is not to be supposed that
it was as easy for this science to strike into； or rather to construct
for itself； that royal road； as it was for logic； in which reason
has only to deal with itself。 On the contrary； I believe that it
must have remained long… chiefly among the Egyptians… in the stage
of blind groping after its true aims and destination； and that it
was revolutionized by the happy idea of one man； who struck out and
determined for all time the path which this science must follow； and
which admits of an indefinite advancement。 The history of this
intellectual revolution… much more important in its results than the
discovery of the passage round the celebrated Cape of Good Hope… and
of its author； has not been preserved。 But Diogenes Laertius； in
naming the supposed discoverer of some of the simplest elements of
geometrical demonstration… elements which； according to the ordinary
opinion； do not even require to be proved… makes it apparent that
the change introduced by the first indication of this new path； must
have seemed of the utmost importance to the mathematicians of that
age； and it has thus been secured against the chance of oblivion。 A
new light must have flashed on the mind of the first man （Thales； or
whatever may have been his name） who demonstrated the properties of
the isosceles triangle。 For he found that it was not sufficient to
meditate on the figure； as it lay before his eyes； or the conception
of it； as it existed in his mind； and thus endeavour to get at the
knowledge of its properties； but that it was necessary to produce
these properties； as it were； by a positive a priori construction； and
that； in order to arrive with certainty at a priori cognition； he must
not attribute to the object any other properties than those which
necessarily followed from that which he had himself； in accordance
with his conception； placed in the object。
A much longer period elapsed before physics entered on the highway
of science。 For it is only about a century and a half since the wise
Bacon gave a new direction to physical studies； or rather… as others
were already on the right track… imparted fresh vigour to the
pursuit of this new direction。 Here； too； as in the case of
mathematics； we find evidence of a rapid intellectual revolution。 In
the remarks which follow I shall confine myself to the empirical
side of natural science。
When Galilei experimented with balls of a definite weight on the
inclined plane； when Torricelli caused the air to sustain a weight
which he had calculated beforehand to be equal to that of a definite
column of water； or when Stahl； at a later period； converted metals
into lime； and reconverted lime into metal； by the addition and
subtraction of certain elements；* a light broke upon all natural
philosophers。 They learned that reason on